Use concatenation to represent adding new nodes
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -6,7 +6,7 @@ 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; lookup-++ˡ; lookup-cast₁)
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open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ; lookup-cast₁; cast-sym)
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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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@ -114,36 +114,27 @@ module Graphs where
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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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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 ∈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 {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
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(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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; g₁[]≡g₂[] = λ idx →
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begin
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lookup ns₁ idx
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≡⟨ g₁[]≡g₂[] _ ⟩
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lookup (cast p₁ ns₂) (idx ↑ˡ n₁)
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≡⟨ lookup-cast₁ p₁ ns₂ _ ⟩
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lookup ns₂ (castᶠ (sym p₁) (idx ↑ˡ n₁))
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≡⟨ g₂[]≡g₃[] _ ⟩
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lookup (cast p₂ ns₃) ((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)
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≡⟨ lookup-cast₁ p₂ _ _ ⟩
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lookup ns₃ (castᶠ (sym p₂) (((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)))
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≡⟨ cong (lookup ns₃) (↑ˡ-assoc (sym p₂) (sym p₁) (sym (+-assoc s₁ n₁ n₂)) idx) ⟩
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lookup ns₃ (castᶠ (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ +ⁿ n₂)))
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≡⟨ sym (lookup-cast₁ (+-assoc s₁ n₁ n₂) _ _) ⟩
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lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ +ⁿ n₂))
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∎
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; e∈g₁⇒e∈g₂ = {!!} -- λ e∈g₁ → e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)
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}
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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 +ⁿ 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 = {!!}
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}
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where
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↑ˡ-assoc : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
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(p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂)
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@ -160,7 +151,7 @@ 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 { (Mk-⊆ n refl _ _) idx → idx ↑ˡ n }
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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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@ -185,9 +176,9 @@ module Graphs where
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relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
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g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
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relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl g₁[]≡g₂[] _) idx =
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trans (g₁[]≡g₂[] idx) (cong (λ vec → lookup vec (idx ↑ˡ n))
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(cast-is-id refl (Graph.nodes g₂)))
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relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
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rewrite cast-is-id refl (Graph.nodes g₂)
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with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
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MonotonicGraphFunction : (Graph → Set) → Set
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MonotonicGraphFunction T = (g₁ : Graph) → Σ Graph (λ g₂ → T g₂ × g₁ ⊆ g₂)
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@ -228,8 +219,9 @@ module Graphs where
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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∈g₁ → x∈xs⇒fx∈fxs (λ e' → ↑ˡ-Edge e' 1) e∈g₁
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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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@ -244,14 +236,8 @@ module Graphs where
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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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; g₁[]≡g₂[] = λ idx →
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begin
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lookup ns idx
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≡⟨ cong (lookup ns) (↑ˡ-identityʳ (sym (+-comm 0 s)) idx) ⟩
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lookup ns (castᶠ (sym (+-comm 0 s)) (idx ↑ˡ 0))
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≡⟨ sym (lookup-cast₁ (+-comm 0 s) _ _) ⟩
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lookup (cast (+-comm 0 s) ns) (idx ↑ˡ 0)
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∎
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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 = {!!}
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}
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)
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