146 lines
6.8 KiB
Agda
146 lines
6.8 KiB
Agda
module Language.Base where
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open import Data.List as List using (List)
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open import Data.Nat using (ℕ; suc)
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open import Data.Product using (Σ; _,_; proj₁)
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open import Data.String as String using (String)
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open import Data.Vec using (Vec; foldr; lookup)
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open import Relation.Binary.PropositionalEquality using (_≡_; refl)
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open import Lattice
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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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infixr 2 _then_
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infix 3 if_then_else_
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infix 3 while_repeat_
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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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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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open import Lattice.MapSet (String._≟_)
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renaming
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( MapSet to StringSet
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; insert to insertˢ
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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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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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