Prove that variables in a program all come from the program's code
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Language.agda
114
Language.agda
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@ -3,12 +3,14 @@ module Language where
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open import Data.Nat using (ℕ; suc; pred)
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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.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; _≡_)
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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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@ -30,18 +32,110 @@ open import Lattice.MapSet String _≟ˢ_
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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 _∈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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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) = insertˢ s emptyˢ
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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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Stmt-vars : Stmt → StringSet
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Stmt-vars (x ← e) = insertˢ x (Expr-vars e)
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Stmt-vars (x ← e) = (singletonˢ x) ⊔ˢ (Expr-vars e)
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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 ∈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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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} {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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-- 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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@ -71,8 +165,7 @@ record Program : Set where
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private
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vars-Set : StringSet
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vars-Set = foldr (λ n → StringSet)
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(λ {k} stmt acc → (Stmt-vars stmt) ⊔ˢ acc) emptyˢ stmts
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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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@ -83,15 +176,18 @@ record Program : Set where
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State : Set
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State = Fin length
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code : State → Stmt
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code = lookup stmts
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states : List State
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states = proj₁ (indices 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 ∈ᵗ (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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@ -553,11 +553,18 @@ open ImplInsert _⊔₂_ using
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; union-preserves-∈₂
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; union-preserves-∉
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; union-preserves-∈k₁
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; union-preserves-∈k₂
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)
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⊔-combines : ∀ {k : A} {v₁ v₂ : B} {m₁ m₂ : Map} → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → (k , v₁ ⊔₂ v₂) ∈ (m₁ ⊔ m₂)
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⊔-combines {k} {v₁} {v₂} {kvs₁ , u₁} {kvs₂ , u₂} k,v₁∈m₁ k,v₂∈m₂ = union-combines u₁ u₂ k,v₁∈m₁ k,v₂∈m₂
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⊔-preserves-∈k₁ : ∀ {k : A} → {m₁ m₂ : Map} → k ∈k m₁ → k ∈k (m₁ ⊔ m₂)
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⊔-preserves-∈k₁ {k} {(l₁ , _)} {(l₂ , _)} k∈km₁ = union-preserves-∈k₁ {l₁ = l₁} {l₂ = l₂} k∈km₁
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⊔-preserves-∈k₂ : ∀ {k : A} → {m₁ m₂ : Map} → k ∈k m₂ → k ∈k (m₁ ⊔ m₂)
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⊔-preserves-∈k₂ {k} {(l₁ , _)} {(l₂ , _)} k∈km₁ = union-preserves-∈k₂ {l₁ = l₁} {l₂ = l₂} k∈km₁
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open ImplInsert _⊓₂_ using
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( restrict-needs-both
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; updates
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@ -6,18 +6,28 @@ open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
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module Lattice.MapSet {a : Level} (A : Set a) (≡-dec-A : Decidable (_≡_ {a} {A})) where
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open import Data.List using (List; map)
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open import Data.Product using (proj₁)
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open import Data.Product using (_,_; proj₁)
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open import Function using (_∘_)
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open import Lattice.Unit using (⊤; tt) renaming (_≈_ to _≈₂_; _⊔_ to _⊔₂_; _⊓_ to _⊓₂_; isLattice to ⊤-isLattice)
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import Lattice.Map
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private module UnitMap = Lattice.Map A ⊤ _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A ⊤-isLattice
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open UnitMap using (Map)
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open UnitMap using
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( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; empty
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open UnitMap
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using (Map; Expr; ⟦_⟧)
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renaming
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( Expr-Provenance to Expr-Provenanceᵐ
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)
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open UnitMap
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using
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( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; _∪_ ; _∩_ ; `_; empty; forget
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; isUnionSemilattice; isIntersectSemilattice; isLattice; lattice
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) public
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; Provenance
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; ⊔-preserves-∈k₁
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; ⊔-preserves-∈k₂
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)
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renaming (_∈k_ to _∈_) public
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open Provenance public
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MapSet : Set a
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MapSet = Map
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@ -27,3 +37,9 @@ to-List = map proj₁ ∘ proj₁
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insert : A → MapSet → MapSet
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insert k = UnitMap.insert k tt
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singleton : A → MapSet
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singleton k = UnitMap.insert k tt empty
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Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈ ⟦ e ⟧ → Provenance k tt e
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Expr-Provenance k e k∈e = let (tt , (prov , _)) = Expr-Provenanceᵐ k e k∈e in prov
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