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stlc_lessimpl10k.v
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Require Import Coq.Unicode.Utf8.
Definition Ty : Set
:= ∀ (Ty : Set)
(nat top bot : Ty)
(arr prod sum : Ty → Ty → Ty)
, Ty.
Definition nat : Ty := λ _ nat _ _ _ _ _ , nat.
Definition top : Ty := λ _ _ top _ _ _ _ , top.
Definition bot : Ty := λ _ _ _ bot _ _ _ , bot.
Definition arr : Ty → Ty → Ty
:= 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).
Definition prod : Ty → Ty → Ty
:= 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).
Definition sum : Ty → Ty → Ty
:= 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).
Definition Con : Set
:= ∀ (Con : Set)
(nil : Con)
(snoc : Con → Ty → Con)
, Con.
Definition nil : Con
:= λ Con nil snoc , nil.
Definition snoc : Con → Ty → Con
:= λ Γ A Con nil snoc , snoc (Γ Con nil snoc) A.
Definition Var : Con → Ty → Set
:= 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.
Definition vz {Γ A} : Var (snoc Γ A) A
:= λ Var vz vs , vz _ _.
Definition vs {Γ B A} : Var Γ A → Var (snoc Γ B) A
:= λ x Var vz vs , vs _ _ _ (x Var vz vs).
Definition Tm : Con → Ty → Set
:= 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.
Definition var {Γ A} : Var Γ A → Tm Γ A
:= λ x Tm var lam app tt pair fst snd left right case zero suc rec ,
var _ _ x.
Definition lam {Γ A B} : Tm (snoc Γ A) B → Tm Γ (arr A B)
:= 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).
Definition app {Γ A B} : Tm Γ (arr A B) → Tm Γ A → Tm Γ B
:= 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).
Definition tt {Γ} : Tm Γ top
:= λ Tm var lam app tt pair fst snd left right case zero suc rec , tt _.
Definition pair {Γ A B} : Tm Γ A → Tm Γ B → Tm Γ (prod A B)
:= 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).
Definition fst {Γ A B} : Tm Γ (prod A B) → Tm Γ A
:= 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).
Definition snd {Γ A B} : Tm Γ (prod A B) → Tm Γ B
:= 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).
Definition left {Γ A B} : Tm Γ A → Tm Γ (sum A B)
:= 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).
Definition right {Γ A B} : Tm Γ B → Tm Γ (sum A B)
:= 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).
Definition case {Γ A B C} : Tm Γ (sum A B) → Tm Γ (arr A C) → Tm Γ (arr B C) → Tm Γ C
:= 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).
Definition zero {Γ} : Tm Γ nat
:= λ Tm var lam app tt pair fst snd left right case zero suc rec , zero _.
Definition suc {Γ} : Tm Γ nat → Tm Γ nat
:= 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).
Definition rec {Γ A} : Tm Γ nat → Tm Γ (arr nat (arr A A)) → Tm Γ A → Tm Γ A
:= 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).
Definition v0 {Γ A} : Tm (snoc Γ A) A
:= var vz.
Definition v1 {Γ A B} : Tm (snoc (snoc Γ A) B) A
:= var (vs vz).
Definition v2 {Γ A B C} : Tm (snoc (snoc (snoc Γ A) B) C) A
:= var (vs (vs vz)).
Definition v3 {Γ A B C D} : Tm (snoc (snoc (snoc (snoc Γ A) B) C) D) A
:= var (vs (vs (vs vz))).
Definition tbool : Ty
:= sum top top.
Definition ttrue {Γ} : Tm Γ tbool
:= left tt.
Definition tfalse {Γ} : Tm Γ tbool
:= right tt.
Definition ifthenelse {Γ A} : Tm Γ (arr tbool (arr A (arr A A)))
:= lam (lam (lam (case v2 (lam v2) (lam v1)))).
Definition times4 {Γ A} : Tm Γ (arr (arr A A) (arr A A))
:= lam (lam (app v1 (app v1 (app v1 (app v1 v0))))).
Definition add {Γ} : Tm Γ (arr nat (arr nat nat))
:= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0)).
