310 lines
14 KiB
Agda
310 lines
14 KiB
Agda
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.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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open import Data.List.Relation.Unary.All using (All; []; _∷_)
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open import Data.List.Relation.Unary.Any as RelAny using ()
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open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁) renaming (_≟_ to _≟ᶠ_)
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open import Data.Fin.Properties using (suc-injective)
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open import Relation.Binary.PropositionalEquality using (cong; _≡_; refl)
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open import Relation.Nullary using (¬_)
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open import Function using (_∘_)
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open import Lattice
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open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs)
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data Expr : Set where
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_+_ : Expr → Expr → Expr
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_-_ : Expr → Expr → Expr
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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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⟨_⟩ : 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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( MapSet to StringSet
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; insert to insertˢ
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; to-List to to-Listˢ
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; empty to emptyˢ
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; singleton to singletonˢ
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; _⊔_ to _⊔ˢ_
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; `_ to `ˢ_
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; _∈_ to _∈ˢ_
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; ⊔-preserves-∈k₁ to ⊔ˢ-preserves-∈k₁
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; ⊔-preserves-∈k₂ to ⊔ˢ-preserves-∈k₂
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)
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data _∈ᵉ_ : String → Expr → Set where
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in⁺₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ + e₂)
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in⁺₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ + e₂)
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in⁻₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ - e₂)
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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 _∈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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Expr-vars (l + r) = Expr-vars l ⊔ˢ Expr-vars r
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Expr-vars (l - r) = Expr-vars l ⊔ˢ Expr-vars r
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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 ∈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 ⟨ 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 ∈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 ∈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 ∈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 ∈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}
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-- {m₂ = Stmts-vars ss'}
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-- (∈⇒∈-Stmts-vars {n} {k} {ss'} {f'} k∈ss')
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-- Creating a new number from a natural number can never create one
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-- equal to one you get from weakening the bounds on another number.
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z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (zero ≡ suc f)
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z≢sf f ()
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z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (mapˡ suc fs)
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z≢mapsfs [] = []
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z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs'
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indices : ∀ (n : ℕ) → Σ (List (Fin n)) Unique
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indices 0 = ([] , empty)
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indices (suc n') =
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let
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(inds' , unids') = indices n'
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in
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( zero ∷ mapˡ suc inds'
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, push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
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)
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indices-complete : ∀ (n : ℕ) (f : Fin n) → f ∈ˡ (proj₁ (indices n))
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indices-complete (suc n') zero = RelAny.here refl
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indices-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (indices-complete n' f'))
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-- Sketch, 'build control flow graph'
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-- -- Create new block, mark it as the current insertion point.
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-- emptyBlock : m Id
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-- currentBlock : m Id
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-- -- Create a new block, and insert the statement into it. Shold restore insertion pont.
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-- createBlock : Stmt → m (Id × Id)
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-- -- Note that the given ID is a successor / predecessor of the given
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-- -- insertion point.
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-- noteSuccessor : Id → m ()
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-- notePredecessor : Id → m ()
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-- noteEdge : Id → Id → m ()
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-- -- Insert the given statment into the current insertion point.
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-- buildCfg : Stmt → m Cfg
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-- buildCfg { bs₁ } = push bs₁
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-- buildCfg (s₁ ; s₂ ) = buildCfg s₁ >> buildCfg s₂
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-- buildCfg (if _ then s₁ else s₂) = do
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-- (b₁ , b₁') ← createBlock s₁
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-- noteSuccessor b₁
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-- (b₂ , b₂') ← createBlock s₂
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-- noteSuccessor b₂
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-- b ← emptyBlock
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-- notePredecessor b₁'
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-- notePredecessor b₂'
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-- buildCfg (while e repeat s) = do
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-- (b₁, b₁') ← createBlock s
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-- noteSuccessor b₁
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-- noteEdge b₁' b₁
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-- b ← emptyBlock
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-- notePredecessor b₁'
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-- For now, just represent the program and CFG as one type, without branching.
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record Program : Set where
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field
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length : ℕ
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stmts : Vec Stmt length
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private
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vars-Set : StringSet
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vars-Set = Stmts-vars stmts
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vars : List String
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vars = to-Listˢ vars-Set
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vars-Unique : Unique vars
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vars-Unique = proj₂ vars-Set
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State : Set
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State = Fin length
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states : List State
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states = proj₁ (indices length)
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states-complete : ∀ (s : State) → s ∈ˡ states
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states-complete = indices-complete length
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states-Unique : Unique states
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states-Unique = proj₂ (indices length)
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code : State → Stmt
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code = lookup stmts
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-- vars-complete : ∀ {k : String} (s : State) → k ∈ᵇ (code s) → k ∈ˡ vars
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-- vars-complete {k} s = ∈⇒∈-Stmts-vars {length} {k} {stmts} {s}
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_≟_ : IsDecidable (_≡_ {_} {State})
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_≟_ = _≟ᶠ_
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-- Computations for incoming and outgoing edges will have to change too
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-- when we support branching etc.
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incoming : State → List State
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incoming
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with length
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... | 0 = (λ ())
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... | suc n' = (λ
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{ zero → []
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; (suc f') → (inject₁ f') ∷ []
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})
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