microsoft/FStarDataSet-V2 · Datasets at Hugging Face

FStar.Buffer.fst

FStar.Buffer.create

val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init))

let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 3, "end_line": 822, "start_col": 0, "start_line": 816 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

init: a -> len: FStar.UInt32.t -> FStar.HyperStack.ST.StackInline (FStar.Buffer.buffer a)

FStar.HyperStack.ST.StackInline

[]

[ "FStar.UInt32.t", "Prims.unit", "Prims._assert", "FStar.Seq.Base.equal", "FStar.Buffer.as_seq", "FStar.Buffer.sel", "FStar.Buffer.buffer", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer._buffer", "FStar.Buffer.MkBuffer", "FStar.UInt32.__uint_to_t", "FStar.HyperStack.ST.reference", "FStar.Buffer.lseq", "FStar.UInt32.v", "FStar.HyperStack.ST.salloc", "FStar.Heap.trivial_preorder", "FStar.Seq.Base.create", "FStar.HyperStack.ST.mstackref" ]

[]

false

true

false

let create #a init len =

let content:reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get () in assert (Seq.equal (as_seq h b) (sel h b)); b

false

FStar.Buffer.fst

FStar.Buffer.index

val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n)))

let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 29, "end_line": 964, "start_col": 0, "start_line": 962 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> FStar.HyperStack.ST.Stack a

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Seq.Base.index", "Prims.op_Addition", "FStar.Buffer.__proj__MkBuffer__item__idx", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.HyperStack.ST.op_Bang", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content" ]

[]

false

true

false

let index #a b n =

let s = !b.content in Seq.index s (v b.idx + v n)

false

FStar.Buffer.fst

FStar.Buffer.to_seq_full

val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b ))

let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 32, "end_line": 956, "start_col": 0, "start_line": 953 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> FStar.HyperStack.ST.ST (FStar.Seq.Base.seq a)

FStar.HyperStack.ST.ST

[]

[ "FStar.Buffer.buffer", "FStar.Seq.Base.slice", "Prims.op_Addition", "FStar.UInt32.v", "FStar.Buffer.__proj__MkBuffer__item__length", "FStar.UInt.uint_t", "FStar.Buffer.__proj__MkBuffer__item__idx", "FStar.Seq.Base.seq", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.HyperStack.ST.op_Bang", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content" ]

[]

false

true

false

let to_seq_full #a b =

let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length)

false

FStar.Buffer.fst

FStar.Buffer.rcreate

val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content)))

let rcreate #a r init len = rcreate_common r init len false

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 59, "end_line": 898, "start_col": 0, "start_line": 898 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

r: FStar.Monotonic.HyperHeap.rid -> init: a -> len: FStar.UInt32.t -> FStar.HyperStack.ST.ST (FStar.Buffer.buffer a)

FStar.HyperStack.ST.ST

[]

[ "FStar.Monotonic.HyperHeap.rid", "FStar.UInt32.t", "FStar.Buffer.rcreate_common", "FStar.Buffer.buffer" ]

[]

false

true

false

let rcreate #a r init len =

rcreate_common r init len false

false

FStar.Buffer.fst

FStar.Buffer.createL

val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b))

let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 3, "end_line": 856, "start_col": 0, "start_line": 848 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 1, "initial_ifuel": 1, "max_fuel": 1, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

init: Prims.list a -> FStar.HyperStack.ST.StackInline (FStar.Buffer.buffer a)

FStar.HyperStack.ST.StackInline

[]

[ "Prims.list", "Prims.unit", "Prims._assert", "FStar.Seq.Base.equal", "FStar.Buffer.as_seq", "FStar.Buffer.sel", "FStar.Buffer.buffer", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer._buffer", "FStar.Buffer.MkBuffer", "FStar.UInt32.__uint_to_t", "FStar.HyperStack.ST.reference", "FStar.Buffer.lseq", "FStar.UInt32.v", "FStar.HyperStack.ST.salloc", "FStar.Heap.trivial_preorder", "FStar.Seq.Base.seq_of_list", "FStar.HyperStack.ST.mstackref", "FStar.Seq.Base.seq", "Prims.eq2", "Prims.nat", "FStar.List.Tot.Base.length", "FStar.Seq.Base.length", "FStar.UInt32.t", "FStar.UInt32.uint_to_t" ]

[]

false

true

false

let createL #a init =

let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content:reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get () in assert (Seq.equal (as_seq h b) (sel h b)); b

false

FStar.Buffer.fst

FStar.Buffer.rfree

val rfree (#a: Type) (b: buffer a) : ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0))

let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 19, "end_line": 921, "start_col": 0, "start_line": 918 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> FStar.HyperStack.ST.ST Prims.unit

FStar.HyperStack.ST.ST

[]

[ "FStar.Buffer.buffer", "FStar.HyperStack.ST.rfree", "FStar.Buffer.lseq", "FStar.UInt32.v", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content", "Prims.unit", "FStar.Monotonic.HyperStack.mem", "Prims.l_and", "FStar.Buffer.live", "FStar.Buffer.freeable", "Prims.b2t", "FStar.Monotonic.HyperStack.is_mm", "FStar.Buffer.max_length", "FStar.Buffer.content", "FStar.HyperStack.ST.is_eternal_region", "FStar.Buffer.frameOf", "Prims.eq2", "FStar.Monotonic.HyperStack.free" ]

[]

false

true

false

let rfree (#a: Type) (b: buffer a) : ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) =

rfree b.content

false

FStar.Buffer.fst

FStar.Buffer.rcreate_common

val rcreate_common (#a: Type) (r: rid) (init: a) (len: UInt32.t) (mm: bool) : ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm))

let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 5, "end_line": 889, "start_col": 8, "start_line": 876 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

r: FStar.Monotonic.HyperHeap.rid -> init: a -> len: FStar.UInt32.t -> mm: Prims.bool -> FStar.HyperStack.ST.ST (FStar.Buffer.buffer a)

FStar.HyperStack.ST.ST

[]

[ "FStar.Monotonic.HyperHeap.rid", "FStar.UInt32.t", "Prims.bool", "Prims.unit", "FStar.Buffer.lemma_upd", "FStar.Buffer.lseq", "FStar.UInt32.v", "Prims._assert", "FStar.Seq.Base.equal", "FStar.Buffer.as_seq", "FStar.Buffer.sel", "FStar.Buffer.buffer", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer._buffer", "FStar.Buffer.MkBuffer", "FStar.UInt32.__uint_to_t", "FStar.HyperStack.ST.reference", "FStar.HyperStack.ST.ralloc_mm", "FStar.Heap.trivial_preorder", "FStar.HyperStack.ST.mmmref", "FStar.HyperStack.ST.ralloc", "FStar.HyperStack.ST.mref", "FStar.Seq.Base.seq", "FStar.Seq.Base.create", "FStar.HyperStack.ST.is_eternal_region", "Prims.l_and", "FStar.Buffer.rcreate_post_common", "Prims.eq2", "FStar.Monotonic.HyperStack.is_mm", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Buffer.__proj__MkBuffer__item__content" ]

[]

false

true

false

let rcreate_common (#a: Type) (r: rid) (init: a) (len: UInt32.t) (mm: bool) : ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) =

let h0 = HST.get () in let s = Seq.create (v len) init in let content:reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get () in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b

false

FStar.Buffer.fst

FStar.Buffer.lemma_aux_0

val lemma_aux_0 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) (tt: Type) (bb: buffer tt) : Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))

let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm ()

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 44, "end_line": 975, "start_col": 8, "start_line": 968 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> z: a -> h0: FStar.Monotonic.HyperStack.mem -> tt: Type0 -> bb: FStar.Buffer.buffer tt -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h0 b /\ FStar.Buffer.live h0 bb /\ FStar.Buffer.disjoint b bb) (ensures FStar.Buffer.live h0 b /\ FStar.Buffer.live h0 bb /\ (let h1 = FStar.Monotonic.HyperStack.upd h0 (MkBuffer?.content b) (FStar.Seq.Base.upd (FStar.Buffer.sel h0 b) (FStar.Buffer.idx b + FStar.UInt32.v n) z) in FStar.Buffer.as_seq h0 bb == FStar.Buffer.as_seq h1 bb))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "FStar.Monotonic.Heap.lemma_distinct_addrs_distinct_mm", "Prims.unit", "FStar.Monotonic.Heap.lemma_distinct_addrs_distinct_preorders", "Prims.l_and", "FStar.Buffer.live", "FStar.Buffer.disjoint", "Prims.squash", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.l_or", "Prims.nat", "FStar.Seq.Base.length", "FStar.Buffer.as_seq", "FStar.Monotonic.HyperStack.upd", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.Seq.Base.upd", "FStar.Buffer.sel", "Prims.op_Addition", "FStar.Buffer.idx", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

true

false

true

false

let lemma_aux_0 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) (tt: Type) (bb: buffer tt) : Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) =

Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm ()

false

FStar.Buffer.fst

FStar.Buffer.lemma_aux

val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))]

let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 48, "end_line": 1006, "start_col": 0, "start_line": 1006 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) ))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> z: a -> h0: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h0 b) (ensures FStar.Buffer.live h0 b /\ FStar.Buffer.modifies_1 b h0 (FStar.Monotonic.HyperStack.upd h0 (MkBuffer?.content b) (FStar.Seq.Base.upd (FStar.Buffer.sel h0 b) (FStar.Buffer.idx b + FStar.UInt32.v n) z) )) [ SMTPat (FStar.Monotonic.HyperStack.upd h0 (MkBuffer?.content b) (FStar.Seq.Base.upd (FStar.Buffer.sel h0 b) (FStar.Buffer.idx b + FStar.UInt32.v n) z) ) ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "FStar.Buffer.lemma_aux_2", "Prims.unit" ]

[]

true

false

true

false

let lemma_aux #a b n z h0 =

lemma_aux_2 b n z h0

false

FStar.Buffer.fst

FStar.Buffer.to_seq

val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) ))

let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 25, "end_line": 945, "start_col": 0, "start_line": 942 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> l: FStar.UInt32.t{FStar.UInt32.v l <= FStar.Buffer.length b} -> FStar.HyperStack.ST.STL (FStar.Seq.Base.seq a)

FStar.HyperStack.ST.STL

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Seq.Base.slice", "Prims.op_Addition", "FStar.UInt.uint_t", "FStar.Buffer.__proj__MkBuffer__item__idx", "FStar.Seq.Base.seq", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.HyperStack.ST.op_Bang", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content" ]

[]

false

true

false

let to_seq #a b l =

let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l)

false

FStar.Buffer.fst

FStar.Buffer.lemma_aux_1

val lemma_aux_1 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) (tt: Type) : Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb: buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))))

let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 58, "end_line": 986, "start_col": 8, "start_line": 978 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 10, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> z: a -> h0: FStar.Monotonic.HyperStack.mem -> tt: Type0 -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h0 b) (ensures FStar.Buffer.live h0 b /\ (forall (bb: FStar.Buffer.buffer tt). FStar.Buffer.live h0 bb /\ FStar.Buffer.disjoint b bb ==> (let h1 = FStar.Monotonic.HyperStack.upd h0 (MkBuffer?.content b) (FStar.Seq.Base.upd (FStar.Buffer.sel h0 b) (FStar.Buffer.idx b + FStar.UInt32.v n) z) in FStar.Buffer.as_seq h0 bb == FStar.Buffer.as_seq h1 bb)))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "FStar.Classical.forall_intro", "Prims.l_imp", "Prims.l_and", "FStar.Buffer.live", "FStar.Buffer.disjoint", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.l_or", "Prims.nat", "FStar.Seq.Base.length", "FStar.Buffer.as_seq", "FStar.Monotonic.HyperStack.upd", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.Seq.Base.upd", "FStar.Buffer.sel", "Prims.op_Addition", "FStar.Buffer.idx", "FStar.Classical.move_requires", "FStar.Buffer.lemma_aux_0", "Prims.unit", "Prims.squash", "Prims.l_Forall", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

false

true

false

let lemma_aux_1 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) (tt: Type) : Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb: buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) =

let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt))

false

FStar.Buffer.fst

FStar.Buffer.offset

val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'})

let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 62, "end_line": 1072, "start_col": 0, "start_line": 1071 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> i: FStar.UInt32.t { FStar.UInt32.v i + FStar.UInt32.v (MkBuffer?.idx b) < Prims.pow2 FStar.UInt32.n /\ FStar.UInt32.v i <= FStar.UInt32.v (MkBuffer?.length b) } -> b': FStar.Buffer.buffer a {FStar.Buffer.includes b b'}

Prims.Tot

[ "total" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThan", "Prims.op_Addition", "FStar.UInt32.v", "FStar.Buffer.__proj__MkBuffer__item__idx", "Prims.pow2", "FStar.UInt32.n", "Prims.op_LessThanOrEqual", "FStar.Buffer.__proj__MkBuffer__item__length", "FStar.Buffer.MkBuffer", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.UInt32.op_Plus_Hat", "FStar.UInt32.op_Subtraction_Hat", "FStar.Buffer.includes" ]

[]

false

let offset #a b i =

MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i)

false

FStar.Buffer.fst

FStar.Buffer.upd

val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z ))

let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 53, "end_line": 1021, "start_col": 0, "start_line": 1013 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t -> z: a -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "FStar.Seq.Properties.upd_slice", "FStar.Buffer.idx", "Prims.op_Addition", "FStar.Buffer.length", "FStar.UInt32.v", "Prims.unit", "FStar.Seq.Base.lemma_eq_intro", "FStar.Buffer.as_seq", "FStar.Seq.Base.slice", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer.lemma_aux", "FStar.HyperStack.ST.op_Colon_Equals", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.Seq.Base.seq", "FStar.Seq.Base.upd", "FStar.Buffer.__proj__MkBuffer__item__idx", "FStar.HyperStack.ST.op_Bang" ]

[]

false

true

false

let upd #a b n z =

let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get () in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z

false

FStar.Buffer.fst

FStar.Buffer.lemma_sub_spec'

val lemma_sub_spec' (#a: Type) (b: buffer a) (i: UInt32.t) (len: UInt32.t{v len <= length b /\ v i + v len <= length b}) (h: _) : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))]

let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 28, "end_line": 1066, "start_col": 0, "start_line": 1058 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> i: FStar.UInt32.t -> len: FStar.UInt32.t { FStar.UInt32.v len <= FStar.Buffer.length b /\ FStar.UInt32.v i + FStar.UInt32.v len <= FStar.Buffer.length b } -> h: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h b) (ensures FStar.Buffer.live h (FStar.Buffer.sub b i len) /\ FStar.Buffer.as_seq h (FStar.Buffer.sub b i len) == FStar.Seq.Base.slice (FStar.Buffer.as_seq h b) (FStar.UInt32.v i) (FStar.UInt32.v i + FStar.UInt32.v len)) [SMTPat (FStar.Buffer.live h (FStar.Buffer.sub b i len))]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "Prims.op_Addition", "FStar.Monotonic.HyperStack.mem", "FStar.Buffer.lemma_sub_spec", "Prims.unit", "FStar.Buffer.live", "Prims.squash", "FStar.Buffer.sub", "Prims.eq2", "FStar.Seq.Base.seq", "FStar.Buffer.as_seq", "FStar.Seq.Base.slice", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let lemma_sub_spec' (#a: Type) (b: buffer a) (i: UInt32.t) (len: UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] =

lemma_sub_spec b i len h

false

FStar.Buffer.fst

FStar.Buffer.op_Array_Access

val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n)))

let op_Array_Access #a b n = index #a b n

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 41, "end_line": 1140, "start_col": 0, "start_line": 1140 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 20, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> FStar.HyperStack.ST.Stack a

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Buffer.index" ]

[]

false

true

false

let ( .() ) #a b n =

index #a b n

false

FStar.Buffer.fst

FStar.Buffer.sub

val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len})

let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 50, "end_line": 1028, "start_col": 0, "start_line": 1026 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b}

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> i: FStar.UInt32.t -> len: FStar.UInt32.t{FStar.UInt32.v i + FStar.UInt32.v len <= FStar.Buffer.length b} -> b': FStar.Buffer.buffer a {FStar.Buffer.includes b b' /\ FStar.Buffer.length b' == FStar.UInt32.v len}

Prims.Tot

[ "total" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThanOrEqual", "Prims.op_Addition", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Buffer.MkBuffer", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.UInt32.op_Plus_Hat", "FStar.Buffer.__proj__MkBuffer__item__idx", "Prims.unit", "Prims._assert", "Prims.op_LessThan", "Prims.pow2", "FStar.UInt32.n", "Prims.l_and", "FStar.Buffer.includes", "Prims.eq2", "Prims.int", "Prims.l_or", "Prims.op_GreaterThanOrEqual", "FStar.UInt.size" ]

[]

false

let sub #a b i len =

assert (v i + v b.idx < pow2 n); MkBuffer b.max_length b.content (i +^ b.idx) len

false

FStar.Buffer.fst

FStar.Buffer.split

val split (#t: _) (b: buffer t) (i: UInt32.t{v i <= length b}) : Tot (buffer t * buffer t)

let split #t (b:buffer t) (i:UInt32.t{v i <= length b}) : Tot (buffer t * buffer t) = sub b 0ul i, offset b i

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 27, "end_line": 1216, "start_col": 0, "start_line": 1215 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let op_Array_Assignment #a b n z = upd #a b n z let lemma_modifies_one_trans_1 (#a:Type) (b:buffer a) (h0:mem) (h1:mem) (h2:mem): Lemma (requires (modifies_one (frameOf b) h0 h1 /\ modifies_one (frameOf b) h1 h2)) (ensures (modifies_one (frameOf b) h0 h2)) [SMTPat (modifies_one (frameOf b) h0 h1); SMTPat (modifies_one (frameOf b) h1 h2)] = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (** Corresponds to memcpy *) val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) == Seq.slice (as_seq h0 b) (v idx_b+v len) (length b) )) let rec blit #t a idx_a b idx_b len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get() in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b)) end (** Corresponds to memset *) val fill: #t:Type -> b:buffer t -> z:t -> len:UInt32.t{v len <= length b} -> Stack unit (requires (fun h -> live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) 0 (v len) == Seq.create (v len) z /\ Seq.slice (as_seq h1 b) (v len) (length b) == Seq.slice (as_seq h0 b) (v len) (length b) )) let rec fill #t b z len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in fill #t b z len'; b.(len') <- z; let h = HST.get() in Seq.snoc_slice_index (as_seq h b) 0 (v len'); Seq.lemma_tail_slice (as_seq h b) (v len') (length b) end; let h1 = HST.get() in Seq.lemma_eq_intro (Seq.slice (as_seq h1 b) 0 (v len)) (Seq.create (v len) z)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer t -> i: FStar.UInt32.t{FStar.UInt32.v i <= FStar.Buffer.length b} -> FStar.Buffer.buffer t * FStar.Buffer.buffer t

Prims.Tot

[ "total" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Pervasives.Native.Mktuple2", "FStar.Buffer.sub", "FStar.UInt32.__uint_to_t", "FStar.Buffer.offset", "FStar.Pervasives.Native.tuple2" ]

[]

false

let split #t (b: buffer t) (i: UInt32.t{v i <= length b}) : Tot (buffer t * buffer t) =

sub b 0ul i, offset b i

false

FStar.Buffer.fst

FStar.Buffer.lemma_offset_spec

val lemma_offset_spec (#a: Type) (b: buffer a) (i: UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) (h: _) : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [ SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]] ]

let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 88, "end_line": 1081, "start_col": 0, "start_line": 1074 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> i: FStar.UInt32.t { FStar.UInt32.v i + FStar.UInt32.v (MkBuffer?.idx b) < Prims.pow2 FStar.UInt32.n /\ FStar.UInt32.v i <= FStar.UInt32.v (MkBuffer?.length b) } -> h: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (ensures FStar.Buffer.as_seq h (FStar.Buffer.offset b i) == FStar.Seq.Base.slice (FStar.Buffer.as_seq h b) (FStar.UInt32.v i) (FStar.Buffer.length b)) [ SMTPatOr [ [SMTPat (FStar.Buffer.as_seq h (FStar.Buffer.offset b i))]; [ SMTPat (FStar.Seq.Base.slice (FStar.Buffer.as_seq h b) (FStar.UInt32.v i) (FStar.Buffer.length b)) ] ] ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThan", "Prims.op_Addition", "FStar.UInt32.v", "FStar.Buffer.__proj__MkBuffer__item__idx", "Prims.pow2", "FStar.UInt32.n", "Prims.op_LessThanOrEqual", "FStar.Buffer.__proj__MkBuffer__item__length", "FStar.Monotonic.HyperStack.mem", "FStar.Seq.Base.lemma_eq_intro", "FStar.Buffer.as_seq", "FStar.Buffer.offset", "FStar.Seq.Base.slice", "FStar.Buffer.length", "Prims.unit", "Prims.l_True", "Prims.squash", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat_or", "Prims.list", "FStar.Pervasives.smt_pat", "Prims.nat", "FStar.Seq.Base.length", "Prims.Nil" ]

[]

true

false

true

false

let lemma_offset_spec (#a: Type) (b: buffer a) (i: UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [ SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]] ] =

Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b))

false

FStar.Buffer.fst

FStar.Buffer.lemma_aux_2

val lemma_aux_2 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) : Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt: Type) (bb: buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))))

let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 55, "end_line": 998, "start_col": 8, "start_line": 990 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0"

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t{FStar.UInt32.v n < FStar.Buffer.length b} -> z: a -> h0: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h0 b) (ensures FStar.Buffer.live h0 b /\ (forall (tt: Type0) (bb: FStar.Buffer.buffer tt). FStar.Buffer.live h0 bb /\ FStar.Buffer.disjoint b bb ==> (let h1 = FStar.Monotonic.HyperStack.upd h0 (MkBuffer?.content b) (FStar.Seq.Base.upd (FStar.Buffer.sel h0 b) (FStar.Buffer.idx b + FStar.UInt32.v n) z) in FStar.Buffer.as_seq h0 bb == FStar.Buffer.as_seq h1 bb)))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThan", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "FStar.Classical.forall_intro", "Prims.l_imp", "FStar.Buffer.live", "Prims.l_and", "Prims.l_Forall", "FStar.Buffer.disjoint", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.l_or", "Prims.nat", "FStar.Seq.Base.length", "FStar.Buffer.as_seq", "FStar.Monotonic.HyperStack.upd", "FStar.Buffer.lseq", "FStar.Buffer.__proj__MkBuffer__item__max_length", "FStar.Heap.trivial_preorder", "FStar.Buffer.__proj__MkBuffer__item__content", "FStar.Seq.Base.upd", "FStar.Buffer.sel", "Prims.op_Addition", "FStar.Buffer.idx", "FStar.Classical.move_requires", "FStar.Buffer.lemma_aux_1", "Prims.unit", "Prims.squash", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

false

true

false

let lemma_aux_2 (#a: Type) (b: buffer a) (n: UInt32.t{v n < length b}) (z: a) (h0: mem) : Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt: Type) (bb: buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) =

let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0))

false

FStar.Buffer.fst

FStar.Buffer.lemma_sub_spec

val lemma_sub_spec (#a: Type) (b: buffer a) (i: UInt32.t) (len: UInt32.t{v len <= length b /\ v i + v len <= length b}) (h: _) : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)]

let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 92, "end_line": 1056, "start_col": 0, "start_line": 1048 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> i: FStar.UInt32.t -> len: FStar.UInt32.t { FStar.UInt32.v len <= FStar.Buffer.length b /\ FStar.UInt32.v i + FStar.UInt32.v len <= FStar.Buffer.length b } -> h: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires FStar.Buffer.live h b) (ensures FStar.Buffer.live h (FStar.Buffer.sub b i len) /\ FStar.Buffer.as_seq h (FStar.Buffer.sub b i len) == FStar.Seq.Base.slice (FStar.Buffer.as_seq h b) (FStar.UInt32.v i) (FStar.UInt32.v i + FStar.UInt32.v len)) [SMTPat (FStar.Buffer.sub b i len); SMTPat (FStar.Buffer.live h b)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "Prims.op_Addition", "FStar.Monotonic.HyperStack.mem", "FStar.Seq.Base.lemma_eq_intro", "FStar.Buffer.as_seq", "FStar.Buffer.sub", "FStar.Seq.Base.slice", "Prims.unit", "FStar.Buffer.live", "Prims.squash", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "FStar.Buffer.includes", "Prims.int", "Prims.l_or", "Prims.op_GreaterThanOrEqual", "FStar.UInt.size", "FStar.UInt32.n", "Prims.Nil" ]

[]

true

false

true

false

let lemma_sub_spec (#a: Type) (b: buffer a) (i: UInt32.t) (len: UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] =

Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len))

false

FStar.Buffer.fst

FStar.Buffer.eq_lemma1

val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))]

let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 76, "end_line": 1094, "start_col": 0, "start_line": 1093 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 1, "max_fuel": 0, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b1: FStar.Buffer.buffer a -> b2: FStar.Buffer.buffer a -> len: FStar.UInt32.t { FStar.UInt32.v len <= FStar.Buffer.length b1 /\ FStar.UInt32.v len <= FStar.Buffer.length b2 } -> h: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires forall (j: Prims.nat). j < FStar.UInt32.v len ==> FStar.Buffer.get h b1 j == FStar.Buffer.get h b2 j) (ensures FStar.Buffer.equal h (FStar.Buffer.sub b1 0ul len) h (FStar.Buffer.sub b2 0ul len)) [SMTPat (FStar.Buffer.equal h (FStar.Buffer.sub b1 0ul len) h (FStar.Buffer.sub b2 0ul len))]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.eqtype", "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "FStar.Seq.Base.lemma_eq_intro", "FStar.Buffer.as_seq", "FStar.Buffer.sub", "FStar.UInt32.__uint_to_t", "Prims.unit" ]

[]

true

false

true

false

let eq_lemma1 #a b1 b2 len h =

Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len))

false

FStar.Buffer.fst

FStar.Buffer.op_Array_Assignment

val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z ))

let op_Array_Assignment #a b n z = upd #a b n z

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 47, "end_line": 1147, "start_col": 0, "start_line": 1147 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 20, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer a -> n: FStar.UInt32.t -> z: a -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "FStar.Buffer.upd", "Prims.unit" ]

[]

false

true

false

let ( .()<- ) #a b n z =

upd #a b n z

false

FStar.Buffer.fst

FStar.Buffer.join

val join (#t: _) (b: buffer t) (b': buffer t { b.max_length == b'.max_length /\ b.content === b'.content /\ idx b + length b == idx b' }) : Tot (buffer t)

let join #t (b:buffer t) (b':buffer t{b.max_length == b'.max_length /\ b.content === b'.content /\ idx b + length b == idx b'}) : Tot (buffer t) = MkBuffer (b.max_length) (b.content) (b.idx) (FStar.UInt32.(b.length +^ b'.length))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 86, "end_line": 1219, "start_col": 0, "start_line": 1218 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let op_Array_Assignment #a b n z = upd #a b n z let lemma_modifies_one_trans_1 (#a:Type) (b:buffer a) (h0:mem) (h1:mem) (h2:mem): Lemma (requires (modifies_one (frameOf b) h0 h1 /\ modifies_one (frameOf b) h1 h2)) (ensures (modifies_one (frameOf b) h0 h2)) [SMTPat (modifies_one (frameOf b) h0 h1); SMTPat (modifies_one (frameOf b) h1 h2)] = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (** Corresponds to memcpy *) val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) == Seq.slice (as_seq h0 b) (v idx_b+v len) (length b) )) let rec blit #t a idx_a b idx_b len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get() in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b)) end (** Corresponds to memset *) val fill: #t:Type -> b:buffer t -> z:t -> len:UInt32.t{v len <= length b} -> Stack unit (requires (fun h -> live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) 0 (v len) == Seq.create (v len) z /\ Seq.slice (as_seq h1 b) (v len) (length b) == Seq.slice (as_seq h0 b) (v len) (length b) )) let rec fill #t b z len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in fill #t b z len'; b.(len') <- z; let h = HST.get() in Seq.snoc_slice_index (as_seq h b) 0 (v len'); Seq.lemma_tail_slice (as_seq h b) (v len') (length b) end; let h1 = HST.get() in Seq.lemma_eq_intro (Seq.slice (as_seq h1 b) 0 (v len)) (Seq.create (v len) z) let split #t (b:buffer t) (i:UInt32.t{v i <= length b}) : Tot (buffer t * buffer t) = sub b 0ul i, offset b i

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer t -> b': FStar.Buffer.buffer t { MkBuffer?.max_length b == MkBuffer?.max_length b' /\ MkBuffer?.content b === MkBuffer?.content b' /\ FStar.Buffer.idx b + FStar.Buffer.length b == FStar.Buffer.idx b' } -> FStar.Buffer.buffer t

Prims.Tot

[ "total" ]

[]

[ "FStar.Buffer.buffer", "Prims.l_and", "Prims.eq2", "FStar.UInt32.t", "FStar.Buffer.__proj__MkBuffer__item__max_length", "Prims.op_Equals_Equals_Equals", "FStar.HyperStack.ST.reference", "FStar.Buffer.lseq", "FStar.UInt32.v", "FStar.Buffer.__proj__MkBuffer__item__content", "Prims.int", "Prims.op_Addition", "FStar.Buffer.idx", "FStar.Buffer.length", "FStar.Buffer.MkBuffer", "FStar.Buffer.__proj__MkBuffer__item__idx", "FStar.UInt32.op_Plus_Hat", "FStar.Buffer.__proj__MkBuffer__item__length" ]

[]

false

let join #t (b: buffer t) (b': buffer t { b.max_length == b'.max_length /\ b.content === b'.content /\ idx b + length b == idx b' }) : Tot (buffer t) =

MkBuffer (b.max_length) (b.content) (b.idx) (let open FStar.UInt32 in b.length +^ b'.length)

false

FStar.Buffer.fst

FStar.Buffer.eq_lemma2

val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))]

let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 66, "end_line": 1112, "start_col": 0, "start_line": 1108 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 20, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b1: FStar.Buffer.buffer a -> b2: FStar.Buffer.buffer a -> len: FStar.UInt32.t { FStar.UInt32.v len <= FStar.Buffer.length b1 /\ FStar.UInt32.v len <= FStar.Buffer.length b2 } -> h: FStar.Monotonic.HyperStack.mem -> FStar.Pervasives.Lemma (requires FStar.Buffer.equal h (FStar.Buffer.sub b1 0ul len) h (FStar.Buffer.sub b2 0ul len)) (ensures forall (j: Prims.nat). j < FStar.UInt32.v len ==> FStar.Buffer.get h b1 j == FStar.Buffer.get h b2 j) [SMTPat (FStar.Buffer.equal h (FStar.Buffer.sub b1 0ul len) h (FStar.Buffer.sub b2 0ul len))]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.eqtype", "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Monotonic.HyperStack.mem", "Prims.cut", "Prims.l_Forall", "Prims.nat", "Prims.l_imp", "Prims.op_LessThan", "Prims.eq2", "FStar.Buffer.get", "FStar.Seq.Base.index", "Prims.unit", "FStar.Seq.Base.seq", "FStar.Seq.Base.length", "FStar.Buffer.sub", "FStar.UInt32.uint_to_t", "FStar.Buffer.as_seq", "FStar.UInt32.__uint_to_t" ]

[]

false

true

false

let eq_lemma2 #a b1 b2 len h =

let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j: nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j: nat). j < v len ==> get h b2 j == Seq.index s2 j)

false

Pulse.JoinComp.fst

Pulse.JoinComp.join_comps

val join_comps (g_then:env) (e_then:st_term) (c_then:comp_st) (e_then_typing:st_typing g_then e_then c_then) (g_else:env) (e_else:st_term) (c_else:comp_st) (e_else_typing:st_typing g_else e_else c_else) (post:post_hint_t) : TacS (c:comp_st & st_typing g_then e_then c & st_typing g_else e_else c) (requires comp_post_matches_hint c_then (Some post) /\ comp_post_matches_hint c_else (Some post) /\ comp_pre c_then == comp_pre c_else) (ensures fun (| c, _, _ |) -> st_comp_of_comp c == st_comp_of_comp c_then /\ comp_post_matches_hint c (Some post))

let join_comps (g_then:env) (e_then:st_term) (c_then:comp_st) (e_then_typing:st_typing g_then e_then c_then) (g_else:env) (e_else:st_term) (c_else:comp_st) (e_else_typing:st_typing g_else e_else c_else) (post:post_hint_t) : TacS (c:comp_st & st_typing g_then e_then c & st_typing g_else e_else c) (requires comp_post_matches_hint c_then (Some post) /\ comp_post_matches_hint c_else (Some post) /\ comp_pre c_then == comp_pre c_else) (ensures fun (| c, _, _ |) -> st_comp_of_comp c == st_comp_of_comp c_then /\ comp_post_matches_hint c (Some post)) = let g = g_then in assert (st_comp_of_comp c_then == st_comp_of_comp c_else); match c_then, c_else with | C_STAtomic _ obs1 _, C_STAtomic _ obs2 _ -> let obs = join_obs obs1 obs2 in let e_then_typing = T_Lift _ _ _ _ e_then_typing (Lift_Observability g_then c_then obs) in let e_else_typing = T_Lift _ _ _ _ e_else_typing (Lift_Observability g_else c_else obs) in (| _, e_then_typing, e_else_typing |) | _ -> (| _, e_then_typing, e_else_typing |)

{ "file_name": "lib/steel/pulse/Pulse.JoinComp.fst", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 41, "end_line": 111, "start_col": 0, "start_line": 82 }

(* Copyright 2023 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Pulse.JoinComp open Pulse.Syntax open Pulse.Typing open Pulse.Typing.Combinators open Pulse.Checker.Pure open Pulse.Checker.Base open Pulse.Checker.Prover module T = FStar.Tactics.V2 module P = Pulse.Syntax.Printer module Metatheory = Pulse.Typing.Metatheory module RU = Pulse.RuntimeUtils (* For now we just create a term with the union, but this could potentially be smarter *) let compute_iname_join (is1 is2 : term) : term = tm_join_inames is1 is2 let lift_atomic_to_st (g : env) (e : st_term) (c : comp_st{C_STAtomic? c}) (d : st_typing g e c) : Pure (c':comp_st & st_typing g e c') (requires True) (ensures fun (| c', _ |) -> st_comp_of_comp c' == st_comp_of_comp c /\ ctag_of_comp_st c' == STT) = let C_STAtomic _ _ c_st = c in let c' = C_ST c_st in let d' : st_typing g e c' = T_Lift g e c c' d (Lift_STAtomic_ST g c) in (| c', d' |) let lift_ghost_to_atomic (g : env) (e : st_term) (c : comp_st{C_STGhost? c}) (d : st_typing g e c) : TacS (c':comp_st & st_typing g e c') (requires True) (ensures fun (| c', _ |) -> st_comp_of_comp c' == st_comp_of_comp c /\ ctag_of_comp_st c' == STT_Atomic /\ tm_emp_inames == C_STAtomic?.inames c') = let C_STGhost c_st = c in let w : non_informative_c g c = get_non_informative_witness g (comp_u c) (comp_res c) in FStar.Tactics.BreakVC.break_vc(); // somehow this proof is unstable, this helps let c' = C_STAtomic tm_emp_inames Neutral c_st in let d' : st_typing g e c' = T_Lift g e c c' d (Lift_Ghost_Neutral g c w) in assert (st_comp_of_comp c' == st_comp_of_comp c); assert (ctag_of_comp_st c' == STT_Atomic); assert (tm_emp_inames == C_STAtomic?.inames c'); (| c', d' |) (* This matches the effects of the two branches, without necessarily matching inames. *) #push-options "--z3rlimit 20" open Pulse.Checker.Base (* NB: g_then and g_else are equal except for containing one extra

{ "checked_file": "/", "dependencies": [ "Pulse.Typing.Metatheory.fsti.checked", "Pulse.Typing.Combinators.fsti.checked", "Pulse.Typing.fst.checked", "Pulse.Syntax.Printer.fsti.checked", "Pulse.Syntax.fst.checked", "Pulse.RuntimeUtils.fsti.checked", "Pulse.Checker.Pure.fsti.checked", "Pulse.Checker.Prover.fsti.checked", "Pulse.Checker.Base.fsti.checked", "prims.fst.checked", "FStar.Tactics.V2.fst.checked", "FStar.Tactics.BreakVC.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked" ], "interface_file": true, "source_file": "Pulse.JoinComp.fst" }

[ { "abbrev": false, "full_module": "Pulse.Checker.Base", "short_module": null }, { "abbrev": true, "full_module": "Pulse.RuntimeUtils", "short_module": "RU" }, { "abbrev": true, "full_module": "Pulse.Typing.Metatheory", "short_module": "Metatheory" }, { "abbrev": true, "full_module": "Pulse.Syntax.Printer", "short_module": "P" }, { "abbrev": true, "full_module": "FStar.Tactics.V2", "short_module": "T" }, { "abbrev": false, "full_module": "Pulse.Checker.Prover", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Checker.Base", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Checker.Pure", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Typing.Combinators", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Typing", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Syntax", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Typing", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Syntax", "short_module": null }, { "abbrev": false, "full_module": "Pulse", "short_module": null }, { "abbrev": false, "full_module": "Pulse", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 20, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

g_then: Pulse.Typing.Env.env -> e_then: Pulse.Syntax.Base.st_term -> c_then: Pulse.Syntax.Base.comp_st -> e_then_typing: Pulse.Typing.st_typing g_then e_then c_then -> g_else: Pulse.Typing.Env.env -> e_else: Pulse.Syntax.Base.st_term -> c_else: Pulse.Syntax.Base.comp_st -> e_else_typing: Pulse.Typing.st_typing g_else e_else c_else -> post: Pulse.Typing.post_hint_t -> Pulse.JoinComp.TacS (FStar.Pervasives.dtuple3 Pulse.Syntax.Base.comp_st (fun c -> Pulse.Typing.st_typing g_then e_then c) (fun c _ -> Pulse.Typing.st_typing g_else e_else c))

Pulse.JoinComp.TacS

[]

[ "Pulse.Typing.Env.env", "Pulse.Syntax.Base.st_term", "Pulse.Syntax.Base.comp_st", "Pulse.Typing.st_typing", "Pulse.Typing.post_hint_t", "FStar.Pervasives.Native.Mktuple2", "Pulse.Syntax.Base.comp", "Pulse.Syntax.Base.term", "Pulse.Syntax.Base.observability", "Pulse.Syntax.Base.st_comp", "FStar.Pervasives.Mkdtuple3", "Pulse.Syntax.Base.C_STAtomic", "Pulse.Syntax.Base.comp_inames", "Pulse.Syntax.Base.st_comp_of_comp", "Pulse.Typing.T_Lift", "Pulse.Typing.Lift_Observability", "Pulse.Typing.join_obs", "FStar.Pervasives.Native.tuple2", "FStar.Pervasives.dtuple3", "Prims.unit", "Prims._assert", "Prims.eq2", "Prims.l_and", "Pulse.Typing.comp_post_matches_hint", "FStar.Pervasives.Native.Some", "Pulse.Syntax.Base.vprop", "Pulse.Syntax.Base.comp_pre" ]

[]

false

true

false

let join_comps (g_then: env) (e_then: st_term) (c_then: comp_st) (e_then_typing: st_typing g_then e_then c_then) (g_else: env) (e_else: st_term) (c_else: comp_st) (e_else_typing: st_typing g_else e_else c_else) (post: post_hint_t) : TacS (c: comp_st & st_typing g_then e_then c & st_typing g_else e_else c) (requires comp_post_matches_hint c_then (Some post) /\ comp_post_matches_hint c_else (Some post) /\ comp_pre c_then == comp_pre c_else) (ensures fun (| c , _ , _ |) -> st_comp_of_comp c == st_comp_of_comp c_then /\ comp_post_matches_hint c (Some post)) =

let g = g_then in assert (st_comp_of_comp c_then == st_comp_of_comp c_else); match c_then, c_else with | C_STAtomic _ obs1 _, C_STAtomic _ obs2 _ -> let obs = join_obs obs1 obs2 in let e_then_typing = T_Lift _ _ _ _ e_then_typing (Lift_Observability g_then c_then obs) in let e_else_typing = T_Lift _ _ _ _ e_else_typing (Lift_Observability g_else c_else obs) in (| _, e_then_typing, e_else_typing |) | _ -> (| _, e_then_typing, e_else_typing |)

false

FStar.Buffer.fst

FStar.Buffer.fill

val fill: #t:Type -> b:buffer t -> z:t -> len:UInt32.t{v len <= length b} -> Stack unit (requires (fun h -> live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) 0 (v len) == Seq.create (v len) z /\ Seq.slice (as_seq h1 b) (v len) (length b) == Seq.slice (as_seq h0 b) (v len) (length b) ))

let rec fill #t b z len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in fill #t b z len'; b.(len') <- z; let h = HST.get() in Seq.snoc_slice_index (as_seq h b) 0 (v len'); Seq.lemma_tail_slice (as_seq h b) (v len') (length b) end; let h1 = HST.get() in Seq.lemma_eq_intro (Seq.slice (as_seq h1 b) 0 (v len)) (Seq.create (v len) z)

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 79, "end_line": 1212, "start_col": 0, "start_line": 1199 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let op_Array_Assignment #a b n z = upd #a b n z let lemma_modifies_one_trans_1 (#a:Type) (b:buffer a) (h0:mem) (h1:mem) (h2:mem): Lemma (requires (modifies_one (frameOf b) h0 h1 /\ modifies_one (frameOf b) h1 h2)) (ensures (modifies_one (frameOf b) h0 h2)) [SMTPat (modifies_one (frameOf b) h0 h1); SMTPat (modifies_one (frameOf b) h1 h2)] = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (** Corresponds to memcpy *) val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) == Seq.slice (as_seq h0 b) (v idx_b+v len) (length b) )) let rec blit #t a idx_a b idx_b len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get() in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b)) end (** Corresponds to memset *) val fill: #t:Type -> b:buffer t -> z:t -> len:UInt32.t{v len <= length b} -> Stack unit (requires (fun h -> live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) 0 (v len) == Seq.create (v len) z /\ Seq.slice (as_seq h1 b) (v len) (length b) ==

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: FStar.Buffer.buffer t -> z: t -> len: FStar.UInt32.t{FStar.UInt32.v len <= FStar.Buffer.length b} -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Seq.Base.lemma_eq_intro", "FStar.Seq.Base.slice", "FStar.Buffer.as_seq", "FStar.Seq.Base.create", "Prims.unit", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.UInt32.op_Equals_Hat", "FStar.UInt32.__uint_to_t", "Prims.bool", "FStar.Seq.Properties.lemma_tail_slice", "FStar.Seq.Properties.snoc_slice_index", "FStar.Buffer.op_Array_Assignment", "FStar.Buffer.fill", "FStar.UInt32.op_Subtraction_Hat" ]

