Library Coq.Classes.SetoidTactics


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.

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.