Package set-finite-thm: Properties of finite sets

Information

name	set-finite-thm
version	1.67
description	Properties of finite sets
author	Joe Leslie-Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory exported on 2017-06-23
checksum	7071fc262666d89cd6104ea49f6e156018b58452
requires	bool function natural pair set-def set-finite-def set-thm
show	Data.Bool Data.Pair Function Number.Natural Set

Files

Package tarball set-finite-thm-1.67.tgz
Theory source file set-finite-thm.thy (included in the package tarball)

Theorems

⊦ finite universe

⊦ infinite universe

⊦ finite universe ⇔ finite universe

⊦ infinite universe ⇔ infinite universe

⊦ ∀a. finite (insert a ∅)

⊦ ∀s. infinite s ⇒ ¬(s = ∅)

⊦ ∀s x. finite s ⇒ finite (delete s x)

⊦ ∀s x. finite (delete s x) ⇔ finite s

⊦ ∀s x. finite (insert x s) ⇔ finite s

⊦ ∀s t. finite s ⇒ finite (s \ t)

⊦ ∀s. finite s ⇒ ∃a. ¬(a ∈ s)

⊦ ∀f s. finite s ⇒ finite (image f s)

⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s

⊦ ∀s t. infinite s ∧ s ⊆ t ⇒ infinite t

⊦ ∀n. finite { m. m | m < n }

⊦ ∀n. finite { m. m | m ≤ n }

⊦ ∀s t. finite (s ∪ t) ⇔ finite s ∧ finite t

⊦ ∀s t. finite s ∧ finite t ⇒ finite (s ∪ t)

⊦ ∀s t. infinite s ∧ finite t ⇒ infinite (s \ t)

⊦ ∀s t. finite s ∨ finite t ⇒ finite (s ∩ t)

⊦ ∀s t. finite s ∧ finite t ⇒ finite (cross s t)

⊦ ∀s. finite s ⇔ ∃a. ∀x. x ∈ s ⇒ x ≤ a

⊦ ∀s. infinite s ⇔ ∀N. ∃n. N ≤ n ∧ n ∈ s

⊦ ∀s. finite s ⇒ finite { t. t | t ⊆ s }

⊦ ∀s. finite { t. t | t ⊆ s } ⇔ finite s

⊦ ∀s. finite s ⇒ (finite (bigUnion s) ⇔ ∀t. t ∈ s ⇒ finite t)

⊦ ∀s. finite (bigUnion s) ⇔ finite s ∧ ∀t. t ∈ s ⇒ finite t

⊦ ∀s p. finite s ⇒ finite { x. x | x ∈ s ∧ p x }

⊦ ∀s t. finite (cross s t) ⇔ s = ∅ ∨ t = ∅ ∨ finite s ∧ finite t

⊦ ∀f. (∀x y. f x = f y ⇒ x = y) ⇒ ∀s. infinite (image f s) ⇔ infinite s

⊦ ∀f s. finite s ⇒ finite { y. y | ∃x. x ∈ s ∧ y = f x }

⊦ ∀f s. finite (image f s) ⇔ ∃t. finite t ∧ t ⊆ s ∧ image f s = image f t

⊦ ∀f s. finite s ∧ s ⊆ bigUnion f ⇒ ∃g. finite g ∧ g ⊆ f ∧ s ⊆ bigUnion g

⊦ ∀s. infinite s ⇒ ∃r. (∀m n. m < n ⇒ r m < r n) ∧ image r universe = s

⊦ ∀f s t.
finite t ∧ t ⊆ image f s ⇔ ∃s'. finite s' ∧ s' ⊆ s ∧ t = image f s'

⊦ ∀f s t.
finite t ∧ t ⊆ image f s ⇒ ∃s'. finite s' ∧ s' ⊆ s ∧ t ⊆ image f s'

⊦ ∀s t. finite s ∧ finite t ⇒ finite { x y. (x, y) | x ∈ s ∧ y ∈ t }

⊦ ∀f.
finite f ⇒
bigUnion { t. t | t ∈ f ∧ ∀u. u ∈ f ⇒ ¬(t ⊂ u) } = bigUnion f

⊦ ∀p.
p ∅ ∧ (∀x s. p s ∧ ¬(x ∈ s) ∧ finite s ⇒ p (insert x s)) ⇒
∀s. finite s ⇒ p s

⊦ ∀p f s.
(∀t. finite t ∧ t ⊆ image f s ⇒ p t) ⇔
∀t. finite t ∧ t ⊆ s ⇒ p (image f t)

⊦ ∀p f s.
(∃t. finite t ∧ t ⊆ image f s ∧ p t) ⇔
∃t. finite t ∧ t ⊆ s ∧ p (image f t)

⊦ ∀f s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
(finite (image f s) ⇔ finite s)

⊦ ∀f s.
infinite s ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
infinite (image f s)

⊦ ∀f A. (∀x y. f x = f y ⇒ x = y) ∧ finite A ⇒ finite { x. x | f x ∈ A }

⊦ ∀f.
finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
bigIntersect f ∈ f

⊦ ∀f.
finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
bigUnion f ∈ f

⊦ ∀f s.
infinite s ∧ finite (image f s) ⇒
∃a. a ∈ s ∧ infinite { x. x | x ∈ s ∧ f x = f a }

⊦ ∀f t.
finite t ∧ (∀y. y ∈ t ⇒ finite { x. x | f x = y }) ⇒
finite { x. x | f x ∈ t }

⊦ ∀f s t.
finite s ∧ (∀x. x ∈ s ⇒ finite (t x)) ⇒
finite { x y. f x y | x ∈ s ∧ y ∈ t x }

⊦ ∀d s t.
finite s ∧ finite t ⇒
finite { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d }

⊦ ∀s t k.
finite s ∧ finite t ⇒
finite { f. f | image f s ⊆ t ∧ { x. x | ¬(f x = k x) } ⊆ s }

⊦ ∀f s.
finite s ∧ s ⊆ bigUnion f ∧ ¬(f = ∅) ∧
(∀t u. t ∈ f ∧ u ∈ f ⇒ t ⊆ u ∨ u ⊆ t) ⇒ ∃t. t ∈ f ∧ s ⊆ t

⊦ ∀f s.
    finite (image f s) ⇔
    ∃t.
      finite t ∧ t ⊆ s ∧ image f s = image f t ∧
      ∀x y. x ∈ t ∧ y ∈ t ⇒ (f x = f y ⇔ x = y)

⊦ ∀f t s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ finite t ⇒
finite { x. x | x ∈ s ∧ f x ∈ t }

⊦ ∀f s t.
finite t ∧ (∀y. y ∈ t ⇒ finite { x. x | x ∈ s ∧ f x = y }) ⇒
finite { x. x | x ∈ s ∧ f x ∈ t }

⊦ ∀r s.
finite s ∧ (∀x. ¬r x x) ∧ (∀x y z. r x y ∧ r y z ⇒ r x z) ∧
(∀x. x ∈ s ⇒ ∃y. y ∈ s ∧ r x y) ⇒ s = ∅

⊦ ∀p f s.
    (∀t. finite t ∧ t ⊆ image f s ⇒ p t) ⇔
    ∀t.
      finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒
      p (image f t)

⊦ ∀p f s.
    (∃t. finite t ∧ t ⊆ image f s ∧ p t) ⇔
    ∃t.
      finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧
      p (image f t)

External Type Operators

→
bool
Data
- Pair
  - ×
Number
- Natural
  - natural
Set
- set

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- Pair
  - ,
  - fst
  - snd
Function
- id
- ∘
Number
- Natural
  - +
  - <
  - ≤
  - bit1
  - max
  - suc
  - zero
Set
- ∅
- bigIntersect
- bigUnion
- cross
- delete
- \
- finite
- fromPredicate
- image
- infinite
- insert
- ∩
- ∈
- ⊂
- ⊆
- ∪
- universe

Assumptions

⊦ ⊤

⊦ finite ∅

⊦ ¬(universe = ∅)

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bigIntersect ∅ = universe

⊦ bigUnion ∅ = ∅

⊦ ∀x. x ∈ universe

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀s. ∅ ⊆ s

⊦ ∀s. s ⊆ s

⊦ ⊥ ⇔ ∀p. p

⊦ fromPredicate (λx. ⊥) = ∅

⊦ cross universe universe = universe

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀m. ¬(m < 0)

⊦ ∀n. n < suc n

⊦ ∀n. n ≤ suc n

⊦ ∀s. ¬(s ⊂ s)

⊦ (¬) = λp. 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 ⇔ t

⊦ ∀t. t ∧ ⊥ ⇔ ⊥

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. t ∧ t ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀t. t ∨ t ⇔ t

⊦ ∀s. s ∩ universe = s

⊦ ∀s. ∅ ∪ s = s

⊦ ∀s. s ∪ ∅ = s

⊦ ∀s. cross s ∅ = ∅

⊦ ∀s. cross ∅ s = ∅

⊦ universe = insert ⊤ (insert ⊥ ∅)

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀s. infinite s ⇔ ¬finite s

⊦ ∀x s. delete s x ⊆ s

⊦ ∀m n. m ≤ max m n

⊦ ∀m n. n ≤ max m n

⊦ ∀s t. s ⊆ s ∪ t

⊦ ∀s t. s ⊆ t ∪ s

⊦ ∀s t. s \ t ⊆ s

⊦ ∀s t. s ∩ t ⊆ s

⊦ ∀s t. t ∩ s ⊆ s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀m. suc m = m + 1