[ "recursion" ]

false

true

false

let rec fill #t b z len =

let h0 = HST.get () in if len =^ 0ul then () else (let len' = len -^ 1ul in fill #t b z len'; b.(len') <- z; let h = HST.get () in Seq.snoc_slice_index (as_seq h b) 0 (v len'); Seq.lemma_tail_slice (as_seq h b) (v len') (length b)); let h1 = HST.get () in Seq.lemma_eq_intro (Seq.slice (as_seq h1 b) 0 (v len)) (Seq.create (v len) z)

false

FStar.Buffer.fst

FStar.Buffer.eqb

val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len))))

let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 11, "end_line": 1128, "start_col": 0, "start_line": 1121 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 20, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b1: FStar.Buffer.buffer a -> b2: FStar.Buffer.buffer a -> len: FStar.UInt32.t { FStar.UInt32.v len <= FStar.Buffer.length b1 /\ FStar.UInt32.v len <= FStar.Buffer.length b2 } -> FStar.HyperStack.ST.ST Prims.bool

FStar.HyperStack.ST.ST

[]

[ "Prims.eqtype", "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.l_and", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.UInt32.op_Equals_Hat", "FStar.UInt32.__uint_to_t", "Prims.bool", "FStar.Buffer.eqb", "Prims.op_Equality", "FStar.Buffer.index", "FStar.UInt32.op_Subtraction_Hat" ]

[ "recursion" ]

false

true

false

let rec eqb #a b1 b2 len =

if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false

false

FStar.Buffer.fst

FStar.Buffer.assignL

val assignL (#a: _) (l: list a) (b: buffer a) : Stack unit (requires (fun h0 -> live h0 b /\ length b = List.Tot.length l)) (ensures (fun h0 _ h1 -> live h1 b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.seq_of_list l))

let rec assignL #a (l: list a) (b: buffer a): Stack unit (requires (fun h0 -> live h0 b /\ length b = List.Tot.length l)) (ensures (fun h0 _ h1 -> live h1 b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.seq_of_list l)) = lemma_seq_of_list_induction l; match l with | [] -> () | hd :: tl -> let b_hd = sub b 0ul 1ul in let b_tl = offset b 1ul in b_hd.(0ul) <- hd; assignL tl b_tl; let h = HST.get () in assert (get h b_hd 0 == hd); assert (as_seq h b_tl == Seq.seq_of_list tl); assert (Seq.equal (as_seq h b) (Seq.append (as_seq h b_hd) (as_seq h b_tl))); assert (Seq.equal (as_seq h b) (Seq.seq_of_list l))

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 57, "end_line": 1464, "start_col": 0, "start_line": 1444 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let op_Array_Assignment #a b n z = upd #a b n z let lemma_modifies_one_trans_1 (#a:Type) (b:buffer a) (h0:mem) (h1:mem) (h2:mem): Lemma (requires (modifies_one (frameOf b) h0 h1 /\ modifies_one (frameOf b) h1 h2)) (ensures (modifies_one (frameOf b) h0 h2)) [SMTPat (modifies_one (frameOf b) h0 h1); SMTPat (modifies_one (frameOf b) h1 h2)] = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (** Corresponds to memcpy *) val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) == Seq.slice (as_seq h0 b) (v idx_b+v len) (length b) )) let rec blit #t a idx_a b idx_b len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get() in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b)) end (** Corresponds to memset *) val fill: #t:Type -> b:buffer t -> z:t -> len:UInt32.t{v len <= length b} -> Stack unit (requires (fun h -> live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) 0 (v len) == Seq.create (v len) z /\ Seq.slice (as_seq h1 b) (v len) (length b) == Seq.slice (as_seq h0 b) (v len) (length b) )) let rec fill #t b z len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in fill #t b z len'; b.(len') <- z; let h = HST.get() in Seq.snoc_slice_index (as_seq h b) 0 (v len'); Seq.lemma_tail_slice (as_seq h b) (v len') (length b) end; let h1 = HST.get() in Seq.lemma_eq_intro (Seq.slice (as_seq h1 b) 0 (v len)) (Seq.create (v len) z) let split #t (b:buffer t) (i:UInt32.t{v i <= length b}) : Tot (buffer t * buffer t) = sub b 0ul i, offset b i let join #t (b:buffer t) (b':buffer t{b.max_length == b'.max_length /\ b.content === b'.content /\ idx b + length b == idx b'}) : Tot (buffer t) = MkBuffer (b.max_length) (b.content) (b.idx) (FStar.UInt32.(b.length +^ b'.length)) val no_upd_lemma_0: #t:Type -> h0:mem -> h1:mem -> b:buffer t -> Lemma (requires (live h0 b /\ modifies_0 h0 h1)) (ensures (live h0 b /\ live h1 b /\ equal h0 b h1 b)) [SMTPat (modifies_0 h0 h1); SMTPat (live h0 b)] let no_upd_lemma_0 #t h0 h1 b = () val no_upd_lemma_1: #t:Type -> #t':Type -> h0:mem -> h1:mem -> a:buffer t -> b:buffer t' -> Lemma (requires (live h0 b /\ disjoint a b /\ modifies_1 a h0 h1)) (ensures (live h0 b /\ live h1 b /\ equal h0 b h1 b)) [SMTPat (modifies_1 a h0 h1); SMTPat (live h0 b)] let no_upd_lemma_1 #t #t' h0 h1 a b = () #reset-options "--z3rlimit 30 --initial_fuel 0 --max_fuel 0" val no_upd_lemma_2: #t:Type -> #t':Type -> #t'':Type -> h0:mem -> h1:mem -> a:buffer t -> a':buffer t' -> b:buffer t'' -> Lemma (requires (live h0 b /\ disjoint a b /\ disjoint a' b /\ modifies_2 a a' h0 h1)) (ensures (live h0 b /\ live h1 b /\ equal h0 b h1 b)) [SMTPat (live h0 b); SMTPat (modifies_2 a a' h0 h1)] let no_upd_lemma_2 #t #t' #t'' h0 h1 a a' b = () val no_upd_lemma_2_1: #t:Type -> #t':Type -> h0:mem -> h1:mem -> a:buffer t -> b:buffer t' -> Lemma (requires (live h0 b /\ disjoint a b /\ modifies_2_1 a h0 h1)) (ensures (live h0 b /\ live h1 b /\ equal h0 b h1 b)) [SMTPat (live h0 b); SMTPat (modifies_2_1 a h0 h1)] let no_upd_lemma_2_1 #t #t' h0 h1 a b = () val no_upd_fresh: #t:Type -> h0:mem -> h1:mem -> a:buffer t -> Lemma (requires (live h0 a /\ fresh_frame h0 h1)) (ensures (live h0 a /\ live h1 a /\ equal h0 a h1 a)) [SMTPat (live h0 a); SMTPat (fresh_frame h0 h1)] let no_upd_fresh #t h0 h1 a = () val no_upd_popped: #t:Type -> h0:mem -> h1:mem -> b:buffer t -> Lemma (requires (live h0 b /\ frameOf b =!= HS.get_tip h0 /\ popped h0 h1)) (ensures (live h0 b /\ live h1 b /\ equal h0 b h1 b)) [SMTPat (live h0 b); SMTPat (popped h0 h1)] let no_upd_popped #t h0 h1 b = () (* Modifies of subset lemmas *) let lemma_modifies_sub_0 h0 h1 : Lemma (requires (h1 == h0)) (ensures (modifies_0 h0 h1)) [SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_sub_1 #t h0 h1 (b:buffer t) : Lemma (requires (h1 == h0)) (ensures (modifies_1 b h0 h1)) [SMTPat (live h0 b); SMTPat (modifies_1 b h0 h1)] = () let lemma_modifies_sub_2 #t #t' h0 h1 (b:buffer t) (b':buffer t') : Lemma (requires (h1 == h0)) (ensures (modifies_2 b b' h0 h1)) [SMTPat (live h0 b); SMTPat (live h0 b'); SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_sub_2_1 #t h0 h1 (b:buffer t) : Lemma (requires (modifies_0 h0 h1 /\ live h0 b)) (ensures (modifies_2_1 b h0 h1)) [SMTPat (live h0 b); SMTPat (modifies_2_1 b h0 h1)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let modifies_subbuffer_1 (#t:Type) h0 h1 (sub:buffer t) (a:buffer t) : Lemma (requires (live h0 a /\ modifies_1 sub h0 h1 /\ includes a sub)) (ensures (modifies_1 a h0 h1 /\ live h1 a)) [SMTPat (modifies_1 sub h0 h1); SMTPat (includes a sub)] = () let modifies_subbuffer_2 (#t:Type) (#t':Type) h0 h1 (sub:buffer t) (a':buffer t') (a:buffer t) : Lemma (requires (live h0 a /\ live h0 a' /\ includes a sub /\ modifies_2 sub a' h0 h1 )) (ensures (modifies_2 a a' h0 h1 /\ modifies_2 a' a h0 h1 /\ live h1 a)) [SMTPat (modifies_2 sub a' h0 h1); SMTPat (includes a sub)] = () let modifies_subbuffer_2' (#t:Type) (#t':Type) h0 h1 (sub:buffer t) (a':buffer t') (a:buffer t) : Lemma (requires (live h0 a /\ live h0 a' /\ includes a sub /\ modifies_2 a' sub h0 h1 )) (ensures (modifies_2 a a' h0 h1 /\ live h1 a)) [SMTPat (modifies_2 a' sub h0 h1); SMTPat (includes a sub)] = () let modifies_subbuffer_2_1 (#t:Type) h0 h1 (sub:buffer t) (a:buffer t) : Lemma (requires (live h0 a /\ includes a sub /\ modifies_2_1 sub h0 h1)) (ensures (modifies_2_1 a h0 h1 /\ live h1 a)) [SMTPat (modifies_2_1 sub h0 h1); SMTPat (includes a sub)] = () let modifies_subbuffer_2_prime (#t:Type) h0 h1 (sub1:buffer t) (sub2:buffer t) (a:buffer t) : Lemma (requires (live h0 a /\ includes a sub1 /\ includes a sub2 /\ modifies_2 sub1 sub2 h0 h1)) (ensures (modifies_1 a h0 h1 /\ live h1 a)) [SMTPat (modifies_2 sub1 sub2 h0 h1); SMTPat (includes a sub1); SMTPat (includes a sub2)] = () let modifies_popped_3_2 (#t:Type) #t' (a:buffer t) (b:buffer t') h0 h1 h2 h3 : Lemma (requires (live h0 a /\ live h0 b /\ fresh_frame h0 h1 /\ popped h2 h3 /\ modifies_3_2 a b h1 h2)) (ensures (modifies_2 a b h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3); SMTPat (modifies_3_2 a b h1 h2)] = () let modifies_popped_2 (#t:Type) #t' (a:buffer t) (b:buffer t') h0 h1 h2 h3 : Lemma (requires (live h0 a /\ live h0 b /\ fresh_frame h0 h1 /\ popped h2 h3 /\ modifies_2 a b h1 h2)) (ensures (modifies_2 a b h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3); SMTPat (modifies_2 a b h1 h2)] = () let modifies_popped_1 (#t:Type) (a:buffer t) h0 h1 h2 h3 : Lemma (requires (live h0 a /\ fresh_frame h0 h1 /\ popped h2 h3 /\ modifies_2_1 a h1 h2)) (ensures (modifies_1 a h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3); SMTPat (modifies_2_1 a h1 h2)] = () let modifies_popped_1' (#t:Type) (a:buffer t) h0 h1 h2 h3 : Lemma (requires (live h0 a /\ fresh_frame h0 h1 /\ popped h2 h3 /\ modifies_1 a h1 h2)) (ensures (modifies_1 a h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3); SMTPat (modifies_1 a h1 h2)] = () let modifies_popped_0 h0 h1 h2 h3 : Lemma (requires (fresh_frame h0 h1 /\ popped h2 h3 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3); SMTPat (modifies_0 h1 h2)] = () let live_popped (#t:Type) (b:buffer t) h0 h1 : Lemma (requires (popped h0 h1 /\ live h0 b /\ frameOf b =!= HS.get_tip h0)) (ensures (live h1 b)) [SMTPat (popped h0 h1); SMTPat (live h0 b)] = () let live_fresh (#t:Type) (b:buffer t) h0 h1 : Lemma (requires (fresh_frame h0 h1 /\ live h0 b)) (ensures (live h1 b)) [SMTPat (fresh_frame h0 h1); SMTPat (live h0 b)] = () let modifies_0_to_2_1_lemma (#t:Type) h0 h1 (b:buffer t) : Lemma (requires (modifies_0 h0 h1 /\ live h0 b)) (ensures (modifies_2_1 b h0 h1)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (live h0 b) ] = () let lemma_modifies_none_push_pop h0 h1 h2 : Lemma (requires (fresh_frame h0 h1 /\ popped h1 h2)) (ensures (modifies_none h0 h2)) = () let lemma_modifies_0_push_pop h0 h1 h2 h3 : Lemma (requires (fresh_frame h0 h1 /\ modifies_0 h1 h2 /\ popped h2 h3)) (ensures (modifies_none h0 h3)) = () let modifies_1_to_2_1_lemma (#t:Type) h0 h1 (b:buffer t) : Lemma (requires (modifies_1 b h0 h1 /\ live h0 b)) (ensures (modifies_2_1 b h0 h1)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (live h0 b) ] = () (* let modifies_1_to_2_lemma (#t:Type) #t' h0 h1 (b:buffer t) (b':buffer t'): Lemma *) (* (requires (modifies_1 b h0 h1 /\ live h0 b)) *) (* (ensures (modifies_2 b b' h0 h1)) *) (* [SMTPat (modifies_2 b b' h0 h1); SMTPat (live h0 b) ] *) (* = () *) let modifies_poppable_0 (h0 h1:mem) : Lemma (requires (modifies_0 h0 h1 /\ HS.poppable h0)) (ensures (HS.poppable h1)) [SMTPat (modifies_0 h0 h1)] = () let modifies_poppable_1 #t (h0 h1:mem) (b:buffer t) : Lemma (requires (modifies_1 b h0 h1 /\ HS.poppable h0)) (ensures (HS.poppable h1)) [SMTPat (modifies_1 b h0 h1)] = () let modifies_poppable_2_1 #t (h0 h1:mem) (b:buffer t) : Lemma (requires (modifies_2_1 b h0 h1 /\ HS.poppable h0)) (ensures (HS.poppable h1)) [SMTPat (modifies_2_1 b h0 h1)] = () let modifies_poppable_2 #t #t' (h0 h1:mem) (b:buffer t) (b':buffer t') : Lemma (requires (modifies_2 b b' h0 h1 /\ HS.poppable h0)) (ensures (HS.poppable h1)) [SMTPat (modifies_2 b' b h0 h1)] = () let modifies_poppable_3_2 #t #t' (h0 h1:mem) (b:buffer t) (b':buffer t') : Lemma (requires (modifies_3_2 b b' h0 h1 /\ HS.poppable h0)) (ensures (HS.poppable h1)) [SMTPat (modifies_3_2 b' b h0 h1)] = () let lemma_fresh_poppable (h0 h1:mem) : Lemma (requires (fresh_frame h0 h1)) (ensures (poppable h1)) [SMTPat (fresh_frame h0 h1)] = () let lemma_equal_domains_popped (h0 h1 h2 h3:mem) : Lemma (requires (equal_domains h0 h1 /\ popped h0 h2 /\ popped h1 h3)) (ensures (equal_domains h2 h3)) = () let lemma_equal_domains (h0 h1 h2 h3:mem) : Lemma (requires (fresh_frame h0 h1 /\ equal_domains h1 h2 /\ popped h2 h3)) (ensures (equal_domains h0 h3)) [SMTPat (fresh_frame h0 h1); SMTPat (equal_domains h1 h2); SMTPat (popped h2 h3)] = () let lemma_equal_domains_2 (h0 h1 h2 h3 h4:mem) : Lemma (requires (fresh_frame h0 h1 /\ modifies_0 h1 h2 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h2) /\ equal_domains h2 h3 /\ popped h3 h4)) (ensures (equal_domains h0 h4)) [SMTPat (fresh_frame h0 h1); SMTPat (modifies_0 h1 h2); SMTPat (popped h3 h4)] = () #reset-options "--z3rlimit 50"

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

l: Prims.list a -> b: FStar.Buffer.buffer a -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Prims.list", "FStar.Buffer.buffer", "Prims.unit", "Prims._assert", "FStar.Seq.Base.equal", "FStar.Buffer.as_seq", "FStar.Seq.Base.seq_of_list", "FStar.Seq.Base.append", "Prims.eq2", "FStar.Seq.Base.seq", "Prims.l_or", "Prims.nat", "FStar.List.Tot.Base.length", "FStar.Seq.Base.length", "FStar.Buffer.length", "FStar.Buffer.get", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer.assignL", "FStar.Buffer.op_Array_Assignment", "FStar.UInt32.__uint_to_t", "FStar.Buffer.includes", "FStar.Buffer.offset", "Prims.l_and", "Prims.int", "Prims.b2t", "Prims.op_GreaterThanOrEqual", "FStar.UInt.size", "FStar.UInt32.v", "FStar.UInt32.uint_to_t", "FStar.UInt32.t", "FStar.Buffer.sub", "FStar.Seq.Properties.lemma_seq_of_list_induction", "FStar.Buffer.live", "Prims.op_Equality", "FStar.Buffer.modifies_1" ]

[ "recursion" ]

false

true

false

let rec assignL #a (l: list a) (b: buffer a) : Stack unit (requires (fun h0 -> live h0 b /\ length b = List.Tot.length l)) (ensures (fun h0 _ h1 -> live h1 b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.seq_of_list l)) =

lemma_seq_of_list_induction l; match l with | [] -> () | hd :: tl -> let b_hd = sub b 0ul 1ul in let b_tl = offset b 1ul in b_hd.(0ul) <- hd; assignL tl b_tl; let h = HST.get () in assert (get h b_hd 0 == hd); assert (as_seq h b_tl == Seq.seq_of_list tl); assert (Seq.equal (as_seq h b) (Seq.append (as_seq h b_hd) (as_seq h b_tl))); assert (Seq.equal (as_seq h b) (Seq.seq_of_list l))

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.lemma_proj_g_pow2_128_eval

val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128)

let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ)

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 48, "end_line": 79, "start_col": 0, "start_line": 72 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128)

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> FStar.Pervasives.Lemma (ensures Spec.Exponentiation.exp_pow2 Spec.P256.mk_p256_concrete_ops Hacl.P256.PrecompTable.proj_g_pow2_64 64 == Hacl.P256.PrecompTable.proj_g_pow2_128)

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.unit", "Prims.nat", "FStar.Pervasives.assert_norm", "Prims.l_and", "Prims.eq2", "Spec.P256.PointOps.felem", "FStar.Pervasives.normalize_term_spec", "Spec.P256.PointOps.proj_point", "Hacl.Spec.PrecompBaseTable256.exp_pow2_rec", "Spec.P256.mk_p256_concrete_ops", "Hacl.P256.PrecompTable.proj_g_pow2_64", "FStar.Pervasives.Native.tuple3", "FStar.Pervasives.normalize_term", "Hacl.Spec.PrecompBaseTable256.exp_pow2_rec_is_exp_pow2" ]

[]

false

true

false

let lemma_proj_g_pow2_128_eval () =

SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX:S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY:S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ:S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ)

false

Benton2004.RHL.fst

Benton2004.RHL.exec_equiv

val exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0

let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 51, "end_line": 54, "start_col": 0, "start_line": 50 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p: Benton2004.RHL.gexp Prims.bool -> p': Benton2004.RHL.gexp Prims.bool -> f: Benton2004.computation -> f': Benton2004.computation -> Prims.GTot Type0

Prims.GTot

[ "sometrivial" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.exec_equiv", "Benton2004.RHL.interp" ]

[]

false

true

let exec_equiv (p p': gexp bool) (f f': computation) : GTot Type0 =

Benton2004.exec_equiv (interp p) (interp p') f f'

false

Benton2004.RHL.fst

Benton2004.RHL.holds_interp

val holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)]

let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 31, "end_line": 48, "start_col": 0, "start_line": 42 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

ge: Benton2004.RHL.gexp Prims.bool -> s1: FStar.DM4F.Heap.IntStoreFixed.heap -> s2: FStar.DM4F.Heap.IntStoreFixed.heap -> FStar.Pervasives.Lemma (ensures Benton2004.Aux.holds (Benton2004.RHL.interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (Benton2004.Aux.holds (Benton2004.RHL.interp ge) s1 s2)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "FStar.DM4F.Heap.IntStoreFixed.heap", "Benton2004.Aux.holds_equiv", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_True", "Prims.squash", "Prims.l_iff", "Benton2004.Aux.holds", "Prims.eq2", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] =

holds_equiv (interp ge) s1 s2

false

Benton2004.RHL.fst

Benton2004.RHL.r_skip

val r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)]

let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p)

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 19, "end_line": 69, "start_col": 0, "start_line": 64 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p: Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (ensures Benton2004.RHL.exec_equiv p p Benton2004.skip Benton2004.skip) [SMTPat (Benton2004.RHL.exec_equiv p p Benton2004.skip Benton2004.skip)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.d_skip", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_True", "Prims.squash", "Benton2004.RHL.exec_equiv", "Benton2004.skip", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] =

d_skip (interp p)

false

Benton2004.RHL.fst

Benton2004.RHL.r_if

val r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') ))

let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 40, "end_line": 212, "start_col": 0, "start_line": 186 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> b': Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> d: Benton2004.computation -> d': Benton2004.computation -> p: Benton2004.RHL.gexp Prims.bool -> p': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv (Benton2004.RHL.r_if_precond_true b b' c c' d d' p p') p' c c' /\ Benton2004.RHL.exec_equiv (Benton2004.RHL.r_if_precond_false b b' c c' d d' p p') p' d d') (ensures Benton2004.RHL.exec_equiv (Benton2004.RHL.r_if_precond b b' c c' d d' p p') p' (Benton2004.ifthenelse b c d) (Benton2004.ifthenelse b' c' d'))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "Benton2004.RHL.holds_r_if_precond", "Prims.unit", "Benton2004.RHL.holds_r_if_precond_false", "Benton2004.RHL.holds_r_if_precond_true", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.RHL.r_if_precond_true", "Benton2004.RHL.r_if_precond_false", "Prims.squash", "Benton2004.RHL.r_if_precond", "Benton2004.ifthenelse", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

true

false

true

false

let r_if (b b': exp bool) (c c' d d': computation) (p p': gexp bool) : Lemma (requires (exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d')) (ensures (exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d'))) =

holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p'

false

Benton2004.RHL.fst

Benton2004.RHL.r_sub

val r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')]

let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 63, "end_line": 262, "start_col": 0, "start_line": 251 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p1: Benton2004.RHL.gexp Prims.bool -> p2: Benton2004.RHL.gexp Prims.bool -> p1': Benton2004.RHL.gexp Prims.bool -> p2': Benton2004.RHL.gexp Prims.bool -> f: Benton2004.computation -> f': Benton2004.computation -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv p1 p2 f f' /\ Benton2004.RHL.included p1' p1 /\ Benton2004.RHL.included p2 p2') (ensures Benton2004.RHL.exec_equiv p1' p2' f f') [ SMTPat (Benton2004.RHL.exec_equiv p1' p2' f f'); SMTPat (Benton2004.RHL.exec_equiv p1 p2 f f') ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.d_csub", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.RHL.included", "Prims.squash", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let r_sub (p1 p2 p1' p2': gexp bool) (f f': computation) : Lemma (requires (exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2')) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] =

d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f'

false

Benton2004.RHL.fst

Benton2004.RHL.holds_interp_flip

val holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))]

let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi)

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 36, "end_line": 333, "start_col": 0, "start_line": 330 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

phi: Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (ensures forall (s1: FStar.DM4F.Heap.IntStoreFixed.heap) (s2: FStar.DM4F.Heap.IntStoreFixed.heap). Benton2004.Aux.holds (Benton2004.RHL.interp (Benton2004.RHL.flip phi)) s1 s2 <==> Benton2004.Aux.holds (Benton2004.flip (Benton2004.RHL.interp phi)) s1 s2) [SMTPat (Benton2004.Aux.holds (Benton2004.RHL.interp (Benton2004.RHL.flip phi)))]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.holds_flip", "FStar.DM4F.Heap.IntStoreFixed.heap", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_True", "Prims.squash", "Prims.l_Forall", "Prims.l_iff", "Benton2004.Aux.holds", "Benton2004.RHL.flip", "Benton2004.flip", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2. holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2 ) [SMTPat (holds (interp (flip phi)))] =

Benton2004.holds_flip (interp phi)

false

Benton2004.RHL.fst

Benton2004.RHL.included_alt

val included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)]

let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true)

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 71, "end_line": 249, "start_col": 0, "start_line": 245 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p1: Benton2004.RHL.gexp Prims.bool -> p2: Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (ensures Benton2004.RHL.included p1 p2 <==> (forall (s1: FStar.DM4F.Heap.IntStoreFixed.heap) (s2: FStar.DM4F.Heap.IntStoreFixed.heap). p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (Benton2004.RHL.included p1 p2)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Prims._assert", "Prims.l_Forall", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.l_iff", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Prims.eq2", "Prims.unit", "Prims.l_True", "Prims.squash", "Benton2004.RHL.included", "Prims.l_imp", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

false

true

false

let included_alt (p1 p2: gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2. p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] =

assert (forall s1 s2. holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2. holds (interp p2) s1 s2 <==> p2 s1 s2 == true)

false

Steel.Channel.Simplex.fst

Steel.Channel.Simplex.send

val send (#p:prot) (c:chan p) (#next:prot{more next}) (x:msg_t next) : SteelT unit (sender c next) (fun _ -> sender c (step next x))

let rec send (#p:prot) (c:chan p) (#next:prot{more next}) (x:msg_t next) : SteelT unit (sender c next) (fun _ -> sender c (step next x)) = let v = send_receive_prelude c in //matching v as vs,vr fails if (fst v).chan_ctr = (snd v).chan_ctr then ( rewrite_slprop (chan_inv_cond (fst v) (snd v)) (pure (fst v == snd v)) (fun _ -> ()); send_available c x (fst v) (snd v) () //TODO: inlining send_availableT here fails ) else ( rewrite_slprop (chan_inv_cond (fst v) (snd v)) (chan_inv_step (snd v) (fst v)) (fun _ -> ()); intro_chan_inv_stepT c.chan_chan (fst v) (snd v); Steel.SpinLock.release c.chan_lock; send c x )

{ "file_name": "lib/steel/Steel.Channel.Simplex.fst", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 5, "end_line": 388, "start_col": 0, "start_line": 371 }

(* Copyright 2020 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Steel.Channel.Simplex module P = Steel.Channel.Protocol open Steel.SpinLock open Steel.Memory open Steel.Effect.Atomic open Steel.Effect open Steel.HigherReference open Steel.FractionalPermission module MRef = Steel.MonotonicHigherReference module H = Steel.HigherReference let sprot = p:prot { more p } noeq type chan_val = { chan_prot : sprot; chan_msg : msg_t chan_prot; chan_ctr : nat } let mref a p = MRef.ref a p let trace_ref (p:prot) = mref (partial_trace_of p) extended_to noeq type chan_t (p:prot) = { send: ref chan_val; recv: ref chan_val; trace: trace_ref p; } let half : perm = half_perm full_perm let step (s:sprot) (x:msg_t s) = step s x let chan_inv_step_p (vrecv vsend:chan_val) : prop = (vsend.chan_prot == step vrecv.chan_prot vrecv.chan_msg /\ vsend.chan_ctr == vrecv.chan_ctr + 1) let chan_inv_step (vrecv vsend:chan_val) : vprop = pure (chan_inv_step_p vrecv vsend) let chan_inv_cond (vsend:chan_val) (vrecv:chan_val) : vprop = if vsend.chan_ctr = vrecv.chan_ctr then pure (vsend == vrecv) else chan_inv_step vrecv vsend let trace_until_prop #p (r:trace_ref p) (vr:chan_val) (tr: partial_trace_of p) : vprop = MRef.pts_to r full_perm tr `star` pure (until tr == step vr.chan_prot vr.chan_msg) let trace_until #p (r:trace_ref p) (vr:chan_val) = h_exists (trace_until_prop r vr) let chan_inv_recv #p (c:chan_t p) (vsend:chan_val) = h_exists (fun (vrecv:chan_val) -> pts_to c.recv half vrecv `star` trace_until c.trace vrecv `star` chan_inv_cond vsend vrecv) let chan_inv #p (c:chan_t p) : vprop = h_exists (fun (vsend:chan_val) -> pts_to c.send half vsend `star` chan_inv_recv c vsend) let intro_chan_inv_cond_eqT (vs vr:chan_val) : Steel unit emp (fun _ -> chan_inv_cond vs vr) (requires fun _ -> vs == vr) (ensures fun _ _ _ -> True) = intro_pure (vs == vs); rewrite_slprop (chan_inv_cond vs vs) (chan_inv_cond vs vr) (fun _ -> ()) let intro_chan_inv_cond_stepT (vs vr:chan_val) : SteelT unit (chan_inv_step vr vs) (fun _ -> chan_inv_cond vs vr) = Steel.Utils.extract_pure (chan_inv_step_p vr vs); rewrite_slprop (chan_inv_step vr vs) (chan_inv_cond vs vr) (fun _ -> ()) let intro_chan_inv_auxT #p (#vs : chan_val) (#vr : chan_val) (c:chan_t p) : SteelT unit (pts_to c.send half vs `star` pts_to c.recv half vr `star` trace_until c.trace vr `star` chan_inv_cond vs vr) (fun _ -> chan_inv c) = intro_exists _ (fun (vr:chan_val) -> pts_to c.recv half vr `star` trace_until c.trace vr `star` chan_inv_cond vs vr); intro_exists _ (fun (vs:chan_val) -> pts_to c.send half vs `star` chan_inv_recv c vs) let intro_chan_inv_stepT #p (c:chan_t p) (vs vr:chan_val) : SteelT unit (pts_to c.send half vs `star` pts_to c.recv half vr `star` trace_until c.trace vr `star` chan_inv_step vr vs) (fun _ -> chan_inv c) = intro_chan_inv_cond_stepT vs vr; intro_chan_inv_auxT c let intro_chan_inv_eqT #p (c:chan_t p) (vs vr:chan_val) : Steel unit (pts_to c.send half vs `star` pts_to c.recv half vr `star` trace_until c.trace vr) (fun _ -> chan_inv c) (requires fun _ -> vs == vr) (ensures fun _ _ _ -> True) = intro_chan_inv_cond_eqT vs vr; intro_chan_inv_auxT c noeq type chan p = { chan_chan : chan_t p; chan_lock : lock (chan_inv chan_chan) } let in_state_prop (p:prot) (vsend:chan_val) : prop = p == step vsend.chan_prot vsend.chan_msg irreducible let next_chan_val (#p:sprot) (x:msg_t p) (vs0:chan_val { in_state_prop p vs0 }) : Tot (vs:chan_val{in_state_prop (step p x) vs /\ chan_inv_step_p vs0 vs}) = { chan_prot = (step vs0.chan_prot vs0.chan_msg); chan_msg = x; chan_ctr = vs0.chan_ctr + 1 } [@@__reduce__] let in_state_slprop (p:prot) (vsend:chan_val) : vprop = pure (in_state_prop p vsend) let in_state (r:ref chan_val) (p:prot) = h_exists (fun (vsend:chan_val) -> pts_to r half vsend `star` in_state_slprop p vsend) let sender #q (c:chan q) (p:prot) = in_state c.chan_chan.send p let receiver #q (c:chan q) (p:prot) = in_state c.chan_chan.recv p let intro_chan_inv #p (c:chan_t p) (v:chan_val) : SteelT unit (pts_to c.send half v `star` pts_to c.recv half v `star` trace_until c.trace v) (fun _ -> chan_inv c) = intro_chan_inv_eqT c v v let chan_val_p (p:prot) = (vs0:chan_val { in_state_prop p vs0 }) let intro_in_state (r:ref chan_val) (p:prot) (v:chan_val_p p) : SteelT unit (pts_to r half v) (fun _ -> in_state r p) = intro_pure (in_state_prop p v); intro_exists v (fun (v:chan_val) -> pts_to r half v `star` in_state_slprop p v) let msg t p = Msg Send unit (fun _ -> p) let init_chan_val (p:prot) = v:chan_val {v.chan_prot == msg unit p} let initial_trace (p:prot) : (q:partial_trace_of p {until q == p}) = { to = p; tr=Waiting p} let intro_trace_until #q (r:trace_ref q) (tr:partial_trace_of q) (v:chan_val) : Steel unit (MRef.pts_to r full_perm tr) (fun _ -> trace_until r v) (requires fun _ -> until tr == step v.chan_prot v.chan_msg) (ensures fun _ _ _ -> True) = intro_pure (until tr == step v.chan_prot v.chan_msg); intro_exists tr (fun (tr:partial_trace_of q) -> MRef.pts_to r full_perm tr `star` pure (until tr == (step v.chan_prot v.chan_msg))); () let chan_t_sr (p:prot) (send recv:ref chan_val) = (c:chan_t p{c.send == send /\ c.recv == recv}) let intro_trace_until_init #p (c:chan_t p) (v:init_chan_val p) : SteelT unit (MRef.pts_to c.trace full_perm (initial_trace p)) (fun _ -> trace_until c.trace v) = intro_pure (until (initial_trace p) == step v.chan_prot v.chan_msg); //TODO: Not sure why I need this rewrite rewrite_slprop (MRef.pts_to c.trace full_perm (initial_trace p) `star` pure (until (initial_trace p) == step v.chan_prot v.chan_msg)) (MRef.pts_to c.trace full_perm (initial_trace p) `star` pure (until (initial_trace p) == step v.chan_prot v.chan_msg)) (fun _ -> ()); intro_exists (initial_trace p) (trace_until_prop c.trace v) let mk_chan (#p:prot) (send recv:ref chan_val) (v:init_chan_val p) : SteelT (chan_t_sr p send recv) (pts_to send half v `star` pts_to recv half v) (fun c -> chan_inv c) = let tr: trace_ref p = MRef.alloc (extended_to #p) (initial_trace p) in let c = Mkchan_t send recv tr in rewrite_slprop (MRef.pts_to tr full_perm (initial_trace p)) (MRef.pts_to c.trace full_perm (initial_trace p)) (fun _ -> ()); intro_trace_until_init c v; rewrite_slprop (pts_to send half v `star` pts_to recv half v) (pts_to c.send half v `star` pts_to c.recv half v) (fun _ -> ()); intro_chan_inv #p c v; let c' : chan_t_sr p send recv = c in rewrite_slprop (chan_inv c) (chan_inv c') (fun _ -> ()); return c' let new_chan (p:prot) : SteelT (chan p) emp (fun c -> sender c p `star` receiver c p) = let q = msg unit p in let v : chan_val = { chan_prot = q; chan_msg = (); chan_ctr = 0 } in let vp : init_chan_val p = v in let send = H.alloc v in let recv = H.alloc v in H.share recv; H.share send; (* TODO: use smt_fallback *) rewrite_slprop (pts_to send (half_perm full_perm) v `star` pts_to send (half_perm full_perm) v `star` pts_to recv (half_perm full_perm) v `star` pts_to recv (half_perm full_perm) v) (pts_to send half vp `star` pts_to send half vp `star` pts_to recv half vp `star` pts_to recv half vp) (fun _ -> ()); let c = mk_chan send recv vp in intro_in_state send p vp; intro_in_state recv p vp; let l = Steel.SpinLock.new_lock (chan_inv c) in let ch = { chan_chan = c; chan_lock = l } in rewrite_slprop (in_state send p) (sender ch p) (fun _ -> ()); rewrite_slprop (in_state recv p) (receiver ch p) (fun _ -> ()); return ch [@@__reduce__] let send_recv_in_sync (r:ref chan_val) (p:prot{more p}) #q (c:chan_t q) (vs vr:chan_val) : vprop = (pts_to c.send half vs `star` pts_to c.recv half vr `star` trace_until c.trace vr `star` pure (vs == vr) `star` in_state r p) [@@__reduce__] let sender_ahead (r:ref chan_val) (p:prot{more p}) #q (c:chan_t q) (vs vr:chan_val) : vprop = (pts_to c.send half vs `star` pts_to c.recv half vr `star` trace_until c.trace vr `star` chan_inv_step vr vs `star` in_state r p) let update_channel (#p:sprot) #q (c:chan_t q) (x:msg_t p) (vs:chan_val) (r:ref chan_val) : SteelT chan_val (pts_to r full_perm vs `star` in_state_slprop p vs) (fun vs' -> pts_to r full_perm vs' `star` (in_state_slprop (step p x) vs' `star` chan_inv_step vs vs')) = elim_pure (in_state_prop p vs); let vs' = next_chan_val x vs in H.write r vs'; intro_pure (in_state_prop (step p x) vs'); intro_pure (chan_inv_step_p vs vs'); return vs' [@@__reduce__] let send_pre_available (p:sprot) #q (c:chan_t q) (vs vr:chan_val) = send_recv_in_sync c.send p c vs vr let gather_r (#p:sprot) (r:ref chan_val) (v:chan_val) : SteelT unit (pts_to r half v `star` in_state r p) (fun _ -> pts_to r full_perm v `star` in_state_slprop p v) = let v' = witness_exists () in H.higher_ref_pts_to_injective_eq #_ #_ #_ #_ #v #_ r; H.gather #_ #_ #half #half #v #v r; rewrite_slprop (pts_to r (sum_perm half half) v) (pts_to r full_perm v) (fun _ -> ()); rewrite_slprop (in_state_slprop p v') (in_state_slprop p v) (fun _ -> ()) let send_available (#p:sprot) #q (cc:chan q) (x:msg_t p) (vs vr:chan_val) (_:unit) : SteelT unit (send_pre_available p #q cc.chan_chan vs vr) (fun _ -> sender cc (step p x)) = Steel.Utils.extract_pure (vs == vr); Steel.Utils.rewrite #_ #(send_recv_in_sync cc.chan_chan.send p cc.chan_chan vs) vr vs; elim_pure (vs == vs); gather_r cc.chan_chan.send vs; let next_vs = update_channel cc.chan_chan x vs cc.chan_chan.send in H.share cc.chan_chan.send; intro_exists next_vs (fun (next_vs:chan_val) -> pts_to cc.chan_chan.send half next_vs `star` in_state_slprop (step p x) next_vs); intro_chan_inv_stepT cc.chan_chan next_vs vs; Steel.SpinLock.release cc.chan_lock let extensible (#p:prot) (x:partial_trace_of p) = P.more x.to let next_msg_t (#p:prot) (x:partial_trace_of p) = P.next_msg_t x.to let next_trace #p (vr:chan_val) (vs:chan_val) (tr:partial_trace_of p) (s:squash (until tr == step vr.chan_prot vr.chan_msg)) (_:squash (chan_inv_step_p vr vs)) : (ts:partial_trace_of p { until ts == step vs.chan_prot vs.chan_msg }) = let msg : next_msg_t tr = vs.chan_msg in assert (extensible tr); extend_partial_trace tr msg let next_trace_st #p (vr:chan_val) (vs:chan_val) (tr:partial_trace_of p) : Steel (extension_of tr) (chan_inv_step vr vs) (fun _ -> emp) (requires fun _ -> until tr == step vr.chan_prot vr.chan_msg) (ensures fun _ ts _ -> until ts == step vs.chan_prot vs.chan_msg) = elim_pure (chan_inv_step_p vr vs); let ts : extension_of tr = next_trace vr vs tr () () in return ts let update_trace #p (r:trace_ref p) (vr:chan_val) (vs:chan_val) : Steel unit (trace_until r vr) (fun _ -> trace_until r vs) (requires fun _ -> chan_inv_step_p vr vs) (ensures fun _ _ _ -> True) = intro_pure (chan_inv_step_p vr vs); let tr = MRef.read_refine r in elim_pure (until tr == step vr.chan_prot vr.chan_msg); let ts : extension_of tr = next_trace_st vr vs tr in MRef.write r ts; intro_pure (until ts == step vs.chan_prot vs.chan_msg); intro_exists ts (fun (ts:partial_trace_of p) -> MRef.pts_to r full_perm ts `star` pure (until ts == step vs.chan_prot vs.chan_msg)) let recv_availableT (#p:sprot) #q (cc:chan q) (vs vr:chan_val) (_:unit) : SteelT (msg_t p) (sender_ahead cc.chan_chan.recv p cc.chan_chan vs vr) (fun x -> receiver cc (step p x)) = elim_pure (chan_inv_step_p vr vs); gather_r cc.chan_chan.recv vr; elim_pure (in_state_prop p vr); H.write cc.chan_chan.recv vs; H.share cc.chan_chan.recv; assert (vs.chan_prot == p); let vs_msg : msg_t p = vs.chan_msg in intro_pure (in_state_prop (step p vs_msg) vs); intro_exists vs (fun (vs:chan_val) -> pts_to cc.chan_chan.recv half vs `star` in_state_slprop (step p vs_msg) vs); update_trace cc.chan_chan.trace vr vs; intro_chan_inv cc.chan_chan vs; Steel.SpinLock.release cc.chan_lock; vs_msg #push-options "--ide_id_info_off" let send_receive_prelude (#p:prot) (cc:chan p) : SteelT (chan_val & chan_val) emp (fun v -> pts_to cc.chan_chan.send half (fst v) `star` pts_to cc.chan_chan.recv half (snd v) `star` trace_until cc.chan_chan.trace (snd v) `star` chan_inv_cond (fst v) (snd v)) = let c = cc.chan_chan in Steel.SpinLock.acquire cc.chan_lock; let vs = read_refine (chan_inv_recv cc.chan_chan) cc.chan_chan.send in let _ = witness_exists () in let vr = H.read cc.chan_chan.recv in rewrite_slprop (trace_until _ _ `star` chan_inv_cond _ _) (trace_until cc.chan_chan.trace vr `star` chan_inv_cond vs vr) (fun _ -> ()); return (vs, vr)

{ "checked_file": "/", "dependencies": [ "Steel.Utils.fst.checked", "Steel.SpinLock.fsti.checked", "Steel.MonotonicHigherReference.fsti.checked", "Steel.Memory.fsti.checked", "Steel.HigherReference.fsti.checked", "Steel.FractionalPermission.fst.checked", "Steel.Effect.Atomic.fsti.checked", "Steel.Effect.fsti.checked", "Steel.Channel.Protocol.fst.checked", "prims.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked" ], "interface_file": true, "source_file": "Steel.Channel.Simplex.fst" }

[ { "abbrev": true, "full_module": "Steel.HigherReference", "short_module": "H" }, { "abbrev": true, "full_module": "Steel.MonotonicHigherReference", "short_module": "MRef" }, { "abbrev": false, "full_module": "Steel.FractionalPermission", "short_module": null }, { "abbrev": false, "full_module": "Steel.HigherReference", "short_module": null }, { "abbrev": false, "full_module": "Steel.Effect.Atomic", "short_module": null }, { "abbrev": false, "full_module": "Steel.SpinLock", "short_module": null }, { "abbrev": true, "full_module": "Steel.Channel.Protocol", "short_module": "P" }, { "abbrev": false, "full_module": "Steel.Effect", "short_module": null }, { "abbrev": false, "full_module": "Steel.Memory", "short_module": null }, { "abbrev": false, "full_module": "Steel.Channel.Protocol", "short_module": null }, { "abbrev": false, "full_module": "Steel.Channel", "short_module": null }, { "abbrev": false, "full_module": "Steel.Channel", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

c: Steel.Channel.Simplex.chan p -> x: Steel.Channel.Protocol.msg_t next -> Steel.Effect.SteelT Prims.unit

