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title: "Path induction: a GADT perspective"
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slug: 2023-10-23-path-induction-gadt-perspective
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date: 2023-10-23
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tags: ["type-theory"]
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tags: ["type-theory", "programming-languages"]
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---
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<details>
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src/content/posts/2024-06-11-logical-relations.lagda.md
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src/content/posts/2024-06-11-logical-relations.lagda.md
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---
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title: "Logical Relations"
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slug: 2024-06-11-path-induction-gadt-perspective
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date: 2024-06-11
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tags: ["programming-languages", "formal-verification"]
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draft: true
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---
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<details>
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<summary>Imports</summary>
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```
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open import Agda.Builtin.Sigma
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open import Data.Bool
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open import Data.Empty
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open import Data.Fin
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open import Data.Maybe
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open import Data.Nat
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open import Data.Product
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open import Data.Sum
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open import Relation.Nullary
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id : {A : Set} → A → A
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id {A} x = x
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```
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</details>
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## Syntax
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```
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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 term : Set where
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`_ : ℕ → term
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`true : term
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`false : term
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`if_then_else_ : term → term → term → term
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`λ[_]_ : (τ : type) → (e : term) → term
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_∙_ : term → term → term
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```
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## Substitution
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```
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data ctx : Set where
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nil : ctx
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cons : ctx → type → ctx
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lookup : ctx → ℕ → Maybe type
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lookup nil _ = nothing
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lookup (cons ctx₁ x) zero = just x
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lookup (cons ctx₁ x) (suc n) = lookup ctx₁ n
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data sub : Set where
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nil : sub
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cons : sub → term → sub
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subst : term → term → term
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subst (` zero) v = v
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subst (` suc x) v = ` x
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subst `true v = `true
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subst `false v = `false
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subst (`if x then x₁ else x₂) v = `if (subst x v) then (subst x₁ v) else (subst x₂ v)
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subst (`λ[ τ ] x) v = `λ[ τ ] subst x v
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subst (x ∙ x₁) v = (subst x v) ∙ (subst x₁ v)
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data value-rel : type → term → Set where
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v-`true : value-rel bool `true
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v-`false : value-rel bool `false
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v-`λ[_]_ : ∀ {τ e} → value-rel τ (`λ[ τ ] e)
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data good-subst : ctx → sub → Set where
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nil : good-subst nil nil
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cons : ∀ {ctx τ γ v}
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→ good-subst ctx γ
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→ value-rel τ v
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→ good-subst (cons ctx τ) γ
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```
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## Semantics
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```
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data step : term → term → Set where
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step-if-1 : ∀ {e₁ e₂} → step (`if `true then e₁ else e₂) e₁
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step-if-2 : ∀ {e₁ e₂} → step (`if `false then e₁ else e₂) e₂
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step-`λ : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)
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data steps : ℕ → term → term → Set where
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zero : ∀ {e} → steps zero e e
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suc : ∀ {e e' e''} → (n : ℕ) → step e e' → steps n e' e'' → steps (suc n) e e''
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data _⊢_∶_ : ctx → term → type → Set where
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type-`true : ∀ {ctx} → ctx ⊢ `true ∶ bool
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type-`false : ∀ {ctx} → ctx ⊢ `false ∶ bool
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type-`ifthenelse : ∀ {ctx e e₁ e₂ τ}
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→ ctx ⊢ e ∶ bool
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→ ctx ⊢ e₁ ∶ τ
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→ ctx ⊢ e₂ ∶ τ
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→ ctx ⊢ (`if e then e₁ else e₂) ∶ τ
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type-`x : ∀ {ctx x}
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→ (p : Is-just (lookup ctx x))
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→ ctx ⊢ (` x) ∶ (to-witness p)
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type-`λ : ∀ {ctx τ τ₂ e}
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→ (cons ctx τ) ⊢ e ∶ τ₂
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→ ctx ⊢ (`λ[ τ ] e) ∶ (τ -→ τ₂)
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type-∙ : ∀ {ctx τ₁ τ₂ e₁ e₂}
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→ ctx ⊢ e₁ ∶ (τ₁ -→ τ₂)
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→ ctx ⊢ e₂ ∶ τ₂
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→ ctx ⊢ (e₁ ∙ e₂) ∶ τ₂
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irreducible : term → Set
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irreducible e = ¬ (∃ λ e' → step e e')
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data term-relation : type → term → Set where
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e-term : ∀ {τ e}
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→ (∀ {n} → (e' : term) → steps n e e' → irreducible e' → value-rel τ e')
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→ term-relation τ e
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type-sound : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Set
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type-sound {Γ} {e} {τ} s = ∀ {n} → (e' : term) → steps n e e' → value-rel τ e' ⊎ ∃ λ e'' → step e' e''
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_⊨_∶_ : (Γ : ctx) → (e : term) → (τ : type) → Set
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_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-relation τ e
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fundamental : ∀ {Γ e τ} → (well-typed : Γ ⊢ e ∶ τ) → type-sound well-typed → Γ ⊨ e ∶ τ
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fundamental {Γ} {e} {τ} well-typed type-sound γ good-sub = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n e e' → irreducible e' → value-rel τ e'
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f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (type-sound e' steps)
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```
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@ -31,7 +31,7 @@ const hasToc = toc ?? false;
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<body>
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<div class="flex-wrapper">
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<LeftNav />
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<div class="sep"></div>
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