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A Theory for Finite Sets
P. J. Windley
University of California, Davis
May 12, 1989
The HOL theory "sets.th" provides a theory of finite sets for HOL. The
theory is taken from chapter 10 of Manna and Waldinger's book "The Logical
Basis for Computer Porgramming, VOL 1." The theory presented there was
formalized inside HOL.
The theory of sets has two constructors EMPTY, which constructs the empty
set and INSERT which constructs a new set from an atom and another set.
There are two induction theorems (and tactics). The first is the standard
structural induction theorem returned from Tom Melham's type package:
set_induction =
|- !P. P EMPTY /\ (!s. P s ==> (!x. P(INSERT x s))) ==> (!s. P s)
The second is more specific to set and is quite useful. It is derived from
the first:
set_induction_2 =
|- !P.
P EMPTY /\ (!s x. ~x IN s ==> P s ==> P(INSERT x s)) ==> (!s. P s)
The two induction tactics are called SET_INDUCT_TAC and SET_INDUCT_2_TAC
respectively.
The following is a list of theorems and definitions contained in the theory
of sets:
SET_DISTINCT = |- !x s. ~(EMPTY = INSERT x s)
SET_CASES_THM = |- !s. (s = EMPTY) \/ (?s' x. s = INSERT x s')
IN =
|- (!x. x IN EMPTY = F) /\
(!x y s. x IN (INSERT y s) = (x = y) \/ x IN s)
COMPONENT = |- !x s. x IN (INSERT x s)
NONEMPTY_MEMBER = |- !s. ~(s = EMPTY) = (?x. x IN s)
ABSORPTION = |- !x s. x IN s ==> (INSERT x s = s)
MEMBER_DECOMP = |- !x s. x IN s ==> (?t. (s = INSERT x t) /\ ~x IN t)
SET_EQ = |- !s t. (s = t) = (!x. x IN s = x IN t)
UNION =
|- (!s. EMPTY UNION s = s) /\
(!x s1 s2.
(INSERT x s1) UNION s2 =
(x IN s2 => s1 UNION s2 | INSERT x(s1 UNION s2)))
IN_UNION = |- !x s1 s2. x IN (s1 UNION s2) = x IN s1 \/ x IN s2
UNION_ASSOC =
|- !s1 s2 s3. (s1 UNION s2) UNION s3 = s1 UNION (s2 UNION s3)
UNION_IDENT = |- !s. s UNION s = s
UNION_SYM = |- !s1 s2. s1 UNION s2 = s2 UNION s1
INTERSECT =
|- (!s. EMPTY INTERSECT s = EMPTY) /\
(!x s1 s2.
(INSERT x s1) INTERSECT s2 =
(x IN s2 => INSERT x(s1 INTERSECT s2) | s1 INTERSECT s2))
IN_INTERSECT = |- !x s1 s2. x IN (s1 INTERSECT s2) = x IN s1 /\ x IN s2
INTERSECT_ASSOC =
|- !s1 s2 s3.
(s1 INTERSECT s2) INTERSECT s3 = s1 INTERSECT (s2 INTERSECT s3)
INTERSECT_IDENT = |- !s. s INTERSECT s = s
INTERSECT_SYM = |- !s1 s2. s1 INTERSECT s2 = s2 INTERSECT s1
UNION_OVER_INTERSECT =
|- !s1 s2 s3.
s1 INTERSECT (s2 UNION s3) =
(s1 INTERSECT s2) UNION (s1 INTERSECT s3)
INTERSECT_OVER_UNION =
|- !s1 s2 s3.
s1 UNION (s2 INTERSECT s3) = (s1 UNION s2) INTERSECT (s1 UNION s3)
DISJOINT = |- !s t. DISJOINT s t = (s INTERSECT t = EMPTY)
DISJOINT_MEMBER = |- !s t. DISJOINT s t = ~(?x. x IN s /\ x IN t)
DELETE =
|- (!x. EMPTY DELETE x = EMPTY) /\
(!x s y.
