setup for bisimulation
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2 changed files with 204 additions and 165 deletions
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@ -330,13 +330,13 @@ judgements:
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* `` ∅ , "s" ⦂ `ℕ ⇒ `ℕ , "z" ⦂ `ℕ ∋ "s" ⦂ `ℕ ⇒ `ℕ ``
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They correspond to the following inherently-typed variables.
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-- \begin{code}
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\begin{code}
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_ : ∅ , `ℕ ⇒ `ℕ , `ℕ ∋ `ℕ
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_ = Z
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_ : ∅ , `ℕ ⇒ `ℕ , `ℕ ∋ `ℕ ⇒ `ℕ
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_ = S Z
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-- \end{code}
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\end{code}
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In the given context, `"z"` is represented by `Z`, and `"s"` by `S Z`.
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### Terms and the typing judgement
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@ -18,7 +18,6 @@ description will be informal. We show how to formalise a few
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of the constructs, but leave the rest as an exercise for the
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reader.
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## Imports
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\begin{code}
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@ -37,17 +36,20 @@ open import Relation.Nullary using (¬_; Dec; yes; no)
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## Syntax
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\begin{code}
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infixl 6 _,_
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infix 4 _⊢_
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infix 4 _∋_
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infixr 5 _⇒_
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infixr 6 _`⊎_
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infixr 7 _`×_
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infixl 5 _,_
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infix 4 ƛ_
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infix 4 μ_
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infixl 5 _·_
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infix 6 S_
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infixr 7 _⇒_
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infixr 8 _`⊎_
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infixr 9 _`×_
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infix 5 ƛ_
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infix 5 μ_
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infixl 7 _·_
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infix 8 `suc_
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infix 9 `_
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infix 9 S_
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data Type : Set where
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`ℕ : Type
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@ -59,7 +61,7 @@ data Type : Set where
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`List : Type → Type
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data Env : Set where
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ε : Env
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∅ : Env
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_,_ : Env → Type → Env
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data _∋_ : Env → Type → Set where
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@ -75,7 +77,7 @@ data _∋_ : Env → Type → Set where
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data _⊢_ : Env → Type → Set where
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⌊_⌋ : ∀ {Γ} {A}
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`_ : ∀ {Γ} {A}
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→ Γ ∋ A
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------
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→ Γ ⊢ A
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@ -95,7 +97,7 @@ data _⊢_ : Env → Type → Set where
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----------
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→ Γ ⊢ `ℕ
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`suc : ∀ {Γ}
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`suc_ : ∀ {Γ}
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→ Γ ⊢ `ℕ
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-------
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→ Γ ⊢ `ℕ
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@ -190,69 +192,69 @@ Simultaneous substitution, a la McBride
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\begin{code}
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ext : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A) → (∀ {A B} → Γ , A ∋ B → Δ , A ∋ B)
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ext σ Z = Z
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ext σ (S x) = S (σ x)
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ext ρ Z = Z
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ext ρ (S x) = S (ρ x)
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rename : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A) → (∀ {A} → Γ ⊢ A → Δ ⊢ A)