Definition mul {Γ} : Tm Γ (arr nat (arr nat nat))
:= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero)).
Definition fact {Γ} : Tm Γ (arr nat nat)
:= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero)).
Require Import Coq.Unicode.Utf8.
Definition Ty1 : Set
:= ∀ (Ty1 : Set)
(nat top bot : Ty1)
(arr prod sum : Ty1 → Ty1 → Ty1)
, Ty1.
Definition nat1 : Ty1 := λ _ nat1 _ _ _ _ _ , nat1.
Definition top1 : Ty1 := λ _ _ top1 _ _ _ _ , top1.
Definition bot1 : Ty1 := λ _ _ _ bot1 _ _ _ , bot1.
Definition arr1 : Ty1 → Ty1 → Ty1
:= 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).
Definition prod1 : Ty1 → Ty1 → Ty1
:= 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).
Definition sum1 : Ty1 → Ty1 → Ty1
:= 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).
Definition Con1 : Set
:= ∀ (Con1 : Set)
(nil : Con1)
(snoc : Con1 → Ty1 → Con1)
, Con1.
Definition nil1 : Con1
:= λ Con1 nil1 snoc , nil1.
Definition snoc1 : Con1 → Ty1 → Con1
:= λ Γ A Con1 nil1 snoc1 , snoc1 (Γ Con1 nil1 snoc1) A.
Definition Var1 : Con1 → Ty1 → Set
:= 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.
Definition vz1 {Γ A} : Var1 (snoc1 Γ A) A
:= λ Var1 vz1 vs , vz1 _ _.
Definition vs1 {Γ B A} : Var1 Γ A → Var1 (snoc1 Γ B) A
:= λ x Var1 vz1 vs1 , vs1 _ _ _ (x Var1 vz1 vs1).
Definition Tm1 : Con1 → Ty1 → Set
:= 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.
Definition var1 {Γ A} : Var1 Γ A → Tm1 Γ A
:= λ x Tm1 var1 lam app tt pair fst snd left right case zero suc rec ,
var1 _ _ x.
Definition lam1 {Γ A B} : Tm1 (snoc1 Γ A) B → Tm1 Γ (arr1 A B)
:= 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).
Definition app1 {Γ A B} : Tm1 Γ (arr1 A B) → Tm1 Γ A → Tm1 Γ B
:= 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).
Definition tt1 {Γ} : Tm1 Γ top1
:= λ Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec , tt1 _.
Definition pair1 {Γ A B} : Tm1 Γ A → Tm1 Γ B → Tm1 Γ (prod1 A B)
:= 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).
Definition fst1 {Γ A B} : Tm1 Γ (prod1 A B) → Tm1 Γ A
:= 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).
Definition snd1 {Γ A B} : Tm1 Γ (prod1 A B) → Tm1 Γ B
:= 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).
Definition left1 {Γ A B} : Tm1 Γ A → Tm1 Γ (sum1 A B)
:= 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).
Definition right1 {Γ A B} : Tm1 Γ B → Tm1 Γ (sum1 A B)
:= 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).
Definition case1 {Γ A B C} : Tm1 Γ (sum1 A B) → Tm1 Γ (arr1 A C) → Tm1 Γ (arr1 B C) → Tm1 Γ C
:= 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).
Definition zero1 {Γ} : Tm1 Γ nat1
:= λ Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec , zero1 _.
Definition suc1 {Γ} : Tm1 Γ nat1 → Tm1 Γ nat1
:= 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).
Definition rec1 {Γ A} : Tm1 Γ nat1 → Tm1 Γ (arr1 nat1 (arr1 A A)) → Tm1 Γ A → Tm1 Γ A
:= 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).
Definition v01 {Γ A} : Tm1 (snoc1 Γ A) A
:= var1 vz1.
Definition v11 {Γ A B} : Tm1 (snoc1 (snoc1 Γ A) B) A
:= var1 (vs1 vz1).
Definition v21 {Γ A B C} : Tm1 (snoc1 (snoc1 (snoc1 Γ A) B) C) A
:= var1 (vs1 (vs1 vz1)).
