Use different graph operations to implement construction
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -52,20 +52,17 @@ record Program : Set where
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field
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rootStmt : Stmt
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private
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buildResult = buildCfg rootStmt empty
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graph : Graph
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graph = proj₁ buildResult
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graph = buildCfg rootStmt
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State : Set
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State = Graph.Index graph
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initialState : State
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initialState = Utils.proj₁ (proj₁ (proj₂ buildResult))
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initialState = proj₁ (buildCfg-input rootStmt)
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finalState : State
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finalState = Utils.proj₂ (proj₁ (proj₂ buildResult))
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finalState = proj₁ (buildCfg-output rootStmt)
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private
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vars-Set : StringSet
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@ -1,8 +1,8 @@
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module Language.Graphs where
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open import Language.Base
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open import Language.Base using (Expr; Stmt; BasicStmt; ⟨_⟩; _then_; if_then_else_; while_repeat_)
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open import Data.Fin as Fin using (Fin; suc; zero; _↑ˡ_; _↑ʳ_)
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open import Data.Fin as Fin using (Fin; suc; zero)
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open import Data.Fin.Properties as FinProp using (suc-injective)
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open import Data.List as List using (List; []; _∷_)
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open import Data.List.Membership.Propositional as ListMem using ()
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@ -15,7 +15,7 @@ open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-s
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst; trans)
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open import Lattice
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open import Utils using (x∈xs⇒fx∈fxs; _⊗_; _,_)
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open import Utils using (x∈xs⇒fx∈fxs; _⊗_; _,_; ∈-cartesianProduct)
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record Graph : Set where
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constructor MkGraph
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@ -31,157 +31,218 @@ record Graph : Set where
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field
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nodes : Vec (List BasicStmt) size
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edges : List Edge
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inputs : List Index
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outputs : List Index
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empty : Graph
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empty = record
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{ size = 0
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; nodes = []
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; edges = []
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_↑ˡ_ : ∀ {n} → (Fin n × Fin n) → ∀ m → (Fin (n Nat.+ m) × Fin (n Nat.+ m))
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_↑ˡ_ (idx₁ , idx₂) m = (idx₁ Fin.↑ˡ m , idx₂ Fin.↑ˡ m)
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_↑ʳ_ : ∀ {m} n → (Fin m × Fin m) → Fin (n Nat.+ m) × Fin (n Nat.+ m)
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_↑ʳ_ n (idx₁ , idx₂) = (n Fin.↑ʳ idx₁ , n Fin.↑ʳ idx₂)
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_↑ˡⁱ_ : ∀ {n} → List (Fin n) → ∀ m → List (Fin (n Nat.+ m))
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_↑ˡⁱ_ l m = List.map (Fin._↑ˡ m) l
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_↑ʳⁱ_ : ∀ {m} n → List (Fin m) → List (Fin (n Nat.+ m))
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_↑ʳⁱ_ n l = List.map (n Fin.↑ʳ_) l
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_↑ˡᵉ_ : ∀ {n} → List (Fin n × Fin n) → ∀ m → List (Fin (n Nat.+ m) × Fin (n Nat.+ m))
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_↑ˡᵉ_ l m = List.map (_↑ˡ m) l
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_↑ʳᵉ_ : ∀ {m} n → List (Fin m × Fin m) → List (Fin (n Nat.+ m) × Fin (n Nat.+ m))
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_↑ʳᵉ_ n l = List.map (n ↑ʳ_) l
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infixl 5 _∙_
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_∙_ : Graph → Graph → Graph
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_∙_ g₁ g₂ = record
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{ size = Graph.size g₁ Nat.+ Graph.size g₂
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; nodes = Graph.nodes g₁ ++ Graph.nodes g₂
