Package relation: Relation operators

Information

Files

• Package tarball relation-1.31.tgz
• Theory file relation.thy (included in the package tarball)

Defined Constants

• Number
  • Natural
    • isSuc
• Relation
  • bigIntersect
  • bigUnion
  • empty
  • fromSet
  • intersect
  • irreflexive
  • measure
  • reflexive
  • subrelation
  • toSet
  • transitive
  • transitiveClosure
  • union
  • universe
  • wellFounded

Theorems

⊦ irreflexive empty

⊦ irreflexive isSuc

⊦ reflexive universe

⊦ transitive empty

⊦ transitive universe

⊦ transitive (<)

⊦ wellFounded empty

⊦ wellFounded (<)

⊦ wellFounded isSuc

⊦ subrelation isSuc (<)

⊦ empty = fromSet Set.∅

⊦ universe = fromSet Set.universe

⊦ transitiveClosure isSuc = (<)

⊦ ∀m. wellFounded (measure m)

⊦ ∀r. transitive (transitiveClosure r)

⊦ ∀r. subrelation r r

⊦ ∀r. subrelation r (transitiveClosure r)

⊦ ∀x y. universe x y

⊦ ∀s. toSet (fromSet s) = s

⊦ ∀r. wellFounded r ⇒ irreflexive r

⊦ ∀r. fromSet (toSet r) = r

⊦ ∀x y. ¬empty x y

⊦ ∀r x. reflexive r ⇒ r x x

⊦ ∀r. reflexive r ⇔ ∀x. r x x

⊦ ∀s. bigIntersect s = fromSet (Set.bigIntersect (Set.image toSet s))

⊦ ∀s. bigUnion s = fromSet (Set.bigUnion (Set.image toSet s))

⊦ ∀r x. irreflexive r ⇒ ¬r x x

⊦ ∀r. irreflexive r ⇔ ∀x. ¬r x x

⊦ ∀m n. isSuc m n ⇔ suc m = n

⊦ ∀r. subrelation isSuc r ∧ transitive r ⇒ subrelation (<) r

⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r

⊦ ∀r s. subrelation r s ⇔ Set.⊆ (toSet r) (toSet s)

⊦ ∀r s. toSet r = toSet s ⇒ r = s

⊦ ∀r s. intersect r s = fromSet (Set.∩ (toSet r) (toSet s))

⊦ ∀r s. union r s = fromSet (Set.∪ (toSet r) (toSet s))

⊦ ∀r m. wellFounded r ⇒ wellFounded (λx y. r (m x) (m y))

⊦ ∀r s.
    subrelation r s ∧ transitive s ⇒ subrelation (transitiveClosure r) s

⊦ ∀r s. subrelation r s ∧ subrelation s r ⇒ r = s

⊦ ∀m x y. measure m x y ⇔ m x < m y

⊦ ∀s x y. fromSet s x y ⇔ Set.∈ (x, y) s

⊦ ∀r. wellFounded r ⇔ ¬∃f. ∀n. r (f (suc n)) (f n)

⊦ ∀r x y. Set.∈ (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. Set.∈ 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

⊦ ∀r. toSet r = { x y. x, y | r x y }

⊦ ∀s x y. bigIntersect s x y ⇔ ∀r. Set.∈ r s ⇒ r x y

⊦ ∀s x y. bigUnion s x y ⇔ ∃r. Set.∈ r s ∧ r x y

⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h x

⊦ ∀r.
    transitiveClosure r =
    bigIntersect { s. s | subrelation r s ∧ transitive s }

⊦ ∀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

⊦ ∀r. transitive r ⇔ ∀x y z. r x y ∧ r y z ⇒ r x z

⊦ ∀m a b. (∀y. measure m y a ⇒ measure m y b) ⇔ m a ≤ m b

⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀r. wellFounded r ⇔ ∀p. (∃x. p x) ⇔ ∃x. p x ∧ ∀y. r y x ⇒ ¬p y

⊦ ∀r. wellFounded r ⇔ ∀p. (∃x. p x) ⇒ ∃x. p x ∧ ∀y. r y x ⇒ ¬p y

⊦ ∀r s.
    wellFounded r ∧ wellFounded s ⇒
    wellFounded (λ(x1, y1) (x2, y2). r x1 x2 ∧ s y1 y2)

⊦ ∀r.
    wellFounded r ⇒
    ∀h.
      (∀f g x. (∀z. r z x ⇒ f z = g z) ⇒ h f x = h g x) ⇒
      ∃!f. ∀x. f x = h f x

⊦ ∀r s.
    wellFounded r ∧ wellFounded s ⇒
    wellFounded (λ(x1, y1) (x2, y2). r x1 x2 ∨ x1 = x2 ∧ s y1 y2)