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rename σ (⌊ n ⌋) = ⌊ σ n ⌋
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rename σ (ƛ N) = ƛ (rename (ext σ) N)
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rename σ (L · M) = (rename σ L) · (rename σ M)
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rename σ (`zero) = `zero
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rename σ (`suc M) = `suc (rename σ M)
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rename σ (`caseℕ L M N) = `caseℕ (rename σ L) (rename σ M) (rename (ext σ) N)
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rename σ (μ N) = μ (rename (ext σ) N)
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rename σ `⟨ M , N ⟩ = `⟨ rename σ M , rename σ N ⟩
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rename σ (`proj₁ L) = `proj₁ (rename σ L)
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rename σ (`proj₂ L) = `proj₂ (rename σ L)
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rename σ (`inj₁ M) = `inj₁ (rename σ M)
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rename σ (`inj₂ N) = `inj₂ (rename σ N)
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rename σ (`case⊎ L M N) = `case⊎ (rename σ L) (rename (ext σ) M) (rename (ext σ) N)
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rename σ `tt = `tt
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rename σ (`case⊥ L) = `case⊥ (rename σ L)
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rename σ `[] = `[]
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rename σ (M `∷ N) = (rename σ M) `∷ (rename σ N)
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rename σ (`caseL L M N) = `caseL (rename σ L) (rename σ M) (rename (ext (ext σ)) N)
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rename σ (`let M N) = `let (rename σ M) (rename (ext σ) N)
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rename ρ (` x) = ` (ρ x)
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rename ρ (ƛ N) = ƛ (rename (ext ρ) N)
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rename ρ (L · M) = (rename ρ L) · (rename ρ M)
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rename ρ (`zero) = `zero
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rename ρ (`suc M) = `suc (rename ρ M)
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rename ρ (`caseℕ L M N) = `caseℕ (rename ρ L) (rename ρ M) (rename (ext ρ) N)
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rename ρ (μ N) = μ (rename (ext ρ) N)
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rename ρ `⟨ M , N ⟩ = `⟨ rename ρ M , rename ρ N ⟩
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rename ρ (`proj₁ L) = `proj₁ (rename ρ L)
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rename ρ (`proj₂ L) = `proj₂ (rename ρ L)
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rename ρ (`inj₁ M) = `inj₁ (rename ρ M)
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rename ρ (`inj₂ N) = `inj₂ (rename ρ N)
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rename ρ (`case⊎ L M N) = `case⊎ (rename ρ L) (rename (ext ρ) M) (rename (ext ρ) N)
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rename ρ `tt = `tt
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rename ρ (`case⊥ L) = `case⊥ (rename ρ L)
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rename ρ `[] = `[]
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rename ρ (M `∷ N) = (rename ρ M) `∷ (rename ρ N)
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rename ρ (`caseL L M N) = `caseL (rename ρ L) (rename ρ M) (rename (ext (ext ρ)) N)
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rename ρ (`let M N) = `let (rename ρ M) (rename (ext ρ) N)
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\end{code}
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## Substitution
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\begin{code}
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exts : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ⊢ A) → (∀ {A B} → Γ , A ∋ B → Δ , A ⊢ B)
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exts ρ Z = ⌊ Z ⌋
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exts ρ (S x) = rename S_ (ρ x)
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exts σ Z = ` Z
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exts σ (S x) = rename S_ (σ x)
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subst : ∀ {Γ Δ} → (∀ {C} → Γ ∋ C → Δ ⊢ C) → (∀ {C} → Γ ⊢ C → Δ ⊢ C)
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subst ρ (⌊ k ⌋) = ρ k
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subst ρ (ƛ N) = ƛ (subst (exts ρ) N)
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subst ρ (L · M) = (subst ρ L) · (subst ρ M)
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subst ρ (`zero) = `zero
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subst ρ (`suc M) = `suc (subst ρ M)
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subst ρ (`caseℕ L M N) = `caseℕ (subst ρ L) (subst ρ M) (subst (exts ρ) N)
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subst ρ (μ N) = μ (subst (exts ρ) N)
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subst ρ `⟨ M , N ⟩ = `⟨ subst ρ M , subst ρ N ⟩
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subst ρ (`proj₁ L) = `proj₁ (subst ρ L)
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subst ρ (`proj₂ L) = `proj₂ (subst ρ L)
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subst ρ (`inj₁ M) = `inj₁ (subst ρ M)
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subst ρ (`inj₂ N) = `inj₂ (subst ρ N)