Definition v31 {Γ A B C D} : Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Γ A) B) C) D) A
:= var1 (vs1 (vs1 (vs1 vz1))).
Definition tbool1 : Ty1
:= sum1 top1 top1.
Definition ttrue1 {Γ} : Tm1 Γ tbool1
:= left1 tt1.
Definition tfalse1 {Γ} : Tm1 Γ tbool1
:= right1 tt1.
Definition ifthenelse1 {Γ A} : Tm1 Γ (arr1 tbool1 (arr1 A (arr1 A A)))
:= lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11)))).
Definition times41 {Γ A} : Tm1 Γ (arr1 (arr1 A A) (arr1 A A))
:= lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01))))).
Definition add1 {Γ} : Tm1 Γ (arr1 nat1 (arr1 nat1 nat1))
:= lam1 (rec1 v01
(lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))
(lam1 v01)).
Definition mul1 {Γ} : Tm1 Γ (arr1 nat1 (arr1 nat1 nat1))
:= lam1 (rec1 v01
(lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))
(lam1 zero1)).
Definition fact1 {Γ} : Tm1 Γ (arr1 nat1 nat1)
:= lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))
(suc1 zero1)).
Require Import Coq.Unicode.Utf8.
Definition Ty2 : Set
:= ∀ (Ty2 : Set)
(nat top bot : Ty2)
(arr prod sum : Ty2 → Ty2 → Ty2)
, Ty2.
Definition nat2 : Ty2 := λ _ nat2 _ _ _ _ _ , nat2.
Definition top2 : Ty2 := λ _ _ top2 _ _ _ _ , top2.
Definition bot2 : Ty2 := λ _ _ _ bot2 _ _ _ , bot2.
Definition arr2 : Ty2 → Ty2 → Ty2
:= 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).
Definition prod2 : Ty2 → Ty2 → Ty2
:= 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).
Definition sum2 : Ty2 → Ty2 → Ty2
:= 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).
Definition Con2 : Set
:= ∀ (Con2 : Set)
(nil : Con2)
(snoc : Con2 → Ty2 → Con2)
, Con2.
Definition nil2 : Con2
:= λ Con2 nil2 snoc , nil2.
Definition snoc2 : Con2 → Ty2 → Con2
:= λ Γ A Con2 nil2 snoc2 , snoc2 (Γ Con2 nil2 snoc2) A.
Definition Var2 : Con2 → Ty2 → Set
:= 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.
Definition vz2 {Γ A} : Var2 (snoc2 Γ A) A
:= λ Var2 vz2 vs , vz2 _ _.
Definition vs2 {Γ B A} : Var2 Γ A → Var2 (snoc2 Γ B) A
:= λ x Var2 vz2 vs2 , vs2 _ _ _ (x Var2 vz2 vs2).
Definition Tm2 : Con2 → Ty2 → Set
:= 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.
Definition var2 {Γ A} : Var2 Γ A → Tm2 Γ A
:= λ x Tm2 var2 lam app tt pair fst snd left right case zero suc rec ,
var2 _ _ x.
Definition lam2 {Γ A B} : Tm2 (snoc2 Γ A) B → Tm2 Γ (arr2 A B)
:= 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).
Definition app2 {Γ A B} : Tm2 Γ (arr2 A B) → Tm2 Γ A → Tm2 Γ B
:= 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).
Definition tt2 {Γ} : Tm2 Γ top2
:= λ Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec , tt2 _.
Definition pair2 {Γ A B} : Tm2 Γ A → Tm2 Γ B → Tm2 Γ (prod2 A B)
:= 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).
Definition fst2 {Γ A B} : Tm2 Γ (prod2 A B) → Tm2 Γ A
:= 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).
Definition snd2 {Γ A B} : Tm2 Γ (prod2 A B) → Tm2 Γ B
:= 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).
Definition left2 {Γ A B} : Tm2 Γ A → Tm2 Γ (sum2 A B)
:= 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).
Definition right2 {Γ A B} : Tm2 Γ B → Tm2 Γ (sum2 A B)
:= 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).