Steel.Effect.SteelT

[]

[ "Steel.Channel.Simplex.prot", "Steel.Channel.Simplex.chan", "Prims.b2t", "Steel.Channel.Protocol.more", "Steel.Channel.Protocol.msg_t", "Prims.op_Equality", "Prims.nat", "Steel.Channel.Simplex.__proj__Mkchan_val__item__chan_ctr", "FStar.Pervasives.Native.fst", "Steel.Channel.Simplex.chan_val", "FStar.Pervasives.Native.snd", "Steel.Channel.Simplex.send_available", "Prims.unit", "Steel.Effect.Atomic.rewrite_slprop", "FStar.Ghost.hide", "FStar.Set.set", "Steel.Memory.iname", "FStar.Set.empty", "Steel.Channel.Simplex.chan_inv_cond", "Steel.Effect.Common.pure", "Prims.eq2", "Steel.Memory.mem", "Prims.bool", "Steel.Channel.Simplex.send", "Steel.SpinLock.release", "Steel.Channel.Simplex.chan_inv", "Steel.Channel.Simplex.__proj__Mkchan__item__chan_chan", "Steel.Channel.Simplex.__proj__Mkchan__item__chan_lock", "Steel.Channel.Simplex.intro_chan_inv_stepT", "Steel.Channel.Simplex.chan_inv_step", "FStar.Pervasives.Native.tuple2", "Steel.Channel.Simplex.send_receive_prelude", "Steel.Channel.Simplex.sender", "Steel.Channel.Simplex.step", "Steel.Effect.Common.vprop" ]

[ "recursion" ]

false

true

false

let rec send (#p: prot) (c: chan p) (#next: prot{more next}) (x: msg_t next) : SteelT unit (sender c next) (fun _ -> sender c (step next x)) =

let v = send_receive_prelude c in if (fst v).chan_ctr = (snd v).chan_ctr then (rewrite_slprop (chan_inv_cond (fst v) (snd v)) (pure (fst v == snd v)) (fun _ -> ()); send_available c x (fst v) (snd v) ()) else (rewrite_slprop (chan_inv_cond (fst v) (snd v)) (chan_inv_step (snd v) (fst v)) (fun _ -> ()); intro_chan_inv_stepT c.chan_chan (fst v) (snd v); Steel.SpinLock.release c.chan_lock; send c x)

false

Benton2004.RHL.fst

Benton2004.RHL.exec_equiv_flip

val exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')]

let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 45, "end_line": 343, "start_col": 0, "start_line": 335 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p: Benton2004.RHL.gexp Prims.bool -> p': Benton2004.RHL.gexp Prims.bool -> f: Benton2004.computation -> f': Benton2004.computation -> FStar.Pervasives.Lemma (ensures Benton2004.RHL.exec_equiv (Benton2004.RHL.flip p) (Benton2004.RHL.flip p') f f' <==> Benton2004.RHL.exec_equiv p p' f' f) [SMTPat (Benton2004.RHL.exec_equiv (Benton2004.RHL.flip p) (Benton2004.RHL.flip p') f f')]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.exec_equiv_flip", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.holds_interp_flip", "Prims.l_True", "Prims.squash", "Prims.l_iff", "Benton2004.RHL.exec_equiv", "Benton2004.RHL.flip", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let exec_equiv_flip (p p': gexp bool) (f f': computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] =

holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f'

false

Benton2004.RHL.fst

Benton2004.RHL.r_seq

val r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]]

let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 61, "end_line": 230, "start_col": 0, "start_line": 214 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p'

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p0: Benton2004.RHL.gexp Prims.bool -> p1: Benton2004.RHL.gexp Prims.bool -> p2: Benton2004.RHL.gexp Prims.bool -> c01: Benton2004.computation -> c01': Benton2004.computation -> c12: Benton2004.computation -> c12': Benton2004.computation -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv p0 p1 c01 c01' /\ Benton2004.RHL.exec_equiv p1 p2 c12 c12' ) (ensures Benton2004.RHL.exec_equiv p0 p2 (Benton2004.seq c01 c12) (Benton2004.seq c01' c12')) [ SMTPatOr [ [ SMTPat (Benton2004.RHL.exec_equiv p0 p2 (Benton2004.seq c01 c12) (Benton2004.seq c01' c12')); SMTPat (Benton2004.RHL.exec_equiv p0 p1 c01 c01') ]; [ SMTPat (Benton2004.RHL.exec_equiv p0 p2 (Benton2004.seq c01 c12) (Benton2004.seq c01' c12')); SMTPat (Benton2004.RHL.exec_equiv p1 p2 c12 c12') ]; [ SMTPat (Benton2004.RHL.exec_equiv p0 p1 c01 c01'); SMTPat (Benton2004.RHL.exec_equiv p1 p2 c12 c12') ] ] ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.d_seq", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Prims.squash", "Benton2004.seq", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat_or", "Prims.list", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let r_seq (p0 p1 p2: gexp bool) (c01 c01' c12 c12': computation) : Lemma (requires (exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12')) (ensures (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12'))) [ SMTPatOr [ [ SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01') ]; [ SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12') ]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')] ] ] =

d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12'

false

Benton2004.RHL.fst

Benton2004.RHL.r_while_terminates

val r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' ))

let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0))

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 127, "end_line": 363, "start_col": 0, "start_line": 345 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f'

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> b': Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> p: Benton2004.RHL.gexp Prims.bool -> s0: FStar.DM4F.Heap.IntStoreFixed.heap -> s0': FStar.DM4F.Heap.IntStoreFixed.heap -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv (Benton2004.RHL.gand p (Benton2004.RHL.gand (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) c c' /\ Benton2004.Aux.holds (Benton2004.RHL.interp (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right)))) s0 s0') (ensures Benton2004.terminates_on (Benton2004.reify_computation (Benton2004.while b c)) s0 <==> Benton2004.terminates_on (Benton2004.reify_computation (Benton2004.while b' c')) s0')

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "FStar.DM4F.Heap.IntStoreFixed.heap", "FStar.Classical.forall_intro", "Prims.nat", "Prims.l_imp", "Prims.l_and", "Benton2004.RHL.included", "Benton2004.RHL.flip", "Benton2004.RHL.geq", "Benton2004.RHL.exp_to_gexp", "Benton2004.RHL.Left", "Benton2004.RHL.Right", "Benton2004.RHL.gand", "Benton2004.RHL.exec_equiv", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Prims.eq2", "FStar.Pervasives.Native.fst", "Benton2004.reify_computation", "Benton2004.while", "Benton2004.terminates_on", "FStar.Classical.move_requires", "Benton2004.RHL.r_while_terminates'", "Prims.unit", "Prims.squash", "Prims.l_iff", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

false

true

false

let r_while_terminates (b b': exp bool) (c c': computation) (p: gexp bool) (s0 s0': heap) : Lemma (requires (exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0')) (ensures (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) =

let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0))

false

Benton2004.RHL.fst

Benton2004.RHL.is_per_gand

val is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2)))

let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 153, "end_line": 442, "start_col": 0, "start_line": 437 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

e1: Benton2004.RHL.gexp Prims.bool -> e2: Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.is_per e1 /\ Benton2004.RHL.is_per e2) (ensures Benton2004.RHL.is_per (Benton2004.RHL.gand e1 e2))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Prims._assert", "Prims.l_Forall", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.l_iff", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Benton2004.RHL.gand", "Prims.l_and", "Prims.unit", "Benton2004.RHL.is_per", "Prims.squash", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

false

true

false

let is_per_gand (e1 e2: gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) =

assert (forall s1 s2. {:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)

false

Benton2004.RHL.fst

Benton2004.RHL.r_sym

val r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')]

let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f'

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 44, "end_line": 460, "start_col": 0, "start_line": 453 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p: Benton2004.RHL.gexp Prims.bool -> p': Benton2004.RHL.gexp Prims.bool -> f: Benton2004.computation -> f': Benton2004.computation -> FStar.Pervasives.Lemma (requires Benton2004.RHL.is_per p /\ Benton2004.RHL.is_per p') (ensures Benton2004.RHL.exec_equiv p p' f f' <==> Benton2004.RHL.exec_equiv p p' f' f) [ SMTPat (Benton2004.RHL.exec_equiv p p' f f'); SMTPat (Benton2004.RHL.is_per p); SMTPat (Benton2004.RHL.is_per p') ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.exec_equiv_sym", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.is_per", "Prims.squash", "Prims.l_iff", "Benton2004.RHL.exec_equiv", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let r_sym (p p': gexp bool) (f f': computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] =

exec_equiv_sym (interp p) (interp p') f f'

false

Benton2004.RHL.fst

Benton2004.RHL.r_while

val r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') ))

let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 28, "end_line": 420, "start_col": 0, "start_line": 393 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> b': Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> p: Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv (Benton2004.RHL.gand p (Benton2004.RHL.gand (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) c c') (ensures Benton2004.RHL.exec_equiv (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) (Benton2004.RHL.gand p (Benton2004.RHL.gnot (Benton2004.RHL.gor (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right)))) (Benton2004.while b c) (Benton2004.while b' c'))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "FStar.Classical.forall_intro_3", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.nat", "Prims.l_imp", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.RHL.gand", "Benton2004.RHL.exp_to_gexp", "Benton2004.RHL.Left", "Benton2004.RHL.Right", "Benton2004.RHL.geq", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Prims.eq2", "FStar.Pervasives.Native.fst", "Benton2004.reify_computation", "Benton2004.while", "Benton2004.RHL.gnot", "Benton2004.RHL.gor", "FStar.Pervasives.Native.snd", "Prims.unit", "FStar.Classical.forall_intro_2", "Prims.l_iff", "Benton2004.terminates_on", "Prims.l_True", "Prims.squash", "Prims.Nil", "FStar.Pervasives.pattern", "FStar.Classical.move_requires", "Benton2004.RHL.r_while_correct", "Benton2004.RHL.r_while_terminates" ]

[ "recursion" ]

false

true

false

let rec r_while (b b': exp bool) (c c': computation) (p: gexp bool) : Lemma (requires (exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c')) (ensures (exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c'))) =

let g (s0 s0': heap) : Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0': heap) (fuel: nat) : Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h

false

Benton2004.RHL.fst

Benton2004.RHL.r_while_correct

val r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel)

let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else ()

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 9, "end_line": 391, "start_col": 0, "start_line": 365 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0))

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> b': Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> p: Benton2004.RHL.gexp Prims.bool -> s0: FStar.DM4F.Heap.IntStoreFixed.heap -> s0': FStar.DM4F.Heap.IntStoreFixed.heap -> fuel: Prims.nat -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv (Benton2004.RHL.gand p (Benton2004.RHL.gand (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) c c' /\ Benton2004.Aux.holds (Benton2004.RHL.interp (Benton2004.RHL.gand p (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right)))) s0 s0' /\ FStar.Pervasives.Native.fst (Benton2004.reify_computation (Benton2004.while b c) fuel s0) == true /\ FStar.Pervasives.Native.fst (Benton2004.reify_computation (Benton2004.while b' c') fuel s0') == true) (ensures Benton2004.Aux.holds (Benton2004.RHL.interp (Benton2004.RHL.gand p (Benton2004.RHL.gnot (Benton2004.RHL.gor (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))))) (FStar.Pervasives.Native.snd (Benton2004.reify_computation (Benton2004.while b c) fuel s0) ) (FStar.Pervasives.Native.snd (Benton2004.reify_computation (Benton2004.while b' c') fuel s0'))) (decreases fuel)

FStar.Pervasives.Lemma

[ "", "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.nat", "FStar.Pervasives.Native.fst", "Benton2004.reify_exp", "Benton2004.RHL.r_while_correct", "Prims.op_Subtraction", "FStar.Pervasives.Native.snd", "Prims.unit", "Benton2004.reified_computation", "Benton2004.reify_computation", "Benton2004.while", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.RHL.gand", "Benton2004.RHL.exp_to_gexp", "Benton2004.RHL.Left", "Benton2004.RHL.Right", "Benton2004.RHL.geq", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Prims.eq2", "Prims.squash", "Benton2004.RHL.gnot", "Benton2004.RHL.gor", "Prims.Nil", "FStar.Pervasives.pattern" ]

[ "recursion" ]

false

true

false

let rec r_while_correct (b b': exp bool) (c c': computation) (p: gexp bool) (s0 s0': heap) (fuel: nat) : Lemma (requires (exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true)) (ensures (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) (decreases fuel) =

let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1)

false

Benton2004.RHL.fst

Benton2004.RHL.d_su1

val d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)]

let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 47, "end_line": 520, "start_col": 0, "start_line": 513 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

c: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' c c) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq Benton2004.skip c) c) [SMTPat (Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq Benton2004.skip c) c)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.computation", "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.d_su1", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.exec_equiv", "Prims.squash", "Benton2004.seq", "Benton2004.skip", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let d_su1 (c: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] =

Benton2004.d_su1 c (interp phi) (interp phi')

false

Benton2004.RHL.fst

Benton2004.RHL.d_su2

val d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)]

let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 47, "end_line": 545, "start_col": 0, "start_line": 538 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

c: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' c c) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq c Benton2004.skip) c) [SMTPat (Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq c Benton2004.skip) c)]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.computation", "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.d_su2", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.exec_equiv", "Prims.squash", "Benton2004.seq", "Benton2004.skip", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let d_su2 (c: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] =

Benton2004.d_su2 c (interp phi) (interp phi')

false

Benton2004.RHL.fst

Benton2004.RHL.r_trans

val r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]]

let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 50, "end_line": 509, "start_col": 0, "start_line": 493 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

p: Benton2004.RHL.gexp Prims.bool -> p': Benton2004.RHL.gexp Prims.bool -> c1: Benton2004.computation -> c2: Benton2004.computation -> c3: Benton2004.computation -> FStar.Pervasives.Lemma (requires Benton2004.RHL.is_per p' /\ Benton2004.RHL.interpolable p /\ Benton2004.RHL.exec_equiv p p' c1 c2 /\ Benton2004.RHL.exec_equiv p p' c2 c3) (ensures Benton2004.RHL.exec_equiv p p' c1 c3) [ SMTPatOr [ [ SMTPat (Benton2004.RHL.exec_equiv p p' c1 c2); SMTPat (Benton2004.RHL.exec_equiv p p' c2 c3); SMTPat (Benton2004.RHL.is_per p'); SMTPat (Benton2004.RHL.interpolable p) ]; [ SMTPat (Benton2004.RHL.exec_equiv p p' c1 c2); SMTPat (Benton2004.RHL.exec_equiv p p' c1 c3); SMTPat (Benton2004.RHL.is_per p'); SMTPat (Benton2004.RHL.interpolable p) ]; [ SMTPat (Benton2004.RHL.exec_equiv p p' c1 c3); SMTPat (Benton2004.RHL.exec_equiv p p' c2 c3); SMTPat (Benton2004.RHL.is_per p'); SMTPat (Benton2004.RHL.interpolable p) ] ] ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.computation", "Benton2004.exec_equiv_trans", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.is_per", "Benton2004.RHL.interpolable", "Benton2004.RHL.exec_equiv", "Prims.squash", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat_or", "Prims.list", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let r_trans (p p': gexp bool) (c1 c2 c3: computation) : Lemma (requires (is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3)) (ensures (exec_equiv p p' c1 c3)) [ SMTPatOr [ [ SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p) ]; [ SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p) ]; [ SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p) ] ] ] =

exec_equiv_trans (interp p) (interp p') c1 c2 c3

false

Benton2004.RHL.fst

Benton2004.RHL.d_su1'

val d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]]

let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 70, "end_line": 536, "start_col": 0, "start_line": 522 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

c: Benton2004.computation -> c': Benton2004.computation -> c'': Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> phi'': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' c Benton2004.skip /\ Benton2004.RHL.exec_equiv phi' phi'' c' c'') (ensures Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq c c') c'') [ SMTPatOr [ [ SMTPat (Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq c c') c''); SMTPat (Benton2004.RHL.exec_equiv phi phi' c Benton2004.skip) ]; [ SMTPat (Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq c c') c''); SMTPat (Benton2004.RHL.exec_equiv phi' phi'' c' c'') ]; [ SMTPat (Benton2004.RHL.exec_equiv phi' phi'' c' c''); SMTPat (Benton2004.RHL.exec_equiv phi phi' c Benton2004.skip) ] ] ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.computation", "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.d_su1'", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.skip", "Prims.squash", "Benton2004.seq", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat_or", "Prims.list", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let d_su1' (c c' c'': computation) (phi phi' phi'': gexp bool) : Lemma (requires (exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'')) (ensures (exec_equiv phi phi'' (seq c c') c'')) [ SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)] ] ] =

Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'')

false

Benton2004.RHL.fst

Benton2004.RHL.d_lu1

val d_lu1 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (ifthenelse b (seq c (while b c)) skip)))

let d_lu1 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (ifthenelse b (seq c (while b c)) skip))) = Benton2004.d_lu1 b c (interp phi) (interp phi')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 49, "end_line": 582, "start_col": 0, "start_line": 575 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'') let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi') let d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] = Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi') let d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3 )) (ensures ( exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)) )) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2))]; [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3)]; [SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3)]; ]] = Benton2004.d_cc b c1 c2 c3 (interp phi) (interp phi') (interp phi'')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> c: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' (Benton2004.while b c) (Benton2004.while b c)) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.while b c) (Benton2004.ifthenelse b (Benton2004.seq c (Benton2004.while b c)) Benton2004.skip))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "Benton2004.d_lu1", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.exec_equiv", "Benton2004.while", "Prims.squash", "Benton2004.ifthenelse", "Benton2004.seq", "Benton2004.skip", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

true

false

true

false

let d_lu1 (b: exp bool) (c: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (ifthenelse b (seq c (while b c)) skip))) =

Benton2004.d_lu1 b c (interp phi) (interp phi')

false

Benton2004.RHL.fst

Benton2004.RHL.d_cc

val d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3 )) (ensures ( exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)) )) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2))]; [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3)]; [SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3)]; ]]

let d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3 )) (ensures ( exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)) )) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2))]; [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3)]; [SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3)]; ]] = Benton2004.d_cc b c1 c2 c3 (interp phi) (interp phi') (interp phi'')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 70, "end_line": 573, "start_col": 0, "start_line": 556 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'') let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi') let d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] = Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> c1: Benton2004.computation -> c2: Benton2004.computation -> c3: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> phi'': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' (Benton2004.ifthenelse b c1 c2) (Benton2004.ifthenelse b c1 c2) /\ Benton2004.RHL.exec_equiv phi' phi'' c3 c3) (ensures Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq (Benton2004.ifthenelse b c1 c2) c3) (Benton2004.ifthenelse b (Benton2004.seq c1 c3) (Benton2004.seq c2 c3))) [ SMTPatOr [ [ SMTPat (Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq (Benton2004.ifthenelse b c1 c2) c3) (Benton2004.ifthenelse b (Benton2004.seq c1 c3) (Benton2004.seq c2 c3))); SMTPat (Benton2004.RHL.exec_equiv phi phi' (Benton2004.ifthenelse b c1 c2) (Benton2004.ifthenelse b c1 c2)) ]; [ SMTPat (Benton2004.RHL.exec_equiv phi phi'' (Benton2004.seq (Benton2004.ifthenelse b c1 c2) c3) (Benton2004.ifthenelse b (Benton2004.seq c1 c3) (Benton2004.seq c2 c3))); SMTPat (Benton2004.RHL.exec_equiv phi' phi'' c3 c3) ]; [ SMTPat (Benton2004.RHL.exec_equiv phi phi' (Benton2004.ifthenelse b c1 c2) (Benton2004.ifthenelse b c1 c2)); SMTPat (Benton2004.RHL.exec_equiv phi' phi'' c3 c3) ] ] ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "Benton2004.d_cc", "Benton2004.RHL.interp", "Prims.unit", "Prims.l_and", "Benton2004.RHL.exec_equiv", "Benton2004.ifthenelse", "Prims.squash", "Benton2004.seq", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat_or", "Prims.list", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'': gexp bool) : Lemma (requires (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3)) (ensures (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)))) [ SMTPatOr [ [ SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)) ]; [ SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3) ]; [ SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3) ] ] ] =

Benton2004.d_cc b c1 c2 c3 (interp phi) (interp phi') (interp phi'')

false

Benton2004.RHL.fst

Benton2004.RHL.d_assoc

val d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))]

let d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] = Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 56, "end_line": 554, "start_col": 0, "start_line": 547 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'') let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

c1: Benton2004.computation -> c2: Benton2004.computation -> c3: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq (Benton2004.seq c1 c2) c3) (Benton2004.seq (Benton2004.seq c1 c2) c3)) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq (Benton2004.seq c1 c2) c3) (Benton2004.seq c1 (Benton2004.seq c2 c3))) [ SMTPat (Benton2004.RHL.exec_equiv phi phi' (Benton2004.seq (Benton2004.seq c1 c2) c3) (Benton2004.seq c1 (Benton2004.seq c2 c3))) ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.computation", "Benton2004.RHL.gexp", "Prims.bool", "Benton2004.d_assoc", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.exec_equiv", "Benton2004.seq", "Prims.squash", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

true

false

true

false

let d_assoc (c1 c2 c3: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] =

Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi')

false

Benton2004.RHL.fst

Benton2004.RHL.d_lu2

val d_lu2 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (while b (seq c (ifthenelse b c skip)))))

let d_lu2 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (while b (seq c (ifthenelse b c skip))))) = Benton2004.d_lu2 b c (interp phi) (interp phi')

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 49, "end_line": 591, "start_col": 0, "start_line": 584 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'') let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi') let d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] = Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi') let d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3 )) (ensures ( exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)) )) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2))]; [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3)]; [SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3)]; ]] = Benton2004.d_cc b c1 c2 c3 (interp phi) (interp phi') (interp phi'') let d_lu1 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (ifthenelse b (seq c (while b c)) skip))) = Benton2004.d_lu1 b c (interp phi) (interp phi')

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> c: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv phi phi' (Benton2004.while b c) (Benton2004.while b c)) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.while b c) (Benton2004.while b (Benton2004.seq c (Benton2004.ifthenelse b c Benton2004.skip))))

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "Benton2004.d_lu2", "Benton2004.RHL.interp", "Prims.unit", "Benton2004.RHL.exec_equiv", "Benton2004.while", "Prims.squash", "Benton2004.seq", "Benton2004.ifthenelse", "Benton2004.skip", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

true

false

true

false

let d_lu2 (b: exp bool) (c: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (while b (seq c (ifthenelse b c skip))))) =

Benton2004.d_lu2 b c (interp phi) (interp phi')

false

Benton2004.RHL.fst

Benton2004.RHL.r_cbl

val r_cbl (b: exp bool) (c c' d : computation) (phi phi' : gexp bool) : Lemma (requires ( exec_equiv (gand phi (exp_to_gexp b Left)) phi' c d /\ exec_equiv (gand phi (gnot (exp_to_gexp b Left))) phi' c' d )) (ensures ( exec_equiv phi phi' (ifthenelse b c c') d ))

let r_cbl (b: exp bool) (c c' d : computation) (phi phi' : gexp bool) : Lemma (requires ( exec_equiv (gand phi (exp_to_gexp b Left)) phi' c d /\ exec_equiv (gand phi (gnot (exp_to_gexp b Left))) phi' c' d )) (ensures ( exec_equiv phi phi' (ifthenelse b c c') d )) = (* NOTE: the following let _ are necessary, and must be stated in this form instead of asserts alone, the latter seeming ineffective *) let _ : squash (forall s1 s2 . holds (interp (gand phi (exp_to_gexp b Left))) s1 s2 <==> holds (interp phi) s1 s2 /\ fst (reify_exp b s1) == true) = assert (forall s1 s2 . holds (interp (gand phi (exp_to_gexp b Left))) s1 s2 <==> holds (interp phi) s1 s2 /\ fst (reify_exp b s1) == true) in let _ : squash (forall s1 s2 . holds (interp (gand phi (gnot (exp_to_gexp b Left)))) s1 s2 <==> holds (interp phi) s1 s2 /\ ~ (fst (reify_exp b s1) == true)) = assert (forall s1 s2 . holds (interp (gand phi (gnot (exp_to_gexp b Left)))) s1 s2 <==> holds (interp phi) s1 s2 /\ ~ (fst (reify_exp b s1) == true)) in ()

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 4, "end_line": 660, "start_col": 0, "start_line": 628 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = () let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else () let holds_interp_flip (phi: gexp bool) : Lemma (forall s1 s2 . holds (interp (flip phi)) s1 s2 <==> holds (Benton2004.flip (interp phi)) s1 s2) [SMTPat (holds (interp (flip phi)))] = Benton2004.holds_flip (interp phi) let exec_equiv_flip (p p': gexp bool) (f f' : computation) : Lemma (exec_equiv (flip p) (flip p') f f' <==> exec_equiv p p' f' f) [SMTPat (exec_equiv (flip p) (flip p') f f')] = holds_interp_flip p; holds_interp_flip p'; exec_equiv_flip (interp p) (interp p') f f' let r_while_terminates (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' )) (ensures ( terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0' )) = let phi = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c = gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)) in let phi_c' = gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) in Classical.forall_intro (Classical.move_requires (r_while_terminates' b b' c c' phi phi_c phi_c' s0 s0')); Classical.forall_intro (Classical.move_requires (r_while_terminates' b' b c' c (flip phi) (flip phi_c) (flip phi_c') s0' s0)) let rec r_while_correct (b b' : exp bool) (c c' : computation) (p: gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true )) (ensures ( holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')) )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel s0') in r_while_correct b b' c c' p s1 s1' (fuel - 1) else () let rec r_while (b b' : exp bool) (c c' : computation) (p: gexp bool) : Lemma (requires ( exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' )) (ensures ( exec_equiv (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right)))) (while b c) (while b' c') )) = let g (s0 s0':heap) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0') ==> (terminates_on (reify_computation (while b c)) s0 <==> terminates_on (reify_computation (while b' c')) s0')) = Classical.move_requires (r_while_terminates b b' c c' p s0) s0' in let h (s0 s0':heap) (fuel:nat) :Lemma ((exec_equiv (gand p (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right))) c c' /\ holds (interp (gand p (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true /\ fst (reify_computation (while b' c') fuel s0') == true) ==> (holds (interp (gand p (gnot (gor (exp_to_gexp b Left) (exp_to_gexp b' Right))))) (snd (reify_computation (while b c) fuel s0)) (snd (reify_computation (while b' c') fuel s0')))) = Classical.move_requires (r_while_correct b b' c c' p s0 s0') fuel in Classical.forall_intro_2 g; Classical.forall_intro_3 h (* Aparte: 4.4 How to prove is_per *) let is_per_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (is_per (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let is_per_gand_exp_to_gexp (b: exp bool) : Lemma (is_per (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () let is_per_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2)) (ensures (is_per (gand e1 e2))) = assert (forall s1 s2 .{:pattern (interp (gand e1 e2) s1 s2)} holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) (* FIXME: holds but not replayable let is_per_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ (forall s1 s2 . ~ (holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2)))) (ensures (is_per (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_sym (p p': gexp bool) (f f' : computation) : Lemma (requires (is_per p /\ is_per p')) (ensures (exec_equiv p p' f f' <==> exec_equiv p p' f' f)) [SMTPat (exec_equiv p p' f f'); SMTPat (is_per p); SMTPat (is_per p')] = exec_equiv_sym (interp p) (interp p') f f' (* Aparte: 4.4 How to prove interpolable *) let interpolable_geq_exp_to_gexp (#t: eqtype) (e: exp t) : Lemma (interpolable (geq (exp_to_gexp e Left) (exp_to_gexp e Right))) = () let interpolable_gand_exp_to_gexp (b: exp bool) : Lemma (interpolable (gand (exp_to_gexp b Left) (exp_to_gexp b Right))) = () (* FIXME: holds but not replayable let interpolable_gand (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gand e1 e2))) = assert (forall s1 s2 . holds (interp (gand e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 /\ holds (interp e2) s1 s2) let interpolable_gor (e1 e2 : gexp bool) : Lemma (requires (is_per e1 /\ is_per e2 /\ interpolable e1 /\ interpolable e2)) (ensures (interpolable (gor e1 e2))) = assert (forall s1 s2 . holds (interp (gor e1 e2)) s1 s2 <==> holds (interp e1) s1 s2 \/ holds (interp e2) s1 s2) *) let r_trans (p p' : gexp bool) (c1 c2 c3 : computation) : Lemma (requires ( is_per p' /\ interpolable p /\ exec_equiv p p' c1 c2 /\ exec_equiv p p' c2 c3 )) (ensures (exec_equiv p p' c1 c3)) [SMTPatOr [ [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c2); SMTPat (exec_equiv p p' c1 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; [SMTPat (exec_equiv p p' c1 c3); SMTPat (exec_equiv p p' c2 c3); SMTPat (is_per p'); SMTPat (interpolable p)]; ]] = exec_equiv_trans (interp p) (interp p') c1 c2 c3 (* 4.2.1 Basic equations *) let d_su1 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq skip c) c)) [SMTPat (exec_equiv phi phi' (seq skip c) c)] = Benton2004.d_su1 c (interp phi) (interp phi') let d_su1' (c c' c'' : computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' c skip /\ exec_equiv phi' phi'' c' c'' )) (ensures (exec_equiv phi phi'' (seq c c') c'')) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi phi' c skip)]; [SMTPat (exec_equiv phi phi'' (seq c c') c''); SMTPat (exec_equiv phi' phi'' c' c'')]; [SMTPat (exec_equiv phi' phi'' c' c''); SMTPat (exec_equiv phi phi' c skip)]; ]] = Benton2004.d_su1' c c' c'' (interp phi) (interp phi') (interp phi'') let d_su2 (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' c c)) (ensures (exec_equiv phi phi' (seq c skip) c)) [SMTPat (exec_equiv phi phi' (seq c skip) c)] = Benton2004.d_su2 c (interp phi) (interp phi') let d_assoc (c1 c2 c3: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq (seq c1 c2) c3))) (ensures (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))) [SMTPat (exec_equiv phi phi' (seq (seq c1 c2) c3) (seq c1 (seq c2 c3)))] = Benton2004.d_assoc c1 c2 c3 (interp phi) (interp phi') let d_cc (b: exp bool) (c1 c2 c3: computation) (phi phi' phi'' : gexp bool) : Lemma (requires ( exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2) /\ exec_equiv phi' phi'' c3 c3 )) (ensures ( exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3)) )) [SMTPatOr [ [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2))]; [SMTPat (exec_equiv phi phi'' (seq (ifthenelse b c1 c2) c3) (ifthenelse b (seq c1 c3) (seq c2 c3))); SMTPat (exec_equiv phi' phi'' c3 c3)]; [SMTPat (exec_equiv phi phi' (ifthenelse b c1 c2) (ifthenelse b c1 c2)); SMTPat (exec_equiv phi' phi'' c3 c3)]; ]] = Benton2004.d_cc b c1 c2 c3 (interp phi) (interp phi') (interp phi'') let d_lu1 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (ifthenelse b (seq c (while b c)) skip))) = Benton2004.d_lu1 b c (interp phi) (interp phi') let d_lu2 (b: exp bool) (c: computation) (phi phi' : gexp bool) : Lemma (requires (exec_equiv phi phi' (while b c) (while b c))) (ensures (exec_equiv phi phi' (while b c) (while b (seq c (ifthenelse b c skip))))) = Benton2004.d_lu2 b c (interp phi) (interp phi') (* 4.2.2 Optimizing Transformations *) (* Falsity *) let r_f (c c' : computation) (phi: gexp bool) : Lemma (ensures ( exec_equiv (gconst false) phi c c' )) = () (* Dead assignment *) let r_dassl (x: var) (e: exp int) (phi: gexp bool) : Lemma (ensures ( exec_equiv (gsubst phi x Left (exp_to_gexp e Left)) phi (assign x e) skip )) = () (* Common branch *)

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> d: Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi': Benton2004.RHL.gexp Prims.bool -> FStar.Pervasives.Lemma (requires Benton2004.RHL.exec_equiv (Benton2004.RHL.gand phi (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left)) phi' c d /\ Benton2004.RHL.exec_equiv (Benton2004.RHL.gand phi (Benton2004.RHL.gnot (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left))) phi' c' d) (ensures Benton2004.RHL.exec_equiv phi phi' (Benton2004.ifthenelse b c c') d)

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "Prims.squash", "Prims.l_Forall", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.l_iff", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Benton2004.RHL.gand", "Benton2004.RHL.gnot", "Benton2004.RHL.exp_to_gexp", "Benton2004.RHL.Left", "Prims.l_and", "Prims.l_not", "Prims.eq2", "FStar.Pervasives.Native.fst", "Benton2004.reify_exp", "Prims._assert", "Prims.unit", "Benton2004.RHL.exec_equiv", "Benton2004.ifthenelse", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

false

true

false

let r_cbl (b: exp bool) (c c' d: computation) (phi phi': gexp bool) : Lemma (requires (exec_equiv (gand phi (exp_to_gexp b Left)) phi' c d /\ exec_equiv (gand phi (gnot (exp_to_gexp b Left))) phi' c' d)) (ensures (exec_equiv phi phi' (ifthenelse b c c') d)) =

let _:squash (forall s1 s2. holds (interp (gand phi (exp_to_gexp b Left))) s1 s2 <==> holds (interp phi) s1 s2 /\ fst (reify_exp b s1) == true) = assert (forall s1 s2. holds (interp (gand phi (exp_to_gexp b Left))) s1 s2 <==> holds (interp phi) s1 s2 /\ fst (reify_exp b s1) == true) in let _:squash (forall s1 s2. holds (interp (gand phi (gnot (exp_to_gexp b Left)))) s1 s2 <==> holds (interp phi) s1 s2 /\ ~(fst (reify_exp b s1) == true)) = assert (forall s1 s2. holds (interp (gand phi (gnot (exp_to_gexp b Left)))) s1 s2 <==> holds (interp phi) s1 s2 /\ ~(fst (reify_exp b s1) == true)) in ()

false

Vale.Curve25519.X64.FastUtil.fst

Vale.Curve25519.X64.FastUtil.va_qcode_Fast_mul1

val va_qcode_Fast_mul1 (va_mods: va_mods_t) (dst_b inA_b: buffer64) : (va_quickCode unit (va_code_Fast_mul1 ()))

let va_qcode_Fast_mul1 (va_mods:va_mods_t) (dst_b:buffer64) (inA_b:buffer64) : (va_quickCode unit (va_code_Fast_mul1 ())) = (qblock va_mods (fun (va_s:va_state) -> let (va_old_s:va_state) = va_s in let (a0:Vale.Def.Types_s.nat64) = Vale.X64.Decls.buffer64_read inA_b 0 (va_get_mem_heaplet 0 va_s) in let (a1:Vale.Def.Types_s.nat64) = Vale.X64.Decls.buffer64_read inA_b 1 (va_get_mem_heaplet 0 va_s) in let (a2:Vale.Def.Types_s.nat64) = Vale.X64.Decls.buffer64_read inA_b 2 (va_get_mem_heaplet 0 va_s) in let (a3:Vale.Def.Types_s.nat64) = Vale.X64.Decls.buffer64_read inA_b 3 (va_get_mem_heaplet 0 va_s) in let (a:Prims.nat) = Vale.Curve25519.Fast_defs.pow2_four a0 a1 a2 a3 in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 91 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> Vale.Arch.Types.xor_lemmas ()) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 93 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 inA_b 0 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 93 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR9) (va_op_dst_opr64_reg64 rR8) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 Secret)) (fun (va_s:va_state) _ -> let (va_arg48:Vale.Def.Types_s.nat64) = va_get_reg64 rRdx va_s in let (va_arg47:Vale.Def.Types_s.nat64) = va_get_reg64 rR8 va_s in let (va_arg46:Vale.Def.Types_s.nat64) = va_get_reg64 rR9 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 93 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg46 va_arg47 va_arg48 a0) (let (old_r8:nat64) = va_get_reg64 rR8 va_s in va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 94 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR8) 0 Secret dst_b 0) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 95 column 10 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Xor64 (va_op_dst_opr64_reg64 rR8) (va_op_opr64_reg64 rR8)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 96 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 inA_b 1 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 96 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR11) (va_op_dst_opr64_reg64 rR10) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 Secret)) (fun (va_s:va_state) _ -> let (va_arg45:Vale.Def.Types_s.nat64) = va_get_reg64 rRdx va_s in let (va_arg44:Vale.Def.Types_s.nat64) = va_get_reg64 rR10 va_s in let (va_arg43:Vale.Def.Types_s.nat64) = va_get_reg64 rR11 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 96 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg43 va_arg44 va_arg45 a1) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 97 column 14 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Add64Wrap (va_op_dst_opr64_reg64 rR10) (va_op_opr64_reg64 rR9)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 98 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR10) 8 Secret dst_b 1) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 99 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 inA_b 2 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 99 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR13) (va_op_dst_opr64_reg64 rRbx) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 Secret)) (fun (va_s:va_state) _ -> let (va_arg42:Vale.Def.Types_s.nat64) = va_get_reg64 rRdx va_s in let (va_arg41:Vale.Def.Types_s.nat64) = va_get_reg64 rRbx va_s in let (va_arg40:Vale.Def.Types_s.nat64) = va_get_reg64 rR13 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 99 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg40 va_arg41 va_arg42 a2) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 100 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Adcx64Wrap (va_op_dst_opr64_reg64 rRbx) (va_op_opr64_reg64 rR11)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 101 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rRbx) 16 Secret dst_b 2) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 102 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 24 inA_b 3 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 102 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rRax) (va_op_dst_opr64_reg64 rR14) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 24 Secret)) (fun (va_s:va_state) _ -> let (va_arg39:Vale.Def.Types_s.nat64) = va_get_reg64 rRdx va_s in let (va_arg38:Vale.Def.Types_s.nat64) = va_get_reg64 rR14 va_s in let (va_arg37:Vale.Def.Types_s.nat64) = va_get_reg64 rRax va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 102 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg37 va_arg38 va_arg39 a3) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 103 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Adcx64Wrap (va_op_dst_opr64_reg64 rR14) (va_op_opr64_reg64 rR13)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 104 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR14) 24 Secret dst_b 3) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 105 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Adcx64Wrap (va_op_dst_opr64_reg64 rRax) (va_op_opr64_reg64 rR8)) (fun (va_s:va_state) _ -> let (carry_bit:Vale.Curve25519.Fast_defs.bit) = Vale.Curve25519.Fast_defs.bool_bit (Vale.X64.Decls.cf (va_get_flags va_s)) in va_qAssert va_range1 "***** PRECONDITION NOT MET AT line 108 column 5 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (carry_bit == 0) (let (va_arg36:prop) = va_mul_nat a (va_get_reg64 rRdx va_s) == 0 + Vale.Curve25519.Fast_defs.pow2_four (va_mul_nat (va_get_reg64 rRdx va_s) a0) (va_mul_nat (va_get_reg64 rRdx va_s) a1) (va_mul_nat (va_get_reg64 rRdx va_s) a2) (va_mul_nat (va_get_reg64 rRdx va_s) a3) in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 109 column 21 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_:unit) -> assert_by_tactic va_arg36 int_canon) (va_QEmpty (())))))))))))))))))))))))))))

{ "file_name": "obj/Vale.Curve25519.X64.FastUtil.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 99, "end_line": 168, "start_col": 0, "start_line": 79 }

module Vale.Curve25519.X64.FastUtil open Vale.Def.Types_s open Vale.Arch.Types open Vale.X64.Machine_s open Vale.X64.Memory open Vale.X64.State open Vale.X64.Decls open Vale.X64.InsBasic open Vale.X64.InsMem open Vale.X64.InsStack open Vale.X64.QuickCode open Vale.X64.QuickCodes open FStar.Tactics open Vale.Curve25519.Fast_defs open Vale.Curve25519.Fast_lemmas_external open Vale.Curve25519.FastUtil_helpers open Vale.X64.CPU_Features_s #reset-options "--z3rlimit 60" //-- Fast_mul1 #push-options "--z3rlimit 600" val va_code_Fast_mul1 : va_dummy:unit -> Tot va_code [@ "opaque_to_smt" va_qattr] let va_code_Fast_mul1 () = (va_Block (va_CCons (va_code_Mem64_lemma ()) (va_CCons (va_code_Mulx64 (va_op_dst_opr64_reg64 rR9) (va_op_dst_opr64_reg64 rR8) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 Secret)) (va_CCons (va_code_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR8) 0 Secret) (va_CCons (va_code_Xor64 (va_op_dst_opr64_reg64 rR8) (va_op_opr64_reg64 rR8)) (va_CCons (va_code_Mem64_lemma ()) (va_CCons (va_code_Mulx64 (va_op_dst_opr64_reg64 rR11) (va_op_dst_opr64_reg64 rR10) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 Secret)) (va_CCons (va_code_Add64Wrap (va_op_dst_opr64_reg64 rR10) (va_op_opr64_reg64 rR9)) (va_CCons (va_code_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR10) 8 Secret) (va_CCons (va_code_Mem64_lemma ()) (va_CCons (va_code_Mulx64 (va_op_dst_opr64_reg64 rR13) (va_op_dst_opr64_reg64 rRbx) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 Secret)) (va_CCons (va_code_Adcx64Wrap (va_op_dst_opr64_reg64 rRbx) (va_op_opr64_reg64 rR11)) (va_CCons (va_code_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rRbx) 16 Secret) (va_CCons (va_code_Mem64_lemma ()) (va_CCons (va_code_Mulx64 (va_op_dst_opr64_reg64 rRax) (va_op_dst_opr64_reg64 rR14) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 24 Secret)) (va_CCons (va_code_Adcx64Wrap (va_op_dst_opr64_reg64 rR14) (va_op_opr64_reg64 rR13)) (va_CCons (va_code_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR14) 24 Secret) (va_CCons (va_code_Adcx64Wrap (va_op_dst_opr64_reg64 rRax) (va_op_opr64_reg64 rR8)) (va_CNil ()))))))))))))))))))) val va_codegen_success_Fast_mul1 : va_dummy:unit -> Tot va_pbool [@ "opaque_to_smt" va_qattr] let va_codegen_success_Fast_mul1 () = (va_pbool_and (va_codegen_success_Mem64_lemma ()) (va_pbool_and (va_codegen_success_Mulx64 (va_op_dst_opr64_reg64 rR9) (va_op_dst_opr64_reg64 rR8) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 Secret)) (va_pbool_and (va_codegen_success_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR8) 0 Secret) (va_pbool_and (va_codegen_success_Xor64 (va_op_dst_opr64_reg64 rR8) (va_op_opr64_reg64 rR8)) (va_pbool_and (va_codegen_success_Mem64_lemma ()) (va_pbool_and (va_codegen_success_Mulx64 (va_op_dst_opr64_reg64 rR11) (va_op_dst_opr64_reg64 rR10) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 Secret)) (va_pbool_and (va_codegen_success_Add64Wrap (va_op_dst_opr64_reg64 rR10) (va_op_opr64_reg64 rR9)) (va_pbool_and (va_codegen_success_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR10) 8 Secret) (va_pbool_and (va_codegen_success_Mem64_lemma ()) (va_pbool_and (va_codegen_success_Mulx64 (va_op_dst_opr64_reg64 rR13) (va_op_dst_opr64_reg64 rRbx) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 Secret)) (va_pbool_and (va_codegen_success_Adcx64Wrap (va_op_dst_opr64_reg64 rRbx) (va_op_opr64_reg64 rR11)) (va_pbool_and (va_codegen_success_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rRbx) 16 Secret) (va_pbool_and (va_codegen_success_Mem64_lemma ()) (va_pbool_and (va_codegen_success_Mulx64 (va_op_dst_opr64_reg64 rRax) (va_op_dst_opr64_reg64 rR14) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 24 Secret)) (va_pbool_and (va_codegen_success_Adcx64Wrap (va_op_dst_opr64_reg64 rR14) (va_op_opr64_reg64 rR13)) (va_pbool_and (va_codegen_success_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR14) 24 Secret) (va_pbool_and (va_codegen_success_Adcx64Wrap (va_op_dst_opr64_reg64 rRax) (va_op_opr64_reg64 rR8)) (va_ttrue ()))))))))))))))))))