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subst ρ (`case⊎ L M N) = `case⊎ (subst ρ L) (subst (exts ρ) M) (subst (exts ρ) N)
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subst ρ `tt = `tt
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subst ρ (`case⊥ L) = `case⊥ (subst ρ L)
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subst ρ `[] = `[]
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subst ρ (M `∷ N) = (subst ρ M) `∷ (subst ρ N)
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subst ρ (`caseL L M N) = `caseL (subst ρ L) (subst ρ M) (subst (exts (exts ρ)) N)
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subst ρ (`let M N) = `let (subst ρ M) (subst (exts ρ) N)
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subst σ (` k) = σ k
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subst σ (ƛ N) = ƛ (subst (exts σ) N)
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subst σ (L · M) = (subst σ L) · (subst σ M)
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subst σ (`zero) = `zero
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subst σ (`suc M) = `suc (subst σ M)
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subst σ (`caseℕ L M N) = `caseℕ (subst σ L) (subst σ M) (subst (exts σ) N)
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subst σ (μ N) = μ (subst (exts σ) N)
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subst σ `⟨ M , N ⟩ = `⟨ subst σ M , subst σ N ⟩
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subst σ (`proj₁ L) = `proj₁ (subst σ L)
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subst σ (`proj₂ L) = `proj₂ (subst σ L)
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subst σ (`inj₁ M) = `inj₁ (subst σ M)
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subst σ (`inj₂ N) = `inj₂ (subst σ N)
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subst σ (`case⊎ L M N) = `case⊎ (subst σ L) (subst (exts σ) M) (subst (exts σ) N)
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subst σ `tt = `tt
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subst σ (`case⊥ L) = `case⊥ (subst σ L)
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subst σ `[] = `[]
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subst σ (M `∷ N) = (subst σ M) `∷ (subst σ N)
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subst σ (`caseL L M N) = `caseL (subst σ L) (subst σ M) (subst (exts (exts σ)) N)
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subst σ (`let M N) = `let (subst σ M) (subst (exts σ) N)
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_[_] : ∀ {Γ A B}
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→ Γ , A ⊢ B
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→ Γ ⊢ A
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------------
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→ Γ ⊢ B
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_[_] {Γ} {A} N V = subst {Γ , A} {Γ} ρ N
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_[_] {Γ} {A} N V = subst {Γ , A} {Γ} σ N
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where
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ρ : ∀ {B} → Γ , A ∋ B → Γ ⊢ B
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ρ Z = V
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ρ (S x) = ⌊ x ⌋
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σ : ∀ {B} → Γ , A ∋ B → Γ ⊢ B
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σ Z = V
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σ (S x) = ` x
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_[_][_] : ∀ {Γ A B C}
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→ Γ , A , B ⊢ C
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@ -260,12 +262,12 @@ _[_][_] : ∀ {Γ A B C}
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→ Γ ⊢ B
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---------------
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→ Γ ⊢ C
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_[_][_] {Γ} {A} {B} N V W = subst {Γ , A , B} {Γ} ρ N
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_[_][_] {Γ} {A} {B} N V W = subst {Γ , A , B} {Γ} σ N
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where
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ρ : ∀ {C} → Γ , A , B ∋ C → Γ ⊢ C
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ρ Z = W
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ρ (S Z) = V
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ρ (S (S x)) = ⌊ x ⌋
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σ : ∀ {C} → Γ , A , B ∋ C → Γ ⊢ C
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σ Z = W
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σ (S Z) = V
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σ (S (S x)) = ` x
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\end{code}
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## Value
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@ -323,172 +325,172 @@ not fixed by the given arguments.
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## Reduction step
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\begin{code}
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infix 2 _⟶_
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infix 2 _—→_
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data _⟶_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
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data _—→_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
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ξ-·₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A ⇒ B} {M : Γ ⊢ A}
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→ L ⟶ L′