(INSERT x s) DELETE y =
((x = y) => s DELETE y | INSERT x(s DELETE y)))
DELETE_MEMBER = |- !s x y. x IN (s DELETE y) = x IN s /\ ~(x = y)
DELETE_DECOMPOSITION = |- !s x. x IN s ==> (s = INSERT x(s DELETE x))
DELETE_ABSORPTION = |- !s x. ~x IN s ==> (s = s DELETE x)
CHOICE = |- !s. CHOICE s = (@x. x IN s)
REST = |- !s. REST s = s DELETE (CHOICE s)
CHOICE_MEMBER = |- !s. ~(s = EMPTY) ==> (CHOICE s) IN s
CHOICE_DECOMPOSITION =
|- !s. ~(s = EMPTY) ==> (s = INSERT(CHOICE s)(REST s))
CHOICE_NON_MEMBER = |- !s. ~(s = EMPTY) ==> ~(CHOICE s) IN (REST s)
DIFF =
|- (!s. s DIFF EMPTY = s) /\
(!s t x. s DIFF (INSERT x t) = (s DELETE x) DIFF t)
DIFF_MEMBER = |- !t s x. x IN (s DIFF t) = x IN s /\ ~x IN t
SUBSET =
|- (!s. EMPTY SUBSET s = T) /\
(!x s t. (INSERT x s) SUBSET t = x IN t /\ s SUBSET t)
SUBSET_MEMBER = |- !s t. s SUBSET t = (!y. y IN s ==> y IN t)
SUBSET_UNION = |- !s t. s SUBSET (s UNION t) /\ t SUBSET (s UNION t)
SUBSET_INTERSECT =
|- !s t. (s INTERSECT t) SUBSET s /\ (s INTERSECT t) SUBSET t
SUBSET_DELETE = |- !s x. ~(s = EMPTY) ==> (s DELETE x) SUBSET s
SUBSET_UNION_ABSORPTION = |- !s t. s SUBSET t = (s UNION t = t)
SUBSET_INTERSECT_ABSORPTION = |- !s t. s SUBSET t = (s INTERSECT t = s)
SUBSET_TRANS =
|- !s1 s2 s3. s1 SUBSET s2 /\ s2 SUBSET s3 ==> s1 SUBSET s3
SUBSET_ANTISYM = |- !s t. s SUBSET t /\ t SUBSET s ==> (s = t)
SUBSET_REFL = |- !s. s SUBSET s
PSUBSET = |- !s t. s PSUBSET t = s SUBSET t /\ ~(s = t)
PSUBSET_TRANS =
|- !s t.
s PSUBSET t = (!x. x IN s ==> x IN t) /\ (?y. y IN t /\ ~y IN s)
PSUBSET_REFL = |- !s. ~s PSUBSET s
PSUBSET_REST = |- !s. ~(s = EMPTY) ==> (REST s) PSUBSET s
MK_SET =
|- (!f. MK_SET EMPTY f = EMPTY) /\
(!x s f.
MK_SET(INSERT x s)f = (f x => INSERT x(MK_SET s f) | MK_SET s f))
MK_SET_MEMBER = |- !s x. x IN (MK_SET s f) = x IN s /\ f x
MK_SET_TRUE = |- !s. MK_SET s(\x. T) = s
MK_SET_FALSE = |- !s. MK_SET s(\x. F) = EMPTY
MK_SET_INTERSECT = |- !s t. s INTERSECT t = MK_SET s(\x. x IN t)
MK_SET_DELETE = |- !s y. s DELETE y = MK_SET s(\x. ~(x = y))
MK_SET_DIFF = |- !t s. s DIFF t = MK_SET s(\x. ~x IN t)
MK_SET_OR =
|- !s f g. MK_SET s(\x. f x \/ g x) = (MK_SET s f) UNION (MK_SET s g)
MK_SET_AND =
|- !s f g.
MK_SET s(\x. f x /\ g x) = (MK_SET s f) INTERSECT (MK_SET s g)
SING = |- !s. SING s = (?x. s = INSERT x EMPTY)
SING_CHOICE = |- !x. CHOICE(INSERT x EMPTY) = x
SING_REST = |- !s. SING s = ~(s = EMPTY) /\ (REST s = EMPTY)
The following are the cardinality results
(the theorem CARD is proved in the file
card.ml composed by Philippe Leveilley):
CARD =
|- (CARD EMPTY = 0) /\
(!x s. CARD(INSERT x s) = (x IN s => CARD s | (CARD s) + 1))
CARD_EQ_0 = |- !s. (CARD s = 0) ==> (s = EMPTY)
CARD_ABSORPTION = |- !s x. x IN s ==> (CARD(INSERT x s) = CARD s)
CARD_INTERSECT =
|- !s t.
(CARD(s INTERSECT t)) <= (CARD s) /\
(CARD(s INTERSECT t)) <= (CARD t)
CARD_UNION =
|- !s t. (CARD(s UNION t)) + (CARD(s INTERSECT t)) = (CARD s) + (CARD t)
CARD_SUBSET = |- !s t. s SUBSET t ==> (CARD s) <= (CARD t)
CARD_PSUBSET = |- !s t. s PSUBSET t ==> (CARD s) < (CARD t)
SING_CARD = |- !s. SING s = (CARD s = 1)
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