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{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Application {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Application
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.Introductions.SingleSubst
open import Tools.Product
import Tools.PropositionalEquality as PE
-- Application of valid terms.
appᵛ : ∀ {F G t u Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
([ΠFG] : Γ ⊩ᵛ⟨ l ⟩ Π F ▹ G / [Γ])
([t] : Γ ⊩ᵛ⟨ l ⟩ t ∷ Π F ▹ G / [Γ] / [ΠFG])
([u] : Γ ⊩ᵛ⟨ l ⟩ u ∷ F / [Γ] / [F])
→ Γ ⊩ᵛ⟨ l ⟩ t ∘ u ∷ G [ u ] / [Γ] / substSΠ {F} {G} {u} [Γ] [F] [ΠFG] [u]
appᵛ {F} {G} {t} {u} [Γ] [F] [ΠFG] [t] [u] {σ = σ} ⊢Δ [σ] =
let [G[u]] = substSΠ {F} {G} {u} [Γ] [F] [ΠFG] [u]
[σF] = proj₁ ([F] ⊢Δ [σ])
[σΠFG] = proj₁ ([ΠFG] ⊢Δ [σ])
[σt] = proj₁ ([t] ⊢Δ [σ])
[σu] = proj₁ ([u] ⊢Δ [σ])
[σG[u]] = proj₁ ([G[u]] ⊢Δ [σ])
[σG[u]]′ = irrelevance′ (singleSubstLift G u) [σG[u]]
in irrelevanceTerm′ (PE.sym (singleSubstLift G u))
[σG[u]]′ [σG[u]]
(appTerm [σF] [σG[u]]′ [σΠFG] [σt] [σu])
, (λ [σ′] [σ≡σ′] →
let [σu′] = convTerm₂ [σF] (proj₁ ([F] ⊢Δ [σ′]))
(proj₂ ([F] ⊢Δ [σ]) [σ′] [σ≡σ′])
(proj₁ ([u] ⊢Δ [σ′]))
in irrelevanceEqTerm′ (PE.sym (singleSubstLift G u))
[σG[u]]′ [σG[u]]
(app-congTerm [σF] [σG[u]]′ [σΠFG]
(proj₂ ([t] ⊢Δ [σ]) [σ′] [σ≡σ′])
[σu] [σu′]
(proj₂ ([u] ⊢Δ [σ]) [σ′] [σ≡σ′])))
-- Application congurence of valid terms.
app-congᵛ : ∀ {F G t u a b Γ l}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ])
([ΠFG] : Γ ⊩ᵛ⟨ l ⟩ Π F ▹ G / [Γ])
([t≡u] : Γ ⊩ᵛ⟨ l ⟩ t ≡ u ∷ Π F ▹ G / [Γ] / [ΠFG])
([a] : Γ ⊩ᵛ⟨ l ⟩ a ∷ F / [Γ] / [F])
([b] : Γ ⊩ᵛ⟨ l ⟩ b ∷ F / [Γ] / [F])
([a≡b] : Γ ⊩ᵛ⟨ l ⟩ a ≡ b ∷ F / [Γ] / [F])
→ Γ ⊩ᵛ⟨ l ⟩ t ∘ a ≡ u ∘ b ∷ G [ a ] / [Γ]
/ substSΠ {F} {G} {a} [Γ] [F] [ΠFG] [a]
app-congᵛ {F} {G} {a = a} [Γ] [F] [ΠFG] [t≡u] [a] [b] [a≡b] ⊢Δ [σ] =
let [σF] = proj₁ ([F] ⊢Δ [σ])
[G[a]] = proj₁ (substSΠ {F} {G} {a} [Γ] [F] [ΠFG] [a] ⊢Δ [σ])
[G[a]]′ = irrelevance′ (singleSubstLift G a) [G[a]]
[σΠFG] = proj₁ ([ΠFG] ⊢Δ [σ])
[σa] = proj₁ ([a] ⊢Δ [σ])
[σb] = proj₁ ([b] ⊢Δ [σ])
in irrelevanceEqTerm′ (PE.sym (singleSubstLift G a)) [G[a]]′ [G[a]]
(app-congTerm [σF] [G[a]]′ [σΠFG] ([t≡u] ⊢Δ [σ])
[σa] [σb] ([a≡b] ⊢Δ [σ]))
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open import Agda.Builtin.Char
open import Agda.Builtin.Sigma
CC = Σ Char λ _ → Char
Test : (c : Char) → Set
Test 'a' = CC
Test _ = CC
test : (c : Char) → Test c
test 'a' .fst = 'a'
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-- Andreas, 2011-04-07
module IrrelevantFin where
data Nat : Set where
zero : Nat
suc : Nat -> Nat
data Fin : Nat -> Set where
zero : .(n : Nat) -> Fin (suc n)
suc : .(n : Nat) -> Fin n -> Fin (suc n)
-- this should not type check, since irrelevant n cannot appear in Fin n
-- or Fin (suc c)
-- note: this is possible in ICC*, but not in Agda! | {
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{-# OPTIONS --without-K #-}
module Proofs where
-- Various general lemmas
open import Level using (Level)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst; cong)
open import Data.Sum using (inj₁; inj₂)
open import Data.Empty
open import Data.Nat.Properties.Simple using ()
----------------------------------------------
-- re-open some sub-files 'public'
open import FiniteFunctions public
open import VectorLemmas public
open import SubstLemmas public
open import LeqLemmas public
open import FinNatLemmas public
open import PathLemmas public
---------------------------------------------
-- Some generally useful functions
-- From Alan Jeffrey's post to Agda list
_⊨_⇒_≡_ : ∀ {I : Set} (F : I → Set) {i j} →
(i ≡ j) → (F i) → (F j) → Set
(F ⊨ refl ⇒ x ≡ y) = (x ≡ y)
xcong : ∀ {I J F G} (f : I → J) (g : ∀ {i} → F i → G (f i)) →
∀ {i j} (i≡j : i ≡ j) {x y} →
(F ⊨ i≡j ⇒ x ≡ y) → (G ⊨ cong f i≡j ⇒ g x ≡ g y)
xcong f g refl refl = refl
congD! : {a b : Level} {A : Set a} {B : A → Set b}
(f : (x : A) → B x) → {x₁ x₂ : A} → (x₂≡x₁ : x₂ ≡ x₁) →
subst B x₂≡x₁ (f x₂) ≡ f x₁
congD! f refl = refl
-- Courtesy of Wolfram Kahl, a dependent cong₂
cong₂D! : {a b c : Level} {A : Set a} {B : A → Set b} {C : Set c}
(f : (x : A) → B x → C)
→ {x₁ x₂ : A} {y₁ : B x₁} {y₂ : B x₂}
→ (x₂≡x₁ : x₂ ≡ x₁) → subst B x₂≡x₁ y₂ ≡ y₁ → f x₁ y₁ ≡ f x₂ y₂
cong₂D! f refl refl = refl
----------------------------------------------
inj-injective : ∀ {A B : Set} {a : A} {b : B} → inj₁ a ≡ inj₂ b → ⊥
inj-injective ()
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{-# OPTIONS --without-K --safe --no-sized-types --no-guardedness #-}
module Agda.Builtin.Reflection where
open import Agda.Builtin.Unit
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat
open import Agda.Builtin.Word
open import Agda.Builtin.List
open import Agda.Builtin.String
open import Agda.Builtin.Char
open import Agda.Builtin.Float
open import Agda.Builtin.Int
open import Agda.Builtin.Sigma
-- Names --
postulate Name : Set
{-# BUILTIN QNAME Name #-}
primitive
primQNameEquality : Name → Name → Bool
primQNameLess : Name → Name → Bool
primShowQName : Name → String
-- Fixity --
data Associativity : Set where
left-assoc : Associativity
right-assoc : Associativity
non-assoc : Associativity
data Precedence : Set where
related : Int → Precedence
unrelated : Precedence
data Fixity : Set where
fixity : Associativity → Precedence → Fixity
{-# BUILTIN ASSOC Associativity #-}
{-# BUILTIN ASSOCLEFT left-assoc #-}
{-# BUILTIN ASSOCRIGHT right-assoc #-}
{-# BUILTIN ASSOCNON non-assoc #-}
{-# BUILTIN PRECEDENCE Precedence #-}
{-# BUILTIN PRECRELATED related #-}
{-# BUILTIN PRECUNRELATED unrelated #-}
{-# BUILTIN FIXITY Fixity #-}
{-# BUILTIN FIXITYFIXITY fixity #-}
{-# COMPILE GHC Associativity = data MAlonzo.RTE.Assoc (MAlonzo.RTE.LeftAssoc | MAlonzo.RTE.RightAssoc | MAlonzo.RTE.NonAssoc) #-}
{-# COMPILE GHC Precedence = data MAlonzo.RTE.Precedence (MAlonzo.RTE.Related | MAlonzo.RTE.Unrelated) #-}
{-# COMPILE GHC Fixity = data MAlonzo.RTE.Fixity (MAlonzo.RTE.Fixity) #-}
{-# COMPILE JS Associativity = function (x,v) { return v[x](); } #-}
{-# COMPILE JS left-assoc = "left-assoc" #-}
{-# COMPILE JS right-assoc = "right-assoc" #-}
{-# COMPILE JS non-assoc = "non-assoc" #-}
{-# COMPILE JS Precedence =
function (x,v) {
if (x === "unrelated") { return v[x](); } else { return v["related"](x); }} #-}
{-# COMPILE JS related = function(x) { return x; } #-}
{-# COMPILE JS unrelated = "unrelated" #-}
{-# COMPILE JS Fixity = function (x,v) { return v["fixity"](x["assoc"], x["prec"]); } #-}
{-# COMPILE JS fixity = function (x) { return function (y) { return { "assoc": x, "prec": y}; }; } #-}
primitive
primQNameFixity : Name → Fixity
-- Metavariables --
postulate Meta : Set
{-# BUILTIN AGDAMETA Meta #-}
primitive
primMetaEquality : Meta → Meta → Bool
primMetaLess : Meta → Meta → Bool
primShowMeta : Meta → String
-- Arguments --
-- Arguments can be (visible), {hidden}, or {{instance}}.
data Visibility : Set where
visible hidden instance′ : Visibility
{-# BUILTIN HIDING Visibility #-}
{-# BUILTIN VISIBLE visible #-}
{-# BUILTIN HIDDEN hidden #-}
{-# BUILTIN INSTANCE instance′ #-}
-- Arguments can be relevant or irrelevant.
data Relevance : Set where
relevant irrelevant : Relevance
{-# BUILTIN RELEVANCE Relevance #-}
{-# BUILTIN RELEVANT relevant #-}
{-# BUILTIN IRRELEVANT irrelevant #-}
data ArgInfo : Set where
arg-info : (v : Visibility) (r : Relevance) → ArgInfo
data Arg (A : Set) : Set where
arg : (i : ArgInfo) (x : A) → Arg A
{-# BUILTIN ARGINFO ArgInfo #-}
{-# BUILTIN ARGARGINFO arg-info #-}
{-# BUILTIN ARG Arg #-}
{-# BUILTIN ARGARG arg #-}
-- Name abstraction --
data Abs (A : Set) : Set where
abs : (s : String) (x : A) → Abs A
{-# BUILTIN ABS Abs #-}
{-# BUILTIN ABSABS abs #-}
-- Literals --
data Literal : Set where
nat : (n : Nat) → Literal
word64 : (n : Word64) → Literal
float : (x : Float) → Literal
char : (c : Char) → Literal
string : (s : String) → Literal
name : (x : Name) → Literal
meta : (x : Meta) → Literal
{-# BUILTIN AGDALITERAL Literal #-}
{-# BUILTIN AGDALITNAT nat #-}
{-# BUILTIN AGDALITWORD64 word64 #-}
{-# BUILTIN AGDALITFLOAT float #-}
{-# BUILTIN AGDALITCHAR char #-}
{-# BUILTIN AGDALITSTRING string #-}
{-# BUILTIN AGDALITQNAME name #-}
{-# BUILTIN AGDALITMETA meta #-}
-- Patterns --
data Pattern : Set where
con : (c : Name) (ps : List (Arg Pattern)) → Pattern
dot : Pattern
var : (s : String) → Pattern
lit : (l : Literal) → Pattern
proj : (f : Name) → Pattern
absurd : Pattern
{-# BUILTIN AGDAPATTERN Pattern #-}
{-# BUILTIN AGDAPATCON con #-}
{-# BUILTIN AGDAPATDOT dot #-}
{-# BUILTIN AGDAPATVAR var #-}
{-# BUILTIN AGDAPATLIT lit #-}
{-# BUILTIN AGDAPATPROJ proj #-}
{-# BUILTIN AGDAPATABSURD absurd #-}
-- Terms --
data Sort : Set
data Clause : Set
data Term : Set
Type = Term
data Term where
var : (x : Nat) (args : List (Arg Term)) → Term
con : (c : Name) (args : List (Arg Term)) → Term
def : (f : Name) (args : List (Arg Term)) → Term
lam : (v : Visibility) (t : Abs Term) → Term
pat-lam : (cs : List Clause) (args : List (Arg Term)) → Term
pi : (a : Arg Type) (b : Abs Type) → Term
agda-sort : (s : Sort) → Term
lit : (l : Literal) → Term
meta : (x : Meta) → List (Arg Term) → Term
unknown : Term
data Sort where
set : (t : Term) → Sort
lit : (n : Nat) → Sort
unknown : Sort
data Clause where
clause : (ps : List (Arg Pattern)) (t : Term) → Clause
absurd-clause : (ps : List (Arg Pattern)) → Clause
{-# BUILTIN AGDASORT Sort #-}
{-# BUILTIN AGDATERM Term #-}
{-# BUILTIN AGDACLAUSE Clause #-}
{-# BUILTIN AGDATERMVAR var #-}
{-# BUILTIN AGDATERMCON con #-}
{-# BUILTIN AGDATERMDEF def #-}
{-# BUILTIN AGDATERMMETA meta #-}
{-# BUILTIN AGDATERMLAM lam #-}
{-# BUILTIN AGDATERMEXTLAM pat-lam #-}
{-# BUILTIN AGDATERMPI pi #-}
{-# BUILTIN AGDATERMSORT agda-sort #-}
{-# BUILTIN AGDATERMLIT lit #-}
{-# BUILTIN AGDATERMUNSUPPORTED unknown #-}
{-# BUILTIN AGDASORTSET set #-}
{-# BUILTIN AGDASORTLIT lit #-}
{-# BUILTIN AGDASORTUNSUPPORTED unknown #-}
{-# BUILTIN AGDACLAUSECLAUSE clause #-}
{-# BUILTIN AGDACLAUSEABSURD absurd-clause #-}
-- Definitions --
data Definition : Set where
function : (cs : List Clause) → Definition
data-type : (pars : Nat) (cs : List Name) → Definition
record-type : (c : Name) (fs : List (Arg Name)) → Definition
data-cons : (d : Name) → Definition
axiom : Definition
prim-fun : Definition
{-# BUILTIN AGDADEFINITION Definition #-}
{-# BUILTIN AGDADEFINITIONFUNDEF function #-}
{-# BUILTIN AGDADEFINITIONDATADEF data-type #-}
{-# BUILTIN AGDADEFINITIONRECORDDEF record-type #-}
{-# BUILTIN AGDADEFINITIONDATACONSTRUCTOR data-cons #-}
{-# BUILTIN AGDADEFINITIONPOSTULATE axiom #-}
{-# BUILTIN AGDADEFINITIONPRIMITIVE prim-fun #-}
-- Errors --
data ErrorPart : Set where
strErr : String → ErrorPart
termErr : Term → ErrorPart
nameErr : Name → ErrorPart
{-# BUILTIN AGDAERRORPART ErrorPart #-}
{-# BUILTIN AGDAERRORPARTSTRING strErr #-}
{-# BUILTIN AGDAERRORPARTTERM termErr #-}
{-# BUILTIN AGDAERRORPARTNAME nameErr #-}
-- TC monad --
postulate
TC : ∀ {a} → Set a → Set a
returnTC : ∀ {a} {A : Set a} → A → TC A
bindTC : ∀ {a b} {A : Set a} {B : Set b} → TC A → (A → TC B) → TC B
unify : Term → Term → TC ⊤
typeError : ∀ {a} {A : Set a} → List ErrorPart → TC A
inferType : Term → TC Type
checkType : Term → Type → TC Term
normalise : Term → TC Term
reduce : Term → TC Term
catchTC : ∀ {a} {A : Set a} → TC A → TC A → TC A
quoteTC : ∀ {a} {A : Set a} → A → TC Term
unquoteTC : ∀ {a} {A : Set a} → Term → TC A
getContext : TC (List (Arg Type))
extendContext : ∀ {a} {A : Set a} → Arg Type → TC A → TC A
inContext : ∀ {a} {A : Set a} → List (Arg Type) → TC A → TC A
freshName : String → TC Name
declareDef : Arg Name → Type → TC ⊤
declarePostulate : Arg Name → Type → TC ⊤
defineFun : Name → List Clause → TC ⊤
getType : Name → TC Type
getDefinition : Name → TC Definition
blockOnMeta : ∀ {a} {A : Set a} → Meta → TC A
commitTC : TC ⊤
isMacro : Name → TC Bool
-- If the argument is 'true' makes the following primitives also normalise
-- their results: inferType, checkType, quoteTC, getType, and getContext
withNormalisation : ∀ {a} {A : Set a} → Bool → TC A → TC A
-- Prints the third argument if the corresponding verbosity level is turned
-- on (with the -v flag to Agda).
debugPrint : String → Nat → List ErrorPart → TC ⊤
-- Fail if the given computation gives rise to new, unsolved
-- "blocking" constraints.
noConstraints : ∀ {a} {A : Set a} → TC A → TC A
-- Run the given TC action and return the first component. Resets to
-- the old TC state if the second component is 'false', or keep the
-- new TC state if it is 'true'.
runSpeculative : ∀ {a} {A : Set a} → TC (Σ A λ _ → Bool) → TC A
{-# BUILTIN AGDATCM TC #-}
{-# BUILTIN AGDATCMRETURN returnTC #-}
{-# BUILTIN AGDATCMBIND bindTC #-}
{-# BUILTIN AGDATCMUNIFY unify #-}
{-# BUILTIN AGDATCMTYPEERROR typeError #-}
{-# BUILTIN AGDATCMINFERTYPE inferType #-}
{-# BUILTIN AGDATCMCHECKTYPE checkType #-}
{-# BUILTIN AGDATCMNORMALISE normalise #-}
{-# BUILTIN AGDATCMREDUCE reduce #-}
{-# BUILTIN AGDATCMCATCHERROR catchTC #-}
{-# BUILTIN AGDATCMQUOTETERM quoteTC #-}
{-# BUILTIN AGDATCMUNQUOTETERM unquoteTC #-}
{-# BUILTIN AGDATCMGETCONTEXT getContext #-}
{-# BUILTIN AGDATCMEXTENDCONTEXT extendContext #-}
{-# BUILTIN AGDATCMINCONTEXT inContext #-}
{-# BUILTIN AGDATCMFRESHNAME freshName #-}
{-# BUILTIN AGDATCMDECLAREDEF declareDef #-}
{-# BUILTIN AGDATCMDECLAREPOSTULATE declarePostulate #-}
{-# BUILTIN AGDATCMDEFINEFUN defineFun #-}
{-# BUILTIN AGDATCMGETTYPE getType #-}
{-# BUILTIN AGDATCMGETDEFINITION getDefinition #-}
{-# BUILTIN AGDATCMBLOCKONMETA blockOnMeta #-}
{-# BUILTIN AGDATCMCOMMIT commitTC #-}
{-# BUILTIN AGDATCMISMACRO isMacro #-}
{-# BUILTIN AGDATCMWITHNORMALISATION withNormalisation #-}
{-# BUILTIN AGDATCMDEBUGPRINT debugPrint #-}
{-# BUILTIN AGDATCMNOCONSTRAINTS noConstraints #-}
{-# BUILTIN AGDATCMRUNSPECULATIVE runSpeculative #-}
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module Where where
{-
id : Set -> Set
id a = a
-}
-- x : (_ : _) -> _
x = id Set3000
where id = \x -> x
y = False
where data False : Set where
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.CircleHSpace
open import homotopy.JoinAssocCubical
open import homotopy.JoinSusp
module homotopy.Hopf where
import homotopy.HopfConstruction
module Hopf = homotopy.HopfConstruction S¹-conn ⊙S¹-hSpace
Hopf : S² → Type₀
Hopf = Hopf.H.f
Hopf-fiber : Hopf north == S¹
Hopf-fiber = idp
-- TODO Turn [Hopf.theorem] into an equivalence
Hopf-total : Σ _ Hopf ≃ S³
Hopf-total =
Σ _ Hopf ≃⟨ coe-equiv Hopf.theorem ⟩
S¹ * S¹ ≃⟨ *-Susp-l S⁰ S¹ ⟩
Susp (S⁰ * S¹) ≃⟨ Susp-emap (*-Bool-l S¹) ⟩
S³ ≃∎
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module Relations where
open import Agda.Builtin.Equality
open import Natural
data _≤_ : ℕ → ℕ → Set where
z≤n : ∀ {n : ℕ} →
zero ≤ n
s≤s : ∀ {m n : ℕ} →
m ≤ n →
(suc m) ≤ (suc n)
infix 4 _≤_
-- example : finished by auto mode
_ : 3 ≤ 5
_ = s≤s {2} {4} (s≤s {1} {3} (s≤s {0} {2} (z≤n {2})))
inv-s≤s : ∀ {m n : ℕ}
→ suc m ≤ suc n
→ m ≤ n
inv-s≤s (s≤s m≤n) = m≤n
inv-z≤n : ∀ {m : ℕ}
→ m ≤ zero
→ m ≡ zero
inv-z≤n z≤n = refl
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-- cj-xu and fredriknordvallforsberg, 2018-04-30
open import Common.IO
data Bool : Set where
true false : Bool
data MyUnit : Set where
tt : MyUnit -- no eta!
HiddenFunType : MyUnit -> Set
HiddenFunType tt = Bool -> Bool
notTooManyArgs : (x : MyUnit) -> HiddenFunType x
notTooManyArgs tt b = b
{- This should not happen when compiling notTooManyArgs:
• Couldn't match expected type ‘GHC.Prim.Any’
with actual type ‘a0 -> b0’
The type variables ‘b0’, ‘a0’ are ambiguous
• The equation(s) for ‘d10’ have two arguments,
but its type ‘T2 -> AgdaAny’ has only one
-}
main : IO MyUnit
main = return tt
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module Data.Num.Nat where
open import Data.Num.Core renaming
( carry to carry'
; carry-lower-bound to carry-lower-bound'
; carry-upper-bound to carry-upper-bound'
)
open import Data.Num.Maximum
open import Data.Num.Bounded
open import Data.Num.Next
open import Data.Num.Increment
open import Data.Num.Continuous
open import Data.Nat
open import Data.Nat.Properties
open import Data.Nat.Properties.Simple
open import Data.Nat.Properties.Extra
open import Data.Nat.DM
open import Data.Fin as Fin
using (Fin; fromℕ≤; inject≤)
renaming (zero to z; suc to s)
open import Data.Fin.Properties using (toℕ-fromℕ≤; bounded)
open import Data.Product
open import Data.Empty using (⊥)
open import Data.Unit using (⊤; tt)
open import Function
open import Relation.Nullary.Decidable
open import Relation.Nullary
open import Relation.Nullary.Negation
open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
open ≤-Reasoning renaming (begin_ to start_; _∎ to _□; _≡⟨_⟩_ to _≈⟨_⟩_)
open DecTotalOrder decTotalOrder using (reflexive) renaming (refl to ≤-refl)
-- left-bounded infinite interval of natural number
data Nat : ℕ → Set where
from : ∀ offset → Nat offset
suc : ∀ {offset} → Nat offset → Nat offset
Nat-toℕ : ∀ {offset} → Nat offset → ℕ
Nat-toℕ (from offset) = offset
Nat-toℕ (suc n) = suc (Nat-toℕ n)
Nat-fromℕ : ∀ offset
→ (n : ℕ)
→ .(offset ≤ n)
→ Nat offset
Nat-fromℕ offset n p with offset ≟ n
Nat-fromℕ offset n p | yes eq = from offset
Nat-fromℕ offset zero p | no ¬eq = from offset
Nat-fromℕ offset (suc n) p | no ¬eq = suc (Nat-fromℕ offset n (≤-pred (≤∧≢⇒< p ¬eq)))
Nat-fromℕ-toℕ : ∀ offset
→ (n : ℕ)
→ (p : offset ≤ n)
→ Nat-toℕ (Nat-fromℕ offset n p) ≡ n
Nat-fromℕ-toℕ offset n p with offset ≟ n
Nat-fromℕ-toℕ offset n p | yes eq = eq
Nat-fromℕ-toℕ .0 zero z≤n | no ¬eq = refl
Nat-fromℕ-toℕ offset (suc n) p | no ¬eq =
begin
suc (Nat-toℕ (Nat-fromℕ offset n (≤-pred (≤∧≢⇒< p ¬eq))))
≡⟨ cong suc (Nat-fromℕ-toℕ offset n (≤-pred (≤∧≢⇒< p ¬eq))) ⟩
suc n
∎
fromNat : ∀ {b d o}
→ {cont : True (Continuous? b (suc d) o)}
→ Nat o
→ Num b (suc d) o
fromNat (from offset) = z ∙
fromNat {cont = cont} (suc nat) = 1+ {cont = cont} (fromNat {cont = cont} nat)
toℕ-fromNat : ∀ b d o
→ (cont : True (Continuous? b (suc d) o))
→ (n : ℕ)
→ (p : o ≤ n)
→ ⟦ fromNat {b} {d} {o} {cont = cont} (Nat-fromℕ o n p) ⟧ ≡ n
toℕ-fromNat b d o cont n p with o ≟ n
toℕ-fromNat b d o cont n p | yes eq = cong (_+_ zero) eq
toℕ-fromNat b d .0 cont zero z≤n | no ¬eq = refl
toℕ-fromNat b d o cont (suc n) p | no ¬eq =
begin
⟦ 1+ {b} {cont = cont} (fromNat (Nat-fromℕ o n (≤-pred (≤∧≢⇒< p ¬eq)))) ⟧
≡⟨ 1+-toℕ {b} (fromNat {cont = cont} (Nat-fromℕ o n (≤-pred (≤∧≢⇒< p ¬eq)))) ⟩
suc ⟦ fromNat {cont = cont} (Nat-fromℕ o n (≤-pred (≤∧≢⇒< p ¬eq))) ⟧
-- mysterious agda bug, this refl must stay here
≡⟨ refl ⟩
suc ⟦ fromNat {cont = cont} (Nat-fromℕ o n (≤-pred (≤∧≢⇒< p ¬eq))) ⟧
≡⟨ cong suc (toℕ-fromNat b d o cont n (≤-pred (≤∧≢⇒< p ¬eq))) ⟩
suc n
∎
-- -- a partial function that only maps ℕ to Continuous Nums
-- fromℕ : ∀ {b d o}
-- → {cond : N+Closed b d o}
-- → ℕ
-- → Num b (suc d) o
-- fromℕ {cond = cond} zero = z ∙
-- fromℕ {cond = cond} (suc n) = {! !}
-- fromℕ {cont = cont} zero = z ∙
-- fromℕ {cont = cont} (suc n) = 1+ {cont = cont} (fromℕ {cont = cont} n)
-- toℕ-fromℕ : ∀ {b d o}
-- → {cont : True (Continuous? b (suc d) o)}
-- → (n : ℕ)
-- → ⟦ fromℕ {cont = cont} n ⟧ ≡ n + o
-- toℕ-fromℕ {_} {_} {_} {cont} zero = refl
-- toℕ-fromℕ {b} {d} {o} {cont} (suc n) =
-- begin
-- ⟦ 1+ {cont = cont} (fromℕ {cont = cont} n) ⟧
-- ≡⟨ 1+-toℕ {cont = cont} (fromℕ {cont = cont} n) ⟩
-- suc ⟦ fromℕ {cont = cont} n ⟧
-- ≡⟨ cong suc (toℕ-fromℕ {cont = cont} n) ⟩
-- suc (n + o)
-- ∎
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------------------------------------------------------------------------
-- The Agda standard library
--
-- This module is DEPRECATED. Please use
-- Data.List.Relation.Binary.BagAndSetEquality directly.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.List.Relation.BagAndSetEquality where
open import Data.List.Relation.Binary.BagAndSetEquality public
{-# WARNING_ON_IMPORT
"Data.List.Relation.BagAndSetEquality was deprecated in v1.0.
Use Data.List.Relation.Binary.BagAndSetEquality instead."
#-}
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open import Categories
open import Monads
module Monads.EM.Functors {a b}{C : Cat {a}{b}}(M : Monad C) where
open import Library
open import Functors
open import Monads.EM M
open Cat C
open Fun
open Monad M
open Alg
open AlgMorph
EML : Fun C EM
EML = record {
OMap = λ X → record {
acar = T X;
astr = λ _ → bind;
alaw1 = sym law2;
alaw2 = law3};
HMap = λ f → record {
amor = bind (comp η f);
ahom = λ {Z} {g} →
proof
comp (bind (comp η f)) (bind g)
≅⟨ sym law3 ⟩
bind (comp (bind (comp η f)) g)
∎};
fid = AlgMorphEq (
proof
bind (comp η iden)
≅⟨ cong bind idr ⟩
bind η
≅⟨ law1 ⟩
iden ∎);
fcomp = λ {_}{_}{_}{f}{g} →
AlgMorphEq (
proof
bind (comp η (comp f g))
≅⟨ cong bind (sym ass) ⟩
bind (comp (comp η f) g)
≅⟨ cong (λ f → bind (comp f g)) (sym law2) ⟩
bind (comp (comp (bind (comp η f)) η) g)
≅⟨ cong bind ass ⟩
bind (comp (bind (comp η f)) (comp η g))
≅⟨ law3 ⟩
comp (bind (comp η f)) (bind (comp η g))
∎)}
EMR : Fun EM C
EMR = record {
OMap = acar;
HMap = amor;
fid = refl;
fcomp = refl}
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{-# OPTIONS --no-positivity-check #-}
open import Prelude
module Implicits.Resolution.Undecidable.Resolution where
open import Data.Fin.Substitution
open import Implicits.Syntax
open import Implicits.Syntax.MetaType
open import Implicits.Substitutions
open import Extensions.ListFirst
infixl 4 _⊢ᵣ_ _⊢_↓_ _⟨_⟩=_
mutual
data _⊢_↓_ {ν} (Δ : ICtx ν) : Type ν → SimpleType ν → Set where
i-simp : ∀ a → Δ ⊢ simpl a ↓ a
i-iabs : ∀ {ρ₁ ρ₂ a} → Δ ⊢ᵣ ρ₁ → Δ ⊢ ρ₂ ↓ a → Δ ⊢ ρ₁ ⇒ ρ₂ ↓ a
i-tabs : ∀ {ρ a} b → Δ ⊢ ρ tp[/tp b ] ↓ a → Δ ⊢ ∀' ρ ↓ a
-- implicit resolution is simply the first rule in the implicit context
-- that yields the queried type
_⟨_⟩=_ : ∀ {ν} → ICtx ν → SimpleType ν → Type ν → Set
Δ ⟨ a ⟩= r = first r ∈ Δ ⇔ (λ r' → Δ ⊢ r' ↓ a)
data _⊢ᵣ_ {ν} (Δ : ICtx ν) : Type ν → Set where
r-simp : ∀ {τ ρ} → Δ ⟨ τ ⟩= ρ → Δ ⊢ᵣ simpl τ
r-iabs : ∀ ρ₁ {ρ₂} → ρ₁ List.∷ Δ ⊢ᵣ ρ₂ → Δ ⊢ᵣ ρ₁ ⇒ ρ₂
r-tabs : ∀ {ρ} → ictx-weaken Δ ⊢ᵣ ρ → Δ ⊢ᵣ ∀' ρ
_⊢ᵣ[_] : ∀ {ν} → (Δ : ICtx ν) → List (Type ν) → Set
_⊢ᵣ[_] Δ ρs = All (λ r → Δ ⊢ᵣ r) ρs
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Library.Data.Natural where
open import Light.Level using (Level ; Setω)
open import Light.Library.Arithmetic using (Arithmetic)
open import Light.Package using (Package)
open import Light.Library.Relation.Binary
using (SelfTransitive ; SelfSymmetric ; Reflexive)
open import Light.Library.Relation.Binary.Decidable using (DecidableSelfBinary)
import Light.Library.Relation.Decidable
open import Light.Library.Relation.Binary.Equality as ≈ using (_≈_)
open import Light.Library.Relation.Binary.Equality.Decidable using (DecidableSelfEquality)
open import Light.Library.Relation using (False)
record Dependencies : Setω where
record Library (dependencies : Dependencies) : Setω where
field
ℓ : Level
ℕ : Set ℓ
⦃ equals ⦄ : DecidableSelfEquality ℕ
zero : ℕ
successor : ℕ → ℕ
predecessor : ∀ (a : ℕ) ⦃ a‐≉‐0 : False (a ≈ zero) ⦄ → ℕ
_+_ _∗_ _//_ _∸_ : ℕ → ℕ → ℕ
⦃ ≈‐transitive ⦄ : SelfTransitive (≈.self‐relation ℕ)
⦃ ≈‐symmetric ⦄ : SelfSymmetric (≈.self‐relation ℕ)
⦃ ≈‐reflexive ⦄ : Reflexive (≈.self‐relation ℕ)
instance arithmetic : Arithmetic ℕ
arithmetic = record { _+_ = _+_ ; _∗_ = _∗_ }
open Library ⦃ ... ⦄ public
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------------------------------------------------------------------------
-- The Agda standard library
--
-- An either-or-both data type
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.These where
open import Level
open import Data.Maybe.Base using (Maybe; just; nothing; maybe′)
open import Data.Sum.Base using (_⊎_; [_,_]′)
open import Function
------------------------------------------------------------------------
-- Re-exporting the datatype and its operations
open import Data.These.Base public
private
variable
a b : Level
A : Set a
B : Set b
------------------------------------------------------------------------
-- Additional operations
-- projections
fromThis : These A B → Maybe A
fromThis = fold just (const nothing) (const ∘′ just)
fromThat : These A B → Maybe B
fromThat = fold (const nothing) just (const just)
leftMost : These A A → A
leftMost = fold id id const
rightMost : These A A → A
rightMost = fold id id (flip const)
mergeThese : (A → A → A) → These A A → A
mergeThese = fold id id
-- deletions
deleteThis : These A B → Maybe (These A B)
deleteThis = fold (const nothing) (just ∘′ that) (const (just ∘′ that))
deleteThat : These A B → Maybe (These A B)
deleteThat = fold (just ∘′ this) (const nothing) (const ∘′ just ∘′ this)
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module Formalization.ClassicalPropositionalLogic.NaturalDeduction where
open import Data.Either as Either using (Left ; Right)
open import Formalization.ClassicalPropositionalLogic.Syntax
open import Functional
import Lvl
import Logic.Propositional as Meta
open import Logic
open import Relator.Equals
open import Relator.Equals.Proofs.Equiv
open import Sets.PredicateSet using (PredSet ; _∈_ ; _∉_ ; _∪_ ; _∪•_ ; _∖_ ; _⊆_ ; _⊇_ ; ∅ ; [≡]-to-[⊆] ; [≡]-to-[⊇]) renaming (•_ to singleton ; _≡_ to _≡ₛ_)
open import Type
private variable ℓₚ ℓ ℓ₁ ℓ₂ : Lvl.Level
data _⊢_ {ℓ ℓₚ} {P : Type{ℓₚ}} : Formulas(P){ℓ} → Formula(P) → Stmt{Lvl.𝐒(ℓₚ Lvl.⊔ ℓ)} where
direct : ∀{Γ} → (Γ ⊆ (Γ ⊢_))
[⊤]-intro : ∀{Γ} → (Γ ⊢ ⊤)
[⊥]-intro : ∀{Γ}{φ} → (Γ ⊢ φ) → (Γ ⊢ (¬ φ)) → (Γ ⊢ ⊥)
[⊥]-elim : ∀{Γ}{φ} → (Γ ⊢ ⊥) → (Γ ⊢ φ)
[¬]-intro : ∀{Γ}{φ} → ((Γ ∪ singleton(φ)) ⊢ ⊥) → (Γ ⊢ (¬ φ))
[¬]-elim : ∀{Γ}{φ} → ((Γ ∪ singleton(¬ φ)) ⊢ ⊥) → (Γ ⊢ φ)
[∧]-intro : ∀{Γ}{φ ψ} → (Γ ⊢ φ) → (Γ ⊢ ψ) → (Γ ⊢ (φ ∧ ψ))
[∧]-elimₗ : ∀{Γ}{φ ψ} → (Γ ⊢ (φ ∧ ψ)) → (Γ ⊢ φ)
[∧]-elimᵣ : ∀{Γ}{φ ψ} → (Γ ⊢ (φ ∧ ψ)) → (Γ ⊢ ψ)
[∨]-introₗ : ∀{Γ}{φ ψ} → (Γ ⊢ φ) → (Γ ⊢ (φ ∨ ψ))
[∨]-introᵣ : ∀{Γ}{φ ψ} → (Γ ⊢ ψ) → (Γ ⊢ (φ ∨ ψ))
[∨]-elim : ∀{Γ}{φ ψ χ} → ((Γ ∪ singleton(φ)) ⊢ χ) → ((Γ ∪ singleton(ψ)) ⊢ χ) → (Γ ⊢ (φ ∨ ψ)) → (Γ ⊢ χ)
[⟶]-intro : ∀{Γ}{φ ψ} → ((Γ ∪ singleton(φ)) ⊢ ψ) → (Γ ⊢ (φ ⟶ ψ))
[⟶]-elim : ∀{Γ}{φ ψ} → (Γ ⊢ φ) → (Γ ⊢ (φ ⟶ ψ)) → (Γ ⊢ ψ)
[⟷]-intro : ∀{Γ}{φ ψ} → ((Γ ∪ singleton(ψ)) ⊢ φ) → ((Γ ∪ singleton(φ)) ⊢ ψ) → (Γ ⊢ (φ ⟷ ψ))
[⟷]-elimₗ : ∀{Γ}{φ ψ} → (Γ ⊢ ψ) → (Γ ⊢ (φ ⟷ ψ)) → (Γ ⊢ φ)
[⟷]-elimᵣ : ∀{Γ}{φ ψ} → (Γ ⊢ φ) → (Γ ⊢ (φ ⟷ ψ)) → (Γ ⊢ ψ)
module _ where
private variable P : Type{ℓₚ}
private variable Γ Γ₁ Γ₂ : Formulas(P){ℓ}
private variable φ ψ : Formula(P)
_⊬_ : Formulas(P){ℓ} → Formula(P) → Stmt
_⊬_ = Meta.¬_ ∘₂ (_⊢_)
weaken-union-singleton : (Γ₁ ⊆ Γ₂) → (((Γ₁ ∪ singleton(φ)) ⊢_) ⊆ ((Γ₂ ∪ singleton(φ)) ⊢_))
weaken : (Γ₁ ⊆ Γ₂) → ((Γ₁ ⊢_) ⊆ (Γ₂ ⊢_))
weaken Γ₁Γ₂ {φ} (direct p) = direct (Γ₁Γ₂ p)
weaken Γ₁Γ₂ {.⊤} [⊤]-intro = [⊤]-intro
weaken Γ₁Γ₂ {.⊥} ([⊥]-intro p q) = [⊥]-intro (weaken Γ₁Γ₂ p) (weaken Γ₁Γ₂ q)
weaken Γ₁Γ₂ {φ} ([⊥]-elim p) = [⊥]-elim (weaken Γ₁Γ₂ p)
weaken Γ₁Γ₂ {.(¬ _)} ([¬]-intro p) = [¬]-intro (weaken-union-singleton Γ₁Γ₂ p)
weaken Γ₁Γ₂ {φ} ([¬]-elim p) = [¬]-elim (weaken-union-singleton Γ₁Γ₂ p)
weaken Γ₁Γ₂ {.(_ ∧ _)} ([∧]-intro p q) = [∧]-intro (weaken Γ₁Γ₂ p) (weaken Γ₁Γ₂ q)
weaken Γ₁Γ₂ {φ} ([∧]-elimₗ p) = [∧]-elimₗ (weaken Γ₁Γ₂ p)
weaken Γ₁Γ₂ {φ} ([∧]-elimᵣ p) = [∧]-elimᵣ (weaken Γ₁Γ₂ p)
weaken Γ₁Γ₂ {.(_ ∨ _)} ([∨]-introₗ p) = [∨]-introₗ (weaken Γ₁Γ₂ p)
weaken Γ₁Γ₂ {.(_ ∨ _)} ([∨]-introᵣ p) = [∨]-introᵣ (weaken Γ₁Γ₂ p)
weaken Γ₁Γ₂ {φ} ([∨]-elim p q r) = [∨]-elim (weaken-union-singleton Γ₁Γ₂ p) (weaken-union-singleton Γ₁Γ₂ q) (weaken Γ₁Γ₂ r)
weaken Γ₁Γ₂ {.(_ ⟶ _)} ([⟶]-intro p) = [⟶]-intro (weaken-union-singleton Γ₁Γ₂ p)
weaken Γ₁Γ₂ {φ} ([⟶]-elim p q) = [⟶]-elim (weaken Γ₁Γ₂ p) (weaken Γ₁Γ₂ q)
weaken Γ₁Γ₂ {.(_ ⟷ _)} ([⟷]-intro p q) = [⟷]-intro (weaken-union-singleton Γ₁Γ₂ p) (weaken-union-singleton Γ₁Γ₂ q)
weaken Γ₁Γ₂ {φ} ([⟷]-elimₗ p q) = [⟷]-elimₗ (weaken Γ₁Γ₂ p) (weaken Γ₁Γ₂ q)
weaken Γ₁Γ₂ {φ} ([⟷]-elimᵣ p q) = [⟷]-elimᵣ (weaken Γ₁Γ₂ p) (weaken Γ₁Γ₂ q)
weaken-union-singleton Γ₁Γ₂ p = weaken (Either.mapLeft Γ₁Γ₂) p
weaken-union : (Γ₁ ⊢_) ⊆ ((Γ₁ ∪ Γ₂) ⊢_)
weaken-union = weaken Either.Left
[⟵]-intro : ((Γ ∪ singleton(φ)) ⊢ ψ) → (Γ ⊢ (ψ ⟵ φ))
[⟵]-intro = [⟶]-intro
[⟵]-elim : (Γ ⊢ φ) → (Γ ⊢ (ψ ⟵ φ)) → (Γ ⊢ ψ)
[⟵]-elim = [⟶]-elim
[¬¬]-elim : (Γ ⊢ ¬(¬ φ)) → (Γ ⊢ φ)
[¬¬]-elim nnφ =
([¬]-elim
([⊥]-intro
(direct(Right [≡]-intro))
(weaken-union nnφ)
)
)
[¬¬]-intro : (Γ ⊢ φ) → (Γ ⊢ ¬(¬ φ))
[¬¬]-intro Γφ =
([¬]-intro
([⊥]-intro
(weaken-union Γφ)
(direct (Right [≡]-intro))
)
)
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{-# OPTIONS --without-K #-}
open import Base
module Homotopy.Skeleton where
private
module Graveyard {i} {A B : Set i} {f : A → B} where
private
data #skeleton₁ : Set i where
#point : A → #skeleton₁
skeleton₁ : Set i
skeleton₁ = #skeleton₁
point : A → skeleton₁
point = #point
postulate -- HIT
link : ∀ a₁ a₂ → f a₁ ≡ f a₂ → point a₁ ≡ point a₂
skeleton₁-rec : ∀ {l} (P : skeleton₁ → Set l)
(point* : ∀ a → P (point a))
(link* : ∀ a₁ a₂ (p : f a₁ ≡ f a₂)
→ transport P (link a₁ a₂ p) (point* a₁) ≡ point* a₂)
→ (x : skeleton₁) → P x
skeleton₁-rec P point* link* (#point a) = point* a
postulate -- HIT
skeleton₁-β-link : ∀ {l} (P : skeleton₁ → Set l)
(point* : ∀ a → P (point a))
(link* : ∀ a₁ a₂ (p : f a₁ ≡ f a₂)
→ transport P (link a₁ a₂ p) (point* a₁) ≡ point* a₂)
a₁ a₂ (p : f a₁ ≡ f a₂)
→ apd (skeleton₁-rec {l} P point* link*) (link a₁ a₂ p)
≡ link* a₁ a₂ p
skeleton₁-rec-nondep : ∀ {l} (P : Set l) (point* : A → P)
(link* : ∀ a₁ a₂ → f a₁ ≡ f a₂ → point* a₁ ≡ point* a₂)
→ (skeleton₁ → P)
skeleton₁-rec-nondep P point* link* (#point a) = point* a
postulate -- HIT
skeleton₁-β-link-nondep : ∀ {l} (P : Set l) (point* : A → P)
(link* : ∀ a₁ a₂ → f a₁ ≡ f a₂ → point* a₁ ≡ point* a₂)
a₁ a₂ (p : f a₁ ≡ f a₂)
→ ap (skeleton₁-rec-nondep P point* link*) (link a₁ a₂ p)
≡ link* a₁ a₂ p
open Graveyard public hiding (skeleton₁)
module _ {i} {A B : Set i} where
skeleton₁ : (A → B) → Set i
skeleton₁ f = Graveyard.skeleton₁ {i} {A} {B} {f}
skeleton₁-lifted : ∀ {f} → skeleton₁ f → B
skeleton₁-lifted {f} = skeleton₁-rec-nondep B f (λ _ _ p → p)
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{-# OPTIONS --cubical-compatible --rewriting --local-confluence-check #-}
open import Agda.Primitive using (Level; _⊔_; Setω; lzero; lsuc)
infix 4 _≡_
data _≡_ {ℓ : Level} {A : Set ℓ} (a : A) : A → Set ℓ where
refl : a ≡ a
{-# BUILTIN REWRITE _≡_ #-}
run : ∀ {ℓ} {A B : Set ℓ} → A ≡ B → A → B
run refl x = x
ap : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) {a₁ a₂} → a₁ ≡ a₂ → f a₁ ≡ f a₂
ap f refl = refl
transport1 : ∀ {a b} {A : Set a} (B : A → Set b) {x y : A} (p : x ≡ y) → B x → B y
transport1 B p = run (ap B p)
ap-const : ∀ {a b} {A : Set a} {B : Set b} (f : B) {a₁ a₂ : A} (p : a₁ ≡ a₂) → ap (\ _ → f) p ≡ refl
ap-const f refl = refl
{-# REWRITE ap-const #-}
ap2 : ∀ {a b rv} {A : Set a} {B : A → Set b} {RV : Set rv} (fn : ∀ a → B a → RV) {a₁ a₂} (pa : a₁ ≡ a₂) {b₁ b₂} (pb : transport1 B pa b₁ ≡ b₂) → fn a₁ b₁ ≡ fn a₂ b₂
ap2 fn refl refl = refl
transport2 : ∀ {a b rv} {A : Set a} {B : A → Set b} (RV : ∀ a → B a → Set rv) {a₁ a₂} (pa : a₁ ≡ a₂) {b₁ b₂} (pb : transport1 B pa b₁ ≡ b₂) → RV a₁ b₁ → RV a₂ b₂
transport2 RV pa pb = run (ap2 RV pa pb)
ap2-const : ∀ {a b rv} {A : Set a} {B : A → Set b} {RV : Set rv} (fn : RV) {a₁ a₂} (pa : a₁ ≡ a₂) {b₁ b₂} (pb : transport1 B pa b₁ ≡ b₂) → ap2 {a} {b} {rv} {A} {B} {RV} (λ _ _ → fn) pa pb ≡ refl
ap2-const fn refl refl = refl
{-# REWRITE ap2-const #-}
ap2-const' : ∀ {a b rv} {A : Set a} {B : A → Set b} {RV : Set rv} (fn : RV) {a₁ a₂} (pa : a₁ ≡ a₂) {b₁ b₂} (pb : transport1 B pa b₁ ≡ b₂) → ap2 {a} {b} {rv} {A} {B} {RV} (λ _ _ → fn) pa pb ≡ refl
ap2-const' {a} {b} {rv} {A} {B} {RV} fn pa pb = {!ap2 {a} {b} {_} {A} {_} {_} (λ _ _ → fn ≡ fn) pa _!}
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module conversion where
open import lib
open import cedille-types
open import ctxt
open import is-free
open import lift
open import rename
open import subst
open import syntax-util
open import general-util
open import to-string
{- Some notes:
-- hnf{TERM} implements erasure as well as normalization.
-- hnf{TYPE} does not descend into terms.
-- definitions are assumed to be in hnf
-}
data unfolding : Set where
no-unfolding : unfolding
unfold : (unfold-all : 𝔹) {- if ff we unfold just the head -}
→ (unfold-lift : 𝔹) {- if tt we unfold lifting types -}
→ (dampen-after-head-beta : 𝔹) {- if tt we will not unfold definitions after a head beta reduction -}
→ (erase : 𝔹) -- if tt erase the term as we unfold
→ unfolding
unfolding-get-erased : unfolding → 𝔹
unfolding-get-erased no-unfolding = ff
unfolding-get-erased (unfold _ _ _ e) = e
unfolding-set-erased : unfolding → 𝔹 → unfolding
unfolding-set-erased no-unfolding e = no-unfolding
unfolding-set-erased (unfold b1 b2 b3 _) e = unfold b1 b2 b3 e
unfold-all : unfolding
unfold-all = unfold tt tt ff tt
unfold-head : unfolding
unfold-head = unfold ff tt ff tt
unfold-head-no-lift : unfolding
unfold-head-no-lift = unfold ff ff ff ff
unfold-head-one : unfolding
unfold-head-one = unfold ff tt tt tt
unfold-dampen : (after-head-beta : 𝔹) → unfolding → unfolding
unfold-dampen _ no-unfolding = no-unfolding
unfold-dampen _ (unfold tt b b' e) = unfold tt b b e -- we do not dampen unfolding when unfolding everywhere
unfold-dampen tt (unfold ff b tt e) = no-unfolding
unfold-dampen tt (unfold ff b ff e) = (unfold ff b ff e)
unfold-dampen ff _ = no-unfolding
unfolding-elab : unfolding → unfolding
unfolding-elab no-unfolding = no-unfolding
unfolding-elab (unfold b b' b'' _) = unfold b b' b'' ff
conv-t : Set → Set
conv-t T = ctxt → T → T → 𝔹
{-# TERMINATING #-}
-- main entry point
-- does not assume erased
conv-term : conv-t term
conv-type : conv-t type
conv-kind : conv-t kind
-- assume erased
conv-terme : conv-t term
conv-argse : conv-t (𝕃 term)
conv-typee : conv-t type
conv-kinde : conv-t kind
-- call hnf, then the conv-X-norm functions
conv-term' : conv-t term
conv-type' : conv-t type
hnf : {ed : exprd} → ctxt → (u : unfolding) → ⟦ ed ⟧ → (is-head : 𝔹) → ⟦ ed ⟧
-- assume head normalized inputs
conv-term-norm : conv-t term
conv-type-norm : conv-t type
conv-kind-norm : conv-t kind
hnf-optClass : ctxt → unfolding → optClass → optClass
-- hnf-tk : ctxt → unfolding → tk → tk
-- does not assume erased
conv-tk : conv-t tk
conv-liftingType : conv-t liftingType
conv-optClass : conv-t optClass
-- conv-optType : conv-t optType
conv-tty* : conv-t (𝕃 tty)
-- assume erased
conv-tke : conv-t tk
conv-liftingTypee : conv-t liftingType
conv-optClasse : conv-t optClass
-- -- conv-optTypee : conv-t optType
conv-ttye* : conv-t (𝕃 tty)
conv-term Γ t t' = conv-terme Γ (erase t) (erase t')
conv-terme Γ t t' with decompose-apps t | decompose-apps t'
conv-terme Γ t t' | Var _ x , args | Var _ x' , args' =
if ctxt-eq-rep Γ x x' && conv-argse Γ args args' then tt else
conv-term' Γ t t'
conv-terme Γ t t' | _ | _ = conv-term' Γ t t'
conv-argse Γ [] [] = tt
conv-argse Γ (a :: args) (a' :: args') = conv-terme Γ a a' && conv-argse Γ args args'
conv-argse Γ _ _ = ff
conv-type Γ t t' = conv-typee Γ (erase t) (erase t')
conv-typee Γ t t' with decompose-tpapps t | decompose-tpapps t'
conv-typee Γ t t' | TpVar _ x , args | TpVar _ x' , args' =
if ctxt-eq-rep Γ x x' && conv-tty* Γ args args' then tt else
conv-type' Γ t t'
conv-typee Γ t t' | _ | _ = conv-type' Γ t t'
conv-kind Γ k k' = conv-kinde Γ (erase k) (erase k')
conv-kinde Γ k k' = conv-kind-norm Γ (hnf Γ unfold-head k tt) (hnf Γ unfold-head k' tt)
conv-term' Γ t t' = conv-term-norm Γ (hnf Γ unfold-head t tt) (hnf Γ unfold-head t' tt)
conv-type' Γ t t' = conv-type-norm Γ (hnf Γ unfold-head t tt) (hnf Γ unfold-head t' tt)
-- is-head is only used in hnf{TYPE}
hnf{TERM} Γ no-unfolding e hd = erase-term e
hnf{TERM} Γ u (Parens _ t _) hd = hnf Γ u t hd
hnf{TERM} Γ u (App t1 Erased t2) hd = hnf Γ u t1 hd
hnf{TERM} Γ u (App t1 NotErased t2) hd with hnf Γ u t1 hd
hnf{TERM} Γ u (App _ NotErased t2) hd | Lam _ _ _ x _ t1 = hnf Γ (unfold-dampen tt u) (subst Γ t2 x t1) hd
hnf{TERM} Γ u (App _ NotErased t2) hd | t1 = App t1 NotErased (hnf Γ (unfold-dampen ff u) t2 hd)
hnf{TERM} Γ u (Lam _ Erased _ _ _ t) hd = hnf Γ u t hd
hnf{TERM} Γ u (Lam _ NotErased _ x oc t) hd with hnf (ctxt-var-decl x Γ) u t hd
hnf{TERM} Γ u (Lam _ NotErased _ x oc t) hd | (App t' NotErased (Var _ x')) with x =string x' && ~ (is-free-in skip-erased x t')
hnf{TERM} Γ u (Lam _ NotErased _ x oc t) hd | (App t' NotErased (Var _ x')) | tt = t' -- eta-contraction
hnf{TERM} Γ u (Lam _ NotErased _ x oc t) hd | (App t' NotErased (Var _ x')) | ff =
Lam posinfo-gen NotErased posinfo-gen x NoClass (App t' NotErased (Var posinfo-gen x'))
hnf{TERM} Γ u (Lam _ NotErased _ x oc t) hd | t' = Lam posinfo-gen NotErased posinfo-gen x NoClass t'
hnf{TERM} Γ u (Let _ (DefTerm _ x _ t) t') hd = hnf Γ u (subst Γ t x t') hd
hnf{TERM} Γ u (Let _ (DefType _ x _ _) t') hd = hnf (ctxt-var-decl x Γ) u t' hd
hnf{TERM} Γ (unfold _ _ _ _) (Var _ x) hd with ctxt-lookup-term-var-def Γ x
hnf{TERM} Γ (unfold _ _ _ _) (Var _ x) hd | nothing = Var posinfo-gen x
hnf{TERM} Γ (unfold ff _ _ e) (Var _ x) hd | just t = erase-if e t -- definitions should be stored in hnf
hnf{TERM} Γ (unfold tt b b' e) (Var _ x) hd | just t = hnf Γ (unfold tt b b' e) t hd -- this might not be fully normalized, only head-normalized
hnf{TERM} Γ u (AppTp t tp) hd = hnf Γ u t hd
hnf{TERM} Γ u (Sigma _ t) hd = hnf Γ u t hd
hnf{TERM} Γ u (Epsilon _ _ _ t) hd = hnf Γ u t hd
hnf{TERM} Γ u (IotaPair _ t1 t2 _ _) hd = hnf Γ u t1 hd
hnf{TERM} Γ u (IotaProj t _ _) hd = hnf Γ u t hd
hnf{TERM} Γ u (Phi _ eq t₁ t₂ _) hd = hnf Γ u t₂ hd
hnf{TERM} Γ u (Rho _ _ _ t _ t') hd = hnf Γ u t' hd
hnf{TERM} Γ u (Chi _ T t') hd = hnf Γ u t' hd
hnf{TERM} Γ u (Delta _ T t') hd = hnf Γ u t' hd
hnf{TERM} Γ u (Theta _ u' t ls) hd = hnf Γ u (lterms-to-term u' t ls) hd
hnf{TERM} Γ u (Beta _ _ (SomeTerm t _)) hd = hnf Γ u t hd
hnf{TERM} Γ u (Beta _ _ NoTerm) hd = id-term
hnf{TERM} Γ u (Open _ _ t) hd = hnf Γ u t hd
hnf{TERM} Γ u x hd = x
hnf{TYPE} Γ no-unfolding e _ = e
hnf{TYPE} Γ u (TpParens _ t _) hd = hnf Γ u t hd
hnf{TYPE} Γ u (NoSpans t _) hd = hnf Γ u t hd
hnf{TYPE} Γ (unfold b b' _ _) (TpVar _ x) ff = TpVar posinfo-gen x
hnf{TYPE} Γ (unfold b b' _ _) (TpVar _ x) tt with ctxt-lookup-type-var-def Γ x
hnf{TYPE} Γ (unfold b b' _ _) (TpVar _ x) tt | just tp = tp
hnf{TYPE} Γ (unfold b b' _ _) (TpVar _ x) tt | nothing = TpVar posinfo-gen x
hnf{TYPE} Γ u (TpAppt tp t) hd with hnf Γ u tp hd
hnf{TYPE} Γ u (TpAppt _ t) hd | TpLambda _ _ x _ tp = hnf Γ u (subst Γ t x tp) hd
hnf{TYPE} Γ u (TpAppt _ t) hd | tp = TpAppt tp (erase-if (unfolding-get-erased u) t)
hnf{TYPE} Γ u (TpApp tp tp') hd with hnf Γ u tp hd
hnf{TYPE} Γ u (TpApp _ tp') hd | TpLambda _ _ x _ tp = hnf Γ u (subst Γ tp' x tp) hd
hnf{TYPE} Γ u (TpApp _ tp') hd | tp with hnf Γ u tp' hd
hnf{TYPE} Γ u (TpApp _ _) hd | tp | tp' = try-pull-lift-types tp tp'
{- given (T1 T2), with T1 and T2 types, see if we can pull a lifting operation from the heads of T1 and T2 to
surround the entire application. If not, just return (T1 T2). -}
where try-pull-lift-types : type → type → type
try-pull-lift-types tp1 tp2 with decompose-tpapps tp1 | decompose-tpapps (hnf Γ u tp2 tt)
try-pull-lift-types tp1 tp2 | Lft _ _ X t l , args1 | Lft _ _ X' t' l' , args2 =
if conv-tty* Γ args1 args2 then
try-pull-term-in Γ t l (length args1) [] []
else
TpApp tp1 tp2
where try-pull-term-in : ctxt → term → liftingType → ℕ → 𝕃 var → 𝕃 liftingType → type
try-pull-term-in Γ t (LiftParens _ l _) n vars ltps = try-pull-term-in Γ t l n vars ltps
try-pull-term-in Γ t (LiftArrow _ l) 0 vars ltps =
recompose-tpapps
(Lft posinfo-gen posinfo-gen X
(Lam* vars (hnf Γ no-unfolding (App t NotErased (App* t' (map (λ v → NotErased , mvar v) vars))) tt))
(LiftArrow* ltps l) , args1)
try-pull-term-in Γ (Lam _ _ _ x _ t) (LiftArrow l1 l2) (suc n) vars ltps =
try-pull-term-in (ctxt-var-decl x Γ) t l2 n (x :: vars) (l1 :: ltps)
try-pull-term-in Γ t (LiftArrow l1 l2) (suc n) vars ltps =
let x = fresh-var "x" (ctxt-binds-var Γ) empty-renamectxt in
try-pull-term-in (ctxt-var-decl x Γ) (App t NotErased (mvar x)) l2 n (x :: vars) (l1 :: ltps)
try-pull-term-in Γ t l n vars ltps = TpApp tp1 tp2
try-pull-lift-types tp1 tp2 | _ | _ = TpApp tp1 tp2
hnf{TYPE} Γ u (Abs _ b _ x atk tp) _ with Abs posinfo-gen b posinfo-gen x atk (hnf (ctxt-var-decl x Γ) u tp ff)
hnf{TYPE} Γ u (Abs _ b _ x atk tp) _ | tp' with to-abs tp'
hnf{TYPE} Γ u (Abs _ _ _ _ _ _) _ | tp'' | just (mk-abs b x atk tt {- x is free in tp -} tp) = Abs posinfo-gen b posinfo-gen x atk tp
hnf{TYPE} Γ u (Abs _ _ _ _ _ _) _ | tp'' | just (mk-abs b x (Tkk k) ff tp) = Abs posinfo-gen b posinfo-gen x (Tkk k) tp
hnf{TYPE} Γ u (Abs _ _ _ _ _ _) _ | tp'' | just (mk-abs b x (Tkt tp') ff tp) = TpArrow tp' b tp
hnf{TYPE} Γ u (Abs _ _ _ _ _ _) _ | tp'' | nothing = tp''
hnf{TYPE} Γ u (TpArrow tp1 arrowtype tp2) _ = TpArrow (hnf Γ u tp1 ff) arrowtype (hnf Γ u tp2 ff)
hnf{TYPE} Γ u (TpEq _ t1 t2 _) _
= TpEq posinfo-gen (erase t1) (erase t2) posinfo-gen
hnf{TYPE} Γ u (TpLambda _ _ x atk tp) _ =
TpLambda posinfo-gen posinfo-gen x (hnf Γ u atk ff) (hnf (ctxt-var-decl x Γ) u tp ff)
hnf{TYPE} Γ u @ (unfold b tt b'' b''') (Lft _ _ y t l) _ =
let t = hnf (ctxt-var-decl y Γ) u t tt in
do-lift Γ (Lft posinfo-gen posinfo-gen y t l) y l (λ t → hnf{TERM} Γ unfold-head t ff) t
-- We need hnf{TYPE} to preserve types' well-kindedness, so we must check if
-- the defined term is being checked against a type and use chi to make sure
-- that wherever it is substituted, the term will have the same directionality.
-- For example, "[e ◂ {a ≃ b} = ρ e' - β] - A (ρ e - a)", would otherwise
-- head-normalize to A (ρ (ρ e' - β) - a), which wouldn't check because it
-- synthesizes the type of "ρ e' - β" (which in turn fails to synthesize the type
-- of "β"). Similar issues could happen if the term is synthesized and it uses a ρ,
-- and then substitutes into a place where it would be checked against a type.
hnf{TYPE} Γ u (TpLet _ (DefTerm _ x ot t) T) hd =
hnf Γ u (subst Γ (Chi posinfo-gen ot t) x T) hd
-- Note that if we ever remove the requirement that type-lambdas have a classifier,
-- we would need to introduce a type-level chi to do the same thing as above.
-- Currently, synthesizing or checking a type should not make a difference.
hnf{TYPE} Γ u (TpLet _ (DefType _ x k T) T') hd = hnf Γ u (subst Γ T x T') hd
hnf{TYPE} Γ u x _ = x
hnf{KIND} Γ no-unfolding e hd = e
hnf{KIND} Γ u (KndParens _ k _) hd = hnf Γ u k hd
hnf{KIND} Γ (unfold _ _ _ _) (KndVar _ x ys) _ with ctxt-lookup-kind-var-def Γ x
... | nothing = KndVar posinfo-gen x ys
... | just (ps , k) = fst $ subst-params-args Γ ps ys k {- do-subst ys ps k
where do-subst : args → params → kind → kind
do-subst (ArgsCons (TermArg _ t) ys) (ParamsCons (Decl _ _ _ x _ _) ps) k = do-subst ys ps (subst Γ t x k)
do-subst (ArgsCons (TypeArg t) ys) (ParamsCons (Decl _ _ _ x _ _) ps) k = do-subst ys ps (subst Γ t x k)
do-subst _ _ k = k -- should not happen -}
hnf{KIND} Γ u (KndPi _ _ x atk k) hd =
if is-free-in check-erased x k then
(KndPi posinfo-gen posinfo-gen x atk k)
else
tk-arrow-kind atk k
hnf{KIND} Γ u x hd = x
hnf{LIFTINGTYPE} Γ u x hd = x
hnf{TK} Γ u (Tkk k) _ = Tkk (hnf Γ u k tt)
hnf{TK} Γ u (Tkt tp) _ = Tkt (hnf Γ u tp ff)
hnf{QUALIF} Γ u x hd = x
hnf{ARG} Γ u x hd = x
hnf-optClass Γ u NoClass = NoClass
hnf-optClass Γ u (SomeClass atk) = SomeClass (hnf Γ u atk ff)
{- this function reduces a term to "head-applicative" normal form,
which avoids unfolding definitions if they would lead to a top-level
lambda-abstraction or top-level application headed by a variable for which we
do not have a (global) definition. -}
{-# TERMINATING #-}
hanf : ctxt → (e : 𝔹) → term → term
hanf Γ e t with hnf Γ (unfolding-set-erased unfold-head-one e) t tt
hanf Γ e t | t' with decompose-apps t'
hanf Γ e t | t' | (Var _ x) , [] = t'
hanf Γ e t | t' | (Var _ x) , args with ctxt-lookup-term-var-def Γ x
hanf Γ e t | t' | (Var _ x) , args | nothing = t'
hanf Γ e t | t' | (Var _ x) , args | just _ = hanf Γ e t'
hanf Γ e t | t' | h , args {- h could be a Lambda if args is [] -} = t
-- unfold across the term-type barrier
hnf-term-type : ctxt → (e : 𝔹) → type → type
hnf-term-type Γ e (TpEq _ t1 t2 _) = TpEq posinfo-gen (hanf Γ e t1) (hanf Γ e t2) posinfo-gen
hnf-term-type Γ e (TpAppt tp t) = hnf Γ (unfolding-set-erased unfold-head e) (TpAppt tp (hanf Γ e t)) tt
hnf-term-type Γ e tp = hnf Γ unfold-head tp tt
conv-term-norm Γ (Var _ x) (Var _ x') = ctxt-eq-rep Γ x x'
-- hnf implements erasure for terms, so we can ignore some subterms for App and Lam cases below
conv-term-norm Γ (App t1 m t2) (App t1' m' t2') = conv-term-norm Γ t1 t1' && conv-term Γ t2 t2'
conv-term-norm Γ (Lam _ l _ x oc t) (Lam _ l' _ x' oc' t') = conv-term (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) t t'
conv-term-norm Γ (Hole _) _ = tt
conv-term-norm Γ _ (Hole _) = tt
-- conv-term-norm Γ (Beta _ _ NoTerm) (Beta _ _ NoTerm) = tt
-- conv-term-norm Γ (Beta _ _ (SomeTerm t _)) (Beta _ _ (SomeTerm t' _)) = conv-term Γ t t'
-- conv-term-norm Γ (Beta _ _ _) (Beta _ _ _) = ff
{- it can happen that a term is equal to a lambda abstraction in head-normal form,
if that lambda-abstraction would eta-contract following some further beta-reductions.
We implement this here by implicitly eta-expanding the variable and continuing
the comparison.
A simple example is
λ v . t ((λ a . a) v) ≃ t
-}
conv-term-norm Γ (Lam _ l _ x oc t) t' =
let x' = fresh-var x (ctxt-binds-var Γ) empty-renamectxt in
conv-term (ctxt-rename x x' Γ) t (App t' NotErased (Var posinfo-gen x'))
conv-term-norm Γ t' (Lam _ l _ x oc t) =
let x' = fresh-var x (ctxt-binds-var Γ) empty-renamectxt in
conv-term (ctxt-rename x x' Γ) (App t' NotErased (Var posinfo-gen x')) t
conv-term-norm Γ _ _ = ff
conv-type-norm Γ (TpVar _ x) (TpVar _ x') = ctxt-eq-rep Γ x x'
conv-type-norm Γ (TpApp t1 t2) (TpApp t1' t2') = conv-type-norm Γ t1 t1' && conv-type Γ t2 t2'
conv-type-norm Γ (TpAppt t1 t2) (TpAppt t1' t2') = conv-type-norm Γ t1 t1' && conv-term Γ t2 t2'
conv-type-norm Γ (Abs _ b _ x atk tp) (Abs _ b' _ x' atk' tp') =
eq-maybeErased b b' && conv-tk Γ atk atk' && conv-type (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) tp tp'
conv-type-norm Γ (TpArrow tp1 a1 tp2) (TpArrow tp1' a2 tp2') = eq-maybeErased a1 a2 && conv-type Γ tp1 tp1' && conv-type Γ tp2 tp2'
conv-type-norm Γ (TpArrow tp1 a tp2) (Abs _ b _ _ (Tkt tp1') tp2') = eq-maybeErased a b && conv-type Γ tp1 tp1' && conv-type Γ tp2 tp2'
conv-type-norm Γ (Abs _ b _ _ (Tkt tp1) tp2) (TpArrow tp1' a tp2') = eq-maybeErased a b && conv-type Γ tp1 tp1' && conv-type Γ tp2 tp2'
conv-type-norm Γ (Iota _ _ x m tp) (Iota _ _ x' m' tp') =
conv-type Γ m m' && conv-type (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) tp tp'
conv-type-norm Γ (TpEq _ t1 t2 _) (TpEq _ t1' t2' _) = conv-term Γ t1 t1' && conv-term Γ t2 t2'
conv-type-norm Γ (Lft _ _ x t l) (Lft _ _ x' t' l') =
conv-liftingType Γ l l' && conv-term (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) t t'
conv-type-norm Γ (TpLambda _ _ x atk tp) (TpLambda _ _ x' atk' tp') =
conv-tk Γ atk atk' && conv-type (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) tp tp'
conv-type-norm Γ _ _ = ff
{- even though hnf turns Pi-kinds where the variable is not free in the body into arrow kinds,
we still need to check off-cases, because normalizing the body of a kind could cause the
bound variable to be erased (hence allowing it to match an arrow kind). -}
conv-kind-norm Γ (KndArrow k k₁) (KndArrow k' k'') = conv-kind Γ k k' && conv-kind Γ k₁ k''
conv-kind-norm Γ (KndArrow k k₁) (KndPi _ _ x (Tkk k') k'') = conv-kind Γ k k' && conv-kind Γ k₁ k''
conv-kind-norm Γ (KndArrow k k₁) _ = ff
conv-kind-norm Γ (KndPi _ _ x (Tkk k₁) k) (KndArrow k' k'') = conv-kind Γ k₁ k' && conv-kind Γ k k''
conv-kind-norm Γ (KndPi _ _ x atk k) (KndPi _ _ x' atk' k'') =
conv-tk Γ atk atk' && conv-kind (ctxt-rename x x' (ctxt-var-decl-if x' Γ)) k k''
conv-kind-norm Γ (KndPi _ _ x (Tkt t) k) (KndTpArrow t' k'') = conv-type Γ t t' && conv-kind Γ k k''
conv-kind-norm Γ (KndPi _ _ x (Tkt t) k) _ = ff
conv-kind-norm Γ (KndPi _ _ x (Tkk k') k) _ = ff
conv-kind-norm Γ (KndTpArrow t k) (KndTpArrow t' k') = conv-type Γ t t' && conv-kind Γ k k'
conv-kind-norm Γ (KndTpArrow t k) (KndPi _ _ x (Tkt t') k') = conv-type Γ t t' && conv-kind Γ k k'
conv-kind-norm Γ (KndTpArrow t k) _ = ff
conv-kind-norm Γ (Star x) (Star x') = tt
conv-kind-norm Γ (Star x) _ = ff
conv-kind-norm Γ _ _ = ff -- should not happen, since the kinds are in hnf
conv-tk Γ tk tk' = conv-tke Γ (erase-tk tk) (erase-tk tk')
conv-tke Γ (Tkk k) (Tkk k') = conv-kind Γ k k'
conv-tke Γ (Tkt t) (Tkt t') = conv-type Γ t t'
conv-tke Γ _ _ = ff
conv-liftingType Γ l l' = conv-liftingTypee Γ (erase l) (erase l')
conv-liftingTypee Γ l l' = conv-kind Γ (liftingType-to-kind l) (liftingType-to-kind l')
conv-optClass Γ NoClass NoClass = tt
conv-optClass Γ (SomeClass x) (SomeClass x') = conv-tk Γ (erase-tk x) (erase-tk x')
conv-optClass Γ _ _ = ff
conv-optClasse Γ NoClass NoClass = tt
conv-optClasse Γ (SomeClass x) (SomeClass x') = conv-tk Γ x x'
conv-optClasse Γ _ _ = ff
-- conv-optType Γ NoType NoType = tt
-- conv-optType Γ (SomeType x) (SomeType x') = conv-type Γ x x'
-- conv-optType Γ _ _ = ff
conv-tty* Γ [] [] = tt
conv-tty* Γ (tterm t :: args) (tterm t' :: args')
= conv-term Γ (erase t) (erase t') && conv-tty* Γ args args'
conv-tty* Γ (ttype t :: args) (ttype t' :: args')
= conv-type Γ (erase t) (erase t') && conv-tty* Γ args args'
conv-tty* Γ _ _ = ff
conv-ttye* Γ [] [] = tt
conv-ttye* Γ (tterm t :: args) (tterm t' :: args') = conv-term Γ t t' && conv-ttye* Γ args args'
conv-ttye* Γ (ttype t :: args) (ttype t' :: args') = conv-type Γ t t' && conv-ttye* Γ args args'
conv-ttye* Γ _ _ = ff
hnf-qualif-term : ctxt → term → term
hnf-qualif-term Γ t = hnf Γ unfold-head (qualif-term Γ t) tt
hnf-qualif-type : ctxt → type → type
hnf-qualif-type Γ t = hnf Γ unfold-head (qualif-type Γ t) tt
hnf-qualif-kind : ctxt → kind → kind
hnf-qualif-kind Γ t = hnf Γ unfold-head (qualif-kind Γ t) tt
ctxt-params-def : params → ctxt → ctxt
ctxt-params-def ps Γ@(mk-ctxt (fn , mn , _ , q) syms i symb-occs d) =
mk-ctxt (fn , mn , ps' , q) syms i symb-occs d
where ps' = qualif-params Γ ps
ctxt-kind-def : posinfo → var → params → kind → ctxt → ctxt
ctxt-kind-def pi v ps2 k Γ@(mk-ctxt (fn , mn , ps1 , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps1 , qualif-insert-params q (mn # v) v ps1)
(trie-insert-append2 syms fn mn v , mn-fn)
(trie-insert i (mn # v) (kind-def (append-params ps1 $ qualif-params Γ ps2) k' , fn , pi))
symb-occs
d
where
k' = hnf Γ unfold-head (qualif-kind Γ k) tt
-- assumption: classifier (i.e. kind) already qualified
ctxt-datatype-def : posinfo → var → params → kind → defDatatype → ctxt → ctxt
ctxt-datatype-def pi v pa k dd Γ@(mk-ctxt (fn , mn , ps , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps , q')
((trie-insert-append2 syms fn mn v) , mn-fn)
(trie-insert i v' (datatype-def pa k , fn , pi))
symb-occs
(trie-insert d v' dd)
where
v' = mn # v
q' = qualif-insert-params q v' v ps
-- assumption: classifier (i.e. kind) already qualified
ctxt-type-def : posinfo → defScope → opacity → var → type → kind → ctxt → ctxt
ctxt-type-def pi s op v t k Γ@(mk-ctxt (fn , mn , ps , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps , q')
((if (s iff localScope) then syms else trie-insert-append2 syms fn mn v) , mn-fn)
(trie-insert i v' (type-def (def-params s ps) op t' k , fn , pi))
symb-occs
d
where
t' = hnf Γ unfold-head (qualif-type Γ t) tt
v' = if s iff localScope then pi % v else mn # v
q' = qualif-insert-params q v' v ps
ctxt-const-def : posinfo → var → type → ctxt → ctxt
ctxt-const-def pi c t Γ@(mk-ctxt mod@(fn , mn , ps , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps , q')
((trie-insert-append2 syms fn mn c) , mn-fn)
(trie-insert i c' (const-def t , fn , pi))
symb-occs
d
where
c' = mn # c
q' = qualif-insert-params q c' c ps
-- assumption: classifier (i.e. type) already qualified
ctxt-term-def : posinfo → defScope → opacity → var → term → type → ctxt → ctxt
ctxt-term-def pi s op v t tp Γ@(mk-ctxt (fn , mn , ps , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps , q')
((if (s iff localScope) then syms else trie-insert-append2 syms fn mn v) , mn-fn)
(trie-insert i v' (term-def (def-params s ps) op t' tp , fn , pi))
symb-occs
d
where
t' = hnf Γ unfold-head (qualif-term Γ t) tt
v' = if s iff localScope then pi % v else mn # v
q' = qualif-insert-params q v' v ps
ctxt-term-udef : posinfo → defScope → opacity → var → term → ctxt → ctxt
ctxt-term-udef pi s op v t Γ@(mk-ctxt (fn , mn , ps , q) (syms , mn-fn) i symb-occs d) = mk-ctxt
(fn , mn , ps , qualif-insert-params q v' v ps)
((if (s iff localScope) then syms else trie-insert-append2 syms fn mn v) , mn-fn)
(trie-insert i v' (term-udef (def-params s ps) op t' , fn , pi))
symb-occs
d
where
t' = hnf Γ unfold-head (qualif-term Γ t) tt
v' = if s iff localScope then pi % v else mn # v
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.types.Paths
open import lib.types.Sigma
open import lib.types.Span
open import lib.types.Pointed
open import lib.types.Pushout
module lib.types.Join where
module _ {i j} (A : Type i) (B : Type j) where
*-span : Span
*-span = span A B (A × B) fst snd
infix 80 _*_
_*_ : Type _
_*_ = Pushout *-span
module JoinElim {i j} {A : Type i} {B : Type j} {k} {P : A * B → Type k}
(left* : (a : A) → P (left a)) (right* : (b : B) → P (right b))
(glue* : (ab : A × B) → left* (fst ab) == right* (snd ab) [ P ↓ glue ab ])
= PushoutElim left* right* glue*
open JoinElim public using () renaming (f to Join-elim)
module JoinRec {i j} {A : Type i} {B : Type j} {k} {D : Type k}
(left* : (a : A) → D) (right* : (b : B) → D)
(glue* : (ab : A × B) → left* (fst ab) == right* (snd ab))
= PushoutRec left* right* glue*
open JoinRec public using () renaming (f to Join-rec)
module _ {i j} (X : Ptd i) (Y : Ptd j) where
*-⊙span : ⊙Span
*-⊙span = ⊙span X Y (X ⊙× Y) ⊙fst ⊙snd
infix 80 _⊙*_
_⊙*_ : Ptd _
_⊙*_ = ⊙Pushout *-⊙span
module _ {i i' j j'} {A : Type i} {A' : Type i'} {B : Type j} {B' : Type j'} where
equiv-* : A ≃ A' → B ≃ B' → A * B ≃ A' * B'
equiv-* eqA eqB = equiv to from to-from from-to where
module To = JoinRec {D = A' * B'} (left ∘ –> eqA) (right ∘ –> eqB) (λ{(a , b) → glue (–> eqA a , –> eqB b)})
module From = JoinRec {D = A * B} (left ∘ <– eqA) (right ∘ <– eqB) (λ{(a , b) → glue (<– eqA a , <– eqB b)})
to : A * B → A' * B'
to = To.f
from : A' * B' → A * B
from = From.f
to-from : ∀ y → to (from y) == y
to-from = Join-elim (ap left ∘ <–-inv-r eqA) (ap right ∘ <–-inv-r eqB) to-from-glue where
to-from-glue : ∀ (ab : A' × B')
→ ap left (<–-inv-r eqA (fst ab)) == ap right (<–-inv-r eqB (snd ab)) [ (λ y → to (from y) == y) ↓ glue ab ]
to-from-glue (a , b) = ↓-app=idf-in $
ap left (<–-inv-r eqA a) ∙' glue (a , b)
=⟨ htpy-natural'-app=cst (λ a → glue (a , b)) (<–-inv-r eqA a) ⟩
glue (–> eqA (<– eqA a) , b)
=⟨ ! $ htpy-natural-cst=app (λ b → glue (–> eqA (<– eqA a) , b)) (<–-inv-r eqB b) ⟩
glue (–> eqA (<– eqA a) , –> eqB (<– eqB b)) ∙ ap right (<–-inv-r eqB b)
=⟨ ! $ To.glue-β (<– eqA a , <– eqB b) |in-ctx (_∙ ap right (<–-inv-r eqB b)) ⟩
ap to (glue (<– eqA a , <– eqB b)) ∙ ap right (<–-inv-r eqB b)
=⟨ ! $ From.glue-β (a , b) |in-ctx (λ p → ap to p ∙ ap right (<–-inv-r eqB b)) ⟩
ap to (ap from (glue (a , b))) ∙ ap right (<–-inv-r eqB b)
=⟨ ! $ ap-∘ to from (glue (a , b)) |in-ctx (_∙ ap right (<–-inv-r eqB b)) ⟩
ap (to ∘ from) (glue (a , b)) ∙ ap right (<–-inv-r eqB b)
∎
from-to : ∀ x → from (to x) == x
from-to = Join-elim (ap left ∘ <–-inv-l eqA) (ap right ∘ <–-inv-l eqB) from-to-glue where
from-to-glue : ∀ (ab : A × B)
→ ap left (<–-inv-l eqA (fst ab)) == ap right (<–-inv-l eqB (snd ab)) [ (λ x → from (to x) == x) ↓ glue ab ]
from-to-glue (a , b) = ↓-app=idf-in $
ap left (<–-inv-l eqA a) ∙' glue (a , b)
=⟨ htpy-natural'-app=cst (λ a → glue (a , b)) (<–-inv-l eqA a) ⟩
glue (<– eqA (–> eqA a) , b)
=⟨ ! $ htpy-natural-cst=app (λ b → glue (<– eqA (–> eqA a) , b)) (<–-inv-l eqB b) ⟩
glue (<– eqA (–> eqA a) , <– eqB (–> eqB b)) ∙ ap right (<–-inv-l eqB b)
=⟨ ! $ From.glue-β (–> eqA a , –> eqB b) |in-ctx (_∙ ap right (<–-inv-l eqB b)) ⟩
ap from (glue (–> eqA a , –> eqB b)) ∙ ap right (<–-inv-l eqB b)
=⟨ ! $ To.glue-β (a , b) |in-ctx (λ p → ap from p ∙ ap right (<–-inv-l eqB b)) ⟩
ap from (ap to (glue (a , b))) ∙ ap right (<–-inv-l eqB b)
=⟨ ! $ ap-∘ from to (glue (a , b)) |in-ctx (_∙ ap right (<–-inv-l eqB b)) ⟩
ap (from ∘ to) (glue (a , b)) ∙ ap right (<–-inv-l eqB b)
∎
module _ {i i' j j'} {X : Ptd i} {X' : Ptd i'} {Y : Ptd j} {Y' : Ptd j'} where
⊙equiv-⊙* : X ⊙≃ X' → Y ⊙≃ Y' → X ⊙* Y ⊙≃ X' ⊙* Y'
⊙equiv-⊙* ⊙eqX ⊙eqY = ⊙≃-in (equiv-* (fst (⊙≃-out ⊙eqX)) (fst (⊙≃-out ⊙eqY))) (ap left (snd (⊙≃-out ⊙eqX)))
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module plfa.part1.Induction where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; cong; sym)
open Eq.≡-Reasoning
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_)
+-assoc : ∀ (m n p : ℕ) → (m + n) + p ≡ m + (n + p)
+-assoc zero n p =
begin
(zero + n) + p
≡⟨⟩
n + p
≡⟨⟩
zero + (n + p)
∎
+-assoc (suc m) n p =
begin
(suc m + n) + p
≡⟨⟩
suc (m + n) + p
≡⟨⟩
suc ((m + n) + p)
-- A relation is said to be a congruence for
-- a given function if it is preserved by applying that function
-- If e is evidence that x ≡ y,
-- then cong f e is evidence that f x ≡ f y,
-- for any function f
-- The correspondence between proof by induction and
-- definition by recursion is one of the most appealing
-- aspects of Agda
-- cong : ∀ (f : A → B) {x y} → x ≡ y → f x ≡ f y
-- ^- suc ^- (m + n) + p ≡ m + (n + p)
-- ----------------------------------------------------------- (=> implies)
-- suc ((m + n) + p) ≡ suc (m + (n + p))
≡⟨ cong suc (+-assoc m n p) ⟩
suc (m + (n + p))
-- cong : ∀ (f : A → B) {x y} → x ≡ y → f x ≡ f y
-- cong f refl = refl
≡⟨⟩
suc m + (n + p)
∎
+-assoc-2 : ∀ (n p : ℕ) → (2 + n) + p ≡ 2 + (n + p)
+-assoc-2 n p =
begin
(2 + n) + p
≡⟨⟩
suc (1 + n) + p
≡⟨⟩
suc ((1 + n) + p)
≡⟨ cong suc (+-assoc-1 n p) ⟩
suc (1 + (n + p))
≡⟨⟩
2 + (n + p)
∎
where
+-assoc-1 : ∀ (n p : ℕ) -> (1 + n) + p ≡ 1 + (n + p)
+-assoc-1 n p =
begin
(1 + n) + p
≡⟨⟩
suc (0 + n) + p
≡⟨⟩
suc ((0 + n) + p)
≡⟨ cong suc (+-assoc-0 n p) ⟩
suc (0 + (n + p))
≡⟨⟩
1 + (n + p)
∎
where
+-assoc-0 : ∀ (n p : ℕ) → (0 + n) + p ≡ 0 + (n + p)
+-assoc-0 n p =
begin
(0 + n) + p
≡⟨⟩
n + p
≡⟨⟩
0 + (n + p)
∎
+-identityᴿ : ∀ (m : ℕ) → m + zero ≡ m
+-identityᴿ zero =
begin
zero + zero
≡⟨⟩
zero
∎
+-identityᴿ (suc m) =
begin
suc m + zero
≡⟨⟩
suc (m + zero)
≡⟨ cong suc (+-identityᴿ m) ⟩
suc m
∎
+-suc : ∀ (m n : ℕ) → m + suc n ≡ suc (m + n)
+-suc zero n =
begin
zero + suc n
≡⟨⟩
suc n
≡⟨⟩
suc (zero + n)
∎
+-suc (suc m) n =
begin
suc m + suc n
≡⟨⟩
suc (m + suc n)
≡⟨ cong suc (+-suc m n) ⟩
suc (suc (m + n))
≡⟨⟩
suc (suc m + n)
∎
+-comm : ∀ (m n : ℕ) → m + n ≡ n + m
+-comm m zero =
begin
m + zero
≡⟨ +-identityᴿ m ⟩
m
≡⟨⟩
zero + m
∎
+-comm m (suc n) =
begin
m + suc n
≡⟨ +-suc m n ⟩
suc (m + n)
≡⟨ cong suc (+-comm m n) ⟩
suc (n + m)
≡⟨⟩
suc n + m
∎
-- Rearranging
-- We can apply associativity to
-- rearrange parentheses however we like
+-rearrange
: ∀ (m n p q : ℕ)
→ (m + n) + (p + q) ≡ m + (n + p) + q
+-rearrange m n p q =
begin
(m + n) + (p + q)
≡⟨ +-assoc m n (p + q) ⟩
m + (n + (p + q))
≡⟨ cong (m +_) (sym (+-assoc n p q)) ⟩
m + ((n + p) + q)
-- +-assoc : (m + n) + p ≡ m + (n + p)
-- sym (+-assoc) : m + (n + p) ≡ (m + n) + p
≡⟨ sym (+-assoc m (n + p) q) ⟩
(m + (n + p)) + q
∎
-- Associativity with rewrite
-- Rewriting avoids not only chains of
-- equations but also the need to invoke cong
+-assoc' : ∀ (m n p : ℕ) → (m + n) + p ≡ m + (n + p)
+-assoc' zero n p = refl
+-assoc' (suc m) n p rewrite +-assoc' m n p = refl
+-identity' : ∀ (n : ℕ) → n + zero ≡ n
+-identity' zero = refl
+-identity' (suc n) rewrite +-identity' n = refl
+-suc' : ∀ (m n : ℕ) → m + suc n ≡ suc (m + n)
+-suc' zero n = refl
+-suc' (suc m) n rewrite +-suc' m n = refl
+-comm' : ∀ (m n : ℕ) → m + n ≡ n + m
+-comm' m zero rewrite +-identity' m = refl
+-comm' m (suc n) rewrite +-suc' m n | +-comm' m n = refl
-- Building proofs interactively
+-assoc'' : ∀ (m n p : ℕ) → (m + n) + p ≡ m + (n + p)
+-assoc'' zero n p = refl
+-assoc'' (suc m) n p rewrite +-assoc'' m n p = refl
-- Exercise
-- Note:
-- sym -- rewrites the left side of the Goal
+-swap : ∀ (m n p : ℕ) → m + (n + p) ≡ n + (m + p)
+-swap zero n p = refl
+-swap (suc m) n p rewrite
+-assoc'' m n p
| +-suc n (m + p)
| +-swap m n p
= refl
-- (suc m + n) * p ≡ suc m * p + n * p
-- p + (m * p + n * p) ≡ p + m * p + n * p
*-distrib-+ : ∀ (m n p : ℕ) → (m + n) * p ≡ m * p + n * p
*-distrib-+ zero n p = refl
*-distrib-+ (suc m) n p rewrite
*-distrib-+ m n p
| sym (+-assoc p (m * p) (n * p))
= refl
-- (n + m * n) * p ≡ n * p + m * (n * p)
*-assoc : ∀ (m n p : ℕ) → (m * n) * p ≡ m * (n * p)
*-assoc zero n p = refl
*-assoc (suc m) n p rewrite
*-distrib-+ n (m * n) p
| *-assoc m n p
= refl
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module ProjectingRecordMeta where
data _==_ {A : Set}(a : A) : A -> Set where
refl : a == a
-- Andreas, Feb/Apr 2011
record Prod (A B : Set) : Set where
constructor _,_
field
fst : A
snd : B
open Prod public
testProj : {A B : Set}(y z : Prod A B) ->
let X : Prod A B
X = _ -- Solution: fst y , snd z
in (C : Set) -> (fst X == fst y -> snd X == snd z -> C) -> C
testProj y z C k = k refl refl
-- ok, Agda handles projections properly during unification
testProj' : {A B : Set}(y z : Prod A B) ->
let X : Prod A B
X = _ -- Solution: fst y , snd z
in Prod (fst X == fst y) (snd X == snd z)
testProj' y z = refl , refl
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------------------------------------------------------------------------
-- The Agda standard library
--
-- "Finite" sets indexed on coinductive "natural" numbers
------------------------------------------------------------------------
module Data.Cofin where
open import Coinduction
open import Data.Conat as Conat using (Coℕ; suc; ∞ℕ)
open import Data.Nat using (ℕ; zero; suc)
open import Data.Fin using (Fin; zero; suc)
------------------------------------------------------------------------
-- The type
-- Note that Cofin ∞ℕ is /not/ finite. Note also that this is not a
-- coinductive type, but it is indexed on a coinductive type.
data Cofin : Coℕ → Set where
zero : ∀ {n} → Cofin (suc n)
suc : ∀ {n} (i : Cofin (♭ n)) → Cofin (suc n)
------------------------------------------------------------------------
-- Some operations
fromℕ : ℕ → Cofin ∞ℕ
fromℕ zero = zero
fromℕ (suc n) = suc (fromℕ n)
toℕ : ∀ {n} → Cofin n → ℕ
toℕ zero = zero
toℕ (suc i) = suc (toℕ i)
fromFin : ∀ {n} → Fin n → Cofin (Conat.fromℕ n)
fromFin zero = zero
fromFin (suc i) = suc (fromFin i)
toFin : ∀ n → Cofin (Conat.fromℕ n) → Fin n
toFin zero ()
toFin (suc n) zero = zero
toFin (suc n) (suc i) = suc (toFin n i)
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module Pi1r where
open import Data.Empty
open import Data.Unit
open import Data.Sum
open import Data.Product
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
open import Groupoid
-- infix 2 _∎ -- equational reasoning for paths
-- infixr 2 _≡⟨_⟩_ -- equational reasoning for paths
infixr 10 _◎_
infixr 30 _⟷_
------------------------------------------------------------------------------
-- Level 0:
-- Types at this level are just plain sets with no interesting path structure.
-- The path structure is defined at levels 1 and beyond.
data U : Set where
ZERO : U
ONE : U
PLUS : U → U → U
TIMES : U → U → U
⟦_⟧ : U → Set
⟦ ZERO ⟧ = ⊥
⟦ ONE ⟧ = ⊤
⟦ PLUS t₁ t₂ ⟧ = ⟦ t₁ ⟧ ⊎ ⟦ t₂ ⟧
⟦ TIMES t₁ t₂ ⟧ = ⟦ t₁ ⟧ × ⟦ t₂ ⟧
-- Programs
-- We use pointed types; programs map a pointed type to another
-- In other words, each program takes one particular value to another; if we
-- want to work on another value, we generally use another program
record U• : Set where
constructor •[_,_]
field
∣_∣ : U
• : ⟦ ∣_∣ ⟧
open U•
-- examples of plain types, values, and pointed types
ONE• : U•
ONE• = •[ ONE , tt ]
BOOL : U
BOOL = PLUS ONE ONE
BOOL² : U
BOOL² = TIMES BOOL BOOL
TRUE : ⟦ BOOL ⟧
TRUE = inj₁ tt
FALSE : ⟦ BOOL ⟧
FALSE = inj₂ tt
BOOL•F : U•
BOOL•F = •[ BOOL , FALSE ]
BOOL•T : U•
BOOL•T = •[ BOOL , TRUE ]
-- The actual programs are the commutative semiring isomorphisms between
-- pointed types.
data _⟷_ : U• → U• → Set where
unite₊ : ∀ {t v} → •[ PLUS ZERO t , inj₂ v ] ⟷ •[ t , v ]
uniti₊ : ∀ {t v} → •[ t , v ] ⟷ •[ PLUS ZERO t , inj₂ v ]
swap1₊ : ∀ {t₁ t₂ v₁} →
•[ PLUS t₁ t₂ , inj₁ v₁ ] ⟷ •[ PLUS t₂ t₁ , inj₂ v₁ ]
swap2₊ : ∀ {t₁ t₂ v₂} →
•[ PLUS t₁ t₂ , inj₂ v₂ ] ⟷ •[ PLUS t₂ t₁ , inj₁ v₂ ]
assocl1₊ : ∀ {t₁ t₂ t₃ v₁} →
•[ PLUS t₁ (PLUS t₂ t₃) , inj₁ v₁ ] ⟷
•[ PLUS (PLUS t₁ t₂) t₃ , inj₁ (inj₁ v₁) ]
assocl2₊ : ∀ {t₁ t₂ t₃ v₂} →
•[ PLUS t₁ (PLUS t₂ t₃) , inj₂ (inj₁ v₂) ] ⟷
•[ PLUS (PLUS t₁ t₂) t₃ , inj₁ (inj₂ v₂) ]
assocl3₊ : ∀ {t₁ t₂ t₃ v₃} →
•[ PLUS t₁ (PLUS t₂ t₃) , inj₂ (inj₂ v₃) ] ⟷
•[ PLUS (PLUS t₁ t₂) t₃ , inj₂ v₃ ]
assocr1₊ : ∀ {t₁ t₂ t₃ v₁} →
•[ PLUS (PLUS t₁ t₂) t₃ , inj₁ (inj₁ v₁) ] ⟷
•[ PLUS t₁ (PLUS t₂ t₃) , inj₁ v₁ ]
assocr2₊ : ∀ {t₁ t₂ t₃ v₂} →
•[ PLUS (PLUS t₁ t₂) t₃ , inj₁ (inj₂ v₂) ] ⟷
•[ PLUS t₁ (PLUS t₂ t₃) , inj₂ (inj₁ v₂) ]
assocr3₊ : ∀ {t₁ t₂ t₃ v₃} →
•[ PLUS (PLUS t₁ t₂) t₃ , inj₂ v₃ ] ⟷
•[ PLUS t₁ (PLUS t₂ t₃) , inj₂ (inj₂ v₃) ]
unite⋆ : ∀ {t v} → •[ TIMES ONE t , (tt , v) ] ⟷ •[ t , v ]
uniti⋆ : ∀ {t v} → •[ t , v ] ⟷ •[ TIMES ONE t , (tt , v) ]
swap⋆ : ∀ {t₁ t₂ v₁ v₂} →
•[ TIMES t₁ t₂ , (v₁ , v₂) ] ⟷ •[ TIMES t₂ t₁ , (v₂ , v₁) ]
assocl⋆ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} →
•[ TIMES t₁ (TIMES t₂ t₃) , (v₁ , (v₂ , v₃)) ] ⟷
•[ TIMES (TIMES t₁ t₂) t₃ , ((v₁ , v₂) , v₃) ]
assocr⋆ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} →
•[ TIMES (TIMES t₁ t₂) t₃ , ((v₁ , v₂) , v₃) ] ⟷
•[ TIMES t₁ (TIMES t₂ t₃) , (v₁ , (v₂ , v₃)) ]
distz : ∀ {t v absurd} →
•[ TIMES ZERO t , (absurd , v) ] ⟷ •[ ZERO , absurd ]
factorz : ∀ {t v absurd} →
•[ ZERO , absurd ] ⟷ •[ TIMES ZERO t , (absurd , v) ]
dist1 : ∀ {t₁ t₂ t₃ v₁ v₃} →
•[ TIMES (PLUS t₁ t₂) t₃ , (inj₁ v₁ , v₃) ] ⟷
•[ PLUS (TIMES t₁ t₃) (TIMES t₂ t₃) , inj₁ (v₁ , v₃) ]
dist2 : ∀ {t₁ t₂ t₃ v₂ v₃} →
•[ TIMES (PLUS t₁ t₂) t₃ , (inj₂ v₂ , v₃) ] ⟷
•[ PLUS (TIMES t₁ t₃) (TIMES t₂ t₃) , inj₂ (v₂ , v₃) ]
factor1 : ∀ {t₁ t₂ t₃ v₁ v₃} →
•[ PLUS (TIMES t₁ t₃) (TIMES t₂ t₃) , inj₁ (v₁ , v₃) ] ⟷
•[ TIMES (PLUS t₁ t₂) t₃ , (inj₁ v₁ , v₃) ]
factor2 : ∀ {t₁ t₂ t₃ v₂ v₃} →
•[ PLUS (TIMES t₁ t₃) (TIMES t₂ t₃) , inj₂ (v₂ , v₃) ] ⟷
•[ TIMES (PLUS t₁ t₂) t₃ , (inj₂ v₂ , v₃) ]
id⟷ : ∀ {t v} → •[ t , v ] ⟷ •[ t , v ]
_◎_ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} → (•[ t₁ , v₁ ] ⟷ •[ t₂ , v₂ ]) →
(•[ t₂ , v₂ ] ⟷ •[ t₃ , v₃ ]) →
(•[ t₁ , v₁ ] ⟷ •[ t₃ , v₃ ])
_⊕1_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(•[ t₁ , v₁ ] ⟷ •[ t₃ , v₃ ]) → (•[ t₂ , v₂ ] ⟷ •[ t₄ , v₄ ]) →
(•[ PLUS t₁ t₂ , inj₁ v₁ ] ⟷ •[ PLUS t₃ t₄ , inj₁ v₃ ])
_⊕2_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(•[ t₁ , v₁ ] ⟷ •[ t₃ , v₃ ]) → (•[ t₂ , v₂ ] ⟷ •[ t₄ , v₄ ]) →
(•[ PLUS t₁ t₂ , inj₂ v₂ ] ⟷ •[ PLUS t₃ t₄ , inj₂ v₄ ])
_⊗_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(•[ t₁ , v₁ ] ⟷ •[ t₃ , v₃ ]) → (•[ t₂ , v₂ ] ⟷ •[ t₄ , v₄ ]) →
(•[ TIMES t₁ t₂ , (v₁ , v₂) ] ⟷ •[ TIMES t₃ t₄ , (v₃ , v₄) ])
loop : ∀ {t v} → •[ t , v ] ⟷ •[ t , v ]
! : {t₁ t₂ : U•} → (t₁ ⟷ t₂) → (t₂ ⟷ t₁)
! unite₊ = uniti₊
! uniti₊ = unite₊
! swap1₊ = swap2₊
! swap2₊ = swap1₊
! assocl1₊ = assocr1₊
! assocl2₊ = assocr2₊
! assocl3₊ = assocr3₊
! assocr1₊ = assocl1₊
! assocr2₊ = assocl2₊
! assocr3₊ = assocl3₊
! unite⋆ = uniti⋆
! uniti⋆ = unite⋆
! swap⋆ = swap⋆
! assocl⋆ = assocr⋆
! assocr⋆ = assocl⋆
! distz = factorz
! factorz = distz
! dist1 = factor1
! dist2 = factor2
! factor1 = dist1
! factor2 = dist2
! id⟷ = id⟷
! (c₁ ◎ c₂) = ! c₂ ◎ ! c₁
! (c₁ ⊕1 c₂) = ! c₁ ⊕1 ! c₂
! (c₁ ⊕2 c₂) = ! c₁ ⊕2 ! c₂
! (c₁ ⊗ c₂) = ! c₁ ⊗ ! c₂
! loop = loop
-- example programs
NOT•T : •[ BOOL , TRUE ] ⟷ •[ BOOL , FALSE ]
NOT•T = swap1₊
NOT•F : •[ BOOL , FALSE ] ⟷ •[ BOOL , TRUE ]
NOT•F = swap2₊
CNOT•Fx : {b : ⟦ BOOL ⟧} →
•[ BOOL² , (FALSE , b) ] ⟷ •[ BOOL² , (FALSE , b) ]
CNOT•Fx = dist2 ◎ ((id⟷ ⊗ NOT•F) ⊕2 id⟷) ◎ factor2
CNOT•TF : •[ BOOL² , (TRUE , FALSE) ] ⟷ •[ BOOL² , (TRUE , TRUE) ]
CNOT•TF = dist1 ◎
((id⟷ ⊗ NOT•F) ⊕1 (id⟷ {TIMES ONE BOOL} {(tt , TRUE)})) ◎
factor1
CNOT•TT : •[ BOOL² , (TRUE , TRUE) ] ⟷ •[ BOOL² , (TRUE , FALSE) ]
CNOT•TT = dist1 ◎
((id⟷ ⊗ NOT•T) ⊕1 (id⟷ {TIMES ONE BOOL} {(tt , TRUE)})) ◎
factor1
-- The evaluation of a program is not done in order to figure out the output
-- value. Both the input and output values are encoded in the type of the
-- program; what the evaluation does is follow the path to constructively
-- reach the ouput value from the input value. Even though programs of the
-- same pointed types are, by definition, observationally equivalent, they
-- may follow different paths. At this point, we simply declare that all such
-- programs are "the same." At the next level, we will weaken this "path
-- irrelevant" equivalence and reason about which paths can be equated to
-- other paths via 2paths etc.
-- Even though individual types are sets, the universe of types is a
-- groupoid. The objects of this groupoid are the pointed types; the
-- morphisms are the programs; and the equivalence of programs is the
-- degenerate observational equivalence that equates every two programs that
-- are extensionally equivalent.
_obs≅_ : {t₁ t₂ : U•} → (c₁ c₂ : t₁ ⟷ t₂) → Set
c₁ obs≅ c₂ = ⊤
UG : 1Groupoid
UG = record
{ set = U•
; _↝_ = _⟷_
; _≈_ = _obs≅_
; id = id⟷
; _∘_ = λ y⟷z x⟷y → x⟷y ◎ y⟷z
; _⁻¹ = !
; lneutr = λ _ → tt
; rneutr = λ _ → tt
; assoc = λ _ _ _ → tt
; equiv = record { refl = tt
; sym = λ _ → tt
; trans = λ _ _ → tt
}
; linv = λ _ → tt
; rinv = λ _ → tt
; ∘-resp-≈ = λ _ _ → tt
}
------------------------------------------------------------------------------
-- Level 1:
-- Types are sets of paths. The paths are defined at the previous level
-- (level 0). At level 1, there is no interesting 2path structure. From
-- the perspective of level 0, we have points with non-trivial paths between
-- them, i.e., we have a groupoid. The paths cross type boundaries, i.e., we
-- have heterogeneous equality
-- types
data 1U : Set where
1ZERO : 1U -- empty set of paths
1ONE : 1U -- a trivial path
1PLUS : 1U → 1U → 1U -- disjoint union of paths
1TIMES : 1U → 1U → 1U -- pairs of paths
PATH : (t₁ t₂ : U•) → 1U -- level 0 paths between values
-- values
data Path (t₁ t₂ : U•) : Set where
path : (c : t₁ ⟷ t₂) → Path t₁ t₂
1⟦_⟧ : 1U → Set
1⟦ 1ZERO ⟧ = ⊥
1⟦ 1ONE ⟧ = ⊤
1⟦ 1PLUS t₁ t₂ ⟧ = 1⟦ t₁ ⟧ ⊎ 1⟦ t₂ ⟧
1⟦ 1TIMES t₁ t₂ ⟧ = 1⟦ t₁ ⟧ × 1⟦ t₂ ⟧
1⟦ PATH t₁ t₂ ⟧ = Path t₁ t₂
-- examples
T⟷F : Set
T⟷F = Path BOOL•T BOOL•F
F⟷T : Set
F⟷T = Path BOOL•F BOOL•T
-- all the following are paths from T to F; we will show below that they
-- are equivalent using 2paths
p₁ p₂ p₃ p₄ p₅ : T⟷F
p₁ = path NOT•T
p₂ = path (id⟷ ◎ NOT•T)
p₃ = path (NOT•T ◎ NOT•F ◎ NOT•T)
p₄ = path (NOT•T ◎ id⟷)
p₅ = path (uniti⋆ ◎ swap⋆ ◎ (NOT•T ⊗ id⟷) ◎ swap⋆ ◎ unite⋆)
p₆ : (T⟷F × T⟷F) ⊎ F⟷T
p₆ = inj₁ (p₁ , p₂)
-- Programs map paths to paths. We also use pointed types.
record 1U• : Set where
constructor 1•[_,_]
field
1∣_∣ : 1U
1• : 1⟦ 1∣_∣ ⟧
open 1U•
data _⇔_ : 1U• → 1U• → Set where
unite₊ : ∀ {t v} → 1•[ 1PLUS 1ZERO t , inj₂ v ] ⇔ 1•[ t , v ]
uniti₊ : ∀ {t v} → 1•[ t , v ] ⇔ 1•[ 1PLUS 1ZERO t , inj₂ v ]
swap1₊ : ∀ {t₁ t₂ v₁} →
1•[ 1PLUS t₁ t₂ , inj₁ v₁ ] ⇔ 1•[ 1PLUS t₂ t₁ , inj₂ v₁ ]
swap2₊ : ∀ {t₁ t₂ v₂} →
1•[ 1PLUS t₁ t₂ , inj₂ v₂ ] ⇔ 1•[ 1PLUS t₂ t₁ , inj₁ v₂ ]
assocl1₊ : ∀ {t₁ t₂ t₃ v₁} →
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₁ v₁ ] ⇔
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₁ (inj₁ v₁) ]
assocl2₊ : ∀ {t₁ t₂ t₃ v₂} →
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₂ (inj₁ v₂) ] ⇔
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₁ (inj₂ v₂) ]
assocl3₊ : ∀ {t₁ t₂ t₃ v₃} →
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₂ (inj₂ v₃) ] ⇔
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₂ v₃ ]
assocr1₊ : ∀ {t₁ t₂ t₃ v₁} →
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₁ (inj₁ v₁) ] ⇔
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₁ v₁ ]
assocr2₊ : ∀ {t₁ t₂ t₃ v₂} →
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₁ (inj₂ v₂) ] ⇔
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₂ (inj₁ v₂) ]
assocr3₊ : ∀ {t₁ t₂ t₃ v₃} →
1•[ 1PLUS (1PLUS t₁ t₂) t₃ , inj₂ v₃ ] ⇔
1•[ 1PLUS t₁ (1PLUS t₂ t₃) , inj₂ (inj₂ v₃) ]
unite⋆ : ∀ {t v} → 1•[ 1TIMES 1ONE t , (tt , v) ] ⇔ 1•[ t , v ]
uniti⋆ : ∀ {t v} → 1•[ t , v ] ⇔ 1•[ 1TIMES 1ONE t , (tt , v) ]
swap⋆ : ∀ {t₁ t₂ v₁ v₂} →
1•[ 1TIMES t₁ t₂ , (v₁ , v₂) ] ⇔ 1•[ 1TIMES t₂ t₁ , (v₂ , v₁) ]
assocl⋆ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} →
1•[ 1TIMES t₁ (1TIMES t₂ t₃) , (v₁ , (v₂ , v₃)) ] ⇔
1•[ 1TIMES (1TIMES t₁ t₂) t₃ , ((v₁ , v₂) , v₃) ]
assocr⋆ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} →
1•[ 1TIMES (1TIMES t₁ t₂) t₃ , ((v₁ , v₂) , v₃) ] ⇔
1•[ 1TIMES t₁ (1TIMES t₂ t₃) , (v₁ , (v₂ , v₃)) ]
distz : ∀ {t v absurd} →
1•[ 1TIMES 1ZERO t , (absurd , v) ] ⇔ 1•[ 1ZERO , absurd ]
factorz : ∀ {t v absurd} →
1•[ 1ZERO , absurd ] ⇔ 1•[ 1TIMES 1ZERO t , (absurd , v) ]
dist1 : ∀ {t₁ t₂ t₃ v₁ v₃} →
1•[ 1TIMES (1PLUS t₁ t₂) t₃ , (inj₁ v₁ , v₃) ] ⇔
1•[ 1PLUS (1TIMES t₁ t₃) (1TIMES t₂ t₃) , inj₁ (v₁ , v₃) ]
dist2 : ∀ {t₁ t₂ t₃ v₂ v₃} →
1•[ 1TIMES (1PLUS t₁ t₂) t₃ , (inj₂ v₂ , v₃) ] ⇔
1•[ 1PLUS (1TIMES t₁ t₃) (1TIMES t₂ t₃) , inj₂ (v₂ , v₃) ]
factor1 : ∀ {t₁ t₂ t₃ v₁ v₃} →
1•[ 1PLUS (1TIMES t₁ t₃) (1TIMES t₂ t₃) , inj₁ (v₁ , v₃) ] ⇔
1•[ 1TIMES (1PLUS t₁ t₂) t₃ , (inj₁ v₁ , v₃) ]
factor2 : ∀ {t₁ t₂ t₃ v₂ v₃} →
1•[ 1PLUS (1TIMES t₁ t₃) (1TIMES t₂ t₃) , inj₂ (v₂ , v₃) ] ⇔
1•[ 1TIMES (1PLUS t₁ t₂) t₃ , (inj₂ v₂ , v₃) ]
id⇔ : ∀ {t v} → 1•[ t , v ] ⇔ 1•[ t , v ]
_◎_ : ∀ {t₁ t₂ t₃ v₁ v₂ v₃} → (1•[ t₁ , v₁ ] ⇔ 1•[ t₂ , v₂ ]) →
(1•[ t₂ , v₂ ] ⇔ 1•[ t₃ , v₃ ]) →
(1•[ t₁ , v₁ ] ⇔ 1•[ t₃ , v₃ ])
_⊕1_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(1•[ t₁ , v₁ ] ⇔ 1•[ t₃ , v₃ ]) → (1•[ t₂ , v₂ ] ⇔ 1•[ t₄ , v₄ ]) →
(1•[ 1PLUS t₁ t₂ , inj₁ v₁ ] ⇔ 1•[ 1PLUS t₃ t₄ , inj₁ v₃ ])
_⊕2_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(1•[ t₁ , v₁ ] ⇔ 1•[ t₃ , v₃ ]) → (1•[ t₂ , v₂ ] ⇔ 1•[ t₄ , v₄ ]) →
(1•[ 1PLUS t₁ t₂ , inj₂ v₂ ] ⇔ 1•[ 1PLUS t₃ t₄ , inj₂ v₄ ])
_⊗_ : ∀ {t₁ t₂ t₃ t₄ v₁ v₂ v₃ v₄} →
(1•[ t₁ , v₁ ] ⇔ 1•[ t₃ , v₃ ]) → (1•[ t₂ , v₂ ] ⇔ 1•[ t₄ , v₄ ]) →
(1•[ 1TIMES t₁ t₂ , (v₁ , v₂) ] ⇔ 1•[ 1TIMES t₃ t₄ , (v₃ , v₄) ])
lidl : ∀ {t₁ t₂} → {c : t₁ ⟷ t₂} →
1•[ PATH t₁ t₂ , path (id⟷ ◎ c) ] ⇔ 1•[ PATH t₁ t₂ , path c ]
lidr : ∀ {t₁ t₂} → {c : t₁ ⟷ t₂} →
1•[ PATH t₁ t₂ , path c ] ⇔ 1•[ PATH t₁ t₂ , path (id⟷ ◎ c) ]
ridl : ∀ {t₁ t₂} → {c : t₁ ⟷ t₂} →
1•[ PATH t₁ t₂ , path (c ◎ id⟷) ] ⇔ 1•[ PATH t₁ t₂ , path c ]
ridr : ∀ {t₁ t₂} → {c : t₁ ⟷ t₂} →
1•[ PATH t₁ t₂ , path c ] ⇔ 1•[ PATH t₁ t₂ , path (c ◎ id⟷) ]
assocl : ∀ {t₁ t₂ t₃ t₄} →
{c₁ : t₁ ⟷ t₂} {c₂ : t₂ ⟷ t₃} {c₃ : t₃ ⟷ t₄} →
1•[ PATH t₁ t₄ , path (c₁ ◎ (c₂ ◎ c₃)) ] ⇔
1•[ PATH t₁ t₄ , path ((c₁ ◎ c₂) ◎ c₃) ]
assocr : ∀ {t₁ t₂ t₃ t₄} →
{c₁ : t₁ ⟷ t₂} {c₂ : t₂ ⟷ t₃} {c₃ : t₃ ⟷ t₄} →
1•[ PATH t₁ t₄ , path ((c₁ ◎ c₂) ◎ c₃) ] ⇔
1•[ PATH t₁ t₄ , path (c₁ ◎ (c₂ ◎ c₃)) ]
unit₊l : ∀ {t v} →
1•[ PATH (•[ PLUS ZERO t , inj₂ v ]) (•[ PLUS ZERO t , inj₂ v ]) ,
path (unite₊ ◎ uniti₊) ] ⇔
1•[ PATH (•[ PLUS ZERO t , inj₂ v ]) (•[ PLUS ZERO t , inj₂ v ]) ,
path id⟷ ]
unit₊r : ∀ {t v} →
1•[ PATH (•[ PLUS ZERO t , inj₂ v ]) (•[ PLUS ZERO t , inj₂ v ]) ,
path id⟷ ] ⇔
1•[ PATH (•[ PLUS ZERO t , inj₂ v ]) (•[ PLUS ZERO t , inj₂ v ]) ,
path (unite₊ ◎ uniti₊) ]
resp◎ : ∀ {t₁ t₂ t₃} →
{c₁ : t₁ ⟷ t₂} {c₂ : t₂ ⟷ t₃} {c₃ : t₁ ⟷ t₂} {c₄ : t₂ ⟷ t₃} →
(1•[ PATH t₁ t₂ , path c₁ ] ⇔ 1•[ PATH t₁ t₂ , path c₃ ]) →
(1•[ PATH t₂ t₃ , path c₂ ] ⇔ 1•[ PATH t₂ t₃ , path c₄ ]) →
1•[ PATH t₁ t₃ , path (c₁ ◎ c₂) ] ⇔ 1•[ PATH t₁ t₃ , path (c₃ ◎ c₄) ]
1! : {t₁ t₂ : 1U•} → (t₁ ⇔ t₂) → (t₂ ⇔ t₁)
1! unite₊ = uniti₊
1! uniti₊ = unite₊
1! swap1₊ = swap2₊
1! swap2₊ = swap1₊
1! assocl1₊ = assocr1₊
1! assocl2₊ = assocr2₊
1! assocl3₊ = assocr3₊
1! assocr1₊ = assocl1₊
1! assocr2₊ = assocl2₊
1! assocr3₊ = assocl3₊
1! unite⋆ = uniti⋆
1! uniti⋆ = unite⋆
1! swap⋆ = swap⋆
1! assocl⋆ = assocr⋆
1! assocr⋆ = assocl⋆
1! distz = factorz
1! factorz = distz
1! dist1 = factor1
1! dist2 = factor2
1! factor1 = dist1
1! factor2 = dist2
1! id⇔ = id⇔
1! (c₁ ◎ c₂) = 1! c₂ ◎ 1! c₁
1! (c₁ ⊕1 c₂) = 1! c₁ ⊕1 1! c₂
1! (c₁ ⊕2 c₂) = 1! c₁ ⊕2 1! c₂
1! (c₁ ⊗ c₂) = 1! c₁ ⊗ 1! c₂
1! (resp◎ c₁ c₂) = resp◎ (1! c₁) (1! c₂)
1! ridl = ridr
1! ridr = ridl
1! lidl = lidr
1! lidr = lidl
1! assocl = assocr
1! assocr = assocl
1! unit₊l = unit₊r
1! unit₊r = unit₊l
1!≡ : {t₁ t₂ : 1U•} → (c : t₁ ⇔ t₂) → 1! (1! c) ≡ c
1!≡ unite₊ = refl
1!≡ uniti₊ = refl
1!≡ swap1₊ = refl
1!≡ swap2₊ = refl
1!≡ assocl1₊ = refl
1!≡ assocl2₊ = refl
1!≡ assocl3₊ = refl
1!≡ assocr1₊ = refl
1!≡ assocr2₊ = refl
1!≡ assocr3₊ = refl
1!≡ unite⋆ = refl
1!≡ uniti⋆ = refl
1!≡ swap⋆ = refl
1!≡ assocl⋆ = refl
1!≡ assocr⋆ = refl
1!≡ distz = refl
1!≡ factorz = refl
1!≡ dist1 = refl
1!≡ dist2 = refl
1!≡ factor1 = refl
1!≡ factor2 = refl
1!≡ id⇔ = refl
1!≡ (c₁ ◎ c₂) = cong₂ (λ c₁ c₂ → c₁ ◎ c₂) (1!≡ c₁) (1!≡ c₂)
1!≡ (c₁ ⊕1 c₂) = cong₂ (λ c₁ c₂ → c₁ ⊕1 c₂) (1!≡ c₁) (1!≡ c₂)
1!≡ (c₁ ⊕2 c₂) = cong₂ (λ c₁ c₂ → c₁ ⊕2 c₂) (1!≡ c₁) (1!≡ c₂)
1!≡ (c₁ ⊗ c₂) = cong₂ (λ c₁ c₂ → c₁ ⊗ c₂) (1!≡ c₁) (1!≡ c₂)
1!≡ lidl = refl
1!≡ lidr = refl
1!≡ ridl = refl
1!≡ ridr = refl
1!≡ (resp◎ c₁ c₂) = cong₂ (λ c₁ c₂ → resp◎ c₁ c₂) (1!≡ c₁) (1!≡ c₂)
1!≡ assocl = refl
1!≡ assocr = refl
1!≡ unit₊l = refl
1!≡ unit₊r = refl
-- sane syntax
α₁ : 1•[ PATH BOOL•T BOOL•F , p₁ ] ⇔ 1•[ PATH BOOL•T BOOL•F , p₁ ]
α₁ = id⇔
α₂ : 1•[ PATH BOOL•T BOOL•F , p₁ ] ⇔ 1•[ PATH BOOL•T BOOL•F , p₂ ]
α₂ = lidr
{--
_≡⟨_⟩_ : {t₁ t₂ : U•} (c₁ : t₁ ⟷ t₂) {c₂ : t₁ ⟷ t₂} {c₃ : t₁ ⟷ t₂} →
(2•[ PATH c₁ , path c₁ ] ⇔ 2•[ PATH c₂ , path c₂ ]) →
(2•[ PATH c₂ , path c₂ ] ⇔ 2•[ PATH c₃ , path c₃ ]) →
(2•[ PATH c₁ , path c₁ ] ⇔ 2•[ PATH c₃ , path c₃ ])
_ ≡⟨ α ⟩ β = α ◎ β
_∎ : {t₁ t₂ : U•} → (c : t₁ ⟷ t₂) →
2•[ PATH c , path c ] ⇔ 2•[ PATH c , path c ]
_∎ c = id⟷
--}
-- example programs
-- level 0 is a groupoid with a non-trivial path equivalence the various inv*
-- rules are not justified by the groupoid proof; they are justified by the
-- need for computational rules. So it is important to have not just a
-- groupoid structure but a groupoid structure that we can compute with. So
-- if we say that we want p ◎ p⁻¹ to be id, we must have computational rules
-- that allow us to derive this for any path p, and similarly for all the
-- other groupoid rules. (cf. The canonicity for 2D type theory by Licata and
-- Harper)
linv : {t₁ t₂ : U•} → (c : t₁ ⟷ t₂) →
1•[ PATH t₁ t₁ , path (c ◎ ! c) ] ⇔ 1•[ PATH t₁ t₁ , path id⟷ ]
linv unite₊ = unit₊l
linv uniti₊ = {!!}
linv swap1₊ = {!!}
linv swap2₊ = {!!}
linv assocl1₊ = {!!}
linv assocl2₊ = {!!}
linv assocl3₊ = {!!}
linv assocr1₊ = {!!}
linv assocr2₊ = {!!}
linv assocr3₊ = {!!}
linv unite⋆ = {!!}
linv uniti⋆ = {!!}
linv swap⋆ = {!!}
linv assocl⋆ = {!!}
linv assocr⋆ = {!!}
linv distz = {!!}
linv factorz = {!!}
linv dist1 = {!!}
linv dist2 = {!!}
linv factor1 = {!!}
linv factor2 = {!!}
linv id⟷ = {!!}
linv (c ◎ c₁) = {!!}
linv (c ⊕1 c₁) = {!!}
linv (c ⊕2 c₁) = {!!}
linv (c ⊗ c₁) = {!!}
linv loop = {!!}
G : 1Groupoid
G = record
{ set = U•
; _↝_ = _⟷_
; _≈_ = λ {t₁} {t₂} c₀ c₁ →
1•[ PATH t₁ t₂ , path c₀ ] ⇔ 1•[ PATH t₁ t₂ , path c₁ ]
; id = id⟷
; _∘_ = λ c₀ c₁ → c₁ ◎ c₀
; _⁻¹ = !
; lneutr = λ _ → ridl
; rneutr = λ _ → lidl
; assoc = λ _ _ _ → assocl
; equiv = record { refl = id⇔
; sym = λ c → 1! c
; trans = λ c₀ c₁ → c₀ ◎ c₁ }
; linv = λ {t₁} {t₂} c → linv c
; rinv = λ {t₁} {t₂} c → {!!}
; ∘-resp-≈ = λ f⟷h g⟷i → resp◎ g⟷i f⟷h
}
------------------------------------------------------------------------------
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{-# OPTIONS --without-K --safe #-}
-- A Categorical WeakInverse induces an Adjoint Equivalence
module Categories.Category.Equivalence.Properties where
open import Level
open import Data.Product using (Σ-syntax; _,_; proj₁)
open import Categories.Adjoint.Equivalence using (⊣Equivalence)
open import Categories.Adjoint
open import Categories.Adjoint.TwoSided using (_⊣⊢_; withZig)
open import Categories.Category
open import Categories.Category.Equivalence using (WeakInverse; StrongEquivalence)
open import Categories.Morphism
import Categories.Morphism.Reasoning as MR
import Categories.Morphism.Properties as MP
open import Categories.Functor renaming (id to idF)
open import Categories.Functor.Properties using ([_]-resp-Iso)
open import Categories.NaturalTransformation using (ntHelper; _∘ᵥ_; _∘ˡ_; _∘ʳ_)
open import Categories.NaturalTransformation.NaturalIsomorphism as ≃
using (NaturalIsomorphism ; unitorˡ; unitorʳ; associator; _ⓘᵥ_; _ⓘˡ_; _ⓘʳ_)
open import Categories.NaturalTransformation.NaturalIsomorphism.Properties using (pointwise-iso)
private
variable
o ℓ e : Level
C D E : Category o ℓ e
module _ {F : Functor C D} {G : Functor D C} (W : WeakInverse F G) where
open WeakInverse W
private
module C = Category C
module D = Category D
module F = Functor F
module G = Functor G
-- adjoint equivalence
F⊣⊢G : F ⊣⊢ G
F⊣⊢G = withZig record
{ unit = ≃.sym G∘F≈id
; counit =
let open D
open HomReasoning
open MR D
open MP D
in record
{ F⇒G = ntHelper record
{ η = λ X → F∘G≈id.⇒.η X ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ X)) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
; commute = λ {X Y} f → begin
(F∘G≈id.⇒.η Y ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ Y)) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ Y))) ∘ F.F₁ (G.F₁ f)
≈⟨ pull-last (F∘G≈id.⇐.commute (F.F₁ (G.F₁ f))) ⟩
F∘G≈id.⇒.η Y ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ Y)) ∘ (F.F₁ (G.F₁ (F.F₁ (G.F₁ f))) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X)))
≈˘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
F∘G≈id.⇒.η Y ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ Y) C.∘ G.F₁ (F.F₁ (G.F₁ f))) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
≈⟨ refl⟩∘⟨ F.F-resp-≈ (G∘F≈id.⇒.commute (G.F₁ f)) ⟩∘⟨refl ⟩
F∘G≈id.⇒.η Y ∘ F.F₁ (G.F₁ f C.∘ G∘F≈id.⇒.η (G.F₀ X)) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
≈⟨ refl⟩∘⟨ F.homomorphism ⟩∘⟨refl ⟩
F∘G≈id.⇒.η Y ∘ (F.F₁ (G.F₁ f) ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ X))) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
≈⟨ center⁻¹ (F∘G≈id.⇒.commute f) Equiv.refl ⟩
(f ∘ F∘G≈id.⇒.η X) ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ X)) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
≈⟨ assoc ⟩
f ∘ F∘G≈id.⇒.η X ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ X)) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ X))
∎
}
; F⇐G = ntHelper record
{ η = λ X → (F∘G≈id.⇒.η (F.F₀ (G.F₀ X)) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ X))) ∘ F∘G≈id.⇐.η X
; commute = λ {X Y} f → begin
((F∘G≈id.⇒.η (F.F₀ (G.F₀ Y)) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ Y))) ∘ F∘G≈id.⇐.η Y) ∘ f
≈⟨ pullʳ (F∘G≈id.⇐.commute f) ⟩
(F∘G≈id.⇒.η (F.F₀ (G.F₀ Y)) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ Y))) ∘ F.F₁ (G.F₁ f) ∘ F∘G≈id.⇐.η X
≈⟨ center (⟺ F.homomorphism) ⟩
F∘G≈id.⇒.η (F.F₀ (G.F₀ Y)) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ Y) C.∘ G.F₁ f) ∘ F∘G≈id.⇐.η X
≈⟨ refl⟩∘⟨ F.F-resp-≈ (G∘F≈id.⇐.commute (G.F₁ f)) ⟩∘⟨refl ⟩
F∘G≈id.⇒.η (F.F₀ (G.F₀ Y)) ∘ F.F₁ (G.F₁ (F.F₁ (G.F₁ f)) C.∘ G∘F≈id.⇐.η (G.F₀ X)) ∘ F∘G≈id.⇐.η X
≈⟨ refl⟩∘⟨ F.homomorphism ⟩∘⟨refl ⟩
F∘G≈id.⇒.η (F.F₀ (G.F₀ Y)) ∘ (F.F₁ (G.F₁ (F.F₁ (G.F₁ f))) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ X))) ∘ F∘G≈id.⇐.η X
≈⟨ center⁻¹ (F∘G≈id.⇒.commute _) Equiv.refl ⟩
(F.F₁ (G.F₁ f) ∘ F∘G≈id.⇒.η (F.F₀ (G.F₀ X))) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ X)) ∘ F∘G≈id.⇐.η X
≈⟨ center Equiv.refl ⟩
F.F₁ (G.F₁ f) ∘ (F∘G≈id.⇒.η (F.F₀ (G.F₀ X)) ∘ F.F₁ (G∘F≈id.⇐.η (G.F₀ X))) ∘ F∘G≈id.⇐.η X
∎
}
; iso = λ X → Iso-∘ (Iso-∘ (Iso-swap (F∘G≈id.iso _)) ([ F ]-resp-Iso (G∘F≈id.iso _)))
(F∘G≈id.iso X)
}
; zig = λ {A} →
let open D
open HomReasoning
open MR D
in begin
(F∘G≈id.⇒.η (F.F₀ A) ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ (F.F₀ A))) ∘ F∘G≈id.⇐.η (F.F₀ (G.F₀ (F.F₀ A))))
∘ F.F₁ (G∘F≈id.⇐.η A)
≈⟨ pull-last (F∘G≈id.⇐.commute (F.F₁ (G∘F≈id.⇐.η A))) ⟩
F∘G≈id.⇒.η (F.F₀ A) ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ (F.F₀ A))) ∘
F.F₁ (G.F₁ (F.F₁ (G∘F≈id.⇐.η A))) ∘ F∘G≈id.⇐.η (F.F₀ A)
≈˘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
F∘G≈id.⇒.η (F.F₀ A) ∘ F.F₁ (G∘F≈id.⇒.η (G.F₀ (F.F₀ A)) C.∘ G.F₁ (F.F₁ (G∘F≈id.⇐.η A))) ∘ F∘G≈id.⇐.η (F.F₀ A)
≈⟨ refl⟩∘⟨ F.F-resp-≈ (G∘F≈id.⇒.commute (G∘F≈id.⇐.η A)) ⟩∘⟨refl ⟩
F∘G≈id.⇒.η (F.F₀ A) ∘ F.F₁ (G∘F≈id.⇐.η A C.∘ G∘F≈id.⇒.η A) ∘ F∘G≈id.⇐.η (F.F₀ A)
≈⟨ refl⟩∘⟨ elimˡ ((F.F-resp-≈ (G∘F≈id.iso.isoˡ _)) ○ F.identity) ⟩
F∘G≈id.⇒.η (F.F₀ A) ∘ F∘G≈id.⇐.η (F.F₀ A)
≈⟨ F∘G≈id.iso.isoʳ _ ⟩
id
∎
}
module F⊣⊢G = _⊣⊢_ F⊣⊢G
module _ {o ℓ e o′ ℓ′ e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′} (SE : StrongEquivalence C D) where
open StrongEquivalence SE
C≅D : ⊣Equivalence C D
C≅D = record
{ L = F
; R = G
; L⊣⊢R = F⊣⊢G weak-inverse
}
module C≅D = ⊣Equivalence C≅D
module _ {F : Functor C D} {G : Functor D C} (F⊣G : F ⊣ G) where
private
module C = Category C
module D = Category D
module F = Functor F
module G = Functor G
open Adjoint F⊣G
-- If the unit and the counit of an adjuction are pointwise isomorphisms, then they form an equivalence of categories.
pointwise-iso-equivalence : (∀ X → Σ[ f ∈ D [ X , F.F₀ (G.F₀ X) ] ] Iso D (counit.η X) f) → (∀ X → Σ[ f ∈ C [ G.F₀ (F.F₀ X) , X ] ] Iso C (unit.η X) f) → WeakInverse F G
pointwise-iso-equivalence counit-iso unit-iso = record
{ F∘G≈id =
let iso X =
let (to , is-iso) = counit-iso X
in record { from = counit.η X ; to = to ; iso = is-iso }
in pointwise-iso iso counit.commute
; G∘F≈id =
let iso X =
let (to , is-iso) = unit-iso X
in record { from = unit.η X ; to = to ; iso = is-iso }
in ≃.sym (pointwise-iso iso unit.commute)
}
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-- Andreas, 2019-08-08, issue #3972 (and #3967)
-- In the presence of an unreachable clause, the serializer crashed on a unsolve meta.
