Module S := FMapList.Make(XOT).
(** The type of dependent maps **)
Definition key := X.t.
Definition OK {A} dom (map : S.t A) := ∀ x, S.In x map <-> Dom.In x dom.
Definition t := fun A dom => { map : S.t A | OK dom map}.
Instance OK_compat A : Proper (Dom.eq ==> @S.Equal A ==> iff) OK.
(** ***  Operations on maps **)
Definition empty : forall A, t A Dom.empty := fun A => exist (OK Dom.empty) (@S.empty A) (empty_OK A).
Definition is_empty (A : Type) dom (m : t A dom) := Dom.equal dom Dom.empty.
Definition mem (A : Type) dom (x : key) (m : t A dom) := Dom.mem x dom.
Definition find (A : Type) dom (x : key) (m : t A dom) := S.find x (proj1_sig m).
Definition domain (A : Type) dom (m : t A dom) := dom.
Definition add {A : Type} {dom : Dom.t} (x : key) (v : A) (m : @t A dom) : @t A (Dom.add x dom) :=
exist (OK (Dom.add x dom)) (S.add x v (proj1_sig m)) (add_OK x v m).
Definition set {A : Type} {dom : Dom.t} (x : key) (v : A) (m : @t A dom) (Hin : Dom.In x dom) :=
exist (OK dom) (S.add x v (proj1_sig m)) (@set_OK _ _ x v m Hin).
Arguments set {A} {dom} x v m _.
Definition singleton {A : Type} (x : key) (v : A) : t A (Dom.singleton x) :=
exist (OK (Dom.singleton x)) (S.add x v (@S.empty A)) (singleton_OK x v).
Definition remove {A : Type} {dom : Dom.t} (x : key) (m : @t A dom) : @t A (Dom.remove x dom) :=
exist (OK (Dom.remove x dom)) (S.remove x (proj1_sig m)) (remove_OK x m).
Definition constant (A : Type) dom (v : A) : t A dom :=
exist (OK dom) (Dom.fold (fun x m => S.add x v m) dom (@S.empty A)) (constant_OK dom v).
(** [fold] **)
Definition fold {A B : Type} (f : key -> A -> B -> B) dom (m : t A dom) (i : B) : B := S.fold f (proj1_sig m) i.
Definition map {A B : Type} (f : A -> B) dom (m : t A dom) : t B dom :=
exist (OK dom) (S.map f (proj1_sig m)) (map_OK f m).
(** [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).

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.