{-# OPTIONS --without-K #-}
module Lambda.Syntax where
open import Equality.Propositional
open import Prelude hiding (Vec)
open import Maybe equality-with-J
infixl 9 _·_
data Tm (n : ℕ) : Set where
con : (i : ℕ) → Tm n
var : (x : Fin n) → Tm n
ƛ : Tm (suc n) → Tm n
_·_ : Tm n → Tm n → Tm n
Vec : ∀ {a} → Set a → ℕ → Set a
Vec A n = Fin n → A
empty : ∀ {a} {A : Set a} → Vec A 0
empty ()
snoc : ∀ {a} {A : Set a} {n} → Vec A n → A → Vec A (suc n)
snoc xs x = [ const x , xs ]
module Closure (Tm : ℕ → Set) where
mutual
Env : ℕ → Set
Env n = Vec Value n
data Value : Set where
con : (i : ℕ) → Value
ƛ : ∀ {n} (t : Tm (suc n)) (ρ : Env n) → Value
infixr 8 _⇾_
mutual
data Ty : Set where
nat : Ty
_⇾_ : (σ τ : ∞Ty) → Ty
record ∞Ty : Set where
coinductive
constructor [_]
field
force : Ty
open ∞Ty public
Ctxt : ℕ → Set
Ctxt n = Vec Ty n
infix 4 _⊢_∈_
data _⊢_∈_ {n} (Γ : Ctxt n) : Tm n → Ty → Set where
con : ∀ {i} → Γ ⊢ con i ∈ nat
var : ∀ {x} → Γ ⊢ var x ∈ Γ x
ƛ : ∀ {t σ τ} →
snoc Γ (force σ) ⊢ t ∈ force τ → Γ ⊢ ƛ t ∈ σ ⇾ τ
_·_ : ∀ {t₁ t₂ σ τ} →
Γ ⊢ t₁ ∈ σ ⇾ τ → Γ ⊢ t₂ ∈ force σ →
Γ ⊢ t₁ · t₂ ∈ force τ
open Closure Tm
mutual
data WF-Value : Ty → Value → Set where
con : ∀ {i} → WF-Value nat (con i)
ƛ : ∀ {n Γ σ τ} {t : Tm (1 + n)} {ρ} →
snoc Γ (force σ) ⊢ t ∈ force τ →
WF-Env Γ ρ →
WF-Value (σ ⇾ τ) (ƛ t ρ)
WF-Env : ∀ {n} → Ctxt n → Env n → Set
WF-Env Γ ρ = ∀ x → WF-Value (Γ x) (ρ x)
WF-MV : Ty → Maybe Value → Set
WF-MV σ v = maybe (WF-Value σ) Prelude.⊥ v
empty-wf : WF-Env empty empty
empty-wf ()
snoc-wf : ∀ {n} {Γ : Ctxt n} {ρ σ v} →
WF-Env Γ ρ → WF-Value σ v → WF-Env (snoc Γ σ) (snoc ρ v)
snoc-wf ρ-wf v-wf fzero = v-wf
snoc-wf ρ-wf v-wf (fsuc x) = ρ-wf x
ω : Tm 0
ω = ƛ (var fzero · var fzero)
Ω : Tm 0
Ω = ω · ω
Ω-well-typed : (τ : Ty) → empty ⊢ Ω ∈ τ
Ω-well-typed τ =
_·_ {σ = σ} {τ = [ τ ]} (ƛ (var · var)) (ƛ (var · var))
where
σ : ∞Ty
force σ = σ ⇾ [ τ ]
Z : Tm 0
Z = ƛ (t · t)
where
t = ƛ (var (fsuc fzero) ·
ƛ (var (fsuc fzero) · var (fsuc fzero) · var fzero))
Z-well-typed :
∀ {σ τ} →
empty ⊢ Z ∈ [ [ [ σ ] ⇾ [ τ ] ] ⇾ [ [ σ ] ⇾ [ τ ] ] ] ⇾
[ [ σ ] ⇾ [ τ ] ]
Z-well-typed {σ = σ} {τ = τ} =
ƛ (_·_ {σ = υ} {τ = [ _ ]}
(ƛ (var · ƛ (var · var · var)))
(ƛ (var · ƛ (var · var · var))))
where
υ : ∞Ty
force υ = υ ⇾ [ [ σ ] ⇾ [ τ ] ]