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101
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stringlengths 1
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| start
stringlengths 6
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https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamMul.lean | Etch.Verification.Stream.mul_spec | [156, 1] | [167, 9] | rw [Stream.eval_valid _ ⟨_, hv⟩, ih _ ((a.mul b).next_wf ⟨q, hv⟩), next_eval_mul_eq _ _ ⟨_, hv⟩,
mul_eval₀_spec _ _ a.strictMono b.strictMono ⟨_, hv⟩] | case pos
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
⊢ eval (mul a b) q = eval a q.1 * eval b q.2 | case pos
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
⊢ Finsupp.filter (fun i => (i, false) < toOrder (mul a b) { val := q, property := hv })
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) +
Finsupp.filter (fun i => toOrder (mul a b) { val := q, property := hv } ≤ (i, false))
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) =
eval a q.1 * eval b q.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
⊢ eval (mul a b) q = eval a q.1 * eval b q.2
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamMul.lean | Etch.Verification.Stream.mul_spec | [156, 1] | [167, 9] | convert Finsupp.filter_pos_add_filter_neg (α := ι) (M := α) .. | case pos
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
⊢ Finsupp.filter (fun i => (i, false) < toOrder (mul a b) { val := q, property := hv })
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) +
Finsupp.filter (fun i => toOrder (mul a b) { val := q, property := hv } ≤ (i, false))
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) =
eval a q.1 * eval b q.2 | case h.e'_2.h.e'_6.h.e'_4.h.a
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
x✝ : ι
⊢ toOrder (mul a b) { val := q, property := hv } ≤ (x✝, false) ↔
¬(x✝, false) < toOrder (mul a b) { val := q, property := hv } | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
⊢ Finsupp.filter (fun i => (i, false) < toOrder (mul a b) { val := q, property := hv })
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) +
Finsupp.filter (fun i => toOrder (mul a b) { val := q, property := hv } ≤ (i, false))
(eval a ↑(mul.valid.fst { val := q, property := hv }) * eval b ↑(mul.valid.snd { val := q, property := hv })) =
eval a q.1 * eval b q.2
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamMul.lean | Etch.Verification.Stream.mul_spec | [156, 1] | [167, 9] | simp | case h.e'_2.h.e'_6.h.e'_4.h.a
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
x✝ : ι
⊢ toOrder (mul a b) { val := q, property := hv } ≤ (x✝, false) ↔
¬(x✝, false) < toOrder (mul a b) { val := q, property := hv } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.h.e'_4.h.a
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : (mul a b).σ
ih : ∀ (y : (mul a b).σ), WfRel (mul a b) y q → eval (mul a b) y = eval a y.1 * eval b y.2
hv : (mul a b).valid q = true
x✝ : ι
⊢ toOrder (mul a b) { val := q, property := hv } ≤ (x✝, false) ↔
¬(x✝, false) < toOrder (mul a b) { val := q, property := hv }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamMul.lean | Etch.Verification.Stream.mul_seek_spec | [169, 1] | [173, 58] | simp only [Finsupp.mul_apply, mul_spec] | ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : { q // (mul a b).valid q = true }
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
⊢ (eval (mul a b) ((mul a b).seek q i)) j = (eval (mul a b) ↑q) j | ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : { q // (mul a b).valid q = true }
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
⊢ (eval a ((mul a b).seek q i).1) j * (eval b ((mul a b).seek q i).2) j = (eval a (↑q).1) j * (eval b (↑q).2) j | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : { q // (mul a b).valid q = true }
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
⊢ (eval (mul a b) ((mul a b).seek q i)) j = (eval (mul a b) ↑q) j
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamMul.lean | Etch.Verification.Stream.mul_seek_spec | [169, 1] | [173, 58] | congr 1 <;> dsimp <;> rw [IsLawful.seek_spec] <;> aesop | ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : { q // (mul a b).valid q = true }
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
⊢ (eval a ((mul a b).seek q i).1) j * (eval b ((mul a b).seek q i).2) j = (eval a (↑q).1) j * (eval b (↑q).2) j | ι : Type
inst : LinearOrder ι
α : Type u
inst_1 : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst_2 : IsStrictLawful a
inst_3 : IsStrictLawful b
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
val : (mul a b).σ
property : (mul a b).valid val = true
⊢ (eval a (a.seek (mul.valid.fst { val := val, property := property }) i)) j = (eval a val.1) j
ι : Type
inst : LinearOrder ι
α : Type u
inst_1 : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst_2 : IsStrictLawful a
inst_3 : IsStrictLawful b
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
val : (mul a b).σ
property : (mul a b).valid val = true
⊢ (eval b (b.seek (mul.valid.snd { val := val, property := property }) i)) j = (eval b val.2) j | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
inst✝³ : LinearOrder ι
α : Type u
inst✝² : NonUnitalNonAssocSemiring α
a b : Stream ι α
inst✝¹ : IsStrictLawful a
inst✝ : IsStrictLawful b
q : { q // (mul a b).valid q = true }
i : Lex (ι × Bool)
j : ι
h : i ≤ (j, false)
⊢ (eval a ((mul a b).seek q i).1) j * (eval b ((mul a b).seek q i).2) j = (eval a (↑q).1) j * (eval b (↑q).2) j
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.min?_mem | [16, 1] | [19, 54] | rw [min?_eq_toList_head?] at h | α : Type u_1
a : α
t : RBNode α
h : RBNode.min? t = some a
⊢ a ∈ t | α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
t : RBNode α
h : RBNode.min? t = some a
