End Elt2.
Section Elt3.
Variable elt elt' elt'' : Type.
Notation oee' := (option elt * option elt')%type.
Definition combine_l (m:t elt)(m':t elt') : t oee' :=
mapi (fun k e => (Some e, find k m')) m.
Definition combine_r (m:t elt)(m':t elt') : t oee' :=
mapi (fun k e' => (find k m, Some e')) m'.
Definition fold_right_pair (A B C:Type)(f:A->B->C->C) :=
List.fold_right (fun p => f (fst p) (snd p)).
Definition combine (m:t elt)(m':t elt') : t oee' :=
let l := combine_l m m' in
let r := combine_r m m' in
fold_right_pair (add (elt:=oee')) r l.

Lemma fold_right_pair_NoDup :
forall l r (Hl: NoDupA (eqk (elt:=oee')) l)
(Hl: NoDupA (eqk (elt:=oee')) r),
NoDupA (eqk (elt:=oee')) (fold_right_pair (add (elt:=oee')) r l).