Package set: Set types

Information

Files

• Package tarball set-1.78.tgz
• Theory source file set.thy (included in the package tarball)

Defined Type Operator

• Set
  • set

Defined Constants

• Set
  • ∅
  • bigIntersect
  • bigUnion
  • bijections
  • choice
  • cross
  • delete
  • \
  • disjoint
  • finite
  • fold
  • fromPredicate
  • hasSize
  • image
  • infinite
  • injections
  • insert
  • ∩
  • ∈
  • ⊂
  • rest
  • singleton
  • size
  • ⊆
  • surjections
  • ∪
  • universe

Theorems

⊦ finite ∅

⊦ finite universe

⊦ infinite universe

⊦ ¬(universe = ∅)

⊦ size ∅ = 0

⊦ bigIntersect ∅ = universe

⊦ bigUnion ∅ = ∅

⊦ bigUnion universe = universe

⊦ ∀x. x ∈ universe

⊦ ∀s. disjoint ∅ s

⊦ ∀s. disjoint s ∅

⊦ ∀s. ∅ ⊆ s

⊦ ∀s. s ⊆ universe

⊦ ∀s. s ⊆ s

⊦ fromPredicate (λx. ⊥) = ∅

⊦ fromPredicate (λx. ⊤) = universe

⊦ hasSize universe 2

⊦ finite universe ⇔ finite universe

⊦ infinite universe ⇔ infinite universe

⊦ cross universe universe = universe

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀a. finite (insert a ∅)

⊦ ∀s. ¬(universe ⊂ s)

⊦ ∀s. ¬(s ⊂ ∅)

⊦ ∀s. ¬(s ⊂ s)

⊦ ∀x. delete ∅ x = ∅

⊦ ∀x. insert x universe = universe

⊦ ∀s. ∅ \ s = ∅

⊦ ∀s. s \ ∅ = s

⊦ ∀s. s \ universe = ∅

⊦ ∀s. s \ s = ∅

⊦ ∀s. image id s = s

⊦ ∀s. ∅ ∩ s = ∅

⊦ ∀s. universe ∩ s = s

⊦ ∀s. s ∩ ∅ = ∅

⊦ ∀s. s ∩ universe = s

⊦ ∀s. s ∩ s = s

⊦ ∀s. ∅ ∪ s = s

⊦ ∀s. universe ∪ s = universe

⊦ ∀s. s ∪ ∅ = s

⊦ ∀s. s ∪ universe = universe

⊦ ∀s. s ∪ s = s

⊦ ∀s. cross s ∅ = ∅

⊦ ∀s. cross ∅ s = ∅

⊦ ∀f. image f ∅ = ∅

⊦ ∀s. image (λx. x) s = s

⊦ universe = insert ⊤ (insert ⊥ ∅)

⊦ size = fold (λx n. suc n) 0

⊦ ∀x. hasSize (insert x ∅) 1

⊦ ∀s. infinite s ⇔ ¬finite s

⊦ ∀s. bigIntersect (insert s ∅) = s

⊦ ∀s. bigUnion (insert s ∅) = s

⊦ ∀x s. x ∈ insert x s

⊦ ∀x s. delete s x ⊆ s

⊦ ∀s t. disjoint s (t \ s)

⊦ ∀s t. s ⊆ s ∪ t

⊦ ∀s t. s ⊆ t ∪ s

⊦ ∀s t. disjoint (t \ s) s

⊦ ∀s t. s \ t ⊆ s

⊦ ∀s t. s ∩ t ⊆ s

⊦ ∀s t. t ∩ s ⊆ s

⊦ finite universe ∧ finite universe ⇒ finite universe

⊦ ∀x. size (insert x ∅) = 1

⊦ ∀s. finite s ⇔ hasSize s (size s)

⊦ ∀s. rest s = delete s (choice s)

⊦ ∀s. infinite s ⇒ ¬(s = ∅)

