Intermediate commit: add while loops and start trying to formalize them.
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Language.agda
246
Language.agda
@ -1,8 +1,9 @@
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module Language where
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open import Data.Nat using (ℕ; suc; pred)
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open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
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open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
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open import Data.Product using (Σ; _,_; proj₁; proj₂)
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open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
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open import Data.Vec using (Vec; foldr; lookup; _∷_)
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open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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@ -23,8 +24,60 @@ data Expr : Set where
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`_ : String → Expr
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#_ : ℕ → Expr
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data BasicStmt : Set where
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_←_ : String → Expr → BasicStmt
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noop : BasicStmt
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data Stmt : Set where
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_←_ : String → Expr → Stmt
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⟨_⟩ : BasicStmt → Stmt
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_then_ : Stmt → Stmt → Stmt
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if_then_else_ : Expr → Stmt → Stmt → Stmt
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while_repeat_ : Expr → Stmt → Stmt
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module Semantics where
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data Value : Set where
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↑ᶻ : ℤ → Value
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Env : Set
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Env = List (String × Value)
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data _∈_ : (String × Value) → Env → Set where
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here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) ∷ ρ)
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there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') ∷ ρ)
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data _,_⇒ᵉ_ : Env → Expr → Value → Set where
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⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n))
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⇒ᵉ-Var : ∀ (ρ : Env) (x : String) (v : Value) → (x , v) ∈ ρ → ρ , (` x) ⇒ᵉ v
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⇒ᵉ-+ : ∀ (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) →
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ρ , e₁ ⇒ᵉ (↑ᶻ z₁) → ρ , e₂ ⇒ᵉ (↑ᶻ z₂) →
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ρ , (e₁ + e₂) ⇒ᵉ (↑ᶻ (z₁ +ᶻ z₂))
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⇒ᵉ-- : ∀ (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) →
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ρ , e₁ ⇒ᵉ (↑ᶻ z₁) → ρ , e₂ ⇒ᵉ (↑ᶻ z₂) →
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ρ , (e₁ - e₂) ⇒ᵉ (↑ᶻ (z₁ -ᶻ z₂))
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data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where
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⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ
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⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) →
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ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) ∷ ρ)
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data _,_⇒ˢ_ : Env → Stmt → Env → Set where
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⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) →
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ρ₁ , bs ⇒ᵇ ρ₂ → ρ₁ , ⟨ bs ⟩ ⇒ˢ ρ₂
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⇒ˢ-then : ∀ (ρ₁ ρ₂ ρ₃ : Env) (s₁ s₂ : Stmt) →
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ρ₁ , s₁ ⇒ˢ ρ₂ → ρ₂ , s₂ ⇒ˢ ρ₃ →
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ρ₁ , (s₁ then s₂) ⇒ˢ ρ₃
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⇒ˢ-if-true : ∀ (ρ₁ ρ₂ : Env) (e : Expr) (z : ℤ) (s₁ s₂ : Stmt) →
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ρ₁ , e ⇒ᵉ (↑ᶻ z) → ¬ z ≡ (+ 0) → ρ₁ , s₁ ⇒ˢ ρ₂ →
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ρ₁ , (if e then s₁ else s₂) ⇒ˢ ρ₂
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⇒ˢ-if-false : ∀ (ρ₁ ρ₂ : Env) (e : Expr) (s₁ s₂ : Stmt) →
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ρ₁ , e ⇒ᵉ (↑ᶻ (+ 0)) → ρ₁ , s₂ ⇒ˢ ρ₂ →
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ρ₁ , (if e then s₁ else s₂) ⇒ˢ ρ₂
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⇒ˢ-while-true : ∀ (ρ₁ ρ₂ ρ₃ : Env) (e : Expr) (z : ℤ) (s : Stmt) →
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ρ₁ , e ⇒ᵉ (↑ᶻ z) → ¬ z ≡ (+ 0) → ρ₁ , s ⇒ˢ ρ₂ → ρ₂ , (while e repeat s) ⇒ˢ ρ₃ →
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ρ₁ , (while e repeat s) ⇒ˢ ρ₃
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⇒ˢ-while-false : ∀ (ρ : Env) (e : Expr) (s : Stmt) →
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ρ , e ⇒ᵉ (↑ᶻ (+ 0)) →
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ρ , (while e repeat s) ⇒ˢ ρ
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open import Lattice.MapSet _≟ˢ_
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renaming
