diff options
Diffstat (limited to 'theories/Classes/Equivalence.v')
| -rw-r--r-- | theories/Classes/Equivalence.v | 38 |
1 files changed, 23 insertions, 15 deletions
diff --git a/theories/Classes/Equivalence.v b/theories/Classes/Equivalence.v index 0e9adf64..c281af80 100644 --- a/theories/Classes/Equivalence.v +++ b/theories/Classes/Equivalence.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -24,7 +24,7 @@ Set Implicit Arguments. Unset Strict Implicit. Generalizable Variables A R eqA B S eqB. -Local Obligation Tactic := simpl_relation. +Local Obligation Tactic := try solve [simpl_relation]. Local Open Scope signature_scope. @@ -56,8 +56,8 @@ Program Instance equiv_symmetric `(sa : Equivalence A) : Symmetric equiv. Program Instance equiv_transitive `(sa : Equivalence A) : Transitive equiv. Next Obligation. - Proof. - transitivity y ; auto. + Proof. intros A R sa x y z Hxy Hyz. + now transitivity y. Qed. (** Use the [substitute] command which substitutes an equivalence in every hypothesis. *) @@ -105,27 +105,35 @@ Section Respecting. (** Here we build an equivalence instance for functions which relates respectful ones only, we do not export it. *) - Definition respecting `(eqa : Equivalence A (R : relation A), eqb : Equivalence B (R' : relation B)) : Type := + Definition respecting `(eqa : Equivalence A (R : relation A), + eqb : Equivalence B (R' : relation B)) : Type := { morph : A -> B | respectful R R' morph morph }. Program Instance respecting_equiv `(eqa : Equivalence A R, eqb : Equivalence B R') : - Equivalence (fun (f g : respecting eqa eqb) => forall (x y : A), R x y -> R' (proj1_sig f x) (proj1_sig g y)). + Equivalence (fun (f g : respecting eqa eqb) => + forall (x y : A), R x y -> R' (proj1_sig f x) (proj1_sig g y)). - Solve Obligations using unfold respecting in * ; simpl_relation ; program_simpl. + Solve Obligations with unfold respecting in * ; simpl_relation ; program_simpl. Next Obligation. - Proof. - unfold respecting in *. program_simpl. transitivity (y y0); auto. apply H0. reflexivity. + Proof. + intros. intros f g h H H' x y Rxy. + unfold respecting in *. program_simpl. transitivity (g y); auto. firstorder. Qed. End Respecting. (** The default equivalence on function spaces, with higher-priority than [eq]. *) -Program Instance pointwise_equivalence {A} `(eqb : Equivalence B eqB) : +Instance pointwise_reflexive {A} `(reflb : Reflexive B eqB) : + Reflexive (pointwise_relation A eqB) | 9. +Proof. firstorder. Qed. +Instance pointwise_symmetric {A} `(symb : Symmetric B eqB) : + Symmetric (pointwise_relation A eqB) | 9. +Proof. firstorder. Qed. +Instance pointwise_transitive {A} `(transb : Transitive B eqB) : + Transitive (pointwise_relation A eqB) | 9. +Proof. firstorder. Qed. +Instance pointwise_equivalence {A} `(eqb : Equivalence B eqB) : Equivalence (pointwise_relation A eqB) | 9. - - Next Obligation. - Proof. - transitivity (y a) ; auto. - Qed. +Proof. split; apply _. Qed. |