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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * Tactics for typeclass-based setoids.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Require Coq.Classes.CRelationClasses Coq.Classes.CMorphisms.
Require Import Coq.Classes.Morphisms Coq.Classes.Morphisms_Prop.
Require Export Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions.
Require Import Coq.Classes.Equivalence Coq.Program.Basics.
Generalizable Variables A R.
Export ProperNotations.
Set Implicit Arguments.
Unset Strict Implicit.
(** 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 a default relation. This is used by setoid_replace to infer
the relation to use on a given type, in a given context.
*)
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). *)
#[global]
Instance equivalence_default `(Equivalence A R) : DefaultRelation R | 4.
Defined.
(** 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.
Local Open 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_proper :=
match goal with
| [ |- @Proper ?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_proper) ;
match goal with
| [ |- (?x ==> ?y) _ _ ] => red_subst_eq_morphism (x ==> y) ; reverse_arrows (x ==> y)
end.
Ltac add_morphism_tactic := default_add_morphism_tactic.
Obligation Tactic := program_simpl.
(* Notation "'Morphism' s t " := (@Proper _ (s%signature) t) (at level 10, s at next level, t at next level). *)
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