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WIP
This commit is contained in:
ajeffrey@roblox.com
parent
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commit
1f4d77bac9
8 changed files with 354 additions and 180 deletions
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@ -40,6 +40,7 @@ postulate
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postulate lookup-insert : ∀ {A} k v (m : KeyMap A) → (lookup k (insert k v m) ≡ just v)
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postulate lookup-empty : ∀ {A} k → (lookup {A} k empty ≡ nothing)
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postulate singleton-insert-empty : ∀ {A} k (v : A) → (singleton k v ≡ insert k v empty)
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data Value : Set where
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object : KeyMap Value → Value
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@ -5,7 +5,7 @@ open import FFI.Data.Maybe using (Maybe; just)
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open import FFI.Data.Vector using (Vector; length; snoc; empty)
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open import Luau.Addr using (Addr)
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open import Luau.Var using (Var)
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open import Luau.Syntax using (Block; Expr; Annotated; FunDec; nil; addr; function_is_end)
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open import Luau.Syntax using (Block; Expr; Annotated; FunDec; nil; function_is_end)
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data HeapValue (a : Annotated) : Set where
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function_is_end : FunDec a → Block a → HeapValue a
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@ -6,6 +6,7 @@ open import Luau.Heap using (Heap; _≡_⊕_↦_; _[_]; function_is_end)
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open import Luau.Substitution using (_[_/_]ᴮ)
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open import Luau.Syntax using (Expr; Stat; Block; nil; addr; var; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name; fun; arg)
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open import Luau.Value using (addr; val)
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open import Luau.Type using (Type)
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data _⊢_⟶ᴮ_⊣_ {a} : Heap a → Block a → Block a → Heap a → Set
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data _⊢_⟶ᴱ_⊣_ {a} : Heap a → Expr a → Expr a → Heap a → Set
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@ -3,52 +3,79 @@ module Luau.StrictMode where
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open import Agda.Builtin.Equality using (_≡_)
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open import Luau.Syntax using (Expr; Stat; Block; yes; nil; addr; var; var_∈_; _⟨_⟩∈_; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name)
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open import Luau.Type using (Type; strict; bot; top; nil; _⇒_; tgt)
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open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
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open import Luau.Heap using (Heap) renaming (_[_] to _[_]ᴴ)
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open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_) renaming (_[_] to _[_]ⱽ)
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open import Luau.TypeCheck(strict) using (_▷_⊢ᴮ_∋_∈_⊣_; _▷_⊢ᴱ_∋_∈_⊣_; var; addr; app; block; return; local)
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open import Luau.TypeCheck(strict) using (_⊢ᴮ_∋_∈_⊣_; _⊢ᴱ_∋_∈_⊣_; var; addr; app; block; return; local; function)
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open import Properties.Equality using (_≢_)
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open import Properties.TypeCheck(strict) using (typeOfᴴ)
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src : Type → Type
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src = Luau.Type.src strict
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data Warningᴱ {Σ Γ S} : ∀ {M T Δ} → (Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → Set
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data Warningᴮ {Σ Γ S} : ∀ {B T Δ} → (Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → Set
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data Warningᴱ (H : Heap yes) {Γ S} : ∀ {M T Δ} → (Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → Set
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data Warningᴮ (H : Heap yes) {Γ S} : ∀ {B T Δ} → (Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → Set
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data Warningᴱ {Σ Γ S} where
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data Warningᴱ H {Γ S} where
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bot : ∀ {M T Δ} {D : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ} →
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bot : ∀ {M T Δ} {D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ} →
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(T ≡ bot) →
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------------
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Warningᴱ H D
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addr : ∀ a T →
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(T ≢ typeOfᴴ(H [ a ]ᴴ)) →
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-------------------------
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Warningᴱ H (addr a T)
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app₁ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