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; edges = (Graph.edges g₁ ↑ˡᵉ Graph.size g₂) List.++
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(Graph.size g₁ ↑ʳᵉ Graph.edges g₂)
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; inputs = (Graph.inputs g₁ ↑ˡⁱ Graph.size g₂) List.++
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(Graph.size g₁ ↑ʳⁱ Graph.inputs g₂)
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; outputs = (Graph.outputs g₁ ↑ˡⁱ Graph.size g₂) List.++
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(Graph.size g₁ ↑ʳⁱ Graph.outputs g₂)
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}
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↑ˡ-Edge : ∀ {n} → (Fin n × Fin n) → ∀ m → (Fin (n Nat.+ m) × Fin (n Nat.+ m))
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↑ˡ-Edge (idx₁ , idx₂) m = (idx₁ ↑ˡ m , idx₂ ↑ˡ m)
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infixl 5 _↦_
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_↦_ : Graph → Graph → Graph
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_↦_ g₁ g₂ = record
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{ size = Graph.size g₁ Nat.+ Graph.size g₂
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; nodes = Graph.nodes g₁ ++ Graph.nodes g₂
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; edges = (Graph.edges g₁ ↑ˡᵉ Graph.size g₂) List.++
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(Graph.size g₁ ↑ʳᵉ Graph.edges g₂) List.++
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(List.cartesianProduct (Graph.outputs g₁ ↑ˡⁱ Graph.size g₂)
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(Graph.size g₁ ↑ʳⁱ Graph.inputs g₂))
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; inputs = Graph.inputs g₁ ↑ˡⁱ Graph.size g₂
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; outputs = Graph.size g₁ ↑ʳⁱ Graph.outputs g₂
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}
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loop : Graph → Graph
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loop g = record
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{ size = Graph.size g
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; nodes = Graph.nodes g
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; edges = Graph.edges g List.++
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List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
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; inputs = Graph.inputs g
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; outputs = Graph.outputs g
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}
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_[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
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_[_] g idx = lookup (Graph.nodes g) idx
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record _⊆_ (g₁ g₂ : Graph) : Set where
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constructor Mk-⊆
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field
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n : ℕ
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sg₂≡sg₁+n : Graph.size g₂ ≡ Graph.size g₁ Nat.+ n
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newNodes : Vec (List BasicStmt) n
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nsg₂≡nsg₁++newNodes : cast sg₂≡sg₁+n (Graph.nodes g₂) ≡ Graph.nodes g₁ ++ newNodes
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e∈g₁⇒e∈g₂ : ∀ {e : Graph.Edge g₁} →
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e ListMem.∈ (Graph.edges g₁) →
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(↑ˡ-Edge e n) ListMem.∈ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
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singleton : List BasicStmt → Graph
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singleton bss = record
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{ size = 1
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; nodes = bss ∷ []
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; edges = []
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; inputs = zero ∷ []
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; outputs = zero ∷ []
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}
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private
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castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
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castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
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buildCfg : Stmt → Graph
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buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ [])
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buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
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buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton []
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buildCfg (while _ repeat s) = loop (buildCfg s ↦ singleton [])
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↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
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↑ˡ-assoc zero p = refl
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↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
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↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
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↑ˡ-Edge-assoc (idx₁ , idx₂) p