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→ L —→ L′
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-----------------
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→ L · M ⟶ L′ · M
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→ L · M —→ L′ · M
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ξ-·₂ : ∀ {Γ A B} {V : Γ ⊢ A ⇒ B} {M M′ : Γ ⊢ A}
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→ Value V
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→ M ⟶ M′
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→ M —→ M′
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-----------------
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→ V · M ⟶ V · M′
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→ V · M —→ V · M′
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β-⇒ : ∀ {Γ A B} {N : Γ , A ⊢ B} {W : Γ ⊢ A}
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→ Value W
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---------------------
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→ (ƛ N) · W ⟶ N [ W ]
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→ (ƛ N) · W —→ N [ W ]
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ξ-suc : ∀ {Γ} {M M′ : Γ ⊢ `ℕ}
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→ M ⟶ M′
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→ M —→ M′
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-------------------
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→ `suc M ⟶ `suc M′
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→ `suc M —→ `suc M′
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ξ-caseℕ : ∀ {Γ A} {L L′ : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
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→ L ⟶ L′
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→ L —→ L′
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-------------------------------
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→ `caseℕ L M N ⟶ `caseℕ L′ M N
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→ `caseℕ L M N —→ `caseℕ L′ M N
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β-ℕ₁ : ∀ {Γ A} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
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-----------------------
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→ `caseℕ `zero M N ⟶ M
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→ `caseℕ `zero M N —→ M
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β-ℕ₂ : ∀ {Γ A} {V : Γ ⊢ `ℕ} {M : Γ ⊢ A} {N : Γ , `ℕ ⊢ A}
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→ Value V
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--------------------------------
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→ `caseℕ (`suc V) M N ⟶ N [ V ]
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→ `caseℕ (`suc V) M N —→ N [ V ]
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β-μ : ∀ {Γ A} {N : Γ , A ⊢ A}
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------------------
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→ μ N ⟶ N [ μ N ]
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→ μ N —→ N [ μ N ]
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ξ-⟨,⟩₁ : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ ⊢ B}
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→ M ⟶ M′
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→ M —→ M′
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--------------------------
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→ `⟨ M , N ⟩ ⟶ `⟨ M′ , N ⟩
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→ `⟨ M , N ⟩ —→ `⟨ M′ , N ⟩
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ξ-⟨,⟩₂ : ∀ {Γ A B} {V : Γ ⊢ A} {N N′ : Γ ⊢ B}
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→ Value V
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→ N ⟶ N′
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→ N —→ N′
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--------------------------
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→ `⟨ V , N ⟩ ⟶ `⟨ V , N′ ⟩
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→ `⟨ V , N ⟩ —→ `⟨ V , N′ ⟩
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ξ-proj₁ : ∀ {Γ A B} {L L′ : Γ ⊢ A `× B}
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→ L ⟶ L′
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→ L —→ L′
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-----------------------
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→ `proj₁ L ⟶ `proj₁ L′
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→ `proj₁ L —→ `proj₁ L′
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ξ-proj₂ : ∀ {Γ A B} {L L′ : Γ ⊢ A `× B}
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→ L ⟶ L′
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→ L —→ L′
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-----------------------
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→ `proj₂ L ⟶ `proj₂ L′
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→ `proj₂ L —→ `proj₂ L′
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β-×₁ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
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→ Value V
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→ Value W
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------------------------
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→ `proj₁ `⟨ V , W ⟩ ⟶ V
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→ `proj₁ `⟨ V , W ⟩ —→ V
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β-×₂ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ B}
