proposition: Lemma sle_cat ks1 ks2 x y : x <=[ks1++ks2] y = if x \in ks1 then x <=[ks1] y else (y \notin ks1) && x <=[ks2] y.
proof: Proof. rewrite /seq_le !index_cat. case X: (x \in ks1); case Y: (y \in ks1)=>//=. - move/negbT/index_memN: Y=>Y. by rewrite Y index_size ltnW // ltn_addr // index_mem. - rewrite -index_mem in Y. apply/negP=>H; move/(leq_ltn_trans H): Y. by rewrite ltnNge leq_addr. by rewrite leq_add2l. Qed.
imports: Coq(ssreflect ssrbool ssrfun) mathcomp(ssrnat eqtype seq path interval order) pcm(options prelude ordtype seqext)