Library Coq.Classes.SetoidTactics

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.
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).
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.