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Warningᴱ H D₁ →
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-----------------
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Warningᴱ H (app D₁ D₂)
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app₂ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
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Warningᴱ H D₂ →
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-----------------
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Warningᴱ H(app D₁ D₂)
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block : ∀ b {B T Δ} {D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ} →
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Warningᴮ H D →
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-----------------
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Warningᴱ H(block b D)
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data Warningᴮ H {Γ S} where
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disagree : ∀ {B T Δ} {D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ} →
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(S ≢ T) →
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-----------
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Warningᴱ(D)
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Warningᴮ H D
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app₁ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Σ ▷ Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Σ ▷ Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
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return : ∀ {M B T U Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁} {D₂ : Γ ⊢ᴮ nil ∋ B ∈ U ⊣ Δ₂} →
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Warningᴱ D₁ →
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-----------------
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Warningᴱ(app D₁ D₂)
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app₂ : ∀ {M N T U Δ₁ Δ₂} {D₁ : Σ ▷ Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁} {D₂ : Σ ▷ Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂} →
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Warningᴱ D₂ →
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-----------------
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Warningᴱ(app D₁ D₂)
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block : ∀ b {B T Δ} {D : Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ} →
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Warningᴮ D →
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-----------------
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Warningᴱ(block b D)
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data Warningᴮ {Σ Γ S} where
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return : ∀ {M B T U Δ₁ Δ₂} {D₁ : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁} {D₂ : Σ ▷ Γ ⊢ᴮ top ∋ B ∈ U ⊣ Δ₂} →
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Warningᴱ(D₁) →
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Warningᴱ H D₁ →
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------------------
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Warningᴮ(return D₁ D₂)
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Warningᴮ H (return D₁ D₂)
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local₁ : ∀ {x M B T U V Δ₁ Δ₂} {D₁ : Σ ▷ Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁} {D₂ : Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂} →
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local₁ : ∀ {x M B T U V Δ₁ Δ₂} {D₁ : Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁} {D₂ : (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂} →
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Warningᴱ(D₁) →
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Warningᴱ H D₁ →
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--------------------
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Warningᴮ(local D₁ D₂)
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Warningᴮ H (local D₁ D₂)
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-- data Warningᴴ {H} : ∀ {V T} → (H ▷ V ∈ T) → Set where
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-- nothing :
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-- -----------------
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-- Warningᴴ(nothing)
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-- function : ∀ f {x B T U V W} {D : (x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ (x ↦ W)} →
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-- Warningᴮ(D) →
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-- --------------------
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-- Warningᴴ(function f D)
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@ -10,7 +10,6 @@ open import Luau.Addr using (Addr)
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open import Luau.Heap using (Heap; HeapValue; function_is_end) renaming (_[_] to _[_]ᴴ)
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open import Luau.Value using (addr; val)
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open import Luau.Type using (Type; Mode; nil; bot; top; _⇒_; tgt)
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open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
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open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_) renaming (_[_] to _[_]ⱽ)
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open import FFI.Data.Vector using (Vector)
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open import FFI.Data.Maybe using (Maybe; just; nothing)
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@ -18,87 +17,70 @@ open import FFI.Data.Maybe using (Maybe; just; nothing)
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src : Type → Type
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src = Luau.Type.src m
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data _▷_⊢ᴮ_∋_∈_⊣_ : AddrCtxt → VarCtxt → Type → Block yes → Type → VarCtxt → Set
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data _▷_⊢ᴱ_∋_∈_⊣_ : AddrCtxt → VarCtxt → Type → Expr yes → Type → VarCtxt → Set
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data _⊢ᴮ_∋_∈_⊣_ : VarCtxt → Type → Block yes → Type → VarCtxt → Set
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data _⊢ᴱ_∋_∈_⊣_ : VarCtxt → Type → Expr yes → Type → VarCtxt → Set
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data _▷_⊢ᴮ_∋_∈_⊣_ where
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data _⊢ᴮ_∋_∈_⊣_ where
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done : ∀ {Σ S Γ} →
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done : ∀ {S Γ} →
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----------------------
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Σ ▷ Γ ⊢ᴮ S ∋ done ∈ nil ⊣ ∅
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Γ ⊢ᴮ S ∋ done ∈ nil ⊣ ∅
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return : ∀ {Σ M B S T U Γ Δ₁ Δ₂} →
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return : ∀ {M B S T U Γ Δ₁ Δ₂} →
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Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁ →
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Σ ▷ Γ ⊢ᴮ top ∋ B ∈ U ⊣ Δ₂ →
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Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ₁ →
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Γ ⊢ᴮ nil ∋ B ∈ U ⊣ Δ₂ →
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---------------------------------
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Σ ▷ Γ ⊢ᴮ S ∋ return M ∙ B ∈ T ⊣ Δ₁
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Γ ⊢ᴮ S ∋ return M ∙ B ∈ T ⊣ Δ₁
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local : ∀ {Σ x M B S T U V Γ Δ₁ Δ₂} →
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local : ∀ {x M B S T U V Γ Δ₁ Δ₂} →
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Σ ▷ Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁ →
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Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂ →
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Γ ⊢ᴱ T ∋ M ∈ U ⊣ Δ₁ →
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(Γ ⊕ x ↦ T) ⊢ᴮ S ∋ B ∈ V ⊣ Δ₂ →
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----------------------------------------------------------
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Σ ▷ Γ ⊢ᴮ S ∋ local var x ∈ T ← M ∙ B ∈ V ⊣ (Δ₁ ⋒ (Δ₂ ⊝ x))
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Γ ⊢ᴮ S ∋ local var x ∈ T ← M ∙ B ∈ V ⊣ (Δ₁ ⋒ (Δ₂ ⊝ x))
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function : ∀ {Σ f x B C S T U V W Γ Δ₁ Δ₂} →
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function : ∀ {f x B C S T U V W Γ Δ₁ Δ₂} →
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Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ C ∈ V ⊣ Δ₁ →
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Σ ▷ (Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ S ∋ B ∈ W ⊣ Δ₂ →
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(Γ ⊕ x ↦ T) ⊢ᴮ U ∋ C ∈ V ⊣ Δ₁ →
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(Γ ⊕ f ↦ (T ⇒ U)) ⊢ᴮ S ∋ B ∈ W ⊣ Δ₂ →
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---------------------------------------------------------------------------------
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Σ ▷ Γ ⊢ᴮ S ∋ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ W ⊣ ((Δ₁ ⊝ x) ⋒ (Δ₂ ⊝ f))
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Γ ⊢ᴮ S ∋ function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B ∈ W ⊣ ((Δ₁ ⊝ x) ⋒ (Δ₂ ⊝ f))
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data _▷_⊢ᴱ_∋_∈_⊣_ where
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data _⊢ᴱ_∋_∈_⊣_ where
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nil : ∀ {Σ S Γ} →
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nil : ∀ {S Γ} →
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----------------------
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Σ ▷ Γ ⊢ᴱ S ∋ nil ∈ nil ⊣ ∅
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Γ ⊢ᴱ S ∋ nil ∈ nil ⊣ ∅
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var : ∀ x {Σ S T Γ} →
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var : ∀ x {S T Γ} →
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T ≡ Γ [ x ]ⱽ →
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----------------------------
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Σ ▷ Γ ⊢ᴱ S ∋ var x ∈ T ⊣ (x ↦ S)
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Γ ⊢ᴱ S ∋ var x ∈ T ⊣ (x ↦ S)
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addr : ∀ a {Σ S T Γ} →
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addr : ∀ a T {S Γ} →
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T ≡ Σ [ a ]ᴬ →
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----------------------------
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Σ ▷ Γ ⊢ᴱ S ∋ addr a ∈ T ⊣ ∅
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-------------------------
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Γ ⊢ᴱ S ∋ (addr a) ∈ T ⊣ ∅
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app : ∀ {Σ M N S T U Γ Δ₁ Δ₂} →
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app : ∀ {M N S T U Γ Δ₁ Δ₂} →
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Σ ▷ Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁ →
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Σ ▷ Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂ →
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Γ ⊢ᴱ (U ⇒ S) ∋ M ∈ T ⊣ Δ₁ →
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Γ ⊢ᴱ (src T) ∋ N ∈ U ⊣ Δ₂ →
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--------------------------------------
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Σ ▷ Γ ⊢ᴱ S ∋ (M $ N) ∈ (tgt T) ⊣ (Δ₁ ⋒ Δ₂)
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Γ ⊢ᴱ S ∋ (M $ N) ∈ (tgt T) ⊣ (Δ₁ ⋒ Δ₂)
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function : ∀ {Σ f x B S T U V Γ Δ} →
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function : ∀ {f x B S T U V Γ Δ} →
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Σ ▷ (Γ ⊕ x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ Δ →
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(Γ ⊕ x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ Δ →
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-----------------------------------------------------------------------
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Σ ▷ Γ ⊢ᴱ S ∋ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
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Γ ⊢ᴱ S ∋ (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U) ⊣ (Δ ⊝ x)
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block : ∀ b {Σ B S T Γ Δ} →
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block : ∀ b {B S T Γ Δ} →
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Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ →
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Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ →
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----------------------------------------------------
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Σ ▷ Γ ⊢ᴱ S ∋ (block b is B end) ∈ T ⊣ Δ
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data _▷_∈_ (Σ : AddrCtxt) : Maybe (HeapValue yes) → Type → Set where
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nothing :
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-----------------
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Σ ▷ nothing ∈ bot
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function : ∀ {f x B T U V W} →
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Σ ▷ (x ↦ T) ⊢ᴮ U ∋ B ∈ V ⊣ (x ↦ W) →
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---------------------------------------------------------
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Σ ▷ just (function f ⟨ var x ∈ T ⟩∈ U is B end) ∈ (T ⇒ U)
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_▷_✓ : AddrCtxt → Heap yes → Set
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(Σ ▷ H ✓) = (∀ a → Σ ▷ (H [ a ]ᴴ) ∈ (Σ [ a ]ᴬ))
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Γ ⊢ᴱ S ∋ (block b is B end) ∈ T ⊣ Δ
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@ -12,6 +12,12 @@ trans refl refl = refl
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cong : ∀ {A B : Set} {a b : A} (f : A → B) → (a ≡ b) → (f a ≡ f b)
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cong f refl = refl
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subst₁ : ∀ {A : Set} {a b : A} (F : A → Set) → (a ≡ b) → (F a) → (F b)
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subst₁ F refl x = x
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subst₂ : ∀ {A B : Set} {a b : A} {c d : B} (F : A → B → Set) → (a ≡ b) → (c ≡ d) → (F a c) → (F b d)
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subst₂ F refl refl x = x
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_≢_ : ∀ {A : Set} → A → A → Set
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(a ≢ b) = ¬(a ≡ b)
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@ -5,73 +5,225 @@ module Properties.StrictMode where
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import Agda.Builtin.Equality.Rewrite
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open import Agda.Builtin.Equality using (_≡_; refl)
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open import FFI.Data.Maybe using (Maybe; just; nothing)
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open import Luau.Heap using (Heap; HeapValue; function_is_end; defn; alloc; ok; next; lookup-next) renaming (_[_] to _[_]ᴴ)
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open import Luau.StrictMode using (Warningᴱ; Warningᴮ; bot; app₁; app₂; block; return; local₁)
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open import Luau.Syntax using (Expr; yes; var_∈_; _⟨_⟩∈_; _$_; addr; nil; block_is_end; done; return; local_←_; _∙_; fun; arg)
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open import Luau.Heap using (Heap; HeapValue; function_is_end; defn; alloc; ok; next; lookup-next) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ)
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open import Luau.StrictMode using (Warningᴱ; Warningᴮ; bot; disagree; addr; app₁; app₂; block; return; local₁)
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open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ)
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open import Luau.Syntax using (Expr; yes; var_∈_; _⟨_⟩∈_; _$_; addr; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg)
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open import Luau.Type using (Type; strict; nil; _⇒_; bot; tgt)
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open import Luau.TypeCheck(strict) using (_▷_⊢ᴮ_∋_∈_⊣_; _▷_⊢ᴱ_∋_∈_⊣_; _▷_∈_; _▷_✓; nil; var; addr; app; function; block; done; return; local; nothing)
|
||||
open import Luau.TypeCheck(strict) using (_⊢ᴮ_∋_∈_⊣_; _⊢ᴱ_∋_∈_⊣_; nil; var; addr; app; function; block; done; return; local)
|
||||
open import Luau.Value using (val; nil; addr)
|
||||
open import Luau.Addr using (_≡ᴬ_)
|
||||
open import Luau.AddrCtxt using (AddrCtxt)
|
||||
open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_; ∅-[]) renaming (_[_] to _[_]ⱽ)
|
||||
open import Luau.VarCtxt using (VarCtxt; ∅)
|
||||
open import Properties.Remember using (remember; _,_)
|
||||
open import Properties.Equality using (cong)
|
||||
open import Properties.TypeCheck(strict) using (typeOfᴱ; typeCheckᴱ)
|
||||
open import Properties.Equality using (sym; cong; trans; subst₁)
|
||||
open import Properties.Dec using (Dec; yes; no)
|
||||
open import Properties.Contradiction using (CONTRADICTION)
|
||||
open import Properties.TypeCheck(strict) using (typeOfᴴ; typeOfᴱ; typeOfᴮ; typeCheckᴱ; typeCheckᴮ)
|
||||
open import Luau.OpSem using (_⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app; function; beta; return; block; done; local; subst)
|
||||
|
||||
{-# REWRITE ∅-[] #-}
|
||||
{-# REWRITE lookup-next #-}
|
||||
|
||||
heap-miss : ∀ {Σ HV T} → (Σ ▷ HV ∈ T) → (HV ≡ nothing) → (T ≡ bot)
|
||||
heap-miss nothing refl = refl
|
||||
src = Luau.Type.src strict
|
||||
|
||||
data ProgressResultᴱ {Σ Γ S M T Δ} (H : Heap yes) (D : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) : Set
|
||||
data ProgressResultᴮ {Σ Γ S B T Δ} (H : Heap yes) (D : Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) : Set
|
||||
data _⊑_ (H : Heap yes) : Heap yes → Set where
|
||||
refl : (H ⊑ H)
|
||||
snoc : ∀ {H′ H″ a V} → (H ⊑ H′) → (H″ ≡ᴴ H′ ⊕ a ↦ V) → (H ⊑ H″)
|
||||
|
||||
data ProgressResultᴱ {Σ Γ S M T Δ} H D where
|
||||
warning-⊑ : ∀ {H H′ Γ Δ S T M} {D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ} → (H ⊑ H′) → (Warningᴱ H′ D) → Warningᴱ H D
|
||||
warning-⊑ = {!!}
|
||||
|
||||
value : ∀ V → (M ≡ val V) → ProgressResultᴱ H D
|
||||
warning : (Warningᴱ D) → ProgressResultᴱ H D
|
||||
step : ∀ {M′ H′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → ProgressResultᴱ H D
|
||||
data TypeOfᴱ-⊑-Result H H′ Γ M : Set where
|
||||
ok : (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M) → TypeOfᴱ-⊑-Result H H′ Γ M
|
||||
warning : (∀ {S} → Warningᴱ H (typeCheckᴱ H Γ S M)) → TypeOfᴱ-⊑-Result H H′ Γ M
|
||||
|
||||
data ProgressResultᴮ {Σ Γ S B T Δ} H D where
|
||||
data TypeOfᴮ-⊑-Result H H′ Γ B : Set where
|
||||
ok : (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B) → TypeOfᴮ-⊑-Result H H′ Γ B
|
||||
warning : (∀ {S} → Warningᴮ H (typeCheckᴮ H Γ S B)) → TypeOfᴮ-⊑-Result H H′ Γ B
|
||||
|
||||
done : (B ≡ done) → ProgressResultᴮ H D
|
||||
return : ∀ V {C} → (B ≡ (return (val V) ∙ C)) → ProgressResultᴮ H D
|
||||
warning : (Warningᴮ D) → ProgressResultᴮ H D
|
||||
step : ∀ {B′ H′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → ProgressResultᴮ H D
|
||||
typeOfᴱ-⊑ : ∀ {H H′ Γ M} → (H ⊑ H′) → (TypeOfᴱ-⊑-Result H H′ Γ M)
|
||||
typeOfᴱ-⊑ = {!!}
|
||||
|
||||
progressᴱ : ∀ {Σ Γ S M T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴱ H D
|
||||
progressᴮ : ∀ {Σ Γ S B T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴮ H D
|
||||
typeOfᴮ-⊑ : ∀ {H H′ Γ B} → (H ⊑ H′) → (TypeOfᴮ-⊑-Result H H′ Γ B)
|
||||
typeOfᴮ-⊑ = {!!}
|
||||
|
||||
progressᴱ H h nil _ = value nil refl
|
||||
progressᴱ H h (var x p) refl = warning (bot p)
|
||||
progressᴱ H h (addr a refl) _ = value (addr a) refl
|
||||
progressᴱ H h (app D₁ D₂) p with progressᴱ H h D₁ p
|
||||
progressᴱ H h (app nil D₂) p | value nil refl = warning (bot refl)
|
||||
progressᴱ H h (app (var _ _) D₂) p | value nil ()
|
||||
progressᴱ H h (app (app _ _) D₂) p | value nil ()
|
||||
progressᴱ H h (app (function _) D₂) p | value nil ()
|
||||
progressᴱ H h (app (block _ _) D₂) p | value nil ()
|
||||
progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl with remember(H [ a ]ᴴ)
|
||||
progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (nothing , r) = warning (bot (cong tgt (heap-miss (h a) r)))
|
||||
progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (just(function f ⟨ var x ∈ S ⟩∈ T is B end) , r) = step (beta r)
|
||||
progressᴱ H h (app D₁ D₂) p | warning W = warning (app₁ W)
|
||||
progressᴱ H h (app D₁ D₂) p | step S = step (app S)
|
||||
progressᴱ H h (function D) _ with alloc H _
|
||||
progressᴱ H h (function D) _ | ok a H′ r = step (function r)
|
||||
progressᴱ H h (block b D) q with progressᴮ H h D q
|
||||
progressᴱ H h (block b D) q | done refl = step done
|
||||
progressᴱ H h (block b D) q | return V refl = step (return refl)
|
||||
progressᴱ H h (block b D) q | warning W = warning (block b W)
|
||||
progressᴱ H h (block b D) q | step S = step (block S)
|
||||
blah : ∀ {H H′ Γ S S′ M} → (H ⊑ H′) → (S ≡ S′) → (Warningᴱ H′ (typeCheckᴱ H′ Γ S′ M)) → (Warningᴱ H (typeCheckᴱ H Γ S M))
|
||||
blah = {!!}
|
||||
|
||||
progressᴮ H h done q = done refl
|
||||
progressᴮ H h (return D₁ D₂) q with progressᴱ H h D₁ q
|
||||
progressᴮ H h (return D₁ D₂) q | value V refl = return V refl
|
||||
progressᴮ H h (return D₁ D₂) q | warning W = warning (return W)
|
||||
progressᴮ H h (return D₁ D₂) q | step S = step (return S)
|
||||
progressᴮ H h (local D₁ D₂) q with progressᴱ H h D₁ q
|
||||
progressᴮ H h (local D₁ D₂) q | value V refl = step subst
|
||||
progressᴮ H h (local D₁ D₂) q | warning W = warning (local₁ W)
|
||||
progressᴮ H h (local D₁ D₂) q | step S = step (local S)
|
||||
progressᴮ H h (function D₁ D₂) q with alloc H _
|
||||
progressᴮ H h (function D₁ D₂) q | ok a H′ r = step (function r)
|
||||
bloz : ∀ {H Γ S S′ M} → (S ≡ S′) → (Warningᴱ H (typeCheckᴱ H Γ S′ M)) → (Warningᴱ H (typeCheckᴱ H Γ S M))
|
||||
bloz = {!!}
|
||||
|
||||
redn-⊑ : ∀ {H H′ M M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (H ⊑ H′)
|
||||
redn-⊑ = {!!}
|
||||
|
||||