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rewrite ↑ˡ-assoc idx₁ p
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rewrite ↑ˡ-assoc idx₂ p = refl
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↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
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f ≡ Fin.cast p (f ↑ˡ 0)
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↑ˡ-identityʳ zero p = refl
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↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
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↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
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e ≡ castᵉ p (↑ˡ-Edge e 0)
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↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
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rewrite sym (↑ˡ-identityʳ idx₁ p)
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rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
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cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
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(e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
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castᵉ p e ListMem.∈ es →
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e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
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cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
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rewrite FinProp.cast-is-id refl idx₁
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rewrite FinProp.cast-is-id refl idx₂ = e∈es
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⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
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⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
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(Mk-⊆ n₁ p₁@refl newNodes₁ nsg₂≡nsg₁++newNodes₁ e∈g₁⇒e∈g₂)
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(Mk-⊆ n₂ p₂@refl newNodes₂ nsg₃≡nsg₂++newNodes₂ e∈g₂⇒e∈g₃)
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rewrite cast-is-id refl ns₂
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rewrite cast-is-id refl ns₃
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with refl ← nsg₂≡nsg₁++newNodes₁
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with refl ← nsg₃≡nsg₂++newNodes₂ =
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record
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{ n = n₁ Nat.+ n₂
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; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
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; newNodes = newNodes₁ ++ newNodes₂
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; nsg₂≡nsg₁++newNodes = ++-assoc (+-assoc s₁ n₁ n₂) ns₁ newNodes₁ newNodes₂
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; e∈g₁⇒e∈g₂ = λ {e} e∈g₁ →
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cast∈⇒∈subst (sym (+-assoc s₁ n₁ n₂)) (+-assoc s₁ n₁ n₂) _ _
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(subst (λ e' → e' ListMem.∈ es₃)
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(↑ˡ-Edge-assoc e (sym (+-assoc s₁ n₁ n₂)))
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(e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)))
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}
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open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
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instance
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IndexRelaxable : Relaxable Graph.Index
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IndexRelaxable = record
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{ relax = λ { (Mk-⊆ n refl _ _ _) idx → idx ↑ˡ n }
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}
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EdgeRelaxable : Relaxable Graph.Edge
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EdgeRelaxable = record
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{ relax = λ g₁⊆g₂ (idx₁ , idx₂) →
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( Relaxable.relax IndexRelaxable g₁⊆g₂ idx₁
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, Relaxable.relax IndexRelaxable g₁⊆g₂ idx₂
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)
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}
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open Relaxable {{...}}
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pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
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pushBasicBlock bss g =
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( record
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{ size = Graph.size g Nat.+ 1
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; nodes = Graph.nodes g ++ (bss ∷ [])
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; edges = List.map (λ e → ↑ˡ-Edge e 1) (Graph.edges g)
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}
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, ( Graph.size g ↑ʳ zero
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, record
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{ n = 1
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; sg₂≡sg₁+n = refl
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; newNodes = (bss ∷ [])