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→ Value V
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→ Value W
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------------------------
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→ `proj₂ `⟨ V , W ⟩ ⟶ W
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→ `proj₂ `⟨ V , W ⟩ —→ W
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ξ-inj₁ : ∀ {Γ A B} {M M′ : Γ ⊢ A}
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→ M ⟶ M′
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→ M —→ M′
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-----------------------------
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→ `inj₁ {B = B} M ⟶ `inj₁ M′
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→ `inj₁ {B = B} M —→ `inj₁ M′
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ξ-inj₂ : ∀ {Γ A B} {N N′ : Γ ⊢ B}
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→ N ⟶ N′
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→ N —→ N′
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-----------------------------
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→ `inj₂ {A = A} N ⟶ `inj₂ N′
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→ `inj₂ {A = A} N —→ `inj₂ N′
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ξ-case⊎ : ∀ {Γ A B C} {L L′ : Γ ⊢ A `⊎ B} {M : Γ , A ⊢ C} {N : Γ , B ⊢ C}
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→ L ⟶ L′
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→ L —→ L′
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-------------------------------
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→ `case⊎ L M N ⟶ `case⊎ L′ M N
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→ `case⊎ L M N —→ `case⊎ L′ M N
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β-⊎₁ : ∀ {Γ A B C} {V : Γ ⊢ A} {M : Γ , A ⊢ C} {N : Γ , B ⊢ C}
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→ Value V
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---------------------------------
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→ `case⊎ (`inj₁ V) M N ⟶ M [ V ]
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→ `case⊎ (`inj₁ V) M N —→ M [ V ]
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β-⊎₂ : ∀ {Γ A B C} {W : Γ ⊢ B} {M : Γ , A ⊢ C} {N : Γ , B ⊢ C}
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→ Value W
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---------------------------------
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→ `case⊎ (`inj₂ W) M N ⟶ N [ W ]
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→ `case⊎ (`inj₂ W) M N —→ N [ W ]
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ξ-case⊥ : ∀ {Γ A} {L L′ : Γ ⊢ `⊥}
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→ L ⟶ L′
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→ L —→ L′
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-------------------------------
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→ `case⊥ {A = A} L ⟶ `case⊥ L′
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→ `case⊥ {A = A} L —→ `case⊥ L′
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ξ-∷₁ : ∀ {Γ A} {M M′ : Γ ⊢ A} {N : Γ ⊢ `List A}
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→ M ⟶ M′
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→ M —→ M′
|
||||
-------------------
|
||||
→ M `∷ N ⟶ M′ `∷ N
|
||||
→ M `∷ N —→ M′ `∷ N
|
||||
|
||||
ξ-∷₂ : ∀ {Γ A} {V : Γ ⊢ A} {N N′ : Γ ⊢ `List A}
|
||||
→ Value V
|
||||
→ N ⟶ N′
|
||||
→ N —→ N′
|
||||
-------------------
|
||||
→ V `∷ N ⟶ V `∷ N′
|
||||
→ V `∷ N —→ V `∷ N′
|
||||
|
||||
ξ-caseL : ∀ {Γ A B} {L L′ : Γ ⊢ `List A} {M : Γ ⊢ B} {N : Γ , A , `List A ⊢ B}
|
||||
→ L ⟶ L′
|
||||
→ L —→ L′
|
||||
-------------------------------
|
||||
→ `caseL L M N ⟶ `caseL L′ M N
|
||||
→ `caseL L M N —→ `caseL L′ M N
|
||||
|
||||
β-List₁ : ∀ {Γ A B} {M : Γ ⊢ B} {N : Γ , A , `List A ⊢ B}
|
||||
---------------------
|
||||
→ `caseL `[] M N ⟶ M
|
||||
→ `caseL `[] M N —→ M
|
||||
|
||||
β-List₂ : ∀ {Γ A B} {V : Γ ⊢ A} {W : Γ ⊢ `List A}
|
||||
{M : Γ ⊢ B} {N : Γ , A , `List A ⊢ B}
|
||||
→ Value V
|
||||
→ Value W
|
||||
-------------------------------------
|
||||
→ `caseL (V `∷ W) M N ⟶ N [ V ][ W ]
|
||||
→ `caseL (V `∷ W) M N —→ N [ V ][ W ]
|
||||
|
||||
ξ-let : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N : Γ , A ⊢ B}
|
||||
→ M ⟶ M′
|
||||
→ M —→ M′
|
||||
-----------------------
|
||||
→ `let M N ⟶ `let M′ N
|
||||
→ `let M N —→ `let M′ N
|
||||
|
||||
β-let : ∀ {Γ A B} {V : Γ ⊢ A} {N : Γ , A ⊢ B}
|
||||
→ Value V
|
||||
---------------------
|
||||
→ `let V N ⟶ N [ V ]
|
||||
→ `let V N —→ N [ V ]
|
||||
\end{code}
|
||||
|
||||
## Reflexive and transitive closure
|
||||
|
||||
\begin{code}
|
||||
infix 2 _⟶*_
|
||||
infix 2 _—↠_
|
||||
infix 1 begin_
|
||||
infixr 2 _⟶⟨_⟩_
|
||||
infixr 2 _—→⟨_⟩_
|
||||
infix 3 _∎
|
||||
|
||||
data _⟶*_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
|
||||
data _—↠_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
|
||||
|
||||
_∎ : ∀ {Γ A} (M : Γ ⊢ A)
|
||||
--------
|
||||
→ M ⟶* M
|
||||
→ M —↠ M
|
||||
|
||||
_⟶⟨_⟩_ : ∀ {Γ A} (L : Γ ⊢ A) {M N : Γ ⊢ A}
|
||||
→ L ⟶ M
|
||||
→ M ⟶* N
|
||||
_—→⟨_⟩_ : ∀ {Γ A} (L : Γ ⊢ A) {M N : Γ ⊢ A}
|
||||
→ L —→ M
|
||||
→ M —↠ N
|
||||
---------
|
||||
→ L ⟶* N
|
||||
→ L —↠ N
|
||||
|
||||
begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A} → (M ⟶* N) → (M ⟶* N)
|
||||
begin M⟶*N = M⟶*N
|
||||
begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A} → (M —↠ N) → (M —↠ N)
|
||||
begin M—↠N = M—↠N
|
||||
\end{code}
|
||||
|
||||
|
||||
|
|
@ -496,33 +498,33 @@ begin M⟶*N = M⟶*N
|
|||
|
||||
Values do not reduce.
