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stlc_lessimpl.stt
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stlc_lessimpl.stt
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Ty : U
= (Ty : U)
(nat top bot : Ty)
(arr prod sum : Ty → Ty → Ty)
→ Ty
nat : Ty = λ _ nat _ _ _ _ _. nat
top : Ty = λ _ _ top _ _ _ _. top
bot : Ty = λ _ _ _ bot _ _ _. bot
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)
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)
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)
Con : U
= (Con : U)
(nil : Con)
(snoc : Con → Ty → Con)
→ Con
nil : Con
= λ Con nil snoc. nil
snoc : Con → Ty → Con
= λ Γ A Con nil snoc. snoc (Γ Con nil snoc) A
Var : Con → Ty → U
= λ Γ A.
(Var : Con → Ty → U)
(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
= λ Var vz vs. vz _ _
vs : {Γ B A} → Var Γ A → Var (snoc Γ B) A
= λ x Var vz vs. vs _ _ _ (x Var vz vs)
Tm : Con → Ty → U
= λ Γ A.
(Tm : Con → Ty → U)
(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
= 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)
= 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
= 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
= 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)
= 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
= 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
= 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)
= 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)
= 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
= 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
= λ Tm var lam app tt pair fst snd left right case zero suc rec. zero _
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)
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)
v0 : {Γ A} → Tm (snoc Γ A) A
= var vz
v1 : {Γ A B} → Tm (snoc (snoc Γ A) B) A
= var (vs vz)
v2 : {Γ A B C} → Tm (snoc (snoc (snoc Γ A) B) C) A
= var (vs (vs vz))
v3 : {Γ A B C D} → Tm (snoc (snoc (snoc (snoc Γ A) B) C) D) A
= var (vs (vs (vs vz)))
tbool : Ty
= sum top top
true : {Γ} → Tm Γ tbool
= left tt
tfalse : {Γ} → Tm Γ tbool
= right tt
ifthenelse : {Γ A} → Tm Γ (arr tbool (arr A (arr A A)))
= lam (lam (lam (case v2 (lam v2) (lam v1))))
times4 : {Γ A} → Tm Γ (arr (arr A A) (arr A A))
= lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))
add : {Γ} → Tm Γ (arr nat (arr nat nat))
= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0))
mul : {Γ} → Tm Γ (arr nat (arr nat nat))
= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero))
fact : {Γ} → Tm Γ (arr nat nat)
= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero))