# Library Coq.MSets.MSetDecide

This file implements a decision procedure for a certain class of propositions involving finite sets.
Require Import Decidable DecidableTypeEx MSetFacts.
First, a version for Weak Sets in functorial presentation
Module WDecideOn (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 MSetLogicalFacts.
Require Export Decidable.
Require 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.
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 MSetLogicalFacts.
Import MSetLogicalFacts.

# 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.
Module MSetDecideAuxiliary.

## Generic Tactics

We begin by defining a few generic, useful tactics.
remove logical hypothesis inter-dependencies (fix 2136). *) 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. (** [if t then t1 else t2] executes [t] and, if it does not fail, then [t1] will be applied to all subgoals produced. If [t] fails, then [t2] is executed. *) Tactic Notation "if" tactic(t) "then" tactic(t1) "else" tactic(t2) := first [ t; first [ t1 | fail 2 ] | t2 ]. Ltac abstract_term t := if (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. (** 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. (** [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 *. (** [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). (** ** Discarding Irrelevant Hypotheses We will want to clear the context of any non-MSet-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 MSet_elt_Prop : Prop -> Prop := | eq_Prop : forall (S : Type) (x y : S), MSet_elt_Prop (x = y) | eq_elt_prop : forall x y, MSet_elt_Prop (E.eq x y) | In_elt_prop : forall x s, MSet_elt_Prop (In x s) | True_elt_prop : MSet_elt_Prop True | False_elt_prop : MSet_elt_Prop False | conj_elt_prop : forall P Q, MSet_elt_Prop P -> MSet_elt_Prop Q -> MSet_elt_Prop (P /\ Q) | disj_elt_prop : forall P Q, MSet_elt_Prop P -> MSet_elt_Prop Q -> MSet_elt_Prop (P \/ Q) | impl_elt_prop : forall P Q, MSet_elt_Prop P -> MSet_elt_Prop Q -> MSet_elt_Prop (P -> Q) | not_elt_prop : forall P, MSet_elt_Prop P -> MSet_elt_Prop (~ P). Inductive MSet_Prop : Prop -> Prop := | elt_MSet_Prop : forall P, MSet_elt_Prop P -> MSet_Prop P | Empty_MSet_Prop : forall s, MSet_Prop (Empty s) | Subset_MSet_Prop : forall s1 s2, MSet_Prop (Subset s1 s2) | Equal_MSet_Prop : forall s1 s2, MSet_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). *) Hint Constructors MSet_elt_Prop MSet_Prop : MSet_Prop. Ltac discard_nonMSet := repeat ( match goal with | H : context [ @Logic.eq ?T ?x ?y ] |- _ => if (change T with E.t in H) then fail else if (change T with t in H) then fail else clear H | H : ?P |- _ => if prop (MSet_Prop P) holds by (auto 100 with MSet_Prop) then fail else clear H end). (** ** Turning Set Operators into Propositional Connectives The lemmas from [MSetFacts] 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]. *) 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. Proof. now split. Qed. Hint Rewrite eq_refl_iff : set_eq_simpl. (** ** Decidability of MSet Propositions *) (** [In] is decidable. *) Lemma dec_In : forall x s, decidable (In x s). Proof. red; intros; generalize (F.mem_iff s x); case (mem x s); intuition. Qed. (** [E.eq] is decidable. *) Lemma dec_eq : forall (x y : E.t), decidable (E.eq x y). Proof. red; intros x y; destruct (E.eq_dec x y); auto. Qed. (** The hint database [MSet_decidability] will be given to the [push_neg] tactic from the module [Negation]. *) Hint Resolve dec_In dec_eq : MSet_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. *) (** The next tactic, [Logic_eq_to_E_eq], mentions the term [E.t]; thus, we must ensure that [E.t] is used in favor of any other convertible but syntactically distinct term. *) 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). (** 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 substMSet := 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); (** Now we eliminate the useless facts we added (because they would likely be very harmful to performance). *) repeat ( match goal with | _: ~ ?P, H : ?P \/ ~ ?P |- _ => clear H end). (** ** 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_MSet_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). (** ** The Core [fsetdec] Auxiliary Tactics *) (** Here is the crux of the proof search. Recursion through [intuition]! (This will terminate if I correctly understand the behavior of [intuition].) *) Ltac fsetdec_rec := progress substMSet; 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_MSet_hypotheses; autorewrite with set_simpl set_eq_simpl in *; push not in * using MSet_decidability; substMSet; assert_decidability; auto; (intuition fsetdec_rec) || fail 1 "because the goal is beyond the scope of this tactic". End MSetDecideAuxiliary. Import MSetDecideAuxiliary. (** * 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]. *) unfold iff in *; (** 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)

# Examples

Module MSetDecideTestCases.
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 MSetDecideTestCases.
End WDecideOn.
Require Import MSetInterface.
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.
Module WDecide (M:WSets) := !WDecideOn M.E M.
Module Decide := WDecide.