proposition: Lemma cat_reducible_range j1 j2 cfs : reducible_in_range j1 j2 cfs -> forall j3, reducible_in_range j2 j3 cfs -> reducible_in_range j1 j3 cfs.
proof: Proof. move=> red_12 j3 red_j23 i j1_le_i i_lt_j3. by case: (ltnP i j2) => [/red_12 | /red_j23]; apply. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq choice) mathcomp(fintype path fingraph) fourcolor(hypermap geometry color chromogram coloring kempe) fourcolor(cfmap cfcolor cfcontract ctree kempetree)