Definition case2 {Γ A B C} : Tm2 Γ (sum2 A B) → Tm2 Γ (arr2 A C) → Tm2 Γ (arr2 B C) → Tm2 Γ C
:= 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).
Definition zero2 {Γ} : Tm2 Γ nat2
:= λ Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec , zero2 _.
Definition suc2 {Γ} : Tm2 Γ nat2 → Tm2 Γ nat2
:= 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).
Definition rec2 {Γ A} : Tm2 Γ nat2 → Tm2 Γ (arr2 nat2 (arr2 A A)) → Tm2 Γ A → Tm2 Γ A
:= 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).
Definition v02 {Γ A} : Tm2 (snoc2 Γ A) A
:= var2 vz2.
Definition v12 {Γ A B} : Tm2 (snoc2 (snoc2 Γ A) B) A
:= var2 (vs2 vz2).
Definition v22 {Γ A B C} : Tm2 (snoc2 (snoc2 (snoc2 Γ A) B) C) A
:= var2 (vs2 (vs2 vz2)).
Definition v32 {Γ A B C D} : Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Γ A) B) C) D) A
:= var2 (vs2 (vs2 (vs2 vz2))).
Definition tbool2 : Ty2
:= sum2 top2 top2.
Definition ttrue2 {Γ} : Tm2 Γ tbool2
:= left2 tt2.
Definition tfalse2 {Γ} : Tm2 Γ tbool2
:= right2 tt2.
Definition ifthenelse2 {Γ A} : Tm2 Γ (arr2 tbool2 (arr2 A (arr2 A A)))
:= lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12)))).
Definition times42 {Γ A} : Tm2 Γ (arr2 (arr2 A A) (arr2 A A))
:= lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02))))).
Definition add2 {Γ} : Tm2 Γ (arr2 nat2 (arr2 nat2 nat2))
:= lam2 (rec2 v02
(lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))
(lam2 v02)).
Definition mul2 {Γ} : Tm2 Γ (arr2 nat2 (arr2 nat2 nat2))
:= lam2 (rec2 v02
(lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))
(lam2 zero2)).
Definition fact2 {Γ} : Tm2 Γ (arr2 nat2 nat2)
:= lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))
(suc2 zero2)).
Require Import Coq.Unicode.Utf8.
Definition Ty3 : Set
:= ∀ (Ty3 : Set)
(nat top bot : Ty3)
(arr prod sum : Ty3 → Ty3 → Ty3)
, Ty3.
Definition nat3 : Ty3 := λ _ nat3 _ _ _ _ _ , nat3.
Definition top3 : Ty3 := λ _ _ top3 _ _ _ _ , top3.
Definition bot3 : Ty3 := λ _ _ _ bot3 _ _ _ , bot3.
Definition arr3 : Ty3 → Ty3 → Ty3
:= 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).
Definition prod3 : Ty3 → Ty3 → Ty3
:= 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).
Definition sum3 : Ty3 → Ty3 → Ty3
:= 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).
Definition Con3 : Set
:= ∀ (Con3 : Set)
(nil : Con3)
(snoc : Con3 → Ty3 → Con3)
, Con3.
Definition nil3 : Con3
:= λ Con3 nil3 snoc , nil3.
Definition snoc3 : Con3 → Ty3 → Con3
:= λ Γ A Con3 nil3 snoc3 , snoc3 (Γ Con3 nil3 snoc3) A.
Definition Var3 : Con3 → Ty3 → Set
:= 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.
Definition vz3 {Γ A} : Var3 (snoc3 Γ A) A
:= λ Var3 vz3 vs , vz3 _ _.
Definition vs3 {Γ B A} : Var3 Γ A → Var3 (snoc3 Γ B) A
:= λ x Var3 vz3 vs3 , vs3 _ _ _ (x Var3 vz3 vs3).
Definition Tm3 : Con3 → Ty3 → Set
:= 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.
Definition var3 {Γ A} : Var3 Γ A → Tm3 Γ A
:= λ x Tm3 var3 lam app tt pair fst snd left right case zero suc rec ,
var3 _ _ x.
