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(* -*- coq-prog-args: ("-emacs-U" "-top" "Coq.Classes.SetoidTactics") -*- *)
(************************************************************************)
(* 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 *)
(************************************************************************)
(* Tactics for typeclass-based setoids.
*
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
* 91405 Orsay, France *)
(* $Id: SetoidTactics.v 11709 2008-12-20 11:42:15Z msozeau $ *)
Require Export Coq.Classes.RelationClasses.
Require Export Coq.Classes.Morphisms.
Require Export Coq.Classes.Morphisms_Prop.
Require Export Coq.Classes.Equivalence.
Require Export Coq.Relations.Relation_Definitions.
Set Implicit Arguments.
Unset Strict Implicit.
(** Setoid relation on a given support: declares a relation as a setoid
for use with rewrite. It helps choosing if a rewrite should be handled
by setoid_rewrite or the regular rewrite using leibniz equality.
Users can declare an [SetoidRelation A RA] anywhere to declare default
relations. This is also done automatically by the [Declare Relation A RA]
commands. *)
Class SetoidRelation A (R : relation A).
Instance impl_setoid_relation : SetoidRelation impl.
Instance iff_setoid_relation : SetoidRelation iff.
(** Default relation on a given support. Can be used by tactics
to find a sensible default relation on any carrier. Users can
declare an [Instance def : DefaultRelation A RA] anywhere to
declare default relations. *)
Class DefaultRelation A (R : relation A).
(** To search for the default relation, just call [default_relation]. *)
Definition default_relation `{DefaultRelation A R} := R.
(** Every [Equivalence] gives a default relation, if no other is given (lowest priority). *)
Instance equivalence_default `(Equivalence A R) : DefaultRelation R | 4.
(** The setoid_replace tactics in Ltac, defined in terms of default relations and
the setoid_rewrite tactic. *)
Ltac setoidreplace H t :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq ; clear Heq | t ].
Ltac setoidreplacein H H' t :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq in H' ; clear Heq | t ].
Ltac setoidreplaceinat H H' t occs :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq in H' at occs ; clear Heq | t ].
Ltac setoidreplaceat H t occs :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq at occs ; clear Heq | t ].
Tactic Notation "setoid_replace" constr(x) "with" constr(y) :=
setoidreplace (default_relation x y) idtac.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"at" int_or_var_list(o) :=
setoidreplaceat (default_relation x y) idtac o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"in" hyp(id) :=
setoidreplacein (default_relation x y) id idtac.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"in" hyp(id)
"at" int_or_var_list(o) :=
setoidreplaceinat (default_relation x y) id idtac o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"by" tactic3(t) :=
setoidreplace (default_relation x y) ltac:t.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"at" int_or_var_list(o)
"by" tactic3(t) :=
setoidreplaceat (default_relation x y) ltac:t o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"in" hyp(id)
"by" tactic3(t) :=
setoidreplacein (default_relation x y) id ltac:t.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"in" hyp(id)
"at" int_or_var_list(o)
"by" tactic3(t) :=
setoidreplaceinat (default_relation x y) id ltac:t o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel) :=
setoidreplace (rel x y) idtac.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"at" int_or_var_list(o) :=
setoidreplaceat (rel x y) idtac o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"by" tactic3(t) :=
setoidreplace (rel x y) ltac:t.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"at" int_or_var_list(o)
"by" tactic3(t) :=
setoidreplaceat (rel x y) ltac:t o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"in" hyp(id) :=
setoidreplacein (rel x y) id idtac.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"in" hyp(id)
"at" int_or_var_list(o) :=
setoidreplaceinat (rel x y) id idtac o.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"in" hyp(id)
"by" tactic3(t) :=
setoidreplacein (rel x y) id ltac:t.
Tactic Notation "setoid_replace" constr(x) "with" constr(y)
"using" "relation" constr(rel)
"in" hyp(id)
"at" int_or_var_list(o)
"by" tactic3(t) :=
setoidreplaceinat (rel x y) id ltac:t o.
(** The [add_morphism_tactic] tactic is run at each [Add Morphism] command before giving the hand back
to the user to discharge the proof. It essentially amounts to unfold the right amount of [respectful] calls
and substitute leibniz equalities. One can redefine it using [Ltac add_morphism_tactic ::= t]. *)
Require Import Coq.Program.Tactics.
Open Local Scope signature_scope.
Ltac red_subst_eq_morphism concl :=
match concl with
| @Logic.eq ?A ==> ?R' => red ; intros ; subst ; red_subst_eq_morphism R'
| ?R ==> ?R' => red ; intros ; red_subst_eq_morphism R'
| _ => idtac
end.
Ltac destruct_morphism :=
match goal with
| [ |- @Morphism ?A ?R ?m ] => red
end.
Ltac reverse_arrows x :=
match x with
| @Logic.eq ?A ==> ?R' => revert_last ; reverse_arrows R'
| ?R ==> ?R' => do 3 revert_last ; reverse_arrows R'
| _ => idtac
end.
Ltac default_add_morphism_tactic :=
unfold flip ; intros ;
(try destruct_morphism) ;
match goal with
| [ |- (?x ==> ?y) _ _ ] => red_subst_eq_morphism (x ==> y) ; reverse_arrows (x ==> y)
end.
Ltac add_morphism_tactic := default_add_morphism_tactic.
Ltac obligation_tactic ::= program_simpl.
|