-- It seems this issue was fixed along #3966: only the ranges of unreachable clauses
-- are now serialized.
open import Agda.Builtin.Equality public
postulate
List : Set → Set
data Coo {A} (xs : List A) : Set where
coo : Coo xs → Coo xs
test : {A : Set} (xs : List A) (z : Coo xs) → Set₁
test xs z = cs xs z refl
where
cs : (xs : List _) -- filling this meta _=A removes the internal error
(z : Coo xs)
(eq : xs ≡ xs)
→ Set₁
cs xs (coo z) refl = Set
cs xs (coo z) eq = Set -- unreachable
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{-# OPTIONS --type-in-type #-}
module functors where
open import prelude
record Category {O : Set} (𝒞[_,_] : O → O → Set) : Set where
constructor 𝒾:_▸:_𝒾▸:_▸𝒾:
infixl 8 _▸_
field
𝒾 : ∀ {x} → 𝒞[ x , x ]
_▸_ : ∀ {x y z} → 𝒞[ x , y ] → 𝒞[ y , z ] → 𝒞[ x , z ]
𝒾▸ : ∀ {x y} (f : 𝒞[ x , y ]) → (𝒾 ▸ f) ≡ f
▸𝒾 : ∀ {x y} (f : 𝒞[ x , y ]) → (f ▸ 𝒾) ≡ f
open Category ⦃...⦄ public
record Functor {A : Set} {𝒜[_,_] : A → A → Set}
{B : Set} {ℬ[_,_] : B → B → Set}
(f : A → B) : Set where
constructor φ:_𝒾:_▸:
field
⦃ cat₁ ⦄ : Category {A} 𝒜[_,_]
⦃ cat₂ ⦄ : Category {B} ℬ[_,_]
φ : ∀ {a b} → 𝒜[ a , b ] → ℬ[ f a , f b ]
functor-id : ∀ {a} → φ {a} 𝒾 ≡ 𝒾
functor-comp : ∀ {A B C} → (f : 𝒜[ A , B ]) (g : 𝒜[ B , C ])
→ φ (f ▸ g) ≡ φ f ▸ φ g
open Functor ⦃...⦄ public
record Nat {A : Set} {𝒜[_,_] : A → A → Set}
{B : Set} {ℬ[_,_] : B → B → Set}
{F : A → B} {G : A → B}
(α : (x : A) → ℬ[ F x , G x ]) : Set where
field
overlap ⦃ fun₁ ⦄ : Functor {A} {𝒜[_,_]} {B} {ℬ[_,_]} F
overlap ⦃ fun₂ ⦄ : Functor {A} {𝒜[_,_]} {B} {ℬ[_,_]} G
nat : ∀ {x y : A} (f : 𝒜[ x , y ]) → let _▹_ = _▸_ ⦃ cat₂ ⦄
in φ f ▹ α y ≡ α x ▹ φ f
module _ {A : Set} {𝒜[_,_] : A → A → Set} where
instance
id-functor : ⦃ Category 𝒜[_,_] ⦄ → Functor id
id-functor = φ: id
𝒾: refl
▸: λ f g → refl
record Monad (f : A → A) : Set where
field
⦃ functor ⦄ : Functor {_} {𝒜[_,_]} {_} {𝒜[_,_]} f
η : ∀ {a} → 𝒜[ a , f a ]
μ : ∀ {a} → 𝒜[ f (f a) , f a ]
open Monad ⦃...⦄ public
{-
record Lift[_⇒_] (pre post : (Set → Set) → Set) (t : (Set → Set) → Set → Set) : Set where
field
⦃ lifted ⦄ : ∀ {f} → post (t f)
lift : {f : Set → Set} {a : Set} → ⦃ pre f ⦄ → f a → t f a
open Lift[_⇒_] ⦃...⦄ public
-}
Cat[_,_] : Set → Set → Set
Cat[ a , b ] = a → b
instance
Cat-cat : Category Cat[_,_]
Cat-cat = record { 𝒾 = id ; _▸_ = λ f g → g ∘ f ; 𝒾▸ = λ _ → refl; ▸𝒾 = λ _ → refl}
instance
≡-cat : ∀ {A} → Category {A} (λ a b → a ≡ b)
𝒾 {{≡-cat}} = refl
_▸_ {{≡-cat}} = trans
𝒾▸ {{≡-cat}} _ = refl
▸𝒾 {{≡-cat}} x≡y rewrite x≡y = refl
instance
×-cat : ∀ {A 𝒜[_,_]} ⦃ 𝒜 : Category {A} 𝒜[_,_] ⦄
{B ℬ[_,_]} ⦃ ℬ : Category {B} ℬ[_,_] ⦄
→ Category {A × B} (λ (a , b) (a′ , b′) → 𝒜[ a , a′ ] × ℬ[ b , b′ ])
𝒾 {{×-cat}} = 𝒾 , 𝒾
_▸_ {{×-cat}} (f , f′ ) ( g , g′ ) = f ▸ g , f′ ▸ g′
𝒾▸ {{×-cat}} (f , g) = 𝒾▸ f ×⁼ 𝒾▸ g
▸𝒾 {{×-cat}} (f , g) = ▸𝒾 f ×⁼ ▸𝒾 g
data _ᵒᵖ {A : Set} (𝒜 : A → A → Set) : A → A → Set where
_ᵗ : ∀ {y x} → 𝒜(y )( x) → (𝒜 ᵒᵖ)(x )( y)
instance
op-cat : {A : Set} {𝒜[_,_] : A → A → Set} ⦃ 𝒜 : Category {A} 𝒜[_,_] ⦄
→ Category (𝒜[_,_] ᵒᵖ)
𝒾 {{op-cat}} = 𝒾 ᵗ
_▸_ {{op-cat}} (f ᵗ) (g ᵗ) = (g ▸ f) ᵗ
𝒾▸ {{op-cat}} (f ᵗ) = _ᵗ ⟨$⟩ ▸𝒾 f
▸𝒾 {{op-cat}} (f ᵗ) = _ᵗ ⟨$⟩ 𝒾▸ f
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{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Properties.Neutral {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped
open import Definition.Typed
open import Definition.Typed.Properties
open import Definition.Typed.Weakening
open import Definition.LogicalRelation
open import Definition.LogicalRelation.ShapeView
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Properties.Reflexivity
open import Definition.LogicalRelation.Properties.Escape
open import Definition.LogicalRelation.Properties.Symmetry
open import Tools.Embedding
open import Tools.Product
import Tools.PropositionalEquality as PE
-- Neutral reflexive types are reducible.
neu : ∀ {l Γ A} (neA : Neutral A)
→ Γ ⊢ A
→ Γ ⊢ A ~ A ∷ U
→ Γ ⊩⟨ l ⟩ A
neu neA A A~A = ne′ _ (idRed:*: A) neA A~A
-- Helper function for reducible neutral equality of a specific type of derivation.
neuEq′ : ∀ {l Γ A B} ([A] : Γ ⊩⟨ l ⟩ne A)
(neA : Neutral A)
(neB : Neutral B)
→ Γ ⊢ A → Γ ⊢ B
→ Γ ⊢ A ~ B ∷ U
→ Γ ⊩⟨ l ⟩ A ≡ B / ne-intr [A]
neuEq′ (noemb (ne K [ ⊢A , ⊢B , D ] neK K≡K)) neA neB A B A~B =
let A≡K = whnfRed* D (ne neA)
in ιx (ne₌ _ (idRed:*: B) neB (PE.subst (λ x → _ ⊢ x ~ _ ∷ _) A≡K A~B))
neuEq′ (emb 0<1 x) neA neB A B A~B = ιx (neuEq′ x neA neB A B A~B)
-- Neutrally equal types are of reducible equality.
neuEq : ∀ {l Γ A B} ([A] : Γ ⊩⟨ l ⟩ A)
(neA : Neutral A)
(neB : Neutral B)
→ Γ ⊢ A → Γ ⊢ B
→ Γ ⊢ A ~ B ∷ U
→ Γ ⊩⟨ l ⟩ A ≡ B / [A]
neuEq [A] neA neB A B A~B =
irrelevanceEq (ne-intr (ne-elim neA [A]))
[A]
(neuEq′ (ne-elim neA [A]) neA neB A B A~B)
mutual
-- Neutral reflexive terms are reducible.
neuTerm : ∀ {l Γ A n} ([A] : Γ ⊩⟨ l ⟩ A) (neN : Neutral n)
→ Γ ⊢ n ∷ A
→ Γ ⊢ n ~ n ∷ A
→ Γ ⊩⟨ l ⟩ n ∷ A / [A]
neuTerm (Uᵣ′ .⁰ 0<1 ⊢Γ) neN n n~n =
Uₜ _ (idRedTerm:*: n) (ne neN) (~-to-≅ₜ n~n) (neu neN (univ n) n~n)
neuTerm (ℕᵣ [ ⊢A , ⊢B , D ]) neN n n~n =
let A≡ℕ = subset* D
n~n′ = ~-conv n~n A≡ℕ
n≡n = ~-to-≅ₜ n~n′
in ιx (ℕₜ _ (idRedTerm:*: (conv n A≡ℕ)) n≡n (ne (neNfₜ neN (conv n A≡ℕ) n~n′)))
neuTerm (ne′ K [ ⊢A , ⊢B , D ] neK K≡K) neN n n~n =
let A≡K = subset* D
in ιx (neₜ _ (idRedTerm:*: (conv n A≡K)) (neNfₜ neN (conv n A≡K)
(~-conv n~n A≡K)))
neuTerm (Πᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) neN n n~n =
let A≡ΠFG = subset* (red D)
in Πₜ _ (idRedTerm:*: (conv n A≡ΠFG)) (ne neN) (~-to-≅ₜ (~-conv n~n A≡ΠFG))
(λ {ρ} [ρ] ⊢Δ [a] [b] [a≡b] →
let A≡ΠFG = subset* (red D)
ρA≡ρΠFG = wkEq [ρ] ⊢Δ (subset* (red D))
G[a]≡G[b] = escapeEq ([G] [ρ] ⊢Δ [b])
(symEq ([G] [ρ] ⊢Δ [a]) ([G] [ρ] ⊢Δ [b])
(G-ext [ρ] ⊢Δ [a] [b] [a≡b]))
a = escapeTerm ([F] [ρ] ⊢Δ) [a]
b = escapeTerm ([F] [ρ] ⊢Δ) [b]
a≡b = escapeTermEq ([F] [ρ] ⊢Δ) [a≡b]
ρn = conv (wkTerm [ρ] ⊢Δ n) ρA≡ρΠFG
neN∘a = ∘ₙ (wkNeutral ρ neN)
neN∘b = ∘ₙ (wkNeutral ρ neN)
in neuEqTerm ([G] [ρ] ⊢Δ [a]) neN∘a neN∘b
(ρn ∘ⱼ a)
(conv (ρn ∘ⱼ b) (≅-eq G[a]≡G[b]))
(~-app (~-wk [ρ] ⊢Δ (~-conv n~n A≡ΠFG)) a≡b))
(λ {ρ} [ρ] ⊢Δ [a] →
let ρA≡ρΠFG = wkEq [ρ] ⊢Δ (subset* (red D))
a = escapeTerm ([F] [ρ] ⊢Δ) [a]
a≡a = escapeTermEq ([F] [ρ] ⊢Δ) (reflEqTerm ([F] [ρ] ⊢Δ) [a])
in neuTerm ([G] [ρ] ⊢Δ [a]) (∘ₙ (wkNeutral ρ neN))
(conv (wkTerm [ρ] ⊢Δ n) ρA≡ρΠFG ∘ⱼ a)
(~-app (~-wk [ρ] ⊢Δ (~-conv n~n A≡ΠFG)) a≡a))
neuTerm (emb′ 0<1 x) neN n n~n = ιx (neuTerm x neN n n~n)
-- Neutrally equal terms are of reducible equality.
neuEqTerm : ∀ {l Γ A n n′} ([A] : Γ ⊩⟨ l ⟩ A)
(neN : Neutral n) (neN′ : Neutral n′)
→ Γ ⊢ n ∷ A
→ Γ ⊢ n′ ∷ A
→ Γ ⊢ n ~ n′ ∷ A
→ Γ ⊩⟨ l ⟩ n ≡ n′ ∷ A / [A]
neuEqTerm (Uᵣ′ .⁰ 0<1 ⊢Γ) neN neN′ n n′ n~n′ =
let [n] = neu neN (univ n) (~-trans n~n′ (~-sym n~n′))
[n′] = neu neN′ (univ n′) (~-trans (~-sym n~n′) n~n′)
in Uₜ₌ _ _ (idRedTerm:*: n) (idRedTerm:*: n′) (ne neN) (ne neN′)
(~-to-≅ₜ n~n′) [n] [n′] (neuEq [n] neN neN′ (univ n) (univ n′) n~n′)
neuEqTerm (ℕᵣ [ ⊢A , ⊢B , D ]) neN neN′ n n′ n~n′ =
let A≡ℕ = subset* D
n~n′₁ = ~-conv n~n′ A≡ℕ
n≡n′ = ~-to-≅ₜ n~n′₁
in ιx (ℕₜ₌ _ _ (idRedTerm:*: (conv n A≡ℕ)) (idRedTerm:*: (conv n′ A≡ℕ))
n≡n′ (ne (neNfₜ₌ neN neN′ n~n′₁)))
neuEqTerm (ne′ K [ ⊢A , ⊢B , D ] neK K≡K) neN neN′ n n′ n~n′ =
let A≡K = subset* D
in ιx (neₜ₌ _ _ (idRedTerm:*: (conv n A≡K)) (idRedTerm:*: (conv n′ A≡K))
(neNfₜ₌ neN neN′ (~-conv n~n′ A≡K)))
neuEqTerm (Πᵣ′ F G [ ⊢A , ⊢B , D ] ⊢F ⊢G A≡A [F] [G] G-ext) neN neN′ n n′ n~n′ =
let [ΠFG] = Πᵣ′ F G [ ⊢A , ⊢B , D ] ⊢F ⊢G A≡A [F] [G] G-ext
A≡ΠFG = subset* D
n~n′₁ = ~-conv n~n′ A≡ΠFG
n≡n′ = ~-to-≅ₜ n~n′₁
n~n = ~-trans n~n′ (~-sym n~n′)
n′~n′ = ~-trans (~-sym n~n′) n~n′
in Πₜ₌ _ _ (idRedTerm:*: (conv n A≡ΠFG)) (idRedTerm:*: (conv n′ A≡ΠFG))
(ne neN) (ne neN′) n≡n′
(neuTerm [ΠFG] neN n n~n) (neuTerm [ΠFG] neN′ n′ n′~n′)
(λ {ρ} [ρ] ⊢Δ [a] →
let ρA≡ρΠFG = wkEq [ρ] ⊢Δ A≡ΠFG
ρn = wkTerm [ρ] ⊢Δ n
ρn′ = wkTerm [ρ] ⊢Δ n′
a = escapeTerm ([F] [ρ] ⊢Δ) [a]
a≡a = escapeTermEq ([F] [ρ] ⊢Δ)
(reflEqTerm ([F] [ρ] ⊢Δ) [a])
neN∙a = ∘ₙ (wkNeutral ρ neN)
neN′∙a′ = ∘ₙ (wkNeutral ρ neN′)
in neuEqTerm ([G] [ρ] ⊢Δ [a]) neN∙a neN′∙a′
(conv ρn ρA≡ρΠFG ∘ⱼ a)
(conv ρn′ ρA≡ρΠFG ∘ⱼ a)
(~-app (~-wk [ρ] ⊢Δ n~n′₁) a≡a))
neuEqTerm (emb′ 0<1 x) neN neN′ n n′ n~n′ = ιx (neuEqTerm x neN neN′ n n′ n~n′)
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{-# OPTIONS --without-K #-}
-- Define all the permutations that occur in Pi
-- These are defined by transport, using univalence
module Permutation where
open import Relation.Binary.PropositionalEquality
using (_≡_; refl)
open import Data.Nat using (_+_;_*_)
open import Data.Fin using (Fin)
open import Relation.Binary using (Setoid)
open import Function.Equality using (_⟨$⟩_)
open import Equiv using (_≃_; id≃; sym≃; trans≃; _●_)
open import ConcretePermutation
open import ConcretePermutationProperties
open import SEquivSCPermEquiv
open import FinEquivTypeEquiv using (module PlusE; module TimesE)
open import EquivEquiv
open import Proofs using (
-- FiniteFunctions
finext
)
infixr 5 _●p_
infix 8 _⊎p_
infixr 7 _×p_
------------------------------------------------------------------------------
-- useful short-hands; these should all be moved elsewhere.
e⇒p : ∀ {m n} → (Fin m ≃ Fin n) → CPerm m n
e⇒p e = _≃S_.g univalence ⟨$⟩ e
p⇒e : ∀ {m n} → CPerm m n → (Fin m ≃ Fin n)
p⇒e p = _≃S_.f univalence ⟨$⟩ p
αu : ∀ {m n} → {e f : Fin m ≃ Fin n} → e ≋ f → p⇒e (e⇒p e) ≋ f
αu e≋f = _≃S_.α univalence e≋f
βu : ∀ {m n} → {p q : CPerm m n} → p ≡ q → e⇒p (p⇒e p) ≡ q
βu p≡q = _≃S_.β univalence p≡q
α₁ : ∀ {m n} → {e : Fin m ≃ Fin n} → p⇒e (e⇒p e) ≋ e
α₁ {e = e} = αu (id≋ {x = e}) -- Agda can infer here, but this helps later?
-- inside an e⇒p, we can freely replace equivalences
-- this expresses the fundamental property that equivalent equivalences
-- map to THE SAME permutation.
≋⇒≡ : ∀ {m n} → {e₁ e₂ : Fin m ≃ Fin n} → e₁ ≋ e₂ → e⇒p e₁ ≡ e⇒p e₂
≋⇒≡ {e₁} {e₂} (eq fwd bwd) = p≡ (finext bwd)
-- combination of above, where we use αu on left/right of ●
right-α-over-● : ∀ {m n o} → (e₁ : Fin n ≃ Fin o) → (e₂ : Fin m ≃ Fin n) →
(e₁ ● (p⇒e (e⇒p e₂))) ≋ (e₁ ● e₂)
right-α-over-● e₁ e₂ = (id≋ {x = e₁}) ◎ (αu {e = e₂} id≋)
left-α-over-● : ∀ {m n o} → (e₁ : Fin n ≃ Fin o) → (e₂ : Fin m ≃ Fin n) →
((p⇒e (e⇒p e₁)) ● e₂) ≋ (e₁ ● e₂)
left-α-over-● e₁ e₂ = (αu {e = e₁} id≋) ◎ (id≋ {x = e₂})
------------------------------------------------------------------------------
-- zero permutation
0p : CPerm 0 0
0p = e⇒p (id≃ {A = Fin 0})
-- identity permutation
idp : ∀ {n} → CPerm n n
idp {n} = e⇒p (id≃ {A = Fin n})
-- disjoint union
_⊎p_ : ∀ {m₁ m₂ n₁ n₂} → CPerm m₁ m₂ → CPerm n₁ n₂ → CPerm (m₁ + n₁) (m₂ + n₂)
p₁ ⊎p p₂ = e⇒p ((p⇒e p₁) +F (p⇒e p₂))
where open PlusE
-- cartesian product
_×p_ : ∀ {m₁ m₂ n₁ n₂} → CPerm m₁ m₂ → CPerm n₁ n₂ → CPerm (m₁ * n₁) (m₂ * n₂)
p₁ ×p p₂ = e⇒p ((p⇒e p₁) *F (p⇒e p₂))
where open TimesE
-- symmetry
symp : ∀ {m n} → CPerm m n → CPerm n m
symp p = e⇒p (sym≃ (p⇒e p))
-- transitivity; note the 'transposition' of the arguments!
_●p_ : ∀ {m₁ m₂ m₃} → CPerm m₂ m₁ → CPerm m₃ m₂ → CPerm m₃ m₁
p₁ ●p p₂ = e⇒p ((p⇒e p₁) ● (p⇒e p₂))
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module PreludeShow where
import RTP -- magic module
import AlonzoPrelude as Prelude
open import PreludeNat
open import PreludeString
import PreludeList
open import PreludeBool
open Prelude
-- open Data.Integer, using (Int, pos, neg)
open PreludeList hiding (_++_)
showInt : Int -> String
showInt = RTP.primShowInt
showNat : Nat -> String
showNat n = showInt (RTP.primNatToInt n)
{-
showNat : Nat -> String
showNat zero = "0"
showNat n = reverseString $ show n
where
digit : Nat -> String
digit 0 = "0"
digit 1 = "1"
digit 2 = "2"
digit 3 = "3"
digit 4 = "4"
digit 5 = "5"
digit 6 = "6"
digit 7 = "7"
digit 8 = "8"
digit 9 = "9"
digit _ = "?"
show : Nat -> String
show zero = ""
show n = digit (mod n 10) ++ show (div n 10)
-}
{-
showInt : Int -> String
showInt (pos n) = showNat n
showInt (neg n) = "-" ++ showNat (suc n)
-}
showChar : Char -> String
showChar = RTP.primShowChar
showBool : Bool -> String
showBool true = "true"
showBool false = "false"
primitive
primShowFloat : Float -> String
showFloat : Float -> String
showFloat = primShowFloat | {
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-- | Trailing inductive copattern matches on the LHS can be savely
-- translated to record expressions on RHS, without jeopardizing
-- termination.
--
{-# LANGUAGE CPP #-}
module Agda.TypeChecking.CompiledClause.ToRHS where
-- import Control.Applicative
import Data.Monoid
import qualified Data.Map as Map
import Data.List (genericReplicate, nubBy, findIndex)
import Data.Function
import Agda.Syntax.Common
import Agda.Syntax.Internal as I
import Agda.TypeChecking.CompiledClause
import Agda.TypeChecking.Coverage
import Agda.TypeChecking.Coverage.SplitTree
import Agda.TypeChecking.Monad
import Agda.TypeChecking.RecordPatterns
import Agda.TypeChecking.Substitute
import Agda.TypeChecking.Pretty
import Agda.Utils.List
import Agda.Utils.Impossible
#include "../../undefined.h"
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module _ where
open import Agda.Builtin.Bool
module M (b : Bool) where
module Inner where
some-boolean : Bool
some-boolean = b
postulate
@0 a-postulate : Bool
@0 A : @0 Bool → Set
A b = Bool
module A where
module M′ = M b
bad : @0 Bool → Bool
bad = A.M′.Inner.some-boolean
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-- MIT License
-- Copyright (c) 2021 Luca Ciccone and Luca Padovani
-- Permission is hereby granted, free of charge, to any person
-- obtaining a copy of this software and associated documentation
-- files (the "Software"), to deal in the Software without
-- restriction, including without limitation the rights to use,
-- copy, modify, merge, publish, distribute, sublicense, and/or sell
-- copies of the Software, and to permit persons to whom the
-- Software is furnished to do so, subject to the following
-- conditions:
-- The above copyright notice and this permission notice shall be
-- included in all copies or substantial portions of the Software.
-- THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
-- EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
-- OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
-- NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
-- HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
-- WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
-- FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
-- OTHER DEALINGS IN THE SOFTWARE.
{-# OPTIONS --guardedness --sized-types #-}
open import Data.Product
open import Data.Empty
open import Data.Sum
open import Data.Vec
open import Data.List as List
open import Data.Unit
open import Data.Fin
open import Data.Bool renaming (Bool to 𝔹)
open import Relation.Unary using (_∈_; _⊆_;_∉_)
open import Relation.Binary.Construct.Closure.ReflexiveTransitive
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
open import Size
open import Codata.Thunk
open import is-lib.InfSys
open import Common using (Message)
module FairSubtyping-IS {𝕋 : Set} (message : Message 𝕋) where
open Message message
open import SessionType message
open import Session message
open import Transitions message
open import Convergence message
open import Divergence message
open import Discriminator message
open import Action message using (Action)
open import Subtyping message
open import FairSubtyping message as FS
open import HasTrace message
open import Compliance message
open import FairCompliance message
open import Trace message
open import FairCompliance-IS message
private
U : Set
U = SessionType × SessionType
data FSubIS-RN : Set where
nil-any end-def : FSubIS-RN
ii oo : FSubIS-RN
data FSubCOIS-RN : Set where
co-conv : FSubCOIS-RN
nil-any-r : FinMetaRule U
nil-any-r .Ctx = SessionType
nil-any-r .comp T =
[] ,
------------------
(nil , T)
end-def-r : FinMetaRule U
end-def-r .Ctx = Σ[ (T , S) ∈ SessionType × SessionType ] End T × Defined S
end-def-r .comp ((T , S) , _) =
[] ,
------------------
(T , S)
ii-r : MetaRule U
ii-r .Ctx = Σ[ (f , g) ∈ Continuation × Continuation ] dom f ⊆ dom g
ii-r .Pos ((f , _) , _) = Σ[ t ∈ 𝕋 ] t ∈ dom f
ii-r .prems ((f , g) , _) (t , _) = f t .force , g t .force
ii-r .conclu ((f , g) , _) = inp f , inp g
oo-r : MetaRule U
oo-r .Ctx = Σ[ (f , g) ∈ Continuation × Continuation ] dom g ⊆ dom f × Witness g
oo-r .Pos ((_ , g) , _) = Σ[ t ∈ 𝕋 ] t ∈ dom g
oo-r .prems ((f , g) , _) (t , _) = f t .force , g t .force
oo-r .conclu ((f , g) , _) = out f , out g
co-conv-r : FinMetaRule U
co-conv-r .Ctx = Σ[ (T , S) ∈ SessionType × SessionType ] T ↓ S
co-conv-r .comp ((T , S) , _) =
[] ,
------------------
(T , S)
FSubIS : IS U
FSubIS .Names = FSubIS-RN
FSubIS .rules nil-any = from nil-any-r
FSubIS .rules end-def = from end-def-r
FSubIS .rules ii = ii-r
FSubIS .rules oo = oo-r
FSubCOIS : IS U
FSubCOIS .Names = FSubCOIS-RN
FSubCOIS .rules co-conv = from co-conv-r
_≤F_ : SessionType → SessionType → Set
T ≤F S = FCoInd⟦ FSubIS , FSubCOIS ⟧ (T , S)
_≤Fᵢ_ : SessionType → SessionType → Set
T ≤Fᵢ S = Ind⟦ FSubIS ∪ FSubCOIS ⟧ (T , S)
_≤Fc_ : SessionType → SessionType → Set
T ≤Fc S = CoInd⟦ FSubIS ⟧ (T , S)
{- Specification using _⊢_ is correct wrt FairSubtypingS -}
FSSpec-⊢ : U → Set
FSSpec-⊢ (T , S) = ∀{R} → R ⊢ T → R ⊢ S
spec-sound : ∀{T S} → FairSubtypingS T S → FSSpec-⊢ (T , S)
spec-sound fs fc = fc-complete (fs (fc-sound fc))
spec-complete : ∀{T S} → FSSpec-⊢ (T , S) → FairSubtypingS T S
spec-complete fs fc = fc-sound (fs (fc-complete fc))
------------------------------------------------------
{- Soundness -}
-- Using bounded coinduction wrt SpecAux
≤Fᵢ->↓ : ∀{S T} → S ≤Fᵢ T → S ↓ T
≤Fᵢ->↓ (fold (inj₁ nil-any , _ , refl , _)) = nil-converges
≤Fᵢ->↓ (fold (inj₁ end-def , (_ , (end , def)) , refl , _)) = end-converges end def
≤Fᵢ->↓ (fold (inj₁ ii , _ , refl , pr)) = converge (pre-conv-inp-back λ x → ↓->preconv (≤Fᵢ->↓ (pr (_ , x))))
≤Fᵢ->↓ (fold (inj₁ oo , (_ , (incl , (t , ok-t))) , refl , pr)) =
converge
λ _ _ → [] , t , none , (_ , incl ok-t , step (out (incl ok-t)) refl) , (_ , ok-t , step (out ok-t) refl) , ≤Fᵢ->↓ (pr (t , ok-t))
≤Fᵢ->↓ (fold (inj₂ co-conv , (_ , conv) , refl , _)) = conv
SpecAux : U → Set
SpecAux (R , T) = Σ[ S ∈ SessionType ] S ≤F T × R ⊢ S
≤Fᵢ->defined : ∀{S T} → Defined S → S ≤Fᵢ T → Defined T
≤Fᵢ->defined def fs = conv->defined def (≤Fᵢ->↓ fs)
spec-bounded-rec : ∀{R S} T → T ≤Fᵢ S → R ⊢ T → R ⊢ᵢ S
spec-bounded-rec _ fs fc =
let _ , reds , succ = con-sound (≤Fᵢ->↓ fs) (fc-sound fc) in
maysucceed->⊢ᵢ reds succ
spec-bounded : SpecAux ⊆ λ (R , S) → R ⊢ᵢ S
spec-bounded (T , fs , fc) = spec-bounded-rec T (fcoind-to-ind fs) fc
spec-cons : SpecAux ⊆ ISF[ FCompIS ] SpecAux
spec-cons {(R , T)} (S , fs , fc) with fc .CoInd⟦_⟧.unfold
spec-cons {(R , T)} (S , fs , fc) | client-end , ((_ , (win , def)) , _) , refl , _ =
client-end , ((R , _) , (win , ≤Fᵢ->defined def (fcoind-to-ind fs))) , refl , λ ()
spec-cons {(out r , _)} ((inp f) , fs , fc) | oi , (((.r , .f) , wit-r) , _) , refl , pr with fs .CoInd⟦_⟧.unfold
... | end-def , (((.(inp f) , _) , (inp e , _)) , _) , refl , _ = ⊥-elim (e _ (proj₂ (fc->defined (pr wit-r))))
... | ii , (((.f , g) , _) , _) , refl , pr' = oi , (_ , wit-r) , refl , λ wit → _ , pr' (_ , proj₂ (fc->defined (pr wit))) , pr wit
spec-cons {(inp r , T)} (out f , fs , fc) | io , (((.r , .f) , wit-f) , _) , refl , pr with fs .CoInd⟦_⟧.unfold
... | end-def , (((.(out f) , _) , (out e , _)) , _) , refl , _ = ⊥-elim (e _ (proj₂ wit-f))
... | oo , (((.f , g) , (incl , wit-g)) , _) , refl , pr' = io , (_ , wit-g) , refl , λ wit → _ , pr' wit , pr (_ , incl (proj₂ wit))
spec-aux-sound : SpecAux ⊆ λ (R , S) → R ⊢ S
spec-aux-sound = bounded-coind[ FCompIS , FCompCOIS ] SpecAux spec-bounded spec-cons
fs-sound : ∀{T S} → T ≤F S → FSSpec-⊢ (T , S)
fs-sound {T} fs fc = spec-aux-sound (T , fs , fc)
{- Soundness & Completeness of Sub wrt ≤Fc -}
≤Fc->sub : ∀{S T} → S ≤Fc T → ∀ {i} → Sub S T i
≤Fc->sub fs with fs .CoInd⟦_⟧.unfold
... | nil-any , _ , refl , _ = nil<:any
... | end-def , (_ , (end , def)) , refl , _ = end<:def end def
... | ii , ((f , g) , incl) , refl , pr = inp<:inp incl λ x → λ where .force → if-def x
where
if-def : (t : 𝕋) → ∀{i} → Sub (f t .force) (g t .force) i
if-def t with t ∈? f
... | yes ok-t = ≤Fc->sub (pr (_ , ok-t))
... | no no-t = subst (λ x → Sub x (g t .force) _) (sym (not-def->nil no-t)) nil<:any
... | oo , (_ , (incl , wit)) , refl , pr = out<:out wit incl λ ok-x → λ where .force → ≤Fc->sub (pr (_ , ok-x))
sub->≤Fc : ∀{S T} → (∀{i} → Sub S T i) → S ≤Fc T
CoInd⟦_⟧.unfold (sub->≤Fc fs) with fs
... | nil<:any = nil-any , _ , refl , λ ()
... | end<:def end def = end-def , (_ , (end , def)) , refl , λ ()
... | inp<:inp incl pr = ii , (_ , incl) , refl , λ (p , _) → sub->≤Fc (pr p .force)
... | out<:out wit incl pr = oo , (_ , (incl , wit)) , refl , λ (_ , ok) → sub->≤Fc (pr ok .force)
{- Auxiliary -}
-- Only premise for rules using sample-cont in ⊢
sample-cont-prem : ∀{f : Continuation}{t R} → R ⊢ f t .force
→ (pos : Σ[ p ∈ 𝕋 ] p ∈ dom (sample-cont t R nil)) → (sample-cont t R nil) (proj₁ pos) .force ⊢ f (proj₁ pos) .force
sample-cont-prem {f} {t} pr (p , ok-p) with p ?= t
... | yes refl = pr
sample-cont-prem {f} {t} pr (p , ()) | no ¬eq
-- Premises using sample-cont-dual in ⊢
sample-cont-prems : ∀{f : Continuation}{t} → t ∉ dom f
→ (pos : Σ[ p ∈ 𝕋 ] p ∈ dom f) → (sample-cont t nil win) (proj₁ pos) .force ⊢ f (proj₁ pos) .force
sample-cont-prems {f} {t} no-t (p , ok-p) with p ?= t
... | yes refl = ⊥-elim (no-t ok-p)
... | no ¬eq = win⊢def ok-p
-- Premises using sample-cont-dual in ⊢
sample-cont-prems' : ∀{f : Continuation}{t R} → R ⊢ f t .force
→ (pos : Σ[ p ∈ 𝕋 ] p ∈ dom f) → (sample-cont t R win) (proj₁ pos) .force ⊢ f (proj₁ pos) .force
sample-cont-prems' {f} {t} pr (p , ok-p) with p ?= t
... | yes refl = pr
... | no ¬eq = win⊢def ok-p
spec-inp->incl : ∀{f g} → FSSpec-⊢ (inp f , inp g) → dom f ⊆ dom g
spec-inp->incl {f} {g} fs {t} ok-t with fs (apply-fcoind oi ((sample-cont t win nil , f) , wit-cont out) (sample-cont-prem {f} {t} (win⊢def ok-t))) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (out e , _)) , _) , refl , _ = ⊥-elim (e t (proj₂ (wit-cont out)))
... | oi , _ , refl , pr = proj₂ (fc->defined (pr (t , proj₂ (wit-cont out))))
spec-out->incl : ∀{f g} → FSSpec-⊢ (out f , out g) → Witness f → dom g ⊆ dom f
spec-out->incl {f} {g} fs wit {t} ok-t with t ∈? f
... | yes ok = ok
... | no no-t with (fs (apply-fcoind io ((sample-cont t nil win , f) , wit) (sample-cont-prems {f} {t} no-t))) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (() , _)) , _) , refl , _
... | io , _ , refl , pr = ⊥-elim (cont-not-def (proj₁ (fc->defined (pr (t , ok-t)))))
spec-out->wit : ∀{f g} → Witness f → FSSpec-⊢ (out f , out g) → Witness g
spec-out->wit {f} {g} wit-f fs with Empty? g
... | inj₂ wit = wit
... | inj₁ e with (fs (apply-fcoind io ((full-cont win , f) , wit-f) λ (_ , ok) → win⊢def ok)) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (() , _)) , _) , refl , _
... | io , ((_ , wit-g) , _) , refl , _ = ⊥-elim (e _ (proj₂ wit-g))
{- Boundedness & Consistency -}
fsspec-cons : FSSpec-⊢ ⊆ ISF[ FSubIS ] FSSpec-⊢
fsspec-cons {nil , T} fs = nil-any , _ , refl , λ ()
fsspec-cons {inp f , nil} fs with (fs (apply-fcoind client-end ((win , _) , (Win-win , inp)) λ ())) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (_ , ())) , _) , refl , _
fsspec-cons {inp f , inp g} fs = ii , ((f , g) , spec-inp->incl fs) , refl ,
λ (p , _) {R} fc-r-f →
let wit = wit-cont (proj₁ (fc->defined fc-r-f)) in
let fc-Or-Ig = fs (apply-fcoind oi ((sample-cont p R nil , f) , wit) (sample-cont-prem {f} {p} fc-r-f)) in
let fc-r-g = ⊢-after-out {sample-cont p R nil} {g} {p} (proj₂ wit) fc-Or-Ig in
subst (λ x → x ⊢ g p .force) (sym force-eq) fc-r-g
fsspec-cons {inp f , out g} fs with Empty? f
... | inj₁ e = end-def , (_ , (inp e , out)) , refl , λ ()
... | inj₂ (t , ok-t) with (fs (apply-fcoind oi ((sample-cont t win nil , f) , wit-cont out) (sample-cont-prem {f} {t} (win⊢def ok-t)))) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (out e , out)) , _) , refl , _ = ⊥-elim (e t (proj₂ (wit-cont out)))
fsspec-cons {out f , nil} fs with (fs (apply-fcoind client-end ((win , _) , (Win-win , out)) λ ())) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (_ , ())) , _) , refl , _
fsspec-cons {out f , inp g} fs with Empty? f
... | inj₁ e = end-def , (_ , (out e , inp)) , refl , λ ()
... | inj₂ (t , ok-t) with (fs (apply-fcoind io ((full-cont win , f) , (t , ok-t)) λ (_ , ok-p) → win⊢def ok-p)) .CoInd⟦_⟧.unfold
... | client-end , ((_ , (() , inp)) , _) , refl , _
fsspec-cons {out f , out g} fs with Empty? f
... | inj₁ e = end-def , (_ , (out e , out)) , refl , λ ()
... | inj₂ (t , ok-t) =
let wit-g = spec-out->wit (t , ok-t) fs in
let incl = spec-out->incl fs (t , ok-t) in
oo , ((f , g) , (incl , wit-g)) , refl , λ (p , ok-p) {R} fc-r-f →
let fc-Ir-Og = fs (apply-fcoind io ((sample-cont p R win , f) , (t , ok-t)) (sample-cont-prems' {f} {p} fc-r-f)) in
let fc-r-g = ⊢-after-in {sample-cont p R win} {g} {p} ok-p fc-Ir-Og in
subst (λ x → x ⊢ g p .force) (sym force-eq) fc-r-g
fsspec->sub : ∀{S T} → FSSpec-⊢ (S , T) → S ≤Fc T
fsspec->sub = coind[ FSubIS ] FSSpec-⊢ fsspec-cons
postulate
not-conv-div : ∀{T S} → ¬ T ↓ S → T ↑ S
fs-convergence : ∀{T S} → FairSubtypingS T S → T ↓ S
fs-convergence {T} {S} fs with Common.excluded-middle {T ↓ S}
fs-convergence {T} {S} fs | yes p = p
fs-convergence {T} {S} fs | no p =
let div = not-conv-div p in
let sub = ≤Fc->sub (fsspec->sub (spec-sound fs)) in
let d-comp = discriminator-compliant sub div in
let ¬d-comp = discriminator-not-compliant sub div in
⊥-elim (¬d-comp (fs d-comp))
fsspec-bounded : ∀{S T} → FSSpec-⊢ (S , T) → S ≤Fᵢ T
fsspec-bounded fs = apply-ind (inj₂ co-conv) (_ , (fs-convergence (spec-complete fs))) λ ()
{- Completeness -}
fs-complete : ∀{S T} → FSSpec-⊢ (S , T) → S ≤F T
fs-complete = bounded-coind[ FSubIS , FSubCOIS ] FSSpec-⊢ fsspec-bounded fsspec-cons | {
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-- Andreas, 2016-10-11, AIM XXIV
-- We cannot bind NATURAL to an abstract version of Nat.
abstract
data ℕ : Set where
zero : ℕ
suc : ℕ → ℕ
{-# BUILTIN NATURAL ℕ #-}
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-- Jesper, 2018-10-29 (comment on #3332): Besides for rewrite, builtin
-- equality is also used for primErase and primForceLemma. But I don't
-- see how it would hurt to have these use a Prop instead of a Set for
-- equality.
{-# OPTIONS --prop #-}
data _≡_ {A : Set} (a : A) : A → Prop where
refl : a ≡ a
{-# BUILTIN EQUALITY _≡_ #-}
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open import Agda.Builtin.Nat
open import Agda.Builtin.Reflection
open import Agda.Builtin.Unit
macro
five : Term → TC ⊤
five hole = unify hole (lit (nat 5))
-- Here you get hole = _X (λ {n} → y {_n})
-- and fail to solve _n.
yellow : ({n : Nat} → Set) → Nat
yellow y = five
-- Here you get hole = _X ⦃ n ⦄ (λ ⦃ n' ⦄ → y ⦃ _n ⦄)
-- and fail to solve _n due to the multiple candidates n and n'.
more-yellow : ⦃ n : Nat ⦄ → (⦃ n : Nat ⦄ → Set) → Nat
more-yellow y = five
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Category.Monoidal.Core using (Monoidal)
open import Categories.Category.Monoidal.Symmetric
open import Data.Sum
module Categories.Category.Monoidal.CompactClosed {o ℓ e} {C : Category o ℓ e} (M : Monoidal C) where
open import Level
open import Categories.Functor.Bifunctor
open import Categories.NaturalTransformation.NaturalIsomorphism
open import Categories.Category.Monoidal.Rigid
open Category C
open Commutation C
record CompactClosed : Set (levelOfTerm M) where
field
symmetric : Symmetric M
rigid : LeftRigid M ⊎ RightRigid M
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{-# OPTIONS --rewriting #-}
data Unit : Set where
unit : Unit
_+_ : Unit → Unit → Unit
unit + x = x
data _≡_ (x : Unit) : Unit → Set where
refl : x ≡ x
{-# BUILTIN REWRITE _≡_ #-}
postulate
f : Unit → Unit
fu : f unit ≡ unit
{-# REWRITE fu #-}
g : Unit → Unit
g unit = unit
data D (h : Unit → Unit) (x : Unit) : Set where
wrap : Unit → D h x
run : ∀ {h x} → D h x → Unit
run (wrap x) = x
postulate
d₁ : ∀ n x y (p : D y n) → x (run p + n) ≡ y (run p + n) → D x n
d₂ : ∀ s n → D s n
d₂ _ _ = wrap unit
d₃ : D (λ _ → unit) unit
d₃ = d₁ _ (λ _ → unit) (λ n → f (g n)) (d₂ (λ n → f (g n)) _) refl
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module Yoneda where
open import Level
open import Data.Product
open import Relation.Binary
import Relation.Binary.SetoidReasoning as SetR
open Setoid renaming (_≈_ to eqSetoid)
open import Basic
open import Category
import Functor
import Nat
open Category.Category
open Functor.Functor
open Nat.Nat
open Nat.Export
yoneda : ∀{C₀ C₁ ℓ} (C : Category C₀ C₁ ℓ) → Functor.Functor C PSh[ C ]
yoneda C = record {
fobj = λ x → Hom[ C ][-, x ] ;
fmapsetoid = λ {A} {B} → record { mapping = λ f → HomNat[ C ][-, f ] ; preserveEq = λ {x} {y} x₁ x₂ → begin⟨ C ⟩
Map.mapping (component HomNat[ C ][-, x ] _) x₂ ≈⟨ refl-hom C ⟩
C [ x ∘ x₂ ] ≈⟨ ≈-composite C x₁ (refl-hom C) ⟩
C [ y ∘ x₂ ] ≈⟨ refl-hom C ⟩
Map.mapping (component HomNat[ C ][-, y ] _) x₂
∎ } ;
preserveId = λ x → leftId C ;
preserveComp = λ x → assoc C
}
YonedaLemma : ∀{C₀ C₁ ℓ} {C : Category C₀ C₁ ℓ} {F : Obj PSh[ C ]} {X : Obj C} → Setoids [ Homsetoid [ op C , Setoids ] (fobj (yoneda C) X) F ≅ LiftSetoid {C₁} {suc (suc ℓ) ⊔ (suc (suc C₁) ⊔ suc C₀)} {ℓ} {C₀ ⊔ C₁ ⊔ ℓ} (fobj F X) ]
YonedaLemma {C₀} {C₁} {ℓ} {C} {F} {X} = record {
map-→ = nat→obj ;
map-← = obj→nat ;
proof = obj→obj≈id , nat→nat≈id }
where
nat→obj : Map.Map (Homsetoid [ op C , Setoids ] (fobj (yoneda C) X) F) (LiftSetoid (fobj F X))
nat→obj = record {
mapping = λ α → lift (Map.mapping (component α X) (id C)) ;
preserveEq = λ x≈y → lift (x≈y (id C)) }
obj→nat : Map.Map (LiftSetoid (fobj F X)) (Homsetoid [ op C , Setoids ] (fobj (yoneda C) X) F)
obj→nat = record {
mapping = λ a → record {
component = component-map a ;
naturality = λ {c} {d} {f} x → SetR.begin⟨ fobj F d ⟩
Map.mapping (Setoids [ component-map a d ∘ fmap (fobj (yoneda C) X) f ]) x SetR.≈⟨ refl (fobj F d) ⟩
Map.mapping (Map.mapping (fmapsetoid F) (C [ x ∘ f ])) (lower a) SetR.≈⟨ (preserveComp F) (lower a) ⟩
Map.mapping (Map.mapping (fmapsetoid F) f) (Map.mapping (Map.mapping (fmapsetoid F) x) (lower a)) SetR.≈⟨ refl (fobj F d) ⟩
Map.mapping (Setoids [ fmap F f ∘ component-map a c ]) x
SetR.∎} ;
preserveEq = λ {x} {y} x≈y f → SetR.begin⟨ fobj F _ ⟩
Map.mapping (component-map x _) f SetR.≈⟨ refl (fobj F _) ⟩
Map.mapping (Map.mapping (fmapsetoid F) f) (lower x) SetR.≈⟨ Map.preserveEq (fmap F f) (lower x≈y) ⟩
Map.mapping (Map.mapping (fmapsetoid F) f) (lower y) SetR.≈⟨ refl (fobj F _) ⟩
Map.mapping (component-map y _) f
SetR.∎}
where
component-map = λ a b → record {
mapping = λ u → Map.mapping (fmap F u) (lower a) ;
preserveEq = λ {x} {y} x≈y → SetR.begin⟨ fobj F b ⟩
Map.mapping (fmap F x) (lower a) SetR.≈⟨ (Functor.preserveEq F x≈y) (lower a) ⟩
Map.mapping (fmap F y) (lower a)
SetR.∎
}
obj→obj≈id : Setoids [ Setoids [ nat→obj ∘ obj→nat ] ≈ id Setoids ]
obj→obj≈id = λ x → lift (SetR.begin⟨ fobj F X ⟩
lower (Map.mapping (Setoids [ nat→obj ∘ obj→nat ]) x) SetR.≈⟨ refl (fobj F X) ⟩
Map.mapping (Map.mapping (fmapsetoid F) (id C)) (lower x) SetR.≈⟨ (preserveId F) (lower x) ⟩
lower x SetR.≈⟨ refl (fobj F X) ⟩
lower (Map.mapping {A = LiftSetoid {ℓ′ = ℓ} (fobj F X)} (id Setoids) x)
SetR.∎)
nat→nat≈id : Setoids [ Setoids [ obj→nat ∘ nat→obj ] ≈ id Setoids ]
nat→nat≈id α f = SetR.begin⟨ fobj F _ ⟩
Map.mapping (component (Map.mapping (Setoids [ obj→nat ∘ nat→obj ]) α) _) f SetR.≈⟨ refl (fobj F _) ⟩
Map.mapping (Setoids [ fmap F f ∘ component α X ]) (id C) SetR.≈⟨ lemma (id C) ⟩
Map.mapping (Setoids [ component α (dom C f) ∘ fmap (fobj (yoneda C) X) f ]) (id C) SetR.≈⟨ Map.preserveEq (component α (dom C f)) (leftId C) ⟩
Map.mapping (component α (dom C f)) f SetR.≈⟨ refl (fobj F _) ⟩
Map.mapping (component (Map.mapping (id Setoids {Homsetoid [ (op C) , Setoids ] _ _}) α) (dom C f)) f
SetR.∎
where
lemma : Setoids [ Setoids [ fmap F f ∘ component α X ] ≈ Setoids [ component α (dom C f) ∘ fmap (fobj (yoneda C) X) f ] ]
lemma = sym-hom Setoids {f = Setoids [ component α (dom C f) ∘ fmap (fobj (yoneda C) X) f ]} {g = Setoids [ fmap F f ∘ component α X ]} (naturality α)
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------------------------------------------------------------------------
-- "Equational" reasoning combinator setup
------------------------------------------------------------------------
{-# OPTIONS --sized-types #-}
open import Prelude
open import Labelled-transition-system
module Similarity.Weak.Equational-reasoning-instances
{ℓ} {lts : LTS ℓ} where
open import Bisimilarity lts
open import Bisimilarity.Weak lts
import Bisimilarity.Weak.Equational-reasoning-instances
open import Equational-reasoning
open import Expansion lts
import Expansion.Equational-reasoning-instances
open import Similarity lts
open import Similarity.Weak lts
instance
reflexive≼ : ∀ {i} → Reflexive [ i ]_≼_
reflexive≼ = is-reflexive reflexive-≼
reflexive≼′ : ∀ {i} → Reflexive [ i ]_≼′_
reflexive≼′ = is-reflexive reflexive-≼′
convert≼≼ : ∀ {i} → Convertible [ i ]_≼_ [ i ]_≼_
convert≼≼ = is-convertible id
convert≼′≼ : ∀ {i} → Convertible _≼′_ [ i ]_≼_
convert≼′≼ = is-convertible λ p≼′q → force p≼′q
convert≼≼′ : ∀ {i} → Convertible [ i ]_≼_ [ i ]_≼′_
convert≼≼′ {i} = is-convertible lemma
where
lemma : ∀ {p q} → [ i ] p ≼ q → [ i ] p ≼′ q
force (lemma p≼q) = p≼q
convert≼′≼′ : ∀ {i} → Convertible [ i ]_≼′_ [ i ]_≼′_
convert≼′≼′ = is-convertible id
convert≤≼ : ∀ {i} → Convertible [ i ]_≤_ [ i ]_≼_
convert≤≼ = is-convertible ≤⇒≼
convert≤′≼ : ∀ {i} → Convertible _≤′_ [ i ]_≼_
convert≤′≼ = is-convertible (convert ∘ ≤⇒≼′)
convert≤≼′ : ∀ {i} → Convertible [ i ]_≤_ [ i ]_≼′_
convert≤≼′ {i} = is-convertible lemma
where
lemma : ∀ {p q} → [ i ] p ≤ q → [ i ] p ≼′ q
force (lemma p≤q) = convert p≤q
convert≤′≼′ : ∀ {i} → Convertible [ i ]_≤′_ [ i ]_≼′_
convert≤′≼′ = is-convertible ≤⇒≼′
convert≈≼ : ∀ {i} → Convertible [ i ]_≈_ [ i ]_≼_
convert≈≼ = is-convertible ≈⇒≼
convert≈′≼ : ∀ {i} → Convertible _≈′_ [ i ]_≼_
convert≈′≼ = is-convertible (convert ∘ ≈⇒≼′)
convert≈≼′ : ∀ {i} → Convertible [ i ]_≈_ [ i ]_≼′_
convert≈≼′ {i} = is-convertible lemma
where
lemma : ∀ {p q} → [ i ] p ≈ q → [ i ] p ≼′ q
force (lemma p≈q) = convert p≈q
convert≈′≼′ : ∀ {i} → Convertible [ i ]_≈′_ [ i ]_≼′_
convert≈′≼′ = is-convertible ≈⇒≼′
convert∼≼ : ∀ {i} → Convertible [ i ]_∼_ [ i ]_≼_
convert∼≼ = is-convertible (≈⇒≼ ∘ convert {a = ℓ})
convert∼′≼ : ∀ {i} → Convertible _∼′_ [ i ]_≼_
convert∼′≼ = is-convertible (convert ∘ ≈⇒≼′ ∘ convert {a = ℓ})
convert∼≼′ : ∀ {i} → Convertible [ i ]_∼_ [ i ]_≼′_
convert∼≼′ {i} = is-convertible lemma
where
lemma : ∀ {p q} → [ i ] p ∼ q → [ i ] p ≼′ q
force (lemma p∼q) = ≈⇒≼ (convert {a = ℓ} p∼q)
convert∼′≼′ : ∀ {i} → Convertible [ i ]_∼′_ [ i ]_≼′_
convert∼′≼′ = is-convertible (≈⇒≼′ ∘ convert {a = ℓ})
trans≼≼ : ∀ {i} → Transitive′ [ i ]_≼_ _≼_
trans≼≼ = is-transitive transitive-≼
trans≼′≼ : Transitive′ _≼′_ _≼_
trans≼′≼ = is-transitive transitive-≼′
trans≼′≼′ : ∀ {i} → Transitive′ [ i ]_≼′_ _≼′_
trans≼′≼′ = is-transitive (λ p≼q → transitive-≼′ p≼q ∘ convert)
trans≼≼′ : ∀ {i} → Transitive′ [ i ]_≼_ _≼′_
trans≼≼′ = is-transitive (λ p≼q → transitive-≼ p≼q ∘ convert)
trans≳≼ : ∀ {i} → Transitive [ i ]_≳_ [ i ]_≼_
trans≳≼ = is-transitive transitive-≳≼
trans≳′≼ : ∀ {i} → Transitive _≳′_ [ i ]_≼_
trans≳′≼ = is-transitive (transitive-≳≼ ∘ convert {a = ℓ})
trans≳′≼′ : ∀ {i} → Transitive [ i ]_≳′_ [ i ]_≼′_
trans≳′≼′ = is-transitive transitive-≳≼′
trans≳≼′ : ∀ {i} → Transitive [ i ]_≳_ [ i ]_≼′_
trans≳≼′ = is-transitive (transitive-≳≼′ ∘ convert {a = ℓ})
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Numbers.Naturals.Semiring
open import Functions
open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Orders.Archimedean
module Fields.FieldOfFractions.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {c : _} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) where
open import Groups.Cyclic.Definition
open import Fields.FieldOfFractions.Setoid I
open import Fields.FieldOfFractions.Group I
open import Fields.FieldOfFractions.Addition I
open import Fields.FieldOfFractions.Ring I
open import Fields.FieldOfFractions.Order I order
open import Fields.FieldOfFractions.Field I
open import Fields.Orders.Partial.Archimedean {F = fieldOfFractions} record { oRing = fieldOfFractionsPOrderedRing }
open import Rings.Orders.Partial.Lemmas pRing
open import Rings.Orders.Total.Lemmas
open Setoid S
open Equivalence eq
open Ring R
open Group additiveGroup
open TotallyOrderedRing order
open SetoidTotalOrder total
private
denomPower : (n : ℕ) → fieldOfFractionsSet.denom (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I } n) ∼ 1R
denomPower zero = reflexive
denomPower (succ n) = transitive identIsIdent (denomPower n)
denomPlus : {a : A} .(a!=0 : a ∼ 0R → False) (n1 n2 : A) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus record { num = n1 ; denom = a ; denomNonzero = a!=0 } record { num = n2 ; denom = a ; denomNonzero = a!=0 }) (record { num = n1 + n2 ; denom = a ; denomNonzero = a!=0 })
denomPlus {a} a!=0 n1 n2 = transitive *Commutative (transitive (*WellDefined reflexive (transitive (+WellDefined *Commutative reflexive) (symmetric *DistributesOver+))) *Associative)
d : (a : fieldOfFractionsSet) → fieldOfFractionsSet.denom a ∼ 0R → False
d record { num = num ; denom = denom ; denomNonzero = denomNonzero } bad = exFalso (denomNonzero bad)
simpPower : (n : ℕ) → Setoid._∼_ fieldOfFractionsSetoid (positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I} n) record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }
simpPower zero = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {Group.0G fieldOfFractionsGroup}
simpPower (succ n) = Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = d (fieldOfFractionsPlus (record { num = 1R ; denom = 1R ; denomNonzero = λ t → IntegralDomain.nontrivial I t }) (positiveEltPower fieldOfFractionsGroup _ n)) }} {record { denomNonzero = λ t → IntegralDomain.nontrivial I (transitive (symmetric identIsIdent) t) }} {record { denomNonzero = IntegralDomain.nontrivial I }} (Group.+WellDefined fieldOfFractionsGroup {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I } n} {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I}}) (simpPower n)) (transitive (transitive (transitive *Commutative (transitive identIsIdent (+WellDefined identIsIdent identIsIdent))) (symmetric identIsIdent)) (*WellDefined (symmetric identIsIdent) reflexive))
lemma : (n : ℕ) {num denom : A} .(d!=0 : denom ∼ 0G → False) → (num * denom) < positiveEltPower additiveGroup 1R n → fieldOfFractionsComparison (record { num = num ; denom = denom ; denomNonzero = d!=0}) record { num = positiveEltPower additiveGroup (Ring.1R R) n ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I }
lemma n {num} {denom} d!=0 numdenom<n with totality 0G denom
... | inl (inl 0<denom) with totality 0G 1R
... | inl (inl 0<1) = {!!}
... | inl (inr x) = exFalso (1<0False order x)
... | inr x = exFalso (IntegralDomain.nontrivial I (symmetric x))
lemma n {num} {denom} d!=0 numdenom<n | inl (inr denom<0) with totality 0G 1R
... | inl (inl 0<1) = {!!}
... | inl (inr 1<0) = exFalso (1<0False order 1<0)
... | inr 0=1 = exFalso (IntegralDomain.nontrivial I (symmetric 0=1))
lemma n {num} {denom} d!=0 numdenom<n | inr 0=denom = exFalso (d!=0 (symmetric 0=denom))
fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField
fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) 0<q with totality 0G denom ,, totality 0G 1R
... | inl (inl 0<denom) ,, inl (inl 0<1) rewrite refl { x = 0} = {!!}
... | inl (inl 0<denom) ,, inl (inr x) = exFalso {!!}
... | inl (inl 0<denom) ,, inr x = exFalso {!!}
... | inl (inr denom<0) ,, inl (inl 0<1) = 0 , {!!}
where
t : {!!}
t = {!!}
... | inl (inr denom<0) ,, inl (inr x) = exFalso {!!}
... | inl (inr denom<0) ,, inr x = exFalso {!!}
... | inr x ,, snd = exFalso (denom!=0 (symmetric x))
--... | N , pr = N , SetoidPartialOrder.<WellDefined fieldOfFractionsOrder {record { denomNonzero = denom!=0 }} {record { denomNonzero = denom!=0 }} {record { denomNonzero = λ t → nonempty (symmetric t) }} {record { denomNonzero = d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) { record { denomNonzero = denom!=0 } }) (Equivalence.symmetric (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = λ t → d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) t }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (simpPower N)) (lemma N denom!=0 pr)
fieldOfFractionsArchimedean' : Archimedean (toGroup R pRing) → Archimedean (toGroup fieldOfFractionsRing fieldOfFractionsPOrderedRing)
fieldOfFractionsArchimedean' arch = archFieldToGrp (reciprocalPositive fieldOfFractionsOrderedRing) (fieldOfFractionsArchimedean arch)
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------------------------------------------------------------------------
-- Vectors parameterised on types in Set₁
------------------------------------------------------------------------
-- I want universe polymorphism.
module Data.Vec1 where
infixr 5 _∷_
open import Data.Nat
open import Data.Vec using (Vec; []; _∷_)
open import Data.Fin
------------------------------------------------------------------------
-- The type
data Vec₁ (a : Set₁) : ℕ → Set₁ where
[] : Vec₁ a zero
_∷_ : ∀ {n} (x : a) (xs : Vec₁ a n) → Vec₁ a (suc n)
------------------------------------------------------------------------
-- Some operations
map : ∀ {a b n} → (a → b) → Vec₁ a n → Vec₁ b n
map f [] = []
map f (x ∷ xs) = f x ∷ map f xs
map₀₁ : ∀ {a b n} → (a → b) → Vec a n → Vec₁ b n
map₀₁ f [] = []
map₀₁ f (x ∷ xs) = f x ∷ map₀₁ f xs
map₁₀ : ∀ {a b n} → (a → b) → Vec₁ a n → Vec b n
map₁₀ f [] = []
map₁₀ f (x ∷ xs) = f x ∷ map₁₀ f xs
lookup : ∀ {a n} → Fin n → Vec₁ a n → a
lookup zero (x ∷ xs) = x
lookup (suc i) (x ∷ xs) = lookup i xs
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open import Common.Prelude
open import Common.Reflection
open import Common.Equality
postulate
trustme : ∀ {a} {A : Set a} {x y : A} → x ≡ y
magic : List (Arg Type) → Term → Tactic
magic _ _ = give (def (quote trustme) [])
id : ∀ {a} {A : Set a} → A → A
id x = x
science : List (Arg Type) → Term → Tactic
science _ _ = give (def (quote id) [])
by-magic : ∀ n → n + 4 ≡ 3
by-magic n = tactic magic
by-science : ∀ n → 0 + n ≡ n
by-science n = tactic science | refl
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module Issue274 where
-- data ⊥ : Set where
record U : Set where
constructor roll
field ap : U → U
-- lemma : U → ⊥
-- lemma (roll u) = lemma (u (roll u))
-- bottom : ⊥
-- bottom = lemma (roll λ x → x)
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{-# OPTIONS --without-K #-}
open import HoTT
module cohomology.Choice where
unchoose : ∀ {i j} {n : ℕ₋₂} {A : Type i} {B : A → Type j}
→ Trunc n (Π A B) → Π A (Trunc n ∘ B)
unchoose = Trunc-rec (Π-level (λ _ → Trunc-level)) (λ f → [_] ∘ f)
has-choice : ∀ {i j} (n : ℕ₋₂) (A : Type i) (B : A → Type j) → Type (lmax i j)
has-choice {i} {j} n A B = is-equiv (unchoose {n = n} {A} {B})
{- Binary Choice -}
module _ {k} where
pick-Bool : ∀ {j} {C : Lift {j = k} Bool → Type j}
→ (C (lift true)) → (C (lift false)) → Π (Lift Bool) C
pick-Bool x y (lift true) = x
pick-Bool x y (lift false) = y
pick-Bool-η : ∀ {j} {C : Lift Bool → Type j} (f : Π (Lift Bool) C)
→ pick-Bool (f (lift true)) (f (lift false)) == f
pick-Bool-η {C = C} f = λ= lemma
where
lemma : (b : Lift Bool) →
pick-Bool {C = C} (f (lift true)) (f (lift false)) b == f b
lemma (lift true) = idp
lemma (lift false) = idp
module _ {i} {n : ℕ₋₂} {A : Lift {j = k} Bool → Type i} where
choose-Bool : Π (Lift Bool) (Trunc n ∘ A) → Trunc n (Π (Lift Bool) A)
choose-Bool f = Trunc-rec Trunc-level
(λ ft → Trunc-rec Trunc-level
(λ ff → [ pick-Bool ft ff ] )
(f (lift false)))
(f (lift true))
choose-Bool-squash : (f : Π (Lift Bool) A)
→ choose-Bool (λ b → [ f b ]) == [ f ]
choose-Bool-squash f = ap [_] (pick-Bool-η f)
private
unc-c : ∀ f → unchoose (choose-Bool f) == f
unc-c f = transport
(λ h → unchoose (choose-Bool h) == h)
(pick-Bool-η f)
(Trunc-elim
{P = λ tft →
unchoose (choose-Bool (pick-Bool tft (f (lift false))))
== pick-Bool tft (f (lift false))}
(λ _ → =-preserves-level _ (Π-level (λ _ → Trunc-level)))
(λ ft → Trunc-elim
{P = λ tff →
unchoose (choose-Bool (pick-Bool [ ft ] tff ))
== pick-Bool [ ft ] tff}
(λ _ → =-preserves-level _ (Π-level (λ _ → Trunc-level)))
(λ ff → ! (pick-Bool-η _))
(f (lift false)))
(f (lift true)))
c-unc : ∀ tg → choose-Bool (unchoose tg) == tg
c-unc = Trunc-elim
{P = λ tg → choose-Bool (unchoose tg) == tg}
(λ _ → =-preserves-level _ Trunc-level)
(λ g → ap [_] (pick-Bool-η g))
Bool-has-choice : has-choice n (Lift Bool) A
Bool-has-choice = is-eq unchoose choose-Bool unc-c c-unc
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{-# OPTIONS --without-K #-}
module Data.Tuple where
open import Data.Tuple.Base public
import Data.Product as P
Pair→× : ∀ {a b A B} → Pair {a} {b} A B → A P.× B
Pair→× (fst , snd) = fst P., snd
×→Pair : ∀ {a b A B} → P._×_ {a} {b} A B → Pair A B
×→Pair (fst P., snd) = fst , snd
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module Type.NbE where
open import Context
open import Type.Core
open import Function
open import Data.Empty
open import Data.Sum.Base
infix 3 _⊢ᵗⁿᵉ_ _⊢ᵗⁿᶠ_ _⊨ᵗ_
infixl 9 _[_]ᵗ
mutual
Starⁿᵉ : Conᵏ -> Set
Starⁿᵉ Θ = Θ ⊢ᵗⁿᵉ ⋆
Starⁿᶠ : Conᵏ -> Set
Starⁿᶠ Θ = Θ ⊢ᵗⁿᶠ ⋆
data _⊢ᵗⁿᵉ_ Θ : Kind -> Set where
Varⁿᵉ : ∀ {σ} -> σ ∈ Θ -> Θ ⊢ᵗⁿᵉ σ
_∙ⁿᵉ_ : ∀ {σ τ} -> Θ ⊢ᵗⁿᵉ σ ⇒ᵏ τ -> Θ ⊢ᵗⁿᶠ σ -> Θ ⊢ᵗⁿᵉ τ
_⇒ⁿᵉ_ : Starⁿᵉ Θ -> Starⁿᵉ Θ -> Starⁿᵉ Θ
πⁿᵉ : ∀ σ -> Starⁿᵉ (Θ ▻ σ) -> Starⁿᵉ Θ
μⁿᵉ : ∀ {κ} -> Θ ⊢ᵗⁿᶠ (κ ⇒ᵏ ⋆) ⇒ᵏ κ ⇒ᵏ ⋆ -> Θ ⊢ᵗⁿᶠ κ -> Starⁿᵉ Θ
data _⊢ᵗⁿᶠ_ Θ : Kind -> Set where
Neⁿᶠ : ∀ {σ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊢ᵗⁿᶠ σ
Lamⁿᶠ_ : ∀ {σ τ} -> Θ ▻ σ ⊢ᵗⁿᶠ τ -> Θ ⊢ᵗⁿᶠ σ ⇒ᵏ τ
mutual
embᵗⁿᵉ : ∀ {Θ σ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊢ᵗ σ
embᵗⁿᵉ (Varⁿᵉ v) = Var v
embᵗⁿᵉ (φ ∙ⁿᵉ α) = embᵗⁿᵉ φ ∙ embᵗⁿᶠ α
embᵗⁿᵉ (α ⇒ⁿᵉ β) = embᵗⁿᵉ α ⇒ embᵗⁿᵉ β
embᵗⁿᵉ (πⁿᵉ σ α) = π σ (embᵗⁿᵉ α)
embᵗⁿᵉ (μⁿᵉ ψ α) = μ (embᵗⁿᶠ ψ) (embᵗⁿᶠ α)
embᵗⁿᶠ : ∀ {Θ σ} -> Θ ⊢ᵗⁿᶠ σ -> Θ ⊢ᵗ σ
embᵗⁿᶠ (Neⁿᶠ α) = embᵗⁿᵉ α
embᵗⁿᶠ (Lamⁿᶠ β) = Lam (embᵗⁿᶠ β)
mutual
renᵗⁿᵉ : ∀ {Θ Ξ σ} -> Θ ⊆ Ξ -> Θ ⊢ᵗⁿᵉ σ -> Ξ ⊢ᵗⁿᵉ σ
renᵗⁿᵉ ι (Varⁿᵉ v) = Varⁿᵉ (renᵛ ι v)
renᵗⁿᵉ ι (φ ∙ⁿᵉ α) = renᵗⁿᵉ ι φ ∙ⁿᵉ renᵗⁿᶠ ι α
renᵗⁿᵉ ι (α ⇒ⁿᵉ β) = renᵗⁿᵉ ι α ⇒ⁿᵉ renᵗⁿᵉ ι β
renᵗⁿᵉ ι (πⁿᵉ σ α) = πⁿᵉ σ (renᵗⁿᵉ (keep ι) α)
renᵗⁿᵉ ι (μⁿᵉ ψ α) = μⁿᵉ (renᵗⁿᶠ ι ψ) (renᵗⁿᶠ ι α)
renᵗⁿᶠ : ∀ {Θ Ξ σ} -> Θ ⊆ Ξ -> Θ ⊢ᵗⁿᶠ σ -> Ξ ⊢ᵗⁿᶠ σ
renᵗⁿᶠ ι (Neⁿᶠ α) = Neⁿᶠ (renᵗⁿᵉ ι α)
renᵗⁿᶠ ι (Lamⁿᶠ β) = Lamⁿᶠ (renᵗⁿᶠ (keep ι) β)
mutual
_⊨ᵗ_ : Conᵏ -> Kind -> Set
Θ ⊨ᵗ σ = Θ ⊢ᵗⁿᵉ σ ⊎ Kripke Θ σ
Kripke : Conᵏ -> Kind -> Set
Kripke Θ ⋆ = ⊥
Kripke Θ (σ ⇒ᵏ τ) = ∀ {Ξ} -> Θ ⊆ Ξ -> Ξ ⊨ᵗ σ -> Ξ ⊨ᵗ τ
Neˢ : ∀ {σ Θ} -> Θ ⊢ᵗⁿᵉ σ -> Θ ⊨ᵗ σ
Neˢ = inj₁
Varˢ : ∀ {σ Θ} -> σ ∈ Θ -> Θ ⊨ᵗ σ
Varˢ = Neˢ ∘ Varⁿᵉ
renᵗˢ : ∀ {σ Θ Δ} -> Θ ⊆ Δ -> Θ ⊨ᵗ σ -> Δ ⊨ᵗ σ
renᵗˢ ι (inj₁ α) = inj₁ (renᵗⁿᵉ ι α)
renᵗˢ {⋆} ι (inj₂ ())
renᵗˢ {σ ⇒ᵏ τ} ι (inj₂ k) = inj₂ λ κ -> k (κ ∘ ι)
readbackᵗ : ∀ {σ Θ} -> Θ ⊨ᵗ σ -> Θ ⊢ᵗⁿᶠ σ
readbackᵗ (inj₁ α) = Neⁿᶠ α
readbackᵗ {⋆} (inj₂ ())
readbackᵗ {σ ⇒ᵏ τ} (inj₂ k) = Lamⁿᶠ (readbackᵗ (k topᵒ (Varˢ vz)))
_∙ˢ_ : ∀ {Θ σ τ} -> Θ ⊨ᵗ σ ⇒ᵏ τ -> Θ ⊨ᵗ σ -> Θ ⊨ᵗ τ
inj₁ f ∙ˢ x = Neˢ (f ∙ⁿᵉ readbackᵗ x)
inj₂ k ∙ˢ x = k idᵒ x
groundⁿᵉ : ∀ {Θ} -> Θ ⊨ᵗ ⋆ -> Θ ⊢ᵗⁿᵉ ⋆
groundⁿᵉ (inj₁ α) = α
groundⁿᵉ (inj₂ ())
environmentᵗˢ : Environment _⊨ᵗ_
environmentᵗˢ = record
{ varᵈ = Varˢ
; renᵈ = renᵗˢ
}
module _ where
open Environment environmentᵗˢ
mutual
evalᵗ : ∀ {Θ Ξ σ} -> Ξ ⊢ᵉ Θ -> Θ ⊢ᵗ σ -> Ξ ⊨ᵗ σ
evalᵗ ρ (Var v) = lookupᵉ v ρ
evalᵗ ρ (Lam β) = inj₂ λ ι α -> evalᵗ (renᵉ ι ρ ▷ α) β
evalᵗ ρ (φ ∙ α) = evalᵗ ρ φ ∙ˢ evalᵗ ρ α
evalᵗ ρ (α ⇒ β) = inj₁ $ groundⁿᵉ (evalᵗ ρ α) ⇒ⁿᵉ groundⁿᵉ (evalᵗ ρ β)
evalᵗ ρ (π σ α) = inj₁ $ πⁿᵉ σ ∘ groundⁿᵉ $ evalᵗ (keepᵉ ρ) α
evalᵗ ρ (μ ψ α) = inj₁ $ μⁿᵉ (nfᵗ ρ ψ) (nfᵗ ρ α)
nfᵗ : ∀ {Θ Ξ σ} -> Ξ ⊢ᵉ Θ -> Θ ⊢ᵗ σ -> Ξ ⊢ᵗⁿᶠ σ
nfᵗ ρ = readbackᵗ ∘ evalᵗ ρ
normalize : ∀ {Θ σ} -> Θ ⊢ᵗ σ -> Θ ⊢ᵗ σ
normalize = embᵗⁿᶠ ∘ nfᵗ idᵉ
_[_]ᵗ : ∀ {Θ σ τ} -> Θ ▻ σ ⊢ᵗ τ -> Θ ⊢ᵗ σ -> Θ ⊢ᵗ τ
β [ α ]ᵗ = normalize $ Lam β ∙ α
{-# DISPLAY normalize (Lam β ∙ α) = β [ α ]ᵗ #-}
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postulate
_→_ : Set
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{-# OPTIONS --safe --no-termination-check #-}
module Issue2442-conflicting where
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
open import lib.NType2
open import lib.Function2
open import lib.types.Group
open import lib.types.Sigma
open import lib.types.Truncation
open import lib.groups.Homomorphism
open import lib.groups.Isomorphism
open import lib.groups.SubgroupProp
module lib.groups.Subgroup where
module _ {i j} {G : Group i} (P : SubgroupProp G j) where
private
module G = Group G
module P = SubgroupProp P
subgroup-struct : GroupStructure P.SubEl
subgroup-struct = record {M} where
module M where
ident : P.SubEl
ident = G.ident , P.ident
inv : P.SubEl → P.SubEl
inv (g , p) = G.inv g , P.inv p
comp : P.SubEl → P.SubEl → P.SubEl
comp (g₁ , p₁) (g₂ , p₂) = G.comp g₁ g₂ , P.comp p₁ p₂
abstract
unit-l : ∀ g → comp ident g == g
unit-l (g , _) = Subtype=-out P.subEl-prop (G.unit-l g)
assoc : ∀ g₁ g₂ g₃ → comp (comp g₁ g₂) g₃ == comp g₁ (comp g₂ g₃)
assoc (g₁ , _) (g₂ , _) (g₃ , _) = Subtype=-out P.subEl-prop (G.assoc g₁ g₂ g₃)
inv-l : ∀ g → comp (inv g) g == ident
inv-l (g , _) = Subtype=-out P.subEl-prop (G.inv-l g)
Subgroup : Group (lmax i j)
Subgroup = group _ SubEl-level subgroup-struct
where abstract SubEl-level = Subtype-level P.subEl-prop G.El-level
module Subgroup {i j} {G : Group i} (P : SubgroupProp G j) where
private
module P = SubgroupProp P
module G = Group G
grp = Subgroup P
open Group grp public
El=-out : ∀ {s₁ s₂ : El} → fst s₁ == fst s₂ → s₁ == s₂
El=-out = Subtype=-out P.subEl-prop
inject : Subgroup P →ᴳ G
inject = record {f = fst; pres-comp = λ _ _ → idp}
inject-lift : ∀ {j} {H : Group j} (φ : H →ᴳ G)
→ Π (Group.El H) (P.prop ∘ GroupHom.f φ)
→ (H →ᴳ Subgroup P)
inject-lift {H = H} φ P-all = record {M} where
module H = Group H
module φ = GroupHom φ
module M where
f : H.El → P.SubEl
f h = φ.f h , P-all h
abstract
pres-comp : ∀ h₁ h₂ → f (H.comp h₁ h₂) == Group.comp (Subgroup P) (f h₁) (f h₂)
pres-comp h₁ h₂ = Subtype=-out P.subEl-prop (φ.pres-comp h₁ h₂)
full-subgroup : ∀ {i j} {G : Group i} {P : SubgroupProp G j}
→ is-fullᴳ P → Subgroup P ≃ᴳ G
full-subgroup {G = G} {P} full = Subgroup.inject P , is-eq _ from to-from from-to where
from : Group.El G → Subgroup.El P
from g = g , full g
abstract
from-to : ∀ p → from (fst p) == p
from-to p = Subtype=-out (SubgroupProp.subEl-prop P) idp
to-from : ∀ g → fst (from g) == g
to-from g = idp
module _ {i} {j} {G : Group i} {H : Group j} (φ : G →ᴳ H) where
private
module G = Group G
module H = Group H
module φ = GroupHom φ
module Ker = Subgroup (ker-propᴳ φ)
Ker = Ker.grp
module Im = Subgroup (im-propᴳ φ)
Im = Im.grp
im-lift : G →ᴳ Im
im-lift = Im.inject-lift φ (λ g → [ g , idp ])
im-lift-is-surj : is-surjᴳ im-lift
im-lift-is-surj (_ , s) = Trunc-fmap
(λ {(g , p) → (g , Subtype=-out (_ , λ _ → Trunc-level) p)}) s
module _ {i j k} {G : Group i} (P : SubgroupProp G j)
(Q : SubgroupProp G k) where
-- FIXME looks like a bad name
Subgroup-fmap-r : P ⊆ᴳ Q → Subgroup P →ᴳ Subgroup Q
Subgroup-fmap-r P⊆Q = group-hom (Σ-fmap-r P⊆Q)
(λ _ _ → Subtype=-out (SubgroupProp.subEl-prop Q) idp)
Subgroup-emap-r : P ⊆ᴳ Q → Q ⊆ᴳ P → Subgroup P ≃ᴳ Subgroup Q
Subgroup-emap-r P⊆Q Q⊆P = Subgroup-fmap-r P⊆Q ,
is-eq _ (Σ-fmap-r Q⊆P)
(λ _ → Subtype=-out (SubgroupProp.subEl-prop Q) idp)
(λ _ → Subtype=-out (SubgroupProp.subEl-prop P) idp)
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module Lemmachine.Resource.Configure where
open import Lemmachine.Request
open import Lemmachine.Response
open import Lemmachine.Resource
open import Lemmachine.Resource.Universe
open import Data.Bool
open import Data.Maybe
open import Data.Product
open import Data.List
open import Data.Function using (const)
private
update : (code : Code) → El code → Resource → Resource
update resourceExists f c = record {
resourceExists = f
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update serviceAvailable f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = f
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update isAuthorized f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = f
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update forbidden f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = f
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update allowMissingPost f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = f
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update malformedRequest f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = f
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update uriTooLong f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = f
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update knownContentType f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = f
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update validContentHeaders f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = f
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update validEntityLength f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = f
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update options f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = f
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update allowedMethods f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = f
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update knownMethods f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = f
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update deleteResource f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = f
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update deleteCompleted f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = f
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update postIsCreate f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = f
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update createPath f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = f
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update processPost f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = f
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update contentTypesProvided f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = f
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update languageAvailable f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = f
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update contentTypesAccepted f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = f
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update charsetsProvided f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = f
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update encodingsProvided f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = f
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update variances f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = f
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update isConflict f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = f
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update multipleChoices f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = f
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update previouslyExisted f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = f
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update movedPermanently f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = f
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update movedTemporarily f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = f
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update lastModified f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = f
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update expires f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = f
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update generateETag f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = f
; finishRequest = Resource.finishRequest c
; body = Resource.body c
}
update finishRequest f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = f
; body = Resource.body c
}
update body f c = record {
resourceExists = Resource.resourceExists c
; serviceAvailable = Resource.serviceAvailable c
; isAuthorized = Resource.isAuthorized c
; forbidden = Resource.forbidden c
; allowMissingPost = Resource.allowMissingPost c
; malformedRequest = Resource.malformedRequest c
; uriTooLong = Resource.uriTooLong c
; knownContentType = Resource.knownContentType c
; validContentHeaders = Resource.validContentHeaders c
; validEntityLength = Resource.validEntityLength c
; options = Resource.options c
; allowedMethods = Resource.allowedMethods c
; knownMethods = Resource.knownMethods c
; deleteResource = Resource.deleteResource c
; deleteCompleted = Resource.deleteCompleted c
; postIsCreate = Resource.postIsCreate c
; createPath = Resource.createPath c
; processPost = Resource.processPost c
; contentTypesProvided = Resource.contentTypesProvided c
; languageAvailable = Resource.languageAvailable c
; contentTypesAccepted = Resource.contentTypesAccepted c
; charsetsProvided = Resource.charsetsProvided c
; encodingsProvided = Resource.encodingsProvided c
; variances = Resource.variances c
; isConflict = Resource.isConflict c
; multipleChoices = Resource.multipleChoices c
; previouslyExisted = Resource.previouslyExisted c
; movedPermanently = Resource.movedPermanently c
; movedTemporarily = Resource.movedTemporarily c
; lastModified = Resource.lastModified c
; expires = Resource.expires c
; generateETag = Resource.generateETag c
; finishRequest = Resource.finishRequest c
; body = f
}
default : Resource
default = record {
resourceExists = const true
; serviceAvailable = const true
; isAuthorized = const true
; forbidden = const false
; allowMissingPost = const false
; malformedRequest = const false
; uriTooLong = const false
; knownContentType = const true
; validContentHeaders = const true
; validEntityLength = const true
; options = const []
; allowedMethods = const (HEAD ∷ GET ∷ [])
; knownMethods = const (HEAD ∷ GET ∷ PUT ∷ DELETE ∷ POST ∷ TRACE ∷ CONNECT ∷ OPTIONS ∷ [])
; deleteResource = const false
; deleteCompleted = const true
; postIsCreate = const false
; createPath = const nothing
; processPost = const false
; contentTypesProvided = const [ "text/html" ]
; languageAvailable = const true
; contentTypesAccepted = const []
; charsetsProvided = const []
; encodingsProvided = const [ "identity" , "defaultEncoder" ]
; variances = const []
; isConflict = const false
; multipleChoices = const false
; previouslyExisted = const false
; movedPermanently = const nothing
; movedTemporarily = const nothing
; lastModified = const nothing
; expires = const nothing
; generateETag = const nothing
; finishRequest = const true
; body = defaultHtml
}
configure : Resource → Hooks → Resource
configure base [] = base
configure base ((code , f) ∷ cfs) = update code f (configure base cfs)
toResource : Hooks → Resource
toResource props = configure default props
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Finite sets, based on AVL trees
------------------------------------------------------------------------
open import Relation.Binary
open import Relation.Binary.PropositionalEquality using (_≡_)
module Data.AVL.Sets
{k ℓ} {Key : Set k} {_<_ : Rel Key ℓ}
(isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_)
where
import Data.AVL as AVL
open import Data.Bool
open import Data.List as List using (List)
open import Data.Maybe as Maybe
open import Data.Product as Prod using (_×_; _,_; proj₁)
open import Data.Unit
open import Function
open import Level
-- The set type. (Note that Set is a reserved word.)
private
open module S = AVL (const ⊤) isStrictTotalOrder
public using () renaming (Tree to ⟨Set⟩)
-- Repackaged functions.
empty : ⟨Set⟩
empty = S.empty
singleton : Key → ⟨Set⟩
singleton k = S.singleton k _
insert : Key → ⟨Set⟩ → ⟨Set⟩
insert k = S.insert k _
delete : Key → ⟨Set⟩ → ⟨Set⟩
delete = S.delete
_∈?_ : Key → ⟨Set⟩ → Bool
_∈?_ = S._∈?_
headTail : ⟨Set⟩ → Maybe (Key × ⟨Set⟩)
headTail s = Maybe.map (Prod.map proj₁ id) (S.headTail s)
initLast : ⟨Set⟩ → Maybe (⟨Set⟩ × Key)
initLast s = Maybe.map (Prod.map id proj₁) (S.initLast s)
fromList : List Key → ⟨Set⟩
fromList = S.fromList ∘ List.map (λ k → (k , _))
toList : ⟨Set⟩ → List Key
toList = List.map proj₁ ∘ S.toList
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module VecAppend where
open import Prelude
add : Nat -> Nat -> Nat
add zero y = y
add (suc x) y = suc (add x y)
append : forall {A m n} -> Vec A m -> Vec A n -> Vec A (add m n)
append xs ys = {!!}
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{-# OPTIONS --without-K --overlapping-instances #-}
open import lib.Basics
open import lib.types.Coproduct
open import lib.types.Truncation
open import lib.types.Sigma
open import lib.types.Empty
open import lib.types.Bool
open import lib.NConnected
open import lib.NType2
module Util.Misc where
transp-↓' : ∀ {k j} {A : Type k} (P : A → Type j) {a₁ a₂ : A}
(p : a₁ == a₂) (y : P a₁) → y == transport P p y [ P ↓ p ]
transp-↓' P p y = from-transp P p idp
Σ-transp : ∀ {i j} {A : Type i} (B : A → Type j) {a a' : A} (p : a == a') (b : B a) →
(a , b) == (a' , transport B p b)
Σ-transp B p b = pair= p (transp-↓' B p b)
equiv-prop : {i j : ULevel} {P : Type i} ⦃ P-prop : is-prop P ⦄ {Q : Type j} ⦃ Q-prop : is-prop Q ⦄ → (P → Q) →
(Q → P) → (P ≃ Q)
equiv-prop f g = f , (record { g = g ; f-g = λ q → prop-has-all-paths (f (g q)) q
; g-f = λ p → prop-has-all-paths (g (f p)) p
; adj = λ p → prop-has-all-paths _ _ })
_↔_ : {i j : ULevel} (A : Type i) (B : Type j) → Type (lmax i j)
A ↔ B = (A → B) × (B → A)
prop-extensionality : {i : ULevel} {P Q : Type i} ⦃ P-prop : is-prop P ⦄ ⦃ Q-prop : is-prop Q ⦄ → (P ↔ Q) → (P == Q)
prop-extensionality e = ua (equiv-prop (fst e) (snd e))
{- Notation for propositional truncation. -}
∥_∥ : {j : ULevel} → (A : Type j) → Type j
∥_∥ = Trunc (-1)
{- This enables do notation for propositional truncation. It works well for any modality, but we will
only need this special case. -}
_>>=_ : {i j : ULevel} {X : Type i} {Y : Type j} → (Trunc (-1) X) → (X → Trunc (-1) Y) → (Trunc (-1) Y)
x >>= f = Trunc-rec f x
{- We also add notation for 0 and 1 truncation -}
∥_∥₀ : {j : ULevel} → (A : Type j) → Type j
∥_∥₀ = Trunc 0
∥_∥₁ : {j : ULevel} → (A : Type j) → Type j
∥_∥₁ = Trunc 1
conn-to-path : {i : ULevel} {A : Type i} ⦃ conn : is-connected 0 A ⦄ → (a b : A) → ∥ a == b ∥
conn-to-path ⦃ conn ⦄ _ _ = contr-center (path-conn conn)
conn-to-inh : {i : ULevel} {A : Type i} ⦃ conn : is-connected 0 A ⦄ → ∥ A ∥
conn-to-inh ⦃ conn ⦄ = Trunc-rec ⦃ raise-level _ ⟨⟩ ⦄ [_] (contr-center conn)
inj-to-embed-allpaths : {i j : ULevel} {A : Type i} {B : Type j} ⦃ Bset : is-set B ⦄ (inc : A → B) (inc-inj : is-inj inc) → (b : B) → has-all-paths (hfiber inc b)
inj-to-embed-allpaths inc inc-inj b (a , p) (a' , q) = pair= (inc-inj _ _ (p ∙ ! q)) prop-has-all-paths-↓
inj-to-embed : {i j : ULevel} {A : Type i} {B : Type j} ⦃ Bset : is-set B ⦄ (inc : A → B) (inc-inj : is-inj inc) → (b : B) → is-prop (hfiber inc b)
inj-to-embed inc inc-inj b = all-paths-is-prop (inj-to-embed-allpaths inc inc-inj b)
dec-img-to-⊔ : {i j : ULevel} {A : Type i} {B : Type j} (inc : A → B) (inc-inj : is-inj inc) (dec-img : (b : B) → Dec (hfiber inc b)) → A ⊔ Σ B (λ b → ¬ (hfiber inc b)) ≃ B
dec-img-to-⊔ {A = A} {B = B} inc inc-inj dec-img = equiv f g f-g g-f
where
g-aux : (b : B) → (Dec (hfiber inc b)) → A ⊔ Σ B (λ b → ¬ (hfiber inc b))
g-aux b (inl x) = inl (fst x)
g-aux b (inr x) = inr (b , x)
g : (b : B) → A ⊔ Σ B (λ b → ¬ (hfiber inc b))
g b = g-aux b (dec-img b)
f : A ⊔ Σ B (λ b → ¬ (hfiber inc b)) → B
f (inl a) = inc a
f (inr (b , p)) = b
g-f-aux : (z : A ⊔ Σ B (λ b → ¬ (hfiber inc b))) (x : Dec (hfiber inc (f z))) → (g-aux (f z) x == z)
g-f-aux (inl a) (inl x) = ap inl (inc-inj _ _ (snd x))
g-f-aux (inl a) (inr x) = ⊥-rec (x (a , idp))
g-f-aux (inr p) (inl x) = ⊥-rec (snd p x)
g-f-aux (inr p) (inr x) = ap inr (pair= idp (λ= λ w → prop-has-all-paths _ _))
g-f : (z : A ⊔ Σ B (λ b → ¬ (hfiber inc b))) → (g (f z) == z)
g-f z = g-f-aux z (dec-img (f z))
f-g-aux : (b : B) → (x : Dec (hfiber inc b)) → (f (g-aux b x) == b)
f-g-aux b (inl x) = snd x
f-g-aux b (inr x) = idp
f-g : (b : B) → (f (g b) == b)
f-g b = f-g-aux b (dec-img b)
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------------------------------------------------------------------------
-- An alternative but equivalent definition of the partiality monad
-- (but only for sets), based on the lifting construction in Lifting
------------------------------------------------------------------------
-- The code in this module is based on a suggestion from Paolo
-- Capriotti.
{-# OPTIONS --erased-cubical --safe #-}
open import Prelude hiding (T; ⊥)
module Lifting.Partiality-monad {a : Level} where
open import Equality.Propositional.Cubical
open import Logical-equivalence using (_⇔_)
open import Bijection equality-with-J using (_↔_)
open import Equivalence equality-with-J as Eq using (_≃_)
open import Function-universe equality-with-J hiding (⊥↔⊥)
open import H-level equality-with-J
import Lifting
open import Omega-cpo
import Partiality-monad.Inductive as I
import Partiality-monad.Inductive.Eliminators as IE
private
module L {A : Set a} = Lifting (Set→ω-cpo A)
-- The partiality monad as an ω-cppo.
partiality-monad : Set a → ω-cppo a a
partiality-monad A = L.cppo {A = A}
-- The partiality monad.
infix 10 _⊥
infix 4 _⊑_
_⊥ : Set a → Type a
A ⊥ = ω-cppo.Carrier (partiality-monad A)
_⊑_ : ∀ {A} → A ⊥ → A ⊥ → Type a
_⊑_ {A = A} = ω-cppo._⊑_ (partiality-monad A)
-- This definition of the partiality monad is isomorphic to the
-- definition in Partiality-monad.Inductive.
private
argsL :
∀ {A} →
L.Rec-args (λ (_ : A ⊥) → ⌞ A ⌟ I.⊥) (λ x y _ → x I.⊑ y)
argsL = record
{ pe = I.never
; po = I.now
; pl = λ _ → I.⨆
; pa = λ x⊑y y⊑x p-x p-y p-x⊑p-y p-y⊑p-x →
subst (const _) (L.antisymmetry x⊑y y⊑x) p-x ≡⟨ subst-const (L.antisymmetry x⊑y y⊑x) ⟩
p-x ≡⟨ I.antisymmetry p-x⊑p-y p-y⊑p-x ⟩∎
p-y ∎
; pp = I.⊥-is-set
; qr = λ _ → I.⊑-refl
; qt = λ _ _ _ _ _ → I.⊑-trans
; qe = λ _ → I.never⊑
; qu = λ _ → I.upper-bound
; ql = λ _ _ _ → I.least-upper-bound
; qm = λ { refl → I.⊑-refl _ }
; q⨆ = λ s →
I.now (proj₁ s 0) I.⊑⟨ I.upper-bound _ 0 ⟩■
I.⨆ ((λ n → I.now (proj₁ s n)) , _) ■
; qp = λ _ _ _ → I.⊑-propositional
}
argsI : ∀ {A} → IE.Arguments-nd a a ⌞ A ⌟
argsI {A} = record
{ P = A ⊥
; Q = _⊑_
; pe = L.never
; po = L.now
; pl = λ _ → L.⨆
; pa = λ _ _ → L.antisymmetry
; ps = L.Carrier-is-set
; qr = λ _ → L.⊑-refl
; qt = λ _ _ → L.⊑-trans
; qe = λ _ → L.never⊑
; qu = λ _ → L.upper-bound
; ql = λ _ _ _ → L.least-upper-bound
; qp = λ _ _ → L.⊑-propositional
}
⊥↔⊥ : ∀ {A} → A ⊥ ↔ ⌞ A ⌟ I.⊥
⊥↔⊥ = record
{ surjection = record
{ logical-equivalence = record
{ to = L.⊥-rec argsL
; from = IE.⊥-rec-nd argsI
}
; right-inverse-of = IE.⊥-rec-⊥ (record
{ pe = L.⊥-rec argsL (IE.⊥-rec-nd argsI I.never) ≡⟨ cong (L.⊥-rec argsL) (IE.⊥-rec-nd-never argsI) ⟩
L.⊥-rec argsL L.never ≡⟨ L.⊥-rec-never _ ⟩∎
I.never ∎
; po = λ x →
L.⊥-rec argsL (IE.⊥-rec-nd argsI (I.now x)) ≡⟨ cong (L.⊥-rec argsL) (IE.⊥-rec-nd-now argsI _) ⟩
L.⊥-rec argsL (L.now x) ≡⟨ L.⊥-rec-now _ _ ⟩∎
I.now x ∎
; pl = λ s ih →
L.⊥-rec argsL (IE.⊥-rec-nd argsI (I.⨆ s)) ≡⟨ cong (L.⊥-rec argsL) (IE.⊥-rec-nd-⨆ argsI _) ⟩
L.⊥-rec argsL (L.⨆ (IE.inc-rec-nd argsI s)) ≡⟨ L.⊥-rec-⨆ _ _ ⟩
I.⨆ (L.inc-rec argsL (IE.inc-rec-nd argsI s)) ≡⟨ cong I.⨆ $ _↔_.to I.equality-characterisation-increasing ih ⟩∎
I.⨆ s ∎
; pp = λ _ → I.⊥-is-set
})
}
; left-inverse-of = L.⊥-rec {Q = λ _ _ _ → ⊤} (record
{ pe = IE.⊥-rec-nd argsI (L.⊥-rec argsL L.never) ≡⟨ cong (IE.⊥-rec-nd argsI) (L.⊥-rec-never _) ⟩
IE.⊥-rec-nd argsI I.never ≡⟨ IE.⊥-rec-nd-never argsI ⟩∎
L.never ∎
; po = λ x →
IE.⊥-rec-nd argsI (L.⊥-rec argsL (L.now x)) ≡⟨ cong (IE.⊥-rec-nd argsI) (L.⊥-rec-now _ _) ⟩
IE.⊥-rec-nd argsI (I.now x) ≡⟨ IE.⊥-rec-nd-now argsI _ ⟩∎
L.now x ∎
; pl = λ s ih →
IE.⊥-rec-nd argsI (L.⊥-rec argsL (L.⨆ s)) ≡⟨ cong (IE.⊥-rec-nd argsI) (L.⊥-rec-⨆ _ _) ⟩
IE.⊥-rec-nd argsI (I.⨆ (L.inc-rec argsL s)) ≡⟨ IE.⊥-rec-nd-⨆ argsI _ ⟩
L.⨆ (IE.inc-rec-nd argsI (L.inc-rec argsL s)) ≡⟨ cong L.⨆ $ _↔_.to L.equality-characterisation-increasing (proj₁ ih) ⟩∎
L.⨆ s ∎
; pa = λ _ _ _ _ _ _ → L.Carrier-is-set _ _
; pp = mono₁ 1 L.Carrier-is-set
; qp = λ _ _ _ _ _ → refl
})
}
⊑≃⊑ : ∀ {A} {x y : A ⊥} →
(x ⊑ y) ≃ (_↔_.to ⊥↔⊥ x I.⊑ _↔_.to ⊥↔⊥ y)
⊑≃⊑ {x = x} {y} =
_↔_.to (Eq.⇔↔≃ ext L.⊑-propositional I.⊑-propositional)
(record
{ to = L.⊑-rec argsL
; from =
_↔_.to ⊥↔⊥ x I.⊑ _↔_.to ⊥↔⊥ y ↝⟨ IE.⊑-rec-nd argsI ⟩
_↔_.from ⊥↔⊥ (_↔_.to ⊥↔⊥ x) ⊑
_↔_.from ⊥↔⊥ (_↔_.to ⊥↔⊥ y) ↝⟨ ≡⇒↝ _ (cong₂ _⊑_ (_↔_.left-inverse-of ⊥↔⊥ _)
(_↔_.left-inverse-of ⊥↔⊥ _)) ⟩□
x ⊑ y □
})
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module container.w where
open import container.w.core public
open import container.w.algebra public
open import container.w.fibration public
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module Preduploid where
open import Relation.Unary using (Pred)
open import Relation.Binary using (REL)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym)
open import Level
data Polarity : Set where
+ : Polarity
⊝ : Polarity
private
variable p q r s : Polarity
record Preduploid o ℓ : Set (suc (o ⊔ ℓ)) where
infix 4 _⇒_
infix 9 _∙_ _∘_ _⊙_
field
Ob : Polarity -> Set o
_⇒_ : REL (Ob p) (Ob q) ℓ
id : forall {A : Ob p} -> A ⇒ A
_∙_ : forall {A : Ob p} {B : Ob +} {C : Ob q} -> B ⇒ C -> A ⇒ B -> A ⇒ C
_∘_ : forall {A : Ob p} {B : Ob ⊝} {C : Ob q} -> B ⇒ C -> A ⇒ B -> A ⇒ C
identity∙ˡ : forall {A : Ob p} {B : Ob +} {f : A ⇒ B} -> id ∙ f ≡ f
identity∙ʳ : forall {A : Ob +} {B : Ob p} {f : A ⇒ B} -> f ∙ id ≡ f
identity∘ˡ : forall {A : Ob p} {B : Ob ⊝} {f : A ⇒ B} -> id ∘ f ≡ f
identity∘ʳ : forall {A : Ob ⊝} {B : Ob p} {f : A ⇒ B} -> f ∘ id ≡ f
assoc∙∙ : forall {A : Ob p} {B : Ob +} {C : Ob +} {D : Ob q}
{f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} -> (h ∙ g) ∙ f ≡ h ∙ (g ∙ f)
assoc∘∘ : forall {A : Ob p} {B : Ob ⊝} {C : Ob ⊝} {D : Ob q}
{f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} -> (h ∘ g) ∘ f ≡ h ∘ (g ∘ f)
assoc∙∘ : forall {A : Ob p} {B : Ob ⊝} {C : Ob +} {D : Ob q}
{f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} -> (h ∙ g) ∘ f ≡ h ∙ (g ∘ f)
π : Ob p -> Polarity
π {p = p} _ = p
_⊙_ : forall {A : Ob p} {B : Ob q} {C : Ob r} -> B ⇒ C -> A ⇒ B -> A ⇒ C
_⊙_ {q = +} = _∙_
_⊙_ {q = ⊝} = _∘_
identityˡ : forall {A : Ob p} {B : Ob q} {f : A ⇒ B} -> id ⊙ f ≡ f
identityˡ {q = +} = identity∙ˡ
identityˡ {q = ⊝} = identity∘ˡ
identityʳ : forall {A : Ob p} {B : Ob q} {f : A ⇒ B} -> f ⊙ id ≡ f
identityʳ {p = +} = identity∙ʳ
identityʳ {p = ⊝} = identity∘ʳ
Linear : forall {C : Ob r} {D : Ob s} -> Pred (C ⇒ D) (o ⊔ ℓ)
Linear {C = C} {D = D} f = forall {p q} {A : Ob p} {B : Ob q} {g : B ⇒ C} {h : A ⇒ B} -> f ⊙ (g ⊙ h) ≡ (f ⊙ g) ⊙ h
Thunkable : forall {A : Ob p} {B : Ob q} -> Pred (A ⇒ B) (o ⊔ ℓ)
Thunkable {A = A} {B = B} f = forall {r s} {C : Ob r} {D : Ob s} {g : B ⇒ C} {h : C ⇒ D} -> h ⊙ (g ⊙ f) ≡ (h ⊙ g) ⊙ f
P-Linear : forall {P : Ob +} {A : Ob p} -> (f : P ⇒ A) -> Linear f
P-Linear f {q = +} = sym assoc∙∙
P-Linear f {q = ⊝} = sym assoc∙∘
N-Thunkable : forall {A : Ob p} {N : Ob ⊝} -> (f : A ⇒ N) -> Thunkable f
N-Thunkable f {r = +} = sym assoc∙∘
N-Thunkable f {r = ⊝} = sym assoc∘∘
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{-# OPTIONS --safe #-}
module Cubical.Algebra.Field.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.SIP
open import Cubical.Relation.Nullary
open import Cubical.Data.Sigma
open import Cubical.Data.Unit
open import Cubical.Data.Empty
open import Cubical.Algebra.Semigroup
open import Cubical.Algebra.Monoid
open import Cubical.Algebra.AbGroup
open import Cubical.Algebra.Ring
hiding (_$_)
open import Cubical.Algebra.CommRing
open import Cubical.Displayed.Base
open import Cubical.Displayed.Auto
open import Cubical.Displayed.Record
open import Cubical.Displayed.Universe
open import Cubical.Reflection.RecordEquiv
open Iso
private
variable
ℓ ℓ' : Level
record IsField {R : Type ℓ}
(0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R)
(_[_]⁻¹ : (x : R) → ¬ (x ≡ 0r) → R) : Type ℓ where
constructor isfield
field
isCommRing : IsCommRing 0r 1r _+_ _·_ -_
·⁻¹≡1 : (x : R) (≢0 : ¬ (x ≡ 0r)) → x · (x [ ≢0 ]⁻¹) ≡ 1r
0≢1 : ¬ (0r ≡ 1r)
open IsCommRing isCommRing public
record FieldStr (A : Type ℓ) : Type (ℓ-suc ℓ) where
constructor fieldstr
field
0r : A
1r : A
_+_ : A → A → A
_·_ : A → A → A
-_ : A → A
_[_]⁻¹ : (x : A) → ¬ (x ≡ 0r) → A
isField : IsField 0r 1r _+_ _·_ -_ _[_]⁻¹
infix 8 -_
infixl 7 _·_
infixl 6 _+_
open IsField isField public
Field : ∀ ℓ → Type (ℓ-suc ℓ)
Field ℓ = TypeWithStr ℓ FieldStr
isSetField : (R : Field ℓ) → isSet ⟨ R ⟩
isSetField R = R .snd .FieldStr.isField .IsField.·IsMonoid .IsMonoid.isSemigroup .IsSemigroup.is-set
makeIsField : {R : Type ℓ} {0r 1r : R} {_+_ _·_ : R → R → R} { -_ : R → R}
{_[_]⁻¹ : (x : R) → ¬ (x ≡ 0r) → R}
(is-setR : isSet R)
(+-assoc : (x y z : R) → x + (y + z) ≡ (x + y) + z)
(+-rid : (x : R) → x + 0r ≡ x)
(+-rinv : (x : R) → x + (- x) ≡ 0r)
(+-comm : (x y : R) → x + y ≡ y + x)
(·-assoc : (x y z : R) → x · (y · z) ≡ (x · y) · z)
(·-rid : (x : R) → x · 1r ≡ x)
(·-rdist-+ : (x y z : R) → x · (y + z) ≡ (x · y) + (x · z))
(·-comm : (x y : R) → x · y ≡ y · x)
(·⁻¹≡1 : (x : R) (≢0 : ¬ (x ≡ 0r)) → x · (x [ ≢0 ]⁻¹) ≡ 1r)
(0≢1 : ¬ (0r ≡ 1r))
→ IsField 0r 1r _+_ _·_ -_ _[_]⁻¹
makeIsField {_+_ = _+_} is-setR +-assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-rdist-+ ·-comm ·⁻¹≡1 0≢1 =
isfield (makeIsCommRing is-setR +-assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-rdist-+ ·-comm) ·⁻¹≡1 0≢1
makeField : {R : Type ℓ} (0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R) (_[_]⁻¹ : (x : R) → ¬ (x ≡ 0r) → R)
(is-setR : isSet R)
(+-assoc : (x y z : R) → x + (y + z) ≡ (x + y) + z)
(+-rid : (x : R) → x + 0r ≡ x)
(+-rinv : (x : R) → x + (- x) ≡ 0r)
(+-comm : (x y : R) → x + y ≡ y + x)
(·-assoc : (x y z : R) → x · (y · z) ≡ (x · y) · z)
(·-rid : (x : R) → x · 1r ≡ x)
(·-rdist-+ : (x y z : R) → x · (y + z) ≡ (x · y) + (x · z))
(·-comm : (x y : R) → x · y ≡ y · x)
(·⁻¹≡1 : (x : R) (≢0 : ¬ (x ≡ 0r)) → x · (x [ ≢0 ]⁻¹) ≡ 1r)
(0≢1 : ¬ (0r ≡ 1r))
→ Field ℓ
makeField 0r 1r _+_ _·_ -_ _[_]⁻¹ is-setR +-assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-rdist-+ ·-comm ·⁻¹≡1 0≢1 =
_ , fieldstr _ _ _ _ _ _ (makeIsField is-setR +-assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-rdist-+ ·-comm ·⁻¹≡1 0≢1)
FieldStr→CommRingStr : {A : Type ℓ} → FieldStr A → CommRingStr A
FieldStr→CommRingStr (fieldstr _ _ _ _ _ _ H) = commringstr _ _ _ _ _ (IsField.isCommRing H)
Field→CommRing : Field ℓ → CommRing ℓ
Field→CommRing (_ , fieldstr _ _ _ _ _ _ H) = _ , commringstr _ _ _ _ _ (IsField.isCommRing H)
record IsFieldHom {A : Type ℓ} {B : Type ℓ'} (R : FieldStr A) (f : A → B) (S : FieldStr B)
: Type (ℓ-max ℓ ℓ')
where
-- Shorter qualified names
private
module R = FieldStr R
module S = FieldStr S
field
pres0 : f R.0r ≡ S.0r
pres1 : f R.1r ≡ S.1r
pres+ : (x y : A) → f (x R.+ y) ≡ f x S.+ f y
pres· : (x y : A) → f (x R.· y) ≡ f x S.· f y
pres- : (x : A) → f (R.- x) ≡ S.- (f x)
pres⁻¹ : (x : A) (≢0 : ¬ (x ≡ R.0r)) (f≢0 : ¬ (f x ≡ S.0r)) → f (x R.[ ≢0 ]⁻¹) ≡ ((f x) S.[ f≢0 ]⁻¹)
unquoteDecl IsFieldHomIsoΣ = declareRecordIsoΣ IsFieldHomIsoΣ (quote IsFieldHom)
FieldHom : (R : Field ℓ) (S : Field ℓ') → Type (ℓ-max ℓ ℓ')
FieldHom R S = Σ[ f ∈ (⟨ R ⟩ → ⟨ S ⟩) ] IsFieldHom (R .snd) f (S .snd)
IsFieldEquiv : {A : Type ℓ} {B : Type ℓ'}
(R : FieldStr A) (e : A ≃ B) (S : FieldStr B) → Type (ℓ-max ℓ ℓ')
IsFieldEquiv R e S = IsFieldHom R (e .fst) S
FieldEquiv : (R : Field ℓ) (S : Field ℓ') → Type (ℓ-max ℓ ℓ')
FieldEquiv R S = Σ[ e ∈ (R .fst ≃ S .fst) ] IsFieldEquiv (R .snd) e (S .snd)
_$_ : {R S : Field ℓ} → (φ : FieldHom R S) → (x : ⟨ R ⟩) → ⟨ S ⟩
φ $ x = φ .fst x
FieldEquiv→FieldHom : {A B : Field ℓ} → FieldEquiv A B → FieldHom A B
FieldEquiv→FieldHom (e , eIsHom) = e .fst , eIsHom
isPropIsField : {R : Type ℓ} (0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R) (_[_]⁻¹ : (x : R) → ¬ (x ≡ 0r) → R)
→ isProp (IsField 0r 1r _+_ _·_ -_ _[_]⁻¹)
isPropIsField 0r 1r _+_ _·_ -_ _[_]⁻¹ (isfield RR RC RD) (isfield SR SC SD) =
λ i → isfield (isPropIsCommRing _ _ _ _ _ RR SR i)
(isPropInv RC SC i) (isProp→⊥ RD SD i)
where
isSetR : isSet _
isSetR = RR .IsCommRing.·IsMonoid .IsMonoid.isSemigroup .IsSemigroup.is-set
isPropInv : isProp ((x : _) → (≢0 : ¬ (x ≡ 0r)) → x · (x [ ≢0 ]⁻¹) ≡ 1r)
isPropInv = isPropΠ2 λ _ _ → isSetR _ _
isProp→⊥ : ∀ {A : Type ℓ} → isProp (A → ⊥)
isProp→⊥ = isPropΠ λ _ → isProp⊥
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module Data.Vec.Any {a} {A : Set a} where
open import Level using (_⊔_)
open import Relation.Nullary
open import Data.Fin
open import Data.Nat as ℕ hiding (_⊔_)
open import Data.Vec as Vec using (Vec; _∷_; [])
open import Relation.Unary renaming (_⊆_ to _⋐_) using (Decidable)
open import Function.Inverse using (_↔_)
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_)
import Relation.Binary.PropositionalEquality as P
open import Relation.Nullary using (¬_; yes; no)
import Relation.Nullary.Decidable as Dec
open import Data.Empty using (⊥-elim)
open import Data.Product using (∃ ; ,_ )
import Data.List.Any as ListAny
data Any {p}(P : A → Set p) : ∀ {n} → Vec A n → Set (a ⊔ p) where
here : ∀ {n x}{xs : Vec A n} → P x → Any P (x ∷ xs)
there : ∀ {n x}{xs : Vec A n} → Any P xs → Any P (x ∷ xs)
open Any public
map : ∀ {p q} {P : A → Set p} {Q : A → Set q} {n} {xs : Vec A n} →
P ⋐ Q → Any P xs → Any Q xs
map g (here px) = here (g px)
map g (there pxs) = there (map g pxs)
tail : ∀ {p} {P : A → Set p}{x n} {xs : Vec A n} → ¬ P x → Any P (x ∷ xs) → Any P xs
tail ¬px (here px) = ⊥-elim (¬px px)
tail ¬px (there pxs) = pxs
any : ∀ {p} {P : A → Set p} → Decidable P → ∀ {n} → Decidable (Any P {n})
any p [] = no λ()
any p (x ∷ xs) with p x
any p (x ∷ xs) | yes px = yes (here px)
any p (x ∷ xs) | no ¬px = Dec.map′ there (tail ¬px) (any p xs)
index : ∀ {p}{P : A → Set p}{n}{xs : Vec A n} → Any P xs → Fin n
index (here px) = zero
index (there pxs) = suc (index pxs)
satisfied : ∀ {p}{P : A → Set p}{n}{xs : Vec A n} → Any P xs → ∃ P
satisfied (here px) = , px
satisfied (there pxs) = satisfied pxs
Any↔ListAny : ∀ {p}{P : A → Set p}{n}{xs : Vec A n} →
Any P xs ↔ ListAny.Any P (Vec.toList xs)
Any↔ListAny {P = P} = record {
to = P.→-to-⟶ to
; from = P.→-to-⟶ from
; inverse-of = record {
left-inverse-of = left-inverse
; right-inverse-of = right-inverse
}
}
where
to : ∀ {n}{xs : Vec A n} → Any P xs → ListAny.Any P (Vec.toList xs)
to (here px) = ListAny.here px
to (there pxs) = ListAny.there (to pxs)
from : ∀ {n}{xs : Vec A n} → ListAny.Any P (Vec.toList xs) → Any P xs
from {xs = []} ()
from {xs = x ∷ xs} (ListAny.here px) = here px
from {xs = x ∷ xs} (ListAny.there pxs) = there (from pxs)
left-inverse : ∀ {n}{xs : Vec A n}(p : Any P xs) → from (to p) P.≡ p
left-inverse (here px) = P.refl
left-inverse (there pxs) = P.cong there (left-inverse pxs)
right-inverse : ∀ {n}{xs : Vec A n}(p : ListAny.Any P (Vec.toList xs)) → to (from p) P.≡ p
right-inverse {xs = []} ()
right-inverse {xs = x ∷ xs} (ListAny.here px) = P.refl
right-inverse {xs = x ∷ xs} (ListAny.there pxs) = P.cong ListAny.there (right-inverse pxs)
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------------------------------------------------------------------------
-- Well-typed substitutions
------------------------------------------------------------------------
module Data.Fin.Substitution.Typed where
open import Data.Fin using (Fin; zero; suc)
open import Data.Fin.Substitution
open import Data.Fin.Substitution.Lemmas
open import Data.Fin.Substitution.ExtraLemmas
open import Data.Nat using (ℕ; zero; suc; _+_)
open import Data.Unit using (⊤; tt)
open import Data.Vec as Vec using (Vec; []; _∷_; map)
open import Data.Vec.All as All
using (All; All₂; []; _∷_; map₂)
open import Data.Vec.All.Properties
using (gmap; gmap₂; gmap₂₁; gmap₂₂)
open import Data.Vec.Properties using (lookup-map)
open import Function as Fun using (_∘_; flip)
open import Relation.Binary.PropositionalEquality as PropEq hiding (trans)
open PropEq.≡-Reasoning
------------------------------------------------------------------------
-- Abstract typing contexts and well-typedness relations
-- Abstract typing contexts over T-types.
--
-- A typing context Ctx T n maps n free variables to T-types
-- containing up to n free variables each.
module Context (T : ℕ → Set) where
infixr 5 _∷_
-- Typing contexts.
data Ctx : ℕ → Set where
[] : Ctx 0
_∷_ : ∀ {n} → T n → Ctx n → Ctx (1 + n)
-- Operations on contexts that require weakening of types.
record WeakenOps : Set where
-- Weakening of types.
field weaken : ∀ {n} → T n → T (1 + n)
-- Convert a context to its vector representation.
toVec : ∀ {n} → Ctx n → Vec (T n) n
toVec [] = []
toVec (a ∷ Γ) = weaken a ∷ map weaken (toVec Γ)
-- Lookup the type of a variable in a context.
lookup : ∀ {n} → Fin n → Ctx n → T n
lookup x = Vec.lookup x ∘ toVec
open Context
-- Abstract typings.
--
-- An abtract typing _⊢_∈_ : Typing Tp₁ Tm Tp₂ is a ternary relation
-- which, in a given Tp₁-context, relates Tm-terms to their Tp₂-types.
Typing : (ℕ → Set) → (ℕ → Set) → (ℕ → Set) → Set₁
Typing Tp₁ Tm Tp₂ = ∀ {n} → Ctx Tp₁ n → Tm n → Tp₂ n → Set
-- Abstract well-formedness.
--
-- An abtract well-formedness relation _⊢_wf : Wf Tp is a binary
-- relation which, in a given Tp-context, asserts the well-formedness
-- of Tp-types.
Wf : (ℕ → Set) → Set₁
Wf Tp = ∀ {n} → Ctx Tp n → Tp n → Set
-- Well-formed contexts.
--
-- A well-formed typing context (Γ wf) is a context Γ in which every
-- participating T-type is well-formed.
module WellFormedContext {T} (_⊢_wf : Wf T) where
infix 4 _wf
infixr 5 _∷_
-- Well-formed typing contexts.
data _wf : ∀ {n} → Ctx T n → Set where
[] : [] wf
_∷_ : ∀ {n a} {Γ : Ctx T n} → Γ ⊢ a wf → Γ wf → a ∷ Γ wf
-- Inversions.
wf-∷₁ : ∀ {n} {Γ : Ctx T n} {a} → a ∷ Γ wf → Γ ⊢ a wf
wf-∷₁ (a-wf ∷ _) = a-wf
wf-∷₂ : ∀ {n} {Γ : Ctx T n} {a} → a ∷ Γ wf → Γ wf
wf-∷₂ (_ ∷ Γ-wf) = Γ-wf
-- Operations on well-formed contexts that require weakening of
-- well-formedness judgments.
record WfWeakenOps (weakenOps : WeakenOps T) : Set₁ where
private module W = WeakenOps weakenOps
field
-- Weakening of well-formedness judgments.
weaken : ∀ {n} {Γ : Ctx T n} {a b} → Γ ⊢ a wf → Γ ⊢ b wf →
(a ∷ Γ) ⊢ W.weaken b wf
-- Convert a well-formed context to its All representation.
toAll : ∀ {n} {Γ : Ctx T n} → Γ wf → All (λ a → Γ ⊢ a wf) (W.toVec Γ)
toAll [] = []
toAll (a-wf ∷ Γ) = weaken a-wf a-wf ∷ gmap (weaken a-wf) (toAll Γ)
-- Lookup the well-formedness proof of a variable in a context.
lookup : ∀ {n} {Γ : Ctx T n} → (x : Fin n) → Γ wf → Γ ⊢ (W.lookup x Γ) wf
lookup x = All.lookup x ∘ toAll
-- Trivial well-formedness.
--
-- This module provides a trivial well-formedness relation and the
-- corresponding trivially well-formed contexts. This is useful when
-- implmenting typed substitutions on types that either lack or do not
-- necessitate a notion well-formedness.
module ⊤-WellFormed {T} (weakenOps : WeakenOps T) where
infix 4 _⊢_wf
-- Trivial well-formedness.
_⊢_wf : Wf T
_ ⊢ _ wf = ⊤
open WellFormedContext _⊢_wf public
-- Trivial well-formedness of contexts.
ctx-wf : ∀ {n} (Γ : Ctx T n) → Γ wf
ctx-wf [] = []
ctx-wf (a ∷ Γ) = tt ∷ ctx-wf Γ
module ⊤-WfWeakenOps where
wfWeakenOps : WfWeakenOps weakenOps
wfWeakenOps = record { weaken = λ _ _ → tt }
open WfWeakenOps public
------------------------------------------------------------------------
-- Abstract well-typed substitutions (i.e. substitution lemmas)
-- Abstract typed substitutions.
record TypedSub (Tp₁ Tp₂ Tm : ℕ → Set) : Set₁ where
infix 4 _⊢_∈_ _⊢_wf
field
_⊢_∈_ : Typing Tp₂ Tm Tp₁ -- the associated typing
_⊢_wf : Wf Tp₂ -- Tp₂-well-formedness
-- Application of Tm-substitutions to (source) Tp₁-types
application : Application Tp₁ Tm
-- Operations on the contexts.
weakenOps : WeakenOps Tp₁
open Application application public using (_/_)
open WeakenOps weakenOps using (toVec)
open WellFormedContext _⊢_wf public
infix 4 _⇒_⊢_
infixr 4 _,_
-- Typed substitutions.
--
-- A typed substitution Γ ⇒ Δ ⊢ σ is a substitution σ which, when
-- applied to something that is well-typed in a source context Γ,
-- yields something well-typed in a well-formed target context Δ.
data _⇒_⊢_ {m n} (Γ : Ctx Tp₁ m) (Δ : Ctx Tp₂ n) : Sub Tm m n → Set where
_,_ : ∀ {σ} → All₂ (λ t a → Δ ⊢ t ∈ (a / σ)) σ (toVec Γ) → Δ wf → Γ ⇒ Δ ⊢ σ
-- Project out the first component of a typed substitution.
proj₁ : ∀ {m n} {Γ : Ctx Tp₁ m} {Δ : Ctx Tp₂ n} {σ : Sub Tm m n} →
Γ ⇒ Δ ⊢ σ → All₂ (λ t a → Δ ⊢ t ∈ (a / σ)) σ (toVec Γ)
proj₁ (σ-wt , _) = σ-wt
-- Abstract extensions of substitutions.
record ExtensionTyped {Tp₁ Tp₂ Tm} (extension : Extension Tm)
(typedSub : TypedSub Tp₁ Tp₂ Tm) : Set where
open TypedSub typedSub
private
module E = Extension extension
module C = WeakenOps weakenOps
field
-- Weakens well-typed Tm-terms.
weaken : ∀ {n} {Δ : Ctx Tp₂ n} {t a b} → Δ ⊢ a wf → Δ ⊢ t ∈ b →
a ∷ Δ ⊢ E.weaken t ∈ C.weaken b
-- Weakening commutes with other substitutions.
weaken-/ : ∀ {m n} {σ : Sub Tm m n} {t} a →
C.weaken (a / σ) ≡ C.weaken a / (t E./∷ σ)
-- Well-typedness implies well-formedness of contexts.
wt-wf : ∀ {n} {Γ : Ctx Tp₂ n} {t a} → Γ ⊢ t ∈ a → Γ wf
infixr 5 _/∷_
-- Extension by a typed term.
_/∷_ : ∀ {m n} {Γ : Ctx Tp₁ m} {Δ : Ctx Tp₂ n} {t a b σ} →
b ∷ Δ ⊢ t ∈ C.weaken (a / σ) → Γ ⇒ Δ ⊢ σ →
a ∷ Γ ⇒ b ∷ Δ ⊢ (t E./∷ σ)
t∈a/σ /∷ (σ-wt , _) = σ-wt′ , wt-wf t∈a/σ
where
b∷Δ-wf = wt-wf t∈a/σ
b-wf = wf-∷₁ b∷Δ-wf
t∈a/σ′ = subst (_⊢_∈_ _ _) (weaken-/ _) t∈a/σ
σ-wt′ =
t∈a/σ′ ∷ gmap₂ (subst (_⊢_∈_ _ _) (weaken-/ _) ∘ weaken b-wf) σ-wt
-- Abstract simple typed substitutions.
record SimpleTyped {Tp Tm} (simple : Simple Tm)
(typedSub : TypedSub Tp Tp Tm) : Set where
open TypedSub typedSub
private
module S = SimpleExt simple
module L₀ = Lemmas₀ (record { simple = simple })
module C = WeakenOps weakenOps
field
extensionTyped : ExtensionTyped (record { weaken = S.weaken }) typedSub
-- Takes variables to well-typed Tms.
var : ∀ {n} {Γ : Ctx Tp n} (x : Fin n) → Γ wf → Γ ⊢ S.var x ∈ C.lookup x Γ
-- Types are invariant under the identity substitution.
id-vanishes : ∀ {n} (a : Tp n) → a / S.id ≡ a
-- Relates weakening of types to weakening of Tms.
/-wk : ∀ {n} {a : Tp n} → a / S.wk ≡ C.weaken a
-- Single-variable substitution is a left-inverse of weakening.
wk-sub-vanishes : ∀ {n t} (a : Tp n) → a / S.wk / S.sub t ≡ a
-- Well-formedness of types implies well-formedness of contexts.
wf-wf : ∀ {n} {Γ : Ctx Tp n} {a} → Γ ⊢ a wf → Γ wf
open ExtensionTyped extensionTyped public
infixl 10 _↑_
-- Lifting.
