Is there a well established Coq graph library for proving simple theorems ?
I would like to learn how to prove simple stuff like: "G1, G2 are isomorphic if and only if their complements are isomorphic".
Are there related/similar examples or tutorials available?
Here's a demonstration.
Allow rewriting with equivalence relations.
Require Import Coq.Setoids.Setoid.
A graph is a set of vertices along with an adjacency relation.
Definition graph : Type := {V : Type & V -> V -> bool}.
A graph from vertices and adjacency.
Definition create : forall V, (V -> V -> bool) -> graph := @existT _ _.
Vertices from a graph.
Definition vertices : graph -> Type := @projT1 _ _.
Adjacency from a graph.
Definition adjacent : forall g1, vertices g1 -> vertices g1 -> bool := @projT2 _ _.
The complement of a graph has the same vertices, but a negated adjacency relation.
Definition complement : graph -> graph := fun g1 => create (vertices g1) (fun v1 v2 => negb (adjacent g1 v1 v2)).
Standard stuff.
Definition injective : forall {t1 t2}, (t1 -> t2) -> Prop := fun t1 t2 f1 => forall x1 x2, f1 x1 = f1 x2 -> x1 = x2.
Definition surjective : forall {t1 t2}, (t1 -> t2) -> Prop := fun t1 t2 f1 => forall x1, exists x2, f1 x2 = x1.
Definition bijective : forall {t1 t2}, (t1 -> t2) -> Prop := fun t1 t2 f1 => injective f1 /\ surjective f1.
Two graphs are isomorphic if there's a bijection between their vertices that preserves adjacency.
Definition isomorphic : graph -> graph -> Prop := fun g1 g2 => exists f1, bijective f1 /\ (forall x1 x2, adjacent g1 x1 x2 = adjacent g2 (f1 x1) (f1 x2)).
Infix "~" := isomorphic (at level 70).
A helpful fact whose proof I leave to you.
Conjecture C1 : forall b1 b2, negb b1 = negb b2 <-> b1 = b2.
Your fact.
Goal forall g1 g2, g1 ~ g2 <-> complement g1 ~ complement g2.
Proof.
Access the components of the graphs.
destruct g1.
destruct g2.
Substitute defined by definition.
unfold isomorphic, complement, adjacent, vertices, create, projT2, projT1.
First-order logic simplifications.
firstorder.
Instanciation.
exists x1.
firstorder.
rewrite C1.
firstorder.
exists x1.
firstorder.
Instanciation.
specialize (H0 x2 x3).
rewrite C1 in H0.
firstorder.
Qed.
Actually, this is a formalization of graphs whose adjacency relation is decidable V -> V -> bool. In intuitionistic logic, not all graphs with a general adjacency relation V -> V -> Prop have the property you want to prove.
Instead of sticking to finite or otherwise decidable graphs, you can also move to classical logic or use double negation translation.