|
||||
\begin{code}
|
||||
Value-lemma : ∀ {Γ A} {M N : Γ ⊢ A} → Value M → ¬ (M ⟶ N)
|
||||
Value-lemma : ∀ {Γ A} {M N : Γ ⊢ A} → Value M → ¬ (M —→ N)
|
||||
Value-lemma Fun ()
|
||||
Value-lemma Zero ()
|
||||
Value-lemma (Suc VM) (ξ-suc M⟶M′) = Value-lemma VM M⟶M′
|
||||
Value-lemma (Pair VM _) (ξ-⟨,⟩₁ M⟶M′) = Value-lemma VM M⟶M′
|
||||
Value-lemma (Pair _ VN) (ξ-⟨,⟩₂ _ N⟶N′) = Value-lemma VN N⟶N′
|
||||
Value-lemma (Inj₁ VM) (ξ-inj₁ M⟶M′) = Value-lemma VM M⟶M′
|
||||
Value-lemma (Inj₂ VN) (ξ-inj₂ N⟶N′) = Value-lemma VN N⟶N′
|
||||
Value-lemma (Suc VM) (ξ-suc M—→M′) = Value-lemma VM M—→M′
|
||||
Value-lemma (Pair VM _) (ξ-⟨,⟩₁ M—→M′) = Value-lemma VM M—→M′
|
||||
Value-lemma (Pair _ VN) (ξ-⟨,⟩₂ _ N—→N′) = Value-lemma VN N—→N′
|
||||
Value-lemma (Inj₁ VM) (ξ-inj₁ M—→M′) = Value-lemma VM M—→M′
|
||||
Value-lemma (Inj₂ VN) (ξ-inj₂ N—→N′) = Value-lemma VN N—→N′
|
||||
Value-lemma TT ()
|
||||
Value-lemma Nil ()
|
||||
Value-lemma (Cons VM _) (ξ-∷₁ M⟶M′) = Value-lemma VM M⟶M′
|
||||
Value-lemma (Cons _ VN) (ξ-∷₂ _ N⟶N′) = Value-lemma VN N⟶N′
|
||||
Value-lemma (Cons VM _) (ξ-∷₁ M—→M′) = Value-lemma VM M—→M′
|
||||
Value-lemma (Cons _ VN) (ξ-∷₂ _ N—→N′) = Value-lemma VN N—→N′
|
||||
\end{code}
|
||||
|
||||
As a corollary, terms that reduce are not values.
|
||||
\begin{code}
|
||||
⟶-corollary : ∀ {Γ A} {M N : Γ ⊢ A} → (M ⟶ N) → ¬ Value M
|
||||
⟶-corollary M⟶N VM = Value-lemma VM M⟶N
|
||||
—→-corollary : ∀ {Γ A} {M N : Γ ⊢ A} → (M —→ N) → ¬ Value M
|
||||
—→-corollary M—→N VM = Value-lemma VM M—→N
|
||||
\end{code}
|
||||
|
||||
|
||||
## Progress
|
||||
|
||||
\begin{code}
|
||||
data Progress {A} (M : ε ⊢ A) : Set where
|
||||
step : ∀ {N : ε ⊢ A}
|
||||
→ M ⟶ N
|
||||
data Progress {A} (M : ∅ ⊢ A) : Set where
|
||||
step : ∀ {N : ∅ ⊢ A}
|
||||
→ M —→ N
|
||||
-------------
|
||||
→ Progress M
|
||||
done :
|
||||
|
|
@ -530,60 +532,60 @@ data Progress {A} (M : ε ⊢ A) : Set where
|
|||
----------
|
||||
→ Progress M
|
||||
|
||||
progress : ∀ {A} → (M : ε ⊢ A) → Progress M
|
||||
progress ⌊ () ⌋
|
||||
progress : ∀ {A} → (M : ∅ ⊢ A) → Progress M
|
||||
progress (` ())
|
||||
progress (ƛ N) = done Fun
|
||||
progress (L · M) with progress L
|
||||
... | step L⟶L′ = step (ξ-·₁ L⟶L′)
|
||||
... | step L—→L′ = step (ξ-·₁ L—→L′)
|
||||
... | done Fun with progress M
|
||||
... | step M⟶M′ = step (ξ-·₂ Fun M⟶M′)
|
||||
... | step M—→M′ = step (ξ-·₂ Fun M—→M′)
