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).

(* Specifications: *)
Lemma fcat_ins' k v s1 s2 : k \notin (map key s2) -> fcat' (ins k v s1) s2 = ins k v (fcat' s1 s2).
Lemma fcat_nil' s : fcat' nil (seq_of s) = s.
Lemma fcat0s s : fcat nil s = s.
Lemma fcats0 s : fcat s nil = s.
Lemma fcat_inss k v s1 s2 : k \notin supp s2 -> fcat (ins k v s1) s2 = ins k v (fcat s1 s2).
Lemma fcat_sins k v s1 s2 : fcat s1 (ins k v s2) = ins k v (fcat s1 s2).
Lemma fcat_rems k s1 s2 : k \notin supp s2 -> fcat (rem k s1) s2 = rem k (fcat s1 s2).
Lemma fcat_srem k s1 s2 : k \notin supp s1 -> fcat s1 (rem k s2) = rem k (fcat s1 s2).
Lemma fnd_fcat k s1 s2 : fnd k (fcat s1 s2) = if k \in supp s2 then fnd k s2 else fnd k s1.
Lemma supp_fcat s1 s2 : supp (fcat s1 s2) =i [predU supp s1 & supp s2].
Lemma supp_eq_ins (s1 s2 : fmap) k1 k2 v1 v2 : path ord k1 (supp s1) -> path ord k2 (supp s2) -> supp (ins k1 v1 s1) =i supp (ins k2 v2 s2) -> k1 = k2 /\ supp s1 =i supp s2.
Lemma fmap_ind2 (P : fmap -> fmap -> Prop) : P nil nil -> (forall k v1 v2 s1 s2, path ord k (supp s1) -> path ord k (supp s2) -> P s1 s2 -> P (ins k v1 s1) (ins k v2 s2)) -> forall s1 s2, supp s1 =i supp s2 -> P s1 s2.