Package base: The standard theory library
Information
| name | base |
| version | 1.214 |
| description | The standard theory library |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| homepage | http://opentheory.gilith.com/?pkg=base |
| haskell-name | opentheory |
| haskell-int-file | haskell.int |
| haskell-src-file | haskell.art |
| haskell-test-file | haskell-test.art |
| haskell-equality-type | "Data.List.list" |
| haskell-equality-type | "Data.Option.option" |
| haskell-equality-type | "Data.Pair.*" |
| haskell-equality-type | "Data.Sum.+" |
| haskell-equality-type | "Number.Natural.natural" |
| haskell-arbitrary-type | "Data.List.list" |
| haskell-arbitrary-type | "Data.Option.option" |
| haskell-arbitrary-type | "Data.Pair.*" |
| haskell-arbitrary-type | "Data.Sum.+" |
| haskell-arbitrary-type | "Number.Natural.natural" |
| checksum | 40d0b8944b5f55e2b1f89cc5773c11784568d21c |
| show | Data.Bool Data.List Data.Option Data.Pair Data.Sum Data.Unit Function Number.Natural Number.Real Relation Set |
Files
- Package tarball base-1.214.tgz
- Theory source file base.thy (included in the package tarball)
Defined Type Operators
- Data
- List
- list
- Option
- option
- Pair
- ×
- Sum
- +
- Unit
- unit
- List
- Number
- Natural
- natural
- Real
- real
- Natural
- Set
- set
Defined Constants
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- @
- []
- all
- any
- case
- concat
- drop
- filter
- foldl
- foldr
- fromSet
- head
- interval
- last
- length
- map
- member
- nth
- nub
- nubReverse
- null
- replicate
- reverse
- tail
- take
- toSet
- unzip
- zip
- zipWith
- Option
- case
- isNone
- isSome
- map
- none
- some
- Pair
- ,
- fst
- snd
- Sum
- case
- destLeft
- destRight
- isLeft
- isRight
- left
- right
- Unit
- ()
- Bool
- Function
- ↑
- const
- flip
- id
- injective
- ∘
- surjective
- Combinator
- Combinator.s
- Combinator.w
- Number
- Natural
- *
- +
- -
- <
- ≤
- >
- ≥
- ↑
- bit0
- bit1
- distance
- div
- even
- factorial
- isSuc
- log
- max
- min
- minimal
- mod
- odd
- pre
- suc
- zero
- Real
- *
- +
- -
- /
- <
- ≤
- >
- ≥
- ↑
- ~
- abs
- fromNatural
- inv
- max
- min
- sup
- Natural
- Relation
- bigIntersect
- bigUnion
- empty
- fromSet
- intersect
- irreflexive
- measure
- reflexive
- subrelation
- toSet
- transitive
- transitiveClosure
- union
- universe
- wellFounded
- Set
- ∅
- bigIntersect
- bigUnion
- bijections
- choice
- cross
- delete
- \
- disjoint
- finite
- fold
- fromPredicate
- hasSize
- image
- infinite
- injections
- insert
- ∩
- ∈
- ⊂
- rest
- singleton
- size
- ⊆
- surjections
- ∪
- universe
Theorems
⊦ ⊤
⊦ null []
⊦ isNone none
⊦ even 0
⊦ irreflexive empty
⊦ irreflexive isSuc
⊦ reflexive universe
⊦ transitive empty
⊦ transitive universe
⊦ transitive (<)
⊦ wellFounded empty
⊦ wellFounded (<)
⊦ wellFounded isSuc
⊦ finite ∅
⊦ finite universe
⊦ infinite universe
⊦ ¬isSome none
⊦ ¬odd 0
⊦ subrelation isSuc (<)
⊦ id = λx. x
⊦ ¬(universe = ∅)
⊦ const = λx y. x
⊦ empty = fromSet ∅
⊦ universe = fromSet universe
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ length [] = 0
⊦ bit0 0 = 0
⊦ size ∅ = 0
⊦ concat [] = []
⊦ nubReverse [] = []
⊦ reverse [] = []
⊦ toSet [] = ∅
⊦ bigIntersect ∅ = universe
⊦ bigUnion ∅ = ∅
⊦ bigUnion universe = universe
⊦ transitiveClosure isSuc = (<)
⊦ map id = id
⊦ map id = id
⊦ ∀a. isSome (some a)
⊦ ∀a. isLeft (left a)
⊦ ∀x. x = x
⊦ ∀x. x ∈ universe
⊦ ∀b. isRight (right b)
⊦ ∀t. t ⇒ t
⊦ ∀v. v = ()
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ∀x. x ≤ x
⊦ ∀l. finite (toSet l)
⊦ ∀s. disjoint ∅ s
⊦ ∀s. disjoint s ∅
⊦ ∀s. ∅ ⊆ s
⊦ ∀s. s ⊆ universe
⊦ ∀s. s ⊆ s
⊦ ∀p. all p []
⊦ ∀m. wellFounded (measure m)
⊦ ∀r. transitive (transitiveClosure r)
⊦ ∀r. subrelation r r
⊦ ⊥ ⇔ ∀p. p
⊦ (minimal n. ⊤) = 0
⊦ fromPredicate (λx. ⊥) = ∅
⊦ fromPredicate (λx. ⊤) = universe
⊦ ∀l. all (λx. ⊤) l
⊦ hasSize universe 2
⊦ finite universe ⇔ finite universe
⊦ infinite universe ⇔ infinite universe
⊦ factorial 0 = 1
⊦ zip [] [] = []
⊦ cross universe universe = universe
⊦ ∀a. ¬isNone (some a)
⊦ ∀a. ¬isRight (left a)
⊦ ∀x. ¬member x []
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀a. finite (insert a ∅)
⊦ ∀x. id x = x
⊦ ∀b. ¬isLeft (right b)
⊦ ∀t. t ∨ ¬t
⊦ ∀m. ¬(m < 0)
⊦ ∀n. ¬(n < n)
⊦ ∀n. 0 < factorial n
⊦ ∀n. 0 < suc n
⊦ ∀n. n < suc n
⊦ ∀n. n ≤ suc n
⊦ ∀s. ¬(universe ⊂ s)
⊦ ∀s. ¬(s ⊂ ∅)
⊦ ∀s. ¬(s ⊂ s)
⊦ ∀p. ¬any p []
⊦ ∀r. subrelation r (transitiveClosure r)
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = λp. p ((select) p)
⊦ ∀x y. universe x y
⊦ ∀a. ∃x. x = a
⊦ ∀a. ∃!x. x = a
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (∃x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ ∀f g. f = g
⊦ ⊤ ⇔ (λp. p) = λp. p
⊦ (∀) = λp. p = λx. ⊤
⊦ Combinator.w = λf x. f x x
⊦ unzip [] = ([], [])
⊦ ∀a. ¬(some a = none)
⊦ ∀a. destLeft (left a) = a
⊦ ∀x. replicate x 0 = []
⊦ ∀x. delete ∅ x = ∅
⊦ ∀x. insert x universe = universe
⊦ ∀b. destRight (right b) = b
⊦ ∀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. ¬(factorial n = 0)
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. n ≤ n * n
⊦ ∀n. even n ∨ odd n
⊦ ∀n. 1 ≤ factorial n
⊦ ∀n. pre (suc n) = n
⊦ ∀n. 0 * n = 0
⊦ ∀m. m * 0 = 0
⊦ ∀n. 0 + n = n
⊦ ∀m. m + 0 = m
⊦ ∀m. m - 0 = m
⊦ ∀n. n - n = 0
⊦ ∀n. distance 0 n = n
⊦ ∀n. distance n 0 = n
⊦ ∀n. distance n n = 0
⊦ ∀n. max 0 n = n
⊦ ∀n. max n 0 = n
⊦ ∀n. max n n = n
⊦ ∀n. min 0 n = 0
⊦ ∀n. min n 0 = 0
⊦ ∀n. min n n = n
⊦ ∀m. interval m 0 = []
⊦ ∀n. id ↑ n = id
⊦ ∀l. reverse (reverse l) = l
⊦ ∀l. [] @ l = l
⊦ ∀l. l @ [] = l
⊦ ∀l. drop 0 l = l
⊦ ∀l. take 0 l = []
⊦ ∀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. f ↑ 0 = id
⊦ ∀f. map f [] = []
⊦ ∀f. map f none = none
⊦ ∀f. image f ∅ = ∅
⊦ ∀x. Combinator.s const x = id
⊦ ∀f. f ∘ id = f
⊦ ∀f. id ∘ f = f
⊦ ∀p. filter p [] = []
⊦ ∀s. toSet (fromSet s) = s
⊦ ∀r. wellFounded r ⇒ irreflexive r
⊦ ∀f. flip (flip f) = f
⊦ ∀r. fromSet (toSet r) = r
⊦ (∘) = λf g x. f (g x)
⊦ flip = λf x y. f y x
⊦ ∀x y. ¬empty x y
⊦ ∀e. ∃fn. fn () = e
⊦ ∀e. ∃!fn. fn () = e
⊦ ∀l. map (λx. x) l = l
⊦ ∀s. image (λx. x) s = s
⊦ universe = insert ⊤ (insert ⊥ ∅)
⊦ size = fold (λx n. suc n) 0
⊦ ∀t1 t2. (let x ← t2 in t1) = t1
⊦ ∀x. hasSize (insert x ∅) 1
⊦ ∀x. last (x :: []) = x
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ ∀n. ¬(even n ∧ odd n)
⊦ ∀n. even (2 * n)
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀n. ¬even n ⇔ odd n
⊦ ∀n. ¬odd n ⇔ even n
⊦ ∀m. m ↑ 0 = 1
⊦ ∀m. m * 1 = m
⊦ ∀n. n ↑ 1 = n
⊦ ∀n. n div 1 = n
⊦ ∀n. n mod 1 = 0
⊦ ∀m. 1 * m = m
⊦ ∀x. 0 + x = x
⊦ ∀l. null l ⇔ l = []
⊦ ∀l. length (reverse l) = length l
⊦ ∀l. nub (nub l) = nub l
⊦ ∀l. nubReverse (nubReverse l) = nubReverse l
⊦ ∀l. toSet (nub l) = toSet l
⊦ ∀l. toSet (nubReverse l) = toSet l
⊦ ∀l. toSet (reverse l) = toSet l
⊦ ∀l. length (nub l) ≤ length l
⊦ ∀l. length (nubReverse l) ≤ length l
⊦ ∀l. size (toSet l) ≤ length l
⊦ ∀l. drop (length l) l = []
⊦ ∀l. take (length l) l = l
⊦ ∀l. case [] (::) l = l
⊦ ∀l. foldr (::) [] l = l
⊦ ∀x. case none some x = x
⊦ ∀s. infinite s ⇔ ¬finite s
⊦ ∀s. bigIntersect (insert s ∅) = s
⊦ ∀s. bigUnion (insert s ∅) = s
⊦ ∀f. f ↑ 1 = f
⊦ ∀x. case left right x = x
⊦ ∀f. zipWith f [] [] = []
⊦ ∃f. injective f ∧ ¬surjective f
⊦ Combinator.s = λf g x. f x (g x)
⊦ ∀x y. const x y = x
⊦ ∀h t. ¬null (h :: t)
⊦ ∀x s. x ∈ insert x s
⊦ ∀x s. delete s x ⊆ s
⊦ ∀x. (select y. y = x) = x
⊦ ∀m n. m ≤ m + n
⊦ ∀m n. m ≤ max m n
⊦ ∀m n. n ≤ m + n
⊦ ∀m n. n ≤ max m n
⊦ ∀m n. min m n ≤ m
⊦ ∀m n. min m n ≤ n
⊦ ∀l. all (λx. ⊥) l ⇔ null l
⊦ ∀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
⊦ ∀p. p () ⇒ ∀x. p x
⊦ ∀p. (∀x. p x) ⇔ p ()
⊦ ∀p. (∃x. p x) ⇔ p ()
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ finite universe ∧ finite universe ⇒ finite universe
⊦ ∀x. size (insert x ∅) = 1
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀n. odd (suc (2 * n))
⊦ ∀n. n < 2 ↑ n
⊦ ∀m. suc m = m + 1
⊦ ∀n. even (suc n) ⇔ ¬even n
⊦ ∀n. odd (suc n) ⇔ ¬odd n
⊦ ∀m. m ≤ 0 ⇔ m = 0
⊦ ∀n. 1 ↑ n = 1
⊦ ∀n. suc n - 1 = n
⊦ ∀x. x ↑ 0 = 1
⊦ ∀x. ~x + x = 0
⊦ ∀x. 1 * x = x
⊦ ∀l. nub l = reverse (nubReverse (reverse l))
⊦ ∀l. length l = 0 ⇔ null l
⊦ ∀l. toSet l = ∅ ⇔ null l
⊦ ∀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
⊦ ∀l. null (concat l) ⇔ all null l
⊦ ∀x. (fst x, snd x) = x
⊦ ∀s. bigUnion (delete s ∅) = bigUnion s
⊦ ∀s. bigUnion (insert ∅ s) = bigUnion s
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀a b. ¬(left a = right b)
⊦ ∀a b. fst (a, b) = a
⊦ ∀a b. snd (a, b) = b
⊦ ∀x n. length (replicate x n) = n
⊦ ∀h t. ¬(h :: t = [])
⊦ ∀h t. head (h :: t) = h
⊦ ∀h t. tail (h :: t) = t
⊦ ∀x s. ¬(insert x s = ∅)
⊦ ∀b f. case b f none = b
⊦ ∀b f. case b f [] = b
⊦ ∀b t. (if b then t else t) = t
⊦ ∀m n. ¬(m + n < m)
⊦ ∀m n. ¬(n + m < m)
⊦ ∀m n. length (interval m n) = n
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ ∀r x. reflexive r ⇒ r x x
⊦ ∀r. reflexive r ⇔ ∀x. r x x
⊦ ∀f b. foldr f b [] = b
⊦ ∀f b. foldl f b [] = b
⊦ ∀n. 0 < n ⇔ ¬(n = 0)
⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀l. length l = 0 ⇔ l = []
⊦ ∀l. reverse l = [] ⇔ l = []
⊦ ∀l. toSet l = ∅ ⇔ l = []
⊦ ∀l. foldl (flip (::)) [] l = reverse l
⊦ ∀s. finite s ⇒ toSet (fromSet s) = s
⊦ ∀f y. (let x ← y in f x) = f y
⊦ ∀x. ∃a b. x = (a, b)
⊦ ∀s. bigIntersect s = fromSet (bigIntersect (image toSet s))
⊦ ∀s. bigUnion s = fromSet (bigUnion (image toSet s))
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀x y. x = y ⇒ y = x
⊦ ∀h t. nth (h :: t) 0 = h
⊦ ∀x s. finite s ⇒ finite (insert x s)
⊦ ∀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 = n * m
⊦ ∀m n. m + n = n + m
⊦ ∀m n. distance m n = distance n m
⊦ ∀m n. max m n = max n m
⊦ ∀m n. min m n = min 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. distance m n ≤ m + n
⊦ ∀m n. distance m (m + n) = n
⊦ ∀m n. m + n - m = n
⊦ ∀m n. m + n - n = m
⊦ ∀m n. distance (m + n) m = n
⊦ ∀m n. n * (m div n) ≤ m
⊦ ∀x y. x > y ⇔ y < x
⊦ ∀x y. x ≥ y ⇔ y ≤ x
⊦ ∀x y. x * y = y * x
⊦ ∀x y. x + y = y + x
⊦ ∀x y. x ≤ y ∨ y ≤ x
⊦ ∀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 l. null (map f l) ⇔ null l
⊦ ∀f l. length (map f l) = length l
⊦ ∀f s. finite s ⇒ finite (image f s)
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∀p l. length (filter p l) ≤ length l
⊦ ∀f x. Combinator.w f x = f x x
⊦ ∀r x. irreflexive r ⇒ ¬r x x
⊦ ∀r. irreflexive r ⇔ ∀x. ¬r x x
⊦ ∅ = { x. x | ⊥ }
⊦ universe = { x. x | ⊤ }
⊦ ∀n. factorial (suc n) = suc n * factorial n
⊦ ∀n. n ↑ 2 = n * n
⊦ ∀n. 2 * n = n + n
⊦ ∀s. finite s ⇒ length (fromSet s) = size s
⊦ ∀s. ¬(s = ∅) ⇒ choice s ∈ s
⊦ ∀x n. null (replicate x n) ⇔ n = 0
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ ∀m n. ¬(m < n ∧ n < m)
⊦ ∀m n. ¬(m < n ∧ n ≤ m)
⊦ ∀m n. ¬(m ≤ n ∧ n < m)
⊦ ∀m n. isSuc m n ⇔ suc m = n
⊦ ∀m n. m < n ⇒ ¬(m = n)
⊦ ∀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
⊦ ∀x y. x < y ⇔ ¬(y ≤ x)
⊦ ∀x y. x - y = x + ~y
⊦ ∀l x. member x l ⇔ x ∈ toSet l
⊦ ∀l x. member x (nub l) ⇔ member x l
⊦ ∀l x. member x (nubReverse l) ⇔ member x l
⊦ ∀l x. member x (reverse l) ⇔ member x l
⊦ ∀l1 l2. foldr (::) l2 l1 = l1 @ l2
⊦ ∀l1 l2. drop (length l1) (l1 @ l2) = l2
⊦ ∀l1 l2. take (length l1) (l1 @ l2) = l1
⊦ ∀l1 l2. zip l1 l2 = zipWith , l1 l2
⊦ ∀x. x = none ∨ ∃a. x = some a
⊦ ∀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 = ∅)
⊦ ∀f a. map f (some a) = some (f a)
⊦ ∀p. (∀b. p b) ⇔ p ⊤ ∧ p ⊥
⊦ ∀p. (∃b. p b) ⇔ p ⊤ ∨ p ⊥
⊦ ∀p. p ⊥ ∧ p ⊤ ⇒ ∀x. p x
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀n. even n ⇔ n mod 2 = 0
⊦ ∀n. ¬(n = 0) ⇒ 0 div n = 0
⊦ ∀n. ¬(n = 0) ⇒ 0 mod n = 0
⊦ ∀n. ¬(n = 0) ⇒ n mod n = 0
⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x
⊦ ∀p. (∀x. ¬p x) ⇔ ¬∃x. p x
⊦ ∀p. (∃x. ¬p x) ⇔ ¬∀x. p x
⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x
⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x
