Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# 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.
Require Import Eqdep.
Set Implicit Arguments.
Inductive JMeq (A:Type) (x:A) : forall B:Type, B -> Prop :=
JMeq_refl : JMeq x x.
Arguments JMeq_refl {A x} , [A] x.
Register JMeq as core.JMeq.type.
Register JMeq_refl as core.JMeq.refl.
#[global]
Hint Resolve JMeq_refl : core.
Definition JMeq_hom {A : Type} (x y : A) := JMeq x y.
Register JMeq_hom as core.JMeq.hom.
Lemma JMeq_sym : forall (A B:Type) (x:A) (y:B), JMeq x y -> JMeq y x.
#[global]
Hint Immediate JMeq_sym : core.
Register JMeq_sym as core.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.
Register JMeq_trans as core.JMeq.trans.
Theorem 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.
Register JMeq_ind as core.JMeq.ind.
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.
Register JMeq_congr as core.JMeq.congr.
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).