Package set-def: Definition of set types
Information
| name | set-def |
| version | 1.44 |
| description | Definition of set types |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-08-06 |
| requires | bool pair |
| show | Data.Bool Data.Pair Set |
Files
- Package tarball set-def-1.44.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
- ×
- Pair
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Pair
- ,
- Bool
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