Package set-thm: Properties of set types

Information

nameset-thm
version1.28
descriptionProperties of set types
authorJoe Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2011-11-27
requiresbool
function
pair
natural
set-def
showData.Bool
Data.Pair
Function
Number.Natural
Set

Files

Theorems

¬(universe = )

bigIntersect = universe

bigUnion =

x. x universe

s. disjoint s

s. disjoint s

s. s

s. s universe

s. s s

fromPredicate (λx. F) =

x. ¬(x )

s. ¬(universe s)

s. ¬(s )

s. ¬(s s)

x. delete x =

x. insert x universe = universe

s. \ s =

s. s \ = s

s. s \ universe =

s. s \ s =

s. image id s = s

s. s s = s

s. s s = s

f. image f =

s. image (λx. x) s = s

universe = insert T (insert F )

s. bigIntersect (insert s ) = s

s. bigUnion (insert s ) = s

x s. x insert x s

x s. delete s x s

s t. disjoint s (t \ s)

s t. disjoint (t \ s) s

s t. s \ t s

s. disjoint s s s =

s. universe s s = universe

s. s s =

x s. ¬(insert x s = )

s t. disjoint s t disjoint t s

s t. s t = t s

s t. s t = t s

s. (x. x s) s = universe

s. s universe x. ¬(x s)

s. (x. x s) ¬(s = )

x y. x insert y x = y

x s. x s insert x s = s

x s. s \ insert x = delete s x

x s. insert x (insert x s) = insert x s

x s. delete (delete s x) x = delete s x

x s. insert x s = insert x s

s x. insert x s x s

s t. s t s t = s

s t. s t s t = t

s t. s \ t = s t

s t. s \ t = s disjoint s t

s t. t (s \ t) = t s

s t. s \ t \ t = s \ t

s t. s \ t t = s t

s u. bigIntersect (insert s u) = s bigIntersect u

s u. bigUnion (insert s u) = s bigUnion u

f s. image f s = s =

f g. f g bigIntersect g bigIntersect f

f g. f g bigUnion f bigUnion g

{ m. m | m < 0 } =

x s. disjoint s (insert x ) ¬(x s)

x s. delete s x = s ¬(x s)

x s. disjoint (insert x ) s ¬(x s)

s t. bigIntersect (insert s (insert t )) = s t

s t. bigUnion (insert s (insert t )) = s t

f x. image f (insert x ) = insert (f x)

f s. image f (bigUnion s) = bigUnion (image (image f) s)

s t. bigIntersect (s t) = bigIntersect s bigIntersect t

s t. bigUnion (s t) = bigUnion s bigUnion t

s t x. s t s insert x t

x s. x s insert x (delete s x) = s

s t. s t t s s = t

s t. s (t \ s) = t s t

s t. t \ s s = t s t

s t. s t t s s = t

(s. s = ) s. s =

(s. universe s = s) s. s universe = s

(s. s = s) s. s = s

(s. universe s = universe) s. s universe = universe

{ s. s | s } = insert

f s x. x s f x image f s

x s. delete (insert x s) x = s ¬(x s)

s t. s t s t ¬(t s)

s. bigIntersect s = universe t. t s t = universe

s. bigUnion s = t. t s t =

x y s. insert x (insert y s) = insert y (insert x s)

x y s. delete (delete s x) y = delete (delete s y) x

x s t. s insert x t delete s x t

x s t. insert x s t = insert x (s t)

a. { x. x | a = x } = insert a

a. { x. x | x = a } = insert a

s c. image (λx. c) s = if s = then else insert c

s t x. disjoint (delete s x) t disjoint (delete t x) s

s t x. s \ insert x t = delete s x \ t

s t x. delete s x t = delete (s t) x

s t u. s t u s \ t u

s t u. s t u s \ u t

t u s. s \ t \ u = s \ (t u)

t u s. s \ t \ u = s \ u \ t

s t u. s t u = s (t u)

s t u. s t u = s (t u)

