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Diffstat (limited to 'theories7/Logic/JMeq.v')
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diff --git a/theories7/Logic/JMeq.v b/theories7/Logic/JMeq.v new file mode 100644 index 00000000..38dfa5e6 --- /dev/null +++ b/theories7/Logic/JMeq.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: JMeq.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** John Major's Equality as proposed by C. Mc Bride *) + +Set Implicit Arguments. + +Inductive JMeq [A:Set;x:A] : (B:Set)B->Prop := + JMeq_refl : (JMeq x x). +Reset JMeq_ind. + +Hints Resolve JMeq_refl. + +Lemma sym_JMeq : (A,B:Set)(x:A)(y:B)(JMeq x y)->(JMeq y x). +NewDestruct 1; Trivial. +Qed. + +Hints Immediate sym_JMeq. + +Lemma trans_JMeq : (A,B,C:Set)(x:A)(y:B)(z:C) + (JMeq x y)->(JMeq y z)->(JMeq x z). +NewDestruct 1; Trivial. +Qed. + +Axiom JMeq_eq : (A:Set)(x,y:A)(JMeq x y)->(x=y). + +Lemma JMeq_ind : (A:Set)(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 : (A:Set)(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_ind_r : (A:Set)(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 : (A:Set)(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. + +(** [JMeq] is equivalent to [(eq_dep Set [X]X)] *) + +Require Eqdep. + +Lemma JMeq_eq_dep : (A,B:Set)(x:A)(y:B)(JMeq x y)->(eq_dep Set [X]X A x B y). +Proof. +NewDestruct 1. +Apply eq_dep_intro. +Qed. + +Lemma eq_dep_JMeq : (A,B:Set)(x:A)(y:B)(eq_dep Set [X]X A x B y)->(JMeq x y). +Proof. +NewDestruct 1. +Apply JMeq_refl. +Qed. |