⊦ ∀l. unzip l = (map fst l, map snd l)
⊦ ∀r. subrelation isSuc r ∧ transitive r ⇒ subrelation (<) r
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y
⊦ ∀a b. some a = some b ⇔ a = b
⊦ ∀a b. left a = left b ⇔ a = b
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀h t. (h :: []) @ t = h :: t
⊦ ∀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
⊦ ∀a b. right a = right b ⇔ a = b
⊦ ∀p q. (q ⇒ p) ⇒ ¬p ⇒ ¬q
⊦ ∀t1 t2. ¬(t1 ⇒ t2) ⇔ t1 ∧ ¬t2
⊦ ∀t1 t2. (¬t1 ⇔ ¬t2) ⇔ t1 ⇔ t2
⊦ ∀t1 t2. ¬t1 ⇒ ¬t2 ⇔ t2 ⇒ t1
⊦ ∀m n. m < n ⇒ m div n = 0
⊦ ∀m n. m < n ⇒ m mod n = m
⊦ ∀m n. m ≤ n ⇒ factorial m ≤ factorial n
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. m < m + n ⇔ 0 < n
⊦ ∀m n. n < m + n ⇔ 0 < m
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. fromNatural m = fromNatural n ⇔ m = n
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀m n. fromNatural m ≤ fromNatural n ⇔ m ≤ n
⊦ ∀m n. m + n = m ⇔ n = 0
⊦ ∀m n. m + n = n ⇔ m = 0
⊦ ∀m n. distance m n = 0 ⇔ m = n
⊦ ∀m n. m + n ≤ m ⇔ n = 0
⊦ ∀m n. n + m ≤ m ⇔ n = 0
⊦ ∀m n. distance (suc m) (suc n) = distance m n
⊦ ∀h t. concat (h :: t) = h @ concat t
⊦ ∀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 l. reverse (map f l) = map f (reverse l)
⊦ ∀f l. toSet (map f l) = image f (toSet l)
⊦ ∀f l. map f l = [] ⇔ l = []
⊦ ∀f s. image f s = ∅ ⇔ s = ∅
⊦ ∀s t. s ⊆ t ⇒ bigIntersect t ⊆ bigIntersect s
⊦ ∀f g. f ⊆ g ⇒ bigUnion f ⊆ bigUnion g
⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r
⊦ ∀r s. subrelation r s ⇔ toSet r ⊆ toSet s
⊦ ∀r s. toSet r = toSet s ⇒ r = s
⊦ { m. m | m < 0 } = ∅
⊦ ∀b f a. case b f (some a) = f a
⊦ ∀n. finite { m. m | m < n }
⊦ ∀n. finite { m. m | m ≤ n }
⊦ ∀n. odd n ⇔ n mod 2 = 1
⊦ ∀n. 0 ↑ n = if n = 0 then 1 else 0
⊦ ∀n. ¬(n = 0) ⇒ n div n = 1
⊦ ∀k. 1 < k ⇒ log k 1 = 0
⊦ ∀x. abs x = if 0 ≤ x then x else ~x
⊦ ∀l. ¬null l ⇒ head l :: tail l = l
⊦ ∀l. ¬(l = []) ⇒ head (reverse l) = last l
⊦ ∀l. ¬(l = []) ⇒ last (reverse l) = head l
⊦ ∀s. finite s ⇒ (size s = 0 ⇔ s = ∅)
⊦ ∀f g a. case f g (left a) = f a
⊦ ∀f g b. case f g (right b) = g b
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀f x y. flip f x y = f y x
⊦ ∀x n. replicate x (suc n) = x :: replicate x n
⊦ ∀h t. take 1 (h :: t) = h :: []
⊦ ∀x s. disjoint s (insert x ∅) ⇔ ¬(x ∈ s)
⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)
⊦ ∀x s. disjoint (insert x ∅) s ⇔ ¬(x ∈ s)
⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2
⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2
⊦ ∀m n. max m n = if m ≤ n then n else m
⊦ ∀m n. min m n = if m ≤ n then m else n
⊦ ∀m n. even (m * n) ⇔ even m ∨ even n
⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n
⊦ ∀m n. odd (m * n) ⇔ odd m ∧ odd n
⊦ ∀m n. m * suc n = m + m * n
⊦ ∀m n. m ↑ suc n = m * m ↑ n
⊦ ∀m n. suc m * n = m * n + n
⊦ ∀m n. ¬(n = 0) ⇒ m mod n < n
⊦ ∀m n. ¬(n = 0) ⇒ m div n ≤ m
⊦ ∀m n. ¬(n = 0) ⇒ m mod n ≤ m
⊦ ∀m n. max (suc m) (suc n) = suc (max m n)
⊦ ∀m n. min (suc m) (suc n) = suc (min m n)
⊦ ∀m n. fromNatural m * fromNatural n = fromNatural (m * n)
⊦ ∀m n. fromNatural m + fromNatural n = fromNatural (m + n)
⊦ ∀m n. map suc (interval m n) = interval (suc m) n
⊦ ∀n. even n ⇔ ∃m. n = 2 * m
⊦ ∀x n. x ↑ suc n = x * x ↑ n
⊦ ∀m n. max m n = if m ≤ n then n else m
⊦ ∀m n. min m n = if m ≤ n then m else n
⊦ ∀l1 l2. null (l1 @ l2) ⇔ null l1 ∧ null l2
⊦ ∀l1 l2. length (l1 @ l2) = length l1 + length l2
⊦ ∀l1 l2. reverse (l1 @ l2) = reverse l2 @ reverse l1
⊦ ∀l1 l2. toSet (l1 @ l2) = toSet l1 ∪ toSet l2
⊦ ∀l1 l2. foldl (flip (::)) l2 l1 = reverse l1 @ l2
⊦ ∀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 n. f ↑ suc n = f ∘ f ↑ n
⊦ ∀f n. f ↑ suc n = f ↑ n ∘ f
⊦ ∀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)
⊦ ∀f g. map f ∘ map g = map (f ∘ g)
⊦ ∀f g. map f ∘ map g = map (f ∘ g)
⊦ ∀s t. bigIntersect (s ∪ t) = bigIntersect s ∩ bigIntersect t
⊦ ∀s t. bigUnion (s ∪ t) = bigUnion s ∪ bigUnion t
⊦ ∀r s. intersect r s = fromSet (toSet r ∩ toSet s)
⊦ ∀r s. union r s = fromSet (toSet r ∪ toSet s)
⊦ ∀r m. wellFounded r ⇒ wellFounded (λx y. r (m x) (m y))
⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)
⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)
⊦ ∀p. (∀a b. p (a, b)) ⇒ ∀x. p x
⊦ ∀a b. ∃f. f ⊥ = a ∧ f ⊤ = b
⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d
⊦ ∀a b. (∀n. a * n ≤ b) ⇔ a = 0
⊦ ∀n. ¬(n = 0) ⇒ pre n = n - 1
⊦ ∀n. hasSize { m. m | m < n } n
⊦ ∀l. l = [] ∨ ∃h t. l = h :: t
⊦ ∀s t x. s ⊆ t ⇒ s ⊆ insert x t
⊦ ∀s. { x. x | x ∈ s } = s
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ ∀f g. (∀x. f x = g x) ⇒ f = g
⊦ ∀p a. (∀x. a = x ⇒ p x) ⇔ p a
⊦ ∀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
⊦ ∀p l. ¬all p l ⇔ any (λx. ¬p x) l
⊦ ∀p l. ¬any p l ⇔ all (λx. ¬p x) l
⊦ ∀p l. ¬all (λx. ¬p x) l ⇔ any p l
⊦ ∀p l. ¬any (λx. ¬p x) l ⇔ all p l
⊦ ∀x. (∃a. x = left a) ∨ ∃b. x = right b
⊦ ∀p. p none ∧ (∀a. p (some a)) ⇒ ∀x. p x
⊦ ∀f g x. Combinator.s f g x = f x (g x)
⊦ ∀p. (∀f. p f) ⇔ ∀f. p (λa b. f (a, b))
⊦ ∀p. (∃f. p f) ⇔ ∃f. p (λa b. f (a, b))
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀h t. last (h :: t) = if null t then h else last t
⊦ ∀h t. reverse (h :: t) = reverse t @ h :: []
⊦ ∀x s. x ∈ s ⇒ insert x (delete s x) = s
⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n
⊦ ∀m n. n ≤ m ⇒ n + (m - n) = m
⊦ ∀m n. n ≤ m ⇒ m - n + n = m
⊦ ∀m n. m < n ∨ n < m ∨ m = n
⊦ ∀m n. odd (m + n) ⇔ ¬(odd m ⇔ odd n)
⊦ ∀m n. odd (m ↑ n) ⇔ odd m ∨ n = 0
⊦ ∀m n. interval m (suc n) = m :: interval (suc m) n
⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n
⊦ ∀n l. n ≤ length l ⇒ length (take n l) = n
⊦ ∀n. odd n ⇔ ∃m. n = suc (2 * m)
⊦ ∀x y. x ≤ y ∧ y ≤ x ⇔ x = y
⊦ ∀l k. foldl (λn x. suc n) k l = length l + k
⊦ ∀l k. foldr (λx n. suc n) k l = length l + k
⊦ ∀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
⊦ ∀s. finite s ⇒ ∀x. member x (fromSet s) ⇔ x ∈ s
⊦ ∀r s.
subrelation r s ∧ transitive s ⇒ subrelation (transitiveClosure r) s
⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b
⊦ ∀r s. subrelation r s ∧ subrelation s r ⇒ r = s
⊦ finite universe ∧ finite universe ⇒
size universe = size universe ↑ size universe
⊦ { s. s | s ⊆ ∅ } = insert ∅ ∅
⊦ ∀c x y. (if ¬c then x else y) = if c then y else x
⊦ ∀p q. (∀x. p ⇒ q) ⇔ (∃x. p) ⇒ ∀x. q
⊦ ∀p q. (∀x. p ∨ q) ⇔ (∀x. p) ∨ ∀x. q
⊦ ∀p q. (∃x. p ∧ q) ⇔ (∃x. p) ∧ ∃x. q
⊦ ∀p q. (∃x. p ⇒ q) ⇔ (∀x. p) ⇒ ∃x. q
⊦ ∀p q. (∃x. p) ∧ (∃x. q) ⇔ ∃x. p ∧ q
⊦ ∀p q. (∀x. p) ∨ (∀x. q) ⇔ ∀x. p ∨ q
⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d
⊦ ∀k n. 1 < k ⇒ ∃m. n ≤ k ↑ m
⊦ ∀n. size { m. m | m < n } = n
⊦ ∀l. ¬null l ⇒ length (tail l) = length l - 1
⊦ ∀s. finite s ⇔ ∃a. ∀x. x ∈ s ⇒ x ≤ a
⊦ ∀s. infinite s ⇔ ∀N. ∃n. N ≤ n ∧ n ∈ s
⊦ ∀f x n. map f (replicate x n) = replicate (f x) n
⊦ ∀f s x. x ∈ s ⇒ f x ∈ image f s
⊦ ∀p l. (∀x. member x l ⇒ p x) ⇔ all p l
⊦ ∀p l. (∃x. member x l ∧ p x) ⇔ any p l
⊦ ∀m x y. measure m x y ⇔ m x < m y
⊦ ∀s x y. fromSet s x y ⇔ (x, y) ∈ s
⊦ ∀r. wellFounded r ⇔ ¬∃f. ∀n. r (f (suc n)) (f n)
⊦ ∀r x y. (x, y) ∈ toSet r ⇔ r x y
⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y
⊦ ∀p. (∃x y. p x y) ⇔ ∃y x. p x y
⊦ ∀x s. delete (insert x s) x = s ⇔ ¬(x ∈ s)
⊦ ∀p q. (∀x. p ⇒ q x) ⇔ p ⇒ ∀x. q x
⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x
⊦ ∀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
⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x
⊦ ∀m n. m < n ⇔ m ≤ n ∧ ¬(m = n)
⊦ ∀m n. even (m ↑ n) ⇔ even m ∧ ¬(n = 0)
⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ ∀m n. ¬(m = 0) ⇒ m * n div m = n
⊦ ∀m n. ¬(m = 0) ⇒ m * n mod m = 0
⊦ ∀k m. 1 < k ⇒ log k (k ↑ m) = m
⊦ ∀l1 l2. last (l1 @ l2) = if null l2 then last l1 else last l2
⊦ ∀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
⊦ ∀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. (∃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. (∃!x. p x) ⇔ ∃x. ∀y. p y ⇔ x = y
⊦ ∀s. bigIntersect s = universe ⇔ ∀t. t ∈ s ⇒ t = universe
⊦ ∀s. bigUnion s = ∅ ⇔ ∀t. t ∈ s ⇒ t = ∅
⊦ ∀p. (∀x y. p x y) ⇔ ∀z. p (fst z) (snd z)
⊦ ∀p. (∃x y. p x y) ⇔ ∃z. p (fst z) (snd z)
⊦ ∀x y z. x = y ∧ y = z ⇒ x = z
⊦ ∀x1 x2 l. last (x1 :: x2 :: l) = last (x2 :: l)
⊦ ∀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 n i. i < n ⇒ nth (replicate x n) i = x
⊦ ∀h t l. last (l @ h :: t) = last (h :: t)
⊦ ∀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 ∅
⊦ ∀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
⊦ ∀x y n. x ≤ y ⇒ x ↑ n ≤ y ↑ n
⊦ ∀m n p. distance m p ≤ distance m n + distance n p
⊦ ∀m n p. m * (n * p) = n * (m * p)
⊦ ∀m n p. m * (n * p) = m * n * p
⊦ ∀m n p. m + (n + p) = m + n + p
⊦ ∀m n p. m ↑ (n * p) = (m ↑ n) ↑ p
⊦ ∀a b n. b < a * n ⇒ b div a < n
⊦ ∀m n p. m + n = m + p ⇔ n = p
⊦ ∀p m n. m + p = n + p ⇔ m = n
⊦ ∀m n p. m + n < m + p ⇔ n < p
⊦ ∀m n p. n + m < p + m ⇔ n < p
⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
⊦ ∀m n p. n + m ≤ p + m ⇔ n ≤ p
⊦ ∀m n p. distance (m + n) (m + p) = distance n p
⊦ ∀p m n. distance (m + p) (n + p) = distance m n
⊦ ∀m n p. (m * n + p) mod n = p mod n
⊦ ∀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
⊦ ∀x y z. y ≤ z ⇒ x + y ≤ x + z
⊦ ∀x y z. x * (y * z) = x * y * z
⊦ ∀x y z. x + (y + z) = x + y + z
⊦ ∀x y z. x ≤ y ∧ y ≤ z ⇒ x ≤ z
⊦ ∀x. ¬(x = 0) ⇒ inv x * x = 1
⊦ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀l1 l2 l3. l1 @ l2 @ l3 = (l1 @ l2) @ l3
⊦ ∀l l1 l2. l @ l1 = l @ l2 ⇔ l1 = l2
⊦ ∀l l1 l2. l1 @ l = l2 @ l ⇔ l1 = l2
⊦ ∀l l1 l2. l @ l1 = l @ l2 ⇒ l1 = l2
⊦ ∀l l1 l2. l1 @ l = l2 @ l ⇒ l1 = l2
⊦ ∀l. ¬null l ⇒ nth l (length l - 1) = last l
⊦ ∀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 }
⊦ ∀s. finite { t. t | t ⊆ s } ⇔ finite s
⊦ ∀f n x. (f ↑ suc n) x = f ((f ↑ n) x)
⊦ ∀f n x. (f ↑ suc n) x = (f ↑ n) (f x)
⊦ ∀f m n. f ↑ (m * n) = (f ↑ m) ↑ n
⊦ ∀f s t. s ⊆ t ⇒ image f s ⊆ image f t
⊦ ∀f g x. isLeft x ⇒ case f g x = f (destLeft x)
⊦ ∀f g x. isRight x ⇒ case f g x = g (destRight x)
⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x
⊦ ∀p l. all p l ⇔ ∀x. x ∈ toSet l ⇒ p x
⊦ ∀p l. any p l ⇔ ∃x. x ∈ toSet l ∧ p x
⊦ ∀f g l. map (f ∘ g) l = map f (map g l)
⊦ ∀f g x. map (f ∘ g) x = map f (map g x)
⊦ ∀f g s. image (f ∘ g) s = image f (image g s)
⊦ ∀p f l. all p (map f l) ⇔ all (p ∘ f) l
⊦ ∀p f l. any p (map f l) ⇔ any (p ∘ f) l
⊦ ∀f g h. f ∘ (g ∘ h) = f ∘ g ∘ h
⊦ ∀f g h. f ∘ g ∘ h = f ∘ (g ∘ h)
⊦ ∀f b l. foldr f b (reverse l) = foldl (flip f) b l
⊦ ∀r s t. subrelation r s ∧ subrelation s t ⇒ subrelation r t
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃!f. ∀x. p x (f x)
⊦ ∀f b l. foldl f b l = foldr (flip f) b (reverse l)
⊦ ∀f b l. foldl f b (reverse l) = foldr (flip f) b l
⊦ ∀p. (∀f. p f) ⇔ ∀g h. p (λa. (g a, h a))
⊦ ∀p. (∃f. p f) ⇔ ∃g h. p (λa. (g a, h a))
⊦ ∀x y. zip (x :: []) (y :: []) = (x, y) :: []
⊦ ∀x y. cross (insert x ∅) (insert y ∅) = insert (x, y) ∅
⊦ ∀x n. toSet (replicate x n) = if n = 0 then ∅ else insert x ∅
⊦ ∀b f h t. case b f (h :: t) = f h t
⊦ ∀b f g. (λx. if b then f x else g x) = if b then f else g
⊦ ∀m n. n < m ⇒ m - suc n = pre (m - n)
⊦ ∀m n. n < m ⇒ suc (m - suc n) = m - n
⊦ ∀m n. n ≤ m ⇒ suc (m - n) = suc m - n
⊦ ∀m n. n ≤ m ⇒ (m - n = 0 ⇔ m = n)
⊦ ∀m n. n ≤ m ⇒ pre (suc m - n) = m - n
⊦ ∀m n. n ≤ m ⇒ suc m - suc n = m - n
⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n
⊦ ∀m n. 0 < m * n ⇔ 0 < m ∧ 0 < n
⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0
⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀m n. max m n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀m n. hasSize universe m ∧ hasSize universe n ⇒ hasSize universe (n ↑ m)
⊦ ∀m n. distance (distance m n) (distance m (n + 1)) = 1
⊦ ∀l1 l2. l1 @ l2 = [] ⇔ l1 = [] ∧ l2 = []
⊦ ∀xs ys. length xs = length ys ⇒ map fst (zip xs ys) = xs
⊦ ∀xs ys. length xs = length ys ⇒ map snd (zip xs ys) = ys
⊦ ∀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
⊦ ∀p. (∀a. p (left a)) ∧ (∀b. p (right b)) ⇒ ∀x. p x
⊦ ∀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. finite s ⇒ toSet (fromSet s) = s ∧ length (fromSet s) = size s
⊦ ∀s. bigUnion { x. insert x ∅ | x ∈ s } = s
⊦ ∀f h t. map f (h :: t) = f h :: map f t
⊦ ∀f x s. image f (insert x s) = insert (f x) (image f s)
⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
⊦ ∀p h t. all p (h :: t) ⇔ p h ∧ all p t
⊦ ∀p h t. any p (h :: t) ⇔ p h ∨ any p t
⊦ ∀p l x. member x (filter p l) ⇔ member x l ∧ p x
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀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
⊦ ∀p l. (∀x. all (p x) l) ⇔ all (λy. ∀x. p x y) l
⊦ ∀p l. (∃x. any (p x) l) ⇔ any (λy. ∃x. p x y) l
⊦ ∀r s. subrelation r (bigIntersect s) ⇔ ∀t. t ∈ s ⇒ subrelation r t
⊦ ∀x s. x ∈ rest s ⇔ x ∈ s ∧ ¬(x = choice s)
⊦ ∀b f. ∃fn. fn none = b ∧ ∀a. fn (some a) = f a
⊦ ∀m n. distance m n = if m ≤ n then n - m else m - n
⊦ ∀m n. n ≤ m ⇒ (even (m - n) ⇔ even m ⇔ even n)
⊦ ∀n m. ¬(n = 0) ⇒ m mod n mod n = m mod n
⊦ ∀m n. ¬(n = 0) ⇒ (m div n = 0 ⇔ m < n)
⊦ ∀m n. m = m * n ⇔ m = 0 ∨ n = 1
⊦ ∀m n. m = n * m ⇔ m = 0 ∨ n = 1
⊦ ∀n x. 0 < x ↑ n ⇔ ¬(x = 0) ∨ n = 0
⊦ ∀m n. m * n = m ⇔ m = 0 ∨ n = 1
⊦ ∀m n. n * m = m ⇔ m = 0 ∨ n = 1
⊦ ∀m n. m ↑ n = 0 ⇔ m = 0 ∧ ¬(n = 0)
⊦ ∀n l. n ≤ length l ⇒ length (drop n l) = length l - n
⊦ ∀n l. n ≤ length l ⇒ n + length (drop n l) = length l
⊦ ∀n l. n ≤ length l ⇒ take n l @ drop n l = l
⊦ ∀x y. ¬(y = 0) ⇒ x / y = x * inv y
⊦ ∀s. finite s ∧ size s ≤ 1 ⇔ ∃a. s ⊆ insert a ∅
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ ∀l xs. unzip l = (xs, []) ⇔ l = [] ∧ xs = []
⊦ ∀l ys. unzip l = ([], ys) ⇔ l = [] ∧ ys = []
⊦ ∀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
⊦ ∀p. (∀f g. p f g) ⇔ ∀h. p (fst ∘ h) (snd ∘ h)
⊦ ∀p. (∃f g. p f g) ⇔ ∃h. p (fst ∘ h) (snd ∘ h)
⊦ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t
⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s
⊦ ∀x m n. replicate x (m + n) = replicate x m @ replicate x n
⊦ ∀x s t. insert x s ⊆ t ⇔ x ∈ t ∧ s ⊆ t
⊦ ∀b t1 t2. (if b then t1 else t2) ⇔ (¬b ∨ t1) ∧ (b ∨ t2)
⊦ ∀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 ∨ q) ∧ (p ∨ r)
⊦ ∀p q r. (p ∨ q) ∧ r ⇔ p ∧ r ∨ q ∧ r
⊦ ∀p q r. p ∨ q ⇒ r ⇔ (p ⇒ r) ∧ (q ⇒ r)
⊦ ∀p q r. p ∧ q ∨ r ⇔ (p ∨ r) ∧ (q ∨ r)
⊦ ∀m n x. m ≤ n ⇒ take m (replicate x n) = replicate x m
⊦ ∀m n i. i < n ⇒ nth (interval m n) i = m + i
⊦ ∀m n p. m < max n p ⇔ m < n ∨ m < p
⊦ ∀m n p. m < min n p ⇔ m < n ∧ m < p
⊦ ∀m n p. m ≤ max n p ⇔ m ≤ n ∨ m ≤ p
⊦ ∀m n p. m ≤ min n p ⇔ m ≤ n ∧ m ≤ p
⊦ ∀m n p. max n p < m ⇔ n < m ∧ p < m
⊦ ∀m n p. min n p < m ⇔ n < m ∨ p < m
⊦ ∀m n p. max n p ≤ m ⇔ n ≤ m ∧ p ≤ m
⊦ ∀m n p. min n p ≤ m ⇔ n ≤ m ∨ p ≤ m
⊦ ∀m n p. m * (n + p) = m * n + m * p
⊦ ∀m n p. m * distance n p = distance (m * n) (m * p)
⊦ ∀m n p. m ↑ (n + p) = m ↑ n * m ↑ p
⊦ ∀m n p. (m + n) * p = m * p + n * p
⊦ ∀p m n. distance m n * p = distance (m * p) (n * p)
⊦ ∀p m n. (m * n) ↑ p = m ↑ p * n ↑ p
⊦ ∀m n p. max (m + n) (m + p) = m + max n p
⊦ ∀m n p. max (n + m) (p + m) = max n p + m
⊦ ∀m n p. min (m + n) (m + p) = m + min n p
⊦ ∀m n p. min (n + m) (p + m) = min n p + m
⊦ ∀n. size { m. m | m ≤ n } = n + 1
⊦ ∀x y z. x * (y + z) = x * y + x * z
⊦ ∀l1 l2 x. member x (l1 @ l2) ⇔ member x l1 ∨ member x l2
⊦ ∀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 m n. f ↑ (m + n) = f ↑ m ∘ f ↑ n
⊦ ∀f l1 l2. map f (l1 @ l2) = map f l1 @ map f l2
⊦ ∀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. (∀ys. ∃xs. map f xs = ys) ⇔ ∀y. ∃x. f x = y
⊦ ∀f. (∀t. ∃s. image f s = t) ⇔ ∀y. ∃x. f x = y
⊦ ∀p l1 l2. all p (l1 @ l2) ⇔ all p l1 ∧ all p l2
⊦ ∀p l1 l2. any p (l1 @ l2) ⇔ any p l1 ∨ any p l2
⊦ ∀p l1 l2. filter p (l1 @ l2) = filter p l1 @ filter p l2
⊦ ∀p l. all p l ⇔ ∀i. i < length l ⇒ p (nth l i)
⊦ ∀p l. any p l ⇔ ∃i. i < length l ∧ p (nth l i)
⊦ ∀p f l. filter p (map f l) = map f (filter (p ∘ f) l)
⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y
⊦ ∀h t.
nubReverse (h :: t) =
if member h t then nubReverse t else h :: nubReverse t
⊦ ∀b f x y. f (if b then x else y) = if b then f x else f y
⊦ ∀b f g x. (if b then f else g) x = if b then f x else g x
⊦ ∀m n. n ≤ m ⇒ (odd (m - n) ⇔ ¬(odd m ⇔ odd n))
⊦ ∀x n. x ↑ n = 1 ⇔ x = 1 ∨ n = 0
⊦ ∀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
⊦ ∀p q l. all p (filter q l) ⇔ all (λx. q x ⇒ p x) l
⊦ ∀p q l. any p (filter q l) ⇔ any (λx. q x ∧ p x) l
⊦ ∀p. (∃!x. p x) ⇔ ∃x. p x ∧ ∀y. p y ⇒ y = x
⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n
⊦ ∀r. toSet r = { x y. (x, y) | r x y }
⊦ ∀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
⊦ ∀x n y. member y (replicate x n) ⇔ y = x ∧ ¬(n = 0)
⊦ ∀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
⊦ ∀l x. member x l ⇔ ∃i. i < length l ∧ x = nth l i
⊦ ∀l. toSet l = image (nth l) { i. i | i < length l }
⊦ ∀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
⊦ ∀f g. (∀x. ∃y. g y = f x) ⇔ ∃h. f = g ∘ h
⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h 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 ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x
⊦ ∀p q. (∀x. p x) ∧ (∀x. q x) ⇔ ∀x. p x ∧ q x
⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x
⊦ ∀r.