s t u. s t t u s u

s t u. s t t u s u

s t u. s t t u s u

s t u. s t t u s u

s t. (x. x s x t) s = t

f s t. s t image f s image f t

f g s. image (f g) s = image f (image g s)

s t. s t = s = t =

s t. cross s t = s = t =

s t. disjoint s t ¬x. x s x t

s t. disjoint s (bigUnion t) x. x t disjoint s x

t f. t bigIntersect f s. s f t s

f x s. image f (insert x s) = insert (f x) (image f s)

s x. x bigIntersect s t. t s x t

s x. x bigUnion s t. t s x t

f t. bigUnion f t s. s f s t

x s. x rest s x s ¬(x = choice s)

p x. x { y. y | p y } p x

(s t. s s t) s t. s t s

(s t. s t s) s t. t s s

x y s. x insert y s x = y x s

x s t. insert x s t x t s t

s t x. x s t x s x t

s t x. x s t x s x t

s t u. s t \ u s t disjoint s u

s t u. s t u s t s u

s t u. disjoint (s t) u disjoint s u disjoint t u

s t u. s t u s u t u

s t u. s (t u) = s t s u

s t u. s t u = (s t) (s u)

s t u. (s t) u = s u t u

s t u. s t u = (s u) (t u)

s t. ¬(s = t) x. x t ¬(x s)

f s t. image f (s t) = image f s image f t

f. (y. x. f x = y) g. y. f (g y) = y

f. (t. s. image f s = t) y. x. f x = y

s P. { x. x | x s P x } s

f s. { x. f x | x s } = image f s

x s t. s delete t x s t ¬(x s)

x s t. disjoint (insert x s) t ¬(x t) disjoint s t

x s. ¬(x s) t. s insert x t s t

s x y. x delete s y x s ¬(x = y)

s x. x s t. s = insert x t ¬(x t)

s t x. x s \ t x s ¬(x t)

s t. s t x. ¬(x s) insert x s t

s. s = x t. s = insert x t ¬(x t)

s. bigIntersect s = universe \ bigUnion { t. universe \ t | t s }

s. bigUnion s = universe \ bigIntersect { t. universe \ t | t s }

y s f. y image f s x. y = f x x s

s t. s t s t a. a t ¬(a s)

x y s t. (x, y) cross s t x s y t

u s. u \ bigIntersect s = bigUnion { t. u \ t | t s }

f s t. s image f t u. u t s = image f u

s t. t bigUnion s = bigUnion { x. t x | x s }

s t. t bigIntersect s = bigIntersect { x. t x | x s }

s t. bigUnion s t = bigUnion { x. x t | x s }

s t. bigIntersect s t = bigIntersect { x. x t | x s }

P. P (a s. ¬(a s) P (insert a s)) s. P s

t. { x y. x, y | x y t x } =

x y s.
    delete (insert x s) y =
    if x = y then delete s y else insert x (delete s y)

x s t. insert x s t = if x t then insert x (s t) else s t

x s t. insert x s t = if x t then s t else insert x (s t)

s t x. insert x s \ t = if x t then s \ t else insert x (s \ t)

f. (x y. f x = f y x = y) g. x. g (f x) = x

s x x'. (x s x' s) (x delete s x' x' delete s x)

s x x'. (x delete s x' x' delete s x) x s x' s

P f s. (y. y image f s P y) x. x s P (f x)

P f s. (y. y image f s P y) x. x s P (f x)

f s. bigIntersect (image f s) = { y. y | x. x s y f x }

f s. bigUnion (image f s) = { y. y | x. x s y f x }

P f s. (t. t image f s P t) t. t s P (image f t)

P f s. (t. t image f s P t) t. t s P (image f t)