_↑_ : ∀ {m n} {Γ : Ctx Tp m} {Δ : Ctx Tp n} {σ} → Γ ⇒ Δ ⊢ σ →
∀ {a} → Δ ⊢ a / σ wf → a ∷ Γ ⇒ a / σ ∷ Δ ⊢ σ S.↑
(σ-wt , Δ-wf) ↑ a/σ-wf = var zero (a/σ-wf ∷ Δ-wf) /∷ (σ-wt , Δ-wf)
-- The identity substitution.
id : ∀ {n} {Γ : Ctx Tp n} → Γ wf → Γ ⇒ Γ ⊢ S.id
id [] = [] , []
id {Γ = a ∷ Γ} (a-wf ∷ Γ-wf) =
subst₂ (λ Δ σ → a ∷ Γ ⇒ Δ ⊢ σ)
(cong (flip _∷_ Γ) (id-vanishes a)) (L₀.id-↑⋆ 1)
(id Γ-wf ↑ subst (_⊢_wf Γ) (sym (id-vanishes a)) a-wf)
-- The first component of the identity substitution.
private
id₁ : ∀ {n} {Γ : Ctx Tp n} → Γ wf →
All₂ (λ t a → Γ ⊢ t ∈ (a / S.id)) S.id (C.toVec Γ)
id₁ = proj₁ ∘ id
-- Weakening.
wk : ∀ {n} {Γ : Ctx Tp n} {a} → Γ ⊢ a wf → Γ ⇒ a ∷ Γ ⊢ S.wk
wk a-wf =
gmap₂₁ (weaken′ a-wf ∘ subst (_⊢_∈_ _ _) (id-vanishes _)) (id₁ Γ-wf) ,
a-wf ∷ Γ-wf
where
weaken′ : ∀ {n} {Γ : Ctx Tp n} {t a b} → Γ ⊢ a wf → Γ ⊢ t ∈ b →
a ∷ Γ ⊢ S.weaken t ∈ b / S.wk
weaken′ a-wf = subst (_⊢_∈_ _ _) (sym /-wk) ∘ weaken a-wf
Γ-wf = wf-wf a-wf
-- Some helper lemmas.
private
wk-sub-vanishes′ : ∀ {n a} {t : Tm n} → a ≡ C.weaken a / S.sub t
wk-sub-vanishes′ {a = a} {t} = begin
a ≡⟨ sym (wk-sub-vanishes a) ⟩
a / S.wk / S.sub t ≡⟨ cong (flip _/_ _) /-wk ⟩
C.weaken a / S.sub t ∎
id-wk-sub-vanishes : ∀ {n a} {t : Tm n} →
a / S.id ≡ C.weaken a / S.sub t
id-wk-sub-vanishes {a = a} {t} = begin
a / S.id ≡⟨ id-vanishes a ⟩
a ≡⟨ wk-sub-vanishes′ ⟩
C.weaken a / S.sub t ∎
-- A substitution which only replaces the first variable.
sub : ∀ {n} {Γ : Ctx Tp n} {t a} → Γ ⊢ t ∈ a → a ∷ Γ ⇒ Γ ⊢ S.sub t
sub t∈a =
t∈a′ ∷ gmap₂₂ (subst (_⊢_∈_ _ _) id-wk-sub-vanishes) (id₁ Γ-wf) , Γ-wf
where
Γ-wf = wt-wf t∈a
t∈a′ = subst (_⊢_∈_ _ _) wk-sub-vanishes′ t∈a
-- A substitution which only changes the type of the first variable.
tsub : ∀ {n} {Γ : Ctx Tp n} {a b} → b ∷ Γ ⊢ S.var zero ∈ C.weaken a →
a ∷ Γ ⇒ b ∷ Γ ⊢ S.id
tsub z∈a = z∈a′ /∷ id (wf-∷₂ (wt-wf z∈a))
where
z∈a′ = subst (_⊢_∈_ _ _) (cong C.weaken (sym (id-vanishes _))) z∈a
-- Abstract typed liftings from Tm₁ to Tm₂.
record LiftTyped {Tp Tm₁ Tm₂} (l : Lift Tm₁ Tm₂)
(typedSub : TypedSub Tp Tp Tm₁)
(_⊢₂_∈_ : Typing Tp Tm₂ Tp) : Set where
open TypedSub typedSub renaming (_⊢_∈_ to _⊢₁_∈_)
private module L = Lift l
-- The underlying well-typed simple Tm₁-substitutions.
field simpleTyped : SimpleTyped L.simple typedSub
open SimpleTyped simpleTyped public
-- Lifts well-typed Tm₁-terms to well-typed Tm₂-terms.
field lift : ∀ {n} {Γ : Ctx Tp n} {t a} → Γ ⊢₁ t ∈ a → Γ ⊢₂ (L.lift t) ∈ a
-- Abstract variable typings.
module VarTyping {Tp} (weakenOps : WeakenOps Tp) (_⊢_wf : Wf Tp) where
open WeakenOps weakenOps
open WellFormedContext _⊢_wf
infix 4 _⊢Var_∈_
-- Abstract reflexive variable typings.
data _⊢Var_∈_ {n} (Γ : Ctx Tp n) : Fin n → Tp n → Set where
var : ∀ x → Γ wf → Γ ⊢Var x ∈ lookup x Γ
-- Abstract typed variable substitutions (renamings).
record TypedVarSubst {Tp} (_⊢_wf : Wf Tp) : Set where
field
application : Application Tp Fin
weakenOps : WeakenOps Tp
open WellFormedContext _⊢_wf
open VarTyping weakenOps _⊢_wf public
typedSub : TypedSub Tp Tp Fin
typedSub = record
{ _⊢_∈_ = _⊢Var_∈_
; _⊢_wf = _⊢_wf
; application = application
; weakenOps = weakenOps
}
open TypedSub typedSub public using () renaming (_⇒_⊢_ to _⇒_⊢Var_)
open Application application using (_/_)
open Lemmas₄ VarLemmas.lemmas₄ using (id; wk; _⊙_)
private module C = WeakenOps weakenOps
field
/-wk : ∀ {n} {a : Tp n} → a / wk ≡ C.weaken a
id-vanishes : ∀ {n} (a : Tp n) → a / id ≡ a
/-⊙ : ∀ {m n k} {σ₁ : Sub Fin m n} {σ₂ : Sub Fin n k} a →
a / σ₁ ⊙ σ₂ ≡ a / σ₁ / σ₂
wf-wf : ∀ {n} {Γ : Ctx Tp n} {a} → Γ ⊢ a wf → Γ wf
appLemmas : AppLemmas Tp Fin
appLemmas = record
{ application = application
; lemmas₄ = VarLemmas.lemmas₄
; id-vanishes = id-vanishes
; /-⊙ = /-⊙
}
open ExtAppLemmas appLemmas using (wk-commutes; wk-sub-vanishes)
open SimpleExt VarLemmas.simple using (extension; _/∷_)
-- Extensions of renamings.
extensionTyped : ExtensionTyped extension typedSub
extensionTyped = record
{ weaken = weaken
; weaken-/ = weaken-/
; wt-wf = wt-wf
}
where
weaken : ∀ {n} {Γ : Ctx Tp n} {x a b} → Γ ⊢ a wf → Γ ⊢Var x ∈ b →
a ∷ Γ ⊢Var suc x ∈ C.weaken b
weaken a-wf (var x Γ-wf) =
subst (_⊢Var_∈_ _ _) (lookup-map x _ _)
(var (suc x) (a-wf ∷ Γ-wf))
weaken-/ : ∀ {m n} {σ : Sub Fin m n} {t} a →
C.weaken (a / σ) ≡ C.weaken a / (t /∷ σ)
weaken-/ {σ = σ} {t} a = begin
C.weaken (a / σ) ≡⟨ sym /-wk ⟩
a / σ / wk ≡⟨ wk-commutes a ⟩
a / wk / (t /∷ σ) ≡⟨ cong₂ _/_ /-wk refl ⟩
C.weaken a / (t /∷ σ) ∎
wt-wf : ∀ {n} {Γ : Ctx Tp n} {x a} → Γ ⊢Var x ∈ a → Γ wf
wt-wf (var x Γ-wf) = Γ-wf
-- Simple typed renamings.
simpleTyped : SimpleTyped VarLemmas.simple typedSub
simpleTyped = record
{ extensionTyped = extensionTyped
; var = var
; id-vanishes = id-vanishes
; /-wk = /-wk
; wk-sub-vanishes = wk-sub-vanishes
; wf-wf = wf-wf
}
open SimpleTyped simpleTyped public
hiding (extensionTyped; var; /-wk; id-vanishes; wk-sub-vanishes; wf-wf)
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Idlemmas {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.Typed.Properties
import Definition.Typed.Weakening as Twk
open import Definition.Typed.EqualityRelation
open import Definition.Typed.RedSteps
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Application
open import Definition.LogicalRelation.Substitution
import Definition.LogicalRelation.Weakening as Lwk
open import Definition.LogicalRelation.Substitution.Properties
import Definition.LogicalRelation.Substitution.Irrelevance as S
open import Definition.LogicalRelation.Substitution.Reflexivity
open import Definition.LogicalRelation.Substitution.Weakening
-- open import Definition.LogicalRelation.Substitution.Introductions.Nat
open import Definition.LogicalRelation.Substitution.Introductions.Empty
open import Definition.LogicalRelation.Substitution.Introductions.Pi
open import Definition.LogicalRelation.Substitution.MaybeEmbed
open import Definition.LogicalRelation.Substitution.Introductions.Cast
-- open import Definition.LogicalRelation.Substitution.Introductions.SingleSubst
open import Definition.LogicalRelation.Substitution.Introductions.Universe
open import Tools.Product
open import Tools.Empty
import Tools.Unit as TU
import Tools.PropositionalEquality as PE
import Data.Nat as Nat
[proj₁cast] : ∀ {A B Γ r}
(⊢Γ : ⊢ Γ)
([A] : Γ ⊩⟨ ι ⁰ ⟩ A ^ [ r , ι ⁰ ])
([B] : Γ ⊩⟨ ι ⁰ ⟩ B ^ [ r , ι ⁰ ])
→ (∀ {t e} → ([t] : Γ ⊩⟨ ι ⁰ ⟩ t ∷ A ^ [ r , ι ⁰ ] / [A]) → (⊢e : Γ ⊢ e ∷ Id (Univ r ⁰) A B ^ [ % , ι ¹ ]) → Γ ⊩⟨ ι ⁰ ⟩ cast ⁰ A B e t ∷ B ^ [ r , ι ⁰ ] / [B])
[proj₁cast] ⊢Γ [A] [B] = proj₁ ([cast] ⊢Γ [A] [B])
[proj₁castext] : ∀ {A A′ B B′ Γ r}
(⊢Γ : ⊢ Γ)
([A] : Γ ⊩⟨ ι ⁰ ⟩ A ^ [ r , ι ⁰ ])
([A′] : Γ ⊩⟨ ι ⁰ ⟩ A′ ^ [ r , ι ⁰ ])
([A≡A′] : Γ ⊩⟨ ι ⁰ ⟩ A ≡ A′ ^ [ r , ι ⁰ ] / [A])
([B] : Γ ⊩⟨ ι ⁰ ⟩ B ^ [ r , ι ⁰ ])
([B′] : Γ ⊩⟨ ι ⁰ ⟩ B′ ^ [ r , ι ⁰ ])
([B≡B′] : Γ ⊩⟨ ι ⁰ ⟩ B ≡ B′ ^ [ r , ι ⁰ ] / [B])
→ (∀ {t t′ e e′} → ([t] : Γ ⊩⟨ ι ⁰ ⟩ t ∷ A ^ [ r , ι ⁰ ] / [A])
→ ([t′] : Γ ⊩⟨ ι ⁰ ⟩ t′ ∷ A′ ^ [ r , ι ⁰ ] / [A′])
→ ([t≡t′] : Γ ⊩⟨ ι ⁰ ⟩ t ≡ t′ ∷ A ^ [ r , ι ⁰ ] / [A])
→ (⊢e : Γ ⊢ e ∷ Id (Univ r ⁰) A B ^ [ % , ι ¹ ])
→ (⊢e′ : Γ ⊢ e′ ∷ Id (Univ r ⁰) A′ B′ ^ [ % , ι ¹ ])
→ Γ ⊩⟨ ι ⁰ ⟩ cast ⁰ A B e t ≡ cast ⁰ A′ B′ e′ t′ ∷ B ^ [ r , ι ⁰ ] / [B])
[proj₁castext] ⊢Γ [A] [A′] [A≡A′] [B] [B′] [B≡B′] = proj₁ ([castext] ⊢Γ [A] [A′] [A≡A′] [B] [B′] [B≡B′])
[nondep] : ∀ {Γ A B l} → Γ ⊩⟨ l ⟩ B ^ [ % , l ]
→ ([A] : ∀ {ρ} {Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ) → Δ ⊩⟨ l ⟩ wk ρ A ^ [ % , l ])
→ ∀ {ρ} {Δ} {a} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩⟨ l ⟩ a ∷ wk ρ A ^ [ % , l ] / [A] [ρ] ⊢Δ)
→ Δ ⊩⟨ l ⟩ wk (lift ρ) (wk1 B) [ a ] ^ [ % , l ]
[nondep] {Γ} {A} {B} {l} [B] [A] {ρ} {Δ} {a} [ρ] ⊢Δ [a] = PE.subst (λ X → Δ ⊩⟨ l ⟩ X ^ [ % , l ]) (Id-subst-lemma ρ B a) (Lwk.wk [ρ] ⊢Δ [B])
[nondepext] : ∀ {Γ A B l} → ([B] : Γ ⊩⟨ l ⟩ B ^ [ % , l ])
→ ([A] : ∀ {ρ} {Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ) → Δ ⊩⟨ l ⟩ wk ρ A ^ [ % , l ])
→ ∀ {ρ} {Δ} {a} {b} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩⟨ l ⟩ a ∷ wk ρ A ^ [ % , l ] / [A] [ρ] ⊢Δ)
→ ([b] : Δ ⊩⟨ l ⟩ b ∷ wk ρ A ^ [ % , l ] / [A] [ρ] ⊢Δ)
→ ([a≡b] : Δ ⊩⟨ l ⟩ a ≡ b ∷ wk ρ A ^ [ % , l ] / [A] [ρ] ⊢Δ)
→ Δ ⊩⟨ l ⟩ wk (lift ρ) (wk1 B) [ a ] ≡ wk (lift ρ) (wk1 B) [ b ] ^ [ % , l ] / [nondep] [B] [A] [ρ] ⊢Δ [a]
[nondepext] {Γ} {A} {B} {l} [B] [A] {ρ} {Δ} {a} {b} [ρ] ⊢Δ [a] [b] [a≡b] =
irrelevanceEq″ (Id-subst-lemma ρ B a) (Id-subst-lemma ρ B b) PE.refl PE.refl
(Lwk.wk [ρ] ⊢Δ [B]) ([nondep] [B] [A] [ρ] ⊢Δ [a]) (reflEq (Lwk.wk [ρ] ⊢Δ [B]))
[nondepext'] : ∀ {Γ A A' B B' l}
→ ([B] : Γ ⊩⟨ l ⟩ B ^ [ % , l ])
→ ([B'] : Γ ⊩⟨ l ⟩ B' ^ [ % , l ])
→ ([B≡B'] : Γ ⊩⟨ l ⟩ B ≡ B' ^ [ % , l ] / [B])
→ ([A] : ∀ {ρ} {Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ) → Δ ⊩⟨ l ⟩ wk ρ A ^ [ % , l ])
→ ([A'] : ∀ {ρ} {Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ) → Δ ⊩⟨ l ⟩ wk ρ A' ^ [ % , l ])
→ ∀ {ρ} {Δ} {a} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩⟨ l ⟩ a ∷ wk ρ A ^ [ % , l ] / [A] [ρ] ⊢Δ)
→ Δ ⊩⟨ l ⟩ wk (lift ρ) (wk1 B) [ a ] ≡ wk (lift ρ) (wk1 B') [ a ] ^ [ % , l ] / [nondep] [B] [A] [ρ] ⊢Δ [a]
[nondepext'] {Γ} {A} {A'} {B} {B'} {l} [B] [B'] [B≡B'] [A] [A'] {ρ} {Δ} {a} [ρ] ⊢Δ [a] =
irrelevanceEq″ (Id-subst-lemma ρ B a) (Id-subst-lemma ρ B' a) PE.refl PE.refl
(Lwk.wk [ρ] ⊢Δ [B]) ([nondep] [B] [A] [ρ] ⊢Δ [a]) (Lwk.wkEq [ρ] ⊢Δ [B] [B≡B'])
module IdTypeU-lemmas
{Γ rF A B F G F₁ G₁}
(⊢Γ : ⊢ Γ)
(⊢A : Γ ⊢ A ^ [ ! , ι ⁰ ])
(⊢ΠFG : Γ ⊢ Π F ^ rF ° ⁰ ▹ G ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D : Γ ⊢ A ⇒* Π F ^ rF ° ⁰ ▹ G ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F : Γ ⊢ F ^ [ rF , ι ⁰ ])
(⊢G : (Γ ∙ F ^ [ rF , ι ⁰ ]) ⊢ G ^ [ ! , ι ⁰ ])
(A≡A : Γ ⊢ (Π F ^ rF ° ⁰ ▹ G ° ⁰ ° ⁰) ≅ (Π F ^ rF ° ⁰ ▹ G ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F ^ [ rF , ι ⁰ ])
([G] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F ^ [ rF , ι ⁰ ] / ([F] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G [ a ] ^ [ ! , ι ⁰ ]))
(G-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F ^ [ rF , ι ⁰ ] / ([F] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F ^ [ rF , ι ⁰ ] / ([F] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F ^ [ rF , ι ⁰ ] / ([F] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G [ a ] ≡ wk (lift ρ) G [ b ] ^ [ ! , ι ⁰ ] / ([G] [ρ] ⊢Δ [a])))
(⊢B : Γ ⊢ B ^ [ ! , ι ⁰ ])
(⊢ΠF₁G₁ : Γ ⊢ Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D₁ : Γ ⊢ B ⇒* Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F₁ : Γ ⊢ F₁ ^ [ rF , ι ⁰ ])
(⊢G₁ : (Γ ∙ F₁ ^ [ rF , ι ⁰ ]) ⊢ G₁ ^ [ ! , ι ⁰ ])
(B≡B : Γ ⊢ (Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰) ≅ (Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F₁] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₁ ^ [ rF , ι ⁰ ])
([G₁] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ a ] ^ [ ! , ι ⁰ ]))
(G₁-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ a ] ≡ wk (lift ρ) G₁ [ b ] ^ [ ! , ι ⁰ ] / ([G₁] [ρ] ⊢Δ [a])))
(recursor₁ : ∀ {ρ Δ}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (Univ rF ⁰) (wk ρ F) (wk ρ F₁) ^ [ % , ι ¹ ])
(recursor₂ : ∀ {ρ Δ x y}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F ^ [ rF , ι ⁰ ] / [F] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G [ x ]) (wk (lift ρ) G₁ [ y ]) ^ [ % , ι ¹ ])
(extrecursor : ∀ {ρ Δ x y x′ y′}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F ^ [ rF , ι ⁰ ] / [F] [ρ] ⊢Δ)
([x′] : Δ ⊩⟨ ι ⁰ ⟩ x′ ∷ wk ρ F ^ [ rF , ι ⁰ ] / [F] [ρ] ⊢Δ)
([x≡x′] : Δ ⊩⟨ ι ⁰ ⟩ x ≡ x′ ∷ wk ρ F ^ [ rF , ι ⁰ ] / [F] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([y′] : Δ ⊩⟨ ι ⁰ ⟩ y′ ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([y≡y′] : Δ ⊩⟨ ι ⁰ ⟩ y ≡ y′ ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G [ x ]) (wk (lift ρ) G₁ [ y ]) ≡ Id (U ⁰) (wk (lift ρ) G [ x′ ]) (wk (lift ρ) G₁ [ y′ ]) ^ [ % , ι ¹ ] / recursor₂ [ρ] ⊢Δ [x] [y])
where
⊢IdFF₁ : Γ ⊢ Id (Univ rF ⁰) F F₁ ^ [ % , ι ¹ ]
⊢IdFF₁ = univ (Idⱼ (univ 0<1 ⊢Γ) (un-univ ⊢F) (un-univ ⊢F₁))
[IdFF₁] : ∀ {ρ Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ¹ ⟩ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ]
[IdFF₁] [ρ] ⊢Δ = recursor₁ [ρ] ⊢Δ
b = λ ρ e x → cast ⁰ (wk ρ F₁) (wk ρ F) (Idsym (Univ rF ⁰) (wk ρ F) (wk ρ F₁) e) x
IdGG₁ = λ ρ e → Π (wk ρ F₁) ^ rF ° ⁰ ▹ Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁) ° ¹ ° ¹
abstract
IdTel₂-prettify : ∀ ρ₁ ρ e a → wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁)) [ a ]
PE.≡ Id (U ⁰) (wk (lift (ρ₁ • ρ)) G [ b (ρ₁ • ρ) (wk ρ₁ e) a ]) (wk (lift (ρ₁ • ρ)) G₁ [ a ])
IdTel₂-prettify ρ₁ ρ e a = let
x₂ = PE.trans (PE.cong (subst (sgSubst a)) (wk-Idsym (lift ρ₁) (Univ rF ⁰) (wk (step ρ) F) (wk (step ρ) F₁) (wk1 e)))
(PE.trans (subst-Idsym (sgSubst a) (Univ rF ⁰) (wk _ (wk (step ρ) F)) (wk _ (wk (step ρ) F₁)) (wk _ (wk1 e)))
(PE.cong₃ (λ X Y Z → Idsym (Univ rF ⁰) X Y Z) (Id-subst-lemma4 ρ ρ₁ F a) (Id-subst-lemma4 ρ ρ₁ F₁ a) (irrelevant-subst′ ρ₁ e a)))
x₁ = PE.cong₂ (λ X Y → X [ Y ]) (Id-subst-lemma3 ρ ρ₁ G a) (PE.cong₃ (λ X Y Z → cast ⁰ X Y Z a) (Id-subst-lemma4 ρ ρ₁ F₁ a) ((Id-subst-lemma4 ρ ρ₁ F a)) x₂)
x = PE.trans (PE.cong (λ X → X [ a ]) (wk-β (wk (lift (step ρ)) G)))
(PE.trans (singleSubstLift (wk (lift (lift ρ₁)) (wk (lift (step ρ)) G)) (wk (lift ρ₁) (b (step ρ) (wk1 e) (var 0)))) x₁)
in PE.cong₂ (λ X Y → Id (U ⁰) X Y) x (PE.cong (λ X → X [ a ]) (wk-comp (lift ρ₁) (lift ρ) G₁))
abstract
[Id] : ∀ {ρ Δ e} →([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ ∀ {ρ₁ Δ₁ a} → ([ρ₁] : ρ₁ Twk.∷ Δ₁ ⊆ Δ) → (⊢Δ₁ : ⊢ Δ₁)
→ ([a] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ₁ (wk ρ F₁) ^ [ rF , ι ⁰ ] / (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)))
→ Δ₁ ⊩⟨ ι ¹ ⟩ wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁)) [ a ] ^ [ % , ι ¹ ]
[Id] {ρ} {Δ} {e} [ρ] ⊢Δ [e] {ρ₁} {Δ₁} {a} [ρ₁] ⊢Δ₁ [a] =
let
[a] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk (ρ₁ • ρ) F₁ ^ [ rF , ι ⁰ ] / [F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁
[a] = irrelevanceTerm′ (wk-comp ρ₁ ρ F₁) PE.refl PE.refl (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)) ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a]
⊢e″ = PE.subst (λ X → Δ₁ ⊢ wk ρ₁ e ∷ X ^ [ % , ι ¹ ]) (wk-comp ρ₁ ρ _) (Twk.wkTerm [ρ₁] ⊢Δ₁ (escapeTerm ([IdFF₁] [ρ] ⊢Δ) [e]))
⊢e′ = Idsymⱼ (univ 0<1 ⊢Δ₁) (un-univ (escape ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) (un-univ (escape ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) ⊢e″
[b] : Δ₁ ⊩⟨ ι ⁰ ⟩ b (ρ₁ • ρ) (wk ρ₁ e) a ∷ wk (ρ₁ • ρ) F ^ [ rF , ι ⁰ ] / [F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁
[b] = [proj₁cast] ⊢Δ₁ ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a] ⊢e′
x = recursor₂ ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁ [b] [a]
in PE.subst (λ X → Δ₁ ⊩⟨ ι ¹ ⟩ X ^ [ % , ι ¹ ]) (PE.sym (IdTel₂-prettify ρ₁ ρ e a)) x
abstract
[Idext] : ∀ {ρ Δ e e′} →([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ ([e′] : Δ ⊩⟨ ι ¹ ⟩ e′ ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ ∀ {ρ₁ Δ₁ a a′} → ([ρ₁] : ρ₁ Twk.∷ Δ₁ ⊆ Δ) → (⊢Δ₁ : ⊢ Δ₁)
→ ([a] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ₁ (wk ρ F₁) ^ [ rF , ι ⁰ ] / (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)))
→ ([a′] : Δ₁ ⊩⟨ ι ⁰ ⟩ a′ ∷ wk ρ₁ (wk ρ F₁) ^ [ rF , ι ⁰ ] / (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)))
→ ([a≡a′] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ≡ a′ ∷ wk ρ₁ (wk ρ F₁) ^ [ rF , ι ⁰ ] / (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)))
→ Δ₁ ⊩⟨ ι ¹ ⟩ wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁)) [ a ]
≡ wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e′) (var 0) ]) (wk (lift ρ) G₁)) [ a′ ] ^ [ % , ι ¹ ]
/ [Id] [ρ] ⊢Δ [e] [ρ₁] ⊢Δ₁ [a]
[Idext] {ρ} {Δ} {e} {e′} [ρ] ⊢Δ [e] [e′] {ρ₁} {Δ₁} {a} {a′} [ρ₁] ⊢Δ₁ [a] [a′] [a≡a′] =
let
[a]₁ : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk (ρ₁ • ρ) F₁ ^ [ rF , ι ⁰ ] / [F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁
[a]₁ = irrelevanceTerm′ (wk-comp ρ₁ ρ F₁) PE.refl PE.refl (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)) ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a]
[a′] = irrelevanceTerm′ (wk-comp ρ₁ ρ F₁) PE.refl PE.refl (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)) ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a′]
[a≡a′] = irrelevanceEqTerm′ (wk-comp ρ₁ ρ F₁) PE.refl PE.refl (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)) ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a≡a′]
⊢e₂ = PE.subst (λ X → Δ₁ ⊢ wk ρ₁ e ∷ X ^ [ % , ι ¹ ]) (wk-comp ρ₁ ρ _) (Twk.wkTerm [ρ₁] ⊢Δ₁ (escapeTerm ([IdFF₁] [ρ] ⊢Δ) [e]))
⊢e₁ = Idsymⱼ (univ 0<1 ⊢Δ₁) (un-univ (escape ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) (un-univ (escape ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) ⊢e₂
⊢e′₂ = PE.subst (λ X → Δ₁ ⊢ wk ρ₁ e′ ∷ X ^ [ % , ι ¹ ]) (wk-comp ρ₁ ρ _) (Twk.wkTerm [ρ₁] ⊢Δ₁ (escapeTerm ([IdFF₁] [ρ] ⊢Δ) [e′]))
⊢e′₁ = Idsymⱼ (univ 0<1 ⊢Δ₁) (un-univ (escape ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) (un-univ (escape ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))) ⊢e′₂
[b] : Δ₁ ⊩⟨ ι ⁰ ⟩ b (ρ₁ • ρ) (wk ρ₁ e) a ∷ wk (ρ₁ • ρ) F ^ [ rF , ι ⁰ ] / [F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁
[b] = [proj₁cast] ⊢Δ₁ ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a]₁ ⊢e₁
[b′] = [proj₁cast] ⊢Δ₁ ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) [a′] ⊢e′₁
[b≡b′] = [proj₁castext] ⊢Δ₁ ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) (reflEq ([F₁] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))
([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁) (reflEq ([F] ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁))
[a]₁ [a′] [a≡a′] ⊢e₁ ⊢e′₁
x₁ = recursor₂ ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁ [b] [a]₁
x = extrecursor ([ρ₁] Twk.•ₜ [ρ]) ⊢Δ₁ [b] [b′] [b≡b′] [a]₁ [a′] [a≡a′]
in irrelevanceEq″ (PE.sym (IdTel₂-prettify ρ₁ ρ e a)) (PE.sym (IdTel₂-prettify ρ₁ ρ e′ a′)) PE.refl PE.refl x₁ ([Id] [ρ] ⊢Δ [e] [ρ₁] ⊢Δ₁ [a]) x
[IdGG₁] : ∀ {ρ Δ e} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ IdGG₁ ρ e ^ [ % , ι ¹ ]
[IdGG₁] {ρ} {Δ} {e} [ρ] ⊢Δ [e] =
let
⊢wkF₁ = escape ([F₁] [ρ] ⊢Δ)
[0] = let ⊢0 = (var (⊢Δ ∙ ⊢wkF₁) here) in
neuTerm (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ escape ([F₁] [ρ] ⊢Δ)) ([F₁] [ρ] ⊢Δ)) (var 0) ⊢0 (~-var ⊢0)
x : Δ ∙ wk ρ F₁ ^ [ rF , ι ⁰ ] ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁) ^ [ % , ι ¹ ]
x = PE.subst (λ X → Δ ∙ wk ρ F₁ ^ [ rF , ι ⁰ ] ⊩⟨ ι ¹ ⟩ X ^ [ % , ι ¹ ]) (wkSingleSubstId _) ([Id] [ρ] ⊢Δ [e] (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₁) [0])
⊢Id = escape x
in
Πᵣ′ rF ⁰ ¹ (<is≤ 0<1) (≡is≤ PE.refl) (wk ρ F₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₁))
(idRed:*: (univ (Πⱼ <is≤ 0<1 ▹ ≡is≤ PE.refl ▹ un-univ ⊢wkF₁ ▹ un-univ ⊢Id)))
⊢wkF₁ ⊢Id (≅-univ (≅ₜ-Π-cong (<is≤ 0<1) (≡is≤ PE.refl) ⊢wkF₁ (≅-un-univ (escapeEqRefl ([F₁] [ρ] ⊢Δ))) (≅-un-univ (escapeEqRefl x))))
(λ [ρ₁] ⊢Δ₁ → emb emb< (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ))) ([Id] [ρ] ⊢Δ [e]) ([Idext] [ρ] ⊢Δ [e] [e])
abstract
[IdGG₁-ext] : ∀ {ρ Δ e e′} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ ([e′] : Δ ⊩⟨ ι ¹ ⟩ e′ ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ ([e≡e′] : Δ ⊩⟨ ι ¹ ⟩ e ≡ e′ ∷ wk ρ (Id (Univ rF ⁰) F F₁) ^ [ % , ι ¹ ] / [IdFF₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ IdGG₁ ρ e ≡ IdGG₁ ρ e′ ^ [ % , ι ¹ ] / [IdGG₁] [ρ] ⊢Δ [e]
[IdGG₁-ext] {ρ} {Δ} {e} {e′} [ρ] ⊢Δ [e] [e′] _ =
let
⊢wkF₁ = escape ([F₁] [ρ] ⊢Δ)
[0] = let ⊢0 = (var (⊢Δ ∙ ⊢wkF₁) here) in
neuTerm (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ escape ([F₁] [ρ] ⊢Δ)) ([F₁] [ρ] ⊢Δ)) (var 0) ⊢0 (~-var ⊢0)
x = [Id] [ρ] ⊢Δ [e′] (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₁) [0]
⊢x = PE.subst (λ X → Δ ∙ wk ρ F₁ ^ [ rF , ι ⁰ ] ⊢ X ^ [ % , ι ¹ ]) (wkSingleSubstId _) (escape x)
x₁ = [Idext] [ρ] ⊢Δ [e′] [e] (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₁) [0] [0] (reflEqTerm (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ escape ([F₁] [ρ] ⊢Δ)) ([F₁] [ρ] ⊢Δ)) [0])
⊢x₁ = PE.subst₂ (λ X Y → Δ ∙ wk ρ F₁ ^ [ rF , ι ⁰ ] ⊢ X ≅ Y ^ [ % , ι ¹ ]) (wkSingleSubstId _) (wkSingleSubstId _) (escapeEq x x₁)
in
Π₌ (wk ρ F₁) (Id (U ⁰) (wk (lift (step ρ)) G [ b (step ρ) (wk1 e′) (var 0) ]) (wk (lift ρ) G₁))
(id (univ (Πⱼ <is≤ 0<1 ▹ ≡is≤ PE.refl ▹ un-univ ⊢wkF₁ ▹ un-univ ⊢x)))
(≅-univ (≅ₜ-Π-cong (<is≤ 0<1) (≡is≤ PE.refl) ⊢wkF₁ (≅-un-univ (escapeEqRefl ([F₁] [ρ] ⊢Δ))) (≅-un-univ (≅-sym ⊢x₁))))
(λ [ρ]₁ ⊢Δ₁ → reflEq (Lwk.wk [ρ]₁ ⊢Δ₁ ([F₁] [ρ] ⊢Δ)))
(λ [ρ₁] ⊢Δ₁ [a] → [Idext] [ρ] ⊢Δ [e] [e′] [ρ₁] ⊢Δ₁ [a] [a] (reflEqTerm (Lwk.wk [ρ₁] ⊢Δ₁ ([F₁] [ρ] ⊢Δ)) [a]))
abstract
IdTel≡IdUΠΠ : ∃ Id (Univ rF ⁰) F F₁ ▹ (IdGG₁ (step id) (var 0)) PE.≡
∃ (Id (Univ rF ⁰) F F₁) ▹ (Π (wk1 F₁) ^ rF ° ⁰ ▹
Id (U ⁰) ((wk1d G) [ cast ⁰ (wk1 (wk1 F₁)) (wk1 (wk1 F)) (Idsym (Univ rF ⁰) (wk1 (wk1 F)) (wk1 (wk1 F₁)) (var 1)) (var 0) ]↑) (wk1d G₁) ° ¹ ° ¹)
IdTel≡IdUΠΠ = PE.cong (λ X → ∃ Id (Univ rF ⁰) F F₁ ▹ Π (wk1 F₁) ^ rF ° ⁰ ▹ Id (U ⁰) X (wk1d G₁) ° ¹ ° ¹)
(PE.trans
(PE.cong₃ (λ X Y Z → X [ cast ⁰ Z Y (Idsym (Univ rF ⁰) Y Z (var 1)) (var 0) ])
(PE.sym (wk-comp (lift (step id)) (lift (step id)) G))
(PE.sym (wk-comp (step id) (step id) F))
(PE.sym (wk-comp (step id) (step id) F₁)))
(PE.sym (wk1d[]-[]↑ (wk1d G) _)))
abstract
wksubst-IdTel : ∀ ρ e → wk (lift ρ) (IdGG₁ (step id) (var 0)) [ e ] PE.≡ IdGG₁ ρ e
wksubst-IdTel ρ e =
let
x₃ = PE.trans (PE.cong (subst (liftSubst (sgSubst e))) (wk-Idsym (lift (lift ρ)) (Univ rF ⁰) (wk (step (step id)) F) (wk (step (step id)) F₁) (var 1)))
(PE.trans (subst-Idsym (liftSubst (sgSubst e)) (Univ rF ⁰) (wk _ (wk (step (step id)) F)) (wk _ (wk (step (step id)) F₁)) (var 1))
(PE.cong₂ (λ X Y → Idsym (Univ rF ⁰) X Y (wk1 e)) (Id-subst-lemma1 ρ F e) (Id-subst-lemma1 ρ F₁ e)))
x₂ = PE.cong₃ (λ X Y Z → cast ⁰ X Y Z (var 0)) (Id-subst-lemma1 ρ F₁ e) (Id-subst-lemma1 ρ F e) x₃
x₁ = PE.cong₂ (λ X Y → X [ Y ]) (Id-subst-lemma2 ρ G e) x₂
x₀ = (PE.trans (PE.cong (subst (liftSubst (sgSubst e))) (wk-β (wk (lift (step (step id))) G)))
(PE.trans (singleSubstLift (wk (lift (lift (lift ρ))) (wk (lift (step (step id))) G)) (wk (lift (lift ρ)) (b (step (step id)) (var 1) (var 0)))) x₁))
in PE.cong₃ (λ X Y Z → Π X ^ rF ° ⁰ ▹ Id (U ⁰) Y Z ° ¹ ° ¹) (irrelevant-subst′ ρ F₁ e) x₀ (cast-subst-lemma4 ρ e G₁)
[IdGG₁0] : Γ ∙ Id (Univ rF ⁰) F F₁ ^ [ % , ι ¹ ] ⊩⟨ ι ¹ ⟩ IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
[IdGG₁0] = let
⊢0 = var (⊢Γ ∙ ⊢IdFF₁) here
[0] = neuTerm ([IdFF₁] (Twk.step Twk.id) (⊢Γ ∙ ⊢IdFF₁)) (var 0) ⊢0 (~-var ⊢0)
in [IdGG₁] (Twk.step Twk.id) (⊢Γ ∙ ⊢IdFF₁) [0]
⊢IdGG₁ : Γ ∙ Id (Univ rF ⁰) F F₁ ^ [ % , ι ¹ ] ⊢ IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
⊢IdGG₁ = escape [IdGG₁0]
⊢∃ : Γ ⊢ ∃ Id (Univ rF ⁰) F F₁ ▹ IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
⊢∃ = univ (∃ⱼ un-univ ⊢IdFF₁ ▹ un-univ ⊢IdGG₁)
∃≡∃ : Γ ⊢ ∃ Id (Univ rF ⁰) F F₁ ▹ IdGG₁ (step id) (var 0) ≅ ∃ Id (Univ rF ⁰) F F₁ ▹ IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
∃≡∃ = (≅-univ (≅ₜ-∃-cong
⊢IdFF₁
(≅-un-univ (escapeEqRefl (PE.subst (λ X → Γ ⊩⟨ ι ¹ ⟩ X ^ [ % , ι ¹ ]) (wk-id (Id (Univ rF ⁰) F F₁)) ([IdFF₁] Twk.id ⊢Γ))))
(≅-un-univ (escapeEqRefl [IdGG₁0]))))
D∃ : Γ ⊢ Id (U ⁰) A B ⇒* ∃ Id (Univ rF ⁰) F F₁ ▹ IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
D∃ = univ⇒* (IdURed*Term′ (un-univ ⊢A) (un-univ ⊢ΠFG) (un-univ⇒* D) (un-univ ⊢B)
⇨∷* IdUΠRed*Term′ (un-univ ⊢F) (un-univ ⊢G) (un-univ ⊢B) (un-univ ⊢ΠF₁G₁) (un-univ⇒* D₁))
⇨* PE.subst (λ X → Γ ⊢ Id (U ⁰) (Π F ^ rF ° ⁰ ▹ G ° ⁰ ° ⁰) (Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰) ⇒* X ^ [ % , ι ¹ ]) (PE.sym IdTel≡IdUΠΠ)
(univ (Id-U-ΠΠ (un-univ ⊢F) (un-univ ⊢G) (un-univ ⊢F₁) (un-univ ⊢G₁))
⇨ id (PE.subst (λ X → Γ ⊢ X ^ [ % , ι ¹ ]) IdTel≡IdUΠΠ ⊢∃))
module IdTypeU-lemmas-2
{Γ rF A₁ A₂ A₃ A₄ F₁ F₂ F₃ F₄ G₁ G₂ G₃ G₄}
(⊢Γ : ⊢ Γ)
(⊢A₁ : Γ ⊢ A₁ ^ [ ! , ι ⁰ ])
(⊢ΠF₁G₁ : Γ ⊢ Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D₁ : Γ ⊢ A₁ ⇒* Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F₁ : Γ ⊢ F₁ ^ [ rF , ι ⁰ ])
(⊢G₁ : (Γ ∙ F₁ ^ [ rF , ι ⁰ ]) ⊢ G₁ ^ [ ! , ι ⁰ ])
(A₁≡A₁ : Γ ⊢ (Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰) ≅ (Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F₁] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₁ ^ [ rF , ι ⁰ ])
([G₁] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ a ] ^ [ ! , ι ⁰ ]))
(G₁-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / ([F₁] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ a ] ≡ wk (lift ρ) G₁ [ b ] ^ [ ! , ι ⁰ ] / ([G₁] [ρ] ⊢Δ [a])))
(⊢A₂ : Γ ⊢ A₂ ^ [ ! , ι ⁰ ])
(⊢ΠF₂G₂ : Γ ⊢ Π F₂ ^ rF ° ⁰ ▹ G₂ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D₂ : Γ ⊢ A₂ ⇒* Π F₂ ^ rF ° ⁰ ▹ G₂ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F₂ : Γ ⊢ F₂ ^ [ rF , ι ⁰ ])
(⊢G₂ : (Γ ∙ F₂ ^ [ rF , ι ⁰ ]) ⊢ G₂ ^ [ ! , ι ⁰ ])
(A₂≡A₂ : Γ ⊢ (Π F₂ ^ rF ° ⁰ ▹ G₂ ° ⁰ ° ⁰) ≅ (Π F₂ ^ rF ° ⁰ ▹ G₂ ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F₂] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₂ ^ [ rF , ι ⁰ ])
([G₂] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / ([F₂] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₂ [ a ] ^ [ ! , ι ⁰ ]))
(G₂-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / ([F₂] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / ([F₂] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / ([F₂] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₂ [ a ] ≡ wk (lift ρ) G₂ [ b ] ^ [ ! , ι ⁰ ] / ([G₂] [ρ] ⊢Δ [a])))
(⊢A₃ : Γ ⊢ A₃ ^ [ ! , ι ⁰ ])
(⊢ΠF₃G₃ : Γ ⊢ Π F₃ ^ rF ° ⁰ ▹ G₃ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D₃ : Γ ⊢ A₃ ⇒* Π F₃ ^ rF ° ⁰ ▹ G₃ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F₃ : Γ ⊢ F₃ ^ [ rF , ι ⁰ ])
(⊢G₃ : (Γ ∙ F₃ ^ [ rF , ι ⁰ ]) ⊢ G₃ ^ [ ! , ι ⁰ ])
(A₃≡A₃ : Γ ⊢ (Π F₃ ^ rF ° ⁰ ▹ G₃ ° ⁰ ° ⁰) ≅ (Π F₃ ^ rF ° ⁰ ▹ G₃ ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F₃] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₃ ^ [ rF , ι ⁰ ])
([G₃] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / ([F₃] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₃ [ a ] ^ [ ! , ι ⁰ ]))
(G₃-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / ([F₃] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / ([F₃] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / ([F₃] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₃ [ a ] ≡ wk (lift ρ) G₃ [ b ] ^ [ ! , ι ⁰ ] / ([G₃] [ρ] ⊢Δ [a])))
(⊢A₄ : Γ ⊢ A₄ ^ [ ! , ι ⁰ ])
(⊢ΠF₄G₄ : Γ ⊢ Π F₄ ^ rF ° ⁰ ▹ G₄ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(D₄ : Γ ⊢ A₄ ⇒* Π F₄ ^ rF ° ⁰ ▹ G₄ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(⊢F₄ : Γ ⊢ F₄ ^ [ rF , ι ⁰ ])
(⊢G₄ : (Γ ∙ F₄ ^ [ rF , ι ⁰ ]) ⊢ G₄ ^ [ ! , ι ⁰ ])
(A₄≡A₄ : Γ ⊢ (Π F₄ ^ rF ° ⁰ ▹ G₄ ° ⁰ ° ⁰) ≅ (Π F₄ ^ rF ° ⁰ ▹ G₄ ° ⁰ ° ⁰) ^ [ ! , ι ⁰ ])
([F₄] : ∀ {ρ} {Δ} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₄ ^ [ rF , ι ⁰ ])
([G₄] : ∀ {ρ} {Δ} {a} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / ([F₄] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₄ [ a ] ^ [ ! , ι ⁰ ]))
(G₄-ext : ∀ {ρ} {Δ} {a} {b} ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / ([F₄] [ρ] ⊢Δ))
([b] : Δ ⊩⟨ ι ⁰ ⟩ b ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / ([F₄] [ρ] ⊢Δ))
([a≡b] : Δ ⊩⟨ ι ⁰ ⟩ a ≡ b ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / ([F₄] [ρ] ⊢Δ))
→ (Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₄ [ a ] ≡ wk (lift ρ) G₄ [ b ] ^ [ ! , ι ⁰ ] / ([G₄] [ρ] ⊢Δ [a])))
(A₁≡A₂ : Γ ⊢ Π F₁ ^ rF ° ⁰ ▹ G₁ ° ⁰ ° ⁰ ≅ Π F₂ ^ rF ° ⁰ ▹ G₂ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
(A₃≡A₄ : Γ ⊢ Π F₃ ^ rF ° ⁰ ▹ G₃ ° ⁰ ° ⁰ ≅ Π F₄ ^ rF ° ⁰ ▹ G₄ ° ⁰ ° ⁰ ^ [ ! , ι ⁰ ])
([F₁≡F₂] : ∀ {ρ Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₁ ≡ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([F₃≡F₄] : ∀ {ρ Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩⟨ ι ⁰ ⟩ wk ρ F₃ ≡ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
([G₁≡G₂] : ∀ {ρ Δ a} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ a ] ≡ wk (lift ρ) G₂ [ a ] ^ [ ! , ι ⁰ ] / [G₁] [ρ] ⊢Δ [a])
([G₃≡G₄] : ∀ {ρ Δ a} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₃ [ a ] ≡ wk (lift ρ) G₄ [ a ] ^ [ ! , ι ⁰ ] / [G₃] [ρ] ⊢Δ [a])
(recursor₁ : ∀ {ρ Δ}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (Univ rF ⁰) (wk ρ F₁) (wk ρ F₃) ^ [ % , ι ¹ ])
(recursor₂ : ∀ {ρ Δ x y}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G₁ [ x ]) (wk (lift ρ) G₃ [ y ]) ^ [ % , ι ¹ ])
(recursor₃ : ∀ {ρ Δ}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (Univ rF ⁰) (wk ρ F₂) (wk ρ F₄) ^ [ % , ι ¹ ])
(recursor₄ : ∀ {ρ Δ x y}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₂] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₄] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G₂ [ x ]) (wk (lift ρ) G₄ [ y ]) ^ [ % , ι ¹ ])
(extrecursor₁ : ∀ {ρ Δ x y x′ y′}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([x′] : Δ ⊩⟨ ι ⁰ ⟩ x′ ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([x≡x′] : Δ ⊩⟨ ι ⁰ ⟩ x ≡ x′ ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
([y′] : Δ ⊩⟨ ι ⁰ ⟩ y′ ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
([y≡y′] : Δ ⊩⟨ ι ⁰ ⟩ y ≡ y′ ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G₁ [ x ]) (wk (lift ρ) G₃ [ y ]) ≡ Id (U ⁰) (wk (lift ρ) G₁ [ x′ ]) (wk (lift ρ) G₃ [ y′ ]) ^ [ % , ι ¹ ] / recursor₂ [ρ] ⊢Δ [x] [y])
(extrecursor₂ : ∀ {ρ Δ x y x′ y′}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([x] : Δ ⊩⟨ ι ⁰ ⟩ x ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₂] [ρ] ⊢Δ)
([x′] : Δ ⊩⟨ ι ⁰ ⟩ x′ ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₂] [ρ] ⊢Δ)
([x≡x′] : Δ ⊩⟨ ι ⁰ ⟩ x ≡ x′ ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₂] [ρ] ⊢Δ)
([y] : Δ ⊩⟨ ι ⁰ ⟩ y ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₄] [ρ] ⊢Δ)
([y′] : Δ ⊩⟨ ι ⁰ ⟩ y′ ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₄] [ρ] ⊢Δ)
([y≡y′] : Δ ⊩⟨ ι ⁰ ⟩ y ≡ y′ ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₄] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G₂ [ x ]) (wk (lift ρ) G₄ [ y ]) ≡ Id (U ⁰) (wk (lift ρ) G₂ [ x′ ]) (wk (lift ρ) G₄ [ y′ ]) ^ [ % , ι ¹ ] / recursor₄ [ρ] ⊢Δ [x] [y])
(eqrecursor₁ : ∀ {ρ Δ}
([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩⟨ ι ¹ ⟩ Id (Univ rF ⁰) (wk ρ F₁) (wk ρ F₃) ≡ Id (Univ rF ⁰) (wk ρ F₂) (wk ρ F₄) ^ [ % , ι ¹ ] / recursor₁ [ρ] ⊢Δ)
(eqrecursor₂ : ∀ {ρ Δ x₁ x₂ x₃ x₄}
→ ([ρ] : ρ Twk.∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([x₁] : Δ ⊩⟨ ι ⁰ ⟩ x₁ ∷ wk ρ F₁ ^ [ rF , ι ⁰ ] / [F₁] [ρ] ⊢Δ)
→ ([x₂] : Δ ⊩⟨ ι ⁰ ⟩ x₂ ∷ wk ρ F₂ ^ [ rF , ι ⁰ ] / [F₂] [ρ] ⊢Δ)
→ ([G₁x₁≡G₂x₂] : Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₁ [ x₁ ] ≡ wk (lift ρ) G₂ [ x₂ ] ^ [ ! , ι ⁰ ] / [G₁] [ρ] ⊢Δ [x₁])
→ ([x₃] : Δ ⊩⟨ ι ⁰ ⟩ x₃ ∷ wk ρ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ] ⊢Δ)
→ ([x₄] : Δ ⊩⟨ ι ⁰ ⟩ x₄ ∷ wk ρ F₄ ^ [ rF , ι ⁰ ] / [F₄] [ρ] ⊢Δ)
→ ([G₃x₃≡G₄x₄] : Δ ⊩⟨ ι ⁰ ⟩ wk (lift ρ) G₃ [ x₃ ] ≡ wk (lift ρ) G₄ [ x₄ ] ^ [ ! , ι ⁰ ] / [G₃] [ρ] ⊢Δ [x₃])
→ Δ ⊩⟨ ι ¹ ⟩ Id (U ⁰) (wk (lift ρ) G₁ [ x₁ ]) (wk (lift ρ) G₃ [ x₃ ]) ≡ Id (U ⁰) (wk (lift ρ) G₂ [ x₂ ]) (wk (lift ρ) G₄ [ x₄ ]) ^ [ % , ι ¹ ] / recursor₂ [ρ] ⊢Δ [x₁] [x₃])
where
module E₁ = IdTypeU-lemmas ⊢Γ ⊢A₁ ⊢ΠF₁G₁ D₁ ⊢F₁ ⊢G₁ A₁≡A₁ [F₁] [G₁] G₁-ext ⊢A₃ ⊢ΠF₃G₃ D₃ ⊢F₃ ⊢G₃ A₃≡A₃ [F₃] [G₃] G₃-ext
recursor₁ recursor₂ extrecursor₁
module E₂ = IdTypeU-lemmas ⊢Γ ⊢A₂ ⊢ΠF₂G₂ D₂ ⊢F₂ ⊢G₂ A₂≡A₂ [F₂] [G₂] G₂-ext ⊢A₄ ⊢ΠF₄G₄ D₄ ⊢F₄ ⊢G₄ A₄≡A₄ [F₄] [G₄] G₄-ext
recursor₃ recursor₄ extrecursor₂
[IdFF₁≡IdFF₂] : ∀ {ρ Δ} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ Δ ⊩⟨ ι ¹ ⟩ wk ρ (Id (Univ rF ⁰) F₁ F₃) ≡ wk ρ (Id (Univ rF ⁰) F₂ F₄) ^ [ % , ι ¹ ] / E₁.[IdFF₁] [ρ] ⊢Δ
[IdFF₁≡IdFF₂] = (λ [ρ] ⊢Δ → irrelevanceEq (recursor₁ [ρ] ⊢Δ) (E₁.[IdFF₁] [ρ] ⊢Δ) (eqrecursor₁ [ρ] ⊢Δ))
abstract
[Ideq] : ∀ {ρ Δ e} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F₁ F₃) ^ [ % , ι ¹ ] / E₁.[IdFF₁] [ρ] ⊢Δ)
→ ∀ {ρ₁ Δ₁ a} → ([ρ₁] : ρ₁ Twk.∷ Δ₁ ⊆ Δ) → (⊢Δ₁ : ⊢ Δ₁)
→ ([a] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ₁ (wk ρ F₃) ^ [ rF , ι ⁰ ] / (Lwk.wk [ρ₁] ⊢Δ₁ ([F₃] [ρ] ⊢Δ)))
→ Δ₁ ⊩⟨ ι ¹ ⟩ wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G₁ [ E₁.b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₃)) [ a ]
≡ wk (lift ρ₁) (Id (U ⁰) (wk (lift (step ρ)) G₂ [ E₂.b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₄)) [ a ] ^ [ % , ι ¹ ]
/ E₁.[Id] [ρ] ⊢Δ [e] [ρ₁] ⊢Δ₁ [a]
[Ideq] {ρ} {Δ} {e} [ρ] ⊢Δ [e] {ρ₁} {Δ₁} {a} [ρ₁] ⊢Δ₁ [a] =
let
ρ′ = ρ₁ • ρ
[ρ′] : ρ′ Twk.∷ Δ₁ ⊆ Γ
[ρ′] = [ρ₁] Twk.•ₜ [ρ]
[a₃] : Δ₁ ⊩⟨ ι ⁰ ⟩ a ∷ wk ρ′ F₃ ^ [ rF , ι ⁰ ] / [F₃] [ρ′] ⊢Δ₁
[a₃] = irrelevanceTerm′ (wk-comp ρ₁ ρ F₃) PE.refl PE.refl (Lwk.wk [ρ₁] ⊢Δ₁ ([F₃] [ρ] ⊢Δ)) ([F₃] [ρ′] ⊢Δ₁) [a]
[a₄] = convTerm₁ ([F₃] [ρ′] ⊢Δ₁) ([F₄] [ρ′] ⊢Δ₁) ([F₃≡F₄] [ρ′] ⊢Δ₁) [a₃]
[a₃≡a₄] = reflEqTerm ([F₃] [ρ′] ⊢Δ₁) [a₃]
⊢e′ = PE.subst (λ X → Δ₁ ⊢ wk ρ₁ e ∷ X ^ [ % , ι ¹ ]) (wk-comp ρ₁ ρ _) (Twk.wkTerm [ρ₁] ⊢Δ₁ (escapeTerm (E₁.[IdFF₁] [ρ] ⊢Δ) [e]))
⊢e₁ = Idsymⱼ (univ 0<1 ⊢Δ₁) (un-univ (escape ([F₁] [ρ′] ⊢Δ₁))) (un-univ (escape ([F₃] [ρ′] ⊢Δ₁))) ⊢e′
⊢e″ = conv ⊢e′ (≅-eq (escapeEq (E₁.[IdFF₁] [ρ′] ⊢Δ₁) ([IdFF₁≡IdFF₂] [ρ′] ⊢Δ₁)))
⊢e₂ = Idsymⱼ (univ 0<1 ⊢Δ₁) (un-univ (escape ([F₂] [ρ′] ⊢Δ₁))) (un-univ (escape ([F₄] [ρ′] ⊢Δ₁))) ⊢e″
[b₁] = [proj₁cast] ⊢Δ₁ ([F₃] [ρ′] ⊢Δ₁) ([F₁] [ρ′] ⊢Δ₁) [a₃] ⊢e₁
[b₂] = [proj₁cast] ⊢Δ₁ ([F₄] [ρ′] ⊢Δ₁) ([F₂] [ρ′] ⊢Δ₁) [a₄] ⊢e₂
[b₁≡b₂] = [proj₁castext] ⊢Δ₁ ([F₃] [ρ′] ⊢Δ₁) ([F₄] [ρ′] ⊢Δ₁) ([F₃≡F₄] [ρ′] ⊢Δ₁)
([F₁] [ρ′] ⊢Δ₁) ([F₂] [ρ′] ⊢Δ₁) ([F₁≡F₂] [ρ′] ⊢Δ₁)
[a₃] [a₄] [a₃≡a₄] ⊢e₁ ⊢e₂
[b₁:F₂] = convTerm₁ ([F₁] [ρ′] ⊢Δ₁) ([F₂] [ρ′] ⊢Δ₁) ([F₁≡F₂] [ρ′] ⊢Δ₁) [b₁]
[b₁≡b₂:F₂] = convEqTerm₁ ([F₁] [ρ′] ⊢Δ₁) ([F₂] [ρ′] ⊢Δ₁) ([F₁≡F₂] [ρ′] ⊢Δ₁) [b₁≡b₂]
[G₁b₁≡G₂b₂] = transEq ([G₁] [ρ′] ⊢Δ₁ [b₁]) ([G₂] [ρ′] ⊢Δ₁ [b₁:F₂]) ([G₂] [ρ′] ⊢Δ₁ [b₂])
([G₁≡G₂] [ρ′] ⊢Δ₁ [b₁]) (G₂-ext [ρ′] ⊢Δ₁ [b₁:F₂] [b₂] [b₁≡b₂:F₂])
[a₃≡a₄:F₄] = convEqTerm₁ ([F₃] [ρ′] ⊢Δ₁) ([F₄] [ρ′] ⊢Δ₁) ([F₃≡F₄] [ρ′] ⊢Δ₁) [a₃≡a₄]
[G₃a₃≡G₄a₄] = transEq ([G₃] [ρ′] ⊢Δ₁ [a₃]) ([G₄] [ρ′] ⊢Δ₁ [a₄]) ([G₄] [ρ′] ⊢Δ₁ [a₄])
([G₃≡G₄] [ρ′] ⊢Δ₁ [a₃]) (G₄-ext [ρ′] ⊢Δ₁ [a₄] [a₄] [a₃≡a₄:F₄])
x₁ = recursor₂ [ρ′] ⊢Δ₁ [b₁] [a₃]
x = eqrecursor₂ [ρ′] ⊢Δ₁ [b₁] [b₂] [G₁b₁≡G₂b₂] [a₃] [a₄] [G₃a₃≡G₄a₄]
in irrelevanceEq″ (PE.sym (E₁.IdTel₂-prettify ρ₁ ρ e a)) (PE.sym (E₂.IdTel₂-prettify ρ₁ ρ e a)) PE.refl PE.refl x₁ (E₁.[Id] [ρ] ⊢Δ [e] [ρ₁] ⊢Δ₁ [a]) x
abstract
[IdGG₁≡IdGG₂] : ∀ {ρ Δ e} → ([ρ] : ρ Twk.∷ Δ ⊆ Γ) → (⊢Δ : ⊢ Δ)
→ ([e] : Δ ⊩⟨ ι ¹ ⟩ e ∷ wk ρ (Id (Univ rF ⁰) F₁ F₃) ^ [ % , ι ¹ ] / E₁.[IdFF₁] [ρ] ⊢Δ)
→ Δ ⊩⟨ ι ¹ ⟩ E₁.IdGG₁ ρ e ≡ E₂.IdGG₁ ρ e ^ [ % , ι ¹ ] / E₁.[IdGG₁] [ρ] ⊢Δ [e]
[IdGG₁≡IdGG₂] {ρ} {Δ} {e} [ρ] ⊢Δ [e] = let
[e]′ = convTerm₁ (E₁.[IdFF₁] [ρ] ⊢Δ) (E₂.[IdFF₁] [ρ] ⊢Δ) ([IdFF₁≡IdFF₂] [ρ] ⊢Δ) [e]
⊢wkF₃ = escape ([F₃] [ρ] ⊢Δ)
⊢wkF₄ = escape ([F₄] [ρ] ⊢Δ)
[0:F₃] : Δ ∙ wk ρ F₃ ^ [ rF , ι ⁰ ] ⊩⟨ ι ⁰ ⟩ var 0 ∷ wk1 (wk ρ F₃) ^ [ rF , ι ⁰ ] / (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₃) ([F₃] [ρ] ⊢Δ))
[0:F₃] = let ⊢0 = (var (⊢Δ ∙ ⊢wkF₃) here) in
neuTerm (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₃) ([F₃] [ρ] ⊢Δ)) (var 0) ⊢0 (~-var ⊢0)
[0:F₄] = let ⊢0 = (var (⊢Δ ∙ ⊢wkF₄) here) in
neuTerm (Lwk.wk (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₄) ([F₄] [ρ] ⊢Δ)) (var 0) ⊢0 (~-var ⊢0)
x = E₁.[Id] [ρ] ⊢Δ [e] (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₃) [0:F₃]
x₁ = [Ideq] [ρ] ⊢Δ [e] (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₃) [0:F₃]
⊢x₁ = PE.subst₂ (λ X Y → Δ ∙ wk ρ F₃ ^ [ rF , ι ⁰ ] ⊢ X ≅ Y ^ [ % , ι ¹ ]) (wkSingleSubstId _) (wkSingleSubstId _) (escapeEq x x₁)
x₂ = E₂.[Id] [ρ] ⊢Δ [e]′ (Twk.step Twk.id) (⊢Δ ∙ ⊢wkF₄) [0:F₄]
⊢x₂ = PE.subst (λ X → Δ ∙ wk ρ F₄ ^ [ rF , ι ⁰ ] ⊢ X ^ [ % , ι ¹ ]) (wkSingleSubstId _) (escape x₂)
in Π₌ (wk ρ F₄) (Id (U ⁰) (wk (lift (step ρ)) G₂ [ E₂.b (step ρ) (wk1 e) (var 0) ]) (wk (lift ρ) G₄))
(id (univ (Πⱼ <is≤ 0<1 ▹ ≡is≤ PE.refl ▹ un-univ ⊢wkF₄ ▹ un-univ ⊢x₂)))
(≅-univ (≅ₜ-Π-cong (<is≤ 0<1) (≡is≤ PE.refl) ⊢wkF₃ (≅-un-univ (escapeEq ([F₃] [ρ] ⊢Δ) ([F₃≡F₄] [ρ] ⊢Δ))) (≅-un-univ ⊢x₁)))
(λ [ρ₁] ⊢Δ₁ → Lwk.wkEq [ρ₁] ⊢Δ₁ ([F₃] [ρ] ⊢Δ) ([F₃≡F₄] [ρ] ⊢Δ))
(λ [ρ₁] ⊢Δ₁ [a] → [Ideq] [ρ] ⊢Δ [e] [ρ₁] ⊢Δ₁ [a])
[IdGG₁≡IdGG₂0] : Γ ∙ Id (Univ rF ⁰) F₁ F₃ ^ [ % , ι ¹ ] ⊩⟨ ι ¹ ⟩ E₁.IdGG₁ (step id) (var 0) ≡ E₂.IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ] / E₁.[IdGG₁0]
[IdGG₁≡IdGG₂0] = let
⊢0 = var (⊢Γ ∙ E₁.⊢IdFF₁) here
[0] = neuTerm (E₁.[IdFF₁] (Twk.step Twk.id) (⊢Γ ∙ E₁.⊢IdFF₁)) (var 0) ⊢0 (~-var ⊢0)
in [IdGG₁≡IdGG₂] (Twk.step Twk.id) (⊢Γ ∙ E₁.⊢IdFF₁) [0]
∃₁≡∃₂ : Γ ⊢ ∃ Id (Univ rF ⁰) F₁ F₃ ▹ E₁.IdGG₁ (step id) (var 0) ≅ ∃ Id (Univ rF ⁰) F₂ F₄ ▹ E₂.IdGG₁ (step id) (var 0) ^ [ % , ι ¹ ]
∃₁≡∃₂ = (≅-univ (≅ₜ-∃-cong
E₁.⊢IdFF₁
(≅-un-univ (PE.subst₂ (λ X Y → Γ ⊢ X ≅ Y ^ [ % , ι ¹ ]) (wk-id (Id (Univ rF ⁰) F₁ F₃)) (wk-id (Id (Univ rF ⁰) F₂ F₄)) (escapeEq (E₁.[IdFF₁] Twk.id ⊢Γ) ([IdFF₁≡IdFF₂] Twk.id ⊢Γ))))
(≅-un-univ (escapeEq E₁.[IdGG₁0] [IdGG₁≡IdGG₂0]))))
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module Numeral.Natural where
import Lvl
open import Type
-- The set of natural numbers (0,..).
-- Positive integers including zero.
data ℕ : Type{Lvl.𝟎} where
𝟎 : ℕ -- Zero
𝐒 : ℕ → ℕ -- Successor function (Intuitively: 𝐒(n) = n+1)
{-# BUILTIN NATURAL ℕ #-}
pattern 𝟏 = ℕ.𝐒(𝟎)
{-# DISPLAY ℕ.𝐒(𝟎) = 𝟏 #-}
-- Limited predecessor function
-- Intuitively: 𝐏(n) = max(0,n-1)
𝐏 : ℕ → ℕ
𝐏(𝟎) = 𝟎
𝐏(𝐒(n)) = n
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------------------------------------------------------------------------
-- Abstract typing contexts
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module Data.Context where
open import Data.Fin using (Fin)
open import Data.Fin.Substitution
open import Data.Fin.Substitution.ExtraLemmas
open import Data.Nat using (ℕ; zero; suc; _+_)
open import Data.Vec as Vec using (Vec; []; _∷_)
open import Relation.Unary using (Pred)
------------------------------------------------------------------------
-- Abstract typing contexts and context extensions.
infixr 5 _∷_
-- Typing contexts.
--
-- A |Ctx T n| is an indexed sequences of T-typed bindings mapping n
-- variables to T-types with 0 to (n - 1) free variables each. Like
-- lists (Data.List) and vectors (Data.Vec), contexts are cons
-- sequences, i.e. new bindings are added to the front (rather than
-- the back, as is more common in the literature). For example, a
-- typing context Γ represented in the usual notation as
--
-- Γ = xᵢ: Aᵢ, ..., x₁: A₁, x₀: A₀
--
-- is represented here by a term |Γ : Ctx Type i| of the form
--
-- Γ = A₀ ∷ A₁ ∷ ... ∷ Aᵢ
--
-- which is consistent with A₀ being the 0-th element of Γ, and with
-- the de Bruijn convention that the 0-th variable corresponds to the
-- closest binding.
data Ctx {ℓ} (T : Pred ℕ ℓ) : ℕ → Set ℓ where
[] : Ctx T zero
_∷_ : ∀ {n} → T n → Ctx T n → Ctx T (suc n)
module _ {ℓ} {T : Pred ℕ ℓ} where
head : ∀ {n} → Ctx T (suc n) → T n
head (t ∷ ts) = t
tail : ∀ {n} → Ctx T (suc n) → Ctx T n
tail (t ∷ ts) = ts
-- Drop the m innermost elements of a context Γ.
drop : ∀ {n} m → Ctx T (m + n) → Ctx T n
drop zero Γ = Γ
drop (suc m) (_ ∷ Γ) = drop m Γ
-- A map function that changes the entries in a context pointwise.
map : ∀ {ℓ₁ ℓ₂} {T₁ : Pred ℕ ℓ₁} {T₂ : Pred ℕ ℓ₂} {n} →
(∀ {k} → T₁ k → T₂ k) → Ctx T₁ n → Ctx T₂ n
map f [] = []
map f (t ∷ Γ) = f t ∷ map f Γ
-- Extensions of typing contexts.
--
-- Context extensions are indexed sequences of bindings that can be
-- concatenated to the front of a typing context. A |CtxExt T m n| is
-- an extension mapping n variables to T-types with m to (n + m - 1)
-- free variables each.
--
-- NOTE. It is tempting to define contexts as just a special case of
-- context extensions, i.e. as
--
-- Ctx T n = CtxExt T zero n
--
-- But this leads to problems when defining e.g. concatenation because
-- of the way context extensions are indexed. This could be remedied
-- by indexing context extensions differently, but then the definition
-- of |mapExt| below becomes difficult. An earlier version of this
-- module contained two different (but equivalent) representations for
-- context extensions, but this complicated (rather than simplified)
-- the development overall.
data CtxExt {ℓ} (T : Pred ℕ ℓ) (m : ℕ) : ℕ → Set ℓ where
[] : CtxExt T m zero
_∷_ : ∀ {l} → T (l + m) → CtxExt T m l → CtxExt T m (suc l)
infixr 5 _++_
-- Concatenation of context extensions with contexts.
_++_ : ∀ {ℓ} {T : Pred ℕ ℓ} {m n} → CtxExt T m n → Ctx T m → Ctx T (n + m)
[] ++ Γ = Γ
(t ∷ Δ) ++ Γ = t ∷ (Δ ++ Γ)
-- A map function that point-wise re-indexes and changes the entries
-- in a context extension.
mapExt : ∀ {ℓ₁ ℓ₂} {T₁ : Pred ℕ ℓ₁} {T₂ : Pred ℕ ℓ₂} {m n k} →
(∀ l → T₁ (l + m) → T₂ (l + n)) → CtxExt T₁ m k → CtxExt T₂ n k
mapExt f [] = []
mapExt f (_∷_ {l} t Γ) = f l t ∷ mapExt (λ l → f l) Γ
-- Operations on contexts that require weakening of types.
module WeakenOps {ℓ} {T : Pred ℕ ℓ} (extension : Extension T) where
-- Weakening of types.
open Extension extension public
-- Convert a context or context extension to its vector representation.
toVec : ∀ {n} → Ctx T n → Vec (T n) n
toVec [] = []
toVec (t ∷ Γ) = weaken t /∷ toVec Γ
extToVec : ∀ {k m n} → CtxExt T m n → Vec (T m) k → Vec (T (n + m)) (n + k)
extToVec [] ts = ts
extToVec (t ∷ Γ) ts = weaken t /∷ extToVec Γ ts
-- Lookup the type of a variable in a context or context extension.
lookup : ∀ {n} → Ctx T n → Fin n → T n
lookup Γ x = Vec.lookup (toVec Γ) x
extLookup : ∀ {k m n} → CtxExt T m n → Vec (T m) k → Fin (n + k) → T (n + m)
extLookup Δ ts x = Vec.lookup (extToVec Δ ts) x
-- Operations on contexts that require substitutions in types.
module SubstOps {ℓ₁ ℓ₂} {T₁ : Pred ℕ ℓ₁} {T₂ : Pred ℕ ℓ₂}
(application : Application T₁ T₂)
(simple : Simple T₂)
where
open Application application public -- Application of T′ substitutions to Ts.
open Simple simple public -- Simple T′ substitutions.
-- Application of substitutions to context extensions.
_E/_ : ∀ {k m n} → CtxExt T₁ m k → Sub T₂ m n → CtxExt T₁ n k
Γ E/ σ = mapExt (λ l t → t / σ ↑⋆ l) Γ
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{-# OPTIONS --prop --without-K --rewriting #-}
module Data.Nat.PredExp2 where
open import Data.Nat
open import Data.Nat.Properties
open import Relation.Nullary
open import Relation.Nullary.Negation
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; _≢_; module ≡-Reasoning)
open import Data.Nat.Log2 using (⌈log₂_⌉)
pred[2^_] : ℕ → ℕ
pred[2^ n ] = pred (2 ^ n)
lemma/2^suc : ∀ n → 2 ^ n + 2 ^ n ≡ 2 ^ suc n
lemma/2^suc n =
begin
2 ^ n + 2 ^ n
≡˘⟨ Eq.cong ((2 ^ n) +_) (*-identityˡ (2 ^ n)) ⟩
2 ^ n + (2 ^ n + 0)
≡⟨⟩
2 ^ n + (2 ^ n + 0 * (2 ^ n))
≡⟨⟩
2 * (2 ^ n)
≡⟨⟩
2 ^ suc n
∎
where open ≡-Reasoning
private
lemma/1≤2^n : ∀ n → 1 ≤ 2 ^ n
lemma/1≤2^n zero = ≤-refl {1}
lemma/1≤2^n (suc n) =
begin
1
≤⟨ s≤s z≤n ⟩
1 + 1
≤⟨ +-mono-≤ (lemma/1≤2^n n) (lemma/1≤2^n n) ⟩
2 ^ n + 2 ^ n
≡⟨ lemma/2^suc n ⟩
2 ^ suc n
∎
where open ≤-Reasoning
lemma/2^n≢0 : ∀ n → 2 ^ n ≢ zero
lemma/2^n≢0 n 2^n≡0 with 2 ^ n | lemma/1≤2^n n
... | zero | ()
pred[2^]-mono : pred[2^_] Preserves _≤_ ⟶ _≤_
pred[2^]-mono m≤n = pred-mono (2^-mono m≤n)
where
2^-mono : (2 ^_) Preserves _≤_ ⟶ _≤_
2^-mono {y = y} z≤n = lemma/1≤2^n y
2^-mono (s≤s m≤n) = *-monoʳ-≤ 2 (2^-mono m≤n)
pred[2^suc[n]] : (n : ℕ) → suc (pred[2^ n ] + pred[2^ n ]) ≡ pred[2^ suc n ]
pred[2^suc[n]] n =
begin
suc (pred[2^ n ] + pred[2^ n ])
≡⟨⟩
suc (pred (2 ^ n) + pred (2 ^ n))
≡˘⟨ +-suc (pred (2 ^ n)) (pred (2 ^ n)) ⟩
pred (2 ^ n) + suc (pred (2 ^ n))
≡⟨ Eq.cong (pred (2 ^ n) +_) (suc[pred[n]]≡n (lemma/2^n≢0 n)) ⟩
pred (2 ^ n) + 2 ^ n
≡⟨ lemma/pred-+ (2 ^ n) (2 ^ n) (lemma/2^n≢0 n) ⟩
pred (2 ^ n + 2 ^ n)
≡⟨ Eq.cong pred (lemma/2^suc n) ⟩
pred (2 ^ suc n)
≡⟨⟩
pred[2^ suc n ]
∎
where
open ≡-Reasoning
lemma/pred-+ : ∀ m n → m ≢ zero → pred m + n ≡ pred (m + n)
lemma/pred-+ zero n m≢zero = contradiction refl m≢zero
lemma/pred-+ (suc m) n m≢zero = refl
pred[2^log₂] : (n : ℕ) → pred[2^ ⌈log₂ suc ⌈ n /2⌉ ⌉ ] ≤ n
pred[2^log₂] n = strong-induction n n ≤-refl
where
strong-induction : (n m : ℕ) → m ≤ n → pred[2^ ⌈log₂ suc ⌈ m /2⌉ ⌉ ] ≤ m
strong-induction n zero h = z≤n
strong-induction n (suc zero) h = s≤s z≤n
strong-induction (suc (suc n)) (suc (suc m)) (s≤s (s≤s h)) =
begin
pred[2^ ⌈log₂ suc ⌈ suc (suc m) /2⌉ ⌉ ]
≡⟨⟩
pred[2^ suc ⌈log₂ ⌈ suc ⌈ suc (suc m) /2⌉ /2⌉ ⌉ ]
≡˘⟨ pred[2^suc[n]] ⌈log₂ ⌈ suc ⌈ suc (suc m) /2⌉ /2⌉ ⌉ ⟩
suc (pred[2^ ⌈log₂ ⌈ suc ⌈ suc (suc m) /2⌉ /2⌉ ⌉ ] + pred[2^ ⌈log₂ ⌈ suc ⌈ suc (suc m) /2⌉ /2⌉ ⌉ ])
≡⟨⟩
suc (pred[2^ ⌈log₂ ⌈ suc (suc ⌈ m /2⌉) /2⌉ ⌉ ] + pred[2^ ⌈log₂ ⌈ suc (suc ⌈ m /2⌉) /2⌉ ⌉ ])
≡⟨⟩
suc (pred[2^ ⌈log₂ suc ⌈ ⌈ m /2⌉ /2⌉ ⌉ ] + pred[2^ ⌈log₂ suc ⌈ ⌈ m /2⌉ /2⌉ ⌉ ])
≤⟨
s≤s (
+-mono-≤
(strong-induction (suc n) ⌈ m /2⌉ (≤-trans (⌊n/2⌋≤n (suc m)) (s≤s h)))
(strong-induction (suc n) ⌈ m /2⌉ (≤-trans (⌊n/2⌋≤n (suc m)) (s≤s h)))
)
⟩
suc (⌈ m /2⌉ + ⌈ m /2⌉)
≡⟨⟩
suc (⌊ suc m /2⌋ + ⌈ m /2⌉)
≤⟨ s≤s (+-monoʳ-≤ ⌊ suc m /2⌋ (⌈n/2⌉-mono (n≤1+n m))) ⟩
suc (⌊ suc m /2⌋ + ⌈ suc m /2⌉)
≡⟨ Eq.cong suc (⌊n/2⌋+⌈n/2⌉≡n (suc m)) ⟩
suc (suc m)
∎
where open ≤-Reasoning
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties of reflexive closures which rely on the K rule
------------------------------------------------------------------------
{-# OPTIONS --safe --with-K #-}
module Relation.Binary.Construct.Closure.Reflexive.Properties.WithK where
open import Data.Empty.Irrelevant using (⊥-elim)
open import Data.Product as Prod
open import Data.Sum.Base as Sum
open import Relation.Binary
open import Relation.Binary.Construct.Closure.Reflexive
open import Relation.Binary.Construct.Closure.Reflexive.Properties public
open import Relation.Binary.PropositionalEquality as PropEq using (_≡_; refl; cong)
open import Relation.Nullary.Negation using (contradiction)
module _ {a ℓ} {A : Set a} {_∼_ : Rel A ℓ} where
irrel : Irrelevant _∼_ → Irreflexive _≡_ _∼_ → Irrelevant (Refl _∼_)
irrel irrel irrefl [ x∼y₁ ] [ x∼y₂ ] = cong [_] (irrel x∼y₁ x∼y₂)
irrel irrel irrefl [ x∼y ] refl = contradiction x∼y (irrefl refl)
irrel irrel irrefl refl [ x∼y ] = contradiction x∼y (irrefl refl)
irrel irrel irrefl refl refl = refl
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module Setoids where
open import Eq
open import Prelude
record Setoid : Set1 where
field
carrier : Set
_≈_ : carrier -> carrier -> Set
equiv : Equiv _≈_
record Datoid : Set1 where
field
setoid : Setoid
_≟_ : forall x y -> Dec (Setoid._≈_ setoid x y)
Setoid-≡ : Set -> Setoid
Setoid-≡ a = record { carrier = a; _≈_ = _≡_; equiv = Equiv-≡ }
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head : #$\forall$# {A n} → Vec A (1 + n) → A
zip : #$\forall$# {A B n} → Vec A n → Vec B n → Vec (A × B) n
take : #$\forall$# {A} m {n} → Vec A (m + n) → Vec A m
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-- Andreas, 2013-02-18 problem with 'with'-display, see also issue 295
-- {-# OPTIONS -v tc.with:50 #-}
module Issue800 where
data ⊤ : Set where
tt : ⊤
data I⊤ : ⊤ → Set where
itt : ∀ r → I⊤ r
bug : ∀ l → ∀ k → I⊤ l → ⊤
bug .l k (itt l) with itt k
... | foo = {! foo!}
{-
Current rewriting:
bug .l l (itt k) | itt .k = ?
Desired rewriting:
bug .l k (itt l) | itt .k = ?
-}
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{-# OPTIONS --type-in-type #-}
Ty : Set
Ty =
(Ty : Set)
(nat top bot : Ty)
(arr prod sum : Ty → Ty → Ty)
→ Ty
nat : Ty; nat = λ _ nat _ _ _ _ _ → nat
top : Ty; top = λ _ _ top _ _ _ _ → top
bot : Ty; bot = λ _ _ _ bot _ _ _ → bot
arr : Ty → Ty → Ty; arr
= λ A B Ty nat top bot arr prod sum →
arr (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
prod : Ty → Ty → Ty; prod
= λ A B Ty nat top bot arr prod sum →
prod (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
sum : Ty → Ty → Ty; sum
= λ A B Ty nat top bot arr prod sum →
sum (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
Con : Set; Con
= (Con : Set)
(nil : Con)
(snoc : Con → Ty → Con)
→ Con
nil : Con; nil
= λ Con nil snoc → nil
snoc : Con → Ty → Con; snoc
= λ Γ A Con nil snoc → snoc (Γ Con nil snoc) A
Var : Con → Ty → Set; Var
= λ Γ A →
(Var : Con → Ty → Set)
(vz : ∀ Γ A → Var (snoc Γ A) A)
(vs : ∀ Γ B A → Var Γ A → Var (snoc Γ B) A)
→ Var Γ A
vz : ∀{Γ A} → Var (snoc Γ A) A; vz
= λ Var vz vs → vz _ _
vs : ∀{Γ B A} → Var Γ A → Var (snoc Γ B) A; vs
= λ x Var vz vs → vs _ _ _ (x Var vz vs)
Tm : Con → Ty → Set; Tm
= λ Γ A →
(Tm : Con → Ty → Set)
(var : ∀ Γ A → Var Γ A → Tm Γ A)
(lam : ∀ Γ A B → Tm (snoc Γ A) B → Tm Γ (arr A B))
(app : ∀ Γ A B → Tm Γ (arr A B) → Tm Γ A → Tm Γ B)
(tt : ∀ Γ → Tm Γ top)
(pair : ∀ Γ A B → Tm Γ A → Tm Γ B → Tm Γ (prod A B))
(fst : ∀ Γ A B → Tm Γ (prod A B) → Tm Γ A)
(snd : ∀ Γ A B → Tm Γ (prod A B) → Tm Γ B)
(left : ∀ Γ A B → Tm Γ A → Tm Γ (sum A B))
(right : ∀ Γ A B → Tm Γ B → Tm Γ (sum A B))
(case : ∀ Γ A B C → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C)
(zero : ∀ Γ → Tm Γ nat)
(suc : ∀ Γ → Tm Γ nat → Tm Γ nat)
(rec : ∀ Γ A → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A)
→ Tm Γ A
var : ∀{Γ A} → Var Γ A → Tm Γ A; var
= λ x Tm var lam app tt pair fst snd left right case zero suc rec →
var _ _ x
lam : ∀{Γ A B} → Tm (snoc Γ A) B → Tm Γ (arr A B); lam
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
lam _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
app : ∀{Γ A B} → Tm Γ (arr A B) → Tm Γ A → Tm Γ B; app
= λ t u Tm var lam app tt pair fst snd left right case zero suc rec →
app _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
tt : ∀{Γ} → Tm Γ top; tt
= λ Tm var lam app tt pair fst snd left right case zero suc rec → tt _
pair : ∀{Γ A B} → Tm Γ A → Tm Γ B → Tm Γ (prod A B); pair
= λ t u Tm var lam app tt pair fst snd left right case zero suc rec →
pair _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
fst : ∀{Γ A B} → Tm Γ (prod A B) → Tm Γ A; fst
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
fst _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
snd : ∀{Γ A B} → Tm Γ (prod A B) → Tm Γ B; snd
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
snd _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
left : ∀{Γ A B} → Tm Γ A → Tm Γ (sum A B); left
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
left _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
right : ∀{Γ A B} → Tm Γ B → Tm Γ (sum A B); right
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
right _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
case : ∀{Γ A B C} → Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C; case
= λ t u v Tm var lam app tt pair fst snd left right case zero suc rec →
case _ _ _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
zero : ∀{Γ} → Tm Γ nat; zero
= λ Tm var lam app tt pair fst snd left right case zero suc rec → zero _
suc : ∀{Γ} → Tm Γ nat → Tm Γ nat; suc
= λ t Tm var lam app tt pair fst snd left right case zero suc rec →
suc _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
rec : ∀{Γ A} → Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A; rec
= λ t u v Tm var lam app tt pair fst snd left right case zero suc rec →
rec _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
v0 : ∀{Γ A} → Tm (snoc Γ A) A; v0
= var vz
v1 : ∀{Γ A B} → Tm (snoc (snoc Γ A) B) A; v1
= var (vs vz)
v2 : ∀{Γ A B C} → Tm (snoc (snoc (snoc Γ A) B) C) A; v2
= var (vs (vs vz))
v3 : ∀{Γ A B C D} → Tm (snoc (snoc (snoc (snoc Γ A) B) C) D) A; v3
= var (vs (vs (vs vz)))
tbool : Ty; tbool
= sum top top
true : ∀{Γ} → Tm Γ tbool; true
= left tt
tfalse : ∀{Γ} → Tm Γ tbool; tfalse
= right tt
ifthenelse : ∀{Γ A} → Tm Γ (arr tbool (arr A (arr A A))); ifthenelse
= lam (lam (lam (case v2 (lam v2) (lam v1))))
times4 : ∀{Γ A} → Tm Γ (arr (arr A A) (arr A A)); times4
= lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))
add : ∀{Γ} → Tm Γ (arr nat (arr nat nat)); add
= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0))
mul : ∀{Γ} → Tm Γ (arr nat (arr nat nat)); mul
= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero))
fact : ∀{Γ} → Tm Γ (arr nat nat); fact
= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero))
{-# OPTIONS --type-in-type #-}
Ty1 : Set
Ty1 =
(Ty1 : Set)
(nat top bot : Ty1)
(arr prod sum : Ty1 → Ty1 → Ty1)
→ Ty1
nat1 : Ty1; nat1 = λ _ nat1 _ _ _ _ _ → nat1
top1 : Ty1; top1 = λ _ _ top1 _ _ _ _ → top1
bot1 : Ty1; bot1 = λ _ _ _ bot1 _ _ _ → bot1
arr1 : Ty1 → Ty1 → Ty1; arr1
= λ A B Ty1 nat1 top1 bot1 arr1 prod sum →
arr1 (A Ty1 nat1 top1 bot1 arr1 prod sum) (B Ty1 nat1 top1 bot1 arr1 prod sum)
prod1 : Ty1 → Ty1 → Ty1; prod1
= λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum →
prod1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum) (B Ty1 nat1 top1 bot1 arr1 prod1 sum)
sum1 : Ty1 → Ty1 → Ty1; sum1
= λ A B Ty1 nat1 top1 bot1 arr1 prod1 sum1 →
sum1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum1) (B Ty1 nat1 top1 bot1 arr1 prod1 sum1)
Con1 : Set; Con1
= (Con1 : Set)
(nil : Con1)
(snoc : Con1 → Ty1 → Con1)
→ Con1
nil1 : Con1; nil1
= λ Con1 nil1 snoc → nil1
snoc1 : Con1 → Ty1 → Con1; snoc1
= λ Γ A Con1 nil1 snoc1 → snoc1 (Γ Con1 nil1 snoc1) A
Var1 : Con1 → Ty1 → Set; Var1
= λ Γ A →
(Var1 : Con1 → Ty1 → Set)
(vz : ∀ Γ A → Var1 (snoc1 Γ A) A)
(vs : ∀ Γ B A → Var1 Γ A → Var1 (snoc1 Γ B) A)
→ Var1 Γ A
vz1 : ∀{Γ A} → Var1 (snoc1 Γ A) A; vz1
= λ Var1 vz1 vs → vz1 _ _
vs1 : ∀{Γ B A} → Var1 Γ A → Var1 (snoc1 Γ B) A; vs1
= λ x Var1 vz1 vs1 → vs1 _ _ _ (x Var1 vz1 vs1)
Tm1 : Con1 → Ty1 → Set; Tm1
= λ Γ A →
(Tm1 : Con1 → Ty1 → Set)
(var : ∀ Γ A → Var1 Γ A → Tm1 Γ A)
(lam : ∀ Γ A B → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B))
(app : ∀ Γ A B → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B)
(tt : ∀ Γ → Tm1 Γ top1)
(pair : ∀ Γ A B → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B))
(fst : ∀ Γ A B → Tm1 Γ (prod1 A B) → Tm1 Γ A)
(snd : ∀ Γ A B → Tm1 Γ (prod1 A B) → Tm1 Γ B)
(left : ∀ Γ A B → Tm1 Γ A → Tm1 Γ (sum1 A B))
(right : ∀ Γ A B → Tm1 Γ B → Tm1 Γ (sum1 A B))
(case : ∀ Γ A B C → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C)
(zero : ∀ Γ → Tm1 Γ nat1)
(suc : ∀ Γ → Tm1 Γ nat1 → Tm1 Γ nat1)
(rec : ∀ Γ A → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A)
→ Tm1 Γ A
var1 : ∀{Γ A} → Var1 Γ A → Tm1 Γ A; var1
= λ x Tm1 var1 lam app tt pair fst snd left right case zero suc rec →
var1 _ _ x
lam1 : ∀{Γ A B} → Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B); lam1
= λ t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec →
lam1 _ _ _ (t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec)
app1 : ∀{Γ A B} → Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B; app1
= λ t u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec →
app1 _ _ _ (t Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
tt1 : ∀{Γ} → Tm1 Γ top1; tt1
= λ Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec → tt1 _
pair1 : ∀{Γ A B} → Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B); pair1
= λ t u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec →
pair1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
fst1 : ∀{Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ A; fst1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec →
fst1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec)
snd1 : ∀{Γ A B} → Tm1 Γ (prod1 A B) → Tm1 Γ B; snd1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec →
snd1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec)
left1 : ∀{Γ A B} → Tm1 Γ A → Tm1 Γ (sum1 A B); left1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec →
left1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec)
right1 : ∀{Γ A B} → Tm1 Γ B → Tm1 Γ (sum1 A B); right1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec →
right1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec)
case1 : ∀{Γ A B C} → Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C; case1
= λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec →
case1 _ _ _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
zero1 : ∀{Γ} → Tm1 Γ nat1; zero1
= λ Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec → zero1 _
suc1 : ∀{Γ} → Tm1 Γ nat1 → Tm1 Γ nat1; suc1
= λ t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec →
suc1 _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec)
rec1 : ∀{Γ A} → Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A; rec1
= λ t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1 →
rec1 _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
v01 : ∀{Γ A} → Tm1 (snoc1 Γ A) A; v01
= var1 vz1
v11 : ∀{Γ A B} → Tm1 (snoc1 (snoc1 Γ A) B) A; v11
= var1 (vs1 vz1)
v21 : ∀{Γ A B C} → Tm1 (snoc1 (snoc1 (snoc1 Γ A) B) C) A; v21
= var1 (vs1 (vs1 vz1))
v31 : ∀{Γ A B C D} → Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Γ A) B) C) D) A; v31
= var1 (vs1 (vs1 (vs1 vz1)))
tbool1 : Ty1; tbool1
= sum1 top1 top1
true1 : ∀{Γ} → Tm1 Γ tbool1; true1
= left1 tt1
tfalse1 : ∀{Γ} → Tm1 Γ tbool1; tfalse1
= right1 tt1
ifthenelse1 : ∀{Γ A} → Tm1 Γ (arr1 tbool1 (arr1 A (arr1 A A))); ifthenelse1
= lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11))))
times41 : ∀{Γ A} → Tm1 Γ (arr1 (arr1 A A) (arr1 A A)); times41
= lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01)))))
add1 : ∀{Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1)); add1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))
(lam1 v01))
mul1 : ∀{Γ} → Tm1 Γ (arr1 nat1 (arr1 nat1 nat1)); mul1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))
(lam1 zero1))
fact1 : ∀{Γ} → Tm1 Γ (arr1 nat1 nat1); fact1
= lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))
(suc1 zero1))
{-# OPTIONS --type-in-type #-}
Ty2 : Set
Ty2 =
(Ty2 : Set)
(nat top bot : Ty2)
(arr prod sum : Ty2 → Ty2 → Ty2)
→ Ty2
nat2 : Ty2; nat2 = λ _ nat2 _ _ _ _ _ → nat2
top2 : Ty2; top2 = λ _ _ top2 _ _ _ _ → top2
bot2 : Ty2; bot2 = λ _ _ _ bot2 _ _ _ → bot2
arr2 : Ty2 → Ty2 → Ty2; arr2
= λ A B Ty2 nat2 top2 bot2 arr2 prod sum →
arr2 (A Ty2 nat2 top2 bot2 arr2 prod sum) (B Ty2 nat2 top2 bot2 arr2 prod sum)
prod2 : Ty2 → Ty2 → Ty2; prod2
= λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum →
prod2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum) (B Ty2 nat2 top2 bot2 arr2 prod2 sum)
sum2 : Ty2 → Ty2 → Ty2; sum2
= λ A B Ty2 nat2 top2 bot2 arr2 prod2 sum2 →
sum2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum2) (B Ty2 nat2 top2 bot2 arr2 prod2 sum2)
Con2 : Set; Con2
= (Con2 : Set)
(nil : Con2)
(snoc : Con2 → Ty2 → Con2)
→ Con2
nil2 : Con2; nil2
= λ Con2 nil2 snoc → nil2
snoc2 : Con2 → Ty2 → Con2; snoc2
= λ Γ A Con2 nil2 snoc2 → snoc2 (Γ Con2 nil2 snoc2) A
Var2 : Con2 → Ty2 → Set; Var2
= λ Γ A →
(Var2 : Con2 → Ty2 → Set)
(vz : ∀ Γ A → Var2 (snoc2 Γ A) A)
(vs : ∀ Γ B A → Var2 Γ A → Var2 (snoc2 Γ B) A)
→ Var2 Γ A
vz2 : ∀{Γ A} → Var2 (snoc2 Γ A) A; vz2
= λ Var2 vz2 vs → vz2 _ _
vs2 : ∀{Γ B A} → Var2 Γ A → Var2 (snoc2 Γ B) A; vs2
= λ x Var2 vz2 vs2 → vs2 _ _ _ (x Var2 vz2 vs2)
Tm2 : Con2 → Ty2 → Set; Tm2
= λ Γ A →
(Tm2 : Con2 → Ty2 → Set)
(var : ∀ Γ A → Var2 Γ A → Tm2 Γ A)
(lam : ∀ Γ A B → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B))
(app : ∀ Γ A B → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B)
(tt : ∀ Γ → Tm2 Γ top2)
(pair : ∀ Γ A B → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B))
(fst : ∀ Γ A B → Tm2 Γ (prod2 A B) → Tm2 Γ A)
(snd : ∀ Γ A B → Tm2 Γ (prod2 A B) → Tm2 Γ B)
(left : ∀ Γ A B → Tm2 Γ A → Tm2 Γ (sum2 A B))
(right : ∀ Γ A B → Tm2 Γ B → Tm2 Γ (sum2 A B))
(case : ∀ Γ A B C → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C)
(zero : ∀ Γ → Tm2 Γ nat2)
(suc : ∀ Γ → Tm2 Γ nat2 → Tm2 Γ nat2)
(rec : ∀ Γ A → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A)
→ Tm2 Γ A
var2 : ∀{Γ A} → Var2 Γ A → Tm2 Γ A; var2
= λ x Tm2 var2 lam app tt pair fst snd left right case zero suc rec →
var2 _ _ x
lam2 : ∀{Γ A B} → Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B); lam2
= λ t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec →
lam2 _ _ _ (t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec)
app2 : ∀{Γ A B} → Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B; app2
= λ t u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec →
app2 _ _ _ (t Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
tt2 : ∀{Γ} → Tm2 Γ top2; tt2
= λ Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec → tt2 _
pair2 : ∀{Γ A B} → Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B); pair2
= λ t u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec →
pair2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
fst2 : ∀{Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ A; fst2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec →
fst2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec)
snd2 : ∀{Γ A B} → Tm2 Γ (prod2 A B) → Tm2 Γ B; snd2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec →
snd2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec)
left2 : ∀{Γ A B} → Tm2 Γ A → Tm2 Γ (sum2 A B); left2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec →
left2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec)
right2 : ∀{Γ A B} → Tm2 Γ B → Tm2 Γ (sum2 A B); right2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec →
right2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec)
case2 : ∀{Γ A B C} → Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C; case2
= λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec →
case2 _ _ _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
zero2 : ∀{Γ} → Tm2 Γ nat2; zero2
= λ Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec → zero2 _
suc2 : ∀{Γ} → Tm2 Γ nat2 → Tm2 Γ nat2; suc2
= λ t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec →
suc2 _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec)
rec2 : ∀{Γ A} → Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A; rec2
= λ t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2 →
rec2 _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
v02 : ∀{Γ A} → Tm2 (snoc2 Γ A) A; v02
= var2 vz2
v12 : ∀{Γ A B} → Tm2 (snoc2 (snoc2 Γ A) B) A; v12
= var2 (vs2 vz2)
v22 : ∀{Γ A B C} → Tm2 (snoc2 (snoc2 (snoc2 Γ A) B) C) A; v22
= var2 (vs2 (vs2 vz2))
v32 : ∀{Γ A B C D} → Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Γ A) B) C) D) A; v32
= var2 (vs2 (vs2 (vs2 vz2)))
tbool2 : Ty2; tbool2
= sum2 top2 top2
true2 : ∀{Γ} → Tm2 Γ tbool2; true2
= left2 tt2
tfalse2 : ∀{Γ} → Tm2 Γ tbool2; tfalse2
= right2 tt2
ifthenelse2 : ∀{Γ A} → Tm2 Γ (arr2 tbool2 (arr2 A (arr2 A A))); ifthenelse2
= lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12))))
times42 : ∀{Γ A} → Tm2 Γ (arr2 (arr2 A A) (arr2 A A)); times42
= lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02)))))
add2 : ∀{Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2)); add2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))
(lam2 v02))
mul2 : ∀{Γ} → Tm2 Γ (arr2 nat2 (arr2 nat2 nat2)); mul2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))
(lam2 zero2))
fact2 : ∀{Γ} → Tm2 Γ (arr2 nat2 nat2); fact2
= lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))
(suc2 zero2))
{-# OPTIONS --type-in-type #-}
Ty3 : Set
Ty3 =
(Ty3 : Set)
(nat top bot : Ty3)
(arr prod sum : Ty3 → Ty3 → Ty3)
→ Ty3
nat3 : Ty3; nat3 = λ _ nat3 _ _ _ _ _ → nat3
top3 : Ty3; top3 = λ _ _ top3 _ _ _ _ → top3
bot3 : Ty3; bot3 = λ _ _ _ bot3 _ _ _ → bot3
arr3 : Ty3 → Ty3 → Ty3; arr3
= λ A B Ty3 nat3 top3 bot3 arr3 prod sum →
arr3 (A Ty3 nat3 top3 bot3 arr3 prod sum) (B Ty3 nat3 top3 bot3 arr3 prod sum)
prod3 : Ty3 → Ty3 → Ty3; prod3
= λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum →
prod3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum) (B Ty3 nat3 top3 bot3 arr3 prod3 sum)
sum3 : Ty3 → Ty3 → Ty3; sum3
= λ A B Ty3 nat3 top3 bot3 arr3 prod3 sum3 →
sum3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum3) (B Ty3 nat3 top3 bot3 arr3 prod3 sum3)
Con3 : Set; Con3
= (Con3 : Set)
(nil : Con3)
(snoc : Con3 → Ty3 → Con3)
→ Con3
nil3 : Con3; nil3
= λ Con3 nil3 snoc → nil3
snoc3 : Con3 → Ty3 → Con3; snoc3
= λ Γ A Con3 nil3 snoc3 → snoc3 (Γ Con3 nil3 snoc3) A
Var3 : Con3 → Ty3 → Set; Var3
= λ Γ A →
(Var3 : Con3 → Ty3 → Set)
(vz : ∀ Γ A → Var3 (snoc3 Γ A) A)
(vs : ∀ Γ B A → Var3 Γ A → Var3 (snoc3 Γ B) A)
→ Var3 Γ A
vz3 : ∀{Γ A} → Var3 (snoc3 Γ A) A; vz3
= λ Var3 vz3 vs → vz3 _ _
vs3 : ∀{Γ B A} → Var3 Γ A → Var3 (snoc3 Γ B) A; vs3
= λ x Var3 vz3 vs3 → vs3 _ _ _ (x Var3 vz3 vs3)
Tm3 : Con3 → Ty3 → Set; Tm3
= λ Γ A →
(Tm3 : Con3 → Ty3 → Set)
(var : ∀ Γ A → Var3 Γ A → Tm3 Γ A)
(lam : ∀ Γ A B → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B))
(app : ∀ Γ A B → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B)
(tt : ∀ Γ → Tm3 Γ top3)
(pair : ∀ Γ A B → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B))
(fst : ∀ Γ A B → Tm3 Γ (prod3 A B) → Tm3 Γ A)
(snd : ∀ Γ A B → Tm3 Γ (prod3 A B) → Tm3 Γ B)
(left : ∀ Γ A B → Tm3 Γ A → Tm3 Γ (sum3 A B))
(right : ∀ Γ A B → Tm3 Γ B → Tm3 Γ (sum3 A B))
(case : ∀ Γ A B C → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C)
(zero : ∀ Γ → Tm3 Γ nat3)
(suc : ∀ Γ → Tm3 Γ nat3 → Tm3 Γ nat3)
(rec : ∀ Γ A → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A)
→ Tm3 Γ A
var3 : ∀{Γ A} → Var3 Γ A → Tm3 Γ A; var3
= λ x Tm3 var3 lam app tt pair fst snd left right case zero suc rec →
var3 _ _ x
lam3 : ∀{Γ A B} → Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B); lam3
= λ t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec →
lam3 _ _ _ (t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec)
app3 : ∀{Γ A B} → Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B; app3
= λ t u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec →
app3 _ _ _ (t Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
tt3 : ∀{Γ} → Tm3 Γ top3; tt3
= λ Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec → tt3 _
pair3 : ∀{Γ A B} → Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B); pair3
= λ t u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec →
pair3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
fst3 : ∀{Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ A; fst3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec →
fst3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec)
snd3 : ∀{Γ A B} → Tm3 Γ (prod3 A B) → Tm3 Γ B; snd3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec →
snd3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec)
left3 : ∀{Γ A B} → Tm3 Γ A → Tm3 Γ (sum3 A B); left3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec →
left3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec)
right3 : ∀{Γ A B} → Tm3 Γ B → Tm3 Γ (sum3 A B); right3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec →
right3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec)
case3 : ∀{Γ A B C} → Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C; case3
= λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec →
case3 _ _ _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
zero3 : ∀{Γ} → Tm3 Γ nat3; zero3
= λ Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec → zero3 _
suc3 : ∀{Γ} → Tm3 Γ nat3 → Tm3 Γ nat3; suc3
= λ t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec →
suc3 _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec)
rec3 : ∀{Γ A} → Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A; rec3
= λ t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3 →
rec3 _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
v03 : ∀{Γ A} → Tm3 (snoc3 Γ A) A; v03
= var3 vz3
v13 : ∀{Γ A B} → Tm3 (snoc3 (snoc3 Γ A) B) A; v13
= var3 (vs3 vz3)
v23 : ∀{Γ A B C} → Tm3 (snoc3 (snoc3 (snoc3 Γ A) B) C) A; v23
= var3 (vs3 (vs3 vz3))
v33 : ∀{Γ A B C D} → Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Γ A) B) C) D) A; v33
= var3 (vs3 (vs3 (vs3 vz3)))
tbool3 : Ty3; tbool3
= sum3 top3 top3
true3 : ∀{Γ} → Tm3 Γ tbool3; true3
= left3 tt3
tfalse3 : ∀{Γ} → Tm3 Γ tbool3; tfalse3
= right3 tt3
ifthenelse3 : ∀{Γ A} → Tm3 Γ (arr3 tbool3 (arr3 A (arr3 A A))); ifthenelse3
= lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13))))
times43 : ∀{Γ A} → Tm3 Γ (arr3 (arr3 A A) (arr3 A A)); times43
= lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03)))))
add3 : ∀{Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3)); add3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))
(lam3 v03))
mul3 : ∀{Γ} → Tm3 Γ (arr3 nat3 (arr3 nat3 nat3)); mul3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))
(lam3 zero3))
fact3 : ∀{Γ} → Tm3 Γ (arr3 nat3 nat3); fact3
= lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))
(suc3 zero3))
{-# OPTIONS --type-in-type #-}
Ty4 : Set
Ty4 =
(Ty4 : Set)
(nat top bot : Ty4)
(arr prod sum : Ty4 → Ty4 → Ty4)
→ Ty4
nat4 : Ty4; nat4 = λ _ nat4 _ _ _ _ _ → nat4
top4 : Ty4; top4 = λ _ _ top4 _ _ _ _ → top4
bot4 : Ty4; bot4 = λ _ _ _ bot4 _ _ _ → bot4
arr4 : Ty4 → Ty4 → Ty4; arr4
= λ A B Ty4 nat4 top4 bot4 arr4 prod sum →
arr4 (A Ty4 nat4 top4 bot4 arr4 prod sum) (B Ty4 nat4 top4 bot4 arr4 prod sum)
prod4 : Ty4 → Ty4 → Ty4; prod4
= λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum →
prod4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum) (B Ty4 nat4 top4 bot4 arr4 prod4 sum)
sum4 : Ty4 → Ty4 → Ty4; sum4
= λ A B Ty4 nat4 top4 bot4 arr4 prod4 sum4 →
sum4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum4) (B Ty4 nat4 top4 bot4 arr4 prod4 sum4)
Con4 : Set; Con4
= (Con4 : Set)
(nil : Con4)
(snoc : Con4 → Ty4 → Con4)
→ Con4
nil4 : Con4; nil4
= λ Con4 nil4 snoc → nil4
snoc4 : Con4 → Ty4 → Con4; snoc4
= λ Γ A Con4 nil4 snoc4 → snoc4 (Γ Con4 nil4 snoc4) A
Var4 : Con4 → Ty4 → Set; Var4
= λ Γ A →
(Var4 : Con4 → Ty4 → Set)
(vz : ∀ Γ A → Var4 (snoc4 Γ A) A)
(vs : ∀ Γ B A → Var4 Γ A → Var4 (snoc4 Γ B) A)
→ Var4 Γ A
vz4 : ∀{Γ A} → Var4 (snoc4 Γ A) A; vz4
= λ Var4 vz4 vs → vz4 _ _
vs4 : ∀{Γ B A} → Var4 Γ A → Var4 (snoc4 Γ B) A; vs4
= λ x Var4 vz4 vs4 → vs4 _ _ _ (x Var4 vz4 vs4)
Tm4 : Con4 → Ty4 → Set; Tm4
= λ Γ A →
(Tm4 : Con4 → Ty4 → Set)
(var : ∀ Γ A → Var4 Γ A → Tm4 Γ A)
(lam : ∀ Γ A B → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B))
(app : ∀ Γ A B → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B)
(tt : ∀ Γ → Tm4 Γ top4)
(pair : ∀ Γ A B → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B))
(fst : ∀ Γ A B → Tm4 Γ (prod4 A B) → Tm4 Γ A)
(snd : ∀ Γ A B → Tm4 Γ (prod4 A B) → Tm4 Γ B)
(left : ∀ Γ A B → Tm4 Γ A → Tm4 Γ (sum4 A B))
(right : ∀ Γ A B → Tm4 Γ B → Tm4 Γ (sum4 A B))
(case : ∀ Γ A B C → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C)
(zero : ∀ Γ → Tm4 Γ nat4)
(suc : ∀ Γ → Tm4 Γ nat4 → Tm4 Γ nat4)
(rec : ∀ Γ A → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A)
→ Tm4 Γ A
var4 : ∀{Γ A} → Var4 Γ A → Tm4 Γ A; var4
= λ x Tm4 var4 lam app tt pair fst snd left right case zero suc rec →
var4 _ _ x
lam4 : ∀{Γ A B} → Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B); lam4
= λ t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec →
lam4 _ _ _ (t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec)
app4 : ∀{Γ A B} → Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B; app4
= λ t u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec →
app4 _ _ _ (t Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
tt4 : ∀{Γ} → Tm4 Γ top4; tt4
= λ Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec → tt4 _
pair4 : ∀{Γ A B} → Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B); pair4
= λ t u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec →
pair4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
fst4 : ∀{Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ A; fst4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec →
fst4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec)
snd4 : ∀{Γ A B} → Tm4 Γ (prod4 A B) → Tm4 Γ B; snd4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec →
snd4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec)
left4 : ∀{Γ A B} → Tm4 Γ A → Tm4 Γ (sum4 A B); left4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec →
left4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec)
right4 : ∀{Γ A B} → Tm4 Γ B → Tm4 Γ (sum4 A B); right4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec →
right4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec)
case4 : ∀{Γ A B C} → Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C; case4
= λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec →
case4 _ _ _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
zero4 : ∀{Γ} → Tm4 Γ nat4; zero4
= λ Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec → zero4 _
suc4 : ∀{Γ} → Tm4 Γ nat4 → Tm4 Γ nat4; suc4
= λ t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec →
suc4 _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec)
rec4 : ∀{Γ A} → Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A; rec4
= λ t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4 →
rec4 _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
v04 : ∀{Γ A} → Tm4 (snoc4 Γ A) A; v04
= var4 vz4
v14 : ∀{Γ A B} → Tm4 (snoc4 (snoc4 Γ A) B) A; v14
= var4 (vs4 vz4)
v24 : ∀{Γ A B C} → Tm4 (snoc4 (snoc4 (snoc4 Γ A) B) C) A; v24
= var4 (vs4 (vs4 vz4))
v34 : ∀{Γ A B C D} → Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Γ A) B) C) D) A; v34
= var4 (vs4 (vs4 (vs4 vz4)))
tbool4 : Ty4; tbool4
= sum4 top4 top4
true4 : ∀{Γ} → Tm4 Γ tbool4; true4
= left4 tt4
tfalse4 : ∀{Γ} → Tm4 Γ tbool4; tfalse4
= right4 tt4
ifthenelse4 : ∀{Γ A} → Tm4 Γ (arr4 tbool4 (arr4 A (arr4 A A))); ifthenelse4
= lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14))))
times44 : ∀{Γ A} → Tm4 Γ (arr4 (arr4 A A) (arr4 A A)); times44
= lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04)))))
add4 : ∀{Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4)); add4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))
(lam4 v04))
mul4 : ∀{Γ} → Tm4 Γ (arr4 nat4 (arr4 nat4 nat4)); mul4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))
(lam4 zero4))
fact4 : ∀{Γ} → Tm4 Γ (arr4 nat4 nat4); fact4
= lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))
(suc4 zero4))
{-# OPTIONS --type-in-type #-}
Ty5 : Set
Ty5 =
(Ty5 : Set)
(nat top bot : Ty5)
(arr prod sum : Ty5 → Ty5 → Ty5)
→ Ty5
nat5 : Ty5; nat5 = λ _ nat5 _ _ _ _ _ → nat5
top5 : Ty5; top5 = λ _ _ top5 _ _ _ _ → top5
bot5 : Ty5; bot5 = λ _ _ _ bot5 _ _ _ → bot5
arr5 : Ty5 → Ty5 → Ty5; arr5
= λ A B Ty5 nat5 top5 bot5 arr5 prod sum →
arr5 (A Ty5 nat5 top5 bot5 arr5 prod sum) (B Ty5 nat5 top5 bot5 arr5 prod sum)
prod5 : Ty5 → Ty5 → Ty5; prod5
= λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum →
prod5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum) (B Ty5 nat5 top5 bot5 arr5 prod5 sum)
sum5 : Ty5 → Ty5 → Ty5; sum5
= λ A B Ty5 nat5 top5 bot5 arr5 prod5 sum5 →
sum5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum5) (B Ty5 nat5 top5 bot5 arr5 prod5 sum5)
Con5 : Set; Con5
= (Con5 : Set)
(nil : Con5)
(snoc : Con5 → Ty5 → Con5)
→ Con5
nil5 : Con5; nil5
= λ Con5 nil5 snoc → nil5
snoc5 : Con5 → Ty5 → Con5; snoc5
= λ Γ A Con5 nil5 snoc5 → snoc5 (Γ Con5 nil5 snoc5) A
Var5 : Con5 → Ty5 → Set; Var5
= λ Γ A →
(Var5 : Con5 → Ty5 → Set)
(vz : ∀ Γ A → Var5 (snoc5 Γ A) A)
(vs : ∀ Γ B A → Var5 Γ A → Var5 (snoc5 Γ B) A)
→ Var5 Γ A
vz5 : ∀{Γ A} → Var5 (snoc5 Γ A) A; vz5
= λ Var5 vz5 vs → vz5 _ _
vs5 : ∀{Γ B A} → Var5 Γ A → Var5 (snoc5 Γ B) A; vs5
= λ x Var5 vz5 vs5 → vs5 _ _ _ (x Var5 vz5 vs5)
Tm5 : Con5 → Ty5 → Set; Tm5
= λ Γ A →
(Tm5 : Con5 → Ty5 → Set)
(var : ∀ Γ A → Var5 Γ A → Tm5 Γ A)
(lam : ∀ Γ A B → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B))
(app : ∀ Γ A B → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B)
(tt : ∀ Γ → Tm5 Γ top5)
(pair : ∀ Γ A B → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B))
(fst : ∀ Γ A B → Tm5 Γ (prod5 A B) → Tm5 Γ A)
(snd : ∀ Γ A B → Tm5 Γ (prod5 A B) → Tm5 Γ B)
(left : ∀ Γ A B → Tm5 Γ A → Tm5 Γ (sum5 A B))
(right : ∀ Γ A B → Tm5 Γ B → Tm5 Γ (sum5 A B))
(case : ∀ Γ A B C → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C)
(zero : ∀ Γ → Tm5 Γ nat5)
(suc : ∀ Γ → Tm5 Γ nat5 → Tm5 Γ nat5)
(rec : ∀ Γ A → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A)
→ Tm5 Γ A
var5 : ∀{Γ A} → Var5 Γ A → Tm5 Γ A; var5
= λ x Tm5 var5 lam app tt pair fst snd left right case zero suc rec →
var5 _ _ x
lam5 : ∀{Γ A B} → Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B); lam5
= λ t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec →
lam5 _ _ _ (t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec)
app5 : ∀{Γ A B} → Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B; app5
= λ t u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec →
app5 _ _ _ (t Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
tt5 : ∀{Γ} → Tm5 Γ top5; tt5
= λ Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec → tt5 _
pair5 : ∀{Γ A B} → Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B); pair5
= λ t u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec →
pair5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
fst5 : ∀{Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ A; fst5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec →
fst5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec)
snd5 : ∀{Γ A B} → Tm5 Γ (prod5 A B) → Tm5 Γ B; snd5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec →
snd5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec)
left5 : ∀{Γ A B} → Tm5 Γ A → Tm5 Γ (sum5 A B); left5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec →
left5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec)
right5 : ∀{Γ A B} → Tm5 Γ B → Tm5 Γ (sum5 A B); right5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec →
right5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec)
case5 : ∀{Γ A B C} → Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C; case5
= λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec →
case5 _ _ _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
zero5 : ∀{Γ} → Tm5 Γ nat5; zero5
= λ Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec → zero5 _
suc5 : ∀{Γ} → Tm5 Γ nat5 → Tm5 Γ nat5; suc5
= λ t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec →
suc5 _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec)
rec5 : ∀{Γ A} → Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A; rec5
= λ t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5 →
rec5 _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
v05 : ∀{Γ A} → Tm5 (snoc5 Γ A) A; v05
= var5 vz5
v15 : ∀{Γ A B} → Tm5 (snoc5 (snoc5 Γ A) B) A; v15
= var5 (vs5 vz5)
v25 : ∀{Γ A B C} → Tm5 (snoc5 (snoc5 (snoc5 Γ A) B) C) A; v25
= var5 (vs5 (vs5 vz5))
v35 : ∀{Γ A B C D} → Tm5 (snoc5 (snoc5 (snoc5 (snoc5 Γ A) B) C) D) A; v35
= var5 (vs5 (vs5 (vs5 vz5)))
tbool5 : Ty5; tbool5
= sum5 top5 top5
true5 : ∀{Γ} → Tm5 Γ tbool5; true5
= left5 tt5
tfalse5 : ∀{Γ} → Tm5 Γ tbool5; tfalse5
= right5 tt5
ifthenelse5 : ∀{Γ A} → Tm5 Γ (arr5 tbool5 (arr5 A (arr5 A A))); ifthenelse5
= lam5 (lam5 (lam5 (case5 v25 (lam5 v25) (lam5 v15))))
times45 : ∀{Γ A} → Tm5 Γ (arr5 (arr5 A A) (arr5 A A)); times45
= lam5 (lam5 (app5 v15 (app5 v15 (app5 v15 (app5 v15 v05)))))
add5 : ∀{Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5)); add5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (suc5 (app5 v15 v05)))))
(lam5 v05))
mul5 : ∀{Γ} → Tm5 Γ (arr5 nat5 (arr5 nat5 nat5)); mul5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (app5 (app5 add5 (app5 v15 v05)) v05))))
(lam5 zero5))
fact5 : ∀{Γ} → Tm5 Γ (arr5 nat5 nat5); fact5
= lam5 (rec5 v05 (lam5 (lam5 (app5 (app5 mul5 (suc5 v15)) v05)))
(suc5 zero5))
{-# OPTIONS --type-in-type #-}
Ty6 : Set
Ty6 =
(Ty6 : Set)
(nat top bot : Ty6)
(arr prod sum : Ty6 → Ty6 → Ty6)
→ Ty6
nat6 : Ty6; nat6 = λ _ nat6 _ _ _ _ _ → nat6
top6 : Ty6; top6 = λ _ _ top6 _ _ _ _ → top6
bot6 : Ty6; bot6 = λ _ _ _ bot6 _ _ _ → bot6
arr6 : Ty6 → Ty6 → Ty6; arr6
= λ A B Ty6 nat6 top6 bot6 arr6 prod sum →
arr6 (A Ty6 nat6 top6 bot6 arr6 prod sum) (B Ty6 nat6 top6 bot6 arr6 prod sum)
prod6 : Ty6 → Ty6 → Ty6; prod6
= λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum →
prod6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum) (B Ty6 nat6 top6 bot6 arr6 prod6 sum)
sum6 : Ty6 → Ty6 → Ty6; sum6
= λ A B Ty6 nat6 top6 bot6 arr6 prod6 sum6 →
sum6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum6) (B Ty6 nat6 top6 bot6 arr6 prod6 sum6)
Con6 : Set; Con6
= (Con6 : Set)
(nil : Con6)
(snoc : Con6 → Ty6 → Con6)
→ Con6
nil6 : Con6; nil6
= λ Con6 nil6 snoc → nil6
snoc6 : Con6 → Ty6 → Con6; snoc6
= λ Γ A Con6 nil6 snoc6 → snoc6 (Γ Con6 nil6 snoc6) A
Var6 : Con6 → Ty6 → Set; Var6
= λ Γ A →
(Var6 : Con6 → Ty6 → Set)
(vz : ∀ Γ A → Var6 (snoc6 Γ A) A)
(vs : ∀ Γ B A → Var6 Γ A → Var6 (snoc6 Γ B) A)
→ Var6 Γ A
vz6 : ∀{Γ A} → Var6 (snoc6 Γ A) A; vz6
= λ Var6 vz6 vs → vz6 _ _
vs6 : ∀{Γ B A} → Var6 Γ A → Var6 (snoc6 Γ B) A; vs6
= λ x Var6 vz6 vs6 → vs6 _ _ _ (x Var6 vz6 vs6)
Tm6 : Con6 → Ty6 → Set; Tm6
= λ Γ A →
(Tm6 : Con6 → Ty6 → Set)
(var : ∀ Γ A → Var6 Γ A → Tm6 Γ A)
(lam : ∀ Γ A B → Tm6 (snoc6 Γ A) B → Tm6 Γ (arr6 A B))
(app : ∀ Γ A B → Tm6 Γ (arr6 A B) → Tm6 Γ A → Tm6 Γ B)
(tt : ∀ Γ → Tm6 Γ top6)
(pair : ∀ Γ A B → Tm6 Γ A → Tm6 Γ B → Tm6 Γ (prod6 A B))
(fst : ∀ Γ A B → Tm6 Γ (prod6 A B) → Tm6 Γ A)
(snd : ∀ Γ A B → Tm6 Γ (prod6 A B) → Tm6 Γ B)
(left : ∀ Γ A B → Tm6 Γ A → Tm6 Γ (sum6 A B))
(right : ∀ Γ A B → Tm6 Γ B → Tm6 Γ (sum6 A B))
(case : ∀ Γ A B C → Tm6 Γ (sum6 A B) → Tm6 Γ (arr6 A C) → Tm6 Γ (arr6 B C) → Tm6 Γ C)
(zero : ∀ Γ → Tm6 Γ nat6)
(suc : ∀ Γ → Tm6 Γ nat6 → Tm6 Γ nat6)
(rec : ∀ Γ A → Tm6 Γ nat6 → Tm6 Γ (arr6 nat6 (arr6 A A)) → Tm6 Γ A → Tm6 Γ A)
→ Tm6 Γ A
var6 : ∀{Γ A} → Var6 Γ A → Tm6 Γ A; var6
= λ x Tm6 var6 lam app tt pair fst snd left right case zero suc rec →
var6 _ _ x
lam6 : ∀{Γ A B} → Tm6 (snoc6 Γ A) B → Tm6 Γ (arr6 A B); lam6
= λ t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec →
lam6 _ _ _ (t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec)
app6 : ∀{Γ A B} → Tm6 Γ (arr6 A B) → Tm6 Γ A → Tm6 Γ B; app6
= λ t u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec →
app6 _ _ _ (t Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
tt6 : ∀{Γ} → Tm6 Γ top6; tt6
= λ Tm6 var6 lam6 app6 tt6 pair fst snd left right case zero suc rec → tt6 _
pair6 : ∀{Γ A B} → Tm6 Γ A → Tm6 Γ B → Tm6 Γ (prod6 A B); pair6
= λ t u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec →
pair6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
fst6 : ∀{Γ A B} → Tm6 Γ (prod6 A B) → Tm6 Γ A; fst6
= λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec →
fst6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec)
snd6 : ∀{Γ A B} → Tm6 Γ (prod6 A B) → Tm6 Γ B; snd6
= λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec →
snd6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec)
left6 : ∀{Γ A B} → Tm6 Γ A → Tm6 Γ (sum6 A B); left6
= λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec →
left6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec)
right6 : ∀{Γ A B} → Tm6 Γ B → Tm6 Γ (sum6 A B); right6
= λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec →
right6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec)
case6 : ∀{Γ A B C} → Tm6 Γ (sum6 A B) → Tm6 Γ (arr6 A C) → Tm6 Γ (arr6 B C) → Tm6 Γ C; case6
= λ t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec →
case6 _ _ _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
zero6 : ∀{Γ} → Tm6 Γ nat6; zero6
= λ Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc rec → zero6 _
suc6 : ∀{Γ} → Tm6 Γ nat6 → Tm6 Γ nat6; suc6
= λ t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec →
suc6 _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec)
rec6 : ∀{Γ A} → Tm6 Γ nat6 → Tm6 Γ (arr6 nat6 (arr6 A A)) → Tm6 Γ A → Tm6 Γ A; rec6
= λ t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6 →
rec6 _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
v06 : ∀{Γ A} → Tm6 (snoc6 Γ A) A; v06
= var6 vz6
v16 : ∀{Γ A B} → Tm6 (snoc6 (snoc6 Γ A) B) A; v16
= var6 (vs6 vz6)
v26 : ∀{Γ A B C} → Tm6 (snoc6 (snoc6 (snoc6 Γ A) B) C) A; v26
= var6 (vs6 (vs6 vz6))
v36 : ∀{Γ A B C D} → Tm6 (snoc6 (snoc6 (snoc6 (snoc6 Γ A) B) C) D) A; v36
= var6 (vs6 (vs6 (vs6 vz6)))
tbool6 : Ty6; tbool6
= sum6 top6 top6
true6 : ∀{Γ} → Tm6 Γ tbool6; true6
= left6 tt6
tfalse6 : ∀{Γ} → Tm6 Γ tbool6; tfalse6
= right6 tt6
ifthenelse6 : ∀{Γ A} → Tm6 Γ (arr6 tbool6 (arr6 A (arr6 A A))); ifthenelse6
= lam6 (lam6 (lam6 (case6 v26 (lam6 v26) (lam6 v16))))
times46 : ∀{Γ A} → Tm6 Γ (arr6 (arr6 A A) (arr6 A A)); times46
= lam6 (lam6 (app6 v16 (app6 v16 (app6 v16 (app6 v16 v06)))))
add6 : ∀{Γ} → Tm6 Γ (arr6 nat6 (arr6 nat6 nat6)); add6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (suc6 (app6 v16 v06)))))
(lam6 v06))
mul6 : ∀{Γ} → Tm6 Γ (arr6 nat6 (arr6 nat6 nat6)); mul6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (app6 (app6 add6 (app6 v16 v06)) v06))))
(lam6 zero6))
fact6 : ∀{Γ} → Tm6 Γ (arr6 nat6 nat6); fact6
= lam6 (rec6 v06 (lam6 (lam6 (app6 (app6 mul6 (suc6 v16)) v06)))
(suc6 zero6))
{-# OPTIONS --type-in-type #-}
Ty7 : Set
Ty7 =
(Ty7 : Set)
(nat top bot : Ty7)
(arr prod sum : Ty7 → Ty7 → Ty7)
→ Ty7
nat7 : Ty7; nat7 = λ _ nat7 _ _ _ _ _ → nat7
top7 : Ty7; top7 = λ _ _ top7 _ _ _ _ → top7
bot7 : Ty7; bot7 = λ _ _ _ bot7 _ _ _ → bot7
arr7 : Ty7 → Ty7 → Ty7; arr7
= λ A B Ty7 nat7 top7 bot7 arr7 prod sum →
arr7 (A Ty7 nat7 top7 bot7 arr7 prod sum) (B Ty7 nat7 top7 bot7 arr7 prod sum)
prod7 : Ty7 → Ty7 → Ty7; prod7
= λ A B Ty7 nat7 top7 bot7 arr7 prod7 sum →
prod7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum) (B Ty7 nat7 top7 bot7 arr7 prod7 sum)
sum7 : Ty7 → Ty7 → Ty7; sum7
= λ A B Ty7 nat7 top7 bot7 arr7 prod7 sum7 →
sum7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum7) (B Ty7 nat7 top7 bot7 arr7 prod7 sum7)
Con7 : Set; Con7
= (Con7 : Set)
(nil : Con7)
(snoc : Con7 → Ty7 → Con7)
→ Con7
nil7 : Con7; nil7
= λ Con7 nil7 snoc → nil7
snoc7 : Con7 → Ty7 → Con7; snoc7
= λ Γ A Con7 nil7 snoc7 → snoc7 (Γ Con7 nil7 snoc7) A
Var7 : Con7 → Ty7 → Set; Var7
= λ Γ A →
(Var7 : Con7 → Ty7 → Set)
(vz : ∀ Γ A → Var7 (snoc7 Γ A) A)
(vs : ∀ Γ B A → Var7 Γ A → Var7 (snoc7 Γ B) A)
→ Var7 Γ A
vz7 : ∀{Γ A} → Var7 (snoc7 Γ A) A; vz7
= λ Var7 vz7 vs → vz7 _ _
vs7 : ∀{Γ B A} → Var7 Γ A → Var7 (snoc7 Γ B) A; vs7
= λ x Var7 vz7 vs7 → vs7 _ _ _ (x Var7 vz7 vs7)
Tm7 : Con7 → Ty7 → Set; Tm7
= λ Γ A →
(Tm7 : Con7 → Ty7 → Set)
(var : ∀ Γ A → Var7 Γ A → Tm7 Γ A)
(lam : ∀ Γ A B → Tm7 (snoc7 Γ A) B → Tm7 Γ (arr7 A B))
(app : ∀ Γ A B → Tm7 Γ (arr7 A B) → Tm7 Γ A → Tm7 Γ B)
(tt : ∀ Γ → Tm7 Γ top7)
(pair : ∀ Γ A B → Tm7 Γ A → Tm7 Γ B → Tm7 Γ (prod7 A B))
(fst : ∀ Γ A B → Tm7 Γ (prod7 A B) → Tm7 Γ A)
(snd : ∀ Γ A B → Tm7 Γ (prod7 A B) → Tm7 Γ B)
(left : ∀ Γ A B → Tm7 Γ A → Tm7 Γ (sum7 A B))
(right : ∀ Γ A B → Tm7 Γ B → Tm7 Γ (sum7 A B))
(case : ∀ Γ A B C → Tm7 Γ (sum7 A B) → Tm7 Γ (arr7 A C) → Tm7 Γ (arr7 B C) → Tm7 Γ C)
(zero : ∀ Γ → Tm7 Γ nat7)
(suc : ∀ Γ → Tm7 Γ nat7 → Tm7 Γ nat7)
(rec : ∀ Γ A → Tm7 Γ nat7 → Tm7 Γ (arr7 nat7 (arr7 A A)) → Tm7 Γ A → Tm7 Γ A)
→ Tm7 Γ A
var7 : ∀{Γ A} → Var7 Γ A → Tm7 Γ A; var7
= λ x Tm7 var7 lam app tt pair fst snd left right case zero suc rec →
var7 _ _ x
lam7 : ∀{Γ A B} → Tm7 (snoc7 Γ A) B → Tm7 Γ (arr7 A B); lam7
= λ t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec →
lam7 _ _ _ (t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec)
app7 : ∀{Γ A B} → Tm7 Γ (arr7 A B) → Tm7 Γ A → Tm7 Γ B; app7
= λ t u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec →
app7 _ _ _ (t Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
tt7 : ∀{Γ} → Tm7 Γ top7; tt7
= λ Tm7 var7 lam7 app7 tt7 pair fst snd left right case zero suc rec → tt7 _
pair7 : ∀{Γ A B} → Tm7 Γ A → Tm7 Γ B → Tm7 Γ (prod7 A B); pair7
= λ t u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec →
pair7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
fst7 : ∀{Γ A B} → Tm7 Γ (prod7 A B) → Tm7 Γ A; fst7
= λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec →
fst7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec)
snd7 : ∀{Γ A B} → Tm7 Γ (prod7 A B) → Tm7 Γ B; snd7
= λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec →
snd7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec)
left7 : ∀{Γ A B} → Tm7 Γ A → Tm7 Γ (sum7 A B); left7
= λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec →
left7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec)
right7 : ∀{Γ A B} → Tm7 Γ B → Tm7 Γ (sum7 A B); right7
= λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec →
right7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec)
case7 : ∀{Γ A B C} → Tm7 Γ (sum7 A B) → Tm7 Γ (arr7 A C) → Tm7 Γ (arr7 B C) → Tm7 Γ C; case7
= λ t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec →
case7 _ _ _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
zero7 : ∀{Γ} → Tm7 Γ nat7; zero7
= λ Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc rec → zero7 _
suc7 : ∀{Γ} → Tm7 Γ nat7 → Tm7 Γ nat7; suc7
= λ t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec →
suc7 _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec)
rec7 : ∀{Γ A} → Tm7 Γ nat7 → Tm7 Γ (arr7 nat7 (arr7 A A)) → Tm7 Γ A → Tm7 Γ A; rec7
= λ t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7 →
rec7 _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
v07 : ∀{Γ A} → Tm7 (snoc7 Γ A) A; v07
= var7 vz7
v17 : ∀{Γ A B} → Tm7 (snoc7 (snoc7 Γ A) B) A; v17
= var7 (vs7 vz7)
v27 : ∀{Γ A B C} → Tm7 (snoc7 (snoc7 (snoc7 Γ A) B) C) A; v27
= var7 (vs7 (vs7 vz7))
v37 : ∀{Γ A B C D} → Tm7 (snoc7 (snoc7 (snoc7 (snoc7 Γ A) B) C) D) A; v37
= var7 (vs7 (vs7 (vs7 vz7)))
tbool7 : Ty7; tbool7
= sum7 top7 top7
true7 : ∀{Γ} → Tm7 Γ tbool7; true7
= left7 tt7
tfalse7 : ∀{Γ} → Tm7 Γ tbool7; tfalse7
= right7 tt7
ifthenelse7 : ∀{Γ A} → Tm7 Γ (arr7 tbool7 (arr7 A (arr7 A A))); ifthenelse7
= lam7 (lam7 (lam7 (case7 v27 (lam7 v27) (lam7 v17))))
times47 : ∀{Γ A} → Tm7 Γ (arr7 (arr7 A A) (arr7 A A)); times47
= lam7 (lam7 (app7 v17 (app7 v17 (app7 v17 (app7 v17 v07)))))
add7 : ∀{Γ} → Tm7 Γ (arr7 nat7 (arr7 nat7 nat7)); add7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (suc7 (app7 v17 v07)))))
(lam7 v07))
mul7 : ∀{Γ} → Tm7 Γ (arr7 nat7 (arr7 nat7 nat7)); mul7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (app7 (app7 add7 (app7 v17 v07)) v07))))
(lam7 zero7))
fact7 : ∀{Γ} → Tm7 Γ (arr7 nat7 nat7); fact7
= lam7 (rec7 v07 (lam7 (lam7 (app7 (app7 mul7 (suc7 v17)) v07)))
(suc7 zero7))
{-# OPTIONS --type-in-type #-}
Ty8 : Set
Ty8 =
(Ty8 : Set)
(nat top bot : Ty8)
(arr prod sum : Ty8 → Ty8 → Ty8)
→ Ty8
nat8 : Ty8; nat8 = λ _ nat8 _ _ _ _ _ → nat8
top8 : Ty8; top8 = λ _ _ top8 _ _ _ _ → top8
bot8 : Ty8; bot8 = λ _ _ _ bot8 _ _ _ → bot8
arr8 : Ty8 → Ty8 → Ty8; arr8
= λ A B Ty8 nat8 top8 bot8 arr8 prod sum →
arr8 (A Ty8 nat8 top8 bot8 arr8 prod sum) (B Ty8 nat8 top8 bot8 arr8 prod sum)
prod8 : Ty8 → Ty8 → Ty8; prod8
= λ A B Ty8 nat8 top8 bot8 arr8 prod8 sum →
prod8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum) (B Ty8 nat8 top8 bot8 arr8 prod8 sum)
sum8 : Ty8 → Ty8 → Ty8; sum8
= λ A B Ty8 nat8 top8 bot8 arr8 prod8 sum8 →
sum8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum8) (B Ty8 nat8 top8 bot8 arr8 prod8 sum8)
Con8 : Set; Con8
= (Con8 : Set)
(nil : Con8)
(snoc : Con8 → Ty8 → Con8)
→ Con8
nil8 : Con8; nil8
= λ Con8 nil8 snoc → nil8
snoc8 : Con8 → Ty8 → Con8; snoc8
= λ Γ A Con8 nil8 snoc8 → snoc8 (Γ Con8 nil8 snoc8) A
Var8 : Con8 → Ty8 → Set; Var8
= λ Γ A →
(Var8 : Con8 → Ty8 → Set)
(vz : ∀ Γ A → Var8 (snoc8 Γ A) A)
(vs : ∀ Γ B A → Var8 Γ A → Var8 (snoc8 Γ B) A)
→ Var8 Γ A
vz8 : ∀{Γ A} → Var8 (snoc8 Γ A) A; vz8
= λ Var8 vz8 vs → vz8 _ _
vs8 : ∀{Γ B A} → Var8 Γ A → Var8 (snoc8 Γ B) A; vs8
= λ x Var8 vz8 vs8 → vs8 _ _ _ (x Var8 vz8 vs8)
Tm8 : Con8 → Ty8 → Set; Tm8
= λ Γ A →
(Tm8 : Con8 → Ty8 → Set)
(var : ∀ Γ A → Var8 Γ A → Tm8 Γ A)
(lam : ∀ Γ A B → Tm8 (snoc8 Γ A) B → Tm8 Γ (arr8 A B))
(app : ∀ Γ A B → Tm8 Γ (arr8 A B) → Tm8 Γ A → Tm8 Γ B)
(tt : ∀ Γ → Tm8 Γ top8)
(pair : ∀ Γ A B → Tm8 Γ A → Tm8 Γ B → Tm8 Γ (prod8 A B))
(fst : ∀ Γ A B → Tm8 Γ (prod8 A B) → Tm8 Γ A)
(snd : ∀ Γ A B → Tm8 Γ (prod8 A B) → Tm8 Γ B)
(left : ∀ Γ A B → Tm8 Γ A → Tm8 Γ (sum8 A B))
(right : ∀ Γ A B → Tm8 Γ B → Tm8 Γ (sum8 A B))
(case : ∀ Γ A B C → Tm8 Γ (sum8 A B) → Tm8 Γ (arr8 A C) → Tm8 Γ (arr8 B C) → Tm8 Γ C)
(zero : ∀ Γ → Tm8 Γ nat8)
(suc : ∀ Γ → Tm8 Γ nat8 → Tm8 Γ nat8)
(rec : ∀ Γ A → Tm8 Γ nat8 → Tm8 Γ (arr8 nat8 (arr8 A A)) → Tm8 Γ A → Tm8 Γ A)
→ Tm8 Γ A
var8 : ∀{Γ A} → Var8 Γ A → Tm8 Γ A; var8
= λ x Tm8 var8 lam app tt pair fst snd left right case zero suc rec →
var8 _ _ x
lam8 : ∀{Γ A B} → Tm8 (snoc8 Γ A) B → Tm8 Γ (arr8 A B); lam8
= λ t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec →
lam8 _ _ _ (t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec)
app8 : ∀{Γ A B} → Tm8 Γ (arr8 A B) → Tm8 Γ A → Tm8 Γ B; app8
= λ t u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec →
app8 _ _ _ (t Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
tt8 : ∀{Γ} → Tm8 Γ top8; tt8
= λ Tm8 var8 lam8 app8 tt8 pair fst snd left right case zero suc rec → tt8 _
pair8 : ∀{Γ A B} → Tm8 Γ A → Tm8 Γ B → Tm8 Γ (prod8 A B); pair8
= λ t u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec →
pair8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
fst8 : ∀{Γ A B} → Tm8 Γ (prod8 A B) → Tm8 Γ A; fst8
= λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec →
fst8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec)
snd8 : ∀{Γ A B} → Tm8 Γ (prod8 A B) → Tm8 Γ B; snd8
= λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec →
snd8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec)
left8 : ∀{Γ A B} → Tm8 Γ A → Tm8 Γ (sum8 A B); left8
= λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec →
left8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec)
right8 : ∀{Γ A B} → Tm8 Γ B → Tm8 Γ (sum8 A B); right8
= λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec →
right8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec)
case8 : ∀{Γ A B C} → Tm8 Γ (sum8 A B) → Tm8 Γ (arr8 A C) → Tm8 Γ (arr8 B C) → Tm8 Γ C; case8
= λ t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec →
case8 _ _ _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
zero8 : ∀{Γ} → Tm8 Γ nat8; zero8
= λ Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc rec → zero8 _
suc8 : ∀{Γ} → Tm8 Γ nat8 → Tm8 Γ nat8; suc8
= λ t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec →
suc8 _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec)
rec8 : ∀{Γ A} → Tm8 Γ nat8 → Tm8 Γ (arr8 nat8 (arr8 A A)) → Tm8 Γ A → Tm8 Γ A; rec8
= λ t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8 →
rec8 _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
v08 : ∀{Γ A} → Tm8 (snoc8 Γ A) A; v08
= var8 vz8
v18 : ∀{Γ A B} → Tm8 (snoc8 (snoc8 Γ A) B) A; v18
= var8 (vs8 vz8)
v28 : ∀{Γ A B C} → Tm8 (snoc8 (snoc8 (snoc8 Γ A) B) C) A; v28
= var8 (vs8 (vs8 vz8))
v38 : ∀{Γ A B C D} → Tm8 (snoc8 (snoc8 (snoc8 (snoc8 Γ A) B) C) D) A; v38
= var8 (vs8 (vs8 (vs8 vz8)))
tbool8 : Ty8; tbool8
= sum8 top8 top8
true8 : ∀{Γ} → Tm8 Γ tbool8; true8
= left8 tt8
tfalse8 : ∀{Γ} → Tm8 Γ tbool8; tfalse8
= right8 tt8
ifthenelse8 : ∀{Γ A} → Tm8 Γ (arr8 tbool8 (arr8 A (arr8 A A))); ifthenelse8
= lam8 (lam8 (lam8 (case8 v28 (lam8 v28) (lam8 v18))))
times48 : ∀{Γ A} → Tm8 Γ (arr8 (arr8 A A) (arr8 A A)); times48
= lam8 (lam8 (app8 v18 (app8 v18 (app8 v18 (app8 v18 v08)))))
add8 : ∀{Γ} → Tm8 Γ (arr8 nat8 (arr8 nat8 nat8)); add8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (suc8 (app8 v18 v08)))))
(lam8 v08))
mul8 : ∀{Γ} → Tm8 Γ (arr8 nat8 (arr8 nat8 nat8)); mul8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (app8 (app8 add8 (app8 v18 v08)) v08))))
(lam8 zero8))
fact8 : ∀{Γ} → Tm8 Γ (arr8 nat8 nat8); fact8
= lam8 (rec8 v08 (lam8 (lam8 (app8 (app8 mul8 (suc8 v18)) v08)))
(suc8 zero8))
{-# OPTIONS --type-in-type #-}
Ty9 : Set
Ty9 =
(Ty9 : Set)
(nat top bot : Ty9)
(arr prod sum : Ty9 → Ty9 → Ty9)
→ Ty9
nat9 : Ty9; nat9 = λ _ nat9 _ _ _ _ _ → nat9
top9 : Ty9; top9 = λ _ _ top9 _ _ _ _ → top9
bot9 : Ty9; bot9 = λ _ _ _ bot9 _ _ _ → bot9
arr9 : Ty9 → Ty9 → Ty9; arr9
= λ A B Ty9 nat9 top9 bot9 arr9 prod sum →
arr9 (A Ty9 nat9 top9 bot9 arr9 prod sum) (B Ty9 nat9 top9 bot9 arr9 prod sum)
prod9 : Ty9 → Ty9 → Ty9; prod9
= λ A B Ty9 nat9 top9 bot9 arr9 prod9 sum →
prod9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum) (B Ty9 nat9 top9 bot9 arr9 prod9 sum)
sum9 : Ty9 → Ty9 → Ty9; sum9
= λ A B Ty9 nat9 top9 bot9 arr9 prod9 sum9 →
sum9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum9) (B Ty9 nat9 top9 bot9 arr9 prod9 sum9)
Con9 : Set; Con9
= (Con9 : Set)
(nil : Con9)
(snoc : Con9 → Ty9 → Con9)
→ Con9
nil9 : Con9; nil9
= λ Con9 nil9 snoc → nil9
snoc9 : Con9 → Ty9 → Con9; snoc9
= λ Γ A Con9 nil9 snoc9 → snoc9 (Γ Con9 nil9 snoc9) A
Var9 : Con9 → Ty9 → Set; Var9
= λ Γ A →
(Var9 : Con9 → Ty9 → Set)
(vz : ∀ Γ A → Var9 (snoc9 Γ A) A)
(vs : ∀ Γ B A → Var9 Γ A → Var9 (snoc9 Γ B) A)
→ Var9 Γ A
vz9 : ∀{Γ A} → Var9 (snoc9 Γ A) A; vz9
= λ Var9 vz9 vs → vz9 _ _
vs9 : ∀{Γ B A} → Var9 Γ A → Var9 (snoc9 Γ B) A; vs9
= λ x Var9 vz9 vs9 → vs9 _ _ _ (x Var9 vz9 vs9)
Tm9 : Con9 → Ty9 → Set; Tm9
= λ Γ A →
(Tm9 : Con9 → Ty9 → Set)
(var : ∀ Γ A → Var9 Γ A → Tm9 Γ A)
(lam : ∀ Γ A B → Tm9 (snoc9 Γ A) B → Tm9 Γ (arr9 A B))
(app : ∀ Γ A B → Tm9 Γ (arr9 A B) → Tm9 Γ A → Tm9 Γ B)
(tt : ∀ Γ → Tm9 Γ top9)
(pair : ∀ Γ A B → Tm9 Γ A → Tm9 Γ B → Tm9 Γ (prod9 A B))
(fst : ∀ Γ A B → Tm9 Γ (prod9 A B) → Tm9 Γ A)
(snd : ∀ Γ A B → Tm9 Γ (prod9 A B) → Tm9 Γ B)
(left : ∀ Γ A B → Tm9 Γ A → Tm9 Γ (sum9 A B))
(right : ∀ Γ A B → Tm9 Γ B → Tm9 Γ (sum9 A B))
(case : ∀ Γ A B C → Tm9 Γ (sum9 A B) → Tm9 Γ (arr9 A C) → Tm9 Γ (arr9 B C) → Tm9 Γ C)
(zero : ∀ Γ → Tm9 Γ nat9)
(suc : ∀ Γ → Tm9 Γ nat9 → Tm9 Γ nat9)
(rec : ∀ Γ A → Tm9 Γ nat9 → Tm9 Γ (arr9 nat9 (arr9 A A)) → Tm9 Γ A → Tm9 Γ A)
→ Tm9 Γ A
var9 : ∀{Γ A} → Var9 Γ A → Tm9 Γ A; var9
= λ x Tm9 var9 lam app tt pair fst snd left right case zero suc rec →
var9 _ _ x
lam9 : ∀{Γ A B} → Tm9 (snoc9 Γ A) B → Tm9 Γ (arr9 A B); lam9
= λ t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec →
lam9 _ _ _ (t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec)
app9 : ∀{Γ A B} → Tm9 Γ (arr9 A B) → Tm9 Γ A → Tm9 Γ B; app9
= λ t u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec →
app9 _ _ _ (t Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
tt9 : ∀{Γ} → Tm9 Γ top9; tt9
= λ Tm9 var9 lam9 app9 tt9 pair fst snd left right case zero suc rec → tt9 _
pair9 : ∀{Γ A B} → Tm9 Γ A → Tm9 Γ B → Tm9 Γ (prod9 A B); pair9
= λ t u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec →
pair9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
fst9 : ∀{Γ A B} → Tm9 Γ (prod9 A B) → Tm9 Γ A; fst9
= λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec →
fst9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec)
snd9 : ∀{Γ A B} → Tm9 Γ (prod9 A B) → Tm9 Γ B; snd9
= λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec →
snd9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec)
left9 : ∀{Γ A B} → Tm9 Γ A → Tm9 Γ (sum9 A B); left9
= λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec →
left9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec)
right9 : ∀{Γ A B} → Tm9 Γ B → Tm9 Γ (sum9 A B); right9
= λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec →
right9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec)
case9 : ∀{Γ A B C} → Tm9 Γ (sum9 A B) → Tm9 Γ (arr9 A C) → Tm9 Γ (arr9 B C) → Tm9 Γ C; case9
= λ t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec →
case9 _ _ _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
zero9 : ∀{Γ} → Tm9 Γ nat9; zero9
= λ Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc rec → zero9 _
suc9 : ∀{Γ} → Tm9 Γ nat9 → Tm9 Γ nat9; suc9
= λ t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec →
suc9 _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec)
rec9 : ∀{Γ A} → Tm9 Γ nat9 → Tm9 Γ (arr9 nat9 (arr9 A A)) → Tm9 Γ A → Tm9 Γ A; rec9
= λ t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9 →
rec9 _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
v09 : ∀{Γ A} → Tm9 (snoc9 Γ A) A; v09
= var9 vz9
v19 : ∀{Γ A B} → Tm9 (snoc9 (snoc9 Γ A) B) A; v19
= var9 (vs9 vz9)
v29 : ∀{Γ A B C} → Tm9 (snoc9 (snoc9 (snoc9 Γ A) B) C) A; v29
= var9 (vs9 (vs9 vz9))
v39 : ∀{Γ A B C D} → Tm9 (snoc9 (snoc9 (snoc9 (snoc9 Γ A) B) C) D) A; v39
= var9 (vs9 (vs9 (vs9 vz9)))
tbool9 : Ty9; tbool9
= sum9 top9 top9
true9 : ∀{Γ} → Tm9 Γ tbool9; true9
= left9 tt9
tfalse9 : ∀{Γ} → Tm9 Γ tbool9; tfalse9
= right9 tt9
ifthenelse9 : ∀{Γ A} → Tm9 Γ (arr9 tbool9 (arr9 A (arr9 A A))); ifthenelse9
= lam9 (lam9 (lam9 (case9 v29 (lam9 v29) (lam9 v19))))
times49 : ∀{Γ A} → Tm9 Γ (arr9 (arr9 A A) (arr9 A A)); times49
= lam9 (lam9 (app9 v19 (app9 v19 (app9 v19 (app9 v19 v09)))))
add9 : ∀{Γ} → Tm9 Γ (arr9 nat9 (arr9 nat9 nat9)); add9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (suc9 (app9 v19 v09)))))
(lam9 v09))
mul9 : ∀{Γ} → Tm9 Γ (arr9 nat9 (arr9 nat9 nat9)); mul9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (app9 (app9 add9 (app9 v19 v09)) v09))))
(lam9 zero9))
fact9 : ∀{Γ} → Tm9 Γ (arr9 nat9 nat9); fact9
= lam9 (rec9 v09 (lam9 (lam9 (app9 (app9 mul9 (suc9 v19)) v09)))
(suc9 zero9))
{-# OPTIONS --type-in-type #-}
Ty10 : Set
Ty10 =
(Ty10 : Set)
(nat top bot : Ty10)
(arr prod sum : Ty10 → Ty10 → Ty10)
→ Ty10
nat10 : Ty10; nat10 = λ _ nat10 _ _ _ _ _ → nat10
top10 : Ty10; top10 = λ _ _ top10 _ _ _ _ → top10
bot10 : Ty10; bot10 = λ _ _ _ bot10 _ _ _ → bot10
arr10 : Ty10 → Ty10 → Ty10; arr10
= λ A B Ty10 nat10 top10 bot10 arr10 prod sum →
arr10 (A Ty10 nat10 top10 bot10 arr10 prod sum) (B Ty10 nat10 top10 bot10 arr10 prod sum)
prod10 : Ty10 → Ty10 → Ty10; prod10
= λ A B Ty10 nat10 top10 bot10 arr10 prod10 sum →
prod10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum) (B Ty10 nat10 top10 bot10 arr10 prod10 sum)
sum10 : Ty10 → Ty10 → Ty10; sum10
= λ A B Ty10 nat10 top10 bot10 arr10 prod10 sum10 →
sum10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum10) (B Ty10 nat10 top10 bot10 arr10 prod10 sum10)
Con10 : Set; Con10
= (Con10 : Set)
(nil : Con10)
(snoc : Con10 → Ty10 → Con10)
→ Con10
nil10 : Con10; nil10
= λ Con10 nil10 snoc → nil10
snoc10 : Con10 → Ty10 → Con10; snoc10
= λ Γ A Con10 nil10 snoc10 → snoc10 (Γ Con10 nil10 snoc10) A
Var10 : Con10 → Ty10 → Set; Var10
= λ Γ A →
(Var10 : Con10 → Ty10 → Set)
(vz : ∀ Γ A → Var10 (snoc10 Γ A) A)
(vs : ∀ Γ B A → Var10 Γ A → Var10 (snoc10 Γ B) A)
→ Var10 Γ A
vz10 : ∀{Γ A} → Var10 (snoc10 Γ A) A; vz10
= λ Var10 vz10 vs → vz10 _ _
vs10 : ∀{Γ B A} → Var10 Γ A → Var10 (snoc10 Γ B) A; vs10
= λ x Var10 vz10 vs10 → vs10 _ _ _ (x Var10 vz10 vs10)
Tm10 : Con10 → Ty10 → Set; Tm10
= λ Γ A →
(Tm10 : Con10 → Ty10 → Set)
(var : ∀ Γ A → Var10 Γ A → Tm10 Γ A)
(lam : ∀ Γ A B → Tm10 (snoc10 Γ A) B → Tm10 Γ (arr10 A B))
(app : ∀ Γ A B → Tm10 Γ (arr10 A B) → Tm10 Γ A → Tm10 Γ B)
(tt : ∀ Γ → Tm10 Γ top10)
(pair : ∀ Γ A B → Tm10 Γ A → Tm10 Γ B → Tm10 Γ (prod10 A B))
(fst : ∀ Γ A B → Tm10 Γ (prod10 A B) → Tm10 Γ A)
(snd : ∀ Γ A B → Tm10 Γ (prod10 A B) → Tm10 Γ B)
(left : ∀ Γ A B → Tm10 Γ A → Tm10 Γ (sum10 A B))
(right : ∀ Γ A B → Tm10 Γ B → Tm10 Γ (sum10 A B))
(case : ∀ Γ A B C → Tm10 Γ (sum10 A B) → Tm10 Γ (arr10 A C) → Tm10 Γ (arr10 B C) → Tm10 Γ C)
(zero : ∀ Γ → Tm10 Γ nat10)
(suc : ∀ Γ → Tm10 Γ nat10 → Tm10 Γ nat10)
(rec : ∀ Γ A → Tm10 Γ nat10 → Tm10 Γ (arr10 nat10 (arr10 A A)) → Tm10 Γ A → Tm10 Γ A)
→ Tm10 Γ A
var10 : ∀{Γ A} → Var10 Γ A → Tm10 Γ A; var10
= λ x Tm10 var10 lam app tt pair fst snd left right case zero suc rec →
var10 _ _ x
lam10 : ∀{Γ A B} → Tm10 (snoc10 Γ A) B → Tm10 Γ (arr10 A B); lam10
= λ t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec →
lam10 _ _ _ (t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec)
app10 : ∀{Γ A B} → Tm10 Γ (arr10 A B) → Tm10 Γ A → Tm10 Γ B; app10
= λ t u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec →
app10 _ _ _ (t Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
tt10 : ∀{Γ} → Tm10 Γ top10; tt10
= λ Tm10 var10 lam10 app10 tt10 pair fst snd left right case zero suc rec → tt10 _
pair10 : ∀{Γ A B} → Tm10 Γ A → Tm10 Γ B → Tm10 Γ (prod10 A B); pair10
= λ t u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec →
pair10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
fst10 : ∀{Γ A B} → Tm10 Γ (prod10 A B) → Tm10 Γ A; fst10
= λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec →
fst10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec)
snd10 : ∀{Γ A B} → Tm10 Γ (prod10 A B) → Tm10 Γ B; snd10
= λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec →
snd10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec)
left10 : ∀{Γ A B} → Tm10 Γ A → Tm10 Γ (sum10 A B); left10
= λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec →
left10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec)
right10 : ∀{Γ A B} → Tm10 Γ B → Tm10 Γ (sum10 A B); right10
= λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec →
right10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec)
case10 : ∀{Γ A B C} → Tm10 Γ (sum10 A B) → Tm10 Γ (arr10 A C) → Tm10 Γ (arr10 B C) → Tm10 Γ C; case10
= λ t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec →
case10 _ _ _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
zero10 : ∀{Γ} → Tm10 Γ nat10; zero10
= λ Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc rec → zero10 _
suc10 : ∀{Γ} → Tm10 Γ nat10 → Tm10 Γ nat10; suc10
= λ t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec →
suc10 _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec)
rec10 : ∀{Γ A} → Tm10 Γ nat10 → Tm10 Γ (arr10 nat10 (arr10 A A)) → Tm10 Γ A → Tm10 Γ A; rec10
= λ t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10 →
rec10 _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
v010 : ∀{Γ A} → Tm10 (snoc10 Γ A) A; v010
= var10 vz10
v110 : ∀{Γ A B} → Tm10 (snoc10 (snoc10 Γ A) B) A; v110
= var10 (vs10 vz10)
v210 : ∀{Γ A B C} → Tm10 (snoc10 (snoc10 (snoc10 Γ A) B) C) A; v210
= var10 (vs10 (vs10 vz10))
v310 : ∀{Γ A B C D} → Tm10 (snoc10 (snoc10 (snoc10 (snoc10 Γ A) B) C) D) A; v310
= var10 (vs10 (vs10 (vs10 vz10)))
tbool10 : Ty10; tbool10
= sum10 top10 top10
true10 : ∀{Γ} → Tm10 Γ tbool10; true10
= left10 tt10
tfalse10 : ∀{Γ} → Tm10 Γ tbool10; tfalse10
= right10 tt10
ifthenelse10 : ∀{Γ A} → Tm10 Γ (arr10 tbool10 (arr10 A (arr10 A A))); ifthenelse10
= lam10 (lam10 (lam10 (case10 v210 (lam10 v210) (lam10 v110))))
times410 : ∀{Γ A} → Tm10 Γ (arr10 (arr10 A A) (arr10 A A)); times410
= lam10 (lam10 (app10 v110 (app10 v110 (app10 v110 (app10 v110 v010)))))
add10 : ∀{Γ} → Tm10 Γ (arr10 nat10 (arr10 nat10 nat10)); add10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (suc10 (app10 v110 v010)))))
(lam10 v010))
mul10 : ∀{Γ} → Tm10 Γ (arr10 nat10 (arr10 nat10 nat10)); mul10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (app10 (app10 add10 (app10 v110 v010)) v010))))
(lam10 zero10))
fact10 : ∀{Γ} → Tm10 Γ (arr10 nat10 nat10); fact10
= lam10 (rec10 v010 (lam10 (lam10 (app10 (app10 mul10 (suc10 v110)) v010)))
(suc10 zero10))
{-# OPTIONS --type-in-type #-}
Ty11 : Set
Ty11 =
(Ty11 : Set)
(nat top bot : Ty11)
(arr prod sum : Ty11 → Ty11 → Ty11)
→ Ty11
nat11 : Ty11; nat11 = λ _ nat11 _ _ _ _ _ → nat11
top11 : Ty11; top11 = λ _ _ top11 _ _ _ _ → top11
bot11 : Ty11; bot11 = λ _ _ _ bot11 _ _ _ → bot11
arr11 : Ty11 → Ty11 → Ty11; arr11
= λ A B Ty11 nat11 top11 bot11 arr11 prod sum →
arr11 (A Ty11 nat11 top11 bot11 arr11 prod sum) (B Ty11 nat11 top11 bot11 arr11 prod sum)
prod11 : Ty11 → Ty11 → Ty11; prod11
= λ A B Ty11 nat11 top11 bot11 arr11 prod11 sum →
prod11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum) (B Ty11 nat11 top11 bot11 arr11 prod11 sum)
sum11 : Ty11 → Ty11 → Ty11; sum11
= λ A B Ty11 nat11 top11 bot11 arr11 prod11 sum11 →
sum11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum11) (B Ty11 nat11 top11 bot11 arr11 prod11 sum11)
Con11 : Set; Con11
= (Con11 : Set)
(nil : Con11)
(snoc : Con11 → Ty11 → Con11)
→ Con11
nil11 : Con11; nil11
= λ Con11 nil11 snoc → nil11
snoc11 : Con11 → Ty11 → Con11; snoc11
= λ Γ A Con11 nil11 snoc11 → snoc11 (Γ Con11 nil11 snoc11) A
Var11 : Con11 → Ty11 → Set; Var11
= λ Γ A →
(Var11 : Con11 → Ty11 → Set)
(vz : ∀ Γ A → Var11 (snoc11 Γ A) A)
(vs : ∀ Γ B A → Var11 Γ A → Var11 (snoc11 Γ B) A)
→ Var11 Γ A
vz11 : ∀{Γ A} → Var11 (snoc11 Γ A) A; vz11
= λ Var11 vz11 vs → vz11 _ _
vs11 : ∀{Γ B A} → Var11 Γ A → Var11 (snoc11 Γ B) A; vs11
= λ x Var11 vz11 vs11 → vs11 _ _ _ (x Var11 vz11 vs11)
Tm11 : Con11 → Ty11 → Set; Tm11
= λ Γ A →
(Tm11 : Con11 → Ty11 → Set)
(var : ∀ Γ A → Var11 Γ A → Tm11 Γ A)
(lam : ∀ Γ A B → Tm11 (snoc11 Γ A) B → Tm11 Γ (arr11 A B))
(app : ∀ Γ A B → Tm11 Γ (arr11 A B) → Tm11 Γ A → Tm11 Γ B)
(tt : ∀ Γ → Tm11 Γ top11)
(pair : ∀ Γ A B → Tm11 Γ A → Tm11 Γ B → Tm11 Γ (prod11 A B))
(fst : ∀ Γ A B → Tm11 Γ (prod11 A B) → Tm11 Γ A)
(snd : ∀ Γ A B → Tm11 Γ (prod11 A B) → Tm11 Γ B)
(left : ∀ Γ A B → Tm11 Γ A → Tm11 Γ (sum11 A B))
(right : ∀ Γ A B → Tm11 Γ B → Tm11 Γ (sum11 A B))
(case : ∀ Γ A B C → Tm11 Γ (sum11 A B) → Tm11 Γ (arr11 A C) → Tm11 Γ (arr11 B C) → Tm11 Γ C)
(zero : ∀ Γ → Tm11 Γ nat11)
(suc : ∀ Γ → Tm11 Γ nat11 → Tm11 Γ nat11)
(rec : ∀ Γ A → Tm11 Γ nat11 → Tm11 Γ (arr11 nat11 (arr11 A A)) → Tm11 Γ A → Tm11 Γ A)
→ Tm11 Γ A
var11 : ∀{Γ A} → Var11 Γ A → Tm11 Γ A; var11
= λ x Tm11 var11 lam app tt pair fst snd left right case zero suc rec →
var11 _ _ x
lam11 : ∀{Γ A B} → Tm11 (snoc11 Γ A) B → Tm11 Γ (arr11 A B); lam11
= λ t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec →
lam11 _ _ _ (t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec)
app11 : ∀{Γ A B} → Tm11 Γ (arr11 A B) → Tm11 Γ A → Tm11 Γ B; app11
= λ t u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec →
app11 _ _ _ (t Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
tt11 : ∀{Γ} → Tm11 Γ top11; tt11
= λ Tm11 var11 lam11 app11 tt11 pair fst snd left right case zero suc rec → tt11 _
pair11 : ∀{Γ A B} → Tm11 Γ A → Tm11 Γ B → Tm11 Γ (prod11 A B); pair11
= λ t u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec →
pair11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
fst11 : ∀{Γ A B} → Tm11 Γ (prod11 A B) → Tm11 Γ A; fst11
= λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec →
fst11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec)
snd11 : ∀{Γ A B} → Tm11 Γ (prod11 A B) → Tm11 Γ B; snd11
= λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec →
snd11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec)
left11 : ∀{Γ A B} → Tm11 Γ A → Tm11 Γ (sum11 A B); left11
= λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec →
left11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec)
right11 : ∀{Γ A B} → Tm11 Γ B → Tm11 Γ (sum11 A B); right11
= λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec →
right11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec)
case11 : ∀{Γ A B C} → Tm11 Γ (sum11 A B) → Tm11 Γ (arr11 A C) → Tm11 Γ (arr11 B C) → Tm11 Γ C; case11
= λ t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec →
case11 _ _ _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
zero11 : ∀{Γ} → Tm11 Γ nat11; zero11
= λ Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc rec → zero11 _
suc11 : ∀{Γ} → Tm11 Γ nat11 → Tm11 Γ nat11; suc11
= λ t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec →
suc11 _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec)
rec11 : ∀{Γ A} → Tm11 Γ nat11 → Tm11 Γ (arr11 nat11 (arr11 A A)) → Tm11 Γ A → Tm11 Γ A; rec11
= λ t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11 →
rec11 _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
v011 : ∀{Γ A} → Tm11 (snoc11 Γ A) A; v011
= var11 vz11
v111 : ∀{Γ A B} → Tm11 (snoc11 (snoc11 Γ A) B) A; v111
= var11 (vs11 vz11)
v211 : ∀{Γ A B C} → Tm11 (snoc11 (snoc11 (snoc11 Γ A) B) C) A; v211
= var11 (vs11 (vs11 vz11))
v311 : ∀{Γ A B C D} → Tm11 (snoc11 (snoc11 (snoc11 (snoc11 Γ A) B) C) D) A; v311
= var11 (vs11 (vs11 (vs11 vz11)))
tbool11 : Ty11; tbool11
= sum11 top11 top11
true11 : ∀{Γ} → Tm11 Γ tbool11; true11
= left11 tt11
tfalse11 : ∀{Γ} → Tm11 Γ tbool11; tfalse11
= right11 tt11
ifthenelse11 : ∀{Γ A} → Tm11 Γ (arr11 tbool11 (arr11 A (arr11 A A))); ifthenelse11
= lam11 (lam11 (lam11 (case11 v211 (lam11 v211) (lam11 v111))))
times411 : ∀{Γ A} → Tm11 Γ (arr11 (arr11 A A) (arr11 A A)); times411
= lam11 (lam11 (app11 v111 (app11 v111 (app11 v111 (app11 v111 v011)))))
add11 : ∀{Γ} → Tm11 Γ (arr11 nat11 (arr11 nat11 nat11)); add11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (suc11 (app11 v111 v011)))))
(lam11 v011))
mul11 : ∀{Γ} → Tm11 Γ (arr11 nat11 (arr11 nat11 nat11)); mul11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (app11 (app11 add11 (app11 v111 v011)) v011))))
(lam11 zero11))
fact11 : ∀{Γ} → Tm11 Γ (arr11 nat11 nat11); fact11
= lam11 (rec11 v011 (lam11 (lam11 (app11 (app11 mul11 (suc11 v111)) v011)))
(suc11 zero11))
{-# OPTIONS --type-in-type #-}
Ty12 : Set
Ty12 =
(Ty12 : Set)
(nat top bot : Ty12)
(arr prod sum : Ty12 → Ty12 → Ty12)
→ Ty12
nat12 : Ty12; nat12 = λ _ nat12 _ _ _ _ _ → nat12
top12 : Ty12; top12 = λ _ _ top12 _ _ _ _ → top12
bot12 : Ty12; bot12 = λ _ _ _ bot12 _ _ _ → bot12
arr12 : Ty12 → Ty12 → Ty12; arr12
= λ A B Ty12 nat12 top12 bot12 arr12 prod sum →
arr12 (A Ty12 nat12 top12 bot12 arr12 prod sum) (B Ty12 nat12 top12 bot12 arr12 prod sum)
prod12 : Ty12 → Ty12 → Ty12; prod12
= λ A B Ty12 nat12 top12 bot12 arr12 prod12 sum →
prod12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum) (B Ty12 nat12 top12 bot12 arr12 prod12 sum)
sum12 : Ty12 → Ty12 → Ty12; sum12
= λ A B Ty12 nat12 top12 bot12 arr12 prod12 sum12 →
sum12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum12) (B Ty12 nat12 top12 bot12 arr12 prod12 sum12)
Con12 : Set; Con12
= (Con12 : Set)
(nil : Con12)
(snoc : Con12 → Ty12 → Con12)
→ Con12
nil12 : Con12; nil12
= λ Con12 nil12 snoc → nil12
snoc12 : Con12 → Ty12 → Con12; snoc12
= λ Γ A Con12 nil12 snoc12 → snoc12 (Γ Con12 nil12 snoc12) A
Var12 : Con12 → Ty12 → Set; Var12
= λ Γ A →
(Var12 : Con12 → Ty12 → Set)
(vz : ∀ Γ A → Var12 (snoc12 Γ A) A)
(vs : ∀ Γ B A → Var12 Γ A → Var12 (snoc12 Γ B) A)
→ Var12 Γ A
vz12 : ∀{Γ A} → Var12 (snoc12 Γ A) A; vz12
= λ Var12 vz12 vs → vz12 _ _
vs12 : ∀{Γ B A} → Var12 Γ A → Var12 (snoc12 Γ B) A; vs12
= λ x Var12 vz12 vs12 → vs12 _ _ _ (x Var12 vz12 vs12)
Tm12 : Con12 → Ty12 → Set; Tm12
= λ Γ A →
(Tm12 : Con12 → Ty12 → Set)
(var : ∀ Γ A → Var12 Γ A → Tm12 Γ A)
(lam : ∀ Γ A B → Tm12 (snoc12 Γ A) B → Tm12 Γ (arr12 A B))
(app : ∀ Γ A B → Tm12 Γ (arr12 A B) → Tm12 Γ A → Tm12 Γ B)
(tt : ∀ Γ → Tm12 Γ top12)
(pair : ∀ Γ A B → Tm12 Γ A → Tm12 Γ B → Tm12 Γ (prod12 A B))
(fst : ∀ Γ A B → Tm12 Γ (prod12 A B) → Tm12 Γ A)
(snd : ∀ Γ A B → Tm12 Γ (prod12 A B) → Tm12 Γ B)
(left : ∀ Γ A B → Tm12 Γ A → Tm12 Γ (sum12 A B))
(right : ∀ Γ A B → Tm12 Γ B → Tm12 Γ (sum12 A B))
(case : ∀ Γ A B C → Tm12 Γ (sum12 A B) → Tm12 Γ (arr12 A C) → Tm12 Γ (arr12 B C) → Tm12 Γ C)
(zero : ∀ Γ → Tm12 Γ nat12)
(suc : ∀ Γ → Tm12 Γ nat12 → Tm12 Γ nat12)
(rec : ∀ Γ A → Tm12 Γ nat12 → Tm12 Γ (arr12 nat12 (arr12 A A)) → Tm12 Γ A → Tm12 Γ A)
→ Tm12 Γ A
var12 : ∀{Γ A} → Var12 Γ A → Tm12 Γ A; var12
= λ x Tm12 var12 lam app tt pair fst snd left right case zero suc rec →
var12 _ _ x
lam12 : ∀{Γ A B} → Tm12 (snoc12 Γ A) B → Tm12 Γ (arr12 A B); lam12
= λ t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec →
lam12 _ _ _ (t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec)
app12 : ∀{Γ A B} → Tm12 Γ (arr12 A B) → Tm12 Γ A → Tm12 Γ B; app12
= λ t u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec →
app12 _ _ _ (t Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
tt12 : ∀{Γ} → Tm12 Γ top12; tt12
= λ Tm12 var12 lam12 app12 tt12 pair fst snd left right case zero suc rec → tt12 _
pair12 : ∀{Γ A B} → Tm12 Γ A → Tm12 Γ B → Tm12 Γ (prod12 A B); pair12
= λ t u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec →
pair12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
fst12 : ∀{Γ A B} → Tm12 Γ (prod12 A B) → Tm12 Γ A; fst12
= λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec →
fst12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec)
snd12 : ∀{Γ A B} → Tm12 Γ (prod12 A B) → Tm12 Γ B; snd12
= λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec →
snd12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec)
left12 : ∀{Γ A B} → Tm12 Γ A → Tm12 Γ (sum12 A B); left12
= λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec →
left12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec)
right12 : ∀{Γ A B} → Tm12 Γ B → Tm12 Γ (sum12 A B); right12
= λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec →
right12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec)
case12 : ∀{Γ A B C} → Tm12 Γ (sum12 A B) → Tm12 Γ (arr12 A C) → Tm12 Γ (arr12 B C) → Tm12 Γ C; case12
= λ t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec →
case12 _ _ _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
zero12 : ∀{Γ} → Tm12 Γ nat12; zero12
= λ Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc rec → zero12 _
suc12 : ∀{Γ} → Tm12 Γ nat12 → Tm12 Γ nat12; suc12
= λ t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec →
suc12 _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec)
rec12 : ∀{Γ A} → Tm12 Γ nat12 → Tm12 Γ (arr12 nat12 (arr12 A A)) → Tm12 Γ A → Tm12 Γ A; rec12
= λ t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12 →
rec12 _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
v012 : ∀{Γ A} → Tm12 (snoc12 Γ A) A; v012
= var12 vz12
v112 : ∀{Γ A B} → Tm12 (snoc12 (snoc12 Γ A) B) A; v112
= var12 (vs12 vz12)
v212 : ∀{Γ A B C} → Tm12 (snoc12 (snoc12 (snoc12 Γ A) B) C) A; v212
= var12 (vs12 (vs12 vz12))
v312 : ∀{Γ A B C D} → Tm12 (snoc12 (snoc12 (snoc12 (snoc12 Γ A) B) C) D) A; v312
= var12 (vs12 (vs12 (vs12 vz12)))
tbool12 : Ty12; tbool12
= sum12 top12 top12
true12 : ∀{Γ} → Tm12 Γ tbool12; true12
= left12 tt12
tfalse12 : ∀{Γ} → Tm12 Γ tbool12; tfalse12
= right12 tt12
ifthenelse12 : ∀{Γ A} → Tm12 Γ (arr12 tbool12 (arr12 A (arr12 A A))); ifthenelse12
= lam12 (lam12 (lam12 (case12 v212 (lam12 v212) (lam12 v112))))
times412 : ∀{Γ A} → Tm12 Γ (arr12 (arr12 A A) (arr12 A A)); times412
= lam12 (lam12 (app12 v112 (app12 v112 (app12 v112 (app12 v112 v012)))))
add12 : ∀{Γ} → Tm12 Γ (arr12 nat12 (arr12 nat12 nat12)); add12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (suc12 (app12 v112 v012)))))
(lam12 v012))
mul12 : ∀{Γ} → Tm12 Γ (arr12 nat12 (arr12 nat12 nat12)); mul12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (app12 (app12 add12 (app12 v112 v012)) v012))))
(lam12 zero12))
fact12 : ∀{Γ} → Tm12 Γ (arr12 nat12 nat12); fact12
= lam12 (rec12 v012 (lam12 (lam12 (app12 (app12 mul12 (suc12 v112)) v012)))
(suc12 zero12))
{-# OPTIONS --type-in-type #-}
Ty13 : Set
Ty13 =
(Ty13 : Set)
(nat top bot : Ty13)
(arr prod sum : Ty13 → Ty13 → Ty13)
→ Ty13
nat13 : Ty13; nat13 = λ _ nat13 _ _ _ _ _ → nat13
top13 : Ty13; top13 = λ _ _ top13 _ _ _ _ → top13
bot13 : Ty13; bot13 = λ _ _ _ bot13 _ _ _ → bot13
arr13 : Ty13 → Ty13 → Ty13; arr13
= λ A B Ty13 nat13 top13 bot13 arr13 prod sum →
arr13 (A Ty13 nat13 top13 bot13 arr13 prod sum) (B Ty13 nat13 top13 bot13 arr13 prod sum)
prod13 : Ty13 → Ty13 → Ty13; prod13
= λ A B Ty13 nat13 top13 bot13 arr13 prod13 sum →
prod13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum) (B Ty13 nat13 top13 bot13 arr13 prod13 sum)
sum13 : Ty13 → Ty13 → Ty13; sum13
= λ A B Ty13 nat13 top13 bot13 arr13 prod13 sum13 →
sum13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum13) (B Ty13 nat13 top13 bot13 arr13 prod13 sum13)
Con13 : Set; Con13
= (Con13 : Set)
(nil : Con13)
(snoc : Con13 → Ty13 → Con13)
→ Con13
nil13 : Con13; nil13
= λ Con13 nil13 snoc → nil13
snoc13 : Con13 → Ty13 → Con13; snoc13
= λ Γ A Con13 nil13 snoc13 → snoc13 (Γ Con13 nil13 snoc13) A
Var13 : Con13 → Ty13 → Set; Var13
= λ Γ A →
(Var13 : Con13 → Ty13 → Set)
(vz : ∀ Γ A → Var13 (snoc13 Γ A) A)
(vs : ∀ Γ B A → Var13 Γ A → Var13 (snoc13 Γ B) A)
→ Var13 Γ A
vz13 : ∀{Γ A} → Var13 (snoc13 Γ A) A; vz13
= λ Var13 vz13 vs → vz13 _ _
vs13 : ∀{Γ B A} → Var13 Γ A → Var13 (snoc13 Γ B) A; vs13
= λ x Var13 vz13 vs13 → vs13 _ _ _ (x Var13 vz13 vs13)
Tm13 : Con13 → Ty13 → Set; Tm13
= λ Γ A →
(Tm13 : Con13 → Ty13 → Set)
(var : ∀ Γ A → Var13 Γ A → Tm13 Γ A)
(lam : ∀ Γ A B → Tm13 (snoc13 Γ A) B → Tm13 Γ (arr13 A B))
(app : ∀ Γ A B → Tm13 Γ (arr13 A B) → Tm13 Γ A → Tm13 Γ B)
(tt : ∀ Γ → Tm13 Γ top13)
(pair : ∀ Γ A B → Tm13 Γ A → Tm13 Γ B → Tm13 Γ (prod13 A B))
(fst : ∀ Γ A B → Tm13 Γ (prod13 A B) → Tm13 Γ A)
(snd : ∀ Γ A B → Tm13 Γ (prod13 A B) → Tm13 Γ B)
(left : ∀ Γ A B → Tm13 Γ A → Tm13 Γ (sum13 A B))
(right : ∀ Γ A B → Tm13 Γ B → Tm13 Γ (sum13 A B))
(case : ∀ Γ A B C → Tm13 Γ (sum13 A B) → Tm13 Γ (arr13 A C) → Tm13 Γ (arr13 B C) → Tm13 Γ C)
(zero : ∀ Γ → Tm13 Γ nat13)
(suc : ∀ Γ → Tm13 Γ nat13 → Tm13 Γ nat13)
(rec : ∀ Γ A → Tm13 Γ nat13 → Tm13 Γ (arr13 nat13 (arr13 A A)) → Tm13 Γ A → Tm13 Γ A)
→ Tm13 Γ A
var13 : ∀{Γ A} → Var13 Γ A → Tm13 Γ A; var13
= λ x Tm13 var13 lam app tt pair fst snd left right case zero suc rec →
var13 _ _ x
lam13 : ∀{Γ A B} → Tm13 (snoc13 Γ A) B → Tm13 Γ (arr13 A B); lam13
= λ t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec →
lam13 _ _ _ (t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec)
app13 : ∀{Γ A B} → Tm13 Γ (arr13 A B) → Tm13 Γ A → Tm13 Γ B; app13
= λ t u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec →
app13 _ _ _ (t Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
tt13 : ∀{Γ} → Tm13 Γ top13; tt13
= λ Tm13 var13 lam13 app13 tt13 pair fst snd left right case zero suc rec → tt13 _
pair13 : ∀{Γ A B} → Tm13 Γ A → Tm13 Γ B → Tm13 Γ (prod13 A B); pair13
= λ t u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec →
pair13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
fst13 : ∀{Γ A B} → Tm13 Γ (prod13 A B) → Tm13 Γ A; fst13
= λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec →
fst13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec)
snd13 : ∀{Γ A B} → Tm13 Γ (prod13 A B) → Tm13 Γ B; snd13
= λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec →
snd13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec)
left13 : ∀{Γ A B} → Tm13 Γ A → Tm13 Γ (sum13 A B); left13
= λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec →
left13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec)
right13 : ∀{Γ A B} → Tm13 Γ B → Tm13 Γ (sum13 A B); right13
= λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec →
right13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec)
case13 : ∀{Γ A B C} → Tm13 Γ (sum13 A B) → Tm13 Γ (arr13 A C) → Tm13 Γ (arr13 B C) → Tm13 Γ C; case13
= λ t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec →
case13 _ _ _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
zero13 : ∀{Γ} → Tm13 Γ nat13; zero13
= λ Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc rec → zero13 _
suc13 : ∀{Γ} → Tm13 Γ nat13 → Tm13 Γ nat13; suc13
= λ t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec →
suc13 _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec)
rec13 : ∀{Γ A} → Tm13 Γ nat13 → Tm13 Γ (arr13 nat13 (arr13 A A)) → Tm13 Γ A → Tm13 Γ A; rec13
= λ t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13 →
rec13 _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
v013 : ∀{Γ A} → Tm13 (snoc13 Γ A) A; v013
= var13 vz13
v113 : ∀{Γ A B} → Tm13 (snoc13 (snoc13 Γ A) B) A; v113
= var13 (vs13 vz13)
v213 : ∀{Γ A B C} → Tm13 (snoc13 (snoc13 (snoc13 Γ A) B) C) A; v213
= var13 (vs13 (vs13 vz13))
v313 : ∀{Γ A B C D} → Tm13 (snoc13 (snoc13 (snoc13 (snoc13 Γ A) B) C) D) A; v313
= var13 (vs13 (vs13 (vs13 vz13)))
tbool13 : Ty13; tbool13
= sum13 top13 top13
true13 : ∀{Γ} → Tm13 Γ tbool13; true13
= left13 tt13
tfalse13 : ∀{Γ} → Tm13 Γ tbool13; tfalse13
= right13 tt13
ifthenelse13 : ∀{Γ A} → Tm13 Γ (arr13 tbool13 (arr13 A (arr13 A A))); ifthenelse13
= lam13 (lam13 (lam13 (case13 v213 (lam13 v213) (lam13 v113))))
times413 : ∀{Γ A} → Tm13 Γ (arr13 (arr13 A A) (arr13 A A)); times413
= lam13 (lam13 (app13 v113 (app13 v113 (app13 v113 (app13 v113 v013)))))
add13 : ∀{Γ} → Tm13 Γ (arr13 nat13 (arr13 nat13 nat13)); add13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (suc13 (app13 v113 v013)))))
(lam13 v013))
mul13 : ∀{Γ} → Tm13 Γ (arr13 nat13 (arr13 nat13 nat13)); mul13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (app13 (app13 add13 (app13 v113 v013)) v013))))
(lam13 zero13))
fact13 : ∀{Γ} → Tm13 Γ (arr13 nat13 nat13); fact13
= lam13 (rec13 v013 (lam13 (lam13 (app13 (app13 mul13 (suc13 v113)) v013)))
(suc13 zero13))
{-# OPTIONS --type-in-type #-}
Ty14 : Set
Ty14 =
(Ty14 : Set)
(nat top bot : Ty14)
(arr prod sum : Ty14 → Ty14 → Ty14)
→ Ty14
nat14 : Ty14; nat14 = λ _ nat14 _ _ _ _ _ → nat14
top14 : Ty14; top14 = λ _ _ top14 _ _ _ _ → top14
bot14 : Ty14; bot14 = λ _ _ _ bot14 _ _ _ → bot14
arr14 : Ty14 → Ty14 → Ty14; arr14
= λ A B Ty14 nat14 top14 bot14 arr14 prod sum →
arr14 (A Ty14 nat14 top14 bot14 arr14 prod sum) (B Ty14 nat14 top14 bot14 arr14 prod sum)
prod14 : Ty14 → Ty14 → Ty14; prod14
= λ A B Ty14 nat14 top14 bot14 arr14 prod14 sum →
prod14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum) (B Ty14 nat14 top14 bot14 arr14 prod14 sum)
sum14 : Ty14 → Ty14 → Ty14; sum14
= λ A B Ty14 nat14 top14 bot14 arr14 prod14 sum14 →
sum14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum14) (B Ty14 nat14 top14 bot14 arr14 prod14 sum14)
Con14 : Set; Con14
= (Con14 : Set)
(nil : Con14)
(snoc : Con14 → Ty14 → Con14)
→ Con14
nil14 : Con14; nil14
= λ Con14 nil14 snoc → nil14
snoc14 : Con14 → Ty14 → Con14; snoc14
= λ Γ A Con14 nil14 snoc14 → snoc14 (Γ Con14 nil14 snoc14) A
Var14 : Con14 → Ty14 → Set; Var14
= λ Γ A →
(Var14 : Con14 → Ty14 → Set)
(vz : ∀ Γ A → Var14 (snoc14 Γ A) A)
(vs : ∀ Γ B A → Var14 Γ A → Var14 (snoc14 Γ B) A)
→ Var14 Γ A
vz14 : ∀{Γ A} → Var14 (snoc14 Γ A) A; vz14
= λ Var14 vz14 vs → vz14 _ _
vs14 : ∀{Γ B A} → Var14 Γ A → Var14 (snoc14 Γ B) A; vs14
= λ x Var14 vz14 vs14 → vs14 _ _ _ (x Var14 vz14 vs14)
Tm14 : Con14 → Ty14 → Set; Tm14
= λ Γ A →
(Tm14 : Con14 → Ty14 → Set)
(var : ∀ Γ A → Var14 Γ A → Tm14 Γ A)
(lam : ∀ Γ A B → Tm14 (snoc14 Γ A) B → Tm14 Γ (arr14 A B))
(app : ∀ Γ A B → Tm14 Γ (arr14 A B) → Tm14 Γ A → Tm14 Γ B)
(tt : ∀ Γ → Tm14 Γ top14)
(pair : ∀ Γ A B → Tm14 Γ A → Tm14 Γ B → Tm14 Γ (prod14 A B))
(fst : ∀ Γ A B → Tm14 Γ (prod14 A B) → Tm14 Γ A)
(snd : ∀ Γ A B → Tm14 Γ (prod14 A B) → Tm14 Γ B)
(left : ∀ Γ A B → Tm14 Γ A → Tm14 Γ (sum14 A B))
(right : ∀ Γ A B → Tm14 Γ B → Tm14 Γ (sum14 A B))
(case : ∀ Γ A B C → Tm14 Γ (sum14 A B) → Tm14 Γ (arr14 A C) → Tm14 Γ (arr14 B C) → Tm14 Γ C)
(zero : ∀ Γ → Tm14 Γ nat14)
(suc : ∀ Γ → Tm14 Γ nat14 → Tm14 Γ nat14)
(rec : ∀ Γ A → Tm14 Γ nat14 → Tm14 Γ (arr14 nat14 (arr14 A A)) → Tm14 Γ A → Tm14 Γ A)
→ Tm14 Γ A
var14 : ∀{Γ A} → Var14 Γ A → Tm14 Γ A; var14
= λ x Tm14 var14 lam app tt pair fst snd left right case zero suc rec →
var14 _ _ x
lam14 : ∀{Γ A B} → Tm14 (snoc14 Γ A) B → Tm14 Γ (arr14 A B); lam14
= λ t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec →
lam14 _ _ _ (t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec)
app14 : ∀{Γ A B} → Tm14 Γ (arr14 A B) → Tm14 Γ A → Tm14 Γ B; app14
= λ t u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec →
app14 _ _ _ (t Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
tt14 : ∀{Γ} → Tm14 Γ top14; tt14
= λ Tm14 var14 lam14 app14 tt14 pair fst snd left right case zero suc rec → tt14 _
pair14 : ∀{Γ A B} → Tm14 Γ A → Tm14 Γ B → Tm14 Γ (prod14 A B); pair14
= λ t u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec →
pair14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
fst14 : ∀{Γ A B} → Tm14 Γ (prod14 A B) → Tm14 Γ A; fst14
= λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec →
fst14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec)
snd14 : ∀{Γ A B} → Tm14 Γ (prod14 A B) → Tm14 Γ B; snd14
= λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec →
snd14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec)
left14 : ∀{Γ A B} → Tm14 Γ A → Tm14 Γ (sum14 A B); left14
= λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec →
left14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec)
right14 : ∀{Γ A B} → Tm14 Γ B → Tm14 Γ (sum14 A B); right14
= λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec →
right14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec)
case14 : ∀{Γ A B C} → Tm14 Γ (sum14 A B) → Tm14 Γ (arr14 A C) → Tm14 Γ (arr14 B C) → Tm14 Γ C; case14
= λ t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec →
case14 _ _ _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
zero14 : ∀{Γ} → Tm14 Γ nat14; zero14
= λ Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc rec → zero14 _
suc14 : ∀{Γ} → Tm14 Γ nat14 → Tm14 Γ nat14; suc14
= λ t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec →
suc14 _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec)
rec14 : ∀{Γ A} → Tm14 Γ nat14 → Tm14 Γ (arr14 nat14 (arr14 A A)) → Tm14 Γ A → Tm14 Γ A; rec14
= λ t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14 →
rec14 _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
v014 : ∀{Γ A} → Tm14 (snoc14 Γ A) A; v014
= var14 vz14
v114 : ∀{Γ A B} → Tm14 (snoc14 (snoc14 Γ A) B) A; v114
= var14 (vs14 vz14)
v214 : ∀{Γ A B C} → Tm14 (snoc14 (snoc14 (snoc14 Γ A) B) C) A; v214
= var14 (vs14 (vs14 vz14))
v314 : ∀{Γ A B C D} → Tm14 (snoc14 (snoc14 (snoc14 (snoc14 Γ A) B) C) D) A; v314
= var14 (vs14 (vs14 (vs14 vz14)))
tbool14 : Ty14; tbool14
= sum14 top14 top14
true14 : ∀{Γ} → Tm14 Γ tbool14; true14
= left14 tt14
tfalse14 : ∀{Γ} → Tm14 Γ tbool14; tfalse14
= right14 tt14
ifthenelse14 : ∀{Γ A} → Tm14 Γ (arr14 tbool14 (arr14 A (arr14 A A))); ifthenelse14
= lam14 (lam14 (lam14 (case14 v214 (lam14 v214) (lam14 v114))))
times414 : ∀{Γ A} → Tm14 Γ (arr14 (arr14 A A) (arr14 A A)); times414
= lam14 (lam14 (app14 v114 (app14 v114 (app14 v114 (app14 v114 v014)))))
add14 : ∀{Γ} → Tm14 Γ (arr14 nat14 (arr14 nat14 nat14)); add14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (suc14 (app14 v114 v014)))))
(lam14 v014))
mul14 : ∀{Γ} → Tm14 Γ (arr14 nat14 (arr14 nat14 nat14)); mul14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (app14 (app14 add14 (app14 v114 v014)) v014))))
(lam14 zero14))
fact14 : ∀{Γ} → Tm14 Γ (arr14 nat14 nat14); fact14
= lam14 (rec14 v014 (lam14 (lam14 (app14 (app14 mul14 (suc14 v114)) v014)))
(suc14 zero14))
{-# OPTIONS --type-in-type #-}
Ty15 : Set
Ty15 =
(Ty15 : Set)
(nat top bot : Ty15)
(arr prod sum : Ty15 → Ty15 → Ty15)
→ Ty15
nat15 : Ty15; nat15 = λ _ nat15 _ _ _ _ _ → nat15
top15 : Ty15; top15 = λ _ _ top15 _ _ _ _ → top15
bot15 : Ty15; bot15 = λ _ _ _ bot15 _ _ _ → bot15
arr15 : Ty15 → Ty15 → Ty15; arr15
= λ A B Ty15 nat15 top15 bot15 arr15 prod sum →
arr15 (A Ty15 nat15 top15 bot15 arr15 prod sum) (B Ty15 nat15 top15 bot15 arr15 prod sum)
prod15 : Ty15 → Ty15 → Ty15; prod15
= λ A B Ty15 nat15 top15 bot15 arr15 prod15 sum →
prod15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum) (B Ty15 nat15 top15 bot15 arr15 prod15 sum)
sum15 : Ty15 → Ty15 → Ty15; sum15
= λ A B Ty15 nat15 top15 bot15 arr15 prod15 sum15 →
sum15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum15) (B Ty15 nat15 top15 bot15 arr15 prod15 sum15)
Con15 : Set; Con15
= (Con15 : Set)
(nil : Con15)
(snoc : Con15 → Ty15 → Con15)
→ Con15
nil15 : Con15; nil15
= λ Con15 nil15 snoc → nil15
snoc15 : Con15 → Ty15 → Con15; snoc15
= λ Γ A Con15 nil15 snoc15 → snoc15 (Γ Con15 nil15 snoc15) A
Var15 : Con15 → Ty15 → Set; Var15
= λ Γ A →
(Var15 : Con15 → Ty15 → Set)
(vz : ∀ Γ A → Var15 (snoc15 Γ A) A)
(vs : ∀ Γ B A → Var15 Γ A → Var15 (snoc15 Γ B) A)
→ Var15 Γ A
vz15 : ∀{Γ A} → Var15 (snoc15 Γ A) A; vz15
= λ Var15 vz15 vs → vz15 _ _
vs15 : ∀{Γ B A} → Var15 Γ A → Var15 (snoc15 Γ B) A; vs15
= λ x Var15 vz15 vs15 → vs15 _ _ _ (x Var15 vz15 vs15)
Tm15 : Con15 → Ty15 → Set; Tm15
= λ Γ A →
(Tm15 : Con15 → Ty15 → Set)
(var : ∀ Γ A → Var15 Γ A → Tm15 Γ A)
(lam : ∀ Γ A B → Tm15 (snoc15 Γ A) B → Tm15 Γ (arr15 A B))
(app : ∀ Γ A B → Tm15 Γ (arr15 A B) → Tm15 Γ A → Tm15 Γ B)
(tt : ∀ Γ → Tm15 Γ top15)
(pair : ∀ Γ A B → Tm15 Γ A → Tm15 Γ B → Tm15 Γ (prod15 A B))
(fst : ∀ Γ A B → Tm15 Γ (prod15 A B) → Tm15 Γ A)
(snd : ∀ Γ A B → Tm15 Γ (prod15 A B) → Tm15 Γ B)
(left : ∀ Γ A B → Tm15 Γ A → Tm15 Γ (sum15 A B))
(right : ∀ Γ A B → Tm15 Γ B → Tm15 Γ (sum15 A B))
(case : ∀ Γ A B C → Tm15 Γ (sum15 A B) → Tm15 Γ (arr15 A C) → Tm15 Γ (arr15 B C) → Tm15 Γ C)
(zero : ∀ Γ → Tm15 Γ nat15)
(suc : ∀ Γ → Tm15 Γ nat15 → Tm15 Γ nat15)
(rec : ∀ Γ A → Tm15 Γ nat15 → Tm15 Γ (arr15 nat15 (arr15 A A)) → Tm15 Γ A → Tm15 Γ A)
→ Tm15 Γ A
var15 : ∀{Γ A} → Var15 Γ A → Tm15 Γ A; var15
= λ x Tm15 var15 lam app tt pair fst snd left right case zero suc rec →
var15 _ _ x
lam15 : ∀{Γ A B} → Tm15 (snoc15 Γ A) B → Tm15 Γ (arr15 A B); lam15
= λ t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec →
lam15 _ _ _ (t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec)
app15 : ∀{Γ A B} → Tm15 Γ (arr15 A B) → Tm15 Γ A → Tm15 Γ B; app15
= λ t u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec →
app15 _ _ _ (t Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
tt15 : ∀{Γ} → Tm15 Γ top15; tt15
= λ Tm15 var15 lam15 app15 tt15 pair fst snd left right case zero suc rec → tt15 _
pair15 : ∀{Γ A B} → Tm15 Γ A → Tm15 Γ B → Tm15 Γ (prod15 A B); pair15
= λ t u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec →
pair15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
fst15 : ∀{Γ A B} → Tm15 Γ (prod15 A B) → Tm15 Γ A; fst15
= λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec →
fst15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec)
snd15 : ∀{Γ A B} → Tm15 Γ (prod15 A B) → Tm15 Γ B; snd15
= λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec →
snd15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec)
left15 : ∀{Γ A B} → Tm15 Γ A → Tm15 Γ (sum15 A B); left15
= λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec →
left15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec)
right15 : ∀{Γ A B} → Tm15 Γ B → Tm15 Γ (sum15 A B); right15
= λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec →
right15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec)
case15 : ∀{Γ A B C} → Tm15 Γ (sum15 A B) → Tm15 Γ (arr15 A C) → Tm15 Γ (arr15 B C) → Tm15 Γ C; case15
= λ t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec →
case15 _ _ _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
zero15 : ∀{Γ} → Tm15 Γ nat15; zero15
= λ Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc rec → zero15 _
suc15 : ∀{Γ} → Tm15 Γ nat15 → Tm15 Γ nat15; suc15
= λ t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec →
suc15 _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec)
rec15 : ∀{Γ A} → Tm15 Γ nat15 → Tm15 Γ (arr15 nat15 (arr15 A A)) → Tm15 Γ A → Tm15 Γ A; rec15
= λ t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15 →
rec15 _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
v015 : ∀{Γ A} → Tm15 (snoc15 Γ A) A; v015
= var15 vz15
v115 : ∀{Γ A B} → Tm15 (snoc15 (snoc15 Γ A) B) A; v115
= var15 (vs15 vz15)
v215 : ∀{Γ A B C} → Tm15 (snoc15 (snoc15 (snoc15 Γ A) B) C) A; v215
= var15 (vs15 (vs15 vz15))
v315 : ∀{Γ A B C D} → Tm15 (snoc15 (snoc15 (snoc15 (snoc15 Γ A) B) C) D) A; v315
= var15 (vs15 (vs15 (vs15 vz15)))
tbool15 : Ty15; tbool15
= sum15 top15 top15
true15 : ∀{Γ} → Tm15 Γ tbool15; true15
= left15 tt15
tfalse15 : ∀{Γ} → Tm15 Γ tbool15; tfalse15
= right15 tt15
ifthenelse15 : ∀{Γ A} → Tm15 Γ (arr15 tbool15 (arr15 A (arr15 A A))); ifthenelse15
= lam15 (lam15 (lam15 (case15 v215 (lam15 v215) (lam15 v115))))
times415 : ∀{Γ A} → Tm15 Γ (arr15 (arr15 A A) (arr15 A A)); times415
= lam15 (lam15 (app15 v115 (app15 v115 (app15 v115 (app15 v115 v015)))))
add15 : ∀{Γ} → Tm15 Γ (arr15 nat15 (arr15 nat15 nat15)); add15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (suc15 (app15 v115 v015)))))
(lam15 v015))
mul15 : ∀{Γ} → Tm15 Γ (arr15 nat15 (arr15 nat15 nat15)); mul15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (app15 (app15 add15 (app15 v115 v015)) v015))))
(lam15 zero15))
fact15 : ∀{Γ} → Tm15 Γ (arr15 nat15 nat15); fact15
= lam15 (rec15 v015 (lam15 (lam15 (app15 (app15 mul15 (suc15 v115)) v015)))
(suc15 zero15))
{-# OPTIONS --type-in-type #-}
Ty16 : Set
Ty16 =
(Ty16 : Set)
(nat top bot : Ty16)
(arr prod sum : Ty16 → Ty16 → Ty16)
→ Ty16
nat16 : Ty16; nat16 = λ _ nat16 _ _ _ _ _ → nat16
top16 : Ty16; top16 = λ _ _ top16 _ _ _ _ → top16
bot16 : Ty16; bot16 = λ _ _ _ bot16 _ _ _ → bot16
arr16 : Ty16 → Ty16 → Ty16; arr16
= λ A B Ty16 nat16 top16 bot16 arr16 prod sum →
arr16 (A Ty16 nat16 top16 bot16 arr16 prod sum) (B Ty16 nat16 top16 bot16 arr16 prod sum)
prod16 : Ty16 → Ty16 → Ty16; prod16
= λ A B Ty16 nat16 top16 bot16 arr16 prod16 sum →
prod16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum) (B Ty16 nat16 top16 bot16 arr16 prod16 sum)
sum16 : Ty16 → Ty16 → Ty16; sum16
= λ A B Ty16 nat16 top16 bot16 arr16 prod16 sum16 →
sum16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum16) (B Ty16 nat16 top16 bot16 arr16 prod16 sum16)
Con16 : Set; Con16
= (Con16 : Set)
(nil : Con16)
(snoc : Con16 → Ty16 → Con16)
→ Con16
nil16 : Con16; nil16
= λ Con16 nil16 snoc → nil16
snoc16 : Con16 → Ty16 → Con16; snoc16
= λ Γ A Con16 nil16 snoc16 → snoc16 (Γ Con16 nil16 snoc16) A
Var16 : Con16 → Ty16 → Set; Var16
= λ Γ A →
(Var16 : Con16 → Ty16 → Set)
(vz : ∀ Γ A → Var16 (snoc16 Γ A) A)
(vs : ∀ Γ B A → Var16 Γ A → Var16 (snoc16 Γ B) A)
→ Var16 Γ A
vz16 : ∀{Γ A} → Var16 (snoc16 Γ A) A; vz16
= λ Var16 vz16 vs → vz16 _ _
vs16 : ∀{Γ B A} → Var16 Γ A → Var16 (snoc16 Γ B) A; vs16
= λ x Var16 vz16 vs16 → vs16 _ _ _ (x Var16 vz16 vs16)
Tm16 : Con16 → Ty16 → Set; Tm16
= λ Γ A →
(Tm16 : Con16 → Ty16 → Set)
(var : ∀ Γ A → Var16 Γ A → Tm16 Γ A)
(lam : ∀ Γ A B → Tm16 (snoc16 Γ A) B → Tm16 Γ (arr16 A B))
(app : ∀ Γ A B → Tm16 Γ (arr16 A B) → Tm16 Γ A → Tm16 Γ B)
(tt : ∀ Γ → Tm16 Γ top16)
(pair : ∀ Γ A B → Tm16 Γ A → Tm16 Γ B → Tm16 Γ (prod16 A B))
(fst : ∀ Γ A B → Tm16 Γ (prod16 A B) → Tm16 Γ A)
(snd : ∀ Γ A B → Tm16 Γ (prod16 A B) → Tm16 Γ B)
(left : ∀ Γ A B → Tm16 Γ A → Tm16 Γ (sum16 A B))
(right : ∀ Γ A B → Tm16 Γ B → Tm16 Γ (sum16 A B))
(case : ∀ Γ A B C → Tm16 Γ (sum16 A B) → Tm16 Γ (arr16 A C) → Tm16 Γ (arr16 B C) → Tm16 Γ C)
(zero : ∀ Γ → Tm16 Γ nat16)
(suc : ∀ Γ → Tm16 Γ nat16 → Tm16 Γ nat16)
(rec : ∀ Γ A → Tm16 Γ nat16 → Tm16 Γ (arr16 nat16 (arr16 A A)) → Tm16 Γ A → Tm16 Γ A)
→ Tm16 Γ A
var16 : ∀{Γ A} → Var16 Γ A → Tm16 Γ A; var16
= λ x Tm16 var16 lam app tt pair fst snd left right case zero suc rec →
var16 _ _ x
lam16 : ∀{Γ A B} → Tm16 (snoc16 Γ A) B → Tm16 Γ (arr16 A B); lam16
= λ t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec →
lam16 _ _ _ (t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec)
app16 : ∀{Γ A B} → Tm16 Γ (arr16 A B) → Tm16 Γ A → Tm16 Γ B; app16
= λ t u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec →
app16 _ _ _ (t Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
tt16 : ∀{Γ} → Tm16 Γ top16; tt16
= λ Tm16 var16 lam16 app16 tt16 pair fst snd left right case zero suc rec → tt16 _
pair16 : ∀{Γ A B} → Tm16 Γ A → Tm16 Γ B → Tm16 Γ (prod16 A B); pair16
= λ t u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec →
pair16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
fst16 : ∀{Γ A B} → Tm16 Γ (prod16 A B) → Tm16 Γ A; fst16
= λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec →
fst16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec)
snd16 : ∀{Γ A B} → Tm16 Γ (prod16 A B) → Tm16 Γ B; snd16
= λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec →
snd16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec)
left16 : ∀{Γ A B} → Tm16 Γ A → Tm16 Γ (sum16 A B); left16
= λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec →
left16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec)
right16 : ∀{Γ A B} → Tm16 Γ B → Tm16 Γ (sum16 A B); right16
= λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec →
right16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec)
case16 : ∀{Γ A B C} → Tm16 Γ (sum16 A B) → Tm16 Γ (arr16 A C) → Tm16 Γ (arr16 B C) → Tm16 Γ C; case16
= λ t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec →
case16 _ _ _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
zero16 : ∀{Γ} → Tm16 Γ nat16; zero16
= λ Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc rec → zero16 _
suc16 : ∀{Γ} → Tm16 Γ nat16 → Tm16 Γ nat16; suc16
= λ t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec →
suc16 _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec)
rec16 : ∀{Γ A} → Tm16 Γ nat16 → Tm16 Γ (arr16 nat16 (arr16 A A)) → Tm16 Γ A → Tm16 Γ A; rec16
= λ t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16 →
rec16 _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
v016 : ∀{Γ A} → Tm16 (snoc16 Γ A) A; v016
= var16 vz16
v116 : ∀{Γ A B} → Tm16 (snoc16 (snoc16 Γ A) B) A; v116
= var16 (vs16 vz16)
v216 : ∀{Γ A B C} → Tm16 (snoc16 (snoc16 (snoc16 Γ A) B) C) A; v216
= var16 (vs16 (vs16 vz16))
v316 : ∀{Γ A B C D} → Tm16 (snoc16 (snoc16 (snoc16 (snoc16 Γ A) B) C) D) A; v316
= var16 (vs16 (vs16 (vs16 vz16)))
tbool16 : Ty16; tbool16
= sum16 top16 top16
true16 : ∀{Γ} → Tm16 Γ tbool16; true16
= left16 tt16
tfalse16 : ∀{Γ} → Tm16 Γ tbool16; tfalse16
= right16 tt16
ifthenelse16 : ∀{Γ A} → Tm16 Γ (arr16 tbool16 (arr16 A (arr16 A A))); ifthenelse16
= lam16 (lam16 (lam16 (case16 v216 (lam16 v216) (lam16 v116))))
times416 : ∀{Γ A} → Tm16 Γ (arr16 (arr16 A A) (arr16 A A)); times416
= lam16 (lam16 (app16 v116 (app16 v116 (app16 v116 (app16 v116 v016)))))
add16 : ∀{Γ} → Tm16 Γ (arr16 nat16 (arr16 nat16 nat16)); add16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (suc16 (app16 v116 v016)))))
(lam16 v016))
mul16 : ∀{Γ} → Tm16 Γ (arr16 nat16 (arr16 nat16 nat16)); mul16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (app16 (app16 add16 (app16 v116 v016)) v016))))
(lam16 zero16))
fact16 : ∀{Γ} → Tm16 Γ (arr16 nat16 nat16); fact16
= lam16 (rec16 v016 (lam16 (lam16 (app16 (app16 mul16 (suc16 v116)) v016)))
(suc16 zero16))
{-# OPTIONS --type-in-type #-}
Ty17 : Set
Ty17 =
(Ty17 : Set)
(nat top bot : Ty17)
(arr prod sum : Ty17 → Ty17 → Ty17)
→ Ty17
nat17 : Ty17; nat17 = λ _ nat17 _ _ _ _ _ → nat17
top17 : Ty17; top17 = λ _ _ top17 _ _ _ _ → top17
bot17 : Ty17; bot17 = λ _ _ _ bot17 _ _ _ → bot17
arr17 : Ty17 → Ty17 → Ty17; arr17
= λ A B Ty17 nat17 top17 bot17 arr17 prod sum →
arr17 (A Ty17 nat17 top17 bot17 arr17 prod sum) (B Ty17 nat17 top17 bot17 arr17 prod sum)
prod17 : Ty17 → Ty17 → Ty17; prod17
= λ A B Ty17 nat17 top17 bot17 arr17 prod17 sum →
prod17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum) (B Ty17 nat17 top17 bot17 arr17 prod17 sum)
sum17 : Ty17 → Ty17 → Ty17; sum17
= λ A B Ty17 nat17 top17 bot17 arr17 prod17 sum17 →
sum17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum17) (B Ty17 nat17 top17 bot17 arr17 prod17 sum17)
Con17 : Set; Con17
= (Con17 : Set)
(nil : Con17)
(snoc : Con17 → Ty17 → Con17)
→ Con17
nil17 : Con17; nil17
= λ Con17 nil17 snoc → nil17
snoc17 : Con17 → Ty17 → Con17; snoc17
= λ Γ A Con17 nil17 snoc17 → snoc17 (Γ Con17 nil17 snoc17) A
Var17 : Con17 → Ty17 → Set; Var17
= λ Γ A →
(Var17 : Con17 → Ty17 → Set)
(vz : ∀ Γ A → Var17 (snoc17 Γ A) A)
(vs : ∀ Γ B A → Var17 Γ A → Var17 (snoc17 Γ B) A)
→ Var17 Γ A
vz17 : ∀{Γ A} → Var17 (snoc17 Γ A) A; vz17
= λ Var17 vz17 vs → vz17 _ _
vs17 : ∀{Γ B A} → Var17 Γ A → Var17 (snoc17 Γ B) A; vs17
= λ x Var17 vz17 vs17 → vs17 _ _ _ (x Var17 vz17 vs17)
Tm17 : Con17 → Ty17 → Set; Tm17
= λ Γ A →
(Tm17 : Con17 → Ty17 → Set)
(var : ∀ Γ A → Var17 Γ A → Tm17 Γ A)
(lam : ∀ Γ A B → Tm17 (snoc17 Γ A) B → Tm17 Γ (arr17 A B))
(app : ∀ Γ A B → Tm17 Γ (arr17 A B) → Tm17 Γ A → Tm17 Γ B)
(tt : ∀ Γ → Tm17 Γ top17)
(pair : ∀ Γ A B → Tm17 Γ A → Tm17 Γ B → Tm17 Γ (prod17 A B))
(fst : ∀ Γ A B → Tm17 Γ (prod17 A B) → Tm17 Γ A)
(snd : ∀ Γ A B → Tm17 Γ (prod17 A B) → Tm17 Γ B)
(left : ∀ Γ A B → Tm17 Γ A → Tm17 Γ (sum17 A B))
(right : ∀ Γ A B → Tm17 Γ B → Tm17 Γ (sum17 A B))
(case : ∀ Γ A B C → Tm17 Γ (sum17 A B) → Tm17 Γ (arr17 A C) → Tm17 Γ (arr17 B C) → Tm17 Γ C)
(zero : ∀ Γ → Tm17 Γ nat17)
(suc : ∀ Γ → Tm17 Γ nat17 → Tm17 Γ nat17)
(rec : ∀ Γ A → Tm17 Γ nat17 → Tm17 Γ (arr17 nat17 (arr17 A A)) → Tm17 Γ A → Tm17 Γ A)
→ Tm17 Γ A
var17 : ∀{Γ A} → Var17 Γ A → Tm17 Γ A; var17
= λ x Tm17 var17 lam app tt pair fst snd left right case zero suc rec →
var17 _ _ x
lam17 : ∀{Γ A B} → Tm17 (snoc17 Γ A) B → Tm17 Γ (arr17 A B); lam17
= λ t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec →
lam17 _ _ _ (t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec)
app17 : ∀{Γ A B} → Tm17 Γ (arr17 A B) → Tm17 Γ A → Tm17 Γ B; app17
= λ t u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec →
app17 _ _ _ (t Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
tt17 : ∀{Γ} → Tm17 Γ top17; tt17
= λ Tm17 var17 lam17 app17 tt17 pair fst snd left right case zero suc rec → tt17 _
pair17 : ∀{Γ A B} → Tm17 Γ A → Tm17 Γ B → Tm17 Γ (prod17 A B); pair17
= λ t u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec →
pair17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
fst17 : ∀{Γ A B} → Tm17 Γ (prod17 A B) → Tm17 Γ A; fst17
= λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec →
fst17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec)
snd17 : ∀{Γ A B} → Tm17 Γ (prod17 A B) → Tm17 Γ B; snd17
= λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec →
snd17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec)
left17 : ∀{Γ A B} → Tm17 Γ A → Tm17 Γ (sum17 A B); left17
= λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec →
left17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec)
right17 : ∀{Γ A B} → Tm17 Γ B → Tm17 Γ (sum17 A B); right17
= λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec →
right17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec)
case17 : ∀{Γ A B C} → Tm17 Γ (sum17 A B) → Tm17 Γ (arr17 A C) → Tm17 Γ (arr17 B C) → Tm17 Γ C; case17
= λ t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec →
case17 _ _ _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
zero17 : ∀{Γ} → Tm17 Γ nat17; zero17
= λ Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc rec → zero17 _
suc17 : ∀{Γ} → Tm17 Γ nat17 → Tm17 Γ nat17; suc17
= λ t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec →
suc17 _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec)
rec17 : ∀{Γ A} → Tm17 Γ nat17 → Tm17 Γ (arr17 nat17 (arr17 A A)) → Tm17 Γ A → Tm17 Γ A; rec17
= λ t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17 →
rec17 _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
v017 : ∀{Γ A} → Tm17 (snoc17 Γ A) A; v017
= var17 vz17
v117 : ∀{Γ A B} → Tm17 (snoc17 (snoc17 Γ A) B) A; v117
= var17 (vs17 vz17)
v217 : ∀{Γ A B C} → Tm17 (snoc17 (snoc17 (snoc17 Γ A) B) C) A; v217
= var17 (vs17 (vs17 vz17))
v317 : ∀{Γ A B C D} → Tm17 (snoc17 (snoc17 (snoc17 (snoc17 Γ A) B) C) D) A; v317
= var17 (vs17 (vs17 (vs17 vz17)))
tbool17 : Ty17; tbool17
= sum17 top17 top17
true17 : ∀{Γ} → Tm17 Γ tbool17; true17
= left17 tt17
tfalse17 : ∀{Γ} → Tm17 Γ tbool17; tfalse17
= right17 tt17
ifthenelse17 : ∀{Γ A} → Tm17 Γ (arr17 tbool17 (arr17 A (arr17 A A))); ifthenelse17
= lam17 (lam17 (lam17 (case17 v217 (lam17 v217) (lam17 v117))))
times417 : ∀{Γ A} → Tm17 Γ (arr17 (arr17 A A) (arr17 A A)); times417
= lam17 (lam17 (app17 v117 (app17 v117 (app17 v117 (app17 v117 v017)))))
add17 : ∀{Γ} → Tm17 Γ (arr17 nat17 (arr17 nat17 nat17)); add17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (suc17 (app17 v117 v017)))))
(lam17 v017))
mul17 : ∀{Γ} → Tm17 Γ (arr17 nat17 (arr17 nat17 nat17)); mul17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (app17 (app17 add17 (app17 v117 v017)) v017))))
(lam17 zero17))
fact17 : ∀{Γ} → Tm17 Γ (arr17 nat17 nat17); fact17
= lam17 (rec17 v017 (lam17 (lam17 (app17 (app17 mul17 (suc17 v117)) v017)))
(suc17 zero17))
{-# OPTIONS --type-in-type #-}
Ty18 : Set
Ty18 =
(Ty18 : Set)
(nat top bot : Ty18)
(arr prod sum : Ty18 → Ty18 → Ty18)
→ Ty18
nat18 : Ty18; nat18 = λ _ nat18 _ _ _ _ _ → nat18
top18 : Ty18; top18 = λ _ _ top18 _ _ _ _ → top18
bot18 : Ty18; bot18 = λ _ _ _ bot18 _ _ _ → bot18
arr18 : Ty18 → Ty18 → Ty18; arr18
= λ A B Ty18 nat18 top18 bot18 arr18 prod sum →
arr18 (A Ty18 nat18 top18 bot18 arr18 prod sum) (B Ty18 nat18 top18 bot18 arr18 prod sum)
prod18 : Ty18 → Ty18 → Ty18; prod18
= λ A B Ty18 nat18 top18 bot18 arr18 prod18 sum →
prod18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum) (B Ty18 nat18 top18 bot18 arr18 prod18 sum)
sum18 : Ty18 → Ty18 → Ty18; sum18
= λ A B Ty18 nat18 top18 bot18 arr18 prod18 sum18 →
sum18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum18) (B Ty18 nat18 top18 bot18 arr18 prod18 sum18)
Con18 : Set; Con18
= (Con18 : Set)
(nil : Con18)
(snoc : Con18 → Ty18 → Con18)
→ Con18
nil18 : Con18; nil18
= λ Con18 nil18 snoc → nil18
snoc18 : Con18 → Ty18 → Con18; snoc18
= λ Γ A Con18 nil18 snoc18 → snoc18 (Γ Con18 nil18 snoc18) A
Var18 : Con18 → Ty18 → Set; Var18
= λ Γ A →
(Var18 : Con18 → Ty18 → Set)
(vz : ∀ Γ A → Var18 (snoc18 Γ A) A)
(vs : ∀ Γ B A → Var18 Γ A → Var18 (snoc18 Γ B) A)
→ Var18 Γ A
vz18 : ∀{Γ A} → Var18 (snoc18 Γ A) A; vz18
= λ Var18 vz18 vs → vz18 _ _
vs18 : ∀{Γ B A} → Var18 Γ A → Var18 (snoc18 Γ B) A; vs18
= λ x Var18 vz18 vs18 → vs18 _ _ _ (x Var18 vz18 vs18)
Tm18 : Con18 → Ty18 → Set; Tm18
= λ Γ A →
(Tm18 : Con18 → Ty18 → Set)
(var : ∀ Γ A → Var18 Γ A → Tm18 Γ A)
(lam : ∀ Γ A B → Tm18 (snoc18 Γ A) B → Tm18 Γ (arr18 A B))
(app : ∀ Γ A B → Tm18 Γ (arr18 A B) → Tm18 Γ A → Tm18 Γ B)
(tt : ∀ Γ → Tm18 Γ top18)
(pair : ∀ Γ A B → Tm18 Γ A → Tm18 Γ B → Tm18 Γ (prod18 A B))
(fst : ∀ Γ A B → Tm18 Γ (prod18 A B) → Tm18 Γ A)
(snd : ∀ Γ A B → Tm18 Γ (prod18 A B) → Tm18 Γ B)
(left : ∀ Γ A B → Tm18 Γ A → Tm18 Γ (sum18 A B))
(right : ∀ Γ A B → Tm18 Γ B → Tm18 Γ (sum18 A B))
(case : ∀ Γ A B C → Tm18 Γ (sum18 A B) → Tm18 Γ (arr18 A C) → Tm18 Γ (arr18 B C) → Tm18 Γ C)
(zero : ∀ Γ → Tm18 Γ nat18)
(suc : ∀ Γ → Tm18 Γ nat18 → Tm18 Γ nat18)
(rec : ∀ Γ A → Tm18 Γ nat18 → Tm18 Γ (arr18 nat18 (arr18 A A)) → Tm18 Γ A → Tm18 Γ A)
→ Tm18 Γ A
var18 : ∀{Γ A} → Var18 Γ A → Tm18 Γ A; var18
= λ x Tm18 var18 lam app tt pair fst snd left right case zero suc rec →
var18 _ _ x
lam18 : ∀{Γ A B} → Tm18 (snoc18 Γ A) B → Tm18 Γ (arr18 A B); lam18
= λ t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec →
lam18 _ _ _ (t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec)
app18 : ∀{Γ A B} → Tm18 Γ (arr18 A B) → Tm18 Γ A → Tm18 Γ B; app18
= λ t u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec →
app18 _ _ _ (t Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
tt18 : ∀{Γ} → Tm18 Γ top18; tt18
= λ Tm18 var18 lam18 app18 tt18 pair fst snd left right case zero suc rec → tt18 _
pair18 : ∀{Γ A B} → Tm18 Γ A → Tm18 Γ B → Tm18 Γ (prod18 A B); pair18
= λ t u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec →
pair18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
fst18 : ∀{Γ A B} → Tm18 Γ (prod18 A B) → Tm18 Γ A; fst18
= λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec →
fst18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec)
snd18 : ∀{Γ A B} → Tm18 Γ (prod18 A B) → Tm18 Γ B; snd18
= λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec →
snd18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec)
left18 : ∀{Γ A B} → Tm18 Γ A → Tm18 Γ (sum18 A B); left18
= λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec →
left18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec)
right18 : ∀{Γ A B} → Tm18 Γ B → Tm18 Γ (sum18 A B); right18
= λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec →
right18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec)
case18 : ∀{Γ A B C} → Tm18 Γ (sum18 A B) → Tm18 Γ (arr18 A C) → Tm18 Γ (arr18 B C) → Tm18 Γ C; case18
= λ t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec →
case18 _ _ _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
zero18 : ∀{Γ} → Tm18 Γ nat18; zero18
= λ Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc rec → zero18 _
suc18 : ∀{Γ} → Tm18 Γ nat18 → Tm18 Γ nat18; suc18
= λ t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec →
suc18 _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec)
rec18 : ∀{Γ A} → Tm18 Γ nat18 → Tm18 Γ (arr18 nat18 (arr18 A A)) → Tm18 Γ A → Tm18 Γ A; rec18
= λ t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18 →
rec18 _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
v018 : ∀{Γ A} → Tm18 (snoc18 Γ A) A; v018
= var18 vz18
v118 : ∀{Γ A B} → Tm18 (snoc18 (snoc18 Γ A) B) A; v118
= var18 (vs18 vz18)
v218 : ∀{Γ A B C} → Tm18 (snoc18 (snoc18 (snoc18 Γ A) B) C) A; v218
= var18 (vs18 (vs18 vz18))
v318 : ∀{Γ A B C D} → Tm18 (snoc18 (snoc18 (snoc18 (snoc18 Γ A) B) C) D) A; v318
= var18 (vs18 (vs18 (vs18 vz18)))
tbool18 : Ty18; tbool18
= sum18 top18 top18
true18 : ∀{Γ} → Tm18 Γ tbool18; true18
= left18 tt18
tfalse18 : ∀{Γ} → Tm18 Γ tbool18; tfalse18
= right18 tt18
ifthenelse18 : ∀{Γ A} → Tm18 Γ (arr18 tbool18 (arr18 A (arr18 A A))); ifthenelse18
= lam18 (lam18 (lam18 (case18 v218 (lam18 v218) (lam18 v118))))
times418 : ∀{Γ A} → Tm18 Γ (arr18 (arr18 A A) (arr18 A A)); times418
= lam18 (lam18 (app18 v118 (app18 v118 (app18 v118 (app18 v118 v018)))))
add18 : ∀{Γ} → Tm18 Γ (arr18 nat18 (arr18 nat18 nat18)); add18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (suc18 (app18 v118 v018)))))
(lam18 v018))
mul18 : ∀{Γ} → Tm18 Γ (arr18 nat18 (arr18 nat18 nat18)); mul18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (app18 (app18 add18 (app18 v118 v018)) v018))))
(lam18 zero18))
fact18 : ∀{Γ} → Tm18 Γ (arr18 nat18 nat18); fact18
= lam18 (rec18 v018 (lam18 (lam18 (app18 (app18 mul18 (suc18 v118)) v018)))
(suc18 zero18))
{-# OPTIONS --type-in-type #-}
Ty19 : Set
Ty19 =
(Ty19 : Set)
(nat top bot : Ty19)
(arr prod sum : Ty19 → Ty19 → Ty19)
→ Ty19
nat19 : Ty19; nat19 = λ _ nat19 _ _ _ _ _ → nat19
top19 : Ty19; top19 = λ _ _ top19 _ _ _ _ → top19
bot19 : Ty19; bot19 = λ _ _ _ bot19 _ _ _ → bot19
arr19 : Ty19 → Ty19 → Ty19; arr19
= λ A B Ty19 nat19 top19 bot19 arr19 prod sum →
arr19 (A Ty19 nat19 top19 bot19 arr19 prod sum) (B Ty19 nat19 top19 bot19 arr19 prod sum)
prod19 : Ty19 → Ty19 → Ty19; prod19
= λ A B Ty19 nat19 top19 bot19 arr19 prod19 sum →
prod19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum) (B Ty19 nat19 top19 bot19 arr19 prod19 sum)
sum19 : Ty19 → Ty19 → Ty19; sum19
= λ A B Ty19 nat19 top19 bot19 arr19 prod19 sum19 →
sum19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum19) (B Ty19 nat19 top19 bot19 arr19 prod19 sum19)
Con19 : Set; Con19
= (Con19 : Set)
(nil : Con19)
(snoc : Con19 → Ty19 → Con19)
→ Con19
nil19 : Con19; nil19
= λ Con19 nil19 snoc → nil19
snoc19 : Con19 → Ty19 → Con19; snoc19
= λ Γ A Con19 nil19 snoc19 → snoc19 (Γ Con19 nil19 snoc19) A
Var19 : Con19 → Ty19 → Set; Var19
= λ Γ A →
(Var19 : Con19 → Ty19 → Set)
(vz : ∀ Γ A → Var19 (snoc19 Γ A) A)
(vs : ∀ Γ B A → Var19 Γ A → Var19 (snoc19 Γ B) A)
→ Var19 Γ A
vz19 : ∀{Γ A} → Var19 (snoc19 Γ A) A; vz19
= λ Var19 vz19 vs → vz19 _ _
vs19 : ∀{Γ B A} → Var19 Γ A → Var19 (snoc19 Γ B) A; vs19
= λ x Var19 vz19 vs19 → vs19 _ _ _ (x Var19 vz19 vs19)
Tm19 : Con19 → Ty19 → Set; Tm19
= λ Γ A →
(Tm19 : Con19 → Ty19 → Set)
(var : ∀ Γ A → Var19 Γ A → Tm19 Γ A)
(lam : ∀ Γ A B → Tm19 (snoc19 Γ A) B → Tm19 Γ (arr19 A B))
(app : ∀ Γ A B → Tm19 Γ (arr19 A B) → Tm19 Γ A → Tm19 Γ B)
(tt : ∀ Γ → Tm19 Γ top19)
(pair : ∀ Γ A B → Tm19 Γ A → Tm19 Γ B → Tm19 Γ (prod19 A B))
(fst : ∀ Γ A B → Tm19 Γ (prod19 A B) → Tm19 Γ A)
(snd : ∀ Γ A B → Tm19 Γ (prod19 A B) → Tm19 Γ B)
(left : ∀ Γ A B → Tm19 Γ A → Tm19 Γ (sum19 A B))
(right : ∀ Γ A B → Tm19 Γ B → Tm19 Γ (sum19 A B))
(case : ∀ Γ A B C → Tm19 Γ (sum19 A B) → Tm19 Γ (arr19 A C) → Tm19 Γ (arr19 B C) → Tm19 Γ C)
(zero : ∀ Γ → Tm19 Γ nat19)
(suc : ∀ Γ → Tm19 Γ nat19 → Tm19 Γ nat19)
(rec : ∀ Γ A → Tm19 Γ nat19 → Tm19 Γ (arr19 nat19 (arr19 A A)) → Tm19 Γ A → Tm19 Γ A)
→ Tm19 Γ A
var19 : ∀{Γ A} → Var19 Γ A → Tm19 Γ A; var19
= λ x Tm19 var19 lam app tt pair fst snd left right case zero suc rec →
var19 _ _ x
lam19 : ∀{Γ A B} → Tm19 (snoc19 Γ A) B → Tm19 Γ (arr19 A B); lam19
= λ t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec →
lam19 _ _ _ (t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec)
app19 : ∀{Γ A B} → Tm19 Γ (arr19 A B) → Tm19 Γ A → Tm19 Γ B; app19
= λ t u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec →
app19 _ _ _ (t Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
tt19 : ∀{Γ} → Tm19 Γ top19; tt19
= λ Tm19 var19 lam19 app19 tt19 pair fst snd left right case zero suc rec → tt19 _
pair19 : ∀{Γ A B} → Tm19 Γ A → Tm19 Γ B → Tm19 Γ (prod19 A B); pair19
= λ t u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec →
pair19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
fst19 : ∀{Γ A B} → Tm19 Γ (prod19 A B) → Tm19 Γ A; fst19
= λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec →
fst19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec)
snd19 : ∀{Γ A B} → Tm19 Γ (prod19 A B) → Tm19 Γ B; snd19
= λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec →
snd19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec)
left19 : ∀{Γ A B} → Tm19 Γ A → Tm19 Γ (sum19 A B); left19
= λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec →
left19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec)
right19 : ∀{Γ A B} → Tm19 Γ B → Tm19 Γ (sum19 A B); right19
= λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec →
right19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec)
case19 : ∀{Γ A B C} → Tm19 Γ (sum19 A B) → Tm19 Γ (arr19 A C) → Tm19 Γ (arr19 B C) → Tm19 Γ C; case19
= λ t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec →
case19 _ _ _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
zero19 : ∀{Γ} → Tm19 Γ nat19; zero19
= λ Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc rec → zero19 _
suc19 : ∀{Γ} → Tm19 Γ nat19 → Tm19 Γ nat19; suc19
= λ t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec →
suc19 _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec)
rec19 : ∀{Γ A} → Tm19 Γ nat19 → Tm19 Γ (arr19 nat19 (arr19 A A)) → Tm19 Γ A → Tm19 Γ A; rec19
= λ t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19 →
rec19 _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
v019 : ∀{Γ A} → Tm19 (snoc19 Γ A) A; v019
= var19 vz19
v119 : ∀{Γ A B} → Tm19 (snoc19 (snoc19 Γ A) B) A; v119
= var19 (vs19 vz19)
v219 : ∀{Γ A B C} → Tm19 (snoc19 (snoc19 (snoc19 Γ A) B) C) A; v219
= var19 (vs19 (vs19 vz19))
v319 : ∀{Γ A B C D} → Tm19 (snoc19 (snoc19 (snoc19 (snoc19 Γ A) B) C) D) A; v319
= var19 (vs19 (vs19 (vs19 vz19)))
tbool19 : Ty19; tbool19
= sum19 top19 top19
true19 : ∀{Γ} → Tm19 Γ tbool19; true19
= left19 tt19
tfalse19 : ∀{Γ} → Tm19 Γ tbool19; tfalse19
= right19 tt19
ifthenelse19 : ∀{Γ A} → Tm19 Γ (arr19 tbool19 (arr19 A (arr19 A A))); ifthenelse19
= lam19 (lam19 (lam19 (case19 v219 (lam19 v219) (lam19 v119))))
times419 : ∀{Γ A} → Tm19 Γ (arr19 (arr19 A A) (arr19 A A)); times419
= lam19 (lam19 (app19 v119 (app19 v119 (app19 v119 (app19 v119 v019)))))
add19 : ∀{Γ} → Tm19 Γ (arr19 nat19 (arr19 nat19 nat19)); add19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (suc19 (app19 v119 v019)))))
(lam19 v019))
mul19 : ∀{Γ} → Tm19 Γ (arr19 nat19 (arr19 nat19 nat19)); mul19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (app19 (app19 add19 (app19 v119 v019)) v019))))
(lam19 zero19))
fact19 : ∀{Γ} → Tm19 Γ (arr19 nat19 nat19); fact19
= lam19 (rec19 v019 (lam19 (lam19 (app19 (app19 mul19 (suc19 v119)) v019)))
(suc19 zero19))
{-# OPTIONS --type-in-type #-}
Ty20 : Set
Ty20 =
(Ty20 : Set)
(nat top bot : Ty20)
(arr prod sum : Ty20 → Ty20 → Ty20)
→ Ty20
nat20 : Ty20; nat20 = λ _ nat20 _ _ _ _ _ → nat20
top20 : Ty20; top20 = λ _ _ top20 _ _ _ _ → top20
bot20 : Ty20; bot20 = λ _ _ _ bot20 _ _ _ → bot20
arr20 : Ty20 → Ty20 → Ty20; arr20
= λ A B Ty20 nat20 top20 bot20 arr20 prod sum →
arr20 (A Ty20 nat20 top20 bot20 arr20 prod sum) (B Ty20 nat20 top20 bot20 arr20 prod sum)
prod20 : Ty20 → Ty20 → Ty20; prod20
= λ A B Ty20 nat20 top20 bot20 arr20 prod20 sum →
prod20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum) (B Ty20 nat20 top20 bot20 arr20 prod20 sum)
sum20 : Ty20 → Ty20 → Ty20; sum20
= λ A B Ty20 nat20 top20 bot20 arr20 prod20 sum20 →
sum20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum20) (B Ty20 nat20 top20 bot20 arr20 prod20 sum20)
Con20 : Set; Con20
= (Con20 : Set)
(nil : Con20)
(snoc : Con20 → Ty20 → Con20)
→ Con20
nil20 : Con20; nil20
= λ Con20 nil20 snoc → nil20
snoc20 : Con20 → Ty20 → Con20; snoc20
= λ Γ A Con20 nil20 snoc20 → snoc20 (Γ Con20 nil20 snoc20) A
Var20 : Con20 → Ty20 → Set; Var20
= λ Γ A →
(Var20 : Con20 → Ty20 → Set)
(vz : ∀ Γ A → Var20 (snoc20 Γ A) A)
(vs : ∀ Γ B A → Var20 Γ A → Var20 (snoc20 Γ B) A)
→ Var20 Γ A
vz20 : ∀{Γ A} → Var20 (snoc20 Γ A) A; vz20
= λ Var20 vz20 vs → vz20 _ _
vs20 : ∀{Γ B A} → Var20 Γ A → Var20 (snoc20 Γ B) A; vs20
= λ x Var20 vz20 vs20 → vs20 _ _ _ (x Var20 vz20 vs20)
Tm20 : Con20 → Ty20 → Set; Tm20
= λ Γ A →
(Tm20 : Con20 → Ty20 → Set)
(var : ∀ Γ A → Var20 Γ A → Tm20 Γ A)
(lam : ∀ Γ A B → Tm20 (snoc20 Γ A) B → Tm20 Γ (arr20 A B))
(app : ∀ Γ A B → Tm20 Γ (arr20 A B) → Tm20 Γ A → Tm20 Γ B)
(tt : ∀ Γ → Tm20 Γ top20)
(pair : ∀ Γ A B → Tm20 Γ A → Tm20 Γ B → Tm20 Γ (prod20 A B))
(fst : ∀ Γ A B → Tm20 Γ (prod20 A B) → Tm20 Γ A)
(snd : ∀ Γ A B → Tm20 Γ (prod20 A B) → Tm20 Γ B)
(left : ∀ Γ A B → Tm20 Γ A → Tm20 Γ (sum20 A B))
(right : ∀ Γ A B → Tm20 Γ B → Tm20 Γ (sum20 A B))
(case : ∀ Γ A B C → Tm20 Γ (sum20 A B) → Tm20 Γ (arr20 A C) → Tm20 Γ (arr20 B C) → Tm20 Γ C)
(zero : ∀ Γ → Tm20 Γ nat20)
(suc : ∀ Γ → Tm20 Γ nat20 → Tm20 Γ nat20)
(rec : ∀ Γ A → Tm20 Γ nat20 → Tm20 Γ (arr20 nat20 (arr20 A A)) → Tm20 Γ A → Tm20 Γ A)
→ Tm20 Γ A
var20 : ∀{Γ A} → Var20 Γ A → Tm20 Γ A; var20
= λ x Tm20 var20 lam app tt pair fst snd left right case zero suc rec →
var20 _ _ x
lam20 : ∀{Γ A B} → Tm20 (snoc20 Γ A) B → Tm20 Γ (arr20 A B); lam20
= λ t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec →
lam20 _ _ _ (t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec)
app20 : ∀{Γ A B} → Tm20 Γ (arr20 A B) → Tm20 Γ A → Tm20 Γ B; app20
= λ t u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec →
app20 _ _ _ (t Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
tt20 : ∀{Γ} → Tm20 Γ top20; tt20
= λ Tm20 var20 lam20 app20 tt20 pair fst snd left right case zero suc rec → tt20 _
pair20 : ∀{Γ A B} → Tm20 Γ A → Tm20 Γ B → Tm20 Γ (prod20 A B); pair20
= λ t u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec →
pair20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
fst20 : ∀{Γ A B} → Tm20 Γ (prod20 A B) → Tm20 Γ A; fst20
= λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec →
fst20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec)
snd20 : ∀{Γ A B} → Tm20 Γ (prod20 A B) → Tm20 Γ B; snd20
= λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec →
snd20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec)
left20 : ∀{Γ A B} → Tm20 Γ A → Tm20 Γ (sum20 A B); left20
= λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec →
left20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec)
right20 : ∀{Γ A B} → Tm20 Γ B → Tm20 Γ (sum20 A B); right20
= λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec →
right20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec)
case20 : ∀{Γ A B C} → Tm20 Γ (sum20 A B) → Tm20 Γ (arr20 A C) → Tm20 Γ (arr20 B C) → Tm20 Γ C; case20
= λ t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec →
case20 _ _ _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
zero20 : ∀{Γ} → Tm20 Γ nat20; zero20
= λ Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc rec → zero20 _
suc20 : ∀{Γ} → Tm20 Γ nat20 → Tm20 Γ nat20; suc20
= λ t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec →
suc20 _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec)
rec20 : ∀{Γ A} → Tm20 Γ nat20 → Tm20 Γ (arr20 nat20 (arr20 A A)) → Tm20 Γ A → Tm20 Γ A; rec20
= λ t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20 →
rec20 _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
v020 : ∀{Γ A} → Tm20 (snoc20 Γ A) A; v020
= var20 vz20
v120 : ∀{Γ A B} → Tm20 (snoc20 (snoc20 Γ A) B) A; v120
= var20 (vs20 vz20)
v220 : ∀{Γ A B C} → Tm20 (snoc20 (snoc20 (snoc20 Γ A) B) C) A; v220
= var20 (vs20 (vs20 vz20))
v320 : ∀{Γ A B C D} → Tm20 (snoc20 (snoc20 (snoc20 (snoc20 Γ A) B) C) D) A; v320
= var20 (vs20 (vs20 (vs20 vz20)))
tbool20 : Ty20; tbool20
= sum20 top20 top20
true20 : ∀{Γ} → Tm20 Γ tbool20; true20
= left20 tt20
tfalse20 : ∀{Γ} → Tm20 Γ tbool20; tfalse20
= right20 tt20
ifthenelse20 : ∀{Γ A} → Tm20 Γ (arr20 tbool20 (arr20 A (arr20 A A))); ifthenelse20
= lam20 (lam20 (lam20 (case20 v220 (lam20 v220) (lam20 v120))))
times420 : ∀{Γ A} → Tm20 Γ (arr20 (arr20 A A) (arr20 A A)); times420
= lam20 (lam20 (app20 v120 (app20 v120 (app20 v120 (app20 v120 v020)))))
add20 : ∀{Γ} → Tm20 Γ (arr20 nat20 (arr20 nat20 nat20)); add20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (suc20 (app20 v120 v020)))))
(lam20 v020))
mul20 : ∀{Γ} → Tm20 Γ (arr20 nat20 (arr20 nat20 nat20)); mul20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (app20 (app20 add20 (app20 v120 v020)) v020))))
(lam20 zero20))
fact20 : ∀{Γ} → Tm20 Γ (arr20 nat20 nat20); fact20
= lam20 (rec20 v020 (lam20 (lam20 (app20 (app20 mul20 (suc20 v120)) v020)))
(suc20 zero20))
{-# OPTIONS --type-in-type #-}
Ty21 : Set
Ty21 =
(Ty21 : Set)
(nat top bot : Ty21)
(arr prod sum : Ty21 → Ty21 → Ty21)
→ Ty21
nat21 : Ty21; nat21 = λ _ nat21 _ _ _ _ _ → nat21
top21 : Ty21; top21 = λ _ _ top21 _ _ _ _ → top21
bot21 : Ty21; bot21 = λ _ _ _ bot21 _ _ _ → bot21
arr21 : Ty21 → Ty21 → Ty21; arr21
= λ A B Ty21 nat21 top21 bot21 arr21 prod sum →
arr21 (A Ty21 nat21 top21 bot21 arr21 prod sum) (B Ty21 nat21 top21 bot21 arr21 prod sum)
prod21 : Ty21 → Ty21 → Ty21; prod21
= λ A B Ty21 nat21 top21 bot21 arr21 prod21 sum →
prod21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum) (B Ty21 nat21 top21 bot21 arr21 prod21 sum)
sum21 : Ty21 → Ty21 → Ty21; sum21
= λ A B Ty21 nat21 top21 bot21 arr21 prod21 sum21 →
sum21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum21) (B Ty21 nat21 top21 bot21 arr21 prod21 sum21)
Con21 : Set; Con21
= (Con21 : Set)
(nil : Con21)
(snoc : Con21 → Ty21 → Con21)
→ Con21
nil21 : Con21; nil21
= λ Con21 nil21 snoc → nil21
snoc21 : Con21 → Ty21 → Con21; snoc21
= λ Γ A Con21 nil21 snoc21 → snoc21 (Γ Con21 nil21 snoc21) A
Var21 : Con21 → Ty21 → Set; Var21
= λ Γ A →
(Var21 : Con21 → Ty21 → Set)
(vz : ∀ Γ A → Var21 (snoc21 Γ A) A)
(vs : ∀ Γ B A → Var21 Γ A → Var21 (snoc21 Γ B) A)
→ Var21 Γ A
vz21 : ∀{Γ A} → Var21 (snoc21 Γ A) A; vz21
= λ Var21 vz21 vs → vz21 _ _
vs21 : ∀{Γ B A} → Var21 Γ A → Var21 (snoc21 Γ B) A; vs21
= λ x Var21 vz21 vs21 → vs21 _ _ _ (x Var21 vz21 vs21)
Tm21 : Con21 → Ty21 → Set; Tm21
= λ Γ A →
(Tm21 : Con21 → Ty21 → Set)
(var : ∀ Γ A → Var21 Γ A → Tm21 Γ A)
(lam : ∀ Γ A B → Tm21 (snoc21 Γ A) B → Tm21 Γ (arr21 A B))
(app : ∀ Γ A B → Tm21 Γ (arr21 A B) → Tm21 Γ A → Tm21 Γ B)
(tt : ∀ Γ → Tm21 Γ top21)
(pair : ∀ Γ A B → Tm21 Γ A → Tm21 Γ B → Tm21 Γ (prod21 A B))
(fst : ∀ Γ A B → Tm21 Γ (prod21 A B) → Tm21 Γ A)
(snd : ∀ Γ A B → Tm21 Γ (prod21 A B) → Tm21 Γ B)
(left : ∀ Γ A B → Tm21 Γ A → Tm21 Γ (sum21 A B))
(right : ∀ Γ A B → Tm21 Γ B → Tm21 Γ (sum21 A B))
(case : ∀ Γ A B C → Tm21 Γ (sum21 A B) → Tm21 Γ (arr21 A C) → Tm21 Γ (arr21 B C) → Tm21 Γ C)
(zero : ∀ Γ → Tm21 Γ nat21)
(suc : ∀ Γ → Tm21 Γ nat21 → Tm21 Γ nat21)
(rec : ∀ Γ A → Tm21 Γ nat21 → Tm21 Γ (arr21 nat21 (arr21 A A)) → Tm21 Γ A → Tm21 Γ A)
→ Tm21 Γ A
var21 : ∀{Γ A} → Var21 Γ A → Tm21 Γ A; var21
= λ x Tm21 var21 lam app tt pair fst snd left right case zero suc rec →
var21 _ _ x
lam21 : ∀{Γ A B} → Tm21 (snoc21 Γ A) B → Tm21 Γ (arr21 A B); lam21
= λ t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec →
lam21 _ _ _ (t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec)
app21 : ∀{Γ A B} → Tm21 Γ (arr21 A B) → Tm21 Γ A → Tm21 Γ B; app21
= λ t u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec →
app21 _ _ _ (t Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
tt21 : ∀{Γ} → Tm21 Γ top21; tt21
= λ Tm21 var21 lam21 app21 tt21 pair fst snd left right case zero suc rec → tt21 _
pair21 : ∀{Γ A B} → Tm21 Γ A → Tm21 Γ B → Tm21 Γ (prod21 A B); pair21
= λ t u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec →
pair21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
fst21 : ∀{Γ A B} → Tm21 Γ (prod21 A B) → Tm21 Γ A; fst21
= λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec →
fst21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec)
snd21 : ∀{Γ A B} → Tm21 Γ (prod21 A B) → Tm21 Γ B; snd21
= λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec →
snd21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec)
left21 : ∀{Γ A B} → Tm21 Γ A → Tm21 Γ (sum21 A B); left21
= λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec →
left21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec)
right21 : ∀{Γ A B} → Tm21 Γ B → Tm21 Γ (sum21 A B); right21
= λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec →
right21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec)
case21 : ∀{Γ A B C} → Tm21 Γ (sum21 A B) → Tm21 Γ (arr21 A C) → Tm21 Γ (arr21 B C) → Tm21 Γ C; case21
= λ t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec →
case21 _ _ _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
zero21 : ∀{Γ} → Tm21 Γ nat21; zero21
= λ Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc rec → zero21 _
suc21 : ∀{Γ} → Tm21 Γ nat21 → Tm21 Γ nat21; suc21
= λ t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec →
suc21 _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec)
rec21 : ∀{Γ A} → Tm21 Γ nat21 → Tm21 Γ (arr21 nat21 (arr21 A A)) → Tm21 Γ A → Tm21 Γ A; rec21
= λ t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21 →
rec21 _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
v021 : ∀{Γ A} → Tm21 (snoc21 Γ A) A; v021
= var21 vz21
v121 : ∀{Γ A B} → Tm21 (snoc21 (snoc21 Γ A) B) A; v121
= var21 (vs21 vz21)
v221 : ∀{Γ A B C} → Tm21 (snoc21 (snoc21 (snoc21 Γ A) B) C) A; v221
= var21 (vs21 (vs21 vz21))
v321 : ∀{Γ A B C D} → Tm21 (snoc21 (snoc21 (snoc21 (snoc21 Γ A) B) C) D) A; v321
= var21 (vs21 (vs21 (vs21 vz21)))
tbool21 : Ty21; tbool21
= sum21 top21 top21
true21 : ∀{Γ} → Tm21 Γ tbool21; true21
= left21 tt21
tfalse21 : ∀{Γ} → Tm21 Γ tbool21; tfalse21
= right21 tt21
ifthenelse21 : ∀{Γ A} → Tm21 Γ (arr21 tbool21 (arr21 A (arr21 A A))); ifthenelse21
= lam21 (lam21 (lam21 (case21 v221 (lam21 v221) (lam21 v121))))
times421 : ∀{Γ A} → Tm21 Γ (arr21 (arr21 A A) (arr21 A A)); times421
= lam21 (lam21 (app21 v121 (app21 v121 (app21 v121 (app21 v121 v021)))))
add21 : ∀{Γ} → Tm21 Γ (arr21 nat21 (arr21 nat21 nat21)); add21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (suc21 (app21 v121 v021)))))
(lam21 v021))
mul21 : ∀{Γ} → Tm21 Γ (arr21 nat21 (arr21 nat21 nat21)); mul21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (app21 (app21 add21 (app21 v121 v021)) v021))))
(lam21 zero21))
fact21 : ∀{Γ} → Tm21 Γ (arr21 nat21 nat21); fact21
= lam21 (rec21 v021 (lam21 (lam21 (app21 (app21 mul21 (suc21 v121)) v021)))
(suc21 zero21))
{-# OPTIONS --type-in-type #-}
Ty22 : Set
Ty22 =
(Ty22 : Set)
(nat top bot : Ty22)
(arr prod sum : Ty22 → Ty22 → Ty22)
→ Ty22
nat22 : Ty22; nat22 = λ _ nat22 _ _ _ _ _ → nat22
top22 : Ty22; top22 = λ _ _ top22 _ _ _ _ → top22
bot22 : Ty22; bot22 = λ _ _ _ bot22 _ _ _ → bot22
arr22 : Ty22 → Ty22 → Ty22; arr22
= λ A B Ty22 nat22 top22 bot22 arr22 prod sum →
arr22 (A Ty22 nat22 top22 bot22 arr22 prod sum) (B Ty22 nat22 top22 bot22 arr22 prod sum)
prod22 : Ty22 → Ty22 → Ty22; prod22
= λ A B Ty22 nat22 top22 bot22 arr22 prod22 sum →
prod22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum) (B Ty22 nat22 top22 bot22 arr22 prod22 sum)
sum22 : Ty22 → Ty22 → Ty22; sum22
= λ A B Ty22 nat22 top22 bot22 arr22 prod22 sum22 →
sum22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum22) (B Ty22 nat22 top22 bot22 arr22 prod22 sum22)
Con22 : Set; Con22
= (Con22 : Set)
(nil : Con22)
(snoc : Con22 → Ty22 → Con22)
→ Con22
nil22 : Con22; nil22
= λ Con22 nil22 snoc → nil22
snoc22 : Con22 → Ty22 → Con22; snoc22
= λ Γ A Con22 nil22 snoc22 → snoc22 (Γ Con22 nil22 snoc22) A
Var22 : Con22 → Ty22 → Set; Var22
= λ Γ A →
(Var22 : Con22 → Ty22 → Set)
(vz : ∀ Γ A → Var22 (snoc22 Γ A) A)
(vs : ∀ Γ B A → Var22 Γ A → Var22 (snoc22 Γ B) A)
→ Var22 Γ A
vz22 : ∀{Γ A} → Var22 (snoc22 Γ A) A; vz22
= λ Var22 vz22 vs → vz22 _ _
vs22 : ∀{Γ B A} → Var22 Γ A → Var22 (snoc22 Γ B) A; vs22
= λ x Var22 vz22 vs22 → vs22 _ _ _ (x Var22 vz22 vs22)
Tm22 : Con22 → Ty22 → Set; Tm22
= λ Γ A →
(Tm22 : Con22 → Ty22 → Set)
(var : ∀ Γ A → Var22 Γ A → Tm22 Γ A)
(lam : ∀ Γ A B → Tm22 (snoc22 Γ A) B → Tm22 Γ (arr22 A B))
(app : ∀ Γ A B → Tm22 Γ (arr22 A B) → Tm22 Γ A → Tm22 Γ B)
(tt : ∀ Γ → Tm22 Γ top22)
(pair : ∀ Γ A B → Tm22 Γ A → Tm22 Γ B → Tm22 Γ (prod22 A B))
(fst : ∀ Γ A B → Tm22 Γ (prod22 A B) → Tm22 Γ A)
(snd : ∀ Γ A B → Tm22 Γ (prod22 A B) → Tm22 Γ B)
(left : ∀ Γ A B → Tm22 Γ A → Tm22 Γ (sum22 A B))
(right : ∀ Γ A B → Tm22 Γ B → Tm22 Γ (sum22 A B))
(case : ∀ Γ A B C → Tm22 Γ (sum22 A B) → Tm22 Γ (arr22 A C) → Tm22 Γ (arr22 B C) → Tm22 Γ C)
(zero : ∀ Γ → Tm22 Γ nat22)
(suc : ∀ Γ → Tm22 Γ nat22 → Tm22 Γ nat22)
(rec : ∀ Γ A → Tm22 Γ nat22 → Tm22 Γ (arr22 nat22 (arr22 A A)) → Tm22 Γ A → Tm22 Γ A)
→ Tm22 Γ A
var22 : ∀{Γ A} → Var22 Γ A → Tm22 Γ A; var22
= λ x Tm22 var22 lam app tt pair fst snd left right case zero suc rec →
var22 _ _ x
lam22 : ∀{Γ A B} → Tm22 (snoc22 Γ A) B → Tm22 Γ (arr22 A B); lam22
= λ t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec →
lam22 _ _ _ (t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec)
app22 : ∀{Γ A B} → Tm22 Γ (arr22 A B) → Tm22 Γ A → Tm22 Γ B; app22
= λ t u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec →
app22 _ _ _ (t Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
tt22 : ∀{Γ} → Tm22 Γ top22; tt22
= λ Tm22 var22 lam22 app22 tt22 pair fst snd left right case zero suc rec → tt22 _
pair22 : ∀{Γ A B} → Tm22 Γ A → Tm22 Γ B → Tm22 Γ (prod22 A B); pair22
= λ t u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec →
pair22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
fst22 : ∀{Γ A B} → Tm22 Γ (prod22 A B) → Tm22 Γ A; fst22
= λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec →
fst22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec)
snd22 : ∀{Γ A B} → Tm22 Γ (prod22 A B) → Tm22 Γ B; snd22
= λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec →
snd22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec)
left22 : ∀{Γ A B} → Tm22 Γ A → Tm22 Γ (sum22 A B); left22
= λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec →
left22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec)
right22 : ∀{Γ A B} → Tm22 Γ B → Tm22 Γ (sum22 A B); right22
= λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec →
right22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec)
case22 : ∀{Γ A B C} → Tm22 Γ (sum22 A B) → Tm22 Γ (arr22 A C) → Tm22 Γ (arr22 B C) → Tm22 Γ C; case22
= λ t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec →
case22 _ _ _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
zero22 : ∀{Γ} → Tm22 Γ nat22; zero22
= λ Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc rec → zero22 _
suc22 : ∀{Γ} → Tm22 Γ nat22 → Tm22 Γ nat22; suc22
= λ t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec →
suc22 _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec)
rec22 : ∀{Γ A} → Tm22 Γ nat22 → Tm22 Γ (arr22 nat22 (arr22 A A)) → Tm22 Γ A → Tm22 Γ A; rec22
= λ t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22 →
rec22 _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
v022 : ∀{Γ A} → Tm22 (snoc22 Γ A) A; v022
= var22 vz22
v122 : ∀{Γ A B} → Tm22 (snoc22 (snoc22 Γ A) B) A; v122
= var22 (vs22 vz22)
v222 : ∀{Γ A B C} → Tm22 (snoc22 (snoc22 (snoc22 Γ A) B) C) A; v222
= var22 (vs22 (vs22 vz22))
v322 : ∀{Γ A B C D} → Tm22 (snoc22 (snoc22 (snoc22 (snoc22 Γ A) B) C) D) A; v322
= var22 (vs22 (vs22 (vs22 vz22)))
tbool22 : Ty22; tbool22
= sum22 top22 top22
true22 : ∀{Γ} → Tm22 Γ tbool22; true22
= left22 tt22
tfalse22 : ∀{Γ} → Tm22 Γ tbool22; tfalse22
= right22 tt22
ifthenelse22 : ∀{Γ A} → Tm22 Γ (arr22 tbool22 (arr22 A (arr22 A A))); ifthenelse22
= lam22 (lam22 (lam22 (case22 v222 (lam22 v222) (lam22 v122))))
times422 : ∀{Γ A} → Tm22 Γ (arr22 (arr22 A A) (arr22 A A)); times422
= lam22 (lam22 (app22 v122 (app22 v122 (app22 v122 (app22 v122 v022)))))
add22 : ∀{Γ} → Tm22 Γ (arr22 nat22 (arr22 nat22 nat22)); add22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (suc22 (app22 v122 v022)))))
(lam22 v022))
mul22 : ∀{Γ} → Tm22 Γ (arr22 nat22 (arr22 nat22 nat22)); mul22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (app22 (app22 add22 (app22 v122 v022)) v022))))
(lam22 zero22))
fact22 : ∀{Γ} → Tm22 Γ (arr22 nat22 nat22); fact22
= lam22 (rec22 v022 (lam22 (lam22 (app22 (app22 mul22 (suc22 v122)) v022)))
(suc22 zero22))
{-# OPTIONS --type-in-type #-}
Ty23 : Set
Ty23 =
(Ty23 : Set)
(nat top bot : Ty23)
(arr prod sum : Ty23 → Ty23 → Ty23)
→ Ty23
nat23 : Ty23; nat23 = λ _ nat23 _ _ _ _ _ → nat23
top23 : Ty23; top23 = λ _ _ top23 _ _ _ _ → top23
bot23 : Ty23; bot23 = λ _ _ _ bot23 _ _ _ → bot23
arr23 : Ty23 → Ty23 → Ty23; arr23
= λ A B Ty23 nat23 top23 bot23 arr23 prod sum →
arr23 (A Ty23 nat23 top23 bot23 arr23 prod sum) (B Ty23 nat23 top23 bot23 arr23 prod sum)
prod23 : Ty23 → Ty23 → Ty23; prod23
= λ A B Ty23 nat23 top23 bot23 arr23 prod23 sum →
prod23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum) (B Ty23 nat23 top23 bot23 arr23 prod23 sum)
sum23 : Ty23 → Ty23 → Ty23; sum23
= λ A B Ty23 nat23 top23 bot23 arr23 prod23 sum23 →
sum23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum23) (B Ty23 nat23 top23 bot23 arr23 prod23 sum23)
Con23 : Set; Con23
= (Con23 : Set)
(nil : Con23)
(snoc : Con23 → Ty23 → Con23)
→ Con23
nil23 : Con23; nil23
= λ Con23 nil23 snoc → nil23
snoc23 : Con23 → Ty23 → Con23; snoc23
= λ Γ A Con23 nil23 snoc23 → snoc23 (Γ Con23 nil23 snoc23) A
Var23 : Con23 → Ty23 → Set; Var23
= λ Γ A →
(Var23 : Con23 → Ty23 → Set)
(vz : ∀ Γ A → Var23 (snoc23 Γ A) A)
(vs : ∀ Γ B A → Var23 Γ A → Var23 (snoc23 Γ B) A)
→ Var23 Γ A
vz23 : ∀{Γ A} → Var23 (snoc23 Γ A) A; vz23
= λ Var23 vz23 vs → vz23 _ _
vs23 : ∀{Γ B A} → Var23 Γ A → Var23 (snoc23 Γ B) A; vs23
= λ x Var23 vz23 vs23 → vs23 _ _ _ (x Var23 vz23 vs23)
Tm23 : Con23 → Ty23 → Set; Tm23
= λ Γ A →
(Tm23 : Con23 → Ty23 → Set)
(var : ∀ Γ A → Var23 Γ A → Tm23 Γ A)
(lam : ∀ Γ A B → Tm23 (snoc23 Γ A) B → Tm23 Γ (arr23 A B))
(app : ∀ Γ A B → Tm23 Γ (arr23 A B) → Tm23 Γ A → Tm23 Γ B)
(tt : ∀ Γ → Tm23 Γ top23)
(pair : ∀ Γ A B → Tm23 Γ A → Tm23 Γ B → Tm23 Γ (prod23 A B))
(fst : ∀ Γ A B → Tm23 Γ (prod23 A B) → Tm23 Γ A)
(snd : ∀ Γ A B → Tm23 Γ (prod23 A B) → Tm23 Γ B)
(left : ∀ Γ A B → Tm23 Γ A → Tm23 Γ (sum23 A B))
(right : ∀ Γ A B → Tm23 Γ B → Tm23 Γ (sum23 A B))
(case : ∀ Γ A B C → Tm23 Γ (sum23 A B) → Tm23 Γ (arr23 A C) → Tm23 Γ (arr23 B C) → Tm23 Γ C)
(zero : ∀ Γ → Tm23 Γ nat23)
(suc : ∀ Γ → Tm23 Γ nat23 → Tm23 Γ nat23)
(rec : ∀ Γ A → Tm23 Γ nat23 → Tm23 Γ (arr23 nat23 (arr23 A A)) → Tm23 Γ A → Tm23 Γ A)
→ Tm23 Γ A
var23 : ∀{Γ A} → Var23 Γ A → Tm23 Γ A; var23
= λ x Tm23 var23 lam app tt pair fst snd left right case zero suc rec →
var23 _ _ x
lam23 : ∀{Γ A B} → Tm23 (snoc23 Γ A) B → Tm23 Γ (arr23 A B); lam23
= λ t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec →
lam23 _ _ _ (t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec)
app23 : ∀{Γ A B} → Tm23 Γ (arr23 A B) → Tm23 Γ A → Tm23 Γ B; app23
= λ t u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec →
app23 _ _ _ (t Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
tt23 : ∀{Γ} → Tm23 Γ top23; tt23
= λ Tm23 var23 lam23 app23 tt23 pair fst snd left right case zero suc rec → tt23 _
pair23 : ∀{Γ A B} → Tm23 Γ A → Tm23 Γ B → Tm23 Γ (prod23 A B); pair23
= λ t u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec →
pair23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
fst23 : ∀{Γ A B} → Tm23 Γ (prod23 A B) → Tm23 Γ A; fst23
= λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec →
fst23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec)
snd23 : ∀{Γ A B} → Tm23 Γ (prod23 A B) → Tm23 Γ B; snd23
= λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec →
snd23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec)
left23 : ∀{Γ A B} → Tm23 Γ A → Tm23 Γ (sum23 A B); left23
= λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec →
left23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec)
right23 : ∀{Γ A B} → Tm23 Γ B → Tm23 Γ (sum23 A B); right23
= λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec →
right23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec)
case23 : ∀{Γ A B C} → Tm23 Γ (sum23 A B) → Tm23 Γ (arr23 A C) → Tm23 Γ (arr23 B C) → Tm23 Γ C; case23
= λ t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec →
case23 _ _ _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
zero23 : ∀{Γ} → Tm23 Γ nat23; zero23
= λ Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc rec → zero23 _
suc23 : ∀{Γ} → Tm23 Γ nat23 → Tm23 Γ nat23; suc23
= λ t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec →
suc23 _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec)
rec23 : ∀{Γ A} → Tm23 Γ nat23 → Tm23 Γ (arr23 nat23 (arr23 A A)) → Tm23 Γ A → Tm23 Γ A; rec23
= λ t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23 →
rec23 _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
v023 : ∀{Γ A} → Tm23 (snoc23 Γ A) A; v023
= var23 vz23
v123 : ∀{Γ A B} → Tm23 (snoc23 (snoc23 Γ A) B) A; v123
= var23 (vs23 vz23)
v223 : ∀{Γ A B C} → Tm23 (snoc23 (snoc23 (snoc23 Γ A) B) C) A; v223
= var23 (vs23 (vs23 vz23))
v323 : ∀{Γ A B C D} → Tm23 (snoc23 (snoc23 (snoc23 (snoc23 Γ A) B) C) D) A; v323
= var23 (vs23 (vs23 (vs23 vz23)))
tbool23 : Ty23; tbool23
= sum23 top23 top23
true23 : ∀{Γ} → Tm23 Γ tbool23; true23
= left23 tt23
tfalse23 : ∀{Γ} → Tm23 Γ tbool23; tfalse23
= right23 tt23
ifthenelse23 : ∀{Γ A} → Tm23 Γ (arr23 tbool23 (arr23 A (arr23 A A))); ifthenelse23
= lam23 (lam23 (lam23 (case23 v223 (lam23 v223) (lam23 v123))))
times423 : ∀{Γ A} → Tm23 Γ (arr23 (arr23 A A) (arr23 A A)); times423
= lam23 (lam23 (app23 v123 (app23 v123 (app23 v123 (app23 v123 v023)))))
add23 : ∀{Γ} → Tm23 Γ (arr23 nat23 (arr23 nat23 nat23)); add23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (suc23 (app23 v123 v023)))))
(lam23 v023))
mul23 : ∀{Γ} → Tm23 Γ (arr23 nat23 (arr23 nat23 nat23)); mul23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (app23 (app23 add23 (app23 v123 v023)) v023))))
(lam23 zero23))
fact23 : ∀{Γ} → Tm23 Γ (arr23 nat23 nat23); fact23
= lam23 (rec23 v023 (lam23 (lam23 (app23 (app23 mul23 (suc23 v123)) v023)))
(suc23 zero23))
{-# OPTIONS --type-in-type #-}
Ty24 : Set
Ty24 =
(Ty24 : Set)
(nat top bot : Ty24)
(arr prod sum : Ty24 → Ty24 → Ty24)
→ Ty24
nat24 : Ty24; nat24 = λ _ nat24 _ _ _ _ _ → nat24
top24 : Ty24; top24 = λ _ _ top24 _ _ _ _ → top24
bot24 : Ty24; bot24 = λ _ _ _ bot24 _ _ _ → bot24
arr24 : Ty24 → Ty24 → Ty24; arr24
= λ A B Ty24 nat24 top24 bot24 arr24 prod sum →
arr24 (A Ty24 nat24 top24 bot24 arr24 prod sum) (B Ty24 nat24 top24 bot24 arr24 prod sum)
prod24 : Ty24 → Ty24 → Ty24; prod24
= λ A B Ty24 nat24 top24 bot24 arr24 prod24 sum →
prod24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum) (B Ty24 nat24 top24 bot24 arr24 prod24 sum)
sum24 : Ty24 → Ty24 → Ty24; sum24
= λ A B Ty24 nat24 top24 bot24 arr24 prod24 sum24 →
sum24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum24) (B Ty24 nat24 top24 bot24 arr24 prod24 sum24)
Con24 : Set; Con24
= (Con24 : Set)
(nil : Con24)
(snoc : Con24 → Ty24 → Con24)
→ Con24
nil24 : Con24; nil24
= λ Con24 nil24 snoc → nil24
snoc24 : Con24 → Ty24 → Con24; snoc24
= λ Γ A Con24 nil24 snoc24 → snoc24 (Γ Con24 nil24 snoc24) A
Var24 : Con24 → Ty24 → Set; Var24
= λ Γ A →
(Var24 : Con24 → Ty24 → Set)
(vz : ∀ Γ A → Var24 (snoc24 Γ A) A)
(vs : ∀ Γ B A → Var24 Γ A → Var24 (snoc24 Γ B) A)
→ Var24 Γ A
vz24 : ∀{Γ A} → Var24 (snoc24 Γ A) A; vz24
= λ Var24 vz24 vs → vz24 _ _
vs24 : ∀{Γ B A} → Var24 Γ A → Var24 (snoc24 Γ B) A; vs24
= λ x Var24 vz24 vs24 → vs24 _ _ _ (x Var24 vz24 vs24)
Tm24 : Con24 → Ty24 → Set; Tm24
= λ Γ A →
(Tm24 : Con24 → Ty24 → Set)
(var : ∀ Γ A → Var24 Γ A → Tm24 Γ A)
(lam : ∀ Γ A B → Tm24 (snoc24 Γ A) B → Tm24 Γ (arr24 A B))
(app : ∀ Γ A B → Tm24 Γ (arr24 A B) → Tm24 Γ A → Tm24 Γ B)
(tt : ∀ Γ → Tm24 Γ top24)
(pair : ∀ Γ A B → Tm24 Γ A → Tm24 Γ B → Tm24 Γ (prod24 A B))
(fst : ∀ Γ A B → Tm24 Γ (prod24 A B) → Tm24 Γ A)
(snd : ∀ Γ A B → Tm24 Γ (prod24 A B) → Tm24 Γ B)
(left : ∀ Γ A B → Tm24 Γ A → Tm24 Γ (sum24 A B))
(right : ∀ Γ A B → Tm24 Γ B → Tm24 Γ (sum24 A B))
(case : ∀ Γ A B C → Tm24 Γ (sum24 A B) → Tm24 Γ (arr24 A C) → Tm24 Γ (arr24 B C) → Tm24 Γ C)
(zero : ∀ Γ → Tm24 Γ nat24)
(suc : ∀ Γ → Tm24 Γ nat24 → Tm24 Γ nat24)
(rec : ∀ Γ A → Tm24 Γ nat24 → Tm24 Γ (arr24 nat24 (arr24 A A)) → Tm24 Γ A → Tm24 Γ A)
→ Tm24 Γ A
var24 : ∀{Γ A} → Var24 Γ A → Tm24 Γ A; var24
= λ x Tm24 var24 lam app tt pair fst snd left right case zero suc rec →
var24 _ _ x
lam24 : ∀{Γ A B} → Tm24 (snoc24 Γ A) B → Tm24 Γ (arr24 A B); lam24
= λ t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec →
lam24 _ _ _ (t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec)
app24 : ∀{Γ A B} → Tm24 Γ (arr24 A B) → Tm24 Γ A → Tm24 Γ B; app24
= λ t u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec →
app24 _ _ _ (t Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
tt24 : ∀{Γ} → Tm24 Γ top24; tt24
= λ Tm24 var24 lam24 app24 tt24 pair fst snd left right case zero suc rec → tt24 _
pair24 : ∀{Γ A B} → Tm24 Γ A → Tm24 Γ B → Tm24 Γ (prod24 A B); pair24
= λ t u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec →
pair24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
fst24 : ∀{Γ A B} → Tm24 Γ (prod24 A B) → Tm24 Γ A; fst24
= λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec →
fst24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec)
snd24 : ∀{Γ A B} → Tm24 Γ (prod24 A B) → Tm24 Γ B; snd24
= λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec →
snd24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec)
left24 : ∀{Γ A B} → Tm24 Γ A → Tm24 Γ (sum24 A B); left24
= λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec →
left24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec)
right24 : ∀{Γ A B} → Tm24 Γ B → Tm24 Γ (sum24 A B); right24
= λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec →
right24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec)
case24 : ∀{Γ A B C} → Tm24 Γ (sum24 A B) → Tm24 Γ (arr24 A C) → Tm24 Γ (arr24 B C) → Tm24 Γ C; case24
= λ t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec →
case24 _ _ _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
zero24 : ∀{Γ} → Tm24 Γ nat24; zero24
= λ Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc rec → zero24 _
suc24 : ∀{Γ} → Tm24 Γ nat24 → Tm24 Γ nat24; suc24
= λ t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec →
suc24 _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec)
rec24 : ∀{Γ A} → Tm24 Γ nat24 → Tm24 Γ (arr24 nat24 (arr24 A A)) → Tm24 Γ A → Tm24 Γ A; rec24
= λ t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24 →
rec24 _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
v024 : ∀{Γ A} → Tm24 (snoc24 Γ A) A; v024
= var24 vz24
v124 : ∀{Γ A B} → Tm24 (snoc24 (snoc24 Γ A) B) A; v124
= var24 (vs24 vz24)
v224 : ∀{Γ A B C} → Tm24 (snoc24 (snoc24 (snoc24 Γ A) B) C) A; v224
= var24 (vs24 (vs24 vz24))
v324 : ∀{Γ A B C D} → Tm24 (snoc24 (snoc24 (snoc24 (snoc24 Γ A) B) C) D) A; v324
= var24 (vs24 (vs24 (vs24 vz24)))
tbool24 : Ty24; tbool24
= sum24 top24 top24
true24 : ∀{Γ} → Tm24 Γ tbool24; true24
= left24 tt24
tfalse24 : ∀{Γ} → Tm24 Γ tbool24; tfalse24
= right24 tt24
ifthenelse24 : ∀{Γ A} → Tm24 Γ (arr24 tbool24 (arr24 A (arr24 A A))); ifthenelse24
= lam24 (lam24 (lam24 (case24 v224 (lam24 v224) (lam24 v124))))
times424 : ∀{Γ A} → Tm24 Γ (arr24 (arr24 A A) (arr24 A A)); times424
= lam24 (lam24 (app24 v124 (app24 v124 (app24 v124 (app24 v124 v024)))))
add24 : ∀{Γ} → Tm24 Γ (arr24 nat24 (arr24 nat24 nat24)); add24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (suc24 (app24 v124 v024)))))
(lam24 v024))
mul24 : ∀{Γ} → Tm24 Γ (arr24 nat24 (arr24 nat24 nat24)); mul24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (app24 (app24 add24 (app24 v124 v024)) v024))))
(lam24 zero24))
fact24 : ∀{Γ} → Tm24 Γ (arr24 nat24 nat24); fact24
= lam24 (rec24 v024 (lam24 (lam24 (app24 (app24 mul24 (suc24 v124)) v024)))
(suc24 zero24))
{-# OPTIONS --type-in-type #-}
Ty25 : Set
Ty25 =
(Ty25 : Set)
(nat top bot : Ty25)
(arr prod sum : Ty25 → Ty25 → Ty25)
→ Ty25
nat25 : Ty25; nat25 = λ _ nat25 _ _ _ _ _ → nat25
top25 : Ty25; top25 = λ _ _ top25 _ _ _ _ → top25
bot25 : Ty25; bot25 = λ _ _ _ bot25 _ _ _ → bot25
arr25 : Ty25 → Ty25 → Ty25; arr25
= λ A B Ty25 nat25 top25 bot25 arr25 prod sum →
arr25 (A Ty25 nat25 top25 bot25 arr25 prod sum) (B Ty25 nat25 top25 bot25 arr25 prod sum)
prod25 : Ty25 → Ty25 → Ty25; prod25
= λ A B Ty25 nat25 top25 bot25 arr25 prod25 sum →
prod25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum) (B Ty25 nat25 top25 bot25 arr25 prod25 sum)
sum25 : Ty25 → Ty25 → Ty25; sum25
= λ A B Ty25 nat25 top25 bot25 arr25 prod25 sum25 →
sum25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum25) (B Ty25 nat25 top25 bot25 arr25 prod25 sum25)
Con25 : Set; Con25
= (Con25 : Set)
(nil : Con25)
(snoc : Con25 → Ty25 → Con25)
→ Con25
nil25 : Con25; nil25
= λ Con25 nil25 snoc → nil25
snoc25 : Con25 → Ty25 → Con25; snoc25
= λ Γ A Con25 nil25 snoc25 → snoc25 (Γ Con25 nil25 snoc25) A
Var25 : Con25 → Ty25 → Set; Var25
= λ Γ A →
(Var25 : Con25 → Ty25 → Set)
(vz : ∀ Γ A → Var25 (snoc25 Γ A) A)
(vs : ∀ Γ B A → Var25 Γ A → Var25 (snoc25 Γ B) A)
→ Var25 Γ A
vz25 : ∀{Γ A} → Var25 (snoc25 Γ A) A; vz25
= λ Var25 vz25 vs → vz25 _ _
vs25 : ∀{Γ B A} → Var25 Γ A → Var25 (snoc25 Γ B) A; vs25
= λ x Var25 vz25 vs25 → vs25 _ _ _ (x Var25 vz25 vs25)
Tm25 : Con25 → Ty25 → Set; Tm25
= λ Γ A →
(Tm25 : Con25 → Ty25 → Set)
(var : ∀ Γ A → Var25 Γ A → Tm25 Γ A)
(lam : ∀ Γ A B → Tm25 (snoc25 Γ A) B → Tm25 Γ (arr25 A B))
(app : ∀ Γ A B → Tm25 Γ (arr25 A B) → Tm25 Γ A → Tm25 Γ B)
(tt : ∀ Γ → Tm25 Γ top25)
(pair : ∀ Γ A B → Tm25 Γ A → Tm25 Γ B → Tm25 Γ (prod25 A B))
(fst : ∀ Γ A B → Tm25 Γ (prod25 A B) → Tm25 Γ A)
(snd : ∀ Γ A B → Tm25 Γ (prod25 A B) → Tm25 Γ B)
(left : ∀ Γ A B → Tm25 Γ A → Tm25 Γ (sum25 A B))
(right : ∀ Γ A B → Tm25 Γ B → Tm25 Γ (sum25 A B))
(case : ∀ Γ A B C → Tm25 Γ (sum25 A B) → Tm25 Γ (arr25 A C) → Tm25 Γ (arr25 B C) → Tm25 Γ C)
(zero : ∀ Γ → Tm25 Γ nat25)
(suc : ∀ Γ → Tm25 Γ nat25 → Tm25 Γ nat25)
(rec : ∀ Γ A → Tm25 Γ nat25 → Tm25 Γ (arr25 nat25 (arr25 A A)) → Tm25 Γ A → Tm25 Γ A)
→ Tm25 Γ A
var25 : ∀{Γ A} → Var25 Γ A → Tm25 Γ A; var25
= λ x Tm25 var25 lam app tt pair fst snd left right case zero suc rec →
var25 _ _ x
lam25 : ∀{Γ A B} → Tm25 (snoc25 Γ A) B → Tm25 Γ (arr25 A B); lam25
= λ t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec →
lam25 _ _ _ (t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec)
app25 : ∀{Γ A B} → Tm25 Γ (arr25 A B) → Tm25 Γ A → Tm25 Γ B; app25
= λ t u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec →
app25 _ _ _ (t Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
tt25 : ∀{Γ} → Tm25 Γ top25; tt25
= λ Tm25 var25 lam25 app25 tt25 pair fst snd left right case zero suc rec → tt25 _
pair25 : ∀{Γ A B} → Tm25 Γ A → Tm25 Γ B → Tm25 Γ (prod25 A B); pair25
= λ t u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec →
pair25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
fst25 : ∀{Γ A B} → Tm25 Γ (prod25 A B) → Tm25 Γ A; fst25
= λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec →
fst25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec)
snd25 : ∀{Γ A B} → Tm25 Γ (prod25 A B) → Tm25 Γ B; snd25
= λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec →
snd25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec)
left25 : ∀{Γ A B} → Tm25 Γ A → Tm25 Γ (sum25 A B); left25
= λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec →
left25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec)
right25 : ∀{Γ A B} → Tm25 Γ B → Tm25 Γ (sum25 A B); right25
= λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec →
right25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec)
case25 : ∀{Γ A B C} → Tm25 Γ (sum25 A B) → Tm25 Γ (arr25 A C) → Tm25 Γ (arr25 B C) → Tm25 Γ C; case25
= λ t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec →
case25 _ _ _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
zero25 : ∀{Γ} → Tm25 Γ nat25; zero25
= λ Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc rec → zero25 _
suc25 : ∀{Γ} → Tm25 Γ nat25 → Tm25 Γ nat25; suc25
= λ t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec →
suc25 _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec)
rec25 : ∀{Γ A} → Tm25 Γ nat25 → Tm25 Γ (arr25 nat25 (arr25 A A)) → Tm25 Γ A → Tm25 Γ A; rec25
= λ t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25 →
rec25 _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
v025 : ∀{Γ A} → Tm25 (snoc25 Γ A) A; v025
= var25 vz25
v125 : ∀{Γ A B} → Tm25 (snoc25 (snoc25 Γ A) B) A; v125
= var25 (vs25 vz25)
v225 : ∀{Γ A B C} → Tm25 (snoc25 (snoc25 (snoc25 Γ A) B) C) A; v225
= var25 (vs25 (vs25 vz25))
v325 : ∀{Γ A B C D} → Tm25 (snoc25 (snoc25 (snoc25 (snoc25 Γ A) B) C) D) A; v325
= var25 (vs25 (vs25 (vs25 vz25)))
tbool25 : Ty25; tbool25
= sum25 top25 top25
true25 : ∀{Γ} → Tm25 Γ tbool25; true25
= left25 tt25
tfalse25 : ∀{Γ} → Tm25 Γ tbool25; tfalse25
= right25 tt25
ifthenelse25 : ∀{Γ A} → Tm25 Γ (arr25 tbool25 (arr25 A (arr25 A A))); ifthenelse25
= lam25 (lam25 (lam25 (case25 v225 (lam25 v225) (lam25 v125))))
times425 : ∀{Γ A} → Tm25 Γ (arr25 (arr25 A A) (arr25 A A)); times425
= lam25 (lam25 (app25 v125 (app25 v125 (app25 v125 (app25 v125 v025)))))
add25 : ∀{Γ} → Tm25 Γ (arr25 nat25 (arr25 nat25 nat25)); add25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (suc25 (app25 v125 v025)))))
(lam25 v025))
mul25 : ∀{Γ} → Tm25 Γ (arr25 nat25 (arr25 nat25 nat25)); mul25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (app25 (app25 add25 (app25 v125 v025)) v025))))
(lam25 zero25))
fact25 : ∀{Γ} → Tm25 Γ (arr25 nat25 nat25); fact25
= lam25 (rec25 v025 (lam25 (lam25 (app25 (app25 mul25 (suc25 v125)) v025)))
(suc25 zero25))
{-# OPTIONS --type-in-type #-}
Ty26 : Set
Ty26 =
(Ty26 : Set)
(nat top bot : Ty26)
(arr prod sum : Ty26 → Ty26 → Ty26)
→ Ty26
nat26 : Ty26; nat26 = λ _ nat26 _ _ _ _ _ → nat26
top26 : Ty26; top26 = λ _ _ top26 _ _ _ _ → top26
bot26 : Ty26; bot26 = λ _ _ _ bot26 _ _ _ → bot26
arr26 : Ty26 → Ty26 → Ty26; arr26
= λ A B Ty26 nat26 top26 bot26 arr26 prod sum →
arr26 (A Ty26 nat26 top26 bot26 arr26 prod sum) (B Ty26 nat26 top26 bot26 arr26 prod sum)
prod26 : Ty26 → Ty26 → Ty26; prod26
= λ A B Ty26 nat26 top26 bot26 arr26 prod26 sum →
prod26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum) (B Ty26 nat26 top26 bot26 arr26 prod26 sum)
sum26 : Ty26 → Ty26 → Ty26; sum26
= λ A B Ty26 nat26 top26 bot26 arr26 prod26 sum26 →
sum26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum26) (B Ty26 nat26 top26 bot26 arr26 prod26 sum26)
Con26 : Set; Con26
= (Con26 : Set)
(nil : Con26)
(snoc : Con26 → Ty26 → Con26)
→ Con26
nil26 : Con26; nil26
= λ Con26 nil26 snoc → nil26
snoc26 : Con26 → Ty26 → Con26; snoc26
= λ Γ A Con26 nil26 snoc26 → snoc26 (Γ Con26 nil26 snoc26) A
Var26 : Con26 → Ty26 → Set; Var26
= λ Γ A →
(Var26 : Con26 → Ty26 → Set)
(vz : ∀ Γ A → Var26 (snoc26 Γ A) A)
(vs : ∀ Γ B A → Var26 Γ A → Var26 (snoc26 Γ B) A)
→ Var26 Γ A
vz26 : ∀{Γ A} → Var26 (snoc26 Γ A) A; vz26
= λ Var26 vz26 vs → vz26 _ _
vs26 : ∀{Γ B A} → Var26 Γ A → Var26 (snoc26 Γ B) A; vs26
= λ x Var26 vz26 vs26 → vs26 _ _ _ (x Var26 vz26 vs26)
Tm26 : Con26 → Ty26 → Set; Tm26
= λ Γ A →
(Tm26 : Con26 → Ty26 → Set)
(var : ∀ Γ A → Var26 Γ A → Tm26 Γ A)
(lam : ∀ Γ A B → Tm26 (snoc26 Γ A) B → Tm26 Γ (arr26 A B))
(app : ∀ Γ A B → Tm26 Γ (arr26 A B) → Tm26 Γ A → Tm26 Γ B)
(tt : ∀ Γ → Tm26 Γ top26)
(pair : ∀ Γ A B → Tm26 Γ A → Tm26 Γ B → Tm26 Γ (prod26 A B))
(fst : ∀ Γ A B → Tm26 Γ (prod26 A B) → Tm26 Γ A)
(snd : ∀ Γ A B → Tm26 Γ (prod26 A B) → Tm26 Γ B)
(left : ∀ Γ A B → Tm26 Γ A → Tm26 Γ (sum26 A B))
(right : ∀ Γ A B → Tm26 Γ B → Tm26 Γ (sum26 A B))
(case : ∀ Γ A B C → Tm26 Γ (sum26 A B) → Tm26 Γ (arr26 A C) → Tm26 Γ (arr26 B C) → Tm26 Γ C)
(zero : ∀ Γ → Tm26 Γ nat26)
(suc : ∀ Γ → Tm26 Γ nat26 → Tm26 Γ nat26)
(rec : ∀ Γ A → Tm26 Γ nat26 → Tm26 Γ (arr26 nat26 (arr26 A A)) → Tm26 Γ A → Tm26 Γ A)
→ Tm26 Γ A
var26 : ∀{Γ A} → Var26 Γ A → Tm26 Γ A; var26
= λ x Tm26 var26 lam app tt pair fst snd left right case zero suc rec →
var26 _ _ x
lam26 : ∀{Γ A B} → Tm26 (snoc26 Γ A) B → Tm26 Γ (arr26 A B); lam26
= λ t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec →
lam26 _ _ _ (t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec)
app26 : ∀{Γ A B} → Tm26 Γ (arr26 A B) → Tm26 Γ A → Tm26 Γ B; app26
= λ t u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec →
app26 _ _ _ (t Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
tt26 : ∀{Γ} → Tm26 Γ top26; tt26
= λ Tm26 var26 lam26 app26 tt26 pair fst snd left right case zero suc rec → tt26 _
pair26 : ∀{Γ A B} → Tm26 Γ A → Tm26 Γ B → Tm26 Γ (prod26 A B); pair26
= λ t u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec →
pair26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
fst26 : ∀{Γ A B} → Tm26 Γ (prod26 A B) → Tm26 Γ A; fst26
= λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec →
fst26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec)
snd26 : ∀{Γ A B} → Tm26 Γ (prod26 A B) → Tm26 Γ B; snd26
= λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec →
snd26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec)
left26 : ∀{Γ A B} → Tm26 Γ A → Tm26 Γ (sum26 A B); left26
= λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec →
left26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec)
