78 lines
2.6 KiB
Agda
78 lines
2.6 KiB
Agda
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module Ahmed.LogicalRelations where
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open import Data.Empty
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open import Data.Unit
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open import Data.Bool
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open import Data.Nat
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open import Data.Fin
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open import Data.Product
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open import Relation.Nullary using (Dec; yes; no)
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data type : Set where
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bool : type
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_-→_ : type → type → type
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data ctx : Set where
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nil : ctx
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_,_ : ctx → type → ctx
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data var : ctx → type → Set where
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zero : ∀ {Γ τ} → var (Γ , τ) τ
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suc : ∀ {Γ τ₁ τ₂} → var Γ τ₂ → var (Γ , τ₁) τ₂
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data term : ctx → type → Set where
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true : ∀ {Γ} → term Γ bool
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false : ∀ {Γ} → term Γ bool
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`_ : ∀ {Γ τ} → var Γ τ → term Γ τ
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λ[_]_ : ∀ {Γ τ₂} → (τ₁ : type) → (e : term (Γ , τ₁) τ₂) → term Γ (τ₁ -→ τ₂)
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_∙_ : ∀ {Γ τ₁ τ₂} → term Γ (τ₁ -→ τ₂) → term Γ τ₁ → term Γ τ₂
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length : ctx → ℕ
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length nil = zero
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length (ctx , _) = suc (length ctx)
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extend : ∀ {Γ Δ}
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→ (∀ {τ} → var Γ τ → var Δ τ)
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→ (∀ {τ₁ τ₂} → var (Γ , τ₂) τ₁ → var (Δ , τ₂) τ₁)
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extend ρ zero = zero
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extend ρ (suc c) = suc (ρ c)
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rename : ∀ {Γ Δ}
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→ (∀ {τ} → var Γ τ → var Δ τ)
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→ (∀ {τ} → term Γ τ → term Δ τ)
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rename ρ true = true
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rename ρ false = false
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rename ρ (` x) = ` (ρ x)
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rename ρ (λ[ τ₁ ] c) = λ[ τ₁ ] rename (extend ρ) c
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rename ρ (c ∙ c₁) = rename ρ c ∙ rename ρ c₁
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extends : ∀ {Γ Δ}
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→ (∀ {τ} → var Γ τ → term Δ τ)
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→ (∀ {τ₁ τ₂} → var (Γ , τ₂) τ₁ → term (Δ , τ₂) τ₁)
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extends σ zero = ` zero
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extends σ (suc x) = rename suc (σ x)
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subst : ∀ {Γ Δ}
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→ (∀ {τ} → var Γ τ → term Δ τ)
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→ (∀ {τ} → term Γ τ → term Δ τ)
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subst ρ true = true
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subst ρ false = false
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subst ρ (` x) = ρ x
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subst ρ (λ[ τ₁ ] x) = λ[ τ₁ ] subst (extends ρ) x
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subst ρ (x ∙ x₁) = subst ρ x ∙ subst ρ x₁
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_[_] : ∀ {Γ τ₁ τ₂} → term (Γ , τ₂) τ₁ → term Γ τ₂ → term Γ τ₁
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_[_] {Γ} {τ₁} {τ₂} e e₁ = subst σ e
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where
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σ : ∀ {τ} → var (Γ , τ₂) τ → term Γ τ
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σ zero = e₁
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σ (suc x) = ` x
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data value : ∀ {Γ τ} → term Γ τ → Set where
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true : ∀ {Γ} → value {Γ} true
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false : ∀ {Γ} → value {Γ} false
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λ[_]_ : ∀ {Γ τ₁ τ₂} {e : term (Γ , τ₁) τ₂} → value (λ[ τ₁ ] e)
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data step : ∀ {Γ τ} → term Γ τ → term Γ τ → Set where
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β-λ : ∀ {Γ τ₁ τ₂ v} → (e : term (Γ , τ₁) τ₂) → value v
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→ step ((λ[ τ₁ ] e) ∙ v) (e [ v ])
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