Fixpoint fcat' (s1 : fmap) (s2 : seq (K * V)) {struct s2} : fmap :=
  if s2 is (k, v)::t then fcat' (ins k v s1) t else s1.Definition fcat s1 s2 := fcat' s1 (seq_of s2).

Lemma fcat_sins k v s1 s2 : fcat s1 (ins k v s2) = ins k v (fcat s1 s2).