⊢ a ∈ t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.min?_mem | [16, 1] | [19, 54] | rw [← mem_toList] | α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ t | α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ toList t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.min?_mem | [16, 1] | [19, 54] | revert h | α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ toList t | α : Type u_1
a : α
t : RBNode α
⊢ List.head? (toList t) = some a → a ∈ toList t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
t : RBNode α
h : List.head? (toList t) = some a
⊢ a ∈ toList t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.min?_mem | [16, 1] | [19, 54] | cases toList t <;> rintro ⟨⟩ | α : Type u_1
a : α
t : RBNode α
⊢ List.head? (toList t) = some a → a ∈ toList t | case cons.refl
α : Type u_1
a : α
t : RBNode α
tail✝ : List α
⊢ a ∈ a :: tail✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
t : RBNode α
⊢ List.head? (toList t) = some a → a ∈ toList t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.min?_mem | [16, 1] | [19, 54] | constructor | case cons.refl
α : Type u_1
a : α
t : RBNode α
tail✝ : List α
⊢ a ∈ a :: tail✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl
α : Type u_1
a : α
t : RBNode α
tail✝ : List α
⊢ a ∈ a :: tail✝
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | rw [min?_eq_toList_head?] at h | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : RBNode.min? t = some a
x : α
hx : x ∈ t
⊢ cmp a x ≠ Ordering.gt | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ t
⊢ cmp a x ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : RBNode.min? t = some a
x : α
hx : x ∈ t
⊢ cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | rw [← mem_toList] at hx | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ t
⊢ cmp a x ≠ Ordering.gt | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
⊢ cmp a x ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ t
⊢ cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | have := ht.toList_sorted | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
⊢ cmp a x ≠ Ordering.gt | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
this : List.Pairwise (cmpLT cmp) (toList t)
⊢ cmp a x ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
⊢ cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | revert h hx this | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
this : List.Pairwise (cmpLT cmp) (toList t)
⊢ cmp a x ≠ Ordering.gt | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
⊢ List.head? (toList t) = some a → x ∈ toList t → List.Pairwise (cmpLT cmp) (toList t) → cmp a x ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : List.head? (toList t) = some a
x : α
hx : x ∈ toList t
this : List.Pairwise (cmpLT cmp) (toList t)
⊢ cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | cases toList t <;> rintro ⟨⟩ (_ | ⟨_, hx⟩) (_ | ⟨h1,h2⟩) | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
⊢ List.head? (toList t) = some a → x ∈ toList t → List.Pairwise (cmpLT cmp) (toList t) → cmp a x ≠ Ordering.gt | case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a a ≠ Ordering.gt
case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a x ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
⊢ List.head? (toList t) = some a → x ∈ toList t → List.Pairwise (cmpLT cmp) (toList t) → cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | rw [OrientedCmp.cmp_refl (cmp := cmp)] | case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a a ≠ Ordering.gt | case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.eq ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a a ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | decide | case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.eq ≠ Ordering.gt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl.head.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
tail✝ : List α
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.eq ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | rw [(h1 _ hx).1] | case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a x ≠ Ordering.gt | case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.lt ≠ Ordering.gt | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ cmp a x ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.min?_le | [21, 1] | [28, 29] | decide | case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.lt ≠ Ordering.gt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl.tail.cons
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
x : α
tail✝ : List α
hx : List.Mem x tail✝
h2 : List.Pairwise (cmpLT cmp) tail✝
h1 : ∀ (a' : α), a' ∈ tail✝ → cmpLT cmp a a'
⊢ Ordering.lt ≠ Ordering.gt
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.max?_mem | [30, 1] | [31, 50] | simpa using min?_mem ((min?_reverse _).trans h) | α : Type u_1
a : α
t : RBNode α
h : RBNode.max? t = some a
⊢ a ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
t : RBNode α
h : RBNode.max? t = some a
⊢ a ∈ t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBNode.Ordered.le_max? | [33, 1] | [35, 70] | simpa using hx | α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : RBNode.max? t = some a
x : α
hx : x ∈ t
⊢ x ∈ reverse t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
a : α
t : RBNode α
inst✝ : TransCmp cmp
ht : Ordered cmp t
h : RBNode.max? t = some a
x : α
hx : x ∈ t
⊢ x ∈ reverse t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.isEmpty_iff_toList_eq_nil | [48, 1] | [49, 83] | obtain ⟨⟨⟩, _⟩ := t <;> simp [toList, isEmpty] | α : Type u_1
cmp : α → α → Ordering
t : RBSet α cmp
⊢ isEmpty t = true ↔ toList t = [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
t : RBSet α cmp
⊢ isEmpty t = true ↔ toList t = []
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_some | [60, 1] | [62, 53] | simp [mem_toList] | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
y : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
⊢ y ∈ t.val ∧ cut y = Ordering.eq ↔ y ∈ toList t ∧ cut y = Ordering.eq | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
y : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
⊢ y ∈ t.val ∧ cut y = Ordering.eq ↔ y ∈ toList t ∧ cut y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | refine Option.ext fun u =>
findP?_some.trans <| .trans (and_congr_left fun h' => ?_) findP?_some.symm | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
⊢ findP? (insert t v) cut = findP? t cut | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ u ∈ toList (insert t v) ↔ u ∈ toList t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
⊢ findP? (insert t v) cut = findP? t cut
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | rw [mem_toList_insert, or_iff_left, and_iff_left] | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ u ∈ toList (insert t v) ↔ u ∈ toList t | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ find? t v ≠ some u
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ ¬u = v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ u ∈ toList (insert t v) ↔ u ∈ toList t
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | exact mt (fun h => by rwa [IsCut.congr (cut := cut) (find?_some_eq_eq h)]) h | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ find? t v ≠ some u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ find? t v ≠ some u
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | rwa [IsCut.congr (cut := cut) (find?_some_eq_eq h)] | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h✝ : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
h : find? t v = some u
⊢ cut v = Ordering.eq | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h✝ : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
h : find? t v = some u
⊢ cut v = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | rintro rfl | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ ¬u = v | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
u : α
h' : cut u = Ordering.eq
h : cut u ≠ Ordering.eq
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ ¬u = v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert_of_ne | [71, 1] | [77, 30] | contradiction | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
u : α
h' : cut u = Ordering.eq
h : cut u ≠ Ordering.eq
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
u : α
h' : cut u = Ordering.eq
h : cut u ≠ Ordering.eq
⊢ False
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.findP?_insert | [79, 1] | [81, 79] | split <;> [exact findP?_insert_of_eq t ‹_›; exact findP?_insert_of_ne t ‹_›] | α : Type u_1
cmp : α → α → Ordering
inst✝¹ : TransCmp cmp
t : RBSet α cmp
v : α
cut : α → Ordering
inst✝ : IsStrictCut cmp cut
⊢ findP? (insert t v) cut = if cut v = Ordering.eq then some v else findP? t cut | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
inst✝¹ : TransCmp cmp
t : RBSet α cmp
v : α
cut : α → Ordering
inst✝ : IsStrictCut cmp cut
⊢ findP? (insert t v) cut = if cut v = Ordering.eq then some v else findP? t cut
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.isEmpty_iff_eq_empty | [83, 1] | [88, 9] | obtain ⟨⟨⟩, _⟩ := t <;>
simp [isEmpty] | α : Type u_1
cmp : α → α → Ordering
t : RBSet α cmp
⊢ isEmpty t = true ↔ t = ∅ | case mk.nil
α : Type u_1
cmp : α → α → Ordering
property✝ : RBNode.WF cmp RBNode.nil
⊢ { val := RBNode.nil, property := property✝ } = ∅
case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ ¬{ val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
t : RBSet α cmp
⊢ isEmpty t = true ↔ t = ∅
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.isEmpty_iff_eq_empty | [83, 1] | [88, 9] | rfl | case mk.nil
α : Type u_1
cmp : α → α → Ordering
property✝ : RBNode.WF cmp RBNode.nil
⊢ { val := RBNode.nil, property := property✝ } = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.nil
α : Type u_1
cmp : α → α → Ordering
property✝ : RBNode.WF cmp RBNode.nil
⊢ { val := RBNode.nil, property := property✝ } = ∅
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.isEmpty_iff_eq_empty | [83, 1] | [88, 9] | change _ ≠ Subtype.mk _ _ | case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ ¬{ val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } = ∅ | case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ { val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } ≠ { val := RBNode.nil, property := ⋯ } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ ¬{ val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } = ∅
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBSet.isEmpty_iff_eq_empty | [83, 1] | [88, 9] | simp | case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ { val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } ≠ { val := RBNode.nil, property := ⋯ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.node
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
property✝ : RBNode.WF cmp (RBNode.node c✝ l✝ v✝ r✝)
⊢ { val := RBNode.node c✝ l✝ v✝ r✝, property := property✝ } ≠ { val := RBNode.nil, property := ⋯ }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some_mem_toList | [141, 1] | [144, 69] | obtain ⟨⟨y, v⟩, h', rfl⟩ := Option.map_eq_some'.1 h | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
v : β
t : RBMap α β cmp
h : find? t x = some v
⊢ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq | case intro.mk.intro
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
t : RBMap α β cmp
y : α
v : β
h' : findEntry? t x = some (y, v)
h : find? t x = some (y, v).snd
⊢ ∃ y_1, (y_1, (y, v).snd) ∈ toList t ∧ cmp x y_1 = Ordering.eq | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
v : β
t : RBMap α β cmp
h : find? t x = some v