Definition lam3 {Γ A B} : Tm3 (snoc3 Γ A) B → Tm3 Γ (arr3 A B)
:= 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).
Definition app3 {Γ A B} : Tm3 Γ (arr3 A B) → Tm3 Γ A → Tm3 Γ B
:= 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).
Definition tt3 {Γ} : Tm3 Γ top3
:= λ Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec , tt3 _.
Definition pair3 {Γ A B} : Tm3 Γ A → Tm3 Γ B → Tm3 Γ (prod3 A B)
:= 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).
Definition fst3 {Γ A B} : Tm3 Γ (prod3 A B) → Tm3 Γ A
:= 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).
Definition snd3 {Γ A B} : Tm3 Γ (prod3 A B) → Tm3 Γ B
:= 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).
Definition left3 {Γ A B} : Tm3 Γ A → Tm3 Γ (sum3 A B)
:= 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).
Definition right3 {Γ A B} : Tm3 Γ B → Tm3 Γ (sum3 A B)
:= 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).
Definition case3 {Γ A B C} : Tm3 Γ (sum3 A B) → Tm3 Γ (arr3 A C) → Tm3 Γ (arr3 B C) → Tm3 Γ C
:= 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).
Definition zero3 {Γ} : Tm3 Γ nat3
:= λ Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec , zero3 _.
Definition suc3 {Γ} : Tm3 Γ nat3 → Tm3 Γ nat3
:= 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).
Definition rec3 {Γ A} : Tm3 Γ nat3 → Tm3 Γ (arr3 nat3 (arr3 A A)) → Tm3 Γ A → Tm3 Γ A
:= 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).
Definition v03 {Γ A} : Tm3 (snoc3 Γ A) A
:= var3 vz3.
Definition v13 {Γ A B} : Tm3 (snoc3 (snoc3 Γ A) B) A
:= var3 (vs3 vz3).
Definition v23 {Γ A B C} : Tm3 (snoc3 (snoc3 (snoc3 Γ A) B) C) A
:= var3 (vs3 (vs3 vz3)).
Definition v33 {Γ A B C D} : Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Γ A) B) C) D) A
:= var3 (vs3 (vs3 (vs3 vz3))).
Definition tbool3 : Ty3
:= sum3 top3 top3.
Definition ttrue3 {Γ} : Tm3 Γ tbool3
:= left3 tt3.
Definition tfalse3 {Γ} : Tm3 Γ tbool3
:= right3 tt3.
Definition ifthenelse3 {Γ A} : Tm3 Γ (arr3 tbool3 (arr3 A (arr3 A A)))
:= lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13)))).
Definition times43 {Γ A} : Tm3 Γ (arr3 (arr3 A A) (arr3 A A))
:= lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03))))).
Definition add3 {Γ} : Tm3 Γ (arr3 nat3 (arr3 nat3 nat3))
:= lam3 (rec3 v03
(lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))
(lam3 v03)).
Definition mul3 {Γ} : Tm3 Γ (arr3 nat3 (arr3 nat3 nat3))
:= lam3 (rec3 v03
(lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))
(lam3 zero3)).
Definition fact3 {Γ} : Tm3 Γ (arr3 nat3 nat3)
:= lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))
(suc3 zero3)).
Require Import Coq.Unicode.Utf8.
Definition Ty4 : Set
:= ∀ (Ty4 : Set)
(nat top bot : Ty4)
(arr prod sum : Ty4 → Ty4 → Ty4)
, Ty4.
Definition nat4 : Ty4 := λ _ nat4 _ _ _ _ _ , nat4.
Definition top4 : Ty4 := λ _ _ top4 _ _ _ _ , top4.
Definition bot4 : Ty4 := λ _ _ _ bot4 _ _ _ , bot4.
Definition arr4 : Ty4 → Ty4 → Ty4
:= 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).
Definition prod4 : Ty4 → Ty4 → Ty4
:= 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).
Definition sum4 : Ty4 → Ty4 → Ty4
:= 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).
Definition Con4 : Set
:= ∀ (Con4 : Set)
(nil : Con4)
(snoc : Con4 → Ty4 → Con4)
, Con4.
Definition nil4 : Con4
:= λ Con4 nil4 snoc , nil4.
Definition snoc4 : Con4 → Ty4 → Con4
:= λ Γ A Con4 nil4 snoc4 , snoc4 (Γ Con4 nil4 snoc4) A.
Definition Var4 : Con4 → Ty4 → Set
:= 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.
Definition vz4 {Γ A} : Var4 (snoc4 Γ A) A
:= λ Var4 vz4 vs , vz4 _ _.
Definition vs4 {Γ B A} : Var4 Γ A → Var4 (snoc4 Γ B) A
:= λ x Var4 vz4 vs4 , vs4 _ _ _ (x Var4 vz4 vs4).
Definition Tm4 : Con4 → Ty4 → Set
:= 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.
Definition var4 {Γ A} : Var4 Γ A → Tm4 Γ A
:= λ x Tm4 var4 lam app tt pair fst snd left right case zero suc rec ,
var4 _ _ x.
Definition lam4 {Γ A B} : Tm4 (snoc4 Γ A) B → Tm4 Γ (arr4 A B)
:= 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).
Definition app4 {Γ A B} : Tm4 Γ (arr4 A B) → Tm4 Γ A → Tm4 Γ B
:= 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).
Definition tt4 {Γ} : Tm4 Γ top4
:= λ Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec , tt4 _.
Definition pair4 {Γ A B} : Tm4 Γ A → Tm4 Γ B → Tm4 Γ (prod4 A B)
:= 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).
Definition fst4 {Γ A B} : Tm4 Γ (prod4 A B) → Tm4 Γ A
:= 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).
Definition snd4 {Γ A B} : Tm4 Γ (prod4 A B) → Tm4 Γ B
:= 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).
Definition left4 {Γ A B} : Tm4 Γ A → Tm4 Γ (sum4 A B)
:= 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).
Definition right4 {Γ A B} : Tm4 Γ B → Tm4 Γ (sum4 A B)
:= 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).
Definition case4 {Γ A B C} : Tm4 Γ (sum4 A B) → Tm4 Γ (arr4 A C) → Tm4 Γ (arr4 B C) → Tm4 Γ C
:= 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).
Definition zero4 {Γ} : Tm4 Γ nat4
:= λ Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec , zero4 _.
Definition suc4 {Γ} : Tm4 Γ nat4 → Tm4 Γ nat4
:= 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).
Definition rec4 {Γ A} : Tm4 Γ nat4 → Tm4 Γ (arr4 nat4 (arr4 A A)) → Tm4 Γ A → Tm4 Γ A
:= 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).
Definition v04 {Γ A} : Tm4 (snoc4 Γ A) A
:= var4 vz4.
Definition v14 {Γ A B} : Tm4 (snoc4 (snoc4 Γ A) B) A
:= var4 (vs4 vz4).
Definition v24 {Γ A B C} : Tm4 (snoc4 (snoc4 (snoc4 Γ A) B) C) A
:= var4 (vs4 (vs4 vz4)).
Definition v34 {Γ A B C D} : Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Γ A) B) C) D) A
:= var4 (vs4 (vs4 (vs4 vz4))).
Definition tbool4 : Ty4
:= sum4 top4 top4.
Definition ttrue4 {Γ} : Tm4 Γ tbool4
:= left4 tt4.
Definition tfalse4 {Γ} : Tm4 Γ tbool4
:= right4 tt4.
Definition ifthenelse4 {Γ A} : Tm4 Γ (arr4 tbool4 (arr4 A (arr4 A A)))
:= lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14)))).
Definition times44 {Γ A} : Tm4 Γ (arr4 (arr4 A A) (arr4 A A))
:= lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04))))).
Definition add4 {Γ} : Tm4 Γ (arr4 nat4 (arr4 nat4 nat4))
:= lam4 (rec4 v04
(lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))
(lam4 v04)).
Definition mul4 {Γ} : Tm4 Γ (arr4 nat4 (arr4 nat4 nat4))
:= lam4 (rec4 v04
(lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))
(lam4 zero4)).