⊦ ∀s. disjoint s s ⇔ s = ∅

⊦ ∀s. hasSize s 0 ⇔ s = ∅

⊦ ∀s. universe ⊆ s ⇔ s = universe

⊦ ∀s. s ⊆ ∅ ⇔ s = ∅

⊦ ∀s. universe \ (universe \ s) = s

⊦ ∀s. bigUnion (delete s ∅) = bigUnion s

⊦ ∀s. bigUnion (insert ∅ s) = bigUnion s

⊦ ∀x s. ¬(insert x s = ∅)

⊦ ∀x s. finite s ⇒ finite (insert x 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 t. disjoint s t ⇔ disjoint t s

⊦ ∀s t. s ∩ t = t ∩ s

⊦ ∀s t. s ∪ t = t ∪ s

⊦ ∀s t. s ⊂ t ⇒ s ⊆ t

⊦ ∀s. (∀x. x ∈ s) ⇔ s = universe

⊦ ∀s. finite s ⇒ ∃a. ¬(a ∈ s)

⊦ ∀f s. finite s ⇒ finite (image f s)

⊦ ∀p x. x ∈ fromPredicate p ⇔ p x

⊦ ∅ = { x. x | ⊥ }

⊦ universe = { x. x | ⊤ }

⊦ ∀s. ¬(s = ∅) ⇒ choice s ∈ s

⊦ ∀s n. hasSize s n ⇒ size s = n

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

⊦ ∀s. s ⊂ universe ⇔ ∃x. ¬(x ∈ s)

⊦ ∀s. (∃x. x ∈ s) ⇔ ¬(s = ∅)

⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y

⊦ ∀x s. x ∈ s ⇔ insert x s = s

⊦ ∀x s. s \ insert x ∅ = delete s x

⊦ ∀x s. insert x (insert x s) = insert x s

⊦ ∀x s. delete (delete s x) x = delete s x

⊦ ∀x s. insert x ∅ ∪ s = insert x s

⊦ ∀s x. insert x ∅ ⊆ s ⇔ x ∈ s

⊦ ∀s t. disjoint s t ⇔ s ∩ t = ∅

⊦ ∀s t. s ⊆ t ⇔ s ∩ t = s

⊦ ∀s t. s ⊆ t ⇔ s ∪ t = t

⊦ ∀s t. s \ t = ∅ ⇔ s ⊆ t

⊦ ∀s t. s \ t = s ⇔ disjoint s t

⊦ ∀s t. t ∪ (s \ t) = t ∪ s

⊦ ∀s t. s \ t \ t = s \ t

⊦ ∀s t. s \ t ∪ t = s ∪ t

⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s

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

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

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

⊦ ∀f s. image f s = ∅ ⇔ s = ∅

⊦ ∀s t. s ⊆ t ⇒ bigIntersect t ⊆ bigIntersect s

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

⊦ { m. m | m < 0 } = ∅

⊦ ∀n. finite { m. m | m < n }

⊦ ∀n. finite { m. m | m ≤ n }

⊦ ∀s. finite s ⇒ (size s = 0 ⇔ s = ∅)

⊦ ∀x s. disjoint s (insert x ∅) ⇔ ¬(x ∈ s)

⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)

⊦ ∀x s. disjoint (insert x ∅) s ⇔ ¬(x ∈ s)

⊦ ∀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. bigIntersect (insert s (insert t ∅)) = s ∩ t

⊦ ∀s t. bigUnion (insert s (insert t ∅)) = s ∪ t

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

⊦ ∀f x. image f (insert x ∅) = insert (f x) ∅

⊦ ∀f s. finite s ⇒ size (image f s) ≤ size s

⊦ ∀f s. image f (bigUnion s) = bigUnion (image (image f) s)

⊦ ∀f s. image f (bigIntersect s) ⊆ bigIntersect (image (image f) s)