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; nsg₂≡nsg₁++newNodes = cast-is-id refl _
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; e∈g₁⇒e∈g₂ = λ e∈g₁ → x∈xs⇒fx∈fxs (λ e → ↑ˡ-Edge e 1) e∈g₁
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}
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)
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)
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pushEmptyBlock : MonotonicGraphFunction Graph.Index
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pushEmptyBlock = pushBasicBlock []
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addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
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addEdges (MkGraph s ns es) es' =
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( record
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{ size = s
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; nodes = ns
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; edges = es' List.++ es
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}
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, record
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{ n = 0
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; sg₂≡sg₁+n = +-comm 0 s
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; newNodes = []
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; nsg₂≡nsg₁++newNodes = cast-sym _ (++-identityʳ (+-comm s 0) ns)
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; e∈g₁⇒e∈g₂ = λ {e} e∈es →
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cast∈⇒∈subst (+-comm s 0) (+-comm 0 s) _ _
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(subst (λ e' → e' ListMem.∈ _)
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(↑ˡ-Edge-identityʳ e (+-comm s 0))
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(ListMemProp.∈-++⁺ʳ es' e∈es))
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}
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)
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buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
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buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
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buildCfg (s₁ then s₂) =
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(buildCfg s₁ ⟨⊗⟩ buildCfg s₂)
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update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → addEdges g ((idx₂ , idx₃) ∷ []) })
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map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → (idx₁ , idx₄) })
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buildCfg (if _ then s₁ else s₂) =
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(buildCfg s₁ ⟨⊗⟩ buildCfg s₂ ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
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update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') →
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addEdges g ((idx , idx₁) ∷ (idx , idx₃) ∷ (idx₂ , idx') ∷ (idx₄ , idx') ∷ []) })
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map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') → (idx , idx') })
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buildCfg (while _ repeat s) =
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(buildCfg s ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
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update (λ { g ((idx₁ , idx₂) , idx , idx') →
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addEdges g ((idx , idx') ∷ (idx , idx₁) ∷ (idx₂ , idx) ∷ []) })
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map (λ { g ((idx₁ , idx₂) , idx , idx') → (idx , idx') })
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-- record _⊆_ (g₁ g₂ : Graph) : Set where
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-- constructor Mk-⊆
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-- field
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-- n : ℕ
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-- sg₂≡sg₁+n : Graph.size g₂ ≡ Graph.size g₁ Nat.+ n
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-- newNodes : Vec (List BasicStmt) n
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-- nsg₂≡nsg₁++newNodes : cast sg₂≡sg₁+n (Graph.nodes g₂) ≡ Graph.nodes g₁ ++ newNodes
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-- e∈g₁⇒e∈g₂ : ∀ {e : Graph.Edge g₁} →
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-- e ListMem.∈ (Graph.edges g₁) →
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-- (↑ˡ-Edge e n) ListMem.∈ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
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--
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-- private
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-- castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
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-- castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
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--
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-- ↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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-- f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
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-- ↑ˡ-assoc zero p = refl
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-- ↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
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--
