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(** John Major's Equality as proposed by Conor McBride

  Reference:

  [McBride] Elimination with a Motive, Proceedings of TYPES 2000,
  LNCS 2277, pp 197-216, 2002.

*)

Set Implicit Arguments.

Unset Elimination Schemes.

Inductive JMeq (A:Type) (x:A) : forall B:Type, B -> Prop :=
    JMeq_refl : JMeq x x.

Set Elimination Schemes.

Arguments JMeq_refl {A x} , [A] x.

Hint Resolve JMeq_refl.

Definition JMeq_hom {A : Type} (x y : A) := JMeq x y.

Lemma JMeq_sym : forall (A B:Type) (x:A) (y:B), JMeq x y -> JMeq y x.
Proof. 
intros; destruct H; trivial.
Qed.

Hint Immediate JMeq_sym.

Lemma JMeq_trans :
 forall (A B C:Type) (x:A) (y:B) (z:C), JMeq x y -> JMeq y z -> JMeq x z.
Proof.
destruct 2; trivial.
Qed.

Axiom JMeq_eq : forall (A:Type) (x y:A), JMeq x y -> x = y.

Lemma JMeq_ind : forall (A:Type) (x:A) (P:A -> Prop),
  P x -> forall y, JMeq x y -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := H'); trivial.
Qed.

Lemma JMeq_rec : forall (A:Type) (x:A) (P:A -> Set),
  P x -> forall y, JMeq x y -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := H'); trivial.
Qed.

Lemma JMeq_rect : forall (A:Type) (x:A) (P:A->Type),
  P x -> forall y, JMeq x y -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := H'); trivial.
Qed.

Lemma JMeq_ind_r : forall (A:Type) (x:A) (P:A -> Prop),
   P x -> forall y, JMeq y x -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := JMeq_sym H'); trivial.
Qed.

Lemma JMeq_rec_r : forall (A:Type) (x:A) (P:A -> Set),
   P x -> forall y, JMeq y x -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := JMeq_sym H'); trivial.
Qed.

Lemma JMeq_rect_r : forall (A:Type) (x:A) (P:A -> Type),
   P x -> forall y, JMeq y x -> P y.
Proof.
intros A x P H y H'; case JMeq_eq with (1 := JMeq_sym H'); trivial.
Qed.

Lemma JMeq_congr :
 forall (A:Type) (x:A) (B:Type) (f:A->B) (y:A), JMeq x y -> f x = f y.
Proof.
intros A x B f y H; case JMeq_eq with (1 := H); trivial.
Qed.

(** [JMeq] is equivalent to [eq_dep Type (fun X => X)] *)

Require Import Eqdep.

Lemma JMeq_eq_dep_id :
 forall (A B:Type) (x:A) (y:B), JMeq x y -> eq_dep Type (fun X => X) A x B y.
Proof.
destruct 1.
apply eq_dep_intro.
Qed.

Lemma eq_dep_id_JMeq :
 forall (A B:Type) (x:A) (y:B), eq_dep Type (fun X => X) A x B y -> JMeq x y.
Proof.
destruct 1.
apply JMeq_refl.
Qed.

(** [eq_dep U P p x q y] is strictly finer than [JMeq (P p) x (P q) y] *)

Lemma eq_dep_JMeq :
 forall U P p x q y, eq_dep U P p x q y -> JMeq x y.
Proof.
destruct 1.
apply JMeq_refl.
Qed.

Lemma eq_dep_strictly_stronger_JMeq :
 exists U P p q x y, JMeq x y /\ ~ eq_dep U P p x q y.
Proof.
exists bool. exists (fun _ => True). exists true. exists false.
exists I. exists I.
split.
trivial.
intro H.
assert (true=false) by (destruct H; reflexivity).
discriminate.
Qed.

(** However, when the dependencies are equal, [JMeq (P p) x (P q) y]
    is as strong as [eq_dep U P p x q y] (this uses [JMeq_eq]) *)

Lemma JMeq_eq_dep : 
  forall U (P:U->Type) p q (x:P p) (y:P q), 
  p = q -> JMeq x y -> eq_dep U P p x q y.
Proof.
intros.
destruct H.
apply JMeq_eq in H0 as ->.
reflexivity.
Qed.


(* Compatibility *)
Notation sym_JMeq := JMeq_sym (only parsing).
Notation trans_JMeq := JMeq_trans (only parsing).