{ "checked_file": "/", "dependencies": [ "Vale.X64.State.fsti.checked", "Vale.X64.QuickCodes.fsti.checked", "Vale.X64.QuickCode.fst.checked", "Vale.X64.Memory.fsti.checked", "Vale.X64.Machine_s.fst.checked", "Vale.X64.InsStack.fsti.checked", "Vale.X64.InsMem.fsti.checked", "Vale.X64.InsBasic.fsti.checked", "Vale.X64.Flags.fsti.checked", "Vale.X64.Decls.fsti.checked", "Vale.X64.CPU_Features_s.fst.checked", "Vale.Def.Types_s.fst.checked", "Vale.Curve25519.FastUtil_helpers.fsti.checked", "Vale.Curve25519.Fast_lemmas_external.fsti.checked", "Vale.Curve25519.Fast_defs.fst.checked", "Vale.Arch.Types.fsti.checked", "prims.fst.checked", "FStar.Tactics.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked" ], "interface_file": true, "source_file": "Vale.Curve25519.X64.FastUtil.fst" }

[ { "abbrev": false, "full_module": "Vale.X64.CPU_Features_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.FastUtil_helpers", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.Fast_lemmas_external", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.Fast_defs", "short_module": null }, { "abbrev": false, "full_module": "FStar.Tactics", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.QuickCodes", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.QuickCode", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsStack", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsMem", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsBasic", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Decls", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.State", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Memory", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Machine_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Arch.Types", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Types_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.CPU_Features_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.Fast_defs", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.QuickCodes", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.QuickCode", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsStack", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsMem", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.InsBasic", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Decls", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.State", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Stack_i", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Memory", "short_module": null }, { "abbrev": false, "full_module": "Vale.X64.Machine_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Arch.HeapImpl", "short_module": null }, { "abbrev": false, "full_module": "Vale.Arch.Types", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Types_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.X64", "short_module": null }, { "abbrev": false, "full_module": "Vale.Curve25519.X64", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 0, "max_fuel": 1, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": true, "smtencoding_l_arith_repr": "native", "smtencoding_nl_arith_repr": "wrapped", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [ "smt.arith.nl=false", "smt.QI.EAGER_THRESHOLD=100", "smt.CASE_SPLIT=3" ], "z3refresh": false, "z3rlimit": 600, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

va_mods: Vale.X64.QuickCode.va_mods_t -> dst_b: Vale.X64.Memory.buffer64 -> inA_b: Vale.X64.Memory.buffer64 -> Vale.X64.QuickCode.va_quickCode Prims.unit (Vale.Curve25519.X64.FastUtil.va_code_Fast_mul1 ())

Prims.Tot

[ "total" ]

[]

[ "Vale.X64.QuickCode.va_mods_t", "Vale.X64.Memory.buffer64", "Vale.X64.QuickCodes.qblock", "Prims.unit", "Prims.Cons", "Vale.X64.Decls.va_code", "Vale.X64.InsMem.va_code_Mem64_lemma", "Vale.X64.InsBasic.va_code_Mulx64", "Vale.X64.Decls.va_op_dst_opr64_reg64", "Vale.X64.Machine_s.rR9", "Vale.X64.Machine_s.rR8", "Vale.X64.Decls.va_opr_code_Mem64", "Vale.X64.Decls.va_op_heaplet_mem_heaplet", "Vale.X64.Decls.va_op_reg64_reg64", "Vale.X64.Machine_s.rRsi", "Vale.Arch.HeapTypes_s.Secret", "Vale.X64.InsMem.va_code_Store64_buffer", "Vale.X64.Decls.va_op_reg_opr64_reg64", "Vale.X64.Machine_s.rRdi", "Vale.X64.InsBasic.va_code_Xor64", "Vale.X64.Decls.va_op_opr64_reg64", "Vale.X64.Machine_s.rR11", "Vale.X64.Machine_s.rR10", "Vale.X64.InsBasic.va_code_Add64Wrap", "Vale.X64.Machine_s.rR13", "Vale.X64.Machine_s.rRbx", "Vale.X64.InsBasic.va_code_Adcx64Wrap", "Vale.X64.Machine_s.rRax", "Vale.X64.Machine_s.rR14", "Prims.Nil", "Vale.X64.Machine_s.precode", "Vale.X64.Decls.ins", "Vale.X64.Decls.ocmp", "Vale.X64.Decls.va_state", "Vale.X64.QuickCodes.va_qPURE", "Prims.pure_post", "Prims.l_and", "Prims.l_True", "Prims.l_Forall", "Prims.l_imp", "Vale.Def.Words_s.nat32", "Prims.eq2", "Vale.Def.Types_s.ixor", "Prims.int", "Vale.Def.Words_s.nat64", "Vale.X64.QuickCodes.va_range1", "Vale.Arch.Types.xor_lemmas", "Vale.X64.QuickCodes.va_QSeq", "Vale.X64.InsMem.va_quick_Mem64_lemma", "Vale.X64.QuickCodes.va_QBind", "Vale.X64.InsBasic.va_quick_Mulx64", "Prims.op_Addition", "Prims.op_Multiply", "Prims.b2t", "Prims.op_LessThan", "Prims.op_Subtraction", "Prims.l_or", "Prims.op_LessThanOrEqual", "Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds", "Vale.X64.InsMem.va_quick_Store64_buffer", "Vale.X64.InsBasic.va_quick_Xor64", "Vale.X64.InsBasic.va_quick_Add64Wrap", "Vale.X64.InsBasic.va_quick_Adcx64Wrap", "Vale.X64.QuickCodes.va_qAssert", "FStar.Tactics.Effect.with_tactic", "Vale.Curve25519.FastUtil_helpers.int_canon", "Prims.squash", "FStar.Tactics.Effect.assert_by_tactic", "Vale.X64.QuickCodes.va_QEmpty", "Prims.prop", "Vale.X64.Decls.va_mul_nat", "Vale.X64.Decls.va_get_reg64", "Vale.X64.Machine_s.rRdx", "Vale.Curve25519.Fast_defs.pow2_four", "Vale.Curve25519.Fast_defs.bit", "Vale.Curve25519.Fast_defs.bool_bit", "Vale.X64.Decls.cf", "Vale.X64.Decls.va_get_flags", "Vale.X64.QuickCodes.quickCodes", "Prims.nat", "Vale.X64.Decls.buffer64_read", "Vale.X64.Decls.va_get_mem_heaplet", "Vale.X64.State.vale_state", "Vale.X64.QuickCode.va_quickCode", "Vale.Curve25519.X64.FastUtil.va_code_Fast_mul1" ]

[]

false

let va_qcode_Fast_mul1 (va_mods: va_mods_t) (dst_b inA_b: buffer64) : (va_quickCode unit (va_code_Fast_mul1 ())) =

(qblock va_mods (fun (va_s: va_state) -> let va_old_s:va_state = va_s in let a0:Vale.Def.Types_s.nat64 = Vale.X64.Decls.buffer64_read inA_b 0 (va_get_mem_heaplet 0 va_s) in let a1:Vale.Def.Types_s.nat64 = Vale.X64.Decls.buffer64_read inA_b 1 (va_get_mem_heaplet 0 va_s) in let a2:Vale.Def.Types_s.nat64 = Vale.X64.Decls.buffer64_read inA_b 2 (va_get_mem_heaplet 0 va_s) in let a3:Vale.Def.Types_s.nat64 = Vale.X64.Decls.buffer64_read inA_b 3 (va_get_mem_heaplet 0 va_s) in let a:Prims.nat = Vale.Curve25519.Fast_defs.pow2_four a0 a1 a2 a3 in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 91 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_: unit) -> Vale.Arch.Types.xor_lemmas ()) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 93 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 inA_b 0 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 93 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR9) (va_op_dst_opr64_reg64 rR8) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 0 Secret)) (fun (va_s: va_state) _ -> let va_arg48:Vale.Def.Types_s.nat64 = va_get_reg64 rRdx va_s in let va_arg47:Vale.Def.Types_s.nat64 = va_get_reg64 rR8 va_s in let va_arg46:Vale.Def.Types_s.nat64 = va_get_reg64 rR9 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 93 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_: unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg46 va_arg47 va_arg48 a0) (let old_r8:nat64 = va_get_reg64 rR8 va_s in va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 94 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR8) 0 Secret dst_b 0) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 95 column 10 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Xor64 (va_op_dst_opr64_reg64 rR8) (va_op_opr64_reg64 rR8)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 96 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 inA_b 1 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 96 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR11) (va_op_dst_opr64_reg64 rR10) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 8 Secret)) (fun (va_s: va_state) _ -> let va_arg45:Vale.Def.Types_s.nat64 = va_get_reg64 rRdx va_s in let va_arg44:Vale.Def.Types_s.nat64 = va_get_reg64 rR10 va_s in let va_arg43:Vale.Def.Types_s.nat64 = va_get_reg64 rR11 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 96 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_: unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg43 va_arg44 va_arg45 a1) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 97 column 14 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Add64Wrap (va_op_dst_opr64_reg64 rR10) (va_op_opr64_reg64 rR9)) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 98 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rR10) 8 Secret dst_b 1) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 99 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 inA_b 2 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 99 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rR13) (va_op_dst_opr64_reg64 rRbx) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 16 Secret)) (fun (va_s: va_state) _ -> let va_arg42:Vale.Def.Types_s.nat64 = va_get_reg64 rRdx va_s in let va_arg41:Vale.Def.Types_s.nat64 = va_get_reg64 rRbx va_s in let va_arg40:Vale.Def.Types_s.nat64 = va_get_reg64 rR13 va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 99 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_: unit) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg40 va_arg41 va_arg42 a2) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 100 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Adcx64Wrap (va_op_dst_opr64_reg64 rRbx) (va_op_opr64_reg64 rR11) ) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 101 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer (va_op_heaplet_mem_heaplet 0) (va_op_reg_opr64_reg64 rRdi) (va_op_reg_opr64_reg64 rRbx) 16 Secret dst_b 2) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 102 column 28 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mem64_lemma (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi) 24 inA_b 3 Secret) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 102 column 11 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Mulx64 (va_op_dst_opr64_reg64 rRax) (va_op_dst_opr64_reg64 rR14) (va_opr_code_Mem64 (va_op_heaplet_mem_heaplet 0) (va_op_reg64_reg64 rRsi ) 24 Secret)) (fun (va_s: va_state) _ -> let va_arg39:Vale.Def.Types_s.nat64 = va_get_reg64 rRdx va_s in let va_arg38:Vale.Def.Types_s.nat64 = va_get_reg64 rR14 va_s in let va_arg37:Vale.Def.Types_s.nat64 = va_get_reg64 rRax va_s in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 102 column 99 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (fun (_: unit ) -> Vale.Curve25519.Fast_lemmas_external.lemma_prod_bounds va_arg37 va_arg38 va_arg39 a3) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 103 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Adcx64Wrap (va_op_dst_opr64_reg64 rR14 ) (va_op_opr64_reg64 rR13 )) (va_QSeq va_range1 "***** PRECONDITION NOT MET AT line 104 column 19 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" (va_quick_Store64_buffer ( va_op_heaplet_mem_heaplet 0 ) ( va_op_reg_opr64_reg64 rRdi ) ( va_op_reg_opr64_reg64 rR14 ) 24 Secret dst_b 3 ) (va_QBind va_range1 "***** PRECONDITION NOT MET AT line 105 column 15 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" ( va_quick_Adcx64Wrap ( va_op_dst_opr64_reg64 rRax ) ( va_op_opr64_reg64 rR8 ) ) ( fun ( va_s: va_state ) _ -> let carry_bit:Vale.Curve25519.Fast_defs.bit = Vale.Curve25519.Fast_defs.bool_bit ( Vale.X64.Decls.cf ( va_get_flags va_s ) ) in va_qAssert va_range1 "***** PRECONDITION NOT MET AT line 108 column 5 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" ( carry_bit == 0 ) ( let va_arg36:prop = va_mul_nat a ( va_get_reg64 rRdx va_s ) == 0 + Vale.Curve25519.Fast_defs.pow2_four ( va_mul_nat ( va_get_reg64 rRdx va_s ) a0 ) ( va_mul_nat ( va_get_reg64 rRdx va_s ) a1 ) ( va_mul_nat ( va_get_reg64 rRdx va_s ) a2 ) ( va_mul_nat ( va_get_reg64 rRdx va_s ) a3 ) in va_qPURE va_range1 "***** PRECONDITION NOT MET AT line 109 column 21 of file /home/gebner/fstar_dataset/projects/hacl-star/vale/code/thirdPartyPorts/rfc7748/curve25519/x64/Vale.Curve25519.X64.FastUtil.vaf *****" ( fun ( _: unit ) -> assert_by_tactic va_arg36 int_canon ) ( va_QEmpty ( () ) ) ) ) )) )))))))))) )))))))))))

false

Benton2004.RHL.fst

Benton2004.RHL.r_while_terminates'

val r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel)

let rec r_while_terminates' (b b' : exp bool) (c c' : computation) (phi phi_c phi_c': gexp bool) (s0 s0' : heap) (fuel: nat) : Lemma (requires ( included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true )) (ensures ( terminates_on (reify_computation (while b' c')) s0' )) (decreases fuel) = let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then begin // let s1 = snd (reify_computation c fuel s0) in assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0 : nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2 : nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g) end else ()

{ "file_name": "examples/rel/Benton2004.RHL.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 13, "end_line": 328, "start_col": 0, "start_line": 269 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module Benton2004.RHL (* Relational Hoare Logic (Section 4) *) let gsubst_gconst (#t: Type0) (n: t) (x: var) (side: pos) (ge' : gexp int): Lemma (forall h1 h2. (gsubst (gconst n) x side ge') h1 h2 == (gconst n) h1 h2) [SMTPat (gsubst (gconst n) x side ge')] = () let gsubst_gvar_same (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gvar x side) x side ge') h1 h2 == ge' h1 h2) [SMTPat (gsubst (gvar x side) x side ge')] = () let gsubst_gvar_other (x x': var) (side side': pos) (ge': gexp int) : Lemma (requires (x <> x' \/ side <> side')) (ensures (forall h1 h2. (gsubst (gvar x side) x' side' ge') h1 h2 == (gvar x side) h1 h2)) [SMTPat (gsubst (gvar x side) x' side' ge')] = () let gsubst_gop (#from #to: Type0) (op: (from -> from -> GTot to)) (ge1 ge2: gexp from) (x: var) (side: pos) (ge': gexp int) : Lemma (forall h1 h2. (gsubst (gop op ge1 ge2) x side ge') h1 h2 == (gop op (gsubst ge1 x side ge') (gsubst ge2 x side ge')) h1 h2) [SMTPat (gsubst (gop op ge1 ge2) x side ge')] = () let holds_interp (ge: gexp bool) (s1 s2: heap) : Lemma (holds (interp ge) s1 s2 <==> ge s1 s2 == true) [SMTPat (holds (interp ge) s1 s2)] = holds_equiv (interp ge) s1 s2 let exec_equiv (p p' : gexp bool) (f f' : computation) : GTot Type0 = Benton2004.exec_equiv (interp p) (interp p') f f' let exec_equiv_elim (p p' : gexp bool) (f f' : computation) : Lemma (requires (exec_equiv p p' f f')) (ensures (Benton2004.exec_equiv (interp p) (interp p') f f')) = () let r_skip (p: gexp bool) : Lemma (exec_equiv p p skip skip) [SMTPat (exec_equiv p p skip skip)] = d_skip (interp p) let exp_to_gexp_const (#t: Type0) (c: t) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (const c) side) h1 h2 == (gconst c) h1 h2) [SMTPat (exp_to_gexp (const c) side)] = () let exp_to_gexp_evar (x: var) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (evar x) side) h1 h2 == (gvar x side) h1 h2) [SMTPat (exp_to_gexp (evar x) side)] = () let exp_to_gexp_eop (#from #to: Type0) (op: (from -> from -> Tot to)) (e1 e2: exp from) (side: pos) : Lemma (forall h1 h2. (exp_to_gexp (eop op e1 e2) side) h1 h2 == (gop op (exp_to_gexp e1 side) (exp_to_gexp e2 side)) h1 h2) [SMTPat (exp_to_gexp (eop op e1 e2) side)] = () #set-options "--z3rlimit 50 --max_fuel 2 --max_ifuel 1 --z3cliopt smt.qi.eager_threshold=100" let holds_gand (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gand b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 /\ holds (interp b2) s1 s2) [SMTPat (holds (interp (gand b1 b2)))] = () let gsubst_gand (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gand b1 b2) x side e) h1 h2 == (gand (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gand b1 b2) x side e)] = () let holds_gor (b1 b2 : gexp bool) : Lemma (forall s1 s2 . holds (interp (gor b1 b2)) s1 s2 <==> holds (interp b1) s1 s2 \/ holds (interp b2) s1 s2) [SMTPat (holds (interp (gor b1 b2)))] = () let gsubst_gor (b1 b2: gexp bool) x side e : Lemma (forall h1 h2. (gsubst (gor b1 b2) x side e) h1 h2 == (gor (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (gor b1 b2) x side e)] = () let holds_gnot (b: gexp bool) : Lemma (forall s1 s2 . holds (interp (gnot b)) s1 s2 <==> ~ (holds (interp b) s1 s2)) [SMTPat (holds (interp (gnot b)))] = () let holds_geq (#t: eqtype) (e1 e2 : gexp t) : Lemma (forall s1 s2 . holds (interp (geq e1 e2)) s1 s2 <==> e1 s1 s2 == e2 s1 s2) [SMTPat (holds (interp (geq e1 e2)))] = () let gsubst_geq (#t: eqtype) (b1 b2: gexp t) x side e : Lemma (forall h1 h2. (gsubst (geq b1 b2) x side e) h1 h2 == (geq (gsubst b1 x side e) (gsubst b2 x side e)) h1 h2) [SMTPat (gsubst (geq b1 b2) x side e)] = () let holds_exp_to_gexp_left (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Left)) s1 s2 <==> fst (reify_exp e s1) == true) [SMTPat (holds (interp (exp_to_gexp e Left)))] = () let holds_exp_to_gexp_right (e: exp bool) : Lemma (forall s1 s2 . holds (interp (exp_to_gexp e Right)) s1 s2 <==> fst (reify_exp e s2) == true) [SMTPat (holds (interp (exp_to_gexp e Right)))] = () let holds_r_if_precond_true (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_true b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == true /\ fst (reify_exp b' s2) == true )) = () let holds_r_if_precond_false (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond_false b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ ( ~ (fst (reify_exp b s1) == true \/ fst (reify_exp b' s2) == true)) )) = () let holds_r_if_precond (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (forall s1 s2 . holds (interp (r_if_precond b b' c c' d d' p p')) s1 s2 <==> ( holds (interp p) s1 s2 /\ fst (reify_exp b s1) == fst (reify_exp b' s2) )) = () let r_if (b b': exp bool) (c c' d d' : computation) (p p' : gexp bool) : Lemma (requires ( exec_equiv (r_if_precond_true b b' c c' d d' p p') p' c c' /\ exec_equiv (r_if_precond_false b b' c c' d d' p p') p' d d' )) (ensures ( exec_equiv (r_if_precond b b' c c' d d' p p') p' (ifthenelse b c d) (ifthenelse b' c' d') )) = holds_r_if_precond_true b b' c c' d d' p p'; holds_r_if_precond_false b b' c c' d d' p p'; holds_r_if_precond b b' c c' d d' p p' let r_seq (p0 p1 p2 : gexp bool) (c01 c01' c12 c12' : computation) : Lemma (requires ( exec_equiv p0 p1 c01 c01' /\ exec_equiv p1 p2 c12 c12' )) (ensures ( exec_equiv p0 p2 (seq c01 c12) (seq c01' c12') )) [SMTPatOr [ [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p0 p1 c01 c01')]; [SMTPat (exec_equiv p0 p2 (seq c01 c12) (seq c01' c12')); SMTPat (exec_equiv p1 p2 c12 c12')]; [SMTPat (exec_equiv p0 p1 c01 c01'); SMTPat (exec_equiv p1 p2 c12 c12')]; ]] = d_seq (interp p0) (interp p1) (interp p2) c01 c01' c12 c12' let r_ass (x y: var) (e e' : exp int) (p: gexp bool) : Lemma (exec_equiv (gsubst (gsubst p x Left (exp_to_gexp e Left)) y Right (exp_to_gexp e' Right)) p (assign x e) (assign y e') ) = () let included_alt (p1 p2 : gexp bool) : Lemma (included p1 p2 <==> (forall s1 s2 . p1 s1 s2 == true ==> p2 s1 s2 == true)) [SMTPat (included p1 p2)] = assert (forall s1 s2 . holds (interp p1) s1 s2 <==> p1 s1 s2 == true); assert (forall s1 s2 . holds (interp p2) s1 s2 <==> p2 s1 s2 == true) let r_sub (p1 p2 p1' p2' : gexp bool) (f f' : computation) : Lemma (requires ( exec_equiv p1 p2 f f' /\ included p1' p1 /\ included p2 p2' )) (ensures (exec_equiv p1' p2' f f')) [SMTPat (exec_equiv p1' p2' f f'); SMTPat (exec_equiv p1 p2 f f')] = d_csub (interp p1) (interp p2) (interp p1') (interp p2') f f' let elim_fuel_monotonic (fc:reified_computation) (s0:heap) (f0 f1:nat) : Lemma (requires f0 <= f1 /\ fst (fc f0 s0)) (ensures fc f0 s0 == fc f1 s0) = ()

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Classical.fsti.checked", "Benton2004.fst.checked" ], "interface_file": true, "source_file": "Benton2004.RHL.fst" }

[ { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "Benton2004", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 2, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [ "smt.qi.eager_threshold=100" ], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

b: Benton2004.exp Prims.bool -> b': Benton2004.exp Prims.bool -> c: Benton2004.computation -> c': Benton2004.computation -> phi: Benton2004.RHL.gexp Prims.bool -> phi_c: Benton2004.RHL.gexp Prims.bool -> phi_c': Benton2004.RHL.gexp Prims.bool -> s0: FStar.DM4F.Heap.IntStoreFixed.heap -> s0': FStar.DM4F.Heap.IntStoreFixed.heap -> fuel: Prims.nat -> FStar.Pervasives.Lemma (requires Benton2004.RHL.included phi (Benton2004.RHL.geq (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right)) /\ Benton2004.RHL.included (Benton2004.RHL.gand phi (Benton2004.RHL.gand (Benton2004.RHL.exp_to_gexp b Benton2004.RHL.Left) (Benton2004.RHL.exp_to_gexp b' Benton2004.RHL.Right))) phi_c /\ Benton2004.RHL.included phi_c' phi /\ Benton2004.RHL.exec_equiv phi_c phi_c' c c' /\ Benton2004.Aux.holds (Benton2004.RHL.interp phi) s0 s0' /\ FStar.Pervasives.Native.fst (Benton2004.reify_computation (Benton2004.while b c) fuel s0) == true) (ensures Benton2004.terminates_on (Benton2004.reify_computation (Benton2004.while b' c')) s0') (decreases fuel)

FStar.Pervasives.Lemma

[ "", "lemma" ]

[]

[ "Benton2004.exp", "Prims.bool", "Benton2004.computation", "Benton2004.RHL.gexp", "FStar.DM4F.Heap.IntStoreFixed.heap", "Prims.nat", "FStar.Pervasives.Native.fst", "Benton2004.reify_exp", "FStar.Classical.forall_intro", "Prims.l_imp", "Prims.eq2", "Benton2004.terminates_on", "FStar.Classical.move_requires", "Prims.unit", "Prims.squash", "Prims.Nil", "FStar.Pervasives.pattern", "Prims._assert", "FStar.Pervasives.Native.tuple2", "Benton2004.reified_computation_elim", "Prims.op_Subtraction", "Prims.int", "Prims.op_Addition", "Benton2004.RHL.r_while_terminates'", "Benton2004.RHL.elim_fuel_monotonic", "Benton2004.Aux.holds", "Benton2004.RHL.interp", "Benton2004.terminates_equiv_reified", "Prims.b2t", "FStar.Pervasives.Native.snd", "Benton2004.RHL.gand", "Benton2004.RHL.exp_to_gexp", "Benton2004.RHL.Left", "Benton2004.RHL.Right", "Benton2004.reified_computation", "Benton2004.reify_computation", "Benton2004.while", "Prims.l_and", "Benton2004.RHL.included", "Benton2004.RHL.geq", "Benton2004.RHL.exec_equiv" ]

[ "recursion" ]

false

true

false

let rec r_while_terminates' (b b': exp bool) (c c': computation) (phi phi_c phi_c': gexp bool) (s0 s0': heap) (fuel: nat) : Lemma (requires (included phi (geq (exp_to_gexp b Left) (exp_to_gexp b' Right)) /\ included (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right))) phi_c /\ included phi_c' phi /\ exec_equiv phi_c phi_c' c c' /\ holds (interp phi) s0 s0' /\ fst (reify_computation (while b c) fuel s0) == true)) (ensures (terminates_on (reify_computation (while b' c')) s0')) (decreases fuel) =

let f = reify_computation (while b c) in let f' = reify_computation (while b' c') in let fc = reify_computation c in let fc' = reify_computation c' in if fst (reify_exp b s0) then (assert (holds (interp (gand phi (gand (exp_to_gexp b Left) (exp_to_gexp b' Right)))) s0 s0'); assert (terminates_on (fc') s0'); let g (fuel0: nat) : Lemma (requires (fst (fc' fuel0 s0') == true)) (ensures (terminates_on (f') s0')) = let s1 = snd (fc fuel s0) in let s1' = snd (fc' fuel0 s0') in let fuel1 = fuel + fuel0 in assert (fst (fc' fuel0 s0')); assert (terminates_equiv_reified (interp phi_c) fc fc'); assert (holds (interp phi_c) s0 s0'); assert (terminates_on fc s0); elim_fuel_monotonic fc s0 fuel fuel1; elim_fuel_monotonic fc' s0' fuel0 fuel1; assert (fc fuel1 s0 == fc fuel s0); assert (fc' fuel1 s0' == fc' fuel0 s0'); r_while_terminates' b b' c c' phi phi_c phi_c' s1 s1' (fuel - 1); let g' (fuel2: nat) : Lemma (requires (fst (f' fuel2 s1') == true)) (ensures (terminates_on (f') s0')) = let fuel3 = fuel0 + fuel2 + 1 in assert (f' (fuel3 - 1) s1' == f' fuel2 s1'); reified_computation_elim fc' fuel0 fuel3 s0'; assert (fc' fuel3 s0' == fc' fuel0 s0'); assert (fst (f' fuel3 s0') == true) in Classical.forall_intro (Classical.move_requires g') in Classical.forall_intro (Classical.move_requires g))

false

Spec.P256.fst

Spec.P256.point_mul_g

val point_mul_g (a: qelem) : proj_point

let point_mul_g (a:qelem) : proj_point = point_mul a base_point

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 63, "end_line": 63, "start_col": 0, "start_line": 63 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.P256.PointOps.qelem -> Spec.P256.PointOps.proj_point

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.qelem", "Spec.P256.point_mul", "Spec.P256.PointOps.base_point", "Spec.P256.PointOps.proj_point" ]

[]

false

true

false

let point_mul_g (a: qelem) : proj_point =

point_mul a base_point

false

Spec.P256.fst

Spec.P256.mk_p256_comm_monoid

val mk_p256_comm_monoid:LE.comm_monoid aff_point

let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; }

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 1, "end_line": 28, "start_col": 0, "start_line": 22 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Lib.Exponentiation.Definition.comm_monoid Spec.P256.PointOps.aff_point

Prims.Tot

[ "total" ]

[]

[ "Lib.Exponentiation.Definition.Mkcomm_monoid", "Spec.P256.PointOps.aff_point", "Spec.P256.PointOps.aff_point_at_inf", "Spec.P256.PointOps.aff_point_add", "Spec.P256.Lemmas.aff_point_at_inf_lemma", "Spec.P256.Lemmas.aff_point_add_assoc_lemma", "Spec.P256.Lemmas.aff_point_add_comm_lemma" ]

[]

false

true

false

let mk_p256_comm_monoid:LE.comm_monoid aff_point =

{ LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma }

false

Spec.P256.fst

Spec.P256.point_double_c

val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid

let point_double_c p = PL.to_aff_point_double_lemma p; point_double p

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 16, "end_line": 49, "start_col": 0, "start_line": 47 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Spec.Exponentiation.sqr_st Spec.P256.PointOps.proj_point Spec.P256.mk_to_p256_comm_monoid

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.proj_point", "Spec.P256.PointOps.point_double", "Prims.unit", "Spec.P256.Lemmas.to_aff_point_double_lemma" ]

[]

false

true

false

let point_double_c p =

PL.to_aff_point_double_lemma p; point_double p

false

Spec.P256.fst

Spec.P256.point_mul

val point_mul (a: qelem) (p: proj_point) : proj_point

let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 42, "end_line": 60, "start_col": 0, "start_line": 59 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; }

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.P256.PointOps.qelem -> p: Spec.P256.PointOps.proj_point -> Spec.P256.PointOps.proj_point

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.qelem", "Spec.P256.PointOps.proj_point", "Spec.Exponentiation.exp_fw", "Spec.P256.mk_p256_concrete_ops" ]

[]

false

true

false

let point_mul (a: qelem) (p: proj_point) : proj_point =

SE.exp_fw mk_p256_concrete_ops p 256 a 4

false

Spec.P256.fst

Spec.P256.point_add_c

val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid

let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 15, "end_line": 44, "start_col": 0, "start_line": 42 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Spec.Exponentiation.mul_st Spec.P256.PointOps.proj_point Spec.P256.mk_to_p256_comm_monoid

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.proj_point", "Spec.P256.PointOps.point_add", "Prims.unit", "Spec.P256.Lemmas.to_aff_point_add_lemma" ]

[]

false

true

false

let point_add_c p q =

PL.to_aff_point_add_lemma p q; point_add p q

false

Spec.P256.fst

Spec.P256.point_mul_double_g

val point_mul_double_g (a1 a2: qelem) (p: proj_point) : proj_point

let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 64, "end_line": 67, "start_col": 0, "start_line": 66 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a1: Spec.P256.PointOps.qelem -> a2: Spec.P256.PointOps.qelem -> p: Spec.P256.PointOps.proj_point -> Spec.P256.PointOps.proj_point

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.qelem", "Spec.P256.PointOps.proj_point", "Spec.Exponentiation.exp_double_fw", "Spec.P256.mk_p256_concrete_ops", "Spec.P256.PointOps.base_point" ]

[]

false

true

false

let point_mul_double_g (a1 a2: qelem) (p: proj_point) : proj_point =

SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5

false

Spec.P256.fst

Spec.P256.point_at_inf_c

val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid

let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 14, "end_line": 39, "start_col": 0, "start_line": 37 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; }

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Spec.Exponentiation.one_st Spec.P256.PointOps.proj_point Spec.P256.mk_to_p256_comm_monoid

Prims.Tot

[ "total" ]

[]

[ "Prims.unit", "Spec.P256.PointOps.point_at_inf", "Spec.P256.Lemmas.to_aff_point_at_infinity_lemma", "Spec.P256.PointOps.proj_point" ]

[]

false

true

false

let point_at_inf_c _ =

PL.to_aff_point_at_infinity_lemma (); point_at_inf

false

Spec.P256.fst

Spec.P256.min_input_length

val min_input_length (a: hash_alg_ecdsa) : nat

let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 43, "end_line": 84, "start_col": 0, "start_line": 83 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.P256.hash_alg_ecdsa -> Prims.nat

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.hash_alg_ecdsa", "Spec.Hash.Definitions.hash_alg", "Prims.l_or", "Prims.eq2", "Spec.Hash.Definitions.SHA2_256", "Spec.Hash.Definitions.SHA2_384", "Spec.Hash.Definitions.SHA2_512", "Prims.nat" ]

[]

false

true

false

let min_input_length (a: hash_alg_ecdsa) : nat =

match a with | NoHash -> 32 | Hash a -> 0

false

Spec.P256.fst

Spec.P256.hash_ecdsa

val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32})

let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 69, "end_line": 95, "start_col": 0, "start_line": 94 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32})

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.P256.hash_alg_ecdsa -> msg_len: Lib.IntTypes.size_nat{msg_len >= Spec.P256.min_input_length a} -> msg: Lib.Sequence.lseq Lib.IntTypes.uint8 msg_len -> r: Lib.ByteSequence.lbytes (match Hash? a with | true -> Spec.Hash.Definitions.hash_length (let Spec.P256.Hash a = a in a) | _ -> msg_len) {Lib.Sequence.length r >= 32}

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.hash_alg_ecdsa", "Lib.IntTypes.size_nat", "Prims.b2t", "Prims.op_GreaterThanOrEqual", "Spec.P256.min_input_length", "Lib.Sequence.lseq", "Lib.IntTypes.uint8", "Spec.Hash.Definitions.hash_alg", "Prims.l_or", "Prims.eq2", "Spec.Hash.Definitions.SHA2_256", "Spec.Hash.Definitions.SHA2_384", "Spec.Hash.Definitions.SHA2_512", "Spec.Agile.Hash.hash", "Lib.ByteSequence.lbytes", "Spec.P256.uu___is_Hash", "Spec.Hash.Definitions.hash_length", "Prims.op_Negation", "Spec.Hash.Definitions.is_shake", "Prims.bool", "Lib.Sequence.length", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC" ]

[]

false

let hash_ecdsa a msg_len msg =

match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg

false

Spec.P256.fst

Spec.P256.ecdsa_verify_msg_as_qelem

val ecdsa_verify_msg_as_qelem (m: qelem) (public_key: lbytes 64) (sign_r sign_s: lbytes 32) : bool

let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 5, "end_line": 134, "start_col": 0, "start_line": 117 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

m: Spec.P256.PointOps.qelem -> public_key: Lib.ByteSequence.lbytes 64 -> sign_r: Lib.ByteSequence.lbytes 32 -> sign_s: Lib.ByteSequence.lbytes 32 -> Prims.bool

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.qelem", "Lib.ByteSequence.lbytes", "Prims.op_Negation", "Prims.op_AmpAmp", "FStar.Pervasives.Native.uu___is_Some", "Spec.P256.PointOps.proj_point", "Prims.bool", "Prims.nat", "Spec.P256.PointOps.is_point_at_inf", "FStar.Pervasives.Native.Mktuple3", "Prims.op_Equality", "Prims.int", "Prims.op_Modulus", "Spec.P256.PointOps.order", "Spec.P256.PointOps.felem", "Spec.P256.PointOps.op_Slash_Percent", "Spec.P256.point_mul_double_g", "FStar.Pervasives.Native.__proj__Some__item__v", "Spec.P256.PointOps.op_Star_Hat", "Spec.P256.PointOps.qinv", "Prims.op_LessThan", "Prims.b2t", "Prims.pow2", "Prims.op_Multiply", "Lib.Sequence.length", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.ByteSequence.nat_from_bytes_be", "FStar.Pervasives.Native.option", "Spec.P256.PointOps.load_point" ]

[]

false

let ecdsa_verify_msg_as_qelem (m: qelem) (public_key: lbytes 64) (sign_r sign_s: lbytes 32) : bool =

let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else let x = _X /% _Z in x % order = r

false

Spec.P256.fst

Spec.P256.ecdsa_sign_msg_as_qelem

val ecdsa_sign_msg_as_qelem (m: qelem) (private_key nonce: lbytes 32) : option (lbytes 64)

let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 71, "end_line": 114, "start_col": 0, "start_line": 98 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

m: Spec.P256.PointOps.qelem -> private_key: Lib.ByteSequence.lbytes 32 -> nonce: Lib.ByteSequence.lbytes 32 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.PointOps.qelem", "Lib.ByteSequence.lbytes", "Prims.op_Negation", "Prims.op_AmpAmp", "FStar.Pervasives.Native.None", "Prims.bool", "Prims.nat", "Prims.op_BarBar", "Prims.op_Equality", "Prims.int", "FStar.Pervasives.Native.Some", "Lib.Sequence.concat", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "FStar.Pervasives.Native.option", "Lib.Sequence.seq", "Lib.IntTypes.int_t", "Prims.l_and", "Prims.eq2", "Lib.Sequence.length", "Prims.l_or", "Prims.b2t", "Prims.op_LessThan", "Prims.pow2", "Prims.op_Multiply", "Lib.ByteSequence.nat_from_intseq_be", "Lib.ByteSequence.nat_to_bytes_be", "Spec.P256.PointOps.op_Star_Hat", "Spec.P256.PointOps.op_Plus_Hat", "Spec.P256.PointOps.qinv", "Prims.op_Modulus", "Spec.P256.PointOps.order", "Spec.P256.PointOps.felem", "Spec.P256.PointOps.op_Slash_Percent", "Spec.P256.PointOps.proj_point", "Spec.P256.point_mul_g", "Lib.ByteSequence.nat_from_bytes_be" ]

[]

false

let ecdsa_sign_msg_as_qelem (m: qelem) (private_key nonce: lbytes 32) : option (lbytes 64) =

let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb)

false

Spec.P256.fst

Spec.P256.validate_public_key

val validate_public_key (pk: lbytes 64) : bool

let validate_public_key (pk:lbytes 64) : bool = Some? (load_point pk)

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 23, "end_line": 202, "start_col": 0, "start_line": 201 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None /// Parsing and Serializing public keys // raw = [ x; y ], 64 bytes // uncompressed = [ 0x04; x; y ], 65 bytes // compressed = [ 0x02 for even `y` and 0x03 for odd `y`; x ], 33 bytes

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

pk: Lib.ByteSequence.lbytes 64 -> Prims.bool

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "FStar.Pervasives.Native.uu___is_Some", "Spec.P256.PointOps.proj_point", "Spec.P256.PointOps.load_point", "Prims.bool" ]

[]

false

let validate_public_key (pk: lbytes 64) : bool =

Some? (load_point pk)

false

Spec.P256.fst

Spec.P256.ecdsa_verification_agile

val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool

let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 66, "end_line": 171, "start_col": 0, "start_line": 168 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

alg: Spec.P256.hash_alg_ecdsa -> msg_len: Lib.IntTypes.size_nat{msg_len >= Spec.P256.min_input_length alg} -> msg: Lib.ByteSequence.lbytes msg_len -> public_key: Lib.ByteSequence.lbytes 64 -> signature_r: Lib.ByteSequence.lbytes 32 -> signature_s: Lib.ByteSequence.lbytes 32 -> Prims.bool

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.hash_alg_ecdsa", "Lib.IntTypes.size_nat", "Prims.b2t", "Prims.op_GreaterThanOrEqual", "Spec.P256.min_input_length", "Lib.ByteSequence.lbytes", "Spec.P256.ecdsa_verify_msg_as_qelem", "Prims.int", "Prims.op_Modulus", "Lib.ByteSequence.nat_from_bytes_be", "Lib.IntTypes.SEC", "Lib.Sequence.sub", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Spec.P256.uu___is_Hash", "Spec.Hash.Definitions.hash_length", "Spec.Hash.Definitions.hash_alg", "Prims.l_or", "Prims.eq2", "Spec.Hash.Definitions.SHA2_256", "Spec.Hash.Definitions.SHA2_384", "Spec.Hash.Definitions.SHA2_512", "Prims.op_Negation", "Spec.Hash.Definitions.is_shake", "Prims.bool", "Spec.P256.PointOps.order", "Lib.Sequence.lseq", "Lib.IntTypes.int_t", "Lib.Sequence.length", "Spec.P256.hash_ecdsa" ]

[]

false

let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s =

let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s

false

Spec.P256.fst

Spec.P256.pk_uncompressed_to_raw

val pk_uncompressed_to_raw (pk: lbytes 65) : option (lbytes 64)

let pk_uncompressed_to_raw (pk:lbytes 65) : option (lbytes 64) = if Lib.RawIntTypes.u8_to_UInt8 pk.[0] <> 0x04uy then None else Some (sub pk 1 64)

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 83, "end_line": 205, "start_col": 0, "start_line": 204 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None /// Parsing and Serializing public keys // raw = [ x; y ], 64 bytes // uncompressed = [ 0x04; x; y ], 65 bytes // compressed = [ 0x02 for even `y` and 0x03 for odd `y`; x ], 33 bytes let validate_public_key (pk:lbytes 64) : bool = Some? (load_point pk)

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

pk: Lib.ByteSequence.lbytes 65 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "Prims.op_disEquality", "FStar.UInt8.t", "Lib.RawIntTypes.u8_to_UInt8", "Lib.Sequence.op_String_Access", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "FStar.UInt8.__uint_to_t", "FStar.Pervasives.Native.None", "Prims.bool", "FStar.Pervasives.Native.Some", "Lib.Sequence.sub", "FStar.Pervasives.Native.option" ]

[]

false

let pk_uncompressed_to_raw (pk: lbytes 65) : option (lbytes 64) =

if Lib.RawIntTypes.u8_to_UInt8 pk.[ 0 ] <> 0x04uy then None else Some (sub pk 1 64)

false

Spec.P256.fst

Spec.P256.ecdh

val ecdh (their_public_key: lbytes 64) (private_key: lbytes 32) : option (lbytes 64)

let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 11, "end_line": 192, "start_col": 0, "start_line": 185 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

their_public_key: Lib.ByteSequence.lbytes 64 -> private_key: Lib.ByteSequence.lbytes 32 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "Prims.op_AmpAmp", "FStar.Pervasives.Native.uu___is_Some", "Spec.P256.PointOps.proj_point", "FStar.Pervasives.Native.Some", "Spec.P256.PointOps.point_store", "Spec.P256.point_mul", "FStar.Pervasives.Native.__proj__Some__item__v", "Prims.bool", "FStar.Pervasives.Native.None", "FStar.Pervasives.Native.option", "Prims.op_LessThan", "Spec.P256.PointOps.order", "Prims.nat", "Prims.b2t", "Prims.pow2", "Prims.op_Multiply", "Lib.Sequence.length", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.ByteSequence.nat_from_bytes_be", "Spec.P256.PointOps.load_point" ]

