Library Coq.Logic.JMeq


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.


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




Hint Resolve JMeq_refl : core.

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.

Hint Immediate JMeq_sym : core.


Lemma JMeq_trans :
 forall (A B C:Type) (x:A) (y:B) (z:C), JMeq x y -> JMeq y z -> JMeq x z.


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.


Lemma JMeq_rec : forall (A:Type) (x:A) (P:A -> Set),
  P x -> forall y, JMeq x y -> P y.

Lemma JMeq_rect : forall (A:Type) (x:A) (P:A->Type),
  P x -> forall y, JMeq x y -> P y.

Lemma JMeq_ind_r : forall (A:Type) (x:A) (P:A -> Prop),
   P x -> forall y, JMeq y x -> P y.

Lemma JMeq_rec_r : forall (A:Type) (x:A) (P:A -> Set),
   P x -> forall y, JMeq y x -> P y.

Lemma JMeq_rect_r : forall (A:Type) (x:A) (P:A -> Type),
   P x -> forall y, JMeq y x -> P y.

Lemma JMeq_congr :
 forall (A:Type) (x:A) (B:Type) (f:A->B) (y:A), JMeq x y -> f x = f y.


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.

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.

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.

Lemma eq_dep_strictly_stronger_JMeq :
 exists U P p q x y, JMeq x y /\ ~ eq_dep U P p x q y.

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.

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