substitutivityᴱ : ∀ {Γ T H M v x} → (T ≡ typeOfᴱ H Γ (val v)) → (typeOfᴱ H (Γ ⊕ x ↦ T) M ≡ typeOfᴱ H Γ (M [ v / x ]ᴱ))
|
||||
substitutivityᴮ : ∀ {Γ T H B v x} → (T ≡ typeOfᴱ H Γ (val v)) → (typeOfᴮ H (Γ ⊕ x ↦ T) B ≡ typeOfᴮ H Γ (B [ v / x ]ᴮ))
|
||||
|
||||
substitutivityᴱ = {!!}
|
||||
substitutivityᴮ = {!!}
|
||||
|
||||
preservationᴱ : ∀ {H H′ M M′ Γ} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (typeOfᴱ H Γ M ≡ typeOfᴱ H′ Γ M′)
|
||||
preservationᴮ : ∀ {H H′ B B′ Γ} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (typeOfᴮ H Γ B ≡ typeOfᴮ H′ Γ B′)
|
||||
|
||||
preservationᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) = refl
|
||||
preservationᴱ (app s) = cong tgt (preservationᴱ s)
|
||||
preservationᴱ (beta {F = f ⟨ var x ∈ S ⟩∈ T} p) = trans (cong tgt (cong typeOfᴴ p)) {!!}
|
||||
preservationᴱ (block s) = preservationᴮ s
|
||||
preservationᴱ (return p) = refl
|
||||
preservationᴱ done = refl
|
||||
|
||||
preservationᴮ (local {x = var x ∈ T} {B = B} s) with typeOfᴮ-⊑ {B = B} (redn-⊑ s)
|
||||
preservationᴮ (local {x = var x ∈ T} s) | ok p = p
|
||||
preservationᴮ (local {x = var x ∈ T} s) | warning W = {!!}
|
||||
preservationᴮ (subst {x = var x ∈ T} {B = B}) = substitutivityᴮ {B = B} {!!}
|
||||
preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) with typeOfᴮ-⊑ {B = B} (snoc refl defn)
|
||||
preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | ok r = trans r (substitutivityᴮ {T = S ⇒ T} {B = B} refl)
|
||||
preservationᴮ (function {F = f ⟨ var x ∈ S ⟩∈ T} {B = B} defn) | warning W = {!!}
|
||||
preservationᴮ (return s) = preservationᴱ s
|
||||
|
||||
reflectᴱ : ∀ {H H′ M M′ S} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ S M′) → Warningᴱ H (typeCheckᴱ H ∅ S M)
|
||||
reflectᴮ : ∀ {H H′ B B′ S} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ S B′) → Warningᴮ H (typeCheckᴮ H ∅ S B)
|
||||
|
||||
reflectᴱ s W with redn-⊑ s
|
||||
reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (addr a _ r) | p = CONTRADICTION (r refl)
|
||||
reflectᴱ (app s) (bot x) | p = {!x!}
|
||||
reflectᴱ (app s) (app₁ W) | p with typeOfᴱ-⊑ p
|
||||
reflectᴱ (app s) (app₁ W) | p | ok q = app₁ (bloz (cong (λ ∙ → ∙ ⇒ _) q) (reflectᴱ s W))
|
||||
reflectᴱ (app s) (app₁ W) | p | warning W′ = app₂ W′
|
||||
reflectᴱ (app s) (app₂ W) | p = app₂ (blah p (cong src (preservationᴱ s)) W)
|
||||
reflectᴱ (beta s) (bot x₁) | p = {!!}
|
||||
reflectᴱ (beta {F = f ⟨ var x ∈ T ⟩∈ U} q) (block _ (disagree x₁)) | p = {!!}
|
||||
reflectᴱ (beta {F = f ⟨ var x ∈ T ⟩∈ U} q) (block _ (local₁ W)) | p = app₂ (bloz (cong src (cong typeOfᴴ q)) W)
|
||||
reflectᴱ (block s) (bot x₁) | p = {!!}
|
||||
reflectᴱ (block s) (block b W) | p = block b (reflectᴮ s W)
|
||||
reflectᴱ (return q) W | p = block _ (return W)
|
||||
reflectᴱ done (bot x) | p = {!!}
|
||||
|
||||
reflectᴮ s = {!!}
|
||||
|
||||
-- reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (bot ())
|
||||
-- reflectᴱ (function defn) (addr a T q) = CONTRADICTION (q refl)
|
||||
-- reflectᴱ (app s) (bot x) = {!x!}
|
||||
-- reflectᴱ (app s) (app₁ W) = app₁ {!reflectᴱ s W!}
|
||||
-- reflectᴱ (app s) (app₂ W) = {!!}
|
||||
-- reflectᴱ (beta x) W = {!!}
|
||||
-- reflectᴱ (block x) W = {!!}
|
||||
-- reflectᴱ (return x) W = {!!}
|
||||
-- reflectᴱ done W = {!!}
|
||||
|
||||
-- heap-miss : ∀ {Σ HV T} → (Σ ▷ HV ∈ T) → (HV ≡ nothing) → (T ≡ bot)
|
||||
-- heap-miss nothing refl = refl
|
||||
|
||||
-- data ProgressResultᴱ {Σ Γ S M T Δ} (H : Heap yes) (D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) : Set
|
||||
-- data ProgressResultᴮ {Σ Γ S B T Δ} (H : Heap yes) (D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) : Set
|
||||
|
||||
-- data ProgressResultᴱ {Σ Γ S M T Δ} H D where
|
||||
|
||||
-- value : ∀ V → (M ≡ val V) → ProgressResultᴱ H D
|
||||
-- warning : (Warningᴱ Σ D) → ProgressResultᴱ H D
|
||||
-- step : ∀ {M′ H′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → ProgressResultᴱ H D
|
||||
|
||||
-- data ProgressResultᴮ {Σ Γ S B T Δ} H D where
|
||||
|
||||
-- done : (B ≡ done) → ProgressResultᴮ H D
|
||||
-- return : ∀ V {C} → (B ≡ (return (val V) ∙ C)) → ProgressResultᴮ H D
|
||||
-- warning : (Warningᴮ Σ D) → ProgressResultᴮ H D
|
||||
-- step : ∀ {B′ H′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → ProgressResultᴮ H D
|
||||
|
||||
-- progressᴱ : ∀ {Σ Γ S M T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴱ H D
|
||||
-- progressᴮ : ∀ {Σ Γ S B T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴮ H D
|
||||
|
||||
-- progressᴱ H h nil _ = value nil refl
|
||||
-- progressᴱ H h (var x p) refl = warning (bot p)
|
||||
-- progressᴱ H h (addr a refl) _ = value (addr a) refl
|
||||
-- progressᴱ H h (app D₁ D₂) p with progressᴱ H h D₁ p
|
||||
-- progressᴱ H h (app nil D₂) p | value nil refl = warning (bot refl)
|
||||
-- progressᴱ H h (app (var _ _) D₂) p | value nil ()
|
||||
-- progressᴱ H h (app (app _ _) D₂) p | value nil ()
|
||||
-- progressᴱ H h (app (function _) D₂) p | value nil ()
|
||||
-- progressᴱ H h (app (block _ _) D₂) p | value nil ()
|
||||
-- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl with remember(H [ a ]ᴴ)
|
||||
-- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (nothing , r) = warning (bot (cong tgt (heap-miss (h a) r)))
|
||||
-- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (just(function f ⟨ var x ∈ S ⟩∈ T is B end) , r) = step (beta r)
|
||||
-- progressᴱ H h (app D₁ D₂) p | warning W = warning (app₁ W)
|
||||
-- progressᴱ H h (app D₁ D₂) p | step S = step (app S)
|
||||
-- progressᴱ H h (function D) _ with alloc H _
|
||||
-- progressᴱ H h (function D) _ | ok a H′ r = step (function r)
|
||||
-- progressᴱ H h (block b D) q with progressᴮ H h D q
|
||||
-- progressᴱ H h (block b D) q | done refl = step done
|
||||
-- progressᴱ H h (block b D) q | return V refl = step (return refl)