right26 : ∀{Γ A B} → Tm26 Γ B → Tm26 Γ (sum26 A B); right26
= λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec →
right26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec)
case26 : ∀{Γ A B C} → Tm26 Γ (sum26 A B) → Tm26 Γ (arr26 A C) → Tm26 Γ (arr26 B C) → Tm26 Γ C; case26
= λ t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec →
case26 _ _ _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
zero26 : ∀{Γ} → Tm26 Γ nat26; zero26
= λ Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc rec → zero26 _
suc26 : ∀{Γ} → Tm26 Γ nat26 → Tm26 Γ nat26; suc26
= λ t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec →
suc26 _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec)
rec26 : ∀{Γ A} → Tm26 Γ nat26 → Tm26 Γ (arr26 nat26 (arr26 A A)) → Tm26 Γ A → Tm26 Γ A; rec26
= λ t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26 →
rec26 _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
v026 : ∀{Γ A} → Tm26 (snoc26 Γ A) A; v026
= var26 vz26
v126 : ∀{Γ A B} → Tm26 (snoc26 (snoc26 Γ A) B) A; v126
= var26 (vs26 vz26)
v226 : ∀{Γ A B C} → Tm26 (snoc26 (snoc26 (snoc26 Γ A) B) C) A; v226
= var26 (vs26 (vs26 vz26))
v326 : ∀{Γ A B C D} → Tm26 (snoc26 (snoc26 (snoc26 (snoc26 Γ A) B) C) D) A; v326
= var26 (vs26 (vs26 (vs26 vz26)))
tbool26 : Ty26; tbool26
= sum26 top26 top26
true26 : ∀{Γ} → Tm26 Γ tbool26; true26
= left26 tt26
tfalse26 : ∀{Γ} → Tm26 Γ tbool26; tfalse26
= right26 tt26
ifthenelse26 : ∀{Γ A} → Tm26 Γ (arr26 tbool26 (arr26 A (arr26 A A))); ifthenelse26
= lam26 (lam26 (lam26 (case26 v226 (lam26 v226) (lam26 v126))))
times426 : ∀{Γ A} → Tm26 Γ (arr26 (arr26 A A) (arr26 A A)); times426
= lam26 (lam26 (app26 v126 (app26 v126 (app26 v126 (app26 v126 v026)))))
add26 : ∀{Γ} → Tm26 Γ (arr26 nat26 (arr26 nat26 nat26)); add26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (suc26 (app26 v126 v026)))))
(lam26 v026))
mul26 : ∀{Γ} → Tm26 Γ (arr26 nat26 (arr26 nat26 nat26)); mul26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (app26 (app26 add26 (app26 v126 v026)) v026))))
(lam26 zero26))
fact26 : ∀{Γ} → Tm26 Γ (arr26 nat26 nat26); fact26
= lam26 (rec26 v026 (lam26 (lam26 (app26 (app26 mul26 (suc26 v126)) v026)))
(suc26 zero26))
{-# OPTIONS --type-in-type #-}
Ty27 : Set
Ty27 =
(Ty27 : Set)
(nat top bot : Ty27)
(arr prod sum : Ty27 → Ty27 → Ty27)
→ Ty27
nat27 : Ty27; nat27 = λ _ nat27 _ _ _ _ _ → nat27
top27 : Ty27; top27 = λ _ _ top27 _ _ _ _ → top27
bot27 : Ty27; bot27 = λ _ _ _ bot27 _ _ _ → bot27
arr27 : Ty27 → Ty27 → Ty27; arr27
= λ A B Ty27 nat27 top27 bot27 arr27 prod sum →
arr27 (A Ty27 nat27 top27 bot27 arr27 prod sum) (B Ty27 nat27 top27 bot27 arr27 prod sum)
prod27 : Ty27 → Ty27 → Ty27; prod27
= λ A B Ty27 nat27 top27 bot27 arr27 prod27 sum →
prod27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum) (B Ty27 nat27 top27 bot27 arr27 prod27 sum)
sum27 : Ty27 → Ty27 → Ty27; sum27
= λ A B Ty27 nat27 top27 bot27 arr27 prod27 sum27 →
sum27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum27) (B Ty27 nat27 top27 bot27 arr27 prod27 sum27)
Con27 : Set; Con27
= (Con27 : Set)
(nil : Con27)
(snoc : Con27 → Ty27 → Con27)
→ Con27
nil27 : Con27; nil27
= λ Con27 nil27 snoc → nil27
snoc27 : Con27 → Ty27 → Con27; snoc27
= λ Γ A Con27 nil27 snoc27 → snoc27 (Γ Con27 nil27 snoc27) A
Var27 : Con27 → Ty27 → Set; Var27
= λ Γ A →
(Var27 : Con27 → Ty27 → Set)
(vz : ∀ Γ A → Var27 (snoc27 Γ A) A)
(vs : ∀ Γ B A → Var27 Γ A → Var27 (snoc27 Γ B) A)
→ Var27 Γ A
vz27 : ∀{Γ A} → Var27 (snoc27 Γ A) A; vz27
= λ Var27 vz27 vs → vz27 _ _
vs27 : ∀{Γ B A} → Var27 Γ A → Var27 (snoc27 Γ B) A; vs27
= λ x Var27 vz27 vs27 → vs27 _ _ _ (x Var27 vz27 vs27)
Tm27 : Con27 → Ty27 → Set; Tm27
= λ Γ A →
(Tm27 : Con27 → Ty27 → Set)
(var : ∀ Γ A → Var27 Γ A → Tm27 Γ A)
(lam : ∀ Γ A B → Tm27 (snoc27 Γ A) B → Tm27 Γ (arr27 A B))
(app : ∀ Γ A B → Tm27 Γ (arr27 A B) → Tm27 Γ A → Tm27 Γ B)
(tt : ∀ Γ → Tm27 Γ top27)
(pair : ∀ Γ A B → Tm27 Γ A → Tm27 Γ B → Tm27 Γ (prod27 A B))
(fst : ∀ Γ A B → Tm27 Γ (prod27 A B) → Tm27 Γ A)
(snd : ∀ Γ A B → Tm27 Γ (prod27 A B) → Tm27 Γ B)
(left : ∀ Γ A B → Tm27 Γ A → Tm27 Γ (sum27 A B))
(right : ∀ Γ A B → Tm27 Γ B → Tm27 Γ (sum27 A B))
(case : ∀ Γ A B C → Tm27 Γ (sum27 A B) → Tm27 Γ (arr27 A C) → Tm27 Γ (arr27 B C) → Tm27 Γ C)
(zero : ∀ Γ → Tm27 Γ nat27)
(suc : ∀ Γ → Tm27 Γ nat27 → Tm27 Γ nat27)
(rec : ∀ Γ A → Tm27 Γ nat27 → Tm27 Γ (arr27 nat27 (arr27 A A)) → Tm27 Γ A → Tm27 Γ A)
→ Tm27 Γ A
var27 : ∀{Γ A} → Var27 Γ A → Tm27 Γ A; var27
= λ x Tm27 var27 lam app tt pair fst snd left right case zero suc rec →
var27 _ _ x
lam27 : ∀{Γ A B} → Tm27 (snoc27 Γ A) B → Tm27 Γ (arr27 A B); lam27
= λ t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec →
lam27 _ _ _ (t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec)
app27 : ∀{Γ A B} → Tm27 Γ (arr27 A B) → Tm27 Γ A → Tm27 Γ B; app27
= λ t u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec →
app27 _ _ _ (t Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
tt27 : ∀{Γ} → Tm27 Γ top27; tt27
= λ Tm27 var27 lam27 app27 tt27 pair fst snd left right case zero suc rec → tt27 _
pair27 : ∀{Γ A B} → Tm27 Γ A → Tm27 Γ B → Tm27 Γ (prod27 A B); pair27
= λ t u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec →
pair27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
fst27 : ∀{Γ A B} → Tm27 Γ (prod27 A B) → Tm27 Γ A; fst27
= λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec →
fst27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec)
snd27 : ∀{Γ A B} → Tm27 Γ (prod27 A B) → Tm27 Γ B; snd27
= λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec →
snd27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec)
left27 : ∀{Γ A B} → Tm27 Γ A → Tm27 Γ (sum27 A B); left27
= λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec →
left27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec)
right27 : ∀{Γ A B} → Tm27 Γ B → Tm27 Γ (sum27 A B); right27
= λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec →
right27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec)
case27 : ∀{Γ A B C} → Tm27 Γ (sum27 A B) → Tm27 Γ (arr27 A C) → Tm27 Γ (arr27 B C) → Tm27 Γ C; case27
= λ t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec →
case27 _ _ _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
zero27 : ∀{Γ} → Tm27 Γ nat27; zero27
= λ Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc rec → zero27 _
suc27 : ∀{Γ} → Tm27 Γ nat27 → Tm27 Γ nat27; suc27
= λ t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec →
suc27 _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec)
rec27 : ∀{Γ A} → Tm27 Γ nat27 → Tm27 Γ (arr27 nat27 (arr27 A A)) → Tm27 Γ A → Tm27 Γ A; rec27
= λ t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27 →
rec27 _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
v027 : ∀{Γ A} → Tm27 (snoc27 Γ A) A; v027
= var27 vz27
v127 : ∀{Γ A B} → Tm27 (snoc27 (snoc27 Γ A) B) A; v127
= var27 (vs27 vz27)
v227 : ∀{Γ A B C} → Tm27 (snoc27 (snoc27 (snoc27 Γ A) B) C) A; v227
= var27 (vs27 (vs27 vz27))
v327 : ∀{Γ A B C D} → Tm27 (snoc27 (snoc27 (snoc27 (snoc27 Γ A) B) C) D) A; v327
= var27 (vs27 (vs27 (vs27 vz27)))
tbool27 : Ty27; tbool27
= sum27 top27 top27
true27 : ∀{Γ} → Tm27 Γ tbool27; true27
= left27 tt27
tfalse27 : ∀{Γ} → Tm27 Γ tbool27; tfalse27
= right27 tt27
ifthenelse27 : ∀{Γ A} → Tm27 Γ (arr27 tbool27 (arr27 A (arr27 A A))); ifthenelse27
= lam27 (lam27 (lam27 (case27 v227 (lam27 v227) (lam27 v127))))
times427 : ∀{Γ A} → Tm27 Γ (arr27 (arr27 A A) (arr27 A A)); times427
= lam27 (lam27 (app27 v127 (app27 v127 (app27 v127 (app27 v127 v027)))))
add27 : ∀{Γ} → Tm27 Γ (arr27 nat27 (arr27 nat27 nat27)); add27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (suc27 (app27 v127 v027)))))
(lam27 v027))
mul27 : ∀{Γ} → Tm27 Γ (arr27 nat27 (arr27 nat27 nat27)); mul27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (app27 (app27 add27 (app27 v127 v027)) v027))))
(lam27 zero27))
fact27 : ∀{Γ} → Tm27 Γ (arr27 nat27 nat27); fact27
= lam27 (rec27 v027 (lam27 (lam27 (app27 (app27 mul27 (suc27 v127)) v027)))
(suc27 zero27))
{-# OPTIONS --type-in-type #-}
Ty28 : Set
Ty28 =
(Ty28 : Set)
(nat top bot : Ty28)
(arr prod sum : Ty28 → Ty28 → Ty28)
→ Ty28
nat28 : Ty28; nat28 = λ _ nat28 _ _ _ _ _ → nat28
top28 : Ty28; top28 = λ _ _ top28 _ _ _ _ → top28
bot28 : Ty28; bot28 = λ _ _ _ bot28 _ _ _ → bot28
arr28 : Ty28 → Ty28 → Ty28; arr28
= λ A B Ty28 nat28 top28 bot28 arr28 prod sum →
arr28 (A Ty28 nat28 top28 bot28 arr28 prod sum) (B Ty28 nat28 top28 bot28 arr28 prod sum)
prod28 : Ty28 → Ty28 → Ty28; prod28
= λ A B Ty28 nat28 top28 bot28 arr28 prod28 sum →
prod28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum) (B Ty28 nat28 top28 bot28 arr28 prod28 sum)
sum28 : Ty28 → Ty28 → Ty28; sum28
= λ A B Ty28 nat28 top28 bot28 arr28 prod28 sum28 →
sum28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum28) (B Ty28 nat28 top28 bot28 arr28 prod28 sum28)
Con28 : Set; Con28
= (Con28 : Set)
(nil : Con28)
(snoc : Con28 → Ty28 → Con28)
→ Con28
nil28 : Con28; nil28
= λ Con28 nil28 snoc → nil28
snoc28 : Con28 → Ty28 → Con28; snoc28
= λ Γ A Con28 nil28 snoc28 → snoc28 (Γ Con28 nil28 snoc28) A
Var28 : Con28 → Ty28 → Set; Var28
= λ Γ A →
(Var28 : Con28 → Ty28 → Set)
(vz : ∀ Γ A → Var28 (snoc28 Γ A) A)
(vs : ∀ Γ B A → Var28 Γ A → Var28 (snoc28 Γ B) A)
→ Var28 Γ A
vz28 : ∀{Γ A} → Var28 (snoc28 Γ A) A; vz28
= λ Var28 vz28 vs → vz28 _ _
vs28 : ∀{Γ B A} → Var28 Γ A → Var28 (snoc28 Γ B) A; vs28
= λ x Var28 vz28 vs28 → vs28 _ _ _ (x Var28 vz28 vs28)
Tm28 : Con28 → Ty28 → Set; Tm28
= λ Γ A →
(Tm28 : Con28 → Ty28 → Set)
(var : ∀ Γ A → Var28 Γ A → Tm28 Γ A)
(lam : ∀ Γ A B → Tm28 (snoc28 Γ A) B → Tm28 Γ (arr28 A B))
(app : ∀ Γ A B → Tm28 Γ (arr28 A B) → Tm28 Γ A → Tm28 Γ B)
(tt : ∀ Γ → Tm28 Γ top28)
(pair : ∀ Γ A B → Tm28 Γ A → Tm28 Γ B → Tm28 Γ (prod28 A B))
(fst : ∀ Γ A B → Tm28 Γ (prod28 A B) → Tm28 Γ A)
(snd : ∀ Γ A B → Tm28 Γ (prod28 A B) → Tm28 Γ B)
(left : ∀ Γ A B → Tm28 Γ A → Tm28 Γ (sum28 A B))
(right : ∀ Γ A B → Tm28 Γ B → Tm28 Γ (sum28 A B))
(case : ∀ Γ A B C → Tm28 Γ (sum28 A B) → Tm28 Γ (arr28 A C) → Tm28 Γ (arr28 B C) → Tm28 Γ C)
(zero : ∀ Γ → Tm28 Γ nat28)
(suc : ∀ Γ → Tm28 Γ nat28 → Tm28 Γ nat28)
(rec : ∀ Γ A → Tm28 Γ nat28 → Tm28 Γ (arr28 nat28 (arr28 A A)) → Tm28 Γ A → Tm28 Γ A)
→ Tm28 Γ A
var28 : ∀{Γ A} → Var28 Γ A → Tm28 Γ A; var28
= λ x Tm28 var28 lam app tt pair fst snd left right case zero suc rec →
var28 _ _ x
lam28 : ∀{Γ A B} → Tm28 (snoc28 Γ A) B → Tm28 Γ (arr28 A B); lam28
= λ t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec →
lam28 _ _ _ (t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec)
app28 : ∀{Γ A B} → Tm28 Γ (arr28 A B) → Tm28 Γ A → Tm28 Γ B; app28
= λ t u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec →
app28 _ _ _ (t Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
tt28 : ∀{Γ} → Tm28 Γ top28; tt28
= λ Tm28 var28 lam28 app28 tt28 pair fst snd left right case zero suc rec → tt28 _
pair28 : ∀{Γ A B} → Tm28 Γ A → Tm28 Γ B → Tm28 Γ (prod28 A B); pair28
= λ t u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec →
pair28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
fst28 : ∀{Γ A B} → Tm28 Γ (prod28 A B) → Tm28 Γ A; fst28
= λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec →
fst28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec)
snd28 : ∀{Γ A B} → Tm28 Γ (prod28 A B) → Tm28 Γ B; snd28
= λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec →
snd28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec)
left28 : ∀{Γ A B} → Tm28 Γ A → Tm28 Γ (sum28 A B); left28
= λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec →
left28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec)
right28 : ∀{Γ A B} → Tm28 Γ B → Tm28 Γ (sum28 A B); right28
= λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec →
right28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec)
case28 : ∀{Γ A B C} → Tm28 Γ (sum28 A B) → Tm28 Γ (arr28 A C) → Tm28 Γ (arr28 B C) → Tm28 Γ C; case28
= λ t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec →
case28 _ _ _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
zero28 : ∀{Γ} → Tm28 Γ nat28; zero28
= λ Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc rec → zero28 _
suc28 : ∀{Γ} → Tm28 Γ nat28 → Tm28 Γ nat28; suc28
= λ t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec →
suc28 _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec)
rec28 : ∀{Γ A} → Tm28 Γ nat28 → Tm28 Γ (arr28 nat28 (arr28 A A)) → Tm28 Γ A → Tm28 Γ A; rec28
= λ t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28 →
rec28 _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
v028 : ∀{Γ A} → Tm28 (snoc28 Γ A) A; v028
= var28 vz28
v128 : ∀{Γ A B} → Tm28 (snoc28 (snoc28 Γ A) B) A; v128
= var28 (vs28 vz28)
v228 : ∀{Γ A B C} → Tm28 (snoc28 (snoc28 (snoc28 Γ A) B) C) A; v228
= var28 (vs28 (vs28 vz28))
v328 : ∀{Γ A B C D} → Tm28 (snoc28 (snoc28 (snoc28 (snoc28 Γ A) B) C) D) A; v328
= var28 (vs28 (vs28 (vs28 vz28)))
tbool28 : Ty28; tbool28
= sum28 top28 top28
true28 : ∀{Γ} → Tm28 Γ tbool28; true28
= left28 tt28
tfalse28 : ∀{Γ} → Tm28 Γ tbool28; tfalse28
= right28 tt28
ifthenelse28 : ∀{Γ A} → Tm28 Γ (arr28 tbool28 (arr28 A (arr28 A A))); ifthenelse28
= lam28 (lam28 (lam28 (case28 v228 (lam28 v228) (lam28 v128))))
times428 : ∀{Γ A} → Tm28 Γ (arr28 (arr28 A A) (arr28 A A)); times428
= lam28 (lam28 (app28 v128 (app28 v128 (app28 v128 (app28 v128 v028)))))
add28 : ∀{Γ} → Tm28 Γ (arr28 nat28 (arr28 nat28 nat28)); add28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (suc28 (app28 v128 v028)))))
(lam28 v028))
mul28 : ∀{Γ} → Tm28 Γ (arr28 nat28 (arr28 nat28 nat28)); mul28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (app28 (app28 add28 (app28 v128 v028)) v028))))
(lam28 zero28))
fact28 : ∀{Γ} → Tm28 Γ (arr28 nat28 nat28); fact28
= lam28 (rec28 v028 (lam28 (lam28 (app28 (app28 mul28 (suc28 v128)) v028)))
(suc28 zero28))
{-# OPTIONS --type-in-type #-}
Ty29 : Set
Ty29 =
(Ty29 : Set)
(nat top bot : Ty29)
(arr prod sum : Ty29 → Ty29 → Ty29)
→ Ty29
nat29 : Ty29; nat29 = λ _ nat29 _ _ _ _ _ → nat29
top29 : Ty29; top29 = λ _ _ top29 _ _ _ _ → top29
bot29 : Ty29; bot29 = λ _ _ _ bot29 _ _ _ → bot29
arr29 : Ty29 → Ty29 → Ty29; arr29
= λ A B Ty29 nat29 top29 bot29 arr29 prod sum →
arr29 (A Ty29 nat29 top29 bot29 arr29 prod sum) (B Ty29 nat29 top29 bot29 arr29 prod sum)
prod29 : Ty29 → Ty29 → Ty29; prod29
= λ A B Ty29 nat29 top29 bot29 arr29 prod29 sum →
prod29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum) (B Ty29 nat29 top29 bot29 arr29 prod29 sum)
sum29 : Ty29 → Ty29 → Ty29; sum29
= λ A B Ty29 nat29 top29 bot29 arr29 prod29 sum29 →
sum29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum29) (B Ty29 nat29 top29 bot29 arr29 prod29 sum29)
Con29 : Set; Con29
= (Con29 : Set)
(nil : Con29)
(snoc : Con29 → Ty29 → Con29)
→ Con29
nil29 : Con29; nil29
= λ Con29 nil29 snoc → nil29
snoc29 : Con29 → Ty29 → Con29; snoc29
= λ Γ A Con29 nil29 snoc29 → snoc29 (Γ Con29 nil29 snoc29) A
Var29 : Con29 → Ty29 → Set; Var29
= λ Γ A →
(Var29 : Con29 → Ty29 → Set)
(vz : ∀ Γ A → Var29 (snoc29 Γ A) A)
(vs : ∀ Γ B A → Var29 Γ A → Var29 (snoc29 Γ B) A)
→ Var29 Γ A
vz29 : ∀{Γ A} → Var29 (snoc29 Γ A) A; vz29
= λ Var29 vz29 vs → vz29 _ _
vs29 : ∀{Γ B A} → Var29 Γ A → Var29 (snoc29 Γ B) A; vs29
= λ x Var29 vz29 vs29 → vs29 _ _ _ (x Var29 vz29 vs29)
Tm29 : Con29 → Ty29 → Set; Tm29
= λ Γ A →
(Tm29 : Con29 → Ty29 → Set)
(var : ∀ Γ A → Var29 Γ A → Tm29 Γ A)
(lam : ∀ Γ A B → Tm29 (snoc29 Γ A) B → Tm29 Γ (arr29 A B))
(app : ∀ Γ A B → Tm29 Γ (arr29 A B) → Tm29 Γ A → Tm29 Γ B)
(tt : ∀ Γ → Tm29 Γ top29)
(pair : ∀ Γ A B → Tm29 Γ A → Tm29 Γ B → Tm29 Γ (prod29 A B))
(fst : ∀ Γ A B → Tm29 Γ (prod29 A B) → Tm29 Γ A)
(snd : ∀ Γ A B → Tm29 Γ (prod29 A B) → Tm29 Γ B)
(left : ∀ Γ A B → Tm29 Γ A → Tm29 Γ (sum29 A B))
(right : ∀ Γ A B → Tm29 Γ B → Tm29 Γ (sum29 A B))
(case : ∀ Γ A B C → Tm29 Γ (sum29 A B) → Tm29 Γ (arr29 A C) → Tm29 Γ (arr29 B C) → Tm29 Γ C)
(zero : ∀ Γ → Tm29 Γ nat29)
(suc : ∀ Γ → Tm29 Γ nat29 → Tm29 Γ nat29)
(rec : ∀ Γ A → Tm29 Γ nat29 → Tm29 Γ (arr29 nat29 (arr29 A A)) → Tm29 Γ A → Tm29 Γ A)
→ Tm29 Γ A
var29 : ∀{Γ A} → Var29 Γ A → Tm29 Γ A; var29
= λ x Tm29 var29 lam app tt pair fst snd left right case zero suc rec →
var29 _ _ x
lam29 : ∀{Γ A B} → Tm29 (snoc29 Γ A) B → Tm29 Γ (arr29 A B); lam29
= λ t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec →
lam29 _ _ _ (t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec)
app29 : ∀{Γ A B} → Tm29 Γ (arr29 A B) → Tm29 Γ A → Tm29 Γ B; app29
= λ t u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec →
app29 _ _ _ (t Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
tt29 : ∀{Γ} → Tm29 Γ top29; tt29
= λ Tm29 var29 lam29 app29 tt29 pair fst snd left right case zero suc rec → tt29 _
pair29 : ∀{Γ A B} → Tm29 Γ A → Tm29 Γ B → Tm29 Γ (prod29 A B); pair29
= λ t u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec →
pair29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
fst29 : ∀{Γ A B} → Tm29 Γ (prod29 A B) → Tm29 Γ A; fst29
= λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec →
fst29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec)
snd29 : ∀{Γ A B} → Tm29 Γ (prod29 A B) → Tm29 Γ B; snd29
= λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec →
snd29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec)
left29 : ∀{Γ A B} → Tm29 Γ A → Tm29 Γ (sum29 A B); left29
= λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec →
left29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec)
right29 : ∀{Γ A B} → Tm29 Γ B → Tm29 Γ (sum29 A B); right29
= λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec →
right29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec)
case29 : ∀{Γ A B C} → Tm29 Γ (sum29 A B) → Tm29 Γ (arr29 A C) → Tm29 Γ (arr29 B C) → Tm29 Γ C; case29
= λ t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec →
case29 _ _ _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
zero29 : ∀{Γ} → Tm29 Γ nat29; zero29
= λ Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc rec → zero29 _
suc29 : ∀{Γ} → Tm29 Γ nat29 → Tm29 Γ nat29; suc29
= λ t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec →
suc29 _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec)
rec29 : ∀{Γ A} → Tm29 Γ nat29 → Tm29 Γ (arr29 nat29 (arr29 A A)) → Tm29 Γ A → Tm29 Γ A; rec29
= λ t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29 →
rec29 _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
v029 : ∀{Γ A} → Tm29 (snoc29 Γ A) A; v029
= var29 vz29
v129 : ∀{Γ A B} → Tm29 (snoc29 (snoc29 Γ A) B) A; v129
= var29 (vs29 vz29)
v229 : ∀{Γ A B C} → Tm29 (snoc29 (snoc29 (snoc29 Γ A) B) C) A; v229
= var29 (vs29 (vs29 vz29))
v329 : ∀{Γ A B C D} → Tm29 (snoc29 (snoc29 (snoc29 (snoc29 Γ A) B) C) D) A; v329
= var29 (vs29 (vs29 (vs29 vz29)))
tbool29 : Ty29; tbool29
= sum29 top29 top29
true29 : ∀{Γ} → Tm29 Γ tbool29; true29
= left29 tt29
tfalse29 : ∀{Γ} → Tm29 Γ tbool29; tfalse29
= right29 tt29
ifthenelse29 : ∀{Γ A} → Tm29 Γ (arr29 tbool29 (arr29 A (arr29 A A))); ifthenelse29
= lam29 (lam29 (lam29 (case29 v229 (lam29 v229) (lam29 v129))))
times429 : ∀{Γ A} → Tm29 Γ (arr29 (arr29 A A) (arr29 A A)); times429
= lam29 (lam29 (app29 v129 (app29 v129 (app29 v129 (app29 v129 v029)))))
add29 : ∀{Γ} → Tm29 Γ (arr29 nat29 (arr29 nat29 nat29)); add29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (suc29 (app29 v129 v029)))))
(lam29 v029))
mul29 : ∀{Γ} → Tm29 Γ (arr29 nat29 (arr29 nat29 nat29)); mul29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (app29 (app29 add29 (app29 v129 v029)) v029))))
(lam29 zero29))
fact29 : ∀{Γ} → Tm29 Γ (arr29 nat29 nat29); fact29
= lam29 (rec29 v029 (lam29 (lam29 (app29 (app29 mul29 (suc29 v129)) v029)))
(suc29 zero29))
{-# OPTIONS --type-in-type #-}
Ty30 : Set
Ty30 =
(Ty30 : Set)
(nat top bot : Ty30)
(arr prod sum : Ty30 → Ty30 → Ty30)
→ Ty30
nat30 : Ty30; nat30 = λ _ nat30 _ _ _ _ _ → nat30
top30 : Ty30; top30 = λ _ _ top30 _ _ _ _ → top30
bot30 : Ty30; bot30 = λ _ _ _ bot30 _ _ _ → bot30
arr30 : Ty30 → Ty30 → Ty30; arr30
= λ A B Ty30 nat30 top30 bot30 arr30 prod sum →
arr30 (A Ty30 nat30 top30 bot30 arr30 prod sum) (B Ty30 nat30 top30 bot30 arr30 prod sum)
prod30 : Ty30 → Ty30 → Ty30; prod30
= λ A B Ty30 nat30 top30 bot30 arr30 prod30 sum →
prod30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum) (B Ty30 nat30 top30 bot30 arr30 prod30 sum)
sum30 : Ty30 → Ty30 → Ty30; sum30
= λ A B Ty30 nat30 top30 bot30 arr30 prod30 sum30 →
sum30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum30) (B Ty30 nat30 top30 bot30 arr30 prod30 sum30)
Con30 : Set; Con30
= (Con30 : Set)
(nil : Con30)
(snoc : Con30 → Ty30 → Con30)
→ Con30
nil30 : Con30; nil30
= λ Con30 nil30 snoc → nil30
snoc30 : Con30 → Ty30 → Con30; snoc30
= λ Γ A Con30 nil30 snoc30 → snoc30 (Γ Con30 nil30 snoc30) A
Var30 : Con30 → Ty30 → Set; Var30
= λ Γ A →
(Var30 : Con30 → Ty30 → Set)
(vz : ∀ Γ A → Var30 (snoc30 Γ A) A)
(vs : ∀ Γ B A → Var30 Γ A → Var30 (snoc30 Γ B) A)
→ Var30 Γ A
vz30 : ∀{Γ A} → Var30 (snoc30 Γ A) A; vz30
= λ Var30 vz30 vs → vz30 _ _
vs30 : ∀{Γ B A} → Var30 Γ A → Var30 (snoc30 Γ B) A; vs30
= λ x Var30 vz30 vs30 → vs30 _ _ _ (x Var30 vz30 vs30)
Tm30 : Con30 → Ty30 → Set; Tm30
= λ Γ A →
(Tm30 : Con30 → Ty30 → Set)
(var : ∀ Γ A → Var30 Γ A → Tm30 Γ A)
(lam : ∀ Γ A B → Tm30 (snoc30 Γ A) B → Tm30 Γ (arr30 A B))
(app : ∀ Γ A B → Tm30 Γ (arr30 A B) → Tm30 Γ A → Tm30 Γ B)
(tt : ∀ Γ → Tm30 Γ top30)
(pair : ∀ Γ A B → Tm30 Γ A → Tm30 Γ B → Tm30 Γ (prod30 A B))
(fst : ∀ Γ A B → Tm30 Γ (prod30 A B) → Tm30 Γ A)
(snd : ∀ Γ A B → Tm30 Γ (prod30 A B) → Tm30 Γ B)
(left : ∀ Γ A B → Tm30 Γ A → Tm30 Γ (sum30 A B))
(right : ∀ Γ A B → Tm30 Γ B → Tm30 Γ (sum30 A B))
(case : ∀ Γ A B C → Tm30 Γ (sum30 A B) → Tm30 Γ (arr30 A C) → Tm30 Γ (arr30 B C) → Tm30 Γ C)
(zero : ∀ Γ → Tm30 Γ nat30)
(suc : ∀ Γ → Tm30 Γ nat30 → Tm30 Γ nat30)
(rec : ∀ Γ A → Tm30 Γ nat30 → Tm30 Γ (arr30 nat30 (arr30 A A)) → Tm30 Γ A → Tm30 Γ A)
→ Tm30 Γ A
var30 : ∀{Γ A} → Var30 Γ A → Tm30 Γ A; var30
= λ x Tm30 var30 lam app tt pair fst snd left right case zero suc rec →
var30 _ _ x
lam30 : ∀{Γ A B} → Tm30 (snoc30 Γ A) B → Tm30 Γ (arr30 A B); lam30
= λ t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec →
lam30 _ _ _ (t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec)
app30 : ∀{Γ A B} → Tm30 Γ (arr30 A B) → Tm30 Γ A → Tm30 Γ B; app30
= λ t u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec →
app30 _ _ _ (t Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
tt30 : ∀{Γ} → Tm30 Γ top30; tt30
= λ Tm30 var30 lam30 app30 tt30 pair fst snd left right case zero suc rec → tt30 _
pair30 : ∀{Γ A B} → Tm30 Γ A → Tm30 Γ B → Tm30 Γ (prod30 A B); pair30
= λ t u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec →
pair30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
fst30 : ∀{Γ A B} → Tm30 Γ (prod30 A B) → Tm30 Γ A; fst30
= λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec →
fst30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec)
snd30 : ∀{Γ A B} → Tm30 Γ (prod30 A B) → Tm30 Γ B; snd30
= λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec →
snd30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec)
left30 : ∀{Γ A B} → Tm30 Γ A → Tm30 Γ (sum30 A B); left30
= λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec →
left30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec)
right30 : ∀{Γ A B} → Tm30 Γ B → Tm30 Γ (sum30 A B); right30
= λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec →
right30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec)
case30 : ∀{Γ A B C} → Tm30 Γ (sum30 A B) → Tm30 Γ (arr30 A C) → Tm30 Γ (arr30 B C) → Tm30 Γ C; case30
= λ t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec →
case30 _ _ _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
zero30 : ∀{Γ} → Tm30 Γ nat30; zero30
= λ Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc rec → zero30 _
suc30 : ∀{Γ} → Tm30 Γ nat30 → Tm30 Γ nat30; suc30
= λ t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec →
suc30 _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec)
rec30 : ∀{Γ A} → Tm30 Γ nat30 → Tm30 Γ (arr30 nat30 (arr30 A A)) → Tm30 Γ A → Tm30 Γ A; rec30
= λ t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30 →
rec30 _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
v030 : ∀{Γ A} → Tm30 (snoc30 Γ A) A; v030
= var30 vz30
v130 : ∀{Γ A B} → Tm30 (snoc30 (snoc30 Γ A) B) A; v130
= var30 (vs30 vz30)
v230 : ∀{Γ A B C} → Tm30 (snoc30 (snoc30 (snoc30 Γ A) B) C) A; v230
= var30 (vs30 (vs30 vz30))
v330 : ∀{Γ A B C D} → Tm30 (snoc30 (snoc30 (snoc30 (snoc30 Γ A) B) C) D) A; v330
= var30 (vs30 (vs30 (vs30 vz30)))
tbool30 : Ty30; tbool30
= sum30 top30 top30
true30 : ∀{Γ} → Tm30 Γ tbool30; true30
= left30 tt30
tfalse30 : ∀{Γ} → Tm30 Γ tbool30; tfalse30
= right30 tt30
ifthenelse30 : ∀{Γ A} → Tm30 Γ (arr30 tbool30 (arr30 A (arr30 A A))); ifthenelse30
= lam30 (lam30 (lam30 (case30 v230 (lam30 v230) (lam30 v130))))
times430 : ∀{Γ A} → Tm30 Γ (arr30 (arr30 A A) (arr30 A A)); times430
= lam30 (lam30 (app30 v130 (app30 v130 (app30 v130 (app30 v130 v030)))))
add30 : ∀{Γ} → Tm30 Γ (arr30 nat30 (arr30 nat30 nat30)); add30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (suc30 (app30 v130 v030)))))
(lam30 v030))
mul30 : ∀{Γ} → Tm30 Γ (arr30 nat30 (arr30 nat30 nat30)); mul30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (app30 (app30 add30 (app30 v130 v030)) v030))))
(lam30 zero30))
fact30 : ∀{Γ} → Tm30 Γ (arr30 nat30 nat30); fact30
= lam30 (rec30 v030 (lam30 (lam30 (app30 (app30 mul30 (suc30 v130)) v030)))
(suc30 zero30))
{-# OPTIONS --type-in-type #-}
Ty31 : Set
Ty31 =
(Ty31 : Set)
(nat top bot : Ty31)
(arr prod sum : Ty31 → Ty31 → Ty31)
→ Ty31
nat31 : Ty31; nat31 = λ _ nat31 _ _ _ _ _ → nat31
top31 : Ty31; top31 = λ _ _ top31 _ _ _ _ → top31
bot31 : Ty31; bot31 = λ _ _ _ bot31 _ _ _ → bot31
arr31 : Ty31 → Ty31 → Ty31; arr31
= λ A B Ty31 nat31 top31 bot31 arr31 prod sum →
arr31 (A Ty31 nat31 top31 bot31 arr31 prod sum) (B Ty31 nat31 top31 bot31 arr31 prod sum)
prod31 : Ty31 → Ty31 → Ty31; prod31
= λ A B Ty31 nat31 top31 bot31 arr31 prod31 sum →
prod31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum) (B Ty31 nat31 top31 bot31 arr31 prod31 sum)
sum31 : Ty31 → Ty31 → Ty31; sum31
= λ A B Ty31 nat31 top31 bot31 arr31 prod31 sum31 →
sum31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum31) (B Ty31 nat31 top31 bot31 arr31 prod31 sum31)
Con31 : Set; Con31
= (Con31 : Set)
(nil : Con31)
(snoc : Con31 → Ty31 → Con31)
→ Con31
nil31 : Con31; nil31
= λ Con31 nil31 snoc → nil31
snoc31 : Con31 → Ty31 → Con31; snoc31
= λ Γ A Con31 nil31 snoc31 → snoc31 (Γ Con31 nil31 snoc31) A
Var31 : Con31 → Ty31 → Set; Var31
= λ Γ A →
(Var31 : Con31 → Ty31 → Set)
(vz : ∀ Γ A → Var31 (snoc31 Γ A) A)
(vs : ∀ Γ B A → Var31 Γ A → Var31 (snoc31 Γ B) A)
→ Var31 Γ A
vz31 : ∀{Γ A} → Var31 (snoc31 Γ A) A; vz31
= λ Var31 vz31 vs → vz31 _ _
vs31 : ∀{Γ B A} → Var31 Γ A → Var31 (snoc31 Γ B) A; vs31
= λ x Var31 vz31 vs31 → vs31 _ _ _ (x Var31 vz31 vs31)
Tm31 : Con31 → Ty31 → Set; Tm31
= λ Γ A →
(Tm31 : Con31 → Ty31 → Set)
(var : ∀ Γ A → Var31 Γ A → Tm31 Γ A)
(lam : ∀ Γ A B → Tm31 (snoc31 Γ A) B → Tm31 Γ (arr31 A B))
(app : ∀ Γ A B → Tm31 Γ (arr31 A B) → Tm31 Γ A → Tm31 Γ B)
(tt : ∀ Γ → Tm31 Γ top31)
(pair : ∀ Γ A B → Tm31 Γ A → Tm31 Γ B → Tm31 Γ (prod31 A B))
(fst : ∀ Γ A B → Tm31 Γ (prod31 A B) → Tm31 Γ A)
(snd : ∀ Γ A B → Tm31 Γ (prod31 A B) → Tm31 Γ B)
(left : ∀ Γ A B → Tm31 Γ A → Tm31 Γ (sum31 A B))
(right : ∀ Γ A B → Tm31 Γ B → Tm31 Γ (sum31 A B))
(case : ∀ Γ A B C → Tm31 Γ (sum31 A B) → Tm31 Γ (arr31 A C) → Tm31 Γ (arr31 B C) → Tm31 Γ C)
(zero : ∀ Γ → Tm31 Γ nat31)
(suc : ∀ Γ → Tm31 Γ nat31 → Tm31 Γ nat31)
(rec : ∀ Γ A → Tm31 Γ nat31 → Tm31 Γ (arr31 nat31 (arr31 A A)) → Tm31 Γ A → Tm31 Γ A)
→ Tm31 Γ A
var31 : ∀{Γ A} → Var31 Γ A → Tm31 Γ A; var31
= λ x Tm31 var31 lam app tt pair fst snd left right case zero suc rec →
var31 _ _ x
lam31 : ∀{Γ A B} → Tm31 (snoc31 Γ A) B → Tm31 Γ (arr31 A B); lam31
= λ t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec →
lam31 _ _ _ (t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec)
app31 : ∀{Γ A B} → Tm31 Γ (arr31 A B) → Tm31 Γ A → Tm31 Γ B; app31
= λ t u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec →
app31 _ _ _ (t Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
tt31 : ∀{Γ} → Tm31 Γ top31; tt31
= λ Tm31 var31 lam31 app31 tt31 pair fst snd left right case zero suc rec → tt31 _
pair31 : ∀{Γ A B} → Tm31 Γ A → Tm31 Γ B → Tm31 Γ (prod31 A B); pair31
= λ t u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec →
pair31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
fst31 : ∀{Γ A B} → Tm31 Γ (prod31 A B) → Tm31 Γ A; fst31
= λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec →
fst31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec)
snd31 : ∀{Γ A B} → Tm31 Γ (prod31 A B) → Tm31 Γ B; snd31
= λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec →
snd31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec)
left31 : ∀{Γ A B} → Tm31 Γ A → Tm31 Γ (sum31 A B); left31
= λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec →
left31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec)
right31 : ∀{Γ A B} → Tm31 Γ B → Tm31 Γ (sum31 A B); right31
= λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec →
right31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec)
case31 : ∀{Γ A B C} → Tm31 Γ (sum31 A B) → Tm31 Γ (arr31 A C) → Tm31 Γ (arr31 B C) → Tm31 Γ C; case31
= λ t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec →
case31 _ _ _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
zero31 : ∀{Γ} → Tm31 Γ nat31; zero31
= λ Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc rec → zero31 _
suc31 : ∀{Γ} → Tm31 Γ nat31 → Tm31 Γ nat31; suc31
= λ t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec →
suc31 _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec)
rec31 : ∀{Γ A} → Tm31 Γ nat31 → Tm31 Γ (arr31 nat31 (arr31 A A)) → Tm31 Γ A → Tm31 Γ A; rec31
= λ t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31 →
rec31 _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
v031 : ∀{Γ A} → Tm31 (snoc31 Γ A) A; v031
= var31 vz31
v131 : ∀{Γ A B} → Tm31 (snoc31 (snoc31 Γ A) B) A; v131
= var31 (vs31 vz31)
v231 : ∀{Γ A B C} → Tm31 (snoc31 (snoc31 (snoc31 Γ A) B) C) A; v231
= var31 (vs31 (vs31 vz31))
v331 : ∀{Γ A B C D} → Tm31 (snoc31 (snoc31 (snoc31 (snoc31 Γ A) B) C) D) A; v331
= var31 (vs31 (vs31 (vs31 vz31)))
tbool31 : Ty31; tbool31
= sum31 top31 top31
true31 : ∀{Γ} → Tm31 Γ tbool31; true31
= left31 tt31
tfalse31 : ∀{Γ} → Tm31 Γ tbool31; tfalse31
= right31 tt31
ifthenelse31 : ∀{Γ A} → Tm31 Γ (arr31 tbool31 (arr31 A (arr31 A A))); ifthenelse31
= lam31 (lam31 (lam31 (case31 v231 (lam31 v231) (lam31 v131))))
times431 : ∀{Γ A} → Tm31 Γ (arr31 (arr31 A A) (arr31 A A)); times431
= lam31 (lam31 (app31 v131 (app31 v131 (app31 v131 (app31 v131 v031)))))
add31 : ∀{Γ} → Tm31 Γ (arr31 nat31 (arr31 nat31 nat31)); add31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (suc31 (app31 v131 v031)))))
(lam31 v031))
mul31 : ∀{Γ} → Tm31 Γ (arr31 nat31 (arr31 nat31 nat31)); mul31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (app31 (app31 add31 (app31 v131 v031)) v031))))
(lam31 zero31))
fact31 : ∀{Γ} → Tm31 Γ (arr31 nat31 nat31); fact31
= lam31 (rec31 v031 (lam31 (lam31 (app31 (app31 mul31 (suc31 v131)) v031)))
(suc31 zero31))
{-# OPTIONS --type-in-type #-}
Ty32 : Set
Ty32 =
(Ty32 : Set)
(nat top bot : Ty32)
(arr prod sum : Ty32 → Ty32 → Ty32)
→ Ty32
nat32 : Ty32; nat32 = λ _ nat32 _ _ _ _ _ → nat32
top32 : Ty32; top32 = λ _ _ top32 _ _ _ _ → top32
bot32 : Ty32; bot32 = λ _ _ _ bot32 _ _ _ → bot32
arr32 : Ty32 → Ty32 → Ty32; arr32
= λ A B Ty32 nat32 top32 bot32 arr32 prod sum →
arr32 (A Ty32 nat32 top32 bot32 arr32 prod sum) (B Ty32 nat32 top32 bot32 arr32 prod sum)
prod32 : Ty32 → Ty32 → Ty32; prod32
= λ A B Ty32 nat32 top32 bot32 arr32 prod32 sum →
prod32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum) (B Ty32 nat32 top32 bot32 arr32 prod32 sum)
sum32 : Ty32 → Ty32 → Ty32; sum32
= λ A B Ty32 nat32 top32 bot32 arr32 prod32 sum32 →
sum32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum32) (B Ty32 nat32 top32 bot32 arr32 prod32 sum32)
Con32 : Set; Con32
= (Con32 : Set)
(nil : Con32)
(snoc : Con32 → Ty32 → Con32)
→ Con32
nil32 : Con32; nil32
= λ Con32 nil32 snoc → nil32
snoc32 : Con32 → Ty32 → Con32; snoc32
= λ Γ A Con32 nil32 snoc32 → snoc32 (Γ Con32 nil32 snoc32) A
Var32 : Con32 → Ty32 → Set; Var32
= λ Γ A →
(Var32 : Con32 → Ty32 → Set)
(vz : ∀ Γ A → Var32 (snoc32 Γ A) A)
(vs : ∀ Γ B A → Var32 Γ A → Var32 (snoc32 Γ B) A)
→ Var32 Γ A
vz32 : ∀{Γ A} → Var32 (snoc32 Γ A) A; vz32
= λ Var32 vz32 vs → vz32 _ _
vs32 : ∀{Γ B A} → Var32 Γ A → Var32 (snoc32 Γ B) A; vs32
= λ x Var32 vz32 vs32 → vs32 _ _ _ (x Var32 vz32 vs32)
Tm32 : Con32 → Ty32 → Set; Tm32
= λ Γ A →
(Tm32 : Con32 → Ty32 → Set)
(var : ∀ Γ A → Var32 Γ A → Tm32 Γ A)
(lam : ∀ Γ A B → Tm32 (snoc32 Γ A) B → Tm32 Γ (arr32 A B))
(app : ∀ Γ A B → Tm32 Γ (arr32 A B) → Tm32 Γ A → Tm32 Γ B)
(tt : ∀ Γ → Tm32 Γ top32)
(pair : ∀ Γ A B → Tm32 Γ A → Tm32 Γ B → Tm32 Γ (prod32 A B))
(fst : ∀ Γ A B → Tm32 Γ (prod32 A B) → Tm32 Γ A)
(snd : ∀ Γ A B → Tm32 Γ (prod32 A B) → Tm32 Γ B)
(left : ∀ Γ A B → Tm32 Γ A → Tm32 Γ (sum32 A B))
(right : ∀ Γ A B → Tm32 Γ B → Tm32 Γ (sum32 A B))
(case : ∀ Γ A B C → Tm32 Γ (sum32 A B) → Tm32 Γ (arr32 A C) → Tm32 Γ (arr32 B C) → Tm32 Γ C)
(zero : ∀ Γ → Tm32 Γ nat32)
(suc : ∀ Γ → Tm32 Γ nat32 → Tm32 Γ nat32)
(rec : ∀ Γ A → Tm32 Γ nat32 → Tm32 Γ (arr32 nat32 (arr32 A A)) → Tm32 Γ A → Tm32 Γ A)
→ Tm32 Γ A
var32 : ∀{Γ A} → Var32 Γ A → Tm32 Γ A; var32
= λ x Tm32 var32 lam app tt pair fst snd left right case zero suc rec →
var32 _ _ x
lam32 : ∀{Γ A B} → Tm32 (snoc32 Γ A) B → Tm32 Γ (arr32 A B); lam32
= λ t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec →
lam32 _ _ _ (t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec)
app32 : ∀{Γ A B} → Tm32 Γ (arr32 A B) → Tm32 Γ A → Tm32 Γ B; app32
= λ t u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec →
app32 _ _ _ (t Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
tt32 : ∀{Γ} → Tm32 Γ top32; tt32
= λ Tm32 var32 lam32 app32 tt32 pair fst snd left right case zero suc rec → tt32 _
pair32 : ∀{Γ A B} → Tm32 Γ A → Tm32 Γ B → Tm32 Γ (prod32 A B); pair32
= λ t u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec →
pair32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
fst32 : ∀{Γ A B} → Tm32 Γ (prod32 A B) → Tm32 Γ A; fst32
= λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec →
fst32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec)
snd32 : ∀{Γ A B} → Tm32 Γ (prod32 A B) → Tm32 Γ B; snd32
= λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec →
snd32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec)
left32 : ∀{Γ A B} → Tm32 Γ A → Tm32 Γ (sum32 A B); left32
= λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec →
left32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec)
right32 : ∀{Γ A B} → Tm32 Γ B → Tm32 Γ (sum32 A B); right32
= λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec →
right32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec)
case32 : ∀{Γ A B C} → Tm32 Γ (sum32 A B) → Tm32 Γ (arr32 A C) → Tm32 Γ (arr32 B C) → Tm32 Γ C; case32
= λ t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec →
case32 _ _ _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
zero32 : ∀{Γ} → Tm32 Γ nat32; zero32
= λ Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc rec → zero32 _
suc32 : ∀{Γ} → Tm32 Γ nat32 → Tm32 Γ nat32; suc32
= λ t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec →
suc32 _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec)
rec32 : ∀{Γ A} → Tm32 Γ nat32 → Tm32 Γ (arr32 nat32 (arr32 A A)) → Tm32 Γ A → Tm32 Γ A; rec32
= λ t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32 →
rec32 _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
v032 : ∀{Γ A} → Tm32 (snoc32 Γ A) A; v032
= var32 vz32
v132 : ∀{Γ A B} → Tm32 (snoc32 (snoc32 Γ A) B) A; v132
= var32 (vs32 vz32)
v232 : ∀{Γ A B C} → Tm32 (snoc32 (snoc32 (snoc32 Γ A) B) C) A; v232
= var32 (vs32 (vs32 vz32))
v332 : ∀{Γ A B C D} → Tm32 (snoc32 (snoc32 (snoc32 (snoc32 Γ A) B) C) D) A; v332
= var32 (vs32 (vs32 (vs32 vz32)))
tbool32 : Ty32; tbool32
= sum32 top32 top32
true32 : ∀{Γ} → Tm32 Γ tbool32; true32
= left32 tt32
tfalse32 : ∀{Γ} → Tm32 Γ tbool32; tfalse32
= right32 tt32
ifthenelse32 : ∀{Γ A} → Tm32 Γ (arr32 tbool32 (arr32 A (arr32 A A))); ifthenelse32
= lam32 (lam32 (lam32 (case32 v232 (lam32 v232) (lam32 v132))))
times432 : ∀{Γ A} → Tm32 Γ (arr32 (arr32 A A) (arr32 A A)); times432
= lam32 (lam32 (app32 v132 (app32 v132 (app32 v132 (app32 v132 v032)))))
add32 : ∀{Γ} → Tm32 Γ (arr32 nat32 (arr32 nat32 nat32)); add32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (suc32 (app32 v132 v032)))))
(lam32 v032))
mul32 : ∀{Γ} → Tm32 Γ (arr32 nat32 (arr32 nat32 nat32)); mul32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (app32 (app32 add32 (app32 v132 v032)) v032))))
(lam32 zero32))
fact32 : ∀{Γ} → Tm32 Γ (arr32 nat32 nat32); fact32
= lam32 (rec32 v032 (lam32 (lam32 (app32 (app32 mul32 (suc32 v132)) v032)))
(suc32 zero32))
{-# OPTIONS --type-in-type #-}
Ty33 : Set
Ty33 =
(Ty33 : Set)
(nat top bot : Ty33)
(arr prod sum : Ty33 → Ty33 → Ty33)
→ Ty33
nat33 : Ty33; nat33 = λ _ nat33 _ _ _ _ _ → nat33
top33 : Ty33; top33 = λ _ _ top33 _ _ _ _ → top33
bot33 : Ty33; bot33 = λ _ _ _ bot33 _ _ _ → bot33
arr33 : Ty33 → Ty33 → Ty33; arr33
= λ A B Ty33 nat33 top33 bot33 arr33 prod sum →
arr33 (A Ty33 nat33 top33 bot33 arr33 prod sum) (B Ty33 nat33 top33 bot33 arr33 prod sum)
prod33 : Ty33 → Ty33 → Ty33; prod33
= λ A B Ty33 nat33 top33 bot33 arr33 prod33 sum →
prod33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum) (B Ty33 nat33 top33 bot33 arr33 prod33 sum)
sum33 : Ty33 → Ty33 → Ty33; sum33
= λ A B Ty33 nat33 top33 bot33 arr33 prod33 sum33 →
sum33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum33) (B Ty33 nat33 top33 bot33 arr33 prod33 sum33)
Con33 : Set; Con33
= (Con33 : Set)
(nil : Con33)
(snoc : Con33 → Ty33 → Con33)
→ Con33
nil33 : Con33; nil33
= λ Con33 nil33 snoc → nil33
snoc33 : Con33 → Ty33 → Con33; snoc33
= λ Γ A Con33 nil33 snoc33 → snoc33 (Γ Con33 nil33 snoc33) A
Var33 : Con33 → Ty33 → Set; Var33
= λ Γ A →
(Var33 : Con33 → Ty33 → Set)
(vz : ∀ Γ A → Var33 (snoc33 Γ A) A)
(vs : ∀ Γ B A → Var33 Γ A → Var33 (snoc33 Γ B) A)
→ Var33 Γ A
vz33 : ∀{Γ A} → Var33 (snoc33 Γ A) A; vz33
= λ Var33 vz33 vs → vz33 _ _
vs33 : ∀{Γ B A} → Var33 Γ A → Var33 (snoc33 Γ B) A; vs33
= λ x Var33 vz33 vs33 → vs33 _ _ _ (x Var33 vz33 vs33)
Tm33 : Con33 → Ty33 → Set; Tm33
= λ Γ A →
(Tm33 : Con33 → Ty33 → Set)
(var : ∀ Γ A → Var33 Γ A → Tm33 Γ A)
(lam : ∀ Γ A B → Tm33 (snoc33 Γ A) B → Tm33 Γ (arr33 A B))
(app : ∀ Γ A B → Tm33 Γ (arr33 A B) → Tm33 Γ A → Tm33 Γ B)
(tt : ∀ Γ → Tm33 Γ top33)
(pair : ∀ Γ A B → Tm33 Γ A → Tm33 Γ B → Tm33 Γ (prod33 A B))
(fst : ∀ Γ A B → Tm33 Γ (prod33 A B) → Tm33 Γ A)
(snd : ∀ Γ A B → Tm33 Γ (prod33 A B) → Tm33 Γ B)
(left : ∀ Γ A B → Tm33 Γ A → Tm33 Γ (sum33 A B))
(right : ∀ Γ A B → Tm33 Γ B → Tm33 Γ (sum33 A B))
(case : ∀ Γ A B C → Tm33 Γ (sum33 A B) → Tm33 Γ (arr33 A C) → Tm33 Γ (arr33 B C) → Tm33 Γ C)
(zero : ∀ Γ → Tm33 Γ nat33)
(suc : ∀ Γ → Tm33 Γ nat33 → Tm33 Γ nat33)
(rec : ∀ Γ A → Tm33 Γ nat33 → Tm33 Γ (arr33 nat33 (arr33 A A)) → Tm33 Γ A → Tm33 Γ A)
→ Tm33 Γ A
var33 : ∀{Γ A} → Var33 Γ A → Tm33 Γ A; var33
= λ x Tm33 var33 lam app tt pair fst snd left right case zero suc rec →
var33 _ _ x
lam33 : ∀{Γ A B} → Tm33 (snoc33 Γ A) B → Tm33 Γ (arr33 A B); lam33
= λ t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec →
lam33 _ _ _ (t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec)
app33 : ∀{Γ A B} → Tm33 Γ (arr33 A B) → Tm33 Γ A → Tm33 Γ B; app33
= λ t u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec →
app33 _ _ _ (t Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
tt33 : ∀{Γ} → Tm33 Γ top33; tt33
= λ Tm33 var33 lam33 app33 tt33 pair fst snd left right case zero suc rec → tt33 _
pair33 : ∀{Γ A B} → Tm33 Γ A → Tm33 Γ B → Tm33 Γ (prod33 A B); pair33
= λ t u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec →
pair33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
fst33 : ∀{Γ A B} → Tm33 Γ (prod33 A B) → Tm33 Γ A; fst33
= λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec →
fst33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec)
snd33 : ∀{Γ A B} → Tm33 Γ (prod33 A B) → Tm33 Γ B; snd33
= λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec →
snd33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec)
left33 : ∀{Γ A B} → Tm33 Γ A → Tm33 Γ (sum33 A B); left33
= λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec →
left33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec)
right33 : ∀{Γ A B} → Tm33 Γ B → Tm33 Γ (sum33 A B); right33
= λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec →
right33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec)
case33 : ∀{Γ A B C} → Tm33 Γ (sum33 A B) → Tm33 Γ (arr33 A C) → Tm33 Γ (arr33 B C) → Tm33 Γ C; case33
= λ t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec →
case33 _ _ _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
zero33 : ∀{Γ} → Tm33 Γ nat33; zero33
= λ Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc rec → zero33 _
suc33 : ∀{Γ} → Tm33 Γ nat33 → Tm33 Γ nat33; suc33
= λ t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec →
suc33 _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec)
rec33 : ∀{Γ A} → Tm33 Γ nat33 → Tm33 Γ (arr33 nat33 (arr33 A A)) → Tm33 Γ A → Tm33 Γ A; rec33
= λ t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33 →
rec33 _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
v033 : ∀{Γ A} → Tm33 (snoc33 Γ A) A; v033
= var33 vz33
v133 : ∀{Γ A B} → Tm33 (snoc33 (snoc33 Γ A) B) A; v133
= var33 (vs33 vz33)
v233 : ∀{Γ A B C} → Tm33 (snoc33 (snoc33 (snoc33 Γ A) B) C) A; v233
= var33 (vs33 (vs33 vz33))
v333 : ∀{Γ A B C D} → Tm33 (snoc33 (snoc33 (snoc33 (snoc33 Γ A) B) C) D) A; v333
= var33 (vs33 (vs33 (vs33 vz33)))
tbool33 : Ty33; tbool33
= sum33 top33 top33
true33 : ∀{Γ} → Tm33 Γ tbool33; true33
= left33 tt33
tfalse33 : ∀{Γ} → Tm33 Γ tbool33; tfalse33
= right33 tt33
ifthenelse33 : ∀{Γ A} → Tm33 Γ (arr33 tbool33 (arr33 A (arr33 A A))); ifthenelse33
= lam33 (lam33 (lam33 (case33 v233 (lam33 v233) (lam33 v133))))
times433 : ∀{Γ A} → Tm33 Γ (arr33 (arr33 A A) (arr33 A A)); times433
= lam33 (lam33 (app33 v133 (app33 v133 (app33 v133 (app33 v133 v033)))))
add33 : ∀{Γ} → Tm33 Γ (arr33 nat33 (arr33 nat33 nat33)); add33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (suc33 (app33 v133 v033)))))
(lam33 v033))
mul33 : ∀{Γ} → Tm33 Γ (arr33 nat33 (arr33 nat33 nat33)); mul33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (app33 (app33 add33 (app33 v133 v033)) v033))))
(lam33 zero33))
fact33 : ∀{Γ} → Tm33 Γ (arr33 nat33 nat33); fact33
= lam33 (rec33 v033 (lam33 (lam33 (app33 (app33 mul33 (suc33 v133)) v033)))
(suc33 zero33))
{-# OPTIONS --type-in-type #-}
Ty34 : Set
Ty34 =
(Ty34 : Set)
(nat top bot : Ty34)
(arr prod sum : Ty34 → Ty34 → Ty34)
→ Ty34
nat34 : Ty34; nat34 = λ _ nat34 _ _ _ _ _ → nat34
top34 : Ty34; top34 = λ _ _ top34 _ _ _ _ → top34
bot34 : Ty34; bot34 = λ _ _ _ bot34 _ _ _ → bot34
arr34 : Ty34 → Ty34 → Ty34; arr34
= λ A B Ty34 nat34 top34 bot34 arr34 prod sum →
arr34 (A Ty34 nat34 top34 bot34 arr34 prod sum) (B Ty34 nat34 top34 bot34 arr34 prod sum)
prod34 : Ty34 → Ty34 → Ty34; prod34
= λ A B Ty34 nat34 top34 bot34 arr34 prod34 sum →
prod34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum) (B Ty34 nat34 top34 bot34 arr34 prod34 sum)
sum34 : Ty34 → Ty34 → Ty34; sum34
= λ A B Ty34 nat34 top34 bot34 arr34 prod34 sum34 →
sum34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum34) (B Ty34 nat34 top34 bot34 arr34 prod34 sum34)
Con34 : Set; Con34
= (Con34 : Set)
(nil : Con34)
(snoc : Con34 → Ty34 → Con34)
→ Con34
nil34 : Con34; nil34
= λ Con34 nil34 snoc → nil34
snoc34 : Con34 → Ty34 → Con34; snoc34
= λ Γ A Con34 nil34 snoc34 → snoc34 (Γ Con34 nil34 snoc34) A
Var34 : Con34 → Ty34 → Set; Var34
= λ Γ A →
(Var34 : Con34 → Ty34 → Set)
(vz : ∀ Γ A → Var34 (snoc34 Γ A) A)
(vs : ∀ Γ B A → Var34 Γ A → Var34 (snoc34 Γ B) A)
→ Var34 Γ A
vz34 : ∀{Γ A} → Var34 (snoc34 Γ A) A; vz34
= λ Var34 vz34 vs → vz34 _ _
vs34 : ∀{Γ B A} → Var34 Γ A → Var34 (snoc34 Γ B) A; vs34
= λ x Var34 vz34 vs34 → vs34 _ _ _ (x Var34 vz34 vs34)
Tm34 : Con34 → Ty34 → Set; Tm34
= λ Γ A →
(Tm34 : Con34 → Ty34 → Set)
(var : ∀ Γ A → Var34 Γ A → Tm34 Γ A)
(lam : ∀ Γ A B → Tm34 (snoc34 Γ A) B → Tm34 Γ (arr34 A B))
(app : ∀ Γ A B → Tm34 Γ (arr34 A B) → Tm34 Γ A → Tm34 Γ B)
(tt : ∀ Γ → Tm34 Γ top34)
(pair : ∀ Γ A B → Tm34 Γ A → Tm34 Γ B → Tm34 Γ (prod34 A B))
(fst : ∀ Γ A B → Tm34 Γ (prod34 A B) → Tm34 Γ A)
(snd : ∀ Γ A B → Tm34 Γ (prod34 A B) → Tm34 Γ B)
(left : ∀ Γ A B → Tm34 Γ A → Tm34 Γ (sum34 A B))
(right : ∀ Γ A B → Tm34 Γ B → Tm34 Γ (sum34 A B))
(case : ∀ Γ A B C → Tm34 Γ (sum34 A B) → Tm34 Γ (arr34 A C) → Tm34 Γ (arr34 B C) → Tm34 Γ C)
(zero : ∀ Γ → Tm34 Γ nat34)
(suc : ∀ Γ → Tm34 Γ nat34 → Tm34 Γ nat34)
(rec : ∀ Γ A → Tm34 Γ nat34 → Tm34 Γ (arr34 nat34 (arr34 A A)) → Tm34 Γ A → Tm34 Γ A)
→ Tm34 Γ A
var34 : ∀{Γ A} → Var34 Γ A → Tm34 Γ A; var34
= λ x Tm34 var34 lam app tt pair fst snd left right case zero suc rec →
var34 _ _ x
lam34 : ∀{Γ A B} → Tm34 (snoc34 Γ A) B → Tm34 Γ (arr34 A B); lam34
= λ t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec →
lam34 _ _ _ (t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec)
app34 : ∀{Γ A B} → Tm34 Γ (arr34 A B) → Tm34 Γ A → Tm34 Γ B; app34
= λ t u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec →
app34 _ _ _ (t Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
tt34 : ∀{Γ} → Tm34 Γ top34; tt34
= λ Tm34 var34 lam34 app34 tt34 pair fst snd left right case zero suc rec → tt34 _
pair34 : ∀{Γ A B} → Tm34 Γ A → Tm34 Γ B → Tm34 Γ (prod34 A B); pair34
= λ t u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec →
pair34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
fst34 : ∀{Γ A B} → Tm34 Γ (prod34 A B) → Tm34 Γ A; fst34
= λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec →
fst34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec)
snd34 : ∀{Γ A B} → Tm34 Γ (prod34 A B) → Tm34 Γ B; snd34
= λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec →
snd34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec)
left34 : ∀{Γ A B} → Tm34 Γ A → Tm34 Γ (sum34 A B); left34
= λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec →
left34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec)
right34 : ∀{Γ A B} → Tm34 Γ B → Tm34 Γ (sum34 A B); right34
= λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec →
right34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec)
case34 : ∀{Γ A B C} → Tm34 Γ (sum34 A B) → Tm34 Γ (arr34 A C) → Tm34 Γ (arr34 B C) → Tm34 Γ C; case34
= λ t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec →
case34 _ _ _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
zero34 : ∀{Γ} → Tm34 Γ nat34; zero34
= λ Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc rec → zero34 _
suc34 : ∀{Γ} → Tm34 Γ nat34 → Tm34 Γ nat34; suc34
= λ t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec →
suc34 _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec)
rec34 : ∀{Γ A} → Tm34 Γ nat34 → Tm34 Γ (arr34 nat34 (arr34 A A)) → Tm34 Γ A → Tm34 Γ A; rec34
= λ t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34 →
rec34 _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
v034 : ∀{Γ A} → Tm34 (snoc34 Γ A) A; v034
= var34 vz34
v134 : ∀{Γ A B} → Tm34 (snoc34 (snoc34 Γ A) B) A; v134
= var34 (vs34 vz34)
v234 : ∀{Γ A B C} → Tm34 (snoc34 (snoc34 (snoc34 Γ A) B) C) A; v234
= var34 (vs34 (vs34 vz34))
v334 : ∀{Γ A B C D} → Tm34 (snoc34 (snoc34 (snoc34 (snoc34 Γ A) B) C) D) A; v334
= var34 (vs34 (vs34 (vs34 vz34)))
tbool34 : Ty34; tbool34
= sum34 top34 top34
true34 : ∀{Γ} → Tm34 Γ tbool34; true34
= left34 tt34
tfalse34 : ∀{Γ} → Tm34 Γ tbool34; tfalse34
= right34 tt34
ifthenelse34 : ∀{Γ A} → Tm34 Γ (arr34 tbool34 (arr34 A (arr34 A A))); ifthenelse34
= lam34 (lam34 (lam34 (case34 v234 (lam34 v234) (lam34 v134))))
times434 : ∀{Γ A} → Tm34 Γ (arr34 (arr34 A A) (arr34 A A)); times434
= lam34 (lam34 (app34 v134 (app34 v134 (app34 v134 (app34 v134 v034)))))
add34 : ∀{Γ} → Tm34 Γ (arr34 nat34 (arr34 nat34 nat34)); add34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (suc34 (app34 v134 v034)))))
(lam34 v034))
mul34 : ∀{Γ} → Tm34 Γ (arr34 nat34 (arr34 nat34 nat34)); mul34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (app34 (app34 add34 (app34 v134 v034)) v034))))
(lam34 zero34))
fact34 : ∀{Γ} → Tm34 Γ (arr34 nat34 nat34); fact34
= lam34 (rec34 v034 (lam34 (lam34 (app34 (app34 mul34 (suc34 v134)) v034)))
(suc34 zero34))
{-# OPTIONS --type-in-type #-}
Ty35 : Set
Ty35 =
(Ty35 : Set)
(nat top bot : Ty35)
(arr prod sum : Ty35 → Ty35 → Ty35)
→ Ty35
nat35 : Ty35; nat35 = λ _ nat35 _ _ _ _ _ → nat35
top35 : Ty35; top35 = λ _ _ top35 _ _ _ _ → top35
bot35 : Ty35; bot35 = λ _ _ _ bot35 _ _ _ → bot35
arr35 : Ty35 → Ty35 → Ty35; arr35
= λ A B Ty35 nat35 top35 bot35 arr35 prod sum →
arr35 (A Ty35 nat35 top35 bot35 arr35 prod sum) (B Ty35 nat35 top35 bot35 arr35 prod sum)
prod35 : Ty35 → Ty35 → Ty35; prod35
= λ A B Ty35 nat35 top35 bot35 arr35 prod35 sum →
prod35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum) (B Ty35 nat35 top35 bot35 arr35 prod35 sum)
sum35 : Ty35 → Ty35 → Ty35; sum35
= λ A B Ty35 nat35 top35 bot35 arr35 prod35 sum35 →
sum35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum35) (B Ty35 nat35 top35 bot35 arr35 prod35 sum35)
Con35 : Set; Con35
= (Con35 : Set)
(nil : Con35)
(snoc : Con35 → Ty35 → Con35)
→ Con35
nil35 : Con35; nil35
= λ Con35 nil35 snoc → nil35
snoc35 : Con35 → Ty35 → Con35; snoc35
= λ Γ A Con35 nil35 snoc35 → snoc35 (Γ Con35 nil35 snoc35) A
Var35 : Con35 → Ty35 → Set; Var35
= λ Γ A →
(Var35 : Con35 → Ty35 → Set)
(vz : ∀ Γ A → Var35 (snoc35 Γ A) A)
(vs : ∀ Γ B A → Var35 Γ A → Var35 (snoc35 Γ B) A)
→ Var35 Γ A
vz35 : ∀{Γ A} → Var35 (snoc35 Γ A) A; vz35
= λ Var35 vz35 vs → vz35 _ _
vs35 : ∀{Γ B A} → Var35 Γ A → Var35 (snoc35 Γ B) A; vs35
= λ x Var35 vz35 vs35 → vs35 _ _ _ (x Var35 vz35 vs35)
Tm35 : Con35 → Ty35 → Set; Tm35
= λ Γ A →
(Tm35 : Con35 → Ty35 → Set)
(var : ∀ Γ A → Var35 Γ A → Tm35 Γ A)
(lam : ∀ Γ A B → Tm35 (snoc35 Γ A) B → Tm35 Γ (arr35 A B))
(app : ∀ Γ A B → Tm35 Γ (arr35 A B) → Tm35 Γ A → Tm35 Γ B)
(tt : ∀ Γ → Tm35 Γ top35)
(pair : ∀ Γ A B → Tm35 Γ A → Tm35 Γ B → Tm35 Γ (prod35 A B))
(fst : ∀ Γ A B → Tm35 Γ (prod35 A B) → Tm35 Γ A)
(snd : ∀ Γ A B → Tm35 Γ (prod35 A B) → Tm35 Γ B)
(left : ∀ Γ A B → Tm35 Γ A → Tm35 Γ (sum35 A B))
(right : ∀ Γ A B → Tm35 Γ B → Tm35 Γ (sum35 A B))
(case : ∀ Γ A B C → Tm35 Γ (sum35 A B) → Tm35 Γ (arr35 A C) → Tm35 Γ (arr35 B C) → Tm35 Γ C)
(zero : ∀ Γ → Tm35 Γ nat35)
(suc : ∀ Γ → Tm35 Γ nat35 → Tm35 Γ nat35)
(rec : ∀ Γ A → Tm35 Γ nat35 → Tm35 Γ (arr35 nat35 (arr35 A A)) → Tm35 Γ A → Tm35 Γ A)
→ Tm35 Γ A
var35 : ∀{Γ A} → Var35 Γ A → Tm35 Γ A; var35
= λ x Tm35 var35 lam app tt pair fst snd left right case zero suc rec →
var35 _ _ x
lam35 : ∀{Γ A B} → Tm35 (snoc35 Γ A) B → Tm35 Γ (arr35 A B); lam35
= λ t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec →
lam35 _ _ _ (t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec)
app35 : ∀{Γ A B} → Tm35 Γ (arr35 A B) → Tm35 Γ A → Tm35 Γ B; app35
= λ t u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec →
app35 _ _ _ (t Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
tt35 : ∀{Γ} → Tm35 Γ top35; tt35
= λ Tm35 var35 lam35 app35 tt35 pair fst snd left right case zero suc rec → tt35 _
pair35 : ∀{Γ A B} → Tm35 Γ A → Tm35 Γ B → Tm35 Γ (prod35 A B); pair35
= λ t u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec →
pair35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
fst35 : ∀{Γ A B} → Tm35 Γ (prod35 A B) → Tm35 Γ A; fst35
= λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec →
fst35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec)
snd35 : ∀{Γ A B} → Tm35 Γ (prod35 A B) → Tm35 Γ B; snd35
= λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec →
snd35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec)
left35 : ∀{Γ A B} → Tm35 Γ A → Tm35 Γ (sum35 A B); left35
= λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec →
left35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec)
right35 : ∀{Γ A B} → Tm35 Γ B → Tm35 Γ (sum35 A B); right35
= λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec →
right35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec)
case35 : ∀{Γ A B C} → Tm35 Γ (sum35 A B) → Tm35 Γ (arr35 A C) → Tm35 Γ (arr35 B C) → Tm35 Γ C; case35
= λ t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec →
case35 _ _ _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
zero35 : ∀{Γ} → Tm35 Γ nat35; zero35
= λ Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc rec → zero35 _
suc35 : ∀{Γ} → Tm35 Γ nat35 → Tm35 Γ nat35; suc35
= λ t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec →
suc35 _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec)
rec35 : ∀{Γ A} → Tm35 Γ nat35 → Tm35 Γ (arr35 nat35 (arr35 A A)) → Tm35 Γ A → Tm35 Γ A; rec35
= λ t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35 →
rec35 _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
v035 : ∀{Γ A} → Tm35 (snoc35 Γ A) A; v035
= var35 vz35
v135 : ∀{Γ A B} → Tm35 (snoc35 (snoc35 Γ A) B) A; v135
= var35 (vs35 vz35)
v235 : ∀{Γ A B C} → Tm35 (snoc35 (snoc35 (snoc35 Γ A) B) C) A; v235
= var35 (vs35 (vs35 vz35))
v335 : ∀{Γ A B C D} → Tm35 (snoc35 (snoc35 (snoc35 (snoc35 Γ A) B) C) D) A; v335
= var35 (vs35 (vs35 (vs35 vz35)))
tbool35 : Ty35; tbool35
= sum35 top35 top35
true35 : ∀{Γ} → Tm35 Γ tbool35; true35
= left35 tt35
tfalse35 : ∀{Γ} → Tm35 Γ tbool35; tfalse35
= right35 tt35
ifthenelse35 : ∀{Γ A} → Tm35 Γ (arr35 tbool35 (arr35 A (arr35 A A))); ifthenelse35
= lam35 (lam35 (lam35 (case35 v235 (lam35 v235) (lam35 v135))))
times435 : ∀{Γ A} → Tm35 Γ (arr35 (arr35 A A) (arr35 A A)); times435
= lam35 (lam35 (app35 v135 (app35 v135 (app35 v135 (app35 v135 v035)))))
add35 : ∀{Γ} → Tm35 Γ (arr35 nat35 (arr35 nat35 nat35)); add35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (suc35 (app35 v135 v035)))))
(lam35 v035))
mul35 : ∀{Γ} → Tm35 Γ (arr35 nat35 (arr35 nat35 nat35)); mul35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (app35 (app35 add35 (app35 v135 v035)) v035))))
(lam35 zero35))
fact35 : ∀{Γ} → Tm35 Γ (arr35 nat35 nat35); fact35
= lam35 (rec35 v035 (lam35 (lam35 (app35 (app35 mul35 (suc35 v135)) v035)))
(suc35 zero35))
{-# OPTIONS --type-in-type #-}
Ty36 : Set
Ty36 =
(Ty36 : Set)
(nat top bot : Ty36)
(arr prod sum : Ty36 → Ty36 → Ty36)
→ Ty36
nat36 : Ty36; nat36 = λ _ nat36 _ _ _ _ _ → nat36
top36 : Ty36; top36 = λ _ _ top36 _ _ _ _ → top36
bot36 : Ty36; bot36 = λ _ _ _ bot36 _ _ _ → bot36
arr36 : Ty36 → Ty36 → Ty36; arr36
= λ A B Ty36 nat36 top36 bot36 arr36 prod sum →
arr36 (A Ty36 nat36 top36 bot36 arr36 prod sum) (B Ty36 nat36 top36 bot36 arr36 prod sum)
prod36 : Ty36 → Ty36 → Ty36; prod36
= λ A B Ty36 nat36 top36 bot36 arr36 prod36 sum →
prod36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum) (B Ty36 nat36 top36 bot36 arr36 prod36 sum)
sum36 : Ty36 → Ty36 → Ty36; sum36
= λ A B Ty36 nat36 top36 bot36 arr36 prod36 sum36 →
sum36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum36) (B Ty36 nat36 top36 bot36 arr36 prod36 sum36)
Con36 : Set; Con36
= (Con36 : Set)
(nil : Con36)
(snoc : Con36 → Ty36 → Con36)
→ Con36
nil36 : Con36; nil36
= λ Con36 nil36 snoc → nil36
snoc36 : Con36 → Ty36 → Con36; snoc36
= λ Γ A Con36 nil36 snoc36 → snoc36 (Γ Con36 nil36 snoc36) A
Var36 : Con36 → Ty36 → Set; Var36
= λ Γ A →
(Var36 : Con36 → Ty36 → Set)
(vz : ∀ Γ A → Var36 (snoc36 Γ A) A)
(vs : ∀ Γ B A → Var36 Γ A → Var36 (snoc36 Γ B) A)
→ Var36 Γ A
vz36 : ∀{Γ A} → Var36 (snoc36 Γ A) A; vz36
= λ Var36 vz36 vs → vz36 _ _
vs36 : ∀{Γ B A} → Var36 Γ A → Var36 (snoc36 Γ B) A; vs36
= λ x Var36 vz36 vs36 → vs36 _ _ _ (x Var36 vz36 vs36)
Tm36 : Con36 → Ty36 → Set; Tm36
= λ Γ A →
(Tm36 : Con36 → Ty36 → Set)
(var : ∀ Γ A → Var36 Γ A → Tm36 Γ A)
(lam : ∀ Γ A B → Tm36 (snoc36 Γ A) B → Tm36 Γ (arr36 A B))
(app : ∀ Γ A B → Tm36 Γ (arr36 A B) → Tm36 Γ A → Tm36 Γ B)
(tt : ∀ Γ → Tm36 Γ top36)
(pair : ∀ Γ A B → Tm36 Γ A → Tm36 Γ B → Tm36 Γ (prod36 A B))
(fst : ∀ Γ A B → Tm36 Γ (prod36 A B) → Tm36 Γ A)
(snd : ∀ Γ A B → Tm36 Γ (prod36 A B) → Tm36 Γ B)
(left : ∀ Γ A B → Tm36 Γ A → Tm36 Γ (sum36 A B))
(right : ∀ Γ A B → Tm36 Γ B → Tm36 Γ (sum36 A B))
(case : ∀ Γ A B C → Tm36 Γ (sum36 A B) → Tm36 Γ (arr36 A C) → Tm36 Γ (arr36 B C) → Tm36 Γ C)
(zero : ∀ Γ → Tm36 Γ nat36)
(suc : ∀ Γ → Tm36 Γ nat36 → Tm36 Γ nat36)
(rec : ∀ Γ A → Tm36 Γ nat36 → Tm36 Γ (arr36 nat36 (arr36 A A)) → Tm36 Γ A → Tm36 Γ A)
→ Tm36 Γ A
var36 : ∀{Γ A} → Var36 Γ A → Tm36 Γ A; var36
= λ x Tm36 var36 lam app tt pair fst snd left right case zero suc rec →
var36 _ _ x
lam36 : ∀{Γ A B} → Tm36 (snoc36 Γ A) B → Tm36 Γ (arr36 A B); lam36
= λ t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec →
lam36 _ _ _ (t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec)
app36 : ∀{Γ A B} → Tm36 Γ (arr36 A B) → Tm36 Γ A → Tm36 Γ B; app36
= λ t u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec →
app36 _ _ _ (t Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
tt36 : ∀{Γ} → Tm36 Γ top36; tt36
= λ Tm36 var36 lam36 app36 tt36 pair fst snd left right case zero suc rec → tt36 _
pair36 : ∀{Γ A B} → Tm36 Γ A → Tm36 Γ B → Tm36 Γ (prod36 A B); pair36
= λ t u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec →
pair36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
fst36 : ∀{Γ A B} → Tm36 Γ (prod36 A B) → Tm36 Γ A; fst36
= λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec →
fst36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec)
snd36 : ∀{Γ A B} → Tm36 Γ (prod36 A B) → Tm36 Γ B; snd36
= λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec →
snd36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec)
left36 : ∀{Γ A B} → Tm36 Γ A → Tm36 Γ (sum36 A B); left36
= λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec →
left36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec)
right36 : ∀{Γ A B} → Tm36 Γ B → Tm36 Γ (sum36 A B); right36
= λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec →
right36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec)
case36 : ∀{Γ A B C} → Tm36 Γ (sum36 A B) → Tm36 Γ (arr36 A C) → Tm36 Γ (arr36 B C) → Tm36 Γ C; case36
= λ t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec →
case36 _ _ _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
zero36 : ∀{Γ} → Tm36 Γ nat36; zero36
= λ Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc rec → zero36 _
suc36 : ∀{Γ} → Tm36 Γ nat36 → Tm36 Γ nat36; suc36
= λ t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec →
suc36 _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec)
rec36 : ∀{Γ A} → Tm36 Γ nat36 → Tm36 Γ (arr36 nat36 (arr36 A A)) → Tm36 Γ A → Tm36 Γ A; rec36
= λ t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36 →
rec36 _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
v036 : ∀{Γ A} → Tm36 (snoc36 Γ A) A; v036
= var36 vz36
v136 : ∀{Γ A B} → Tm36 (snoc36 (snoc36 Γ A) B) A; v136
= var36 (vs36 vz36)
v236 : ∀{Γ A B C} → Tm36 (snoc36 (snoc36 (snoc36 Γ A) B) C) A; v236
= var36 (vs36 (vs36 vz36))
v336 : ∀{Γ A B C D} → Tm36 (snoc36 (snoc36 (snoc36 (snoc36 Γ A) B) C) D) A; v336
= var36 (vs36 (vs36 (vs36 vz36)))
tbool36 : Ty36; tbool36
= sum36 top36 top36
true36 : ∀{Γ} → Tm36 Γ tbool36; true36
= left36 tt36
tfalse36 : ∀{Γ} → Tm36 Γ tbool36; tfalse36
= right36 tt36
ifthenelse36 : ∀{Γ A} → Tm36 Γ (arr36 tbool36 (arr36 A (arr36 A A))); ifthenelse36
= lam36 (lam36 (lam36 (case36 v236 (lam36 v236) (lam36 v136))))
times436 : ∀{Γ A} → Tm36 Γ (arr36 (arr36 A A) (arr36 A A)); times436
= lam36 (lam36 (app36 v136 (app36 v136 (app36 v136 (app36 v136 v036)))))
add36 : ∀{Γ} → Tm36 Γ (arr36 nat36 (arr36 nat36 nat36)); add36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (suc36 (app36 v136 v036)))))
(lam36 v036))
mul36 : ∀{Γ} → Tm36 Γ (arr36 nat36 (arr36 nat36 nat36)); mul36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (app36 (app36 add36 (app36 v136 v036)) v036))))
(lam36 zero36))
fact36 : ∀{Γ} → Tm36 Γ (arr36 nat36 nat36); fact36
= lam36 (rec36 v036 (lam36 (lam36 (app36 (app36 mul36 (suc36 v136)) v036)))
(suc36 zero36))
{-# OPTIONS --type-in-type #-}
Ty37 : Set
Ty37 =
(Ty37 : Set)
(nat top bot : Ty37)
(arr prod sum : Ty37 → Ty37 → Ty37)
→ Ty37
nat37 : Ty37; nat37 = λ _ nat37 _ _ _ _ _ → nat37
top37 : Ty37; top37 = λ _ _ top37 _ _ _ _ → top37
bot37 : Ty37; bot37 = λ _ _ _ bot37 _ _ _ → bot37
arr37 : Ty37 → Ty37 → Ty37; arr37
= λ A B Ty37 nat37 top37 bot37 arr37 prod sum →
arr37 (A Ty37 nat37 top37 bot37 arr37 prod sum) (B Ty37 nat37 top37 bot37 arr37 prod sum)
prod37 : Ty37 → Ty37 → Ty37; prod37
= λ A B Ty37 nat37 top37 bot37 arr37 prod37 sum →
prod37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum) (B Ty37 nat37 top37 bot37 arr37 prod37 sum)
sum37 : Ty37 → Ty37 → Ty37; sum37
= λ A B Ty37 nat37 top37 bot37 arr37 prod37 sum37 →
sum37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum37) (B Ty37 nat37 top37 bot37 arr37 prod37 sum37)
Con37 : Set; Con37
= (Con37 : Set)
(nil : Con37)
(snoc : Con37 → Ty37 → Con37)
→ Con37
nil37 : Con37; nil37
= λ Con37 nil37 snoc → nil37
snoc37 : Con37 → Ty37 → Con37; snoc37
= λ Γ A Con37 nil37 snoc37 → snoc37 (Γ Con37 nil37 snoc37) A
Var37 : Con37 → Ty37 → Set; Var37
= λ Γ A →
(Var37 : Con37 → Ty37 → Set)
(vz : ∀ Γ A → Var37 (snoc37 Γ A) A)
(vs : ∀ Γ B A → Var37 Γ A → Var37 (snoc37 Γ B) A)
→ Var37 Γ A
vz37 : ∀{Γ A} → Var37 (snoc37 Γ A) A; vz37
= λ Var37 vz37 vs → vz37 _ _
vs37 : ∀{Γ B A} → Var37 Γ A → Var37 (snoc37 Γ B) A; vs37
= λ x Var37 vz37 vs37 → vs37 _ _ _ (x Var37 vz37 vs37)
Tm37 : Con37 → Ty37 → Set; Tm37
= λ Γ A →
(Tm37 : Con37 → Ty37 → Set)
(var : ∀ Γ A → Var37 Γ A → Tm37 Γ A)
(lam : ∀ Γ A B → Tm37 (snoc37 Γ A) B → Tm37 Γ (arr37 A B))
(app : ∀ Γ A B → Tm37 Γ (arr37 A B) → Tm37 Γ A → Tm37 Γ B)
(tt : ∀ Γ → Tm37 Γ top37)
(pair : ∀ Γ A B → Tm37 Γ A → Tm37 Γ B → Tm37 Γ (prod37 A B))
(fst : ∀ Γ A B → Tm37 Γ (prod37 A B) → Tm37 Γ A)
(snd : ∀ Γ A B → Tm37 Γ (prod37 A B) → Tm37 Γ B)
(left : ∀ Γ A B → Tm37 Γ A → Tm37 Γ (sum37 A B))
(right : ∀ Γ A B → Tm37 Γ B → Tm37 Γ (sum37 A B))
(case : ∀ Γ A B C → Tm37 Γ (sum37 A B) → Tm37 Γ (arr37 A C) → Tm37 Γ (arr37 B C) → Tm37 Γ C)
(zero : ∀ Γ → Tm37 Γ nat37)
(suc : ∀ Γ → Tm37 Γ nat37 → Tm37 Γ nat37)
(rec : ∀ Γ A → Tm37 Γ nat37 → Tm37 Γ (arr37 nat37 (arr37 A A)) → Tm37 Γ A → Tm37 Γ A)
→ Tm37 Γ A
var37 : ∀{Γ A} → Var37 Γ A → Tm37 Γ A; var37
= λ x Tm37 var37 lam app tt pair fst snd left right case zero suc rec →
var37 _ _ x
lam37 : ∀{Γ A B} → Tm37 (snoc37 Γ A) B → Tm37 Γ (arr37 A B); lam37
= λ t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec →
lam37 _ _ _ (t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec)
app37 : ∀{Γ A B} → Tm37 Γ (arr37 A B) → Tm37 Γ A → Tm37 Γ B; app37
= λ t u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec →
app37 _ _ _ (t Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
tt37 : ∀{Γ} → Tm37 Γ top37; tt37
= λ Tm37 var37 lam37 app37 tt37 pair fst snd left right case zero suc rec → tt37 _
pair37 : ∀{Γ A B} → Tm37 Γ A → Tm37 Γ B → Tm37 Γ (prod37 A B); pair37
= λ t u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec →
pair37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
fst37 : ∀{Γ A B} → Tm37 Γ (prod37 A B) → Tm37 Γ A; fst37
= λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec →
fst37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec)
snd37 : ∀{Γ A B} → Tm37 Γ (prod37 A B) → Tm37 Γ B; snd37
= λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec →
snd37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec)
left37 : ∀{Γ A B} → Tm37 Γ A → Tm37 Γ (sum37 A B); left37
= λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec →
left37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec)
right37 : ∀{Γ A B} → Tm37 Γ B → Tm37 Γ (sum37 A B); right37
= λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec →
right37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec)
case37 : ∀{Γ A B C} → Tm37 Γ (sum37 A B) → Tm37 Γ (arr37 A C) → Tm37 Γ (arr37 B C) → Tm37 Γ C; case37
= λ t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec →
case37 _ _ _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
zero37 : ∀{Γ} → Tm37 Γ nat37; zero37
= λ Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc rec → zero37 _
suc37 : ∀{Γ} → Tm37 Γ nat37 → Tm37 Γ nat37; suc37
= λ t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec →
suc37 _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec)
rec37 : ∀{Γ A} → Tm37 Γ nat37 → Tm37 Γ (arr37 nat37 (arr37 A A)) → Tm37 Γ A → Tm37 Γ A; rec37
= λ t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37 →
rec37 _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
v037 : ∀{Γ A} → Tm37 (snoc37 Γ A) A; v037
= var37 vz37
v137 : ∀{Γ A B} → Tm37 (snoc37 (snoc37 Γ A) B) A; v137
= var37 (vs37 vz37)
v237 : ∀{Γ A B C} → Tm37 (snoc37 (snoc37 (snoc37 Γ A) B) C) A; v237
= var37 (vs37 (vs37 vz37))
v337 : ∀{Γ A B C D} → Tm37 (snoc37 (snoc37 (snoc37 (snoc37 Γ A) B) C) D) A; v337
= var37 (vs37 (vs37 (vs37 vz37)))
tbool37 : Ty37; tbool37
= sum37 top37 top37
true37 : ∀{Γ} → Tm37 Γ tbool37; true37
= left37 tt37
tfalse37 : ∀{Γ} → Tm37 Γ tbool37; tfalse37
= right37 tt37
ifthenelse37 : ∀{Γ A} → Tm37 Γ (arr37 tbool37 (arr37 A (arr37 A A))); ifthenelse37
= lam37 (lam37 (lam37 (case37 v237 (lam37 v237) (lam37 v137))))
times437 : ∀{Γ A} → Tm37 Γ (arr37 (arr37 A A) (arr37 A A)); times437
= lam37 (lam37 (app37 v137 (app37 v137 (app37 v137 (app37 v137 v037)))))
add37 : ∀{Γ} → Tm37 Γ (arr37 nat37 (arr37 nat37 nat37)); add37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (suc37 (app37 v137 v037)))))
(lam37 v037))
mul37 : ∀{Γ} → Tm37 Γ (arr37 nat37 (arr37 nat37 nat37)); mul37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (app37 (app37 add37 (app37 v137 v037)) v037))))
(lam37 zero37))
fact37 : ∀{Γ} → Tm37 Γ (arr37 nat37 nat37); fact37
= lam37 (rec37 v037 (lam37 (lam37 (app37 (app37 mul37 (suc37 v137)) v037)))
(suc37 zero37))
{-# OPTIONS --type-in-type #-}
Ty38 : Set
Ty38 =
(Ty38 : Set)
(nat top bot : Ty38)
(arr prod sum : Ty38 → Ty38 → Ty38)
→ Ty38
nat38 : Ty38; nat38 = λ _ nat38 _ _ _ _ _ → nat38
top38 : Ty38; top38 = λ _ _ top38 _ _ _ _ → top38
bot38 : Ty38; bot38 = λ _ _ _ bot38 _ _ _ → bot38
arr38 : Ty38 → Ty38 → Ty38; arr38
= λ A B Ty38 nat38 top38 bot38 arr38 prod sum →
arr38 (A Ty38 nat38 top38 bot38 arr38 prod sum) (B Ty38 nat38 top38 bot38 arr38 prod sum)
prod38 : Ty38 → Ty38 → Ty38; prod38
= λ A B Ty38 nat38 top38 bot38 arr38 prod38 sum →
prod38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum) (B Ty38 nat38 top38 bot38 arr38 prod38 sum)
sum38 : Ty38 → Ty38 → Ty38; sum38
= λ A B Ty38 nat38 top38 bot38 arr38 prod38 sum38 →
sum38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum38) (B Ty38 nat38 top38 bot38 arr38 prod38 sum38)
Con38 : Set; Con38
= (Con38 : Set)
(nil : Con38)
(snoc : Con38 → Ty38 → Con38)
→ Con38
nil38 : Con38; nil38
= λ Con38 nil38 snoc → nil38
snoc38 : Con38 → Ty38 → Con38; snoc38
= λ Γ A Con38 nil38 snoc38 → snoc38 (Γ Con38 nil38 snoc38) A
Var38 : Con38 → Ty38 → Set; Var38
= λ Γ A →
(Var38 : Con38 → Ty38 → Set)
(vz : ∀ Γ A → Var38 (snoc38 Γ A) A)
(vs : ∀ Γ B A → Var38 Γ A → Var38 (snoc38 Γ B) A)
→ Var38 Γ A
vz38 : ∀{Γ A} → Var38 (snoc38 Γ A) A; vz38
= λ Var38 vz38 vs → vz38 _ _
vs38 : ∀{Γ B A} → Var38 Γ A → Var38 (snoc38 Γ B) A; vs38
= λ x Var38 vz38 vs38 → vs38 _ _ _ (x Var38 vz38 vs38)
Tm38 : Con38 → Ty38 → Set; Tm38
= λ Γ A →
(Tm38 : Con38 → Ty38 → Set)
(var : ∀ Γ A → Var38 Γ A → Tm38 Γ A)
(lam : ∀ Γ A B → Tm38 (snoc38 Γ A) B → Tm38 Γ (arr38 A B))
(app : ∀ Γ A B → Tm38 Γ (arr38 A B) → Tm38 Γ A → Tm38 Γ B)
(tt : ∀ Γ → Tm38 Γ top38)
(pair : ∀ Γ A B → Tm38 Γ A → Tm38 Γ B → Tm38 Γ (prod38 A B))
(fst : ∀ Γ A B → Tm38 Γ (prod38 A B) → Tm38 Γ A)
(snd : ∀ Γ A B → Tm38 Γ (prod38 A B) → Tm38 Γ B)
(left : ∀ Γ A B → Tm38 Γ A → Tm38 Γ (sum38 A B))
(right : ∀ Γ A B → Tm38 Γ B → Tm38 Γ (sum38 A B))
(case : ∀ Γ A B C → Tm38 Γ (sum38 A B) → Tm38 Γ (arr38 A C) → Tm38 Γ (arr38 B C) → Tm38 Γ C)
(zero : ∀ Γ → Tm38 Γ nat38)
(suc : ∀ Γ → Tm38 Γ nat38 → Tm38 Γ nat38)
(rec : ∀ Γ A → Tm38 Γ nat38 → Tm38 Γ (arr38 nat38 (arr38 A A)) → Tm38 Γ A → Tm38 Γ A)
→ Tm38 Γ A
var38 : ∀{Γ A} → Var38 Γ A → Tm38 Γ A; var38
= λ x Tm38 var38 lam app tt pair fst snd left right case zero suc rec →
var38 _ _ x
lam38 : ∀{Γ A B} → Tm38 (snoc38 Γ A) B → Tm38 Γ (arr38 A B); lam38
= λ t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec →
lam38 _ _ _ (t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec)
app38 : ∀{Γ A B} → Tm38 Γ (arr38 A B) → Tm38 Γ A → Tm38 Γ B; app38
= λ t u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec →
app38 _ _ _ (t Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
tt38 : ∀{Γ} → Tm38 Γ top38; tt38
= λ Tm38 var38 lam38 app38 tt38 pair fst snd left right case zero suc rec → tt38 _
pair38 : ∀{Γ A B} → Tm38 Γ A → Tm38 Γ B → Tm38 Γ (prod38 A B); pair38
= λ t u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec →
pair38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
fst38 : ∀{Γ A B} → Tm38 Γ (prod38 A B) → Tm38 Γ A; fst38
= λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec →
fst38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec)
snd38 : ∀{Γ A B} → Tm38 Γ (prod38 A B) → Tm38 Γ B; snd38
= λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec →
snd38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec)
left38 : ∀{Γ A B} → Tm38 Γ A → Tm38 Γ (sum38 A B); left38
= λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec →
left38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec)
right38 : ∀{Γ A B} → Tm38 Γ B → Tm38 Γ (sum38 A B); right38
= λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec →
right38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec)
case38 : ∀{Γ A B C} → Tm38 Γ (sum38 A B) → Tm38 Γ (arr38 A C) → Tm38 Γ (arr38 B C) → Tm38 Γ C; case38
= λ t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec →
case38 _ _ _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
zero38 : ∀{Γ} → Tm38 Γ nat38; zero38
= λ Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc rec → zero38 _
suc38 : ∀{Γ} → Tm38 Γ nat38 → Tm38 Γ nat38; suc38
= λ t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec →
suc38 _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec)
rec38 : ∀{Γ A} → Tm38 Γ nat38 → Tm38 Γ (arr38 nat38 (arr38 A A)) → Tm38 Γ A → Tm38 Γ A; rec38
= λ t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38 →
rec38 _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
v038 : ∀{Γ A} → Tm38 (snoc38 Γ A) A; v038
= var38 vz38
v138 : ∀{Γ A B} → Tm38 (snoc38 (snoc38 Γ A) B) A; v138
= var38 (vs38 vz38)
v238 : ∀{Γ A B C} → Tm38 (snoc38 (snoc38 (snoc38 Γ A) B) C) A; v238
= var38 (vs38 (vs38 vz38))
v338 : ∀{Γ A B C D} → Tm38 (snoc38 (snoc38 (snoc38 (snoc38 Γ A) B) C) D) A; v338
= var38 (vs38 (vs38 (vs38 vz38)))
tbool38 : Ty38; tbool38
= sum38 top38 top38
true38 : ∀{Γ} → Tm38 Γ tbool38; true38
= left38 tt38
tfalse38 : ∀{Γ} → Tm38 Γ tbool38; tfalse38
= right38 tt38
ifthenelse38 : ∀{Γ A} → Tm38 Γ (arr38 tbool38 (arr38 A (arr38 A A))); ifthenelse38
= lam38 (lam38 (lam38 (case38 v238 (lam38 v238) (lam38 v138))))
times438 : ∀{Γ A} → Tm38 Γ (arr38 (arr38 A A) (arr38 A A)); times438
= lam38 (lam38 (app38 v138 (app38 v138 (app38 v138 (app38 v138 v038)))))
add38 : ∀{Γ} → Tm38 Γ (arr38 nat38 (arr38 nat38 nat38)); add38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (suc38 (app38 v138 v038)))))
(lam38 v038))
mul38 : ∀{Γ} → Tm38 Γ (arr38 nat38 (arr38 nat38 nat38)); mul38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (app38 (app38 add38 (app38 v138 v038)) v038))))
(lam38 zero38))
fact38 : ∀{Γ} → Tm38 Γ (arr38 nat38 nat38); fact38
= lam38 (rec38 v038 (lam38 (lam38 (app38 (app38 mul38 (suc38 v138)) v038)))
(suc38 zero38))
{-# OPTIONS --type-in-type #-}
Ty39 : Set
Ty39 =
(Ty39 : Set)
(nat top bot : Ty39)
(arr prod sum : Ty39 → Ty39 → Ty39)
→ Ty39
nat39 : Ty39; nat39 = λ _ nat39 _ _ _ _ _ → nat39
top39 : Ty39; top39 = λ _ _ top39 _ _ _ _ → top39
bot39 : Ty39; bot39 = λ _ _ _ bot39 _ _ _ → bot39
arr39 : Ty39 → Ty39 → Ty39; arr39
= λ A B Ty39 nat39 top39 bot39 arr39 prod sum →
arr39 (A Ty39 nat39 top39 bot39 arr39 prod sum) (B Ty39 nat39 top39 bot39 arr39 prod sum)
prod39 : Ty39 → Ty39 → Ty39; prod39
= λ A B Ty39 nat39 top39 bot39 arr39 prod39 sum →
prod39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum) (B Ty39 nat39 top39 bot39 arr39 prod39 sum)
sum39 : Ty39 → Ty39 → Ty39; sum39
= λ A B Ty39 nat39 top39 bot39 arr39 prod39 sum39 →
sum39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum39) (B Ty39 nat39 top39 bot39 arr39 prod39 sum39)
Con39 : Set; Con39
= (Con39 : Set)
(nil : Con39)
(snoc : Con39 → Ty39 → Con39)
→ Con39
nil39 : Con39; nil39
= λ Con39 nil39 snoc → nil39
snoc39 : Con39 → Ty39 → Con39; snoc39
= λ Γ A Con39 nil39 snoc39 → snoc39 (Γ Con39 nil39 snoc39) A
Var39 : Con39 → Ty39 → Set; Var39
= λ Γ A →
(Var39 : Con39 → Ty39 → Set)
(vz : ∀ Γ A → Var39 (snoc39 Γ A) A)
(vs : ∀ Γ B A → Var39 Γ A → Var39 (snoc39 Γ B) A)
→ Var39 Γ A
vz39 : ∀{Γ A} → Var39 (snoc39 Γ A) A; vz39
= λ Var39 vz39 vs → vz39 _ _
vs39 : ∀{Γ B A} → Var39 Γ A → Var39 (snoc39 Γ B) A; vs39
= λ x Var39 vz39 vs39 → vs39 _ _ _ (x Var39 vz39 vs39)
Tm39 : Con39 → Ty39 → Set; Tm39
= λ Γ A →
(Tm39 : Con39 → Ty39 → Set)
(var : ∀ Γ A → Var39 Γ A → Tm39 Γ A)
(lam : ∀ Γ A B → Tm39 (snoc39 Γ A) B → Tm39 Γ (arr39 A B))
(app : ∀ Γ A B → Tm39 Γ (arr39 A B) → Tm39 Γ A → Tm39 Γ B)
(tt : ∀ Γ → Tm39 Γ top39)
(pair : ∀ Γ A B → Tm39 Γ A → Tm39 Γ B → Tm39 Γ (prod39 A B))
(fst : ∀ Γ A B → Tm39 Γ (prod39 A B) → Tm39 Γ A)
(snd : ∀ Γ A B → Tm39 Γ (prod39 A B) → Tm39 Γ B)
(left : ∀ Γ A B → Tm39 Γ A → Tm39 Γ (sum39 A B))
(right : ∀ Γ A B → Tm39 Γ B → Tm39 Γ (sum39 A B))
(case : ∀ Γ A B C → Tm39 Γ (sum39 A B) → Tm39 Γ (arr39 A C) → Tm39 Γ (arr39 B C) → Tm39 Γ C)
(zero : ∀ Γ → Tm39 Γ nat39)
(suc : ∀ Γ → Tm39 Γ nat39 → Tm39 Γ nat39)
(rec : ∀ Γ A → Tm39 Γ nat39 → Tm39 Γ (arr39 nat39 (arr39 A A)) → Tm39 Γ A → Tm39 Γ A)
→ Tm39 Γ A
var39 : ∀{Γ A} → Var39 Γ A → Tm39 Γ A; var39
= λ x Tm39 var39 lam app tt pair fst snd left right case zero suc rec →
var39 _ _ x
lam39 : ∀{Γ A B} → Tm39 (snoc39 Γ A) B → Tm39 Γ (arr39 A B); lam39
= λ t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec →
lam39 _ _ _ (t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec)
app39 : ∀{Γ A B} → Tm39 Γ (arr39 A B) → Tm39 Γ A → Tm39 Γ B; app39
= λ t u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec →
app39 _ _ _ (t Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
tt39 : ∀{Γ} → Tm39 Γ top39; tt39
= λ Tm39 var39 lam39 app39 tt39 pair fst snd left right case zero suc rec → tt39 _
pair39 : ∀{Γ A B} → Tm39 Γ A → Tm39 Γ B → Tm39 Γ (prod39 A B); pair39
= λ t u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec →
pair39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
fst39 : ∀{Γ A B} → Tm39 Γ (prod39 A B) → Tm39 Γ A; fst39
= λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec →
fst39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec)
snd39 : ∀{Γ A B} → Tm39 Γ (prod39 A B) → Tm39 Γ B; snd39
= λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec →
snd39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec)
left39 : ∀{Γ A B} → Tm39 Γ A → Tm39 Γ (sum39 A B); left39
= λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec →
left39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec)
right39 : ∀{Γ A B} → Tm39 Γ B → Tm39 Γ (sum39 A B); right39
= λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec →
right39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec)
case39 : ∀{Γ A B C} → Tm39 Γ (sum39 A B) → Tm39 Γ (arr39 A C) → Tm39 Γ (arr39 B C) → Tm39 Γ C; case39
= λ t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec →
case39 _ _ _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
zero39 : ∀{Γ} → Tm39 Γ nat39; zero39
= λ Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc rec → zero39 _
suc39 : ∀{Γ} → Tm39 Γ nat39 → Tm39 Γ nat39; suc39
= λ t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec →
suc39 _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec)
rec39 : ∀{Γ A} → Tm39 Γ nat39 → Tm39 Γ (arr39 nat39 (arr39 A A)) → Tm39 Γ A → Tm39 Γ A; rec39
= λ t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39 →
rec39 _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
v039 : ∀{Γ A} → Tm39 (snoc39 Γ A) A; v039
= var39 vz39
v139 : ∀{Γ A B} → Tm39 (snoc39 (snoc39 Γ A) B) A; v139
= var39 (vs39 vz39)
v239 : ∀{Γ A B C} → Tm39 (snoc39 (snoc39 (snoc39 Γ A) B) C) A; v239
= var39 (vs39 (vs39 vz39))
v339 : ∀{Γ A B C D} → Tm39 (snoc39 (snoc39 (snoc39 (snoc39 Γ A) B) C) D) A; v339
= var39 (vs39 (vs39 (vs39 vz39)))
tbool39 : Ty39; tbool39
= sum39 top39 top39
true39 : ∀{Γ} → Tm39 Γ tbool39; true39
= left39 tt39
tfalse39 : ∀{Γ} → Tm39 Γ tbool39; tfalse39
= right39 tt39
ifthenelse39 : ∀{Γ A} → Tm39 Γ (arr39 tbool39 (arr39 A (arr39 A A))); ifthenelse39
= lam39 (lam39 (lam39 (case39 v239 (lam39 v239) (lam39 v139))))
times439 : ∀{Γ A} → Tm39 Γ (arr39 (arr39 A A) (arr39 A A)); times439
= lam39 (lam39 (app39 v139 (app39 v139 (app39 v139 (app39 v139 v039)))))
add39 : ∀{Γ} → Tm39 Γ (arr39 nat39 (arr39 nat39 nat39)); add39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (suc39 (app39 v139 v039)))))
(lam39 v039))
mul39 : ∀{Γ} → Tm39 Γ (arr39 nat39 (arr39 nat39 nat39)); mul39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (app39 (app39 add39 (app39 v139 v039)) v039))))
(lam39 zero39))
fact39 : ∀{Γ} → Tm39 Γ (arr39 nat39 nat39); fact39
= lam39 (rec39 v039 (lam39 (lam39 (app39 (app39 mul39 (suc39 v139)) v039)))
(suc39 zero39))
{-# OPTIONS --type-in-type #-}
Ty40 : Set
Ty40 =
(Ty40 : Set)
(nat top bot : Ty40)
(arr prod sum : Ty40 → Ty40 → Ty40)
→ Ty40
nat40 : Ty40; nat40 = λ _ nat40 _ _ _ _ _ → nat40
top40 : Ty40; top40 = λ _ _ top40 _ _ _ _ → top40
bot40 : Ty40; bot40 = λ _ _ _ bot40 _ _ _ → bot40
arr40 : Ty40 → Ty40 → Ty40; arr40
= λ A B Ty40 nat40 top40 bot40 arr40 prod sum →
arr40 (A Ty40 nat40 top40 bot40 arr40 prod sum) (B Ty40 nat40 top40 bot40 arr40 prod sum)
prod40 : Ty40 → Ty40 → Ty40; prod40
= λ A B Ty40 nat40 top40 bot40 arr40 prod40 sum →
prod40 (A Ty40 nat40 top40 bot40 arr40 prod40 sum) (B Ty40 nat40 top40 bot40 arr40 prod40 sum)
sum40 : Ty40 → Ty40 → Ty40; sum40
= λ A B Ty40 nat40 top40 bot40 arr40 prod40 sum40 →
sum40 (A Ty40 nat40 top40 bot40 arr40 prod40 sum40) (B Ty40 nat40 top40 bot40 arr40 prod40 sum40)
Con40 : Set; Con40
= (Con40 : Set)
(nil : Con40)
(snoc : Con40 → Ty40 → Con40)
→ Con40
nil40 : Con40; nil40
= λ Con40 nil40 snoc → nil40
snoc40 : Con40 → Ty40 → Con40; snoc40
= λ Γ A Con40 nil40 snoc40 → snoc40 (Γ Con40 nil40 snoc40) A
Var40 : Con40 → Ty40 → Set; Var40
= λ Γ A →
(Var40 : Con40 → Ty40 → Set)
(vz : ∀ Γ A → Var40 (snoc40 Γ A) A)
(vs : ∀ Γ B A → Var40 Γ A → Var40 (snoc40 Γ B) A)
→ Var40 Γ A
vz40 : ∀{Γ A} → Var40 (snoc40 Γ A) A; vz40
= λ Var40 vz40 vs → vz40 _ _
vs40 : ∀{Γ B A} → Var40 Γ A → Var40 (snoc40 Γ B) A; vs40
= λ x Var40 vz40 vs40 → vs40 _ _ _ (x Var40 vz40 vs40)
Tm40 : Con40 → Ty40 → Set; Tm40
= λ Γ A →
(Tm40 : Con40 → Ty40 → Set)
(var : ∀ Γ A → Var40 Γ A → Tm40 Γ A)
(lam : ∀ Γ A B → Tm40 (snoc40 Γ A) B → Tm40 Γ (arr40 A B))
(app : ∀ Γ A B → Tm40 Γ (arr40 A B) → Tm40 Γ A → Tm40 Γ B)
(tt : ∀ Γ → Tm40 Γ top40)
(pair : ∀ Γ A B → Tm40 Γ A → Tm40 Γ B → Tm40 Γ (prod40 A B))
(fst : ∀ Γ A B → Tm40 Γ (prod40 A B) → Tm40 Γ A)
(snd : ∀ Γ A B → Tm40 Γ (prod40 A B) → Tm40 Γ B)
(left : ∀ Γ A B → Tm40 Γ A → Tm40 Γ (sum40 A B))
(right : ∀ Γ A B → Tm40 Γ B → Tm40 Γ (sum40 A B))
(case : ∀ Γ A B C → Tm40 Γ (sum40 A B) → Tm40 Γ (arr40 A C) → Tm40 Γ (arr40 B C) → Tm40 Γ C)
(zero : ∀ Γ → Tm40 Γ nat40)
(suc : ∀ Γ → Tm40 Γ nat40 → Tm40 Γ nat40)
(rec : ∀ Γ A → Tm40 Γ nat40 → Tm40 Γ (arr40 nat40 (arr40 A A)) → Tm40 Γ A → Tm40 Γ A)
→ Tm40 Γ A
var40 : ∀{Γ A} → Var40 Γ A → Tm40 Γ A; var40
= λ x Tm40 var40 lam app tt pair fst snd left right case zero suc rec →
var40 _ _ x
lam40 : ∀{Γ A B} → Tm40 (snoc40 Γ A) B → Tm40 Γ (arr40 A B); lam40
= λ t Tm40 var40 lam40 app tt pair fst snd left right case zero suc rec →
lam40 _ _ _ (t Tm40 var40 lam40 app tt pair fst snd left right case zero suc rec)
app40 : ∀{Γ A B} → Tm40 Γ (arr40 A B) → Tm40 Γ A → Tm40 Γ B; app40
= λ t u Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec →
app40 _ _ _ (t Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec)
(u Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec)
tt40 : ∀{Γ} → Tm40 Γ top40; tt40
= λ Tm40 var40 lam40 app40 tt40 pair fst snd left right case zero suc rec → tt40 _
pair40 : ∀{Γ A B} → Tm40 Γ A → Tm40 Γ B → Tm40 Γ (prod40 A B); pair40
= λ t u Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec →
pair40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec)
(u Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec)
fst40 : ∀{Γ A B} → Tm40 Γ (prod40 A B) → Tm40 Γ A; fst40
= λ t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd left right case zero suc rec →
fst40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd left right case zero suc rec)
snd40 : ∀{Γ A B} → Tm40 Γ (prod40 A B) → Tm40 Γ B; snd40
= λ t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left right case zero suc rec →
snd40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left right case zero suc rec)
left40 : ∀{Γ A B} → Tm40 Γ A → Tm40 Γ (sum40 A B); left40
= λ t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right case zero suc rec →
left40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right case zero suc rec)
right40 : ∀{Γ A B} → Tm40 Γ B → Tm40 Γ (sum40 A B); right40
= λ t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case zero suc rec →
right40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case zero suc rec)
case40 : ∀{Γ A B C} → Tm40 Γ (sum40 A B) → Tm40 Γ (arr40 A C) → Tm40 Γ (arr40 B C) → Tm40 Γ C; case40
= λ t u v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec →
case40 _ _ _ _
(t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
(u Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
(v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
zero40 : ∀{Γ} → Tm40 Γ nat40; zero40
= λ Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc rec → zero40 _
suc40 : ∀{Γ} → Tm40 Γ nat40 → Tm40 Γ nat40; suc40
= λ t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec →
suc40 _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec)
rec40 : ∀{Γ A} → Tm40 Γ nat40 → Tm40 Γ (arr40 nat40 (arr40 A A)) → Tm40 Γ A → Tm40 Γ A; rec40
= λ t u v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40 →
rec40 _ _
(t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
(u Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
(v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
v040 : ∀{Γ A} → Tm40 (snoc40 Γ A) A; v040
= var40 vz40
v140 : ∀{Γ A B} → Tm40 (snoc40 (snoc40 Γ A) B) A; v140
= var40 (vs40 vz40)
v240 : ∀{Γ A B C} → Tm40 (snoc40 (snoc40 (snoc40 Γ A) B) C) A; v240
= var40 (vs40 (vs40 vz40))
v340 : ∀{Γ A B C D} → Tm40 (snoc40 (snoc40 (snoc40 (snoc40 Γ A) B) C) D) A; v340
= var40 (vs40 (vs40 (vs40 vz40)))
tbool40 : Ty40; tbool40
= sum40 top40 top40
true40 : ∀{Γ} → Tm40 Γ tbool40; true40
= left40 tt40
tfalse40 : ∀{Γ} → Tm40 Γ tbool40; tfalse40
= right40 tt40
ifthenelse40 : ∀{Γ A} → Tm40 Γ (arr40 tbool40 (arr40 A (arr40 A A))); ifthenelse40
= lam40 (lam40 (lam40 (case40 v240 (lam40 v240) (lam40 v140))))
times440 : ∀{Γ A} → Tm40 Γ (arr40 (arr40 A A) (arr40 A A)); times440
= lam40 (lam40 (app40 v140 (app40 v140 (app40 v140 (app40 v140 v040)))))
add40 : ∀{Γ} → Tm40 Γ (arr40 nat40 (arr40 nat40 nat40)); add40
= lam40 (rec40 v040
(lam40 (lam40 (lam40 (suc40 (app40 v140 v040)))))
(lam40 v040))
mul40 : ∀{Γ} → Tm40 Γ (arr40 nat40 (arr40 nat40 nat40)); mul40
= lam40 (rec40 v040
(lam40 (lam40 (lam40 (app40 (app40 add40 (app40 v140 v040)) v040))))
(lam40 zero40))
fact40 : ∀{Γ} → Tm40 Γ (arr40 nat40 nat40); fact40
= lam40 (rec40 v040 (lam40 (lam40 (app40 (app40 mul40 (suc40 v140)) v040)))
(suc40 zero40))
{-# OPTIONS --type-in-type #-}
Ty41 : Set
Ty41 =
(Ty41 : Set)
(nat top bot : Ty41)
(arr prod sum : Ty41 → Ty41 → Ty41)
→ Ty41
nat41 : Ty41; nat41 = λ _ nat41 _ _ _ _ _ → nat41
top41 : Ty41; top41 = λ _ _ top41 _ _ _ _ → top41
bot41 : Ty41; bot41 = λ _ _ _ bot41 _ _ _ → bot41
arr41 : Ty41 → Ty41 → Ty41; arr41
= λ A B Ty41 nat41 top41 bot41 arr41 prod sum →
arr41 (A Ty41 nat41 top41 bot41 arr41 prod sum) (B Ty41 nat41 top41 bot41 arr41 prod sum)
prod41 : Ty41 → Ty41 → Ty41; prod41
= λ A B Ty41 nat41 top41 bot41 arr41 prod41 sum →
prod41 (A Ty41 nat41 top41 bot41 arr41 prod41 sum) (B Ty41 nat41 top41 bot41 arr41 prod41 sum)
sum41 : Ty41 → Ty41 → Ty41; sum41
= λ A B Ty41 nat41 top41 bot41 arr41 prod41 sum41 →
sum41 (A Ty41 nat41 top41 bot41 arr41 prod41 sum41) (B Ty41 nat41 top41 bot41 arr41 prod41 sum41)
Con41 : Set; Con41
= (Con41 : Set)
(nil : Con41)
(snoc : Con41 → Ty41 → Con41)
→ Con41
nil41 : Con41; nil41
= λ Con41 nil41 snoc → nil41
snoc41 : Con41 → Ty41 → Con41; snoc41
= λ Γ A Con41 nil41 snoc41 → snoc41 (Γ Con41 nil41 snoc41) A
Var41 : Con41 → Ty41 → Set; Var41
= λ Γ A →
(Var41 : Con41 → Ty41 → Set)
(vz : ∀ Γ A → Var41 (snoc41 Γ A) A)
(vs : ∀ Γ B A → Var41 Γ A → Var41 (snoc41 Γ B) A)
→ Var41 Γ A
vz41 : ∀{Γ A} → Var41 (snoc41 Γ A) A; vz41
= λ Var41 vz41 vs → vz41 _ _
vs41 : ∀{Γ B A} → Var41 Γ A → Var41 (snoc41 Γ B) A; vs41
= λ x Var41 vz41 vs41 → vs41 _ _ _ (x Var41 vz41 vs41)
Tm41 : Con41 → Ty41 → Set; Tm41
= λ Γ A →
(Tm41 : Con41 → Ty41 → Set)
(var : ∀ Γ A → Var41 Γ A → Tm41 Γ A)
(lam : ∀ Γ A B → Tm41 (snoc41 Γ A) B → Tm41 Γ (arr41 A B))
(app : ∀ Γ A B → Tm41 Γ (arr41 A B) → Tm41 Γ A → Tm41 Γ B)
(tt : ∀ Γ → Tm41 Γ top41)
(pair : ∀ Γ A B → Tm41 Γ A → Tm41 Γ B → Tm41 Γ (prod41 A B))
(fst : ∀ Γ A B → Tm41 Γ (prod41 A B) → Tm41 Γ A)
(snd : ∀ Γ A B → Tm41 Γ (prod41 A B) → Tm41 Γ B)
(left : ∀ Γ A B → Tm41 Γ A → Tm41 Γ (sum41 A B))
(right : ∀ Γ A B → Tm41 Γ B → Tm41 Γ (sum41 A B))
(case : ∀ Γ A B C → Tm41 Γ (sum41 A B) → Tm41 Γ (arr41 A C) → Tm41 Γ (arr41 B C) → Tm41 Γ C)
(zero : ∀ Γ → Tm41 Γ nat41)
(suc : ∀ Γ → Tm41 Γ nat41 → Tm41 Γ nat41)
(rec : ∀ Γ A → Tm41 Γ nat41 → Tm41 Γ (arr41 nat41 (arr41 A A)) → Tm41 Γ A → Tm41 Γ A)
→ Tm41 Γ A
var41 : ∀{Γ A} → Var41 Γ A → Tm41 Γ A; var41
= λ x Tm41 var41 lam app tt pair fst snd left right case zero suc rec →
var41 _ _ x
lam41 : ∀{Γ A B} → Tm41 (snoc41 Γ A) B → Tm41 Γ (arr41 A B); lam41
= λ t Tm41 var41 lam41 app tt pair fst snd left right case zero suc rec →
lam41 _ _ _ (t Tm41 var41 lam41 app tt pair fst snd left right case zero suc rec)
app41 : ∀{Γ A B} → Tm41 Γ (arr41 A B) → Tm41 Γ A → Tm41 Γ B; app41
= λ t u Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec →
app41 _ _ _ (t Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec)
(u Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec)
tt41 : ∀{Γ} → Tm41 Γ top41; tt41
= λ Tm41 var41 lam41 app41 tt41 pair fst snd left right case zero suc rec → tt41 _
pair41 : ∀{Γ A B} → Tm41 Γ A → Tm41 Γ B → Tm41 Γ (prod41 A B); pair41
= λ t u Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec →
pair41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec)
(u Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec)
fst41 : ∀{Γ A B} → Tm41 Γ (prod41 A B) → Tm41 Γ A; fst41
= λ t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd left right case zero suc rec →
fst41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd left right case zero suc rec)
snd41 : ∀{Γ A B} → Tm41 Γ (prod41 A B) → Tm41 Γ B; snd41
= λ t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left right case zero suc rec →
snd41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left right case zero suc rec)
left41 : ∀{Γ A B} → Tm41 Γ A → Tm41 Γ (sum41 A B); left41
= λ t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right case zero suc rec →
left41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right case zero suc rec)
right41 : ∀{Γ A B} → Tm41 Γ B → Tm41 Γ (sum41 A B); right41
= λ t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case zero suc rec →
right41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case zero suc rec)
case41 : ∀{Γ A B C} → Tm41 Γ (sum41 A B) → Tm41 Γ (arr41 A C) → Tm41 Γ (arr41 B C) → Tm41 Γ C; case41
= λ t u v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec →
case41 _ _ _ _
(t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
(u Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
(v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
zero41 : ∀{Γ} → Tm41 Γ nat41; zero41
= λ Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc rec → zero41 _
suc41 : ∀{Γ} → Tm41 Γ nat41 → Tm41 Γ nat41; suc41
= λ t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec →
suc41 _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec)
rec41 : ∀{Γ A} → Tm41 Γ nat41 → Tm41 Γ (arr41 nat41 (arr41 A A)) → Tm41 Γ A → Tm41 Γ A; rec41
= λ t u v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41 →
rec41 _ _
(t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
(u Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
(v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
v041 : ∀{Γ A} → Tm41 (snoc41 Γ A) A; v041
= var41 vz41
v141 : ∀{Γ A B} → Tm41 (snoc41 (snoc41 Γ A) B) A; v141
= var41 (vs41 vz41)
v241 : ∀{Γ A B C} → Tm41 (snoc41 (snoc41 (snoc41 Γ A) B) C) A; v241
= var41 (vs41 (vs41 vz41))
v341 : ∀{Γ A B C D} → Tm41 (snoc41 (snoc41 (snoc41 (snoc41 Γ A) B) C) D) A; v341
= var41 (vs41 (vs41 (vs41 vz41)))
tbool41 : Ty41; tbool41
= sum41 top41 top41
true41 : ∀{Γ} → Tm41 Γ tbool41; true41
= left41 tt41
tfalse41 : ∀{Γ} → Tm41 Γ tbool41; tfalse41
= right41 tt41
ifthenelse41 : ∀{Γ A} → Tm41 Γ (arr41 tbool41 (arr41 A (arr41 A A))); ifthenelse41
= lam41 (lam41 (lam41 (case41 v241 (lam41 v241) (lam41 v141))))
times441 : ∀{Γ A} → Tm41 Γ (arr41 (arr41 A A) (arr41 A A)); times441
= lam41 (lam41 (app41 v141 (app41 v141 (app41 v141 (app41 v141 v041)))))
add41 : ∀{Γ} → Tm41 Γ (arr41 nat41 (arr41 nat41 nat41)); add41
= lam41 (rec41 v041
(lam41 (lam41 (lam41 (suc41 (app41 v141 v041)))))
(lam41 v041))
mul41 : ∀{Γ} → Tm41 Γ (arr41 nat41 (arr41 nat41 nat41)); mul41
= lam41 (rec41 v041
(lam41 (lam41 (lam41 (app41 (app41 add41 (app41 v141 v041)) v041))))
(lam41 zero41))
fact41 : ∀{Γ} → Tm41 Γ (arr41 nat41 nat41); fact41
= lam41 (rec41 v041 (lam41 (lam41 (app41 (app41 mul41 (suc41 v141)) v041)))
(suc41 zero41))
{-# OPTIONS --type-in-type #-}
Ty42 : Set
Ty42 =
(Ty42 : Set)
(nat top bot : Ty42)
(arr prod sum : Ty42 → Ty42 → Ty42)
→ Ty42
nat42 : Ty42; nat42 = λ _ nat42 _ _ _ _ _ → nat42
top42 : Ty42; top42 = λ _ _ top42 _ _ _ _ → top42
bot42 : Ty42; bot42 = λ _ _ _ bot42 _ _ _ → bot42
arr42 : Ty42 → Ty42 → Ty42; arr42
= λ A B Ty42 nat42 top42 bot42 arr42 prod sum →
arr42 (A Ty42 nat42 top42 bot42 arr42 prod sum) (B Ty42 nat42 top42 bot42 arr42 prod sum)
prod42 : Ty42 → Ty42 → Ty42; prod42
= λ A B Ty42 nat42 top42 bot42 arr42 prod42 sum →
prod42 (A Ty42 nat42 top42 bot42 arr42 prod42 sum) (B Ty42 nat42 top42 bot42 arr42 prod42 sum)
sum42 : Ty42 → Ty42 → Ty42; sum42
= λ A B Ty42 nat42 top42 bot42 arr42 prod42 sum42 →
sum42 (A Ty42 nat42 top42 bot42 arr42 prod42 sum42) (B Ty42 nat42 top42 bot42 arr42 prod42 sum42)
Con42 : Set; Con42
= (Con42 : Set)
(nil : Con42)
(snoc : Con42 → Ty42 → Con42)
→ Con42
nil42 : Con42; nil42
= λ Con42 nil42 snoc → nil42
snoc42 : Con42 → Ty42 → Con42; snoc42
= λ Γ A Con42 nil42 snoc42 → snoc42 (Γ Con42 nil42 snoc42) A
Var42 : Con42 → Ty42 → Set; Var42
= λ Γ A →
(Var42 : Con42 → Ty42 → Set)
(vz : ∀ Γ A → Var42 (snoc42 Γ A) A)
(vs : ∀ Γ B A → Var42 Γ A → Var42 (snoc42 Γ B) A)
→ Var42 Γ A
vz42 : ∀{Γ A} → Var42 (snoc42 Γ A) A; vz42
= λ Var42 vz42 vs → vz42 _ _
vs42 : ∀{Γ B A} → Var42 Γ A → Var42 (snoc42 Γ B) A; vs42
= λ x Var42 vz42 vs42 → vs42 _ _ _ (x Var42 vz42 vs42)
Tm42 : Con42 → Ty42 → Set; Tm42
= λ Γ A →
(Tm42 : Con42 → Ty42 → Set)
(var : ∀ Γ A → Var42 Γ A → Tm42 Γ A)
(lam : ∀ Γ A B → Tm42 (snoc42 Γ A) B → Tm42 Γ (arr42 A B))
(app : ∀ Γ A B → Tm42 Γ (arr42 A B) → Tm42 Γ A → Tm42 Γ B)
(tt : ∀ Γ → Tm42 Γ top42)
(pair : ∀ Γ A B → Tm42 Γ A → Tm42 Γ B → Tm42 Γ (prod42 A B))
(fst : ∀ Γ A B → Tm42 Γ (prod42 A B) → Tm42 Γ A)
(snd : ∀ Γ A B → Tm42 Γ (prod42 A B) → Tm42 Γ B)
(left : ∀ Γ A B → Tm42 Γ A → Tm42 Γ (sum42 A B))
(right : ∀ Γ A B → Tm42 Γ B → Tm42 Γ (sum42 A B))
(case : ∀ Γ A B C → Tm42 Γ (sum42 A B) → Tm42 Γ (arr42 A C) → Tm42 Γ (arr42 B C) → Tm42 Γ C)
(zero : ∀ Γ → Tm42 Γ nat42)
(suc : ∀ Γ → Tm42 Γ nat42 → Tm42 Γ nat42)
(rec : ∀ Γ A → Tm42 Γ nat42 → Tm42 Γ (arr42 nat42 (arr42 A A)) → Tm42 Γ A → Tm42 Γ A)
→ Tm42 Γ A
var42 : ∀{Γ A} → Var42 Γ A → Tm42 Γ A; var42
= λ x Tm42 var42 lam app tt pair fst snd left right case zero suc rec →
var42 _ _ x
lam42 : ∀{Γ A B} → Tm42 (snoc42 Γ A) B → Tm42 Γ (arr42 A B); lam42
= λ t Tm42 var42 lam42 app tt pair fst snd left right case zero suc rec →
lam42 _ _ _ (t Tm42 var42 lam42 app tt pair fst snd left right case zero suc rec)
app42 : ∀{Γ A B} → Tm42 Γ (arr42 A B) → Tm42 Γ A → Tm42 Γ B; app42
= λ t u Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec →
app42 _ _ _ (t Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec)
(u Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec)
tt42 : ∀{Γ} → Tm42 Γ top42; tt42
= λ Tm42 var42 lam42 app42 tt42 pair fst snd left right case zero suc rec → tt42 _
pair42 : ∀{Γ A B} → Tm42 Γ A → Tm42 Γ B → Tm42 Γ (prod42 A B); pair42
= λ t u Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec →
pair42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec)
(u Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec)
fst42 : ∀{Γ A B} → Tm42 Γ (prod42 A B) → Tm42 Γ A; fst42
= λ t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd left right case zero suc rec →
fst42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd left right case zero suc rec)
snd42 : ∀{Γ A B} → Tm42 Γ (prod42 A B) → Tm42 Γ B; snd42
= λ t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left right case zero suc rec →
snd42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left right case zero suc rec)
left42 : ∀{Γ A B} → Tm42 Γ A → Tm42 Γ (sum42 A B); left42
= λ t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right case zero suc rec →
left42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right case zero suc rec)
right42 : ∀{Γ A B} → Tm42 Γ B → Tm42 Γ (sum42 A B); right42
= λ t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case zero suc rec →
right42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case zero suc rec)
case42 : ∀{Γ A B C} → Tm42 Γ (sum42 A B) → Tm42 Γ (arr42 A C) → Tm42 Γ (arr42 B C) → Tm42 Γ C; case42
= λ t u v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec →
case42 _ _ _ _
(t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
(u Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
(v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
zero42 : ∀{Γ} → Tm42 Γ nat42; zero42
= λ Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc rec → zero42 _
suc42 : ∀{Γ} → Tm42 Γ nat42 → Tm42 Γ nat42; suc42
= λ t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec →
suc42 _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec)
rec42 : ∀{Γ A} → Tm42 Γ nat42 → Tm42 Γ (arr42 nat42 (arr42 A A)) → Tm42 Γ A → Tm42 Γ A; rec42
= λ t u v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42 →
rec42 _ _
(t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
(u Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
(v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
v042 : ∀{Γ A} → Tm42 (snoc42 Γ A) A; v042
= var42 vz42
v142 : ∀{Γ A B} → Tm42 (snoc42 (snoc42 Γ A) B) A; v142
= var42 (vs42 vz42)
v242 : ∀{Γ A B C} → Tm42 (snoc42 (snoc42 (snoc42 Γ A) B) C) A; v242
= var42 (vs42 (vs42 vz42))
v342 : ∀{Γ A B C D} → Tm42 (snoc42 (snoc42 (snoc42 (snoc42 Γ A) B) C) D) A; v342
= var42 (vs42 (vs42 (vs42 vz42)))
tbool42 : Ty42; tbool42
= sum42 top42 top42
true42 : ∀{Γ} → Tm42 Γ tbool42; true42
= left42 tt42
tfalse42 : ∀{Γ} → Tm42 Γ tbool42; tfalse42
= right42 tt42
ifthenelse42 : ∀{Γ A} → Tm42 Γ (arr42 tbool42 (arr42 A (arr42 A A))); ifthenelse42
= lam42 (lam42 (lam42 (case42 v242 (lam42 v242) (lam42 v142))))
times442 : ∀{Γ A} → Tm42 Γ (arr42 (arr42 A A) (arr42 A A)); times442
= lam42 (lam42 (app42 v142 (app42 v142 (app42 v142 (app42 v142 v042)))))
add42 : ∀{Γ} → Tm42 Γ (arr42 nat42 (arr42 nat42 nat42)); add42
= lam42 (rec42 v042
(lam42 (lam42 (lam42 (suc42 (app42 v142 v042)))))
(lam42 v042))
mul42 : ∀{Γ} → Tm42 Γ (arr42 nat42 (arr42 nat42 nat42)); mul42
= lam42 (rec42 v042
(lam42 (lam42 (lam42 (app42 (app42 add42 (app42 v142 v042)) v042))))
(lam42 zero42))
fact42 : ∀{Γ} → Tm42 Γ (arr42 nat42 nat42); fact42
= lam42 (rec42 v042 (lam42 (lam42 (app42 (app42 mul42 (suc42 v142)) v042)))
(suc42 zero42))
{-# OPTIONS --type-in-type #-}
Ty43 : Set
Ty43 =
(Ty43 : Set)
(nat top bot : Ty43)
(arr prod sum : Ty43 → Ty43 → Ty43)
→ Ty43
nat43 : Ty43; nat43 = λ _ nat43 _ _ _ _ _ → nat43
top43 : Ty43; top43 = λ _ _ top43 _ _ _ _ → top43
bot43 : Ty43; bot43 = λ _ _ _ bot43 _ _ _ → bot43
arr43 : Ty43 → Ty43 → Ty43; arr43
= λ A B Ty43 nat43 top43 bot43 arr43 prod sum →
arr43 (A Ty43 nat43 top43 bot43 arr43 prod sum) (B Ty43 nat43 top43 bot43 arr43 prod sum)
prod43 : Ty43 → Ty43 → Ty43; prod43
= λ A B Ty43 nat43 top43 bot43 arr43 prod43 sum →
prod43 (A Ty43 nat43 top43 bot43 arr43 prod43 sum) (B Ty43 nat43 top43 bot43 arr43 prod43 sum)
sum43 : Ty43 → Ty43 → Ty43; sum43
= λ A B Ty43 nat43 top43 bot43 arr43 prod43 sum43 →
sum43 (A Ty43 nat43 top43 bot43 arr43 prod43 sum43) (B Ty43 nat43 top43 bot43 arr43 prod43 sum43)
Con43 : Set; Con43
= (Con43 : Set)
(nil : Con43)
(snoc : Con43 → Ty43 → Con43)
→ Con43
nil43 : Con43; nil43
= λ Con43 nil43 snoc → nil43
snoc43 : Con43 → Ty43 → Con43; snoc43
= λ Γ A Con43 nil43 snoc43 → snoc43 (Γ Con43 nil43 snoc43) A
Var43 : Con43 → Ty43 → Set; Var43
= λ Γ A →
(Var43 : Con43 → Ty43 → Set)
(vz : ∀ Γ A → Var43 (snoc43 Γ A) A)
(vs : ∀ Γ B A → Var43 Γ A → Var43 (snoc43 Γ B) A)
→ Var43 Γ A
vz43 : ∀{Γ A} → Var43 (snoc43 Γ A) A; vz43
= λ Var43 vz43 vs → vz43 _ _
vs43 : ∀{Γ B A} → Var43 Γ A → Var43 (snoc43 Γ B) A; vs43
= λ x Var43 vz43 vs43 → vs43 _ _ _ (x Var43 vz43 vs43)
Tm43 : Con43 → Ty43 → Set; Tm43
= λ Γ A →
(Tm43 : Con43 → Ty43 → Set)
(var : ∀ Γ A → Var43 Γ A → Tm43 Γ A)
(lam : ∀ Γ A B → Tm43 (snoc43 Γ A) B → Tm43 Γ (arr43 A B))
(app : ∀ Γ A B → Tm43 Γ (arr43 A B) → Tm43 Γ A → Tm43 Γ B)
(tt : ∀ Γ → Tm43 Γ top43)
(pair : ∀ Γ A B → Tm43 Γ A → Tm43 Γ B → Tm43 Γ (prod43 A B))
