Investigations on graph-theoretical constructions in Homotopy type theory

{-# 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-∑  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 ⟩   = ( , 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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