|
||||
-- progressᴱ H h (block b D) q | warning W = warning (block b W)
|
||||
-- progressᴱ H h (block b D) q | step S = step (block S)
|
||||
|
||||
-- progressᴮ H h done q = done refl
|
||||
-- progressᴮ H h (return D₁ D₂) q with progressᴱ H h D₁ q
|
||||
-- progressᴮ H h (return D₁ D₂) q | value V refl = return V refl
|
||||
-- progressᴮ H h (return D₁ D₂) q | warning W = warning (return W)
|
||||
-- progressᴮ H h (return D₁ D₂) q | step S = step (return S)
|
||||
-- progressᴮ H h (local D₁ D₂) q with progressᴱ H h D₁ q
|
||||
-- progressᴮ H h (local D₁ D₂) q | value V refl = step subst
|
||||
-- progressᴮ H h (local D₁ D₂) q | warning W = warning (local₁ W)
|
||||
-- progressᴮ H h (local D₁ D₂) q | step S = step (local S)
|
||||
-- progressᴮ H h (function D₁ D₂) q with alloc H _
|
||||
-- progressᴮ H h (function D₁ D₂) q | ok a H′ r = step (function r)
|
||||
|
||||
import FFI.Data.Aeson
|
||||
{-# REWRITE FFI.Data.Aeson.singleton-insert-empty #-}
|
||||
|
||||
_≡ᵀ_ : (T U : Type) → Dec (T ≡ U)
|
||||
_≡ᵀ_ = {!!}
|
||||
|
||||
-- data LookupResult {Σ V S} (D : Σ ▷ V ∈ S) : Set where
|
||||
|
||||
-- function : ∀ f {x B T U W} →
|
||||
-- (S ≡ (T ⇒ U)) →
|
||||
-- (V ≡ just(function f ⟨ var x ∈ T ⟩∈ U is B end)) →
|
||||
-- (Σ ▷ (x ↦ T) ⊢ᴮ U ∋ B ∈ U ⊣ (x ↦ W)) →
|
||||
-- LookupResult D
|
||||
|
||||
-- warningᴴ :
|
||||
-- Warningᴴ(D) →
|
||||
-- LookupResult D
|
||||
|
||||
-- lookup : ∀ {Σ V T} (D : Σ ▷ V ∈ T) → LookupResult D
|
||||
-- lookup nothing = warningᴴ nothing
|
||||
-- lookup (function f {U = U} {V = V} D) with U ≡ᵀ V
|
||||
-- lookup (function f D) | yes refl = function f refl refl D
|
||||
-- lookup (function f D) | no p = warningᴴ (function f (disagree p))
|
||||
|
||||
-- data PreservationResultᴮ {Σ S Δ T H B} (D : ∅ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) M′ H′ : Set where
|
||||
|
||||
-- ok : ∀ {Δ′} → (∅ ⊢ᴮ S ∋ M′ ∈ T ⊣ Δ′) → PreservationResultᴮ D M′ H′
|
||||
-- warning : Warningᴮ Σ D → PreservationResultᴮ D M′ H′
|
||||
|
||||
-- data PreservationResultᴱ {Σ S Δ T M} (D : ∅ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) M′ : Set where
|
||||
|
||||
-- ok : ∀ {Δ′} → (∅ ⊢ᴱ S ∋ M′ ∈ T ⊣ Δ′) → PreservationResultᴱ D M′ H′
|
||||
-- warning : Warningᴱ Σ D → PreservationResultᴱ D M′ H′
|
||||
|
||||
-- preservationᴱ : ∀ {Σ S Δ T H H′ M M′} → (D : ∅ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → (s : H ⊢ M ⟶ᴱ M′ ⊣ H′) → PreservationResultᴱ D M′ H′
|
||||
-- preservationᴱ {S = S} {T = T} h D s with S ≡ᵀ T
|
||||
-- preservationᴱ h D s | no p = warning (disagree p)
|
||||
-- preservationᴱ h (app D₁ D₂) (app s) | yes refl with preservationᴱ h D₁ s
|
||||
-- preservationᴱ h (app D₁ D₂) (app s) | yes refl | ok h′ D₁′ = ok h′ (app D₁′ {!D₂!})
|
||||
-- preservationᴱ h (app D₁ D₂) (app s) | yes refl | warning W = warning (app₁ W)
|
||||
-- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl with lookup (h a)
|
||||
-- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | function f r₁ r₂ D₁ with trans p r₁ | trans (sym q) r₂
|
||||
-- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | function f r₁ r₂ D₁ | refl | refl = ok h (block f (local D₂ D₁))
|
||||
-- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | warningᴴ W = warningᴴ a W
|
||||
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) with subst₂ (λ X Y → _ ▷ X ∈ Y) q (sym p) (h a)
|
||||
-- -- preservationᴱ {S = S} h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} {V = V′} D with S ≡ᵀ U′ | U′ ≡ᵀ V′
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} {V = V′} D | yes refl | yes refl = ok h (block _ (local D₂ D))
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} D | no r | _ = warningᴱ (disagree r)
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} D | yes refl | no r = warningᴴ a {!function f (disagree r)!} -- (subst₁ Warningᴴ {!!} {!(function f (disagree r))!}) -- (function f (disagree r))
|
||||
|
||||
-- -- with src T ≡ᵀ U -- = ok h (block f (local {!D₂!} {!!}))
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} D₁ D₂) (beta {F = f ⟨ var x ∈ _ ⟩∈ R} p) | yes refl = ok h (block f (local D₂ {!!})) -- with src T ≡ᵀ S -- = ok h (block f (local {!D₂!} {!!}))
|
||||
-- -- preservationᴱ h (app {T = T} {U = U} D₁ D₂) (beta {F = f ⟨ var x ∈ S ⟩∈ R} p) | no q = warning (app₀ {!q!}) -- with src T ≡ᵀ S -- = ok h (block f (local {!D₂!} {!!}))
|
||||
-- preservationᴱ h D s | yes refl = {!!}
|
||||
-- preservationᴱ h (function D) (function p) = {!x!}
|
||||
-- preservationᴱ h (block b D) (block s) = {!!}
|
||||
-- preservationᴱ h (block b D) (return p) = {!x!}
|
||||
-- preservationᴱ h (block b D) done = {!!}
|
||||
|
|
|
|||
|
|
@ -5,13 +5,14 @@ module Properties.TypeCheck (m : Mode) where
|
|||
open import Agda.Builtin.Equality using (_≡_; refl)
|
||||
open import FFI.Data.Maybe using (Maybe; just; nothing)
|
||||
open import FFI.Data.Either using (Either)
|
||||
open import Luau.TypeCheck(m) using (_▷_⊢ᴱ_∋_∈_⊣_; _▷_⊢ᴮ_∋_∈_⊣_; nil; var; addr; app; function; block; done; return; local)
|
||||
open import Luau.TypeCheck(m) using (_⊢ᴱ_∋_∈_⊣_; _⊢ᴮ_∋_∈_⊣_; nil; var; addr; app; function; block; done; return; local)
|
||||
open import Luau.Syntax using (Block; Expr; yes; nil; var; addr; _$_; function_is_end; block_is_end; _∙_; return; done; local_←_; _⟨_⟩; _⟨_⟩∈_; var_∈_; name; fun; arg)
|
||||
open import Luau.Type using (Type; nil; top; _⇒_; tgt)
|
||||
open import Luau.Type using (Type; nil; top; bot; _⇒_; tgt)
|
||||
open import Luau.VarCtxt using (VarCtxt; ∅; _↦_; _⊕_↦_; _⋒_; _⊝_; ⊕-[]) renaming (_[_] to _[_]ⱽ)
|
||||
open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
|
||||
open import Luau.Addr using (Addr)
|
||||
open import Luau.Var using (Var; _≡ⱽ_)
|
||||
open import Luau.AddrCtxt using (AddrCtxt) renaming (_[_] to _[_]ᴬ)
|
||||
