97 lines
3.4 KiB
Agda
97 lines
3.4 KiB
Agda
module Combinatory
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-- "A correct-by-construction conversion from lambda calculus to combinatory logic", Wouter Swierstra
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data Unit : U where
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MkUnit : Unit
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infixr 7 _::_
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data List : U -> U where
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Nil : {A : U} -> List A
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_::_ : {A : U} -> A -> List A -> List A
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-- prj/menagerie/papers/combinatory
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infixr 6 _~>_
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data Type : U where
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ι : Type
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_~>_ : Type -> Type -> Type
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A : U
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A = Unit
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Val : Type -> U
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Val ι = A
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Val (x ~> y) = Val x -> Val y
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Ctx : U
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Ctx = List Type
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data Ref : Type -> Ctx -> U where
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Here : {σ : Type} {Γ : Ctx} -> Ref σ (σ :: Γ)
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There : {σ τ : Type} {Γ : Ctx} -> Ref σ Γ -> Ref σ (τ :: Γ)
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data Term : Ctx -> Type -> U where
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App : {Γ : Ctx} {σ τ : Type} -> Term Γ (σ ~> τ) -> Term Γ σ -> Term Γ τ
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Lam : {Γ : Ctx} {σ τ : Type} -> Term (σ :: Γ) τ -> Term Γ (σ ~> τ)
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Var : {Γ : Ctx} {σ : Type} -> Ref σ Γ → Term Γ σ
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infixr 7 _:::_
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data Env : Ctx -> U where
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ENil : Env Nil
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_:::_ : {Γ : Ctx} {σ : Type} → Val σ → Env Γ → Env (σ :: Γ)
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-- TODO there is a problem here with coverage checking
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-- I suspect something is being split before it's ready
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-- lookup : {σ : Type} {Γ : Ctx} → Ref σ Γ → Env Γ → Val σ
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-- lookup Here (x ::: y) = x
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-- lookup (There i) (x ::: env) = lookup i env
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lookup2 : {σ : Type} {Γ : Ctx} → Env Γ → Ref σ Γ → Val σ
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lookup2 (x ::: y) Here = x
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lookup2 (x ::: env) (There i) = lookup2 env i
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-- TODO MixFix - this was ⟦_⟧
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eval : {Γ : Ctx} {σ : Type} → Term Γ σ → (Env Γ → Val σ)
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-- there was a unification error in direct application
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eval (App t u) env = (eval t env) (eval u env)
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eval (Lam t) env = \ x => eval t (x ::: env)
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eval (Var i) env = lookup2 env i
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data Comb : (Γ : Ctx) → (u : Type) → (Env Γ → Val u) → U where
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S : {Γ : Ctx} {σ τ τ' : Type} → Comb Γ ((σ ~> τ ~> τ') ~> (σ ~> τ) ~> (σ ~> τ')) (\ env => \ f g x => (f x) (g x))
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K : {Γ : Ctx} {σ τ : Type} → Comb Γ (σ ~> (τ ~> σ)) (\ env => \ x y => x)
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I : {Γ : Ctx} {σ : Type} → Comb Γ (σ ~> σ) (\ env => \ x => x)
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B : {Γ : Ctx} {σ τ τ' : Type} → Comb Γ ((τ ~> τ') ~> (σ ~> τ) ~> (σ ~> τ')) (\ env => \ f g x => f (g x))
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C : {Γ : Ctx} {σ τ τ' : Type} → Comb Γ ((σ ~> τ ~> τ') ~> τ ~> (σ ~> τ')) (\ env => \ f g x => (f x) g)
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CVar : {Γ : Ctx} {σ : Type} → (i : Ref σ Γ) → Comb Γ σ (\ env => lookup2 env i)
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CApp : {Γ : Ctx} {σ τ : Type} {f : _} {x : _} → Comb Γ (σ ~> τ) f → Comb Γ σ x → Comb Γ τ (\ env => (f env) (x env))
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sapp : {Γ : Ctx} {σ τ ρ : Type} {f : _} {x : _} →
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Comb Γ (σ ~> τ ~> ρ) f →
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Comb Γ (σ ~> τ) x →
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Comb Γ (σ ~> ρ) (\ env y => (f env y) (x env y))
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sapp (CApp K t) I = t
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sapp (CApp K t) (CApp K u) = CApp K (CApp t u)
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-- was out of pattern because of unexpanded lets.
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sapp (CApp K t) u = CApp (CApp B t) u
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sapp t (CApp K u) = CApp (CApp C t) u
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sapp t u = CApp (CApp S t) u
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abs : {Γ : Ctx} {σ τ : Type} {f : _} → Comb (σ :: Γ) τ f → Comb Γ (σ ~> τ) (\ env x => f (x ::: env))
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abs S = CApp K S
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abs K = CApp K K
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abs I = CApp K I
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abs B = CApp K B
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abs C = CApp K C
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abs (CApp t u) = sapp (abs t) (abs u)
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-- lookup2 was getting stuck, needed to re-eval the types in the rewritten env.
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abs (CVar Here) = I
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abs (CVar (There i)) = CApp K (CVar i)
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translate : {Γ : Ctx} {σ : Type} → (tm : Term Γ σ) → Comb Γ σ (eval tm)
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translate (App t u) = CApp (translate t) (translate u)
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translate (Lam t) = abs (translate t)
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translate (Var i) = CVar i
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