Package relation-thm: Properties of relation operators
Information
| name | relation-thm |
| version | 1.8 |
| description | Properties of relation operators |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2012-06-08 |
| requires | bool pair relation-def set |
| show | Data.Bool Data.Pair Relation Set |
Files
- Package tarball relation-thm-1.8.tgz
- Theory file relation-thm.thy (included in the package tarball)
Theorems
⊦ irreflexive empty
⊦ reflexive universe
⊦ transitive empty
⊦ transitive universe
⊦ ∀r. transitive (transitiveClosure r)
⊦ ∀r. subrelation r r
⊦ ∀r. subrelation r (transitiveClosure r)
⊦ ∀x y. universe x y
⊦ ∀s. toSet (fromSet s) = s
⊦ ∀r. fromSet (toSet r) = r
⊦ ∀x y. ¬empty x y
⊦ ∀r x. reflexive r ⇒ r x x
⊦ ∀r x. irreflexive r ⇒ ¬r x x
⊦ ∀r s. toSet r = toSet s ⇒ r = s
⊦ ∀r s.
subrelation r s ∧ transitive s ⇒ subrelation (transitiveClosure r) s
⊦ ∀r s. subrelation r s ∧ subrelation s r ⇒ r = s
⊦ ∀r x y. (x, y) ∈ toSet r ⇔ r x y
⊦ ∀r s t. subrelation r s ∧ subrelation s t ⇒ subrelation r t
⊦ ∀r s. subrelation r (bigIntersect s) ⇔ ∀t. t ∈ s ⇒ subrelation r t
⊦ ∀r s. subrelation r s ⇔ ∀x y. r x y ⇒ s x y
⊦ ∀r s. (∀x y. r x y ⇔ s x y) ⇒ r = s
⊦ ∀s x y. bigIntersect s x y ⇔ ∀r. r ∈ s ⇒ r x y
⊦ ∀s x y. bigUnion s x y ⇔ ∃r. r ∈ s ∧ r x y
⊦ ∀r s x y. intersect r s x y ⇔ r x y ∧ s x y
⊦ ∀r s x y. union r s x y ⇔ r x y ∨ s x y
Input Type Operators
- →
- bool
- Data
- Pair
- ×
- Pair
- Set
- set
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Pair
- ,
- Bool
- Relation
- bigIntersect
- bigUnion
- empty
- fromSet
- intersect
- irreflexive
- reflexive
- subrelation
- toSet
- transitive
- transitiveClosure
- union
- universe
- Set
- ∅
- bigIntersect
- bigUnion
- fromPredicate
- image
- ∩
- ∈
- ⊆
- ∪
- universe
Assumptions
⊦ ⊤
⊦ empty = fromSet ∅
⊦ universe = fromSet universe
⊦ ¬⊥ ⇔ ⊤
⊦ ∀x. x ∈ universe
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ ∀x. ¬(x ∈ ∅)
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ⊥ ∧ t ⇔ ⊥
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. t ∧ ⊤ ⇔ t
⊦ ∀t. ⊥ ⇒ t ⇔ ⊤
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀r. reflexive r ⇔ ∀x. r x x
⊦ ∀s. bigIntersect s = fromSet (bigIntersect (image toSet s))
⊦ ∀s. bigUnion s = fromSet (bigUnion (image toSet s))
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∀r. irreflexive r ⇔ ∀x. ¬r x x
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀r s. subrelation r s ⇔ toSet r ⊆ toSet s
⊦ ∀r s. intersect r s = fromSet (toSet r ∩ toSet s)
⊦ ∀r s. union r s = fromSet (toSet r ∪ toSet s)
⊦ ∀p. (∀xy. p xy) ⇔ ∀x y. p (x, y)
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇒ s = t
⊦ ∀s x y. fromSet s x y ⇔ (x, y) ∈ s
⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r
⊦ ∀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
⊦ ∀t f. t ⊆ bigIntersect f ⇔ ∀s. s ∈ f ⇒ t ⊆ s
⊦ ∀s x. x ∈ bigIntersect s ⇔ ∀t. t ∈ s ⇒ x ∈ t
⊦ ∀s x. x ∈ bigUnion s ⇔ ∃t. t ∈ s ∧ x ∈ t
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ ∀s t x. x ∈ s ∩ t ⇔ x ∈ s ∧ x ∈ t
⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t
⊦ ∀r. toSet r = { x y. (x, y) | r x y }
⊦ ∀r.
transitiveClosure r =
bigIntersect { s. s | subrelation r s ∧ transitive s }
⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s
⊦ ∀r. transitive r ⇔ ∀x y z. r x y ∧ r y z ⇒ r x z
⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b
⊦ ∀p f s. (∀y. y ∈ image f s ⇒ p y) ⇔ ∀x. x ∈ s ⇒ p (f x)