[]

false

let ecdh (their_public_key: lbytes 64) (private_key: lbytes 32) : option (lbytes 64) =

let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None

false

Spec.P256.fst

Spec.P256.ecdsa_signature_agile

val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64)

let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 47, "end_line": 156, "start_col": 0, "start_line": 153 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64)

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

alg: Spec.P256.hash_alg_ecdsa -> msg_len: Lib.IntTypes.size_nat{msg_len >= Spec.P256.min_input_length alg} -> msg: Lib.ByteSequence.lbytes msg_len -> private_key: Lib.ByteSequence.lbytes 32 -> nonce: Lib.ByteSequence.lbytes 32 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Spec.P256.hash_alg_ecdsa", "Lib.IntTypes.size_nat", "Prims.b2t", "Prims.op_GreaterThanOrEqual", "Spec.P256.min_input_length", "Lib.ByteSequence.lbytes", "Spec.P256.ecdsa_sign_msg_as_qelem", "Prims.int", "Prims.op_Modulus", "Lib.ByteSequence.nat_from_bytes_be", "Lib.IntTypes.SEC", "Lib.Sequence.sub", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Spec.P256.uu___is_Hash", "Spec.Hash.Definitions.hash_length", "Spec.Hash.Definitions.hash_alg", "Prims.l_or", "Prims.eq2", "Spec.Hash.Definitions.SHA2_256", "Spec.Hash.Definitions.SHA2_384", "Spec.Hash.Definitions.SHA2_512", "Prims.op_Negation", "Spec.Hash.Definitions.is_shake", "Prims.bool", "Spec.P256.PointOps.order", "Lib.Sequence.lseq", "Lib.IntTypes.int_t", "Lib.Sequence.length", "Spec.P256.hash_ecdsa", "FStar.Pervasives.Native.option" ]

[]

false

let ecdsa_signature_agile alg msg_len msg private_key nonce =

let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce

false

Spec.P256.fst

Spec.P256.pk_uncompressed_from_raw

val pk_uncompressed_from_raw (pk: lbytes 64) : lbytes 65

let pk_uncompressed_from_raw (pk:lbytes 64) : lbytes 65 = concat (create 1 (u8 0x04)) pk

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 32, "end_line": 208, "start_col": 0, "start_line": 207 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None /// Parsing and Serializing public keys // raw = [ x; y ], 64 bytes // uncompressed = [ 0x04; x; y ], 65 bytes // compressed = [ 0x02 for even `y` and 0x03 for odd `y`; x ], 33 bytes let validate_public_key (pk:lbytes 64) : bool = Some? (load_point pk) let pk_uncompressed_to_raw (pk:lbytes 65) : option (lbytes 64) = if Lib.RawIntTypes.u8_to_UInt8 pk.[0] <> 0x04uy then None else Some (sub pk 1 64)

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

pk: Lib.ByteSequence.lbytes 64 -> Lib.ByteSequence.lbytes 65

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "Lib.Sequence.concat", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.Sequence.create", "Lib.IntTypes.u8" ]

[]

false

let pk_uncompressed_from_raw (pk: lbytes 64) : lbytes 65 =

concat (create 1 (u8 0x04)) pk

false

Spec.P256.fst

Spec.P256.secret_to_public

val secret_to_public (private_key: lbytes 32) : option (lbytes 64)

let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 11, "end_line": 182, "start_col": 0, "start_line": 176 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

private_key: Lib.ByteSequence.lbytes 32 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "FStar.Pervasives.Native.Some", "Spec.P256.PointOps.point_store", "Spec.P256.PointOps.proj_point", "Spec.P256.point_mul_g", "Prims.bool", "FStar.Pervasives.Native.None", "FStar.Pervasives.Native.option", "Prims.op_AmpAmp", "Prims.op_LessThan", "Spec.P256.PointOps.order", "Prims.nat", "Prims.b2t", "Prims.pow2", "Prims.op_Multiply", "Lib.Sequence.length", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.ByteSequence.nat_from_bytes_be" ]

[]

false

let secret_to_public (private_key: lbytes 32) : option (lbytes 64) =

let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None

false

Spec.P256.fst

Spec.P256.pk_compressed_to_raw

val pk_compressed_to_raw (pk: lbytes 33) : option (lbytes 64)

let pk_compressed_to_raw (pk:lbytes 33) : option (lbytes 64) = let pk_x = sub pk 1 32 in match (aff_point_decompress pk) with | Some (x, y) -> Some (concat #_ #32 #32 pk_x (nat_to_bytes_be 32 y)) | None -> None

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 16, "end_line": 214, "start_col": 0, "start_line": 210 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None /// Parsing and Serializing public keys // raw = [ x; y ], 64 bytes // uncompressed = [ 0x04; x; y ], 65 bytes // compressed = [ 0x02 for even `y` and 0x03 for odd `y`; x ], 33 bytes let validate_public_key (pk:lbytes 64) : bool = Some? (load_point pk) let pk_uncompressed_to_raw (pk:lbytes 65) : option (lbytes 64) = if Lib.RawIntTypes.u8_to_UInt8 pk.[0] <> 0x04uy then None else Some (sub pk 1 64) let pk_uncompressed_from_raw (pk:lbytes 64) : lbytes 65 = concat (create 1 (u8 0x04)) pk

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

pk: Lib.ByteSequence.lbytes 33 -> FStar.Pervasives.Native.option (Lib.ByteSequence.lbytes 64)

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "Spec.P256.PointOps.aff_point_decompress", "Prims.nat", "FStar.Pervasives.Native.Some", "Lib.Sequence.concat", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.ByteSequence.nat_to_bytes_be", "FStar.Pervasives.Native.None", "FStar.Pervasives.Native.option", "Lib.Sequence.lseq", "Lib.IntTypes.int_t", "Prims.l_and", "Prims.eq2", "FStar.Seq.Base.seq", "Lib.Sequence.to_seq", "FStar.Seq.Base.slice", "Prims.op_Addition", "Prims.l_Forall", "Prims.b2t", "Prims.op_LessThan", "Prims.l_or", "FStar.Seq.Base.index", "Lib.Sequence.index", "Lib.Sequence.sub" ]

[]

false

let pk_compressed_to_raw (pk: lbytes 33) : option (lbytes 64) =

let pk_x = sub pk 1 32 in match (aff_point_decompress pk) with | Some (x, y) -> Some (concat #_ #32 #32 pk_x (nat_to_bytes_be 32 y)) | None -> None

false

MiniValeSemantics.fst

MiniValeSemantics.inst_Triple

val inst_Triple:with_wps codes_Triple

let inst_Triple : with_wps codes_Triple = //A typeclass instance for our program QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq (inst_Move (OReg Rbx) (OReg Rax)) ( QSeq 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{ "file_name": "examples/metatheory/MiniValeSemantics.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 68, "end_line": 945, "start_col": 0, "start_line": 675 }

(* Copyright 2008-2019 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. Authors: C. Hawblitzel, N. Swamy *) module MiniValeSemantics (* This is a highly-simplified model of Vale/F*, based on Section 3.1-3.3 of the paper of the POPL '19 paper. It is derived from the QuickRegs1 code in the popl-artifact-submit branch of Vale. *) /// We use this tag to mark certain definitions /// and control normalization based on it irreducible let qattr = () /// 2^64 let pow2_64 = 0x10000000000000000 /// Natural numbers representable in 64 bits type nat64 = i:int{0 <= i /\ i < pow2_64} /// We have 4 registers type reg = | Rax | Rbx | Rcx | Rdx /// An operand is either a register or a constant type operand = | OReg: r:reg -> operand | OConst: n:nat64 -> operand /// Only 2 instructions here: /// A move or an add type ins = | Mov64: dst:operand -> src:operand -> ins | Add64: dst:operand -> src:operand -> ins /// A program is /// - a single instruction /// - a block of instructions /// - or a while loop type code = | Ins: ins:ins -> code | Block: block:list code -> code | WhileLessThan: src1:operand -> src2:operand -> whileBody:code -> code /// The state of a program is the register file /// holiding a 64-bit value for each register type state = reg -> nat64 /// fuel: To prove the termination of while loops, we're going to /// instrument while loops with fuel type fuel = nat /// Evaluating an operand: /// -- marked for reduction /// -- Registers evaluated by state lookup [@@qattr] let eval_operand (o:operand) (s:state) : nat64 = match o with | OReg r -> s r | OConst n -> n /// updating a register state `s` at `r` with `v` [@@qattr] let update_reg (s:state) (r:reg) (v:nat64) : state = fun r' -> if r = r' then v else s r' /// updating a register state `s` at `r` with `s' r` [@@qattr] let update_state (r:reg) (s' s:state) : state = update_reg s r (s' r) // We don't have an "ok" flag, so errors just result an arbitrary state: assume val unknown_state (s:state) : state (*** A basic semantics using a big-step interpreter ***) /// Instruction evaluation: /// only some operands are valid let eval_ins (ins:ins) (s:state) : state = match ins with | Mov64 (OConst _) _ -> unknown_state s | Mov64 (OReg dst) src -> update_reg s dst (eval_operand src s) | Add64 (OConst _) _ -> unknown_state s | Add64 (OReg dst) src -> update_reg s dst ((s dst + eval_operand src s) % 0x10000000000000000) /// eval_code: /// A fueled big-step interpreter /// While lops return None when we're out of fuel let rec eval_code (c:code) (f:fuel) (s:state) : option state = match c with | Ins ins -> Some (eval_ins ins s) | Block cs -> eval_codes cs f s | WhileLessThan src1 src2 body -> if f = 0 then None else if eval_operand src1 s < eval_operand src2 s then match eval_code body f s with | None -> None | Some s -> eval_code c (f - 1) s else Some s and eval_codes (cs:list code) (f:fuel) (s:state) : option state = match cs with | [] -> Some s | c::cs -> match eval_code c f s with | None -> None | Some s -> eval_codes cs f s (*** END OF TRUSTED SEMANTICS ***) //////////////////////////////////////////////////////////////////////////////// /// 1. We prove that increasing the fuel is irrelevant to terminating executions val increase_fuel (c:code) (s0:state) (f0:fuel) (sN:state) (fN:fuel) : Lemma (requires eval_code c f0 s0 == Some sN /\ f0 <= fN) (ensures eval_code c fN s0 == Some sN) (decreases %[f0; c]) val increase_fuels (c:list code) (s0:state) (f0:fuel) (sN:state) (fN:fuel) : Lemma (requires eval_code (Block c) f0 s0 == Some sN /\ f0 <= fN) (ensures eval_code (Block c) fN s0 == Some sN) (decreases %[f0; c]) let rec increase_fuel (c:code) (s0:state) (f0:fuel) (sN:state) (fN:fuel) = match c with | Ins ins -> () | Block l -> increase_fuels l s0 f0 sN fN | WhileLessThan src1 src2 body -> if eval_operand src1 s0 < eval_operand src2 s0 then match eval_code body f0 s0 with | None -> () | Some s1 -> increase_fuel body s0 f0 s1 fN; increase_fuel c s1 (f0 - 1) sN (fN - 1) else () and increase_fuels (c:list code) (s0:state) (f0:fuel) (sN:state) (fN:fuel) = match c with | [] -> () | h::t -> let Some s1 = eval_code h f0 s0 in increase_fuel h s0 f0 s1 fN; increase_fuels t s1 f0 sN fN /// 2. We can compute the fuel needed to run a sequential composition /// as the max of the fuel to compute each piece of code in it let lemma_merge (c:code) (cs:list code) (s0:state) (f0:fuel) (sM:state) (fM:fuel) (sN:state) : Ghost fuel (requires eval_code c f0 s0 == Some sM /\ eval_code (Block cs) fM sM == Some sN) (ensures fun fN -> eval_code (Block (c::cs)) fN s0 == Some sN) = let f = if f0 > fM then f0 else fM in increase_fuel c s0 f0 sM f; increase_fuel (Block cs) sM fM sN f; f ///////////////////////////////////////////////////////////////// // Now, we're going to define a verification-condition generator // // The main idea is that we're going to: // // 1. define a kind of typeclass, that associates with a // piece of code a weakest-precondition rule for it // // 2. Define a WP-generator that computes WPs for each of the // control constructs of the language, given a program // represented as the raw code packaged with their typeclass // instances for computing their WPs ///////////////////////////////////////////////////////////////// [@@qattr] let t_post = state -> Type0 [@@qattr] let t_pre = state -> Type0 /// t_wp: The type of weakest preconditions let t_wp = t_post -> t_pre /// c `has_wp` wp: The main judgment in our program logic let has_wp (c:code) (wp:t_wp) : Type = k:t_post -> //for any post-condition s0:state -> //and initial state Ghost (state * fuel) (requires wp k s0) //Given the precondition (ensures fun (sM, f0) -> //we can compute the fuel f0 needed so that eval_code c f0 s0 == Some sM /\ //eval_code with that fuel returns sM k sM) //and the post-condition is true on sM /// An abbreviation for a thunked lemma let t_lemma (pre:Type0) (post:Type0) = unit -> Lemma (requires pre) (ensures post) /// `with_wp` : A typeclass for code packaged with its wp [@@qattr] noeq type with_wp : code -> Type = | QProc: c:code -> wp:t_wp -> hasWp:has_wp c wp -> with_wp c /// `with_wps`: A typclass for lists of code values packages with their wps noeq type with_wps : list code -> Type = | QEmpty: //empty list with_wps [] | QSeq: //cons #c:code -> #cs:list code -> hd:with_wp c -> tl:with_wps cs -> with_wps (c::cs) | QLemma: //augmenting an instruction sequence with a lemma #cs:list code -> pre:Type0 -> post:Type0 -> t_lemma pre post -> with_wps cs -> with_wps cs [@@qattr] let rec vc_gen (cs:list code) (qcs:with_wps cs) (k:t_post) : Tot (state -> Tot Type0 (decreases qcs)) = fun s0 -> match qcs with | QEmpty -> k s0 //no instructions; prove the postcondition right away | QSeq qc qcs -> // let pre_tl = //compute the VC generator for the tail, a precondition qc.wp (vc_gen (Cons?.tl cs) qcs k) s0 // in // qc.wp pre_tl s0 //apply the wp-generator to the precondition for the tail | QLemma pre post _ qcs -> pre /\ //prove the precondition of the lemma (post ==> vc_gen cs qcs k s0) //and assume its postcondition to verify the program /// The vc-generator is sound let rec vc_sound (cs:list code) (qcs:with_wps cs) (k:state -> Type0) (s0:state) : Ghost (state * fuel) (requires vc_gen cs qcs k s0) (ensures fun (sN, fN) -> eval_code (Block cs) fN s0 == Some sN /\ k sN) = match qcs with | QEmpty -> (s0, 0) | QSeq qc qcs -> let Cons c cs' = cs in let (sM, fM) = qc.hasWp (vc_gen cs' qcs k) s0 in let (sN, fN) = vc_sound cs' qcs k sM in let fN' = lemma_merge c cs' s0 fM sM fN sN in (sN, fN') | QLemma pre post lem qcs' -> lem (); vc_sound cs qcs' k s0 let vc_sound' (cs:list code) (qcs:with_wps cs) : has_wp (Block cs) (vc_gen cs qcs) = vc_sound cs qcs (*** Instances of with_wp ***) //////////////////////////////////////////////////////////////////////////////// //Instance for Mov //////////////////////////////////////////////////////////////////////////////// let lemma_Move (s0:state) (dst:operand) (src:operand) : Ghost (state * fuel) (requires OReg? dst) (ensures fun (sM, fM) -> eval_code (Ins (Mov64 dst src)) fM s0 == Some sM /\ eval_operand dst sM == eval_operand src s0 /\ sM == update_state (OReg?.r dst) sM s0 ) = let Some sM = eval_code (Ins (Mov64 dst src)) 0 s0 in (sM, 0) [@@qattr] let wp_Move (dst:operand) (src:operand) (k:state -> Type0) (s0:state) : Type0 = OReg? dst /\ (forall (x:nat64). let sM = update_reg s0 (OReg?.r dst) x in eval_operand dst sM == eval_operand src s0 ==> k sM ) let hasWp_Move (dst:operand) (src:operand) (k:state -> Type0) (s0:state) : Ghost (state * fuel) (requires wp_Move dst src k s0) (ensures fun (sM, f0) -> eval_code (Ins (Mov64 dst src)) f0 s0 == Some sM /\ k sM) = lemma_Move s0 dst src [@@qattr] let inst_Move (dst:operand) (src:operand) : with_wp (Ins (Mov64 dst src)) = QProc (Ins (Mov64 dst src)) (wp_Move dst src) (hasWp_Move dst src) //////////////////////////////////////////////////////////////////////////////// //Instance for Add //////////////////////////////////////////////////////////////////////////////// let lemma_Add (s0:state) (dst:operand) (src:operand) : Ghost (state * fuel) (requires OReg? dst /\ eval_operand dst s0 + eval_operand src s0 < pow2_64) (ensures fun (sM, fM) -> eval_code (Ins (Add64 dst src)) fM s0 == Some sM /\ eval_operand dst sM == eval_operand dst s0 + eval_operand src s0 /\ sM == update_state (OReg?.r dst) sM s0 ) = let Some sM = eval_code (Ins (Add64 dst src)) 0 s0 in (sM, 0) [@@qattr] let wp_Add (dst:operand) (src:operand) (k:state -> Type0) (s0:state) : Type0 = OReg? dst /\ eval_operand dst s0 + eval_operand src s0 < pow2_64 /\ (forall (x:nat64). let sM = update_reg s0 (OReg?.r dst) x in eval_operand dst sM == eval_operand dst s0 + eval_operand src s0 ==> k sM ) let hasWp_Add (dst:operand) (src:operand) (k:state -> Type0) (s0:state) : Ghost (state * fuel) (requires wp_Add dst src k s0) (ensures fun (sM, f0) -> eval_code (Ins (Add64 dst src)) f0 s0 == Some sM /\ k sM) = lemma_Add s0 dst src [@@qattr] let inst_Add (dst:operand) (src:operand) : with_wp (Ins (Add64 dst src)) = QProc (Ins (Add64 dst src)) (wp_Add dst src) (hasWp_Add dst src) //////////////////////////////////////////////////////////////////////////////// //Running the VC generator using the F* normalizer //////////////////////////////////////////////////////////////////////////////// unfold let normal_steps : list string = [ `%OReg?; `%OReg?.r; `%QProc?.wp; ] unfold let normal (x:Type0) : Type0 = norm [nbe; iota; zeta; simplify; primops; delta_attr [`%qattr]; delta_only normal_steps] x let vc_sound_norm (cs:list code) (qcs:with_wps cs) (k:state -> Type0) (s0:state) : Ghost (state * fuel) (requires normal (vc_gen cs qcs k s0)) (ensures fun (sN, fN) -> eval_code (Block cs) fN s0 == Some sN /\ k sN) = vc_sound cs qcs k s0 //////////////////////////////////////////////////////////////////////////////// // Verifying a simple program //////////////////////////////////////////////////////////////////////////////// [@@qattr] let codes_Triple : list code = [Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //1 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //2 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //3 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //4 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //5 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //6 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //7 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //8 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //9 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //10 Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; Ins (Mov64 (OReg Rbx) (OReg Rax)); //mov rbx rax; //11 Ins (Add64 (OReg Rax) (OReg Rbx)); //add rax rbx; Ins (Add64 (OReg Rbx) (OReg Rax))] //add rbx rax

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.FunctionalExtensionality.fsti.checked" ], "interface_file": false, "source_file": "MiniValeSemantics.fst" }

[ { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

MiniValeSemantics.with_wps MiniValeSemantics.codes_Triple

Prims.Tot

[ "total" ]

[]

[ "MiniValeSemantics.QSeq", "MiniValeSemantics.Ins", "MiniValeSemantics.Mov64", "MiniValeSemantics.OReg", "MiniValeSemantics.Rbx", "MiniValeSemantics.Rax", "Prims.Cons", "MiniValeSemantics.code", "MiniValeSemantics.Add64", "Prims.Nil", "MiniValeSemantics.inst_Move", "MiniValeSemantics.inst_Add", "MiniValeSemantics.QEmpty" ]

[]

false

true

false

let inst_Triple:with_wps codes_Triple =

QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax)) (QSeq (inst_Move (OReg Rbx) (OReg Rax) ) (QSeq (inst_Move (OReg Rbx) (OReg Rax) ) (QSeq (inst_Move (OReg Rbx ) (OReg Rax )) (QSeq (inst_Move ( OReg Rbx ) ( OReg Rax ) ) (QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq 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OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) 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inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Move ( OReg Rbx ) ( OReg Rax ) ) ( QSeq ( inst_Add ( OReg Rax ) ( OReg Rbx ) ) ( QSeq ( inst_Add ( OReg Rbx ) ( OReg Rax ) ) ( QEmpty ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )) )))))))))) )))))))))))))

false

Spec.P256.fst

Spec.P256.pk_compressed_from_raw

val pk_compressed_from_raw (pk: lbytes 64) : lbytes 33

let pk_compressed_from_raw (pk:lbytes 64) : lbytes 33 = let pk_x = sub pk 0 32 in let pk_y = sub pk 32 32 in let is_pk_y_odd = nat_from_bytes_be pk_y % 2 = 1 in // <==> pk_y % 2 = 1 let pk0 = if is_pk_y_odd then u8 0x03 else u8 0x02 in concat (create 1 pk0) pk_x

{ "file_name": "specs/Spec.P256.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 28, "end_line": 221, "start_col": 0, "start_line": 216 }

module Spec.P256 open FStar.Mul open Lib.IntTypes open Lib.Sequence open Lib.ByteSequence open Spec.Hash.Definitions module M = Lib.NatMod module LE = Lib.Exponentiation module SE = Spec.Exponentiation module PL = Spec.P256.Lemmas include Spec.P256.PointOps #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" (** https://eprint.iacr.org/2013/816.pdf *) /// Elliptic curve scalar multiplication let mk_p256_comm_monoid : LE.comm_monoid aff_point = { LE.one = aff_point_at_inf; LE.mul = aff_point_add; LE.lemma_one = PL.aff_point_at_inf_lemma; LE.lemma_mul_assoc = PL.aff_point_add_assoc_lemma; LE.lemma_mul_comm = PL.aff_point_add_comm_lemma; } let mk_to_p256_comm_monoid : SE.to_comm_monoid proj_point = { SE.a_spec = aff_point; SE.comm_monoid = mk_p256_comm_monoid; SE.refl = to_aff_point; } val point_at_inf_c: SE.one_st proj_point mk_to_p256_comm_monoid let point_at_inf_c _ = PL.to_aff_point_at_infinity_lemma (); point_at_inf val point_add_c : SE.mul_st proj_point mk_to_p256_comm_monoid let point_add_c p q = PL.to_aff_point_add_lemma p q; point_add p q val point_double_c : SE.sqr_st proj_point mk_to_p256_comm_monoid let point_double_c p = PL.to_aff_point_double_lemma p; point_double p let mk_p256_concrete_ops : SE.concrete_ops proj_point = { SE.to = mk_to_p256_comm_monoid; SE.one = point_at_inf_c; SE.mul = point_add_c; SE.sqr = point_double_c; } // [a]P let point_mul (a:qelem) (p:proj_point) : proj_point = SE.exp_fw mk_p256_concrete_ops p 256 a 4 // [a]G let point_mul_g (a:qelem) : proj_point = point_mul a base_point // [a1]G + [a2]P let point_mul_double_g (a1 a2:qelem) (p:proj_point) : proj_point = SE.exp_double_fw mk_p256_concrete_ops base_point 256 a1 p a2 5 /// ECDSA over the P256 elliptic curve type hash_alg_ecdsa = | NoHash | Hash of (a:hash_alg{a == SHA2_256 \/ a == SHA2_384 \/ a == SHA2_512}) let _: squash (inversion hash_alg_ecdsa) = allow_inversion hash_alg_ecdsa let _: squash (pow2 32 < pow2 61 /\ pow2 32 < pow2 125) = Math.Lemmas.pow2_lt_compat 61 32; Math.Lemmas.pow2_lt_compat 125 32 let min_input_length (a:hash_alg_ecdsa) : nat = match a with | NoHash -> 32 | Hash a -> 0 val hash_ecdsa: a:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length a} -> msg:lseq uint8 msg_len -> Tot (r:lbytes (if Hash? a then hash_length (match a with Hash a -> a) else msg_len){length r >= 32}) let hash_ecdsa a msg_len msg = match a with | NoHash -> msg | Hash a -> Spec.Agile.Hash.hash a msg let ecdsa_sign_msg_as_qelem (m:qelem) (private_key nonce:lbytes 32) : option (lbytes 64) = let k_q = nat_from_bytes_be nonce in let d_a = nat_from_bytes_be private_key in let is_privkey_valid = 0 < d_a && d_a < order in let is_nonce_valid = 0 < k_q && k_q < order in if not (is_privkey_valid && is_nonce_valid) then None else begin let _X, _Y, _Z = point_mul_g k_q in let x = _X /% _Z in let r = x % order in let kinv = qinv k_q in let s = kinv *^ (m +^ r *^ d_a) in let rb = nat_to_bytes_be 32 r in let sb = nat_to_bytes_be 32 s in if r = 0 || s = 0 then None else Some (concat #_ #32 #32 rb sb) end let ecdsa_verify_msg_as_qelem (m:qelem) (public_key:lbytes 64) (sign_r sign_s:lbytes 32) : bool = let pk = load_point public_key in let r = nat_from_bytes_be sign_r in let s = nat_from_bytes_be sign_s in let is_r_valid = 0 < r && r < order in let is_s_valid = 0 < s && s < order in if not (Some? pk && is_r_valid && is_s_valid) then false else begin let sinv = qinv s in let u1 = sinv *^ m in let u2 = sinv *^ r in let _X, _Y, _Z = point_mul_double_g u1 u2 (Some?.v pk) in if is_point_at_inf (_X, _Y, _Z) then false else begin let x = _X /% _Z in x % order = r end end (* _Z <> 0 q < prime < 2 * q let x = _X /% _Z in x % q = r <==> 1. x = r <==> _X = r *% _Z 2. x - q = r <==> _X = (r + q) *% _Z *) val ecdsa_signature_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> private_key:lbytes 32 -> nonce:lbytes 32 -> option (lbytes 64) let ecdsa_signature_agile alg msg_len msg private_key nonce = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_sign_msg_as_qelem m_q private_key nonce val ecdsa_verification_agile: alg:hash_alg_ecdsa -> msg_len:size_nat{msg_len >= min_input_length alg} -> msg:lbytes msg_len -> public_key:lbytes 64 -> signature_r:lbytes 32 -> signature_s:lbytes 32 -> bool let ecdsa_verification_agile alg msg_len msg public_key signature_r signature_s = let hashM = hash_ecdsa alg msg_len msg in let m_q = nat_from_bytes_be (sub hashM 0 32) % order in ecdsa_verify_msg_as_qelem m_q public_key signature_r signature_s /// ECDH over the P256 elliptic curve let secret_to_public (private_key:lbytes 32) : option (lbytes 64) = let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if is_sk_valid then let pk = point_mul_g sk in Some (point_store pk) else None let ecdh (their_public_key:lbytes 64) (private_key:lbytes 32) : option (lbytes 64) = let pk = load_point their_public_key in let sk = nat_from_bytes_be private_key in let is_sk_valid = 0 < sk && sk < order in if Some? pk && is_sk_valid then let ss = point_mul sk (Some?.v pk) in Some (point_store ss) else None /// Parsing and Serializing public keys // raw = [ x; y ], 64 bytes // uncompressed = [ 0x04; x; y ], 65 bytes // compressed = [ 0x02 for even `y` and 0x03 for odd `y`; x ], 33 bytes let validate_public_key (pk:lbytes 64) : bool = Some? (load_point pk) let pk_uncompressed_to_raw (pk:lbytes 65) : option (lbytes 64) = if Lib.RawIntTypes.u8_to_UInt8 pk.[0] <> 0x04uy then None else Some (sub pk 1 64) let pk_uncompressed_from_raw (pk:lbytes 64) : lbytes 65 = concat (create 1 (u8 0x04)) pk let pk_compressed_to_raw (pk:lbytes 33) : option (lbytes 64) = let pk_x = sub pk 1 32 in match (aff_point_decompress pk) with | Some (x, y) -> Some (concat #_ #32 #32 pk_x (nat_to_bytes_be 32 y)) | None -> None

{ "checked_file": "/", "dependencies": [ "Spec.P256.PointOps.fst.checked", "Spec.P256.Lemmas.fsti.checked", "Spec.Hash.Definitions.fst.checked", "Spec.Exponentiation.fsti.checked", "Spec.Agile.Hash.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.RawIntTypes.fsti.checked", "Lib.NatMod.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.ByteSequence.fsti.checked", "FStar.UInt8.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.Math.Lemmas.fst.checked" ], "interface_file": false, "source_file": "Spec.P256.fst" }

[ { "abbrev": false, "full_module": "Spec.P256.PointOps", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "PL" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.NatMod", "short_module": "M" }, { "abbrev": false, "full_module": "Spec.Hash.Definitions", "short_module": null }, { "abbrev": false, "full_module": "Lib.ByteSequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.Sequence", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "Spec", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

pk: Lib.ByteSequence.lbytes 64 -> Lib.ByteSequence.lbytes 33

Prims.Tot

[ "total" ]

[]

[ "Lib.ByteSequence.lbytes", "Lib.Sequence.concat", "Lib.IntTypes.uint_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.Sequence.create", "Lib.IntTypes.int_t", "Lib.IntTypes.u8", "Prims.bool", "Prims.op_Equality", "Prims.int", "Prims.op_Modulus", "Lib.ByteSequence.nat_from_bytes_be", "Lib.Sequence.lseq", "Prims.l_and", "Prims.eq2", "FStar.Seq.Base.seq", "Lib.Sequence.to_seq", "FStar.Seq.Base.slice", "Prims.op_Addition", "Prims.l_Forall", "Prims.nat", "Prims.b2t", "Prims.op_LessThan", "Prims.l_or", "FStar.Seq.Base.index", "Lib.Sequence.index", "Lib.Sequence.sub" ]

[]

false

let pk_compressed_from_raw (pk: lbytes 64) : lbytes 33 =

let pk_x = sub pk 0 32 in let pk_y = sub pk 32 32 in let is_pk_y_odd = nat_from_bytes_be pk_y % 2 = 1 in let pk0 = if is_pk_y_odd then u8 0x03 else u8 0x02 in concat (create 1 pk0) pk_x

false

LowParse.Tot.Bytes.fst

LowParse.Tot.Bytes.parse_all_bytes

val parse_all_bytes : LowParse.Spec.Base.tot_parser LowParse.Spec.Bytes.parse_all_bytes_kind FStar.Bytes.bytes

let parse_all_bytes = tot_parse_all_bytes

{ "file_name": "src/lowparse/LowParse.Tot.Bytes.fst", "git_rev": "00217c4a89f5ba56002ba9aa5b4a9d5903bfe9fa", "git_url": "https://github.com/project-everest/everparse.git", "project_name": "everparse" }

{ "end_col": 41, "end_line": 7, "start_col": 0, "start_line": 7 }

module LowParse.Tot.Bytes include LowParse.Spec.Bytes include LowParse.Tot.Combinators include LowParse.Tot.Int

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "LowParse.Tot.Int.fst.checked", "LowParse.Tot.Combinators.fst.checked", "LowParse.Spec.Bytes.fst.checked", "FStar.Pervasives.fsti.checked" ], "interface_file": false, "source_file": "LowParse.Tot.Bytes.fst" }

[ { "abbrev": false, "full_module": "LowParse.Tot.Int", "short_module": null }, { "abbrev": false, "full_module": "LowParse.Tot.Combinators", "short_module": null }, { "abbrev": false, "full_module": "LowParse.Spec.Bytes", "short_module": null }, { "abbrev": false, "full_module": "LowParse.Tot", "short_module": null }, { "abbrev": false, "full_module": "LowParse.Tot", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

LowParse.Spec.Base.tot_parser LowParse.Spec.Bytes.parse_all_bytes_kind FStar.Bytes.bytes

Prims.Tot

[ "total" ]

[]

[ "LowParse.Spec.Bytes.tot_parse_all_bytes" ]

[]

false

true

false

let parse_all_bytes =

tot_parse_all_bytes

false

SteelNull.fst

SteelNull.main

val main: Prims.unit -> SteelT Int32.t emp (fun _ -> emp)

let main () : SteelT Int32.t emp (fun _ -> emp) = let r = null #UInt32.t in if is_null r then return 0l else return 1l

{ "file_name": "share/steel/tests/krml/SteelNull.fst", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 13, "end_line": 12, "start_col": 0, "start_line": 7 }

module SteelNull open Steel.Effect.Atomic open Steel.Effect open Steel.Reference

{ "checked_file": "/", "dependencies": [ "Steel.Reference.fsti.checked", "Steel.Effect.Atomic.fsti.checked", "Steel.Effect.fsti.checked", "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.fsti.checked", "FStar.Int32.fsti.checked" ], "interface_file": false, "source_file": "SteelNull.fst" }

[ { "abbrev": false, "full_module": "Steel.Reference", "short_module": null }, { "abbrev": false, "full_module": "Steel.Effect", "short_module": null }, { "abbrev": false, "full_module": "Steel.Effect.Atomic", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> Steel.Effect.SteelT FStar.Int32.t

Steel.Effect.SteelT

[]

[ "Prims.unit", "Steel.Reference.is_null", "FStar.UInt32.t", "Steel.Effect.Atomic.return", "FStar.Int32.t", "FStar.Ghost.hide", "FStar.Set.set", "Steel.Memory.iname", "FStar.Set.empty", "FStar.Algebra.CommMonoid.Equiv.__proj__CM__item__mult", "Steel.Effect.Common.vprop", "Steel.Effect.Common.req", "Steel.Effect.Common.rm", "FStar.Algebra.CommMonoid.Equiv.__proj__CM__item__unit", "Steel.Effect.Common.emp", "FStar.Int32.__int_to_t", "Prims.bool", "Steel.Reference.ref", "Steel.Reference.null" ]

[]

false

true

false

let main () : SteelT Int32.t emp (fun _ -> emp) =

let r = null #UInt32.t in if is_null r then return 0l else return 1l

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.define_struct

val define_struct (n: string) (#tf #tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0

let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 32, "end_line": 12, "start_col": 0, "start_line": 11 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

n: Prims.string -> fields: Pulse.C.Types.Fields.nonempty_field_description_t tf -> Type0

Prims.Tot

[ "total" ]

[]

[ "Prims.string", "Prims.squash", "FStar.Pervasives.norm", "Pulse.C.Typestring.norm_typestring", "Prims.eq2", "Pulse.C.Typestring.mk_string_t", "Pulse.C.Types.Fields.nonempty_field_description_t", "Pulse.C.Types.Struct.define_struct0" ]

[]

false

true

let define_struct (n: string) (#tf #tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 =

define_struct0 tn #tf n fields

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.struct_t

val struct_t (#tf: Type0) (n: string) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0

let struct_t (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = struct_t0 tn #tf n fields

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 27, "end_line": 19, "start_col": 0, "start_line": 18 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields // To be extracted as: struct t [@@noextract_to "krml"] // primitive val struct_t0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

n: Prims.string -> fields: Pulse.C.Types.Fields.nonempty_field_description_t tf -> Type0

Prims.Tot

[ "total" ]

[]

[ "Prims.string", "Prims.squash", "FStar.Pervasives.norm", "Pulse.C.Typestring.norm_typestring", "Prims.eq2", "Pulse.C.Typestring.mk_string_t", "Pulse.C.Types.Fields.nonempty_field_description_t", "Pulse.C.Types.Struct.struct_t0" ]

[]

false

true

let struct_t (#tf: Type0) (n: string) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 =

struct_t0 tn #tf n fields

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.struct

val struct (#tf: Type0) (n: string) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields))

let struct (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) = struct0 tn #tf n fields

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 25, "end_line": 92, "start_col": 0, "start_line": 91 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields // To be extracted as: struct t [@@noextract_to "krml"] // primitive val struct_t0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let struct_t (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = struct_t0 tn #tf n fields val struct_set_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (f: field_t fields) (v: fields.fd_type f) (s: struct_t0 tn n fields) : GTot (struct_t0 tn n fields) val struct_get_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) : GTot (fields.fd_type field) val struct_eq (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s1 s2: struct_t0 tn n fields) : Ghost prop (requires True) (ensures (fun y -> (y <==> (s1 == s2)) /\ (y <==> (forall field . struct_get_field s1 field == struct_get_field s2 field)) )) val struct_get_field_same (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) : Lemma (struct_get_field (struct_set_field field v s) field == v) [SMTPat (struct_get_field (struct_set_field field v s) field)] val struct_get_field_other (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (requires (field' <> field)) (ensures (struct_get_field (struct_set_field field v s) field' == struct_get_field s field')) let struct_get_field_pat (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')] = if field' = field then () else struct_get_field_other s field v field' [@@noextract_to "krml"] // proof-only val struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) inline_for_extraction

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

n: Prims.string -> fields: Pulse.C.Types.Fields.nonempty_field_description_t tf -> Pulse.C.Types.Base.typedef (Pulse.C.Types.Struct.struct_t0 tn n fields)

Prims.Tot

[ "total" ]

[]

[ "Prims.string", "Prims.squash", "FStar.Pervasives.norm", "Pulse.C.Typestring.norm_typestring", "Prims.eq2", "Pulse.C.Typestring.mk_string_t", "Pulse.C.Types.Fields.nonempty_field_description_t", "Pulse.C.Types.Struct.struct0", "Pulse.C.Types.Base.typedef", "Pulse.C.Types.Struct.struct_t0" ]

[]

false

let struct (#tf: Type0) (n: string) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) =

struct0 tn #tf n fields

false

Vale.Def.Words.Two.fst

Vale.Def.Words.Two.nat_to_two_to_nat

val nat_to_two_to_nat (n1 n2:nat32) : Lemma (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)) == Mktwo n1 n2) [SMTPat (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)))]

let nat_to_two_to_nat (n1 n2:nat32) : Lemma (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)) == Mktwo n1 n2) = let n = n1 + n2 * pow2_32 in assert_norm (two_to_nat 32 (Mktwo n1 n2) == int_to_natN pow2_64 n); assert (0 <= n); assert (n < pow2_64); assert (two_to_nat 32 (Mktwo n1 n2) == n); assert_norm (pow2_norm 32 == pow2_32); //assert_norm (pow2_norm (2 * 32) == pow2_64); assert_norm (nat_to_two 32 n == Mktwo (n % pow2_32) ((n / pow2_32) % pow2_32)); lemma_fundamental_div_mod n; ()

{ "file_name": "vale/code/lib/util/Vale.Def.Words.Two.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 4, "end_line": 22, "start_col": 0, "start_line": 10 }

module Vale.Def.Words.Two open FStar.Mul open Vale.Lib.Meta let lemma_fundamental_div_mod (x:nat64) : Lemma (x = x % pow2_32 + pow2_32 * ((x / pow2_32) % pow2_32)) = ()

{ "checked_file": "/", "dependencies": [ "Vale.Lib.Meta.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked" ], "interface_file": true, "source_file": "Vale.Def.Words.Two.fst" }

[ { "abbrev": false, "full_module": "Vale.Lib.Meta", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words.Two_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Types_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 0, "max_fuel": 1, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": true, "smtencoding_l_arith_repr": "native", "smtencoding_nl_arith_repr": "wrapped", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [ "smt.arith.nl=false", "smt.QI.EAGER_THRESHOLD=100", "smt.CASE_SPLIT=3" ], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

n1: Vale.Def.Words_s.nat32 -> n2: Vale.Def.Words_s.nat32 -> FStar.Pervasives.Lemma (ensures Vale.Def.Words.Two_s.nat_to_two 32 (Vale.Def.Words.Two_s.two_to_nat 32 (Vale.Def.Words_s.Mktwo n1 n2)) == Vale.Def.Words_s.Mktwo n1 n2) [ SMTPat (Vale.Def.Words.Two_s.nat_to_two 32 (Vale.Def.Words.Two_s.two_to_nat 32 (Vale.Def.Words_s.Mktwo n1 n2))) ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Vale.Def.Words_s.nat32", "Prims.unit", "Vale.Def.Words.Two.lemma_fundamental_div_mod", "FStar.Pervasives.assert_norm", "Prims.eq2", "Vale.Def.Words_s.two", "Vale.Def.Words_s.natN", "Prims.pow2", "Vale.Def.Words.Two_s.nat_to_two", "Vale.Def.Words_s.Mktwo", "Prims.op_Modulus", "Vale.Def.Words_s.pow2_32", "Prims.op_Division", "Prims.int", "Vale.Def.Words_s.pow2_norm", "Prims._assert", "Vale.Def.Words.Two_s.two_to_nat", "Prims.b2t", "Prims.op_LessThan", "Vale.Def.Words_s.pow2_64", "Prims.op_LessThanOrEqual", "Vale.Def.Words_s.int_to_natN", "Prims.op_Addition", "FStar.Mul.op_Star", "Prims.l_True", "Prims.squash", "Prims.Nil", "FStar.Pervasives.pattern" ]

[]

true

false

true

false

let nat_to_two_to_nat (n1 n2: nat32) : Lemma (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)) == Mktwo n1 n2) =

let n = n1 + n2 * pow2_32 in assert_norm (two_to_nat 32 (Mktwo n1 n2) == int_to_natN pow2_64 n); assert (0 <= n); assert (n < pow2_64); assert (two_to_nat 32 (Mktwo n1 n2) == n); assert_norm (pow2_norm 32 == pow2_32); assert_norm (nat_to_two 32 n == Mktwo (n % pow2_32) ((n / pow2_32) % pow2_32)); lemma_fundamental_div_mod n; ()

false

Vale.Def.Words.Two.fst

Vale.Def.Words.Two.two_to_nat_to_two

val two_to_nat_to_two (n:nat64) : Lemma (two_to_nat 32 (nat_to_two 32 n) == n) [SMTPat (two_to_nat 32 (nat_to_two 32 n))]

let two_to_nat_to_two (n:nat64) = let n1 = n % (pow2_32) in let n2 = (n/(pow2_32)) % (pow2_32) in let n_f = two_to_nat 32 (Mktwo n1 n2) in assert_norm (n == n1 + n2 * pow2_32)

{ "file_name": "vale/code/lib/util/Vale.Def.Words.Two.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 38, "end_line": 28, "start_col": 0, "start_line": 24 }

module Vale.Def.Words.Two open FStar.Mul open Vale.Lib.Meta let lemma_fundamental_div_mod (x:nat64) : Lemma (x = x % pow2_32 + pow2_32 * ((x / pow2_32) % pow2_32)) = () let nat_to_two_to_nat (n1 n2:nat32) : Lemma (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)) == Mktwo n1 n2) = let n = n1 + n2 * pow2_32 in assert_norm (two_to_nat 32 (Mktwo n1 n2) == int_to_natN pow2_64 n); assert (0 <= n); assert (n < pow2_64); assert (two_to_nat 32 (Mktwo n1 n2) == n); assert_norm (pow2_norm 32 == pow2_32); //assert_norm (pow2_norm (2 * 32) == pow2_64); assert_norm (nat_to_two 32 n == Mktwo (n % pow2_32) ((n / pow2_32) % pow2_32)); lemma_fundamental_div_mod n; ()

