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_ins' k v s1 s2 : k \notin (map key s2) -> fcat' (ins k v s1) s2 = ins k v (fcat' s1 s2).