⊢ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some_mem_toList | [141, 1] | [144, 69] | exact ⟨_, findEntry?_some_mem_toList h', findEntry?_some_eq_eq h'⟩ | case intro.mk.intro
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
t : RBMap α β cmp
y : α
v : β
h' : findEntry? t x = some (y, v)
h : find? t x = some (y, v).snd
⊢ ∃ y_1, (y_1, (y, v).snd) ∈ toList t ∧ cmp x y_1 = Ordering.eq | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
t : RBMap α β cmp
y : α
v : β
h' : findEntry? t x = some (y, v)
h : find? t x = some (y, v).snd
⊢ ∃ y_1, (y_1, (y, v).snd) ∈ toList t ∧ cmp x y_1 = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | simp [find?, findEntry?_some] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ find? t x = some v ↔ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) ↔ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ find? t x = some v ↔ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | constructor | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) ↔ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq | case mp
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) → ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq
case mpr
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq) → ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) ↔ ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | rintro ⟨_, h, rfl⟩ | case mp
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) → ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq | case mp.intro.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
w✝ : α × β
h : w✝ ∈ toList t ∧ cmp x w✝.fst = Ordering.eq
⊢ ∃ y, (y, w✝.snd) ∈ toList t ∧ cmp x y = Ordering.eq | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v) → ∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | exact ⟨_, h⟩ | case mp.intro.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
w✝ : α × β
h : w✝ ∈ toList t ∧ cmp x w✝.fst = Ordering.eq
⊢ ∃ y, (y, w✝.snd) ∈ toList t ∧ cmp x y = Ordering.eq | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
w✝ : α × β
h : w✝ ∈ toList t ∧ cmp x w✝.fst = Ordering.eq
⊢ ∃ y, (y, w✝.snd) ∈ toList t ∧ cmp x y = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | rintro ⟨b, h⟩ | case mpr
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq) → ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v | case mpr.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
b : α
h : (b, v) ∈ toList t ∧ cmp x b = Ordering.eq
⊢ ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ (∃ y, (y, v) ∈ toList t ∧ cmp x y = Ordering.eq) → ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_some | [165, 1] | [169, 37] | exact ⟨_, h, rfl⟩ | case mpr.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
b : α
h : (b, v) ∈ toList t ∧ cmp x b = Ordering.eq
⊢ ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
x : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
b : α
h : (b, v) ∈ toList t ∧ cmp x b = Ordering.eq
⊢ ∃ a, (a ∈ toList t ∧ cmp x a.fst = Ordering.eq) ∧ a.snd = v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.contains_iff_find? | [174, 1] | [176, 63] | simp [contains_iff_findEntry?, find?, and_comm, exists_comm] | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
t : RBMap α β cmp
⊢ contains t x = true ↔ ∃ v, find? t x = some v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
x : α
t : RBMap α β cmp
⊢ contains t x = true ↔ ∃ v, find? t x = some v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.findEntry?_congr | [190, 1] | [191, 90] | simp [findEntry?, TransCmp.cmp_congr_left' h] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k₁ k₂ : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k₁ k₂ = Ordering.eq
⊢ findEntry? t k₁ = findEntry? t k₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k₁ k₂ : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k₁ k₂ = Ordering.eq
⊢ findEntry? t k₁ = findEntry? t k₂
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_congr | [193, 1] | [194, 69] | simp [find?, findEntry?_congr _ h] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k₁ k₂ : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k₁ k₂ = Ordering.eq
⊢ find? t k₁ = find? t k₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k₁ k₂ : α
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k₁ k₂ = Ordering.eq
⊢ find? t k₁ = find? t k₂
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert_of_eq | [199, 1] | [200, 88] | rw [find?, findEntry?_insert_of_eq _ h] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k = Ordering.eq
⊢ find? (insert t k v) k' = some v | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k = Ordering.eq
⊢ Option.map (fun x => x.snd) (some (k, v)) = some v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k = Ordering.eq
⊢ find? (insert t k v) k' = some v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert_of_eq | [199, 1] | [200, 88] | rfl | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k = Ordering.eq
⊢ Option.map (fun x => x.snd) (some (k, v)) = some v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k = Ordering.eq
⊢ Option.map (fun x => x.snd) (some (k, v)) = some v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert_of_ne | [205, 1] | [206, 89] | simp [find?, findEntry?_insert_of_ne _ h] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k ≠ Ordering.eq
⊢ find? (insert t k v) k' = find? t k' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k ≠ Ordering.eq
⊢ find? (insert t k v) k' = find? t k'
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert | [212, 1] | [214, 77] | split <;> [exact find?_insert_of_eq t ‹_›; exact find?_insert_of_ne t ‹_›] | α : Type u_1
cmp : α → α → Ordering
β : Type u_2
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