|
||||
... | done VM = step (β-⇒ VM)
|
||||
progress (`zero) = done Zero
|
||||
progress (`suc M) with progress M
|
||||
... | step M⟶M′ = step (ξ-suc M⟶M′)
|
||||
... | step M—→M′ = step (ξ-suc M—→M′)
|
||||
... | done VM = done (Suc VM)
|
||||
progress (`caseℕ L M N) with progress L
|
||||
... | step L⟶L′ = step (ξ-caseℕ L⟶L′)
|
||||
... | step L—→L′ = step (ξ-caseℕ L—→L′)
|
||||
... | done Zero = step β-ℕ₁
|
||||
... | done (Suc VL) = step (β-ℕ₂ VL)
|
||||
progress (μ N) = step β-μ
|
||||
progress `⟨ M , N ⟩ with progress M
|
||||
... | step M⟶M′ = step (ξ-⟨,⟩₁ M⟶M′)
|
||||
... | step M—→M′ = step (ξ-⟨,⟩₁ M—→M′)
|
||||
... | done VM with progress N
|
||||
... | step N⟶N′ = step (ξ-⟨,⟩₂ VM N⟶N′)
|
||||
... | step N—→N′ = step (ξ-⟨,⟩₂ VM N—→N′)
|
||||
... | done VN = done (Pair VM VN)
|
||||
progress (`proj₁ L) with progress L
|
||||
... | step L⟶L′ = step (ξ-proj₁ L⟶L′)
|
||||
... | step L—→L′ = step (ξ-proj₁ L—→L′)
|
||||
... | done (Pair VM VN) = step (β-×₁ VM VN)
|
||||
progress (`proj₂ L) with progress L
|
||||
... | step L⟶L′ = step (ξ-proj₂ L⟶L′)
|
||||
... | step L—→L′ = step (ξ-proj₂ L—→L′)
|
||||
... | done (Pair VM VN) = step (β-×₂ VM VN)
|
||||
progress (`inj₁ M) with progress M
|
||||
... | step M⟶M′ = step (ξ-inj₁ M⟶M′)
|
||||
... | step M—→M′ = step (ξ-inj₁ M—→M′)
|
||||
... | done VM = done (Inj₁ VM)
|
||||
progress (`inj₂ N) with progress N
|
||||
... | step N⟶N′ = step (ξ-inj₂ N⟶N′)
|
||||
... | step N—→N′ = step (ξ-inj₂ N—→N′)
|
||||
... | done VN = done (Inj₂ VN)
|
||||
progress (`case⊎ L M N) with progress L
|
||||
... | step L⟶L′ = step (ξ-case⊎ L⟶L′)
|
||||
... | step L—→L′ = step (ξ-case⊎ L—→L′)
|
||||
... | done (Inj₁ VV) = step (β-⊎₁ VV)
|
||||
... | done (Inj₂ VW) = step (β-⊎₂ VW)
|
||||
progress (`tt) = done TT
|
||||
progress (`case⊥ {A = A} L) with progress L
|
||||
... | step L⟶L′ = step (ξ-case⊥ {A = A} L⟶L′)
|
||||
... | step L—→L′ = step (ξ-case⊥ {A = A} L—→L′)
|
||||
... | done ()
|
||||
progress (`[]) = done Nil
|
||||
progress (M `∷ N) with progress M
|
||||
... | step M⟶M′ = step (ξ-∷₁ M⟶M′)
|
||||
... | step M—→M′ = step (ξ-∷₁ M—→M′)
|
||||
... | done VM with progress N
|
||||
... | step N⟶N′ = step (ξ-∷₂ VM N⟶N′)
|
||||
... | step N—→N′ = step (ξ-∷₂ VM N—→N′)
|
||||
... | done VN = done (Cons VM VN)
|
||||
progress (`caseL L M N) with progress L
|
||||
... | step L⟶L′ = step (ξ-caseL L⟶L′)
