Global Instance type_Prop : type Prop :=
{ equal := iff
; proper := fun _ => True
}.
Global Instance typeOk_Prop : typeOk type_Prop.

(* Specifications: *)
(** NOTE: These should fit into a larger picture, e.g. lattices or monoids **)
(** And/Conjunction **)
Lemma and_True_iff : forall P, (P /\ True) <-> P.
Lemma impl_iff
: forall P Q R S : Prop,
(P <-> R) ->
(P -> (Q <-> S)) ->
((P -> Q) <-> (R -> S)).
Lemma forall_impl : forall {T} (P Q : T -> Prop),
(forall x, P x -> Q x) ->
(forall x, P x) -> (forall x, Q x).
(** Exists **)
Lemma exists_iff : forall T P Q,
(forall x,
P x <-> Q x) ->
((exists x : T, P x) <-> (exists x : T, Q x)).
Lemma exists_impl : forall {T} (P Q : T -> Prop),
(forall x, P x -> Q x) ->
(exists x, P x) -> (exists x, Q x).
Lemma iff_eq : forall (P Q : Prop), P = Q -> (P <-> Q).