⊢ find? (insert t k v) k' = if cmp k' k = Ordering.eq then some v else find? t k' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
cmp : α → α → Ordering
β : Type u_2
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
⊢ find? (insert t k v) k' = if cmp k' k = Ordering.eq then some v else find? t k'
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert_self | [221, 9] | [222, 106] | rw [find?_insert_of_eq] | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ find? (insert t k v) k = some v | case h
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ cmp k k = Ordering.eq | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ find? (insert t k v) k = some v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.find?_insert_self | [221, 9] | [222, 106] | rw [OrientedCmp.cmp_refl (cmp := cmp)] | case h
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ cmp k k = Ordering.eq | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
⊢ cmp k k = Ordering.eq
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.findD_insert | [224, 1] | [226, 49] | simp only [findD, find?_insert] | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
d : β
⊢ findD (insert t k v) k' d = if cmp k' k = Ordering.eq then v else findD t k' d | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
d : β
⊢ Option.getD (if cmp k' k = Ordering.eq then some v else find? t k') d =
if cmp k' k = Ordering.eq then v else Option.getD (find? t k') d | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
d : β
⊢ findD (insert t k v) k' d = if cmp k' k = Ordering.eq then v else findD t k' d
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.findD_insert | [224, 1] | [226, 49] | split <;> rfl | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
d : β
⊢ Option.getD (if cmp k' k = Ordering.eq then some v else find? t k') d =
if cmp k' k = Ordering.eq then v else Option.getD (find? t k') d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
inst✝ : TransCmp cmp
t : RBMap α β cmp
k : α
v : β
k' : α
d : β
⊢ Option.getD (if cmp k' k = Ordering.eq then some v else find? t k') d =
if cmp k' k = Ordering.eq then v else Option.getD (find? t k') d
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.findD_insert_self | [228, 9] | [229, 52] | simp [findD] | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
d : β
⊢ findD (insert t k v) k d = v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
d : β
⊢ findD (insert t k v) k d = v
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/Verification/RBMap.lean | Std.RBMap.findD_insert_of_ne | [231, 1] | [232, 74] | simp [findD_insert, h] | α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k ≠ Ordering.eq
d : β
⊢ findD (insert t k v) k' d = findD t k' d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
cmp : α → α → Ordering
k' k : α
v : β
inst✝ : TransCmp cmp
t : RBMap α β cmp
h : cmp k' k ≠ Ordering.eq
d : β
⊢ findD (insert t k v) k' d = findD t k' d
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval₀_map | [34, 1] | [37, 21] | dsimp [Stream.eval₀] | ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
q : { q // s.valid q = true }
inst✝¹ : Zero α
inst✝ : Zero β
f : α → β
hf : f 0 = 0
⊢ eval₀ (map f s) q = Finsupp.mapRange f hf (eval₀ s q) | ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
q : { q // s.valid q = true }
inst✝¹ : Zero α
inst✝ : Zero β
f : α → β
hf : f 0 = 0
⊢ (if h₂ : s.ready q = true then fun₀ | s.index q => f (s.value { val := q, property := h₂ }) else 0) =
Finsupp.mapRange f hf
(if h₂ : s.ready q = true then fun₀ | s.index q => s.value { val := q, property := h₂ } else 0) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
q : { q // s.valid q = true }
inst✝¹ : Zero α
inst✝ : Zero β
f : α → β
hf : f 0 = 0
⊢ eval₀ (map f s) q = Finsupp.mapRange f hf (eval₀ s q)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval₀_map | [34, 1] | [37, 21] | split_ifs <;> simp | ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
q : { q // s.valid q = true }
inst✝¹ : Zero α
inst✝ : Zero β
f : α → β
hf : f 0 = 0
⊢ (if h₂ : s.ready q = true then fun₀ | s.index q => f (s.value { val := q, property := h₂ }) else 0) =
Finsupp.mapRange f hf
(if h₂ : s.ready q = true then fun₀ | s.index q => s.value { val := q, property := h₂ } else 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
q : { q // s.valid q = true }
inst✝¹ : Zero α
inst✝ : Zero β
f : α → β
hf : f 0 = 0
⊢ (if h₂ : s.ready q = true then fun₀ | s.index q => f (s.value { val := q, property := h₂ }) else 0) =
Finsupp.mapRange f hf
(if h₂ : s.ready q = true then fun₀ | s.index q => s.value { val := q, property := h₂ } else 0)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.map_zero | [47, 1] | [51, 10] | ext <;> try rfl | ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ map f 0 = 0 | case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ HEq (map f 0).value 0.value | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ map f 0 = 0
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.map_zero | [47, 1] | [51, 10] | rw [heq_iff_eq] | case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ HEq (map f 0).value 0.value | case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ (map f 0).value = 0.value | Please generate a tactic in lean4 to solve the state.
STATE:
case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ HEq (map f 0).value 0.value
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.map_zero | [47, 1] | [51, 10] | funext ⟨q, h⟩ | case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ (map f 0).value = 0.value | case value.h
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
q : { x // (map f 0).valid x = true }
h : (map f 0).ready q = true
⊢ (map f 0).value { val := q, property := h } = 0.value { val := q, property := h } | Please generate a tactic in lean4 to solve the state.
STATE:
case value
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ (map f 0).value = 0.value
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.map_zero | [47, 1] | [51, 10] | cases h | case value.h
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
q : { x // (map f 0).valid x = true }
h : (map f 0).ready q = true
⊢ (map f 0).value { val := q, property := h } = 0.value { val := q, property := h } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case value.h
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
q : { x // (map f 0).valid x = true }
h : (map f 0).ready q = true
⊢ (map f 0).value { val := q, property := h } = 0.value { val := q, property := h }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.map_zero | [47, 1] | [51, 10] | rfl | case ready
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ HEq (map f 0).ready 0.ready | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ready
ι : Type
α : Type u_2
a b s : Stream ι α
β : Type u_1
f : α → β
⊢ HEq (map f 0).ready 0.ready
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.ready'_val | [64, 1] | [65, 34] | simp [Stream.ready', x.prop] | ι : Type
α : Type u_1
a b s : Stream ι α
x : { q // s.valid q = true }
⊢ ready' s ↑x ↔ s.ready x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
x : { q // s.valid q = true }
⊢ ready' s ↑x ↔ s.ready x = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.coeLex_le_iff | [91, 1] | [92, 26] | simp [Prod.Lex.le_iff'] | ι : Type
α : Type ?u.4823
a b s : Stream ι α
inst✝ : Preorder ι
x y : StreamOrder ι
⊢ coeLex x ≤ coeLex y ↔ x ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type ?u.4823
a b s : Stream ι α
inst✝ : Preorder ι
x y : StreamOrder ι
⊢ coeLex x ≤ coeLex y ↔ x ≤ y
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.coeLex_lt_iff | [95, 1] | [96, 26] | simp [Prod.Lex.lt_iff'] | ι : Type
α : Type ?u.6775
a b s : Stream ι α
inst✝ : Preorder ι
x y : StreamOrder ι
⊢ coeLex x < coeLex y ↔ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type ?u.6775
a b s : Stream ι α
inst✝ : Preorder ι
x y : StreamOrder ι
⊢ coeLex x < coeLex y ↔ x < y
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.coeLex_injective | [98, 1] | [101, 9] | rw [Prod.mk.injEq .., Prod.mk.injEq ..] | ι : Type
α : Type ?u.7590
a b s : Stream ι α
x₁ : ι
y₁ : Bool
x₂ : ι
y₂ : Bool
⊢ coeLex (x₁, y₁) = coeLex (x₂, y₂) → (x₁, y₁) = (x₂, y₂) | ι : Type
α : Type ?u.7590
a b s : Stream ι α
x₁ : ι
y₁ : Bool
x₂ : ι
y₂ : Bool
⊢ ↑(x₁, y₁).1 = ↑(x₂, y₂).1 ∧ (x₁, y₁).2 = (x₂, y₂).2 → x₁ = x₂ ∧ y₁ = y₂ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type ?u.7590
a b s : Stream ι α
x₁ : ι
y₁ : Bool
x₂ : ι
y₂ : Bool
⊢ coeLex (x₁, y₁) = coeLex (x₂, y₂) → (x₁, y₁) = (x₂, y₂)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.coeLex_injective | [98, 1] | [101, 9] | simp | ι : Type
α : Type ?u.7590
a b s : Stream ι α
x₁ : ι
y₁ : Bool
x₂ : ι
y₂ : Bool
⊢ ↑(x₁, y₁).1 = ↑(x₂, y₂).1 ∧ (x₁, y₁).2 = (x₂, y₂).2 → x₁ = x₂ ∧ y₁ = y₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type ?u.7590
a b s : Stream ι α
x₁ : ι
y₁ : Bool
x₂ : ι
y₂ : Bool
⊢ ↑(x₁, y₁).1 = ↑(x₂, y₂).1 ∧ (x₁, y₁).2 = (x₂, y₂).2 → x₁ = x₂ ∧ y₁ = y₂
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.index'_invalid | [118, 1] | [119, 21] | simpa | ι : Type
α : Type u_1
a b s : Stream ι α
x : s.σ
h : s.valid x = false
⊢ ¬s.valid x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
x : s.σ
h : s.valid x = false
⊢ ¬s.valid x = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.toOrder'_val | [152, 1] | [153, 72] | simp [Stream.toOrder'] | ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ toOrder' s ↑q = (index' s ↑q, s.ready q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ toOrder' s ↑q = (index' s ↑q, s.ready q)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.coeLex_toOrder | [155, 1] | [157, 37] | simp [coeLex, Stream.toOrder'_val] | ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ toOrder' s ↑q = coeLex (toOrder s q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ toOrder' s ↑q = coeLex (toOrder s q)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.toOrder'_val_snd | [160, 1] | [161, 66] | simp [Stream.toOrder'_val] | ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ (toOrder' s ↑q).2 = s.ready q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
q : { q // s.valid q = true }
⊢ (toOrder' s ↑q).2 = s.ready q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.index'_lt_top_iff | [165, 1] | [170, 13] | rw [Stream.index'] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
⊢ index' s q < ⊤ ↔ s.valid q = true | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
⊢ (if h : s.valid q = true then ↑(s.index { val := q, property := h }) else ⊤) < ⊤ ↔ s.valid q = true | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
⊢ index' s q < ⊤ ↔ s.valid q = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.index'_lt_top_iff | [165, 1] | [170, 13] | split_ifs | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
⊢ (if h : s.valid q = true then ↑(s.index { val := q, property := h }) else ⊤) < ⊤ ↔ s.valid q = true | case pos
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : s.valid q = true
⊢ ↑(s.index { val := q, property := h✝ }) < ⊤ ↔ s.valid q = true
case neg
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : ¬s.valid q = true
⊢ ⊤ < ⊤ ↔ s.valid q = true | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
⊢ (if h : s.valid q = true then ↑(s.index { val := q, property := h }) else ⊤) < ⊤ ↔ s.valid q = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.index'_lt_top_iff | [165, 1] | [170, 13] | simpa [WithTop.coe_lt_top] | case pos
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : s.valid q = true
⊢ ↑(s.index { val := q, property := h✝ }) < ⊤ ↔ s.valid q = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : s.valid q = true
⊢ ↑(s.index { val := q, property := h✝ }) < ⊤ ↔ s.valid q = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.index'_lt_top_iff | [165, 1] | [170, 13] | simp [*] | case neg
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : ¬s.valid q = true
⊢ ⊤ < ⊤ ↔ s.valid q = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
q : s.σ
h✝ : ¬s.valid q = true
⊢ ⊤ < ⊤ ↔ s.valid q = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.get_index' | [173, 1] | [176, 42] | simpa using h | ι : Type
α : Type ?u.13400
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
⊢ s.valid x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type ?u.13400
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
⊢ s.valid x = true
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.get_index' | [173, 1] | [176, 42] | generalize_proofs hq | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
⊢ Option.get (index' s x) h = s.index { val := x, property := ⋯ } | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
hq : s.valid x = true
⊢ Option.get (index' s x) h = s.index { val := x, property := hq } | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
⊢ Option.get (index' s x) h = s.index { val := x, property := ⋯ }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.get_index' | [173, 1] | [176, 42] | simp_rw [Stream.index', hq, Option.get] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
hq : s.valid x = true
⊢ Option.get (index' s x) h = s.index { val := x, property := hq } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : PartialOrder ι
x : s.σ
h : Option.isSome (index' s x) = true
hq : s.valid x = true
⊢ Option.get (index' s x) h = s.index { val := x, property := hq }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.wf_valid_iff | [219, 1] | [222, 79] | simp only [Stream.seek'_val q.prop, Stream.coeLex_toOrder, coeLex_lt_iff] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
wf_rel : s.σ → s.σ → Prop
q : { x // s.valid x = true }
i : Lex (ι × Bool)
⊢ wf_rel (s.seek q i) ↑q ∨ i < Stream.toOrder s q ∧ s.seek q i = ↑q ↔
wf_rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
wf_rel : s.σ → s.σ → Prop
q : { x // s.valid x = true }
i : Lex (ι × Bool)
⊢ wf_rel (s.seek q i) ↑q ∨ i < Stream.toOrder s q ∧ s.seek q i = ↑q ↔
wf_rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.IsBounded.mk' | [225, 1] | [233, 36] | simp only [wf_valid_iff] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
h :
∃ wf_rel,
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (s.seek q i) ↑q ∨ i < Stream.toOrder s q ∧ s.seek q i = ↑q | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
h :
∃ wf_rel,
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
h :
∃ wf_rel,
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (s.seek q i) ↑q ∨ i < Stream.toOrder s q ∧ s.seek q i = ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.IsBounded.mk' | [225, 1] | [233, 36] | rcases h with ⟨wfr, hr⟩ | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
h :
∃ wf_rel,
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q | case intro
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
wfr : WellFoundedRelation s.σ
hr :
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
h :
∃ wf_rel,
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.IsBounded.mk' | [225, 1] | [233, 36] | exact ⟨wfr, fun q i => hr q i⟩ | case intro
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
wfr : WellFoundedRelation s.σ
hr :
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝ : Preorder ι
wfr : WellFoundedRelation s.σ
hr :
∀ (q : s.σ) (i : StreamOrder ι),
WellFoundedRelation.rel (Stream.seek' s q i) q ∨ coeLex i < Stream.toOrder' s q ∧ Stream.seek' s q i = q
⊢ ∃ wf_rel,
∀ (q : { x // s.valid x = true }) (i : Lex (ι × Bool)),
WellFoundedRelation.rel (Stream.seek' s (↑q) i) ↑q ∨ coeLex i < Stream.toOrder' s ↑q ∧ Stream.seek' s (↑q) i = ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | rw [← wf_valid_iff (q := ⟨q, hq⟩)] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : s.valid q = true
⊢ WfRel s (seek' s q i) q ∨ coeLex i < toOrder' s q ∧ seek' s q i = q | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : s.valid q = true
⊢ WfRel s (s.seek { val := q, property := hq } i) ↑{ val := q, property := hq } ∨
i < toOrder s { val := q, property := hq } ∧ s.seek { val := q, property := hq } i = ↑{ val := q, property := hq } | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : s.valid q = true
⊢ WfRel s (seek' s q i) q ∨ coeLex i < toOrder' s q ∧ seek' s q i = q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | exact s.wf_valid ⟨q, hq⟩ i | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : s.valid q = true
⊢ WfRel s (s.seek { val := q, property := hq } i) ↑{ val := q, property := hq } ∨
i < toOrder s { val := q, property := hq } ∧ s.seek { val := q, property := hq } i = ↑{ val := q, property := hq } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : s.valid q = true
⊢ WfRel s (s.seek { val := q, property := hq } i) ↑{ val := q, property := hq } ∨
i < toOrder s { val := q, property := hq } ∧ s.seek { val := q, property := hq } i = ↑{ val := q, property := hq }
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | right | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ WfRel s (seek' s q i) q ∨ coeLex i < toOrder' s q ∧ seek' s q i = q | case h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q ∧ seek' s q i = q | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ WfRel s (seek' s q i) q ∨ coeLex i < toOrder' s q ∧ seek' s q i = q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | constructor | case h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q ∧ seek' s q i = q | case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q
case h.right
