Theory: set

Parents


Type constants


Term constants


Axioms


Definitions

set_TY_DEF
|- ?rep. TYPE_DEFINITION (\p. T) rep
set_ISO_DEF
|- (!a. SPEC (CHF a) = a) /\ (!r. (\p. T) r = CHF (SPEC r) = r)
IN_DEF
|- !x s. x IN s = CHF s x
GSPEC_DEF
|- !f. GSPEC f = SPEC (\x. ?y. (x,T) = f y)
EMPTY_DEF
|- {} = SPEC (\x. F)
UNIV_DEF
|- UNIV = SPEC (\x. T)
SUBSET_DEF
|- !s t. s SUBSET t = (!x. x IN s ==> x IN t)
PSUBSET_DEF
|- !s t. s PSUBSET t = s SUBSET t /\ ~(s = t)
UNION_DEF
|- !s t. s UNION t = {x | x IN s \/ x IN t}
INTER_DEF
|- !s t. s INTER t = {x | x IN s /\ x IN t}
DISJOINT_DEF
|- !s t. DISJOINT s t = s INTER t = {}
DIFF_DEF
|- !s t. s DIFF t = {x | x IN s /\ ~(x IN t)}
INSERT_DEF
|- !x s. x INSERT s = {y | (y = x) \/ y IN s}
DELETE_DEF
|- !s x. s DELETE x = s DIFF {x}
CHOICE_DEF
|- !s. ~(s = {}) ==> CHOICE s IN s
REST_DEF
|- !s. REST s = s DELETE CHOICE s
SING_DEF
|- !s. SING s = (?x. s = {x})
IMAGE_DEF
|- !f s. IMAGE f s = {f x | x IN s}
INJ_DEF
|- !f s t.
     INJ f s t =
     (!x. x IN s ==> f x IN t) /\
     (!x y. x IN s /\ y IN s ==> (f x = f y) ==> (x = y))
SURJ_DEF
|- !f s t.
     SURJ f s t =
     (!x. x IN s ==> f x IN t) /\ (!x. x IN t ==> (?y. y IN s /\ (f y = x)))
BIJ_DEF
|- !f s t. BIJ f s t = INJ f s t /\ SURJ f s t
LINV_DEF
|- !f s t. INJ f s t ==> (!x. x IN s ==> (LINV f s (f x) = x))
RINV_DEF
|- !f s t. SURJ f s t ==> (!x. x IN t ==> (f (RINV f s x) = x))
FINITE_DEF
|- !s. FINITE s = (!P. P {} /\ (!s. P s ==> (!e. P (e INSERT s))) ==> P s)
CARD_DEF
|- (CARD {} = 0) /\
   (!s.
     FINITE s ==>
     (!x. CARD (x INSERT s) = ((x IN s) => (CARD s) | (SUC (CARD s)))))
INFINITE_DEF
|- !s. INFINITE s = ~(FINITE s)

