(** [combine] **)
Definition Scombine {A B C : Type} (f : A -> B -> C) (g : A -> C) (h : B -> C) (m₁ : S.t A) (m₂ : S.t B) : S.t C :=
Dom.fold (fun x acc => match S.find x m₁, S.find x m₂ with
| Some v₁, Some v₂ => S.add x (f v₁ v₂) acc
| Some v, None => S.add x (g v) acc
| None, Some v => S.add x (h v) acc
| None, None => acc
end)
(Dom.union (S.fold (fun x _ acc => Dom.add x acc) m₁ Dom.empty)
(S.fold (fun x _ acc => Dom.add x acc) m₂ Dom.empty)) (S.empty C).

(* Specifications: *)
Lemma Sdom_aux : forall A x (m : S.t A) s,
Dom.In x (S.fold (fun x _ acc => Dom.add x acc) m s) <-> S.In x m ∨ Dom.In x s.
Corollary Sdom : forall A x (m : S.t A), Dom.In x (S.fold (fun x _ acc => Dom.add x acc) m Dom.empty) <-> S.In x m.
Lemma Scombine_spec_aux : forall A B C (f : A -> B -> C) g h x v m₁ m₂ l m, NoDupA X.eq l ->
S.find (elt:=C) x
(fold_left
(fun (acc : S.t C) (x : Dom.elt) => match S.find x m₁, S.find x m₂ with
| Some v₁, Some v₂ => S.add x (f v₁ v₂) acc
| Some v, None => S.add x (g v) acc
| None, Some v => S.add x (h v) acc
| None, None => acc
end) l m) = Some v
<-> InA X.eq x l ∧ ((exists v₁ v₂, S.find x m₁ = Some v₁ ∧ S.find x m₂ = Some v₂ ∧ v = f v₁ v₂)
∨ (exists v₁, S.find x m₁ = Some v₁ ∧ S.find x m₂ = None ∧ v = g v₁)
∨ (exists v₂, S.find x m₁ = None ∧ S.find x m₂ = Some v₂ ∧ v = h v₂))
∨ ((InA X.eq x l ∧ S.find x m₁ = None ∧ S.find x m₂ = None ∨ ¬InA X.eq x l) ∧ S.find x m = Some v).
Theorem Scombine_spec : forall A B C (f : A -> B -> C) g h x v m₁ m₂,
S.find x (Scombine f g h m₁ m₂) = Some v <->
(exists v₁ v₂, S.find x m₁ = Some v₁ ∧ S.find x m₂ = Some v₂ ∧ v = f v₁ v₂)
∨ (exists v₁, S.find x m₁ = Some v₁ ∧ S.find x m₂ = None ∧ v = g v₁)
∨ (exists v₂, S.find x m₁ = None ∧ S.find x m₂ = Some v₂ ∧ v = h v₂).
Lemma combine_OK : forall A B C (f : A -> B -> C) g₁ g₂ dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t B dom₂),
OK (Dom.union dom₁ dom₂) (Scombine f g₁ g₂ (proj1_sig m₁) (proj1_sig m₂)).