⊦ ∀r.
    (∀x. ¬r x x) ∧ (∀x y z. r x y ∧ r y z ⇒ r x z) ∧
    (∀x. Set.finite { y. y | r y x }) ⇒ wellFounded r

⊦ ∀r s.
    wellFounded r ∧ (∀a. wellFounded (s a)) ⇒
    wellFounded (λ(x1, y1) (x2, y2). r x1 x2 ∨ x1 = x2 ∧ s x1 y1 y2)

Input Type Operators

• →
• bool
• Data
  • Pair
    • ×
• Number
  • Natural
    • natural
• Set
  • Set.set

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∃!
    • ∨
    • ¬
    • cond
    • F
    • T
  • Pair
    • ,
• Function
  • id
• Number
  • Natural
    • +
    • <
    • ≤
    • suc
    • zero
• Set
  • Set.∅
  • Set.bigIntersect
  • Set.bigUnion
  • Set.finite
  • Set.fromPredicate
  • Set.image
  • Set.infinite
  • Set.insert
  • Set.∩
  • Set.∈
  • Set.⊆
  • Set.∪
  • Set.universe

Assumptions

⊦ T

⊦ Set.infinite Set.universe

⊦ ¬F ⇔ T

⊦ ¬T ⇔ F

⊦ ∀x. Set.∈ x Set.universe

⊦ ∀t. t ⇒ t

⊦ F ⇔ ∀p. p

⊦ ∀x. ¬Set.∈ x Set.∅

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ (¬) = λp. p ⇒ F

⊦ ∀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. F ∧ t ⇔ F

⊦ ∀t. T ∧ t ⇔ t

⊦ ∀t. t ∧ F ⇔ F

⊦ ∀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

⊦ ∀m. m < 0 ⇔ F

⊦ ∀m. m + 0 = m

⊦ ∀t. (F ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ F) ⇔ ¬t

⊦ ∀t. t ⇒ F ⇔ ¬t

⊦ ∀s. Set.infinite s ⇔ ¬Set.finite s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀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)

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

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

⊦ ∀p x. Set.∈ x (Set.fromPredicate p) ⇔ p x

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = 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

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m < suc n ⇔ m < n

⊦ ∀s t. Set.finite t ∧ Set.⊆ s t ⇒ Set.finite s

⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2

⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (p1, p2)

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀m n. m < n ∨ n < m ∨ m = n

⊦ ∀s t. Set.⊆ s t ∧ Set.⊆ t s ⇒ s = t

⊦ ∀PAIR'. ∃fn. ∀a0 a1. fn (a0, a1) = PAIR' a0 a1

⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d

⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x

⊦ ∀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

⊦ ∀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

⊦ ∀p q. (∀x. p x) ∨ q ⇔ ∀x. p x ∨ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

⊦ ∀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

⊦ ∀s t u. Set.⊆ s t ∧ Set.⊆ t u ⇒ Set.⊆ s u

⊦ ∀s t. Set.⊆ s t ⇔ ∀x. Set.∈ x s ⇒ Set.∈ x t

⊦ ∀s t. (∀x. Set.∈ x s ⇔ Set.∈ x t) ⇒ s = t

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

⊦ ∀t f. Set.⊆ t (Set.bigIntersect f) ⇔ ∀s. Set.∈ s f ⇒ Set.⊆ t s

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

⊦ ∀s x. Set.∈ x (Set.bigIntersect s) ⇔ ∀t. Set.∈ t s ⇒ Set.∈ x t

⊦ ∀s x. Set.∈ x (Set.bigUnion s) ⇔ ∃t. Set.∈ t s ∧ Set.∈ x t

⊦ ∀p x. Set.∈ x { y. y | p y } ⇔ p x

⊦ ∀x y s. Set.∈ x (Set.insert y s) ⇔ x = y ∨ Set.∈ x s

⊦ ∀s t x. Set.∈ x (Set.∩ s t) ⇔ Set.∈ x s ∧ Set.∈ x t

⊦ ∀s t x. Set.∈ x (Set.∪ s t) ⇔ Set.∈ x s ∨ Set.∈ x t

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

⊦ ∀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 ⇒ 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

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

⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b

⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∧ q1 ⇒ p2 ∧ q2

⊦ ∀p1 p2 q1 q2. (p2 ⇒ p1) ∧ (q1 ⇒ q2) ⇒ (p1 ⇒ q1) ⇒ p2 ⇒ q2

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

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

⊦ ∀p f s. (∀y. Set.∈ y (Set.image f s) ⇒ p y) ⇔ ∀x. Set.∈ x s ⇒ p (f x)

OpenTheory package summary generated by opentheory 1.1 (release 20120129)