Definition fact4 {Γ} : Tm4 Γ (arr4 nat4 nat4)
:= lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))
(suc4 zero4)).
Require Import Coq.Unicode.Utf8.
Definition Ty5 : Set
:= ∀ (Ty5 : Set)
(nat top bot : Ty5)
(arr prod sum : Ty5 → Ty5 → Ty5)
, Ty5.
Definition nat5 : Ty5 := λ _ nat5 _ _ _ _ _ , nat5.
Definition top5 : Ty5 := λ _ _ top5 _ _ _ _ , top5.
Definition bot5 : Ty5 := λ _ _ _ bot5 _ _ _ , bot5.
Definition arr5 : Ty5 → Ty5 → Ty5
:= 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).
Definition prod5 : Ty5 → Ty5 → Ty5
:= 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).
Definition sum5 : Ty5 → Ty5 → Ty5
:= 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).
Definition Con5 : Set
:= ∀ (Con5 : Set)
(nil : Con5)
(snoc : Con5 → Ty5 → Con5)
, Con5.
Definition nil5 : Con5
:= λ Con5 nil5 snoc , nil5.
Definition snoc5 : Con5 → Ty5 → Con5
:= λ Γ A Con5 nil5 snoc5 , snoc5 (Γ Con5 nil5 snoc5) A.
Definition Var5 : Con5 → Ty5 → Set
:= 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.
Definition vz5 {Γ A} : Var5 (snoc5 Γ A) A
:= λ Var5 vz5 vs , vz5 _ _.
Definition vs5 {Γ B A} : Var5 Γ A → Var5 (snoc5 Γ B) A
:= λ x Var5 vz5 vs5 , vs5 _ _ _ (x Var5 vz5 vs5).
Definition Tm5 : Con5 → Ty5 → Set
:= 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.
Definition var5 {Γ A} : Var5 Γ A → Tm5 Γ A
:= λ x Tm5 var5 lam app tt pair fst snd left right case zero suc rec ,
var5 _ _ x.
Definition lam5 {Γ A B} : Tm5 (snoc5 Γ A) B → Tm5 Γ (arr5 A B)
:= 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).
Definition app5 {Γ A B} : Tm5 Γ (arr5 A B) → Tm5 Γ A → Tm5 Γ B
:= 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).
Definition tt5 {Γ} : Tm5 Γ top5
:= λ Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec , tt5 _.
Definition pair5 {Γ A B} : Tm5 Γ A → Tm5 Γ B → Tm5 Γ (prod5 A B)
:= 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).
Definition fst5 {Γ A B} : Tm5 Γ (prod5 A B) → Tm5 Γ A
:= 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).
Definition snd5 {Γ A B} : Tm5 Γ (prod5 A B) → Tm5 Γ B
:= 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).
Definition left5 {Γ A B} : Tm5 Γ A → Tm5 Γ (sum5 A B)
:= 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).
Definition right5 {Γ A B} : Tm5 Γ B → Tm5 Γ (sum5 A B)
:= 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).
Definition case5 {Γ A B C} : Tm5 Γ (sum5 A B) → Tm5 Γ (arr5 A C) → Tm5 Γ (arr5 B C) → Tm5 Γ C
:= 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).
Definition zero5 {Γ} : Tm5 Γ nat5
:= λ Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec , zero5 _.
Definition suc5 {Γ} : Tm5 Γ nat5 → Tm5 Γ nat5
:= 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).
Definition rec5 {Γ A} : Tm5 Γ nat5 → Tm5 Γ (arr5 nat5 (arr5 A A)) → Tm5 Γ A → Tm5 Γ A
:= 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).
Definition v05 {Γ A} : Tm5 (snoc5 Γ A) A
:= var5 vz5.
Definition v15 {Γ A B} : Tm5 (snoc5 (snoc5 Γ A) B) A
:= var5 (vs5 vz5).
Definition v25 {Γ A B C} : Tm5 (snoc5 (snoc5 (snoc5 Γ A) B) C) A
:= var5 (vs5 (vs5 vz5)).