Investigations on graph-theoretical constructions in Homotopy type theory

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

module lib.graph-transformations.Inv
  where
  open import foundations.Core

  open import lib.graph-definitions.Graph
  open import lib.graph-homomorphisms.Hom
  open import lib.graph-homomorphisms.classes.Isomorphisms

  private
    variable
      ℓ : Level

  -- This functor changes the direction on edges.
  Inv : Graph ℓ → Graph ℓ
  Inv  g@(graph Ng Eg Ng-set Eg-set)
    = graph Ng (λ x y → Edge g y x) Ng-set λ x y → Eg-set y x

  open import lib.graph-relations.Isomorphic

  -- Let us prove that given a graph G, G ≅ Inv (G).
  Inv-is-idempotent : (G : Graph ℓ) → G ≡ Inv (Inv (G))
  Inv-is-idempotent G = idp

  Hom-Inv-from
    : ∀ {G H : Graph ℓ}
    → Hom G H
    → Hom (Inv G) (Inv H)
  Hom-Inv-from h = hom (Hom.α h) λ x y e → Hom.β h y x e

Latest changes

(2022-12-28)(57c278b4) Last updated: 2021-09-16 15:00:00 by jonathan.cubides
(2022-07-06)(d3a4a8cf) minors by jonathan.cubides
(2022-01-26)(4aef326b) [ reports ] added some revisions by jonathan.cubides
(2021-12-20)(049db6a8) Added code of cubical experiments. by jonathan.cubides
(2021-12-20)(961730c9) [ html ] regular update by jonathan.cubides
(2021-12-20)(e0ef9faa) Fixed compilation and format, remove hidden parts by jonathan.cubides
(2021-12-20)(5120e5d1) Added cubical experiment to the master by jonathan.cubides
(2021-12-17)(828fdd0a) More revisions added for CPP by jonathan.cubides
(2021-12-15)(0d6a99d8) Fixed some broken links and descriptions by jonathan.cubides
(2021-12-15)(662a1f2d) [ .gitignore ] add by jonathan.cubides
(2021-12-15)(0630ce66) Minor fixes by jonathan.cubides
(2021-12-13)(04f10eba) Fixed a lot of details by jonathan.cubides
(2021-12-10)(24195c9f) [ .gitignore ] ignore .zip and arxiv related files by jonathan.cubides
(2021-12-09)(538d2859) minor fixes before dinner by jonathan.cubides
(2021-12-09)(36a1a69f) [ planar.pdf ] w.i.p revisions to share on arxiv first by jonathan.cubides