(fst : ∀ Γ A B → Tm43 Γ (prod43 A B) → Tm43 Γ A)
(snd : ∀ Γ A B → Tm43 Γ (prod43 A B) → Tm43 Γ B)
(left : ∀ Γ A B → Tm43 Γ A → Tm43 Γ (sum43 A B))
(right : ∀ Γ A B → Tm43 Γ B → Tm43 Γ (sum43 A B))
(case : ∀ Γ A B C → Tm43 Γ (sum43 A B) → Tm43 Γ (arr43 A C) → Tm43 Γ (arr43 B C) → Tm43 Γ C)
(zero : ∀ Γ → Tm43 Γ nat43)
(suc : ∀ Γ → Tm43 Γ nat43 → Tm43 Γ nat43)
(rec : ∀ Γ A → Tm43 Γ nat43 → Tm43 Γ (arr43 nat43 (arr43 A A)) → Tm43 Γ A → Tm43 Γ A)
→ Tm43 Γ A
var43 : ∀{Γ A} → Var43 Γ A → Tm43 Γ A; var43
= λ x Tm43 var43 lam app tt pair fst snd left right case zero suc rec →
var43 _ _ x
lam43 : ∀{Γ A B} → Tm43 (snoc43 Γ A) B → Tm43 Γ (arr43 A B); lam43
= λ t Tm43 var43 lam43 app tt pair fst snd left right case zero suc rec →
lam43 _ _ _ (t Tm43 var43 lam43 app tt pair fst snd left right case zero suc rec)
app43 : ∀{Γ A B} → Tm43 Γ (arr43 A B) → Tm43 Γ A → Tm43 Γ B; app43
= λ t u Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec →
app43 _ _ _ (t Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec)
(u Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec)
tt43 : ∀{Γ} → Tm43 Γ top43; tt43
= λ Tm43 var43 lam43 app43 tt43 pair fst snd left right case zero suc rec → tt43 _
pair43 : ∀{Γ A B} → Tm43 Γ A → Tm43 Γ B → Tm43 Γ (prod43 A B); pair43
= λ t u Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec →
pair43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec)
(u Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec)
fst43 : ∀{Γ A B} → Tm43 Γ (prod43 A B) → Tm43 Γ A; fst43
= λ t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd left right case zero suc rec →
fst43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd left right case zero suc rec)
snd43 : ∀{Γ A B} → Tm43 Γ (prod43 A B) → Tm43 Γ B; snd43
= λ t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left right case zero suc rec →
snd43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left right case zero suc rec)
left43 : ∀{Γ A B} → Tm43 Γ A → Tm43 Γ (sum43 A B); left43
= λ t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right case zero suc rec →
left43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right case zero suc rec)
right43 : ∀{Γ A B} → Tm43 Γ B → Tm43 Γ (sum43 A B); right43
= λ t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case zero suc rec →
right43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case zero suc rec)
case43 : ∀{Γ A B C} → Tm43 Γ (sum43 A B) → Tm43 Γ (arr43 A C) → Tm43 Γ (arr43 B C) → Tm43 Γ C; case43
= λ t u v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec →
case43 _ _ _ _
(t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
(u Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
(v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
zero43 : ∀{Γ} → Tm43 Γ nat43; zero43
= λ Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc rec → zero43 _
suc43 : ∀{Γ} → Tm43 Γ nat43 → Tm43 Γ nat43; suc43
= λ t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec →
suc43 _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec)
rec43 : ∀{Γ A} → Tm43 Γ nat43 → Tm43 Γ (arr43 nat43 (arr43 A A)) → Tm43 Γ A → Tm43 Γ A; rec43
= λ t u v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43 →
rec43 _ _
(t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
(u Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
(v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
v043 : ∀{Γ A} → Tm43 (snoc43 Γ A) A; v043
= var43 vz43
v143 : ∀{Γ A B} → Tm43 (snoc43 (snoc43 Γ A) B) A; v143
= var43 (vs43 vz43)
v243 : ∀{Γ A B C} → Tm43 (snoc43 (snoc43 (snoc43 Γ A) B) C) A; v243
= var43 (vs43 (vs43 vz43))
v343 : ∀{Γ A B C D} → Tm43 (snoc43 (snoc43 (snoc43 (snoc43 Γ A) B) C) D) A; v343
= var43 (vs43 (vs43 (vs43 vz43)))
tbool43 : Ty43; tbool43
= sum43 top43 top43
true43 : ∀{Γ} → Tm43 Γ tbool43; true43
= left43 tt43
tfalse43 : ∀{Γ} → Tm43 Γ tbool43; tfalse43
= right43 tt43
ifthenelse43 : ∀{Γ A} → Tm43 Γ (arr43 tbool43 (arr43 A (arr43 A A))); ifthenelse43
= lam43 (lam43 (lam43 (case43 v243 (lam43 v243) (lam43 v143))))
times443 : ∀{Γ A} → Tm43 Γ (arr43 (arr43 A A) (arr43 A A)); times443
= lam43 (lam43 (app43 v143 (app43 v143 (app43 v143 (app43 v143 v043)))))
add43 : ∀{Γ} → Tm43 Γ (arr43 nat43 (arr43 nat43 nat43)); add43
= lam43 (rec43 v043
(lam43 (lam43 (lam43 (suc43 (app43 v143 v043)))))
(lam43 v043))
mul43 : ∀{Γ} → Tm43 Γ (arr43 nat43 (arr43 nat43 nat43)); mul43
= lam43 (rec43 v043
(lam43 (lam43 (lam43 (app43 (app43 add43 (app43 v143 v043)) v043))))
(lam43 zero43))
fact43 : ∀{Γ} → Tm43 Γ (arr43 nat43 nat43); fact43
= lam43 (rec43 v043 (lam43 (lam43 (app43 (app43 mul43 (suc43 v143)) v043)))
(suc43 zero43))
{-# OPTIONS --type-in-type #-}
Ty44 : Set
Ty44 =
(Ty44 : Set)
(nat top bot : Ty44)
(arr prod sum : Ty44 → Ty44 → Ty44)
→ Ty44
nat44 : Ty44; nat44 = λ _ nat44 _ _ _ _ _ → nat44
top44 : Ty44; top44 = λ _ _ top44 _ _ _ _ → top44
bot44 : Ty44; bot44 = λ _ _ _ bot44 _ _ _ → bot44
arr44 : Ty44 → Ty44 → Ty44; arr44
= λ A B Ty44 nat44 top44 bot44 arr44 prod sum →
arr44 (A Ty44 nat44 top44 bot44 arr44 prod sum) (B Ty44 nat44 top44 bot44 arr44 prod sum)
prod44 : Ty44 → Ty44 → Ty44; prod44
= λ A B Ty44 nat44 top44 bot44 arr44 prod44 sum →
prod44 (A Ty44 nat44 top44 bot44 arr44 prod44 sum) (B Ty44 nat44 top44 bot44 arr44 prod44 sum)
sum44 : Ty44 → Ty44 → Ty44; sum44
= λ A B Ty44 nat44 top44 bot44 arr44 prod44 sum44 →
sum44 (A Ty44 nat44 top44 bot44 arr44 prod44 sum44) (B Ty44 nat44 top44 bot44 arr44 prod44 sum44)
Con44 : Set; Con44
= (Con44 : Set)
(nil : Con44)
(snoc : Con44 → Ty44 → Con44)
→ Con44
nil44 : Con44; nil44
= λ Con44 nil44 snoc → nil44
snoc44 : Con44 → Ty44 → Con44; snoc44
= λ Γ A Con44 nil44 snoc44 → snoc44 (Γ Con44 nil44 snoc44) A
Var44 : Con44 → Ty44 → Set; Var44
= λ Γ A →
(Var44 : Con44 → Ty44 → Set)
(vz : ∀ Γ A → Var44 (snoc44 Γ A) A)
(vs : ∀ Γ B A → Var44 Γ A → Var44 (snoc44 Γ B) A)
→ Var44 Γ A
vz44 : ∀{Γ A} → Var44 (snoc44 Γ A) A; vz44
= λ Var44 vz44 vs → vz44 _ _
vs44 : ∀{Γ B A} → Var44 Γ A → Var44 (snoc44 Γ B) A; vs44
= λ x Var44 vz44 vs44 → vs44 _ _ _ (x Var44 vz44 vs44)
Tm44 : Con44 → Ty44 → Set; Tm44
= λ Γ A →
(Tm44 : Con44 → Ty44 → Set)
(var : ∀ Γ A → Var44 Γ A → Tm44 Γ A)
(lam : ∀ Γ A B → Tm44 (snoc44 Γ A) B → Tm44 Γ (arr44 A B))
(app : ∀ Γ A B → Tm44 Γ (arr44 A B) → Tm44 Γ A → Tm44 Γ B)
(tt : ∀ Γ → Tm44 Γ top44)
(pair : ∀ Γ A B → Tm44 Γ A → Tm44 Γ B → Tm44 Γ (prod44 A B))
(fst : ∀ Γ A B → Tm44 Γ (prod44 A B) → Tm44 Γ A)
(snd : ∀ Γ A B → Tm44 Γ (prod44 A B) → Tm44 Γ B)
(left : ∀ Γ A B → Tm44 Γ A → Tm44 Γ (sum44 A B))
(right : ∀ Γ A B → Tm44 Γ B → Tm44 Γ (sum44 A B))
(case : ∀ Γ A B C → Tm44 Γ (sum44 A B) → Tm44 Γ (arr44 A C) → Tm44 Γ (arr44 B C) → Tm44 Γ C)
(zero : ∀ Γ → Tm44 Γ nat44)
(suc : ∀ Γ → Tm44 Γ nat44 → Tm44 Γ nat44)
(rec : ∀ Γ A → Tm44 Γ nat44 → Tm44 Γ (arr44 nat44 (arr44 A A)) → Tm44 Γ A → Tm44 Γ A)
→ Tm44 Γ A
var44 : ∀{Γ A} → Var44 Γ A → Tm44 Γ A; var44
= λ x Tm44 var44 lam app tt pair fst snd left right case zero suc rec →
var44 _ _ x
lam44 : ∀{Γ A B} → Tm44 (snoc44 Γ A) B → Tm44 Γ (arr44 A B); lam44
= λ t Tm44 var44 lam44 app tt pair fst snd left right case zero suc rec →
lam44 _ _ _ (t Tm44 var44 lam44 app tt pair fst snd left right case zero suc rec)
app44 : ∀{Γ A B} → Tm44 Γ (arr44 A B) → Tm44 Γ A → Tm44 Γ B; app44
= λ t u Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec →
app44 _ _ _ (t Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec)
(u Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec)
tt44 : ∀{Γ} → Tm44 Γ top44; tt44
= λ Tm44 var44 lam44 app44 tt44 pair fst snd left right case zero suc rec → tt44 _
pair44 : ∀{Γ A B} → Tm44 Γ A → Tm44 Γ B → Tm44 Γ (prod44 A B); pair44
= λ t u Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec →
pair44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec)
(u Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec)
fst44 : ∀{Γ A B} → Tm44 Γ (prod44 A B) → Tm44 Γ A; fst44
= λ t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd left right case zero suc rec →
fst44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd left right case zero suc rec)
snd44 : ∀{Γ A B} → Tm44 Γ (prod44 A B) → Tm44 Γ B; snd44
= λ t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left right case zero suc rec →
snd44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left right case zero suc rec)
left44 : ∀{Γ A B} → Tm44 Γ A → Tm44 Γ (sum44 A B); left44
= λ t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right case zero suc rec →
left44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right case zero suc rec)
right44 : ∀{Γ A B} → Tm44 Γ B → Tm44 Γ (sum44 A B); right44
= λ t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case zero suc rec →
right44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case zero suc rec)
case44 : ∀{Γ A B C} → Tm44 Γ (sum44 A B) → Tm44 Γ (arr44 A C) → Tm44 Γ (arr44 B C) → Tm44 Γ C; case44
= λ t u v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec →
case44 _ _ _ _
(t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
(u Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
(v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
zero44 : ∀{Γ} → Tm44 Γ nat44; zero44
= λ Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc rec → zero44 _
suc44 : ∀{Γ} → Tm44 Γ nat44 → Tm44 Γ nat44; suc44
= λ t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec →
suc44 _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec)
rec44 : ∀{Γ A} → Tm44 Γ nat44 → Tm44 Γ (arr44 nat44 (arr44 A A)) → Tm44 Γ A → Tm44 Γ A; rec44
= λ t u v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44 →
rec44 _ _
(t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
(u Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
(v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
v044 : ∀{Γ A} → Tm44 (snoc44 Γ A) A; v044
= var44 vz44
v144 : ∀{Γ A B} → Tm44 (snoc44 (snoc44 Γ A) B) A; v144
= var44 (vs44 vz44)
v244 : ∀{Γ A B C} → Tm44 (snoc44 (snoc44 (snoc44 Γ A) B) C) A; v244
= var44 (vs44 (vs44 vz44))
v344 : ∀{Γ A B C D} → Tm44 (snoc44 (snoc44 (snoc44 (snoc44 Γ A) B) C) D) A; v344
= var44 (vs44 (vs44 (vs44 vz44)))
tbool44 : Ty44; tbool44
= sum44 top44 top44
true44 : ∀{Γ} → Tm44 Γ tbool44; true44
= left44 tt44
tfalse44 : ∀{Γ} → Tm44 Γ tbool44; tfalse44
= right44 tt44
ifthenelse44 : ∀{Γ A} → Tm44 Γ (arr44 tbool44 (arr44 A (arr44 A A))); ifthenelse44
= lam44 (lam44 (lam44 (case44 v244 (lam44 v244) (lam44 v144))))
times444 : ∀{Γ A} → Tm44 Γ (arr44 (arr44 A A) (arr44 A A)); times444
= lam44 (lam44 (app44 v144 (app44 v144 (app44 v144 (app44 v144 v044)))))
add44 : ∀{Γ} → Tm44 Γ (arr44 nat44 (arr44 nat44 nat44)); add44
= lam44 (rec44 v044
(lam44 (lam44 (lam44 (suc44 (app44 v144 v044)))))
(lam44 v044))
mul44 : ∀{Γ} → Tm44 Γ (arr44 nat44 (arr44 nat44 nat44)); mul44
= lam44 (rec44 v044
(lam44 (lam44 (lam44 (app44 (app44 add44 (app44 v144 v044)) v044))))
(lam44 zero44))
fact44 : ∀{Γ} → Tm44 Γ (arr44 nat44 nat44); fact44
= lam44 (rec44 v044 (lam44 (lam44 (app44 (app44 mul44 (suc44 v144)) v044)))
(suc44 zero44))
{-# OPTIONS --type-in-type #-}
Ty45 : Set
Ty45 =
(Ty45 : Set)
(nat top bot : Ty45)
(arr prod sum : Ty45 → Ty45 → Ty45)
→ Ty45
nat45 : Ty45; nat45 = λ _ nat45 _ _ _ _ _ → nat45
top45 : Ty45; top45 = λ _ _ top45 _ _ _ _ → top45
bot45 : Ty45; bot45 = λ _ _ _ bot45 _ _ _ → bot45
arr45 : Ty45 → Ty45 → Ty45; arr45
= λ A B Ty45 nat45 top45 bot45 arr45 prod sum →
arr45 (A Ty45 nat45 top45 bot45 arr45 prod sum) (B Ty45 nat45 top45 bot45 arr45 prod sum)
prod45 : Ty45 → Ty45 → Ty45; prod45
= λ A B Ty45 nat45 top45 bot45 arr45 prod45 sum →
prod45 (A Ty45 nat45 top45 bot45 arr45 prod45 sum) (B Ty45 nat45 top45 bot45 arr45 prod45 sum)
sum45 : Ty45 → Ty45 → Ty45; sum45
= λ A B Ty45 nat45 top45 bot45 arr45 prod45 sum45 →
sum45 (A Ty45 nat45 top45 bot45 arr45 prod45 sum45) (B Ty45 nat45 top45 bot45 arr45 prod45 sum45)
Con45 : Set; Con45
= (Con45 : Set)
(nil : Con45)
(snoc : Con45 → Ty45 → Con45)
→ Con45
nil45 : Con45; nil45
= λ Con45 nil45 snoc → nil45
snoc45 : Con45 → Ty45 → Con45; snoc45
= λ Γ A Con45 nil45 snoc45 → snoc45 (Γ Con45 nil45 snoc45) A
Var45 : Con45 → Ty45 → Set; Var45
= λ Γ A →
(Var45 : Con45 → Ty45 → Set)
(vz : ∀ Γ A → Var45 (snoc45 Γ A) A)
(vs : ∀ Γ B A → Var45 Γ A → Var45 (snoc45 Γ B) A)
→ Var45 Γ A
vz45 : ∀{Γ A} → Var45 (snoc45 Γ A) A; vz45
= λ Var45 vz45 vs → vz45 _ _
vs45 : ∀{Γ B A} → Var45 Γ A → Var45 (snoc45 Γ B) A; vs45
= λ x Var45 vz45 vs45 → vs45 _ _ _ (x Var45 vz45 vs45)
Tm45 : Con45 → Ty45 → Set; Tm45
= λ Γ A →
(Tm45 : Con45 → Ty45 → Set)
(var : ∀ Γ A → Var45 Γ A → Tm45 Γ A)
(lam : ∀ Γ A B → Tm45 (snoc45 Γ A) B → Tm45 Γ (arr45 A B))
(app : ∀ Γ A B → Tm45 Γ (arr45 A B) → Tm45 Γ A → Tm45 Γ B)
(tt : ∀ Γ → Tm45 Γ top45)
(pair : ∀ Γ A B → Tm45 Γ A → Tm45 Γ B → Tm45 Γ (prod45 A B))
(fst : ∀ Γ A B → Tm45 Γ (prod45 A B) → Tm45 Γ A)
(snd : ∀ Γ A B → Tm45 Γ (prod45 A B) → Tm45 Γ B)
(left : ∀ Γ A B → Tm45 Γ A → Tm45 Γ (sum45 A B))
(right : ∀ Γ A B → Tm45 Γ B → Tm45 Γ (sum45 A B))
(case : ∀ Γ A B C → Tm45 Γ (sum45 A B) → Tm45 Γ (arr45 A C) → Tm45 Γ (arr45 B C) → Tm45 Γ C)
(zero : ∀ Γ → Tm45 Γ nat45)
(suc : ∀ Γ → Tm45 Γ nat45 → Tm45 Γ nat45)
(rec : ∀ Γ A → Tm45 Γ nat45 → Tm45 Γ (arr45 nat45 (arr45 A A)) → Tm45 Γ A → Tm45 Γ A)
→ Tm45 Γ A
var45 : ∀{Γ A} → Var45 Γ A → Tm45 Γ A; var45
= λ x Tm45 var45 lam app tt pair fst snd left right case zero suc rec →
var45 _ _ x
lam45 : ∀{Γ A B} → Tm45 (snoc45 Γ A) B → Tm45 Γ (arr45 A B); lam45
= λ t Tm45 var45 lam45 app tt pair fst snd left right case zero suc rec →
lam45 _ _ _ (t Tm45 var45 lam45 app tt pair fst snd left right case zero suc rec)
app45 : ∀{Γ A B} → Tm45 Γ (arr45 A B) → Tm45 Γ A → Tm45 Γ B; app45
= λ t u Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec →
app45 _ _ _ (t Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec)
(u Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec)
tt45 : ∀{Γ} → Tm45 Γ top45; tt45
= λ Tm45 var45 lam45 app45 tt45 pair fst snd left right case zero suc rec → tt45 _
pair45 : ∀{Γ A B} → Tm45 Γ A → Tm45 Γ B → Tm45 Γ (prod45 A B); pair45
= λ t u Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec →
pair45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec)
(u Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec)
fst45 : ∀{Γ A B} → Tm45 Γ (prod45 A B) → Tm45 Γ A; fst45
= λ t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd left right case zero suc rec →
fst45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd left right case zero suc rec)
snd45 : ∀{Γ A B} → Tm45 Γ (prod45 A B) → Tm45 Γ B; snd45
= λ t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left right case zero suc rec →
snd45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left right case zero suc rec)
left45 : ∀{Γ A B} → Tm45 Γ A → Tm45 Γ (sum45 A B); left45
= λ t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right case zero suc rec →
left45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right case zero suc rec)
right45 : ∀{Γ A B} → Tm45 Γ B → Tm45 Γ (sum45 A B); right45
= λ t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case zero suc rec →
right45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case zero suc rec)
case45 : ∀{Γ A B C} → Tm45 Γ (sum45 A B) → Tm45 Γ (arr45 A C) → Tm45 Γ (arr45 B C) → Tm45 Γ C; case45
= λ t u v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec →
case45 _ _ _ _
(t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
(u Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
(v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
zero45 : ∀{Γ} → Tm45 Γ nat45; zero45
= λ Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc rec → zero45 _
suc45 : ∀{Γ} → Tm45 Γ nat45 → Tm45 Γ nat45; suc45
= λ t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec →
suc45 _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec)
rec45 : ∀{Γ A} → Tm45 Γ nat45 → Tm45 Γ (arr45 nat45 (arr45 A A)) → Tm45 Γ A → Tm45 Γ A; rec45
= λ t u v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45 →
rec45 _ _
(t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
(u Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
(v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
v045 : ∀{Γ A} → Tm45 (snoc45 Γ A) A; v045
= var45 vz45
v145 : ∀{Γ A B} → Tm45 (snoc45 (snoc45 Γ A) B) A; v145
= var45 (vs45 vz45)
v245 : ∀{Γ A B C} → Tm45 (snoc45 (snoc45 (snoc45 Γ A) B) C) A; v245
= var45 (vs45 (vs45 vz45))
v345 : ∀{Γ A B C D} → Tm45 (snoc45 (snoc45 (snoc45 (snoc45 Γ A) B) C) D) A; v345
= var45 (vs45 (vs45 (vs45 vz45)))
tbool45 : Ty45; tbool45
= sum45 top45 top45
true45 : ∀{Γ} → Tm45 Γ tbool45; true45
= left45 tt45
tfalse45 : ∀{Γ} → Tm45 Γ tbool45; tfalse45
= right45 tt45
ifthenelse45 : ∀{Γ A} → Tm45 Γ (arr45 tbool45 (arr45 A (arr45 A A))); ifthenelse45
= lam45 (lam45 (lam45 (case45 v245 (lam45 v245) (lam45 v145))))
times445 : ∀{Γ A} → Tm45 Γ (arr45 (arr45 A A) (arr45 A A)); times445
= lam45 (lam45 (app45 v145 (app45 v145 (app45 v145 (app45 v145 v045)))))
add45 : ∀{Γ} → Tm45 Γ (arr45 nat45 (arr45 nat45 nat45)); add45
= lam45 (rec45 v045
(lam45 (lam45 (lam45 (suc45 (app45 v145 v045)))))
(lam45 v045))
mul45 : ∀{Γ} → Tm45 Γ (arr45 nat45 (arr45 nat45 nat45)); mul45
= lam45 (rec45 v045
(lam45 (lam45 (lam45 (app45 (app45 add45 (app45 v145 v045)) v045))))
(lam45 zero45))
fact45 : ∀{Γ} → Tm45 Γ (arr45 nat45 nat45); fact45
= lam45 (rec45 v045 (lam45 (lam45 (app45 (app45 mul45 (suc45 v145)) v045)))
(suc45 zero45))
{-# OPTIONS --type-in-type #-}
Ty46 : Set
Ty46 =
(Ty46 : Set)
(nat top bot : Ty46)
(arr prod sum : Ty46 → Ty46 → Ty46)
→ Ty46
nat46 : Ty46; nat46 = λ _ nat46 _ _ _ _ _ → nat46
top46 : Ty46; top46 = λ _ _ top46 _ _ _ _ → top46
bot46 : Ty46; bot46 = λ _ _ _ bot46 _ _ _ → bot46
arr46 : Ty46 → Ty46 → Ty46; arr46
= λ A B Ty46 nat46 top46 bot46 arr46 prod sum →
arr46 (A Ty46 nat46 top46 bot46 arr46 prod sum) (B Ty46 nat46 top46 bot46 arr46 prod sum)
prod46 : Ty46 → Ty46 → Ty46; prod46
= λ A B Ty46 nat46 top46 bot46 arr46 prod46 sum →
prod46 (A Ty46 nat46 top46 bot46 arr46 prod46 sum) (B Ty46 nat46 top46 bot46 arr46 prod46 sum)
sum46 : Ty46 → Ty46 → Ty46; sum46
= λ A B Ty46 nat46 top46 bot46 arr46 prod46 sum46 →
sum46 (A Ty46 nat46 top46 bot46 arr46 prod46 sum46) (B Ty46 nat46 top46 bot46 arr46 prod46 sum46)
Con46 : Set; Con46
= (Con46 : Set)
(nil : Con46)
(snoc : Con46 → Ty46 → Con46)
→ Con46
nil46 : Con46; nil46
= λ Con46 nil46 snoc → nil46
snoc46 : Con46 → Ty46 → Con46; snoc46
= λ Γ A Con46 nil46 snoc46 → snoc46 (Γ Con46 nil46 snoc46) A
Var46 : Con46 → Ty46 → Set; Var46
= λ Γ A →
(Var46 : Con46 → Ty46 → Set)
(vz : ∀ Γ A → Var46 (snoc46 Γ A) A)
(vs : ∀ Γ B A → Var46 Γ A → Var46 (snoc46 Γ B) A)
→ Var46 Γ A
vz46 : ∀{Γ A} → Var46 (snoc46 Γ A) A; vz46
= λ Var46 vz46 vs → vz46 _ _
vs46 : ∀{Γ B A} → Var46 Γ A → Var46 (snoc46 Γ B) A; vs46
= λ x Var46 vz46 vs46 → vs46 _ _ _ (x Var46 vz46 vs46)
Tm46 : Con46 → Ty46 → Set; Tm46
= λ Γ A →
(Tm46 : Con46 → Ty46 → Set)
(var : ∀ Γ A → Var46 Γ A → Tm46 Γ A)
(lam : ∀ Γ A B → Tm46 (snoc46 Γ A) B → Tm46 Γ (arr46 A B))
(app : ∀ Γ A B → Tm46 Γ (arr46 A B) → Tm46 Γ A → Tm46 Γ B)
(tt : ∀ Γ → Tm46 Γ top46)
(pair : ∀ Γ A B → Tm46 Γ A → Tm46 Γ B → Tm46 Γ (prod46 A B))
(fst : ∀ Γ A B → Tm46 Γ (prod46 A B) → Tm46 Γ A)
(snd : ∀ Γ A B → Tm46 Γ (prod46 A B) → Tm46 Γ B)
(left : ∀ Γ A B → Tm46 Γ A → Tm46 Γ (sum46 A B))
(right : ∀ Γ A B → Tm46 Γ B → Tm46 Γ (sum46 A B))
(case : ∀ Γ A B C → Tm46 Γ (sum46 A B) → Tm46 Γ (arr46 A C) → Tm46 Γ (arr46 B C) → Tm46 Γ C)
(zero : ∀ Γ → Tm46 Γ nat46)
(suc : ∀ Γ → Tm46 Γ nat46 → Tm46 Γ nat46)
(rec : ∀ Γ A → Tm46 Γ nat46 → Tm46 Γ (arr46 nat46 (arr46 A A)) → Tm46 Γ A → Tm46 Γ A)
→ Tm46 Γ A
var46 : ∀{Γ A} → Var46 Γ A → Tm46 Γ A; var46
= λ x Tm46 var46 lam app tt pair fst snd left right case zero suc rec →
var46 _ _ x
lam46 : ∀{Γ A B} → Tm46 (snoc46 Γ A) B → Tm46 Γ (arr46 A B); lam46
= λ t Tm46 var46 lam46 app tt pair fst snd left right case zero suc rec →
lam46 _ _ _ (t Tm46 var46 lam46 app tt pair fst snd left right case zero suc rec)
app46 : ∀{Γ A B} → Tm46 Γ (arr46 A B) → Tm46 Γ A → Tm46 Γ B; app46
= λ t u Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec →
app46 _ _ _ (t Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec)
(u Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec)
tt46 : ∀{Γ} → Tm46 Γ top46; tt46
= λ Tm46 var46 lam46 app46 tt46 pair fst snd left right case zero suc rec → tt46 _
pair46 : ∀{Γ A B} → Tm46 Γ A → Tm46 Γ B → Tm46 Γ (prod46 A B); pair46
= λ t u Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec →
pair46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec)
(u Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec)
fst46 : ∀{Γ A B} → Tm46 Γ (prod46 A B) → Tm46 Γ A; fst46
= λ t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd left right case zero suc rec →
fst46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd left right case zero suc rec)
snd46 : ∀{Γ A B} → Tm46 Γ (prod46 A B) → Tm46 Γ B; snd46
= λ t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left right case zero suc rec →
snd46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left right case zero suc rec)
left46 : ∀{Γ A B} → Tm46 Γ A → Tm46 Γ (sum46 A B); left46
= λ t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right case zero suc rec →
left46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right case zero suc rec)
right46 : ∀{Γ A B} → Tm46 Γ B → Tm46 Γ (sum46 A B); right46
= λ t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case zero suc rec →
right46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case zero suc rec)
case46 : ∀{Γ A B C} → Tm46 Γ (sum46 A B) → Tm46 Γ (arr46 A C) → Tm46 Γ (arr46 B C) → Tm46 Γ C; case46
= λ t u v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec →
case46 _ _ _ _
(t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
(u Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
(v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
zero46 : ∀{Γ} → Tm46 Γ nat46; zero46
= λ Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc rec → zero46 _
suc46 : ∀{Γ} → Tm46 Γ nat46 → Tm46 Γ nat46; suc46
= λ t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec →
suc46 _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec)
rec46 : ∀{Γ A} → Tm46 Γ nat46 → Tm46 Γ (arr46 nat46 (arr46 A A)) → Tm46 Γ A → Tm46 Γ A; rec46
= λ t u v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46 →
rec46 _ _
(t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
(u Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
(v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
v046 : ∀{Γ A} → Tm46 (snoc46 Γ A) A; v046
= var46 vz46
v146 : ∀{Γ A B} → Tm46 (snoc46 (snoc46 Γ A) B) A; v146
= var46 (vs46 vz46)
v246 : ∀{Γ A B C} → Tm46 (snoc46 (snoc46 (snoc46 Γ A) B) C) A; v246
= var46 (vs46 (vs46 vz46))
v346 : ∀{Γ A B C D} → Tm46 (snoc46 (snoc46 (snoc46 (snoc46 Γ A) B) C) D) A; v346
= var46 (vs46 (vs46 (vs46 vz46)))
tbool46 : Ty46; tbool46
= sum46 top46 top46
true46 : ∀{Γ} → Tm46 Γ tbool46; true46
= left46 tt46
tfalse46 : ∀{Γ} → Tm46 Γ tbool46; tfalse46
= right46 tt46
ifthenelse46 : ∀{Γ A} → Tm46 Γ (arr46 tbool46 (arr46 A (arr46 A A))); ifthenelse46
= lam46 (lam46 (lam46 (case46 v246 (lam46 v246) (lam46 v146))))
times446 : ∀{Γ A} → Tm46 Γ (arr46 (arr46 A A) (arr46 A A)); times446
= lam46 (lam46 (app46 v146 (app46 v146 (app46 v146 (app46 v146 v046)))))
add46 : ∀{Γ} → Tm46 Γ (arr46 nat46 (arr46 nat46 nat46)); add46
= lam46 (rec46 v046
(lam46 (lam46 (lam46 (suc46 (app46 v146 v046)))))
(lam46 v046))
mul46 : ∀{Γ} → Tm46 Γ (arr46 nat46 (arr46 nat46 nat46)); mul46
= lam46 (rec46 v046
(lam46 (lam46 (lam46 (app46 (app46 add46 (app46 v146 v046)) v046))))
(lam46 zero46))
fact46 : ∀{Γ} → Tm46 Γ (arr46 nat46 nat46); fact46
= lam46 (rec46 v046 (lam46 (lam46 (app46 (app46 mul46 (suc46 v146)) v046)))
(suc46 zero46))
{-# OPTIONS --type-in-type #-}
Ty47 : Set
Ty47 =
(Ty47 : Set)
(nat top bot : Ty47)
(arr prod sum : Ty47 → Ty47 → Ty47)
→ Ty47
nat47 : Ty47; nat47 = λ _ nat47 _ _ _ _ _ → nat47
top47 : Ty47; top47 = λ _ _ top47 _ _ _ _ → top47
bot47 : Ty47; bot47 = λ _ _ _ bot47 _ _ _ → bot47
arr47 : Ty47 → Ty47 → Ty47; arr47
= λ A B Ty47 nat47 top47 bot47 arr47 prod sum →
arr47 (A Ty47 nat47 top47 bot47 arr47 prod sum) (B Ty47 nat47 top47 bot47 arr47 prod sum)
prod47 : Ty47 → Ty47 → Ty47; prod47
= λ A B Ty47 nat47 top47 bot47 arr47 prod47 sum →
prod47 (A Ty47 nat47 top47 bot47 arr47 prod47 sum) (B Ty47 nat47 top47 bot47 arr47 prod47 sum)
sum47 : Ty47 → Ty47 → Ty47; sum47
= λ A B Ty47 nat47 top47 bot47 arr47 prod47 sum47 →
sum47 (A Ty47 nat47 top47 bot47 arr47 prod47 sum47) (B Ty47 nat47 top47 bot47 arr47 prod47 sum47)
Con47 : Set; Con47
= (Con47 : Set)
(nil : Con47)
(snoc : Con47 → Ty47 → Con47)
→ Con47
nil47 : Con47; nil47
= λ Con47 nil47 snoc → nil47
snoc47 : Con47 → Ty47 → Con47; snoc47
= λ Γ A Con47 nil47 snoc47 → snoc47 (Γ Con47 nil47 snoc47) A
Var47 : Con47 → Ty47 → Set; Var47
= λ Γ A →
(Var47 : Con47 → Ty47 → Set)
(vz : ∀ Γ A → Var47 (snoc47 Γ A) A)
(vs : ∀ Γ B A → Var47 Γ A → Var47 (snoc47 Γ B) A)
→ Var47 Γ A
vz47 : ∀{Γ A} → Var47 (snoc47 Γ A) A; vz47
= λ Var47 vz47 vs → vz47 _ _
vs47 : ∀{Γ B A} → Var47 Γ A → Var47 (snoc47 Γ B) A; vs47
= λ x Var47 vz47 vs47 → vs47 _ _ _ (x Var47 vz47 vs47)
Tm47 : Con47 → Ty47 → Set; Tm47
= λ Γ A →
(Tm47 : Con47 → Ty47 → Set)
(var : ∀ Γ A → Var47 Γ A → Tm47 Γ A)
(lam : ∀ Γ A B → Tm47 (snoc47 Γ A) B → Tm47 Γ (arr47 A B))
(app : ∀ Γ A B → Tm47 Γ (arr47 A B) → Tm47 Γ A → Tm47 Γ B)
(tt : ∀ Γ → Tm47 Γ top47)
(pair : ∀ Γ A B → Tm47 Γ A → Tm47 Γ B → Tm47 Γ (prod47 A B))
(fst : ∀ Γ A B → Tm47 Γ (prod47 A B) → Tm47 Γ A)
(snd : ∀ Γ A B → Tm47 Γ (prod47 A B) → Tm47 Γ B)
(left : ∀ Γ A B → Tm47 Γ A → Tm47 Γ (sum47 A B))
(right : ∀ Γ A B → Tm47 Γ B → Tm47 Γ (sum47 A B))
(case : ∀ Γ A B C → Tm47 Γ (sum47 A B) → Tm47 Γ (arr47 A C) → Tm47 Γ (arr47 B C) → Tm47 Γ C)
(zero : ∀ Γ → Tm47 Γ nat47)
(suc : ∀ Γ → Tm47 Γ nat47 → Tm47 Γ nat47)
(rec : ∀ Γ A → Tm47 Γ nat47 → Tm47 Γ (arr47 nat47 (arr47 A A)) → Tm47 Γ A → Tm47 Γ A)
→ Tm47 Γ A
var47 : ∀{Γ A} → Var47 Γ A → Tm47 Γ A; var47
= λ x Tm47 var47 lam app tt pair fst snd left right case zero suc rec →
var47 _ _ x
lam47 : ∀{Γ A B} → Tm47 (snoc47 Γ A) B → Tm47 Γ (arr47 A B); lam47
= λ t Tm47 var47 lam47 app tt pair fst snd left right case zero suc rec →
lam47 _ _ _ (t Tm47 var47 lam47 app tt pair fst snd left right case zero suc rec)
app47 : ∀{Γ A B} → Tm47 Γ (arr47 A B) → Tm47 Γ A → Tm47 Γ B; app47
= λ t u Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec →
app47 _ _ _ (t Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec)
(u Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec)
tt47 : ∀{Γ} → Tm47 Γ top47; tt47
= λ Tm47 var47 lam47 app47 tt47 pair fst snd left right case zero suc rec → tt47 _
pair47 : ∀{Γ A B} → Tm47 Γ A → Tm47 Γ B → Tm47 Γ (prod47 A B); pair47
= λ t u Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec →
pair47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec)
(u Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec)
fst47 : ∀{Γ A B} → Tm47 Γ (prod47 A B) → Tm47 Γ A; fst47
= λ t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd left right case zero suc rec →
fst47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd left right case zero suc rec)
snd47 : ∀{Γ A B} → Tm47 Γ (prod47 A B) → Tm47 Γ B; snd47
= λ t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left right case zero suc rec →
snd47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left right case zero suc rec)
left47 : ∀{Γ A B} → Tm47 Γ A → Tm47 Γ (sum47 A B); left47
= λ t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right case zero suc rec →
left47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right case zero suc rec)
right47 : ∀{Γ A B} → Tm47 Γ B → Tm47 Γ (sum47 A B); right47
= λ t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case zero suc rec →
right47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case zero suc rec)
case47 : ∀{Γ A B C} → Tm47 Γ (sum47 A B) → Tm47 Γ (arr47 A C) → Tm47 Γ (arr47 B C) → Tm47 Γ C; case47
= λ t u v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec →
case47 _ _ _ _
(t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
(u Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
(v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
zero47 : ∀{Γ} → Tm47 Γ nat47; zero47
= λ Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc rec → zero47 _
suc47 : ∀{Γ} → Tm47 Γ nat47 → Tm47 Γ nat47; suc47
= λ t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec →
suc47 _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec)
rec47 : ∀{Γ A} → Tm47 Γ nat47 → Tm47 Γ (arr47 nat47 (arr47 A A)) → Tm47 Γ A → Tm47 Γ A; rec47
= λ t u v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47 →
rec47 _ _
(t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
(u Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
(v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
v047 : ∀{Γ A} → Tm47 (snoc47 Γ A) A; v047
= var47 vz47
v147 : ∀{Γ A B} → Tm47 (snoc47 (snoc47 Γ A) B) A; v147
= var47 (vs47 vz47)
v247 : ∀{Γ A B C} → Tm47 (snoc47 (snoc47 (snoc47 Γ A) B) C) A; v247
= var47 (vs47 (vs47 vz47))
v347 : ∀{Γ A B C D} → Tm47 (snoc47 (snoc47 (snoc47 (snoc47 Γ A) B) C) D) A; v347
= var47 (vs47 (vs47 (vs47 vz47)))
tbool47 : Ty47; tbool47
= sum47 top47 top47
true47 : ∀{Γ} → Tm47 Γ tbool47; true47
= left47 tt47
tfalse47 : ∀{Γ} → Tm47 Γ tbool47; tfalse47
= right47 tt47
ifthenelse47 : ∀{Γ A} → Tm47 Γ (arr47 tbool47 (arr47 A (arr47 A A))); ifthenelse47
= lam47 (lam47 (lam47 (case47 v247 (lam47 v247) (lam47 v147))))
times447 : ∀{Γ A} → Tm47 Γ (arr47 (arr47 A A) (arr47 A A)); times447
= lam47 (lam47 (app47 v147 (app47 v147 (app47 v147 (app47 v147 v047)))))
add47 : ∀{Γ} → Tm47 Γ (arr47 nat47 (arr47 nat47 nat47)); add47
= lam47 (rec47 v047
(lam47 (lam47 (lam47 (suc47 (app47 v147 v047)))))
(lam47 v047))
mul47 : ∀{Γ} → Tm47 Γ (arr47 nat47 (arr47 nat47 nat47)); mul47
= lam47 (rec47 v047
(lam47 (lam47 (lam47 (app47 (app47 add47 (app47 v147 v047)) v047))))
(lam47 zero47))
fact47 : ∀{Γ} → Tm47 Γ (arr47 nat47 nat47); fact47
= lam47 (rec47 v047 (lam47 (lam47 (app47 (app47 mul47 (suc47 v147)) v047)))
(suc47 zero47))
{-# OPTIONS --type-in-type #-}
Ty48 : Set
Ty48 =
(Ty48 : Set)
(nat top bot : Ty48)
(arr prod sum : Ty48 → Ty48 → Ty48)
→ Ty48
nat48 : Ty48; nat48 = λ _ nat48 _ _ _ _ _ → nat48
top48 : Ty48; top48 = λ _ _ top48 _ _ _ _ → top48
bot48 : Ty48; bot48 = λ _ _ _ bot48 _ _ _ → bot48
arr48 : Ty48 → Ty48 → Ty48; arr48
= λ A B Ty48 nat48 top48 bot48 arr48 prod sum →
arr48 (A Ty48 nat48 top48 bot48 arr48 prod sum) (B Ty48 nat48 top48 bot48 arr48 prod sum)
prod48 : Ty48 → Ty48 → Ty48; prod48
= λ A B Ty48 nat48 top48 bot48 arr48 prod48 sum →
prod48 (A Ty48 nat48 top48 bot48 arr48 prod48 sum) (B Ty48 nat48 top48 bot48 arr48 prod48 sum)
sum48 : Ty48 → Ty48 → Ty48; sum48
= λ A B Ty48 nat48 top48 bot48 arr48 prod48 sum48 →
sum48 (A Ty48 nat48 top48 bot48 arr48 prod48 sum48) (B Ty48 nat48 top48 bot48 arr48 prod48 sum48)
Con48 : Set; Con48
= (Con48 : Set)
(nil : Con48)
(snoc : Con48 → Ty48 → Con48)
→ Con48
nil48 : Con48; nil48
= λ Con48 nil48 snoc → nil48
snoc48 : Con48 → Ty48 → Con48; snoc48
= λ Γ A Con48 nil48 snoc48 → snoc48 (Γ Con48 nil48 snoc48) A
Var48 : Con48 → Ty48 → Set; Var48
= λ Γ A →
(Var48 : Con48 → Ty48 → Set)
(vz : ∀ Γ A → Var48 (snoc48 Γ A) A)
(vs : ∀ Γ B A → Var48 Γ A → Var48 (snoc48 Γ B) A)
→ Var48 Γ A
vz48 : ∀{Γ A} → Var48 (snoc48 Γ A) A; vz48
= λ Var48 vz48 vs → vz48 _ _
vs48 : ∀{Γ B A} → Var48 Γ A → Var48 (snoc48 Γ B) A; vs48
= λ x Var48 vz48 vs48 → vs48 _ _ _ (x Var48 vz48 vs48)
Tm48 : Con48 → Ty48 → Set; Tm48
= λ Γ A →
(Tm48 : Con48 → Ty48 → Set)
(var : ∀ Γ A → Var48 Γ A → Tm48 Γ A)
(lam : ∀ Γ A B → Tm48 (snoc48 Γ A) B → Tm48 Γ (arr48 A B))
(app : ∀ Γ A B → Tm48 Γ (arr48 A B) → Tm48 Γ A → Tm48 Γ B)
(tt : ∀ Γ → Tm48 Γ top48)
(pair : ∀ Γ A B → Tm48 Γ A → Tm48 Γ B → Tm48 Γ (prod48 A B))
(fst : ∀ Γ A B → Tm48 Γ (prod48 A B) → Tm48 Γ A)
(snd : ∀ Γ A B → Tm48 Γ (prod48 A B) → Tm48 Γ B)
(left : ∀ Γ A B → Tm48 Γ A → Tm48 Γ (sum48 A B))
(right : ∀ Γ A B → Tm48 Γ B → Tm48 Γ (sum48 A B))
(case : ∀ Γ A B C → Tm48 Γ (sum48 A B) → Tm48 Γ (arr48 A C) → Tm48 Γ (arr48 B C) → Tm48 Γ C)
(zero : ∀ Γ → Tm48 Γ nat48)
(suc : ∀ Γ → Tm48 Γ nat48 → Tm48 Γ nat48)
(rec : ∀ Γ A → Tm48 Γ nat48 → Tm48 Γ (arr48 nat48 (arr48 A A)) → Tm48 Γ A → Tm48 Γ A)
→ Tm48 Γ A
var48 : ∀{Γ A} → Var48 Γ A → Tm48 Γ A; var48
= λ x Tm48 var48 lam app tt pair fst snd left right case zero suc rec →
var48 _ _ x
lam48 : ∀{Γ A B} → Tm48 (snoc48 Γ A) B → Tm48 Γ (arr48 A B); lam48
= λ t Tm48 var48 lam48 app tt pair fst snd left right case zero suc rec →
lam48 _ _ _ (t Tm48 var48 lam48 app tt pair fst snd left right case zero suc rec)
app48 : ∀{Γ A B} → Tm48 Γ (arr48 A B) → Tm48 Γ A → Tm48 Γ B; app48
= λ t u Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec →
app48 _ _ _ (t Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec)
(u Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec)
tt48 : ∀{Γ} → Tm48 Γ top48; tt48
= λ Tm48 var48 lam48 app48 tt48 pair fst snd left right case zero suc rec → tt48 _
pair48 : ∀{Γ A B} → Tm48 Γ A → Tm48 Γ B → Tm48 Γ (prod48 A B); pair48
= λ t u Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec →
pair48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec)
(u Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec)
fst48 : ∀{Γ A B} → Tm48 Γ (prod48 A B) → Tm48 Γ A; fst48
= λ t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd left right case zero suc rec →
fst48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd left right case zero suc rec)
snd48 : ∀{Γ A B} → Tm48 Γ (prod48 A B) → Tm48 Γ B; snd48
= λ t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left right case zero suc rec →
snd48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left right case zero suc rec)
left48 : ∀{Γ A B} → Tm48 Γ A → Tm48 Γ (sum48 A B); left48
= λ t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right case zero suc rec →
left48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right case zero suc rec)
right48 : ∀{Γ A B} → Tm48 Γ B → Tm48 Γ (sum48 A B); right48
= λ t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case zero suc rec →
right48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case zero suc rec)
case48 : ∀{Γ A B C} → Tm48 Γ (sum48 A B) → Tm48 Γ (arr48 A C) → Tm48 Γ (arr48 B C) → Tm48 Γ C; case48
= λ t u v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec →
case48 _ _ _ _
(t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
(u Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
(v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
zero48 : ∀{Γ} → Tm48 Γ nat48; zero48
= λ Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc rec → zero48 _
suc48 : ∀{Γ} → Tm48 Γ nat48 → Tm48 Γ nat48; suc48
= λ t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec →
suc48 _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec)
rec48 : ∀{Γ A} → Tm48 Γ nat48 → Tm48 Γ (arr48 nat48 (arr48 A A)) → Tm48 Γ A → Tm48 Γ A; rec48
= λ t u v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48 →
rec48 _ _
(t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
(u Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
(v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
v048 : ∀{Γ A} → Tm48 (snoc48 Γ A) A; v048
= var48 vz48
v148 : ∀{Γ A B} → Tm48 (snoc48 (snoc48 Γ A) B) A; v148
= var48 (vs48 vz48)
v248 : ∀{Γ A B C} → Tm48 (snoc48 (snoc48 (snoc48 Γ A) B) C) A; v248
= var48 (vs48 (vs48 vz48))
v348 : ∀{Γ A B C D} → Tm48 (snoc48 (snoc48 (snoc48 (snoc48 Γ A) B) C) D) A; v348
= var48 (vs48 (vs48 (vs48 vz48)))
tbool48 : Ty48; tbool48
= sum48 top48 top48
true48 : ∀{Γ} → Tm48 Γ tbool48; true48
= left48 tt48
tfalse48 : ∀{Γ} → Tm48 Γ tbool48; tfalse48
= right48 tt48
ifthenelse48 : ∀{Γ A} → Tm48 Γ (arr48 tbool48 (arr48 A (arr48 A A))); ifthenelse48
= lam48 (lam48 (lam48 (case48 v248 (lam48 v248) (lam48 v148))))
times448 : ∀{Γ A} → Tm48 Γ (arr48 (arr48 A A) (arr48 A A)); times448
= lam48 (lam48 (app48 v148 (app48 v148 (app48 v148 (app48 v148 v048)))))
add48 : ∀{Γ} → Tm48 Γ (arr48 nat48 (arr48 nat48 nat48)); add48
= lam48 (rec48 v048
(lam48 (lam48 (lam48 (suc48 (app48 v148 v048)))))
(lam48 v048))
mul48 : ∀{Γ} → Tm48 Γ (arr48 nat48 (arr48 nat48 nat48)); mul48
= lam48 (rec48 v048
(lam48 (lam48 (lam48 (app48 (app48 add48 (app48 v148 v048)) v048))))
(lam48 zero48))
fact48 : ∀{Γ} → Tm48 Γ (arr48 nat48 nat48); fact48
= lam48 (rec48 v048 (lam48 (lam48 (app48 (app48 mul48 (suc48 v148)) v048)))
(suc48 zero48))
{-# OPTIONS --type-in-type #-}
Ty49 : Set
Ty49 =
(Ty49 : Set)
(nat top bot : Ty49)
(arr prod sum : Ty49 → Ty49 → Ty49)
→ Ty49
nat49 : Ty49; nat49 = λ _ nat49 _ _ _ _ _ → nat49
top49 : Ty49; top49 = λ _ _ top49 _ _ _ _ → top49
bot49 : Ty49; bot49 = λ _ _ _ bot49 _ _ _ → bot49
arr49 : Ty49 → Ty49 → Ty49; arr49
= λ A B Ty49 nat49 top49 bot49 arr49 prod sum →
arr49 (A Ty49 nat49 top49 bot49 arr49 prod sum) (B Ty49 nat49 top49 bot49 arr49 prod sum)
prod49 : Ty49 → Ty49 → Ty49; prod49
= λ A B Ty49 nat49 top49 bot49 arr49 prod49 sum →
prod49 (A Ty49 nat49 top49 bot49 arr49 prod49 sum) (B Ty49 nat49 top49 bot49 arr49 prod49 sum)
sum49 : Ty49 → Ty49 → Ty49; sum49
= λ A B Ty49 nat49 top49 bot49 arr49 prod49 sum49 →
sum49 (A Ty49 nat49 top49 bot49 arr49 prod49 sum49) (B Ty49 nat49 top49 bot49 arr49 prod49 sum49)
Con49 : Set; Con49
= (Con49 : Set)
(nil : Con49)
(snoc : Con49 → Ty49 → Con49)
→ Con49
nil49 : Con49; nil49
= λ Con49 nil49 snoc → nil49
snoc49 : Con49 → Ty49 → Con49; snoc49
= λ Γ A Con49 nil49 snoc49 → snoc49 (Γ Con49 nil49 snoc49) A
Var49 : Con49 → Ty49 → Set; Var49
= λ Γ A →
(Var49 : Con49 → Ty49 → Set)
(vz : ∀ Γ A → Var49 (snoc49 Γ A) A)
(vs : ∀ Γ B A → Var49 Γ A → Var49 (snoc49 Γ B) A)
→ Var49 Γ A
vz49 : ∀{Γ A} → Var49 (snoc49 Γ A) A; vz49
= λ Var49 vz49 vs → vz49 _ _
vs49 : ∀{Γ B A} → Var49 Γ A → Var49 (snoc49 Γ B) A; vs49
= λ x Var49 vz49 vs49 → vs49 _ _ _ (x Var49 vz49 vs49)
Tm49 : Con49 → Ty49 → Set; Tm49
= λ Γ A →
(Tm49 : Con49 → Ty49 → Set)
(var : ∀ Γ A → Var49 Γ A → Tm49 Γ A)
(lam : ∀ Γ A B → Tm49 (snoc49 Γ A) B → Tm49 Γ (arr49 A B))
(app : ∀ Γ A B → Tm49 Γ (arr49 A B) → Tm49 Γ A → Tm49 Γ B)
(tt : ∀ Γ → Tm49 Γ top49)
(pair : ∀ Γ A B → Tm49 Γ A → Tm49 Γ B → Tm49 Γ (prod49 A B))
(fst : ∀ Γ A B → Tm49 Γ (prod49 A B) → Tm49 Γ A)
(snd : ∀ Γ A B → Tm49 Γ (prod49 A B) → Tm49 Γ B)
(left : ∀ Γ A B → Tm49 Γ A → Tm49 Γ (sum49 A B))
(right : ∀ Γ A B → Tm49 Γ B → Tm49 Γ (sum49 A B))
(case : ∀ Γ A B C → Tm49 Γ (sum49 A B) → Tm49 Γ (arr49 A C) → Tm49 Γ (arr49 B C) → Tm49 Γ C)
(zero : ∀ Γ → Tm49 Γ nat49)
(suc : ∀ Γ → Tm49 Γ nat49 → Tm49 Γ nat49)
(rec : ∀ Γ A → Tm49 Γ nat49 → Tm49 Γ (arr49 nat49 (arr49 A A)) → Tm49 Γ A → Tm49 Γ A)
→ Tm49 Γ A
var49 : ∀{Γ A} → Var49 Γ A → Tm49 Γ A; var49
= λ x Tm49 var49 lam app tt pair fst snd left right case zero suc rec →
var49 _ _ x
lam49 : ∀{Γ A B} → Tm49 (snoc49 Γ A) B → Tm49 Γ (arr49 A B); lam49
= λ t Tm49 var49 lam49 app tt pair fst snd left right case zero suc rec →
lam49 _ _ _ (t Tm49 var49 lam49 app tt pair fst snd left right case zero suc rec)
app49 : ∀{Γ A B} → Tm49 Γ (arr49 A B) → Tm49 Γ A → Tm49 Γ B; app49
= λ t u Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec →
app49 _ _ _ (t Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec)
(u Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec)
tt49 : ∀{Γ} → Tm49 Γ top49; tt49
= λ Tm49 var49 lam49 app49 tt49 pair fst snd left right case zero suc rec → tt49 _
pair49 : ∀{Γ A B} → Tm49 Γ A → Tm49 Γ B → Tm49 Γ (prod49 A B); pair49
= λ t u Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec →
pair49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec)
(u Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec)
fst49 : ∀{Γ A B} → Tm49 Γ (prod49 A B) → Tm49 Γ A; fst49
= λ t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd left right case zero suc rec →
fst49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd left right case zero suc rec)
snd49 : ∀{Γ A B} → Tm49 Γ (prod49 A B) → Tm49 Γ B; snd49
= λ t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left right case zero suc rec →
snd49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left right case zero suc rec)
left49 : ∀{Γ A B} → Tm49 Γ A → Tm49 Γ (sum49 A B); left49
= λ t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right case zero suc rec →
left49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right case zero suc rec)
right49 : ∀{Γ A B} → Tm49 Γ B → Tm49 Γ (sum49 A B); right49
= λ t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case zero suc rec →
right49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case zero suc rec)
case49 : ∀{Γ A B C} → Tm49 Γ (sum49 A B) → Tm49 Γ (arr49 A C) → Tm49 Γ (arr49 B C) → Tm49 Γ C; case49
= λ t u v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec →
case49 _ _ _ _
(t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
(u Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
(v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
zero49 : ∀{Γ} → Tm49 Γ nat49; zero49
= λ Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc rec → zero49 _
suc49 : ∀{Γ} → Tm49 Γ nat49 → Tm49 Γ nat49; suc49
= λ t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec →
suc49 _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec)
rec49 : ∀{Γ A} → Tm49 Γ nat49 → Tm49 Γ (arr49 nat49 (arr49 A A)) → Tm49 Γ A → Tm49 Γ A; rec49
= λ t u v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49 →
rec49 _ _
(t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
(u Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
(v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
v049 : ∀{Γ A} → Tm49 (snoc49 Γ A) A; v049
= var49 vz49
v149 : ∀{Γ A B} → Tm49 (snoc49 (snoc49 Γ A) B) A; v149
= var49 (vs49 vz49)
v249 : ∀{Γ A B C} → Tm49 (snoc49 (snoc49 (snoc49 Γ A) B) C) A; v249
= var49 (vs49 (vs49 vz49))
v349 : ∀{Γ A B C D} → Tm49 (snoc49 (snoc49 (snoc49 (snoc49 Γ A) B) C) D) A; v349
= var49 (vs49 (vs49 (vs49 vz49)))
tbool49 : Ty49; tbool49
= sum49 top49 top49
true49 : ∀{Γ} → Tm49 Γ tbool49; true49
= left49 tt49
tfalse49 : ∀{Γ} → Tm49 Γ tbool49; tfalse49
= right49 tt49
ifthenelse49 : ∀{Γ A} → Tm49 Γ (arr49 tbool49 (arr49 A (arr49 A A))); ifthenelse49
= lam49 (lam49 (lam49 (case49 v249 (lam49 v249) (lam49 v149))))
times449 : ∀{Γ A} → Tm49 Γ (arr49 (arr49 A A) (arr49 A A)); times449
= lam49 (lam49 (app49 v149 (app49 v149 (app49 v149 (app49 v149 v049)))))
add49 : ∀{Γ} → Tm49 Γ (arr49 nat49 (arr49 nat49 nat49)); add49
= lam49 (rec49 v049
(lam49 (lam49 (lam49 (suc49 (app49 v149 v049)))))
(lam49 v049))
mul49 : ∀{Γ} → Tm49 Γ (arr49 nat49 (arr49 nat49 nat49)); mul49
= lam49 (rec49 v049
(lam49 (lam49 (lam49 (app49 (app49 add49 (app49 v149 v049)) v049))))
(lam49 zero49))
fact49 : ∀{Γ} → Tm49 Γ (arr49 nat49 nat49); fact49
= lam49 (rec49 v049 (lam49 (lam49 (app49 (app49 mul49 (suc49 v149)) v049)))
(suc49 zero49))
{-# OPTIONS --type-in-type #-}
Ty50 : Set
Ty50 =
(Ty50 : Set)
(nat top bot : Ty50)
(arr prod sum : Ty50 → Ty50 → Ty50)
→ Ty50
nat50 : Ty50; nat50 = λ _ nat50 _ _ _ _ _ → nat50
top50 : Ty50; top50 = λ _ _ top50 _ _ _ _ → top50
bot50 : Ty50; bot50 = λ _ _ _ bot50 _ _ _ → bot50
arr50 : Ty50 → Ty50 → Ty50; arr50
= λ A B Ty50 nat50 top50 bot50 arr50 prod sum →
arr50 (A Ty50 nat50 top50 bot50 arr50 prod sum) (B Ty50 nat50 top50 bot50 arr50 prod sum)
prod50 : Ty50 → Ty50 → Ty50; prod50
= λ A B Ty50 nat50 top50 bot50 arr50 prod50 sum →
prod50 (A Ty50 nat50 top50 bot50 arr50 prod50 sum) (B Ty50 nat50 top50 bot50 arr50 prod50 sum)
sum50 : Ty50 → Ty50 → Ty50; sum50
= λ A B Ty50 nat50 top50 bot50 arr50 prod50 sum50 →
sum50 (A Ty50 nat50 top50 bot50 arr50 prod50 sum50) (B Ty50 nat50 top50 bot50 arr50 prod50 sum50)
Con50 : Set; Con50
= (Con50 : Set)
(nil : Con50)
(snoc : Con50 → Ty50 → Con50)
→ Con50
nil50 : Con50; nil50
= λ Con50 nil50 snoc → nil50
snoc50 : Con50 → Ty50 → Con50; snoc50
= λ Γ A Con50 nil50 snoc50 → snoc50 (Γ Con50 nil50 snoc50) A
Var50 : Con50 → Ty50 → Set; Var50
= λ Γ A →
(Var50 : Con50 → Ty50 → Set)
(vz : ∀ Γ A → Var50 (snoc50 Γ A) A)
(vs : ∀ Γ B A → Var50 Γ A → Var50 (snoc50 Γ B) A)
→ Var50 Γ A
vz50 : ∀{Γ A} → Var50 (snoc50 Γ A) A; vz50
= λ Var50 vz50 vs → vz50 _ _
vs50 : ∀{Γ B A} → Var50 Γ A → Var50 (snoc50 Γ B) A; vs50
= λ x Var50 vz50 vs50 → vs50 _ _ _ (x Var50 vz50 vs50)
Tm50 : Con50 → Ty50 → Set; Tm50
= λ Γ A →
(Tm50 : Con50 → Ty50 → Set)
(var : ∀ Γ A → Var50 Γ A → Tm50 Γ A)
(lam : ∀ Γ A B → Tm50 (snoc50 Γ A) B → Tm50 Γ (arr50 A B))
(app : ∀ Γ A B → Tm50 Γ (arr50 A B) → Tm50 Γ A → Tm50 Γ B)
(tt : ∀ Γ → Tm50 Γ top50)
(pair : ∀ Γ A B → Tm50 Γ A → Tm50 Γ B → Tm50 Γ (prod50 A B))
(fst : ∀ Γ A B → Tm50 Γ (prod50 A B) → Tm50 Γ A)
(snd : ∀ Γ A B → Tm50 Γ (prod50 A B) → Tm50 Γ B)
(left : ∀ Γ A B → Tm50 Γ A → Tm50 Γ (sum50 A B))
(right : ∀ Γ A B → Tm50 Γ B → Tm50 Γ (sum50 A B))
(case : ∀ Γ A B C → Tm50 Γ (sum50 A B) → Tm50 Γ (arr50 A C) → Tm50 Γ (arr50 B C) → Tm50 Γ C)
(zero : ∀ Γ → Tm50 Γ nat50)
(suc : ∀ Γ → Tm50 Γ nat50 → Tm50 Γ nat50)
(rec : ∀ Γ A → Tm50 Γ nat50 → Tm50 Γ (arr50 nat50 (arr50 A A)) → Tm50 Γ A → Tm50 Γ A)
→ Tm50 Γ A
var50 : ∀{Γ A} → Var50 Γ A → Tm50 Γ A; var50
= λ x Tm50 var50 lam app tt pair fst snd left right case zero suc rec →
var50 _ _ x
lam50 : ∀{Γ A B} → Tm50 (snoc50 Γ A) B → Tm50 Γ (arr50 A B); lam50
= λ t Tm50 var50 lam50 app tt pair fst snd left right case zero suc rec →
lam50 _ _ _ (t Tm50 var50 lam50 app tt pair fst snd left right case zero suc rec)
app50 : ∀{Γ A B} → Tm50 Γ (arr50 A B) → Tm50 Γ A → Tm50 Γ B; app50
= λ t u Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec →
app50 _ _ _ (t Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec)
(u Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec)
tt50 : ∀{Γ} → Tm50 Γ top50; tt50
= λ Tm50 var50 lam50 app50 tt50 pair fst snd left right case zero suc rec → tt50 _
pair50 : ∀{Γ A B} → Tm50 Γ A → Tm50 Γ B → Tm50 Γ (prod50 A B); pair50
= λ t u Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec →
pair50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec)
(u Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec)
fst50 : ∀{Γ A B} → Tm50 Γ (prod50 A B) → Tm50 Γ A; fst50
= λ t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd left right case zero suc rec →
fst50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd left right case zero suc rec)
snd50 : ∀{Γ A B} → Tm50 Γ (prod50 A B) → Tm50 Γ B; snd50
= λ t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left right case zero suc rec →
snd50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left right case zero suc rec)
left50 : ∀{Γ A B} → Tm50 Γ A → Tm50 Γ (sum50 A B); left50
= λ t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right case zero suc rec →
left50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right case zero suc rec)
right50 : ∀{Γ A B} → Tm50 Γ B → Tm50 Γ (sum50 A B); right50
= λ t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case zero suc rec →
right50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case zero suc rec)
case50 : ∀{Γ A B C} → Tm50 Γ (sum50 A B) → Tm50 Γ (arr50 A C) → Tm50 Γ (arr50 B C) → Tm50 Γ C; case50
= λ t u v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec →
case50 _ _ _ _
(t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
(u Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
(v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
zero50 : ∀{Γ} → Tm50 Γ nat50; zero50
= λ Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc rec → zero50 _
suc50 : ∀{Γ} → Tm50 Γ nat50 → Tm50 Γ nat50; suc50
= λ t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec →
suc50 _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec)
rec50 : ∀{Γ A} → Tm50 Γ nat50 → Tm50 Γ (arr50 nat50 (arr50 A A)) → Tm50 Γ A → Tm50 Γ A; rec50
= λ t u v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50 →
rec50 _ _
(t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
(u Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
(v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
v050 : ∀{Γ A} → Tm50 (snoc50 Γ A) A; v050
= var50 vz50
v150 : ∀{Γ A B} → Tm50 (snoc50 (snoc50 Γ A) B) A; v150
= var50 (vs50 vz50)
v250 : ∀{Γ A B C} → Tm50 (snoc50 (snoc50 (snoc50 Γ A) B) C) A; v250
= var50 (vs50 (vs50 vz50))
v350 : ∀{Γ A B C D} → Tm50 (snoc50 (snoc50 (snoc50 (snoc50 Γ A) B) C) D) A; v350
= var50 (vs50 (vs50 (vs50 vz50)))
tbool50 : Ty50; tbool50
= sum50 top50 top50
true50 : ∀{Γ} → Tm50 Γ tbool50; true50
= left50 tt50
tfalse50 : ∀{Γ} → Tm50 Γ tbool50; tfalse50
= right50 tt50
ifthenelse50 : ∀{Γ A} → Tm50 Γ (arr50 tbool50 (arr50 A (arr50 A A))); ifthenelse50
= lam50 (lam50 (lam50 (case50 v250 (lam50 v250) (lam50 v150))))
times450 : ∀{Γ A} → Tm50 Γ (arr50 (arr50 A A) (arr50 A A)); times450
= lam50 (lam50 (app50 v150 (app50 v150 (app50 v150 (app50 v150 v050)))))
add50 : ∀{Γ} → Tm50 Γ (arr50 nat50 (arr50 nat50 nat50)); add50
= lam50 (rec50 v050
(lam50 (lam50 (lam50 (suc50 (app50 v150 v050)))))
(lam50 v050))
mul50 : ∀{Γ} → Tm50 Γ (arr50 nat50 (arr50 nat50 nat50)); mul50
= lam50 (rec50 v050
(lam50 (lam50 (lam50 (app50 (app50 add50 (app50 v150 v050)) v050))))
(lam50 zero50))
fact50 : ∀{Γ} → Tm50 Γ (arr50 nat50 nat50); fact50
= lam50 (rec50 v050 (lam50 (lam50 (app50 (app50 mul50 (suc50 v150)) v050)))
(suc50 zero50))
{-# OPTIONS --type-in-type #-}
Ty51 : Set
Ty51 =
(Ty51 : Set)
(nat top bot : Ty51)
(arr prod sum : Ty51 → Ty51 → Ty51)
→ Ty51
nat51 : Ty51; nat51 = λ _ nat51 _ _ _ _ _ → nat51
top51 : Ty51; top51 = λ _ _ top51 _ _ _ _ → top51
bot51 : Ty51; bot51 = λ _ _ _ bot51 _ _ _ → bot51
arr51 : Ty51 → Ty51 → Ty51; arr51
= λ A B Ty51 nat51 top51 bot51 arr51 prod sum →
arr51 (A Ty51 nat51 top51 bot51 arr51 prod sum) (B Ty51 nat51 top51 bot51 arr51 prod sum)
prod51 : Ty51 → Ty51 → Ty51; prod51
= λ A B Ty51 nat51 top51 bot51 arr51 prod51 sum →
prod51 (A Ty51 nat51 top51 bot51 arr51 prod51 sum) (B Ty51 nat51 top51 bot51 arr51 prod51 sum)
sum51 : Ty51 → Ty51 → Ty51; sum51
= λ A B Ty51 nat51 top51 bot51 arr51 prod51 sum51 →
sum51 (A Ty51 nat51 top51 bot51 arr51 prod51 sum51) (B Ty51 nat51 top51 bot51 arr51 prod51 sum51)
Con51 : Set; Con51
= (Con51 : Set)
(nil : Con51)
(snoc : Con51 → Ty51 → Con51)
→ Con51
nil51 : Con51; nil51
= λ Con51 nil51 snoc → nil51
snoc51 : Con51 → Ty51 → Con51; snoc51
= λ Γ A Con51 nil51 snoc51 → snoc51 (Γ Con51 nil51 snoc51) A
Var51 : Con51 → Ty51 → Set; Var51
= λ Γ A →
(Var51 : Con51 → Ty51 → Set)
(vz : ∀ Γ A → Var51 (snoc51 Γ A) A)
(vs : ∀ Γ B A → Var51 Γ A → Var51 (snoc51 Γ B) A)
→ Var51 Γ A
vz51 : ∀{Γ A} → Var51 (snoc51 Γ A) A; vz51
= λ Var51 vz51 vs → vz51 _ _
vs51 : ∀{Γ B A} → Var51 Γ A → Var51 (snoc51 Γ B) A; vs51
= λ x Var51 vz51 vs51 → vs51 _ _ _ (x Var51 vz51 vs51)
Tm51 : Con51 → Ty51 → Set; Tm51
= λ Γ A →
(Tm51 : Con51 → Ty51 → Set)
(var : ∀ Γ A → Var51 Γ A → Tm51 Γ A)
(lam : ∀ Γ A B → Tm51 (snoc51 Γ A) B → Tm51 Γ (arr51 A B))
(app : ∀ Γ A B → Tm51 Γ (arr51 A B) → Tm51 Γ A → Tm51 Γ B)
(tt : ∀ Γ → Tm51 Γ top51)
(pair : ∀ Γ A B → Tm51 Γ A → Tm51 Γ B → Tm51 Γ (prod51 A B))
(fst : ∀ Γ A B → Tm51 Γ (prod51 A B) → Tm51 Γ A)
(snd : ∀ Γ A B → Tm51 Γ (prod51 A B) → Tm51 Γ B)
(left : ∀ Γ A B → Tm51 Γ A → Tm51 Γ (sum51 A B))
(right : ∀ Γ A B → Tm51 Γ B → Tm51 Γ (sum51 A B))
(case : ∀ Γ A B C → Tm51 Γ (sum51 A B) → Tm51 Γ (arr51 A C) → Tm51 Γ (arr51 B C) → Tm51 Γ C)
(zero : ∀ Γ → Tm51 Γ nat51)
(suc : ∀ Γ → Tm51 Γ nat51 → Tm51 Γ nat51)
(rec : ∀ Γ A → Tm51 Γ nat51 → Tm51 Γ (arr51 nat51 (arr51 A A)) → Tm51 Γ A → Tm51 Γ A)
→ Tm51 Γ A
var51 : ∀{Γ A} → Var51 Γ A → Tm51 Γ A; var51
= λ x Tm51 var51 lam app tt pair fst snd left right case zero suc rec →
var51 _ _ x
lam51 : ∀{Γ A B} → Tm51 (snoc51 Γ A) B → Tm51 Γ (arr51 A B); lam51
= λ t Tm51 var51 lam51 app tt pair fst snd left right case zero suc rec →
lam51 _ _ _ (t Tm51 var51 lam51 app tt pair fst snd left right case zero suc rec)
app51 : ∀{Γ A B} → Tm51 Γ (arr51 A B) → Tm51 Γ A → Tm51 Γ B; app51
= λ t u Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec →
app51 _ _ _ (t Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec)
(u Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec)
tt51 : ∀{Γ} → Tm51 Γ top51; tt51
= λ Tm51 var51 lam51 app51 tt51 pair fst snd left right case zero suc rec → tt51 _
pair51 : ∀{Γ A B} → Tm51 Γ A → Tm51 Γ B → Tm51 Γ (prod51 A B); pair51
= λ t u Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec →
pair51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec)
(u Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec)
fst51 : ∀{Γ A B} → Tm51 Γ (prod51 A B) → Tm51 Γ A; fst51
= λ t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd left right case zero suc rec →
fst51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd left right case zero suc rec)
snd51 : ∀{Γ A B} → Tm51 Γ (prod51 A B) → Tm51 Γ B; snd51
= λ t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left right case zero suc rec →
snd51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left right case zero suc rec)
left51 : ∀{Γ A B} → Tm51 Γ A → Tm51 Γ (sum51 A B); left51
= λ t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right case zero suc rec →
left51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right case zero suc rec)
right51 : ∀{Γ A B} → Tm51 Γ B → Tm51 Γ (sum51 A B); right51
= λ t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case zero suc rec →
right51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case zero suc rec)
case51 : ∀{Γ A B C} → Tm51 Γ (sum51 A B) → Tm51 Γ (arr51 A C) → Tm51 Γ (arr51 B C) → Tm51 Γ C; case51
= λ t u v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec →
case51 _ _ _ _
(t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
(u Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
(v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
zero51 : ∀{Γ} → Tm51 Γ nat51; zero51
= λ Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc rec → zero51 _
suc51 : ∀{Γ} → Tm51 Γ nat51 → Tm51 Γ nat51; suc51
= λ t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec →
suc51 _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec)
rec51 : ∀{Γ A} → Tm51 Γ nat51 → Tm51 Γ (arr51 nat51 (arr51 A A)) → Tm51 Γ A → Tm51 Γ A; rec51
= λ t u v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51 →
rec51 _ _
(t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
(u Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
(v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
v051 : ∀{Γ A} → Tm51 (snoc51 Γ A) A; v051
= var51 vz51
v151 : ∀{Γ A B} → Tm51 (snoc51 (snoc51 Γ A) B) A; v151
= var51 (vs51 vz51)
v251 : ∀{Γ A B C} → Tm51 (snoc51 (snoc51 (snoc51 Γ A) B) C) A; v251
= var51 (vs51 (vs51 vz51))
v351 : ∀{Γ A B C D} → Tm51 (snoc51 (snoc51 (snoc51 (snoc51 Γ A) B) C) D) A; v351
= var51 (vs51 (vs51 (vs51 vz51)))
tbool51 : Ty51; tbool51
= sum51 top51 top51
true51 : ∀{Γ} → Tm51 Γ tbool51; true51
= left51 tt51
tfalse51 : ∀{Γ} → Tm51 Γ tbool51; tfalse51
= right51 tt51
ifthenelse51 : ∀{Γ A} → Tm51 Γ (arr51 tbool51 (arr51 A (arr51 A A))); ifthenelse51
= lam51 (lam51 (lam51 (case51 v251 (lam51 v251) (lam51 v151))))
times451 : ∀{Γ A} → Tm51 Γ (arr51 (arr51 A A) (arr51 A A)); times451
= lam51 (lam51 (app51 v151 (app51 v151 (app51 v151 (app51 v151 v051)))))
add51 : ∀{Γ} → Tm51 Γ (arr51 nat51 (arr51 nat51 nat51)); add51
= lam51 (rec51 v051
(lam51 (lam51 (lam51 (suc51 (app51 v151 v051)))))
(lam51 v051))
mul51 : ∀{Γ} → Tm51 Γ (arr51 nat51 (arr51 nat51 nat51)); mul51
= lam51 (rec51 v051
(lam51 (lam51 (lam51 (app51 (app51 add51 (app51 v151 v051)) v051))))
(lam51 zero51))
fact51 : ∀{Γ} → Tm51 Γ (arr51 nat51 nat51); fact51
= lam51 (rec51 v051 (lam51 (lam51 (app51 (app51 mul51 (suc51 v151)) v051)))
(suc51 zero51))
{-# OPTIONS --type-in-type #-}
Ty52 : Set
Ty52 =
(Ty52 : Set)
(nat top bot : Ty52)
(arr prod sum : Ty52 → Ty52 → Ty52)
→ Ty52
nat52 : Ty52; nat52 = λ _ nat52 _ _ _ _ _ → nat52
top52 : Ty52; top52 = λ _ _ top52 _ _ _ _ → top52
bot52 : Ty52; bot52 = λ _ _ _ bot52 _ _ _ → bot52
arr52 : Ty52 → Ty52 → Ty52; arr52
= λ A B Ty52 nat52 top52 bot52 arr52 prod sum →
arr52 (A Ty52 nat52 top52 bot52 arr52 prod sum) (B Ty52 nat52 top52 bot52 arr52 prod sum)
prod52 : Ty52 → Ty52 → Ty52; prod52
= λ A B Ty52 nat52 top52 bot52 arr52 prod52 sum →
prod52 (A Ty52 nat52 top52 bot52 arr52 prod52 sum) (B Ty52 nat52 top52 bot52 arr52 prod52 sum)
sum52 : Ty52 → Ty52 → Ty52; sum52
= λ A B Ty52 nat52 top52 bot52 arr52 prod52 sum52 →
sum52 (A Ty52 nat52 top52 bot52 arr52 prod52 sum52) (B Ty52 nat52 top52 bot52 arr52 prod52 sum52)
Con52 : Set; Con52
= (Con52 : Set)
(nil : Con52)
(snoc : Con52 → Ty52 → Con52)
→ Con52
nil52 : Con52; nil52
= λ Con52 nil52 snoc → nil52
snoc52 : Con52 → Ty52 → Con52; snoc52
= λ Γ A Con52 nil52 snoc52 → snoc52 (Γ Con52 nil52 snoc52) A
Var52 : Con52 → Ty52 → Set; Var52
= λ Γ A →
(Var52 : Con52 → Ty52 → Set)
(vz : ∀ Γ A → Var52 (snoc52 Γ A) A)
(vs : ∀ Γ B A → Var52 Γ A → Var52 (snoc52 Γ B) A)
→ Var52 Γ A
vz52 : ∀{Γ A} → Var52 (snoc52 Γ A) A; vz52
= λ Var52 vz52 vs → vz52 _ _
vs52 : ∀{Γ B A} → Var52 Γ A → Var52 (snoc52 Γ B) A; vs52
= λ x Var52 vz52 vs52 → vs52 _ _ _ (x Var52 vz52 vs52)
Tm52 : Con52 → Ty52 → Set; Tm52
= λ Γ A →
(Tm52 : Con52 → Ty52 → Set)
(var : ∀ Γ A → Var52 Γ A → Tm52 Γ A)
(lam : ∀ Γ A B → Tm52 (snoc52 Γ A) B → Tm52 Γ (arr52 A B))
(app : ∀ Γ A B → Tm52 Γ (arr52 A B) → Tm52 Γ A → Tm52 Γ B)
(tt : ∀ Γ → Tm52 Γ top52)
(pair : ∀ Γ A B → Tm52 Γ A → Tm52 Γ B → Tm52 Γ (prod52 A B))
(fst : ∀ Γ A B → Tm52 Γ (prod52 A B) → Tm52 Γ A)
(snd : ∀ Γ A B → Tm52 Γ (prod52 A B) → Tm52 Γ B)
(left : ∀ Γ A B → Tm52 Γ A → Tm52 Γ (sum52 A B))
(right : ∀ Γ A B → Tm52 Γ B → Tm52 Γ (sum52 A B))
(case : ∀ Γ A B C → Tm52 Γ (sum52 A B) → Tm52 Γ (arr52 A C) → Tm52 Γ (arr52 B C) → Tm52 Γ C)
(zero : ∀ Γ → Tm52 Γ nat52)
(suc : ∀ Γ → Tm52 Γ nat52 → Tm52 Γ nat52)
(rec : ∀ Γ A → Tm52 Γ nat52 → Tm52 Γ (arr52 nat52 (arr52 A A)) → Tm52 Γ A → Tm52 Γ A)
→ Tm52 Γ A
var52 : ∀{Γ A} → Var52 Γ A → Tm52 Γ A; var52
= λ x Tm52 var52 lam app tt pair fst snd left right case zero suc rec →
var52 _ _ x
lam52 : ∀{Γ A B} → Tm52 (snoc52 Γ A) B → Tm52 Γ (arr52 A B); lam52
= λ t Tm52 var52 lam52 app tt pair fst snd left right case zero suc rec →
lam52 _ _ _ (t Tm52 var52 lam52 app tt pair fst snd left right case zero suc rec)
app52 : ∀{Γ A B} → Tm52 Γ (arr52 A B) → Tm52 Γ A → Tm52 Γ B; app52
= λ t u Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec →
app52 _ _ _ (t Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec)
(u Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec)
tt52 : ∀{Γ} → Tm52 Γ top52; tt52
= λ Tm52 var52 lam52 app52 tt52 pair fst snd left right case zero suc rec → tt52 _
pair52 : ∀{Γ A B} → Tm52 Γ A → Tm52 Γ B → Tm52 Γ (prod52 A B); pair52
= λ t u Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec →
pair52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec)
(u Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec)
fst52 : ∀{Γ A B} → Tm52 Γ (prod52 A B) → Tm52 Γ A; fst52
= λ t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd left right case zero suc rec →
fst52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd left right case zero suc rec)
snd52 : ∀{Γ A B} → Tm52 Γ (prod52 A B) → Tm52 Γ B; snd52
= λ t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left right case zero suc rec →
snd52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left right case zero suc rec)
left52 : ∀{Γ A B} → Tm52 Γ A → Tm52 Γ (sum52 A B); left52
= λ t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right case zero suc rec →
left52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right case zero suc rec)
right52 : ∀{Γ A B} → Tm52 Γ B → Tm52 Γ (sum52 A B); right52
= λ t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case zero suc rec →
right52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case zero suc rec)
case52 : ∀{Γ A B C} → Tm52 Γ (sum52 A B) → Tm52 Γ (arr52 A C) → Tm52 Γ (arr52 B C) → Tm52 Γ C; case52
= λ t u v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec →
case52 _ _ _ _
(t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
(u Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
(v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
zero52 : ∀{Γ} → Tm52 Γ nat52; zero52
= λ Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc rec → zero52 _
suc52 : ∀{Γ} → Tm52 Γ nat52 → Tm52 Γ nat52; suc52
= λ t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec →
suc52 _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec)
rec52 : ∀{Γ A} → Tm52 Γ nat52 → Tm52 Γ (arr52 nat52 (arr52 A A)) → Tm52 Γ A → Tm52 Γ A; rec52
= λ t u v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52 →
rec52 _ _
(t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
(u Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
(v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
v052 : ∀{Γ A} → Tm52 (snoc52 Γ A) A; v052
= var52 vz52
v152 : ∀{Γ A B} → Tm52 (snoc52 (snoc52 Γ A) B) A; v152
= var52 (vs52 vz52)
v252 : ∀{Γ A B C} → Tm52 (snoc52 (snoc52 (snoc52 Γ A) B) C) A; v252
= var52 (vs52 (vs52 vz52))
v352 : ∀{Γ A B C D} → Tm52 (snoc52 (snoc52 (snoc52 (snoc52 Γ A) B) C) D) A; v352
= var52 (vs52 (vs52 (vs52 vz52)))
tbool52 : Ty52; tbool52
= sum52 top52 top52
true52 : ∀{Γ} → Tm52 Γ tbool52; true52
= left52 tt52
tfalse52 : ∀{Γ} → Tm52 Γ tbool52; tfalse52
= right52 tt52
ifthenelse52 : ∀{Γ A} → Tm52 Γ (arr52 tbool52 (arr52 A (arr52 A A))); ifthenelse52
= lam52 (lam52 (lam52 (case52 v252 (lam52 v252) (lam52 v152))))
times452 : ∀{Γ A} → Tm52 Γ (arr52 (arr52 A A) (arr52 A A)); times452
= lam52 (lam52 (app52 v152 (app52 v152 (app52 v152 (app52 v152 v052)))))
add52 : ∀{Γ} → Tm52 Γ (arr52 nat52 (arr52 nat52 nat52)); add52
= lam52 (rec52 v052
(lam52 (lam52 (lam52 (suc52 (app52 v152 v052)))))
(lam52 v052))
mul52 : ∀{Γ} → Tm52 Γ (arr52 nat52 (arr52 nat52 nat52)); mul52
= lam52 (rec52 v052
(lam52 (lam52 (lam52 (app52 (app52 add52 (app52 v152 v052)) v052))))
(lam52 zero52))
fact52 : ∀{Γ} → Tm52 Γ (arr52 nat52 nat52); fact52
= lam52 (rec52 v052 (lam52 (lam52 (app52 (app52 mul52 (suc52 v152)) v052)))
(suc52 zero52))
{-# OPTIONS --type-in-type #-}
Ty53 : Set
Ty53 =
(Ty53 : Set)
(nat top bot : Ty53)
(arr prod sum : Ty53 → Ty53 → Ty53)
→ Ty53
nat53 : Ty53; nat53 = λ _ nat53 _ _ _ _ _ → nat53
top53 : Ty53; top53 = λ _ _ top53 _ _ _ _ → top53
bot53 : Ty53; bot53 = λ _ _ _ bot53 _ _ _ → bot53
arr53 : Ty53 → Ty53 → Ty53; arr53
= λ A B Ty53 nat53 top53 bot53 arr53 prod sum →
arr53 (A Ty53 nat53 top53 bot53 arr53 prod sum) (B Ty53 nat53 top53 bot53 arr53 prod sum)
prod53 : Ty53 → Ty53 → Ty53; prod53
= λ A B Ty53 nat53 top53 bot53 arr53 prod53 sum →
prod53 (A Ty53 nat53 top53 bot53 arr53 prod53 sum) (B Ty53 nat53 top53 bot53 arr53 prod53 sum)
sum53 : Ty53 → Ty53 → Ty53; sum53
= λ A B Ty53 nat53 top53 bot53 arr53 prod53 sum53 →
sum53 (A Ty53 nat53 top53 bot53 arr53 prod53 sum53) (B Ty53 nat53 top53 bot53 arr53 prod53 sum53)
Con53 : Set; Con53
= (Con53 : Set)
(nil : Con53)
(snoc : Con53 → Ty53 → Con53)
→ Con53
nil53 : Con53; nil53
= λ Con53 nil53 snoc → nil53
snoc53 : Con53 → Ty53 → Con53; snoc53
= λ Γ A Con53 nil53 snoc53 → snoc53 (Γ Con53 nil53 snoc53) A
Var53 : Con53 → Ty53 → Set; Var53
= λ Γ A →
(Var53 : Con53 → Ty53 → Set)
(vz : ∀ Γ A → Var53 (snoc53 Γ A) A)
(vs : ∀ Γ B A → Var53 Γ A → Var53 (snoc53 Γ B) A)
→ Var53 Γ A
vz53 : ∀{Γ A} → Var53 (snoc53 Γ A) A; vz53
= λ Var53 vz53 vs → vz53 _ _
vs53 : ∀{Γ B A} → Var53 Γ A → Var53 (snoc53 Γ B) A; vs53
= λ x Var53 vz53 vs53 → vs53 _ _ _ (x Var53 vz53 vs53)
Tm53 : Con53 → Ty53 → Set; Tm53
= λ Γ A →
(Tm53 : Con53 → Ty53 → Set)
(var : ∀ Γ A → Var53 Γ A → Tm53 Γ A)
(lam : ∀ Γ A B → Tm53 (snoc53 Γ A) B → Tm53 Γ (arr53 A B))
(app : ∀ Γ A B → Tm53 Γ (arr53 A B) → Tm53 Γ A → Tm53 Γ B)
(tt : ∀ Γ → Tm53 Γ top53)
(pair : ∀ Γ A B → Tm53 Γ A → Tm53 Γ B → Tm53 Γ (prod53 A B))
(fst : ∀ Γ A B → Tm53 Γ (prod53 A B) → Tm53 Γ A)
(snd : ∀ Γ A B → Tm53 Γ (prod53 A B) → Tm53 Γ B)
(left : ∀ Γ A B → Tm53 Γ A → Tm53 Γ (sum53 A B))
(right : ∀ Γ A B → Tm53 Γ B → Tm53 Γ (sum53 A B))
(case : ∀ Γ A B C → Tm53 Γ (sum53 A B) → Tm53 Γ (arr53 A C) → Tm53 Γ (arr53 B C) → Tm53 Γ C)
(zero : ∀ Γ → Tm53 Γ nat53)
(suc : ∀ Γ → Tm53 Γ nat53 → Tm53 Γ nat53)
(rec : ∀ Γ A → Tm53 Γ nat53 → Tm53 Γ (arr53 nat53 (arr53 A A)) → Tm53 Γ A → Tm53 Γ A)
→ Tm53 Γ A
var53 : ∀{Γ A} → Var53 Γ A → Tm53 Γ A; var53
= λ x Tm53 var53 lam app tt pair fst snd left right case zero suc rec →
var53 _ _ x
lam53 : ∀{Γ A B} → Tm53 (snoc53 Γ A) B → Tm53 Γ (arr53 A B); lam53
= λ t Tm53 var53 lam53 app tt pair fst snd left right case zero suc rec →
lam53 _ _ _ (t Tm53 var53 lam53 app tt pair fst snd left right case zero suc rec)
app53 : ∀{Γ A B} → Tm53 Γ (arr53 A B) → Tm53 Γ A → Tm53 Γ B; app53
= λ t u Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec →
app53 _ _ _ (t Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec)
(u Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec)
tt53 : ∀{Γ} → Tm53 Γ top53; tt53
= λ Tm53 var53 lam53 app53 tt53 pair fst snd left right case zero suc rec → tt53 _
pair53 : ∀{Γ A B} → Tm53 Γ A → Tm53 Γ B → Tm53 Γ (prod53 A B); pair53
= λ t u Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec →
pair53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec)
(u Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec)
fst53 : ∀{Γ A B} → Tm53 Γ (prod53 A B) → Tm53 Γ A; fst53
= λ t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd left right case zero suc rec →
fst53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd left right case zero suc rec)
snd53 : ∀{Γ A B} → Tm53 Γ (prod53 A B) → Tm53 Γ B; snd53
= λ t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left right case zero suc rec →
snd53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left right case zero suc rec)
left53 : ∀{Γ A B} → Tm53 Γ A → Tm53 Γ (sum53 A B); left53
= λ t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right case zero suc rec →
left53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right case zero suc rec)
right53 : ∀{Γ A B} → Tm53 Γ B → Tm53 Γ (sum53 A B); right53
= λ t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case zero suc rec →
right53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case zero suc rec)
case53 : ∀{Γ A B C} → Tm53 Γ (sum53 A B) → Tm53 Γ (arr53 A C) → Tm53 Γ (arr53 B C) → Tm53 Γ C; case53
= λ t u v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec →
case53 _ _ _ _
(t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
(u Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
(v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
zero53 : ∀{Γ} → Tm53 Γ nat53; zero53
= λ Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc rec → zero53 _
suc53 : ∀{Γ} → Tm53 Γ nat53 → Tm53 Γ nat53; suc53
= λ t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec →
suc53 _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec)
rec53 : ∀{Γ A} → Tm53 Γ nat53 → Tm53 Γ (arr53 nat53 (arr53 A A)) → Tm53 Γ A → Tm53 Γ A; rec53
= λ t u v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53 →
rec53 _ _
(t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
(u Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
(v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
v053 : ∀{Γ A} → Tm53 (snoc53 Γ A) A; v053
= var53 vz53
v153 : ∀{Γ A B} → Tm53 (snoc53 (snoc53 Γ A) B) A; v153
= var53 (vs53 vz53)
v253 : ∀{Γ A B C} → Tm53 (snoc53 (snoc53 (snoc53 Γ A) B) C) A; v253
= var53 (vs53 (vs53 vz53))
v353 : ∀{Γ A B C D} → Tm53 (snoc53 (snoc53 (snoc53 (snoc53 Γ A) B) C) D) A; v353
= var53 (vs53 (vs53 (vs53 vz53)))
tbool53 : Ty53; tbool53
= sum53 top53 top53
true53 : ∀{Γ} → Tm53 Γ tbool53; true53
= left53 tt53
tfalse53 : ∀{Γ} → Tm53 Γ tbool53; tfalse53
= right53 tt53
ifthenelse53 : ∀{Γ A} → Tm53 Γ (arr53 tbool53 (arr53 A (arr53 A A))); ifthenelse53
= lam53 (lam53 (lam53 (case53 v253 (lam53 v253) (lam53 v153))))
times453 : ∀{Γ A} → Tm53 Γ (arr53 (arr53 A A) (arr53 A A)); times453
= lam53 (lam53 (app53 v153 (app53 v153 (app53 v153 (app53 v153 v053)))))
add53 : ∀{Γ} → Tm53 Γ (arr53 nat53 (arr53 nat53 nat53)); add53
= lam53 (rec53 v053
(lam53 (lam53 (lam53 (suc53 (app53 v153 v053)))))
(lam53 v053))
mul53 : ∀{Γ} → Tm53 Γ (arr53 nat53 (arr53 nat53 nat53)); mul53
= lam53 (rec53 v053
(lam53 (lam53 (lam53 (app53 (app53 add53 (app53 v153 v053)) v053))))
(lam53 zero53))
fact53 : ∀{Γ} → Tm53 Γ (arr53 nat53 nat53); fact53
= lam53 (rec53 v053 (lam53 (lam53 (app53 (app53 mul53 (suc53 v153)) v053)))
(suc53 zero53))
{-# OPTIONS --type-in-type #-}
Ty54 : Set
Ty54 =
(Ty54 : Set)
(nat top bot : Ty54)
(arr prod sum : Ty54 → Ty54 → Ty54)
→ Ty54
nat54 : Ty54; nat54 = λ _ nat54 _ _ _ _ _ → nat54
top54 : Ty54; top54 = λ _ _ top54 _ _ _ _ → top54
bot54 : Ty54; bot54 = λ _ _ _ bot54 _ _ _ → bot54
arr54 : Ty54 → Ty54 → Ty54; arr54
= λ A B Ty54 nat54 top54 bot54 arr54 prod sum →
arr54 (A Ty54 nat54 top54 bot54 arr54 prod sum) (B Ty54 nat54 top54 bot54 arr54 prod sum)
prod54 : Ty54 → Ty54 → Ty54; prod54
= λ A B Ty54 nat54 top54 bot54 arr54 prod54 sum →
prod54 (A Ty54 nat54 top54 bot54 arr54 prod54 sum) (B Ty54 nat54 top54 bot54 arr54 prod54 sum)
sum54 : Ty54 → Ty54 → Ty54; sum54
= λ A B Ty54 nat54 top54 bot54 arr54 prod54 sum54 →
sum54 (A Ty54 nat54 top54 bot54 arr54 prod54 sum54) (B Ty54 nat54 top54 bot54 arr54 prod54 sum54)
Con54 : Set; Con54
= (Con54 : Set)
(nil : Con54)
(snoc : Con54 → Ty54 → Con54)
→ Con54
nil54 : Con54; nil54
= λ Con54 nil54 snoc → nil54
snoc54 : Con54 → Ty54 → Con54; snoc54
= λ Γ A Con54 nil54 snoc54 → snoc54 (Γ Con54 nil54 snoc54) A
Var54 : Con54 → Ty54 → Set; Var54
= λ Γ A →
(Var54 : Con54 → Ty54 → Set)
(vz : ∀ Γ A → Var54 (snoc54 Γ A) A)
(vs : ∀ Γ B A → Var54 Γ A → Var54 (snoc54 Γ B) A)
→ Var54 Γ A
vz54 : ∀{Γ A} → Var54 (snoc54 Γ A) A; vz54
= λ Var54 vz54 vs → vz54 _ _
vs54 : ∀{Γ B A} → Var54 Γ A → Var54 (snoc54 Γ B) A; vs54
= λ x Var54 vz54 vs54 → vs54 _ _ _ (x Var54 vz54 vs54)
Tm54 : Con54 → Ty54 → Set; Tm54
= λ Γ A →
(Tm54 : Con54 → Ty54 → Set)
(var : ∀ Γ A → Var54 Γ A → Tm54 Γ A)
(lam : ∀ Γ A B → Tm54 (snoc54 Γ A) B → Tm54 Γ (arr54 A B))
(app : ∀ Γ A B → Tm54 Γ (arr54 A B) → Tm54 Γ A → Tm54 Γ B)
(tt : ∀ Γ → Tm54 Γ top54)
(pair : ∀ Γ A B → Tm54 Γ A → Tm54 Γ B → Tm54 Γ (prod54 A B))
(fst : ∀ Γ A B → Tm54 Γ (prod54 A B) → Tm54 Γ A)
(snd : ∀ Γ A B → Tm54 Γ (prod54 A B) → Tm54 Γ B)
(left : ∀ Γ A B → Tm54 Γ A → Tm54 Γ (sum54 A B))
(right : ∀ Γ A B → Tm54 Γ B → Tm54 Γ (sum54 A B))
(case : ∀ Γ A B C → Tm54 Γ (sum54 A B) → Tm54 Γ (arr54 A C) → Tm54 Γ (arr54 B C) → Tm54 Γ C)
(zero : ∀ Γ → Tm54 Γ nat54)
(suc : ∀ Γ → Tm54 Γ nat54 → Tm54 Γ nat54)
(rec : ∀ Γ A → Tm54 Γ nat54 → Tm54 Γ (arr54 nat54 (arr54 A A)) → Tm54 Γ A → Tm54 Γ A)
→ Tm54 Γ A
var54 : ∀{Γ A} → Var54 Γ A → Tm54 Γ A; var54
= λ x Tm54 var54 lam app tt pair fst snd left right case zero suc rec →
var54 _ _ x
lam54 : ∀{Γ A B} → Tm54 (snoc54 Γ A) B → Tm54 Γ (arr54 A B); lam54
= λ t Tm54 var54 lam54 app tt pair fst snd left right case zero suc rec →
lam54 _ _ _ (t Tm54 var54 lam54 app tt pair fst snd left right case zero suc rec)
app54 : ∀{Γ A B} → Tm54 Γ (arr54 A B) → Tm54 Γ A → Tm54 Γ B; app54
= λ t u Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec →
app54 _ _ _ (t Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec)
(u Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec)
tt54 : ∀{Γ} → Tm54 Γ top54; tt54
= λ Tm54 var54 lam54 app54 tt54 pair fst snd left right case zero suc rec → tt54 _
pair54 : ∀{Γ A B} → Tm54 Γ A → Tm54 Γ B → Tm54 Γ (prod54 A B); pair54
= λ t u Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec →
pair54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec)
(u Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec)
fst54 : ∀{Γ A B} → Tm54 Γ (prod54 A B) → Tm54 Γ A; fst54
= λ t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd left right case zero suc rec →
fst54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd left right case zero suc rec)
snd54 : ∀{Γ A B} → Tm54 Γ (prod54 A B) → Tm54 Γ B; snd54
= λ t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left right case zero suc rec →
snd54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left right case zero suc rec)
left54 : ∀{Γ A B} → Tm54 Γ A → Tm54 Γ (sum54 A B); left54
= λ t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right case zero suc rec →
left54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right case zero suc rec)
right54 : ∀{Γ A B} → Tm54 Γ B → Tm54 Γ (sum54 A B); right54
= λ t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case zero suc rec →
right54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case zero suc rec)
case54 : ∀{Γ A B C} → Tm54 Γ (sum54 A B) → Tm54 Γ (arr54 A C) → Tm54 Γ (arr54 B C) → Tm54 Γ C; case54
= λ t u v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec →
case54 _ _ _ _
(t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
(u Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
(v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
zero54 : ∀{Γ} → Tm54 Γ nat54; zero54
= λ Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc rec → zero54 _
suc54 : ∀{Γ} → Tm54 Γ nat54 → Tm54 Γ nat54; suc54
= λ t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec →
suc54 _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec)
rec54 : ∀{Γ A} → Tm54 Γ nat54 → Tm54 Γ (arr54 nat54 (arr54 A A)) → Tm54 Γ A → Tm54 Γ A; rec54
= λ t u v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54 →
rec54 _ _
(t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
(u Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
(v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
v054 : ∀{Γ A} → Tm54 (snoc54 Γ A) A; v054
= var54 vz54
v154 : ∀{Γ A B} → Tm54 (snoc54 (snoc54 Γ A) B) A; v154
= var54 (vs54 vz54)
v254 : ∀{Γ A B C} → Tm54 (snoc54 (snoc54 (snoc54 Γ A) B) C) A; v254
= var54 (vs54 (vs54 vz54))
v354 : ∀{Γ A B C D} → Tm54 (snoc54 (snoc54 (snoc54 (snoc54 Γ A) B) C) D) A; v354
= var54 (vs54 (vs54 (vs54 vz54)))
tbool54 : Ty54; tbool54
= sum54 top54 top54
true54 : ∀{Γ} → Tm54 Γ tbool54; true54
= left54 tt54
tfalse54 : ∀{Γ} → Tm54 Γ tbool54; tfalse54
= right54 tt54
ifthenelse54 : ∀{Γ A} → Tm54 Γ (arr54 tbool54 (arr54 A (arr54 A A))); ifthenelse54
= lam54 (lam54 (lam54 (case54 v254 (lam54 v254) (lam54 v154))))
times454 : ∀{Γ A} → Tm54 Γ (arr54 (arr54 A A) (arr54 A A)); times454
= lam54 (lam54 (app54 v154 (app54 v154 (app54 v154 (app54 v154 v054)))))
add54 : ∀{Γ} → Tm54 Γ (arr54 nat54 (arr54 nat54 nat54)); add54
= lam54 (rec54 v054
(lam54 (lam54 (lam54 (suc54 (app54 v154 v054)))))
(lam54 v054))
mul54 : ∀{Γ} → Tm54 Γ (arr54 nat54 (arr54 nat54 nat54)); mul54
= lam54 (rec54 v054
(lam54 (lam54 (lam54 (app54 (app54 add54 (app54 v154 v054)) v054))))
(lam54 zero54))
fact54 : ∀{Γ} → Tm54 Γ (arr54 nat54 nat54); fact54
= lam54 (rec54 v054 (lam54 (lam54 (app54 (app54 mul54 (suc54 v154)) v054)))
(suc54 zero54))
{-# OPTIONS --type-in-type #-}
Ty55 : Set
Ty55 =
(Ty55 : Set)
(nat top bot : Ty55)
(arr prod sum : Ty55 → Ty55 → Ty55)
→ Ty55
nat55 : Ty55; nat55 = λ _ nat55 _ _ _ _ _ → nat55
top55 : Ty55; top55 = λ _ _ top55 _ _ _ _ → top55
bot55 : Ty55; bot55 = λ _ _ _ bot55 _ _ _ → bot55
arr55 : Ty55 → Ty55 → Ty55; arr55
= λ A B Ty55 nat55 top55 bot55 arr55 prod sum →
arr55 (A Ty55 nat55 top55 bot55 arr55 prod sum) (B Ty55 nat55 top55 bot55 arr55 prod sum)
prod55 : Ty55 → Ty55 → Ty55; prod55
= λ A B Ty55 nat55 top55 bot55 arr55 prod55 sum →
prod55 (A Ty55 nat55 top55 bot55 arr55 prod55 sum) (B Ty55 nat55 top55 bot55 arr55 prod55 sum)
sum55 : Ty55 → Ty55 → Ty55; sum55
= λ A B Ty55 nat55 top55 bot55 arr55 prod55 sum55 →
sum55 (A Ty55 nat55 top55 bot55 arr55 prod55 sum55) (B Ty55 nat55 top55 bot55 arr55 prod55 sum55)
Con55 : Set; Con55
= (Con55 : Set)
(nil : Con55)
(snoc : Con55 → Ty55 → Con55)
→ Con55
nil55 : Con55; nil55
= λ Con55 nil55 snoc → nil55
snoc55 : Con55 → Ty55 → Con55; snoc55
= λ Γ A Con55 nil55 snoc55 → snoc55 (Γ Con55 nil55 snoc55) A
Var55 : Con55 → Ty55 → Set; Var55
= λ Γ A →
(Var55 : Con55 → Ty55 → Set)
(vz : ∀ Γ A → Var55 (snoc55 Γ A) A)
(vs : ∀ Γ B A → Var55 Γ A → Var55 (snoc55 Γ B) A)
→ Var55 Γ A
vz55 : ∀{Γ A} → Var55 (snoc55 Γ A) A; vz55
= λ Var55 vz55 vs → vz55 _ _
vs55 : ∀{Γ B A} → Var55 Γ A → Var55 (snoc55 Γ B) A; vs55
= λ x Var55 vz55 vs55 → vs55 _ _ _ (x Var55 vz55 vs55)
Tm55 : Con55 → Ty55 → Set; Tm55
= λ Γ A →
(Tm55 : Con55 → Ty55 → Set)
(var : ∀ Γ A → Var55 Γ A → Tm55 Γ A)
(lam : ∀ Γ A B → Tm55 (snoc55 Γ A) B → Tm55 Γ (arr55 A B))
(app : ∀ Γ A B → Tm55 Γ (arr55 A B) → Tm55 Γ A → Tm55 Γ B)
(tt : ∀ Γ → Tm55 Γ top55)
(pair : ∀ Γ A B → Tm55 Γ A → Tm55 Γ B → Tm55 Γ (prod55 A B))
(fst : ∀ Γ A B → Tm55 Γ (prod55 A B) → Tm55 Γ A)
(snd : ∀ Γ A B → Tm55 Γ (prod55 A B) → Tm55 Γ B)
(left : ∀ Γ A B → Tm55 Γ A → Tm55 Γ (sum55 A B))
(right : ∀ Γ A B → Tm55 Γ B → Tm55 Γ (sum55 A B))
(case : ∀ Γ A B C → Tm55 Γ (sum55 A B) → Tm55 Γ (arr55 A C) → Tm55 Γ (arr55 B C) → Tm55 Γ C)
(zero : ∀ Γ → Tm55 Γ nat55)
(suc : ∀ Γ → Tm55 Γ nat55 → Tm55 Γ nat55)
(rec : ∀ Γ A → Tm55 Γ nat55 → Tm55 Γ (arr55 nat55 (arr55 A A)) → Tm55 Γ A → Tm55 Γ A)
→ Tm55 Γ A
var55 : ∀{Γ A} → Var55 Γ A → Tm55 Γ A; var55
= λ x Tm55 var55 lam app tt pair fst snd left right case zero suc rec →
var55 _ _ x
lam55 : ∀{Γ A B} → Tm55 (snoc55 Γ A) B → Tm55 Γ (arr55 A B); lam55
= λ t Tm55 var55 lam55 app tt pair fst snd left right case zero suc rec →
lam55 _ _ _ (t Tm55 var55 lam55 app tt pair fst snd left right case zero suc rec)
app55 : ∀{Γ A B} → Tm55 Γ (arr55 A B) → Tm55 Γ A → Tm55 Γ B; app55
= λ t u Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec →
app55 _ _ _ (t Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec)
(u Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec)
tt55 : ∀{Γ} → Tm55 Γ top55; tt55
= λ Tm55 var55 lam55 app55 tt55 pair fst snd left right case zero suc rec → tt55 _
pair55 : ∀{Γ A B} → Tm55 Γ A → Tm55 Γ B → Tm55 Γ (prod55 A B); pair55
= λ t u Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec →
pair55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec)
(u Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec)
fst55 : ∀{Γ A B} → Tm55 Γ (prod55 A B) → Tm55 Γ A; fst55
= λ t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd left right case zero suc rec →
fst55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd left right case zero suc rec)
snd55 : ∀{Γ A B} → Tm55 Γ (prod55 A B) → Tm55 Γ B; snd55
= λ t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left right case zero suc rec →
snd55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left right case zero suc rec)
left55 : ∀{Γ A B} → Tm55 Γ A → Tm55 Γ (sum55 A B); left55
= λ t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right case zero suc rec →
left55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right case zero suc rec)
right55 : ∀{Γ A B} → Tm55 Γ B → Tm55 Γ (sum55 A B); right55
= λ t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case zero suc rec →
right55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case zero suc rec)
case55 : ∀{Γ A B C} → Tm55 Γ (sum55 A B) → Tm55 Γ (arr55 A C) → Tm55 Γ (arr55 B C) → Tm55 Γ C; case55
= λ t u v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec →
case55 _ _ _ _
(t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
(u Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
(v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
zero55 : ∀{Γ} → Tm55 Γ nat55; zero55
= λ Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc rec → zero55 _
suc55 : ∀{Γ} → Tm55 Γ nat55 → Tm55 Γ nat55; suc55
= λ t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec →
suc55 _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec)
rec55 : ∀{Γ A} → Tm55 Γ nat55 → Tm55 Γ (arr55 nat55 (arr55 A A)) → Tm55 Γ A → Tm55 Γ A; rec55
= λ t u v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55 →
rec55 _ _
(t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
(u Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
(v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
v055 : ∀{Γ A} → Tm55 (snoc55 Γ A) A; v055
= var55 vz55
v155 : ∀{Γ A B} → Tm55 (snoc55 (snoc55 Γ A) B) A; v155
= var55 (vs55 vz55)
v255 : ∀{Γ A B C} → Tm55 (snoc55 (snoc55 (snoc55 Γ A) B) C) A; v255
= var55 (vs55 (vs55 vz55))
v355 : ∀{Γ A B C D} → Tm55 (snoc55 (snoc55 (snoc55 (snoc55 Γ A) B) C) D) A; v355
= var55 (vs55 (vs55 (vs55 vz55)))
tbool55 : Ty55; tbool55
= sum55 top55 top55
true55 : ∀{Γ} → Tm55 Γ tbool55; true55
= left55 tt55
tfalse55 : ∀{Γ} → Tm55 Γ tbool55; tfalse55
= right55 tt55
ifthenelse55 : ∀{Γ A} → Tm55 Γ (arr55 tbool55 (arr55 A (arr55 A A))); ifthenelse55
= lam55 (lam55 (lam55 (case55 v255 (lam55 v255) (lam55 v155))))
times455 : ∀{Γ A} → Tm55 Γ (arr55 (arr55 A A) (arr55 A A)); times455
= lam55 (lam55 (app55 v155 (app55 v155 (app55 v155 (app55 v155 v055)))))
add55 : ∀{Γ} → Tm55 Γ (arr55 nat55 (arr55 nat55 nat55)); add55
= lam55 (rec55 v055
(lam55 (lam55 (lam55 (suc55 (app55 v155 v055)))))
(lam55 v055))
mul55 : ∀{Γ} → Tm55 Γ (arr55 nat55 (arr55 nat55 nat55)); mul55
= lam55 (rec55 v055
(lam55 (lam55 (lam55 (app55 (app55 add55 (app55 v155 v055)) v055))))
(lam55 zero55))
fact55 : ∀{Γ} → Tm55 Γ (arr55 nat55 nat55); fact55
= lam55 (rec55 v055 (lam55 (lam55 (app55 (app55 mul55 (suc55 v155)) v055)))
(suc55 zero55))
{-# OPTIONS --type-in-type #-}
Ty56 : Set
Ty56 =
(Ty56 : Set)
(nat top bot : Ty56)
(arr prod sum : Ty56 → Ty56 → Ty56)
→ Ty56
nat56 : Ty56; nat56 = λ _ nat56 _ _ _ _ _ → nat56
top56 : Ty56; top56 = λ _ _ top56 _ _ _ _ → top56
bot56 : Ty56; bot56 = λ _ _ _ bot56 _ _ _ → bot56
arr56 : Ty56 → Ty56 → Ty56; arr56
= λ A B Ty56 nat56 top56 bot56 arr56 prod sum →
arr56 (A Ty56 nat56 top56 bot56 arr56 prod sum) (B Ty56 nat56 top56 bot56 arr56 prod sum)
prod56 : Ty56 → Ty56 → Ty56; prod56
= λ A B Ty56 nat56 top56 bot56 arr56 prod56 sum →
prod56 (A Ty56 nat56 top56 bot56 arr56 prod56 sum) (B Ty56 nat56 top56 bot56 arr56 prod56 sum)
sum56 : Ty56 → Ty56 → Ty56; sum56
= λ A B Ty56 nat56 top56 bot56 arr56 prod56 sum56 →
sum56 (A Ty56 nat56 top56 bot56 arr56 prod56 sum56) (B Ty56 nat56 top56 bot56 arr56 prod56 sum56)
Con56 : Set; Con56
= (Con56 : Set)
(nil : Con56)
(snoc : Con56 → Ty56 → Con56)
→ Con56
nil56 : Con56; nil56
= λ Con56 nil56 snoc → nil56
snoc56 : Con56 → Ty56 → Con56; snoc56
= λ Γ A Con56 nil56 snoc56 → snoc56 (Γ Con56 nil56 snoc56) A
Var56 : Con56 → Ty56 → Set; Var56
= λ Γ A →
(Var56 : Con56 → Ty56 → Set)
(vz : ∀ Γ A → Var56 (snoc56 Γ A) A)
(vs : ∀ Γ B A → Var56 Γ A → Var56 (snoc56 Γ B) A)
→ Var56 Γ A
vz56 : ∀{Γ A} → Var56 (snoc56 Γ A) A; vz56
= λ Var56 vz56 vs → vz56 _ _
vs56 : ∀{Γ B A} → Var56 Γ A → Var56 (snoc56 Γ B) A; vs56
= λ x Var56 vz56 vs56 → vs56 _ _ _ (x Var56 vz56 vs56)
Tm56 : Con56 → Ty56 → Set; Tm56
= λ Γ A →
(Tm56 : Con56 → Ty56 → Set)
(var : ∀ Γ A → Var56 Γ A → Tm56 Γ A)
(lam : ∀ Γ A B → Tm56 (snoc56 Γ A) B → Tm56 Γ (arr56 A B))
(app : ∀ Γ A B → Tm56 Γ (arr56 A B) → Tm56 Γ A → Tm56 Γ B)
(tt : ∀ Γ → Tm56 Γ top56)
(pair : ∀ Γ A B → Tm56 Γ A → Tm56 Γ B → Tm56 Γ (prod56 A B))
(fst : ∀ Γ A B → Tm56 Γ (prod56 A B) → Tm56 Γ A)
(snd : ∀ Γ A B → Tm56 Γ (prod56 A B) → Tm56 Γ B)
(left : ∀ Γ A B → Tm56 Γ A → Tm56 Γ (sum56 A B))
(right : ∀ Γ A B → Tm56 Γ B → Tm56 Γ (sum56 A B))
(case : ∀ Γ A B C → Tm56 Γ (sum56 A B) → Tm56 Γ (arr56 A C) → Tm56 Γ (arr56 B C) → Tm56 Γ C)
(zero : ∀ Γ → Tm56 Γ nat56)
(suc : ∀ Γ → Tm56 Γ nat56 → Tm56 Γ nat56)
(rec : ∀ Γ A → Tm56 Γ nat56 → Tm56 Γ (arr56 nat56 (arr56 A A)) → Tm56 Γ A → Tm56 Γ A)
→ Tm56 Γ A
var56 : ∀{Γ A} → Var56 Γ A → Tm56 Γ A; var56
= λ x Tm56 var56 lam app tt pair fst snd left right case zero suc rec →
var56 _ _ x
lam56 : ∀{Γ A B} → Tm56 (snoc56 Γ A) B → Tm56 Γ (arr56 A B); lam56
= λ t Tm56 var56 lam56 app tt pair fst snd left right case zero suc rec →
lam56 _ _ _ (t Tm56 var56 lam56 app tt pair fst snd left right case zero suc rec)
app56 : ∀{Γ A B} → Tm56 Γ (arr56 A B) → Tm56 Γ A → Tm56 Γ B; app56
= λ t u Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec →
app56 _ _ _ (t Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec)
(u Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec)
tt56 : ∀{Γ} → Tm56 Γ top56; tt56
= λ Tm56 var56 lam56 app56 tt56 pair fst snd left right case zero suc rec → tt56 _
pair56 : ∀{Γ A B} → Tm56 Γ A → Tm56 Γ B → Tm56 Γ (prod56 A B); pair56
= λ t u Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec →
pair56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec)
(u Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec)
fst56 : ∀{Γ A B} → Tm56 Γ (prod56 A B) → Tm56 Γ A; fst56
= λ t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd left right case zero suc rec →
fst56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd left right case zero suc rec)
snd56 : ∀{Γ A B} → Tm56 Γ (prod56 A B) → Tm56 Γ B; snd56
= λ t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left right case zero suc rec →
snd56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left right case zero suc rec)
left56 : ∀{Γ A B} → Tm56 Γ A → Tm56 Γ (sum56 A B); left56
= λ t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right case zero suc rec →
left56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right case zero suc rec)
right56 : ∀{Γ A B} → Tm56 Γ B → Tm56 Γ (sum56 A B); right56
= λ t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case zero suc rec →
right56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case zero suc rec)
case56 : ∀{Γ A B C} → Tm56 Γ (sum56 A B) → Tm56 Γ (arr56 A C) → Tm56 Γ (arr56 B C) → Tm56 Γ C; case56
= λ t u v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec →
case56 _ _ _ _
(t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
(u Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
(v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
zero56 : ∀{Γ} → Tm56 Γ nat56; zero56
= λ Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc rec → zero56 _
suc56 : ∀{Γ} → Tm56 Γ nat56 → Tm56 Γ nat56; suc56
= λ t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec →
suc56 _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec)
rec56 : ∀{Γ A} → Tm56 Γ nat56 → Tm56 Γ (arr56 nat56 (arr56 A A)) → Tm56 Γ A → Tm56 Γ A; rec56
= λ t u v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56 →
rec56 _ _
(t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
(u Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
(v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
v056 : ∀{Γ A} → Tm56 (snoc56 Γ A) A; v056
= var56 vz56
v156 : ∀{Γ A B} → Tm56 (snoc56 (snoc56 Γ A) B) A; v156
= var56 (vs56 vz56)
v256 : ∀{Γ A B C} → Tm56 (snoc56 (snoc56 (snoc56 Γ A) B) C) A; v256
= var56 (vs56 (vs56 vz56))
v356 : ∀{Γ A B C D} → Tm56 (snoc56 (snoc56 (snoc56 (snoc56 Γ A) B) C) D) A; v356
= var56 (vs56 (vs56 (vs56 vz56)))
tbool56 : Ty56; tbool56
= sum56 top56 top56
true56 : ∀{Γ} → Tm56 Γ tbool56; true56
= left56 tt56
tfalse56 : ∀{Γ} → Tm56 Γ tbool56; tfalse56
= right56 tt56
ifthenelse56 : ∀{Γ A} → Tm56 Γ (arr56 tbool56 (arr56 A (arr56 A A))); ifthenelse56
= lam56 (lam56 (lam56 (case56 v256 (lam56 v256) (lam56 v156))))
times456 : ∀{Γ A} → Tm56 Γ (arr56 (arr56 A A) (arr56 A A)); times456
= lam56 (lam56 (app56 v156 (app56 v156 (app56 v156 (app56 v156 v056)))))
add56 : ∀{Γ} → Tm56 Γ (arr56 nat56 (arr56 nat56 nat56)); add56
= lam56 (rec56 v056
(lam56 (lam56 (lam56 (suc56 (app56 v156 v056)))))
(lam56 v056))
mul56 : ∀{Γ} → Tm56 Γ (arr56 nat56 (arr56 nat56 nat56)); mul56
= lam56 (rec56 v056
(lam56 (lam56 (lam56 (app56 (app56 add56 (app56 v156 v056)) v056))))
(lam56 zero56))
fact56 : ∀{Γ} → Tm56 Γ (arr56 nat56 nat56); fact56
= lam56 (rec56 v056 (lam56 (lam56 (app56 (app56 mul56 (suc56 v156)) v056)))
(suc56 zero56))
{-# OPTIONS --type-in-type #-}
Ty57 : Set
Ty57 =
(Ty57 : Set)
(nat top bot : Ty57)
(arr prod sum : Ty57 → Ty57 → Ty57)
→ Ty57
nat57 : Ty57; nat57 = λ _ nat57 _ _ _ _ _ → nat57
top57 : Ty57; top57 = λ _ _ top57 _ _ _ _ → top57
bot57 : Ty57; bot57 = λ _ _ _ bot57 _ _ _ → bot57
arr57 : Ty57 → Ty57 → Ty57; arr57
= λ A B Ty57 nat57 top57 bot57 arr57 prod sum →
arr57 (A Ty57 nat57 top57 bot57 arr57 prod sum) (B Ty57 nat57 top57 bot57 arr57 prod sum)
prod57 : Ty57 → Ty57 → Ty57; prod57
= λ A B Ty57 nat57 top57 bot57 arr57 prod57 sum →
prod57 (A Ty57 nat57 top57 bot57 arr57 prod57 sum) (B Ty57 nat57 top57 bot57 arr57 prod57 sum)
sum57 : Ty57 → Ty57 → Ty57; sum57
= λ A B Ty57 nat57 top57 bot57 arr57 prod57 sum57 →
sum57 (A Ty57 nat57 top57 bot57 arr57 prod57 sum57) (B Ty57 nat57 top57 bot57 arr57 prod57 sum57)
Con57 : Set; Con57
= (Con57 : Set)
(nil : Con57)
(snoc : Con57 → Ty57 → Con57)
→ Con57
nil57 : Con57; nil57
= λ Con57 nil57 snoc → nil57
snoc57 : Con57 → Ty57 → Con57; snoc57
= λ Γ A Con57 nil57 snoc57 → snoc57 (Γ Con57 nil57 snoc57) A
Var57 : Con57 → Ty57 → Set; Var57
= λ Γ A →
(Var57 : Con57 → Ty57 → Set)
(vz : ∀ Γ A → Var57 (snoc57 Γ A) A)
(vs : ∀ Γ B A → Var57 Γ A → Var57 (snoc57 Γ B) A)
→ Var57 Γ A
vz57 : ∀{Γ A} → Var57 (snoc57 Γ A) A; vz57
= λ Var57 vz57 vs → vz57 _ _
vs57 : ∀{Γ B A} → Var57 Γ A → Var57 (snoc57 Γ B) A; vs57
= λ x Var57 vz57 vs57 → vs57 _ _ _ (x Var57 vz57 vs57)
Tm57 : Con57 → Ty57 → Set; Tm57
= λ Γ A →
(Tm57 : Con57 → Ty57 → Set)
(var : ∀ Γ A → Var57 Γ A → Tm57 Γ A)
(lam : ∀ Γ A B → Tm57 (snoc57 Γ A) B → Tm57 Γ (arr57 A B))
(app : ∀ Γ A B → Tm57 Γ (arr57 A B) → Tm57 Γ A → Tm57 Γ B)
(tt : ∀ Γ → Tm57 Γ top57)
(pair : ∀ Γ A B → Tm57 Γ A → Tm57 Γ B → Tm57 Γ (prod57 A B))
(fst : ∀ Γ A B → Tm57 Γ (prod57 A B) → Tm57 Γ A)
(snd : ∀ Γ A B → Tm57 Γ (prod57 A B) → Tm57 Γ B)
(left : ∀ Γ A B → Tm57 Γ A → Tm57 Γ (sum57 A B))
(right : ∀ Γ A B → Tm57 Γ B → Tm57 Γ (sum57 A B))
(case : ∀ Γ A B C → Tm57 Γ (sum57 A B) → Tm57 Γ (arr57 A C) → Tm57 Γ (arr57 B C) → Tm57 Γ C)
(zero : ∀ Γ → Tm57 Γ nat57)
(suc : ∀ Γ → Tm57 Γ nat57 → Tm57 Γ nat57)
(rec : ∀ Γ A → Tm57 Γ nat57 → Tm57 Γ (arr57 nat57 (arr57 A A)) → Tm57 Γ A → Tm57 Γ A)
→ Tm57 Γ A
var57 : ∀{Γ A} → Var57 Γ A → Tm57 Γ A; var57
= λ x Tm57 var57 lam app tt pair fst snd left right case zero suc rec →
var57 _ _ x
lam57 : ∀{Γ A B} → Tm57 (snoc57 Γ A) B → Tm57 Γ (arr57 A B); lam57
= λ t Tm57 var57 lam57 app tt pair fst snd left right case zero suc rec →
lam57 _ _ _ (t Tm57 var57 lam57 app tt pair fst snd left right case zero suc rec)
app57 : ∀{Γ A B} → Tm57 Γ (arr57 A B) → Tm57 Γ A → Tm57 Γ B; app57
= λ t u Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec →
app57 _ _ _ (t Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec)
(u Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec)
tt57 : ∀{Γ} → Tm57 Γ top57; tt57
= λ Tm57 var57 lam57 app57 tt57 pair fst snd left right case zero suc rec → tt57 _
pair57 : ∀{Γ A B} → Tm57 Γ A → Tm57 Γ B → Tm57 Γ (prod57 A B); pair57
= λ t u Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec →
pair57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec)
(u Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec)
fst57 : ∀{Γ A B} → Tm57 Γ (prod57 A B) → Tm57 Γ A; fst57
= λ t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd left right case zero suc rec →
fst57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd left right case zero suc rec)
snd57 : ∀{Γ A B} → Tm57 Γ (prod57 A B) → Tm57 Γ B; snd57
= λ t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left right case zero suc rec →
snd57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left right case zero suc rec)
left57 : ∀{Γ A B} → Tm57 Γ A → Tm57 Γ (sum57 A B); left57
= λ t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right case zero suc rec →
left57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right case zero suc rec)
right57 : ∀{Γ A B} → Tm57 Γ B → Tm57 Γ (sum57 A B); right57
= λ t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case zero suc rec →
right57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case zero suc rec)
case57 : ∀{Γ A B C} → Tm57 Γ (sum57 A B) → Tm57 Γ (arr57 A C) → Tm57 Γ (arr57 B C) → Tm57 Γ C; case57
= λ t u v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec →
case57 _ _ _ _
(t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
(u Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
(v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
zero57 : ∀{Γ} → Tm57 Γ nat57; zero57
= λ Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc rec → zero57 _
suc57 : ∀{Γ} → Tm57 Γ nat57 → Tm57 Γ nat57; suc57
= λ t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec →
suc57 _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec)
rec57 : ∀{Γ A} → Tm57 Γ nat57 → Tm57 Γ (arr57 nat57 (arr57 A A)) → Tm57 Γ A → Tm57 Γ A; rec57
= λ t u v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57 →
rec57 _ _
(t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
(u Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
(v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
v057 : ∀{Γ A} → Tm57 (snoc57 Γ A) A; v057
= var57 vz57
v157 : ∀{Γ A B} → Tm57 (snoc57 (snoc57 Γ A) B) A; v157
= var57 (vs57 vz57)
v257 : ∀{Γ A B C} → Tm57 (snoc57 (snoc57 (snoc57 Γ A) B) C) A; v257
= var57 (vs57 (vs57 vz57))
v357 : ∀{Γ A B C D} → Tm57 (snoc57 (snoc57 (snoc57 (snoc57 Γ A) B) C) D) A; v357
= var57 (vs57 (vs57 (vs57 vz57)))
tbool57 : Ty57; tbool57
= sum57 top57 top57
true57 : ∀{Γ} → Tm57 Γ tbool57; true57
= left57 tt57
tfalse57 : ∀{Γ} → Tm57 Γ tbool57; tfalse57
= right57 tt57
ifthenelse57 : ∀{Γ A} → Tm57 Γ (arr57 tbool57 (arr57 A (arr57 A A))); ifthenelse57
= lam57 (lam57 (lam57 (case57 v257 (lam57 v257) (lam57 v157))))
times457 : ∀{Γ A} → Tm57 Γ (arr57 (arr57 A A) (arr57 A A)); times457
= lam57 (lam57 (app57 v157 (app57 v157 (app57 v157 (app57 v157 v057)))))
add57 : ∀{Γ} → Tm57 Γ (arr57 nat57 (arr57 nat57 nat57)); add57
= lam57 (rec57 v057
(lam57 (lam57 (lam57 (suc57 (app57 v157 v057)))))
(lam57 v057))
mul57 : ∀{Γ} → Tm57 Γ (arr57 nat57 (arr57 nat57 nat57)); mul57
= lam57 (rec57 v057
(lam57 (lam57 (lam57 (app57 (app57 add57 (app57 v157 v057)) v057))))
(lam57 zero57))
fact57 : ∀{Γ} → Tm57 Γ (arr57 nat57 nat57); fact57
= lam57 (rec57 v057 (lam57 (lam57 (app57 (app57 mul57 (suc57 v157)) v057)))
(suc57 zero57))
{-# OPTIONS --type-in-type #-}
Ty58 : Set
Ty58 =
(Ty58 : Set)
(nat top bot : Ty58)
(arr prod sum : Ty58 → Ty58 → Ty58)
→ Ty58
nat58 : Ty58; nat58 = λ _ nat58 _ _ _ _ _ → nat58
top58 : Ty58; top58 = λ _ _ top58 _ _ _ _ → top58
bot58 : Ty58; bot58 = λ _ _ _ bot58 _ _ _ → bot58
arr58 : Ty58 → Ty58 → Ty58; arr58
= λ A B Ty58 nat58 top58 bot58 arr58 prod sum →
arr58 (A Ty58 nat58 top58 bot58 arr58 prod sum) (B Ty58 nat58 top58 bot58 arr58 prod sum)
prod58 : Ty58 → Ty58 → Ty58; prod58
= λ A B Ty58 nat58 top58 bot58 arr58 prod58 sum →
prod58 (A Ty58 nat58 top58 bot58 arr58 prod58 sum) (B Ty58 nat58 top58 bot58 arr58 prod58 sum)
sum58 : Ty58 → Ty58 → Ty58; sum58
= λ A B Ty58 nat58 top58 bot58 arr58 prod58 sum58 →
sum58 (A Ty58 nat58 top58 bot58 arr58 prod58 sum58) (B Ty58 nat58 top58 bot58 arr58 prod58 sum58)
Con58 : Set; Con58
= (Con58 : Set)
(nil : Con58)
(snoc : Con58 → Ty58 → Con58)
→ Con58
nil58 : Con58; nil58
= λ Con58 nil58 snoc → nil58
snoc58 : Con58 → Ty58 → Con58; snoc58
= λ Γ A Con58 nil58 snoc58 → snoc58 (Γ Con58 nil58 snoc58) A
Var58 : Con58 → Ty58 → Set; Var58
= λ Γ A →
(Var58 : Con58 → Ty58 → Set)
(vz : ∀ Γ A → Var58 (snoc58 Γ A) A)
(vs : ∀ Γ B A → Var58 Γ A → Var58 (snoc58 Γ B) A)
→ Var58 Γ A
vz58 : ∀{Γ A} → Var58 (snoc58 Γ A) A; vz58
= λ Var58 vz58 vs → vz58 _ _
vs58 : ∀{Γ B A} → Var58 Γ A → Var58 (snoc58 Γ B) A; vs58
= λ x Var58 vz58 vs58 → vs58 _ _ _ (x Var58 vz58 vs58)
Tm58 : Con58 → Ty58 → Set; Tm58
= λ Γ A →
(Tm58 : Con58 → Ty58 → Set)
(var : ∀ Γ A → Var58 Γ A → Tm58 Γ A)
(lam : ∀ Γ A B → Tm58 (snoc58 Γ A) B → Tm58 Γ (arr58 A B))
(app : ∀ Γ A B → Tm58 Γ (arr58 A B) → Tm58 Γ A → Tm58 Γ B)
(tt : ∀ Γ → Tm58 Γ top58)
(pair : ∀ Γ A B → Tm58 Γ A → Tm58 Γ B → Tm58 Γ (prod58 A B))
(fst : ∀ Γ A B → Tm58 Γ (prod58 A B) → Tm58 Γ A)
(snd : ∀ Γ A B → Tm58 Γ (prod58 A B) → Tm58 Γ B)
(left : ∀ Γ A B → Tm58 Γ A → Tm58 Γ (sum58 A B))
(right : ∀ Γ A B → Tm58 Γ B → Tm58 Γ (sum58 A B))
(case : ∀ Γ A B C → Tm58 Γ (sum58 A B) → Tm58 Γ (arr58 A C) → Tm58 Γ (arr58 B C) → Tm58 Γ C)
(zero : ∀ Γ → Tm58 Γ nat58)
(suc : ∀ Γ → Tm58 Γ nat58 → Tm58 Γ nat58)
(rec : ∀ Γ A → Tm58 Γ nat58 → Tm58 Γ (arr58 nat58 (arr58 A A)) → Tm58 Γ A → Tm58 Γ A)
→ Tm58 Γ A
var58 : ∀{Γ A} → Var58 Γ A → Tm58 Γ A; var58
= λ x Tm58 var58 lam app tt pair fst snd left right case zero suc rec →
var58 _ _ x
lam58 : ∀{Γ A B} → Tm58 (snoc58 Γ A) B → Tm58 Γ (arr58 A B); lam58
= λ t Tm58 var58 lam58 app tt pair fst snd left right case zero suc rec →
lam58 _ _ _ (t Tm58 var58 lam58 app tt pair fst snd left right case zero suc rec)
app58 : ∀{Γ A B} → Tm58 Γ (arr58 A B) → Tm58 Γ A → Tm58 Γ B; app58
= λ t u Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec →
app58 _ _ _ (t Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec)
(u Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec)
tt58 : ∀{Γ} → Tm58 Γ top58; tt58
= λ Tm58 var58 lam58 app58 tt58 pair fst snd left right case zero suc rec → tt58 _
pair58 : ∀{Γ A B} → Tm58 Γ A → Tm58 Γ B → Tm58 Γ (prod58 A B); pair58
= λ t u Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec →
pair58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec)
(u Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec)
fst58 : ∀{Γ A B} → Tm58 Γ (prod58 A B) → Tm58 Γ A; fst58
= λ t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd left right case zero suc rec →
fst58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd left right case zero suc rec)
snd58 : ∀{Γ A B} → Tm58 Γ (prod58 A B) → Tm58 Γ B; snd58
= λ t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left right case zero suc rec →
snd58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left right case zero suc rec)
left58 : ∀{Γ A B} → Tm58 Γ A → Tm58 Γ (sum58 A B); left58
= λ t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right case zero suc rec →
left58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right case zero suc rec)
right58 : ∀{Γ A B} → Tm58 Γ B → Tm58 Γ (sum58 A B); right58
= λ t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case zero suc rec →
right58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case zero suc rec)
case58 : ∀{Γ A B C} → Tm58 Γ (sum58 A B) → Tm58 Γ (arr58 A C) → Tm58 Γ (arr58 B C) → Tm58 Γ C; case58
= λ t u v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec →
case58 _ _ _ _
(t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
(u Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
(v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
zero58 : ∀{Γ} → Tm58 Γ nat58; zero58
= λ Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc rec → zero58 _
suc58 : ∀{Γ} → Tm58 Γ nat58 → Tm58 Γ nat58; suc58
= λ t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec →
suc58 _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec)
rec58 : ∀{Γ A} → Tm58 Γ nat58 → Tm58 Γ (arr58 nat58 (arr58 A A)) → Tm58 Γ A → Tm58 Γ A; rec58
= λ t u v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58 →
rec58 _ _
(t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
(u Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
