proposition: Lemma fcat_ins' k v s1 s2 : k \notin (map key s2) -> fcat' (ins k v s1) s2 = ins k v (fcat' s1 s2).
proof: Proof. move=>H; elim: s2 k v s1 H=>[|[k2 v2] s2 IH] k1 v1 s1 //=. rewrite inE negb_or; case/andP=>H1 H2. by rewrite -IH // ins_ins eq_sym (negbTE H1). Qed.
imports: Coq(ssreflect ssrbool ssrfun) mathcomp(ssrnat eqtype seq path) pcm(options)