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@ -47,9 +100,9 @@ data _∈ᵉ_ : String → Expr → Set where
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in⁻₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ - e₂)
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here : ∀ {k : String} → k ∈ᵉ (` k)
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data _∈ᵗ_ : String → Stmt → Set where
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in←₁ : ∀ {k : String} {e : Expr} → k ∈ᵗ (k ← e)
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in←₂ : ∀ {k k' : String} {e : Expr} → k ∈ᵉ e → k ∈ᵗ (k' ← e)
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data _∈ᵇ_ : String → BasicStmt → Set where
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in←₁ : ∀ {k : String} {e : Expr} → k ∈ᵇ (k ← e)
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in←₂ : ∀ {k k' : String} {e : Expr} → k ∈ᵉ e → k ∈ᵇ (k' ← e)
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private
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Expr-vars : Expr → StringSet
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@ -58,84 +111,91 @@ private
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Expr-vars (` s) = singletonˢ s
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Expr-vars (# _) = emptyˢ
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∈-Expr-vars⇒∈ : ∀ {k : String} (e : Expr) → k ∈ˢ (Expr-vars e) → k ∈ᵉ e
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∈-Expr-vars⇒∈ {k} (e₁ + e₂) k∈vs
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with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
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... | in₁ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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... | in₂ _ (single k,tt∈vs₂) = (in⁺₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
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... | bothᵘ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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∈-Expr-vars⇒∈ {k} (e₁ - e₂) k∈vs
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with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
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... | in₁ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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... | in₂ _ (single k,tt∈vs₂) = (in⁻₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
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... | bothᵘ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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∈-Expr-vars⇒∈ {k} (` k) (RelAny.here refl) = here
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-- ∈-Expr-vars⇒∈ : ∀ {k : String} (e : Expr) → k ∈ˢ (Expr-vars e) → k ∈ᵉ e
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-- ∈-Expr-vars⇒∈ {k} (e₁ + e₂) k∈vs
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-- with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
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-- ... | in₁ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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-- ... | in₂ _ (single k,tt∈vs₂) = (in⁺₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
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-- ... | bothᵘ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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-- ∈-Expr-vars⇒∈ {k} (e₁ - e₂) k∈vs
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-- with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs
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-- ... | in₁ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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-- ... | in₂ _ (single k,tt∈vs₂) = (in⁻₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂)))
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-- ... | bothᵘ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁)))
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-- ∈-Expr-vars⇒∈ {k} (` k) (RelAny.here refl) = here
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∈⇒∈-Expr-vars : ∀ {k : String} {e : Expr} → k ∈ᵉ e → k ∈ˢ (Expr-vars e)
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∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₁ k∈e₁) =
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⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
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{m₂ = Expr-vars e₂}
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(∈⇒∈-Expr-vars k∈e₁)
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∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₂ k∈e₂) =
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⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
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{m₂ = Expr-vars e₂}
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(∈⇒∈-Expr-vars k∈e₂)
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∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₁ k∈e₁) =
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⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
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{m₂ = Expr-vars e₂}
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(∈⇒∈-Expr-vars k∈e₁)
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∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₂ k∈e₂) =
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⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
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{m₂ = Expr-vars e₂}
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(∈⇒∈-Expr-vars k∈e₂)
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∈⇒∈-Expr-vars here = RelAny.here refl
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-- ∈⇒∈-Expr-vars : ∀ {k : String} {e : Expr} → k ∈ᵉ e → k ∈ˢ (Expr-vars e)
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-- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₁ k∈e₁) =
