(** Includes minor tweaks (mostly bug fixes?) by Brian Aydemir. *)
(** SCW 06/06/17: updated to use Coq MSets instead of FSets. This version is still more powerful than Coq.MSets.MSetDecide fsetdec_rec / fsetdec_body are the big differences *)
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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)
(* $Id: FSetDecide.v 11699 2008-12-18 11:49:08Z letouzey $ *)
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(* FSetDecide.v                                               *)
(*                                                            *)
(* Author: Aaron Bohannon                                     *)
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(** This file implements a decision procedure for a certain class of propositions involving finite sets.  *)
(** First, a version for Weak Sets in functorial presentation *)
Module WDecide_fun (E : DecidableType)(Import M : WSetsOn E).
Module F :=  MSetFacts.WFactsOn E M.
(** * Overview This functor defines the tactic [fsetdec], which will solve any valid goal of the form << forall s1 ... sn, forall x1 ... xm, P1 -> ... -> Pk -> P >> where [P]'s are defined by the grammar: <<  P ::= | Q | Empty F | Subset F F' | Equal F F'  Q ::= | E.eq X X' | In X F | Q /\ Q' | Q \/ Q' | Q -> Q' | Q <-> Q' | ~ Q | True | False  F ::= | S | empty | singleton X | add X F | remove X F | union F F' | inter F F' | diff F F'  X ::= x1 | ... | xm S ::= s1 | ... | sn  >>  The tactic will also work on some goals that vary slightly from the above form: - The variables and hypotheses may be mixed in any order and may have already been introduced into the context.  Moreover, there may be additional, unrelated hypotheses mixed in (these will be ignored). - A conjunction of hypotheses will be handled as easily as separate hypotheses, i.e., [P1 /\ P2 -> P] can be solved iff [P1 -> P2 -> P] can be solved. - [fsetdec] should solve any goal if the MSet-related hypotheses are contradictory. - [fsetdec] will first perform any necessary zeta and beta reductions and will invoke [subst] to eliminate any Coq equalities between finite sets or their elements. - If [E.eq] is convertible with Coq's equality, it will not matter which one is used in the hypotheses or conclusion. - The tactic can solve goals where the finite sets or set elements are expressed by Coq terms that are more complicated than variables.  However, non-local definitions are not expanded, and Coq equalities between non-variable terms are not used.  For example, this goal will be solved: << forall (f : t -> t), forall (g : elt -> elt), forall (s1 s2 : t), forall (x1 x2 : elt), Equal s1 (f s2) -> E.eq x1 (g (g x2)) -> In x1 s1 -> In (g (g x2)) (f s2) >> This one will not be solved: << forall (f : t -> t), forall (g : elt -> elt), forall (s1 s2 : t), forall (x1 x2 : elt), Equal s1 (f s2) -> E.eq x1 (g x2) -> In x1 s1 -> g x2 = g (g x2) -> In (g (g x2)) (f s2) >> *)
(** * Facts and Tactics for Propositional Logic These lemmas and tactics are in a module so that they do not affect the namespace if you import the enclosing module [Decide]. *)
Module FSetLogicalFacts.
Export Decidable.
Export Setoid.
(** ** Lemmas and Tactics About Decidable Propositions *)
(** ** Propositional Equivalences Involving Negation These are all written with the unfolded form of negation, since I am not sure if setoid rewriting will always perform conversion. *)
(** ** Tactics for Negations *)
Tactic Notation "fold" "any" "not" :=
repeat (
match goal with
| H: context [?P -> False] |- _ =>
fold (~ P) in H
| |- context [?P -> False] =>
fold (~ P)
end).
(** [push not using db] will pushes all negations to the leaves of propositions in the goal, using the lemmas in [db] to assist in checking the decidability of the propositions involved.  If [using db] is omitted, then [core] will be used.  Additional versions are provided to manipulate the hypotheses or the hypotheses and goal together.  XXX: This tactic and the similar subsequent ones should have been defined using [autorewrite]. However, dealing with multiples rewrite sites and side-conditions is done more cleverly with the following explicit analysis of goals. *)
Ltac or_not_l_iff P Q tac :=
(rewrite (or_not_l_iff_1 P Q) by tac) ||
(rewrite (or_not_l_iff_2 P Q) by tac).
Ltac or_not_r_iff P Q tac :=
(rewrite (or_not_r_iff_1 P Q) by tac) ||
(rewrite (or_not_r_iff_2 P Q) by tac).
Ltac or_not_l_iff_in P Q H tac :=
(rewrite (or_not_l_iff_1 P Q) in H by tac) ||
(rewrite (or_not_l_iff_2 P Q) in H by tac).
Ltac or_not_r_iff_in P Q H tac :=
(rewrite (or_not_r_iff_1 P Q) in H by tac) ||
(rewrite (or_not_r_iff_2 P Q) in H by tac).
Tactic Notation "push" "not" "using" ident(db) :=
let dec := solve_decidable using db in
unfold not, iff;
repeat (
match goal with
| |- context [True -> False] => rewrite not_true_iff
| |- context [False -> False] => rewrite not_false_iff
| |- context [(?P -> False) -> False] => rewrite (not_not_iff P) by dec
| |- context [(?P -> False) -> (?Q -> False)] =>
rewrite (contrapositive P Q) by dec
| |- context [(?P -> False) \/ ?Q] => or_not_l_iff P Q dec
| |- context [?P \/ (?Q -> False)] => or_not_r_iff P Q dec
| |- context [(?P -> False) -> ?Q] => rewrite (imp_not_l P Q) by dec
| |- context [?P \/ ?Q -> False] => rewrite (not_or_iff P Q)
| |- context [?P /\ ?Q -> False] => rewrite (not_and_iff P Q)
| |- context [(?P -> ?Q) -> False] => rewrite (not_imp_iff P Q) by dec
end);
fold any not.
Tactic Notation "push" "not" :=
push not using core.
Tactic Notation
"push" "not" "in" "*" "|-" "using" ident(db) :=
let dec := solve_decidable using db in
unfold not, iff in * |-;
repeat (
match goal with
| H: context [True -> False] |- _ => rewrite not_true_iff in H
| H: context [False -> False] |- _ => rewrite not_false_iff in H
| H: context [(?P -> False) -> False] |- _ =>
rewrite (not_not_iff P) in H by dec
| H: context [(?P -> False) -> (?Q -> False)] |- _ =>
rewrite (contrapositive P Q) in H by dec
| H: context [(?P -> False) \/ ?Q] |- _ => or_not_l_iff_in P Q H dec
| H: context [?P \/ (?Q -> False)] |- _ => or_not_r_iff_in P Q H dec
| H: context [(?P -> False) -> ?Q] |- _ =>
rewrite (imp_not_l P Q) in H by dec
| H: context [?P \/ ?Q -> False] |- _ => rewrite (not_or_iff P Q) in H
| H: context [?P /\ ?Q -> False] |- _ => rewrite (not_and_iff P Q) in H
| H: context [(?P -> ?Q) -> False] |- _ =>
rewrite (not_imp_iff P Q) in H by dec
end);
fold any not.
Tactic Notation "push" "not" "in" "*" "|-"  :=
push not in * |- using core.
Tactic Notation "push" "not" "in" "*" "using" ident(db) :=
push not using db; push not in * |- using db.
Tactic Notation "push" "not" "in" "*" :=
push not in * using core.

(** A simple test case to see how this works.  *)
Lemma test_push : forall P Q R : Prop,
decidable P ->
decidable Q ->
(~ True) ->
(~ False) ->
(~ ~ P) ->
(~ (P /\ Q) -> ~ R) ->
((P /\ Q) \/ ~ R) ->
(~ (P /\ Q) \/ R) ->
(R \/ ~ (P /\ Q)) ->
(~ R \/ (P /\ Q)) ->
(~ P -> R) ->
(~ ((R -> P) \/ (Q -> R))) ->
(~ (P /\ R)) ->
(~ (P -> R)) ->
True.