⊦ ∀s t. bigIntersect (s ∪ t) = bigIntersect s ∩ bigIntersect t

⊦ ∀s t. bigUnion (s ∪ t) = bigUnion s ∪ bigUnion t

⊦ ∀n. hasSize { m. m | m < n } n

⊦ ∀s t x. s ⊆ t ⇒ s ⊆ insert x t

⊦ ∀s. { x. x | x ∈ s } = s

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

⊦ ∀s n. hasSize s n ⇔ finite s ∧ size s = n

⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇔ s = t

⊦ ∀s t. s ∪ (t \ s) = t ⇔ s ⊆ t

⊦ ∀s t. t \ s ∪ s = t ⇔ s ⊆ t

⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇒ s = t

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

⊦ finite universe ∧ finite universe ⇒
  size universe = size universe ↑ size universe

⊦ { s. s | s ⊆ ∅ } = insert ∅ ∅

⊦ ∀n. size { m. m | m < n } = n

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

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

⊦ ∀f s x. x ∈ s ⇒ f x ∈ image f s

⊦ ∀x s. delete (insert x s) x = s ⇔ ¬(x ∈ s)

⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ¬(s = t)

⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ¬(t ⊆ s)

⊦ ∀a b. a ⊂ b ∧ finite b ⇒ size a < size b

⊦ ∀a b. a ⊆ b ∧ finite b ⇒ size a ≤ size b

⊦ ∀s t. image fst (cross s t) = if t = ∅ then ∅ else s

⊦ ∀s t. image snd (cross s t) = if s = ∅ then ∅ else t

⊦ ∀s. bigIntersect s = universe ⇔ ∀t. t ∈ s ⇒ t = universe

⊦ ∀s. bigUnion s = ∅ ⇔ ∀t. t ∈ s ⇒ t = ∅

⊦ ∀x y s. insert x (insert y s) = insert y (insert x s)

⊦ ∀x y s. delete (delete s x) y = delete (delete s y) x

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

⊦ ∀x s t. insert x s ∪ t = insert x (s ∪ t)

⊦ ∀a. { x. x | x = a } = insert a ∅

⊦ ∀s c. image (λx. c) s = if s = ∅ then ∅ else insert c ∅

⊦ ∀s t x. disjoint (delete s x) t ⇔ disjoint (delete t x) s

⊦ ∀s t x. s \ insert x t = delete s x \ t

⊦ ∀s t x. delete s x ∩ t = delete (s ∩ t) x

⊦ ∀s t u. s ⊆ t ∪ u ⇔ s \ t ⊆ u

⊦ ∀s t u. s ⊆ t ∪ u ⇔ s \ u ⊆ t

⊦ ∀t u s. s \ t \ u = s \ (t ∪ u)

⊦ ∀t u s. s \ t \ u = s \ u \ t

⊦ ∀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 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

⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇒ s = t

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

⊦ ∀f s t. s ⊆ t ⇒ image f s ⊆ image f t

⊦ ∀f g s. image (f ∘ g) s = image f (image g s)

⊦ ∀x y. cross (insert x ∅) (insert y ∅) = insert (x, y) ∅

⊦ ∀m n. hasSize universe m ∧ hasSize universe n ⇒ hasSize universe (n ↑ m)

⊦ ∀s t. s ∪ t = ∅ ⇔ s = ∅ ∧ t = ∅

⊦ ∀s t. cross s t = ∅ ⇔ s = ∅ ∨ t = ∅

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

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

⊦ ∀n. hasSize { m. m | m ≤ n } (n + 1)

⊦ ∀s t. disjoint s t ⇔ ¬∃x. x ∈ s ∧ x ∈ t

⊦ ∀s t. disjoint s (bigUnion t) ⇔ ∀x. x ∈ t ⇒ disjoint s x

⊦ ∀t u. t ⊆ bigIntersect u ⇔ ∀s. s ∈ u ⇒ t ⊆ s

⊦ ∀s. bigUnion { x. insert x ∅ | x ∈ s } = s

⊦ ∀f x s. image f (insert x s) = insert (f x) (image f s)

