Investigations on graph-theoretical constructions in Homotopy type theory

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

module lib.graph-homomorphisms.classes.Injective
  where
  open import foundations.Core
  open import lib.graph-definitions.Graph
  open import lib.graph-homomorphisms.Hom
  open Hom
  open import lib.graph-homomorphisms.classes.EdgeInjective

  private
    variable
      ℓ ℓ₁ ℓ₂ : Level

  module _ {G : Graph ℓ₁}{H : Graph ℓ₂}
    where

    isNodeInj : Hom G H → Type (ℓ₁ ⊔ ℓ₂)
    isNodeInj  (hom α _) = ∀ {x y : Node G} → (α x ≡ α y) → x ≡ y
    _is-node-inj = isNodeInj

    being-node-inj-is-prop : (h : Hom G H) → (isNodeInj h) is-prop
    being-node-inj-is-prop h
      = pi-is-prop-implicit (λ x → pi-is-prop-implicit (λ y →
        pi-is-prop (λ _ → Node-is-set G _ _)))

    isHomInj : Hom G H → Type (ℓ₁ ⊔ ℓ₂)
    isHomInj h = isNodeInj h × isEdgeInj h
    _is-hom-inj = isHomInj

    being-hom-inj-is-prop : (h : Hom G H) → (isHomInj h) is-prop
    being-hom-inj-is-prop h = ∑-prop (being-node-inj-is-prop h)
                                     (λ _ → being-edge-inj-is-prop h)

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