Package set: Set types

Information

nameset
version1.75
descriptionSet types
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
checksum3a9a8a2fa08b83583fb4e0ae5b8efb8bc4990e82
requiresbool
function
natural
pair
showData.Bool
Data.Pair
Function
Number.Natural
Set

Files

Defined Type Operator

Defined Constants

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. 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

⊦ ∀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)

⊦ ∀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

⊦ ∀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)

⊦ ∀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 }

⊦ ∀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 }

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

⊦ ∀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'

⊦ ∀s t. finite s ∧ finite t ⇒ finite { x y. (x, y) | x ∈ s ∧ y ∈ t }

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

⊦ ∀p x t. (∀s. s ⊆ insert x t ⇒ p s) ⇔ ∀s. s ⊆ t ⇒ p s ∧ p (insert x s)

⊦ ∀p x t. (∃s. s ⊆ insert x t ∧ p s) ⇔ ∃s. s ⊆ t ∧ (p s ∨ p (insert x s))

⊦ ∀s f.
    cross s (bigIntersect f) =
    if f = ∅ then cross s universe
    else bigIntersect { t. cross s t | t ∈ f }

⊦ ∀f s.
    cross (bigIntersect f) s =
    if f = ∅ then cross universe s
    else bigIntersect { t. cross t s | t ∈ f }

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

⊦ ∀s t s' t'. cross s t ⊆ cross s' t' ⇔ s = ∅ ∨ t = ∅ ∨ s ⊆ s' ∧ t ⊆ t'

⊦ ∀f.
    finite f ⇒
    bigUnion { t. t | t ∈ f ∧ ∀u. u ∈ f ⇒ ¬(t ⊂ u) } = bigUnion f

⊦ ∀p f q. (∀z. z ∈ { x. f x | p x } ⇒ q z) ⇔ ∀x. p x ⇒ q (f x)

⊦ ∀p f q. (∃z. z ∈ { x. f x | p x } ∧ q z) ⇔ ∃x. p x ∧ q (f x)

⊦ ∀p t u. (∀s. s ⊆ t ∪ u ⇒ p s) ⇔ ∀t' u'. t' ⊆ t ∧ u' ⊆ u ⇒ p (t' ∪ u')

⊦ ∀p t u. (∃s. s ⊆ t ∪ u ∧ p s) ⇔ ∃t' u'. t' ⊆ t ∧ u' ⊆ u ∧ p (t' ∪ u')

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

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

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

⊦ ∀f s t.
    (∀x y. f x = f y ⇒ x = y) ⇒ image f (s ∩ t) = image f s ∩ image f t

⊦ ∀f. (∀k. ∃t. { x. x | f x ∈ t } = k) ⇔ ∀x y. f x = f y ⇒ x = y

⊦ ∀p f. bigIntersect { x. f x | p x } = { a. a | ∀x. p x ⇒ a ∈ f x }

⊦ ∀p f. bigUnion { x. f x | p x } = { a. a | ∃x. p x ∧ a ∈ f x }

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

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

⊦ ∀f s.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
    (finite (image f s) ⇔ finite s)

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

⊦ ∀f A. (∀x y. f x = f y ⇒ x = y) ∧ finite A ⇒ finite { x. x | f x ∈ A }

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

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

⊦ ∀s f. finite s ∧ image f s = s ⇒ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y

⊦ ∀f s a.
    (∀x. x ∈ s ∧ f x = f a ⇒ x = a) ⇒
    image f (delete s a) = delete (image f s) (f a)

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

⊦ ∀f.
    finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
    bigIntersect f ∈ f

⊦ ∀f.
    finite f ∧ ¬(f = ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
    bigUnion f ∈ f

⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀p. image f { x. x | p (f x) } = { x. x | p x }

⊦ ∀s t.
    finite s ∧ finite t ⇒
    size { x y. (x, y) | x ∈ s ∧ y ∈ t } = size s * size t

⊦ ∀s n.
    hasSize s n ⇒
    ∃f. (∀m. m < n ⇒ f m ∈ s) ∧ ∀x. x ∈ s ⇒ ∃!m. m < n ∧ f m = x

⊦ ∀f s.
    finite s ⇒
    (size (image f s) = size s ⇔ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y)

