Definition mapf (m : finMap K U) : finMap K V := 
  let: FinMap _ pf := m in FinMap (sorted_map pf).

(* Specifications: *)
Lemma mapf_ins k v s : mapf (ins k v s) = ins k (f v) (mapf s).
Lemma mapf_fcat s1 s2 : mapf (fcat s1 s2) = fcat (mapf s1) (mapf s2).
Lemma mapf_disjL s1 s2 s : mapf s1 = mapf s2 -> disj s1 s = disj s2 s.
Lemma mapf_disj s1 s2 s1' s2' : mapf s1 = mapf s2 -> mapf s1' = mapf s2' -> disj s1 s1' = disj s2 s2'.