Local Open Scope positive_scope.
(** * Properties of coloring maps *)
(** ** Definition of a set of colors *)
Definition colors := S.t.
(** ** A coloring is complete if every vertex is colored *)
Definition coloring_complete (palette: colors) (g: graph) (f: coloring) :=
(forall i, M.In i g -> M.In i f) /\ coloring_ok palette g f.
Example ex_graph :=
mk_graph [ (6,4); (4,5); (4,3); (3,2); (5,2); (1,2); (1,5) ]%positive.
Definition two_colors: colors := SP.of_list [1; 2]%positive.
Definition three_colors: colors := SP.of_list [1; 2; 3]%positive.
Local Open Scope positive_scope.
(* Example of 3-coloring the example graph and proof with ok_coloring *)
Example ex_coloring :=
fold_right (fun p m => M.add (fst p) (snd p) m) (@M.empty _) [(6,1);(3,1);(5,1);(4,2);(2,2);(1,3)].
(* Proving that an explicit graph coloring is a valid coloring *)
Example ex_coloring_good : coloring_ok three_colors ex_graph ex_coloring.

(** ** A set is extensionally equal to folding over its elements *)
Lemma set_elemeum s : S.Equal s (fold_right S.add S.empty (S.elements s)).