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stlc.agda
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stlc.agda
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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%))