Package set: Set types
Information
| name | set |
| version | 1.70 |
| description | Set types |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| checksum | 97ad851e12492552c60989607e0cd134c66771ae |
| requires | bool function natural pair |
| show | Data.Bool Data.Pair Function Number.Natural Set |
Files
- Package tarball set-1.70.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 ∅ = ∅
⊦ ∀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
⊦ 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
⊦ ∀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 = ∅
⊦ ∀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)
⊦ ∀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
⊦ ∀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. 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
⊦ ∀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)
⊦ ∀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
⊦ ∀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 }
⊦ ∀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)
⊦ ∀u s. u \ bigIntersect s = bigUnion { t. u \ t | t ∈ s }
⊦ ∀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 }
⊦ ∀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 } = ∅
⊦ ∀FINITE'.
FINITE' ∅ ∧ (∀x s. FINITE' s ⇒ FINITE' (insert x s)) ⇒
∀a. finite a ⇒ FINITE' 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 s. (∀x y. f x = f y ⇒ x = y) ∧ infinite s ⇒ infinite (image f 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)
⊦ ∀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
⊦ ∀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 }
⊦ ∀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))
⊦ ∀f s a.
(∀x. f x = f a ⇒ x = a) ⇒
image f (delete s a) = delete (image f s) (f a)
⊦ ∀f. (∀s t. image f s = image f t ⇒ s = t) ⇔ ∀x y. f x = f y ⇒ x = y
⊦ ∀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
⊦ ∀f s t.
(∀x y. f x = f y ⇒ x = y) ⇒ image f (s \ t) = image f s \ image f t
⊦ ∀f s t.
(∀x y. f x = f y ⇒ x = y) ⇒ image f (s ∩ t) = image f s ∩ image f t
⊦ ∀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 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 t. (∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ image f s = t
⊦ ∀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
⊦ ∀f s t.
(∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
∃g. ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y
⊦ ∀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.
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 }
⊦ ∀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 }
⊦ ∀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
⊦ ∀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)
⊦ ∀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 A s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ finite A ⇒
finite { x. x | x ∈ s ∧ f x ∈ A }
⊦ ∀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
⊦ ∀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 }
⊦ ∀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)
⊦ ∀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
⊦ ∀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 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)
⊦ ∀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 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
- →
- bool
- Data
- Pair
- ×
- Pair
- Number
- Natural
- natural
- Natural
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- Pair
- ,
- Bool
- Function
- id
- ∘
- Number
- Natural
- *
- +
- -
- <
- ≤
- ↑
- bit0
- bit1
- max
- suc
- zero
- Natural
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
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = 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 ⇔ ⊥
⊦ ∀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
⊦ ∀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