transitiveClosure r =
bigIntersect { s. s | subrelation r s ∧ transitive s }
⊦ ∀f x y. zipWith f (x :: []) (y :: []) = f x y :: []
⊦ ∀f. (λx. f x) = λ(a, b). f (a, b)
⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
⊦ ∀a b c. (∀n. a * n ≤ b * n + c) ⇔ a ≤ b
⊦ ∀m n. ¬(n = 0) ⇒ (m div n) * n + m mod n = m
⊦ ∀m n. m * n = 1 ⇔ m = 1 ∧ n = 1
⊦ ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ 0 ≤ x * y
⊦ ∀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
⊦ ∀p. (∃n. p n) ⇔ ∃n. p n ∧ ∀m. m < n ⇒ ¬p m
⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l
⊦ ∀f b h t. foldr f b (h :: t) = f h (foldr f b t)
⊦ ∀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
⊦ ∀f b h t. foldl f b (h :: t) = foldl f (f b h) t
⊦ ∀h t n. n < length t ⇒ nth (h :: t) (suc n) = nth t n
⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t
⊦ ∀m n x. m ≤ n ⇒ drop m (replicate x n) = replicate x (n - m)
⊦ ∀m n p. n ≤ m ⇒ m + p - (n + p) = m - n
⊦ ∀m n p. p ≤ n ⇒ m + n - (m + p) = n - p
⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p
⊦ ∀m n p. m * p = n * p ⇔ m = n ∨ p = 0
⊦ ∀x y n. x ↑ n = y ↑ n ⇔ x = y ∨ n = 0
⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p
⊦ ∀m n p. m * p ≤ n * p ⇔ m ≤ n ∨ p = 0
⊦ ∀x y n. x ↑ n ≤ y ↑ n ⇔ x ≤ y ∨ n = 0
⊦ ∀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
⊦ ∀f l i. i < length l ⇒ nth (map f l) i = f (nth l i)
⊦ ∀p n. p n ∧ (∀m. m < n ⇒ ¬p m) ⇒ (minimal) p = n
⊦ ∀p. (∃n. p n) ⇔ p ((minimal) p) ∧ ∀m. m < (minimal) p ⇒ ¬p m
⊦ ∀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
⊦ ∀m n. 0 < m ∧ m ≤ n ⇒ 2 * (n mod m) ≤ n
⊦ ∀k n. 1 < k ∧ ¬(n = 0) ⇒ k ↑ log k n ≤ n
⊦ ∀s t. finite s ∧ t ⊆ s ⇒ size (s \ t) = size s - size t
⊦ ∀s p. finite s ⇒ finite { x. x | x ∈ s ∧ p x }
⊦ ∀f l y. member y (map f l) ⇔ ∃x. y = f x ∧ member x l
⊦ ∀f g l. all (λx. f x = g x) l ⇒ map f l = map g l
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀p. (∀x. p (f x)) ⇔ ∀y. p y
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀p. (∃x. p (f x)) ⇔ ∃y. p y
⊦ ∀p q l. all (λx. p x ∧ q x) l ⇔ all p l ∧ all q l
⊦ ∀p q l. all (λx. p x ⇒ q x) l ∧ all p l ⇒ all q l
⊦ ∀s. bigIntersect s = { x. x | ∀u. u ∈ s ⇒ x ∈ u }
⊦ ∀s. bigUnion s = { x. x | ∃u. u ∈ s ∧ x ∈ u }
⊦ ∀r. transitive r ⇔ ∀x y z. r x y ∧ r y z ⇒ r x z
⊦ ∀f b l1 l2. foldr f b (l1 @ l2) = foldr f (foldr f b l2) l1
⊦ ∀f b l1 l2. foldl f b (l1 @ l2) = foldl f (foldl f b l1) l2
⊦ cond = λt t1 t2. select x. ((t ⇔ ⊤) ⇒ x = t1) ∧ ((t ⇔ ⊥) ⇒ x = t2)
⊦ ∀a b n. ¬(a = 0) ⇒ (n ≤ b div a ⇔ a * n ≤ b)
⊦ ∀a b n. ¬(a = 0) ⇒ (b div a < n ⇔ b < a * n)
⊦ ∀m n p. ¬(p = 0) ⇒ m * (n div p) ≤ m * n div p
⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p
⊦ ∀m n p. m * p < n * p ⇔ m < n ∧ ¬(p = 0)
⊦ ∀x y n. x ↑ n < y ↑ n ⇔ x < y ∧ ¬(n = 0)
⊦ ∀x y n. x < y ∧ ¬(n = 0) ⇒ x ↑ n < y ↑ n
⊦ ∀m n p. ¬(m = 0) ∧ n < p ⇒ m * n < m * p
⊦ ∀m n p. ¬(p = 0) ∧ m ≤ n ⇒ m div p ≤ n div p
⊦ ∀m n p. ¬(p = 0) ∧ p ≤ m ⇒ n div m ≤ n div p
⊦ ∀a b n. ¬(a = 0) ∧ b ≤ a * n ⇒ b div a ≤ n
⊦ ∀k m n. 1 < k ∧ k ↑ m ≤ n ⇒ m ≤ log k n
⊦ ∀m n. ¬(n = 0) ⇒ (m mod n = 0 ⇔ ∃q. m = q * n)
⊦ ∀l1 l2 n. length l1 = n ∧ length l2 = n ⇒ length (zip l1 l2) = n
⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ∃a. a ∈ t ∧ ¬(a ∈ s)
⊦ ∀f g. ∃fn. (∀a. fn (left a) = f a) ∧ ∀b. fn (right b) = g b
⊦ ∀f. (λ(x, y). f x y) = λp. f (fst p) (snd p)
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2
⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'
⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t
⊦ ∀x s. insert x s = { y. y | y = x ∨ y ∈ s }
⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∧ q1 ⇒ p2 ∧ q2
⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∨ q1 ⇒ p2 ∨ q2
⊦ ∀p1 p2 q1 q2. (p2 ⇒ p1) ∧ (q1 ⇒ q2) ⇒ (p1 ⇒ q1) ⇒ p2 ⇒ q2
⊦ ∀m n p q. distance m p ≤ distance (m + n) (p + q) + distance n q
⊦ ∀m n p q. m = n + q * p ⇒ m mod p = n mod p
⊦ ∀m n p q. m < n ∧ p < q ⇒ m * p < n * q
⊦ ∀m n p q. m < p ∧ n < q ⇒ m + n < p + q
⊦ ∀m n p q. m < p ∧ n ≤ q ⇒ m + n < p + q
⊦ ∀m n p q. m ≤ n ∧ p ≤ q ⇒ m * p ≤ n * q
⊦ ∀m n p q. m ≤ p ∧ n < q ⇒ m + n < p + q
⊦ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q
⊦ ∀m n p q. distance (m + n) (p + q) ≤ distance m p + distance n q
⊦ ∀m n p q. distance m n + distance n p ≤ q ⇒ distance m p ≤ q
⊦ ∀k m. 1 < k ⇒ log k (k ↑ (m + 1) - 1) = m
⊦ ∀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
⊦ ∀p l. toSet (filter p l) = toSet l \ { x. x | ¬p x }
⊦ ∀m a b. (∀y. measure m y a ⇒ measure m y b) ⇔ m a ≤ m b
⊦ ∀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
⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y
⊦ ∀p. (∀f. p f) ⇔ ∀f. p (λ(a, b). f a b)
⊦ ∀p. (∃f. p f) ⇔ ∃f. p (λ(a, b). f a b)
⊦ ∀n h t. n ≤ length t ⇒ take (suc n) (h :: t) = h :: take n t
⊦ ∀m n p. n ≤ m ⇒ (m - n) * p = m * p - n * p
⊦ ∀m n p. p ≤ n ⇒ m * (n - p) = m * n - m * p
⊦ ∀m n p. distance m n = p ⇔ m + p = n ∨ n + p = m
⊦ ∀m n p. distance m n ≤ p ⇔ m ≤ n + p ∧ n ≤ m + p
⊦ ∀m n p. n + p ≤ m ⇒ m - (n + p) = m - n - p
⊦ ∀n. (∃k m. odd m ∧ n = 2 ↑ k * m) ⇔ ¬(n = 0)
⊦ ∀p h t. filter p (h :: t) = if p h then h :: filter p t else filter p t
⊦ ∀x l. member x l ⇔ ∃l1 l2. ¬member x l1 ∧ l = l1 @ x :: l2
⊦ ∀x s. finite s ⇒ size (delete s x) = if x ∈ s then size s - 1 else size s