P f. { x. f x | P x } = image f { x. x | P x }

x y s.
    insert x (insert y s) = insert y (insert x s)
    insert x (insert x s) = insert x s

n. { m. m | m < suc n } = insert n { m. m | m < n }

P a s. (x. x insert a s P x) P a x. x s P x

P a s. (x. x insert a s P x) P a x. x s P x

P s. (x. x bigUnion s P x) t x. t s x t P x

P s. (x. x bigUnion s P x) t x. t s x t P x

P a b. (a, b) { x y. x, y | P x y } P a b

P. { p. p | P p } = { a b. a, b | P (a, b) }

f s a.
    (x. f x = f a x = a)
    image f (delete s a) = delete (image f s) (f a)

f. (s t. image f s = image f t s = t) x y. f x = f y x = y

p f q. (z. z { x. f x | p x } q z) x. p x q (f x)

p f q. (z. z { x. f x | p x } q z) x. p x q (f x)

f s t.
    (x y. f x = f y x = y) image f (s \ t) = image f s \ image f t

f s t.
    (x y. f x = f y x = y) image f (s t) = image f s image f t

p f. bigIntersect { x. f x | p x } = { a. a | x. p x a f x }

p f. bigUnion { x. f x | p x } = { a. a | x. p x a f x }

f s t. (y. y t x. f x = y) (x. f x t x s) image f s = t

f. (y. x. f x = y) P. image f { x. x | P (f x) } = { x. x | P x }

d t.
    { f. f | (x. x f x t) x. ¬(x ) f x = d } =
    insert (λx. d)

f s.
    (x y. x s y s f x = f y x = y) g. x. x s g (f x) = x

f u v.
    (t. t v s. s u image f s = t)
    y. y v x. x u f x = y

f s.
    t.
      t s image f s = image f t
      x y. x t y t f x = f y x = y

p f q. (z. z { x y. f x y | p x y } q z) x y. p x y q (f x y)

p f q. (z. z { x y. f x y | p x y } q z) x y. p x y q (f x y)

f s t.
    (y. y t x. x s f x = y)
    g. y. y t g y s f (g y) = y

p f.
    bigIntersect { x y. f x y | p x y } =
    { a. a | x y. p x y a f x y }

p f.
    bigUnion { x y. f x y | p x y } = { a. a | x y. p x y a f x y }

(P. (x. x P x) T)
  P a s. (x. x insert a s P x) P a x. x s P x

(P. (x. x P x) F)
  P a s. (x. x insert a s P x) P a x. x s P x

f.
    (x y. f x = f y x = y) (y. x. f x = y)
    g. (y. f (g y) = y) x. g (f x) = x

a t.
    { s. s | s insert a t } =
    { s. s | s t } image (λs. insert a s) { s. s | s t }

p f q.
    (z. z { w x y. f w x y | p w x y } q z)
    w x y. p w x y q (f w x y)

p f q.
    (z. z { w x y. f w x y | p w x y } q z)
    w x y. p w x y q (f w x y)

p f.
    bigIntersect { x y z. f x y z | p x y z } =
    { a. a | x y z. p x y z a f x y z }

p f.
    bigUnion { x y z. f x y z | p x y z } =
    { a. a | x y z. p x y z a f x y z }

f u.
    (s t. s u t u image f s = image f t s = t)
    x y. x u y u f x = f y x = y

f s.
    bigIntersect { x. bigUnion (f x) | x s } =
    bigUnion { g. bigIntersect { x. g x | x s } | x. x s g x f x }

s t a.
    { x y. x, y | x insert a s y t x } =
    image ((,) a) (t a) { x y. x, y | x s y t x }

s.
    { t. t | t s } =
    image (λp. { x. x | p x })
      { p. p | (x. x s p x universe) x. ¬(x s) (p x F) }

p q r.
    p q = q p p q r = p (q r) p (q r) = q (p r)
    p p = p p (p q) = p q

p q r.
    p q = q p p q r = p (q r) p (q r) = q (p r)
    p p = p p (p q) = p q

f s t.
    (x. x s f x t)
    ((x y. x s y s f x = f y x = y)
     (y. y t x. x s f x = y)
     g.
       (y. y t g y s) (y. y t f (g y) = y)
       x. x s g (f x) = x)

d a s t.
    { f. f |
      (x. x insert a s f x t) x. ¬(x insert a s) f x = d } =
    image (λ(b, g) x. if x = a then b else g x)
      (cross t { f. f | (x. x s f x t) x. ¬(x s) f x = d })

