Library Coq.MSets.MSetDecide
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 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.
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.
A simple test case to see how this works.
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.
A simple test case to see how this works.
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.
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.
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.
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 ] |-
_ =>
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 (
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.
Decidability of MSet Propositions
In is decidable.
E.eq is decidable.
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.
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.
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_nonMSet and assert_decidability tactics to
do a much better job.
decompose records;
discard_nonMSet;
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 MSet_elt_Prop, noting
that any MSet_elt_Prop is decidable.
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.
fsetdec is not intended to solve this directly.
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.