open import Luau.Heap using (Heap; HeapValue; function_is_end) renaming (_[_] to _[_]ᴴ)
|
||||
open import Properties.Dec using (yes; no)
|
||||
open import Properties.Equality using (_≢_; sym; trans; cong)
|
||||
open import Properties.Remember using (remember; _,_)
|
||||
|
|
@ -19,47 +20,51 @@ open import Properties.Remember using (remember; _,_)
|
|||
src : Type → Type
|
||||
src = Luau.Type.src m
|
||||
|
||||
typeOfᴱ : AddrCtxt → VarCtxt → (Expr yes) → Type
|
||||
typeOfᴮ : AddrCtxt → VarCtxt → (Block yes) → Type
|
||||
typeOfᴴ : Maybe(HeapValue yes) → Type
|
||||
typeOfᴴ nothing = bot
|
||||
typeOfᴴ (just function f ⟨ var x ∈ S ⟩∈ T is B end) = (S ⇒ T)
|
||||
|
||||
typeOfᴱ Σ Γ nil = nil
|
||||
typeOfᴱ Σ Γ (var x) = Γ [ x ]ⱽ
|
||||
typeOfᴱ Σ Γ (addr a) = Σ [ a ]ᴬ
|
||||
typeOfᴱ Σ Γ (M $ N) = tgt(typeOfᴱ Σ Γ M)
|
||||
typeOfᴱ Σ Γ (function f ⟨ var x ∈ S ⟩∈ T is B end) = S ⇒ T
|
||||
typeOfᴱ Σ Γ (block b is B end) = typeOfᴮ Σ Γ B
|
||||
typeOfᴱ : Heap yes → VarCtxt → (Expr yes) → Type
|
||||
typeOfᴮ : Heap yes → VarCtxt → (Block yes) → Type
|
||||
|
||||
typeOfᴮ Σ Γ (function f ⟨ var x ∈ S ⟩∈ T is C end ∙ B) = typeOfᴮ Σ (Γ ⊕ f ↦ (S ⇒ T)) B
|
||||
typeOfᴮ Σ Γ (local var x ∈ T ← M ∙ B) = typeOfᴮ Σ (Γ ⊕ x ↦ T) B
|
||||
typeOfᴮ Σ Γ (return M ∙ B) = typeOfᴱ Σ Γ M
|
||||
typeOfᴮ Σ Γ done = nil
|
||||
typeOfᴱ H Γ nil = nil
|
||||
typeOfᴱ H Γ (var x) = Γ [ x ]ⱽ
|
||||
typeOfᴱ H Γ (addr a) = typeOfᴴ (H [ a ]ᴴ)
|
||||
typeOfᴱ H Γ (M $ N) = tgt(typeOfᴱ H Γ M)
|
||||
typeOfᴱ H Γ (function f ⟨ var x ∈ S ⟩∈ T is B end) = S ⇒ T
|
||||
typeOfᴱ H Γ (block b is B end) = typeOfᴮ H Γ B
|
||||
|
||||
contextOfᴱ : AddrCtxt → VarCtxt → Type → (Expr yes) → VarCtxt
|
||||
contextOfᴮ : AddrCtxt → VarCtxt → Type → (Block yes) → VarCtxt
|
||||
typeOfᴮ H Γ (function f ⟨ var x ∈ S ⟩∈ T is C end ∙ B) = typeOfᴮ H (Γ ⊕ f ↦ (S ⇒ T)) B
|
||||
typeOfᴮ H Γ (local var x ∈ T ← M ∙ B) = typeOfᴮ H (Γ ⊕ x ↦ T) B
|
||||
typeOfᴮ H Γ (return M ∙ B) = typeOfᴱ H Γ M
|
||||
typeOfᴮ H Γ done = nil
|
||||
|
||||
contextOfᴱ Σ Γ S nil = ∅
|
||||
contextOfᴱ Σ Γ S (var x) = (x ↦ S)
|
||||
contextOfᴱ Σ Γ S (addr a) = ∅
|
||||
contextOfᴱ Σ Γ S (M $ N) = (contextOfᴱ Σ Γ (U ⇒ S) M) ⋒ (contextOfᴱ Σ Γ (src T) N) where T = typeOfᴱ Σ Γ M; U = typeOfᴱ Σ Γ N
|
||||
contextOfᴱ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = (contextOfᴮ Σ (Γ ⊕ x ↦ T) U B) ⊝ x
|
||||
contextOfᴱ Σ Γ S (block b is B end) = (contextOfᴮ Σ Γ S B)
|
||||
contextOfᴱ : Heap yes → VarCtxt → Type → (Expr yes) → VarCtxt
|
||||
contextOfᴮ : Heap yes → VarCtxt → Type → (Block yes) → VarCtxt
|
||||
|
||||
contextOfᴮ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = ((contextOfᴮ Σ (Γ ⊕ x ↦ T) U C) ⊝ x) ⋒ ((contextOfᴮ Σ (Γ ⊕ f ↦ (T ⇒ U)) S B) ⊝ f)
|
||||
contextOfᴮ Σ Γ S (local var x ∈ T ← M ∙ B) = (contextOfᴱ Σ Γ T M) ⋒ ((contextOfᴮ Σ (Γ ⊕ x ↦ T)S B) ⊝ x)
|
||||
contextOfᴮ Σ Γ S (return M ∙ B) = (contextOfᴱ Σ Γ S M)
|
||||
contextOfᴮ Σ Γ S done = ∅
|
||||
contextOfᴱ H Γ S nil = ∅
|
||||
contextOfᴱ H Γ S (var x) = (x ↦ S)
|
||||
contextOfᴱ H Γ S (addr a) = ∅
|
||||
contextOfᴱ H Γ S (M $ N) = (contextOfᴱ H Γ (U ⇒ S) M) ⋒ (contextOfᴱ H Γ (src T) N) where T = typeOfᴱ H Γ M; U = typeOfᴱ H Γ N
|
||||
contextOfᴱ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = (contextOfᴮ H (Γ ⊕ x ↦ T) U B) ⊝ x
|
||||
contextOfᴱ H Γ S (block b is B end) = (contextOfᴮ H Γ S B)
|
||||
|
||||
typeCheckᴱ : ∀ Σ Γ S M → (Σ ▷ Γ ⊢ᴱ S ∋ M ∈ (typeOfᴱ Σ Γ M) ⊣ (contextOfᴱ Σ Γ S M))
|
||||
typeCheckᴮ : ∀ Σ Γ S B → (Σ ▷ Γ ⊢ᴮ S ∋ B ∈ (typeOfᴮ Σ Γ B) ⊣ (contextOfᴮ Σ Γ S B))
|
||||
contextOfᴮ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = ((contextOfᴮ H (Γ ⊕ x ↦ T) U C) ⊝ x) ⋒ ((contextOfᴮ H (Γ ⊕ f ↦ (T ⇒ U)) S B) ⊝ f)
|
||||
contextOfᴮ H Γ S (local var x ∈ T ← M ∙ B) = (contextOfᴱ H Γ T M) ⋒ ((contextOfᴮ H (Γ ⊕ x ↦ T)S B) ⊝ x)
|
||||
contextOfᴮ H Γ S (return M ∙ B) = (contextOfᴱ H Γ S M)
|
||||
contextOfᴮ H Γ S done = ∅
|
||||
|
||||
typeCheckᴱ Σ Γ S nil = nil
|
||||
typeCheckᴱ Σ Γ S (var x) = var x refl
|
||||
typeCheckᴱ Σ Γ S (addr a) = addr a refl
|
||||
typeCheckᴱ Σ Γ S (M $ N) = app (typeCheckᴱ Σ Γ (typeOfᴱ Σ Γ N ⇒ S) M) (typeCheckᴱ Σ Γ (src (typeOfᴱ Σ Γ M)) N)
|
||||
typeCheckᴱ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = function(typeCheckᴮ Σ (Γ ⊕ x ↦ T) U B)
|
||||
typeCheckᴱ Σ Γ S (block b is B end) = block b (typeCheckᴮ Σ Γ S B)
|
||||
typeCheckᴱ : ∀ H Γ S M → (Γ ⊢ᴱ S ∋ M ∈ (typeOfᴱ H Γ M) ⊣ (contextOfᴱ H Γ S M))
|
||||
typeCheckᴮ : ∀ H Γ S B → (Γ ⊢ᴮ S ∋ B ∈ (typeOfᴮ H Γ B) ⊣ (contextOfᴮ H Γ S B))
|
||||
|
||||
typeCheckᴮ Σ Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = function(typeCheckᴮ Σ (Γ ⊕ x ↦ T) U C) (typeCheckᴮ Σ (Γ ⊕ f ↦ (T ⇒ U)) S B)
|
||||
typeCheckᴮ Σ Γ S (local var x ∈ T ← M ∙ B) = local (typeCheckᴱ Σ Γ T M) (typeCheckᴮ Σ (Γ ⊕ x ↦ T) S B)
|
||||
typeCheckᴮ Σ Γ S (return M ∙ B) = return (typeCheckᴱ Σ Γ S M) (typeCheckᴮ Σ Γ top B)
|
||||
typeCheckᴮ Σ Γ S done = done
|
||||
typeCheckᴱ H Γ S nil = nil
|
||||
typeCheckᴱ H Γ S (var x) = var x refl
|
||||
typeCheckᴱ H Γ S (addr a) = addr a (typeOfᴴ (H [ a ]ᴴ))
|
||||
typeCheckᴱ H Γ S (M $ N) = app (typeCheckᴱ H Γ (typeOfᴱ H Γ N ⇒ S) M) (typeCheckᴱ H Γ (src (typeOfᴱ H Γ M)) N)
|
||||
typeCheckᴱ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is B end) = function(typeCheckᴮ H (Γ ⊕ x ↦ T) U B)
|
||||
typeCheckᴱ H Γ S (block b is B end) = block b (typeCheckᴮ H Γ S B)
|
||||
|
||||
typeCheckᴮ H Γ S (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) = function(typeCheckᴮ H (Γ ⊕ x ↦ T) U C) (typeCheckᴮ H (Γ ⊕ f ↦ (T ⇒ U)) S B)
|
||||
typeCheckᴮ H Γ S (local var x ∈ T ← M ∙ B) = local (typeCheckᴱ H Γ T M) (typeCheckᴮ H (Γ ⊕ x ↦ T) S B)
|
||||
typeCheckᴮ H Γ S (return M ∙ B) = return (typeCheckᴱ H Γ S M) (typeCheckᴮ H Γ nil B)
|
||||
typeCheckᴮ H Γ S done = done
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue