# Library Coq.FSets.FSetDecide

This file implements a decision procedure for a certain
class of propositions involving finite sets.

First, a version for Weak Sets in functorial presentation

# Overview

This functor defines the tactic fsetdec, which will solve any valid goal of the formforall s1 ... sn, forall x1 ... xm, P1 -> ... -> Pk -> Pwhere 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 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 FSet-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.## 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.

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.

pull not using db will pull as many negations as
possible toward the top of the 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.

Tactic Notation "pull" "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 -> False) /\ (?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

| |- context [(?Q -> False) /\ ?P] =>

rewrite <- (not_imp_rev_iff P Q) by dec

end);

fold any not.

Tactic Notation "pull" "not" :=

pull not using core.

Tactic Notation

"pull" "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 -> False) /\ (?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

| H: context [(?Q -> False) /\ ?P] |- _ =>

rewrite <- (not_imp_rev_iff P Q) in H by dec

end);

fold any not.

Tactic Notation "pull" "not" "in" "*" "|-" :=

pull not in * |- using core.

Tactic Notation "pull" "not" "in" "*" "using" ident(db) :=

pull not using db; pull not in * |- using db.

Tactic Notation "pull" "not" "in" "*" :=

pull not in * using core.

A simple test case to see how this works.

Lemma test_pull : 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) ->

(~ R /\ P) ->

True.

End FSetLogicalFacts.

Import FSetLogicalFacts.

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) ->

(~ R /\ P) ->

True.

End FSetLogicalFacts.

Import FSetLogicalFacts.

# Auxiliary Tactics

Again, these lemmas and tactics are in a module so that they do not affect the namespace if you import the enclosing module Decide.## Generic Tactics

We begin by defining a few generic, useful tactics.Ltac no_logical_interdep :=

match goal with

| H : ?P |- _ =>

match type of P with

| Prop =>

match goal with H' : context [ H ] |- _ => clear dependent H' end

| _ => fail

end; no_logical_interdep

| _ => idtac

end.

Ltac abstract_term t :=

tryif (is_var t) then fail "no need to abstract a variable"

else (let x := fresh "x" in set (x := t) in *; try clearbody x).

Ltac abstract_elements :=

repeat

(match goal with

| |- context [ singleton ?t ] => abstract_term t

| _ : context [ singleton ?t ] |- _ => abstract_term t

| |- context [ add ?t _ ] => abstract_term t

| _ : context [ add ?t _ ] |- _ => abstract_term t

| |- context [ remove ?t _ ] => abstract_term t

| _ : context [ remove ?t _ ] |- _ => abstract_term t

| |- context [ In ?t _ ] => abstract_term t

| _ : context [ In ?t _ ] |- _ => abstract_term t

end).

prop P holds by t succeeds (but does not modify the
goal or context) if the proposition P can be proved by
t in the current context. Otherwise, the tactic
fails.

Tactic Notation "prop" constr(P) "holds" "by" tactic(t) :=

let H := fresh in

assert P as H by t;

clear H.

let H := fresh in

assert P as H by t;

clear H.

This tactic acts just like assert ... by ... but will
fail if the context already contains the proposition.

Tactic Notation "assert" "new" constr(e) "by" tactic(t) :=

match goal with

| H: e |- _ => fail 1

| _ => assert e by t

end.

match goal with

| H: e |- _ => fail 1

| _ => assert e by t

end.

subst++ is similar to subst except that

- it never fails (as subst does on recursive equations),
- it substitutes locally defined variable for their definitions,
- it performs beta reductions everywhere, which may arise after substituting a locally defined function for its definition.

Tactic Notation "subst" "++" :=

repeat (

match goal with

| x : _ |- _ => subst x

end);

cbv zeta beta in *.

repeat (

match goal with

| x : _ |- _ => subst x

end);

cbv zeta beta in *.

decompose records calls decompose record H on every
relevant hypothesis H.

Tactic Notation "decompose" "records" :=

repeat (

match goal with

| H: _ |- _ => progress (decompose record H); clear H

end).

repeat (

match goal with

| H: _ |- _ => progress (decompose record H); clear H

end).

## Discarding Irrelevant Hypotheses

We will want to clear the context of any non-FSet-related hypotheses in order to increase the speed of the tactic. To do this, we will need to be able to decide which are relevant. We do this by making a simple inductive definition classifying the propositions of interest.Inductive FSet_elt_Prop : Prop -> Prop :=

| eq_Prop : forall (S : Type) (x y : S),

FSet_elt_Prop (x = y)

| eq_elt_prop : forall x y,

FSet_elt_Prop (E.eq x y)

| In_elt_prop : forall x s,

FSet_elt_Prop (In x s)

| True_elt_prop :

FSet_elt_Prop True

| False_elt_prop :

FSet_elt_Prop False

| conj_elt_prop : forall P Q,

FSet_elt_Prop P ->

FSet_elt_Prop Q ->

FSet_elt_Prop (P /\ Q)

| disj_elt_prop : forall P Q,

FSet_elt_Prop P ->

FSet_elt_Prop Q ->

FSet_elt_Prop (P \/ Q)

| impl_elt_prop : forall P Q,

FSet_elt_Prop P ->

FSet_elt_Prop Q ->

FSet_elt_Prop (P -> Q)

| not_elt_prop : forall P,

FSet_elt_Prop P ->

FSet_elt_Prop (~ P).

Inductive FSet_Prop : Prop -> Prop :=

| elt_FSet_Prop : forall P,

FSet_elt_Prop P ->

FSet_Prop P

| Empty_FSet_Prop : forall s,

FSet_Prop (Empty s)

| Subset_FSet_Prop : forall s1 s2,

FSet_Prop (Subset s1 s2)

| Equal_FSet_Prop : forall s1 s2,

FSet_Prop (Equal s1 s2).

Here is the tactic that will throw away hypotheses that
are not useful (for the intended scope of the fsetdec
tactic).

#[global]

Hint Constructors FSet_elt_Prop FSet_Prop : FSet_Prop.

Ltac discard_nonFSet :=

repeat (

match goal with

| H : context [ @Logic.eq ?T ?x ?y ] |- _ =>

tryif (change T with E.t in H) then fail

else tryif (change T with t in H) then fail

else clear H

| H : ?P |- _ =>

tryif prop (FSet_Prop P) holds by

(auto 100 with FSet_Prop)

then fail

else clear H

end).

Hint Constructors FSet_elt_Prop FSet_Prop : FSet_Prop.

Ltac discard_nonFSet :=

repeat (

match goal with

| H : context [ @Logic.eq ?T ?x ?y ] |- _ =>

tryif (change T with E.t in H) then fail

else tryif (change T with t in H) then fail

else clear H

| H : ?P |- _ =>

tryif prop (FSet_Prop P) holds by

(auto 100 with FSet_Prop)

then fail

else clear H

end).

## Turning Set Operators into Propositional Connectives

The lemmas from FSetFacts will be used to break down set operations into propositional formulas built over the predicates In and E.eq applied only to variables. We are going to use them with autorewrite.Global Hint Rewrite

F.empty_iff F.singleton_iff F.add_iff F.remove_iff

F.union_iff F.inter_iff F.diff_iff

: set_simpl.

Lemma eq_refl_iff (x : E.t) : E.eq x x <-> True.

Global Hint Rewrite eq_refl_iff : set_eq_simpl.

E.eq is decidable.

The hint database FSet_decidability will be given to
the push_neg tactic from the module Negation.

#[global]

Hint Resolve dec_In dec_eq : FSet_decidability.

Hint Resolve dec_In dec_eq : FSet_decidability.

## Normalizing Propositions About Equality

We have to deal with the fact that E.eq may be convertible with Coq's equality. Thus, we will find the following tactics useful to replace one form with the other everywhere.
Ltac change_to_E_t :=

repeat (

match goal with

| H : ?T |- _ =>

progress (change T with E.t in H);

repeat (

match goal with

| J : _ |- _ => progress (change T with E.t in J)

| |- _ => progress (change T with E.t)

end )

| H : forall x : ?T, _ |- _ =>

progress (change T with E.t in H);

repeat (

match goal with

| J : _ |- _ => progress (change T with E.t in J)

| |- _ => progress (change T with E.t)

end )

end).

repeat (

match goal with

| H : ?T |- _ =>

progress (change T with E.t in H);

repeat (

match goal with

| J : _ |- _ => progress (change T with E.t in J)

| |- _ => progress (change T with E.t)

end )

| H : forall x : ?T, _ |- _ =>

progress (change T with E.t in H);

repeat (

match goal with

| J : _ |- _ => progress (change T with E.t in J)

| |- _ => progress (change T with E.t)

end )

end).

These two tactics take us from Coq's built-in equality
to E.eq (and vice versa) when possible.

Ltac Logic_eq_to_E_eq :=

repeat (

match goal with

| H: _ |- _ =>

progress (change (@Logic.eq E.t) with E.eq in H)

| |- _ =>

progress (change (@Logic.eq E.t) with E.eq)

end).

Ltac E_eq_to_Logic_eq :=

repeat (

match goal with

| H: _ |- _ =>

progress (change E.eq with (@Logic.eq E.t) in H)

| |- _ =>

progress (change E.eq with (@Logic.eq E.t))

end).

This tactic works like the built-in tactic subst, but
at the level of set element equality (which may not be
the convertible with Coq's equality).

Ltac substFSet :=

repeat (

match goal with

| H: E.eq ?x ?x |- _ => clear H

| H: E.eq ?x ?y |- _ => rewrite H in *; clear H

end);

autorewrite with set_eq_simpl in *.

repeat (

match goal with

| H: E.eq ?x ?x |- _ => clear H

| H: E.eq ?x ?y |- _ => rewrite H in *; clear H

end);

autorewrite with set_eq_simpl in *.

## Considering Decidability of Base Propositions

This tactic adds assertions about the decidability of E.eq and In to the context. This is necessary for the completeness of the fsetdec tactic. However, in order to minimize the cost of proof search, we should be careful to not add more than we need. Once negations have been pushed to the leaves of the propositions, we only need to worry about decidability for those base propositions that appear in a negated form.
Ltac assert_decidability :=

We actually don't want these rules to fire if the
syntactic context in the patterns below is trivially
empty, but we'll just do some clean-up at the
afterward.

repeat (

match goal with

| H: context [~ E.eq ?x ?y] |- _ =>

assert new (E.eq x y \/ ~ E.eq x y) by (apply dec_eq)

| H: context [~ In ?x ?s] |- _ =>

assert new (In x s \/ ~ In x s) by (apply dec_In)

| |- context [~ E.eq ?x ?y] =>

assert new (E.eq x y \/ ~ E.eq x y) by (apply dec_eq)

| |- context [~ In ?x ?s] =>

assert new (In x s \/ ~ In x s) by (apply dec_In)

end);

match goal with

| H: context [~ E.eq ?x ?y] |- _ =>

assert new (E.eq x y \/ ~ E.eq x y) by (apply dec_eq)

| H: context [~ In ?x ?s] |- _ =>

assert new (In x s \/ ~ In x s) by (apply dec_In)

| |- context [~ E.eq ?x ?y] =>

assert new (E.eq x y \/ ~ E.eq x y) by (apply dec_eq)

| |- context [~ In ?x ?s] =>

assert new (In x s \/ ~ In x s) by (apply dec_In)

end);

Now we eliminate the useless facts we added (because
they would likely be very harmful to performance).

## Handling Empty, Subset, and Equal

This tactic instantiates universally quantified hypotheses (which arise from the unfolding of Empty, Subset, and Equal) for each of the set element expressions that is involved in some membership or equality fact. Then it throws away those hypotheses, which should no longer be needed.
Ltac inst_FSet_hypotheses :=

repeat (

match goal with

| H : forall a : E.t, _,

_ : context [ In ?x _ ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ In ?x _ ] =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _,

_ : context [ E.eq ?x _ ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ E.eq ?x _ ] =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _,

_ : context [ E.eq _ ?x ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ E.eq _ ?x ] =>

let P := type of (H x) in

assert new P by (exact (H x))

end);

repeat (

match goal with

| H : forall a : E.t, _ |- _ =>

clear H

end).

repeat (

match goal with

| H : forall a : E.t, _,

_ : context [ In ?x _ ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ In ?x _ ] =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _,

_ : context [ E.eq ?x _ ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ E.eq ?x _ ] =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _,

_ : context [ E.eq _ ?x ] |- _ =>

let P := type of (H x) in

assert new P by (exact (H x))

| H : forall a : E.t, _

|- context [ E.eq _ ?x ] =>

let P := type of (H x) in

assert new P by (exact (H x))

end);

repeat (

match goal with

| H : forall a : E.t, _ |- _ =>

clear H

end).

## The Core fsetdec Auxiliary Tactics

Ltac fsetdec_rec := progress substFSet; intuition fsetdec_rec.

If we add unfold Empty, Subset, Equal in *; intros; to
the beginning of this tactic, it will satisfy the same
specification as the fsetdec tactic; however, it will
be much slower than necessary without the pre-processing
done by the wrapper tactic fsetdec.

Ltac fsetdec_body :=

autorewrite with set_eq_simpl in *;

inst_FSet_hypotheses;

autorewrite with set_simpl set_eq_simpl in *;

push not in * using FSet_decidability;

substFSet;

assert_decidability;

auto;

(intuition fsetdec_rec) ||

fail 1

"because the goal is beyond the scope of this tactic".

End FSetDecideAuxiliary.

Import FSetDecideAuxiliary.

autorewrite with set_eq_simpl in *;

inst_FSet_hypotheses;

autorewrite with set_simpl set_eq_simpl in *;

push not in * using FSet_decidability;

substFSet;

assert_decidability;

auto;

(intuition fsetdec_rec) ||

fail 1

"because the goal is beyond the scope of this tactic".

End FSetDecideAuxiliary.

Import FSetDecideAuxiliary.

# The fsetdec Tactic

Here is the top-level tactic (the only one intended for clients of this library). It's specification is given at the top of the file.
Ltac fsetdec :=

We first unfold any occurrences of iff.

We fold occurrences of not because it is better for
intros to leave us with a goal of ~ P than a goal of
False.

fold any not; intros;

We don't care about the value of elements : complex ones are
abstracted as new variables (avoiding potential dependencies,
see bug 2464)

abstract_elements;

We remove dependencies to logical hypothesis. This way,
later "clear" will work nicely (see bug 2136)

no_logical_interdep;

Now we decompose conjunctions, which will allow the
discard_nonFSet and assert_decidability tactics to
do a much better job.

decompose records;

discard_nonFSet;

discard_nonFSet;

We unfold these defined propositions on finite sets. If
our goal was one of them, then have one more item to
introduce now.

We now want to get rid of all uses of = in favor of
E.eq. However, the best way to eliminate a = is in
the context is with subst, so we will try that first.
In fact, we may as well convert uses of E.eq into =
when possible before we do subst so that we can even
more mileage out of it. Then we will convert all
remaining uses of = back to E.eq when possible. We
use change_to_E_t to ensure that we have a canonical
name for set elements, so that Logic_eq_to_E_eq will
work properly.

change_to_E_t; E_eq_to_Logic_eq; subst++; Logic_eq_to_E_eq;

The next optimization is to swap a negated goal with a
negated hypothesis when possible. Any swap will improve
performance by eliminating the total number of
negations, but we will get the maximum benefit if we
swap the goal with a hypotheses mentioning the same set
element, so we try that first. If we reach the fourth
branch below, we attempt any swap. However, to maintain
completeness of this tactic, we can only perform such a
swap with a decidable proposition; hence, we first test
whether the hypothesis is an FSet_elt_Prop, noting
that any FSet_elt_Prop is decidable.

pull not using FSet_decidability;

unfold not in *;

match goal with

| H: (In ?x ?r) -> False |- (In ?x ?s) -> False =>

contradict H; fsetdec_body

| H: (In ?x ?r) -> False |- (E.eq ?x ?y) -> False =>

contradict H; fsetdec_body

| H: (In ?x ?r) -> False |- (E.eq ?y ?x) -> False =>

contradict H; fsetdec_body

| H: ?P -> False |- ?Q -> False =>

tryif prop (FSet_elt_Prop P) holds by

(auto 100 with FSet_Prop)

then (contradict H; fsetdec_body)

else fsetdec_body

| |- _ =>

fsetdec_body

end.

unfold not in *;

match goal with

| H: (In ?x ?r) -> False |- (In ?x ?s) -> False =>

contradict H; fsetdec_body

| H: (In ?x ?r) -> False |- (E.eq ?x ?y) -> False =>

contradict H; fsetdec_body

| H: (In ?x ?r) -> False |- (E.eq ?y ?x) -> False =>

contradict H; fsetdec_body

| H: ?P -> False |- ?Q -> False =>

tryif prop (FSet_elt_Prop P) holds by

(auto 100 with FSet_Prop)

then (contradict H; fsetdec_body)

else fsetdec_body

| |- _ =>

fsetdec_body

end.

Module FSetDecideTestCases.

Lemma test_eq_trans_1 : forall x y z s,

E.eq x y ->

~ ~ E.eq z y ->

In x s ->

In z s.

Lemma test_eq_trans_2 : forall x y z r s,

In x (singleton y) ->

~ In z r ->

~ ~ In z (add y r) ->

In x s ->

In z s.

Lemma test_eq_neq_trans_1 : forall w x y z s,

E.eq x w ->

~ ~ E.eq x y ->

~ E.eq y z ->

In w s ->

In w (remove z s).

Lemma test_eq_neq_trans_2 : forall w x y z r1 r2 s,

In x (singleton w) ->

~ In x r1 ->

In x (add y r1) ->

In y r2 ->

In y (remove z r2) ->

In w s ->

In w (remove z s).

Lemma test_In_singleton : forall x,

In x (singleton x).

Lemma test_add_In : forall x y s,

In x (add y s) ->

~ E.eq x y ->

In x s.

Lemma test_Subset_add_remove : forall x s,

s [<=] (add x (remove x s)).

Lemma test_eq_disjunction : forall w x y z,

In w (add x (add y (singleton z))) ->

E.eq w x \/ E.eq w y \/ E.eq w z.

Lemma test_not_In_disj : forall x y s1 s2 s3 s4,

~ In x (union s1 (union s2 (union s3 (add y s4)))) ->

~ (In x s1 \/ In x s4 \/ E.eq y x).

Lemma test_not_In_conj : forall x y s1 s2 s3 s4,

~ In x (union s1 (union s2 (union s3 (add y s4)))) ->

~ In x s1 /\ ~ In x s4 /\ ~ E.eq y x.

Lemma test_iff_conj : forall a x s s',

(In a s' <-> E.eq x a \/ In a s) ->

(In a s' <-> In a (add x s)).

Lemma test_set_ops_1 : forall x q r s,

(singleton x) [<=] s ->

Empty (union q r) ->

Empty (inter (diff s q) (diff s r)) ->

~ In x s.

Lemma eq_chain_test : forall x1 x2 x3 x4 s1 s2 s3 s4,

Empty s1 ->

In x2 (add x1 s1) ->

In x3 s2 ->

~ In x3 (remove x2 s2) ->

~ In x4 s3 ->

In x4 (add x3 s3) ->

In x1 s4 ->

Subset (add x4 s4) s4.

Lemma test_too_complex : forall x y z r s,

E.eq x y ->

(In x (singleton y) -> r [<=] s) ->

In z r ->

In z s.

fsetdec is not intended to solve this directly.

Lemma function_test_1 :

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).

Lemma function_test_2 :

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).

fsetdec is not intended to solve this directly.

Lemma test_baydemir :

forall (f : t -> t),

forall (s : t),

forall (x y : elt),

In x (add y (f s)) ->

~ E.eq x y ->

In x (f s).

End FSetDecideTestCases.

End WDecide_fun.

Require Import FSetInterface.

Now comes variants for self-contained weak sets and for full sets.
For these variants, only one argument is necessary. Thanks to
the subtyping WS<=S, the Decide functor which is meant to be
used on modules (M:S) can simply be an alias of WDecide.