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-- ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
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-- {m₂ = Expr-vars e₂}
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-- (∈⇒∈-Expr-vars k∈e₁)
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-- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₂ k∈e₂) =
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-- ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
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-- {m₂ = Expr-vars e₂}
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-- (∈⇒∈-Expr-vars k∈e₂)
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-- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₁ k∈e₁) =
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-- ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁}
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-- {m₂ = Expr-vars e₂}
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-- (∈⇒∈-Expr-vars k∈e₁)
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-- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₂ k∈e₂) =
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-- ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁}
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-- {m₂ = Expr-vars e₂}
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-- (∈⇒∈-Expr-vars k∈e₂)
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-- ∈⇒∈-Expr-vars here = RelAny.here refl
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BasicStmt-vars : BasicStmt → StringSet
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BasicStmt-vars (x ← e) = (singletonˢ x) ⊔ˢ (Expr-vars e)
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BasicStmt-vars noop = emptyˢ
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Stmt-vars : Stmt → StringSet
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Stmt-vars (x ← e) = (singletonˢ x) ⊔ˢ (Expr-vars e)
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Stmt-vars ⟨ bs ⟩ = BasicStmt-vars bs
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Stmt-vars (s₁ then s₂) = (Stmt-vars s₁) ⊔ˢ (Stmt-vars s₂)
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Stmt-vars (if e then s₁ else s₂) = ((Expr-vars e) ⊔ˢ (Stmt-vars s₁)) ⊔ˢ (Stmt-vars s₂)
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Stmt-vars (while e repeat s) = (Expr-vars e) ⊔ˢ (Stmt-vars s)
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∈-Stmt-vars⇒∈ : ∀ {k : String} (s : Stmt) → k ∈ˢ (Stmt-vars s) → k ∈ᵗ s
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∈-Stmt-vars⇒∈ {k} (k' ← e) k∈vs
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with Expr-Provenance k ((`ˢ (singletonˢ k')) ∪ (`ˢ (Expr-vars e))) k∈vs
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... | in₁ (single (RelAny.here refl)) _ = in←₁
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... | in₂ _ (single k,tt∈vs') = in←₂ (∈-Expr-vars⇒∈ e (forget k,tt∈vs'))
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... | bothᵘ (single (RelAny.here refl)) _ = in←₁
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-- ∈-Stmt-vars⇒∈ : ∀ {k : String} (s : Stmt) → k ∈ˢ (Stmt-vars s) → k ∈ᵇ s
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-- ∈-Stmt-vars⇒∈ {k} (k' ← e) k∈vs
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-- with Expr-Provenance k ((`ˢ (singletonˢ k')) ∪ (`ˢ (Expr-vars e))) k∈vs
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-- ... | in₁ (single (RelAny.here refl)) _ = in←₁
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-- ... | in₂ _ (single k,tt∈vs') = in←₂ (∈-Expr-vars⇒∈ e (forget k,tt∈vs'))
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-- ... | bothᵘ (single (RelAny.here refl)) _ = in←₁
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∈⇒∈-Stmt-vars : ∀ {k : String} {s : Stmt} → k ∈ᵗ s → k ∈ˢ (Stmt-vars s)
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∈⇒∈-Stmt-vars {k} {k ← e} in←₁ =
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⊔ˢ-preserves-∈k₁ {m₁ = singletonˢ k}
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{m₂ = Expr-vars e}
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(RelAny.here refl)
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∈⇒∈-Stmt-vars {k} {k' ← e} (in←₂ k∈e) =
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⊔ˢ-preserves-∈k₂ {m₁ = singletonˢ k'}
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{m₂ = Expr-vars e}
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(∈⇒∈-Expr-vars k∈e)
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-- ∈⇒∈-Stmt-vars : ∀ {k : String} {s : Stmt} → k ∈ᵇ s → k ∈ˢ (Stmt-vars s)
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-- ∈⇒∈-Stmt-vars {k} {k ← e} in←₁ =
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-- ⊔ˢ-preserves-∈k₁ {m₁ = singletonˢ k}
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-- {m₂ = Expr-vars e}
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-- (RelAny.here refl)
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-- ∈⇒∈-Stmt-vars {k} {k' ← e} (in←₂ k∈e) =
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-- ⊔ˢ-preserves-∈k₂ {m₁ = singletonˢ k'}
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-- {m₂ = Expr-vars e}
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-- (∈⇒∈-Expr-vars k∈e)
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Stmts-vars : ∀ {n : ℕ} → Vec Stmt n → StringSet
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Stmts-vars = foldr (λ n → StringSet)
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(λ {k} stmt acc → (Stmt-vars stmt) ⊔ˢ acc) emptyˢ
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∈-Stmts-vars⇒∈ : ∀ {n : ℕ} {k : String} (ss : Vec Stmt n) →
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k ∈ˢ (Stmts-vars ss) → Σ (Fin n) (λ f → k ∈ᵗ lookup ss f)
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∈-Stmts-vars⇒∈ {suc n'} {k} (s ∷ ss') k∈vss