Theorems

SPECIFICATION
|- !f x. x IN SPEC f = f x
EXTENSION
|- !s t. (s = t) = (!x. x IN s = x IN t)
NOT_EQUAL_SETS
|- !s t. ~(s = t) = (?x. x IN t = ~(x IN s))
NUM_SET_WOP
|- !s. (?n. n IN s) = (?n. n IN s /\ (!m. m IN s ==> n <= m))
GSPECIFICATION
|- !sp v. v IN GSPEC sp = (?y. (v,T) = sp y)
SET_MINIMUM
|- !s M. (?x. x IN s) = (?x. x IN s /\ (!y. y IN s ==> M x <= M y))
NOT_IN_EMPTY
|- !x. ~(x IN {})
MEMBER_NOT_EMPTY
|- !s. (?x. x IN s) = ~(s = {})
IN_UNIV
|- !x. x IN UNIV
UNIV_NOT_EMPTY
|- ~(UNIV = {})
EMPTY_NOT_UNIV
|- ~({} = UNIV)
EQ_UNIV
|- (!x. x IN s) = s = UNIV
SUBSET_TRANS
|- !s t u. s SUBSET t /\ t SUBSET u ==> s SUBSET u
SUBSET_REFL
|- !s. s SUBSET s
SUBSET_ANTISYM
|- !s t. s SUBSET t /\ t SUBSET s ==> (s = t)
EMPTY_SUBSET
|- !s. {} SUBSET s
SUBSET_EMPTY
|- !s. s SUBSET {} = s = {}
SUBSET_UNIV
|- !s. s SUBSET UNIV
UNIV_SUBSET
|- !s. UNIV SUBSET s = s = UNIV
PSUBSET_TRANS
|- !s t u. s PSUBSET t /\ t PSUBSET u ==> s PSUBSET u
PSUBSET_IRREFL
|- !s. ~(s PSUBSET s)
NOT_PSUBSET_EMPTY
|- !s. ~(s PSUBSET {})
NOT_UNIV_PSUBSET
|- !s. ~(UNIV PSUBSET s)
PSUBSET_UNIV
|- !s. s PSUBSET UNIV = (?x. ~(x IN s))
IN_UNION
|- !s t x. x IN s UNION t = x IN s \/ x IN t
UNION_ASSOC
|- !s t u. (s UNION t) UNION u = s UNION t UNION u
UNION_IDEMPOT
|- !s. s UNION s = s
UNION_COMM
|- !s t. s UNION t = t UNION s
SUBSET_UNION
|- (!s t. s SUBSET s UNION t) /\ (!s t. s SUBSET t UNION s)
SUBSET_UNION_ABSORPTION
|- !s t. s SUBSET t = s UNION t = t
UNION_EMPTY
|- (!s. {} UNION s = s) /\ (!s. s UNION {} = s)
UNION_UNIV
|- (!s. UNIV UNION s = UNIV) /\ (!s. s UNION UNIV = UNIV)
EMPTY_UNION
|- !s t. (s UNION t = {}) = (s = {}) /\ (t = {})
IN_INTER
|- !s t x. x IN s INTER t = x IN s /\ x IN t
INTER_ASSOC
|- !s t u. (s INTER t) INTER u = s INTER t INTER u
INTER_IDEMPOT
|- !s. s INTER s = s
INTER_COMM
|- !s t. s INTER t = t INTER s
INTER_SUBSET
|- (!s t. s INTER t SUBSET s) /\ (!s t. t INTER s SUBSET s)
SUBSET_INTER_ABSORPTION
|- !s t. s SUBSET t = s INTER t = s
INTER_EMPTY
|- (!s. {} INTER s = {}) /\ (!s. s INTER {} = {})
INTER_UNIV
|- (!s. UNIV INTER s = s) /\ (!s. s INTER UNIV = s)
UNION_OVER_INTER
|- !s t u. s INTER (t UNION u) = s INTER t UNION s INTER u
INTER_OVER_UNION
|- !s t u. s UNION t INTER u = (s UNION t) INTER (s UNION u)
IN_DISJOINT
|- !s t. DISJOINT s t = ~(?x. x IN s /\ x IN t)
DISJOINT_SYM
|- !s t. DISJOINT s t = DISJOINT t s
DISJOINT_EMPTY
|- !s. DISJOINT {} s /\ DISJOINT s {}
DISJOINT_EMPTY_REFL
|- !s. (s = {}) = DISJOINT s s
DISJOINT_UNION
|- !s t u. DISJOINT (s UNION t) u = DISJOINT s u /\ DISJOINT t u
IN_DIFF
|- !s t x. x IN s DIFF t = x IN s /\ ~(x IN t)
DIFF_EMPTY
|- !s. s DIFF {} = s
EMPTY_DIFF
|- !s. {} DIFF s = {}
DIFF_UNIV
|- !s. s DIFF UNIV = {}
DIFF_DIFF
|- !s t. (s DIFF t) DIFF t = s DIFF t
DIFF_EQ_EMPTY
|- !s. s DIFF s = {}
IN_INSERT
|- !x y s. x IN y INSERT s = (x = y) \/ x IN s
COMPONENT
|- !x. x IN x INSERT s
SET_CASES
|- !s. (s = {}) \/ (?x t. (s = x INSERT t) /\ ~(x IN t))
DECOMPOSITION
|- !s x. x IN s = (?t. (s = x INSERT t) /\ ~(x IN t))
ABSORPTION
|- !x s. x IN s = x INSERT s = s
INSERT_INSERT
|- !x s. x INSERT x INSERT s = x INSERT s