(v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
v058 : ∀{Γ A} → Tm58 (snoc58 Γ A) A; v058
= var58 vz58
v158 : ∀{Γ A B} → Tm58 (snoc58 (snoc58 Γ A) B) A; v158
= var58 (vs58 vz58)
v258 : ∀{Γ A B C} → Tm58 (snoc58 (snoc58 (snoc58 Γ A) B) C) A; v258
= var58 (vs58 (vs58 vz58))
v358 : ∀{Γ A B C D} → Tm58 (snoc58 (snoc58 (snoc58 (snoc58 Γ A) B) C) D) A; v358
= var58 (vs58 (vs58 (vs58 vz58)))
tbool58 : Ty58; tbool58
= sum58 top58 top58
true58 : ∀{Γ} → Tm58 Γ tbool58; true58
= left58 tt58
tfalse58 : ∀{Γ} → Tm58 Γ tbool58; tfalse58
= right58 tt58
ifthenelse58 : ∀{Γ A} → Tm58 Γ (arr58 tbool58 (arr58 A (arr58 A A))); ifthenelse58
= lam58 (lam58 (lam58 (case58 v258 (lam58 v258) (lam58 v158))))
times458 : ∀{Γ A} → Tm58 Γ (arr58 (arr58 A A) (arr58 A A)); times458
= lam58 (lam58 (app58 v158 (app58 v158 (app58 v158 (app58 v158 v058)))))
add58 : ∀{Γ} → Tm58 Γ (arr58 nat58 (arr58 nat58 nat58)); add58
= lam58 (rec58 v058
(lam58 (lam58 (lam58 (suc58 (app58 v158 v058)))))
(lam58 v058))
mul58 : ∀{Γ} → Tm58 Γ (arr58 nat58 (arr58 nat58 nat58)); mul58
= lam58 (rec58 v058
(lam58 (lam58 (lam58 (app58 (app58 add58 (app58 v158 v058)) v058))))
(lam58 zero58))
fact58 : ∀{Γ} → Tm58 Γ (arr58 nat58 nat58); fact58
= lam58 (rec58 v058 (lam58 (lam58 (app58 (app58 mul58 (suc58 v158)) v058)))
(suc58 zero58))
{-# OPTIONS --type-in-type #-}
Ty59 : Set
Ty59 =
(Ty59 : Set)
(nat top bot : Ty59)
(arr prod sum : Ty59 → Ty59 → Ty59)
→ Ty59
nat59 : Ty59; nat59 = λ _ nat59 _ _ _ _ _ → nat59
top59 : Ty59; top59 = λ _ _ top59 _ _ _ _ → top59
bot59 : Ty59; bot59 = λ _ _ _ bot59 _ _ _ → bot59
arr59 : Ty59 → Ty59 → Ty59; arr59
= λ A B Ty59 nat59 top59 bot59 arr59 prod sum →
arr59 (A Ty59 nat59 top59 bot59 arr59 prod sum) (B Ty59 nat59 top59 bot59 arr59 prod sum)
prod59 : Ty59 → Ty59 → Ty59; prod59
= λ A B Ty59 nat59 top59 bot59 arr59 prod59 sum →
prod59 (A Ty59 nat59 top59 bot59 arr59 prod59 sum) (B Ty59 nat59 top59 bot59 arr59 prod59 sum)
sum59 : Ty59 → Ty59 → Ty59; sum59
= λ A B Ty59 nat59 top59 bot59 arr59 prod59 sum59 →
sum59 (A Ty59 nat59 top59 bot59 arr59 prod59 sum59) (B Ty59 nat59 top59 bot59 arr59 prod59 sum59)
Con59 : Set; Con59
= (Con59 : Set)
(nil : Con59)
(snoc : Con59 → Ty59 → Con59)
→ Con59
nil59 : Con59; nil59
= λ Con59 nil59 snoc → nil59
snoc59 : Con59 → Ty59 → Con59; snoc59
= λ Γ A Con59 nil59 snoc59 → snoc59 (Γ Con59 nil59 snoc59) A
Var59 : Con59 → Ty59 → Set; Var59
= λ Γ A →
(Var59 : Con59 → Ty59 → Set)
(vz : ∀ Γ A → Var59 (snoc59 Γ A) A)
(vs : ∀ Γ B A → Var59 Γ A → Var59 (snoc59 Γ B) A)
→ Var59 Γ A
vz59 : ∀{Γ A} → Var59 (snoc59 Γ A) A; vz59
= λ Var59 vz59 vs → vz59 _ _
vs59 : ∀{Γ B A} → Var59 Γ A → Var59 (snoc59 Γ B) A; vs59
= λ x Var59 vz59 vs59 → vs59 _ _ _ (x Var59 vz59 vs59)
Tm59 : Con59 → Ty59 → Set; Tm59
= λ Γ A →
(Tm59 : Con59 → Ty59 → Set)
(var : ∀ Γ A → Var59 Γ A → Tm59 Γ A)
(lam : ∀ Γ A B → Tm59 (snoc59 Γ A) B → Tm59 Γ (arr59 A B))
(app : ∀ Γ A B → Tm59 Γ (arr59 A B) → Tm59 Γ A → Tm59 Γ B)
(tt : ∀ Γ → Tm59 Γ top59)
(pair : ∀ Γ A B → Tm59 Γ A → Tm59 Γ B → Tm59 Γ (prod59 A B))
(fst : ∀ Γ A B → Tm59 Γ (prod59 A B) → Tm59 Γ A)
(snd : ∀ Γ A B → Tm59 Γ (prod59 A B) → Tm59 Γ B)
(left : ∀ Γ A B → Tm59 Γ A → Tm59 Γ (sum59 A B))
(right : ∀ Γ A B → Tm59 Γ B → Tm59 Γ (sum59 A B))
(case : ∀ Γ A B C → Tm59 Γ (sum59 A B) → Tm59 Γ (arr59 A C) → Tm59 Γ (arr59 B C) → Tm59 Γ C)
(zero : ∀ Γ → Tm59 Γ nat59)
(suc : ∀ Γ → Tm59 Γ nat59 → Tm59 Γ nat59)
(rec : ∀ Γ A → Tm59 Γ nat59 → Tm59 Γ (arr59 nat59 (arr59 A A)) → Tm59 Γ A → Tm59 Γ A)
→ Tm59 Γ A
var59 : ∀{Γ A} → Var59 Γ A → Tm59 Γ A; var59
= λ x Tm59 var59 lam app tt pair fst snd left right case zero suc rec →
var59 _ _ x
lam59 : ∀{Γ A B} → Tm59 (snoc59 Γ A) B → Tm59 Γ (arr59 A B); lam59
= λ t Tm59 var59 lam59 app tt pair fst snd left right case zero suc rec →
lam59 _ _ _ (t Tm59 var59 lam59 app tt pair fst snd left right case zero suc rec)
app59 : ∀{Γ A B} → Tm59 Γ (arr59 A B) → Tm59 Γ A → Tm59 Γ B; app59
= λ t u Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec →
app59 _ _ _ (t Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec)
(u Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec)
tt59 : ∀{Γ} → Tm59 Γ top59; tt59
= λ Tm59 var59 lam59 app59 tt59 pair fst snd left right case zero suc rec → tt59 _
pair59 : ∀{Γ A B} → Tm59 Γ A → Tm59 Γ B → Tm59 Γ (prod59 A B); pair59
= λ t u Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec →
pair59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec)
(u Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec)
fst59 : ∀{Γ A B} → Tm59 Γ (prod59 A B) → Tm59 Γ A; fst59
= λ t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd left right case zero suc rec →
fst59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd left right case zero suc rec)
snd59 : ∀{Γ A B} → Tm59 Γ (prod59 A B) → Tm59 Γ B; snd59
= λ t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left right case zero suc rec →
snd59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left right case zero suc rec)
left59 : ∀{Γ A B} → Tm59 Γ A → Tm59 Γ (sum59 A B); left59
= λ t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right case zero suc rec →
left59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right case zero suc rec)
right59 : ∀{Γ A B} → Tm59 Γ B → Tm59 Γ (sum59 A B); right59
= λ t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case zero suc rec →
right59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case zero suc rec)
case59 : ∀{Γ A B C} → Tm59 Γ (sum59 A B) → Tm59 Γ (arr59 A C) → Tm59 Γ (arr59 B C) → Tm59 Γ C; case59
= λ t u v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec →
case59 _ _ _ _
(t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
(u Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
(v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
zero59 : ∀{Γ} → Tm59 Γ nat59; zero59
= λ Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc rec → zero59 _
suc59 : ∀{Γ} → Tm59 Γ nat59 → Tm59 Γ nat59; suc59
= λ t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec →
suc59 _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec)
rec59 : ∀{Γ A} → Tm59 Γ nat59 → Tm59 Γ (arr59 nat59 (arr59 A A)) → Tm59 Γ A → Tm59 Γ A; rec59
= λ t u v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59 →
rec59 _ _
(t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
(u Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
(v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
v059 : ∀{Γ A} → Tm59 (snoc59 Γ A) A; v059
= var59 vz59
v159 : ∀{Γ A B} → Tm59 (snoc59 (snoc59 Γ A) B) A; v159
= var59 (vs59 vz59)
v259 : ∀{Γ A B C} → Tm59 (snoc59 (snoc59 (snoc59 Γ A) B) C) A; v259
= var59 (vs59 (vs59 vz59))
v359 : ∀{Γ A B C D} → Tm59 (snoc59 (snoc59 (snoc59 (snoc59 Γ A) B) C) D) A; v359
= var59 (vs59 (vs59 (vs59 vz59)))
tbool59 : Ty59; tbool59
= sum59 top59 top59
true59 : ∀{Γ} → Tm59 Γ tbool59; true59
= left59 tt59
tfalse59 : ∀{Γ} → Tm59 Γ tbool59; tfalse59
= right59 tt59
ifthenelse59 : ∀{Γ A} → Tm59 Γ (arr59 tbool59 (arr59 A (arr59 A A))); ifthenelse59
= lam59 (lam59 (lam59 (case59 v259 (lam59 v259) (lam59 v159))))
times459 : ∀{Γ A} → Tm59 Γ (arr59 (arr59 A A) (arr59 A A)); times459
= lam59 (lam59 (app59 v159 (app59 v159 (app59 v159 (app59 v159 v059)))))
add59 : ∀{Γ} → Tm59 Γ (arr59 nat59 (arr59 nat59 nat59)); add59
= lam59 (rec59 v059
(lam59 (lam59 (lam59 (suc59 (app59 v159 v059)))))
(lam59 v059))
mul59 : ∀{Γ} → Tm59 Γ (arr59 nat59 (arr59 nat59 nat59)); mul59
= lam59 (rec59 v059
(lam59 (lam59 (lam59 (app59 (app59 add59 (app59 v159 v059)) v059))))
(lam59 zero59))
fact59 : ∀{Γ} → Tm59 Γ (arr59 nat59 nat59); fact59
= lam59 (rec59 v059 (lam59 (lam59 (app59 (app59 mul59 (suc59 v159)) v059)))
(suc59 zero59))
{-# OPTIONS --type-in-type #-}
Ty60 : Set
Ty60 =
(Ty60 : Set)
(nat top bot : Ty60)
(arr prod sum : Ty60 → Ty60 → Ty60)
→ Ty60
nat60 : Ty60; nat60 = λ _ nat60 _ _ _ _ _ → nat60
top60 : Ty60; top60 = λ _ _ top60 _ _ _ _ → top60
bot60 : Ty60; bot60 = λ _ _ _ bot60 _ _ _ → bot60
arr60 : Ty60 → Ty60 → Ty60; arr60
= λ A B Ty60 nat60 top60 bot60 arr60 prod sum →
arr60 (A Ty60 nat60 top60 bot60 arr60 prod sum) (B Ty60 nat60 top60 bot60 arr60 prod sum)
prod60 : Ty60 → Ty60 → Ty60; prod60
= λ A B Ty60 nat60 top60 bot60 arr60 prod60 sum →
prod60 (A Ty60 nat60 top60 bot60 arr60 prod60 sum) (B Ty60 nat60 top60 bot60 arr60 prod60 sum)
sum60 : Ty60 → Ty60 → Ty60; sum60
= λ A B Ty60 nat60 top60 bot60 arr60 prod60 sum60 →
sum60 (A Ty60 nat60 top60 bot60 arr60 prod60 sum60) (B Ty60 nat60 top60 bot60 arr60 prod60 sum60)
Con60 : Set; Con60
= (Con60 : Set)
(nil : Con60)
(snoc : Con60 → Ty60 → Con60)
→ Con60
nil60 : Con60; nil60
= λ Con60 nil60 snoc → nil60
snoc60 : Con60 → Ty60 → Con60; snoc60
= λ Γ A Con60 nil60 snoc60 → snoc60 (Γ Con60 nil60 snoc60) A
Var60 : Con60 → Ty60 → Set; Var60
= λ Γ A →
(Var60 : Con60 → Ty60 → Set)
(vz : ∀ Γ A → Var60 (snoc60 Γ A) A)
(vs : ∀ Γ B A → Var60 Γ A → Var60 (snoc60 Γ B) A)
→ Var60 Γ A
vz60 : ∀{Γ A} → Var60 (snoc60 Γ A) A; vz60
= λ Var60 vz60 vs → vz60 _ _
vs60 : ∀{Γ B A} → Var60 Γ A → Var60 (snoc60 Γ B) A; vs60
= λ x Var60 vz60 vs60 → vs60 _ _ _ (x Var60 vz60 vs60)
Tm60 : Con60 → Ty60 → Set; Tm60
= λ Γ A →
(Tm60 : Con60 → Ty60 → Set)
(var : ∀ Γ A → Var60 Γ A → Tm60 Γ A)
(lam : ∀ Γ A B → Tm60 (snoc60 Γ A) B → Tm60 Γ (arr60 A B))
(app : ∀ Γ A B → Tm60 Γ (arr60 A B) → Tm60 Γ A → Tm60 Γ B)
(tt : ∀ Γ → Tm60 Γ top60)
(pair : ∀ Γ A B → Tm60 Γ A → Tm60 Γ B → Tm60 Γ (prod60 A B))
(fst : ∀ Γ A B → Tm60 Γ (prod60 A B) → Tm60 Γ A)
(snd : ∀ Γ A B → Tm60 Γ (prod60 A B) → Tm60 Γ B)
(left : ∀ Γ A B → Tm60 Γ A → Tm60 Γ (sum60 A B))
(right : ∀ Γ A B → Tm60 Γ B → Tm60 Γ (sum60 A B))
(case : ∀ Γ A B C → Tm60 Γ (sum60 A B) → Tm60 Γ (arr60 A C) → Tm60 Γ (arr60 B C) → Tm60 Γ C)
(zero : ∀ Γ → Tm60 Γ nat60)
(suc : ∀ Γ → Tm60 Γ nat60 → Tm60 Γ nat60)
(rec : ∀ Γ A → Tm60 Γ nat60 → Tm60 Γ (arr60 nat60 (arr60 A A)) → Tm60 Γ A → Tm60 Γ A)
→ Tm60 Γ A
var60 : ∀{Γ A} → Var60 Γ A → Tm60 Γ A; var60
= λ x Tm60 var60 lam app tt pair fst snd left right case zero suc rec →
var60 _ _ x
lam60 : ∀{Γ A B} → Tm60 (snoc60 Γ A) B → Tm60 Γ (arr60 A B); lam60
= λ t Tm60 var60 lam60 app tt pair fst snd left right case zero suc rec →
lam60 _ _ _ (t Tm60 var60 lam60 app tt pair fst snd left right case zero suc rec)
app60 : ∀{Γ A B} → Tm60 Γ (arr60 A B) → Tm60 Γ A → Tm60 Γ B; app60
= λ t u Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec →
app60 _ _ _ (t Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec)
(u Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec)
tt60 : ∀{Γ} → Tm60 Γ top60; tt60
= λ Tm60 var60 lam60 app60 tt60 pair fst snd left right case zero suc rec → tt60 _
pair60 : ∀{Γ A B} → Tm60 Γ A → Tm60 Γ B → Tm60 Γ (prod60 A B); pair60
= λ t u Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec →
pair60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec)
(u Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec)
fst60 : ∀{Γ A B} → Tm60 Γ (prod60 A B) → Tm60 Γ A; fst60
= λ t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd left right case zero suc rec →
fst60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd left right case zero suc rec)
snd60 : ∀{Γ A B} → Tm60 Γ (prod60 A B) → Tm60 Γ B; snd60
= λ t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left right case zero suc rec →
snd60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left right case zero suc rec)
left60 : ∀{Γ A B} → Tm60 Γ A → Tm60 Γ (sum60 A B); left60
= λ t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right case zero suc rec →
left60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right case zero suc rec)
right60 : ∀{Γ A B} → Tm60 Γ B → Tm60 Γ (sum60 A B); right60
= λ t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case zero suc rec →
right60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case zero suc rec)
case60 : ∀{Γ A B C} → Tm60 Γ (sum60 A B) → Tm60 Γ (arr60 A C) → Tm60 Γ (arr60 B C) → Tm60 Γ C; case60
= λ t u v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec →
case60 _ _ _ _
(t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
(u Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
(v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
zero60 : ∀{Γ} → Tm60 Γ nat60; zero60
= λ Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc rec → zero60 _
suc60 : ∀{Γ} → Tm60 Γ nat60 → Tm60 Γ nat60; suc60
= λ t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec →
suc60 _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec)
rec60 : ∀{Γ A} → Tm60 Γ nat60 → Tm60 Γ (arr60 nat60 (arr60 A A)) → Tm60 Γ A → Tm60 Γ A; rec60
= λ t u v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60 →
rec60 _ _
(t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
(u Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
(v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
v060 : ∀{Γ A} → Tm60 (snoc60 Γ A) A; v060
= var60 vz60
v160 : ∀{Γ A B} → Tm60 (snoc60 (snoc60 Γ A) B) A; v160
= var60 (vs60 vz60)
v260 : ∀{Γ A B C} → Tm60 (snoc60 (snoc60 (snoc60 Γ A) B) C) A; v260
= var60 (vs60 (vs60 vz60))
v360 : ∀{Γ A B C D} → Tm60 (snoc60 (snoc60 (snoc60 (snoc60 Γ A) B) C) D) A; v360
= var60 (vs60 (vs60 (vs60 vz60)))
tbool60 : Ty60; tbool60
= sum60 top60 top60
true60 : ∀{Γ} → Tm60 Γ tbool60; true60
= left60 tt60
tfalse60 : ∀{Γ} → Tm60 Γ tbool60; tfalse60
= right60 tt60
ifthenelse60 : ∀{Γ A} → Tm60 Γ (arr60 tbool60 (arr60 A (arr60 A A))); ifthenelse60
= lam60 (lam60 (lam60 (case60 v260 (lam60 v260) (lam60 v160))))
times460 : ∀{Γ A} → Tm60 Γ (arr60 (arr60 A A) (arr60 A A)); times460
= lam60 (lam60 (app60 v160 (app60 v160 (app60 v160 (app60 v160 v060)))))
add60 : ∀{Γ} → Tm60 Γ (arr60 nat60 (arr60 nat60 nat60)); add60
= lam60 (rec60 v060
(lam60 (lam60 (lam60 (suc60 (app60 v160 v060)))))
(lam60 v060))
mul60 : ∀{Γ} → Tm60 Γ (arr60 nat60 (arr60 nat60 nat60)); mul60
= lam60 (rec60 v060
(lam60 (lam60 (lam60 (app60 (app60 add60 (app60 v160 v060)) v060))))
(lam60 zero60))
fact60 : ∀{Γ} → Tm60 Γ (arr60 nat60 nat60); fact60
= lam60 (rec60 v060 (lam60 (lam60 (app60 (app60 mul60 (suc60 v160)) v060)))
(suc60 zero60))
{-# OPTIONS --type-in-type #-}
Ty61 : Set
Ty61 =
(Ty61 : Set)
(nat top bot : Ty61)
(arr prod sum : Ty61 → Ty61 → Ty61)
→ Ty61
nat61 : Ty61; nat61 = λ _ nat61 _ _ _ _ _ → nat61
top61 : Ty61; top61 = λ _ _ top61 _ _ _ _ → top61
bot61 : Ty61; bot61 = λ _ _ _ bot61 _ _ _ → bot61
arr61 : Ty61 → Ty61 → Ty61; arr61
= λ A B Ty61 nat61 top61 bot61 arr61 prod sum →
arr61 (A Ty61 nat61 top61 bot61 arr61 prod sum) (B Ty61 nat61 top61 bot61 arr61 prod sum)
prod61 : Ty61 → Ty61 → Ty61; prod61
= λ A B Ty61 nat61 top61 bot61 arr61 prod61 sum →
prod61 (A Ty61 nat61 top61 bot61 arr61 prod61 sum) (B Ty61 nat61 top61 bot61 arr61 prod61 sum)
sum61 : Ty61 → Ty61 → Ty61; sum61
= λ A B Ty61 nat61 top61 bot61 arr61 prod61 sum61 →
sum61 (A Ty61 nat61 top61 bot61 arr61 prod61 sum61) (B Ty61 nat61 top61 bot61 arr61 prod61 sum61)
Con61 : Set; Con61
= (Con61 : Set)
(nil : Con61)
(snoc : Con61 → Ty61 → Con61)
→ Con61
nil61 : Con61; nil61
= λ Con61 nil61 snoc → nil61
snoc61 : Con61 → Ty61 → Con61; snoc61
= λ Γ A Con61 nil61 snoc61 → snoc61 (Γ Con61 nil61 snoc61) A
Var61 : Con61 → Ty61 → Set; Var61
= λ Γ A →
(Var61 : Con61 → Ty61 → Set)
(vz : ∀ Γ A → Var61 (snoc61 Γ A) A)
(vs : ∀ Γ B A → Var61 Γ A → Var61 (snoc61 Γ B) A)
→ Var61 Γ A
vz61 : ∀{Γ A} → Var61 (snoc61 Γ A) A; vz61
= λ Var61 vz61 vs → vz61 _ _
vs61 : ∀{Γ B A} → Var61 Γ A → Var61 (snoc61 Γ B) A; vs61
= λ x Var61 vz61 vs61 → vs61 _ _ _ (x Var61 vz61 vs61)
Tm61 : Con61 → Ty61 → Set; Tm61
= λ Γ A →
(Tm61 : Con61 → Ty61 → Set)
(var : ∀ Γ A → Var61 Γ A → Tm61 Γ A)
(lam : ∀ Γ A B → Tm61 (snoc61 Γ A) B → Tm61 Γ (arr61 A B))
(app : ∀ Γ A B → Tm61 Γ (arr61 A B) → Tm61 Γ A → Tm61 Γ B)
(tt : ∀ Γ → Tm61 Γ top61)
(pair : ∀ Γ A B → Tm61 Γ A → Tm61 Γ B → Tm61 Γ (prod61 A B))
(fst : ∀ Γ A B → Tm61 Γ (prod61 A B) → Tm61 Γ A)
(snd : ∀ Γ A B → Tm61 Γ (prod61 A B) → Tm61 Γ B)
(left : ∀ Γ A B → Tm61 Γ A → Tm61 Γ (sum61 A B))
(right : ∀ Γ A B → Tm61 Γ B → Tm61 Γ (sum61 A B))
(case : ∀ Γ A B C → Tm61 Γ (sum61 A B) → Tm61 Γ (arr61 A C) → Tm61 Γ (arr61 B C) → Tm61 Γ C)
(zero : ∀ Γ → Tm61 Γ nat61)
(suc : ∀ Γ → Tm61 Γ nat61 → Tm61 Γ nat61)
(rec : ∀ Γ A → Tm61 Γ nat61 → Tm61 Γ (arr61 nat61 (arr61 A A)) → Tm61 Γ A → Tm61 Γ A)
→ Tm61 Γ A
var61 : ∀{Γ A} → Var61 Γ A → Tm61 Γ A; var61
= λ x Tm61 var61 lam app tt pair fst snd left right case zero suc rec →
var61 _ _ x
lam61 : ∀{Γ A B} → Tm61 (snoc61 Γ A) B → Tm61 Γ (arr61 A B); lam61
= λ t Tm61 var61 lam61 app tt pair fst snd left right case zero suc rec →
lam61 _ _ _ (t Tm61 var61 lam61 app tt pair fst snd left right case zero suc rec)
app61 : ∀{Γ A B} → Tm61 Γ (arr61 A B) → Tm61 Γ A → Tm61 Γ B; app61
= λ t u Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec →
app61 _ _ _ (t Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec)
(u Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec)
tt61 : ∀{Γ} → Tm61 Γ top61; tt61
= λ Tm61 var61 lam61 app61 tt61 pair fst snd left right case zero suc rec → tt61 _
pair61 : ∀{Γ A B} → Tm61 Γ A → Tm61 Γ B → Tm61 Γ (prod61 A B); pair61
= λ t u Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec →
pair61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec)
(u Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec)
fst61 : ∀{Γ A B} → Tm61 Γ (prod61 A B) → Tm61 Γ A; fst61
= λ t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd left right case zero suc rec →
fst61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd left right case zero suc rec)
snd61 : ∀{Γ A B} → Tm61 Γ (prod61 A B) → Tm61 Γ B; snd61
= λ t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left right case zero suc rec →
snd61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left right case zero suc rec)
left61 : ∀{Γ A B} → Tm61 Γ A → Tm61 Γ (sum61 A B); left61
= λ t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right case zero suc rec →
left61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right case zero suc rec)
right61 : ∀{Γ A B} → Tm61 Γ B → Tm61 Γ (sum61 A B); right61
= λ t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case zero suc rec →
right61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case zero suc rec)
case61 : ∀{Γ A B C} → Tm61 Γ (sum61 A B) → Tm61 Γ (arr61 A C) → Tm61 Γ (arr61 B C) → Tm61 Γ C; case61
= λ t u v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec →
case61 _ _ _ _
(t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
(u Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
(v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
zero61 : ∀{Γ} → Tm61 Γ nat61; zero61
= λ Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc rec → zero61 _
suc61 : ∀{Γ} → Tm61 Γ nat61 → Tm61 Γ nat61; suc61
= λ t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec →
suc61 _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec)
rec61 : ∀{Γ A} → Tm61 Γ nat61 → Tm61 Γ (arr61 nat61 (arr61 A A)) → Tm61 Γ A → Tm61 Γ A; rec61
= λ t u v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61 →
rec61 _ _
(t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
(u Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
(v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
v061 : ∀{Γ A} → Tm61 (snoc61 Γ A) A; v061
= var61 vz61
v161 : ∀{Γ A B} → Tm61 (snoc61 (snoc61 Γ A) B) A; v161
= var61 (vs61 vz61)
v261 : ∀{Γ A B C} → Tm61 (snoc61 (snoc61 (snoc61 Γ A) B) C) A; v261
= var61 (vs61 (vs61 vz61))
v361 : ∀{Γ A B C D} → Tm61 (snoc61 (snoc61 (snoc61 (snoc61 Γ A) B) C) D) A; v361
= var61 (vs61 (vs61 (vs61 vz61)))
tbool61 : Ty61; tbool61
= sum61 top61 top61
true61 : ∀{Γ} → Tm61 Γ tbool61; true61
= left61 tt61
tfalse61 : ∀{Γ} → Tm61 Γ tbool61; tfalse61
= right61 tt61
ifthenelse61 : ∀{Γ A} → Tm61 Γ (arr61 tbool61 (arr61 A (arr61 A A))); ifthenelse61
= lam61 (lam61 (lam61 (case61 v261 (lam61 v261) (lam61 v161))))
times461 : ∀{Γ A} → Tm61 Γ (arr61 (arr61 A A) (arr61 A A)); times461
= lam61 (lam61 (app61 v161 (app61 v161 (app61 v161 (app61 v161 v061)))))
add61 : ∀{Γ} → Tm61 Γ (arr61 nat61 (arr61 nat61 nat61)); add61
= lam61 (rec61 v061
(lam61 (lam61 (lam61 (suc61 (app61 v161 v061)))))
(lam61 v061))
mul61 : ∀{Γ} → Tm61 Γ (arr61 nat61 (arr61 nat61 nat61)); mul61
= lam61 (rec61 v061
(lam61 (lam61 (lam61 (app61 (app61 add61 (app61 v161 v061)) v061))))
(lam61 zero61))
fact61 : ∀{Γ} → Tm61 Γ (arr61 nat61 nat61); fact61
= lam61 (rec61 v061 (lam61 (lam61 (app61 (app61 mul61 (suc61 v161)) v061)))
(suc61 zero61))
{-# OPTIONS --type-in-type #-}
Ty62 : Set
Ty62 =
(Ty62 : Set)
(nat top bot : Ty62)
(arr prod sum : Ty62 → Ty62 → Ty62)
→ Ty62
nat62 : Ty62; nat62 = λ _ nat62 _ _ _ _ _ → nat62
top62 : Ty62; top62 = λ _ _ top62 _ _ _ _ → top62
bot62 : Ty62; bot62 = λ _ _ _ bot62 _ _ _ → bot62
arr62 : Ty62 → Ty62 → Ty62; arr62
= λ A B Ty62 nat62 top62 bot62 arr62 prod sum →
arr62 (A Ty62 nat62 top62 bot62 arr62 prod sum) (B Ty62 nat62 top62 bot62 arr62 prod sum)
prod62 : Ty62 → Ty62 → Ty62; prod62
= λ A B Ty62 nat62 top62 bot62 arr62 prod62 sum →
prod62 (A Ty62 nat62 top62 bot62 arr62 prod62 sum) (B Ty62 nat62 top62 bot62 arr62 prod62 sum)
sum62 : Ty62 → Ty62 → Ty62; sum62
= λ A B Ty62 nat62 top62 bot62 arr62 prod62 sum62 →
sum62 (A Ty62 nat62 top62 bot62 arr62 prod62 sum62) (B Ty62 nat62 top62 bot62 arr62 prod62 sum62)
Con62 : Set; Con62
= (Con62 : Set)
(nil : Con62)
(snoc : Con62 → Ty62 → Con62)
→ Con62
nil62 : Con62; nil62
= λ Con62 nil62 snoc → nil62
snoc62 : Con62 → Ty62 → Con62; snoc62
= λ Γ A Con62 nil62 snoc62 → snoc62 (Γ Con62 nil62 snoc62) A
Var62 : Con62 → Ty62 → Set; Var62
= λ Γ A →
(Var62 : Con62 → Ty62 → Set)
(vz : ∀ Γ A → Var62 (snoc62 Γ A) A)
(vs : ∀ Γ B A → Var62 Γ A → Var62 (snoc62 Γ B) A)
→ Var62 Γ A
vz62 : ∀{Γ A} → Var62 (snoc62 Γ A) A; vz62
= λ Var62 vz62 vs → vz62 _ _
vs62 : ∀{Γ B A} → Var62 Γ A → Var62 (snoc62 Γ B) A; vs62
= λ x Var62 vz62 vs62 → vs62 _ _ _ (x Var62 vz62 vs62)
Tm62 : Con62 → Ty62 → Set; Tm62
= λ Γ A →
(Tm62 : Con62 → Ty62 → Set)
(var : ∀ Γ A → Var62 Γ A → Tm62 Γ A)
(lam : ∀ Γ A B → Tm62 (snoc62 Γ A) B → Tm62 Γ (arr62 A B))
(app : ∀ Γ A B → Tm62 Γ (arr62 A B) → Tm62 Γ A → Tm62 Γ B)
(tt : ∀ Γ → Tm62 Γ top62)
(pair : ∀ Γ A B → Tm62 Γ A → Tm62 Γ B → Tm62 Γ (prod62 A B))
(fst : ∀ Γ A B → Tm62 Γ (prod62 A B) → Tm62 Γ A)
(snd : ∀ Γ A B → Tm62 Γ (prod62 A B) → Tm62 Γ B)
(left : ∀ Γ A B → Tm62 Γ A → Tm62 Γ (sum62 A B))
(right : ∀ Γ A B → Tm62 Γ B → Tm62 Γ (sum62 A B))
(case : ∀ Γ A B C → Tm62 Γ (sum62 A B) → Tm62 Γ (arr62 A C) → Tm62 Γ (arr62 B C) → Tm62 Γ C)
(zero : ∀ Γ → Tm62 Γ nat62)
(suc : ∀ Γ → Tm62 Γ nat62 → Tm62 Γ nat62)
(rec : ∀ Γ A → Tm62 Γ nat62 → Tm62 Γ (arr62 nat62 (arr62 A A)) → Tm62 Γ A → Tm62 Γ A)
→ Tm62 Γ A
var62 : ∀{Γ A} → Var62 Γ A → Tm62 Γ A; var62
= λ x Tm62 var62 lam app tt pair fst snd left right case zero suc rec →
var62 _ _ x
lam62 : ∀{Γ A B} → Tm62 (snoc62 Γ A) B → Tm62 Γ (arr62 A B); lam62
= λ t Tm62 var62 lam62 app tt pair fst snd left right case zero suc rec →
lam62 _ _ _ (t Tm62 var62 lam62 app tt pair fst snd left right case zero suc rec)
app62 : ∀{Γ A B} → Tm62 Γ (arr62 A B) → Tm62 Γ A → Tm62 Γ B; app62
= λ t u Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec →
app62 _ _ _ (t Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec)
(u Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec)
tt62 : ∀{Γ} → Tm62 Γ top62; tt62
= λ Tm62 var62 lam62 app62 tt62 pair fst snd left right case zero suc rec → tt62 _
pair62 : ∀{Γ A B} → Tm62 Γ A → Tm62 Γ B → Tm62 Γ (prod62 A B); pair62
= λ t u Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec →
pair62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec)
(u Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec)
fst62 : ∀{Γ A B} → Tm62 Γ (prod62 A B) → Tm62 Γ A; fst62
= λ t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd left right case zero suc rec →
fst62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd left right case zero suc rec)
snd62 : ∀{Γ A B} → Tm62 Γ (prod62 A B) → Tm62 Γ B; snd62
= λ t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left right case zero suc rec →
snd62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left right case zero suc rec)
left62 : ∀{Γ A B} → Tm62 Γ A → Tm62 Γ (sum62 A B); left62
= λ t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right case zero suc rec →
left62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right case zero suc rec)
right62 : ∀{Γ A B} → Tm62 Γ B → Tm62 Γ (sum62 A B); right62
= λ t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case zero suc rec →
right62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case zero suc rec)
case62 : ∀{Γ A B C} → Tm62 Γ (sum62 A B) → Tm62 Γ (arr62 A C) → Tm62 Γ (arr62 B C) → Tm62 Γ C; case62
= λ t u v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec →
case62 _ _ _ _
(t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
(u Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
(v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
zero62 : ∀{Γ} → Tm62 Γ nat62; zero62
= λ Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc rec → zero62 _
suc62 : ∀{Γ} → Tm62 Γ nat62 → Tm62 Γ nat62; suc62
= λ t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec →
suc62 _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec)
rec62 : ∀{Γ A} → Tm62 Γ nat62 → Tm62 Γ (arr62 nat62 (arr62 A A)) → Tm62 Γ A → Tm62 Γ A; rec62
= λ t u v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62 →
rec62 _ _
(t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
(u Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
(v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
v062 : ∀{Γ A} → Tm62 (snoc62 Γ A) A; v062
= var62 vz62
v162 : ∀{Γ A B} → Tm62 (snoc62 (snoc62 Γ A) B) A; v162
= var62 (vs62 vz62)
v262 : ∀{Γ A B C} → Tm62 (snoc62 (snoc62 (snoc62 Γ A) B) C) A; v262
= var62 (vs62 (vs62 vz62))
v362 : ∀{Γ A B C D} → Tm62 (snoc62 (snoc62 (snoc62 (snoc62 Γ A) B) C) D) A; v362
= var62 (vs62 (vs62 (vs62 vz62)))
tbool62 : Ty62; tbool62
= sum62 top62 top62
true62 : ∀{Γ} → Tm62 Γ tbool62; true62
= left62 tt62
tfalse62 : ∀{Γ} → Tm62 Γ tbool62; tfalse62
= right62 tt62
ifthenelse62 : ∀{Γ A} → Tm62 Γ (arr62 tbool62 (arr62 A (arr62 A A))); ifthenelse62
= lam62 (lam62 (lam62 (case62 v262 (lam62 v262) (lam62 v162))))
times462 : ∀{Γ A} → Tm62 Γ (arr62 (arr62 A A) (arr62 A A)); times462
= lam62 (lam62 (app62 v162 (app62 v162 (app62 v162 (app62 v162 v062)))))
add62 : ∀{Γ} → Tm62 Γ (arr62 nat62 (arr62 nat62 nat62)); add62
= lam62 (rec62 v062
(lam62 (lam62 (lam62 (suc62 (app62 v162 v062)))))
(lam62 v062))
mul62 : ∀{Γ} → Tm62 Γ (arr62 nat62 (arr62 nat62 nat62)); mul62
= lam62 (rec62 v062
(lam62 (lam62 (lam62 (app62 (app62 add62 (app62 v162 v062)) v062))))
(lam62 zero62))
fact62 : ∀{Γ} → Tm62 Γ (arr62 nat62 nat62); fact62
= lam62 (rec62 v062 (lam62 (lam62 (app62 (app62 mul62 (suc62 v162)) v062)))
(suc62 zero62))
{-# OPTIONS --type-in-type #-}
Ty63 : Set
Ty63 =
(Ty63 : Set)
(nat top bot : Ty63)
(arr prod sum : Ty63 → Ty63 → Ty63)
→ Ty63
nat63 : Ty63; nat63 = λ _ nat63 _ _ _ _ _ → nat63
top63 : Ty63; top63 = λ _ _ top63 _ _ _ _ → top63
bot63 : Ty63; bot63 = λ _ _ _ bot63 _ _ _ → bot63
arr63 : Ty63 → Ty63 → Ty63; arr63
= λ A B Ty63 nat63 top63 bot63 arr63 prod sum →
arr63 (A Ty63 nat63 top63 bot63 arr63 prod sum) (B Ty63 nat63 top63 bot63 arr63 prod sum)
prod63 : Ty63 → Ty63 → Ty63; prod63
= λ A B Ty63 nat63 top63 bot63 arr63 prod63 sum →
prod63 (A Ty63 nat63 top63 bot63 arr63 prod63 sum) (B Ty63 nat63 top63 bot63 arr63 prod63 sum)
sum63 : Ty63 → Ty63 → Ty63; sum63
= λ A B Ty63 nat63 top63 bot63 arr63 prod63 sum63 →
sum63 (A Ty63 nat63 top63 bot63 arr63 prod63 sum63) (B Ty63 nat63 top63 bot63 arr63 prod63 sum63)
Con63 : Set; Con63
= (Con63 : Set)
(nil : Con63)
(snoc : Con63 → Ty63 → Con63)
→ Con63
nil63 : Con63; nil63
= λ Con63 nil63 snoc → nil63
snoc63 : Con63 → Ty63 → Con63; snoc63
= λ Γ A Con63 nil63 snoc63 → snoc63 (Γ Con63 nil63 snoc63) A
Var63 : Con63 → Ty63 → Set; Var63
= λ Γ A →
(Var63 : Con63 → Ty63 → Set)
(vz : ∀ Γ A → Var63 (snoc63 Γ A) A)
(vs : ∀ Γ B A → Var63 Γ A → Var63 (snoc63 Γ B) A)
→ Var63 Γ A
vz63 : ∀{Γ A} → Var63 (snoc63 Γ A) A; vz63
= λ Var63 vz63 vs → vz63 _ _
vs63 : ∀{Γ B A} → Var63 Γ A → Var63 (snoc63 Γ B) A; vs63
= λ x Var63 vz63 vs63 → vs63 _ _ _ (x Var63 vz63 vs63)
Tm63 : Con63 → Ty63 → Set; Tm63
= λ Γ A →
(Tm63 : Con63 → Ty63 → Set)
(var : ∀ Γ A → Var63 Γ A → Tm63 Γ A)
(lam : ∀ Γ A B → Tm63 (snoc63 Γ A) B → Tm63 Γ (arr63 A B))
(app : ∀ Γ A B → Tm63 Γ (arr63 A B) → Tm63 Γ A → Tm63 Γ B)
(tt : ∀ Γ → Tm63 Γ top63)
(pair : ∀ Γ A B → Tm63 Γ A → Tm63 Γ B → Tm63 Γ (prod63 A B))
(fst : ∀ Γ A B → Tm63 Γ (prod63 A B) → Tm63 Γ A)
(snd : ∀ Γ A B → Tm63 Γ (prod63 A B) → Tm63 Γ B)
(left : ∀ Γ A B → Tm63 Γ A → Tm63 Γ (sum63 A B))
(right : ∀ Γ A B → Tm63 Γ B → Tm63 Γ (sum63 A B))
(case : ∀ Γ A B C → Tm63 Γ (sum63 A B) → Tm63 Γ (arr63 A C) → Tm63 Γ (arr63 B C) → Tm63 Γ C)
(zero : ∀ Γ → Tm63 Γ nat63)
(suc : ∀ Γ → Tm63 Γ nat63 → Tm63 Γ nat63)
(rec : ∀ Γ A → Tm63 Γ nat63 → Tm63 Γ (arr63 nat63 (arr63 A A)) → Tm63 Γ A → Tm63 Γ A)
→ Tm63 Γ A
var63 : ∀{Γ A} → Var63 Γ A → Tm63 Γ A; var63
= λ x Tm63 var63 lam app tt pair fst snd left right case zero suc rec →
var63 _ _ x
lam63 : ∀{Γ A B} → Tm63 (snoc63 Γ A) B → Tm63 Γ (arr63 A B); lam63
= λ t Tm63 var63 lam63 app tt pair fst snd left right case zero suc rec →
lam63 _ _ _ (t Tm63 var63 lam63 app tt pair fst snd left right case zero suc rec)
app63 : ∀{Γ A B} → Tm63 Γ (arr63 A B) → Tm63 Γ A → Tm63 Γ B; app63
= λ t u Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec →
app63 _ _ _ (t Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec)
(u Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec)
tt63 : ∀{Γ} → Tm63 Γ top63; tt63
= λ Tm63 var63 lam63 app63 tt63 pair fst snd left right case zero suc rec → tt63 _
pair63 : ∀{Γ A B} → Tm63 Γ A → Tm63 Γ B → Tm63 Γ (prod63 A B); pair63
= λ t u Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec →
pair63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec)
(u Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec)
fst63 : ∀{Γ A B} → Tm63 Γ (prod63 A B) → Tm63 Γ A; fst63
= λ t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd left right case zero suc rec →
fst63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd left right case zero suc rec)
snd63 : ∀{Γ A B} → Tm63 Γ (prod63 A B) → Tm63 Γ B; snd63
= λ t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left right case zero suc rec →
snd63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left right case zero suc rec)
left63 : ∀{Γ A B} → Tm63 Γ A → Tm63 Γ (sum63 A B); left63
= λ t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right case zero suc rec →
left63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right case zero suc rec)
right63 : ∀{Γ A B} → Tm63 Γ B → Tm63 Γ (sum63 A B); right63
= λ t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case zero suc rec →
right63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case zero suc rec)
case63 : ∀{Γ A B C} → Tm63 Γ (sum63 A B) → Tm63 Γ (arr63 A C) → Tm63 Γ (arr63 B C) → Tm63 Γ C; case63
= λ t u v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec →
case63 _ _ _ _
(t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
(u Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
(v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
zero63 : ∀{Γ} → Tm63 Γ nat63; zero63
= λ Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc rec → zero63 _
suc63 : ∀{Γ} → Tm63 Γ nat63 → Tm63 Γ nat63; suc63
= λ t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec →
suc63 _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec)
rec63 : ∀{Γ A} → Tm63 Γ nat63 → Tm63 Γ (arr63 nat63 (arr63 A A)) → Tm63 Γ A → Tm63 Γ A; rec63
= λ t u v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63 →
rec63 _ _
(t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
(u Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
(v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
v063 : ∀{Γ A} → Tm63 (snoc63 Γ A) A; v063
= var63 vz63
v163 : ∀{Γ A B} → Tm63 (snoc63 (snoc63 Γ A) B) A; v163
= var63 (vs63 vz63)
v263 : ∀{Γ A B C} → Tm63 (snoc63 (snoc63 (snoc63 Γ A) B) C) A; v263
= var63 (vs63 (vs63 vz63))
v363 : ∀{Γ A B C D} → Tm63 (snoc63 (snoc63 (snoc63 (snoc63 Γ A) B) C) D) A; v363
= var63 (vs63 (vs63 (vs63 vz63)))
tbool63 : Ty63; tbool63
= sum63 top63 top63
true63 : ∀{Γ} → Tm63 Γ tbool63; true63
= left63 tt63
tfalse63 : ∀{Γ} → Tm63 Γ tbool63; tfalse63
= right63 tt63
ifthenelse63 : ∀{Γ A} → Tm63 Γ (arr63 tbool63 (arr63 A (arr63 A A))); ifthenelse63
= lam63 (lam63 (lam63 (case63 v263 (lam63 v263) (lam63 v163))))
times463 : ∀{Γ A} → Tm63 Γ (arr63 (arr63 A A) (arr63 A A)); times463
= lam63 (lam63 (app63 v163 (app63 v163 (app63 v163 (app63 v163 v063)))))
add63 : ∀{Γ} → Tm63 Γ (arr63 nat63 (arr63 nat63 nat63)); add63
= lam63 (rec63 v063
(lam63 (lam63 (lam63 (suc63 (app63 v163 v063)))))
(lam63 v063))
mul63 : ∀{Γ} → Tm63 Γ (arr63 nat63 (arr63 nat63 nat63)); mul63
= lam63 (rec63 v063
(lam63 (lam63 (lam63 (app63 (app63 add63 (app63 v163 v063)) v063))))
(lam63 zero63))
fact63 : ∀{Γ} → Tm63 Γ (arr63 nat63 nat63); fact63
= lam63 (rec63 v063 (lam63 (lam63 (app63 (app63 mul63 (suc63 v163)) v063)))
(suc63 zero63))
{-# OPTIONS --type-in-type #-}
Ty64 : Set
Ty64 =
(Ty64 : Set)
(nat top bot : Ty64)
(arr prod sum : Ty64 → Ty64 → Ty64)
→ Ty64
nat64 : Ty64; nat64 = λ _ nat64 _ _ _ _ _ → nat64
top64 : Ty64; top64 = λ _ _ top64 _ _ _ _ → top64
bot64 : Ty64; bot64 = λ _ _ _ bot64 _ _ _ → bot64
arr64 : Ty64 → Ty64 → Ty64; arr64
= λ A B Ty64 nat64 top64 bot64 arr64 prod sum →
arr64 (A Ty64 nat64 top64 bot64 arr64 prod sum) (B Ty64 nat64 top64 bot64 arr64 prod sum)
prod64 : Ty64 → Ty64 → Ty64; prod64
= λ A B Ty64 nat64 top64 bot64 arr64 prod64 sum →
prod64 (A Ty64 nat64 top64 bot64 arr64 prod64 sum) (B Ty64 nat64 top64 bot64 arr64 prod64 sum)
sum64 : Ty64 → Ty64 → Ty64; sum64
= λ A B Ty64 nat64 top64 bot64 arr64 prod64 sum64 →
sum64 (A Ty64 nat64 top64 bot64 arr64 prod64 sum64) (B Ty64 nat64 top64 bot64 arr64 prod64 sum64)
Con64 : Set; Con64
= (Con64 : Set)
(nil : Con64)
(snoc : Con64 → Ty64 → Con64)
→ Con64
nil64 : Con64; nil64
= λ Con64 nil64 snoc → nil64
snoc64 : Con64 → Ty64 → Con64; snoc64
= λ Γ A Con64 nil64 snoc64 → snoc64 (Γ Con64 nil64 snoc64) A
Var64 : Con64 → Ty64 → Set; Var64
= λ Γ A →
(Var64 : Con64 → Ty64 → Set)
(vz : ∀ Γ A → Var64 (snoc64 Γ A) A)
(vs : ∀ Γ B A → Var64 Γ A → Var64 (snoc64 Γ B) A)
→ Var64 Γ A
vz64 : ∀{Γ A} → Var64 (snoc64 Γ A) A; vz64
= λ Var64 vz64 vs → vz64 _ _
vs64 : ∀{Γ B A} → Var64 Γ A → Var64 (snoc64 Γ B) A; vs64
= λ x Var64 vz64 vs64 → vs64 _ _ _ (x Var64 vz64 vs64)
Tm64 : Con64 → Ty64 → Set; Tm64
= λ Γ A →
(Tm64 : Con64 → Ty64 → Set)
(var : ∀ Γ A → Var64 Γ A → Tm64 Γ A)
(lam : ∀ Γ A B → Tm64 (snoc64 Γ A) B → Tm64 Γ (arr64 A B))
(app : ∀ Γ A B → Tm64 Γ (arr64 A B) → Tm64 Γ A → Tm64 Γ B)
(tt : ∀ Γ → Tm64 Γ top64)
(pair : ∀ Γ A B → Tm64 Γ A → Tm64 Γ B → Tm64 Γ (prod64 A B))
(fst : ∀ Γ A B → Tm64 Γ (prod64 A B) → Tm64 Γ A)
(snd : ∀ Γ A B → Tm64 Γ (prod64 A B) → Tm64 Γ B)
(left : ∀ Γ A B → Tm64 Γ A → Tm64 Γ (sum64 A B))
(right : ∀ Γ A B → Tm64 Γ B → Tm64 Γ (sum64 A B))
(case : ∀ Γ A B C → Tm64 Γ (sum64 A B) → Tm64 Γ (arr64 A C) → Tm64 Γ (arr64 B C) → Tm64 Γ C)
(zero : ∀ Γ → Tm64 Γ nat64)
(suc : ∀ Γ → Tm64 Γ nat64 → Tm64 Γ nat64)
(rec : ∀ Γ A → Tm64 Γ nat64 → Tm64 Γ (arr64 nat64 (arr64 A A)) → Tm64 Γ A → Tm64 Γ A)
→ Tm64 Γ A
var64 : ∀{Γ A} → Var64 Γ A → Tm64 Γ A; var64
= λ x Tm64 var64 lam app tt pair fst snd left right case zero suc rec →
var64 _ _ x
lam64 : ∀{Γ A B} → Tm64 (snoc64 Γ A) B → Tm64 Γ (arr64 A B); lam64
= λ t Tm64 var64 lam64 app tt pair fst snd left right case zero suc rec →
lam64 _ _ _ (t Tm64 var64 lam64 app tt pair fst snd left right case zero suc rec)
app64 : ∀{Γ A B} → Tm64 Γ (arr64 A B) → Tm64 Γ A → Tm64 Γ B; app64
= λ t u Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec →
app64 _ _ _ (t Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec)
(u Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec)
tt64 : ∀{Γ} → Tm64 Γ top64; tt64
= λ Tm64 var64 lam64 app64 tt64 pair fst snd left right case zero suc rec → tt64 _
pair64 : ∀{Γ A B} → Tm64 Γ A → Tm64 Γ B → Tm64 Γ (prod64 A B); pair64
= λ t u Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec →
pair64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec)
(u Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec)
fst64 : ∀{Γ A B} → Tm64 Γ (prod64 A B) → Tm64 Γ A; fst64
= λ t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd left right case zero suc rec →
fst64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd left right case zero suc rec)
snd64 : ∀{Γ A B} → Tm64 Γ (prod64 A B) → Tm64 Γ B; snd64
= λ t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left right case zero suc rec →
snd64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left right case zero suc rec)
left64 : ∀{Γ A B} → Tm64 Γ A → Tm64 Γ (sum64 A B); left64
= λ t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right case zero suc rec →
left64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right case zero suc rec)
right64 : ∀{Γ A B} → Tm64 Γ B → Tm64 Γ (sum64 A B); right64
= λ t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case zero suc rec →
right64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case zero suc rec)
case64 : ∀{Γ A B C} → Tm64 Γ (sum64 A B) → Tm64 Γ (arr64 A C) → Tm64 Γ (arr64 B C) → Tm64 Γ C; case64
= λ t u v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec →
case64 _ _ _ _
(t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
(u Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
(v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
zero64 : ∀{Γ} → Tm64 Γ nat64; zero64
= λ Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc rec → zero64 _
suc64 : ∀{Γ} → Tm64 Γ nat64 → Tm64 Γ nat64; suc64
= λ t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec →
suc64 _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec)
rec64 : ∀{Γ A} → Tm64 Γ nat64 → Tm64 Γ (arr64 nat64 (arr64 A A)) → Tm64 Γ A → Tm64 Γ A; rec64
= λ t u v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64 →
rec64 _ _
(t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
(u Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
(v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
v064 : ∀{Γ A} → Tm64 (snoc64 Γ A) A; v064
= var64 vz64
v164 : ∀{Γ A B} → Tm64 (snoc64 (snoc64 Γ A) B) A; v164
= var64 (vs64 vz64)
v264 : ∀{Γ A B C} → Tm64 (snoc64 (snoc64 (snoc64 Γ A) B) C) A; v264
= var64 (vs64 (vs64 vz64))
v364 : ∀{Γ A B C D} → Tm64 (snoc64 (snoc64 (snoc64 (snoc64 Γ A) B) C) D) A; v364
= var64 (vs64 (vs64 (vs64 vz64)))
tbool64 : Ty64; tbool64
= sum64 top64 top64
true64 : ∀{Γ} → Tm64 Γ tbool64; true64
= left64 tt64
tfalse64 : ∀{Γ} → Tm64 Γ tbool64; tfalse64
= right64 tt64
ifthenelse64 : ∀{Γ A} → Tm64 Γ (arr64 tbool64 (arr64 A (arr64 A A))); ifthenelse64
= lam64 (lam64 (lam64 (case64 v264 (lam64 v264) (lam64 v164))))
times464 : ∀{Γ A} → Tm64 Γ (arr64 (arr64 A A) (arr64 A A)); times464
= lam64 (lam64 (app64 v164 (app64 v164 (app64 v164 (app64 v164 v064)))))
add64 : ∀{Γ} → Tm64 Γ (arr64 nat64 (arr64 nat64 nat64)); add64
= lam64 (rec64 v064
(lam64 (lam64 (lam64 (suc64 (app64 v164 v064)))))
(lam64 v064))
mul64 : ∀{Γ} → Tm64 Γ (arr64 nat64 (arr64 nat64 nat64)); mul64
= lam64 (rec64 v064
(lam64 (lam64 (lam64 (app64 (app64 add64 (app64 v164 v064)) v064))))
(lam64 zero64))
fact64 : ∀{Γ} → Tm64 Γ (arr64 nat64 nat64); fact64
= lam64 (rec64 v064 (lam64 (lam64 (app64 (app64 mul64 (suc64 v164)) v064)))
(suc64 zero64))
{-# OPTIONS --type-in-type #-}
Ty65 : Set
Ty65 =
(Ty65 : Set)
(nat top bot : Ty65)
(arr prod sum : Ty65 → Ty65 → Ty65)
→ Ty65
nat65 : Ty65; nat65 = λ _ nat65 _ _ _ _ _ → nat65
top65 : Ty65; top65 = λ _ _ top65 _ _ _ _ → top65
bot65 : Ty65; bot65 = λ _ _ _ bot65 _ _ _ → bot65
arr65 : Ty65 → Ty65 → Ty65; arr65
= λ A B Ty65 nat65 top65 bot65 arr65 prod sum →
arr65 (A Ty65 nat65 top65 bot65 arr65 prod sum) (B Ty65 nat65 top65 bot65 arr65 prod sum)
prod65 : Ty65 → Ty65 → Ty65; prod65
= λ A B Ty65 nat65 top65 bot65 arr65 prod65 sum →
prod65 (A Ty65 nat65 top65 bot65 arr65 prod65 sum) (B Ty65 nat65 top65 bot65 arr65 prod65 sum)
sum65 : Ty65 → Ty65 → Ty65; sum65
= λ A B Ty65 nat65 top65 bot65 arr65 prod65 sum65 →
sum65 (A Ty65 nat65 top65 bot65 arr65 prod65 sum65) (B Ty65 nat65 top65 bot65 arr65 prod65 sum65)
Con65 : Set; Con65
= (Con65 : Set)
(nil : Con65)
(snoc : Con65 → Ty65 → Con65)
→ Con65
nil65 : Con65; nil65
= λ Con65 nil65 snoc → nil65
snoc65 : Con65 → Ty65 → Con65; snoc65
= λ Γ A Con65 nil65 snoc65 → snoc65 (Γ Con65 nil65 snoc65) A
Var65 : Con65 → Ty65 → Set; Var65
= λ Γ A →
(Var65 : Con65 → Ty65 → Set)
(vz : ∀ Γ A → Var65 (snoc65 Γ A) A)
(vs : ∀ Γ B A → Var65 Γ A → Var65 (snoc65 Γ B) A)
→ Var65 Γ A
vz65 : ∀{Γ A} → Var65 (snoc65 Γ A) A; vz65
= λ Var65 vz65 vs → vz65 _ _
vs65 : ∀{Γ B A} → Var65 Γ A → Var65 (snoc65 Γ B) A; vs65
= λ x Var65 vz65 vs65 → vs65 _ _ _ (x Var65 vz65 vs65)
Tm65 : Con65 → Ty65 → Set; Tm65
= λ Γ A →
(Tm65 : Con65 → Ty65 → Set)
(var : ∀ Γ A → Var65 Γ A → Tm65 Γ A)
(lam : ∀ Γ A B → Tm65 (snoc65 Γ A) B → Tm65 Γ (arr65 A B))
(app : ∀ Γ A B → Tm65 Γ (arr65 A B) → Tm65 Γ A → Tm65 Γ B)
(tt : ∀ Γ → Tm65 Γ top65)
(pair : ∀ Γ A B → Tm65 Γ A → Tm65 Γ B → Tm65 Γ (prod65 A B))
(fst : ∀ Γ A B → Tm65 Γ (prod65 A B) → Tm65 Γ A)
(snd : ∀ Γ A B → Tm65 Γ (prod65 A B) → Tm65 Γ B)
(left : ∀ Γ A B → Tm65 Γ A → Tm65 Γ (sum65 A B))
(right : ∀ Γ A B → Tm65 Γ B → Tm65 Γ (sum65 A B))
(case : ∀ Γ A B C → Tm65 Γ (sum65 A B) → Tm65 Γ (arr65 A C) → Tm65 Γ (arr65 B C) → Tm65 Γ C)
(zero : ∀ Γ → Tm65 Γ nat65)
(suc : ∀ Γ → Tm65 Γ nat65 → Tm65 Γ nat65)
(rec : ∀ Γ A → Tm65 Γ nat65 → Tm65 Γ (arr65 nat65 (arr65 A A)) → Tm65 Γ A → Tm65 Γ A)
→ Tm65 Γ A
var65 : ∀{Γ A} → Var65 Γ A → Tm65 Γ A; var65
= λ x Tm65 var65 lam app tt pair fst snd left right case zero suc rec →
var65 _ _ x
lam65 : ∀{Γ A B} → Tm65 (snoc65 Γ A) B → Tm65 Γ (arr65 A B); lam65
= λ t Tm65 var65 lam65 app tt pair fst snd left right case zero suc rec →
lam65 _ _ _ (t Tm65 var65 lam65 app tt pair fst snd left right case zero suc rec)
app65 : ∀{Γ A B} → Tm65 Γ (arr65 A B) → Tm65 Γ A → Tm65 Γ B; app65
= λ t u Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec →
app65 _ _ _ (t Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec)
(u Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec)
tt65 : ∀{Γ} → Tm65 Γ top65; tt65
= λ Tm65 var65 lam65 app65 tt65 pair fst snd left right case zero suc rec → tt65 _
pair65 : ∀{Γ A B} → Tm65 Γ A → Tm65 Γ B → Tm65 Γ (prod65 A B); pair65
= λ t u Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec →
pair65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec)
(u Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec)
fst65 : ∀{Γ A B} → Tm65 Γ (prod65 A B) → Tm65 Γ A; fst65
= λ t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd left right case zero suc rec →
fst65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd left right case zero suc rec)
snd65 : ∀{Γ A B} → Tm65 Γ (prod65 A B) → Tm65 Γ B; snd65
= λ t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left right case zero suc rec →
snd65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left right case zero suc rec)
left65 : ∀{Γ A B} → Tm65 Γ A → Tm65 Γ (sum65 A B); left65
= λ t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right case zero suc rec →
left65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right case zero suc rec)
right65 : ∀{Γ A B} → Tm65 Γ B → Tm65 Γ (sum65 A B); right65
= λ t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case zero suc rec →
right65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case zero suc rec)
case65 : ∀{Γ A B C} → Tm65 Γ (sum65 A B) → Tm65 Γ (arr65 A C) → Tm65 Γ (arr65 B C) → Tm65 Γ C; case65
= λ t u v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec →
case65 _ _ _ _
(t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
(u Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
(v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
zero65 : ∀{Γ} → Tm65 Γ nat65; zero65
= λ Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc rec → zero65 _
suc65 : ∀{Γ} → Tm65 Γ nat65 → Tm65 Γ nat65; suc65
= λ t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec →
suc65 _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec)
rec65 : ∀{Γ A} → Tm65 Γ nat65 → Tm65 Γ (arr65 nat65 (arr65 A A)) → Tm65 Γ A → Tm65 Γ A; rec65
= λ t u v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65 →
rec65 _ _
(t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
(u Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
(v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
v065 : ∀{Γ A} → Tm65 (snoc65 Γ A) A; v065
= var65 vz65
v165 : ∀{Γ A B} → Tm65 (snoc65 (snoc65 Γ A) B) A; v165
= var65 (vs65 vz65)
v265 : ∀{Γ A B C} → Tm65 (snoc65 (snoc65 (snoc65 Γ A) B) C) A; v265
= var65 (vs65 (vs65 vz65))
v365 : ∀{Γ A B C D} → Tm65 (snoc65 (snoc65 (snoc65 (snoc65 Γ A) B) C) D) A; v365
= var65 (vs65 (vs65 (vs65 vz65)))
tbool65 : Ty65; tbool65
= sum65 top65 top65
true65 : ∀{Γ} → Tm65 Γ tbool65; true65
= left65 tt65
tfalse65 : ∀{Γ} → Tm65 Γ tbool65; tfalse65
= right65 tt65
ifthenelse65 : ∀{Γ A} → Tm65 Γ (arr65 tbool65 (arr65 A (arr65 A A))); ifthenelse65
= lam65 (lam65 (lam65 (case65 v265 (lam65 v265) (lam65 v165))))
times465 : ∀{Γ A} → Tm65 Γ (arr65 (arr65 A A) (arr65 A A)); times465
= lam65 (lam65 (app65 v165 (app65 v165 (app65 v165 (app65 v165 v065)))))
add65 : ∀{Γ} → Tm65 Γ (arr65 nat65 (arr65 nat65 nat65)); add65
= lam65 (rec65 v065
(lam65 (lam65 (lam65 (suc65 (app65 v165 v065)))))
(lam65 v065))
mul65 : ∀{Γ} → Tm65 Γ (arr65 nat65 (arr65 nat65 nat65)); mul65
= lam65 (rec65 v065
(lam65 (lam65 (lam65 (app65 (app65 add65 (app65 v165 v065)) v065))))
(lam65 zero65))
fact65 : ∀{Γ} → Tm65 Γ (arr65 nat65 nat65); fact65
= lam65 (rec65 v065 (lam65 (lam65 (app65 (app65 mul65 (suc65 v165)) v065)))
(suc65 zero65))
{-# OPTIONS --type-in-type #-}
Ty66 : Set
Ty66 =
(Ty66 : Set)
(nat top bot : Ty66)
(arr prod sum : Ty66 → Ty66 → Ty66)
→ Ty66
nat66 : Ty66; nat66 = λ _ nat66 _ _ _ _ _ → nat66
top66 : Ty66; top66 = λ _ _ top66 _ _ _ _ → top66
bot66 : Ty66; bot66 = λ _ _ _ bot66 _ _ _ → bot66
arr66 : Ty66 → Ty66 → Ty66; arr66
= λ A B Ty66 nat66 top66 bot66 arr66 prod sum →
arr66 (A Ty66 nat66 top66 bot66 arr66 prod sum) (B Ty66 nat66 top66 bot66 arr66 prod sum)
prod66 : Ty66 → Ty66 → Ty66; prod66
= λ A B Ty66 nat66 top66 bot66 arr66 prod66 sum →
prod66 (A Ty66 nat66 top66 bot66 arr66 prod66 sum) (B Ty66 nat66 top66 bot66 arr66 prod66 sum)
sum66 : Ty66 → Ty66 → Ty66; sum66
= λ A B Ty66 nat66 top66 bot66 arr66 prod66 sum66 →
sum66 (A Ty66 nat66 top66 bot66 arr66 prod66 sum66) (B Ty66 nat66 top66 bot66 arr66 prod66 sum66)
Con66 : Set; Con66
= (Con66 : Set)
(nil : Con66)
(snoc : Con66 → Ty66 → Con66)
→ Con66
nil66 : Con66; nil66
= λ Con66 nil66 snoc → nil66
snoc66 : Con66 → Ty66 → Con66; snoc66
= λ Γ A Con66 nil66 snoc66 → snoc66 (Γ Con66 nil66 snoc66) A
Var66 : Con66 → Ty66 → Set; Var66
= λ Γ A →
(Var66 : Con66 → Ty66 → Set)
(vz : ∀ Γ A → Var66 (snoc66 Γ A) A)
(vs : ∀ Γ B A → Var66 Γ A → Var66 (snoc66 Γ B) A)
→ Var66 Γ A
vz66 : ∀{Γ A} → Var66 (snoc66 Γ A) A; vz66
= λ Var66 vz66 vs → vz66 _ _
vs66 : ∀{Γ B A} → Var66 Γ A → Var66 (snoc66 Γ B) A; vs66
= λ x Var66 vz66 vs66 → vs66 _ _ _ (x Var66 vz66 vs66)
Tm66 : Con66 → Ty66 → Set; Tm66
= λ Γ A →
(Tm66 : Con66 → Ty66 → Set)
(var : ∀ Γ A → Var66 Γ A → Tm66 Γ A)
(lam : ∀ Γ A B → Tm66 (snoc66 Γ A) B → Tm66 Γ (arr66 A B))
(app : ∀ Γ A B → Tm66 Γ (arr66 A B) → Tm66 Γ A → Tm66 Γ B)
(tt : ∀ Γ → Tm66 Γ top66)
(pair : ∀ Γ A B → Tm66 Γ A → Tm66 Γ B → Tm66 Γ (prod66 A B))
(fst : ∀ Γ A B → Tm66 Γ (prod66 A B) → Tm66 Γ A)
(snd : ∀ Γ A B → Tm66 Γ (prod66 A B) → Tm66 Γ B)
(left : ∀ Γ A B → Tm66 Γ A → Tm66 Γ (sum66 A B))
(right : ∀ Γ A B → Tm66 Γ B → Tm66 Γ (sum66 A B))
(case : ∀ Γ A B C → Tm66 Γ (sum66 A B) → Tm66 Γ (arr66 A C) → Tm66 Γ (arr66 B C) → Tm66 Γ C)
(zero : ∀ Γ → Tm66 Γ nat66)
(suc : ∀ Γ → Tm66 Γ nat66 → Tm66 Γ nat66)
(rec : ∀ Γ A → Tm66 Γ nat66 → Tm66 Γ (arr66 nat66 (arr66 A A)) → Tm66 Γ A → Tm66 Γ A)
→ Tm66 Γ A
var66 : ∀{Γ A} → Var66 Γ A → Tm66 Γ A; var66
= λ x Tm66 var66 lam app tt pair fst snd left right case zero suc rec →
var66 _ _ x
lam66 : ∀{Γ A B} → Tm66 (snoc66 Γ A) B → Tm66 Γ (arr66 A B); lam66
= λ t Tm66 var66 lam66 app tt pair fst snd left right case zero suc rec →
lam66 _ _ _ (t Tm66 var66 lam66 app tt pair fst snd left right case zero suc rec)
app66 : ∀{Γ A B} → Tm66 Γ (arr66 A B) → Tm66 Γ A → Tm66 Γ B; app66
= λ t u Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec →
app66 _ _ _ (t Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec)
(u Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec)
tt66 : ∀{Γ} → Tm66 Γ top66; tt66
= λ Tm66 var66 lam66 app66 tt66 pair fst snd left right case zero suc rec → tt66 _
pair66 : ∀{Γ A B} → Tm66 Γ A → Tm66 Γ B → Tm66 Γ (prod66 A B); pair66
= λ t u Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec →
pair66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec)
(u Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec)
fst66 : ∀{Γ A B} → Tm66 Γ (prod66 A B) → Tm66 Γ A; fst66
= λ t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd left right case zero suc rec →
fst66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd left right case zero suc rec)
snd66 : ∀{Γ A B} → Tm66 Γ (prod66 A B) → Tm66 Γ B; snd66
= λ t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left right case zero suc rec →
snd66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left right case zero suc rec)
left66 : ∀{Γ A B} → Tm66 Γ A → Tm66 Γ (sum66 A B); left66
= λ t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right case zero suc rec →
left66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right case zero suc rec)
right66 : ∀{Γ A B} → Tm66 Γ B → Tm66 Γ (sum66 A B); right66
= λ t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case zero suc rec →
right66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case zero suc rec)
case66 : ∀{Γ A B C} → Tm66 Γ (sum66 A B) → Tm66 Γ (arr66 A C) → Tm66 Γ (arr66 B C) → Tm66 Γ C; case66
= λ t u v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec →
case66 _ _ _ _
(t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
(u Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
(v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
zero66 : ∀{Γ} → Tm66 Γ nat66; zero66
= λ Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc rec → zero66 _
suc66 : ∀{Γ} → Tm66 Γ nat66 → Tm66 Γ nat66; suc66
= λ t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec →
suc66 _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec)
rec66 : ∀{Γ A} → Tm66 Γ nat66 → Tm66 Γ (arr66 nat66 (arr66 A A)) → Tm66 Γ A → Tm66 Γ A; rec66
= λ t u v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66 →
rec66 _ _
(t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
(u Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
(v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
v066 : ∀{Γ A} → Tm66 (snoc66 Γ A) A; v066
= var66 vz66
v166 : ∀{Γ A B} → Tm66 (snoc66 (snoc66 Γ A) B) A; v166
= var66 (vs66 vz66)
v266 : ∀{Γ A B C} → Tm66 (snoc66 (snoc66 (snoc66 Γ A) B) C) A; v266
= var66 (vs66 (vs66 vz66))
v366 : ∀{Γ A B C D} → Tm66 (snoc66 (snoc66 (snoc66 (snoc66 Γ A) B) C) D) A; v366
= var66 (vs66 (vs66 (vs66 vz66)))
tbool66 : Ty66; tbool66
= sum66 top66 top66
true66 : ∀{Γ} → Tm66 Γ tbool66; true66
= left66 tt66
tfalse66 : ∀{Γ} → Tm66 Γ tbool66; tfalse66
= right66 tt66
ifthenelse66 : ∀{Γ A} → Tm66 Γ (arr66 tbool66 (arr66 A (arr66 A A))); ifthenelse66
= lam66 (lam66 (lam66 (case66 v266 (lam66 v266) (lam66 v166))))
times466 : ∀{Γ A} → Tm66 Γ (arr66 (arr66 A A) (arr66 A A)); times466
= lam66 (lam66 (app66 v166 (app66 v166 (app66 v166 (app66 v166 v066)))))
add66 : ∀{Γ} → Tm66 Γ (arr66 nat66 (arr66 nat66 nat66)); add66
= lam66 (rec66 v066
(lam66 (lam66 (lam66 (suc66 (app66 v166 v066)))))
(lam66 v066))
mul66 : ∀{Γ} → Tm66 Γ (arr66 nat66 (arr66 nat66 nat66)); mul66
= lam66 (rec66 v066
(lam66 (lam66 (lam66 (app66 (app66 add66 (app66 v166 v066)) v066))))
(lam66 zero66))
fact66 : ∀{Γ} → Tm66 Γ (arr66 nat66 nat66); fact66
= lam66 (rec66 v066 (lam66 (lam66 (app66 (app66 mul66 (suc66 v166)) v066)))
(suc66 zero66))
{-# OPTIONS --type-in-type #-}
Ty67 : Set
Ty67 =
(Ty67 : Set)
(nat top bot : Ty67)
(arr prod sum : Ty67 → Ty67 → Ty67)
→ Ty67
nat67 : Ty67; nat67 = λ _ nat67 _ _ _ _ _ → nat67
top67 : Ty67; top67 = λ _ _ top67 _ _ _ _ → top67
bot67 : Ty67; bot67 = λ _ _ _ bot67 _ _ _ → bot67
arr67 : Ty67 → Ty67 → Ty67; arr67
= λ A B Ty67 nat67 top67 bot67 arr67 prod sum →
arr67 (A Ty67 nat67 top67 bot67 arr67 prod sum) (B Ty67 nat67 top67 bot67 arr67 prod sum)
prod67 : Ty67 → Ty67 → Ty67; prod67
= λ A B Ty67 nat67 top67 bot67 arr67 prod67 sum →
prod67 (A Ty67 nat67 top67 bot67 arr67 prod67 sum) (B Ty67 nat67 top67 bot67 arr67 prod67 sum)
sum67 : Ty67 → Ty67 → Ty67; sum67
= λ A B Ty67 nat67 top67 bot67 arr67 prod67 sum67 →
sum67 (A Ty67 nat67 top67 bot67 arr67 prod67 sum67) (B Ty67 nat67 top67 bot67 arr67 prod67 sum67)
Con67 : Set; Con67
= (Con67 : Set)
(nil : Con67)
(snoc : Con67 → Ty67 → Con67)
→ Con67
nil67 : Con67; nil67
= λ Con67 nil67 snoc → nil67
snoc67 : Con67 → Ty67 → Con67; snoc67
= λ Γ A Con67 nil67 snoc67 → snoc67 (Γ Con67 nil67 snoc67) A
Var67 : Con67 → Ty67 → Set; Var67
= λ Γ A →
(Var67 : Con67 → Ty67 → Set)
(vz : ∀ Γ A → Var67 (snoc67 Γ A) A)
(vs : ∀ Γ B A → Var67 Γ A → Var67 (snoc67 Γ B) A)
→ Var67 Γ A
vz67 : ∀{Γ A} → Var67 (snoc67 Γ A) A; vz67
= λ Var67 vz67 vs → vz67 _ _
vs67 : ∀{Γ B A} → Var67 Γ A → Var67 (snoc67 Γ B) A; vs67
= λ x Var67 vz67 vs67 → vs67 _ _ _ (x Var67 vz67 vs67)
Tm67 : Con67 → Ty67 → Set; Tm67
= λ Γ A →
(Tm67 : Con67 → Ty67 → Set)
(var : ∀ Γ A → Var67 Γ A → Tm67 Γ A)
(lam : ∀ Γ A B → Tm67 (snoc67 Γ A) B → Tm67 Γ (arr67 A B))
(app : ∀ Γ A B → Tm67 Γ (arr67 A B) → Tm67 Γ A → Tm67 Γ B)
(tt : ∀ Γ → Tm67 Γ top67)
(pair : ∀ Γ A B → Tm67 Γ A → Tm67 Γ B → Tm67 Γ (prod67 A B))
(fst : ∀ Γ A B → Tm67 Γ (prod67 A B) → Tm67 Γ A)
(snd : ∀ Γ A B → Tm67 Γ (prod67 A B) → Tm67 Γ B)
(left : ∀ Γ A B → Tm67 Γ A → Tm67 Γ (sum67 A B))
(right : ∀ Γ A B → Tm67 Γ B → Tm67 Γ (sum67 A B))
(case : ∀ Γ A B C → Tm67 Γ (sum67 A B) → Tm67 Γ (arr67 A C) → Tm67 Γ (arr67 B C) → Tm67 Γ C)
(zero : ∀ Γ → Tm67 Γ nat67)
(suc : ∀ Γ → Tm67 Γ nat67 → Tm67 Γ nat67)
(rec : ∀ Γ A → Tm67 Γ nat67 → Tm67 Γ (arr67 nat67 (arr67 A A)) → Tm67 Γ A → Tm67 Γ A)
→ Tm67 Γ A
var67 : ∀{Γ A} → Var67 Γ A → Tm67 Γ A; var67
= λ x Tm67 var67 lam app tt pair fst snd left right case zero suc rec →
var67 _ _ x
lam67 : ∀{Γ A B} → Tm67 (snoc67 Γ A) B → Tm67 Γ (arr67 A B); lam67
= λ t Tm67 var67 lam67 app tt pair fst snd left right case zero suc rec →
lam67 _ _ _ (t Tm67 var67 lam67 app tt pair fst snd left right case zero suc rec)
app67 : ∀{Γ A B} → Tm67 Γ (arr67 A B) → Tm67 Γ A → Tm67 Γ B; app67
= λ t u Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec →
app67 _ _ _ (t Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec)
(u Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec)
tt67 : ∀{Γ} → Tm67 Γ top67; tt67
= λ Tm67 var67 lam67 app67 tt67 pair fst snd left right case zero suc rec → tt67 _
pair67 : ∀{Γ A B} → Tm67 Γ A → Tm67 Γ B → Tm67 Γ (prod67 A B); pair67
= λ t u Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec →
pair67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec)
(u Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec)
fst67 : ∀{Γ A B} → Tm67 Γ (prod67 A B) → Tm67 Γ A; fst67
= λ t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd left right case zero suc rec →
fst67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd left right case zero suc rec)
snd67 : ∀{Γ A B} → Tm67 Γ (prod67 A B) → Tm67 Γ B; snd67
= λ t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left right case zero suc rec →
snd67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left right case zero suc rec)
left67 : ∀{Γ A B} → Tm67 Γ A → Tm67 Γ (sum67 A B); left67
= λ t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right case zero suc rec →
left67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right case zero suc rec)
right67 : ∀{Γ A B} → Tm67 Γ B → Tm67 Γ (sum67 A B); right67
= λ t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case zero suc rec →
right67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case zero suc rec)
case67 : ∀{Γ A B C} → Tm67 Γ (sum67 A B) → Tm67 Γ (arr67 A C) → Tm67 Γ (arr67 B C) → Tm67 Γ C; case67
= λ t u v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec →
case67 _ _ _ _
(t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
(u Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
(v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
zero67 : ∀{Γ} → Tm67 Γ nat67; zero67
= λ Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc rec → zero67 _
suc67 : ∀{Γ} → Tm67 Γ nat67 → Tm67 Γ nat67; suc67
= λ t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec →
suc67 _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec)
rec67 : ∀{Γ A} → Tm67 Γ nat67 → Tm67 Γ (arr67 nat67 (arr67 A A)) → Tm67 Γ A → Tm67 Γ A; rec67
= λ t u v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67 →
rec67 _ _
(t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
(u Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
(v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
v067 : ∀{Γ A} → Tm67 (snoc67 Γ A) A; v067
= var67 vz67
v167 : ∀{Γ A B} → Tm67 (snoc67 (snoc67 Γ A) B) A; v167
= var67 (vs67 vz67)
v267 : ∀{Γ A B C} → Tm67 (snoc67 (snoc67 (snoc67 Γ A) B) C) A; v267
= var67 (vs67 (vs67 vz67))
v367 : ∀{Γ A B C D} → Tm67 (snoc67 (snoc67 (snoc67 (snoc67 Γ A) B) C) D) A; v367
= var67 (vs67 (vs67 (vs67 vz67)))
tbool67 : Ty67; tbool67
= sum67 top67 top67
true67 : ∀{Γ} → Tm67 Γ tbool67; true67
= left67 tt67
tfalse67 : ∀{Γ} → Tm67 Γ tbool67; tfalse67
= right67 tt67
ifthenelse67 : ∀{Γ A} → Tm67 Γ (arr67 tbool67 (arr67 A (arr67 A A))); ifthenelse67
= lam67 (lam67 (lam67 (case67 v267 (lam67 v267) (lam67 v167))))
times467 : ∀{Γ A} → Tm67 Γ (arr67 (arr67 A A) (arr67 A A)); times467
= lam67 (lam67 (app67 v167 (app67 v167 (app67 v167 (app67 v167 v067)))))
add67 : ∀{Γ} → Tm67 Γ (arr67 nat67 (arr67 nat67 nat67)); add67
= lam67 (rec67 v067
(lam67 (lam67 (lam67 (suc67 (app67 v167 v067)))))
(lam67 v067))
mul67 : ∀{Γ} → Tm67 Γ (arr67 nat67 (arr67 nat67 nat67)); mul67
= lam67 (rec67 v067
(lam67 (lam67 (lam67 (app67 (app67 add67 (app67 v167 v067)) v067))))
(lam67 zero67))
fact67 : ∀{Γ} → Tm67 Γ (arr67 nat67 nat67); fact67
= lam67 (rec67 v067 (lam67 (lam67 (app67 (app67 mul67 (suc67 v167)) v067)))
(suc67 zero67))
{-# OPTIONS --type-in-type #-}
Ty68 : Set
Ty68 =
(Ty68 : Set)
(nat top bot : Ty68)
(arr prod sum : Ty68 → Ty68 → Ty68)
→ Ty68
nat68 : Ty68; nat68 = λ _ nat68 _ _ _ _ _ → nat68
top68 : Ty68; top68 = λ _ _ top68 _ _ _ _ → top68
bot68 : Ty68; bot68 = λ _ _ _ bot68 _ _ _ → bot68
arr68 : Ty68 → Ty68 → Ty68; arr68
= λ A B Ty68 nat68 top68 bot68 arr68 prod sum →
arr68 (A Ty68 nat68 top68 bot68 arr68 prod sum) (B Ty68 nat68 top68 bot68 arr68 prod sum)
prod68 : Ty68 → Ty68 → Ty68; prod68
= λ A B Ty68 nat68 top68 bot68 arr68 prod68 sum →
prod68 (A Ty68 nat68 top68 bot68 arr68 prod68 sum) (B Ty68 nat68 top68 bot68 arr68 prod68 sum)
sum68 : Ty68 → Ty68 → Ty68; sum68
= λ A B Ty68 nat68 top68 bot68 arr68 prod68 sum68 →
sum68 (A Ty68 nat68 top68 bot68 arr68 prod68 sum68) (B Ty68 nat68 top68 bot68 arr68 prod68 sum68)
Con68 : Set; Con68
= (Con68 : Set)
(nil : Con68)
(snoc : Con68 → Ty68 → Con68)
→ Con68
nil68 : Con68; nil68
= λ Con68 nil68 snoc → nil68
snoc68 : Con68 → Ty68 → Con68; snoc68
= λ Γ A Con68 nil68 snoc68 → snoc68 (Γ Con68 nil68 snoc68) A
Var68 : Con68 → Ty68 → Set; Var68
= λ Γ A →
(Var68 : Con68 → Ty68 → Set)
(vz : ∀ Γ A → Var68 (snoc68 Γ A) A)
(vs : ∀ Γ B A → Var68 Γ A → Var68 (snoc68 Γ B) A)
→ Var68 Γ A
vz68 : ∀{Γ A} → Var68 (snoc68 Γ A) A; vz68
= λ Var68 vz68 vs → vz68 _ _
vs68 : ∀{Γ B A} → Var68 Γ A → Var68 (snoc68 Γ B) A; vs68
= λ x Var68 vz68 vs68 → vs68 _ _ _ (x Var68 vz68 vs68)
Tm68 : Con68 → Ty68 → Set; Tm68
= λ Γ A →
(Tm68 : Con68 → Ty68 → Set)
(var : ∀ Γ A → Var68 Γ A → Tm68 Γ A)
(lam : ∀ Γ A B → Tm68 (snoc68 Γ A) B → Tm68 Γ (arr68 A B))
(app : ∀ Γ A B → Tm68 Γ (arr68 A B) → Tm68 Γ A → Tm68 Γ B)
(tt : ∀ Γ → Tm68 Γ top68)
(pair : ∀ Γ A B → Tm68 Γ A → Tm68 Γ B → Tm68 Γ (prod68 A B))
(fst : ∀ Γ A B → Tm68 Γ (prod68 A B) → Tm68 Γ A)
(snd : ∀ Γ A B → Tm68 Γ (prod68 A B) → Tm68 Γ B)
(left : ∀ Γ A B → Tm68 Γ A → Tm68 Γ (sum68 A B))
(right : ∀ Γ A B → Tm68 Γ B → Tm68 Γ (sum68 A B))
(case : ∀ Γ A B C → Tm68 Γ (sum68 A B) → Tm68 Γ (arr68 A C) → Tm68 Γ (arr68 B C) → Tm68 Γ C)
(zero : ∀ Γ → Tm68 Γ nat68)
(suc : ∀ Γ → Tm68 Γ nat68 → Tm68 Γ nat68)
(rec : ∀ Γ A → Tm68 Γ nat68 → Tm68 Γ (arr68 nat68 (arr68 A A)) → Tm68 Γ A → Tm68 Γ A)
→ Tm68 Γ A
var68 : ∀{Γ A} → Var68 Γ A → Tm68 Γ A; var68
= λ x Tm68 var68 lam app tt pair fst snd left right case zero suc rec →
var68 _ _ x
lam68 : ∀{Γ A B} → Tm68 (snoc68 Γ A) B → Tm68 Γ (arr68 A B); lam68
= λ t Tm68 var68 lam68 app tt pair fst snd left right case zero suc rec →
lam68 _ _ _ (t Tm68 var68 lam68 app tt pair fst snd left right case zero suc rec)
app68 : ∀{Γ A B} → Tm68 Γ (arr68 A B) → Tm68 Γ A → Tm68 Γ B; app68
= λ t u Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec →
app68 _ _ _ (t Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec)
(u Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec)
tt68 : ∀{Γ} → Tm68 Γ top68; tt68
= λ Tm68 var68 lam68 app68 tt68 pair fst snd left right case zero suc rec → tt68 _
pair68 : ∀{Γ A B} → Tm68 Γ A → Tm68 Γ B → Tm68 Γ (prod68 A B); pair68
= λ t u Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec →
pair68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec)
(u Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec)
fst68 : ∀{Γ A B} → Tm68 Γ (prod68 A B) → Tm68 Γ A; fst68
= λ t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd left right case zero suc rec →
fst68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd left right case zero suc rec)
snd68 : ∀{Γ A B} → Tm68 Γ (prod68 A B) → Tm68 Γ B; snd68
= λ t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left right case zero suc rec →
snd68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left right case zero suc rec)
left68 : ∀{Γ A B} → Tm68 Γ A → Tm68 Γ (sum68 A B); left68
= λ t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right case zero suc rec →
left68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right case zero suc rec)
right68 : ∀{Γ A B} → Tm68 Γ B → Tm68 Γ (sum68 A B); right68
= λ t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case zero suc rec →
right68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case zero suc rec)
case68 : ∀{Γ A B C} → Tm68 Γ (sum68 A B) → Tm68 Γ (arr68 A C) → Tm68 Γ (arr68 B C) → Tm68 Γ C; case68
= λ t u v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec →
case68 _ _ _ _
(t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
(u Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
(v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
zero68 : ∀{Γ} → Tm68 Γ nat68; zero68
= λ Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc rec → zero68 _
suc68 : ∀{Γ} → Tm68 Γ nat68 → Tm68 Γ nat68; suc68
= λ t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec →
suc68 _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec)
rec68 : ∀{Γ A} → Tm68 Γ nat68 → Tm68 Γ (arr68 nat68 (arr68 A A)) → Tm68 Γ A → Tm68 Γ A; rec68
= λ t u v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68 →
rec68 _ _
(t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
(u Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
(v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
v068 : ∀{Γ A} → Tm68 (snoc68 Γ A) A; v068
= var68 vz68
v168 : ∀{Γ A B} → Tm68 (snoc68 (snoc68 Γ A) B) A; v168
= var68 (vs68 vz68)
v268 : ∀{Γ A B C} → Tm68 (snoc68 (snoc68 (snoc68 Γ A) B) C) A; v268
= var68 (vs68 (vs68 vz68))
v368 : ∀{Γ A B C D} → Tm68 (snoc68 (snoc68 (snoc68 (snoc68 Γ A) B) C) D) A; v368
= var68 (vs68 (vs68 (vs68 vz68)))
tbool68 : Ty68; tbool68
= sum68 top68 top68
true68 : ∀{Γ} → Tm68 Γ tbool68; true68
= left68 tt68
tfalse68 : ∀{Γ} → Tm68 Γ tbool68; tfalse68
= right68 tt68
ifthenelse68 : ∀{Γ A} → Tm68 Γ (arr68 tbool68 (arr68 A (arr68 A A))); ifthenelse68
= lam68 (lam68 (lam68 (case68 v268 (lam68 v268) (lam68 v168))))
times468 : ∀{Γ A} → Tm68 Γ (arr68 (arr68 A A) (arr68 A A)); times468
= lam68 (lam68 (app68 v168 (app68 v168 (app68 v168 (app68 v168 v068)))))
add68 : ∀{Γ} → Tm68 Γ (arr68 nat68 (arr68 nat68 nat68)); add68
= lam68 (rec68 v068
(lam68 (lam68 (lam68 (suc68 (app68 v168 v068)))))
(lam68 v068))
mul68 : ∀{Γ} → Tm68 Γ (arr68 nat68 (arr68 nat68 nat68)); mul68
= lam68 (rec68 v068
(lam68 (lam68 (lam68 (app68 (app68 add68 (app68 v168 v068)) v068))))
(lam68 zero68))
fact68 : ∀{Γ} → Tm68 Γ (arr68 nat68 nat68); fact68
= lam68 (rec68 v068 (lam68 (lam68 (app68 (app68 mul68 (suc68 v168)) v068)))
(suc68 zero68))
{-# OPTIONS --type-in-type #-}
Ty69 : Set
Ty69 =
(Ty69 : Set)
(nat top bot : Ty69)
(arr prod sum : Ty69 → Ty69 → Ty69)
→ Ty69
nat69 : Ty69; nat69 = λ _ nat69 _ _ _ _ _ → nat69
top69 : Ty69; top69 = λ _ _ top69 _ _ _ _ → top69
bot69 : Ty69; bot69 = λ _ _ _ bot69 _ _ _ → bot69
arr69 : Ty69 → Ty69 → Ty69; arr69
= λ A B Ty69 nat69 top69 bot69 arr69 prod sum →
arr69 (A Ty69 nat69 top69 bot69 arr69 prod sum) (B Ty69 nat69 top69 bot69 arr69 prod sum)
prod69 : Ty69 → Ty69 → Ty69; prod69
= λ A B Ty69 nat69 top69 bot69 arr69 prod69 sum →
prod69 (A Ty69 nat69 top69 bot69 arr69 prod69 sum) (B Ty69 nat69 top69 bot69 arr69 prod69 sum)
sum69 : Ty69 → Ty69 → Ty69; sum69
= λ A B Ty69 nat69 top69 bot69 arr69 prod69 sum69 →
sum69 (A Ty69 nat69 top69 bot69 arr69 prod69 sum69) (B Ty69 nat69 top69 bot69 arr69 prod69 sum69)
Con69 : Set; Con69
= (Con69 : Set)
(nil : Con69)
(snoc : Con69 → Ty69 → Con69)
→ Con69
nil69 : Con69; nil69
= λ Con69 nil69 snoc → nil69
snoc69 : Con69 → Ty69 → Con69; snoc69
= λ Γ A Con69 nil69 snoc69 → snoc69 (Γ Con69 nil69 snoc69) A
Var69 : Con69 → Ty69 → Set; Var69
= λ Γ A →
(Var69 : Con69 → Ty69 → Set)
(vz : ∀ Γ A → Var69 (snoc69 Γ A) A)
(vs : ∀ Γ B A → Var69 Γ A → Var69 (snoc69 Γ B) A)
→ Var69 Γ A
vz69 : ∀{Γ A} → Var69 (snoc69 Γ A) A; vz69
= λ Var69 vz69 vs → vz69 _ _
vs69 : ∀{Γ B A} → Var69 Γ A → Var69 (snoc69 Γ B) A; vs69
= λ x Var69 vz69 vs69 → vs69 _ _ _ (x Var69 vz69 vs69)
Tm69 : Con69 → Ty69 → Set; Tm69
= λ Γ A →
(Tm69 : Con69 → Ty69 → Set)
(var : ∀ Γ A → Var69 Γ A → Tm69 Γ A)
(lam : ∀ Γ A B → Tm69 (snoc69 Γ A) B → Tm69 Γ (arr69 A B))
(app : ∀ Γ A B → Tm69 Γ (arr69 A B) → Tm69 Γ A → Tm69 Γ B)
(tt : ∀ Γ → Tm69 Γ top69)
(pair : ∀ Γ A B → Tm69 Γ A → Tm69 Γ B → Tm69 Γ (prod69 A B))
(fst : ∀ Γ A B → Tm69 Γ (prod69 A B) → Tm69 Γ A)
(snd : ∀ Γ A B → Tm69 Γ (prod69 A B) → Tm69 Γ B)
(left : ∀ Γ A B → Tm69 Γ A → Tm69 Γ (sum69 A B))
(right : ∀ Γ A B → Tm69 Γ B → Tm69 Γ (sum69 A B))
(case : ∀ Γ A B C → Tm69 Γ (sum69 A B) → Tm69 Γ (arr69 A C) → Tm69 Γ (arr69 B C) → Tm69 Γ C)
(zero : ∀ Γ → Tm69 Γ nat69)
(suc : ∀ Γ → Tm69 Γ nat69 → Tm69 Γ nat69)
(rec : ∀ Γ A → Tm69 Γ nat69 → Tm69 Γ (arr69 nat69 (arr69 A A)) → Tm69 Γ A → Tm69 Γ A)
→ Tm69 Γ A
var69 : ∀{Γ A} → Var69 Γ A → Tm69 Γ A; var69
= λ x Tm69 var69 lam app tt pair fst snd left right case zero suc rec →
var69 _ _ x
lam69 : ∀{Γ A B} → Tm69 (snoc69 Γ A) B → Tm69 Γ (arr69 A B); lam69
= λ t Tm69 var69 lam69 app tt pair fst snd left right case zero suc rec →
lam69 _ _ _ (t Tm69 var69 lam69 app tt pair fst snd left right case zero suc rec)
app69 : ∀{Γ A B} → Tm69 Γ (arr69 A B) → Tm69 Γ A → Tm69 Γ B; app69
= λ t u Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec →
app69 _ _ _ (t Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec)
(u Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec)
tt69 : ∀{Γ} → Tm69 Γ top69; tt69
= λ Tm69 var69 lam69 app69 tt69 pair fst snd left right case zero suc rec → tt69 _
pair69 : ∀{Γ A B} → Tm69 Γ A → Tm69 Γ B → Tm69 Γ (prod69 A B); pair69
= λ t u Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec →
pair69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec)
(u Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec)
fst69 : ∀{Γ A B} → Tm69 Γ (prod69 A B) → Tm69 Γ A; fst69
= λ t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd left right case zero suc rec →
fst69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd left right case zero suc rec)
snd69 : ∀{Γ A B} → Tm69 Γ (prod69 A B) → Tm69 Γ B; snd69
= λ t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left right case zero suc rec →
snd69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left right case zero suc rec)
left69 : ∀{Γ A B} → Tm69 Γ A → Tm69 Γ (sum69 A B); left69
= λ t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right case zero suc rec →
left69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right case zero suc rec)
right69 : ∀{Γ A B} → Tm69 Γ B → Tm69 Γ (sum69 A B); right69
= λ t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case zero suc rec →
right69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case zero suc rec)
case69 : ∀{Γ A B C} → Tm69 Γ (sum69 A B) → Tm69 Γ (arr69 A C) → Tm69 Γ (arr69 B C) → Tm69 Γ C; case69
= λ t u v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec →
case69 _ _ _ _
(t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
(u Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
(v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
zero69 : ∀{Γ} → Tm69 Γ nat69; zero69
= λ Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc rec → zero69 _
suc69 : ∀{Γ} → Tm69 Γ nat69 → Tm69 Γ nat69; suc69
= λ t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec →
suc69 _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec)
rec69 : ∀{Γ A} → Tm69 Γ nat69 → Tm69 Γ (arr69 nat69 (arr69 A A)) → Tm69 Γ A → Tm69 Γ A; rec69
= λ t u v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69 →
rec69 _ _
(t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
(u Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
(v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
v069 : ∀{Γ A} → Tm69 (snoc69 Γ A) A; v069
= var69 vz69
v169 : ∀{Γ A B} → Tm69 (snoc69 (snoc69 Γ A) B) A; v169
= var69 (vs69 vz69)
v269 : ∀{Γ A B C} → Tm69 (snoc69 (snoc69 (snoc69 Γ A) B) C) A; v269
= var69 (vs69 (vs69 vz69))
v369 : ∀{Γ A B C D} → Tm69 (snoc69 (snoc69 (snoc69 (snoc69 Γ A) B) C) D) A; v369
= var69 (vs69 (vs69 (vs69 vz69)))
tbool69 : Ty69; tbool69
= sum69 top69 top69
true69 : ∀{Γ} → Tm69 Γ tbool69; true69
= left69 tt69
tfalse69 : ∀{Γ} → Tm69 Γ tbool69; tfalse69
= right69 tt69
ifthenelse69 : ∀{Γ A} → Tm69 Γ (arr69 tbool69 (arr69 A (arr69 A A))); ifthenelse69
= lam69 (lam69 (lam69 (case69 v269 (lam69 v269) (lam69 v169))))
times469 : ∀{Γ A} → Tm69 Γ (arr69 (arr69 A A) (arr69 A A)); times469
= lam69 (lam69 (app69 v169 (app69 v169 (app69 v169 (app69 v169 v069)))))
add69 : ∀{Γ} → Tm69 Γ (arr69 nat69 (arr69 nat69 nat69)); add69
= lam69 (rec69 v069
(lam69 (lam69 (lam69 (suc69 (app69 v169 v069)))))
(lam69 v069))
mul69 : ∀{Γ} → Tm69 Γ (arr69 nat69 (arr69 nat69 nat69)); mul69
= lam69 (rec69 v069
(lam69 (lam69 (lam69 (app69 (app69 add69 (app69 v169 v069)) v069))))
(lam69 zero69))
fact69 : ∀{Γ} → Tm69 Γ (arr69 nat69 nat69); fact69
= lam69 (rec69 v069 (lam69 (lam69 (app69 (app69 mul69 (suc69 v169)) v069)))
(suc69 zero69))
{-# OPTIONS --type-in-type #-}
Ty70 : Set
Ty70 =
(Ty70 : Set)
(nat top bot : Ty70)
(arr prod sum : Ty70 → Ty70 → Ty70)
→ Ty70
nat70 : Ty70; nat70 = λ _ nat70 _ _ _ _ _ → nat70
top70 : Ty70; top70 = λ _ _ top70 _ _ _ _ → top70
bot70 : Ty70; bot70 = λ _ _ _ bot70 _ _ _ → bot70
arr70 : Ty70 → Ty70 → Ty70; arr70
= λ A B Ty70 nat70 top70 bot70 arr70 prod sum →
arr70 (A Ty70 nat70 top70 bot70 arr70 prod sum) (B Ty70 nat70 top70 bot70 arr70 prod sum)
prod70 : Ty70 → Ty70 → Ty70; prod70
= λ A B Ty70 nat70 top70 bot70 arr70 prod70 sum →
prod70 (A Ty70 nat70 top70 bot70 arr70 prod70 sum) (B Ty70 nat70 top70 bot70 arr70 prod70 sum)
sum70 : Ty70 → Ty70 → Ty70; sum70
= λ A B Ty70 nat70 top70 bot70 arr70 prod70 sum70 →
sum70 (A Ty70 nat70 top70 bot70 arr70 prod70 sum70) (B Ty70 nat70 top70 bot70 arr70 prod70 sum70)
Con70 : Set; Con70
= (Con70 : Set)
(nil : Con70)
(snoc : Con70 → Ty70 → Con70)
→ Con70
nil70 : Con70; nil70
= λ Con70 nil70 snoc → nil70
snoc70 : Con70 → Ty70 → Con70; snoc70
= λ Γ A Con70 nil70 snoc70 → snoc70 (Γ Con70 nil70 snoc70) A
Var70 : Con70 → Ty70 → Set; Var70
= λ Γ A →
(Var70 : Con70 → Ty70 → Set)
(vz : ∀ Γ A → Var70 (snoc70 Γ A) A)
(vs : ∀ Γ B A → Var70 Γ A → Var70 (snoc70 Γ B) A)
→ Var70 Γ A
vz70 : ∀{Γ A} → Var70 (snoc70 Γ A) A; vz70
= λ Var70 vz70 vs → vz70 _ _
vs70 : ∀{Γ B A} → Var70 Γ A → Var70 (snoc70 Γ B) A; vs70
= λ x Var70 vz70 vs70 → vs70 _ _ _ (x Var70 vz70 vs70)
Tm70 : Con70 → Ty70 → Set; Tm70
= λ Γ A →
(Tm70 : Con70 → Ty70 → Set)
(var : ∀ Γ A → Var70 Γ A → Tm70 Γ A)
(lam : ∀ Γ A B → Tm70 (snoc70 Γ A) B → Tm70 Γ (arr70 A B))
(app : ∀ Γ A B → Tm70 Γ (arr70 A B) → Tm70 Γ A → Tm70 Γ B)
(tt : ∀ Γ → Tm70 Γ top70)
(pair : ∀ Γ A B → Tm70 Γ A → Tm70 Γ B → Tm70 Γ (prod70 A B))
(fst : ∀ Γ A B → Tm70 Γ (prod70 A B) → Tm70 Γ A)
(snd : ∀ Γ A B → Tm70 Γ (prod70 A B) → Tm70 Γ B)
(left : ∀ Γ A B → Tm70 Γ A → Tm70 Γ (sum70 A B))
(right : ∀ Γ A B → Tm70 Γ B → Tm70 Γ (sum70 A B))
(case : ∀ Γ A B C → Tm70 Γ (sum70 A B) → Tm70 Γ (arr70 A C) → Tm70 Γ (arr70 B C) → Tm70 Γ C)
(zero : ∀ Γ → Tm70 Γ nat70)
(suc : ∀ Γ → Tm70 Γ nat70 → Tm70 Γ nat70)
(rec : ∀ Γ A → Tm70 Γ nat70 → Tm70 Γ (arr70 nat70 (arr70 A A)) → Tm70 Γ A → Tm70 Γ A)
→ Tm70 Γ A
var70 : ∀{Γ A} → Var70 Γ A → Tm70 Γ A; var70
= λ x Tm70 var70 lam app tt pair fst snd left right case zero suc rec →
var70 _ _ x
lam70 : ∀{Γ A B} → Tm70 (snoc70 Γ A) B → Tm70 Γ (arr70 A B); lam70
= λ t Tm70 var70 lam70 app tt pair fst snd left right case zero suc rec →
lam70 _ _ _ (t Tm70 var70 lam70 app tt pair fst snd left right case zero suc rec)
app70 : ∀{Γ A B} → Tm70 Γ (arr70 A B) → Tm70 Γ A → Tm70 Γ B; app70
= λ t u Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec →
app70 _ _ _ (t Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec)
(u Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec)
tt70 : ∀{Γ} → Tm70 Γ top70; tt70
= λ Tm70 var70 lam70 app70 tt70 pair fst snd left right case zero suc rec → tt70 _
pair70 : ∀{Γ A B} → Tm70 Γ A → Tm70 Γ B → Tm70 Γ (prod70 A B); pair70
= λ t u Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec →
pair70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec)
(u Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec)
fst70 : ∀{Γ A B} → Tm70 Γ (prod70 A B) → Tm70 Γ A; fst70
= λ t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd left right case zero suc rec →
fst70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd left right case zero suc rec)
snd70 : ∀{Γ A B} → Tm70 Γ (prod70 A B) → Tm70 Γ B; snd70
= λ t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left right case zero suc rec →
snd70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left right case zero suc rec)
left70 : ∀{Γ A B} → Tm70 Γ A → Tm70 Γ (sum70 A B); left70
= λ t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right case zero suc rec →
left70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right case zero suc rec)
right70 : ∀{Γ A B} → Tm70 Γ B → Tm70 Γ (sum70 A B); right70
= λ t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case zero suc rec →
right70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case zero suc rec)
case70 : ∀{Γ A B C} → Tm70 Γ (sum70 A B) → Tm70 Γ (arr70 A C) → Tm70 Γ (arr70 B C) → Tm70 Γ C; case70
= λ t u v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec →
case70 _ _ _ _
(t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
(u Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
(v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
zero70 : ∀{Γ} → Tm70 Γ nat70; zero70
= λ Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc rec → zero70 _
suc70 : ∀{Γ} → Tm70 Γ nat70 → Tm70 Γ nat70; suc70
= λ t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec →
suc70 _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec)
rec70 : ∀{Γ A} → Tm70 Γ nat70 → Tm70 Γ (arr70 nat70 (arr70 A A)) → Tm70 Γ A → Tm70 Γ A; rec70
= λ t u v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70 →
rec70 _ _
(t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
(u Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
(v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
v070 : ∀{Γ A} → Tm70 (snoc70 Γ A) A; v070
= var70 vz70
v170 : ∀{Γ A B} → Tm70 (snoc70 (snoc70 Γ A) B) A; v170
= var70 (vs70 vz70)
v270 : ∀{Γ A B C} → Tm70 (snoc70 (snoc70 (snoc70 Γ A) B) C) A; v270
= var70 (vs70 (vs70 vz70))
v370 : ∀{Γ A B C D} → Tm70 (snoc70 (snoc70 (snoc70 (snoc70 Γ A) B) C) D) A; v370
= var70 (vs70 (vs70 (vs70 vz70)))
tbool70 : Ty70; tbool70
= sum70 top70 top70
true70 : ∀{Γ} → Tm70 Γ tbool70; true70
= left70 tt70
tfalse70 : ∀{Γ} → Tm70 Γ tbool70; tfalse70
= right70 tt70
ifthenelse70 : ∀{Γ A} → Tm70 Γ (arr70 tbool70 (arr70 A (arr70 A A))); ifthenelse70
= lam70 (lam70 (lam70 (case70 v270 (lam70 v270) (lam70 v170))))
times470 : ∀{Γ A} → Tm70 Γ (arr70 (arr70 A A) (arr70 A A)); times470
= lam70 (lam70 (app70 v170 (app70 v170 (app70 v170 (app70 v170 v070)))))
add70 : ∀{Γ} → Tm70 Γ (arr70 nat70 (arr70 nat70 nat70)); add70
= lam70 (rec70 v070
(lam70 (lam70 (lam70 (suc70 (app70 v170 v070)))))
(lam70 v070))
mul70 : ∀{Γ} → Tm70 Γ (arr70 nat70 (arr70 nat70 nat70)); mul70
= lam70 (rec70 v070
(lam70 (lam70 (lam70 (app70 (app70 add70 (app70 v170 v070)) v070))))
(lam70 zero70))
fact70 : ∀{Γ} → Tm70 Γ (arr70 nat70 nat70); fact70
= lam70 (rec70 v070 (lam70 (lam70 (app70 (app70 mul70 (suc70 v170)) v070)))
(suc70 zero70))
{-# OPTIONS --type-in-type #-}
Ty71 : Set
Ty71 =
(Ty71 : Set)
(nat top bot : Ty71)
(arr prod sum : Ty71 → Ty71 → Ty71)
→ Ty71
nat71 : Ty71; nat71 = λ _ nat71 _ _ _ _ _ → nat71
top71 : Ty71; top71 = λ _ _ top71 _ _ _ _ → top71
bot71 : Ty71; bot71 = λ _ _ _ bot71 _ _ _ → bot71
arr71 : Ty71 → Ty71 → Ty71; arr71
= λ A B Ty71 nat71 top71 bot71 arr71 prod sum →
arr71 (A Ty71 nat71 top71 bot71 arr71 prod sum) (B Ty71 nat71 top71 bot71 arr71 prod sum)
prod71 : Ty71 → Ty71 → Ty71; prod71
= λ A B Ty71 nat71 top71 bot71 arr71 prod71 sum →
prod71 (A Ty71 nat71 top71 bot71 arr71 prod71 sum) (B Ty71 nat71 top71 bot71 arr71 prod71 sum)
sum71 : Ty71 → Ty71 → Ty71; sum71
= λ A B Ty71 nat71 top71 bot71 arr71 prod71 sum71 →
sum71 (A Ty71 nat71 top71 bot71 arr71 prod71 sum71) (B Ty71 nat71 top71 bot71 arr71 prod71 sum71)
Con71 : Set; Con71
= (Con71 : Set)
(nil : Con71)
(snoc : Con71 → Ty71 → Con71)
→ Con71
nil71 : Con71; nil71
= λ Con71 nil71 snoc → nil71
snoc71 : Con71 → Ty71 → Con71; snoc71
= λ Γ A Con71 nil71 snoc71 → snoc71 (Γ Con71 nil71 snoc71) A
Var71 : Con71 → Ty71 → Set; Var71
= λ Γ A →
(Var71 : Con71 → Ty71 → Set)
(vz : ∀ Γ A → Var71 (snoc71 Γ A) A)
(vs : ∀ Γ B A → Var71 Γ A → Var71 (snoc71 Γ B) A)
→ Var71 Γ A
vz71 : ∀{Γ A} → Var71 (snoc71 Γ A) A; vz71
= λ Var71 vz71 vs → vz71 _ _
vs71 : ∀{Γ B A} → Var71 Γ A → Var71 (snoc71 Γ B) A; vs71
= λ x Var71 vz71 vs71 → vs71 _ _ _ (x Var71 vz71 vs71)
Tm71 : Con71 → Ty71 → Set; Tm71
= λ Γ A →
(Tm71 : Con71 → Ty71 → Set)
(var : ∀ Γ A → Var71 Γ A → Tm71 Γ A)
(lam : ∀ Γ A B → Tm71 (snoc71 Γ A) B → Tm71 Γ (arr71 A B))
(app : ∀ Γ A B → Tm71 Γ (arr71 A B) → Tm71 Γ A → Tm71 Γ B)
(tt : ∀ Γ → Tm71 Γ top71)
(pair : ∀ Γ A B → Tm71 Γ A → Tm71 Γ B → Tm71 Γ (prod71 A B))
(fst : ∀ Γ A B → Tm71 Γ (prod71 A B) → Tm71 Γ A)
(snd : ∀ Γ A B → Tm71 Γ (prod71 A B) → Tm71 Γ B)
(left : ∀ Γ A B → Tm71 Γ A → Tm71 Γ (sum71 A B))
(right : ∀ Γ A B → Tm71 Γ B → Tm71 Γ (sum71 A B))
(case : ∀ Γ A B C → Tm71 Γ (sum71 A B) → Tm71 Γ (arr71 A C) → Tm71 Γ (arr71 B C) → Tm71 Γ C)
(zero : ∀ Γ → Tm71 Γ nat71)
(suc : ∀ Γ → Tm71 Γ nat71 → Tm71 Γ nat71)
(rec : ∀ Γ A → Tm71 Γ nat71 → Tm71 Γ (arr71 nat71 (arr71 A A)) → Tm71 Γ A → Tm71 Γ A)
→ Tm71 Γ A
var71 : ∀{Γ A} → Var71 Γ A → Tm71 Γ A; var71
= λ x Tm71 var71 lam app tt pair fst snd left right case zero suc rec →
var71 _ _ x
lam71 : ∀{Γ A B} → Tm71 (snoc71 Γ A) B → Tm71 Γ (arr71 A B); lam71
= λ t Tm71 var71 lam71 app tt pair fst snd left right case zero suc rec →
lam71 _ _ _ (t Tm71 var71 lam71 app tt pair fst snd left right case zero suc rec)
app71 : ∀{Γ A B} → Tm71 Γ (arr71 A B) → Tm71 Γ A → Tm71 Γ B; app71
= λ t u Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec →
app71 _ _ _ (t Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec)
(u Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec)
tt71 : ∀{Γ} → Tm71 Γ top71; tt71
= λ Tm71 var71 lam71 app71 tt71 pair fst snd left right case zero suc rec → tt71 _
pair71 : ∀{Γ A B} → Tm71 Γ A → Tm71 Γ B → Tm71 Γ (prod71 A B); pair71
= λ t u Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec →
pair71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec)
(u Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec)
fst71 : ∀{Γ A B} → Tm71 Γ (prod71 A B) → Tm71 Γ A; fst71
= λ t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd left right case zero suc rec →
fst71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd left right case zero suc rec)
snd71 : ∀{Γ A B} → Tm71 Γ (prod71 A B) → Tm71 Γ B; snd71
= λ t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left right case zero suc rec →
snd71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left right case zero suc rec)
left71 : ∀{Γ A B} → Tm71 Γ A → Tm71 Γ (sum71 A B); left71
= λ t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right case zero suc rec →
left71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right case zero suc rec)
right71 : ∀{Γ A B} → Tm71 Γ B → Tm71 Γ (sum71 A B); right71
= λ t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case zero suc rec →
right71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case zero suc rec)
case71 : ∀{Γ A B C} → Tm71 Γ (sum71 A B) → Tm71 Γ (arr71 A C) → Tm71 Γ (arr71 B C) → Tm71 Γ C; case71
= λ t u v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec →
case71 _ _ _ _
(t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
(u Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
(v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
zero71 : ∀{Γ} → Tm71 Γ nat71; zero71
= λ Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc rec → zero71 _
suc71 : ∀{Γ} → Tm71 Γ nat71 → Tm71 Γ nat71; suc71
= λ t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec →
suc71 _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec)
rec71 : ∀{Γ A} → Tm71 Γ nat71 → Tm71 Γ (arr71 nat71 (arr71 A A)) → Tm71 Γ A → Tm71 Γ A; rec71
= λ t u v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71 →
rec71 _ _
(t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
(u Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
(v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
v071 : ∀{Γ A} → Tm71 (snoc71 Γ A) A; v071
= var71 vz71
v171 : ∀{Γ A B} → Tm71 (snoc71 (snoc71 Γ A) B) A; v171
= var71 (vs71 vz71)
v271 : ∀{Γ A B C} → Tm71 (snoc71 (snoc71 (snoc71 Γ A) B) C) A; v271
= var71 (vs71 (vs71 vz71))
v371 : ∀{Γ A B C D} → Tm71 (snoc71 (snoc71 (snoc71 (snoc71 Γ A) B) C) D) A; v371
= var71 (vs71 (vs71 (vs71 vz71)))
tbool71 : Ty71; tbool71
= sum71 top71 top71
true71 : ∀{Γ} → Tm71 Γ tbool71; true71
= left71 tt71
tfalse71 : ∀{Γ} → Tm71 Γ tbool71; tfalse71
= right71 tt71
ifthenelse71 : ∀{Γ A} → Tm71 Γ (arr71 tbool71 (arr71 A (arr71 A A))); ifthenelse71
= lam71 (lam71 (lam71 (case71 v271 (lam71 v271) (lam71 v171))))
times471 : ∀{Γ A} → Tm71 Γ (arr71 (arr71 A A) (arr71 A A)); times471
= lam71 (lam71 (app71 v171 (app71 v171 (app71 v171 (app71 v171 v071)))))
add71 : ∀{Γ} → Tm71 Γ (arr71 nat71 (arr71 nat71 nat71)); add71
= lam71 (rec71 v071
(lam71 (lam71 (lam71 (suc71 (app71 v171 v071)))))
(lam71 v071))
mul71 : ∀{Γ} → Tm71 Γ (arr71 nat71 (arr71 nat71 nat71)); mul71
= lam71 (rec71 v071
(lam71 (lam71 (lam71 (app71 (app71 add71 (app71 v171 v071)) v071))))
(lam71 zero71))
fact71 : ∀{Γ} → Tm71 Γ (arr71 nat71 nat71); fact71
= lam71 (rec71 v071 (lam71 (lam71 (app71 (app71 mul71 (suc71 v171)) v071)))
(suc71 zero71))
{-# OPTIONS --type-in-type #-}
Ty72 : Set
Ty72 =
(Ty72 : Set)
(nat top bot : Ty72)
(arr prod sum : Ty72 → Ty72 → Ty72)
→ Ty72
nat72 : Ty72; nat72 = λ _ nat72 _ _ _ _ _ → nat72
top72 : Ty72; top72 = λ _ _ top72 _ _ _ _ → top72
bot72 : Ty72; bot72 = λ _ _ _ bot72 _ _ _ → bot72
arr72 : Ty72 → Ty72 → Ty72; arr72
= λ A B Ty72 nat72 top72 bot72 arr72 prod sum →
arr72 (A Ty72 nat72 top72 bot72 arr72 prod sum) (B Ty72 nat72 top72 bot72 arr72 prod sum)
prod72 : Ty72 → Ty72 → Ty72; prod72
= λ A B Ty72 nat72 top72 bot72 arr72 prod72 sum →
prod72 (A Ty72 nat72 top72 bot72 arr72 prod72 sum) (B Ty72 nat72 top72 bot72 arr72 prod72 sum)
sum72 : Ty72 → Ty72 → Ty72; sum72
= λ A B Ty72 nat72 top72 bot72 arr72 prod72 sum72 →
sum72 (A Ty72 nat72 top72 bot72 arr72 prod72 sum72) (B Ty72 nat72 top72 bot72 arr72 prod72 sum72)
Con72 : Set; Con72
= (Con72 : Set)
(nil : Con72)
(snoc : Con72 → Ty72 → Con72)
→ Con72
nil72 : Con72; nil72
= λ Con72 nil72 snoc → nil72
snoc72 : Con72 → Ty72 → Con72; snoc72
= λ Γ A Con72 nil72 snoc72 → snoc72 (Γ Con72 nil72 snoc72) A
Var72 : Con72 → Ty72 → Set; Var72
= λ Γ A →
(Var72 : Con72 → Ty72 → Set)
(vz : ∀ Γ A → Var72 (snoc72 Γ A) A)
(vs : ∀ Γ B A → Var72 Γ A → Var72 (snoc72 Γ B) A)
→ Var72 Γ A
vz72 : ∀{Γ A} → Var72 (snoc72 Γ A) A; vz72
= λ Var72 vz72 vs → vz72 _ _
vs72 : ∀{Γ B A} → Var72 Γ A → Var72 (snoc72 Γ B) A; vs72
= λ x Var72 vz72 vs72 → vs72 _ _ _ (x Var72 vz72 vs72)
Tm72 : Con72 → Ty72 → Set; Tm72
= λ Γ A →
(Tm72 : Con72 → Ty72 → Set)
(var : ∀ Γ A → Var72 Γ A → Tm72 Γ A)
(lam : ∀ Γ A B → Tm72 (snoc72 Γ A) B → Tm72 Γ (arr72 A B))
(app : ∀ Γ A B → Tm72 Γ (arr72 A B) → Tm72 Γ A → Tm72 Γ B)
(tt : ∀ Γ → Tm72 Γ top72)
(pair : ∀ Γ A B → Tm72 Γ A → Tm72 Γ B → Tm72 Γ (prod72 A B))
(fst : ∀ Γ A B → Tm72 Γ (prod72 A B) → Tm72 Γ A)
(snd : ∀ Γ A B → Tm72 Γ (prod72 A B) → Tm72 Γ B)
(left : ∀ Γ A B → Tm72 Γ A → Tm72 Γ (sum72 A B))
(right : ∀ Γ A B → Tm72 Γ B → Tm72 Γ (sum72 A B))
(case : ∀ Γ A B C → Tm72 Γ (sum72 A B) → Tm72 Γ (arr72 A C) → Tm72 Γ (arr72 B C) → Tm72 Γ C)
(zero : ∀ Γ → Tm72 Γ nat72)
(suc : ∀ Γ → Tm72 Γ nat72 → Tm72 Γ nat72)
(rec : ∀ Γ A → Tm72 Γ nat72 → Tm72 Γ (arr72 nat72 (arr72 A A)) → Tm72 Γ A → Tm72 Γ A)
→ Tm72 Γ A
var72 : ∀{Γ A} → Var72 Γ A → Tm72 Γ A; var72
= λ x Tm72 var72 lam app tt pair fst snd left right case zero suc rec →
var72 _ _ x
lam72 : ∀{Γ A B} → Tm72 (snoc72 Γ A) B → Tm72 Γ (arr72 A B); lam72
= λ t Tm72 var72 lam72 app tt pair fst snd left right case zero suc rec →
lam72 _ _ _ (t Tm72 var72 lam72 app tt pair fst snd left right case zero suc rec)
app72 : ∀{Γ A B} → Tm72 Γ (arr72 A B) → Tm72 Γ A → Tm72 Γ B; app72
= λ t u Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec →
app72 _ _ _ (t Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec)
(u Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec)
tt72 : ∀{Γ} → Tm72 Γ top72; tt72
= λ Tm72 var72 lam72 app72 tt72 pair fst snd left right case zero suc rec → tt72 _
pair72 : ∀{Γ A B} → Tm72 Γ A → Tm72 Γ B → Tm72 Γ (prod72 A B); pair72
= λ t u Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec →
pair72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec)
(u Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec)
fst72 : ∀{Γ A B} → Tm72 Γ (prod72 A B) → Tm72 Γ A; fst72
= λ t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd left right case zero suc rec →
fst72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd left right case zero suc rec)
snd72 : ∀{Γ A B} → Tm72 Γ (prod72 A B) → Tm72 Γ B; snd72
= λ t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left right case zero suc rec →
snd72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left right case zero suc rec)
left72 : ∀{Γ A B} → Tm72 Γ A → Tm72 Γ (sum72 A B); left72
= λ t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right case zero suc rec →
left72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right case zero suc rec)
right72 : ∀{Γ A B} → Tm72 Γ B → Tm72 Γ (sum72 A B); right72
= λ t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case zero suc rec →
right72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case zero suc rec)
case72 : ∀{Γ A B C} → Tm72 Γ (sum72 A B) → Tm72 Γ (arr72 A C) → Tm72 Γ (arr72 B C) → Tm72 Γ C; case72
= λ t u v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec →
case72 _ _ _ _
(t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
(u Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
(v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
zero72 : ∀{Γ} → Tm72 Γ nat72; zero72
= λ Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc rec → zero72 _
suc72 : ∀{Γ} → Tm72 Γ nat72 → Tm72 Γ nat72; suc72
= λ t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec →
suc72 _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec)
rec72 : ∀{Γ A} → Tm72 Γ nat72 → Tm72 Γ (arr72 nat72 (arr72 A A)) → Tm72 Γ A → Tm72 Γ A; rec72
= λ t u v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72 →
rec72 _ _
(t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
(u Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
(v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
v072 : ∀{Γ A} → Tm72 (snoc72 Γ A) A; v072
= var72 vz72
v172 : ∀{Γ A B} → Tm72 (snoc72 (snoc72 Γ A) B) A; v172
= var72 (vs72 vz72)
v272 : ∀{Γ A B C} → Tm72 (snoc72 (snoc72 (snoc72 Γ A) B) C) A; v272
= var72 (vs72 (vs72 vz72))
v372 : ∀{Γ A B C D} → Tm72 (snoc72 (snoc72 (snoc72 (snoc72 Γ A) B) C) D) A; v372
= var72 (vs72 (vs72 (vs72 vz72)))
tbool72 : Ty72; tbool72
= sum72 top72 top72
true72 : ∀{Γ} → Tm72 Γ tbool72; true72
= left72 tt72
tfalse72 : ∀{Γ} → Tm72 Γ tbool72; tfalse72
= right72 tt72
ifthenelse72 : ∀{Γ A} → Tm72 Γ (arr72 tbool72 (arr72 A (arr72 A A))); ifthenelse72
= lam72 (lam72 (lam72 (case72 v272 (lam72 v272) (lam72 v172))))
times472 : ∀{Γ A} → Tm72 Γ (arr72 (arr72 A A) (arr72 A A)); times472
= lam72 (lam72 (app72 v172 (app72 v172 (app72 v172 (app72 v172 v072)))))
add72 : ∀{Γ} → Tm72 Γ (arr72 nat72 (arr72 nat72 nat72)); add72
= lam72 (rec72 v072
(lam72 (lam72 (lam72 (suc72 (app72 v172 v072)))))
(lam72 v072))
mul72 : ∀{Γ} → Tm72 Γ (arr72 nat72 (arr72 nat72 nat72)); mul72
= lam72 (rec72 v072
(lam72 (lam72 (lam72 (app72 (app72 add72 (app72 v172 v072)) v072))))
(lam72 zero72))
fact72 : ∀{Γ} → Tm72 Γ (arr72 nat72 nat72); fact72
= lam72 (rec72 v072 (lam72 (lam72 (app72 (app72 mul72 (suc72 v172)) v072)))
(suc72 zero72))
{-# OPTIONS --type-in-type #-}
Ty73 : Set
Ty73 =
(Ty73 : Set)
(nat top bot : Ty73)
(arr prod sum : Ty73 → Ty73 → Ty73)
→ Ty73
nat73 : Ty73; nat73 = λ _ nat73 _ _ _ _ _ → nat73
top73 : Ty73; top73 = λ _ _ top73 _ _ _ _ → top73
bot73 : Ty73; bot73 = λ _ _ _ bot73 _ _ _ → bot73
arr73 : Ty73 → Ty73 → Ty73; arr73
= λ A B Ty73 nat73 top73 bot73 arr73 prod sum →
arr73 (A Ty73 nat73 top73 bot73 arr73 prod sum) (B Ty73 nat73 top73 bot73 arr73 prod sum)
prod73 : Ty73 → Ty73 → Ty73; prod73
= λ A B Ty73 nat73 top73 bot73 arr73 prod73 sum →
prod73 (A Ty73 nat73 top73 bot73 arr73 prod73 sum) (B Ty73 nat73 top73 bot73 arr73 prod73 sum)
sum73 : Ty73 → Ty73 → Ty73; sum73
= λ A B Ty73 nat73 top73 bot73 arr73 prod73 sum73 →
sum73 (A Ty73 nat73 top73 bot73 arr73 prod73 sum73) (B Ty73 nat73 top73 bot73 arr73 prod73 sum73)
Con73 : Set; Con73
= (Con73 : Set)
(nil : Con73)
(snoc : Con73 → Ty73 → Con73)
→ Con73
nil73 : Con73; nil73
= λ Con73 nil73 snoc → nil73
snoc73 : Con73 → Ty73 → Con73; snoc73
= λ Γ A Con73 nil73 snoc73 → snoc73 (Γ Con73 nil73 snoc73) A
Var73 : Con73 → Ty73 → Set; Var73
= λ Γ A →
(Var73 : Con73 → Ty73 → Set)
(vz : ∀ Γ A → Var73 (snoc73 Γ A) A)
(vs : ∀ Γ B A → Var73 Γ A → Var73 (snoc73 Γ B) A)
→ Var73 Γ A
vz73 : ∀{Γ A} → Var73 (snoc73 Γ A) A; vz73
= λ Var73 vz73 vs → vz73 _ _
vs73 : ∀{Γ B A} → Var73 Γ A → Var73 (snoc73 Γ B) A; vs73
= λ x Var73 vz73 vs73 → vs73 _ _ _ (x Var73 vz73 vs73)
Tm73 : Con73 → Ty73 → Set; Tm73
= λ Γ A →
(Tm73 : Con73 → Ty73 → Set)
(var : ∀ Γ A → Var73 Γ A → Tm73 Γ A)
(lam : ∀ Γ A B → Tm73 (snoc73 Γ A) B → Tm73 Γ (arr73 A B))
(app : ∀ Γ A B → Tm73 Γ (arr73 A B) → Tm73 Γ A → Tm73 Γ B)
(tt : ∀ Γ → Tm73 Γ top73)
(pair : ∀ Γ A B → Tm73 Γ A → Tm73 Γ B → Tm73 Γ (prod73 A B))
(fst : ∀ Γ A B → Tm73 Γ (prod73 A B) → Tm73 Γ A)
(snd : ∀ Γ A B → Tm73 Γ (prod73 A B) → Tm73 Γ B)
(left : ∀ Γ A B → Tm73 Γ A → Tm73 Γ (sum73 A B))
(right : ∀ Γ A B → Tm73 Γ B → Tm73 Γ (sum73 A B))
(case : ∀ Γ A B C → Tm73 Γ (sum73 A B) → Tm73 Γ (arr73 A C) → Tm73 Γ (arr73 B C) → Tm73 Γ C)
(zero : ∀ Γ → Tm73 Γ nat73)
(suc : ∀ Γ → Tm73 Γ nat73 → Tm73 Γ nat73)
(rec : ∀ Γ A → Tm73 Γ nat73 → Tm73 Γ (arr73 nat73 (arr73 A A)) → Tm73 Γ A → Tm73 Γ A)
→ Tm73 Γ A
var73 : ∀{Γ A} → Var73 Γ A → Tm73 Γ A; var73
= λ x Tm73 var73 lam app tt pair fst snd left right case zero suc rec →
var73 _ _ x
lam73 : ∀{Γ A B} → Tm73 (snoc73 Γ A) B → Tm73 Γ (arr73 A B); lam73
= λ t Tm73 var73 lam73 app tt pair fst snd left right case zero suc rec →
lam73 _ _ _ (t Tm73 var73 lam73 app tt pair fst snd left right case zero suc rec)
app73 : ∀{Γ A B} → Tm73 Γ (arr73 A B) → Tm73 Γ A → Tm73 Γ B; app73
= λ t u Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec →
app73 _ _ _ (t Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec)
(u Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec)
tt73 : ∀{Γ} → Tm73 Γ top73; tt73
= λ Tm73 var73 lam73 app73 tt73 pair fst snd left right case zero suc rec → tt73 _
pair73 : ∀{Γ A B} → Tm73 Γ A → Tm73 Γ B → Tm73 Γ (prod73 A B); pair73
= λ t u Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec →
pair73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec)
(u Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec)
fst73 : ∀{Γ A B} → Tm73 Γ (prod73 A B) → Tm73 Γ A; fst73
= λ t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd left right case zero suc rec →
fst73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd left right case zero suc rec)
snd73 : ∀{Γ A B} → Tm73 Γ (prod73 A B) → Tm73 Γ B; snd73
= λ t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left right case zero suc rec →
snd73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left right case zero suc rec)
left73 : ∀{Γ A B} → Tm73 Γ A → Tm73 Γ (sum73 A B); left73
= λ t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right case zero suc rec →
left73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right case zero suc rec)
right73 : ∀{Γ A B} → Tm73 Γ B → Tm73 Γ (sum73 A B); right73
= λ t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case zero suc rec →
right73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case zero suc rec)
case73 : ∀{Γ A B C} → Tm73 Γ (sum73 A B) → Tm73 Γ (arr73 A C) → Tm73 Γ (arr73 B C) → Tm73 Γ C; case73
= λ t u v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec →
case73 _ _ _ _
(t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
(u Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
(v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
zero73 : ∀{Γ} → Tm73 Γ nat73; zero73
= λ Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc rec → zero73 _
suc73 : ∀{Γ} → Tm73 Γ nat73 → Tm73 Γ nat73; suc73
= λ t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec →
suc73 _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec)
rec73 : ∀{Γ A} → Tm73 Γ nat73 → Tm73 Γ (arr73 nat73 (arr73 A A)) → Tm73 Γ A → Tm73 Γ A; rec73
= λ t u v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73 →
rec73 _ _
(t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
(u Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
(v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
v073 : ∀{Γ A} → Tm73 (snoc73 Γ A) A; v073
= var73 vz73
v173 : ∀{Γ A B} → Tm73 (snoc73 (snoc73 Γ A) B) A; v173
= var73 (vs73 vz73)
v273 : ∀{Γ A B C} → Tm73 (snoc73 (snoc73 (snoc73 Γ A) B) C) A; v273
= var73 (vs73 (vs73 vz73))
v373 : ∀{Γ A B C D} → Tm73 (snoc73 (snoc73 (snoc73 (snoc73 Γ A) B) C) D) A; v373
= var73 (vs73 (vs73 (vs73 vz73)))
tbool73 : Ty73; tbool73
= sum73 top73 top73
true73 : ∀{Γ} → Tm73 Γ tbool73; true73
= left73 tt73
tfalse73 : ∀{Γ} → Tm73 Γ tbool73; tfalse73
= right73 tt73
ifthenelse73 : ∀{Γ A} → Tm73 Γ (arr73 tbool73 (arr73 A (arr73 A A))); ifthenelse73
= lam73 (lam73 (lam73 (case73 v273 (lam73 v273) (lam73 v173))))
times473 : ∀{Γ A} → Tm73 Γ (arr73 (arr73 A A) (arr73 A A)); times473
= lam73 (lam73 (app73 v173 (app73 v173 (app73 v173 (app73 v173 v073)))))
add73 : ∀{Γ} → Tm73 Γ (arr73 nat73 (arr73 nat73 nat73)); add73
= lam73 (rec73 v073
(lam73 (lam73 (lam73 (suc73 (app73 v173 v073)))))
(lam73 v073))
mul73 : ∀{Γ} → Tm73 Γ (arr73 nat73 (arr73 nat73 nat73)); mul73
= lam73 (rec73 v073
(lam73 (lam73 (lam73 (app73 (app73 add73 (app73 v173 v073)) v073))))
(lam73 zero73))
fact73 : ∀{Γ} → Tm73 Γ (arr73 nat73 nat73); fact73
= lam73 (rec73 v073 (lam73 (lam73 (app73 (app73 mul73 (suc73 v173)) v073)))
(suc73 zero73))
{-# OPTIONS --type-in-type #-}
Ty74 : Set
Ty74 =
(Ty74 : Set)
(nat top bot : Ty74)
(arr prod sum : Ty74 → Ty74 → Ty74)
→ Ty74
nat74 : Ty74; nat74 = λ _ nat74 _ _ _ _ _ → nat74
top74 : Ty74; top74 = λ _ _ top74 _ _ _ _ → top74
bot74 : Ty74; bot74 = λ _ _ _ bot74 _ _ _ → bot74
arr74 : Ty74 → Ty74 → Ty74; arr74
= λ A B Ty74 nat74 top74 bot74 arr74 prod sum →
arr74 (A Ty74 nat74 top74 bot74 arr74 prod sum) (B Ty74 nat74 top74 bot74 arr74 prod sum)
prod74 : Ty74 → Ty74 → Ty74; prod74
= λ A B Ty74 nat74 top74 bot74 arr74 prod74 sum →
prod74 (A Ty74 nat74 top74 bot74 arr74 prod74 sum) (B Ty74 nat74 top74 bot74 arr74 prod74 sum)
sum74 : Ty74 → Ty74 → Ty74; sum74
= λ A B Ty74 nat74 top74 bot74 arr74 prod74 sum74 →
sum74 (A Ty74 nat74 top74 bot74 arr74 prod74 sum74) (B Ty74 nat74 top74 bot74 arr74 prod74 sum74)
Con74 : Set; Con74
= (Con74 : Set)
(nil : Con74)
(snoc : Con74 → Ty74 → Con74)
→ Con74
nil74 : Con74; nil74
= λ Con74 nil74 snoc → nil74
snoc74 : Con74 → Ty74 → Con74; snoc74
= λ Γ A Con74 nil74 snoc74 → snoc74 (Γ Con74 nil74 snoc74) A
Var74 : Con74 → Ty74 → Set; Var74
= λ Γ A →
(Var74 : Con74 → Ty74 → Set)
(vz : ∀ Γ A → Var74 (snoc74 Γ A) A)
(vs : ∀ Γ B A → Var74 Γ A → Var74 (snoc74 Γ B) A)
→ Var74 Γ A
vz74 : ∀{Γ A} → Var74 (snoc74 Γ A) A; vz74
= λ Var74 vz74 vs → vz74 _ _
vs74 : ∀{Γ B A} → Var74 Γ A → Var74 (snoc74 Γ B) A; vs74
= λ x Var74 vz74 vs74 → vs74 _ _ _ (x Var74 vz74 vs74)
Tm74 : Con74 → Ty74 → Set; Tm74
= λ Γ A →
(Tm74 : Con74 → Ty74 → Set)
(var : ∀ Γ A → Var74 Γ A → Tm74 Γ A)
(lam : ∀ Γ A B → Tm74 (snoc74 Γ A) B → Tm74 Γ (arr74 A B))
(app : ∀ Γ A B → Tm74 Γ (arr74 A B) → Tm74 Γ A → Tm74 Γ B)
(tt : ∀ Γ → Tm74 Γ top74)
(pair : ∀ Γ A B → Tm74 Γ A → Tm74 Γ B → Tm74 Γ (prod74 A B))
(fst : ∀ Γ A B → Tm74 Γ (prod74 A B) → Tm74 Γ A)
(snd : ∀ Γ A B → Tm74 Γ (prod74 A B) → Tm74 Γ B)
(left : ∀ Γ A B → Tm74 Γ A → Tm74 Γ (sum74 A B))
(right : ∀ Γ A B → Tm74 Γ B → Tm74 Γ (sum74 A B))
(case : ∀ Γ A B C → Tm74 Γ (sum74 A B) → Tm74 Γ (arr74 A C) → Tm74 Γ (arr74 B C) → Tm74 Γ C)
(zero : ∀ Γ → Tm74 Γ nat74)
(suc : ∀ Γ → Tm74 Γ nat74 → Tm74 Γ nat74)
(rec : ∀ Γ A → Tm74 Γ nat74 → Tm74 Γ (arr74 nat74 (arr74 A A)) → Tm74 Γ A → Tm74 Γ A)
→ Tm74 Γ A
var74 : ∀{Γ A} → Var74 Γ A → Tm74 Γ A; var74
= λ x Tm74 var74 lam app tt pair fst snd left right case zero suc rec →
var74 _ _ x
lam74 : ∀{Γ A B} → Tm74 (snoc74 Γ A) B → Tm74 Γ (arr74 A B); lam74
= λ t Tm74 var74 lam74 app tt pair fst snd left right case zero suc rec →
lam74 _ _ _ (t Tm74 var74 lam74 app tt pair fst snd left right case zero suc rec)
app74 : ∀{Γ A B} → Tm74 Γ (arr74 A B) → Tm74 Γ A → Tm74 Γ B; app74
= λ t u Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec →
app74 _ _ _ (t Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec)
(u Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec)
tt74 : ∀{Γ} → Tm74 Γ top74; tt74
= λ Tm74 var74 lam74 app74 tt74 pair fst snd left right case zero suc rec → tt74 _
pair74 : ∀{Γ A B} → Tm74 Γ A → Tm74 Γ B → Tm74 Γ (prod74 A B); pair74
= λ t u Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec →
pair74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec)
(u Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec)
fst74 : ∀{Γ A B} → Tm74 Γ (prod74 A B) → Tm74 Γ A; fst74
= λ t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd left right case zero suc rec →
fst74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd left right case zero suc rec)
snd74 : ∀{Γ A B} → Tm74 Γ (prod74 A B) → Tm74 Γ B; snd74
= λ t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left right case zero suc rec →
snd74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left right case zero suc rec)
left74 : ∀{Γ A B} → Tm74 Γ A → Tm74 Γ (sum74 A B); left74
= λ t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right case zero suc rec →
left74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right case zero suc rec)
right74 : ∀{Γ A B} → Tm74 Γ B → Tm74 Γ (sum74 A B); right74
= λ t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case zero suc rec →
right74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case zero suc rec)
case74 : ∀{Γ A B C} → Tm74 Γ (sum74 A B) → Tm74 Γ (arr74 A C) → Tm74 Γ (arr74 B C) → Tm74 Γ C; case74
= λ t u v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec →
case74 _ _ _ _
(t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
(u Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
(v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
zero74 : ∀{Γ} → Tm74 Γ nat74; zero74
= λ Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc rec → zero74 _
suc74 : ∀{Γ} → Tm74 Γ nat74 → Tm74 Γ nat74; suc74
= λ t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec →
suc74 _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec)
rec74 : ∀{Γ A} → Tm74 Γ nat74 → Tm74 Γ (arr74 nat74 (arr74 A A)) → Tm74 Γ A → Tm74 Γ A; rec74
= λ t u v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74 →
rec74 _ _
(t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
(u Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
(v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
v074 : ∀{Γ A} → Tm74 (snoc74 Γ A) A; v074
= var74 vz74
v174 : ∀{Γ A B} → Tm74 (snoc74 (snoc74 Γ A) B) A; v174
= var74 (vs74 vz74)
v274 : ∀{Γ A B C} → Tm74 (snoc74 (snoc74 (snoc74 Γ A) B) C) A; v274
= var74 (vs74 (vs74 vz74))
v374 : ∀{Γ A B C D} → Tm74 (snoc74 (snoc74 (snoc74 (snoc74 Γ A) B) C) D) A; v374
= var74 (vs74 (vs74 (vs74 vz74)))
tbool74 : Ty74; tbool74
= sum74 top74 top74
true74 : ∀{Γ} → Tm74 Γ tbool74; true74
= left74 tt74
tfalse74 : ∀{Γ} → Tm74 Γ tbool74; tfalse74
= right74 tt74
ifthenelse74 : ∀{Γ A} → Tm74 Γ (arr74 tbool74 (arr74 A (arr74 A A))); ifthenelse74
= lam74 (lam74 (lam74 (case74 v274 (lam74 v274) (lam74 v174))))
times474 : ∀{Γ A} → Tm74 Γ (arr74 (arr74 A A) (arr74 A A)); times474
= lam74 (lam74 (app74 v174 (app74 v174 (app74 v174 (app74 v174 v074)))))
add74 : ∀{Γ} → Tm74 Γ (arr74 nat74 (arr74 nat74 nat74)); add74
= lam74 (rec74 v074
(lam74 (lam74 (lam74 (suc74 (app74 v174 v074)))))
(lam74 v074))
mul74 : ∀{Γ} → Tm74 Γ (arr74 nat74 (arr74 nat74 nat74)); mul74
= lam74 (rec74 v074
(lam74 (lam74 (lam74 (app74 (app74 add74 (app74 v174 v074)) v074))))
(lam74 zero74))
fact74 : ∀{Γ} → Tm74 Γ (arr74 nat74 nat74); fact74
= lam74 (rec74 v074 (lam74 (lam74 (app74 (app74 mul74 (suc74 v174)) v074)))
(suc74 zero74))
{-# OPTIONS --type-in-type #-}
Ty75 : Set
Ty75 =
(Ty75 : Set)
(nat top bot : Ty75)
(arr prod sum : Ty75 → Ty75 → Ty75)
→ Ty75
nat75 : Ty75; nat75 = λ _ nat75 _ _ _ _ _ → nat75
top75 : Ty75; top75 = λ _ _ top75 _ _ _ _ → top75
bot75 : Ty75; bot75 = λ _ _ _ bot75 _ _ _ → bot75
arr75 : Ty75 → Ty75 → Ty75; arr75
= λ A B Ty75 nat75 top75 bot75 arr75 prod sum →
arr75 (A Ty75 nat75 top75 bot75 arr75 prod sum) (B Ty75 nat75 top75 bot75 arr75 prod sum)
prod75 : Ty75 → Ty75 → Ty75; prod75
= λ A B Ty75 nat75 top75 bot75 arr75 prod75 sum →
prod75 (A Ty75 nat75 top75 bot75 arr75 prod75 sum) (B Ty75 nat75 top75 bot75 arr75 prod75 sum)
sum75 : Ty75 → Ty75 → Ty75; sum75
= λ A B Ty75 nat75 top75 bot75 arr75 prod75 sum75 →
sum75 (A Ty75 nat75 top75 bot75 arr75 prod75 sum75) (B Ty75 nat75 top75 bot75 arr75 prod75 sum75)
Con75 : Set; Con75
= (Con75 : Set)
(nil : Con75)
(snoc : Con75 → Ty75 → Con75)
→ Con75
nil75 : Con75; nil75
= λ Con75 nil75 snoc → nil75
snoc75 : Con75 → Ty75 → Con75; snoc75
= λ Γ A Con75 nil75 snoc75 → snoc75 (Γ Con75 nil75 snoc75) A
Var75 : Con75 → Ty75 → Set; Var75
= λ Γ A →
(Var75 : Con75 → Ty75 → Set)
(vz : ∀ Γ A → Var75 (snoc75 Γ A) A)
(vs : ∀ Γ B A → Var75 Γ A → Var75 (snoc75 Γ B) A)
→ Var75 Γ A
vz75 : ∀{Γ A} → Var75 (snoc75 Γ A) A; vz75
= λ Var75 vz75 vs → vz75 _ _
vs75 : ∀{Γ B A} → Var75 Γ A → Var75 (snoc75 Γ B) A; vs75
= λ x Var75 vz75 vs75 → vs75 _ _ _ (x Var75 vz75 vs75)
Tm75 : Con75 → Ty75 → Set; Tm75
= λ Γ A →
(Tm75 : Con75 → Ty75 → Set)
(var : ∀ Γ A → Var75 Γ A → Tm75 Γ A)
(lam : ∀ Γ A B → Tm75 (snoc75 Γ A) B → Tm75 Γ (arr75 A B))
(app : ∀ Γ A B → Tm75 Γ (arr75 A B) → Tm75 Γ A → Tm75 Γ B)
(tt : ∀ Γ → Tm75 Γ top75)
(pair : ∀ Γ A B → Tm75 Γ A → Tm75 Γ B → Tm75 Γ (prod75 A B))
(fst : ∀ Γ A B → Tm75 Γ (prod75 A B) → Tm75 Γ A)
(snd : ∀ Γ A B → Tm75 Γ (prod75 A B) → Tm75 Γ B)
(left : ∀ Γ A B → Tm75 Γ A → Tm75 Γ (sum75 A B))
(right : ∀ Γ A B → Tm75 Γ B → Tm75 Γ (sum75 A B))
(case : ∀ Γ A B C → Tm75 Γ (sum75 A B) → Tm75 Γ (arr75 A C) → Tm75 Γ (arr75 B C) → Tm75 Γ C)
(zero : ∀ Γ → Tm75 Γ nat75)
(suc : ∀ Γ → Tm75 Γ nat75 → Tm75 Γ nat75)
(rec : ∀ Γ A → Tm75 Γ nat75 → Tm75 Γ (arr75 nat75 (arr75 A A)) → Tm75 Γ A → Tm75 Γ A)
→ Tm75 Γ A
var75 : ∀{Γ A} → Var75 Γ A → Tm75 Γ A; var75
= λ x Tm75 var75 lam app tt pair fst snd left right case zero suc rec →
var75 _ _ x
lam75 : ∀{Γ A B} → Tm75 (snoc75 Γ A) B → Tm75 Γ (arr75 A B); lam75
= λ t Tm75 var75 lam75 app tt pair fst snd left right case zero suc rec →
lam75 _ _ _ (t Tm75 var75 lam75 app tt pair fst snd left right case zero suc rec)
app75 : ∀{Γ A B} → Tm75 Γ (arr75 A B) → Tm75 Γ A → Tm75 Γ B; app75
= λ t u Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec →
app75 _ _ _ (t Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec)
(u Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec)
tt75 : ∀{Γ} → Tm75 Γ top75; tt75
= λ Tm75 var75 lam75 app75 tt75 pair fst snd left right case zero suc rec → tt75 _
pair75 : ∀{Γ A B} → Tm75 Γ A → Tm75 Γ B → Tm75 Γ (prod75 A B); pair75
= λ t u Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec →
pair75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec)
(u Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec)
fst75 : ∀{Γ A B} → Tm75 Γ (prod75 A B) → Tm75 Γ A; fst75
= λ t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd left right case zero suc rec →
fst75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd left right case zero suc rec)
snd75 : ∀{Γ A B} → Tm75 Γ (prod75 A B) → Tm75 Γ B; snd75
= λ t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left right case zero suc rec →
snd75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left right case zero suc rec)
left75 : ∀{Γ A B} → Tm75 Γ A → Tm75 Γ (sum75 A B); left75
= λ t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right case zero suc rec →
left75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right case zero suc rec)
right75 : ∀{Γ A B} → Tm75 Γ B → Tm75 Γ (sum75 A B); right75
= λ t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case zero suc rec →
right75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case zero suc rec)
case75 : ∀{Γ A B C} → Tm75 Γ (sum75 A B) → Tm75 Γ (arr75 A C) → Tm75 Γ (arr75 B C) → Tm75 Γ C; case75
= λ t u v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec →
case75 _ _ _ _
(t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
(u Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
(v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
zero75 : ∀{Γ} → Tm75 Γ nat75; zero75
= λ Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc rec → zero75 _
suc75 : ∀{Γ} → Tm75 Γ nat75 → Tm75 Γ nat75; suc75
= λ t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec →
suc75 _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec)
rec75 : ∀{Γ A} → Tm75 Γ nat75 → Tm75 Γ (arr75 nat75 (arr75 A A)) → Tm75 Γ A → Tm75 Γ A; rec75
= λ t u v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75 →
rec75 _ _
(t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
(u Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
(v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
v075 : ∀{Γ A} → Tm75 (snoc75 Γ A) A; v075
= var75 vz75
v175 : ∀{Γ A B} → Tm75 (snoc75 (snoc75 Γ A) B) A; v175
= var75 (vs75 vz75)
v275 : ∀{Γ A B C} → Tm75 (snoc75 (snoc75 (snoc75 Γ A) B) C) A; v275
= var75 (vs75 (vs75 vz75))
v375 : ∀{Γ A B C D} → Tm75 (snoc75 (snoc75 (snoc75 (snoc75 Γ A) B) C) D) A; v375
= var75 (vs75 (vs75 (vs75 vz75)))
tbool75 : Ty75; tbool75
= sum75 top75 top75
true75 : ∀{Γ} → Tm75 Γ tbool75; true75
= left75 tt75
tfalse75 : ∀{Γ} → Tm75 Γ tbool75; tfalse75
= right75 tt75
ifthenelse75 : ∀{Γ A} → Tm75 Γ (arr75 tbool75 (arr75 A (arr75 A A))); ifthenelse75
= lam75 (lam75 (lam75 (case75 v275 (lam75 v275) (lam75 v175))))
times475 : ∀{Γ A} → Tm75 Γ (arr75 (arr75 A A) (arr75 A A)); times475
= lam75 (lam75 (app75 v175 (app75 v175 (app75 v175 (app75 v175 v075)))))
add75 : ∀{Γ} → Tm75 Γ (arr75 nat75 (arr75 nat75 nat75)); add75
= lam75 (rec75 v075
(lam75 (lam75 (lam75 (suc75 (app75 v175 v075)))))
(lam75 v075))
mul75 : ∀{Γ} → Tm75 Γ (arr75 nat75 (arr75 nat75 nat75)); mul75
= lam75 (rec75 v075
(lam75 (lam75 (lam75 (app75 (app75 add75 (app75 v175 v075)) v075))))
(lam75 zero75))
fact75 : ∀{Γ} → Tm75 Γ (arr75 nat75 nat75); fact75
= lam75 (rec75 v075 (lam75 (lam75 (app75 (app75 mul75 (suc75 v175)) v075)))
(suc75 zero75))
{-# OPTIONS --type-in-type #-}
Ty76 : Set
Ty76 =
(Ty76 : Set)
(nat top bot : Ty76)
(arr prod sum : Ty76 → Ty76 → Ty76)
→ Ty76
nat76 : Ty76; nat76 = λ _ nat76 _ _ _ _ _ → nat76
top76 : Ty76; top76 = λ _ _ top76 _ _ _ _ → top76
bot76 : Ty76; bot76 = λ _ _ _ bot76 _ _ _ → bot76
arr76 : Ty76 → Ty76 → Ty76; arr76
= λ A B Ty76 nat76 top76 bot76 arr76 prod sum →
arr76 (A Ty76 nat76 top76 bot76 arr76 prod sum) (B Ty76 nat76 top76 bot76 arr76 prod sum)
prod76 : Ty76 → Ty76 → Ty76; prod76
= λ A B Ty76 nat76 top76 bot76 arr76 prod76 sum →
prod76 (A Ty76 nat76 top76 bot76 arr76 prod76 sum) (B Ty76 nat76 top76 bot76 arr76 prod76 sum)
sum76 : Ty76 → Ty76 → Ty76; sum76
= λ A B Ty76 nat76 top76 bot76 arr76 prod76 sum76 →
sum76 (A Ty76 nat76 top76 bot76 arr76 prod76 sum76) (B Ty76 nat76 top76 bot76 arr76 prod76 sum76)
Con76 : Set; Con76
= (Con76 : Set)
(nil : Con76)
(snoc : Con76 → Ty76 → Con76)
→ Con76
nil76 : Con76; nil76
= λ Con76 nil76 snoc → nil76
snoc76 : Con76 → Ty76 → Con76; snoc76
= λ Γ A Con76 nil76 snoc76 → snoc76 (Γ Con76 nil76 snoc76) A
Var76 : Con76 → Ty76 → Set; Var76
= λ Γ A →
(Var76 : Con76 → Ty76 → Set)
(vz : ∀ Γ A → Var76 (snoc76 Γ A) A)
(vs : ∀ Γ B A → Var76 Γ A → Var76 (snoc76 Γ B) A)
→ Var76 Γ A
vz76 : ∀{Γ A} → Var76 (snoc76 Γ A) A; vz76
= λ Var76 vz76 vs → vz76 _ _
vs76 : ∀{Γ B A} → Var76 Γ A → Var76 (snoc76 Γ B) A; vs76
= λ x Var76 vz76 vs76 → vs76 _ _ _ (x Var76 vz76 vs76)
Tm76 : Con76 → Ty76 → Set; Tm76
= λ Γ A →
(Tm76 : Con76 → Ty76 → Set)
(var : ∀ Γ A → Var76 Γ A → Tm76 Γ A)
(lam : ∀ Γ A B → Tm76 (snoc76 Γ A) B → Tm76 Γ (arr76 A B))
(app : ∀ Γ A B → Tm76 Γ (arr76 A B) → Tm76 Γ A → Tm76 Γ B)
(tt : ∀ Γ → Tm76 Γ top76)
(pair : ∀ Γ A B → Tm76 Γ A → Tm76 Γ B → Tm76 Γ (prod76 A B))
(fst : ∀ Γ A B → Tm76 Γ (prod76 A B) → Tm76 Γ A)
(snd : ∀ Γ A B → Tm76 Γ (prod76 A B) → Tm76 Γ B)
(left : ∀ Γ A B → Tm76 Γ A → Tm76 Γ (sum76 A B))
(right : ∀ Γ A B → Tm76 Γ B → Tm76 Γ (sum76 A B))
(case : ∀ Γ A B C → Tm76 Γ (sum76 A B) → Tm76 Γ (arr76 A C) → Tm76 Γ (arr76 B C) → Tm76 Γ C)
(zero : ∀ Γ → Tm76 Γ nat76)
(suc : ∀ Γ → Tm76 Γ nat76 → Tm76 Γ nat76)
(rec : ∀ Γ A → Tm76 Γ nat76 → Tm76 Γ (arr76 nat76 (arr76 A A)) → Tm76 Γ A → Tm76 Γ A)
→ Tm76 Γ A
var76 : ∀{Γ A} → Var76 Γ A → Tm76 Γ A; var76
= λ x Tm76 var76 lam app tt pair fst snd left right case zero suc rec →
var76 _ _ x
lam76 : ∀{Γ A B} → Tm76 (snoc76 Γ A) B → Tm76 Γ (arr76 A B); lam76
= λ t Tm76 var76 lam76 app tt pair fst snd left right case zero suc rec →
lam76 _ _ _ (t Tm76 var76 lam76 app tt pair fst snd left right case zero suc rec)
app76 : ∀{Γ A B} → Tm76 Γ (arr76 A B) → Tm76 Γ A → Tm76 Γ B; app76
= λ t u Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec →
app76 _ _ _ (t Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec)
(u Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec)
tt76 : ∀{Γ} → Tm76 Γ top76; tt76
= λ Tm76 var76 lam76 app76 tt76 pair fst snd left right case zero suc rec → tt76 _
pair76 : ∀{Γ A B} → Tm76 Γ A → Tm76 Γ B → Tm76 Γ (prod76 A B); pair76
= λ t u Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec →
pair76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec)
(u Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec)
fst76 : ∀{Γ A B} → Tm76 Γ (prod76 A B) → Tm76 Γ A; fst76
= λ t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd left right case zero suc rec →
fst76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd left right case zero suc rec)
snd76 : ∀{Γ A B} → Tm76 Γ (prod76 A B) → Tm76 Γ B; snd76
= λ t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left right case zero suc rec →
snd76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left right case zero suc rec)
left76 : ∀{Γ A B} → Tm76 Γ A → Tm76 Γ (sum76 A B); left76
= λ t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right case zero suc rec →
left76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right case zero suc rec)
right76 : ∀{Γ A B} → Tm76 Γ B → Tm76 Γ (sum76 A B); right76
= λ t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case zero suc rec →
right76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case zero suc rec)
case76 : ∀{Γ A B C} → Tm76 Γ (sum76 A B) → Tm76 Γ (arr76 A C) → Tm76 Γ (arr76 B C) → Tm76 Γ C; case76
= λ t u v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec →
case76 _ _ _ _
(t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
(u Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
(v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
zero76 : ∀{Γ} → Tm76 Γ nat76; zero76
= λ Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc rec → zero76 _
suc76 : ∀{Γ} → Tm76 Γ nat76 → Tm76 Γ nat76; suc76
= λ t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec →
suc76 _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec)
rec76 : ∀{Γ A} → Tm76 Γ nat76 → Tm76 Γ (arr76 nat76 (arr76 A A)) → Tm76 Γ A → Tm76 Γ A; rec76
= λ t u v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76 →
rec76 _ _
(t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
(u Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
(v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
v076 : ∀{Γ A} → Tm76 (snoc76 Γ A) A; v076
= var76 vz76
v176 : ∀{Γ A B} → Tm76 (snoc76 (snoc76 Γ A) B) A; v176
= var76 (vs76 vz76)
v276 : ∀{Γ A B C} → Tm76 (snoc76 (snoc76 (snoc76 Γ A) B) C) A; v276
= var76 (vs76 (vs76 vz76))
v376 : ∀{Γ A B C D} → Tm76 (snoc76 (snoc76 (snoc76 (snoc76 Γ A) B) C) D) A; v376
= var76 (vs76 (vs76 (vs76 vz76)))
tbool76 : Ty76; tbool76
= sum76 top76 top76
true76 : ∀{Γ} → Tm76 Γ tbool76; true76
= left76 tt76
tfalse76 : ∀{Γ} → Tm76 Γ tbool76; tfalse76
= right76 tt76
ifthenelse76 : ∀{Γ A} → Tm76 Γ (arr76 tbool76 (arr76 A (arr76 A A))); ifthenelse76
= lam76 (lam76 (lam76 (case76 v276 (lam76 v276) (lam76 v176))))
times476 : ∀{Γ A} → Tm76 Γ (arr76 (arr76 A A) (arr76 A A)); times476
= lam76 (lam76 (app76 v176 (app76 v176 (app76 v176 (app76 v176 v076)))))
add76 : ∀{Γ} → Tm76 Γ (arr76 nat76 (arr76 nat76 nat76)); add76
= lam76 (rec76 v076
(lam76 (lam76 (lam76 (suc76 (app76 v176 v076)))))
(lam76 v076))
mul76 : ∀{Γ} → Tm76 Γ (arr76 nat76 (arr76 nat76 nat76)); mul76
= lam76 (rec76 v076
(lam76 (lam76 (lam76 (app76 (app76 add76 (app76 v176 v076)) v076))))
(lam76 zero76))
fact76 : ∀{Γ} → Tm76 Γ (arr76 nat76 nat76); fact76
= lam76 (rec76 v076 (lam76 (lam76 (app76 (app76 mul76 (suc76 v176)) v076)))
(suc76 zero76))
{-# OPTIONS --type-in-type #-}
Ty77 : Set
Ty77 =
(Ty77 : Set)
(nat top bot : Ty77)
(arr prod sum : Ty77 → Ty77 → Ty77)
→ Ty77
nat77 : Ty77; nat77 = λ _ nat77 _ _ _ _ _ → nat77
top77 : Ty77; top77 = λ _ _ top77 _ _ _ _ → top77
bot77 : Ty77; bot77 = λ _ _ _ bot77 _ _ _ → bot77
arr77 : Ty77 → Ty77 → Ty77; arr77
= λ A B Ty77 nat77 top77 bot77 arr77 prod sum →
arr77 (A Ty77 nat77 top77 bot77 arr77 prod sum) (B Ty77 nat77 top77 bot77 arr77 prod sum)
prod77 : Ty77 → Ty77 → Ty77; prod77
= λ A B Ty77 nat77 top77 bot77 arr77 prod77 sum →
prod77 (A Ty77 nat77 top77 bot77 arr77 prod77 sum) (B Ty77 nat77 top77 bot77 arr77 prod77 sum)
sum77 : Ty77 → Ty77 → Ty77; sum77
= λ A B Ty77 nat77 top77 bot77 arr77 prod77 sum77 →
sum77 (A Ty77 nat77 top77 bot77 arr77 prod77 sum77) (B Ty77 nat77 top77 bot77 arr77 prod77 sum77)
Con77 : Set; Con77
= (Con77 : Set)
(nil : Con77)
(snoc : Con77 → Ty77 → Con77)
→ Con77
nil77 : Con77; nil77
= λ Con77 nil77 snoc → nil77
snoc77 : Con77 → Ty77 → Con77; snoc77
= λ Γ A Con77 nil77 snoc77 → snoc77 (Γ Con77 nil77 snoc77) A
Var77 : Con77 → Ty77 → Set; Var77
= λ Γ A →
(Var77 : Con77 → Ty77 → Set)
(vz : ∀ Γ A → Var77 (snoc77 Γ A) A)
(vs : ∀ Γ B A → Var77 Γ A → Var77 (snoc77 Γ B) A)
→ Var77 Γ A
vz77 : ∀{Γ A} → Var77 (snoc77 Γ A) A; vz77
= λ Var77 vz77 vs → vz77 _ _
vs77 : ∀{Γ B A} → Var77 Γ A → Var77 (snoc77 Γ B) A; vs77
= λ x Var77 vz77 vs77 → vs77 _ _ _ (x Var77 vz77 vs77)
Tm77 : Con77 → Ty77 → Set; Tm77
= λ Γ A →
(Tm77 : Con77 → Ty77 → Set)
(var : ∀ Γ A → Var77 Γ A → Tm77 Γ A)
(lam : ∀ Γ A B → Tm77 (snoc77 Γ A) B → Tm77 Γ (arr77 A B))
(app : ∀ Γ A B → Tm77 Γ (arr77 A B) → Tm77 Γ A → Tm77 Γ B)
(tt : ∀ Γ → Tm77 Γ top77)
(pair : ∀ Γ A B → Tm77 Γ A → Tm77 Γ B → Tm77 Γ (prod77 A B))
(fst : ∀ Γ A B → Tm77 Γ (prod77 A B) → Tm77 Γ A)
(snd : ∀ Γ A B → Tm77 Γ (prod77 A B) → Tm77 Γ B)
(left : ∀ Γ A B → Tm77 Γ A → Tm77 Γ (sum77 A B))
(right : ∀ Γ A B → Tm77 Γ B → Tm77 Γ (sum77 A B))
(case : ∀ Γ A B C → Tm77 Γ (sum77 A B) → Tm77 Γ (arr77 A C) → Tm77 Γ (arr77 B C) → Tm77 Γ C)
(zero : ∀ Γ → Tm77 Γ nat77)
(suc : ∀ Γ → Tm77 Γ nat77 → Tm77 Γ nat77)
(rec : ∀ Γ A → Tm77 Γ nat77 → Tm77 Γ (arr77 nat77 (arr77 A A)) → Tm77 Γ A → Tm77 Γ A)
→ Tm77 Γ A
var77 : ∀{Γ A} → Var77 Γ A → Tm77 Γ A; var77
= λ x Tm77 var77 lam app tt pair fst snd left right case zero suc rec →
var77 _ _ x
lam77 : ∀{Γ A B} → Tm77 (snoc77 Γ A) B → Tm77 Γ (arr77 A B); lam77
= λ t Tm77 var77 lam77 app tt pair fst snd left right case zero suc rec →
lam77 _ _ _ (t Tm77 var77 lam77 app tt pair fst snd left right case zero suc rec)
app77 : ∀{Γ A B} → Tm77 Γ (arr77 A B) → Tm77 Γ A → Tm77 Γ B; app77
= λ t u Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec →
app77 _ _ _ (t Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec)
(u Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec)
tt77 : ∀{Γ} → Tm77 Γ top77; tt77
= λ Tm77 var77 lam77 app77 tt77 pair fst snd left right case zero suc rec → tt77 _
pair77 : ∀{Γ A B} → Tm77 Γ A → Tm77 Γ B → Tm77 Γ (prod77 A B); pair77
= λ t u Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec →
pair77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec)
(u Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec)
fst77 : ∀{Γ A B} → Tm77 Γ (prod77 A B) → Tm77 Γ A; fst77
= λ t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd left right case zero suc rec →
fst77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd left right case zero suc rec)
snd77 : ∀{Γ A B} → Tm77 Γ (prod77 A B) → Tm77 Γ B; snd77
= λ t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left right case zero suc rec →
snd77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left right case zero suc rec)
left77 : ∀{Γ A B} → Tm77 Γ A → Tm77 Γ (sum77 A B); left77
= λ t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right case zero suc rec →
left77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right case zero suc rec)
right77 : ∀{Γ A B} → Tm77 Γ B → Tm77 Γ (sum77 A B); right77
= λ t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case zero suc rec →
right77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case zero suc rec)
case77 : ∀{Γ A B C} → Tm77 Γ (sum77 A B) → Tm77 Γ (arr77 A C) → Tm77 Γ (arr77 B C) → Tm77 Γ C; case77
= λ t u v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec →
case77 _ _ _ _
(t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
(u Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
(v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
zero77 : ∀{Γ} → Tm77 Γ nat77; zero77
= λ Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc rec → zero77 _
suc77 : ∀{Γ} → Tm77 Γ nat77 → Tm77 Γ nat77; suc77
= λ t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec →
suc77 _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec)
rec77 : ∀{Γ A} → Tm77 Γ nat77 → Tm77 Γ (arr77 nat77 (arr77 A A)) → Tm77 Γ A → Tm77 Γ A; rec77
= λ t u v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77 →
rec77 _ _
(t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
(u Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
(v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
v077 : ∀{Γ A} → Tm77 (snoc77 Γ A) A; v077
= var77 vz77
v177 : ∀{Γ A B} → Tm77 (snoc77 (snoc77 Γ A) B) A; v177
= var77 (vs77 vz77)
v277 : ∀{Γ A B C} → Tm77 (snoc77 (snoc77 (snoc77 Γ A) B) C) A; v277
= var77 (vs77 (vs77 vz77))
v377 : ∀{Γ A B C D} → Tm77 (snoc77 (snoc77 (snoc77 (snoc77 Γ A) B) C) D) A; v377
= var77 (vs77 (vs77 (vs77 vz77)))
tbool77 : Ty77; tbool77
= sum77 top77 top77
true77 : ∀{Γ} → Tm77 Γ tbool77; true77
= left77 tt77
tfalse77 : ∀{Γ} → Tm77 Γ tbool77; tfalse77
= right77 tt77
ifthenelse77 : ∀{Γ A} → Tm77 Γ (arr77 tbool77 (arr77 A (arr77 A A))); ifthenelse77
= lam77 (lam77 (lam77 (case77 v277 (lam77 v277) (lam77 v177))))
times477 : ∀{Γ A} → Tm77 Γ (arr77 (arr77 A A) (arr77 A A)); times477
= lam77 (lam77 (app77 v177 (app77 v177 (app77 v177 (app77 v177 v077)))))
add77 : ∀{Γ} → Tm77 Γ (arr77 nat77 (arr77 nat77 nat77)); add77
= lam77 (rec77 v077
(lam77 (lam77 (lam77 (suc77 (app77 v177 v077)))))
(lam77 v077))
mul77 : ∀{Γ} → Tm77 Γ (arr77 nat77 (arr77 nat77 nat77)); mul77
= lam77 (rec77 v077
(lam77 (lam77 (lam77 (app77 (app77 add77 (app77 v177 v077)) v077))))
(lam77 zero77))
fact77 : ∀{Γ} → Tm77 Γ (arr77 nat77 nat77); fact77
= lam77 (rec77 v077 (lam77 (lam77 (app77 (app77 mul77 (suc77 v177)) v077)))
(suc77 zero77))
{-# OPTIONS --type-in-type #-}
Ty78 : Set
Ty78 =
(Ty78 : Set)
(nat top bot : Ty78)
(arr prod sum : Ty78 → Ty78 → Ty78)
→ Ty78
nat78 : Ty78; nat78 = λ _ nat78 _ _ _ _ _ → nat78
top78 : Ty78; top78 = λ _ _ top78 _ _ _ _ → top78
bot78 : Ty78; bot78 = λ _ _ _ bot78 _ _ _ → bot78
arr78 : Ty78 → Ty78 → Ty78; arr78
= λ A B Ty78 nat78 top78 bot78 arr78 prod sum →
arr78 (A Ty78 nat78 top78 bot78 arr78 prod sum) (B Ty78 nat78 top78 bot78 arr78 prod sum)
prod78 : Ty78 → Ty78 → Ty78; prod78
= λ A B Ty78 nat78 top78 bot78 arr78 prod78 sum →
prod78 (A Ty78 nat78 top78 bot78 arr78 prod78 sum) (B Ty78 nat78 top78 bot78 arr78 prod78 sum)
sum78 : Ty78 → Ty78 → Ty78; sum78
= λ A B Ty78 nat78 top78 bot78 arr78 prod78 sum78 →
sum78 (A Ty78 nat78 top78 bot78 arr78 prod78 sum78) (B Ty78 nat78 top78 bot78 arr78 prod78 sum78)
Con78 : Set; Con78
= (Con78 : Set)
(nil : Con78)
(snoc : Con78 → Ty78 → Con78)
→ Con78
nil78 : Con78; nil78
= λ Con78 nil78 snoc → nil78
snoc78 : Con78 → Ty78 → Con78; snoc78
= λ Γ A Con78 nil78 snoc78 → snoc78 (Γ Con78 nil78 snoc78) A
Var78 : Con78 → Ty78 → Set; Var78
= λ Γ A →
(Var78 : Con78 → Ty78 → Set)
(vz : ∀ Γ A → Var78 (snoc78 Γ A) A)
(vs : ∀ Γ B A → Var78 Γ A → Var78 (snoc78 Γ B) A)
→ Var78 Γ A
vz78 : ∀{Γ A} → Var78 (snoc78 Γ A) A; vz78
= λ Var78 vz78 vs → vz78 _ _
vs78 : ∀{Γ B A} → Var78 Γ A → Var78 (snoc78 Γ B) A; vs78
= λ x Var78 vz78 vs78 → vs78 _ _ _ (x Var78 vz78 vs78)
Tm78 : Con78 → Ty78 → Set; Tm78
= λ Γ A →
(Tm78 : Con78 → Ty78 → Set)
(var : ∀ Γ A → Var78 Γ A → Tm78 Γ A)
(lam : ∀ Γ A B → Tm78 (snoc78 Γ A) B → Tm78 Γ (arr78 A B))
(app : ∀ Γ A B → Tm78 Γ (arr78 A B) → Tm78 Γ A → Tm78 Γ B)
(tt : ∀ Γ → Tm78 Γ top78)
(pair : ∀ Γ A B → Tm78 Γ A → Tm78 Γ B → Tm78 Γ (prod78 A B))
(fst : ∀ Γ A B → Tm78 Γ (prod78 A B) → Tm78 Γ A)
(snd : ∀ Γ A B → Tm78 Γ (prod78 A B) → Tm78 Γ B)
(left : ∀ Γ A B → Tm78 Γ A → Tm78 Γ (sum78 A B))
(right : ∀ Γ A B → Tm78 Γ B → Tm78 Γ (sum78 A B))
(case : ∀ Γ A B C → Tm78 Γ (sum78 A B) → Tm78 Γ (arr78 A C) → Tm78 Γ (arr78 B C) → Tm78 Γ C)
(zero : ∀ Γ → Tm78 Γ nat78)
(suc : ∀ Γ → Tm78 Γ nat78 → Tm78 Γ nat78)
(rec : ∀ Γ A → Tm78 Γ nat78 → Tm78 Γ (arr78 nat78 (arr78 A A)) → Tm78 Γ A → Tm78 Γ A)
→ Tm78 Γ A
var78 : ∀{Γ A} → Var78 Γ A → Tm78 Γ A; var78
= λ x Tm78 var78 lam app tt pair fst snd left right case zero suc rec →
var78 _ _ x
lam78 : ∀{Γ A B} → Tm78 (snoc78 Γ A) B → Tm78 Γ (arr78 A B); lam78
= λ t Tm78 var78 lam78 app tt pair fst snd left right case zero suc rec →
lam78 _ _ _ (t Tm78 var78 lam78 app tt pair fst snd left right case zero suc rec)
app78 : ∀{Γ A B} → Tm78 Γ (arr78 A B) → Tm78 Γ A → Tm78 Γ B; app78
= λ t u Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec →
app78 _ _ _ (t Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec)
(u Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec)
tt78 : ∀{Γ} → Tm78 Γ top78; tt78
= λ Tm78 var78 lam78 app78 tt78 pair fst snd left right case zero suc rec → tt78 _
pair78 : ∀{Γ A B} → Tm78 Γ A → Tm78 Γ B → Tm78 Γ (prod78 A B); pair78
= λ t u Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec →
pair78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec)
(u Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec)
fst78 : ∀{Γ A B} → Tm78 Γ (prod78 A B) → Tm78 Γ A; fst78
= λ t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd left right case zero suc rec →
fst78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd left right case zero suc rec)
snd78 : ∀{Γ A B} → Tm78 Γ (prod78 A B) → Tm78 Γ B; snd78
= λ t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left right case zero suc rec →
snd78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left right case zero suc rec)
left78 : ∀{Γ A B} → Tm78 Γ A → Tm78 Γ (sum78 A B); left78
= λ t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right case zero suc rec →
left78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right case zero suc rec)
right78 : ∀{Γ A B} → Tm78 Γ B → Tm78 Γ (sum78 A B); right78
= λ t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case zero suc rec →
right78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case zero suc rec)
case78 : ∀{Γ A B C} → Tm78 Γ (sum78 A B) → Tm78 Γ (arr78 A C) → Tm78 Γ (arr78 B C) → Tm78 Γ C; case78
= λ t u v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec →
case78 _ _ _ _
(t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
(u Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
(v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
zero78 : ∀{Γ} → Tm78 Γ nat78; zero78
= λ Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc rec → zero78 _
suc78 : ∀{Γ} → Tm78 Γ nat78 → Tm78 Γ nat78; suc78
= λ t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec →
suc78 _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec)
rec78 : ∀{Γ A} → Tm78 Γ nat78 → Tm78 Γ (arr78 nat78 (arr78 A A)) → Tm78 Γ A → Tm78 Γ A; rec78
= λ t u v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78 →
rec78 _ _
(t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
(u Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
(v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
v078 : ∀{Γ A} → Tm78 (snoc78 Γ A) A; v078
= var78 vz78
v178 : ∀{Γ A B} → Tm78 (snoc78 (snoc78 Γ A) B) A; v178
= var78 (vs78 vz78)
v278 : ∀{Γ A B C} → Tm78 (snoc78 (snoc78 (snoc78 Γ A) B) C) A; v278
= var78 (vs78 (vs78 vz78))
v378 : ∀{Γ A B C D} → Tm78 (snoc78 (snoc78 (snoc78 (snoc78 Γ A) B) C) D) A; v378
= var78 (vs78 (vs78 (vs78 vz78)))
tbool78 : Ty78; tbool78
= sum78 top78 top78
true78 : ∀{Γ} → Tm78 Γ tbool78; true78
= left78 tt78
tfalse78 : ∀{Γ} → Tm78 Γ tbool78; tfalse78
= right78 tt78
ifthenelse78 : ∀{Γ A} → Tm78 Γ (arr78 tbool78 (arr78 A (arr78 A A))); ifthenelse78
= lam78 (lam78 (lam78 (case78 v278 (lam78 v278) (lam78 v178))))
times478 : ∀{Γ A} → Tm78 Γ (arr78 (arr78 A A) (arr78 A A)); times478
= lam78 (lam78 (app78 v178 (app78 v178 (app78 v178 (app78 v178 v078)))))
add78 : ∀{Γ} → Tm78 Γ (arr78 nat78 (arr78 nat78 nat78)); add78
= lam78 (rec78 v078
(lam78 (lam78 (lam78 (suc78 (app78 v178 v078)))))
(lam78 v078))
mul78 : ∀{Γ} → Tm78 Γ (arr78 nat78 (arr78 nat78 nat78)); mul78
= lam78 (rec78 v078
(lam78 (lam78 (lam78 (app78 (app78 add78 (app78 v178 v078)) v078))))
(lam78 zero78))
fact78 : ∀{Γ} → Tm78 Γ (arr78 nat78 nat78); fact78
= lam78 (rec78 v078 (lam78 (lam78 (app78 (app78 mul78 (suc78 v178)) v078)))
(suc78 zero78))
{-# OPTIONS --type-in-type #-}
Ty79 : Set
Ty79 =
(Ty79 : Set)
(nat top bot : Ty79)
(arr prod sum : Ty79 → Ty79 → Ty79)
→ Ty79
nat79 : Ty79; nat79 = λ _ nat79 _ _ _ _ _ → nat79
top79 : Ty79; top79 = λ _ _ top79 _ _ _ _ → top79
bot79 : Ty79; bot79 = λ _ _ _ bot79 _ _ _ → bot79
arr79 : Ty79 → Ty79 → Ty79; arr79
= λ A B Ty79 nat79 top79 bot79 arr79 prod sum →
arr79 (A Ty79 nat79 top79 bot79 arr79 prod sum) (B Ty79 nat79 top79 bot79 arr79 prod sum)
prod79 : Ty79 → Ty79 → Ty79; prod79
= λ A B Ty79 nat79 top79 bot79 arr79 prod79 sum →
prod79 (A Ty79 nat79 top79 bot79 arr79 prod79 sum) (B Ty79 nat79 top79 bot79 arr79 prod79 sum)
sum79 : Ty79 → Ty79 → Ty79; sum79
= λ A B Ty79 nat79 top79 bot79 arr79 prod79 sum79 →
sum79 (A Ty79 nat79 top79 bot79 arr79 prod79 sum79) (B Ty79 nat79 top79 bot79 arr79 prod79 sum79)
Con79 : Set; Con79
= (Con79 : Set)
(nil : Con79)
(snoc : Con79 → Ty79 → Con79)
→ Con79
nil79 : Con79; nil79
= λ Con79 nil79 snoc → nil79
snoc79 : Con79 → Ty79 → Con79; snoc79
= λ Γ A Con79 nil79 snoc79 → snoc79 (Γ Con79 nil79 snoc79) A
Var79 : Con79 → Ty79 → Set; Var79
= λ Γ A →
(Var79 : Con79 → Ty79 → Set)
(vz : ∀ Γ A → Var79 (snoc79 Γ A) A)
(vs : ∀ Γ B A → Var79 Γ A → Var79 (snoc79 Γ B) A)
→ Var79 Γ A
vz79 : ∀{Γ A} → Var79 (snoc79 Γ A) A; vz79
= λ Var79 vz79 vs → vz79 _ _
vs79 : ∀{Γ B A} → Var79 Γ A → Var79 (snoc79 Γ B) A; vs79
= λ x Var79 vz79 vs79 → vs79 _ _ _ (x Var79 vz79 vs79)
Tm79 : Con79 → Ty79 → Set; Tm79
= λ Γ A →
(Tm79 : Con79 → Ty79 → Set)
(var : ∀ Γ A → Var79 Γ A → Tm79 Γ A)
(lam : ∀ Γ A B → Tm79 (snoc79 Γ A) B → Tm79 Γ (arr79 A B))
(app : ∀ Γ A B → Tm79 Γ (arr79 A B) → Tm79 Γ A → Tm79 Γ B)
(tt : ∀ Γ → Tm79 Γ top79)
(pair : ∀ Γ A B → Tm79 Γ A → Tm79 Γ B → Tm79 Γ (prod79 A B))
(fst : ∀ Γ A B → Tm79 Γ (prod79 A B) → Tm79 Γ A)
(snd : ∀ Γ A B → Tm79 Γ (prod79 A B) → Tm79 Γ B)
(left : ∀ Γ A B → Tm79 Γ A → Tm79 Γ (sum79 A B))
(right : ∀ Γ A B → Tm79 Γ B → Tm79 Γ (sum79 A B))
(case : ∀ Γ A B C → Tm79 Γ (sum79 A B) → Tm79 Γ (arr79 A C) → Tm79 Γ (arr79 B C) → Tm79 Γ C)
(zero : ∀ Γ → Tm79 Γ nat79)
(suc : ∀ Γ → Tm79 Γ nat79 → Tm79 Γ nat79)
(rec : ∀ Γ A → Tm79 Γ nat79 → Tm79 Γ (arr79 nat79 (arr79 A A)) → Tm79 Γ A → Tm79 Γ A)
→ Tm79 Γ A
var79 : ∀{Γ A} → Var79 Γ A → Tm79 Γ A; var79
= λ x Tm79 var79 lam app tt pair fst snd left right case zero suc rec →
var79 _ _ x
lam79 : ∀{Γ A B} → Tm79 (snoc79 Γ A) B → Tm79 Γ (arr79 A B); lam79
= λ t Tm79 var79 lam79 app tt pair fst snd left right case zero suc rec →
lam79 _ _ _ (t Tm79 var79 lam79 app tt pair fst snd left right case zero suc rec)
app79 : ∀{Γ A B} → Tm79 Γ (arr79 A B) → Tm79 Γ A → Tm79 Γ B; app79
= λ t u Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec →
app79 _ _ _ (t Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec)
(u Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec)
tt79 : ∀{Γ} → Tm79 Γ top79; tt79
= λ Tm79 var79 lam79 app79 tt79 pair fst snd left right case zero suc rec → tt79 _
pair79 : ∀{Γ A B} → Tm79 Γ A → Tm79 Γ B → Tm79 Γ (prod79 A B); pair79
= λ t u Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec →
pair79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec)
(u Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec)
fst79 : ∀{Γ A B} → Tm79 Γ (prod79 A B) → Tm79 Γ A; fst79
= λ t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd left right case zero suc rec →
fst79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd left right case zero suc rec)
snd79 : ∀{Γ A B} → Tm79 Γ (prod79 A B) → Tm79 Γ B; snd79
= λ t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left right case zero suc rec →
snd79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left right case zero suc rec)
left79 : ∀{Γ A B} → Tm79 Γ A → Tm79 Γ (sum79 A B); left79
= λ t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right case zero suc rec →
left79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right case zero suc rec)
right79 : ∀{Γ A B} → Tm79 Γ B → Tm79 Γ (sum79 A B); right79
= λ t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case zero suc rec →
right79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case zero suc rec)
case79 : ∀{Γ A B C} → Tm79 Γ (sum79 A B) → Tm79 Γ (arr79 A C) → Tm79 Γ (arr79 B C) → Tm79 Γ C; case79
= λ t u v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec →
case79 _ _ _ _
(t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
(u Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
(v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
zero79 : ∀{Γ} → Tm79 Γ nat79; zero79
= λ Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc rec → zero79 _
suc79 : ∀{Γ} → Tm79 Γ nat79 → Tm79 Γ nat79; suc79
= λ t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec →
suc79 _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec)
rec79 : ∀{Γ A} → Tm79 Γ nat79 → Tm79 Γ (arr79 nat79 (arr79 A A)) → Tm79 Γ A → Tm79 Γ A; rec79
= λ t u v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79 →
rec79 _ _
(t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
(u Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
(v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
v079 : ∀{Γ A} → Tm79 (snoc79 Γ A) A; v079
= var79 vz79
v179 : ∀{Γ A B} → Tm79 (snoc79 (snoc79 Γ A) B) A; v179
= var79 (vs79 vz79)
v279 : ∀{Γ A B C} → Tm79 (snoc79 (snoc79 (snoc79 Γ A) B) C) A; v279
= var79 (vs79 (vs79 vz79))
v379 : ∀{Γ A B C D} → Tm79 (snoc79 (snoc79 (snoc79 (snoc79 Γ A) B) C) D) A; v379
= var79 (vs79 (vs79 (vs79 vz79)))
tbool79 : Ty79; tbool79
= sum79 top79 top79
true79 : ∀{Γ} → Tm79 Γ tbool79; true79
= left79 tt79
tfalse79 : ∀{Γ} → Tm79 Γ tbool79; tfalse79
= right79 tt79
ifthenelse79 : ∀{Γ A} → Tm79 Γ (arr79 tbool79 (arr79 A (arr79 A A))); ifthenelse79
= lam79 (lam79 (lam79 (case79 v279 (lam79 v279) (lam79 v179))))
times479 : ∀{Γ A} → Tm79 Γ (arr79 (arr79 A A) (arr79 A A)); times479
= lam79 (lam79 (app79 v179 (app79 v179 (app79 v179 (app79 v179 v079)))))
add79 : ∀{Γ} → Tm79 Γ (arr79 nat79 (arr79 nat79 nat79)); add79
= lam79 (rec79 v079
(lam79 (lam79 (lam79 (suc79 (app79 v179 v079)))))
(lam79 v079))
mul79 : ∀{Γ} → Tm79 Γ (arr79 nat79 (arr79 nat79 nat79)); mul79
= lam79 (rec79 v079
(lam79 (lam79 (lam79 (app79 (app79 add79 (app79 v179 v079)) v079))))
(lam79 zero79))
fact79 : ∀{Γ} → Tm79 Γ (arr79 nat79 nat79); fact79
= lam79 (rec79 v079 (lam79 (lam79 (app79 (app79 mul79 (suc79 v179)) v079)))
(suc79 zero79))
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module Structure.Operator.IntegralDomain where
open import Functional
import Lvl
open import Logic
open import Logic.Propositional
open import Logic.Predicate
open import Structure.Setoid
open import Structure.Operator.Properties
open import Structure.Operator.Ring
open import Type
-- Rng with no non-zero zero divisors.
record Domain {ℓ ℓₑ} {T : Type{ℓ}} ⦃ _ : Equiv{ℓₑ}(T) ⦄ (_+_ : T → T → T) (_⋅_ : T → T → T) ⦃ rng : Rng(_+_)(_⋅_) ⦄ : Stmt{ℓ Lvl.⊔ ℓₑ} where
constructor intro
open Rng(rng)
field
no-zero-divisors : ∀{x y} → (x ⋅ y ≡ 𝟎) → ((x ≡ 𝟎) ∨ (y ≡ 𝟎))
zero-zero-divisorₗ : ∀{x} → ZeroDivisorₗ(x) → (x ≡ 𝟎)
zero-zero-divisorₗ {x} ([∃]-intro y ⦃ [∧]-intro y𝟎 xy𝟎 ⦄) = [∨]-elim id ([⊥]-elim ∘ y𝟎) (no-zero-divisors xy𝟎)
zero-zero-divisorᵣ : ∀{x} → ZeroDivisorᵣ(x) → (x ≡ 𝟎)
zero-zero-divisorᵣ {x} ([∃]-intro y ⦃ [∧]-intro y𝟎 xy𝟎 ⦄) = [∨]-elim ([⊥]-elim ∘ y𝟎) id (no-zero-divisors xy𝟎)
zero-zero-divisor : ∀{x} → ZeroDivisor(x) → (x ≡ 𝟎)
zero-zero-divisor {x} ([∃]-intro y ⦃ [∧]-intro y𝟎 ([∧]-intro xy𝟎 yx𝟎) ⦄) = [∨]-elim id ([⊥]-elim ∘ y𝟎) (no-zero-divisors xy𝟎)
-- Non-trivial commutative ring and domain.
record IntegralDomain {ℓ ℓₑ} {T : Type{ℓ}} ⦃ _ : Equiv{ℓₑ}(T) ⦄ (_+_ : T → T → T) (_⋅_ : T → T → T) : Stmt{ℓ Lvl.⊔ ℓₑ} where
constructor intro
field
⦃ ring ⦄ : Ring(_+_)(_⋅_)
⦃ domain ⦄ : Domain(_+_)(_⋅_)
⦃ [⋅]-commutativity ⦄ : Commutativity(_⋅_)
open Ring (ring) public
open Domain(domain) public
field
⦃ distinct-identities ⦄ : DistinctIdentities
record IntegralDomainObject {ℓ ℓₑ} : Stmt{Lvl.𝐒(ℓ Lvl.⊔ ℓₑ)} where
constructor intro
field
{T} : Type{ℓ}
⦃ equiv ⦄ : Equiv{ℓₑ}(T)
_+_ : T → T → T
_⋅_ : T → T → T
⦃ integralDomain ⦄ : IntegralDomain(_+_)(_⋅_)
open IntegralDomain(integralDomain) public
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-- Andreas, 2019-03-17, issue #3638
-- Making rewriting in interactive goals work in parametrized modules.
{-# OPTIONS --rewriting #-}
-- {-# OPTIONS -v rewriting:100 #-}
-- {-# OPTIONS -v tc.sig.param:60 #-}
-- {-# OPTIONS -v interactive.meta:10 #-}
open import Agda.Builtin.Equality
{-# BUILTIN REWRITE _≡_ #-}
module _ (A : Set) where
postulate
a b c : A
eq : a ≡ b
rew : c ≡ b
{-# REWRITE rew #-}
goal : a ≡ c
goal = {! eq !} -- C-u C-u C-c C-.
-- Displays:
-- Goal: a ≡ c
-- Have: a ≡ b
-- Expected: Rewrite rule rew should be applied.
-- Goal: a ≡ b
-- Works now.
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module Common.Sum where
open import Common.Level
data _⊎_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
inj₁ : (x : A) → A ⊎ B
inj₂ : (y : B) → A ⊎ B
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module Sessions.Semantics.Runtime {E : Set} (delay : E) where
open import Prelude
open import Relation.Unary hiding (Empty; _∈_)
open import Relation.Unary.PredicateTransformer using (Pt)
open import Relation.Ternary.Separation.Construct.Market
open import Relation.Ternary.Separation.Construct.Product
open import Relation.Ternary.Separation.Morphisms
open import Relation.Ternary.Separation.Monad
open import Sessions.Syntax.Types
open import Sessions.Syntax.Values
open import Sessions.Syntax.Expr
open import Sessions.Semantics.Commands
open import Relation.Ternary.Separation.Monad
open import Relation.Ternary.Separation.Monad.Error
open import Relation.Ternary.Separation.Monad.State
open StateWithErr {C = RCtx} E
open ExceptMonad {A = RCtx} E
open Monads.Monad {{...}}
private
module _ {C : Set} {{ r : RawSep C }} {u} {{ _ : IsUnitalSep r u }} where
-- open Monads.Monad (err-monad {A = C}) public
open Monads using (str) public
module _ where
data _⇜_ : SType → SType → Pred RCtx 0ℓ where
emp : ∀ {α} → (α ⇜ α) ε
cons : ∀ {a} → ∀[ Val a ✴ (β ⇜ γ) ⇒ ((a ¿ β) ⇜ γ) ]
_⇝_ = flip _⇜_
private
-- It is crucial for type-safety that this is evident
send-lemma : ∀[ ((a ! β) ⇜ γ) ⇒ Empty (γ ≡ a ! β) ]
send-lemma emp = emp refl
record Link (α γ : SType) Φ : Set where
constructor link
field
{β₁ β₂} : SType
duals : β₂ ≡ β₁ ⁻¹
buffers : (α ⇜ β₁ ✴ β₂ ⇝ γ) Φ
revLink : ∀[ Link α γ ⇒ Link γ α ]
revLink (link refl buffers) = link (sym dual-involutive) (✴-swap buffers)
push : ∀[ Val a ✴ γ ⇜ (a ¿ β) ⇒ γ ⇜ β ]
push (v ×⟨ σ₁ ⟩ emp) = cons (v ×⟨ σ₁ ⟩ emp)
push (v ×⟨ σ₁ ⟩ cons (w ×⟨ σ₂ ⟩ b)) with ⊎-assoc σ₂ (⊎-comm σ₁)
... | _ , σ₃ , σ₄ with push (v ×⟨ ⊎-comm σ₄ ⟩ b)
... | b' = cons (w ×⟨ σ₃ ⟩ b')
pull : ∀[ γ ⇝ (a ¿ β) ⇒ Except E (Val a ✴ γ ⇝ β) ]
pull emp = error delay
pull (cons (v ×⟨ σ ⟩ vs)) = return (v ×⟨ σ ⟩ vs)
send-into : ∀[ Val a ✴ Link α (a ! β) ⇒ Link α β ]
send-into (v ×⟨ σ ⟩ link {x ¿ β₁} refl (px ×⟨ σ₁ ⟩ emp)) rewrite ⊎-id⁻ʳ σ₁ =
link refl ((push (v ×⟨ σ ⟩ px)) ×⟨ ⊎-idʳ ⟩ emp)
recvₗ : ∀[ Link (a ¿ β) γ ⇒ Except E (Val a ✴ Link β γ) ]
recvₗ c@(link refl (bₗ ×⟨ τ ⟩ bᵣ)) = do
v ×⟨ σ ⟩ l ← mapM (pull bₗ &⟨ τ ⟩ bᵣ) ✴-assocᵣ
return (v ×⟨ σ ⟩ link refl l)
recvᵣ : ∀[ Link γ (a ¿ β) ⇒ Except E (Val a ✴ Link γ β) ]
recvᵣ l = do
v ×⟨ σ ⟩ l' ← recvₗ (revLink l)
return (v ×⟨ σ ⟩ revLink l')
data Recipient : SType → Set where
rec : Recipient (a ¿ β)
end : Recipient end
data Channel : Runtype → Pred RCtx 0ℓ where
twosided : ∀[ Link α β ⇒ Channel (chan α β) ]
onesided : ∀[ end ⇝ β ⇒ Channel (endp β) ]
Channels' = Allstar Channel
Channels = uncurry Channels'
emptyLink : ε[ Link α (α ⁻¹) ]
emptyLink = link refl (emp ×⟨ ⊎-∙ ⟩ emp)
emptyChannel : ε[ Channel (chan α (α ⁻¹)) ]
emptyChannel = twosided (link refl (emp ×⟨ ⊎-∙ ⟩ emp))
flipChan : ∀ {τ} → ∀[ Channel τ ⇒ Channel (flipped τ) ]
flipChan (twosided x) = twosided (revLink x)
flipChan (onesided x) = onesided x
{- Updating a known endpoint of a channel type -}
_≔ₑ_ : ∀ {τ} → End α τ → SType → Runtype
(endp β ∷ [] , divide lr []) ≔ₑ α = chan α β
(endp β ∷ [] , divide rl []) ≔ₑ α = chan β α
( [] , to-left _) ≔ₑ α = endp α
onesided-recipient : ∀ {Φ} → (end ⇝ α) Φ → Recipient α
onesided-recipient emp = end
onesided-recipient (cons x) = rec
{- Receiving on any receiving end of a channel -}
chan-receive : ∀ {τ} → (e : End (a ¿ α) τ) →
∀[ Channel τ ⇒ Except E (Val a ✴ Channel (e ≔ₑ α)) ]
chan-receive (._ , divide lr []) (twosided l) = do
v ×⟨ σ ⟩ l' ← recvₗ l
return (v ×⟨ σ ⟩ twosided l')
chan-receive (._ , divide rl []) (twosided l) = do
v ×⟨ σ ⟩ l' ← recvᵣ l
return (v ×⟨ σ ⟩ twosided l')
chan-receive (fst₁ , to-left []) (onesided b) = do
v ×⟨ σ ⟩ b' ← pull b
return (v ×⟨ σ ⟩ onesided b')
{- Sending on any sending end of a channel -}
chan-send : ∀ {τ} → (e : End (a ! α) τ) →
∀[ Channel τ ⇒ Val a ─✴ Channel (e ≔ₑ α) ]
app (chan-send (_ , divide lr []) (twosided l)) v σ =
let l' = send-into (v ×⟨ ⊎-comm σ ⟩ revLink l)
in twosided (revLink l')
app (chan-send (_ , divide rl []) (twosided l)) v σ =
let l' = send-into (v ×⟨ ⊎-comm σ ⟩ l)
in twosided l'
-- cannot be onesided
app (chan-send (_ , to-left [] ) (onesided b)) v σ with onesided-recipient b
... | ()
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module Issue22 where
postulate
D : Set
_≡_ : D → D → Set
d e : D
foo : ∀ x → x ≡ x
foo x = bar
where
postulate
d≡e : d ≡ e
postulate
bar : x ≡ x
{-# ATP prove bar d≡e #-}
-- $ apia Bug.agda
-- An internal error has occurred. Please report this as a bug.
-- Location of the error: src/Apia/Utils/AgdaAPI/Interface.hs:307
-- The error occurs because the dead code analysis removes `d≡e` from
-- the the interfase file.
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module Terminal where
open import Agda.Builtin.Word
open import BasicIO
open import Data.Bool
open import Data.Char hiding (show)
open import Data.List hiding (_++_)
open import Data.String hiding (show)
open import Data.Unit
open import ByteCount
open import Int
open import Function
open import Pair
open import Show
{-# FOREIGN GHC import System.Posix.IO #-}
{-# FOREIGN GHC import System.Posix.Terminal #-}
{-# FOREIGN GHC import System.Posix.Types #-}
data TerminalState : Set where
immediately : TerminalState
whenDrained : TerminalState
whenFlushed : TerminalState
{-# COMPILE GHC TerminalState =
data TerminalState (Immediately | WhenDrained | WhenFlushed)
#-}
data TerminalMode : Set where
interruptOnBreak : TerminalMode
mapCRtoLF : TerminalMode
ignoreBreak : TerminalMode
ignoreCR : TerminalMode
ignoreParityErrors : TerminalMode
mapLFtoCR : TerminalMode
checkParity : TerminalMode
stripHighBit : TerminalMode
startStopInput : TerminalMode
startStopOutput : TerminalMode
markParityErrors : TerminalMode
processOutput : TerminalMode
localMode : TerminalMode
readEnable : TerminalMode
twoStopBits : TerminalMode
hangupOnClose : TerminalMode
enableParity : TerminalMode
oddParity : TerminalMode
enableEcho : TerminalMode
echoErase : TerminalMode
echoKill : TerminalMode
echoLF : TerminalMode
processInput : TerminalMode
extendedFunctions : TerminalMode
keyboardInterrupts : TerminalMode
noFlushOnInterrupt : TerminalMode
backgroundWriteInterrupt : TerminalMode
{-# COMPILE GHC TerminalMode =
data TerminalMode
( InterruptOnBreak
| MapCRtoLF
| IgnoreBreak
| IgnoreCR
| IgnoreParityErrors
| MapLFtoCR
| CheckParity
| StripHighBit
| StartStopInput
| StartStopOutput
| MarkParityErrors
| ProcessOutput
| LocalMode
| ReadEnable
| TwoStopBits
| HangupOnClose
| EnableParity
| OddParity
| EnableEcho
| EchoErase
| EchoKill
| EchoLF
| ProcessInput
| ExtendedFunctions
| KeyboardInterrupts
| NoFlushOnInterrupt
| BackgroundWriteInterrupt
)
#-}
postulate
Fd : Set
TerminalAttributes : Set
stdInput : Fd
stdOutput : Fd
getTerminalAttributes : Fd → IO TerminalAttributes
setTerminalAttributes : Fd → TerminalAttributes → TerminalState → IO ⊤
withoutMode : TerminalAttributes → TerminalMode → TerminalAttributes
inputTime : TerminalAttributes → Int
withTime : TerminalAttributes → Int → TerminalAttributes
minInput : TerminalAttributes → Int
withMinInput : TerminalAttributes → Int → TerminalAttributes
fdRead : Fd → ByteCount → IO (List Char , ByteCount)
fdWrite : Fd → List Char → IO ByteCount
{-# COMPILE GHC Fd = type Fd #-}
{-# COMPILE GHC TerminalAttributes = type TerminalAttributes #-}
{-# COMPILE GHC stdInput = stdInput #-}
{-# COMPILE GHC stdOutput = stdOutput #-}
{-# COMPILE GHC getTerminalAttributes = getTerminalAttributes #-}
{-# COMPILE GHC setTerminalAttributes = setTerminalAttributes #-}
{-# COMPILE GHC withoutMode = withoutMode #-}
{-# COMPILE GHC inputTime = inputTime #-}
{-# COMPILE GHC withTime = withTime #-}
{-# COMPILE GHC minInput = minInput #-}
{-# COMPILE GHC withMinInput = withMinInput #-}
{-# COMPILE GHC fdRead = fdRead #-}
{-# COMPILE GHC fdWrite = fdWrite #-}
clearScreen : String
clearScreen = "\^[[2J"
showCursor : String
showCursor = "\^[[?25h"
hideCursor : String
hideCursor = "\^[[?25l"
altScreenEnable : String
altScreenEnable = "\^[[?1049h"
altScreenDisable : String
altScreenDisable = "\^[[?1049l"
withUpdatedAttributes :
{A : Set} → (TerminalAttributes → TerminalAttributes) → IO A → IO A
withUpdatedAttributes {A} updateFn actions =
bracket (getTerminalAttributes stdOutput) setAttrs updateAndRun
where
setAttrs : TerminalAttributes → IO ⊤
setAttrs attrs = setTerminalAttributes stdOutput attrs immediately
updateAndRun : TerminalAttributes → IO A
updateAndRun attrs = do
_ ← setAttrs (updateFn attrs)
actions
readMaxChars : ByteCount
readMaxChars = mkCSize (primWord64FromNat 1024)
termRead : IO String
termRead = fdRead stdInput readMaxChars >>= (return ∘ fromList ∘ fst)
termWrite : String → IO ByteCount
termWrite = fdWrite stdOutput ∘ toList
printAttrs : TerminalAttributes → IO ⊤
printAttrs attrs =
do _ ← termWrite (formatField "inputTime" (inputTime attrs))
_ ← termWrite (formatField "minInput" (minInput attrs))
return tt
where
formatField : String → Int → String
formatField name value = name ++ " = " ++ show value ++ "\n"
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-- Deriving TreeEncoding. Note: only for non-dependent datatypes.
module Tactic.Deriving.TreeEncoding where
open import Prelude
open import Tactic.Reflection
open import Tactic.Deriving
open import Tactic.Reflection
open import Tactic.Reflection.Quote
open import Data.TreeRep
open import Container.Traversable
private
mapIx : ∀ {a b} {A : Set a} {B : Set b} → (Nat → A → B) → List A → List B
mapIx f [] = []
mapIx f (x ∷ xs) = f 0 x ∷ mapIx (f ∘ suc) xs
-- encode (ci x₁ .. xn) = node i (treeEncode x₁) .. (treeEncode xn)
encodeClause : Nat → Nat → Name → TC Clause
encodeClause np i c = do
args ← drop np <$> argTel c
let xs = reverse (mapIx const args)
return (clause [ vArg (con c (map (var "x" <$_) args)) ]
(con₂ (quote node)
(lit (nat i))
(quoteList (map (λ i → def₁ (quote treeEncode) (var i [])) xs))))
quoteListP : List Pattern → Pattern
quoteListP = foldr (λ p ps → con₂ (quote List._∷_) p ps) (con₀ (quote List.[]))
qAp : Term → Term → Term
qAp f x = def₂ (quote _<*>′_) f x
-- decode (node i x₁ .. xn) = ⦇ cᵢ (treeDecode x₁) .. (treeDecode xn) ⦈
decodeClause : Nat → Nat → Name → TC Clause
decodeClause np i c = do
args ← drop np <$> argTel c
let xs = reverse (mapIx const args)
pure (clause [ vArg (con₂ (quote node) (lit (nat i)) (quoteListP (map (λ _ → var "x") args))) ]
(foldl qAp (con₁ (quote just) (con₀ c))
(map (λ i → def₁ (quote treeDecode) (var i [])) xs)))
encodeClauses : Name → TC (List Clause)
encodeClauses d = do
cs ← getConstructors d
np ← getParameters d
traverse id (mapIx (encodeClause np) cs)
decodeClauses : Name → TC (List Clause)
decodeClauses d = do
cs ← getConstructors d
np ← getParameters d
cs ← traverse id (mapIx (decodeClause np) cs)
pure (cs ++ clause (vArg (var "_") ∷ []) (con₀ (quote nothing)) ∷ [])
proofClause : Nat → Nat → Name → TC Clause
proofClause np i c = do
args ← drop np <$> argTel c
let xs = reverse (mapIx const args)
pure (clause [ vArg (con c (map (var "x" <$_) args)) ]
(foldl (λ eq eq₁ → def₂ (quote _=*=′_) eq eq₁)
(con₀ (quote refl))
(map (λ i → def₁ (quote isTreeEmbedding) (var i [])) xs)))
proofClauses : Name → TC (List Clause)
proofClauses d = do
cs ← getConstructors d
np ← getParameters d
traverse id (mapIx (proofClause np) cs)
makeProjection : Name → Clause → Clause
makeProjection f (clause ps b) = clause (vArg (proj f) ∷ ps) b
makeProjection f (absurd-clause ps) = absurd-clause (vArg (proj f) ∷ ps)
instanceClauses : Name → TC (List Clause)
instanceClauses d = do
enc ← encodeClauses d
dec ← decodeClauses d
prf ← proofClauses d
pure (map (makeProjection (quote TreeEncoding.treeEncode)) enc ++
map (makeProjection (quote TreeEncoding.treeDecode)) dec ++
map (makeProjection (quote TreeEncoding.isTreeEmbedding)) prf)
deriveTreeEncoding : Name → Name → TC ⊤
deriveTreeEncoding iname dname = do
declareDef (iArg iname) =<< instanceType dname (quote TreeEncoding)
defineFun iname =<< instanceClauses dname
-- unquoteDecl EncodeList = deriveTreeEncoding EncodeList (quote List)
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{-# OPTIONS --without-K --safe #-}
module Math.Combinatorics.IntegerFunction.Properties.Lemma where
open import Data.Integer
open import Data.Integer.Solver
open import Relation.Binary.PropositionalEquality
lemma₁ : ∀ a b c d → a * b * (c * d) ≡ (a * c) * (b * d)
lemma₁ = solve 4 (λ a b c d → a :* b :* (c :* d) := (a :* c) :* (b :* d) ) refl
where open +-*-Solver
lemma₂ : ∀ n → n ≡ 1ℤ + (n - 1ℤ)
lemma₂ = solve 1 (λ n → n := con 1ℤ :+ (n :- con 1ℤ)) refl
where open +-*-Solver
lemma₃ : ∀ m n → m ≡ n + (m - n)
lemma₃ = solve 2 (λ m n → m := n :+ (m :- n)) refl
where open +-*-Solver
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Algebra.CommRing where
open import Cubical.Algebra.CommRing.Base public
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{-
A parameterized family of structures S can be combined into a single structure:
X ↦ (a : A) → S a X
-}
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Structures.Relational.Parameterized where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure
open import Cubical.Foundations.RelationalStructure
open import Cubical.Functions.FunExtEquiv
open import Cubical.Data.Sigma
open import Cubical.HITs.PropositionalTruncation
open import Cubical.HITs.SetQuotients
open import Cubical.Structures.Parameterized
private
variable
ℓ ℓ₀ ℓ₁ ℓ₁' ℓ₁'' : Level
-- Structured relations
module _ (A : Type ℓ₀) where
ParamRelStr : {S : A → Type ℓ → Type ℓ₁}
→ (∀ a → StrRel (S a) ℓ₁')
→ StrRel (ParamStructure A S) (ℓ-max ℓ₀ ℓ₁')
ParamRelStr ρ R s t =
(a : A) → ρ a R (s a) (t a)
paramSuitableRel : {S : A → Type ℓ → Type ℓ₁} {ρ : ∀ a → StrRel (S a) ℓ₁'}
→ (∀ a → SuitableStrRel (S a) (ρ a))
→ SuitableStrRel (ParamStructure A S) (ParamRelStr ρ)
paramSuitableRel {ρ = ρ} θ .quo (X , f) R r .fst .fst a =
θ a .quo (X , f a) R (r a) .fst .fst
paramSuitableRel {ρ = ρ} θ .quo (X , f) R r .fst .snd a =
θ a .quo (X , f a) R (r a) .fst .snd
paramSuitableRel {ρ = ρ} θ .quo (X , f) R r .snd (q , c) i .fst a =
θ a .quo (X , f a) R (r a) .snd (q a , c a) i .fst
paramSuitableRel {ρ = ρ} θ .quo (X , f) R r .snd (q , c) i .snd a =
θ a .quo (X , f a) R (r a) .snd (q a , c a) i .snd
paramSuitableRel {ρ = ρ} θ .symmetric R r a =
θ a .symmetric R (r a)
paramSuitableRel {ρ = ρ} θ .transitive R R' r r' a =
θ a .transitive R R' (r a) (r' a)
paramSuitableRel {ρ = ρ} θ .set setX =
isSetΠ λ a → θ a .set setX
paramSuitableRel {ρ = ρ} θ .prop propR s t =
isPropΠ λ a → θ a .prop propR (s a) (t a)
paramRelMatchesEquiv : {S : A → Type ℓ → Type ℓ₁}
(ρ : ∀ a → StrRel (S a) ℓ₁') {ι : ∀ a → StrEquiv (S a) ℓ₁''}
→ (∀ a → StrRelMatchesEquiv (ρ a) (ι a))
→ StrRelMatchesEquiv (ParamRelStr ρ) (ParamEquivStr A ι)
paramRelMatchesEquiv ρ μ _ _ e = equivΠCod λ a → μ a _ _ e
paramRelAction : {S : A → Type ℓ → Type ℓ₁} {ρ : ∀ a → StrRel (S a) ℓ₁'}
→ (∀ a → StrRelAction (ρ a))
→ StrRelAction (ParamRelStr ρ)
paramRelAction α .actStr f s a = α a .actStr f (s a)
paramRelAction α .actStrId s = funExt λ a → α a .actStrId (s a)
paramRelAction α .actRel h _ _ r a = α a .actRel h _ _ (r a)
-- Detransitivity of ParamRelStr would depend on choice in general, so
-- we don't get positivity
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Some code related to the W type that relies on the K rule
------------------------------------------------------------------------
{-# OPTIONS --with-K --safe #-}
module Data.W.WithK where
open import Data.Product
open import Data.Container.Core
open import Data.W
open import Agda.Builtin.Equality
module _ {s p} {C : Container s p}
{s : Shape C} {f : Position C s → W C} where
sup-injective₂ : ∀ {g} → sup (s , f) ≡ sup (s , g) → f ≡ g
sup-injective₂ refl = refl
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{-# OPTIONS --cubical --safe #-}
module Cubical.Data.Strict2Group.Explicit where
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021 Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import Util.PKCS
open import Util.Prelude
import Yasm.Base as YB
import Yasm.System as YS
-- This module provides a single import for all Yasm modules
module Yasm.Yasm
(ℓ-PeerState : Level)
(ℓ-VSFP : Level)
(parms : YB.SystemTypeParameters ℓ-PeerState)
(iiah : YB.SystemInitAndHandlers ℓ-PeerState parms)
(ValidSenderForPK : YS.WithInitAndHandlers.ValidSenderForPK-type ℓ-PeerState ℓ-VSFP parms iiah)
(ValidSenderForPK-stable : YS.WithInitAndHandlers.ValidSenderForPK-stable-type ℓ-PeerState ℓ-VSFP parms iiah ValidSenderForPK)
where
open YB.SystemTypeParameters parms
open YB.SystemInitAndHandlers iiah
open import Yasm.Base public
open import Yasm.Types public
open import Yasm.System ℓ-PeerState ℓ-VSFP parms public
open import Yasm.Properties ℓ-PeerState ℓ-VSFP parms iiah ValidSenderForPK ValidSenderForPK-stable public
open WithInitAndHandlers iiah public
open import Util.FunctionOverride PeerId _≟PeerId_ public
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-- There was a bug where Confuse and confusing were considered
-- projection-like even though they have absurd clauses.
module _ (A : Set) where
data D : Set where
d : D
data P : D → Set where
Confuse : (d : D) (p : P d) → Set
Confuse x ()
confusing : (d : D) (p : P d) → Confuse d p
confusing x ()
test : (x : P d) → Set
test x with confusing d x
test x | e = A
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open import Oscar.Prelude
open import Oscar.Class
open import Oscar.Class.Unit
module Oscar.Class.Leftunit where
module Leftunit
{𝔞 𝔟} {𝔄 : Ø 𝔞} {𝔅 : Ø 𝔟}
{𝔢} {𝔈 : Ø 𝔢}
{𝔞𝔟}
(_↤_ : 𝔅 → 𝔄 → Ø 𝔞𝔟) (let _↤_ = _↤_; infix 4 _↤_)
(ε : 𝔈)
(_◃_ : 𝔈 → 𝔄 → 𝔅) (let _◃_ = _◃_; infix 16 _◃_)
(x : 𝔄)
= ℭLASS (ε , _◃_ , _↤_) (ε ◃ x ↤ x)
module _
{𝔞} {𝔄 : Ø 𝔞} {𝔢} {𝔈 : Ø 𝔢} {ℓ}
{_↦_ : 𝔄 → 𝔄 → Ø ℓ}
{ε : 𝔈}
{_◃_ : 𝔈 → 𝔄 → 𝔄}
{x : 𝔄}
where
leftunit = Leftunit.method _↦_ ε _◃_ x
open import Oscar.Class.Reflexivity
open import Oscar.Class.Surjection
open import Oscar.Class.Smap
module Leftunit,smaparrow
{𝔵₁ 𝔵₂ 𝔭₁ 𝔭₂ 𝔯 𝔭̇₁₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}
(ℜ : π̂² 𝔯 𝔛₁)
(𝔓₁ : π̂ 𝔭₁ 𝔛₂)
(𝔓₂ : π̂ 𝔭₂ 𝔛₂)
(ε : Reflexivity.type ℜ)
(surjection : Surjection.type 𝔛₁ 𝔛₂)
(smaparrow : Smaparrow.type ℜ 𝔓₁ 𝔓₂ surjection surjection)
(𝔓̇₁₂ : ∀ {x} → 𝔓₁ (surjection x) → 𝔓₂ (surjection x) → Ø 𝔭̇₁₂)
where
class = ∀ {x} {p : 𝔓₁ (surjection x)} → Leftunit.class (flip 𝔓̇₁₂) ε smaparrow p
type = ∀ {x} {p : 𝔓₁ (surjection x)} → Leftunit.type (flip 𝔓̇₁₂) ε smaparrow p
method : ⦃ _ : class ⦄ → type
method {x} {p} = Leftunit.method (flip 𝔓̇₁₂) ε smaparrow p
module Leftunit,smaparrow!
{𝔵₁ 𝔵₂ 𝔭₁ 𝔭₂ 𝔯 𝔭̇₁₂} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}
(ℜ : π̂² 𝔯 𝔛₁)
(𝔓₁ : π̂ 𝔭₁ 𝔛₂)
(𝔓₂ : π̂ 𝔭₂ 𝔛₂)
⦃ _ : Reflexivity.class ℜ ⦄
⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄
⦃ _ : Smaparrow!.class ℜ 𝔓₁ 𝔓₂ ⦄
(𝔓̇₁₂ : ∀ {x} → 𝔓₁ (surjection x) → 𝔓₂ (surjection x) → Ø 𝔭̇₁₂)
= Leftunit,smaparrow ℜ 𝔓₁ 𝔓₂ ε surjection smaparrow 𝔓̇₁₂
module Leftunit,smaphomarrow
{𝔵₁ 𝔵₂ 𝔭 𝔯 𝔭̇} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}
(ℜ : π̂² 𝔯 𝔛₁)
(𝔓 : π̂ 𝔭 𝔛₂)
(ε : Reflexivity.type ℜ)
(surjection : Surjection.type 𝔛₁ 𝔛₂)
(smaparrow : Smaphomarrow.type ℜ 𝔓 surjection)
(𝔓̇ : ∀ {x} → 𝔓 (surjection x) → 𝔓 (surjection x) → Ø 𝔭̇)
= Leftunit,smaparrow ℜ 𝔓 𝔓 ε surjection smaparrow 𝔓̇
module Leftunit,smaphomarrow!
{𝔵₁ 𝔵₂ 𝔭 𝔯 𝔭̇} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}
(ℜ : π̂² 𝔯 𝔛₁)
(𝔓 : π̂ 𝔭 𝔛₂)
⦃ _ : Reflexivity.class ℜ ⦄
⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄
⦃ _ : Smaphomarrow!.class ℜ 𝔓 ⦄
(𝔓̇ : ∀ {x} → 𝔓 (surjection x) → 𝔓 (surjection x) → Ø 𝔭̇)
= Leftunit,smaphomarrow ℜ 𝔓 ε surjection smaparrow 𝔓̇
open import Oscar.Class.HasEquivalence
module Leftunit,equivalence,smaphomarrow!
{𝔵₁ 𝔵₂ 𝔭 𝔯 𝔭̇} {𝔛₁ : Ø 𝔵₁} {𝔛₂ : Ø 𝔵₂}
(ℜ : π̂² 𝔯 𝔛₁)
(𝔓 : π̂ 𝔭 𝔛₂)
⦃ _ : Reflexivity.class ℜ ⦄
⦃ _ : Surjection.class 𝔛₁ 𝔛₂ ⦄
⦃ _ : Smaphomarrow!.class ℜ 𝔓 ⦄
⦃ _ : ∀ {x} → HasEquivalence (𝔓 x) 𝔭̇ ⦄
= Leftunit,smaphomarrow! ℜ 𝔓 _≈_
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-- There was a problem with module instantiation if a definition
-- was in scope under more than one name. For instance, constructors
-- or non-private local modules being open publicly. In this case
-- the module instantiation incorrectly generated two separate names
-- for this definition.
module Issue263 where
module M where
data D : Set where
d : D
-- 'M.D.d' is in scope both as 'd' and 'D.d'
module E where
postulate X : Set
open E public
-- 'M.E.X' is in scope as 'X' and 'E.X'
module M′ = M
bar = M′.E.X -- this panicked
foo = M′.D.d -- and this as well
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-- Basic intuitionistic propositional calculus, without ∨ or ⊥.
-- Kripke-style semantics with contexts as concrete worlds, and glueing for α and ▻.
-- Implicit syntax.
module BasicIPC.Semantics.KripkeConcreteGluedImplicit where
open import BasicIPC.Syntax.Common public
open import Common.Semantics public
open ConcreteWorlds (Ty) public
-- Partial intuitionistic Kripke models.
record Model : Set₁ where
infix 3 _⊩ᵅ_
field
-- Forcing for atomic propositions; monotonic.
_⊩ᵅ_ : World → Atom → Set
mono⊩ᵅ : ∀ {P w w′} → w ≤ w′ → w ⊩ᵅ P → w′ ⊩ᵅ P
open Model {{…}} public
module ImplicitSyntax
(_[⊢]_ : Cx Ty → Ty → Set)
(mono[⊢] : ∀ {A Γ Γ′} → Γ ⊆ Γ′ → Γ [⊢] A → Γ′ [⊢] A)
where
-- Forcing in a particular world of a particular model.
module _ {{_ : Model}} where
infix 3 _⊩_
_⊩_ : World → Ty → Set
w ⊩ α P = Glue (unwrap w [⊢] (α P)) (w ⊩ᵅ P)
w ⊩ A ▻ B = Glue (unwrap w [⊢] (A ▻ B)) (∀ {w′} → w ≤ w′ → w′ ⊩ A → w′ ⊩ B)
w ⊩ A ∧ B = w ⊩ A × w ⊩ B
w ⊩ ⊤ = 𝟙
infix 3 _⊩⋆_
_⊩⋆_ : World → Cx Ty → Set
w ⊩⋆ ∅ = 𝟙
w ⊩⋆ Ξ , A = w ⊩⋆ Ξ × w ⊩ A
-- Monotonicity with respect to context inclusion.
module _ {{_ : Model}} where
mono⊩ : ∀ {A w w′} → w ≤ w′ → w ⊩ A → w′ ⊩ A
mono⊩ {α P} ξ s = mono[⊢] (unwrap≤ ξ) (syn s) ⅋ mono⊩ᵅ ξ (sem s)
mono⊩ {A ▻ B} ξ s = mono[⊢] (unwrap≤ ξ) (syn s) ⅋ λ ξ′ → sem s (trans≤ ξ ξ′)
mono⊩ {A ∧ B} ξ s = mono⊩ {A} ξ (π₁ s) , mono⊩ {B} ξ (π₂ s)
mono⊩ {⊤} ξ s = ∙
mono⊩⋆ : ∀ {Ξ w w′} → w ≤ w′ → w ⊩⋆ Ξ → w′ ⊩⋆ Ξ
mono⊩⋆ {∅} ξ ∙ = ∙
mono⊩⋆ {Ξ , A} ξ (ts , t) = mono⊩⋆ {Ξ} ξ ts , mono⊩ {A} ξ t
-- Additional useful equipment.
module _ {{_ : Model}} where
_⟪$⟫_ : ∀ {A B w} → w ⊩ A ▻ B → w ⊩ A → w ⊩ B
s ⟪$⟫ a = sem s refl≤ a
⟪S⟫ : ∀ {A B C w} → w ⊩ A ▻ B ▻ C → w ⊩ A ▻ B → w ⊩ A → w ⊩ C
⟪S⟫ s₁ s₂ a = (s₁ ⟪$⟫ a) ⟪$⟫ (s₂ ⟪$⟫ a)
-- Forcing in a particular world of a particular model, for sequents.
module _ {{_ : Model}} where
infix 3 _⊩_⇒_
_⊩_⇒_ : World → Cx Ty → Ty → Set
w ⊩ Γ ⇒ A = w ⊩⋆ Γ → w ⊩ A
infix 3 _⊩_⇒⋆_
_⊩_⇒⋆_ : World → Cx Ty → Cx Ty → Set
w ⊩ Γ ⇒⋆ Ξ = w ⊩⋆ Γ → w ⊩⋆ Ξ
-- Entailment, or forcing in all worlds of all models, for sequents.
infix 3 _⊨_
_⊨_ : Cx Ty → Ty → Set₁
Γ ⊨ A = ∀ {{_ : Model}} {w : World} → w ⊩ Γ ⇒ A
infix 3 _⊨⋆_
_⊨⋆_ : Cx Ty → Cx Ty → Set₁
Γ ⊨⋆ Ξ = ∀ {{_ : Model}} {w : World} → w ⊩ Γ ⇒⋆ Ξ
-- Additional useful equipment, for sequents.
module _ {{_ : Model}} where
lookup : ∀ {A Γ w} → A ∈ Γ → w ⊩ Γ ⇒ A
lookup top (γ , a) = a
lookup (pop i) (γ , b) = lookup i γ
_⟦$⟧_ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ▻ B → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ B
(s₁ ⟦$⟧ s₂) γ = s₁ γ ⟪$⟫ s₂ γ
⟦S⟧ : ∀ {A B C Γ w} → w ⊩ Γ ⇒ A ▻ B ▻ C → w ⊩ Γ ⇒ A ▻ B → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ C
⟦S⟧ s₁ s₂ a γ = ⟪S⟫ (s₁ γ) (s₂ γ) (a γ)
_⟦,⟧_ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A → w ⊩ Γ ⇒ B → w ⊩ Γ ⇒ A ∧ B
(a ⟦,⟧ b) γ = a γ , b γ
⟦π₁⟧ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ∧ B → w ⊩ Γ ⇒ A
⟦π₁⟧ s γ = π₁ (s γ)
⟦π₂⟧ : ∀ {A B Γ w} → w ⊩ Γ ⇒ A ∧ B → w ⊩ Γ ⇒ B
⟦π₂⟧ s γ = π₂ (s γ)
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.types.Group
open import lib.types.Pi
open import lib.types.Sigma
open import lib.types.Truncation
open import lib.groups.GroupProduct
open import lib.groups.Homomorphisms
module lib.groups.TruncationGroup where
module _ {i} {El : Type i} (GS : GroupStructure El) where
Trunc-group-struct : GroupStructure (Trunc 0 El)
Trunc-group-struct = record {
ident = [ ident GS ];
inv = Trunc-fmap (inv GS);
comp = _⊗_;
unitl = t-unitl;
unitr = t-unitr;
assoc = t-assoc;
invl = t-invl;
invr = t-invr}
where
open GroupStructure
infix 80 _⊗_
_⊗_ = Trunc-fmap2 (comp GS)
abstract
t-unitl : (t : Trunc 0 El) → [ ident GS ] ⊗ t == t
t-unitl = Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(ap [_] ∘ unitl GS)
t-unitr : (t : Trunc 0 El) → t ⊗ [ ident GS ] == t
t-unitr = Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(ap [_] ∘ unitr GS)
t-assoc : (t₁ t₂ t₃ : Trunc 0 El) → (t₁ ⊗ t₂) ⊗ t₃ == t₁ ⊗ (t₂ ⊗ t₃)
t-assoc = Trunc-elim
(λ _ → Π-level (λ _ → Π-level (λ _ → =-preserves-level _ Trunc-level)))
(λ a → Trunc-elim
(λ _ → Π-level (λ _ → =-preserves-level _ Trunc-level))
(λ b → Trunc-elim
(λ _ → =-preserves-level _ Trunc-level)
(λ c → ap [_] (assoc GS a b c))))
t-invl : (t : Trunc 0 El) → Trunc-fmap (inv GS) t ⊗ t == [ ident GS ]
t-invl = Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(ap [_] ∘ invl GS)
t-invr : (t : Trunc 0 El) → t ⊗ Trunc-fmap (inv GS) t == [ ident GS ]
t-invr = Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(ap [_] ∘ invr GS)
Trunc-group : Group i
Trunc-group = record {
El = Trunc 0 El;
El-level = Trunc-level;
group-struct = Trunc-group-struct }
Trunc-group-× : ∀ {i j} {A : Type i} {B : Type j}
(GS : GroupStructure A) (HS : GroupStructure B)
→ Trunc-group (×-group-struct GS HS) == Trunc-group GS ×ᴳ Trunc-group HS
Trunc-group-× GS HS = group-ua
(record {
f = Trunc-rec (×-level Trunc-level Trunc-level)
(λ {(a , b) → ([ a ] , [ b ])});
pres-comp =
Trunc-elim
(λ _ → (Π-level (λ _ → =-preserves-level _
(×-level Trunc-level Trunc-level))))
(λ a → Trunc-elim
(λ _ → =-preserves-level _ (×-level Trunc-level Trunc-level))
(λ b → idp))} ,
is-eq _
(uncurry (Trunc-rec (→-level Trunc-level)
(λ a → Trunc-rec Trunc-level (λ b → [ a , b ]))))
(uncurry (Trunc-elim
(λ _ → Π-level (λ _ → =-preserves-level _
(×-level Trunc-level Trunc-level)))
(λ a → Trunc-elim
(λ _ → =-preserves-level _
(×-level Trunc-level Trunc-level))
(λ b → idp))))
(Trunc-elim (λ _ → =-preserves-level _ Trunc-level) (λ _ → idp)))
Trunc-group-hom : ∀ {i j} {A : Type i} {B : Type j}
{GS : GroupStructure A} {HS : GroupStructure B} (f : A → B)
→ ((a₁ a₂ : A) → f (GroupStructure.comp GS a₁ a₂)
== GroupStructure.comp HS (f a₁) (f a₂))
→ (Trunc-group GS →ᴳ Trunc-group HS)
Trunc-group-hom {A = A} {GS = GS} {HS = HS} f p =
record {f = Trunc-fmap f; pres-comp = pres-comp}
where
abstract
pres-comp : (t₁ t₂ : Trunc 0 A) →
Trunc-fmap f (Trunc-fmap2 (GroupStructure.comp GS) t₁ t₂)
== Trunc-fmap2 (GroupStructure.comp HS)
(Trunc-fmap f t₁) (Trunc-fmap f t₂)
pres-comp =
Trunc-elim (λ _ → Π-level (λ _ → =-preserves-level _ Trunc-level))
(λ a₁ → Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(λ a₂ → ap [_] (p a₁ a₂)))
Trunc-group-iso : ∀ {i} {A B : Type i}
{GS : GroupStructure A} {HS : GroupStructure B} (f : A → B)
→ ((a₁ a₂ : A) → f (GroupStructure.comp GS a₁ a₂)
== GroupStructure.comp HS (f a₁) (f a₂))
→ is-equiv f → Trunc-group GS ≃ᴳ Trunc-group HS
Trunc-group-iso f pres-comp ie =
(Trunc-group-hom f pres-comp ,
is-eq _ (Trunc-fmap (is-equiv.g ie))
(Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(λ b → ap [_] (is-equiv.f-g ie b)))
(Trunc-elim (λ _ → =-preserves-level _ Trunc-level)
(λ a → ap [_] (is-equiv.g-f ie a))))
Trunc-group-abelian : ∀ {i} {A : Type i} (GS : GroupStructure A)
→ ((a₁ a₂ : A) → GroupStructure.comp GS a₁ a₂ == GroupStructure.comp GS a₂ a₁)
→ is-abelian (Trunc-group GS)
Trunc-group-abelian GS ab =
Trunc-elim (λ _ → Π-level (λ _ → =-preserves-level _ Trunc-level)) $
λ a₁ → Trunc-elim (λ _ → =-preserves-level _ Trunc-level) $
λ a₂ → ap [_] (ab a₁ a₂)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- This module is DEPRECATED. Please use the
-- Relation.Binary.Reasoning.MultiSetoid module directly.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Relation.Binary.SetoidReasoning where
open import Relation.Binary.Reasoning.MultiSetoid public
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module Prelude.Bool where
data Bool : Set where
true : Bool
false : Bool
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN TRUE true #-}
{-# BUILTIN FALSE false #-}
not : Bool -> Bool
not true = false
not false = true
notnot : Bool -> Bool
notnot true = not (not true)
notnot false = not (not false)
infix 90 if_then_else_
infix 90 if'_then_else_
if_then_else_ : ∀{ P : Bool -> Set} -> (b : Bool) -> P true -> P false -> P b
if true then a else b = a
if false then a else b = b
if'_then_else_ : ∀{ P : Set} -> (b : Bool) -> P -> P -> P
if' true then a else b = a
if' false then a else b = b
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------------------------------------------------------------------------
-- Many properties which hold for _∼_ also hold for _∼_ on₁ f
------------------------------------------------------------------------
open import Relation.Binary
module Relation.Binary.On {A B : Set} (f : B → A) where
open import Data.Function
open import Data.Product
implies : ∀ ≈ ∼ → ≈ ⇒ ∼ → (≈ on₁ f) ⇒ (∼ on₁ f)
implies _ _ impl = impl
reflexive : ∀ ∼ → Reflexive ∼ → Reflexive (∼ on₁ f)
reflexive _ refl = refl
irreflexive : ∀ ≈ ∼ → Irreflexive ≈ ∼ → Irreflexive (≈ on₁ f) (∼ on₁ f)
irreflexive _ _ irrefl = irrefl
symmetric : ∀ ∼ → Symmetric ∼ → Symmetric (∼ on₁ f)
symmetric _ sym = sym
transitive : ∀ ∼ → Transitive ∼ → Transitive (∼ on₁ f)
transitive _ trans = trans
antisymmetric : ∀ ≈ ≤ →
Antisymmetric ≈ ≤ → Antisymmetric (≈ on₁ f) (≤ on₁ f)
antisymmetric _ _ antisym = antisym
asymmetric : ∀ < → Asymmetric < → Asymmetric (< on₁ f)
asymmetric _ asym = asym
respects : ∀ ∼ P → P Respects ∼ → (P ∘₀ f) Respects (∼ on₁ f)
respects _ _ resp = resp
respects₂ : ∀ ∼₁ ∼₂ → ∼₁ Respects₂ ∼₂ → (∼₁ on₁ f) Respects₂ (∼₂ on₁ f)
respects₂ _ _ (resp₁ , resp₂) =
((λ {_} {_} {_} → resp₁) , λ {_} {_} {_} → resp₂)
decidable : ∀ ∼ → Decidable ∼ → Decidable (∼ on₁ f)
decidable _ dec = λ x y → dec (f x) (f y)
total : ∀ ∼ → Total ∼ → Total (∼ on₁ f)
total _ tot = λ x y → tot (f x) (f y)
trichotomous : ∀ ≈ < →
Trichotomous ≈ < → Trichotomous (≈ on₁ f) (< on₁ f)
trichotomous _ _ compare = λ x y → compare (f x) (f y)
isEquivalence : ∀ {≈} → IsEquivalence ≈ → IsEquivalence (≈ on₁ f)
isEquivalence {≈} eq = record
{ refl = reflexive ≈ Eq.refl
; sym = symmetric ≈ Eq.sym
; trans = transitive ≈ Eq.trans
}
where module Eq = IsEquivalence eq
isPreorder : ∀ {≈ ∼} → IsPreorder ≈ ∼ → IsPreorder (≈ on₁ f) (∼ on₁ f)
isPreorder {≈} {∼} pre = record
{ isEquivalence = isEquivalence Pre.isEquivalence
; reflexive = implies ≈ ∼ Pre.reflexive
; trans = transitive ∼ Pre.trans
; ∼-resp-≈ = respects₂ ∼ ≈ Pre.∼-resp-≈
}
where module Pre = IsPreorder pre
isDecEquivalence : ∀ {≈} →
IsDecEquivalence ≈ → IsDecEquivalence (≈ on₁ f)
isDecEquivalence {≈} dec = record
{ isEquivalence = isEquivalence Dec.isEquivalence
; _≟_ = decidable ≈ Dec._≟_
}
where module Dec = IsDecEquivalence dec
isPartialOrder : ∀ {≈ ≤} →
IsPartialOrder ≈ ≤ →
IsPartialOrder (≈ on₁ f) (≤ on₁ f)
isPartialOrder {≈} {≤} po = record
{ isPreorder = isPreorder Po.isPreorder
; antisym = antisymmetric ≈ ≤ Po.antisym
}
where module Po = IsPartialOrder po
isStrictPartialOrder : ∀ {≈ <} →
IsStrictPartialOrder ≈ < →
IsStrictPartialOrder (≈ on₁ f) (< on₁ f)
isStrictPartialOrder {≈} {<} spo = record
{ isEquivalence = isEquivalence Spo.isEquivalence
; irrefl = irreflexive ≈ < Spo.irrefl
; trans = transitive < Spo.trans
; <-resp-≈ = respects₂ < ≈ Spo.<-resp-≈
}
where module Spo = IsStrictPartialOrder spo
isTotalOrder : ∀ {≈ ≤} →
IsTotalOrder ≈ ≤ →
IsTotalOrder (≈ on₁ f) (≤ on₁ f)
isTotalOrder {≈} {≤} to = record
{ isPartialOrder = isPartialOrder To.isPartialOrder
; total = total ≤ To.total
}
where module To = IsTotalOrder to
isDecTotalOrder : ∀ {≈ ≤} →
IsDecTotalOrder ≈ ≤ →
IsDecTotalOrder (≈ on₁ f) (≤ on₁ f)
isDecTotalOrder {≈} {≤} dec = record
{ isTotalOrder = isTotalOrder Dec.isTotalOrder
; _≟_ = decidable ≈ Dec._≟_
; _≤?_ = decidable ≤ Dec._≤?_
}
where module Dec = IsDecTotalOrder dec
isStrictTotalOrder : ∀ {≈ <} →
IsStrictTotalOrder ≈ < →
IsStrictTotalOrder (≈ on₁ f) (< on₁ f)
isStrictTotalOrder {≈} {<} sto = record
{ isEquivalence = isEquivalence Sto.isEquivalence
; trans = transitive < Sto.trans
; compare = trichotomous ≈ < Sto.compare
; <-resp-≈ = respects₂ < ≈ Sto.<-resp-≈
}
where module Sto = IsStrictTotalOrder sto
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{-# OPTIONS --cubical --omega-in-omega #-}
open import Agda.Primitive.Cubical
open import Agda.Builtin.Bool
-- With --omega-in-omega we are allowed to split on Setω datatypes.
-- Andrea 22/05/2020: in the future we might be allowed even without --omega-in-omega.
-- This test makes sure the interval I is still special cased and splitting is forbidden.
bad : I → Bool
bad i0 = true
bad i1 = false
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{-# OPTIONS --without-K #-}
module sets.fin.properties where
open import sum
open import decidable
open import equality
open import function.core
open import function.extensionality
open import function.isomorphism
open import function.overloading
open import sets.core
open import sets.nat.core
hiding (_≟_; pred)
open import sets.nat.ordering
open import sets.fin.core
open import sets.empty
open import sets.properties
open import hott.level.core
open import hott.level.closure
open import hott.level.sets
pred : ∀ {n}(i : Fin (suc n))
→ ¬ (i ≡ zero) → Fin n
pred zero u = ⊥-elim (u refl)
pred (suc j) _ = j
pred-β : ∀ {n}(i : Fin (suc n))
→ (u : ¬ (i ≡ zero))
→ suc (pred i u) ≡ i
pred-β zero u = ⊥-elim (u refl)
pred-β (suc i) u = refl
pred-inj : ∀ {n}{i j : Fin (suc n)}
→ (u : ¬ (i ≡ zero))
→ (v : ¬ (j ≡ zero))
→ pred i u ≡ pred j v
→ i ≡ j
pred-inj {n} {zero} u v p = ⊥-elim (u refl)
pred-inj {n} {suc i} {zero} u v p = ⊥-elim (v refl)
pred-inj {n} {suc i} {suc j} u v p = ap suc p
toℕ : ∀ {n} → Fin n → ℕ
toℕ zero = 0
toℕ (suc i) = suc (toℕ i)
toℕ-sound : ∀ {n}(i : Fin n) → suc (toℕ i) ≤ n
toℕ-sound {zero} ()
toℕ-sound {suc n} zero = s≤s z≤n
toℕ-sound {suc n} (suc i) = s≤s (toℕ-sound i)
fromℕ : ∀ {n i} → (suc i ≤ n) → Fin n
fromℕ {zero} ()
fromℕ {suc n} {0} _ = zero
fromℕ {suc n} {suc i} (s≤s p) = suc (fromℕ p)
toℕ-iso : ∀ {n} → Fin n ≅ (Σ ℕ λ i → suc i ≤ n)
toℕ-iso {n} = record
{ to = λ i → toℕ i , toℕ-sound i
; from = λ { (_ , p) → fromℕ p }
; iso₁ = α
; iso₂ = λ { (i , p) → unapΣ (β i p , h1⇒prop ≤-level _ _) } }
where
α : ∀ {n} (i : Fin n) → fromℕ (toℕ-sound i) ≡ i
α zero = refl
α (suc i) = ap suc (α i)
β : ∀ {n} (i : ℕ)(p : suc i ≤ n) → toℕ (fromℕ p) ≡ i
β {0} _ ()
β {suc n} 0 _ = refl
β {suc n} (suc i) (s≤s p) = ap suc (β i p)
toℕ-inj : ∀ {n} → injective (toℕ {n = n})
toℕ-inj p = iso⇒inj toℕ-iso (unapΣ (p , h1⇒prop ≤-level _ _))
#_ : ∀ {n} i {p : True (suc i ≤? n)} → Fin n
#_ {n} i {p} = fromℕ (witness p)
transpose : ∀ {n} → Fin n → Fin n → Fin n → Fin n
transpose i j k with i ≟ k
... | yes _ = j
... | no _ with j ≟ k
... | yes _ = i
... | no _ = k
transpose-β₂ : ∀ {n}(i j : Fin n)
→ transpose i j j ≡ i
transpose-β₂ i j with i ≟ j
... | yes p = sym p
... | no u with j ≟ j
... | yes p = refl
... | no v = ⊥-elim (v refl)
abstract
transpose-invol : ∀ {n}(i j k : Fin n)
→ transpose i j (transpose i j k) ≡ k
transpose-invol i j k with i ≟ k
transpose-invol i j k | yes p with i ≟ j
transpose-invol i j k | yes p | yes p' = sym p' · p
transpose-invol i j k | yes p | no u with j ≟ j
transpose-invol i j k | yes p | no u | yes p' = p
transpose-invol i j k | yes p | no u | no v = ⊥-elim (v refl)
transpose-invol i j k | no u with j ≟ k
transpose-invol i j k | no u | yes p with i ≟ i
transpose-invol i j k | no u | yes p | yes p' = p
transpose-invol i j k | no u | yes p | no v = ⊥-elim (v refl)
transpose-invol i j k | no u | no v with i ≟ k
transpose-invol i j k | no u | no v | yes p = ⊥-elim (u p)
transpose-invol i j k | no u | no v | no w with j ≟ k
transpose-invol i j k | no u | no v | no w | yes p = ⊥-elim (v p)
transpose-invol i j k | no u | no v | no w | no z = refl
transpose-iso : ∀ {n}(i j : Fin n) → Fin n ≅ Fin n
transpose-iso i j = record
{ to = transpose i j
; from = transpose i j
; iso₁ = transpose-invol i j
; iso₂ = transpose-invol i j }
fin-remove₀-iso : ∀ {n} → (Fin (suc n) minus zero) ≅ Fin n
fin-remove₀-iso = record
{ to = uncurry pred
; from = λ i → suc i , λ p → fin-disj _ (sym p)
; iso₁ = λ { (i , u) → unapΣ (pred-β _ u , h1⇒prop ¬-h1 _ _) }
; iso₂ = λ _ → refl }
fin-remove-iso : ∀ {n}(i : Fin (suc n))
→ (Σ (Fin (suc n)) λ j → ¬ (j ≡ i))
≅ Fin n
fin-remove-iso {n} i = begin
(Σ (Fin (suc n)) λ j → ¬ (j ≡ i))
≅⟨ ( Σ-ap-iso (transpose-iso zero i) λ _
→ Π-ap-iso lem λ _ → refl≅ ) ⟩
(Σ (Fin (suc n)) λ j → ¬ (j ≡ zero))
≅⟨ fin-remove₀-iso ⟩
Fin n
∎
where
open ≅-Reasoning
lem : ∀ {x} → (x ≡ i) ≅ (transpose zero i x ≡ zero)
lem {x} = begin
(x ≡ i)
≅⟨ iso≡ (transpose-iso zero i) ⟩
( transpose zero i x ≡ transpose zero i i )
≅⟨ trans≡-iso' (transpose-β₂ zero i) ⟩
( transpose zero i x ≡ zero )
∎ where open ≅-Reasoning
transpose-inj : ∀ {n m}(i j : Fin m)
→ (f : Fin n → Fin m)
→ injective f
→ injective (transpose i j ∘' f)
transpose-inj i j f inj = inj ∘' iso⇒inj (transpose-iso i j)
transpose-inj-iso' : ∀ {n} (i j : Fin n)
→ (Fin n ↣ Fin n)
≅ (Fin n ↣ Fin n)
transpose-inj-iso' {n} i j
= Σ-ap-iso (Π-ap-iso refl≅ λ _ → tiso) λ f
→ mk-prop-iso (inj-level f (fin-set _))
(inj-level _ (fin-set _))
(transpose-inj i j f)
(λ inj p → inj (ap (apply tiso) p))
where
tiso : Fin n ≅ Fin n
tiso = transpose-iso i j
transpose-inj-iso : ∀ {n} (i j : Fin n)
→ (Fin n ↣ Fin n)
≅ (Fin n ↣ Fin n)
transpose-inj-iso {n} i j
= Σ-ap-iso (Π-ap-iso tiso λ _ → refl≅) λ f
→ mk-prop-iso (inj-level f (fin-set _))
(inj-level _ (fin-set _))
(λ inj → iso⇒inj tiso ∘' inj)
(λ inj p → iso⇒inj tiso (inj (
ap f (_≅_.iso₁ tiso _)
· p
· sym (ap f (_≅_.iso₁ tiso _)))))
where
tiso : Fin n ≅ Fin n
tiso = transpose-iso i j
inj-nonsurj : ∀ {n i}{A : Set i}
→ (f : A → Fin (suc n))
→ injective f
→ {i : Fin (suc n)}
→ ((x : A) → ¬ (f x ≡ i))
→ A ↣ Fin n
inj-nonsurj {n}{i}{A} f inj {z} u = g , g-inj
where
f' : A → Σ (Fin (suc n)) λ k → ¬ (k ≡ z)
f' i = f i , u i
inj' : injective f'
inj' p = inj (ap proj₁ p)
abstract
φ : (Σ (Fin (suc n)) λ k → ¬ (k ≡ z)) ≅ Fin n
φ = fin-remove-iso z
g : A → Fin n
g = apply φ ∘' f'
g-inj : injective g
g-inj p = inj' (iso⇒inj φ p)
preimage : ∀ {i n}{A : Set i}
→ (f : Fin n → A)
→ (dec : (x y : A) → Dec (x ≡ y))
→ (x : A)
→ ( (Σ (Fin n) λ i → f i ≡ x)
⊎ ((i : Fin n) → ¬ (f i ≡ x)) )
preimage {n = 0} f dec x = inj₂ (λ ())
preimage {n = suc n} f dec x with dec (f zero) x
... | yes p = inj₁ (zero , p)
... | no u with preimage (λ i → f (suc i)) dec x
... | inj₁ (i , p) = inj₁ (suc i , p)
... | inj₂ v = inj₂ λ { zero → u ; (suc i) → v i }
fin-inj-remove₀ : ∀ {n m}
→ (f : Fin (suc n) ↣ Fin (suc m))
→ (apply f zero ≡ zero)
→ Fin n ↣ Fin m
fin-inj-remove₀ {n}{m} (f , inj) p = g , g-inj
where
nz : {i : Fin n} → ¬ (f (suc i) ≡ zero)
nz q = fin-disj _ (inj (p · sym q))
g : Fin n → Fin m
g i = pred (f (suc i)) nz
g-inj : injective g
g-inj q = fin-suc-inj (inj (pred-inj nz nz q))
fin-inj-remove : ∀ {n m}
→ (Fin (suc n) ↣ Fin (suc m))
→ (Fin n ↣ Fin m)
fin-inj-remove {n}{m} (f , f-inj)
= fin-inj-remove₀
( transpose zero (f zero) ∘' f
, transpose-inj zero (f zero) f f-inj)
( transpose-β₂ zero (f zero))
fin-inj-add : ∀ {n m}
→ (Fin n ↣ Fin m)
→ (Fin (suc n) ↣ Fin (suc m))
fin-inj-add {n}{m} (f , f-inj) = g , g-inj
where
g : Fin (suc n) → Fin (suc m)
g zero = zero
g (suc i) = suc (f i)
g-inj : injective g
g-inj {zero} {zero} p = refl
g-inj {zero} {suc x'} p = ⊥-elim (fin-disj _ p)
g-inj {suc x} {zero} p = ⊥-elim (fin-disj _ (sym p))
g-inj {suc x} {suc x'} p = ap suc (f-inj (fin-suc-inj p))
fin-lt : ∀ {n m}(f : Fin n ↣ Fin m) → n ≤ m
fin-lt {0} _ = z≤n
fin-lt {suc n} {0} (f , _) with f zero
... | ()
fin-lt {suc n}{suc m} f = s≤s (fin-lt (fin-inj-remove f))
inj⇒retr : ∀ {n}(f : Fin n → Fin n)
→ injective f
→ retraction f
inj⇒retr {0} f inj ()
inj⇒retr {suc n} f inj y with preimage f _≟_ y
... | inj₁ t = t
... | inj₂ u = ⊥-elim (suc≰ (fin-lt (inj-nonsurj f inj u)))
inj⇒iso : ∀ {n}(f : Fin n → Fin n)
→ injective f
→ Fin n ≅ Fin n
inj⇒iso f inj = inj+retr⇒iso f inj (inj⇒retr f inj)
Fin-inj : ∀ {n m} → Fin n ≅ Fin m → n ≡ m
Fin-inj {n}{m} isoF
= antisym≤ (fin-lt (_ , iso⇒inj isoF))
(fin-lt (_ , iso⇒inj (sym≅ isoF)))
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module StateSizedIO.IOObject where
open import Data.Product
open import Size
open import SizedIO.Base
open import StateSizedIO.Object
-- ---
-- ---
-- --- FILE IS DELETED !!!
-- ---
-- ---
-- An IO object is like a simple object,
-- but the method returns IO applied to the result type of a simple object
-- which means the method returns an IO program which when terminating
-- returns the result of the simple object
{- NOTE IOObject is now replaced by IOObjectˢ as defined in
StateSizedIO.Base
-}
{-
module _ (ioi : IOInterface) (let C = Command ioi) (let R = Response ioi)
(oi : Interfaceˢ) (let S = Stateˢ oi) (let M = Methodˢ oi)
(let Rt = Resultˢ oi) (let next = nextˢ oi)
where
record IOObject (i : Size) (s : S) : Set where
coinductive
field
method : ∀{j : Size< i} (m : M s) → IO ioi ∞ (Σ[ r ∈ Rt s m ] IOObject j (next s m r) )
open IOObject public
-}
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module A.B where
{-# NON_TERMINATING #-}
easy : (A : Set) → A
easy = easy
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data D : Set where
zero : D
suc : D → D
postulate
f : D → D
{-# COMPILE GHC f = \ x -> x #-}
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{-# OPTIONS --cubical --safe #-}
module Algebra.Construct.Free.Semilattice.Eliminators where
open import Algebra.Construct.Free.Semilattice.Definition
open import Prelude
open import Algebra
record _⇘_ {a p} (A : Type a) (P : 𝒦 A → Type p) : Type (a ℓ⊔ p) where
no-eta-equality
constructor elim
field
⟦_⟧-set : ∀ {xs} → isSet (P xs)
⟦_⟧[] : P []
⟦_⟧_∷_⟨_⟩ : ∀ x xs → P xs → P (x ∷ xs)
private z = ⟦_⟧[]; f = ⟦_⟧_∷_⟨_⟩
field
⟦_⟧-com : ∀ x y xs pxs →
f x (y ∷ xs) (f y xs pxs) ≡[ i ≔ P (com x y xs i) ]≡
f y (x ∷ xs) (f x xs pxs)
⟦_⟧-dup : ∀ x xs pxs →
f x (x ∷ xs) (f x xs pxs) ≡[ i ≔ P (dup x xs i) ]≡
f x xs pxs
⟦_⟧⇓ : ∀ xs → P xs
⟦ [] ⟧⇓ = z
⟦ x ∷ xs ⟧⇓ = f x xs ⟦ xs ⟧⇓
⟦ com x y xs i ⟧⇓ = ⟦_⟧-com x y xs ⟦ xs ⟧⇓ i
⟦ dup x xs i ⟧⇓ = ⟦_⟧-dup x xs ⟦ xs ⟧⇓ i
⟦ trunc xs ys x y i j ⟧⇓ =
isOfHLevel→isOfHLevelDep 2
(λ xs → ⟦_⟧-set {xs})
⟦ xs ⟧⇓ ⟦ ys ⟧⇓
(cong ⟦_⟧⇓ x) (cong ⟦_⟧⇓ y)
(trunc xs ys x y)
i j
{-# INLINE ⟦_⟧⇓ #-}
open _⇘_ public
infixr 0 ⇘-syntax
⇘-syntax = _⇘_
syntax ⇘-syntax A (λ xs → Pxs) = xs ∈𝒦 A ⇒ Pxs
record _⇲_ {a p} (A : Type a) (P : 𝒦 A → Type p) : Type (a ℓ⊔ p) where
no-eta-equality
constructor elim-prop
field
∥_∥-prop : ∀ {xs} → isProp (P xs)
∥_∥[] : P []
∥_∥_∷_⟨_⟩ : ∀ x xs → P xs → P (x ∷ xs)
private z = ∥_∥[]; f = ∥_∥_∷_⟨_⟩
∥_∥⇑ = elim
(λ {xs} → isProp→isSet (∥_∥-prop {xs}))
z f
(λ x y xs pxs → toPathP (∥_∥-prop (transp (λ i → P (com x y xs i)) i0 (f x (y ∷ xs) (f y xs pxs))) (f y (x ∷ xs) (f x xs pxs))))
(λ x xs pxs → toPathP (∥_∥-prop (transp (λ i → P (dup x xs i)) i0 (f x (x ∷ xs) (f x xs pxs))) (f x xs pxs) ))
∥_∥⇓ = ⟦ ∥_∥⇑ ⟧⇓
{-# INLINE ∥_∥⇑ #-}
{-# INLINE ∥_∥⇓ #-}
open _⇲_ public
elim-prop-syntax : ∀ {a p} → (A : Type a) → (𝒦 A → Type p) → Type (a ℓ⊔ p)
elim-prop-syntax = _⇲_
syntax elim-prop-syntax A (λ xs → Pxs) = xs ∈𝒦 A ⇒∥ Pxs ∥
record _↘∥_∥ {a p} (A : Type a) (P : 𝒦 A → Type p) : Type (a ℓ⊔ p) where
no-eta-equality
constructor elim-to-prop
field
∣_∣[] : P []
∣_∣_∷_⟨_⟩ : ∀ x xs → P xs → P (x ∷ xs)
private z = ∣_∣[]; f = ∣_∣_∷_⟨_⟩
open import HITs.PropositionalTruncation.Sugar
open import HITs.PropositionalTruncation
∣_∣⇑ : xs ∈𝒦 A ⇒∥ ∥ P xs ∥ ∥
∣_∣⇑ = elim-prop squash ∣ z ∣ λ x xs ∣Pxs∣ → f x xs ∥$∥ ∣Pxs∣
∣_∣⇓ = ∥ ∣_∣⇑ ∥⇓
open _↘∥_∥ public
elim-to-prop-syntax : ∀ {a p} → (A : Type a) → (𝒦 A → Type p) → Type (a ℓ⊔ p)
elim-to-prop-syntax = _↘∥_∥
syntax elim-to-prop-syntax A (λ xs → Pxs) = xs ∈𝒦 A ⇒∣ Pxs ∣
infixr 0 _↘_
record _↘_ {a b} (A : Type a) (B : Type b) : Type (a ℓ⊔ b) where
no-eta-equality
constructor rec
field
[_]-set : isSet B
[_]_∷_ : A → B → B
[_][] : B
private f = [_]_∷_; z = [_][]
field
[_]-dup : ∀ x xs → f x (f x xs) ≡ f x xs
[_]-com : ∀ x y xs → f x (f y xs) ≡ f y (f x xs)
[_]↑ = elim
[_]-set
z
(λ x _ xs → f x xs)
(λ x y xs → [_]-com x y)
(λ x xs → [_]-dup x)
[_]↓ = ⟦ [_]↑ ⟧⇓
{-# INLINE [_]↑ #-}
{-# INLINE [_]↓ #-}
open _↘_ public
module _ {a p} {A : Type a} {P : 𝒦 A → Type p} where
𝒦-elim-prop : (∀ {xs} → isProp (P xs)) →
(∀ x xs → P xs → P (x ∷ xs)) →
(P []) →
∀ xs → P xs
𝒦-elim-prop isPropB f n = go
where
go : ∀ xs → P xs
go [] = n
go (x ∷ xs) = f x xs (go xs)
go (com x y xs j) = toPathP {A = λ i → P (com x y xs i)} (isPropB (transp (λ i → P (com x y xs i)) i0 (f x (y ∷ xs) (f y xs (go xs)))) (f y (x ∷ xs) (f x xs (go xs)))) j
go (dup x xs j) = toPathP {A = λ i → P (dup x xs i)} (isPropB (transp (λ i → P (dup x xs i)) i0 (f x (x ∷ xs) (f x xs (go xs)))) (f x xs (go xs)) ) j
go (trunc xs ys x y i j) =
isOfHLevel→isOfHLevelDep 2
(λ xs → isProp→isSet (isPropB {xs}))
(go xs) (go ys)
(cong go x) (cong go y)
(trunc xs ys x y)
i j
module _ {a b} {A : Type a} {B : Type b} where
𝒦-rec : isSet B →
(f : A → B → B) →
(n : B) →
(fdup : ∀ x xs → f x (f x xs) ≡ f x xs) →
(fcom : ∀ x y xs → f x (f y xs) ≡ f y (f x xs)) →
𝒦 A →
B
𝒦-rec isSetB f n fdup fcom = go
where
go : 𝒦 A → B
go [] = n
go (x ∷ xs) = f x (go xs)
go (com x y xs i) = fcom x y (go xs) i
go (dup x xs i) = fdup x (go xs) i
go (trunc xs ys x y i j) =
isOfHLevel→isOfHLevelDep 2
(λ xs → isSetB)
(go xs) (go ys)
(cong go x) (cong go y)
(trunc xs ys x y)
i j
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Instantiates the ring solver, using the natural numbers as the
-- coefficient "ring"
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Algebra
import Algebra.Operations.Semiring as SemiringOps
open import Data.Maybe.Base using (Maybe; just; nothing; map)
module Algebra.Solver.Ring.NaturalCoefficients
{r₁ r₂}
(R : CommutativeSemiring r₁ r₂)
(dec : let open CommutativeSemiring R
open SemiringOps semiring in
∀ m n → Maybe (m × 1# ≈ n × 1#)) where
import Algebra.Solver.Ring
open import Algebra.Solver.Ring.AlmostCommutativeRing
open import Data.Nat.Base as ℕ
open import Data.Product using (module Σ)
open import Function
open CommutativeSemiring R
open SemiringOps semiring
open import Relation.Binary.Reasoning.Setoid setoid
private
-- The coefficient "ring".
ℕ-ring : RawRing _
ℕ-ring = record
{ Carrier = ℕ
; _+_ = ℕ._+_
; _*_ = ℕ._*_
; -_ = id
; 0# = 0
; 1# = 1
}
-- There is a homomorphism from ℕ to R.
--
-- Note that _×′_ is used rather than _×_. If _×_ were used, then
-- Function.Related.TypeIsomorphisms.test would fail to type-check.
homomorphism :
ℕ-ring -Raw-AlmostCommutative⟶ fromCommutativeSemiring R
homomorphism = record
{ ⟦_⟧ = λ n → n ×′ 1#
; +-homo = ×′-homo-+ 1#
; *-homo = ×′1-homo-*
; -‿homo = λ _ → refl
; 0-homo = refl
; 1-homo = refl
}
-- Equality of certain expressions can be decided.
dec′ : ∀ m n → Maybe (m ×′ 1# ≈ n ×′ 1#)
dec′ m n = map to (dec m n)
where
to : m × 1# ≈ n × 1# → m ×′ 1# ≈ n ×′ 1#
to m≈n = begin
m ×′ 1# ≈⟨ sym $ ×≈×′ m 1# ⟩
m × 1# ≈⟨ m≈n ⟩
n × 1# ≈⟨ ×≈×′ n 1# ⟩
n ×′ 1# ∎
-- The instantiation.
open Algebra.Solver.Ring _ _ homomorphism dec′ public
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module BBHeap.Complete.Alternative {A : Set}(_≤_ : A → A → Set) where
open import BBHeap _≤_
open import Bound.Lower A
open import BTree.Equality {A} renaming (_≃_ to _≃'_)
open import BTree.Complete.Alternative {A} renaming (_⋘_ to _⋘'_ ; _⋙_ to _⋙'_ ; _⋗_ to _⋗'_)
lemma-forget≃ : {b b' : Bound}{l : BBHeap b}{r : BBHeap b'} → l ≃ r → forget l ≃' forget r
lemma-forget≃ ≃lf = ≃lf
lemma-forget≃ (≃nd {x = x} {x' = x'} _ _ _ _ l≃r l'≃r' l≃l') = ≃nd x x' (lemma-forget≃ l≃r) (lemma-forget≃ l≃l') (lemma-forget≃ l'≃r')
lemma-forget⋗ : {b b' : Bound}{l : BBHeap b}{r : BBHeap b'} → l ⋗ r → forget l ⋗' forget r
lemma-forget⋗ (⋗lf {x = x} _) = ⋗lf x
lemma-forget⋗ (⋗nd {x = x} {x' = x'} _ _ _ _ l≃r l'≃r' l⋗l') = ⋗nd x x' (lemma-forget≃ l≃r) (lemma-forget≃ l'≃r') (lemma-forget⋗ l⋗l')
mutual
lemma-forget⋘ : {b b' : Bound}{l : BBHeap b}{r : BBHeap b'} → l ⋘ r → forget l ⋘' forget r
lemma-forget⋘ lf⋘ = lf⋘
lemma-forget⋘ (ll⋘ {x = x} {x' = x'} _ _ l⋘r l'⋘r' l'≃r' r≃l') = ll⋘ x x' (lemma-forget⋘ l⋘r) (lemma-forget≃ l'≃r') (lemma-forget≃ r≃l')
lemma-forget⋘ (lr⋘ {x = x} {x' = x'} _ _ l⋙r l'⋘r' l'≃r' l⋗l') = lr⋘ x x' (lemma-forget⋙ l⋙r) (lemma-forget≃ l'≃r') (lemma-forget⋗ l⋗l')
lemma-forget⋙ : {b b' : Bound}{l : BBHeap b}{r : BBHeap b'} → l ⋙ r → forget l ⋙' forget r
lemma-forget⋙ (⋙lf {x = x} _) = ⋙lf x
lemma-forget⋙ (⋙rl {x = x} {x' = x'} _ _ l⋘r l≃r l'⋘r' l⋗r') = ⋙rl x x' (lemma-forget≃ l≃r) (lemma-forget⋘ l'⋘r') (lemma-forget⋗ l⋗r')
lemma-forget⋙ (⋙rr {x = x} {x' = x'} _ _ l⋘r l≃r l'⋙r' l≃l') = ⋙rr x x' (lemma-forget≃ l≃r) (lemma-forget⋙ l'⋙r') (lemma-forget≃ l≃l')
lemma-bbheap-complete : {b : Bound}(h : BBHeap b) → Complete (forget h)
lemma-bbheap-complete leaf = leaf
lemma-bbheap-complete (left {x = x} {l = l} {r = r} _ l⋘r) = left x (lemma-bbheap-complete l) (lemma-bbheap-complete r) (lemma-forget⋘ l⋘r)
lemma-bbheap-complete (right {x = x} {l = l} {r = r} _ l⋙r) = right x (lemma-bbheap-complete l) (lemma-bbheap-complete r) (lemma-forget⋙ l⋙r)
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------------------------------------------------------------------------------
-- Existential quantifier on the inductive PA universe
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module PA.Inductive.Existential where
infix 2 ∃
------------------------------------------------------------------------------
-- PA universe
open import PA.Inductive.Base.Core
-- The existential quantifier type on M.
data ∃ (A : ℕ → Set) : Set where
_,_ : (x : ℕ) → A x → ∃ A
-- Sugar syntax for the existential quantifier.
syntax ∃ (λ x → e) = ∃[ x ] e
-- 2012-03-05: We avoid to use the existential elimination or the
-- existential projections because we use pattern matching (and the
-- Agda's with constructor).
-- The existential elimination.
--
-- NB. We do not use the usual type theory elimination with two
-- projections because we are working in first-order logic where we do
-- not need extract a witness from an existence proof.
-- ∃-elim : {A : ℕ → Set}{B : Set} → ∃ A → (∀ {x} → A x → B) → B
-- ∃-elim (_ , Ax) h = h Ax
-- The existential proyections.
-- ∃-proj₁ : ∀ {A} → ∃ A → M
-- ∃-proj₁ (x , _) = x
-- ∃-proj₂ : ∀ {A} → (h : ∃ A) → A (∃-proj₁ h)
-- ∃-proj₂ (_ , Ax) = Ax
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to setoid list membership
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.List.Membership.Setoid.Properties where
open import Algebra.FunctionProperties using (Op₂; Selective)
open import Data.Fin using (Fin; zero; suc)
open import Data.List
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
import Data.List.Relation.Unary.Any.Properties as Any
import Data.List.Membership.Setoid as Membership
import Data.List.Relation.Binary.Equality.Setoid as Equality
open import Data.Nat using (suc; z≤n; s≤s; _≤_; _<_)
open import Data.Nat.Properties using (≤-trans; n≤1+n)
open import Data.Product as Prod using (∃; _×_; _,_ ; ∃₂)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Function using (_$_; flip; _∘_; id)
open import Relation.Binary hiding (Decidable)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Unary using (Decidable; Pred)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Nullary.Negation using (contradiction)
open Setoid using (Carrier)
------------------------------------------------------------------------
-- Equality properties
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S
open Equality S
open Membership S
-- _∈_ respects the underlying equality
∈-resp-≈ : ∀ {xs} → (_∈ xs) Respects _≈_
∈-resp-≈ x≈y x∈xs = Any.map (trans (sym x≈y)) x∈xs
∉-resp-≈ : ∀ {xs} → (_∉ xs) Respects _≈_
∉-resp-≈ v≈w v∉xs w∈xs = v∉xs (∈-resp-≈ (sym v≈w) w∈xs)
∈-resp-≋ : ∀ {x} → (x ∈_) Respects _≋_
∈-resp-≋ = Any.lift-resp (flip trans)
∉-resp-≋ : ∀ {x} → (x ∉_) Respects _≋_
∉-resp-≋ xs≋ys v∉xs v∈ys = v∉xs (∈-resp-≋ (≋-sym xs≋ys) v∈ys)
------------------------------------------------------------------------
-- mapWith∈
module _ {c₁ c₂ ℓ₁ ℓ₂} (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where
open Setoid S₁ renaming (Carrier to A₁; _≈_ to _≈₁_; refl to refl₁)
open Setoid S₂ renaming (Carrier to A₂; _≈_ to _≈₂_; refl to refl₂)
open Equality S₁ using ([]; _∷_) renaming (_≋_ to _≋₁_)
open Equality S₂ using () renaming (_≋_ to _≋₂_)
open Membership S₁
mapWith∈-cong : ∀ {xs ys} → xs ≋₁ ys →
(f : ∀ {x} → x ∈ xs → A₂) →
(g : ∀ {y} → y ∈ ys → A₂) →
(∀ {x y} → x ≈₁ y → (x∈xs : x ∈ xs) (y∈ys : y ∈ ys) →
f x∈xs ≈₂ g y∈ys) →
mapWith∈ xs f ≋₂ mapWith∈ ys g
mapWith∈-cong [] f g cong = []
mapWith∈-cong (x≈y ∷ xs≋ys) f g cong =
cong x≈y (here refl₁) (here refl₁) ∷
mapWith∈-cong xs≋ys (f ∘ there) (g ∘ there)
(λ x≈y x∈xs y∈ys → cong x≈y (there x∈xs) (there y∈ys))
mapWith∈≗map : ∀ f xs → mapWith∈ xs (λ {x} _ → f x) ≋₂ map f xs
mapWith∈≗map f [] = []
mapWith∈≗map f (x ∷ xs) = refl₂ ∷ mapWith∈≗map f xs
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S
open Membership S
length-mapWith∈ : ∀ {a} {A : Set a} xs {f : ∀ {x} → x ∈ xs → A} →
length (mapWith∈ xs f) ≡ length xs
length-mapWith∈ [] = P.refl
length-mapWith∈ (x ∷ xs) = P.cong suc (length-mapWith∈ xs)
------------------------------------------------------------------------
-- map
module _ {c₁ c₂ ℓ₁ ℓ₂} (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where
open Setoid S₁ renaming (Carrier to A₁; _≈_ to _≈₁_; refl to refl₁)
open Setoid S₂ renaming (Carrier to A₂; _≈_ to _≈₂_)
private module M₁ = Membership S₁; open M₁ using (find) renaming (_∈_ to _∈₁_)
private module M₂ = Membership S₂; open M₂ using () renaming (_∈_ to _∈₂_)
∈-map⁺ : ∀ {f} → f Preserves _≈₁_ ⟶ _≈₂_ → ∀ {v xs} →
v ∈₁ xs → f v ∈₂ map f xs
∈-map⁺ pres x∈xs = Any.map⁺ (Any.map pres x∈xs)
∈-map⁻ : ∀ {v xs f} → v ∈₂ map f xs →
∃ λ x → x ∈₁ xs × v ≈₂ f x
∈-map⁻ x∈map = find (Any.map⁻ x∈map)
map-∷= : ∀ {f} (f≈ : f Preserves _≈₁_ ⟶ _≈₂_)
{xs x v} → (x∈xs : x ∈₁ xs) →
map f (x∈xs M₁.∷= v) ≡ ∈-map⁺ f≈ x∈xs M₂.∷= f v
map-∷= f≈ (here x≈y) = P.refl
map-∷= f≈ (there x∈xs) = P.cong (_ ∷_) (map-∷= f≈ x∈xs)
------------------------------------------------------------------------
-- _++_
module _ {c ℓ} (S : Setoid c ℓ) where
open Membership S using (_∈_)
open Setoid S
open Equality S using (_≋_; _∷_; ≋-refl)
∈-++⁺ˡ : ∀ {v xs ys} → v ∈ xs → v ∈ xs ++ ys
∈-++⁺ˡ = Any.++⁺ˡ
∈-++⁺ʳ : ∀ {v} xs {ys} → v ∈ ys → v ∈ xs ++ ys
∈-++⁺ʳ = Any.++⁺ʳ
∈-++⁻ : ∀ {v} xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)
∈-++⁻ = Any.++⁻
∈-insert : ∀ xs {ys v w} → v ≈ w → v ∈ xs ++ [ w ] ++ ys
∈-insert xs = Any.++-insert xs
∈-∃++ : ∀ {v xs} → v ∈ xs → ∃₂ λ ys zs → ∃ λ w →
v ≈ w × xs ≋ ys ++ [ w ] ++ zs
∈-∃++ (here px) = [] , _ , _ , px , ≋-refl
∈-∃++ (there {d} v∈xs) with ∈-∃++ v∈xs
... | hs , _ , _ , v≈v′ , eq = d ∷ hs , _ , _ , v≈v′ , refl ∷ eq
------------------------------------------------------------------------
-- concat
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S using (_≈_)
open Membership S using (_∈_)
open Equality S using (≋-setoid)
open Membership ≋-setoid using (find) renaming (_∈_ to _∈ₗ_)
∈-concat⁺ : ∀ {v xss} → Any (v ∈_) xss → v ∈ concat xss
∈-concat⁺ = Any.concat⁺
∈-concat⁻ : ∀ {v} xss → v ∈ concat xss → Any (v ∈_) xss
∈-concat⁻ = Any.concat⁻
∈-concat⁺′ : ∀ {v vs xss} → v ∈ vs → vs ∈ₗ xss → v ∈ concat xss
∈-concat⁺′ v∈vs = ∈-concat⁺ ∘ Any.map (flip (∈-resp-≋ S) v∈vs)
∈-concat⁻′ : ∀ {v} xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ₗ xss
∈-concat⁻′ xss v∈c[xss] with find (∈-concat⁻ xss v∈c[xss])
... | xs , t , s = xs , s , t
------------------------------------------------------------------------
-- applyUpTo
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S using (_≈_; refl)
open Membership S using (_∈_)
∈-applyUpTo⁺ : ∀ f {i n} → i < n → f i ∈ applyUpTo f n
∈-applyUpTo⁺ f = Any.applyUpTo⁺ f refl
∈-applyUpTo⁻ : ∀ {v} f {n} → v ∈ applyUpTo f n →
∃ λ i → i < n × v ≈ f i
∈-applyUpTo⁻ = Any.applyUpTo⁻
------------------------------------------------------------------------
-- tabulate
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S using (_≈_; refl) renaming (Carrier to A)
open Membership S using (_∈_)
∈-tabulate⁺ : ∀ {n} {f : Fin n → A} i → f i ∈ tabulate f
∈-tabulate⁺ i = Any.tabulate⁺ i refl
∈-tabulate⁻ : ∀ {n} {f : Fin n → A} {v} →
v ∈ tabulate f → ∃ λ i → v ≈ f i
∈-tabulate⁻ = Any.tabulate⁻
------------------------------------------------------------------------
-- filter
module _ {c ℓ p} (S : Setoid c ℓ) {P : Pred (Carrier S) p}
(P? : Decidable P) (resp : P Respects (Setoid._≈_ S)) where
open Setoid S using (_≈_; sym)
open Membership S using (_∈_)
∈-filter⁺ : ∀ {v xs} → v ∈ xs → P v → v ∈ filter P? xs
∈-filter⁺ {xs = x ∷ _} (here v≈x) Pv with P? x
... | yes _ = here v≈x
... | no ¬Px = contradiction (resp v≈x Pv) ¬Px
∈-filter⁺ {xs = x ∷ _} (there v∈xs) Pv with P? x
... | yes _ = there (∈-filter⁺ v∈xs Pv)
... | no _ = ∈-filter⁺ v∈xs Pv
∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v
∈-filter⁻ {xs = []} ()
∈-filter⁻ {xs = x ∷ xs} v∈f[x∷xs] with P? x
... | no _ = Prod.map there id (∈-filter⁻ v∈f[x∷xs])
... | yes Px with v∈f[x∷xs]
... | here v≈x = here v≈x , resp (sym v≈x) Px
... | there v∈fxs = Prod.map there id (∈-filter⁻ v∈fxs)
------------------------------------------------------------------------
-- length
module _ {c ℓ} (S : Setoid c ℓ) where
open Membership S using (_∈_)
∈-length : ∀ {x xs} → x ∈ xs → 1 ≤ length xs
∈-length (here px) = s≤s z≤n
∈-length (there x∈xs) = ≤-trans (∈-length x∈xs) (n≤1+n _)
------------------------------------------------------------------------
-- lookup
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S using (refl)
open Membership S using (_∈_)
∈-lookup : ∀ xs i → lookup xs i ∈ xs
∈-lookup [] ()
∈-lookup (x ∷ xs) zero = here refl
∈-lookup (x ∷ xs) (suc i) = there (∈-lookup xs i)
------------------------------------------------------------------------
-- foldr
module _ {c ℓ} (S : Setoid c ℓ) {_•_ : Op₂ (Carrier S)} where
open Setoid S using (_≈_; refl; sym; trans)
open Membership S using (_∈_)
foldr-selective : Selective _≈_ _•_ → ∀ e xs →
(foldr _•_ e xs ≈ e) ⊎ (foldr _•_ e xs ∈ xs)
foldr-selective •-sel i [] = inj₁ refl
foldr-selective •-sel i (x ∷ xs) with •-sel x (foldr _•_ i xs)
... | inj₁ x•f≈x = inj₂ (here x•f≈x)
... | inj₂ x•f≈f with foldr-selective •-sel i xs
... | inj₁ f≈i = inj₁ (trans x•f≈f f≈i)
... | inj₂ f∈xs = inj₂ (∈-resp-≈ S (sym x•f≈f) (there f∈xs))
------------------------------------------------------------------------
-- _∷=_
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S
open Membership S
∈-∷=⁺-updated : ∀ {xs x v} (x∈xs : x ∈ xs) → v ∈ (x∈xs ∷= v)
∈-∷=⁺-updated (here px) = here refl
∈-∷=⁺-updated (there pxs) = there (∈-∷=⁺-updated pxs)
∈-∷=⁺-untouched : ∀ {xs x y v} (x∈xs : x ∈ xs) → (¬ x ≈ y) → y ∈ xs → y ∈ (x∈xs ∷= v)
∈-∷=⁺-untouched (here x≈z) x≉y (here y≈z) = contradiction (trans x≈z (sym y≈z)) x≉y
∈-∷=⁺-untouched (here x≈z) x≉y (there y∈xs) = there y∈xs
∈-∷=⁺-untouched (there x∈xs) x≉y (here y≈z) = here y≈z
∈-∷=⁺-untouched (there x∈xs) x≉y (there y∈xs) = there (∈-∷=⁺-untouched x∈xs x≉y y∈xs)
∈-∷=⁻ : ∀ {xs x y v} (x∈xs : x ∈ xs) → (¬ y ≈ v) → y ∈ (x∈xs ∷= v) → y ∈ xs
∈-∷=⁻ (here x≈z) y≉v (here y≈v) = contradiction y≈v y≉v
∈-∷=⁻ (here x≈z) y≉v (there y∈) = there y∈
∈-∷=⁻ (there x∈xs) y≉v (here y≈z) = here y≈z
∈-∷=⁻ (there x∈xs) y≉v (there y∈) = there (∈-∷=⁻ x∈xs y≉v y∈)
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