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with Expr-Provenance k ((`ˢ (Stmt-vars s)) ∪ (`ˢ (Stmts-vars ss'))) k∈vss
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... | in₁ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))
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... | in₂ _ (single k,tt∈vss') =
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let
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(f' , k∈s') = ∈-Stmts-vars⇒∈ ss' (forget k,tt∈vss')
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in
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(suc f' , k∈s')
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... | bothᵘ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))
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-- ∈-Stmts-vars⇒∈ : ∀ {n : ℕ} {k : String} (ss : Vec Stmt n) →
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-- k ∈ˢ (Stmts-vars ss) → Σ (Fin n) (λ f → k ∈ᵇ lookup ss f)
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-- ∈-Stmts-vars⇒∈ {suc n'} {k} (s ∷ ss') k∈vss
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-- with Expr-Provenance k ((`ˢ (Stmt-vars s)) ∪ (`ˢ (Stmts-vars ss'))) k∈vss
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-- ... | in₁ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))
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-- ... | in₂ _ (single k,tt∈vss') =
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-- let
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-- (f' , k∈s') = ∈-Stmts-vars⇒∈ ss' (forget k,tt∈vss')
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-- in
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-- (suc f' , k∈s')
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-- ... | bothᵘ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs))
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∈⇒∈-Stmts-vars : ∀ {n : ℕ} {k : String} {ss : Vec Stmt n} {f : Fin n} →
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k ∈ᵗ lookup ss f → k ∈ˢ (Stmts-vars ss)
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∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {zero} k∈s =
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⊔ˢ-preserves-∈k₁ {m₁ = Stmt-vars s}
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{m₂ = Stmts-vars ss'}
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(∈⇒∈-Stmt-vars k∈s)
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∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {suc f'} k∈ss' =
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⊔ˢ-preserves-∈k₂ {m₁ = Stmt-vars s}
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{m₂ = Stmts-vars ss'}
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(∈⇒∈-Stmts-vars {n} {k} {ss'} {f'} k∈ss')
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-- ∈⇒∈-Stmts-vars : ∀ {n : ℕ} {k : String} {ss : Vec Stmt n} {f : Fin n} →
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-- k ∈ᵇ lookup ss f → k ∈ˢ (Stmts-vars ss)
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-- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {zero} k∈s =
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-- ⊔ˢ-preserves-∈k₁ {m₁ = Stmt-vars s}
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-- {m₂ = Stmts-vars ss'}
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-- (∈⇒∈-Stmt-vars k∈s)
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-- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {suc f'} k∈ss' =
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-- ⊔ˢ-preserves-∈k₂ {m₁ = Stmt-vars s}
|
||||
-- {m₂ = Stmts-vars ss'}
|
||||
-- (∈⇒∈-Stmts-vars {n} {k} {ss'} {f'} k∈ss')
|
||||
|
||||
-- Creating a new number from a natural number can never create one
|
||||
-- equal to one you get from weakening the bounds on another number.
|
||||
@ -160,6 +220,44 @@ private
|
||||
indices-complete (suc n') zero = RelAny.here refl
|
||||
indices-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (indices-complete n' f'))
|
||||
|
||||
-- Sketch, 'build control flow graph'
|
||||
|
||||
-- -- Create new block, mark it as the current insertion point.
|
||||
-- emptyBlock : m Id
|
||||
|
||||
-- currentBlock : m Id
|
||||
|
||||
-- -- Create a new block, and insert the statement into it. Shold restore insertion pont.
|
||||
-- createBlock : Stmt → m (Id × Id)
|
||||
|
||||
-- -- Note that the given ID is a successor / predecessor of the given
|
||||
-- -- insertion point.
|
||||
-- noteSuccessor : Id → m ()
|
||||
-- notePredecessor : Id → m ()
|
||||
-- noteEdge : Id → Id → m ()
|
||||
|
||||
-- -- Insert the given statment into the current insertion point.
|
||||
-- buildCfg : Stmt → m Cfg
|
||||
-- buildCfg { bs₁ } = push bs₁
|
||||
-- buildCfg (s₁ ; s₂ ) = buildCfg s₁ >> buildCfg s₂
|
||||
-- buildCfg (if _ then s₁ else s₂) = do
|
||||
-- (b₁ , b₁') ← createBlock s₁
|
||||
-- noteSuccessor b₁
|
||||
|
||||
-- (b₂ , b₂') ← createBlock s₂
|
||||
-- noteSuccessor b₂
|
||||
|
||||
-- b ← emptyBlock
|
||||
-- notePredecessor b₁'
|
||||
-- notePredecessor b₂'
|
||||
-- buildCfg (while e repeat s) = do
|
||||
-- (b₁, b₁') ← createBlock s
|
||||
-- noteSuccessor b₁
|
||||
-- noteEdge b₁' b₁
|
||||
|
||||
-- b ← emptyBlock
|
||||
-- notePredecessor b₁'
|
||||
|
||||
|
||||
-- For now, just represent the program and CFG as one type, without branching.
|
||||
record Program : Set where
|
||||
@ -192,8 +290,8 @@ record Program : Set where
|
||||
code : State → Stmt
|
||||
code = lookup stmts
|
||||
|
||||
vars-complete : ∀ {k : String} (s : State) → k ∈ᵗ (code s) → k ∈ˡ vars
|
||||
vars-complete {k} s = ∈⇒∈-Stmts-vars {length} {k} {stmts} {s}
|
||||
-- vars-complete : ∀ {k : String} (s : State) → k ∈ᵇ (code s) → k ∈ˡ vars
|
||||
-- vars-complete {k} s = ∈⇒∈-Stmts-vars {length} {k} {stmts} {s}
|
||||
|
||||
_≟_ : IsDecidable (_≡_ {_} {State})
|
||||
_≟_ = _≟ᶠ_
|
||||
|
||||
Loading…
Reference in New Issue
Block a user