⊦ ∀k n. 1 < k ∧ ¬(n = 0) ⇒ (log k n = 0 ⇔ n < k)
⊦ ∀k n. 1 < k ∧ ¬(n = 0) ∧ n < k ⇒ log k n = 0
⊦ ∀l n. length l = suc n ⇔ ∃h t. l = h :: t ∧ length t = n
⊦ ∀s x. delete s x = { y. y | y ∈ s ∧ ¬(y = x) }
⊦ ∀s t. s \ t = { x. x | x ∈ s ∧ ¬(x ∈ t) }
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)
⊦ ∀t. { x y. (x, y) | x ∈ ∅ ∧ y ∈ t x } = ∅
⊦ ∀p. p ∅ ∧ (∀x s. p s ⇒ p (insert x s)) ⇒ ∀a. finite a ⇒ p a
⊦ ∀p. (∀(a, b). p a b) ⇔ ∀a b. p a b
⊦ ∀p. (∃(a, b). p a b) ⇔ ∃a b. p a b
⊦ ∀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)
⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)
⊦ ∀n m p. ¬(n = 0) ⇒ m * (p mod n) mod n = m * p mod n
⊦ ∀n m p. ¬(n = 0) ⇒ (m mod n) * p mod n = m * p mod n
⊦ ∀n m p. ¬(n = 0) ⇒ (m mod n) ↑ p mod n = m ↑ p mod n
⊦ ∀a b n. ¬(n = 0) ⇒ (a * n + b) div n = a + b div n
⊦ ∀m n p. ¬(m * p = 0) ⇒ m * n div m * p = n div p
⊦ ∀m n p. ¬(n * p = 0) ⇒ m div n div p = m div n * p
⊦ ∀m n p. ¬(n * p = 0) ⇒ m mod n * p mod n = m mod n
⊦ ∀m n p. ¬(p = 0) ∧ m + p ≤ n ⇒ m div p < n div p
⊦ ∀m n l. m + n ≤ length l ⇒ drop (m + n) l = drop m (drop n l)
⊦ ∀m n. (∃q. m = n * q) ⇔ if n = 0 then m = 0 else m mod n = 0
⊦ ∀n l i. n + i < length l ⇒ nth (drop n l) i = nth l (n + i)
⊦ ∀n l i. n ≤ length l ∧ i < n ⇒ nth (take n l) i = nth l i
⊦ ∀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
⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'
⊦ ∀l xs ys. unzip l = (xs, ys) ⇔ length xs = length ys ∧ l = zip xs ys
⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x
⊦ ∀p. (select (a, b). p a b) = select x. p (fst x) (snd x)
⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m div n = q
⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m mod n = r
⊦ ∀k n. 1 < k ∧ ¬(n = 0) ⇒ n < k ↑ (log k n + 1)
⊦ ∀l1 l1' l2 l2'. length l1 = length l1' ∧ l1 @ l2 = l1' @ l2' ⇒ l1 = l1'
⊦ ∀l1 l1' l2 l2'. length l2 = length l2' ∧ l1 @ l2 = l1' @ l2' ⇒ l2 = l2'
⊦ ∀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
⊦ ∀p q l. (∀x. member x l ∧ p x ⇒ q x) ∧ all p l ⇒ all q l
⊦ ∀p q l. (∀x. member x l ∧ p x ⇒ q x) ∧ any p l ⇒ any q l
⊦ ∀p r. (∀x. p x ⇒ ∃y. r x y) ⇔ ∃f. ∀x. p x ⇒ r x (f x)
⊦ ∀s t. (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ bigUnion s ⊆ bigUnion t
⊦ ∀s t. (∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ x ⊆ y) ⇒ bigIntersect s ⊆ bigIntersect t
⊦ ∀f l1 l2 n. length l1 = n ∧ length l2 = n ⇒ length (zipWith f l1 l2) = n
⊦ ∀a b n. ¬(a = 0) ⇒ (n < b div a ⇔ a * (n + 1) ≤ b)
⊦ ∀a b n. ¬(a = 0) ⇒ (b div a ≤ n ⇔ b < a * (n + 1))
⊦ ∀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
⊦ ∀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
⊦ ∀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
⊦ ∀p. (∃b. ∀n. p n ≤ b) ⇔ ∃a b. ∀n. n * p n ≤ a * n + b
⊦ ∀f. (λt. f t) = λ(x, y, z). f (x, y, z)
⊦ ∀b p q r s. (p ⇒ q) ∧ (r ⇒ s) ⇒ (if b then p else r) ⇒ if b then q else s
⊦ ∀n m p. ¬(n = 0) ⇒ (m mod n) * (p mod n) mod n = m * p mod n
⊦ ∀n a b. ¬(n = 0) ⇒ (a mod n + b mod n) mod n = (a + b) mod n
⊦ ∀m n p. ¬(m * p = 0) ⇒ m * n mod m * p = m * (n mod p)
⊦ ∀m n p. ¬(n * p = 0) ⇒ m div n mod p = m mod n * p div n
⊦ ∀k n m. k ↑ m ≤ n ∧ n < k ↑ (m + 1) ⇒ log k n = m
⊦ ∀s t. cross s t = { x y. (x, y) | x ∈ s ∧ y ∈ t }
⊦ ∀f g. (∀x y. g x = g y ⇒ f x = f y) ⇔ ∃h. f = h ∘ g
⊦ ∀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 }
⊦ ∀k n1 n2. 1 < k ∧ ¬(n1 = 0) ∧ n1 ≤ n2 ⇒ log k n1 ≤ log k n2
⊦ ∀k m n. 1 < k ∧ ¬(n = 0) ∧ n < k ↑ m ⇒ log k n < m
⊦ ∀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 a b. ¬(n = 0) ⇒ (suc a mod n = suc b mod n ⇔ a mod n = b mod n)
⊦ ∀m n l. m + n ≤ length l ⇒ take (m + n) l = take m l @ take n (drop m l)
⊦ ∀n. { m. m | m < suc n } = insert 0 { m. suc m | m < n }
⊦ ∀l1 l2.
length l1 = length l2 ∧ (∀i. i < length l1 ⇒ nth l1 i = nth l2 i) ⇒
l1 = l2
⊦ ∀x y xs ys.
length xs = length ys ⇒ zip (x :: xs) (y :: ys) = (x, y) :: zip xs ys
⊦ ∀s x. ¬(s = ∅) ∧ (∃m. ∀x. x ∈ s ⇒ x ≤ m) ∧ x ∈ s ⇒ x ≤ sup s
⊦ ∀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))
⊦ ∀x p m n. ¬(p = 0) ⇒ x mod p ↑ m mod p ↑ n = x mod p ↑ min m n
⊦ ∀s f.
cross s (bigIntersect f) =
if f = ∅ then cross s universe
else bigIntersect { t. cross s t | t ∈ f }
⊦ ∀p x y. p (distance x y) ⇔ ∀d. (x = y + d ⇒ p d) ∧ (y = x + d ⇒ p d)
⊦ ∀p. (∃n. p n) ∧ (∃m. ∀n. p n ⇒ n ≤ m) ⇔ ∃m. p m ∧ ∀n. p n ⇒ n ≤ m
⊦ ∀f s.
cross (bigIntersect f) s =
if f = ∅ then cross universe s
else bigIntersect { t. cross t s | t ∈ f }
⊦ ∀f. (∀l1 l2. map f l1 = map f l2 ⇒ l1 = l2) ⇔ ∀x y. f x = f y ⇒ x = y
⊦ ∀f. (∀s t. image f s = image f t ⇒ s = t) ⇔ ∀x y. f x = f y ⇒ x = y
⊦ ∀p f g. (∀x. p x ⇒ ∃y. g y = f x) ⇔ ∃h. ∀x. p x ⇒ f x = g (h x)
⊦ ∀a b c d. ¬(b = 0) ∧ b * c < (a + 1) * d ⇒ c div d ≤ a div b
⊦ ∀k p. 1 < k ∧ p 0 ∧ (∀n. ¬(n = 0) ∧ p (n div k) ⇒ p n) ⇒ ∀n. p n
⊦ ∀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
⊦ ∀f. (λ(x, y, z). f x y z) = λt. f (fst t) (fst (snd t)) (snd (snd t))
⊦ ∀p. (∀(a, b, c). p a b c) ⇔ ∀a b c. p a b c
⊦ ∀p. (∃(a, b, c). p a b c) ⇔ ∃a b c. p a b c
⊦ ∀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 m. (∀n. m ≤ n ⇒ p n) ∧ (∀n. n < m ∧ p (n + 1) ⇒ p n) ⇒ ∀n. p n
⊦ ∀p q. (∃b. ∀i. p i ≤ q i + b) ⇔ ∃b n. ∀i. n ≤ i ⇒ p i ≤ q i + b
⊦ ∀k n.
