Package set-def: Definition of set types

Information

name	set-def
version	1.37
description	Definition of set types
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2012-06-16
requires	bool
show	Data.Bool Data.Pair Set

Files

Package tarball set-def-1.37.tgz
Theory file set-def.thy (included in the package tarball)

Defined Type Operator

Set
- set

Defined Constants

Set
- ∅
- bigIntersect
- bigUnion
- bijections
- choice
- cross
- delete
- \
- disjoint
- fromPredicate
- image
- injections
- insert
- ∩
- ∈
- ⊂
- rest
- singleton
- ⊆
- surjections
- ∪
- universe

Theorems

⊦ ∀s. rest s = delete s (choice s)

⊦ ∀p x. x ∈ fromPredicate p ⇔ p x

⊦ ∅ = { x. x | ⊥ }

⊦ universe = { x. x | ⊤ }

⊦ ∀s. ¬(s = ∅) ⇒ choice s ∈ s

⊦ ∀s. singleton s ⇔ ∃x. s = insert x ∅

⊦ ∀s t. disjoint s t ⇔ s ∩ t = ∅

⊦ ∀s t. bijections s t = injections s t ∩ surjections s t

⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ¬(s = t)

⊦ ∀s t. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t

⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇒ s = t

⊦ ∀s. bigIntersect s = { x. x | ∀u. u ∈ s ⇒ x ∈ u }

⊦ ∀s. bigUnion s = { x. x | ∃u. u ∈ s ∧ x ∈ u }

⊦ ∀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) }

⊦ ∀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 }

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

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

Input Type Operators

→
bool
Data
- Pair
  - ×

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - ⊥
  - ⊤
- Pair
  - ,

Assumptions

⊦ ⊤

⊦ (∃) = 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

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀t₁ t₂. ¬t₁ ⇒ ¬t₂ ⇔ t₂ ⇒ t₁

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g