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-- ↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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-- ↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
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-- ↑ˡ-Edge-assoc (idx₁ , idx₂) p
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-- rewrite ↑ˡ-assoc idx₁ p
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-- rewrite ↑ˡ-assoc idx₂ p = refl
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--
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-- ↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
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-- f ≡ Fin.cast p (f ↑ˡ 0)
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-- ↑ˡ-identityʳ zero p = refl
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-- ↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
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--
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-- ↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
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-- e ≡ castᵉ p (↑ˡ-Edge e 0)
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-- ↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
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-- rewrite sym (↑ˡ-identityʳ idx₁ p)
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-- rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
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--
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-- cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
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-- (e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
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-- castᵉ p e ListMem.∈ es →
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-- e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
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-- cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
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-- rewrite FinProp.cast-is-id refl idx₁
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-- rewrite FinProp.cast-is-id refl idx₂ = e∈es
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--
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-- ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
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-- ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
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-- (Mk-⊆ n₁ p₁@refl newNodes₁ nsg₂≡nsg₁++newNodes₁ e∈g₁⇒e∈g₂)
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-- (Mk-⊆ n₂ p₂@refl newNodes₂ nsg₃≡nsg₂++newNodes₂ e∈g₂⇒e∈g₃)
|
||||
-- rewrite cast-is-id refl ns₂
|
||||
-- rewrite cast-is-id refl ns₃
|
||||
-- with refl ← nsg₂≡nsg₁++newNodes₁
|
||||
-- with refl ← nsg₃≡nsg₂++newNodes₂ =
|
||||
-- record
|
||||
-- { n = n₁ Nat.+ n₂
|
||||
-- ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
|
||||
-- ; newNodes = newNodes₁ ++ newNodes₂
|
||||
-- ; nsg₂≡nsg₁++newNodes = ++-assoc (+-assoc s₁ n₁ n₂) ns₁ newNodes₁ newNodes₂
|
||||
-- ; e∈g₁⇒e∈g₂ = λ {e} e∈g₁ →
|
||||
-- cast∈⇒∈subst (sym (+-assoc s₁ n₁ n₂)) (+-assoc s₁ n₁ n₂) _ _
|
||||
-- (subst (λ e' → e' ListMem.∈ es₃)
|
||||
-- (↑ˡ-Edge-assoc e (sym (+-assoc s₁ n₁ n₂)))
|
||||
-- (e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)))
|
||||
-- }
|
||||
--
|
||||
-- open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
|
||||
--
|
||||
-- instance
|
||||
-- IndexRelaxable : Relaxable Graph.Index
|
||||
-- IndexRelaxable = record
|
||||
-- { relax = λ { (Mk-⊆ n refl _ _ _) idx → idx ↑ˡ n }
|
||||
-- }
|
||||
--
|
||||
-- EdgeRelaxable : Relaxable Graph.Edge
|
||||
-- EdgeRelaxable = record
|
||||
-- { relax = λ g₁⊆g₂ (idx₁ , idx₂) →
|
||||
-- ( Relaxable.relax IndexRelaxable g₁⊆g₂ idx₁
|
||||
-- , Relaxable.relax IndexRelaxable g₁⊆g₂ idx₂
|
||||
-- )
|
||||
-- }
|
||||
--
|
||||
-- open Relaxable {{...}}
|
||||
--
|
||||
-- pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
|
||||
-- pushBasicBlock bss g =
|
||||
-- ( record
|
||||
-- { size = Graph.size g Nat.+ 1
|
||||
-- ; nodes = Graph.nodes g ++ (bss ∷ [])
|
||||
-- ; edges = List.map (λ e → ↑ˡ-Edge e 1) (Graph.edges g)
|
||||
-- }
|
||||
-- , ( Graph.size g ↑ʳ zero
|
||||
-- , record
|
||||
-- { n = 1
|
||||
-- ; sg₂≡sg₁+n = refl
|
||||
-- ; newNodes = (bss ∷ [])
|
||||
-- ; nsg₂≡nsg₁++newNodes = cast-is-id refl _
|
||||
-- ; e∈g₁⇒e∈g₂ = λ e∈g₁ → x∈xs⇒fx∈fxs (λ e → ↑ˡ-Edge e 1) e∈g₁
|
||||
-- }
|
||||
-- )
|
||||
-- )
|
||||
--
|
||||
-- pushEmptyBlock : MonotonicGraphFunction Graph.Index
|
||||
-- pushEmptyBlock = pushBasicBlock []
|
||||
--
|
||||
-- addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
|
||||
-- addEdges (MkGraph s ns es) es' =
|
||||
-- ( record
|
||||
-- { size = s
|
||||
-- ; nodes = ns
|
||||
-- ; edges = es' List.++ es
|
||||
-- }
|
||||
-- , record
|
||||
-- { n = 0
|
||||
-- ; sg₂≡sg₁+n = +-comm 0 s
|
||||
-- ; newNodes = []
|
||||
-- ; nsg₂≡nsg₁++newNodes = cast-sym _ (++-identityʳ (+-comm s 0) ns)
|
||||
-- ; e∈g₁⇒e∈g₂ = λ {e} e∈es →
|
||||
-- cast∈⇒∈subst (+-comm s 0) (+-comm 0 s) _ _
|
||||
-- (subst (λ e' → e' ListMem.∈ _)
|
||||
-- (↑ˡ-Edge-identityʳ e (+-comm s 0))
|
||||
-- (ListMemProp.∈-++⁺ʳ es' e∈es))
|
||||
-- }
|
||||
-- )
|
||||
--
|
||||
-- buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
|
||||
-- buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
|
||||
-- buildCfg (s₁ then s₂) =
|
||||
-- (buildCfg s₁ ⟨⊗⟩ buildCfg s₂)
|