⊦ ∀f s.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ finite s ⇒
    size (image f s) = size s

⊦ ∀p.
    p ∅ ∧ (∀s. finite s ∧ ¬(s = ∅) ⇒ ∃x. x ∈ s ∧ (p (delete s x) ⇒ p s)) ⇒
    ∀s. finite s ⇒ p s

⊦ ∀f s n.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇒
    (hasSize (image f s) n ⇔ hasSize s n)

⊦ ∀f s n.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ hasSize s n ⇒
    hasSize (image f s) n

⊦ ∀d t.
    { f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
    insert (λx. d) ∅

⊦ ∀s t m n.
    hasSize s m ∧ hasSize t n ⇒
    hasSize { x y. (x, y) | x ∈ s ∧ y ∈ t } (m * n)

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

⊦ ∀f u v.
    (∀t. t ⊆ v ⇒ ∃s. s ⊆ u ∧ image f s = t) ⇔
    ∀y. y ∈ v ⇒ ∃x. x ∈ u ∧ f x = y

⊦ ∀f s.
    ∃t.
      t ⊆ s ∧ image f s = image f t ∧
      ∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y

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

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

⊦ ∀p f q. (∀z. z ∈ { x y. f x y | p x y } ⇒ q z) ⇔ ∀x y. p x y ⇒ q (f x y)

⊦ ∀p f q. (∃z. z ∈ { x y. f x y | p x y } ∧ q z) ⇔ ∃x y. p x y ∧ q (f x y)

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

⊦ ∀s1 t1 s2 t2.
    cross s1 t1 = cross s2 t2 ⇔
    (s1 = ∅ ∨ t1 = ∅) ∧ (s2 = ∅ ∨ t2 = ∅) ∨ s1 = s2 ∧ t1 = t2

⊦ ∀f s t.
    (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
    ∃g. ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y

⊦ ∀f.
    (∀t t'. { x. x | f x ∈ t } = { x. x | f x ∈ t' } ⇒ t = t') ⇔
    image f universe = universe

⊦ ∀p f.
    bigIntersect { x y. f x y | p x y } =
    { a. a | ∀x y. p x y ⇒ a ∈ f x y }

⊦ ∀p f.
    bigUnion { x y. f x y | p x y } = { a. a | ∃x y. p x y ∧ a ∈ f x y }

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

⊦ ∀f s t.
    (∀x y. x ∈ s ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒
    image f (s \ t) = image f s \ image f t

⊦ ∀f s t.
    finite s ∧ (∀x. x ∈ s ⇒ f x ∈ t) ∧ (∀y. y ∈ t ⇒ ∃!x. x ∈ s ∧ f x = y) ⇒
    size t = size s

⊦ ∀d s t.
    finite s ∧ finite t ⇒
    finite { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d }

⊦ ∀s t f.
    finite s ∧ size s = size t ∧ image f s = t ⇒
    ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y

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

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

⊦ ∀f s.
    ¬(s = ∅) ∧
    (∀x y. x ∈ bigUnion s ∧ y ∈ bigUnion s ∧ f x = f y ⇒ x = y) ⇒
    image f (bigIntersect s) = bigIntersect (image (image f) s)

⊦ ∀a t.
    { s. s | s ⊆ insert a t } =
    { s. s | s ⊆ t } ∪ image (λs. insert a s) { s. s | s ⊆ t }

⊦ ∀f s a.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ a ∈ s ⇒
    image f (delete s a) = delete (image f s) (f a)

⊦ ∀f s t.
    s ⊆ image f t ⇔
    ∃u. u ⊆ t ∧ (∀x y. x ∈ u ∧ y ∈ u ∧ f x = f y ⇒ x = y) ∧ s = image f u

⊦ ∀s t.
    finite s ∧ finite t ∧ size s ≤ size t ⇒
    ∃f. image f s ⊆ t ∧ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y

⊦ ∀s t m n.
    hasSize s m ∧ (∀x. x ∈ s ⇒ finite (t x) ∧ size (t x) ≤ n) ⇒
    size (bigUnion { x. t x | x ∈ s }) ≤ m * n

⊦ ∀f s t.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ t ⊆ s ⇒
    image f (s \ t) = image f s \ image f t