Input Type Operators

Input Constants

Assumptions

T

id = λx. x

¬F T

¬T F

t. t t

F p. p

t. t ¬t

(¬) = λp. p F

() = λP. P ((select) P)

t. (x. t) t

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx. T

t. ¬¬t t

t. (T t) t

t. (t T) t

t. F t F

t. T t t

t. t F F

t. t T t

t. t t t

t. F t T

t. T t t

t. t T T

t. F t t

t. T t T

t. t F t

t. t T T

t. t t t

m. m < 0 F

t. (F t) ¬t

t. (t F) ¬t

t. t F ¬t

() = λp q. p q p

t. (t T) (t F)

s. rest s = delete s (choice s)

t1 t2. (if F then t1 else t2) = t2

t1 t2. (if T then t1 else t2) = t1

p x. p x p ((select) p)

p. x y. p = (x, y)

x y. x = y y = x

x y. x = y y = x

t1 t2. t1 t2 t2 t1

t1 t2. t1 t2 t2 t1

a b. (a b) a b

p x. x fromPredicate p p x

= { x. x | F }

universe = { x. x | T }

() = λp q. (λf. f p q) = λf. f T T

P. ¬(x. P x) x. ¬P x

P. ¬(x. P x) x. ¬P x

() = λP. q. (x. P x q) q

t1 t2. ¬(t1 t2) t1 ¬t2

t1 t2. ¬t1 ¬t2 t2 t1

s t. disjoint s t s t =

f g x. (f g) x = f (g x)

t1 t2. ¬(t1 t2) ¬t1 ¬t2

t1 t2. ¬(t1 t2) ¬t1 ¬t2

P. (p. P p) p1 p2. P (p1, p2)

P. (p. P p) p1 p2. P (p1, p2)

f g. (x. f x = g x) f = g

P a. (x. a = x P x) P a

P a. (x. x = a P x) P a

() = λp q. r. (p r) (q r) r

PAIR'. fn. a0 a1. fn (a0, a1) = PAIR' a0 a1

P. (x y. P x y) y x. P x y

P. (x y. P x y) y x. P x y

P Q. (x. P Q x) P x. Q x

P Q. P (x. Q x) x. P Q x

P Q. P (x. Q x) x. P Q x

P Q. P (x. Q x) x. P Q x

m n. m < suc n m = n m < n

s t. s t s t ¬(s = t)

P Q. (x. P x Q) (x. P x) Q

P Q. (x. P x) Q x. P x Q

P Q. (x. P x) Q x. P x Q

x y z. x = y y = z x = z

t1 t2 t3. (t1 t2) t3 t1 t2 t3

t1 t2 t3. (t1 t2) t3 t1 t2 t3

s t. s t x. x s x t

s t. (x. x s x t) s = t

P. (x. y. P x y) y. x. P x (y x)

p q r. p (q r) p q p r

p q r. p q r (p q) (p r)

p q r. (p q) r p r q r

P Q. (x. P x Q x) (x. P x) x. Q x

P Q. (x. P x Q x) (x. P x) x. Q x

s. bigIntersect s = { x. x | u. u s x u }

s. bigUnion s = { x. x | u. u s x u }

x y a b. (x, y) = (a, b) x = a y = b

x s. insert x s = { y. y | y = x y s }

s t. s t = { x. x | x s x t }

s t. s t = { x. x | x s x t }

s x. delete s x = { y. y | y s ¬(y = x) }

s t. s \ t = { x. x | x s ¬(x t) }

P c x y. P (if c then x else y) (c P x) (¬c P y)

f s. image f s = { y. y | x. x s y = f x }

s t. cross s t = { x y. x, y | x s y t }