diff options
author | msozeau <msozeau@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2008-02-14 12:56:21 +0000 |
---|---|---|
committer | msozeau <msozeau@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2008-02-14 12:56:21 +0000 |
commit | bb3560885d943baef87b7f99a5d646942f0fb288 (patch) | |
tree | e5eaa2c4dc00c2a2d0d53a561e3ff910a0139906 /theories/Classes/Equivalence.v | |
parent | f3e1ed674ebf3281e65f871d366dce38cf980539 (diff) |
Backtrack changes on eauto, move specialized version of eauto in
class_tactics for further customizations. Add Equivalence.v for
typeclass definitions on equivalences and clrewrite.v test-suite for
clrewrite. Debugging the tactic I found missing morphisms that are now
in Morphisms.v and removed some that made proof search fail or take too
long, not sure it's complete however.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10565 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Classes/Equivalence.v')
-rw-r--r-- | theories/Classes/Equivalence.v | 157 |
1 files changed, 157 insertions, 0 deletions
diff --git a/theories/Classes/Equivalence.v b/theories/Classes/Equivalence.v new file mode 100644 index 000000000..bd525e035 --- /dev/null +++ b/theories/Classes/Equivalence.v @@ -0,0 +1,157 @@ +(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *) +(************************************************************************) +(* 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 *) +(************************************************************************) + +(* Typeclass-based setoids, tactics and standard instances. + TODO: explain clrewrite, clsubstitute and so on. + + Author: Matthieu Sozeau + Institution: LRI, CNRS UMR 8623 - UniversitĂcopyright Paris Sud + 91405 Orsay, France *) + +(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *) + +Require Import Coq.Program.Program. +Require Import Coq.Classes.Init. + +Set Implicit Arguments. +Unset Strict Implicit. + +Require Export Coq.Classes.Relations. +Require Export Coq.Classes.Morphisms. + +Definition equiv [ Equivalence A R ] : relation A := R. + +(** Shortcuts to make proof search possible (unification won't unfold equiv). *) + +Definition equivalence_refl [ sa : Equivalence A ] : Reflexive equiv. +Proof. eauto with typeclass_instances. Qed. + +Definition equivalence_sym [ sa : Equivalence A ] : Symmetric equiv. +Proof. eauto with typeclass_instances. Qed. + +Definition equivalence_trans [ sa : Equivalence A ] : Transitive equiv. +Proof. eauto with typeclass_instances. Qed. + +(** Overloaded notations for setoid equivalence and inequivalence. Not to be confused with [eq] and [=]. *) + +(** Subset objects should be first coerced to their underlying type, but that notation doesn't work in the standard case then. *) +(* Notation " x == y " := (equiv (x :>) (y :>)) (at level 70, no associativity) : type_scope. *) + +Notation " x == y " := (equiv x y) (at level 70, no associativity) : type_scope. + +Notation " x =/= y " := (complement equiv x y) (at level 70, no associativity) : type_scope. + +(** Use the [clsubstitute] command which substitutes an equality in every hypothesis. *) + +Ltac clsubst H := + match type of H with + ?x == ?y => clsubstitute H ; clear H x + end. + +Ltac clsubst_nofail := + match goal with + | [ H : ?x == ?y |- _ ] => clsubst H ; clsubst_nofail + | _ => idtac + end. + +(** [subst*] will try its best at substituting every equality in the goal. *) + +Tactic Notation "clsubst" "*" := clsubst_nofail. + +Lemma nequiv_equiv_trans : forall [ Equivalence A ] (x y z : A), x =/= y -> y == z -> x =/= z. +Proof with auto. + intros; intro. + assert(z == y) by relation_sym. + assert(x == y) by relation_trans. + contradiction. +Qed. + +Lemma equiv_nequiv_trans : forall [ Equivalence A ] (x y z : A), x == y -> y =/= z -> x =/= z. +Proof. + intros; intro. + assert(y == x) by relation_sym. + assert(y == z) by relation_trans. + contradiction. +Qed. + +Open Scope type_scope. + +Ltac equiv_simplify_one := + match goal with + | [ H : ?x == ?x |- _ ] => clear H + | [ H : ?x == ?y |- _ ] => clsubst H + | [ |- ?x =/= ?y ] => let name:=fresh "Hneq" in intro name + end. + +Ltac equiv_simplify := repeat equiv_simplify_one. + +Ltac equivify_tac := + match goal with + | [ s : Equivalence ?A, H : ?R ?x ?y |- _ ] => change R with (@equiv A R s) in H + | [ s : Equivalence ?A |- context C [ ?R ?x ?y ] ] => change (R x y) with (@equiv A R s x y) + end. + +Ltac equivify := repeat equivify_tac. + +(** Every equivalence relation gives rise to a morphism, as it is transitive and symmetric. *) + +Instance [ sa : Equivalence A ] => equiv_morphism : ? Morphism (equiv ++> equiv ++> iff) equiv := + respect := respect. + +(** The partial application too as it is reflexive. *) + +Instance [ sa : Equivalence A ] (x : A) => + equiv_partial_app_morphism : ? Morphism (equiv ++> iff) (equiv x) := + respect := respect. + +Definition type_eq : relation Type := + fun x y => x = y. + +Program Instance type_equivalence : Equivalence Type type_eq. + + Solve Obligations using constructor ; unfold type_eq ; program_simpl. + +Ltac morphism_tac := try red ; unfold arrow ; intros ; program_simpl ; try tauto. + +Ltac obligations_tactic ::= morphism_tac. + +(** These are morphisms used to rewrite at the top level of a proof, + using [iff_impl_id_morphism] if the proof is in [Prop] and + [eq_arrow_id_morphism] if it is in Type. *) + +Program Instance iff_impl_id_morphism : ? Morphism (iff ++> impl) id. + +(* Program Instance eq_arrow_id_morphism : ? Morphism (eq +++> arrow) id. *) + +(* Definition compose_respect (A B C : Type) (R : relation (A -> B)) (R' : relation (B -> C)) *) +(* (x y : A -> C) : Prop := forall (f : A -> B) (g : B -> C), R f f -> R' g g. *) + +(* Program Instance (A B C : Type) (R : relation (A -> B)) (R' : relation (B -> C)) *) +(* [ mg : ? Morphism R' g ] [ mf : ? Morphism R f ] => *) +(* compose_morphism : ? Morphism (compose_respect R R') (g o f). *) + +(* Next Obligation. *) +(* Proof. *) +(* apply (respect (m0:=mg)). *) +(* apply (respect (m0:=mf)). *) +(* assumption. *) +(* Qed. *) + +(** Partial equivs don't require reflexivity so we can build a partial equiv on the function space. *) + +Class PartialEquivalence (carrier : Type) (pequiv : relation carrier) := + pequiv_prf :> PER carrier pequiv. + +(** Overloaded notation for partial equiv equivalence. *) + +(* Infix "=~=" := pequiv (at level 70, no associativity) : type_scope. *) + +(** Reset the default Program tactic. *) + +Ltac obligations_tactic ::= program_simpl. |