⊦ ∀s x. x ∈ bigIntersect s ⇔ ∀t. t ∈ s ⇒ x ∈ t

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

⊦ ∀f t. bigUnion f ⊆ t ⇔ ∀s. s ∈ f ⇒ s ⊆ t

⊦ ∀u s. (∃t. t ∈ u ∧ t ⊆ s) ⇒ bigIntersect u ⊆ s

⊦ ∀x s. x ∈ rest s ⇔ x ∈ s ∧ ¬(x = choice s)

⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x

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

⊦ ∀x s t. insert x s ⊆ t ⇔ x ∈ t ∧ s ⊆ t

⊦ ∀n. size { m. m | m ≤ n } = n + 1

⊦ ∀s t x. x ∈ s ∩ t ⇔ x ∈ s ∧ x ∈ t

⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t

⊦ ∀s t u. s ⊆ t \ u ⇔ s ⊆ t ∧ disjoint s u

⊦ ∀s t u. s ⊆ t ∩ u ⇔ s ⊆ t ∧ s ⊆ u

⊦ ∀s t u. disjoint (s ∪ t) u ⇔ disjoint s u ∧ disjoint t u

⊦ ∀s t u. s ∪ t ⊆ u ⇔ s ⊆ u ∧ t ⊆ u

⊦ ∀s t u. s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u

⊦ ∀s t u. s ∪ t ∩ u = (s ∪ t) ∩ (s ∪ u)

⊦ ∀s t u. (s ∪ t) ∩ u = s ∩ u ∪ t ∩ u

⊦ ∀s t u. s ∩ t ∪ u = (s ∪ u) ∩ (t ∪ u)

⊦ ∀s t. ¬(s = t) ⇔ ∃x. x ∈ t ⇔ ¬(x ∈ s)

⊦ ∀s t u. cross s (t \ u) = cross s t \ cross s u

⊦ ∀s t u. cross s (t ∩ u) = cross s t ∩ cross s u

⊦ ∀s t u. cross s (t ∪ u) = cross s t ∪ cross s u

⊦ ∀s t u. cross (t \ u) s = cross t s \ cross u s

⊦ ∀s t u. cross (t ∩ u) s = cross t s ∩ cross u s

⊦ ∀s t u. cross (t ∪ u) s = cross t s ∪ cross u s

⊦ ∀f s t. image f (s ∪ t) = image f s ∪ image f t

⊦ ∀f s t. image f (s ∩ t) ⊆ image f s ∩ image f t

⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∃g. ∀y. f (g y) = y

⊦ ∀f. (∀t. ∃s. image f s = t) ⇔ ∀y. ∃x. f x = y

⊦ ∀s x. s ⊆ insert x ∅ ⇔ s = ∅ ∨ s = insert x ∅

⊦ ∀s p. { x. x | x ∈ s ∧ p x } ⊆ s

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

⊦ ∀x s t. s ⊆ delete t x ⇔ s ⊆ t ∧ ¬(x ∈ s)

⊦ ∀x s t. disjoint (insert x s) t ⇔ ¬(x ∈ t) ∧ disjoint s t

⊦ ∀x s. ¬(x ∈ s) ⇒ ∀t. s ⊆ insert x t ⇔ s ⊆ t

⊦ ∀s x y. x ∈ delete s y ⇔ x ∈ s ∧ ¬(x = y)

⊦ ∀s x. x ∈ s ⇔ ∃t. s = insert x t ∧ ¬(x ∈ t)

⊦ ∀s t x. x ∈ s \ t ⇔ x ∈ s ∧ ¬(x ∈ t)

⊦ ∀s t. s ⊂ t ⇔ ∃x. ¬(x ∈ s) ∧ insert x s ⊆ t

⊦ ∀s. s = ∅ ∨ ∃x t. s = insert x t ∧ ¬(x ∈ t)

⊦ ∀f s t. finite t ∧ s ⊆ image f t ⇒ size s ≤ size t

⊦ ∀s t. finite s ∧ finite t ⇒ size (s ∪ t) ≤ size s + size t

⊦ ∀s t. finite t ∧ s ⊆ t ⇒ (size s = size t ⇔ s = t)

