Investigations on graph-theoretical constructions in Homotopy type theory

{-# OPTIONS --without-K --exact-split   #-}

module lib.graph-walks.Walk.isSet
  where
  open import foundations.Core

  open import lib.graph-definitions.Graph
  open Graph
  open import lib.graph-walks.Walk renaming (length to len)

  module _ {ℓ : Level}(G : Graph ℓ) where
    open import foundations.NaturalsType
    open import lib.graph-walks.Walk.Equality
    open import lib.graph-walks.Walk.SigmaWalks

    Walk-∑-is-set : ∀ {x y} → isSet ((∑[ n ] Walk-∑ G n x y))
    Walk-∑-is-set {x = x}{y} = ∑-set nat-is-set helper
      where
      helper : ∀ {x y} → (n : ℕ) → isSet (Walk-∑ G n x y)
      helper zero     = prop-is-set (Node-is-set G _ _)
      helper (succ n)
        = ∑-set (Node-is-set G) (λ x' → ∑-set (Edge-is-set G _ _) (λ e → helper n))

    Walk-is-set : ∀ {x y} → isSet (Walk G x y)
    Walk-is-set = is-set-equiv-to-set (≃-sym (Walk-is-equiv-to-Walk-∑ G)) Walk-∑-is-set

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