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Add proof of node preservation for adding edges.

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-08 22:43:07 -07:00
parent 85fdf544b9
commit 78252b6c9e
1 changed files with 27 additions and 15 deletions
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42
Language.agda
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@ -117,7 +117,7 @@ module Graphs where
g₁[]≡g₂[] : ∀ (idx : Graph.Index g₁) → g₁[]≡g₂[] : ∀ (idx : Graph.Index g₁) →
lookup (Graph.nodes g₁) idx ≡ lookup (Graph.nodes g₁) idx ≡
lookup (cast sg₂≡sg₁+n (Graph.nodes g₂)) (idx ↑ˡ n) lookup (cast sg₂≡sg₁+n (Graph.nodes g₂)) (idx ↑ˡ n)
e∈g₁⇒e∈g₂ : ∀ (e : Graph.Edge g₁) → e∈g₁⇒e∈g₂ : ∀ {e : Graph.Edge g₁} →
e ∈ˡ (Graph.edges g₁) → e ∈ˡ (Graph.edges g₁) →
(↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂)) (↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
@ -137,23 +137,22 @@ module Graphs where
lookup (cast p₂ ns₃) ((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂) lookup (cast p₂ ns₃) ((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)
≡⟨ lookup-cast₁ p₂ _ _ ⟩ ≡⟨ lookup-cast₁ p₂ _ _ ⟩
lookup ns₃ (castᶠ (sym p₂) (((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂))) lookup ns₃ (castᶠ (sym p₂) (((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)))
≡⟨ cong (lookup ns₃) (flatten-casts (sym p₂) (sym p₁) (sym (+-assoc s₁ n₁ n₂)) idx) ⟩ ≡⟨ cong (lookup ns₃) (↑ˡ-assoc (sym p₂) (sym p₁) (sym (+-assoc s₁ n₁ n₂)) idx) ⟩
lookup ns₃ (castᶠ (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ +ⁿ n₂))) lookup ns₃ (castᶠ (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ +ⁿ n₂)))
≡⟨ sym (lookup-cast₁ (+-assoc s₁ n₁ n₂) _ _) ⟩ ≡⟨ sym (lookup-cast₁ (+-assoc s₁ n₁ n₂) _ _) ⟩
lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ +ⁿ n₂)) lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ +ⁿ n₂))
∎ ∎
; e∈g₁⇒e∈g₂ = {!!} ; e∈g₁⇒e∈g₂ = {!!} -- λ e∈g₁ → e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)
} }
where where
flatten-casts : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ} ↑ˡ-assoc : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
(p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂) (p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂)
(r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃) (r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
(idx : Fin s₁) → (idx : Fin s₁) →
castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂)) castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂))
flatten-casts refl refl r zero = refl ↑ˡ-assoc refl refl r zero = refl
flatten-casts {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx') ↑ˡ-assoc {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
rewrite flatten-casts refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl rewrite ↑ˡ-assoc refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl
record Relaxable (T : Graph → Set) : Set where record Relaxable (T : Graph → Set) : Set where
field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂ field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂
@ -230,25 +229,38 @@ module Graphs where
{ n = 1 { n = 1
; sg₂≡sg₁+n = refl ; sg₂≡sg₁+n = refl
; 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 ∷ []))))) ; 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 ∷ [])))))
; e∈g₁⇒e∈g₂ = λ e e∈g₁ → x∈xs⇒fx∈fxs (λ e' → ↑ˡ-Edge e' 1) e∈g₁ ; e∈g₁⇒e∈g₂ = λ e∈g₁ → x∈xs⇒fx∈fxs (λ e' → ↑ˡ-Edge e' 1) e∈g₁
} }
) )
) )
addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g') addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
addEdges g es = addEdges (MkGraph s ns es) es' =
( record ( record
{ size = Graph.size g { size = s
; nodes = Graph.nodes g ; nodes = ns
; edges = es ++ˡ Graph.edges g ; edges = es' ++ˡ es
} }
, record , record
{ n = 0 { n = 0
; sg₂≡sg₁+n = +-comm 0 (Graph.size g) ; sg₂≡sg₁+n = +-comm 0 s
; g₁[]≡g₂[] = {!!} ; g₁[]≡g₂[] = λ idx →
begin
lookup ns idx
≡⟨ cong (lookup ns) (↑ˡ-identityʳ (sym (+-comm 0 s)) idx) ⟩
lookup ns (castᶠ (sym (+-comm 0 s)) (idx ↑ˡ 0))
≡⟨ sym (lookup-cast₁ (+-comm 0 s) _ _) ⟩
lookup (cast (+-comm 0 s) ns) (idx ↑ˡ 0)
∎
; e∈g₁⇒e∈g₂ = {!!} ; e∈g₁⇒e∈g₂ = {!!}
} }
) )
where
↑ˡ-identityʳ : ∀ {s} (p : s +ⁿ 0 ≡ s) (idx : Fin s) →
idx ≡ castᶠ p (idx ↑ˡ 0)
↑ˡ-identityʳ p zero = refl
↑ˡ-identityʳ {suc s'} p (suc f')
rewrite sym (↑ˡ-identityʳ (+-comm s' 0) f') = refl
pushEmptyBlock : MonotonicGraphFunction Graph.Index pushEmptyBlock : MonotonicGraphFunction Graph.Index
pushEmptyBlock = pushBasicBlock [] pushEmptyBlock = pushBasicBlock []