⊦ ∀a b. finite b ∧ a ⊆ b ∧ size a = size b ⇒ a = b

⊦ ∀a b. finite b ∧ a ⊆ b ∧ size b ≤ size a ⇒ a = b

⊦ ∀s t. finite s ∧ finite t ⇒ size (cross s t) = size s * size t

⊦ ∀n s. finite s ⇒ n ≤ size s ⇔ ∃t. t ⊆ s ∧ hasSize t n

⊦ ∀s n. finite s ∧ n ≤ size s ⇒ ∃t. t ⊆ s ∧ hasSize t n

⊦ ∀s n. (finite s ⇒ n ≤ size s) ⇒ ∃t. t ⊆ s ∧ hasSize t n

⊦ ∀a. finite a ⇔ a = ∅ ∨ ∃x s. a = insert x s ∧ finite s

⊦ ∀s. bigIntersect s = universe \ bigUnion { t. universe \ t | t ∈ s }

⊦ ∀s. bigUnion s = universe \ bigIntersect { t. universe \ t | t ∈ s }

⊦ ∀x s.
    finite s ⇒ size (insert x s) = if x ∈ s then size s else suc (size s)

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

⊦ ∀s t. finite s ∧ t ⊆ s ⇒ size (s \ t) = size s - size t

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

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

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

⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ∃a. a ∈ t ∧ ¬(a ∈ s)

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

⊦ ∀x s. insert x s = { y. y | y = x ∨ y ∈ s }

⊦ ∀s₁ s₂ t₁ t₂. cross s₁ t₁ ∩ cross s₂ t₂ = cross (s₁ ∩ s₂) (t₁ ∩ t₂)

⊦ ∀s t. s ∩ t = { x. x | x ∈ s ∧ x ∈ t }

⊦ ∀s t. s ∪ t = { x. x | x ∈ s ∨ x ∈ t }

⊦ ∀s t. finite s ∧ finite t ⇒ size (s ∪ t) = size s + size (t \ s)

⊦ ∀s t m n. hasSize s m ∧ hasSize t n ⇒ hasSize (cross s t) (m * n)

⊦ ∀s₁ t₁ s₂ t₂.
    disjoint (cross s₁ t₁) (cross s₂ t₂) ⇔ disjoint s₁ s₂ ∨ disjoint t₁ t₂

⊦ ∀s t s' t'. s ⊆ s' ∧ t ⊆ t' ⇒ cross s t ⊆ cross s' t'

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

⊦ ∀u s. u \ bigIntersect s = bigUnion { t. u \ t | t ∈ s }

⊦ ∀s f. cross s (bigUnion f) = bigUnion { t. cross s t | t ∈ f }

⊦ ∀f s t. s ⊆ image f t ⇔ ∃u. u ⊆ t ∧ s = image f u

⊦ ∀s t. t ∩ bigUnion s = bigUnion { x. t ∩ x | x ∈ s }

⊦ ∀s t. t ∪ bigIntersect s = bigIntersect { x. t ∪ x | x ∈ s }

⊦ ∀s t. bigUnion s \ t = bigUnion { x. x \ t | x ∈ s }

⊦ ∀s t. bigUnion s ∩ t = bigUnion { x. x ∩ t | x ∈ s }

⊦ ∀s t. bigIntersect s ∪ t = bigIntersect { x. x ∪ t | x ∈ s }

⊦ ∀f t. cross (bigUnion f) t = bigUnion { s. cross s t | s ∈ f }

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

⊦ ∀x s. finite s ⇒ size (delete s x) = if x ∈ s then size s - 1 else size s

⊦ ∀s x. delete s x = { y. y | y ∈ s ∧ ¬(y = x) }

⊦ ∀s t. s \ t = { x. x | x ∈ s ∧ ¬(x ∈ t) }

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

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

⊦ ∀x y s.
    delete (insert x s) y =
    if x = y then delete s y else insert x (delete s y)

