#[global]
Hint Resolve compat_P_aux : core.
Definition filter :
forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{s' : t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
Definition for_all :
forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{compat_P E.eq P -> For_all P s} + {compat_P E.eq P -> ~ For_all P s}.
Definition exists_ :
forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{compat_P E.eq P -> Exists P s} + {compat_P E.eq P -> ~ Exists P s}.
Definition partition :
forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
{partition : t * t |
let (s1, s2) := partition in
compat_P E.eq P ->
For_all P s1 /\
For_all (fun x => ~ P x) s2 /\
(forall x : elt, In x s <-> In x s1 \/ In x s2)}.
Definition choose_aux: forall s : t,
{ x : elt | M.choose s = Some x } + { M.choose s = None }.
Definition choose : forall s : t, {x : elt | In x s} + {Empty s}.

Lemma choose_equal : forall s s', Equal s s' ->
match choose s, choose s' with
| inleft (exist _ x _), inleft (exist _ x' _) => E.eq x x'
| inright _, inright _  => True
| _, _                        => False
end.