⊦ ∀s. s ⊆ ∅ ⇔ s = ∅

⊦ ∀a b. fst (a, b) = a

⊦ ∀a b. snd (a, b) = b

⊦ ∀x s. ¬(insert x s = ∅)

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x s. finite s ⇒ finite (insert x s)

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀s t. s ∩ t = t ∩ s

⊦ ∀s t. s ∪ t = t ∪ s

⊦ ∀s t. s ⊂ t ⇒ s ⊆ t

⊦ ∀p x. x ∈ fromPredicate p ⇔ p x

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m n. m < suc n ⇔ m ≤ n

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀s. (∃x. x ∈ s) ⇔ ¬(s = ∅)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y

⊦ ∀x s. x ∈ s ⇔ insert x s = s

⊦ ∀x s. insert x ∅ ∪ s = insert x s

⊦ ∀t₁ t₂. ¬(t₁ ⇒ t₂) ⇔ t₁ ∧ ¬t₂

⊦ ∀t₁ t₂. ¬t₁ ⇒ ¬t₂ ⇔ t₂ ⇒ t₁

⊦ ∀s t. s ⊆ t ⇔ s ∩ t = s

⊦ ∀s t. s ⊆ t ⇔ s ∪ t = t

⊦ ∀s u. bigIntersect (insert s u) = s ∩ bigIntersect u

⊦ ∀s u. bigUnion (insert s u) = s ∪ bigUnion u

⊦ ∀f g. f ⊆ g ⇒ bigUnion f ⊆ bigUnion g

⊦ { m. m | m < 0 } = ∅

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)

⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∀x. x = a ⇒ p x) ⇔ p a

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀x s. x ∈ s ⇒ insert x (delete s x) = s

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇔ s = t

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀f s x. x ∈ s ⇒ f x ∈ image f s

⊦ ∀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

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n

⊦ ∀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 s t. s ⊆ insert x t ⇔ delete s x ⊆ t

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀t₁ t₂ t₃. (t₁ ∧ t₂) ∧ t₃ ⇔ t₁ ∧ t₂ ∧ t₃

⊦ ∀t₁ t₂ t₃. (t₁ ∨ t₂) ∨ t₃ ⇔ t₁ ∨ t₂ ∨ t₃

⊦ ∀m n p. m < n ∧ n < p ⇒ m < p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀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. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t

⊦ ∀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)

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀s x. x ∈ bigUnion s ⇔ ∃t. t ∈ s ∧ x ∈ t

⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x

⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s

⊦ ∀p q r. p ∨ q ⇒ r ⇔ (p ⇒ r) ∧ (q ⇒ r)

⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀s p. { x. x | x ∈ s ∧ p x } ⊆ s

⊦ ∀f s. { x. f x | x ∈ s } = image f s

⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n

⊦ ∀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

⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x

⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀p. (∃n. p n) ⇔ ∃n. p n ∧ ∀m. m < n ⇒ ¬p m

⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀t. { x y. (x, y) | x ∈ ∅ ∧ y ∈ t x } = ∅

⊦ ∀p. p ∅ ∧ (∀x s. p s ⇒ p (insert x s)) ⇒ ∀a. finite a ⇒ p a

⊦ ∀s t. (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ bigUnion s ⊆ bigUnion t

⊦ ∀f s. image f s = { y. y | ∃x. x ∈ s ∧ y = f x }

⊦ ∀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)

⊦ ∀s t. cross s t = { x y. (x, y) | x ∈ s ∧ y ∈ t }

⊦ ∀f s. bigUnion (image f s) = { y. y | ∃x. x ∈ s ∧ y ∈ f x }

⊦ ∀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 b. (a, b) ∈ { x y. (x, y) | p x y } ⇔ p a b

⊦ ∀p. (∃n. p n) ∧ (∃m. ∀n. p n ⇒ n ≤ m) ⇔ ∃m. p m ∧ ∀n. p n ⇒ n ≤ m

⊦ ∀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)

⊦ ∀p f. bigUnion { x. f x | p x } = { a. a | ∃x. p x ∧ a ∈ f x }

⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅

⊦ ∀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.
s ⊆ image f t ⇔
∃u. u ⊆ t ∧ (∀x y. x ∈ u ∧ y ∈ u ∧ f x = f y ⇒ x = y) ∧ s = image f u

⊦ ∀p f s.
(∃t. t ⊆ image f s ∧ p t) ⇔
∃t. t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧ p (image f t)

⊦ ∀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 ⇔ ⊥) }

⊦ ∀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 })