proposition: Lemma trivial_coloringP (G : sgraph) (A : {set G}) : coloring (trivial_coloring A) A.
proof: Proof. apply/andP; split; last by apply/forall_inP=> ? /imsetP[x xA ->]; exact: stable1. suff -> : trivial_coloring A = preim_partition id A by apply: preim_partitionP. have E x : x \in A -> [set x] = [set y in A | x == y]. by move=> xA; apply/setP => y; rewrite !inE eq_sym andb_idl // => /eqP<-. by apply/setP => P; apply/imsetP/imsetP => -[x xA ->]; exists x => //; rewrite E. Qed.
imports: mathcomp(all_ssreflect) GraphTheory(preliminaries bij digraph) GraphTheory(sgraph dom partition)