||||
-- update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → addEdges g ((idx₂ , idx₃) ∷ []) })
|
||||
-- map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → (idx₁ , idx₄) })
|
||||
-- buildCfg (if _ then s₁ else s₂) =
|
||||
-- (buildCfg s₁ ⟨⊗⟩ buildCfg s₂ ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
|
||||
-- update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') →
|
||||
-- addEdges g ((idx , idx₁) ∷ (idx , idx₃) ∷ (idx₂ , idx') ∷ (idx₄ , idx') ∷ []) })
|
||||
-- map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') → (idx , idx') })
|
||||
-- buildCfg (while _ repeat s) =
|
||||
-- (buildCfg s ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
|
||||
-- update (λ { g ((idx₁ , idx₂) , idx , idx') →
|
||||
-- addEdges g ((idx , idx') ∷ (idx , idx₁) ∷ (idx₂ , idx) ∷ []) })
|
||||
-- map (λ { g ((idx₁ , idx₂) , idx , idx') → (idx , idx') })
|
||||
|
|
|
|||
|
|
@ -5,48 +5,24 @@ open import Language.Semantics
|
|||
open import Language.Graphs
|
||||
open import Language.Traces
|
||||
|
||||
open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
|
||||
open import Utils using (_⊗_; _,_)
|
||||
open Relaxable {{...}}
|
||||
open import Data.Fin as Fin using (zero)
|
||||
open import Data.List using (_∷_; [])
|
||||
open import Data.Product using (Σ; _,_)
|
||||
|
||||
open import Data.Fin using (zero)
|
||||
open import Data.List using (List; _∷_; [])
|
||||
open import Data.Vec using (_∷_; [])
|
||||
open import Data.Vec.Properties using (cast-is-id; lookup-++ˡ; lookup-++ʳ)
|
||||
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; subst)
|
||||
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
|
||||
|
||||
relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
|
||||
g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
|
||||
relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
|
||||
rewrite cast-is-id refl (Graph.nodes g₂)
|
||||
with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
|
||||
buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
|
||||
buildCfg-input ⟨ bs₁ ⟩ = (zero , refl)
|
||||
buildCfg-input (s₁ then s₂)
|
||||
with (idx , p) ← buildCfg-input s₁ rewrite p = (_ , refl)
|
||||
buildCfg-input (if _ then s₁ else s₂) = (zero , refl)
|
||||
buildCfg-input (while _ repeat s)
|
||||
with (idx , p) ← buildCfg-input s rewrite p = (_ , refl)
|
||||
|
||||
instance
|
||||
NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
|
||||
MonotonicPredicate (λ g n → g [ n ] ≡ bss)
|
||||
NodeEqualsMonotonic = record
|
||||
{ relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
|
||||
trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
|
||||
}
|
||||
|
||||
pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
|
||||
pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)
|
||||
|
||||
TransformsEnv : ∀ (ρ₁ ρ₂ : Env) → DependentPredicate (Graph.Index ⊗ Graph.Index)
|
||||
TransformsEnv ρ₁ ρ₂ g (idx₁ , idx₂) = Trace {g} idx₁ idx₂ ρ₁ ρ₂
|
||||
|
||||
instance
|
||||
TransformsEnvMonotonic : ∀ {ρ₁ ρ₂ : Env} → MonotonicPredicate (TransformsEnv ρ₁ ρ₂)
|
||||
TransformsEnvMonotonic = {!!}
|
||||
|
||||
buildCfg-sufficient : ∀ {ρ₁ ρ₂ : Env} {s : Stmt} → ρ₁ , s ⇒ˢ ρ₂ → Always (TransformsEnv ρ₁ ρ₂) (buildCfg s)
|
||||
buildCfg-sufficient {ρ₁} {ρ₂} {⟨ bs ⟩} (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
|
||||
pushBasicBlock-works (bs ∷ [])
|
||||
map-reason
|
||||
(λ g idx g[idx]≡[bs] → Trace-single (subst (ρ₁ ,_⇒ᵇˢ ρ₂)
|
||||
(sym g[idx]≡[bs])
|
||||
(ρ₁,bs⇒ρ₂ ∷ [])))
|
||||
buildCfg-sufficient {ρ₁} {ρ₂} {s₁ then s₂} (⇒ˢ-then ρ₁ ρ ρ₂ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
|
||||
(buildCfg-sufficient ρ₁,s₁⇒ρ₂ ⟨⊗⟩-reason buildCfg-sufficient ρ₂,s₂⇒ρ₃)
|
||||
update-reason {!!}
|
||||
map-reason {!!}
|
||||
buildCfg-output : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.outputs g ≡ idx ∷ [])
|
||||
buildCfg-output ⟨ bs₁ ⟩ = (zero , refl)
|
||||
buildCfg-output (s₁ then s₂)
|
||||
with (idx , p) ← buildCfg-output s₂ rewrite p = (_ , refl)
|
||||
buildCfg-output (if _ then s₁ else s₂) = (_ , refl)
|
||||
buildCfg-output (while _ repeat s)
|
||||
with (idx , p) ← buildCfg-output s rewrite p = (_ , refl)
|
||||
|
|
|
|||
10
Utils.agda
10
Utils.agda
|
|
@ -1,9 +1,11 @@
|
|||
module Utils where
|
||||
|
||||
open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
|
||||
open import Data.Product as Prod using ()
|
||||
open import Data.Nat using (ℕ; suc)
|
||||
open import Data.List using (List; []; _∷_; _++_) renaming (map to mapˡ)
|
||||
open import Data.List using (List; cartesianProduct; []; _∷_; _++_) renaming (map to mapˡ)
|
||||
open import Data.List.Membership.Propositional using (_∈_)
|
||||
open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
|
||||
open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
|
||||
open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
|
||||
open import Function.Definitions using (Injective)
|
||||
|
|
@ -78,3 +80,9 @@ proj₁ (v , _) = v
|
|||
|
||||
proj₂ : ∀ {a p q} {A : Set a} {P : A → Set p} {Q : A → Set q} {a : A} → (P ⊗ Q) a → Q a
|
||||
proj₂ (_ , v) = v
|
||||
|
||||
∈-cartesianProduct : ∀ {a b} {A : Set a} {B : Set b}
|
||||
{x : A} {xs : List A} {y : B} {ys : List B} →
|
||||
x ∈ xs → y ∈ ys → (x Prod., y) ∈ cartesianProduct xs ys
|
||||
∈-cartesianProduct {x = x} (here refl) y∈ys = ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (x Prod.,_) y∈ys)
|
||||
∈-cartesianProduct {x = x} {xs = x' ∷ _} {ys = ys} (there x∈rest) y∈ys = ListMemProp.∈-++⁺ʳ (mapˡ (x' Prod.,_) ys) (∈-cartesianProduct x∈rest y∈ys)
|
||||
|
|
|
|||
Loading…
Reference in New Issue