diff options
Diffstat (limited to 'theories/Logic/JMeq.v')
| -rw-r--r-- | theories/Logic/JMeq.v | 35 |
1 files changed, 22 insertions, 13 deletions
diff --git a/theories/Logic/JMeq.v b/theories/Logic/JMeq.v index 4d365e32..6a723e43 100644 --- a/theories/Logic/JMeq.v +++ b/theories/Logic/JMeq.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: JMeq.v 6009 2004-08-03 17:42:55Z herbelin $ i*) +(*i $Id: JMeq.v 9077 2006-08-24 08:44:32Z herbelin $ i*) (** John Major's Equality as proposed by Conor McBride @@ -19,56 +19,65 @@ Set Implicit Arguments. -Inductive JMeq (A:Set) (x:A) : forall B:Set, B -> Prop := +Inductive JMeq (A:Type) (x:A) : forall B:Type, B -> Prop := JMeq_refl : JMeq x x. -Reset JMeq_ind. +Reset JMeq_rect. Hint Resolve JMeq_refl. -Lemma sym_JMeq : forall (A B:Set) (x:A) (y:B), JMeq x y -> JMeq y x. +Lemma sym_JMeq : forall (A B:Type) (x:A) (y:B), JMeq x y -> JMeq y x. destruct 1; trivial. Qed. Hint Immediate sym_JMeq. Lemma trans_JMeq : - forall (A B C:Set) (x:A) (y:B) (z:C), JMeq x y -> JMeq y z -> JMeq x z. + forall (A B C:Type) (x:A) (y:B) (z:C), JMeq x y -> JMeq y z -> JMeq x z. destruct 1; trivial. Qed. -Axiom JMeq_eq : forall (A:Set) (x y:A), JMeq x y -> x = y. +Axiom JMeq_eq : forall (A:Type) (x y:A), JMeq x y -> x = y. -Lemma JMeq_ind : forall (A:Set) (x y:A) (P:A -> Prop), P x -> JMeq x y -> P y. +Lemma JMeq_ind : forall (A:Type) (x y:A) (P:A -> Prop), P x -> JMeq x y -> P y. intros A x y P H H'; case JMeq_eq with (1 := H'); trivial. Qed. -Lemma JMeq_rec : forall (A:Set) (x y:A) (P:A -> Set), P x -> JMeq x y -> P y. +Lemma JMeq_rec : forall (A:Type) (x y:A) (P:A -> Set), P x -> JMeq x y -> P y. +intros A x y P H H'; case JMeq_eq with (1 := H'); trivial. +Qed. + +Lemma JMeq_rect : forall (A:Type) (x y:A) (P:A->Type), P x -> JMeq x y -> P y. intros A x y P H H'; case JMeq_eq with (1 := H'); trivial. Qed. Lemma JMeq_ind_r : - forall (A:Set) (x y:A) (P:A -> Prop), P y -> JMeq x y -> P x. + forall (A:Type) (x y:A) (P:A -> Prop), P y -> JMeq x y -> P x. intros A x y P H H'; case JMeq_eq with (1 := sym_JMeq H'); trivial. Qed. Lemma JMeq_rec_r : - forall (A:Set) (x y:A) (P:A -> Set), P y -> JMeq x y -> P x. + forall (A:Type) (x y:A) (P:A -> Set), P y -> JMeq x y -> P x. +intros A x y P H H'; case JMeq_eq with (1 := sym_JMeq H'); trivial. +Qed. + +Lemma JMeq_rect_r : + forall (A:Type) (x y:A) (P:A -> Type), P y -> JMeq x y -> P x. intros A x y P H H'; case JMeq_eq with (1 := sym_JMeq H'); trivial. Qed. -(** [JMeq] is equivalent to [(eq_dep Set [X]X)] *) +(** [JMeq] is equivalent to [(eq_dep Type [X]X)] *) Require Import Eqdep. Lemma JMeq_eq_dep : - forall (A B:Set) (x:A) (y:B), JMeq x y -> eq_dep Set (fun X => X) A x B y. + 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_JMeq : - forall (A B:Set) (x:A) (y:B), eq_dep Set (fun X => X) A x B y -> JMeq x y. + 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. |