⊦ ∀s t m n.
    hasSize s m ∧ (∀x. x ∈ s ⇒ hasSize (t x) n) ⇒
    hasSize { x y. (x, y) | x ∈ s ∧ y ∈ t x } (m * n)

⊦ ∀f g s.
    ¬(g = ∅) ∧ (∀t. t ∈ delete g s ⇒ { x. x | f x ∈ image f t } ⊆ t) ⇒
    image f (bigIntersect g) = bigIntersect (image (image f) g)

⊦ ∀f s.
    finite s ∧ s ⊆ bigUnion f ∧ ¬(f = ∅) ∧
    (∀t u. t ∈ f ∧ u ∈ f ⇒ t ⊆ u ∨ u ⊆ t) ⇒ ∃t. t ∈ f ∧ s ⊆ t

⊦ ∀p f q.
    (∀z. z ∈ { w x y. f w x y | p w x y } ⇒ q z) ⇔
    ∀w x y. p w x y ⇒ q (f w x y)

⊦ ∀p f q.
    (∃z. z ∈ { w x y. f w x y | p w x y } ∧ q z) ⇔
    ∃w x y. p w x y ∧ q (f w x y)

⊦ ∀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 }

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

⊦ ∀f n.
    (∀t u. t ∈ f ∧ u ∈ f ⇒ t ⊆ u ∨ u ⊆ t) ∧
    (∀t. t ∈ f ⇒ finite t ∧ size t ≤ n) ⇒
    finite (bigUnion f) ∧ size (bigUnion f) ≤ n

⊦ ∀p f.
    bigIntersect { x y z. f x y z | p x y z } =
    { a. a | ∀x y z. p x y z ⇒ a ∈ f x y z }

⊦ ∀p f.
    bigUnion { x y z. f x y z | p x y z } =
    { a. a | ∃x y z. p x y z ∧ a ∈ f x y z }

⊦ ∀d s t.
    finite s ∧ finite t ⇒
    size { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d } =
    size t ↑ size s

⊦ ∀p f s.
    (∀t. t ⊆ image f s ⇒ p t) ⇔
    ∀t. t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒ p (image f t)

⊦ ∀p f s.
    (∃t. t ⊆ image f s ∧ p t) ⇔
    ∃t. t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧ p (image f t)

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

⊦ ∀f u.
    (∀s t. s ⊆ u ∧ t ⊆ u ∧ image f s = image f t ⇒ s = t) ⇔
    ∀x y. x ∈ u ∧ y ∈ u ∧ f x = f y ⇒ x = y

⊦ ∀f s.
    bigIntersect { x. bigUnion (f x) | x ∈ s } =
    bigUnion { g. bigIntersect { x. g x | x ∈ s } | ∀x. x ∈ s ⇒ g x ∈ f x }

⊦ ∀f s.
    bigUnion { x. bigIntersect (f x) | x ∈ s } =
    bigIntersect { g. bigUnion { x. g x | x ∈ s } | ∀x. x ∈ s ⇒ g x ∈ f x }

⊦ ∀r s.
    finite s ∧ (∀x. ¬r x x) ∧ (∀x y z. r x y ∧ r y z ⇒ r x z) ∧
    (∀x. x ∈ s ⇒ ∃y. y ∈ s ∧ r x y) ⇒ s = ∅

⊦ ∀d s t m n.
    hasSize s m ∧ hasSize t n ⇒
    hasSize { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d }
      (n ↑ m)

⊦ ∀s t f g n.
    (∀x. x ∈ s ⇒ f x ∈ t ∧ g (f x) = x) ∧
    (∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y) ∧ hasSize s n ⇒ hasSize t n

⊦ ∀s t f g.
    (∀x. x ∈ s ⇒ f x ∈ t ∧ g (f x) = x) ∧
    (∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y) ⇒ ∀n. hasSize s n ⇔ hasSize t n

⊦ ∀s f.
    finite s ∧ image f s ⊆ s ⇒
    ((∀y. y ∈ s ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
     ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y)

⊦ ∀f s t.
    { x. x | f x ∈ image f s } ⊆ s ∨ { x. x | f x ∈ image f t } ⊆ t ⇒
    image f (s ∩ t) = image f s ∩ image f t

