| name | base |
| version | 1.24 |
| description | The standard theory library |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| show | Data.Bool Data.List Data.Option Data.Pair Data.Sum Data.Unit Function Number.Natural Number.Real as Real Relation Set |
⊦ T
⊦ wellFounded (<)
⊦ finite ∅
⊦ finite universe
⊦ infinite universe
⊦ wellFounded (λx y. F)
⊦ id = λx. x
⊦ ¬(∅ = universe)
⊦ ¬(universe = ∅)
⊦ size ∅ = 0
⊦ bigIntersect ∅ = universe
⊦ bigUnion ∅ = ∅
⊦ map id = id
⊦ ∀x. x = x
⊦ ∀x. x ∈ universe
⊦ ∀v. v = ()
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ∀x. Real.≤ x x
⊦ ∀l. finite (toSet l)
⊦ ∀s. ∅ ⊆ s
⊦ ∀s. s ⊆ universe
⊦ ∀s. s ⊆ s
⊦ ∀m. wellFounded (measure m)
⊦ F ⇔ ∀p. p
⊦ fromPredicate (λx. F) = ∅
⊦ ∀l. all (λx. T) l
⊦ hasSize universe 2
⊦ 1 = suc 0
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀a. finite (insert a ∅)
⊦ ∀x. id x = x
⊦ ∀t. t ∨ ¬t
⊦ ∀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)
⊦ (~) = λp. p ⇒ F
⊦ (∃) = λP. P ((select) P)
⊦ ∀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
⊦ T ⇔ (λp. p) = λp. p
⊦ (∀) = λp. p = λx. T
⊦ ∀a'. ¬(none = some a')
⊦ ∀x. destLeft (left x) = x
⊦ ∀x. x = x ⇔ T
⊦ ∀x. delete ∅ x = ∅
⊦ ∀x. insert x universe = universe
⊦ ∀y. destRight (right y) = y
⊦ ∀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
⊦ ∀m. m * 0 = 0
⊦ ∀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
⊦ ∀l. reverse (reverse 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 = s
⊦ ∀s. s ∪ s = s
⊦ ∀e. ∃fn. fn () = e
⊦ ∀e. ∃!fn. fn () = e
⊦ ∀l. map (λx. x) l = l
⊦ ∀s. image (λx. x) s = s
⊦ universe = insert T (insert F ∅)
⊦ 2 = suc 1
⊦ ∀t1 t2. (let x ← t2 ∈ t1) = t1
⊦ ∀x. hasSize (insert x ∅) 1
⊦ ∀n. ¬(even n ∧ odd n)
⊦ ∀n. even (2 * n)
⊦ ∀n. bit0 n = n + n
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀n. ¬even n ⇔ odd n
⊦ ∀n. ¬odd n ⇔ even n
⊦ ∀n. n div 1 = n
⊦ ∀n. exp n 1 = n
⊦ ∀n. n mod 1 = 0
⊦ ∀x. Real.+ 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
⊦ ∀x. case.none.some none some x = x
⊦ ∀s. infinite s ⇔ ¬finite s
⊦ ∀s. bigIntersect (insert s ∅) = s
⊦ ∀s. bigUnion (insert s ∅) = s
⊦ ∀f. zipWith f [] [] = []
⊦ ∃f. injective f ∧ ¬surjective f
⊦ ∀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
⊦ ∀s t. disjoint s (t - s)
⊦ ∀s t. disjoint (t - s) s
⊦ ∀s t. s - t ⊆ s
⊦ ∀P. P () ⇒ ∀x. P x
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀x. size (insert x ∅) = 1
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀n. odd (suc (2 * n))
⊦ ∀n. bit1 n = suc (n + n)
⊦ ∀m. suc m = m + 1
⊦ ∀n. exp 1 n = 1
⊦ ∀n. suc n - 1 = n
⊦ ∀x. Real.+ (Real.~ x) x = 0
⊦ ∀x. Real.* 1 x = x
⊦ ∀l. nub l = reverse (nubReverse (reverse l))
⊦ ∀s. finite s ⇔ hasSize s (size s)
⊦ ∀s. rest s = delete s (choice s)
⊦ ∀s. infinite s ⇒ ¬(s = ∅)
⊦ ∀s. s = ∅ ⇔ disjoint s s
⊦ ∀s. hasSize s 0 ⇔ s = ∅
⊦ ∀s. universe ⊆ s ⇔ s = universe
⊦ ∀s. s ⊆ ∅ ⇔ s = ∅
⊦ ∀s. disjoint ∅ s ∧ disjoint s ∅
⊦ ∀l. null (concat l) ⇔ all null l
⊦ ∀x. (fst x, snd x) = x
⊦ ∀x y. fst (x, y) = x
⊦ ∀x y. snd (x, y) = y
⊦ ∀h t. ¬(h :: t = [])
⊦ ∀h t. head (h :: t) = h
⊦ ∀h t. tail (h :: t) = t
⊦ ∀x s. ¬(∅ = insert x s)
⊦ ∀x s. ¬(insert x s = ∅)
⊦ ∀b t. (if b then t else t) = t
⊦ ∀n x. length (replicate n x) = n
⊦ ∀m n. length (interval m n) = n
⊦ ∀p x. p x ⇒ p ((select) p)
⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀n. 0 < n ⇔ ¬(n = 0)
⊦ ∀l. length l = 0 ⇔ l = []
⊦ ∀l. toSet l = ∅ ⇔ l = []
⊦ ∀s. finite s ⇒ toSet (fromSet s) = s
⊦ ∀f y. (let x ← y ∈ f x) = f y
⊦ ∀p. ∃x y. p = (x, y)
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀x y. x = y ⇒ y = x
⊦ ∀h t. nth 0 (h :: t) = h
⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b
⊦ ∀n m. m > n ⇔ n < m
⊦ ∀n m. 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. Real.> x y ⇔ Real.< y x
⊦ ∀x y. Real.≥ x y ⇔ Real.≤ y x
⊦ ∀x y. Real.* x y = Real.* y x
⊦ ∀x y. Real.+ x y = Real.+ y x
⊦ ∀x y. Real.≤ x y ∨ Real.≤ y x
⊦ ∀l f. length (map f l) = length l
⊦ ∀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. (∀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
⊦ ∀p l. length (filter p l) ≤ length l
⊦ ∀p l. toSet (filter p l) ⊆ toSet l
⊦ ∀<< x. wellFounded << ⇒ ¬<< x x
⊦ ∅ = { x. x | F }
⊦ universe = { x. x | T }
⊦ ∀n. exp n 2 = n * n
⊦ ∀n. 2 * n = n + n
⊦ ∀s. size s = fold (λx n. suc n) s 0
⊦ ∀s. finite s ⇒ length (fromSet s) = size s
⊦ ∀s. ¬(s = ∅) ⇒ choice s ∈ s
⊦ ∀m. measure m = λx y. m x < m y
⊦ ∀f g. (f o g) = λx. f (g x)
⊦ ∀x l. member x l ⇔ x ∈ toSet l
⊦ ∀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. ¬(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. Real.< x y ⇔ ¬Real.≤ y x
⊦ ∀x y. Real.- x y = Real.+ x (Real.~ y)
⊦ ∀x y. Real./ x y = Real.* x (Real.inv y)
⊦ ∀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
⊦ ∀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 = ∅)
⊦ ∀P. (∀b. P b) ⇔ P T ∧ P F
⊦ ∀P. (∃b. P b) ⇔ P T ∨ P F
⊦ ∀P. P F ∧ P T ⇒ ∀x. P x
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀n. even n ⇔ n mod 2 = 0
⊦ ∀n. ¬(n = 0) ⇒ 0 div n = 0
⊦ ∀n. ¬(n = 0) ⇒ 0 mod n = 0
⊦ ∀l. ¬(l = []) ⇒ last l ∈ toSet l
⊦ ∀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
⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q
⊦ (isNone none ⇔ T) ∧ ∀a. isNone (some a) ⇔ F
⊦ (isSome none ⇔ F) ∧ ∀a. isSome (some a) ⇔ T
⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y
⊦ ∀a a'. some a = some a' ⇔ a = a'
⊦ ∀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. (B ⇒ A) ⇒ ¬A ⇒ ¬B
⊦ ∀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. m < m + n ⇔ 0 < n
⊦ ∀m n. n < m + n ⇔ 0 < m
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. Real.fromNatural m = Real.fromNatural n ⇔ m = n
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀m n. Real.≤ (Real.fromNatural m) (Real.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
⊦ ∀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 u. bigIntersect (insert s u) = s ∩ bigIntersect u
⊦ ∀s u. bigUnion (insert s u) = s ∪ bigUnion u
⊦ ∀f l. toSet (map f l) = image f (toSet l)
⊦ ∀f l. map f l = [] ⇔ l = []
⊦ ∀f s. image f s = ∅ ⇔ s = ∅
⊦ ∀f g. f ⊆ g ⇒ bigIntersect g ⊆ bigIntersect f
⊦ ∀f g. f ⊆ g ⇒ bigUnion f ⊆ bigUnion g
⊦ { m. m | m < 0 } = ∅
⊦ ∀n. finite { m. m | m < n }
⊦ ∀n. finite { m. m | m ≤ n }
⊦ ∀n. odd n ⇔ n mod 2 = 1
⊦ ∀n. exp 0 n = if n = 0 then 1 else 0
⊦ ∀n. ¬(n = 0) ⇒ n div n = 1
⊦ ∀x. Real.abs x = if Real.≤ 0 x then x else Real.~ x
⊦ ∀s. finite s ⇒ (size s = 0 ⇔ s = ∅)
⊦ ∀f. (id o f) = f ∧ (f o id) = f
⊦ ∀f g x. (f o g) x = f (g x)
⊦ ∀x s. disjoint s (insert x ∅) ⇔ ¬(x ∈ s)
⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)
⊦ ∀x s. disjoint (insert x ∅) s ⇔ ¬(x ∈ s)
⊦ ∀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. ¬(n = 0) ⇒ m mod n < n
⊦ ∀m n. ¬(n = 0) ⇒ m div n ≤ m
⊦ ∀m n. ¬(n = 0) ⇒ m mod n ≤ m
⊦ ∀m n.
