Try using index-based comparisons
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -6,18 +6,20 @@ 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; _∷_; []; _++_; cast)
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open import Data.Vec.Properties using (++-assoc; ++-identityʳ)
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open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ)
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open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id)
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open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_)
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open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ)
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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.List.Relation.Unary.Any.Properties using (++⁺ʳ)
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open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_; _↑ˡ_) 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 (subst; cong; _≡_; sym; refl)
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open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
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open import Relation.Nullary using (¬_)
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open import Function using (_∘_)
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open Eq.≡-Reasoning using (begin_; step-≡; _∎)
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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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@ -101,47 +103,29 @@ module Graphs where
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nodes : Vec (List BasicStmt) size
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edges : List Edge
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Graph-build-≡ : ∀ (g₁ g₂ : Graph) (p : Graph.size g₁ ≡ Graph.size g₂) →
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(cast p (Graph.nodes g₁) ≡ Graph.nodes g₂) →
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(subst (λ s → List (Fin s × Fin s)) p (Graph.edges g₁) ≡ Graph.edges g₂) →
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g₁ ≡ g₂
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Graph-build-≡ g₁ g₂ refl cns₁≡ns₂ refl
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rewrite cast-is-id refl (Graph.nodes g₁)
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rewrite cns₁≡ns₂ = refl
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↑ˡ-Edge : ∀ {n} → Fin n × Fin n → ∀ m → Fin (n +ⁿ m) × Fin (n +ⁿ m)
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↑ˡ-Edge : ∀ {n} → (Fin n × Fin n) → ∀ m → (Fin (n +ⁿ m) × Fin (n +ⁿ m))
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↑ˡ-Edge (idx₁ , idx₂) m = (idx₁ ↑ˡ m , idx₂ ↑ˡ m)
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↑ʳ-Edge : ∀ {n} m → Fin n × Fin n → Fin (m +ⁿ n) × Fin (m +ⁿ n)
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↑ʳ-Edge m (idx₁ , idx₂) = (m ↑ʳ idx₁ , m ↑ʳ idx₂)
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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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_∙_ : Graph → Graph → Graph
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_∙_ (MkGraph s₁ ns₁ es₁) (MkGraph s₂ ns₂ es₂) = MkGraph
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(s₁ +ⁿ s₂)
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(ns₁ ++ ns₂)
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(
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let
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edges₁ = mapˡ (λ e → ↑ˡ-Edge e s₂) es₁
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edges₂ = mapˡ (↑ʳ-Edge s₁) es₂
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in
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edges₁ ++ˡ edges₂
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)
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∙-assoc : ∀ (g₁ g₂ g₃ : Graph) → g₁ ∙ (g₂ ∙ g₃) ≡ (g₁ ∙ g₂) ∙ g₃
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∙-assoc = {!!}
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∙-zero : ∀ (g : Graph) → g ∙ (MkGraph 0 [] []) ≡ g
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∙-zero (MkGraph s ns es) = Graph-build-≡ _ _ (+-comm s 0) (++-identityʳ (+-comm s 0) ns) {!!}
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_⊆_ : Graph → Graph → Set
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_⊆_ g₁ g₂ = Σ Graph (λ g' → g₁ ∙ g' ≡ g₂)
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⊆-refl : ∀ (g : Graph) → g ⊆ g
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⊆-refl g = (MkGraph 0 [] [] , ∙-zero g)
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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 +ⁿ n
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g₁[]≡g₂[] : ∀ (idx : Graph.Index g₁) →
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lookup (Graph.nodes g₁) idx ≡
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lookup (cast sg₂≡sg₁+n (Graph.nodes g₂)) (idx ↑ˡ n)
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e∈g₁⇒e∈g₂ : ∀ (e : Graph.Edge g₁) →
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e ∈ˡ (Graph.edges g₁) →
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(↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
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⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
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⊆-trans {g₁} {g₂} {g₃} (g₁₂ , refl) (g₂₃ , refl) = ((g₁₂ ∙ g₂₃) , ∙-assoc g₁ g₁₂ g₂₃)
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⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃} (Mk-⊆ n₁ p₁@refl g₁[]≡g₂[] e∈g₁⇒e∈g₂) (Mk-⊆ n₂ p₂@refl g₂[]≡g₃[] e∈g₂⇒e∈g₃) = record
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{ n = n₁ +ⁿ n₂
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; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
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}
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record Relaxable (T : Graph → Set) : Set where
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field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂
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@ -149,7 +133,7 @@ module Graphs where
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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 { (g' , refl) idx → idx ↑ˡ (Graph.size g') }
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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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@ -199,19 +183,36 @@ module Graphs where
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module Construction where
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pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
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pushBasicBlock bss g₁ =
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let
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g' : Graph
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g' = record
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{ size = 1
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; nodes = bss ∷ []
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; edges = []
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pushBasicBlock bss g =
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( record
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{ size = Graph.size g +ⁿ 1
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; nodes = Graph.nodes g ++ (bss ∷ [])
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; edges = mapˡ (λ e → ↑ˡ-Edge e 1) (Graph.edges g)
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}
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in
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(g₁ ∙ g' , (Graph.size g₁ ↑ʳ zero , (g' , refl)))
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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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; g₁[]≡g₂[] = λ idx → trans (sym (lookup-++ˡ (Graph.nodes g) (bss ∷ []) idx)) (sym (cong (λ vec → lookup vec (idx ↑ˡ 1)) (cast-is-id refl (Graph.nodes g ++ (bss ∷ [])))))
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; e∈g₁⇒e∈g₂ = λ e 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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addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
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addEdges g es = {!!}
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addEdges g es =
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( record
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{ size = Graph.size g
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; nodes = Graph.nodes g
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; edges = es ++ˡ Graph.edges g
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}
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, record
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{ n = 0
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; sg₂≡sg₁+n = +-comm 0 (Graph.size g)
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; g₁[]≡g₂[] = {!!}
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; e∈g₁⇒e∈g₂ = {!!}
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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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