{ "checked_file": "/", "dependencies": [ "Vale.Lib.Meta.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked" ], "interface_file": true, "source_file": "Vale.Def.Words.Two.fst" }

[ { "abbrev": false, "full_module": "Vale.Lib.Meta", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words.Two_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Types_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 0, "max_fuel": 1, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": true, "smtencoding_l_arith_repr": "native", "smtencoding_nl_arith_repr": "wrapped", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [ "smt.arith.nl=false", "smt.QI.EAGER_THRESHOLD=100", "smt.CASE_SPLIT=3" ], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

n: Vale.Def.Words_s.nat64 -> FStar.Pervasives.Lemma (ensures Vale.Def.Words.Two_s.two_to_nat 32 (Vale.Def.Words.Two_s.nat_to_two 32 n) == n) [SMTPat (Vale.Def.Words.Two_s.two_to_nat 32 (Vale.Def.Words.Two_s.nat_to_two 32 n))]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Vale.Def.Words_s.nat64", "FStar.Pervasives.assert_norm", "Prims.eq2", "Prims.int", "Prims.op_Addition", "FStar.Mul.op_Star", "Vale.Def.Words_s.pow2_32", "Vale.Def.Words_s.natN", "Prims.pow2", "Prims.op_Multiply", "Vale.Def.Words.Two_s.two_to_nat", "Vale.Def.Words_s.Mktwo", "Prims.op_Modulus", "Prims.op_Division", "Prims.unit" ]

[]

true

false

true

false

let two_to_nat_to_two (n: nat64) =

let n1 = n % (pow2_32) in let n2 = (n / (pow2_32)) % (pow2_32) in let n_f = two_to_nat 32 (Mktwo n1 n2) in assert_norm (n == n1 + n2 * pow2_32)

false

Vale.Def.Words.Two.fst

Vale.Def.Words.Two.two_to_nat_32_injective

val two_to_nat_32_injective (_:unit) : Lemma (forall (x x':two (natN (pow2_norm 32))).{:pattern two_to_nat 32 x; two_to_nat 32 x'} two_to_nat 32 x == two_to_nat 32 x' ==> x == x')

let two_to_nat_32_injective () = generic_injective_proof (two_to_nat 32) (nat_to_two 32) (fun x -> nat_to_two_to_nat x.lo x.hi)

{ "file_name": "vale/code/lib/util/Vale.Def.Words.Two.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 96, "end_line": 31, "start_col": 0, "start_line": 30 }

module Vale.Def.Words.Two open FStar.Mul open Vale.Lib.Meta let lemma_fundamental_div_mod (x:nat64) : Lemma (x = x % pow2_32 + pow2_32 * ((x / pow2_32) % pow2_32)) = () let nat_to_two_to_nat (n1 n2:nat32) : Lemma (nat_to_two 32 (two_to_nat 32 (Mktwo n1 n2)) == Mktwo n1 n2) = let n = n1 + n2 * pow2_32 in assert_norm (two_to_nat 32 (Mktwo n1 n2) == int_to_natN pow2_64 n); assert (0 <= n); assert (n < pow2_64); assert (two_to_nat 32 (Mktwo n1 n2) == n); assert_norm (pow2_norm 32 == pow2_32); //assert_norm (pow2_norm (2 * 32) == pow2_64); assert_norm (nat_to_two 32 n == Mktwo (n % pow2_32) ((n / pow2_32) % pow2_32)); lemma_fundamental_div_mod n; () let two_to_nat_to_two (n:nat64) = let n1 = n % (pow2_32) in let n2 = (n/(pow2_32)) % (pow2_32) in let n_f = two_to_nat 32 (Mktwo n1 n2) in assert_norm (n == n1 + n2 * pow2_32)

{ "checked_file": "/", "dependencies": [ "Vale.Lib.Meta.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked" ], "interface_file": true, "source_file": "Vale.Def.Words.Two.fst" }

[ { "abbrev": false, "full_module": "Vale.Lib.Meta", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words.Two_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Types_s", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "Vale.Def.Words", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 0, "max_fuel": 1, "max_ifuel": 1, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": true, "smtencoding_l_arith_repr": "native", "smtencoding_nl_arith_repr": "wrapped", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [ "smt.arith.nl=false", "smt.QI.EAGER_THRESHOLD=100", "smt.CASE_SPLIT=3" ], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> FStar.Pervasives.Lemma (ensures forall (x: Vale.Def.Words_s.two (Vale.Def.Words_s.natN (Vale.Def.Words_s.pow2_norm 32))) (x': Vale.Def.Words_s.two (Vale.Def.Words_s.natN (Vale.Def.Words_s.pow2_norm 32))). {:pattern Vale.Def.Words.Two_s.two_to_nat 32 x; Vale.Def.Words.Two_s.two_to_nat 32 x'} Vale.Def.Words.Two_s.two_to_nat 32 x == Vale.Def.Words.Two_s.two_to_nat 32 x' ==> x == x')

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.unit", "Vale.Lib.Meta.generic_injective_proof", "Vale.Def.Words_s.two", "Vale.Def.Words_s.nat32", "Vale.Def.Words_s.natN", "Prims.pow2", "FStar.Mul.op_Star", "Vale.Def.Words.Two_s.two_to_nat", "Vale.Def.Words.Two_s.nat_to_two", "Vale.Def.Words.Two.nat_to_two_to_nat", "Vale.Def.Words_s.__proj__Mktwo__item__lo", "Vale.Def.Words_s.__proj__Mktwo__item__hi" ]

[]

false

true

false

let two_to_nat_32_injective () =

generic_injective_proof (two_to_nat 32) (nat_to_two 32) (fun x -> nat_to_two_to_nat x.lo x.hi)

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_64_lseq_lemma

val proj_g_pow2_64_lseq_lemma: unit -> Lemma (point_inv_seq proj_g_pow2_64_lseq /\ S.to_aff_point (from_mont_point (as_point_nat_seq proj_g_pow2_64_lseq)) == g_pow2_64)

let proj_g_pow2_64_lseq_lemma () = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); proj_g_pow2_64_lemma (); SPTK.proj_point_to_list_lemma proj_g_pow2_64

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 46, "end_line": 171, "start_col": 0, "start_line": 168 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) inline_for_extraction noextract let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64) inline_for_extraction noextract let proj_g_pow2_128_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_128) inline_for_extraction noextract let proj_g_pow2_192_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_192) let proj_g_pow2_64_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); Seq.seq_of_list proj_g_pow2_64_list let proj_g_pow2_128_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_128); Seq.seq_of_list proj_g_pow2_128_list let proj_g_pow2_192_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_192); Seq.seq_of_list proj_g_pow2_192_list val proj_g_pow2_64_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_64 == pow_point (pow2 64) g_aff) let proj_g_pow2_64_lemma () = lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_64_lemma S.mk_p256_concrete_ops S.base_point val proj_g_pow2_128_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_128 == pow_point (pow2 128) g_aff) let proj_g_pow2_128_lemma () = lemma_proj_g_pow2_128_eval (); lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_128_lemma S.mk_p256_concrete_ops S.base_point val proj_g_pow2_192_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_192 == pow_point (pow2 192) g_aff) let proj_g_pow2_192_lemma () = lemma_proj_g_pow2_192_eval (); lemma_proj_g_pow2_128_eval (); lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_192_lemma S.mk_p256_concrete_ops S.base_point

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> FStar.Pervasives.Lemma (ensures Hacl.Impl.P256.Point.point_inv_seq Hacl.P256.PrecompTable.proj_g_pow2_64_lseq /\ Spec.P256.PointOps.to_aff_point (Hacl.Impl.P256.Point.from_mont_point (Hacl.Impl.P256.Point.as_point_nat_seq Hacl.P256.PrecompTable.proj_g_pow2_64_lseq)) == Hacl.P256.PrecompTable.g_pow2_64)

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.unit", "Hacl.Spec.P256.PrecompTable.proj_point_to_list_lemma", "Hacl.P256.PrecompTable.proj_g_pow2_64", "Hacl.P256.PrecompTable.proj_g_pow2_64_lemma", "FStar.Pervasives.normalize_term_spec", "Hacl.Spec.P256.PrecompTable.point_list", "Hacl.Spec.P256.PrecompTable.proj_point_to_list" ]

[]

true

false

true

false

let proj_g_pow2_64_lseq_lemma () =

normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); proj_g_pow2_64_lemma (); SPTK.proj_point_to_list_lemma proj_g_pow2_64

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_128_list

val proj_g_pow2_128_list:SPTK.point_list

let proj_g_pow2_128_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_128)

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 58, "end_line": 121, "start_col": 0, "start_line": 120 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) inline_for_extraction noextract let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64)

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Hacl.Spec.P256.PrecompTable.point_list

Prims.Tot

[ "total" ]

[]

[ "FStar.Pervasives.normalize_term", "Hacl.Spec.P256.PrecompTable.point_list", "Hacl.Spec.P256.PrecompTable.proj_point_to_list", "Hacl.P256.PrecompTable.proj_g_pow2_128" ]

[]

false

true

false

let proj_g_pow2_128_list:SPTK.point_list =

normalize_term (SPTK.proj_point_to_list proj_g_pow2_128)

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.struct_get_field_pat

val struct_get_field_pat (#tn #tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')]

let struct_get_field_pat (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')] = if field' = field then () else struct_get_field_other s field v field'

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 46, "end_line": 84, "start_col": 0, "start_line": 70 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields // To be extracted as: struct t [@@noextract_to "krml"] // primitive val struct_t0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let struct_t (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = struct_t0 tn #tf n fields val struct_set_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (f: field_t fields) (v: fields.fd_type f) (s: struct_t0 tn n fields) : GTot (struct_t0 tn n fields) val struct_get_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) : GTot (fields.fd_type field) val struct_eq (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s1 s2: struct_t0 tn n fields) : Ghost prop (requires True) (ensures (fun y -> (y <==> (s1 == s2)) /\ (y <==> (forall field . struct_get_field s1 field == struct_get_field s2 field)) )) val struct_get_field_same (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) : Lemma (struct_get_field (struct_set_field field v s) field == v) [SMTPat (struct_get_field (struct_set_field field v s) field)] val struct_get_field_other (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (requires (field' <> field)) (ensures (struct_get_field (struct_set_field field v s) field' == struct_get_field s field'))

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

s: Pulse.C.Types.Struct.struct_t0 tn n fields -> field: Pulse.C.Types.Fields.field_t fields -> v: Mkfield_description_t?.fd_type fields field -> field': Pulse.C.Types.Fields.field_t fields -> FStar.Pervasives.Lemma (ensures Pulse.C.Types.Struct.struct_get_field (Pulse.C.Types.Struct.struct_set_field field v s) field' == (match field' = field with | true -> v | _ -> Pulse.C.Types.Struct.struct_get_field s field')) [ SMTPat (Pulse.C.Types.Struct.struct_get_field (Pulse.C.Types.Struct.struct_set_field field v s) field') ]

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.string", "Pulse.C.Types.Fields.nonempty_field_description_t", "Pulse.C.Types.Struct.struct_t0", "Pulse.C.Types.Fields.field_t", "Pulse.C.Types.Fields.__proj__Mkfield_description_t__item__fd_type", "Prims.op_Equality", "Prims.bool", "Pulse.C.Types.Struct.struct_get_field_other", "Prims.unit", "Prims.l_True", "Prims.squash", "Prims.eq2", "Pulse.C.Types.Struct.struct_get_field", "Pulse.C.Types.Struct.struct_set_field", "Prims.Cons", "FStar.Pervasives.pattern", "FStar.Pervasives.smt_pat", "Prims.Nil" ]

[]

false

true

false

let struct_get_field_pat (#tn #tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')] =

if field' = field then () else struct_get_field_other s field v field'

false

Hacl.HPKE.Curve51_CP256_SHA256.fsti

Hacl.HPKE.Curve51_CP256_SHA256.cs

val cs:S.ciphersuite

let cs:S.ciphersuite = (DH.DH_Curve25519, Hash.SHA2_256, S.Seal AEAD.CHACHA20_POLY1305, Hash.SHA2_256)

{ "file_name": "code/hpke/Hacl.HPKE.Curve51_CP256_SHA256.fsti", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 102, "end_line": 10, "start_col": 0, "start_line": 10 }

module Hacl.HPKE.Curve51_CP256_SHA256 open Hacl.Impl.HPKE module S = Spec.Agile.HPKE module DH = Spec.Agile.DH module AEAD = Spec.Agile.AEAD module Hash = Spec.Agile.Hash

{ "checked_file": "/", "dependencies": [ "Spec.Agile.HPKE.fsti.checked", "Spec.Agile.Hash.fsti.checked", "Spec.Agile.DH.fst.checked", "Spec.Agile.AEAD.fsti.checked", "prims.fst.checked", "Hacl.Impl.HPKE.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked" ], "interface_file": false, "source_file": "Hacl.HPKE.Curve51_CP256_SHA256.fsti" }

[ { "abbrev": true, "full_module": "Spec.Agile.Hash", "short_module": "Hash" }, { "abbrev": true, "full_module": "Spec.Agile.AEAD", "short_module": "AEAD" }, { "abbrev": true, "full_module": "Spec.Agile.DH", "short_module": "DH" }, { "abbrev": true, "full_module": "Spec.Agile.HPKE", "short_module": "S" }, { "abbrev": false, "full_module": "Hacl.Impl.HPKE", "short_module": null }, { "abbrev": false, "full_module": "Hacl.HPKE", "short_module": null }, { "abbrev": false, "full_module": "Hacl.HPKE", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Spec.Agile.HPKE.ciphersuite

Prims.Tot

[ "total" ]

[]

[ "FStar.Pervasives.Native.Mktuple4", "Spec.Agile.DH.algorithm", "Spec.Agile.HPKE.hash_algorithm", "Spec.Agile.HPKE.aead", "Spec.Hash.Definitions.hash_alg", "Spec.Agile.DH.DH_Curve25519", "Spec.Hash.Definitions.SHA2_256", "Spec.Agile.HPKE.Seal", "Spec.Agile.AEAD.CHACHA20_POLY1305" ]

[]

false

true

false

let cs:S.ciphersuite =

(DH.DH_Curve25519, Hash.SHA2_256, S.Seal AEAD.CHACHA20_POLY1305, Hash.SHA2_256)

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.struct_field1

val struct_field1 (#tn #tf t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t') (sq_t': squash (t' == fields.fd_type field)) (sq_td': squash (td' == fields.fd_typedef field)) : stt (ref td') (pts_to r v) (fun r' -> (pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field)) ** has_struct_field r field r')

let struct_field1 (#tn: Type0) (#tf: Type0) (t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t') (sq_t': squash (t' == fields.fd_type field)) (sq_td': squash (td' == fields.fd_typedef field)) : stt (ref td') (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field) ** has_struct_field r field r') = struct_field0 t' r field td'

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 30, "end_line": 268, "start_col": 0, "start_line": 253 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields // To be extracted as: struct t [@@noextract_to "krml"] // primitive val struct_t0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let struct_t (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = struct_t0 tn #tf n fields val struct_set_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (f: field_t fields) (v: fields.fd_type f) (s: struct_t0 tn n fields) : GTot (struct_t0 tn n fields) val struct_get_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) : GTot (fields.fd_type field) val struct_eq (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s1 s2: struct_t0 tn n fields) : Ghost prop (requires True) (ensures (fun y -> (y <==> (s1 == s2)) /\ (y <==> (forall field . struct_get_field s1 field == struct_get_field s2 field)) )) val struct_get_field_same (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) : Lemma (struct_get_field (struct_set_field field v s) field == v) [SMTPat (struct_get_field (struct_set_field field v s) field)] val struct_get_field_other (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (requires (field' <> field)) (ensures (struct_get_field (struct_set_field field v s) field' == struct_get_field s field')) let struct_get_field_pat (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')] = if field' = field then () else struct_get_field_other s field v field' [@@noextract_to "krml"] // proof-only val struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) inline_for_extraction [@@noextract_to "krml"; norm_field_attr] // proof-only let struct (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) = struct0 tn #tf n fields val struct_get_field_unknown (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) (field: field_t fields) : Lemma (struct_get_field (unknown (struct0 tn n fields)) field == unknown (fields.fd_typedef field)) [SMTPat (struct_get_field (unknown (struct0 tn n fields)) field)] val struct_get_field_uninitialized (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) (field: field_t fields) : Lemma (struct_get_field (uninitialized (struct0 tn n fields)) field == uninitialized (fields.fd_typedef field)) [SMTPat (struct_get_field (uninitialized (struct0 tn n fields)) field)] val has_struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : Tot vprop val has_struct_field_prop (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost unit (has_struct_field r field r') (fun _ -> has_struct_field r field r' ** pure ( t' == fields.fd_type field /\ td' == fields.fd_typedef field )) val has_struct_field_dup (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost unit (has_struct_field r field r') (fun _ -> has_struct_field r field r' ** has_struct_field r field r') val has_struct_field_inj (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t1: Type0) (#td1: typedef t1) (r1: ref td1) (#t2: Type0) (#td2: typedef t2) (r2: ref td2) : stt_ghost (squash (t1 == t2 /\ td1 == td2)) (has_struct_field r field r1 ** has_struct_field r field r2) (fun _ -> has_struct_field r field r1 ** has_struct_field r field r2 ** ref_equiv r1 (coerce_eq () r2)) val has_struct_field_equiv_from (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r1: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') (r2: ref (struct0 tn n fields)) : stt_ghost unit (ref_equiv r1 r2 ** has_struct_field r1 field r') (fun _ -> ref_equiv r1 r2 ** has_struct_field r2 field r') val has_struct_field_equiv_to (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r1' r2': ref td') : stt_ghost unit (ref_equiv r1' r2' ** has_struct_field r field r1') (fun _ -> ref_equiv r1' r2' ** has_struct_field r field r2') val ghost_struct_field_focus (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost (squash ( t' == fields.fd_type field /\ td' == fields.fd_typedef field )) (has_struct_field r field r' ** pts_to r v) (fun _ -> has_struct_field r field r' ** pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (Ghost.hide (coerce_eq () (struct_get_field v field)))) val ghost_struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) : stt_ghost (Ghost.erased (ref (fields.fd_typedef field))) (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field) ** has_struct_field r field r') [@@noextract_to "krml"] // primitive val struct_field0 (#tn: Type0) (#tf: Type0) (t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t' { t' == fields.fd_type field /\ td' == fields.fd_typedef field }) : stt (ref td') (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (Ghost.hide (coerce_eq () (struct_get_field v field))) ** has_struct_field r field r') inline_for_extraction

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

t': Type0 -> r: Pulse.C.Types.Base.ref (Pulse.C.Types.Struct.struct0 tn n fields) -> field: Pulse.C.Types.Fields.field_t fields -> td': Pulse.C.Types.Base.typedef t' -> sq_t': Prims.squash (t' == Mkfield_description_t?.fd_type fields field) -> sq_td': Prims.squash (td' == Mkfield_description_t?.fd_typedef fields field) -> Pulse.Lib.Core.stt (Pulse.C.Types.Base.ref td') (Pulse.C.Types.Base.pts_to r v) (fun r' -> (Pulse.C.Types.Base.pts_to r (FStar.Ghost.hide (Pulse.C.Types.Struct.struct_set_field field (Pulse.C.Types.Base.unknown (Mkfield_description_t?.fd_typedef fields field)) (FStar.Ghost.reveal v))) ** Pulse.C.Types.Base.pts_to r' (FStar.Ghost.hide (Pulse.C.Types.Struct.struct_get_field (FStar.Ghost.reveal v) field))) ** Pulse.C.Types.Struct.has_struct_field r field r')

Prims.Tot

[ "total" ]

[]

[ "Prims.string", "Pulse.C.Types.Fields.nonempty_field_description_t", "FStar.Ghost.erased", "Pulse.C.Types.Struct.struct_t0", "Pulse.C.Types.Base.ref", "Pulse.C.Types.Struct.struct0", "Pulse.C.Types.Fields.field_t", "Pulse.C.Types.Base.typedef", "Prims.squash", "Prims.eq2", "Pulse.C.Types.Fields.__proj__Mkfield_description_t__item__fd_type", "Pulse.C.Types.Fields.__proj__Mkfield_description_t__item__fd_typedef", "Pulse.C.Types.Struct.struct_field0", "Pulse.Lib.Core.stt", "Pulse.C.Types.Base.pts_to", "Pulse.Lib.Core.op_Star_Star", "FStar.Ghost.hide", "Pulse.C.Types.Struct.struct_set_field", "Pulse.C.Types.Base.unknown", "FStar.Ghost.reveal", "Pulse.C.Types.Struct.struct_get_field", "Pulse.C.Types.Struct.has_struct_field", "Pulse.Lib.Core.vprop" ]

[]

false

let struct_field1 (#tn #tf t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t') (sq_t': squash (t' == fields.fd_type field)) (sq_td': squash (td' == fields.fd_typedef field)) : stt (ref td') (pts_to r v) (fun r' -> (pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field)) ** has_struct_field r field r') =

struct_field0 t' r field td'

false

FStar.Buffer.fst

FStar.Buffer.blit

val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) == Seq.slice (as_seq h0 b) (v idx_b+v len) (length b) ))

let rec blit #t a idx_a b idx_b len = let h0 = HST.get () in if len =^ 0ul then () else begin let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get() in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b)) end

{ "file_name": "ulib/legacy/FStar.Buffer.fst", "git_rev": "10183ea187da8e8c426b799df6c825e24c0767d3", "git_url": "https://github.com/FStarLang/FStar.git", "project_name": "FStar" }

{ "end_col": 7, "end_line": 1186, "start_col": 0, "start_line": 1173 }

(* Copyright 2008-2018 Microsoft Research Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *) module FStar.Buffer open FStar.Seq open FStar.UInt32 module Int32 = FStar.Int32 open FStar.HyperStack open FStar.HyperStack.ST open FStar.Ghost module HS = FStar.HyperStack module HST = FStar.HyperStack.ST #set-options "--initial_fuel 0 --max_fuel 0" //17-01-04 usage? move to UInt? let lemma_size (x:int) : Lemma (requires (UInt.size x n)) (ensures (x >= 0)) [SMTPat (UInt.size x n)] = () let lseq (a: Type) (l: nat) : Type = (s: seq a { Seq.length s == l } ) (* Buffer general type, fully implemented on FStar's arrays *) noeq private type _buffer (a:Type) = | MkBuffer: max_length:UInt32.t -> content:reference (lseq a (v max_length)) -> idx:UInt32.t -> length:UInt32.t{v idx + v length <= v max_length} -> _buffer a (* Exposed buffer type *) type buffer (a:Type) = _buffer a (* Ghost getters for specifications *) // TODO: remove `contains` after replacing all uses with `live` let contains #a h (b:buffer a) : GTot Type0 = HS.contains h b.content let unused_in #a (b:buffer a) h : GTot Type0 = HS.unused_in b.content h (* In most cases `as_seq` should be used instead of this one. *) let sel #a h (b:buffer a) : GTot (seq a) = HS.sel h b.content let max_length #a (b:buffer a) : GTot nat = v b.max_length let length #a (b:buffer a) : GTot nat = v b.length let idx #a (b:buffer a) : GTot nat = v b.idx //17-01-04 rename to container or ref? let content #a (b:buffer a) : GTot (reference (lseq a (max_length b))) = b.content (* Lifting from buffer to reference *) let as_ref #a (b:buffer a) = as_ref (content b) let as_addr #a (b:buffer a) = as_addr (content b) let frameOf #a (b:buffer a) : GTot HS.rid = HS.frameOf (content b) (* Liveliness condition, necessary for any computation on the buffer *) let live #a (h:mem) (b:buffer a) : GTot Type0 = HS.contains h b.content let unmapped_in #a (b:buffer a) (h:mem) : GTot Type0 = unused_in b h val recall: #a:Type -> b:buffer a{is_eternal_region (frameOf b) /\ not (is_mm b.content)} -> Stack unit (requires (fun m -> True)) (ensures (fun m0 _ m1 -> m0 == m1 /\ live m1 b)) let recall #a b = recall b.content (* Ghostly access an element of the array, or the full underlying sequence *) let as_seq #a h (b:buffer a) : GTot (s:seq a{Seq.length s == length b}) = Seq.slice (sel h b) (idx b) (idx b + length b) let get #a h (b:buffer a) (i:nat{i < length b}) : GTot a = Seq.index (as_seq h b) i (* Equality predicate on buffer contents, without quantifiers *) //17-01-04 revise comment? rename? let equal #a h (b:buffer a) h' (b':buffer a) : GTot Type0 = as_seq h b == as_seq h' b' (* y is included in x / x contains y *) let includes #a (x:buffer a) (y:buffer a) : GTot Type0 = x.max_length == y.max_length /\ x.content === y.content /\ idx y >= idx x /\ idx x + length x >= idx y + length y let includes_live #a h (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y)) (ensures (live h x <==> live h y)) = () let includes_as_seq #a h1 h2 (x: buffer a) (y: buffer a) : Lemma (requires (x `includes` y /\ as_seq h1 x == as_seq h2 x)) (ensures (as_seq h1 y == as_seq h2 y)) = Seq.slice_slice (sel h1 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y); Seq.slice_slice (sel h2 x) (idx x) (idx x + length x) (idx y - idx x) (idx y - idx x + length y) let includes_trans #a (x y z: buffer a) : Lemma (requires (x `includes` y /\ y `includes` z)) (ensures (x `includes` z)) = () (* Disjointness between two buffers *) let disjoint #a #a' (x:buffer a) (y:buffer a') : GTot Type0 = frameOf x =!= frameOf y \/ as_addr x =!= as_addr y \/ (a == a' /\ as_addr x == as_addr y /\ frameOf x == frameOf y /\ x.max_length == y.max_length /\ (idx x + length x <= idx y \/ idx y + length y <= idx x)) (* Disjointness is symmetric *) let lemma_disjoint_symm #a #a' (x:buffer a) (y:buffer a') : Lemma (requires True) (ensures (disjoint x y <==> disjoint y x)) [SMTPat (disjoint x y)] = () let lemma_disjoint_sub #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint subx y); SMTPat (includes x subx)] = () let lemma_disjoint_sub' #a #a' (x:buffer a) (subx:buffer a) (y:buffer a') : Lemma (requires (includes x subx /\ disjoint x y)) (ensures (disjoint subx y)) [SMTPat (disjoint y subx); SMTPat (includes x subx)] = () val lemma_live_disjoint: #a:Type -> #a':Type -> h:mem -> b:buffer a -> b':buffer a' -> Lemma (requires (live h b /\ b' `unused_in` h)) (ensures (disjoint b b')) [SMTPat (disjoint b b'); SMTPat (live h b)] let lemma_live_disjoint #a #a' h b b' = () (* Heterogeneous buffer type *) noeq type abuffer = | Buff: #t:Type -> b:buffer t -> abuffer (* let empty : TSet.set abuffer = TSet.empty #abuffer *) let only #t (b:buffer t) : Tot (TSet.set abuffer) = FStar.TSet.singleton (Buff #t b) (* let op_Plus_Plus #a s (b:buffer a) : Tot (TSet.set abuffer) = TSet.union s (only b) *) (* let op_Plus_Plus_Plus set1 set2 : Tot (TSet.set abuffer) = FStar.TSet.union set1 set2 *) let op_Bang_Bang = TSet.singleton let op_Plus_Plus = TSet.union (* Maps a set of buffer to the set of their references *) assume val arefs: TSet.set abuffer -> Tot (Set.set nat) assume Arefs_def: forall (x:nat) (s:TSet.set abuffer). {:pattern (Set.mem x (arefs s))} Set.mem x (arefs s) <==> (exists (y:abuffer). TSet.mem y s /\ as_addr y.b == x) val lemma_arefs_1: s:TSet.set abuffer -> Lemma (requires (s == TSet.empty #abuffer)) (ensures (arefs s == Set.empty #nat)) [SMTPat (arefs s)] let lemma_arefs_1 s = Set.lemma_equal_intro (arefs s) (Set.empty) val lemma_arefs_2: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires True) (ensures (arefs (s1 ++ s2) == Set.union (arefs s1) (arefs s2))) [SMTPatOr [ [SMTPat (arefs (s2 ++ s1))]; [SMTPat (arefs (s1 ++ s2))] ]] let lemma_arefs_2 s1 s2 = Set.lemma_equal_intro (arefs (s1 ++ s2)) (Set.union (arefs s1) (arefs s2)) val lemma_arefs_3: s1:TSet.set abuffer -> s2:TSet.set abuffer -> Lemma (requires (TSet.subset s1 s2)) (ensures (Set.subset (arefs s1) (arefs s2))) let lemma_arefs_3 s1 s2 = () (* General disjointness predicate between a buffer and a set of heterogeneous buffers *) let disjoint_from_bufs #a (b:buffer a) (bufs:TSet.set abuffer) = forall b'. TSet.mem b' bufs ==> disjoint b b'.b (* General disjointness predicate between a buffer and a set of heterogeneous references *) let disjoint_from_refs #a (b:buffer a) (set:Set.set nat) = ~(Set.mem (as_addr b) set) (* Similar but specialized disjointness predicates *) let disjoint_1 a b = disjoint a b let disjoint_2 a b b' = disjoint a b /\ disjoint a b' let disjoint_3 a b b' b'' = disjoint a b /\ disjoint a b' /\ disjoint a b'' let disjoint_4 a b b' b'' b''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' let disjoint_5 a b b' b'' b''' b'''' = disjoint a b /\ disjoint a b' /\ disjoint a b'' /\ disjoint a b''' /\ disjoint a b'''' let disjoint_ref_1 (#t:Type) (#u:Type) (a:buffer t) (r:reference u) = frameOf a =!= HS.frameOf r \/ as_addr a =!= HS.as_addr r let disjoint_ref_2 a r r' = disjoint_ref_1 a r /\ disjoint_ref_1 a r' let disjoint_ref_3 a r r' r'' = disjoint_ref_1 a r /\ disjoint_ref_2 a r' r'' let disjoint_ref_4 a r r' r'' r''' = disjoint_ref_1 a r /\ disjoint_ref_3 a r' r'' r''' let disjoint_ref_5 a r r' r'' r''' r'''' = disjoint_ref_1 a r /\ disjoint_ref_4 a r' r'' r''' r'''' val disjoint_only_lemma: #a:Type -> #a':Type -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint b b')) (ensures (disjoint_from_bufs b (only b'))) let disjoint_only_lemma #a #a' b b' = () (* Fully general modifies clause *) let modifies_bufs_and_refs (bufs:TSet.set abuffer) (refs:Set.set nat) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (Set.union (arefs bufs) refs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs /\ disjoint_from_refs b refs) ==> equal h b h' b /\ live h' b))) (* Fully general modifies clause for buffer sets *) let modifies_buffers (bufs:TSet.set abuffer) h h' : GTot Type0 = (forall rid. Set.mem rid (Map.domain (HS.get_hmap h)) ==> (HS.modifies_ref rid (arefs bufs) h h' /\ (forall (#a:Type) (b:buffer a). {:pattern (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs)} (frameOf b == rid /\ live h b /\ disjoint_from_bufs b bufs ==> equal h b h' b /\ live h' b)))) (* General modifies clause for buffers only *) let modifies_bufs rid buffs h h' = modifies_ref rid (arefs buffs) h h' /\ (forall (#a:Type) (b:buffer a). (frameOf b == rid /\ live h b /\ disjoint_from_bufs b buffs) ==> equal h b h' b /\ live h' b) let modifies_none h h' = HS.get_tip h' == HS.get_tip h /\ HS.modifies_transitively Set.empty h h' (* Specialized clauses for small numbers of buffers *) let modifies_buf_0 rid h h' = modifies_ref rid (Set.empty #nat) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb) ==> equal h bb h' bb /\ live h' bb) let modifies_buf_1 (#t:Type) rid (b:buffer t) h h' = //would be good to drop the rid argument on these, since they can be computed from the buffers modifies_ref rid (Set.singleton (Heap.addr_of (as_ref b))) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb) ==> equal h bb h' bb /\ live h' bb) let to_set_2 (n1:nat) (n2:nat) :Set.set nat = Set.union (Set.singleton n1) (Set.singleton n2) let modifies_buf_2 (#t:Type) (#t':Type) rid (b:buffer t) (b':buffer t') h h' = modifies_ref rid (to_set_2 (as_addr b) (as_addr b')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_3 (n1:nat) (n2:nat) (n3:nat) :Set.set nat = Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3) let modifies_buf_3 (#t:Type) (#t':Type) (#t'':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') h h' = modifies_ref rid (to_set_3 (as_addr b) (as_addr b') (as_addr b'')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb) ==> equal h bb h' bb /\ live h' bb) let to_set_4 (n1:nat) (n2:nat) (n3:nat) (n4:nat) :Set.set nat = Set.union (Set.union (Set.union (Set.singleton n1) (Set.singleton n2)) (Set.singleton n3)) (Set.singleton n4) let modifies_buf_4 (#t:Type) (#t':Type) (#t'':Type) (#t''':Type) rid (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') h h' = modifies_ref rid (to_set_4 (as_addr b) (as_addr b') (as_addr b'') (as_addr b''')) h h' /\ (forall (#tt:Type) (bb:buffer tt). (frameOf bb == rid /\ live h bb /\ disjoint b bb /\ disjoint b' bb /\ disjoint b'' bb /\ disjoint b''' bb) ==> equal h bb h' bb /\ live h' bb) (* General lemmas for the modifies_bufs clause *) let lemma_modifies_bufs_trans rid bufs h0 h1 h2 : Lemma (requires (modifies_bufs rid bufs h0 h1 /\ modifies_bufs rid bufs h1 h2)) (ensures (modifies_bufs rid bufs h0 h2)) [SMTPat (modifies_bufs rid bufs h0 h1); SMTPat (modifies_bufs rid bufs h1 h2)] = () let lemma_modifies_bufs_sub rid bufs subbufs h0 h1 : Lemma (requires (TSet.subset subbufs bufs /\ modifies_bufs rid subbufs h0 h1)) (ensures (modifies_bufs rid bufs h0 h1)) [SMTPat (modifies_bufs rid subbufs h0 h1); SMTPat (TSet.subset subbufs bufs)] = () val lemma_modifies_bufs_subset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (disjoint_from_bufs b (bufs ++ (only b')) )) (ensures (disjoint_from_bufs b bufs)) [SMTPat (modifies_bufs (HS.get_tip h0) (bufs ++ (only b')) h0 h1); SMTPat (live h0 b)] let lemma_modifies_bufs_subset #a #a' h0 h1 bufs b b' = () val lemma_modifies_bufs_superset: #a:Type -> #a':Type -> h0:mem -> h1:mem -> bufs:TSet.set abuffer -> b:buffer a -> b':buffer a' -> Lemma (requires (b' `unused_in` h0 /\ live h0 b /\ disjoint_from_bufs b bufs)) (ensures (disjoint_from_bufs b (bufs ++ (only b')))) [SMTPat (modifies_bufs (HS.get_tip h0) bufs h0 h1); SMTPat (b' `unmapped_in` h0); SMTPat (live h0 b)] let lemma_modifies_bufs_superset #a #a' h0 h1 bufs b b' = () (* Specialized lemmas *) let modifies_trans_0_0 (rid:rid) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_0 rid h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_1_0 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_0_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1 (#t:Type) (rid:rid) (b:buffer t) (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_1 rid b h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_1_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_1 rid b' h1 h2)] = () let modifies_trans_2_0 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 rid h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_0 rid h1 h2)] = () let modifies_trans_2_1 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_2_1' (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b' b h0 h1 /\ modifies_buf_1 rid b h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b' b h0 h1); SMTPat (modifies_buf_1 rid b h1 h2)] = () let modifies_trans_0_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_0 rid h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_0 rid h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_1_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_1 rid b h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_1 rid b h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_2_2 (#t:Type) (#t':Type) (rid:rid) (b:buffer t) (b':buffer t') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_2 rid b b' h1 h2)) (ensures (modifies_buf_2 rid b b' h0 h2)) [SMTPat (modifies_buf_2 rid b b' h0 h1); SMTPat (modifies_buf_2 rid b b' h1 h2)] = () let modifies_trans_3_3 (#t #t' #t'':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_buf_3 rid b b' b'' h1 h2)) (ensures (modifies_buf_3 rid b b' b'' h0 h2)) [SMTPat (modifies_buf_3 rid b b' b'' h0 h1); SMTPat (modifies_buf_3 rid b b' b'' h1 h2)] = () let modifies_trans_4_4 (#t #t' #t'' #t''':Type) (rid:rid) (b:buffer t) (b':buffer t') (b'':buffer t'') (b''':buffer t''') (h0 h1 h2:mem) : Lemma (requires (modifies_buf_4 rid b b' b'' b''' h0 h1 /\ modifies_buf_4 rid b b' b'' b''' h1 h2)) (ensures (modifies_buf_4 rid b b' b'' b''' h0 h2)) [SMTPat (modifies_buf_4 rid b b' b'' b''' h0 h1); SMTPat (modifies_buf_4 rid b b' b'' b''' h1 h2)] = () (* TODO: complete with specialized versions of every general lemma *) (* Modifies clauses that do not change the shape of the HyperStack ((HS.get_tip h1) = (HS.get_tip h0)) *) (* NB: those clauses are made abstract in order to make verification faster // Lemmas follow to allow the programmer to make use of the real definition // of those predicates in a general setting *) let modifies_0 (h0 h1:mem) :Type0 = modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* This one is very generic: it says // * - some references have changed in the frame of b, but // * - among all buffers in this frame, b is the only one that changed. *) let modifies_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 let modifies_2_1 (#a:Type) (b:buffer a) (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))) let modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))) let modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))) let modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') (h0 h1:mem) :Type0 = HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))) let modifies_region (rid:rid) (bufs:TSet.set abuffer) (h0 h1:mem) :Type0 = modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1 (* Lemmas introducing the 'modifies' predicates *) let lemma_intro_modifies_0 h0 h1 : Lemma (requires (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_0 h0 h1)) = () let lemma_intro_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_1 b h0 h1)) = () let lemma_intro_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) (ensures (modifies_2_1 b h0 h1)) = () let lemma_intro_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 ))))) (ensures (modifies_2 b b' h0 h1)) = () let lemma_intro_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1))))) (ensures (modifies_3 b b' b'' h0 h1)) = () let lemma_intro_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1))))) (ensures (modifies_3_2 b b' h0 h1)) = () let lemma_intro_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) (ensures (modifies_region rid bufs h0 h1)) = () (* Lemmas revealing the content of the specialized modifies clauses in order to // be able to generalize them if needs be. *) let lemma_reveal_modifies_0 h0 h1 : Lemma (requires (modifies_0 h0 h1)) (ensures (modifies_one (HS.get_tip h0) h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_1 b h0 h1)) (ensures (let rid = frameOf b in modifies_one rid h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () let lemma_reveal_modifies_2_1 (#a:Type) (b:buffer a) h0 h1 : Lemma (requires (modifies_2_1 b h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in ((rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 ))))) = () let lemma_reveal_modifies_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid =!= rid' /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 )) ))) = () let lemma_reveal_modifies_3 (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 : Lemma (requires (modifies_3 b b' b'' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in let rid'' = frameOf b'' in ((rid == rid' /\ rid' == rid'' /\ modifies_buf_3 rid b b' b'' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid =!= rid' /\ rid' == rid'' /\ modifies_buf_2 rid' b' b'' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid'')) h0 h1 ) \/ (rid == rid'' /\ rid' =!= rid'' /\ modifies_buf_2 rid b b'' h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton rid')) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= rid'' /\ rid =!= rid'' /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton rid'')) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid'' b'' h0 h1)) ))) = () let lemma_reveal_modifies_3_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires (modifies_3_2 b b' h0 h1)) (ensures ( HS.get_tip h0 == HS.get_tip h1 /\ (let rid = frameOf b in let rid' = frameOf b' in ((rid == rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_one rid h0 h1) \/ (rid == rid' /\ rid' =!= HS.get_tip h0 /\ modifies_buf_2 rid b b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid == HS.get_tip h0 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ HS.modifies (Set.union (Set.singleton rid') (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' == HS.get_tip h0 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ HS.modifies (Set.union (Set.singleton rid) (Set.singleton (HS.get_tip h0))) h0 h1 ) \/ (rid =!= rid' /\ rid' =!= HS.get_tip h0 /\ rid =!= HS.get_tip h0 /\ HS.modifies (Set.union (Set.union (Set.singleton rid) (Set.singleton rid')) (Set.singleton (HS.get_tip h0))) h0 h1 /\ modifies_buf_1 rid b h0 h1 /\ modifies_buf_1 rid' b' h0 h1 /\ modifies_buf_0 (HS.get_tip h0) h0 h1)) ))) = () let lemma_reveal_modifies_region (rid:rid) bufs h0 h1 : Lemma (requires (modifies_region rid bufs h0 h1)) (ensures (modifies_one rid h0 h1 /\ modifies_bufs rid bufs h0 h1 /\ HS.get_tip h0 == HS.get_tip h1)) = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (* Stack effect specific lemmas *) let lemma_stack_1 (#a:Type) (b:buffer a) h0 h1 h2 h3 : Lemma (requires (live h0 b /\ fresh_frame h0 h1 /\ modifies_1 b h1 h2 /\ popped h2 h3)) (ensures (modifies_buf_1 (frameOf b) b h0 h3)) [SMTPat (modifies_1 b h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () let lemma_stack_2 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 h3 : Lemma (requires (live h0 b /\ live h0 b' /\ fresh_frame h0 h1 /\ modifies_2 b b' h1 h2 /\ popped h2 h3)) (ensures (modifies_2 b b' h0 h3)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (fresh_frame h0 h1); SMTPat (popped h2 h3)] = () (* Specialized modifies clauses lemmas + associated SMTPatterns. Those are critical for // verification as the specialized modifies clauses are abstract from outside the // module *) (** Commutativity lemmas *) let lemma_modifies_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_2 b b' h0 h1 <==> modifies_2 b' b h0 h1)) [SMTPat (modifies_2 b b' h0 h1)] = () let lemma_modifies_3_2_comm (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 : Lemma (requires True) (ensures (modifies_3_2 b b' h0 h1 <==> modifies_3_2 b' b h0 h1)) [SMTPat (modifies_3_2 b b' h0 h1)] = () (* TODO: add commutativity lemmas for modifies_3 *) #reset-options "--z3rlimit 20" (** Transitivity lemmas *) let lemma_modifies_0_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_1 b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_2_1 b h0 h1 /\ modifies_2_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_2_1 b h1 h2)] = () let lemma_modifies_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) (* TODO: Make the following work and merge with the following lemma *) (* [SMTPatOr [ *) (* [SMTPat (modifies_2 b b' h0 h1); *) (* SMTPat (modifies_2 b' b h0 h1)]]; *) (* SMTPat (modifies_2 b' b h1 h2)] *) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_2 b b' h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () #reset-options "--z3rlimit 40" let lemma_modifies_3_trans (#a:Type) (#a':Type) (#a'':Type) (b:buffer a) (b':buffer a') (b'':buffer a'') h0 h1 h2 : Lemma (requires (modifies_3 b b' b'' h0 h1 /\ modifies_3 b b' b'' h1 h2)) (ensures (modifies_3 b b' b'' h0 h2)) (* TODO: add the appropriate SMTPatOr patterns so as not to rewrite X times the same lemma *) [SMTPat (modifies_3 b b' b'' h0 h1); SMTPat (modifies_3 b b' b'' h1 h2)] = () #reset-options "--z3rlimit 200" let lemma_modifies_3_2_trans (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b b' h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b b' h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () let lemma_modifies_3_2_trans' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (modifies_3_2 b' b h0 h1 /\ modifies_3_2 b b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_3_2 b' b h0 h1); SMTPat (modifies_3_2 b b' h1 h2)] = () #reset-options "--z3rlimit 20" (* Specific modifies clause lemmas *) val lemma_modifies_0_0: h0:mem -> h1:mem -> h2:mem -> Lemma (requires (modifies_0 h0 h1 /\ modifies_0 h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_0 h1 h2)] let lemma_modifies_0_0 h0 h1 h2 = () #reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_0 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ modifies_0 h1 h2)) (ensures (live h2 b /\ modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_0 h1 h2)] = () let lemma_modifies_0_1 (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_0_1' (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) [SMTPat (modifies_0 h0 h1); SMTPat (modifies_1 b h1 h2)] = () #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" let lemma_modifies_1_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_2 b b' h0 h2 /\ modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = if frameOf b = frameOf b' then modifies_trans_1_1' (frameOf b) b b' h0 h1 h2 else () #reset-options "--z3rlimit 200 --initial_fuel 0 --max_fuel 0" let lemma_modifies_0_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b b' h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_0_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ b' `unused_in` h0 /\ modifies_0 h0 h1 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_2 b' b h1 h2); SMTPat (modifies_0 h0 h1)] = () let lemma_modifies_1_2 (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_2 b' b h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_2'' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b b' h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b b' h1 h2)] = () let lemma_modifies_1_2''' (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_1 b h0 h1 /\ modifies_2 b' b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_2 b' b h1 h2)] = () let lemma_modifies_1_1_prime (#t:Type) (#t':Type) (b:buffer t) (b':buffer t') h0 h1 h2 : Lemma (requires (live h0 b /\ modifies_1 b h0 h1 /\ b' `unused_in` h0 /\ live h1 b' /\ modifies_1 b' h1 h2)) (ensures (modifies_2_1 b h0 h2)) [SMTPat (modifies_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () let lemma_modifies_2_1 (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b b' h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b b' h0 h2)) [SMTPat (modifies_2 b b' h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2 b' b h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_2 b' b h0 h2)) [SMTPat (modifies_2 b' b h0 h1); SMTPat (modifies_1 b h1 h2)] = () let lemma_modifies_2_1'' (#a:Type) (#a':Type) (b:buffer a) (b':buffer a') h0 h1 h2 : Lemma (requires (live h0 b /\ live h0 b' /\ modifies_2_1 b h0 h1 /\ modifies_1 b' h1 h2)) (ensures (modifies_3_2 b b' h0 h2)) [SMTPat (modifies_2_1 b h0 h1); SMTPat (modifies_1 b' h1 h2)] = () (* TODO: lemmas for modifies_3 *) let lemma_modifies_0_unalloc (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (b `unused_in` h0 /\ frameOf b == HS.get_tip h0 /\ modifies_0 h0 h1 /\ modifies_1 b h1 h2)) (ensures (modifies_0 h0 h2)) = () let lemma_modifies_none_1_trans (#a:Type) (b:buffer a) h0 h1 h2 : Lemma (requires (modifies_none h0 h1 /\ live h0 b /\ modifies_1 b h1 h2)) (ensures (modifies_1 b h0 h2)) = () let lemma_modifies_0_none_trans h0 h1 h2 : Lemma (requires (modifies_0 h0 h1 /\ modifies_none h1 h2)) (ensures (modifies_0 h0 h2)) = () #reset-options "--initial_fuel 0 --max_fuel 0" (** Concrete getters and setters *) val create: #a:Type -> init:a -> len:UInt32.t -> StackInline (buffer a) (requires (fun h -> True)) (ensures (fun (h0:mem) b h1 -> b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ frameOf b == HS.get_tip h0 /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.create (v len) init)) let create #a init len = let content: reference (lseq a (v len)) = salloc (Seq.create (v len) init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" unfold let p (#a:Type0) (init:list a) : GTot Type0 = normalize (0 < FStar.List.Tot.length init) /\ normalize (FStar.List.Tot.length init <= UInt.max_int 32) unfold let q (#a:Type0) (len:nat) (buf:buffer a) : GTot Type0 = normalize (length buf == len) (** Concrete getters and setters *) val createL: #a:Type0 -> init:list a -> StackInline (buffer a) (requires (fun h -> p #a init)) (ensures (fun (h0:mem) b h1 -> let len = FStar.List.Tot.length init in len > 0 /\ b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == len /\ frameOf b == (HS.get_tip h0) /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ modifies_0 h0 h1 /\ as_seq h1 b == Seq.seq_of_list init /\ q #a len b)) #set-options "--initial_fuel 1 --max_fuel 1" //the normalize_term (length init) in the pre-condition will be unfolded //whereas the L.length init below will not let createL #a init = let len = UInt32.uint_to_t (FStar.List.Tot.length init) in let s = Seq.seq_of_list init in let content: reference (lseq a (v len)) = salloc (Seq.seq_of_list init) in let b = MkBuffer len content 0ul len in let h = HST.get() in assert (Seq.equal (as_seq h b) (sel h b)); b #reset-options "--initial_fuel 0 --max_fuel 0" let lemma_upd (#a:Type) (h:mem) (x:reference a{live_region h (HS.frameOf x)}) (v:a) : Lemma (requires True) (ensures (Map.domain (HS.get_hmap h) == Map.domain (HS.get_hmap (upd h x v)))) = let m = HS.get_hmap h in let m' = Map.upd m (HS.frameOf x) (Heap.upd (Map.sel m (HS.frameOf x)) (HS.as_ref x) v) in Set.lemma_equal_intro (Map.domain m) (Map.domain m') unfold let rcreate_post_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (b:buffer a) (h0 h1:mem) :Type0 = b `unused_in` h0 /\ live h1 b /\ idx b == 0 /\ length b == v len /\ Map.domain (HS.get_hmap h1) == Map.domain (HS.get_hmap h0) /\ HS.get_tip h1 == HS.get_tip h0 /\ modifies (Set.singleton r) h0 h1 /\ modifies_ref r Set.empty h0 h1 /\ as_seq h1 b == Seq.create (v len) init private let rcreate_common (#a:Type) (r:rid) (init:a) (len:UInt32.t) (mm:bool) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm b.content == mm)) = let h0 = HST.get() in let s = Seq.create (v len) init in let content: reference (lseq a (v len)) = if mm then ralloc_mm r s else ralloc r s in let b = MkBuffer len content 0ul len in let h1 = HST.get() in assert (Seq.equal (as_seq h1 b) (sel h1 b)); lemma_upd h0 content s; b (** This function allocates a buffer in an "eternal" region, i.e. a region where memory is // * automatically-managed. One does not need to call rfree on such a buffer. It // * translates to C as a call to malloc and assumes a conservative garbage // * collector is running. *) val rcreate: #a:Type -> r:rid -> init:a -> len:UInt32.t -> ST (buffer a) (requires (fun h -> is_eternal_region r)) (ensures (fun (h0:mem) b h1 -> rcreate_post_common r init len b h0 h1 /\ ~(is_mm b.content))) let rcreate #a r init len = rcreate_common r init len false (** This predicate tells whether a buffer can be `rfree`d. The only way to produce it should be `rcreate_mm`, and the only way to consume it should be `rfree.` Rationale: a buffer can be `rfree`d only if it is the result of `rcreate_mm`. Subbuffers should not. *) let freeable (#a: Type) (b: buffer a) : GTot Type0 = is_mm b.content /\ is_eternal_region (frameOf b) /\ idx b == 0 (** This function allocates a buffer into a manually-managed buffer in a heap * region, meaning that the client must call rfree in order to avoid memory * leaks. It translates to C as a straight malloc. *) let rcreate_mm (#a:Type) (r:rid) (init:a) (len:UInt32.t) :ST (buffer a) (requires (fun h0 -> is_eternal_region r)) (ensures (fun h0 b h1 -> rcreate_post_common r init len b h0 h1 /\ is_mm (content b) /\ freeable b)) = rcreate_common r init len true #reset-options (** This function frees a buffer allocated with `rcreate_mm`. It translates to C as a regular free. *) let rfree (#a:Type) (b:buffer a) :ST unit (requires (fun h0 -> live h0 b /\ freeable b)) (ensures (fun h0 _ h1 -> is_mm (content b) /\ is_eternal_region (frameOf b) /\ h1 == HS.free (content b) h0)) = rfree b.content (* #reset-options "--z3rlimit 100 --initial_fuel 0 --max_fuel 0" *) (* val create_null: #a:Type -> init:a -> len:UInt32.t -> Stack (buffer a) *) (* (requires (fun h -> True)) *) (* (ensures (fun h0 b h1 -> length b = UInt32.v len /\ h0 == h1)) *) (* let create_null #a init len = *) (* push_frame(); *) (* let r = create init len in *) (* pop_frame(); *) (* r *) #reset-options "--initial_fuel 0 --max_fuel 0" // ocaml-only, used for conversions to Platform.bytes val to_seq: #a:Type -> b:buffer a -> l:UInt32.t{v l <= length b} -> STL (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ Seq.length r == v l (*/\ r == as_seq #a h1 b *) )) let to_seq #a b l = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v l) // ocaml-only, used for conversions to Platform.bytes val to_seq_full: #a:Type -> b:buffer a -> ST (seq a) (requires (fun h -> live h b)) (ensures (fun h0 r h1 -> h0 == h1 /\ live h1 b /\ r == as_seq #a h1 b )) let to_seq_full #a b = let s = !b.content in let i = v b.idx in Seq.slice s i (i + v b.length) val index: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> live h0 b /\ h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let index #a b n = let s = !b.content in Seq.index s (v b.idx + v n) (** REMARK: the proof of this lemma relies crucially on the `a == a'` condition // in `disjoint`, and on the pattern in `Seq.slice_upd` *) private let lemma_aux_0 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) (bb:buffer tt) :Lemma (requires (live h0 b /\ live h0 bb /\ disjoint b bb)) (ensures (live h0 b /\ live h0 bb /\ (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb))) = Heap.lemma_distinct_addrs_distinct_preorders (); Heap.lemma_distinct_addrs_distinct_mm () #set-options "--z3rlimit 10" private let lemma_aux_1 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) (tt:Type) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_0 b n z h0 tt)) #reset-options "--initial_fuel 0 --max_fuel 0" private let lemma_aux_2 (#a:Type) (b:buffer a) (n:UInt32.t{v n < length b}) (z:a) (h0:mem) :Lemma (requires (live h0 b)) (ensures (live h0 b /\ (forall (tt:Type) (bb:buffer tt). (live h0 bb /\ disjoint b bb) ==> (let h1 = HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z) in as_seq h0 bb == as_seq h1 bb)))) = let open FStar.Classical in forall_intro (move_requires (lemma_aux_1 b n z h0)) private val lemma_aux: #a:Type -> b:buffer a -> n:UInt32.t{v n < length b} -> z:a -> h0:mem -> Lemma (requires (live h0 b)) (ensures (live h0 b /\ modifies_1 b h0 (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z)) )) [SMTPat (HS.upd h0 b.content (Seq.upd (sel h0 b) (idx b + v n) z))] let lemma_aux #a b n z h0 = lemma_aux_2 b n z h0 val upd: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let upd #a b n z = let h0 = HST.get () in let s0 = !b.content in let s = Seq.upd s0 (v b.idx + v n) z in b.content := s; lemma_aux b n z h0; let h = HST.get() in Seq.lemma_eq_intro (as_seq h b) (Seq.slice s (idx b) (idx b + length b)); Seq.upd_slice s0 (idx b) (idx b + length b) (v n) z val sub: #a:Type -> b:buffer a -> i:UInt32.t -> len:UInt32.t{v i + v len <= length b} -> Tot (b':buffer a{b `includes` b' /\ length b' == v len}) let sub #a b i len = assert (v i + v b.idx < pow2 n); // was formerly a precondition MkBuffer b.max_length b.content (i +^ b.idx) len let sub_sub (#a: Type) (b: buffer a) (i1: UInt32.t) (len1: UInt32.t{v i1 + v len1 <= length b}) (i2: UInt32.t) (len2: UInt32.t {v i2 + v len2 <= v len1}) : Lemma (ensures (sub (sub b i1 len1) i2 len2 == sub b (i1 +^ i2) len2)) = () let sub_zero_length (#a: Type) (b: buffer a) : Lemma (ensures (sub b (UInt32.uint_to_t 0) (UInt32.uint_to_t (length b)) == b)) = () let lemma_sub_spec (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (sub b i len); SMTPat (live h b)] = Seq.lemma_eq_intro (as_seq h (sub b i len)) (Seq.slice (as_seq h b) (v i) (v i + v len)) let lemma_sub_spec' (#a:Type) (b:buffer a) (i:UInt32.t) (len:UInt32.t{v len <= length b /\ v i + v len <= length b}) h : Lemma (requires (live h b)) (ensures (live h (sub b i len) /\ as_seq h (sub b i len) == Seq.slice (as_seq h b) (v i) (v i + v len))) [SMTPat (live h (sub b i len))] = lemma_sub_spec b i len h val offset: #a:Type -> b:buffer a -> i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length} -> Tot (b':buffer a{b `includes` b'}) let offset #a b i = MkBuffer b.max_length b.content (i +^ b.idx) (b.length -^ i) let lemma_offset_spec (#a:Type) (b:buffer a) (i:UInt32.t{v i + v b.idx < pow2 n /\ v i <= v b.length}) h : Lemma (requires True) (ensures (as_seq h (offset b i) == Seq.slice (as_seq h b) (v i) (length b))) [SMTPatOr [[SMTPat (as_seq h (offset b i))]; [SMTPat (Seq.slice (as_seq h b) (v i) (length b))]]] = Seq.lemma_eq_intro (as_seq h (offset b i)) (Seq.slice (as_seq h b) (v i) (length b)) private val eq_lemma1: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) (ensures equal h (sub b1 0ul len) h (sub b2 0ul len)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma1 #a b1 b2 len h = Seq.lemma_eq_intro (as_seq h (sub b1 0ul len)) (as_seq h (sub b2 0ul len)) #reset-options "--z3rlimit 20" private val eq_lemma2: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> h:mem -> Lemma (requires equal h (sub b1 0ul len) h (sub b2 0ul len)) (ensures (forall (j:nat). j < v len ==> get h b1 j == get h b2 j)) [SMTPat (equal h (sub b1 0ul len) h (sub b2 0ul len))] let eq_lemma2 #a b1 b2 len h = let s1 = as_seq h (sub b1 0ul len) in let s2 = as_seq h (sub b2 0ul len) in cut (forall (j:nat). j < v len ==> get h b1 j == Seq.index s1 j); cut (forall (j:nat). j < v len ==> get h b2 j == Seq.index s2 j) (** Corresponds to memcmp for `eqtype` *) val eqb: #a:eqtype -> b1:buffer a -> b2:buffer a -> len:UInt32.t{v len <= length b1 /\ v len <= length b2} -> ST bool (requires (fun h -> live h b1 /\ live h b2)) (ensures (fun h0 z h1 -> h1 == h0 /\ (z <==> equal h0 (sub b1 0ul len) h0 (sub b2 0ul len)))) let rec eqb #a b1 b2 len = if len =^ 0ul then true else let len' = len -^ 1ul in if index b1 len' = index b2 len' then eqb b1 b2 len' else false (** // Defining operators for buffer accesses as specified at // https://github.com/FStarLang/FStar/wiki/Parsing-and-operator-precedence // *) (* JP: if the [val] is not specified, there's an issue with these functions // * taking an extra unification parameter at extraction-time... *) val op_Array_Access: #a:Type -> b:buffer a -> n:UInt32.t{v n<length b} -> Stack a (requires (fun h -> live h b)) (ensures (fun h0 z h1 -> h1 == h0 /\ z == Seq.index (as_seq h0 b) (v n))) let op_Array_Access #a b n = index #a b n val op_Array_Assignment: #a:Type -> b:buffer a -> n:UInt32.t -> z:a -> Stack unit (requires (fun h -> live h b /\ v n < length b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h1 b /\ v n < length b /\ modifies_1 b h0 h1 /\ as_seq h1 b == Seq.upd (as_seq h0 b) (v n) z )) let op_Array_Assignment #a b n z = upd #a b n z let lemma_modifies_one_trans_1 (#a:Type) (b:buffer a) (h0:mem) (h1:mem) (h2:mem): Lemma (requires (modifies_one (frameOf b) h0 h1 /\ modifies_one (frameOf b) h1 h2)) (ensures (modifies_one (frameOf b) h0 h2)) [SMTPat (modifies_one (frameOf b) h0 h1); SMTPat (modifies_one (frameOf b) h1 h2)] = () #reset-options "--z3rlimit 100 --max_fuel 0 --max_ifuel 0 --initial_fuel 0 --initial_ifuel 0" (** Corresponds to memcpy *) val blit: #t:Type -> a:buffer t -> idx_a:UInt32.t{v idx_a <= length a} -> b:buffer t{disjoint a b} -> idx_b:UInt32.t{v idx_b <= length b} -> len:UInt32.t{v idx_a + v len <= length a /\ v idx_b + v len <= length b} -> Stack unit (requires (fun h -> live h a /\ live h b)) (ensures (fun h0 _ h1 -> live h0 b /\ live h0 a /\ live h1 b /\ live h1 a /\ modifies_1 b h0 h1 /\ Seq.slice (as_seq h1 b) (v idx_b) (v idx_b + v len) == Seq.slice (as_seq h0 a) (v idx_a) (v idx_a + v len) /\ Seq.slice (as_seq h1 b) 0 (v idx_b) == Seq.slice (as_seq h0 b) 0 (v idx_b) /\ Seq.slice (as_seq h1 b) (v idx_b+v len) (length b) ==

