DanilaFe/agda-spa
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Translate informal proof of (node) transitivity into formal one.

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-08 20:57:08 -07:00
parent 4f14a7b765
commit 85fdf544b9
1 changed files with 28 additions and 25 deletions
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53
Language.agda
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@ -6,7 +6,7 @@ open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
open import Data.String using (String) renaming (_≟_ to _≟ˢ_) open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂) open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_; cast) open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_; cast)
open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ) open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ; lookup-cast₁)
open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id) open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id)
open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_) open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_)
open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ) open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ)
@ -19,7 +19,7 @@ open import Data.Fin.Properties using (suc-injective)
open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl) open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
open import Relation.Nullary using (¬_) open import Relation.Nullary using (¬_)
open import Function using (_∘_) open import Function using (_∘_)
open Eq.≡-Reasoning using (begin_; step-≡; _∎) open Eq.≡-Reasoning
open import Lattice open import Lattice
open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs; _⊗_; _,_) open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs; _⊗_; _,_)
@ -121,36 +121,39 @@ module Graphs where
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))
flatten-casts : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
(p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂) (r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
(idx : Fin s₁) →
castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂))
flatten-casts refl refl r zero = refl
flatten-casts {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
rewrite flatten-casts refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl
⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃ ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃} ⊆-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 (Mk-⊆ n₁ p₁@refl g₁[]≡g₂[] e∈g₁⇒e∈g₂) (Mk-⊆ n₂ p₂@refl g₂[]≡g₃[] e∈g₂⇒e∈g₃) = record
{ n = n₁ +ⁿ n₂ { n = n₁ +ⁿ n₂
; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂ ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
; g₁[]≡g₂[] = {!!} ; g₁[]≡g₂[] = λ idx →
begin
lookup ns₁ idx
≡⟨ g₁[]≡g₂[] _ ⟩
lookup (cast p₁ ns₂) (idx ↑ˡ n₁)
≡⟨ lookup-cast₁ p₁ ns₂ _ ⟩
lookup ns₂ (castᶠ (sym p₁) (idx ↑ˡ n₁))
≡⟨ g₂[]≡g₃[] _ ⟩
lookup (cast p₂ ns₃) ((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)
≡⟨ lookup-cast₁ p₂ _ _ ⟩
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) ⟩
lookup ns₃ (castᶠ (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ +ⁿ n₂)))
≡⟨ sym (lookup-cast₁ (+-assoc s₁ n₁ n₂) _ _) ⟩
lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ +ⁿ n₂))
∎
; e∈g₁⇒e∈g₂ = {!!} ; e∈g₁⇒e∈g₂ = {!!}
-- lookup ns₁ idx ≡ lookup (cast p₁ ns₂) (idx ↑ˡ n) -- by g₁[]≡g₂[]
-- lookup (cast p₁ ns₂) (idx ↑ˡ n) ≡ lookup ns₂ (Fin.cast (sym p₁) (idx ↑ˡ n)) -- by lookup-cast₁
-- -------- s₁ + n₁ → s₂
-- ---------- Fin (s₁ + n₁)
-- ----------------------------- Fin s₂
-- lookup ns₂ (Fin.cast (sym p₁) (idx ↑ˡ n)) ≡ lookup (cast p₂ ns₃) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ n₂)) -- by g₂[]≡g₃[]
-- lookup (cast p₂ ns₃) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ n₂)) ≡ lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ n₂))) -- by lookup-cast₂
-- lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ n₂))) ≡ lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) -- by flatten-casts
--
-- lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) ≡ lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ + n₂)) ∎
} }
where
flatten-casts : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
(p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂)
(r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
(idx : Fin s₁) →
castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂))
flatten-casts refl refl r zero = refl
flatten-casts {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
rewrite flatten-casts 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