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ seek' s q i = q | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q ∧ seek' s q i = q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | rw [Prod.Lex.lt_iff'] | case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q | case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1 ∨ (coeLex i).1 = (toOrder' s q).1 ∧ (coeLex i).2 < (toOrder' s q).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ coeLex i < toOrder' s q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | left | case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1 ∨ (coeLex i).1 = (toOrder' s q).1 ∧ (coeLex i).2 < (toOrder' s q).2 | case h.left.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1 ∨ (coeLex i).1 = (toOrder' s q).1 ∧ (coeLex i).2 < (toOrder' s q).2
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | simp [hq] | case h.left.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ (coeLex i).1 < (toOrder' s q).1
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.wf_valid' | [236, 1] | [247, 16] | simp [hq] | case h.right
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ seek' s q i = q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : s.σ
i : StreamOrder ι
hq : ¬s.valid q = true
⊢ seek' s q i = q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.next_wf | [253, 1] | [255, 26] | rw [Stream.advance_val] | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ WfRel s (advance s ↑q) ↑q | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ WfRel s (s.seek q (toOrder s q)) ↑q | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ WfRel s (advance s ↑q) ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.next_wf | [253, 1] | [255, 26] | exact s.progress rfl.le | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ WfRel s (s.seek q (toOrder s q)) ↑q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ WfRel s (s.seek q (toOrder s q)) ↑q
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_zero | [275, 1] | [276, 30] | ext | ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
⊢ eval 0 = 0 | case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (eval 0 x✝) a✝ = (0 x✝) a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
⊢ eval 0 = 0
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_zero | [275, 1] | [276, 30] | rw [Stream.eval] | case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (eval 0 x✝) a✝ = (0 x✝) a✝ | case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (if h : 0.valid x✝ = true then
let_fun this := ⋯;
eval₀ 0 { val := x✝, property := h } + eval 0 (advance 0 x✝)
else 0)
a✝ =
(0 x✝) a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (eval 0 x✝) a✝ = (0 x✝) a✝
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_zero | [275, 1] | [276, 30] | simp | case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (if h : 0.valid x✝ = true then
let_fun this := ⋯;
eval₀ 0 { val := x✝, property := h } + eval 0 (advance 0 x✝)
else 0)
a✝ =
(0 x✝) a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
ι : Type
α : Type u_1
a b s : Stream ι α
inst✝¹ : Preorder ι
inst✝ : AddZeroClass α
x✝ : 0.σ
a✝ : ι
⊢ (if h : 0.valid x✝ = true then
let_fun this := ⋯;
eval₀ 0 { val := x✝, property := h } + eval 0 (advance 0 x✝)
else 0)
a✝ =
(0 x✝) a✝
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_invalid | [279, 1] | [281, 67] | rwa [Stream.eval, dif_neg] | ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
q : s.σ
h : ¬s.valid q = true
⊢ eval s q = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
q : s.σ
h : ¬s.valid q = true
⊢ eval s q = 0
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_valid | [284, 1] | [286, 78] | rw [Stream.eval, dif_pos] | ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ eval s ↑q = eval₀ s q + eval s (advance s ↑q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
q : { q // s.valid q = true }
⊢ eval s ↑q = eval₀ s q + eval s (advance s ↑q)
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_map_id | [289, 1] | [290, 47] | subst f | ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
f : α → α
hf : f = id
s : Stream ι α
inst✝ : IsBounded s
⊢ eval (map f s) = eval s | ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
⊢ eval (map id s) = eval s | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
f : α → α
hf : f = id
s : Stream ι α
inst✝ : IsBounded s
⊢ eval (map f s) = eval s
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_map_id | [289, 1] | [290, 47] | rfl | ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
⊢ eval (map id s) = eval s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_1
a b s✝ : Stream ι α
inst✝² : Preorder ι
inst✝¹ : AddZeroClass α
s : Stream ι α
inst✝ : IsBounded s
⊢ eval (map id s) = eval s
TACTIC:
|
https://github.com/kovach/etch.git | b9e66fe99c33dc1edd926626e598ba00d5d78627 | Etch/StreamFusion/Proofs/StreamProof.lean | Etch.Verification.Stream.eval_map | [292, 1] | [302, 49] | apply s.wf.induction q | ι : Type
α : Type u_2
a b s✝ : Stream ι α
inst✝³ : Preorder ι
β : Type u_1
inst✝² : AddCommMonoid α
inst✝¹ : AddCommMonoid β
f : α →+ β
s : Stream ι α
inst✝ : IsBounded s
q : s.σ
⊢ eval (map (⇑f) s) q = (Finsupp.mapRange.addMonoidHom f) (eval s q) | ι : Type
α : Type u_2
a b s✝ : Stream ι α
inst✝³ : Preorder ι
β : Type u_1
inst✝² : AddCommMonoid α
inst✝¹ : AddCommMonoid β
f : α →+ β
s : Stream ι α
inst✝ : IsBounded s
q : s.σ
⊢ ∀ (x : s.σ),
(∀ (y : s.σ), WfRel s y x → eval (map (⇑f) s) y = (Finsupp.mapRange.addMonoidHom f) (eval s y)) →
eval (map (⇑f) s) x = (Finsupp.mapRange.addMonoidHom f) (eval s x) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type
α : Type u_2
a b s✝ : Stream ι α
inst✝³ : Preorder ι
β : Type u_1
inst✝² : AddCommMonoid α
inst✝¹ : AddCommMonoid β
f : α →+ β
s : Stream ι α
inst✝ : IsBounded s
q : s.σ
⊢ eval (map (⇑f) s) q = (Finsupp.mapRange.addMonoidHom f) (eval s q)
TACTIC:
|
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