⊦ ∀x s t. insert x s ∩ t = if x ∈ t then insert x (s ∩ t) else s ∩ t

⊦ ∀x s t. insert x s ∪ t = if x ∈ t then s ∪ t else insert x (s ∪ t)

⊦ ∀s t x. insert x s \ t = if x ∈ t then s \ t else insert x (s \ t)

⊦ ∀s. finite s ⇒ size { t. t | t ⊆ s } = 2 ↑ size s

⊦ ∀f. (∀x y. f x = f y ⇒ x = y) ⇔ ∃g. ∀x. g (f x) = x

⊦ ∀s n. hasSize s n ⇒ hasSize { t. t | t ⊆ s } (2 ↑ n)

⊦ ∀s t.
    finite s ∧ finite t ⇒ (size (s ∪ t) = size s + size t ⇔ disjoint s t)

⊦ ∀s t. finite s ∧ finite t ∧ disjoint s t ⇒ size (s ∪ t) = size s + size t

⊦ ∀u s. u \ bigUnion s = u ∩ bigIntersect { t. u \ t | t ∈ s }

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

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

⊦ ∀s x x'. (x ∈ s ⇔ x' ∈ s) ⇒ (x ∈ delete s x' ⇔ x' ∈ delete s x)

⊦ ∀s x x'. (x ∈ delete s x' ⇔ x' ∈ delete s x) ⇔ x ∈ s ⇔ x' ∈ s

⊦ ∀s n. hasSize s (suc n) ⇔ ¬(s = ∅) ∧ ∀a. a ∈ s ⇒ hasSize (delete s a) n

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

⊦ ∀f s. image f s = { 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

⊦ ∀p. bigUnion { x. insert x ∅ | p x } = { x. x | p 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)

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

⊦ ∀s t. finite s ∧ finite t ⇒ size (s ∪ t) = size s + size t - size (s ∩ t)

⊦ ∀s t. finite s ∧ finite t ⇒ size (s ∪ t) + size (s ∩ t) = size s + size t

⊦ ∀s t.
    finite s ∧ finite t ∧ size (s ∪ t) < size s + size t ⇒ ¬disjoint s t

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

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

⊦ ∀f s. bigIntersect (image f s) = { y. y | ∀x. x ∈ s ⇒ y ∈ f x }

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

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

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

⊦ ∀s n. hasSize s (suc n) ⇔ ∃a t. hasSize t n ∧ ¬(a ∈ t) ∧ s = insert a t

⊦ ∀s t m n.
    hasSize s m ∧ hasSize t n ∧ disjoint s t ⇒ hasSize (s ∪ t) (m + n)

⊦ ∀s t m n. hasSize s m ∧ hasSize t n ∧ t ⊆ s ⇒ hasSize (s \ t) (m - n)

⊦ ∀p f. { x. f x | p x } = image f { x. x | p x }

⊦ ∀n. { m. m | m < suc n } = insert n { m. m | m < n }

⊦ ∀s t u. finite u ∧ disjoint s t ∧ s ∪ t = u ⇒ size s + size t = size u

⊦ ∀p a s. (∀x. x ∈ insert a s ⇒ p x) ⇔ p a ∧ ∀x. x ∈ s ⇒ p x

⊦ ∀p a s. (∃x. x ∈ insert a s ∧ p x) ⇔ p a ∨ ∃x. x ∈ s ∧ p x

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

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

⊦ ∀s f.
    s ∩ bigIntersect f =
    if f = ∅ then s else bigIntersect { t. s ∩ t | t ∈ f }

⊦ ∀f s.
    bigIntersect f ∩ s =
    if f = ∅ then s else bigIntersect { t. t ∩ s | t ∈ f }

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

⊦ ∀p. { x. x | p x } = { a b. (a, b) | p (a, b) }

⊦ ∀n. { m. m | m < suc n } = insert 0 { m. suc m | m < n }

⊦ ∀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'