INSERT_COMM
|- !x y s. x INSERT y INSERT s = y INSERT x INSERT s
INSERT_UNIV
|- !x. x INSERT UNIV = UNIV
NOT_INSERT_EMPTY
|- !x s. ~(x INSERT s = {})
NOT_EMPTY_INSERT
|- !x s. ~({} = x INSERT s)
INSERT_UNION
|- !x s t.
     (x INSERT s) UNION t = ((x IN t) => (s UNION t) | (x INSERT (s UNION t)))
INSERT_UNION_EQ
|- !x s t. (x INSERT s) UNION t = x INSERT (s UNION t)
INSERT_INTER
|- !x s t.
     (x INSERT s) INTER t = ((x IN t) => (x INSERT s INTER t) | (s INTER t))
DISJOINT_INSERT
|- !x s t. DISJOINT (x INSERT s) t = DISJOINT s t /\ ~(x IN t)
INSERT_SUBSET
|- !x s t. x INSERT s SUBSET t = x IN t /\ s SUBSET t
SUBSET_INSERT
|- !x s. ~(x IN s) ==> (!t. s SUBSET x INSERT t = s SUBSET t)
INSERT_DIFF
|- !s t x.
     (x INSERT s) DIFF t = ((x IN t) => (s DIFF t) | (x INSERT (s DIFF t)))
IN_DELETE
|- !s x y. x IN s DELETE y = x IN s /\ ~(x = y)
DELETE_NON_ELEMENT
|- !x s. ~(x IN s) = s DELETE x = s
IN_DELETE_EQ
|- !s x x'. (x IN s = x' IN s) = x IN s DELETE x' = x' IN s DELETE x
EMPTY_DELETE
|- !x. {} DELETE x = {}
DELETE_DELETE
|- !x s. (s DELETE x) DELETE x = s DELETE x
DELETE_COMM
|- !x y s. (s DELETE x) DELETE y = (s DELETE y) DELETE x
DELETE_SUBSET
|- !x s. s DELETE x SUBSET s
SUBSET_DELETE
|- !x s t. s SUBSET t DELETE x = ~(x IN s) /\ s SUBSET t
SUBSET_INSERT_DELETE
|- !x s t. s SUBSET x INSERT t = s DELETE x SUBSET t
DIFF_INSERT
|- !s t x. s DIFF (x INSERT t) = (s DELETE x) DIFF t
PSUBSET_INSERT_SUBSET
|- !s t. s PSUBSET t = (?x. ~(x IN s) /\ x INSERT s SUBSET t)
PSUBSET_MEMBER
|- !s t. s PSUBSET t = s SUBSET t /\ (?y. y IN t /\ ~(y IN s))
DELETE_INSERT
|- !x y s.
     (x INSERT s) DELETE y =
     ((x = y) => (s DELETE y) | (x INSERT (s DELETE y)))
INSERT_DELETE
|- !x s. x IN s ==> (x INSERT (s DELETE x) = s)
DELETE_INTER
|- !s t x. (s DELETE x) INTER t = s INTER t DELETE x
DISJOINT_DELETE_SYM
|- !s t x. DISJOINT (s DELETE x) t = DISJOINT (t DELETE x) s
CHOICE_NOT_IN_REST
|- !s. ~(CHOICE s IN REST s)
CHOICE_INSERT_REST
|- !s. ~(s = {}) ==> (CHOICE s INSERT REST s = s)
REST_SUBSET
|- !s. REST s SUBSET s
REST_PSUBSET
|- !s. ~(s = {}) ==> REST s PSUBSET s
SING
|- !x. SING {x}
IN_SING
|- !x y. x IN {y} = x = y
NOT_SING_EMPTY
|- !x. ~({x} = {})
NOT_EMPTY_SING
|- !x. ~({} = {x})
EQUAL_SING
|- !x y. ({x} = {y}) = x = y
DISJOINT_SING_EMPTY
|- !x. DISJOINT {x} {}
INSERT_SING_UNION
|- !s x. x INSERT s = {x} UNION s
SING_DELETE
|- !x. {x} DELETE x = {}
DELETE_EQ_SING
|- !s x. x IN s ==> ((s DELETE x = {}) = s = {x})
CHOICE_SING
|- !x. CHOICE {x} = x
REST_SING
|- !x. REST {x} = {}
SING_IFF_EMPTY_REST
|- !s. SING s = ~(s = {}) /\ (REST s = {})
IN_IMAGE
|- !y s f. y IN IMAGE f s = (?x. (y = f x) /\ x IN s)
IMAGE_IN
|- !x s. x IN s ==> (!f. f x IN IMAGE f s)
IMAGE_EMPTY
|- !f. IMAGE f {} = {}
IMAGE_ID
|- !s. IMAGE (\x. x) s = s
IMAGE_COMPOSE
|- !f g s. IMAGE (f o g) s = IMAGE f (IMAGE g s)
IMAGE_INSERT
|- !f x s. IMAGE f (x INSERT s) = f x INSERT IMAGE f s
IMAGE_DELETE
|- !f x s. ~(x IN s) ==> (IMAGE f (s DELETE x) = IMAGE f s)
IMAGE_UNION
|- !f s t. IMAGE f (s UNION t) = IMAGE f s UNION IMAGE f t
IMAGE_SUBSET
|- !s t. s SUBSET t ==> (!f. IMAGE f s SUBSET IMAGE f t)
