lib.graph-walks.Walk.SigmaWalks.md.
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{-# OPTIONS --without-K --exact-split #-} module lib.graph-walks.Walk.SigmaWalks where open import foundations.Core open import lib.graph-definitions.Graph open import lib.graph-walks.Walk renaming (length to len) open Graph module _ {ℓ : Level} (G : Graph ℓ) where
Walk-∑ : ℕ → Node G → Node G → Type ℓ Walk-∑ 0 x y = x ≡ y Walk-∑ (succ n) a c = ∑[ b ] (Edge G a b) × Walk-∑ n b c to-∑ : ∀ {x y} → Walk G x y → ∑[ n ] Walk-∑ n x y to-∑ ⟨ x ⟩ = (0 , idp) to-∑ (e ⊙ w) with (to-∑ w) ... | (n , q) = succ n , _ , e , q from-∑ : ∀ {x y} → ∑[ n ] Walk-∑ n x y → Walk G x y from-∑ (zero , idp) = ⟨ _ ⟩ from-∑ (succ n , (b , (e , w))) = e ⊙ from-∑ (n , w) Walk-is-equiv-to-Walk-∑ : ∀ {x y} → Walk G x y ≃ (∑[ n ] Walk-∑ n x y) Walk-is-equiv-to-Walk-∑ = qinv-≃ to-∑ (from-∑ , h1 , h2) where h1 : ∀ {x y} → (q : ∑[ n ] Walk-∑ n x y) → to-∑ (from-∑ q) ≡ q h1 (zero , idp) = idp h1 (succ n , (b , (e , w))) with to-∑ (from-∑ (n , w)) | h1 (n , w) ... | .n , .w | idp = idp h2 : ∀ {x y} → (w : Walk G x y) → from-∑ (to-∑ w) ≡ w h2 ⟨ _ ⟩ = idp h2 (e ⊙ w) = ap (e ⊙_) (h2 w)
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