{ "checked_file": "/", "dependencies": [ "prims.fst.checked", "FStar.UInt32.fsti.checked", "FStar.UInt.fsti.checked", "FStar.TSet.fsti.checked", "FStar.Set.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Map.fsti.checked", "FStar.List.Tot.fst.checked", "FStar.Int32.fsti.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked", "FStar.Heap.fst.checked", "FStar.Ghost.fsti.checked", "FStar.Classical.fsti.checked" ], "interface_file": false, "source_file": "FStar.Buffer.fst" }

[ { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "HST" }, { "abbrev": true, "full_module": "FStar.HyperStack", "short_module": "HS" }, { "abbrev": false, "full_module": "FStar.Ghost", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": true, "full_module": "FStar.Int32", "short_module": "Int32" }, { "abbrev": false, "full_module": "FStar.UInt32", "short_module": null }, { "abbrev": false, "full_module": "FStar.Seq", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: FStar.Buffer.buffer t -> idx_a: FStar.UInt32.t{FStar.UInt32.v idx_a <= FStar.Buffer.length a} -> b: FStar.Buffer.buffer t {FStar.Buffer.disjoint a b} -> idx_b: FStar.UInt32.t{FStar.UInt32.v idx_b <= FStar.Buffer.length b} -> len: FStar.UInt32.t { FStar.UInt32.v idx_a + FStar.UInt32.v len <= FStar.Buffer.length a /\ FStar.UInt32.v idx_b + FStar.UInt32.v len <= FStar.Buffer.length b } -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "FStar.Buffer.buffer", "FStar.UInt32.t", "Prims.b2t", "Prims.op_LessThanOrEqual", "FStar.UInt32.v", "FStar.Buffer.length", "FStar.Buffer.disjoint", "Prims.l_and", "Prims.op_Addition", "FStar.UInt32.op_Equals_Hat", "FStar.UInt32.__uint_to_t", "Prims.unit", "Prims.bool", "FStar.Seq.Properties.cons_head_tail", "FStar.Seq.Base.slice", "FStar.Buffer.as_seq", "FStar.Seq.Properties.snoc_slice_index", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "FStar.Buffer.op_Array_Assignment", "FStar.UInt32.op_Plus_Hat", "FStar.Buffer.op_Array_Access", "FStar.Buffer.blit", "FStar.UInt32.op_Subtraction_Hat" ]

[ "recursion" ]

false

true

false

let rec blit #t a idx_a b idx_b len =

let h0 = HST.get () in if len =^ 0ul then () else let len' = len -^ 1ul in blit #t a idx_a b idx_b len'; let z = a.(idx_a +^ len') in b.(idx_b +^ len') <- z; let h1 = HST.get () in Seq.snoc_slice_index (as_seq h1 b) (v idx_b) (v idx_b + v len'); Seq.cons_head_tail (Seq.slice (as_seq h0 b) (v idx_b + v len') (length b)); Seq.cons_head_tail (Seq.slice (as_seq h1 b) (v idx_b + v len') (length b))

false

Pulse.C.Types.Struct.fsti

Pulse.C.Types.Struct.struct_field

val struct_field: #tn: Type0 -> #tf: Type0 -> #n: string -> #fields: nonempty_field_description_t tf -> #v: Ghost.erased (struct_t0 tn n fields) -> r: ref (struct0 tn n fields) -> field: field_t fields -> #t': Type0 -> #td': typedef t' -> (#[norm_fields ()] sq_t': squash (t' == fields.fd_type field)) -> (#[norm_fields ()] sq_td': squash (td' == fields.fd_typedef field)) -> Prims.unit -> stt (ref td') (pts_to r v) (fun r' -> (pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field)) ** has_struct_field r field r')

let struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (# [ norm_fields () ] sq_t': squash (t' == fields.fd_type field)) (# [ norm_fields () ] sq_td': squash (td' == fields.fd_typedef field)) () : stt (ref td') (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field) ** has_struct_field r field r') = struct_field0 t' r field td'

{ "file_name": "share/steel/examples/pulse/lib/c/Pulse.C.Types.Struct.fsti", "git_rev": "f984200f79bdc452374ae994a5ca837496476c41", "git_url": "https://github.com/FStarLang/steel.git", "project_name": "steel" }

{ "end_col": 7, "end_line": 291, "start_col": 0, "start_line": 271 }

module Pulse.C.Types.Struct open Pulse.Lib.Pervasives include Pulse.C.Types.Fields open Pulse.C.Typestring // To be extracted as: struct t { fields ... } [@@noextract_to "krml"] // primitive val define_struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let define_struct (n: string) (#tf: Type0) (#tn: Type0) (#[solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = define_struct0 tn #tf n fields // To be extracted as: struct t [@@noextract_to "krml"] // primitive val struct_t0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot Type0 inline_for_extraction [@@noextract_to "krml"] let struct_t (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot Type0 = struct_t0 tn #tf n fields val struct_set_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (f: field_t fields) (v: fields.fd_type f) (s: struct_t0 tn n fields) : GTot (struct_t0 tn n fields) val struct_get_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) : GTot (fields.fd_type field) val struct_eq (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s1 s2: struct_t0 tn n fields) : Ghost prop (requires True) (ensures (fun y -> (y <==> (s1 == s2)) /\ (y <==> (forall field . struct_get_field s1 field == struct_get_field s2 field)) )) val struct_get_field_same (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) : Lemma (struct_get_field (struct_set_field field v s) field == v) [SMTPat (struct_get_field (struct_set_field field v s) field)] val struct_get_field_other (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (requires (field' <> field)) (ensures (struct_get_field (struct_set_field field v s) field' == struct_get_field s field')) let struct_get_field_pat (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (s: struct_t0 tn n fields) (field: field_t fields) (v: fields.fd_type field) (field': field_t fields) : Lemma (struct_get_field (struct_set_field field v s) field' == (if field' = field then v else struct_get_field s field')) [SMTPat (struct_get_field (struct_set_field field v s) field')] = if field' = field then () else struct_get_field_other s field v field' [@@noextract_to "krml"] // proof-only val struct0 (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) inline_for_extraction [@@noextract_to "krml"; norm_field_attr] // proof-only let struct (#tf: Type0) (n: string) (#tn: Type0) (# [solve_mk_string_t ()] prf: squash (norm norm_typestring (mk_string_t n == tn))) (fields: nonempty_field_description_t tf) : Tot (typedef (struct_t0 tn n fields)) = struct0 tn #tf n fields val struct_get_field_unknown (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) (field: field_t fields) : Lemma (struct_get_field (unknown (struct0 tn n fields)) field == unknown (fields.fd_typedef field)) [SMTPat (struct_get_field (unknown (struct0 tn n fields)) field)] val struct_get_field_uninitialized (tn: Type0) (#tf: Type0) (n: string) (fields: nonempty_field_description_t tf) (field: field_t fields) : Lemma (struct_get_field (uninitialized (struct0 tn n fields)) field == uninitialized (fields.fd_typedef field)) [SMTPat (struct_get_field (uninitialized (struct0 tn n fields)) field)] val has_struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : Tot vprop val has_struct_field_prop (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost unit (has_struct_field r field r') (fun _ -> has_struct_field r field r' ** pure ( t' == fields.fd_type field /\ td' == fields.fd_typedef field )) val has_struct_field_dup (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost unit (has_struct_field r field r') (fun _ -> has_struct_field r field r' ** has_struct_field r field r') val has_struct_field_inj (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t1: Type0) (#td1: typedef t1) (r1: ref td1) (#t2: Type0) (#td2: typedef t2) (r2: ref td2) : stt_ghost (squash (t1 == t2 /\ td1 == td2)) (has_struct_field r field r1 ** has_struct_field r field r2) (fun _ -> has_struct_field r field r1 ** has_struct_field r field r2 ** ref_equiv r1 (coerce_eq () r2)) val has_struct_field_equiv_from (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r1: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') (r2: ref (struct0 tn n fields)) : stt_ghost unit (ref_equiv r1 r2 ** has_struct_field r1 field r') (fun _ -> ref_equiv r1 r2 ** has_struct_field r2 field r') val has_struct_field_equiv_to (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r1' r2': ref td') : stt_ghost unit (ref_equiv r1' r2' ** has_struct_field r field r1') (fun _ -> ref_equiv r1' r2' ** has_struct_field r field r2') val ghost_struct_field_focus (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (r': ref td') : stt_ghost (squash ( t' == fields.fd_type field /\ td' == fields.fd_typedef field )) (has_struct_field r field r' ** pts_to r v) (fun _ -> has_struct_field r field r' ** pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (Ghost.hide (coerce_eq () (struct_get_field v field)))) val ghost_struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) : stt_ghost (Ghost.erased (ref (fields.fd_typedef field))) (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field) ** has_struct_field r field r') [@@noextract_to "krml"] // primitive val struct_field0 (#tn: Type0) (#tf: Type0) (t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t' { t' == fields.fd_type field /\ td' == fields.fd_typedef field }) : stt (ref td') (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (Ghost.hide (coerce_eq () (struct_get_field v field))) ** has_struct_field r field r') inline_for_extraction [@@noextract_to "krml"] // primitive let struct_field1 (#tn: Type0) (#tf: Type0) (t': Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (td': typedef t') (sq_t': squash (t' == fields.fd_type field)) (sq_td': squash (td' == fields.fd_typedef field)) : stt (ref td') (pts_to r v) (fun r' -> pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field) ** has_struct_field r field r') = struct_field0 t' r field td'

{ "checked_file": "/", "dependencies": [ "Pulse.Lib.Pervasives.fst.checked", "Pulse.C.Typestring.fsti.checked", "Pulse.C.Types.Fields.fsti.checked", "prims.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Ghost.fsti.checked" ], "interface_file": false, "source_file": "Pulse.C.Types.Struct.fsti" }

[ { "abbrev": false, "full_module": "Pulse.C.Typestring", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types.Fields", "short_module": null }, { "abbrev": false, "full_module": "Pulse.Lib.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "Pulse.C.Types", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 2, "initial_ifuel": 1, "max_fuel": 8, "max_ifuel": 2, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": true, "z3cliopt": [], "z3refresh": false, "z3rlimit": 5, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

r: Pulse.C.Types.Base.ref (Pulse.C.Types.Struct.struct0 tn n fields) -> field: Pulse.C.Types.Fields.field_t fields -> _: Prims.unit -> Pulse.Lib.Core.stt (Pulse.C.Types.Base.ref td') (Pulse.C.Types.Base.pts_to r v) (fun r' -> (Pulse.C.Types.Base.pts_to r (FStar.Ghost.hide (Pulse.C.Types.Struct.struct_set_field field (Pulse.C.Types.Base.unknown (Mkfield_description_t?.fd_typedef fields field)) (FStar.Ghost.reveal v))) ** Pulse.C.Types.Base.pts_to r' (FStar.Ghost.hide (Pulse.C.Types.Struct.struct_get_field (FStar.Ghost.reveal v) field))) ** Pulse.C.Types.Struct.has_struct_field r field r')

Prims.Tot

[ "total" ]

[]

[ "Prims.string", "Pulse.C.Types.Fields.nonempty_field_description_t", "FStar.Ghost.erased", "Pulse.C.Types.Struct.struct_t0", "Pulse.C.Types.Base.ref", "Pulse.C.Types.Struct.struct0", "Pulse.C.Types.Fields.field_t", "Pulse.C.Types.Base.typedef", "Prims.squash", "Prims.eq2", "Pulse.C.Types.Fields.__proj__Mkfield_description_t__item__fd_type", "Pulse.C.Types.Fields.__proj__Mkfield_description_t__item__fd_typedef", "Prims.unit", "Pulse.C.Types.Struct.struct_field0", "Pulse.Lib.Core.stt", "Pulse.C.Types.Base.pts_to", "Pulse.Lib.Core.op_Star_Star", "FStar.Ghost.hide", "Pulse.C.Types.Struct.struct_set_field", "Pulse.C.Types.Base.unknown", "FStar.Ghost.reveal", "Pulse.C.Types.Struct.struct_get_field", "Pulse.C.Types.Struct.has_struct_field", "Pulse.Lib.Core.vprop" ]

[]

false

let struct_field (#tn: Type0) (#tf: Type0) (#n: string) (#fields: nonempty_field_description_t tf) (#v: Ghost.erased (struct_t0 tn n fields)) (r: ref (struct0 tn n fields)) (field: field_t fields) (#t': Type0) (#td': typedef t') (#[norm_fields ()] sq_t': squash (t' == fields.fd_type field)) (#[norm_fields ()] sq_td': squash (td' == fields.fd_typedef field)) () : stt (ref td') (pts_to r v) (fun r' -> (pts_to r (struct_set_field field (unknown (fields.fd_typedef field)) v) ** pts_to r' (struct_get_field v field)) ** has_struct_field r field r') =

struct_field0 t' r field td'

false

Hacl.Impl.Frodo.KEM.KeyGen.fst

Hacl.Impl.Frodo.KEM.KeyGen.clear_matrix2

val clear_matrix2: a:FP.frodo_alg -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h s_matrix /\ live h e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1)

let clear_matrix2 a s_matrix e_matrix = clear_matrix s_matrix; clear_matrix e_matrix

{ "file_name": "code/frodo/Hacl.Impl.Frodo.KEM.KeyGen.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 23, "end_line": 147, "start_col": 0, "start_line": 145 }

module Hacl.Impl.Frodo.KEM.KeyGen open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open LowStar.Buffer open Lib.IntTypes open Lib.Buffer open Hacl.Impl.Matrix open Hacl.Impl.Frodo.Params open Hacl.Impl.Frodo.KEM open Hacl.Impl.Frodo.Pack open Hacl.Impl.Frodo.Sample open Hacl.Frodo.Random module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module M = Spec.Matrix module FP = Spec.Frodo.Params module S = Spec.Frodo.KEM.KeyGen #set-options "--z3rlimit 100 --fuel 0 --ifuel 0" inline_for_extraction noextract val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len)) let frodo_shake_r a c seed_se output_len r = push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame () inline_for_extraction noextract val frodo_mul_add_as_plus_e: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_a /\ live h s_matrix /\ live h e_matrix /\ live h b_matrix /\ disjoint b_matrix seed_a /\ disjoint b_matrix e_matrix /\ disjoint b_matrix s_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b_matrix) h0 h1 /\ as_matrix h1 b_matrix == S.frodo_mul_add_as_plus_e a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix = FP.params_n_sqr a; push_frame(); let a_matrix = matrix_create (params_n a) (params_n a) in frodo_gen_matrix gen_a (params_n a) seed_a a_matrix; matrix_mul_s a_matrix s_matrix b_matrix; matrix_add b_matrix e_matrix; pop_frame() inline_for_extraction noextract val frodo_mul_add_as_plus_e_pack: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b:lbytes (publicmatrixbytes_len a) -> Stack unit (requires fun h -> live h seed_a /\ live h b /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_a b /\ disjoint b s_matrix /\ disjoint b e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b) h0 h1 /\ as_seq h1 b == S.frodo_mul_add_as_plus_e_pack a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e_pack a gen_a seed_a s_matrix e_matrix b = push_frame (); let b_matrix = matrix_create (params_n a) params_nbar in frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix; frodo_pack (params_logq a) b_matrix b; pop_frame () inline_for_extraction noextract val get_s_e_matrices: a:FP.frodo_alg -> seed_se:lbytes (crypto_bytes a) -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_se /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_se s_matrix /\ disjoint seed_se e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1 /\ (as_matrix h1 s_matrix, as_matrix h1 e_matrix) == S.get_s_e_matrices a (as_seq h0 seed_se)) let get_s_e_matrices a seed_se s_matrix e_matrix = push_frame (); [@inline_let] let s_bytes_len = secretmatrixbytes_len a in let r = create (2ul *! s_bytes_len) (u8 0) in frodo_shake_r a (u8 0x5f) seed_se (2ul *! s_bytes_len) r; frodo_sample_matrix a (params_n a) params_nbar (sub r 0ul s_bytes_len) s_matrix; frodo_sample_matrix a (params_n a) params_nbar (sub r s_bytes_len s_bytes_len) e_matrix; pop_frame () inline_for_extraction noextract val clear_matrix2: a:FP.frodo_alg -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h s_matrix /\ live h e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1)

{ "checked_file": "/", "dependencies": [ "Spec.Matrix.fst.checked", "Spec.Frodo.Params.fst.checked", "Spec.Frodo.KEM.KeyGen.fst.checked", "prims.fst.checked", "LowStar.Buffer.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Impl.Matrix.fst.checked", "Hacl.Impl.Frodo.Sample.fst.checked", "Hacl.Impl.Frodo.Params.fst.checked", "Hacl.Impl.Frodo.Pack.fst.checked", "Hacl.Impl.Frodo.KEM.fst.checked", "Hacl.Frodo.Random.fsti.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": false, "source_file": "Hacl.Impl.Frodo.KEM.KeyGen.fst" }

[ { "abbrev": true, "full_module": "Spec.Frodo.KEM.KeyGen", "short_module": "S" }, { "abbrev": true, "full_module": "Spec.Frodo.Params", "short_module": "FP" }, { "abbrev": true, "full_module": "Spec.Matrix", "short_module": "M" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Hacl.Frodo.Random", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Sample", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Pack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Params", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Matrix", "short_module": null }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "LowStar.Buffer", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.Frodo.Params.frodo_alg -> s_matrix: Hacl.Impl.Matrix.matrix_t (Hacl.Impl.Frodo.Params.params_n a) Hacl.Impl.Frodo.Params.params_nbar -> e_matrix: Hacl.Impl.Matrix.matrix_t (Hacl.Impl.Frodo.Params.params_n a) Hacl.Impl.Frodo.Params.params_nbar -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Spec.Frodo.Params.frodo_alg", "Hacl.Impl.Matrix.matrix_t", "Hacl.Impl.Frodo.Params.params_n", "Hacl.Impl.Frodo.Params.params_nbar", "Hacl.Impl.Frodo.KEM.clear_matrix", "Prims.unit" ]

[]

false

true

false

let clear_matrix2 a s_matrix e_matrix =

clear_matrix s_matrix; clear_matrix e_matrix

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_192_lemma

val proj_g_pow2_192_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_192 == pow_point (pow2 192) g_aff)

let proj_g_pow2_192_lemma () = lemma_proj_g_pow2_192_eval (); lemma_proj_g_pow2_128_eval (); lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_192_lemma S.mk_p256_concrete_ops S.base_point

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 61, "end_line": 165, "start_col": 0, "start_line": 161 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) inline_for_extraction noextract let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64) inline_for_extraction noextract let proj_g_pow2_128_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_128) inline_for_extraction noextract let proj_g_pow2_192_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_192) let proj_g_pow2_64_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); Seq.seq_of_list proj_g_pow2_64_list let proj_g_pow2_128_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_128); Seq.seq_of_list proj_g_pow2_128_list let proj_g_pow2_192_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_192); Seq.seq_of_list proj_g_pow2_192_list val proj_g_pow2_64_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_64 == pow_point (pow2 64) g_aff) let proj_g_pow2_64_lemma () = lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_64_lemma S.mk_p256_concrete_ops S.base_point val proj_g_pow2_128_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_128 == pow_point (pow2 128) g_aff) let proj_g_pow2_128_lemma () = lemma_proj_g_pow2_128_eval (); lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_128_lemma S.mk_p256_concrete_ops S.base_point val proj_g_pow2_192_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_192 == pow_point (pow2 192) g_aff)

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> FStar.Pervasives.Lemma (ensures Spec.P256.PointOps.to_aff_point Hacl.P256.PrecompTable.proj_g_pow2_192 == Hacl.P256.PrecompTable.pow_point (Prims.pow2 192) Hacl.P256.PrecompTable.g_aff)

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.unit", "Hacl.Spec.PrecompBaseTable256.a_pow2_192_lemma", "Spec.P256.PointOps.proj_point", "Spec.P256.mk_p256_concrete_ops", "Spec.P256.PointOps.base_point", "Hacl.P256.PrecompTable.lemma_proj_g_pow2_64_eval", "Hacl.P256.PrecompTable.lemma_proj_g_pow2_128_eval", "Hacl.P256.PrecompTable.lemma_proj_g_pow2_192_eval" ]

[]

true

false

true

false

let proj_g_pow2_192_lemma () =

lemma_proj_g_pow2_192_eval (); lemma_proj_g_pow2_128_eval (); lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_192_lemma S.mk_p256_concrete_ops S.base_point

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_64_list

val proj_g_pow2_64_list:SPTK.point_list

let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64)

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 57, "end_line": 117, "start_col": 0, "start_line": 116 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64)

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Hacl.Spec.P256.PrecompTable.point_list

Prims.Tot

[ "total" ]

[]

[ "FStar.Pervasives.normalize_term", "Hacl.Spec.P256.PrecompTable.point_list", "Hacl.Spec.P256.PrecompTable.proj_point_to_list", "Hacl.P256.PrecompTable.proj_g_pow2_64" ]

[]

false

true

false

let proj_g_pow2_64_list:SPTK.point_list =

normalize_term (SPTK.proj_point_to_list proj_g_pow2_64)

false

Hacl.Impl.Frodo.KEM.KeyGen.fst

Hacl.Impl.Frodo.KEM.KeyGen.frodo_shake_r

val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len))

let frodo_shake_r a c seed_se output_len r = push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame ()

{ "file_name": "code/frodo/Hacl.Impl.Frodo.KEM.KeyGen.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 14, "end_line": 53, "start_col": 0, "start_line": 41 }

module Hacl.Impl.Frodo.KEM.KeyGen open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open LowStar.Buffer open Lib.IntTypes open Lib.Buffer open Hacl.Impl.Matrix open Hacl.Impl.Frodo.Params open Hacl.Impl.Frodo.KEM open Hacl.Impl.Frodo.Pack open Hacl.Impl.Frodo.Sample open Hacl.Frodo.Random module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module M = Spec.Matrix module FP = Spec.Frodo.Params module S = Spec.Frodo.KEM.KeyGen #set-options "--z3rlimit 100 --fuel 0 --ifuel 0" inline_for_extraction noextract val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len))

{ "checked_file": "/", "dependencies": [ "Spec.Matrix.fst.checked", "Spec.Frodo.Params.fst.checked", "Spec.Frodo.KEM.KeyGen.fst.checked", "prims.fst.checked", "LowStar.Buffer.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Impl.Matrix.fst.checked", "Hacl.Impl.Frodo.Sample.fst.checked", "Hacl.Impl.Frodo.Params.fst.checked", "Hacl.Impl.Frodo.Pack.fst.checked", "Hacl.Impl.Frodo.KEM.fst.checked", "Hacl.Frodo.Random.fsti.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": false, "source_file": "Hacl.Impl.Frodo.KEM.KeyGen.fst" }

[ { "abbrev": true, "full_module": "Spec.Frodo.KEM.KeyGen", "short_module": "S" }, { "abbrev": true, "full_module": "Spec.Frodo.Params", "short_module": "FP" }, { "abbrev": true, "full_module": "Spec.Matrix", "short_module": "M" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Hacl.Frodo.Random", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Sample", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Pack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Params", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Matrix", "short_module": null }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "LowStar.Buffer", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.Frodo.Params.frodo_alg -> c: Lib.IntTypes.uint8 -> seed_se: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_bytes a) -> output_len: Lib.IntTypes.size_t -> r: Hacl.Impl.Matrix.lbytes output_len -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Spec.Frodo.Params.frodo_alg", "Lib.IntTypes.uint8", "Hacl.Impl.Matrix.lbytes", "Hacl.Impl.Frodo.Params.crypto_bytes", "Lib.IntTypes.size_t", "FStar.HyperStack.ST.pop_frame", "Prims.unit", "Hacl.Impl.Frodo.KEM.clear_words_u8", "Lib.IntTypes.op_Plus_Bang", "Lib.IntTypes.U32", "Lib.IntTypes.PUB", "FStar.UInt32.__uint_to_t", "Hacl.Impl.Frodo.Params.frodo_shake", "Lib.Sequence.eq_intro", "Prims.op_Addition", "Lib.IntTypes.v", "Lib.Sequence.concat", "Lib.Sequence.create", "Lib.Buffer.as_seq", "Lib.Buffer.MUT", "Lib.Sequence.sub", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "Lib.Buffer.update_sub", "Lib.Buffer.op_Array_Assignment", "Lib.Buffer.lbuffer_t", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.IntTypes.add", "FStar.UInt32.uint_to_t", "FStar.UInt32.t", "Lib.Buffer.create", "Lib.IntTypes.u8", "Lib.Buffer.lbuffer", "FStar.HyperStack.ST.push_frame" ]

[]

false

true

false

let frodo_shake_r a c seed_se output_len r =

push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame ()

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_64_lemma

val proj_g_pow2_64_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_64 == pow_point (pow2 64) g_aff)

let proj_g_pow2_64_lemma () = lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_64_lemma S.mk_p256_concrete_ops S.base_point