1 < k ∧ ¬(n = 0) ⇒ log k n = if n < k then 0 else log k (n div k) + 1
⊦ ∀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)
⊦ ∀m n p. ¬(m = 0) ⇒ m ↑ n mod m ↑ p = if p ≤ n ∨ m = 1 then 0 else m ↑ n
⊦ ∀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 h1 h2 t1 t2.
length t1 = length t2 ⇒
zipWith f (h1 :: t1) (h2 :: t2) = f h1 h2 :: zipWith f t1 t2
⊦ ∀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
⊦ ∀x m n. x ↑ m = x ↑ n ⇔ if x = 0 then m = 0 ⇔ n = 0 else x = 1 ∨ m = n
⊦ ∀x m n. x ↑ m ≤ x ↑ n ⇔ if x = 0 then m = 0 ⇒ n = 0 else x = 1 ∨ m ≤ n
⊦ ∀m n p q r s.
distance m n ≤ r ∧ distance p q ≤ s ⇒
distance m p ≤ distance n q + (r + s)
⊦ ∀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
⊦ unzip = foldr (λ(x, y) (xs, ys). (x :: xs, y :: ys)) ([], [])
⊦ ∀l1 l2 k.
k < length l1 + length l2 ⇒
nth (l1 @ l2) k =
if k < length l1 then nth l1 k else nth l2 (k - length l1)
⊦ ∀s m.
¬(s = ∅) ∧ (∃m. ∀x. x ∈ s ⇒ x ≤ m) ∧ (∀x. x ∈ s ⇒ x ≤ m) ⇒ sup s ≤ m
⊦ ∀f g s t.
image (λ(x, y). (f x, g y)) (cross s t) = cross (image f s) (image g 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
⊦ ∀x1 x2 y1 y2.
length x1 = length y1 ∧ length x2 = length y2 ⇒
zip (x1 @ x2) (y1 @ y2) = zip x1 y1 @ zip x2 y2
⊦ ∀l1 l2 n i.
length l1 = n ∧ length l2 = n ∧ i < n ⇒
nth (zip l1 l2) i = (nth l1 i, nth l2 i)
⊦ ∀f. (∀y. ∃x. f x = y) ⇔ ∀p. image f { x. x | p (f x) } = { x. x | p x }
⊦ ∀p.
(∀m. p m m) ∧ (∀m n. m ≤ n ∧ p m n ⇒ p m (suc n)) ⇒ ∀m n. m ≤ n ⇒ p m n
⊦ ∀s t.
finite s ∧ finite t ⇒
size { x y. (x, y) | x ∈ s ∧ y ∈ t } = size s * size t
⊦ ∀f s.
infinite s ∧ finite (image f s) ⇒
∃a. a ∈ s ∧ infinite { x. x | x ∈ s ∧ f x = f a }
⊦ ∀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
⊦ ∀h x y t xs ys.
unzip (h :: t) = (x :: xs, y :: ys) ⇔ h = (x, y) ∧ unzip t = (xs, ys)
⊦ ∀p.
p ∅ ∧ (∀s. finite s ∧ ¬(s = ∅) ⇒ ∃x. x ∈ s ∧ (p (delete s x) ⇒ p s)) ⇒
∀s. finite s ⇒ p s
⊦ ∀m n p.
¬(m = 0) ⇒
m ↑ n div m ↑ p = if p ≤ n then m ↑ (n - p) else if m = 1 then 1 else 0
⊦ ∀x m n. x ↑ m < x ↑ n ⇔ 2 ≤ x ∧ m < n ∨ x = 0 ∧ ¬(m = 0) ∧ n = 0
⊦ ∀k n m. 1 < k ∧ ¬(n = 0) ⇒ (log k n = m ⇔ k ↑ m ≤ n ∧ n < k ↑ (m + 1))
⊦ ∀n l i.
n ≤ length l ∧ i < length l ⇒
nth l i = if i < n then nth (take n l) i else nth (drop n l) (i - n)
⊦ ∀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
⊦ ∀x n l1 l2.
l1 @ l2 = replicate x n ⇔
replicate x (length l1) = l1 ∧ replicate x (length l2) = l2 ∧
length l1 + length l2 = n
⊦ ∀a b c d. b * c < (a + 1) * d ∧ a * d < (c + 1) * b ⇒ a div b = c div d
⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅
⊦ ∀k n1 n2.
1 < k ∧ ¬(n1 = 0) ∧ ¬(n2 = 0) ⇒
log k (n1 * n2) ≤ log k n1 + (log k n2 + 1)
⊦ ∀s t m n.
hasSize s m ∧ hasSize t n ⇒
hasSize { x y. (x, y) | x ∈ s ∧ y ∈ t } (m * n)
⊦ ∀f l1 l2 n i.
length l1 = n ∧ length l2 = n ∧ i < n ⇒
nth (zipWith f l1 l2) i = f (nth l1 i) (nth l2 i)
⊦ ∀f s.
(∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ⇔ ∃g. ∀x. x ∈ s ⇒ g (f x) = x
⊦ ∀r s.
wellFounded r ∧ wellFounded s ⇒
wellFounded (λ(x1, y1) (x2, y2). r x1 x2 ∧ s y1 y2)
⊦ ∀a b n.
¬(n = 0) ⇒
((a + b) mod n = a mod n + b mod n ⇔ (a + b) div n = a div n + b div n)
⊦ ∀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)
⊦ ∀p q. (∃a b. p a b) ∧ (∀a b. p a b ⇒ q (a, b)) ⇒ q (select (a, b). p a b)
⊦ ∀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 g.
(∀x y. p x ∧ p y ∧ g x = g y ⇒ f x = f y) ⇔ ∃h. ∀x. p x ⇒ f x = h (g x)
⊦ ∀f x1 x2 y1 y2.
length x1 = length y1 ∧ length x2 = length y2 ⇒
zipWith f (x1 @ x2) (y1 @ y2) = zipWith f x1 y1 @ zipWith f x2 y2
⊦ ∀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
⊦ ∀p.
(∃x. p x) ∧ (∃m. ∀x. p x ⇒ x ≤ m) ⇒
∃s. (∀x. p x ⇒ x ≤ s) ∧ ∀m. (∀x. p x ⇒ x ≤ m) ⇒ s ≤ m
⊦ ∀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.
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.
(∀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) ⇒ wellFounded r
⊦ ∀p a b.
p 0 0 = 0 ∧ (∀m n. p m n ≤ a * (m + n) + b) ⇒
∃c. ∀m n. p m n ≤ c * (m + n)
⊦ ∀d s t.
finite s ∧ finite t ⇒
finite { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d }
⊦ ∀s t k.
finite s ∧ finite t ⇒
finite { f. f | image f s ⊆ t ∧ { x. x | ¬(f x = k x) } ⊆ s }
⊦ ∀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)
⊦ ∀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. finite { y. y | r y x }) ⇒ wellFounded r
⊦ ∀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
⊦ ∀r p.
(∀x y. p x y ⇒ p y x) ∧ (∀x y. r x y ∨ r y x) ∧ (∀x y. r x y ⇒ p x y) ⇒
∀x y. p x y
⊦ ∀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
⊦ ∀r s.
wellFounded r ∧ (∀a. wellFounded (s a)) ⇒
wellFounded (λ(x1, y1) (x2, y2). r x1 x2 ∨ x1 = x2 ∧ s x1 y1 y2)
⊦ ∀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 s.
finite (image f s) ⇔
∃t.
finite t ∧ t ⊆ s ∧ image f s = image f t ∧
∀x y. x ∈ t ∧ y ∈ t ⇒ (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
⊦ ∀g f b l1 l2.
(∀s. g b s = s) ∧ (∀x s1 s2. g (f x s1) s2 = f x (g s1 s2)) ⇒
foldr f b (l1 @ l2) = g (foldr f b l1) (foldr f b l2)
⊦ ∀g f b l1 l2.
(∀s. g s b = s) ∧ (∀s1 s2 x. g s1 (f s2 x) = f (g s1 s2) x) ⇒
foldl f b (l1 @ l2) = g (foldl f b l1) (foldl f b l2)
⊦ ∀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 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)
⊦ ∀r.
wellFounded r ⇒
∀h s.
(∀f g x.
(∀z. r z x ⇒ f z = g z ∧ s z (f z)) ⇒
h f x = h g x ∧ s x (h f x)) ⇒ ∃f. ∀x. f x = h f x
⊦ ∀r.
wellFounded r ⇒
∀h.
(∀f g x. (∀z. r z x ⇒ f z = g z) ⇒ h f x = h g x) ⇒
∀f g. (∀x. f x = h f x) ∧ (∀x. g x = h g x) ⇒ f = g
⊦ ∀r.
(∀h.
(∀f g x. (∀z. r z x ⇒ (f z ⇔ g z)) ⇒ (h f x ⇔ h g x)) ⇒
∀f g. (∀x. f x ⇔ h f x) ∧ (∀x. g x ⇔ h g x) ⇒ f = g) ⇒ wellFounded r
⊦ ∀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 }
⊦ ∀r.
wellFounded r ⇒
∀p.
(∀f g x y. (∀z. r z x ⇒ f z = g z) ⇒ (p f x y ⇔ p g x y)) ∧
(∀f x. (∀z. r z x ⇒ p f z (f z)) ⇒ ∃y. p f x y) ⇒ ∃f. ∀x. p f x (f 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 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. (λ(a, b). f (a, b)) = f) ∧ (∀f. (λ(a, b, c). f (a, b, c)) = f) ∧
∀f. (λ(a, b, c, d). f (a, b, c, d)) = f
⊦ ∀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
- ind
External Constants
- =
- select
Assumptions
⊦ AXIOM OF EXTENSIONALITY
⊦ AXIOM OF CHOICE
⊦ AXIOM OF INFINITY