⊦ ∀s t f g.
    (finite s ∨ finite t) ∧ (∀x. x ∈ s ⇒ f x ∈ t ∧ g (f x) = x) ∧
    (∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y) ⇒ size s = size t

⊦ ∀s t.
    finite s ∧ finite t ∧ size s = size t ⇒
    ∃f g.
      (∀x. x ∈ s ⇒ f x ∈ t ∧ g (f x) = x) ∧
      ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y

⊦ ∀s t a.
    { x y. (x, y) | x ∈ insert a s ∧ y ∈ t x } =
    image (, a) (t a) ∪ { x y. (x, y) | x ∈ s ∧ y ∈ t x }

⊦ ∀s.
    { t. t | t ⊆ s } =
    image (λp. { x. x | p x })
      { p. p | (∀x. x ∈ s ⇒ p x ∈ universe) ∧ ∀x. ¬(x ∈ s) ⇒ (p x ⇔ ⊥) }

⊦ ∀p q r.
    p ∩ q = q ∩ p ∧ p ∩ q ∩ r = p ∩ (q ∩ r) ∧ p ∩ (q ∩ r) = q ∩ (p ∩ r) ∧
    p ∩ p = p ∧ p ∩ (p ∩ q) = p ∩ q

⊦ ∀p q r.
    p ∪ q = q ∪ p ∧ p ∪ q ∪ r = p ∪ (q ∪ r) ∧ p ∪ (q ∪ r) = q ∪ (p ∪ r) ∧
    p ∪ p = p ∧ p ∪ (p ∪ q) = p ∪ q

⊦ ∀p f s.
    (∀t. finite t ∧ t ⊆ image f s ⇒ p t) ⇔
    ∀t.
      finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ⇒
      p (image f t)

⊦ ∀p f s.
    (∃t. finite t ∧ t ⊆ image f s ∧ p t) ⇔
    ∃t.
      finite t ∧ t ⊆ s ∧ (∀x y. x ∈ t ∧ y ∈ t ∧ f x = f y ⇒ x = y) ∧
      p (image f t)

⊦ ∀f s u.
    (∀t1 t2.
       t1 ⊆ u ∧ t2 ⊆ u ∧
       { x. x | x ∈ s ∧ f x ∈ t1 } = { x. x | x ∈ s ∧ f x ∈ t2 } ⇒
       t1 = t2) ⇔ u ⊆ image f s

⊦ ∀f g s u.
    ¬(g = ∅) ∧ (∀t. t ∈ g ⇒ t ⊆ u) ∧
    (∀t. t ∈ delete g s ⇒ { x. x | x ∈ u ∧ f x ∈ image f t } ⊆ t) ⇒
    image f (bigIntersect g) = bigIntersect (image (image f) g)

⊦ ∀s t f.
    finite s ∧ finite t ∧ size s = size t ∧ image f s ⊆ t ⇒
    ((∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
     ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y)

⊦ ∀s t m n.
    hasSize s m ∧ (∀x. x ∈ s ⇒ hasSize (t x) n) ∧
    (∀x y. x ∈ s ∧ y ∈ s ∧ ¬(x = y) ⇒ disjoint (t x) (t y)) ⇒
    hasSize (bigUnion { x. t x | x ∈ s }) (m * n)

⊦ ∀f b.
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ⇒
    fold f b ∅ = b ∧
    ∀x s.
      finite s ⇒
      fold f b (insert x s) =
      if x ∈ s then fold f b s else f x (fold f b s)

⊦ ∀f s u.
    (∀k. k ⊆ s ⇒ ∃t. t ⊆ u ∧ { x. x | x ∈ s ∧ f x ∈ t } = k) ⇔
    image f s ⊆ u ∧ ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y

⊦ ∀f b.
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ⇒
    fold f b ∅ = b ∧
    ∀x s.
      finite s ⇒
      fold f b s =
      if x ∈ s then f x (fold f b (delete s x)) else fold f b (delete s x)

⊦ ∀f g.
    cross (bigIntersect f) (bigIntersect g) =
    if f = ∅ then bigIntersect { t. cross universe t | t ∈ g }
    else if g = ∅ then bigIntersect { s. cross s universe | s ∈ f }
    else bigIntersect { s t. cross s t | s ∈ f ∧ t ∈ g }