IMAGE_INTER
|- !f s t. IMAGE f (s INTER t) SUBSET IMAGE f s INTER IMAGE f t
INJ_ID
|- !s. INJ (\x. x) s s
INJ_COMPOSE
|- !f g s t u. INJ f s t /\ INJ g t u ==> INJ (g o f) s u
INJ_EMPTY
|- !f. (!s. INJ f {} s) /\ (!s. INJ f s {} = s = {})
SURJ_ID
|- !s. SURJ (\x. x) s s
SURJ_COMPOSE
|- !f g s t u. SURJ f s t /\ SURJ g t u ==> SURJ (g o f) s u
SURJ_EMPTY
|- !f. (!s. SURJ f {} s = s = {}) /\ (!s. SURJ f s {} = s = {})
IMAGE_SURJ
|- !f s t. SURJ f s t = IMAGE f s = t
BIJ_ID
|- !s. BIJ (\x. x) s s
BIJ_EMPTY
|- !f. (!s. BIJ f {} s = s = {}) /\ (!s. BIJ f s {} = s = {})
BIJ_COMPOSE
|- !f g s t u. BIJ f s t /\ BIJ g t u ==> BIJ (g o f) s u
FINITE_EMPTY
|- FINITE {}
FINITE_INDUCT
|- !P.
     P {} /\ (!s. FINITE s /\ P s ==> (!e. ~(e IN s) ==> P (e INSERT s))) ==>
     (!s. FINITE s ==> P s)
FINITE_INSERT
|- !x s. FINITE (x INSERT s) = FINITE s
FINITE_DELETE
|- !x s. FINITE (s DELETE x) = FINITE s
FINITE_UNION
|- !s t. FINITE (s UNION t) = FINITE s /\ FINITE t
INTER_FINITE
|- !s. FINITE s ==> (!t. FINITE (s INTER t))
SUBSET_FINITE
|- !s. FINITE s ==> (!t. t SUBSET s ==> FINITE t)
PSUBSET_FINITE
|- !s. FINITE s ==> (!t. t PSUBSET s ==> FINITE t)
FINITE_DIFF
|- !s. FINITE s ==> (!t. FINITE (s DIFF t))
FINITE_SING
|- !x. FINITE {x}
SING_FINITE
|- !s. SING s ==> FINITE s
IMAGE_FINITE
|- !s. FINITE s ==> (!f. FINITE (IMAGE f s))
CARD_EMPTY
|- CARD {} = 0
CARD_INSERT
|- !s.
     FINITE s ==>
     (!x. CARD (x INSERT s) = ((x IN s) => (CARD s) | (SUC (CARD s))))
CARD_EQ_0
|- !s. FINITE s ==> ((CARD s = 0) = s = {})
CARD_DELETE
|- !s.
     FINITE s ==>
     (!x. CARD (s DELETE x) = ((x IN s) => (CARD s - 1) | (CARD s)))
CARD_INTER_LESS_EQ
|- !s. FINITE s ==> (!t. CARD (s INTER t) <= CARD s)
CARD_UNION
|- !s.
     FINITE s ==>
     (!t.
       FINITE t ==> (CARD (s UNION t) + CARD (s INTER t) = CARD s + CARD t))
CARD_SUBSET
|- !s. FINITE s ==> (!t. t SUBSET s ==> CARD t <= CARD s)
CARD_PSUBSET
|- !s. FINITE s ==> (!t. t PSUBSET s ==> CARD t < CARD s)
CARD_SING
|- !x. CARD {x} = 1
SING_IFF_CARD1
|- !s. SING s = (CARD s = 1) /\ FINITE s
CARD_DIFF
|- !t.
     FINITE t ==>
     (!s. FINITE s ==> (CARD (s DIFF t) = CARD s - CARD (s INTER t)))
LESS_CARD_DIFF
|- !t. FINITE t ==> (!s. FINITE s ==> CARD t < CARD s ==> 0 < CARD (s DIFF t))
NOT_IN_FINITE
|- INFINITE UNIV = (!s. FINITE s ==> (?x. ~(x IN s)))
IMAGE_11_INFINITE
|- !f.
     (!x y. (f x = f y) ==> (x = y)) ==>
     (!s. INFINITE s ==> INFINITE (IMAGE f s))
INFINITE_SUBSET
|- !s. INFINITE s ==> (!t. s SUBSET t ==> INFINITE t)
IN_INFINITE_NOT_FINITE
|- !s t. INFINITE s /\ FINITE t ==> (?x. x IN s /\ ~(x IN t))
INFINITE_UNIV
|- INFINITE UNIV =
   (?f. (!x y. (f x = f y) ==> (x = y)) /\ (?y. !x. ~(f x = y)))
FINITE_PSUBSET_INFINITE
|- !s. INFINITE s = (!t. FINITE t ==> t SUBSET s ==> t PSUBSET s)
FINITE_PSUBSET_UNIV
|- INFINITE UNIV = (!s. FINITE s ==> s PSUBSET UNIV)
INFINITE_DIFF_FINITE
|- !s t. INFINITE s /\ FINITE t ==> ~(s DIFF t = {})
FINITE_ISO_NUM
|- !s.
     FINITE s ==>
     (?f.
       (!n m. n < CARD s /\ m < CARD s ==> (f n = f m) ==> (n = m)) /\
       (s = {f n | n < CARD s}))