|
||||
... | step L—→L′ = step (ξ-caseL L—→L′)
|
||||
... | done Nil = step β-List₁
|
||||
... | done (Cons VV VW) = step (β-List₂ VV VW)
|
||||
progress (`let M N) with progress M
|
||||
... | step M⟶M′ = step (ξ-let M⟶M′)
|
||||
... | step M—→M′ = step (ξ-let M—→M′)
|
||||
... | done VM = step (β-let VM)
|
||||
\end{code}
|
||||
|
||||
|
|
@ -591,27 +593,64 @@ progress (`let M N) with progress M
|
|||
## Normalise
|
||||
|
||||
\begin{code}
|
||||
Gas : Set
|
||||
Gas = ℕ
|
||||
data Gas : Set where
|
||||
gas : ℕ → Gas
|
||||
|
||||
data Normalise {A} (M : ε ⊢ A) : Set where
|
||||
normal : ∀ {N : ε ⊢ A}
|
||||
data Finished {Γ A} (N : Γ ⊢ A) : Set where
|
||||
|
||||
done :
|
||||
Value N
|
||||
----------
|
||||
→ Finished N
|
||||
|
||||
out-of-gas :
|
||||
----------
|
||||
Finished N
|
||||
|
||||
data Steps : ∀ {A} → ∅ ⊢ A → Set where
|
||||
|
||||
steps : ∀ {A} {L N : ∅ ⊢ A}
|
||||
→ L —↠ N
|
||||
→ Finished N
|
||||
----------
|
||||
→ Steps L
|
||||
|
||||
eval : ∀ {A}
|
||||
→ Gas
|
||||
→ M ⟶* N
|
||||
→ (L : ∅ ⊢ A)
|
||||
-----------
|
||||
→ Normalise M
|
||||
|
||||
normalise : ∀ {A} → ℕ → (L : ε ⊢ A) → Normalise L
|
||||
normalise zero L = normal zero (L ∎)
|
||||
normalise (suc g) L with progress L
|
||||
... | done VL = normal (suc zero) (L ∎)
|
||||
... | step {M} L⟶M with normalise g M
|
||||
... | normal h M⟶*N = normal (suc h) (L ⟶⟨ L⟶M ⟩ M⟶*N)
|
||||
→ Steps L
|
||||
eval (gas zero) L = steps (L ∎) out-of-gas
|
||||
eval (gas (suc m)) L with progress L
|
||||
... | done VL = steps (L ∎) (done VL)
|
||||
... | step {M} L—→M with eval (gas m) M
|
||||
... | steps M—↠N fin = steps (L —→⟨ L—→M ⟩ M—↠N) fin
|
||||
\end{code}
|
||||
|
||||
|
||||
## Bisimulation
|
||||
|
||||
\begin{code}
|
||||
data _~_ : ∀ {Γ A} → Γ ⊢ A → Γ ⊢ A → Set where
|
||||
|
||||
~` : ∀ {Γ A} {x : Γ ∋ A}
|
||||
-------------
|
||||
→ (` x) ~ (` x)
|
||||
|
||||
~ƛ : ∀ {Γ A B} {N N′ : Γ , A ⊢ B}
|
||||
→ N ~ N′
|
||||
--------------
|
||||
→ (ƛ N) ~ (ƛ N′)
|
||||
|
||||
~· : ∀ {Γ A B} {L L′ : Γ ⊢ A ⇒ B} {M M′ : Γ ⊢ A}
|
||||
→ L ~ L′
|
||||
→ M ~ M′
|
||||
-------------------
|
||||
→ (L · M) ~ (L′ · M′)
|
||||
|
||||
~let : ∀ {Γ A B} {M M′ : Γ ⊢ A} {N N′ : Γ , A ⊢ B}
|
||||
→ M ~ M′
|
||||
→ N ~ N′
|
||||
--------------------------
|
||||
→ (`let M N) ~ ((ƛ N′) · M′)
|
||||
\end{code}
|
||||
|
|
|
|||
Loading…
Reference in a new issue