⊦ ∀s t.
    finite s ∧ finite t ∧ size s = size t ⇒
    ∃f.
      (∀x. x ∈ s ⇒ f x ∈ t) ∧ (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ∧
      ∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y

⊦ ∀f s t u.
    { x. x | x ∈ u ∧ f x ∈ image f s } ⊆ s ∧ t ⊆ u ∨
    { x. x | x ∈ u ∧ f x ∈ image f t } ⊆ t ∧ s ⊆ u ⇒
    image f (s ∩ t) = image f s ∩ image f t

⊦ ∀f g b s.
    finite s ∧ (∀x. x ∈ s ⇒ f x = g x) ∧
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ∧
    (∀x y s. ¬(x = y) ⇒ g x (g y s) = g y (g x s)) ⇒
    fold f b s = fold g b s

⊦ ∀f s t.
    (∀x. x ∈ s ⇒ f x ∈ t) ⇒
    ((∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧
     (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
     ∃g.
       (∀y. y ∈ t ⇒ g y ∈ s) ∧ (∀y. y ∈ t ⇒ f (g y) = y) ∧
       ∀x. x ∈ s ⇒ g (f x) = x)

⊦ ∀d a s t.
    { f. f |
      (∀x. x ∈ insert a s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ insert a s) ⇒ f x = d } =
    image (λ(b, g) x. if x = a then b else g x)
      (cross t { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d })

External Type Operators

External Constants

Assumptions

⊦ ⊤

⊦ id = λx. x

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀m. ¬(m < 0)

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀a. ∃x. x = a

⊦ ∀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 ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊥ ⇔ ⊥

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. t ∧ t ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀t. t ∨ t ⇔ t

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 * n = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m - 0 = m

⊦ ∀n. n - n = 0

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m ↑ 0 = 1

⊦ ∀m n. m ≤ max m n

⊦ ∀m n. n ≤ m + n

⊦ ∀m n. n ≤ max m n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀m. suc m = m + 1

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀n. suc n - 1 = n

⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2

⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1

⊦ ∀a b. fst (a, b) = a

⊦ ∀a b. snd (a, b) = b

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b

⊦ ∀m n. m * n = n * m

⊦ ∀m n. m + n = n + m

⊦ ∀m n. m = n ⇒ m ≤ n

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀m n. m + n - n = m

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m n. m < suc n ⇔ m ≤ n

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀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 + n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀m n. suc m < suc n ⇔ m < n

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀m n. m + n = m ⇔ n = 0

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2

⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2

⊦ ∀m n. m * suc n = m + m * n

⊦ ∀m n. m ↑ suc n = m * m ↑ n

⊦ ∀m n. suc m * n = m * n + n

⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)

⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∀x. x = a ⇒ p x) ⇔ p a

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

⊦ ∀p a. (∃x. x = a ∧ p x) ⇔ p a

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

⊦ ∀m n. n ≤ m ⇒ m - n + n = m

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀p. (∃x y. p x y) ⇔ ∃y x. p x y

⊦ ∀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

⊦ ∀m n. m < n ⇔ m ≤ n ∧ ¬(m = n)

⊦ ∀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

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

⊦ ∀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 + p) = m + n + p

⊦ ∀m n p. m + n = m + p ⇔ n = p

⊦ ∀m n p. m < n ∧ n < p ⇒ m < p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n < p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀m n. n < m ⇒ suc (m - suc n) = m - n

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

⊦ ∀p q r. p ∧ (q ∨ r) ⇔ p ∧ q ∨ p ∧ r

⊦ ∀p q r. p ∨ q ∧ r ⇔ (p ∨ q) ∧ (p ∨ r)

⊦ ∀p q r. (p ∨ q) ∧ r ⇔ p ∧ r ∨ q ∧ r

⊦ ∀p q r. p ∨ q ⇒ r ⇔ (p ⇒ r) ∧ (q ⇒ r)

⊦ (∃!) = 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

⊦ ∀p. (∃n. p n) ⇔ ∃n. p n ∧ ∀m. m < n ⇒ ¬p m

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

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

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

⊦ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q

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

⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'

⊦ ∀p. (∃n. p n) ∧ (∃m. ∀n. p n ⇒ n ≤ m) ⇔ ∃m. p m ∧ ∀n. p n ⇒ n ≤ m