Investigations on graph-theoretical constructions in Homotopy type theory

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

module lib.graph-embeddings.Map.Spherical
  where
  open import foundations.Core
  open import lib.graph-definitions.Graph
  open import lib.graph-transformations.U
  open Graph

  module _ {ℓ : Level} (G : Graph ℓ)
    where
    open import lib.graph-embeddings.Map
    open import lib.graph-embeddings.Map.Face.Walk.Homotopy
    open import lib.graph-walks.Walk
    open import lib.graph-walks.Walk.QuasiSimple
    open quasi-simple-walks {ℓ = ℓ}{U G}

    isSphericalMap : Map G → Type (lsuc ℓ)
    isSphericalMap 𝕄
      = (x y : Node (U G))
      → (w₁ w₂ : Walk (U G) x y)
      → w₁ is-quasi-simple
      → w₂ is-quasi-simple
      → ∥ w₁ ∼⟨ 𝕄 ⟩∼ w₂ ∥

    isSphericalMap-is-prop : (𝕄 : Map G) → isProp (isSphericalMap 𝕄)
    isSphericalMap-is-prop 𝕄 =
       pi-is-prop
        $ λ _ → pi-is-prop
          $ λ _ → pi-is-prop
            $ λ _ → pi-is-prop
             $ λ _ → pi-is-prop
               $ λ _ → pi-is-prop
                 $ λ _ → truncated-is-prop

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