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 60, "end_line": 146, "start_col": 0, "start_line": 144 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) inline_for_extraction noextract let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64) inline_for_extraction noextract let proj_g_pow2_128_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_128) inline_for_extraction noextract let proj_g_pow2_192_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_192) let proj_g_pow2_64_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); Seq.seq_of_list proj_g_pow2_64_list let proj_g_pow2_128_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_128); Seq.seq_of_list proj_g_pow2_128_list let proj_g_pow2_192_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_192); Seq.seq_of_list proj_g_pow2_192_list val proj_g_pow2_64_lemma: unit -> Lemma (S.to_aff_point proj_g_pow2_64 == pow_point (pow2 64) g_aff)

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

_: Prims.unit -> FStar.Pervasives.Lemma (ensures Spec.P256.PointOps.to_aff_point Hacl.P256.PrecompTable.proj_g_pow2_64 == Hacl.P256.PrecompTable.pow_point (Prims.pow2 64) Hacl.P256.PrecompTable.g_aff)

FStar.Pervasives.Lemma

[ "lemma" ]

[]

[ "Prims.unit", "Hacl.Spec.PrecompBaseTable256.a_pow2_64_lemma", "Spec.P256.PointOps.proj_point", "Spec.P256.mk_p256_concrete_ops", "Spec.P256.PointOps.base_point", "Hacl.P256.PrecompTable.lemma_proj_g_pow2_64_eval" ]

[]

true

false

true

false

let proj_g_pow2_64_lemma () =

lemma_proj_g_pow2_64_eval (); SPT256.a_pow2_64_lemma S.mk_p256_concrete_ops S.base_point

false

Hacl.P256.PrecompTable.fst

Hacl.P256.PrecompTable.proj_g_pow2_192_lseq

val proj_g_pow2_192_lseq : LSeq.lseq uint64 12

let proj_g_pow2_192_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_192); Seq.seq_of_list proj_g_pow2_192_list

{ "file_name": "code/ecdsap256/Hacl.P256.PrecompTable.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 38, "end_line": 138, "start_col": 0, "start_line": 136 }

module Hacl.P256.PrecompTable open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open Lib.IntTypes open Lib.Buffer module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module LE = Lib.Exponentiation module SE = Spec.Exponentiation module SPT = Hacl.Spec.PrecompBaseTable module SPT256 = Hacl.Spec.PrecompBaseTable256 module SPTK = Hacl.Spec.P256.PrecompTable module S = Spec.P256 module SL = Spec.P256.Lemmas open Hacl.Impl.P256.Point include Hacl.Impl.P256.Group #set-options "--z3rlimit 50 --fuel 0 --ifuel 0" let proj_point_to_list p = SPTK.proj_point_to_list_lemma p; SPTK.proj_point_to_list p let lemma_refl x = SPTK.proj_point_to_list_lemma x //----------------- inline_for_extraction noextract let proj_g_pow2_64 : S.proj_point = [@inline_let] let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in [@inline_let] let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in [@inline_let] let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in (rX, rY, rZ) val lemma_proj_g_pow2_64_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops S.base_point 64 == proj_g_pow2_64) let lemma_proj_g_pow2_64_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops S.base_point 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64); let rX : S.felem = 0x000931f4ae428a4ad81ee0aa89cf5247ce85d4dd696c61b4bb9d4761e57b7fbe in let rY : S.felem = 0x7e88e5e6a142d5c2269f21a158e82ab2c79fcecb26e397b96fd5b9fbcd0a69a5 in let rZ : S.felem = 0x02626dc2dd5e06cd19de5e6afb6c5dbdd3e41dc1472e7b8ef11eb0662e41c44b in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_128 : S.proj_point = [@inline_let] let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in [@inline_let] let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in [@inline_let] let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in (rX, rY, rZ) val lemma_proj_g_pow2_128_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64 == proj_g_pow2_128) let lemma_proj_g_pow2_128_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_64 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64); let rX : S.felem = 0x04c3aaf6c6c00704e96eda89461d63fd2c97ee1e6786fc785e6afac7aa92f9b1 in let rY : S.felem = 0x14f1edaeb8e9c8d4797d164a3946c7ff50a7c8cd59139a4dbce354e6e4df09c3 in let rZ : S.felem = 0x80119ced9a5ce83c4e31f8de1a38f89d5f9ff9f637dca86d116a4217f83e55d2 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) inline_for_extraction noextract let proj_g_pow2_192 : S.proj_point = [@inline_let] let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in [@inline_let] let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in [@inline_let] let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in (rX, rY, rZ) val lemma_proj_g_pow2_192_eval : unit -> Lemma (SE.exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64 == proj_g_pow2_192) let lemma_proj_g_pow2_192_eval () = SPT256.exp_pow2_rec_is_exp_pow2 S.mk_p256_concrete_ops proj_g_pow2_128 64; let qX, qY, qZ = normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) in normalize_term_spec (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64); let rX : S.felem = 0xc762a9c8ae1b2f7434ff8da70fe105e0d4f188594989f193de0dbdbf5f60cb9a in let rY : S.felem = 0x1eddaf51836859e1369f1ae8d9ab02e4123b6f151d9b796e297a38fa5613d9bc in let rZ : S.felem = 0xcb433ab3f67815707e398dc7910cc4ec6ea115360060fc73c35b53dce02e2c72 in assert_norm (qX == rX /\ qY == rY /\ qZ == rZ) // let proj_g_pow2_64 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops S.base_point 64) // let proj_g_pow2_128 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_64 64) // let proj_g_pow2_192 : S.proj_point = // normalize_term (SPT256.exp_pow2_rec S.mk_p256_concrete_ops proj_g_pow2_128 64) inline_for_extraction noextract let proj_g_pow2_64_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_64) inline_for_extraction noextract let proj_g_pow2_128_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_128) inline_for_extraction noextract let proj_g_pow2_192_list : SPTK.point_list = normalize_term (SPTK.proj_point_to_list proj_g_pow2_192) let proj_g_pow2_64_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_64); Seq.seq_of_list proj_g_pow2_64_list let proj_g_pow2_128_lseq : LSeq.lseq uint64 12 = normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_128); Seq.seq_of_list proj_g_pow2_128_list

{ "checked_file": "/", "dependencies": [ "Spec.P256.Lemmas.fsti.checked", "Spec.P256.fst.checked", "Spec.Exponentiation.fsti.checked", "prims.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Exponentiation.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Spec.PrecompBaseTable256.fsti.checked", "Hacl.Spec.PrecompBaseTable.fsti.checked", "Hacl.Spec.P256.PrecompTable.fsti.checked", "Hacl.Impl.P256.Point.fsti.checked", "Hacl.Impl.P256.Group.fst.checked", "FStar.UInt32.fsti.checked", "FStar.Seq.fst.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.List.Tot.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": true, "source_file": "Hacl.P256.PrecompTable.fst" }

[ { "abbrev": true, "full_module": "Spec.P256.Lemmas", "short_module": "SL" }, { "abbrev": true, "full_module": "Hacl.Spec.P256.PrecompTable", "short_module": "SPTK" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable256", "short_module": "SPT256" }, { "abbrev": true, "full_module": "Lib.Exponentiation", "short_module": "LE" }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Group", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.P256.Point", "short_module": null }, { "abbrev": true, "full_module": "Hacl.Spec.P256.Montgomery", "short_module": "SM" }, { "abbrev": true, "full_module": "Spec.P256", "short_module": "S" }, { "abbrev": true, "full_module": "Hacl.Spec.PrecompBaseTable", "short_module": "SPT" }, { "abbrev": true, "full_module": "Hacl.Impl.Exponentiation.Definitions", "short_module": "BE" }, { "abbrev": true, "full_module": "Spec.Exponentiation", "short_module": "SE" }, { "abbrev": true, "full_module": "Lib.Exponentiation.Definition", "short_module": "LE" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "Hacl.P256", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 50, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

Lib.Sequence.lseq Lib.IntTypes.uint64 12

Prims.Tot

[ "total" ]

[]

[ "FStar.Seq.Base.seq_of_list", "Lib.IntTypes.uint64", "Hacl.P256.PrecompTable.proj_g_pow2_192_list", "Prims.unit", "FStar.Pervasives.normalize_term_spec", "Hacl.Spec.P256.PrecompTable.point_list", "Hacl.Spec.P256.PrecompTable.proj_point_to_list", "Hacl.P256.PrecompTable.proj_g_pow2_192", "Lib.Sequence.lseq" ]

[]

false

let proj_g_pow2_192_lseq:LSeq.lseq uint64 12 =

normalize_term_spec (SPTK.proj_point_to_list proj_g_pow2_192); Seq.seq_of_list proj_g_pow2_192_list

false

Hacl.Impl.Frodo.KEM.KeyGen.fst

Hacl.Impl.Frodo.KEM.KeyGen.crypto_kem_keypair_

val crypto_kem_keypair_: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> coins:lbytes (size 2 *! crypto_bytes a +! bytes_seed_a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h coins /\ disjoint pk sk /\ disjoint coins sk /\ disjoint coins pk) (ensures fun h0 _ h1 -> modifies (loc pk |+| loc sk) h0 h1 /\ (as_seq h1 pk, as_seq h1 sk) == S.crypto_kem_keypair_ a gen_a (as_seq h0 coins))

let crypto_kem_keypair_ a gen_a coins pk sk = FP.expand_crypto_secretkeybytes a; FP.expand_crypto_secretkeybytes a; let h0 = ST.get () in let s = sub coins 0ul (crypto_bytes a) in let seed_se = sub coins (crypto_bytes a) (crypto_bytes a) in let z = sub coins (2ul *! crypto_bytes a) bytes_seed_a in let seed_a = sub pk 0ul bytes_seed_a in frodo_shake a bytes_seed_a z bytes_seed_a seed_a; let b_bytes = sub pk bytes_seed_a (publicmatrixbytes_len a) in let s_bytes = sub sk (crypto_bytes a +! crypto_publickeybytes a) (secretmatrixbytes_len a) in frodo_mul_add_as_plus_e_pack_shake a gen_a seed_a seed_se b_bytes s_bytes; let h1 = ST.get () in LSeq.lemma_concat2 (v bytes_seed_a) (as_seq h1 seed_a) (v (publicmatrixbytes_len a)) (as_seq h1 b_bytes) (as_seq h1 pk); crypto_kem_sk a s pk sk

{ "file_name": "code/frodo/Hacl.Impl.Frodo.KEM.KeyGen.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 25, "end_line": 280, "start_col": 0, "start_line": 263 }

module Hacl.Impl.Frodo.KEM.KeyGen open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open LowStar.Buffer open Lib.IntTypes open Lib.Buffer open Hacl.Impl.Matrix open Hacl.Impl.Frodo.Params open Hacl.Impl.Frodo.KEM open Hacl.Impl.Frodo.Pack open Hacl.Impl.Frodo.Sample open Hacl.Frodo.Random module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module M = Spec.Matrix module FP = Spec.Frodo.Params module S = Spec.Frodo.KEM.KeyGen #set-options "--z3rlimit 100 --fuel 0 --ifuel 0" inline_for_extraction noextract val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len)) let frodo_shake_r a c seed_se output_len r = push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame () inline_for_extraction noextract val frodo_mul_add_as_plus_e: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_a /\ live h s_matrix /\ live h e_matrix /\ live h b_matrix /\ disjoint b_matrix seed_a /\ disjoint b_matrix e_matrix /\ disjoint b_matrix s_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b_matrix) h0 h1 /\ as_matrix h1 b_matrix == S.frodo_mul_add_as_plus_e a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix = FP.params_n_sqr a; push_frame(); let a_matrix = matrix_create (params_n a) (params_n a) in frodo_gen_matrix gen_a (params_n a) seed_a a_matrix; matrix_mul_s a_matrix s_matrix b_matrix; matrix_add b_matrix e_matrix; pop_frame() inline_for_extraction noextract val frodo_mul_add_as_plus_e_pack: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b:lbytes (publicmatrixbytes_len a) -> Stack unit (requires fun h -> live h seed_a /\ live h b /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_a b /\ disjoint b s_matrix /\ disjoint b e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b) h0 h1 /\ as_seq h1 b == S.frodo_mul_add_as_plus_e_pack a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e_pack a gen_a seed_a s_matrix e_matrix b = push_frame (); let b_matrix = matrix_create (params_n a) params_nbar in frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix; frodo_pack (params_logq a) b_matrix b; pop_frame () inline_for_extraction noextract val get_s_e_matrices: a:FP.frodo_alg -> seed_se:lbytes (crypto_bytes a) -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_se /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_se s_matrix /\ disjoint seed_se e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1 /\ (as_matrix h1 s_matrix, as_matrix h1 e_matrix) == S.get_s_e_matrices a (as_seq h0 seed_se)) let get_s_e_matrices a seed_se s_matrix e_matrix = push_frame (); [@inline_let] let s_bytes_len = secretmatrixbytes_len a in let r = create (2ul *! s_bytes_len) (u8 0) in frodo_shake_r a (u8 0x5f) seed_se (2ul *! s_bytes_len) r; frodo_sample_matrix a (params_n a) params_nbar (sub r 0ul s_bytes_len) s_matrix; frodo_sample_matrix a (params_n a) params_nbar (sub r s_bytes_len s_bytes_len) e_matrix; pop_frame () inline_for_extraction noextract val clear_matrix2: a:FP.frodo_alg -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h s_matrix /\ live h e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1) let clear_matrix2 a s_matrix e_matrix = clear_matrix s_matrix; clear_matrix e_matrix inline_for_extraction noextract val frodo_mul_add_as_plus_e_pack_shake: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> seed_se:lbytes (crypto_bytes a) -> b:lbytes (publicmatrixbytes_len a) -> s:lbytes (secretmatrixbytes_len a) -> Stack unit (requires fun h -> live h seed_a /\ live h seed_se /\ live h s /\ live h b /\ disjoint b s /\ disjoint seed_a b /\ disjoint seed_a s /\ disjoint seed_se b /\ disjoint seed_se s) (ensures fun h0 _ h1 -> modifies (loc s |+| loc b) h0 h1 /\ (as_seq h1 b, as_seq h1 s) == S.frodo_mul_add_as_plus_e_pack_shake a gen_a (as_seq h0 seed_a) (as_seq h0 seed_se)) let frodo_mul_add_as_plus_e_pack_shake a gen_a seed_a seed_se b s = push_frame (); let s_matrix = matrix_create (params_n a) params_nbar in let e_matrix = matrix_create (params_n a) params_nbar in get_s_e_matrices a seed_se s_matrix e_matrix; frodo_mul_add_as_plus_e_pack a gen_a seed_a s_matrix e_matrix b; matrix_to_lbytes s_matrix s; clear_matrix2 a s_matrix e_matrix; pop_frame () inline_for_extraction noextract val crypto_kem_sk1: a:FP.frodo_alg -> s:lbytes (crypto_bytes a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a -! bytes_pkhash a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h s /\ disjoint pk sk /\ disjoint sk s) (ensures fun h0 _ h1 -> modifies (loc sk) h0 h1 /\ (let s_bytes = LSeq.sub (as_seq h0 sk) (v (crypto_bytes a) + v (crypto_publickeybytes a)) (v (secretmatrixbytes_len a)) in as_seq h1 sk == LSeq.concat (LSeq.concat (as_seq h0 s) (as_seq h0 pk)) s_bytes)) let crypto_kem_sk1 a s pk sk = let h1 = ST.get () in FP.expand_crypto_secretkeybytes a; let s_pk_len = crypto_bytes a +! crypto_publickeybytes a in [@inline_let] let sm_len = secretmatrixbytes_len a in let slen1 = crypto_secretkeybytes a -! bytes_pkhash a in let s_bytes = sub sk s_pk_len sm_len in update_sub sk 0ul (crypto_bytes a) s; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); update_sub sk (crypto_bytes a) (crypto_publickeybytes a) pk; let h3 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h3 sk) 0 (v (crypto_bytes a))) (as_seq h1 s); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v (crypto_bytes a)) (v (crypto_publickeybytes a))) (as_seq h1 pk); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); LSeq.lemma_concat3 (v (crypto_bytes a)) (as_seq h1 s) (v (crypto_publickeybytes a)) (as_seq h1 pk) (v sm_len) (as_seq h1 s_bytes) (as_seq h3 sk) inline_for_extraction noextract val crypto_kem_sk: a:FP.frodo_alg -> s:lbytes (crypto_bytes a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h s /\ disjoint pk sk /\ disjoint sk s) (ensures fun h0 _ h1 -> modifies (loc sk) h0 h1 /\ (let s_bytes = LSeq.sub (as_seq h0 sk) (v (crypto_bytes a) + v (crypto_publickeybytes a)) (v (secretmatrixbytes_len a)) in as_seq h1 sk == S.crypto_kem_sk a (as_seq h0 s) (as_seq h0 pk) s_bytes)) let crypto_kem_sk a s pk sk = FP.expand_crypto_secretkeybytes a; let slen1 = crypto_secretkeybytes a -! bytes_pkhash a in let sk_p = sub sk 0ul slen1 in crypto_kem_sk1 a s pk sk_p; let h0 = ST.get () in update_sub_f h0 sk slen1 (bytes_pkhash a) (fun h -> FP.frodo_shake a (v (crypto_publickeybytes a)) (as_seq h0 pk) (v (bytes_pkhash a))) (fun _ -> frodo_shake a (crypto_publickeybytes a) pk (bytes_pkhash a) (sub sk slen1 (bytes_pkhash a))); let h1 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h0 sk) 0 (v slen1)) (LSeq.sub (as_seq h1 sk) 0 (v slen1)); LSeq.lemma_concat2 (v slen1) (LSeq.sub (as_seq h0 sk) 0 (v slen1)) (v (bytes_pkhash a)) (LSeq.sub (as_seq h1 sk) (v slen1) (v (bytes_pkhash a))) (as_seq h1 sk) inline_for_extraction noextract val crypto_kem_keypair_: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> coins:lbytes (size 2 *! crypto_bytes a +! bytes_seed_a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h coins /\ disjoint pk sk /\ disjoint coins sk /\ disjoint coins pk) (ensures fun h0 _ h1 -> modifies (loc pk |+| loc sk) h0 h1 /\ (as_seq h1 pk, as_seq h1 sk) == S.crypto_kem_keypair_ a gen_a (as_seq h0 coins))

{ "checked_file": "/", "dependencies": [ "Spec.Matrix.fst.checked", "Spec.Frodo.Params.fst.checked", "Spec.Frodo.KEM.KeyGen.fst.checked", "prims.fst.checked", "LowStar.Buffer.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Impl.Matrix.fst.checked", "Hacl.Impl.Frodo.Sample.fst.checked", "Hacl.Impl.Frodo.Params.fst.checked", "Hacl.Impl.Frodo.Pack.fst.checked", "Hacl.Impl.Frodo.KEM.fst.checked", "Hacl.Frodo.Random.fsti.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": false, "source_file": "Hacl.Impl.Frodo.KEM.KeyGen.fst" }

[ { "abbrev": true, "full_module": "Spec.Frodo.KEM.KeyGen", "short_module": "S" }, { "abbrev": true, "full_module": "Spec.Frodo.Params", "short_module": "FP" }, { "abbrev": true, "full_module": "Spec.Matrix", "short_module": "M" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Hacl.Frodo.Random", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Sample", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Pack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Params", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Matrix", "short_module": null }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "LowStar.Buffer", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.Frodo.Params.frodo_alg -> gen_a: Spec.Frodo.Params.frodo_gen_a{Hacl.Impl.Frodo.Params.is_supported gen_a} -> coins: Hacl.Impl.Matrix.lbytes (Lib.IntTypes.size 2 *! Hacl.Impl.Frodo.Params.crypto_bytes a +! Hacl.Impl.Frodo.Params.bytes_seed_a) -> pk: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_publickeybytes a) -> sk: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_secretkeybytes a) -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Spec.Frodo.Params.frodo_alg", "Spec.Frodo.Params.frodo_gen_a", "Prims.b2t", "Hacl.Impl.Frodo.Params.is_supported", "Hacl.Impl.Matrix.lbytes", "Lib.IntTypes.op_Plus_Bang", "Lib.IntTypes.U32", "Lib.IntTypes.PUB", "Lib.IntTypes.op_Star_Bang", "Lib.IntTypes.size", "Hacl.Impl.Frodo.Params.crypto_bytes", "Hacl.Impl.Frodo.Params.bytes_seed_a", "Hacl.Impl.Frodo.Params.crypto_publickeybytes", "Hacl.Impl.Frodo.Params.crypto_secretkeybytes", "Hacl.Impl.Frodo.KEM.KeyGen.crypto_kem_sk", "Prims.unit", "Lib.Sequence.lemma_concat2", "Lib.IntTypes.uint8", "Lib.IntTypes.v", "Lib.Buffer.as_seq", "Lib.Buffer.MUT", "Hacl.Impl.Frodo.Params.publicmatrixbytes_len", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "Hacl.Impl.Frodo.KEM.KeyGen.frodo_mul_add_as_plus_e_pack_shake", "Lib.Buffer.lbuffer_t", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Hacl.Impl.Frodo.Params.secretmatrixbytes_len", "Lib.Buffer.sub", "Hacl.Impl.Frodo.Params.frodo_shake", "FStar.UInt32.__uint_to_t", "Spec.Frodo.Params.expand_crypto_secretkeybytes" ]

[]

false

true

false

let crypto_kem_keypair_ a gen_a coins pk sk =

FP.expand_crypto_secretkeybytes a; FP.expand_crypto_secretkeybytes a; let h0 = ST.get () in let s = sub coins 0ul (crypto_bytes a) in let seed_se = sub coins (crypto_bytes a) (crypto_bytes a) in let z = sub coins (2ul *! crypto_bytes a) bytes_seed_a in let seed_a = sub pk 0ul bytes_seed_a in frodo_shake a bytes_seed_a z bytes_seed_a seed_a; let b_bytes = sub pk bytes_seed_a (publicmatrixbytes_len a) in let s_bytes = sub sk (crypto_bytes a +! crypto_publickeybytes a) (secretmatrixbytes_len a) in frodo_mul_add_as_plus_e_pack_shake a gen_a seed_a seed_se b_bytes s_bytes; let h1 = ST.get () in LSeq.lemma_concat2 (v bytes_seed_a) (as_seq h1 seed_a) (v (publicmatrixbytes_len a)) (as_seq h1 b_bytes) (as_seq h1 pk); crypto_kem_sk a s pk sk

false

Hacl.Impl.Frodo.KEM.KeyGen.fst

Hacl.Impl.Frodo.KEM.KeyGen.crypto_kem_sk1

val crypto_kem_sk1: a:FP.frodo_alg -> s:lbytes (crypto_bytes a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a -! bytes_pkhash a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h s /\ disjoint pk sk /\ disjoint sk s) (ensures fun h0 _ h1 -> modifies (loc sk) h0 h1 /\ (let s_bytes = LSeq.sub (as_seq h0 sk) (v (crypto_bytes a) + v (crypto_publickeybytes a)) (v (secretmatrixbytes_len a)) in as_seq h1 sk == LSeq.concat (LSeq.concat (as_seq h0 s) (as_seq h0 pk)) s_bytes))

let crypto_kem_sk1 a s pk sk = let h1 = ST.get () in FP.expand_crypto_secretkeybytes a; let s_pk_len = crypto_bytes a +! crypto_publickeybytes a in [@inline_let] let sm_len = secretmatrixbytes_len a in let slen1 = crypto_secretkeybytes a -! bytes_pkhash a in let s_bytes = sub sk s_pk_len sm_len in update_sub sk 0ul (crypto_bytes a) s; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); update_sub sk (crypto_bytes a) (crypto_publickeybytes a) pk; let h3 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h3 sk) 0 (v (crypto_bytes a))) (as_seq h1 s); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v (crypto_bytes a)) (v (crypto_publickeybytes a))) (as_seq h1 pk); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); LSeq.lemma_concat3 (v (crypto_bytes a)) (as_seq h1 s) (v (crypto_publickeybytes a)) (as_seq h1 pk) (v sm_len) (as_seq h1 s_bytes) (as_seq h3 sk)

{ "file_name": "code/frodo/Hacl.Impl.Frodo.KEM.KeyGen.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 18, "end_line": 215, "start_col": 0, "start_line": 194 }

module Hacl.Impl.Frodo.KEM.KeyGen open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open LowStar.Buffer open Lib.IntTypes open Lib.Buffer open Hacl.Impl.Matrix open Hacl.Impl.Frodo.Params open Hacl.Impl.Frodo.KEM open Hacl.Impl.Frodo.Pack open Hacl.Impl.Frodo.Sample open Hacl.Frodo.Random module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module M = Spec.Matrix module FP = Spec.Frodo.Params module S = Spec.Frodo.KEM.KeyGen #set-options "--z3rlimit 100 --fuel 0 --ifuel 0" inline_for_extraction noextract val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len)) let frodo_shake_r a c seed_se output_len r = push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame () inline_for_extraction noextract val frodo_mul_add_as_plus_e: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_a /\ live h s_matrix /\ live h e_matrix /\ live h b_matrix /\ disjoint b_matrix seed_a /\ disjoint b_matrix e_matrix /\ disjoint b_matrix s_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b_matrix) h0 h1 /\ as_matrix h1 b_matrix == S.frodo_mul_add_as_plus_e a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix = FP.params_n_sqr a; push_frame(); let a_matrix = matrix_create (params_n a) (params_n a) in frodo_gen_matrix gen_a (params_n a) seed_a a_matrix; matrix_mul_s a_matrix s_matrix b_matrix; matrix_add b_matrix e_matrix; pop_frame() inline_for_extraction noextract val frodo_mul_add_as_plus_e_pack: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b:lbytes (publicmatrixbytes_len a) -> Stack unit (requires fun h -> live h seed_a /\ live h b /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_a b /\ disjoint b s_matrix /\ disjoint b e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b) h0 h1 /\ as_seq h1 b == S.frodo_mul_add_as_plus_e_pack a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix)) let frodo_mul_add_as_plus_e_pack a gen_a seed_a s_matrix e_matrix b = push_frame (); let b_matrix = matrix_create (params_n a) params_nbar in frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix; frodo_pack (params_logq a) b_matrix b; pop_frame () inline_for_extraction noextract val get_s_e_matrices: a:FP.frodo_alg -> seed_se:lbytes (crypto_bytes a) -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_se /\ live h s_matrix /\ live h e_matrix /\ disjoint seed_se s_matrix /\ disjoint seed_se e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1 /\ (as_matrix h1 s_matrix, as_matrix h1 e_matrix) == S.get_s_e_matrices a (as_seq h0 seed_se)) let get_s_e_matrices a seed_se s_matrix e_matrix = push_frame (); [@inline_let] let s_bytes_len = secretmatrixbytes_len a in let r = create (2ul *! s_bytes_len) (u8 0) in frodo_shake_r a (u8 0x5f) seed_se (2ul *! s_bytes_len) r; frodo_sample_matrix a (params_n a) params_nbar (sub r 0ul s_bytes_len) s_matrix; frodo_sample_matrix a (params_n a) params_nbar (sub r s_bytes_len s_bytes_len) e_matrix; pop_frame () inline_for_extraction noextract val clear_matrix2: a:FP.frodo_alg -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h s_matrix /\ live h e_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc s_matrix |+| loc e_matrix) h0 h1) let clear_matrix2 a s_matrix e_matrix = clear_matrix s_matrix; clear_matrix e_matrix inline_for_extraction noextract val frodo_mul_add_as_plus_e_pack_shake: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> seed_se:lbytes (crypto_bytes a) -> b:lbytes (publicmatrixbytes_len a) -> s:lbytes (secretmatrixbytes_len a) -> Stack unit (requires fun h -> live h seed_a /\ live h seed_se /\ live h s /\ live h b /\ disjoint b s /\ disjoint seed_a b /\ disjoint seed_a s /\ disjoint seed_se b /\ disjoint seed_se s) (ensures fun h0 _ h1 -> modifies (loc s |+| loc b) h0 h1 /\ (as_seq h1 b, as_seq h1 s) == S.frodo_mul_add_as_plus_e_pack_shake a gen_a (as_seq h0 seed_a) (as_seq h0 seed_se)) let frodo_mul_add_as_plus_e_pack_shake a gen_a seed_a seed_se b s = push_frame (); let s_matrix = matrix_create (params_n a) params_nbar in let e_matrix = matrix_create (params_n a) params_nbar in get_s_e_matrices a seed_se s_matrix e_matrix; frodo_mul_add_as_plus_e_pack a gen_a seed_a s_matrix e_matrix b; matrix_to_lbytes s_matrix s; clear_matrix2 a s_matrix e_matrix; pop_frame () inline_for_extraction noextract val crypto_kem_sk1: a:FP.frodo_alg -> s:lbytes (crypto_bytes a) -> pk:lbytes (crypto_publickeybytes a) -> sk:lbytes (crypto_secretkeybytes a -! bytes_pkhash a) -> Stack unit (requires fun h -> live h pk /\ live h sk /\ live h s /\ disjoint pk sk /\ disjoint sk s) (ensures fun h0 _ h1 -> modifies (loc sk) h0 h1 /\ (let s_bytes = LSeq.sub (as_seq h0 sk) (v (crypto_bytes a) + v (crypto_publickeybytes a)) (v (secretmatrixbytes_len a)) in as_seq h1 sk == LSeq.concat (LSeq.concat (as_seq h0 s) (as_seq h0 pk)) s_bytes))

{ "checked_file": "/", "dependencies": [ "Spec.Matrix.fst.checked", "Spec.Frodo.Params.fst.checked", "Spec.Frodo.KEM.KeyGen.fst.checked", "prims.fst.checked", "LowStar.Buffer.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Impl.Matrix.fst.checked", "Hacl.Impl.Frodo.Sample.fst.checked", "Hacl.Impl.Frodo.Params.fst.checked", "Hacl.Impl.Frodo.Pack.fst.checked", "Hacl.Impl.Frodo.KEM.fst.checked", "Hacl.Frodo.Random.fsti.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": false, "source_file": "Hacl.Impl.Frodo.KEM.KeyGen.fst" }

[ { "abbrev": true, "full_module": "Spec.Frodo.KEM.KeyGen", "short_module": "S" }, { "abbrev": true, "full_module": "Spec.Frodo.Params", "short_module": "FP" }, { "abbrev": true, "full_module": "Spec.Matrix", "short_module": "M" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Hacl.Frodo.Random", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Sample", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Pack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Params", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Matrix", "short_module": null }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "LowStar.Buffer", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.Frodo.Params.frodo_alg -> s: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_bytes a) -> pk: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_publickeybytes a) -> sk: Hacl.Impl.Matrix.lbytes (Hacl.Impl.Frodo.Params.crypto_secretkeybytes a -! Hacl.Impl.Frodo.Params.bytes_pkhash a) -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Spec.Frodo.Params.frodo_alg", "Hacl.Impl.Matrix.lbytes", "Hacl.Impl.Frodo.Params.crypto_bytes", "Hacl.Impl.Frodo.Params.crypto_publickeybytes", "Lib.IntTypes.op_Subtraction_Bang", "Lib.IntTypes.U32", "Lib.IntTypes.PUB", "Hacl.Impl.Frodo.Params.crypto_secretkeybytes", "Hacl.Impl.Frodo.Params.bytes_pkhash", "Lib.Sequence.lemma_concat3", "Lib.IntTypes.uint8", "Lib.IntTypes.v", "Lib.Buffer.as_seq", "Lib.Buffer.MUT", "Prims.unit", "Lib.Sequence.eq_intro", "Lib.Sequence.sub", "FStar.Monotonic.HyperStack.mem", "FStar.HyperStack.ST.get", "Lib.Buffer.update_sub", "FStar.UInt32.__uint_to_t", "Lib.Buffer.lbuffer_t", "Lib.IntTypes.int_t", "Lib.IntTypes.U8", "Lib.IntTypes.SEC", "Lib.Buffer.sub", "Hacl.Impl.Frodo.Params.secretmatrixbytes_len", "Lib.IntTypes.op_Plus_Bang", "Spec.Frodo.Params.expand_crypto_secretkeybytes" ]

[]

false

true

false

let crypto_kem_sk1 a s pk sk =

let h1 = ST.get () in FP.expand_crypto_secretkeybytes a; let s_pk_len = crypto_bytes a +! crypto_publickeybytes a in [@@ inline_let ]let sm_len = secretmatrixbytes_len a in let slen1 = crypto_secretkeybytes a -! bytes_pkhash a in let s_bytes = sub sk s_pk_len sm_len in update_sub sk 0ul (crypto_bytes a) s; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); update_sub sk (crypto_bytes a) (crypto_publickeybytes a) pk; let h3 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h3 sk) 0 (v (crypto_bytes a))) (as_seq h1 s); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v (crypto_bytes a)) (v (crypto_publickeybytes a))) (as_seq h1 pk); LSeq.eq_intro (LSeq.sub (as_seq h3 sk) (v s_pk_len) (v sm_len)) (as_seq h1 s_bytes); LSeq.lemma_concat3 (v (crypto_bytes a)) (as_seq h1 s) (v (crypto_publickeybytes a)) (as_seq h1 pk) (v sm_len) (as_seq h1 s_bytes) (as_seq h3 sk)

false

Hacl.Impl.Frodo.KEM.KeyGen.fst

Hacl.Impl.Frodo.KEM.KeyGen.frodo_mul_add_as_plus_e

val frodo_mul_add_as_plus_e: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_a /\ live h s_matrix /\ live h e_matrix /\ live h b_matrix /\ disjoint b_matrix seed_a /\ disjoint b_matrix e_matrix /\ disjoint b_matrix s_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b_matrix) h0 h1 /\ as_matrix h1 b_matrix == S.frodo_mul_add_as_plus_e a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix))

let frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix = FP.params_n_sqr a; push_frame(); let a_matrix = matrix_create (params_n a) (params_n a) in frodo_gen_matrix gen_a (params_n a) seed_a a_matrix; matrix_mul_s a_matrix s_matrix b_matrix; matrix_add b_matrix e_matrix; pop_frame()

{ "file_name": "code/frodo/Hacl.Impl.Frodo.KEM.KeyGen.fst", "git_rev": "eb1badfa34c70b0bbe0fe24fe0f49fb1295c7872", "git_url": "https://github.com/project-everest/hacl-star.git", "project_name": "hacl-star" }

{ "end_col": 13, "end_line": 80, "start_col": 0, "start_line": 73 }

module Hacl.Impl.Frodo.KEM.KeyGen open FStar.HyperStack open FStar.HyperStack.ST open FStar.Mul open LowStar.Buffer open Lib.IntTypes open Lib.Buffer open Hacl.Impl.Matrix open Hacl.Impl.Frodo.Params open Hacl.Impl.Frodo.KEM open Hacl.Impl.Frodo.Pack open Hacl.Impl.Frodo.Sample open Hacl.Frodo.Random module ST = FStar.HyperStack.ST module LSeq = Lib.Sequence module M = Spec.Matrix module FP = Spec.Frodo.Params module S = Spec.Frodo.KEM.KeyGen #set-options "--z3rlimit 100 --fuel 0 --ifuel 0" inline_for_extraction noextract val frodo_shake_r: a:FP.frodo_alg -> c:uint8 -> seed_se:lbytes (crypto_bytes a) -> output_len:size_t -> r:lbytes output_len -> Stack unit (requires fun h -> live h seed_se /\ live h r /\ disjoint seed_se r) (ensures fun h0 _ h1 -> modifies (loc r) h0 h1 /\ as_seq h1 r == S.frodo_shake_r a c (as_seq h0 seed_se) (v output_len)) let frodo_shake_r a c seed_se output_len r = push_frame (); let h0 = ST.get () in let shake_input_seed_se = create (1ul +! crypto_bytes a) (u8 0) in shake_input_seed_se.(0ul) <- c; update_sub shake_input_seed_se 1ul (crypto_bytes a) seed_se; let h2 = ST.get () in LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 0 1) (LSeq.create 1 c); LSeq.eq_intro (LSeq.sub (as_seq h2 shake_input_seed_se) 1 (v (crypto_bytes a))) (as_seq h0 seed_se); LSeq.eq_intro (LSeq.concat (LSeq.create 1 c) (as_seq h0 seed_se)) (as_seq h2 shake_input_seed_se); frodo_shake a (1ul +! crypto_bytes a) shake_input_seed_se output_len r; clear_words_u8 shake_input_seed_se; pop_frame () inline_for_extraction noextract val frodo_mul_add_as_plus_e: a:FP.frodo_alg -> gen_a:FP.frodo_gen_a{is_supported gen_a} -> seed_a:lbytes bytes_seed_a -> s_matrix:matrix_t (params_n a) params_nbar -> e_matrix:matrix_t (params_n a) params_nbar -> b_matrix:matrix_t (params_n a) params_nbar -> Stack unit (requires fun h -> live h seed_a /\ live h s_matrix /\ live h e_matrix /\ live h b_matrix /\ disjoint b_matrix seed_a /\ disjoint b_matrix e_matrix /\ disjoint b_matrix s_matrix /\ disjoint s_matrix e_matrix) (ensures fun h0 _ h1 -> modifies (loc b_matrix) h0 h1 /\ as_matrix h1 b_matrix == S.frodo_mul_add_as_plus_e a gen_a (as_seq h0 seed_a) (as_matrix h0 s_matrix) (as_matrix h0 e_matrix))

{ "checked_file": "/", "dependencies": [ "Spec.Matrix.fst.checked", "Spec.Frodo.Params.fst.checked", "Spec.Frodo.KEM.KeyGen.fst.checked", "prims.fst.checked", "LowStar.Buffer.fst.checked", "Lib.Sequence.fsti.checked", "Lib.IntTypes.fsti.checked", "Lib.Buffer.fsti.checked", "Hacl.Impl.Matrix.fst.checked", "Hacl.Impl.Frodo.Sample.fst.checked", "Hacl.Impl.Frodo.Params.fst.checked", "Hacl.Impl.Frodo.Pack.fst.checked", "Hacl.Impl.Frodo.KEM.fst.checked", "Hacl.Frodo.Random.fsti.checked", "FStar.UInt32.fsti.checked", "FStar.Pervasives.Native.fst.checked", "FStar.Pervasives.fsti.checked", "FStar.Mul.fst.checked", "FStar.HyperStack.ST.fsti.checked", "FStar.HyperStack.fst.checked" ], "interface_file": false, "source_file": "Hacl.Impl.Frodo.KEM.KeyGen.fst" }

[ { "abbrev": true, "full_module": "Spec.Frodo.KEM.KeyGen", "short_module": "S" }, { "abbrev": true, "full_module": "Spec.Frodo.Params", "short_module": "FP" }, { "abbrev": true, "full_module": "Spec.Matrix", "short_module": "M" }, { "abbrev": true, "full_module": "Lib.Sequence", "short_module": "LSeq" }, { "abbrev": true, "full_module": "FStar.HyperStack.ST", "short_module": "ST" }, { "abbrev": false, "full_module": "Hacl.Frodo.Random", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Sample", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Pack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.Params", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Matrix", "short_module": null }, { "abbrev": false, "full_module": "Lib.Buffer", "short_module": null }, { "abbrev": false, "full_module": "Lib.IntTypes", "short_module": null }, { "abbrev": false, "full_module": "LowStar.Buffer", "short_module": null }, { "abbrev": false, "full_module": "FStar.Mul", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack.ST", "short_module": null }, { "abbrev": false, "full_module": "FStar.HyperStack", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "Hacl.Impl.Frodo.KEM", "short_module": null }, { "abbrev": false, "full_module": "FStar.Pervasives", "short_module": null }, { "abbrev": false, "full_module": "Prims", "short_module": null }, { "abbrev": false, "full_module": "FStar", "short_module": null } ]

{ "detail_errors": false, "detail_hint_replay": false, "initial_fuel": 0, "initial_ifuel": 0, "max_fuel": 0, "max_ifuel": 0, "no_plugins": false, "no_smt": false, "no_tactics": false, "quake_hi": 1, "quake_keep": false, "quake_lo": 1, "retry": false, "reuse_hint_for": null, "smtencoding_elim_box": false, "smtencoding_l_arith_repr": "boxwrap", "smtencoding_nl_arith_repr": "boxwrap", "smtencoding_valid_elim": false, "smtencoding_valid_intro": true, "tcnorm": true, "trivial_pre_for_unannotated_effectful_fns": false, "z3cliopt": [], "z3refresh": false, "z3rlimit": 100, "z3rlimit_factor": 1, "z3seed": 0, "z3smtopt": [], "z3version": "4.8.5" }

false

a: Spec.Frodo.Params.frodo_alg -> gen_a: Spec.Frodo.Params.frodo_gen_a{Hacl.Impl.Frodo.Params.is_supported gen_a} -> seed_a: Hacl.Impl.Matrix.lbytes Hacl.Impl.Frodo.Params.bytes_seed_a -> s_matrix: Hacl.Impl.Matrix.matrix_t (Hacl.Impl.Frodo.Params.params_n a) Hacl.Impl.Frodo.Params.params_nbar -> e_matrix: Hacl.Impl.Matrix.matrix_t (Hacl.Impl.Frodo.Params.params_n a) Hacl.Impl.Frodo.Params.params_nbar -> b_matrix: Hacl.Impl.Matrix.matrix_t (Hacl.Impl.Frodo.Params.params_n a) Hacl.Impl.Frodo.Params.params_nbar -> FStar.HyperStack.ST.Stack Prims.unit

FStar.HyperStack.ST.Stack

[]

[ "Spec.Frodo.Params.frodo_alg", "Spec.Frodo.Params.frodo_gen_a", "Prims.b2t", "Hacl.Impl.Frodo.Params.is_supported", "Hacl.Impl.Matrix.lbytes", "Hacl.Impl.Frodo.Params.bytes_seed_a", "Hacl.Impl.Matrix.matrix_t", "Hacl.Impl.Frodo.Params.params_n", "Hacl.Impl.Frodo.Params.params_nbar", "FStar.HyperStack.ST.pop_frame", "Prims.unit", "Hacl.Impl.Matrix.matrix_add", "Hacl.Impl.Matrix.matrix_mul_s", "Hacl.Impl.Frodo.Params.frodo_gen_matrix", "Lib.Buffer.lbuffer_t", "Lib.Buffer.MUT", "Lib.IntTypes.int_t", "Lib.IntTypes.U16", "Lib.IntTypes.SEC", "Lib.IntTypes.mul", "Lib.IntTypes.U32", "Lib.IntTypes.PUB", "Hacl.Impl.Matrix.matrix_create", "FStar.HyperStack.ST.push_frame", "Spec.Frodo.Params.params_n_sqr" ]

[]

false

true

false

let frodo_mul_add_as_plus_e a gen_a seed_a s_matrix e_matrix b_matrix =

FP.params_n_sqr a; push_frame (); let a_matrix = matrix_create (params_n a) (params_n a) in frodo_gen_matrix gen_a (params_n a) seed_a a_matrix; matrix_mul_s a_matrix s_matrix b_matrix; matrix_add b_matrix e_matrix; pop_frame ()

false