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module STLC where | |
-- Simple substitution à la Conor, as described in http://strictlypositive.org/ren-sub.pdf | |
id : {A : Set} → A → A | |
id x = x | |
data Type : Set where | |
* : Type | |
_⇒_ : (S T : Type) → Type | |
data Context : Set where | |
ε : Context | |
_,_ : (Γ : Context) (S : Type) → Context | |
data _∋_ : Context → Type → Set where | |
here : ∀ {Γ S} → (Γ , S) ∋ S | |
there : ∀ {Γ S T} (i : Γ ∋ S) → (Γ , T) ∋ S | |
data Term : Context → Type → Set where | |
var : ∀ {Γ S} (v : Γ ∋ S) → Term Γ S | |
lam : ∀ {Γ S T} (t : Term (Γ , S) T) → Term Γ (S ⇒ T) | |
app : ∀ {Γ S T} (f : Term Γ (S ⇒ T)) (x : Term Γ S) → Term Γ T | |
record Kit (_◆_ : Context → Type → Set) : Set where | |
constructor kit | |
field | |
variable : ∀ {Γ T} → Γ ∋ T → Γ ◆ T | |
term : ∀ {Γ T} → Γ ◆ T → Term Γ T | |
weaken : ∀ {Γ S T} → Γ ◆ T → (Γ , S) ◆ T | |
open Kit | |
lift : ∀ {Γ Δ S T _◆_} → Kit _◆_ → (∀ {X} → Γ ∋ X → Δ ◆ X) → (Γ , S) ∋ T → (Δ , S) ◆ T | |
lift K τ here = variable K here | |
lift K τ (there v) = weaken K (τ v) | |
traverse : ∀ {Γ Δ T _◆_} → Kit _◆_ → (∀ {X} → Γ ∋ X → Δ ◆ X) → Term Γ T → Term Δ T | |
traverse K τ (var v) = term K (τ v) | |
traverse K τ (lam t) = lam (traverse K (lift K τ) t) | |
traverse K τ (app f x) = app (traverse K τ f) (traverse K τ x) | |
rename : ∀ {Γ Δ T} → (∀ {X} → Γ ∋ X → Δ ∋ X) → Term Γ T → Term Δ T | |
rename p t = traverse (kit id var there) p t | |
subst : ∀ {Γ Δ T} → (∀ {X} → Γ ∋ X → Term Δ X) → Term Γ T → Term Δ T | |
subst σ t = traverse (kit var id (rename there)) σ t | |
subst₁ : ∀ {Γ S T} → Term Γ S → Term (Γ , S) T → Term Γ T | |
subst₁ {Γ} {S} t = subst f | |
where | |
f : ∀ {X} → (Γ , S) ∋ X → Term Γ X | |
f here = t | |
f (there x) = var x | |
data Env : Context → Set where | |
ε : Env ε | |
_,_ : ∀ {Γ T} (xs : Env Γ) (x : Term Γ T) → Env (Γ , T) | |
-- World's slowest full substitution | |
subst′ : ∀ {Γ T} → Env Γ → Term Γ T → Term ε T | |
subst′ ε t = t | |
subst′ (xs , x) t = subst′ xs (subst₁ x t) |
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