Real.* (Real.fromNatural m) (Real.fromNatural n) =
Real.fromNatural (m * n)
⊦ ∀m n.
Real.+ (Real.fromNatural m) (Real.fromNatural n) =
Real.fromNatural (m + n)
⊦ ∀n. even n ⇔ ∃m. n = 2 * m
⊦ ∀m n. Real.max m n = if Real.≤ m n then n else m
⊦ ∀m n. Real.min m n = if Real.≤ m n then m else n
⊦ ∀l n. n < length l ⇒ member (nth n l) l
⊦ ∀l m. null (l @ m) ⇔ null l ∧ null m
⊦ ∀l m. length (l @ m) = length l + length m
⊦ ∀l m. reverse (l @ m) = reverse m @ reverse l
⊦ ∀l1 l2. toSet (l1 @ l2) = toSet l1 ∪ toSet 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 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
⊦ ∀<< m. wellFounded << ⇒ wellFounded (λx x'. << (m x) (m x'))
⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (p1, p2)
⊦ ∀P. (∃p. P p) ⇔ ∃p1 p2. P (p1, p2)
⊦ ∀P. (∀x y. P (x, y)) ⇒ ∀p. P p
⊦ finite ∅ ∧ ∀x s. finite s ⇒ finite (insert x s)
⊦ ∀a b. ∃f. f F = a ∧ f T = 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
⊦ ∀f g. f = g ⇔ ∀x. f x = g x
⊦ ∀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 ⇔ exists (λx. ¬P x) l
⊦ ∀P l. ¬exists P l ⇔ all (λx. ¬P x) l
⊦ ∀P. P none ∧ (∀a. P (some a)) ⇒ ∀x. P x
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ (even 0 ⇔ T) ∧ ∀n. even (suc n) ⇔ ¬even n
⊦ (odd 0 ⇔ F) ∧ ∀n. odd (suc n) ⇔ ¬odd n
⊦ ∀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 (exp m n) ⇔ odd m ∨ n = 0
⊦ ∀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. Real.≤ x y ∧ Real.≤ y x ⇔ x = y
⊦ ∀l i. i < length l ⇒ nth i l ∈ toSet l
⊦ ∀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
⊦ ∀PAIR'. ∃fn. ∀a0 a1. fn (a0, a1) = PAIR' a0 a1
⊦ (null [] ⇔ T) ∧ ∀h t. null (h :: t) ⇔ F
⊦ (∀s. ∅ ∩ s = ∅) ∧ ∀s. s ∩ ∅ = ∅
⊦ (∀s. universe ∩ s = s) ∧ ∀s. s ∩ universe = s
⊦ (∀s. ∅ ∪ s = s) ∧ ∀s. s ∪ ∅ = s
⊦ (∀s. universe ∪ s = universe) ∧ ∀s. s ∪ universe = universe
⊦ ∀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
⊦ ∀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
⊦ ∀P l. (∀x. member x l ⇒ P x) ⇔ all P l
⊦ ∀P l. (∃x. P x ∧ member x l) ⇔ exists P l
⊦ ∀<<. wellFounded << ⇔ ¬∃s. ∀n. << (s (suc n)) (s n)
⊦ ∀P. (∀x y. P x y) ⇔ ∀y x. P x y
⊦ ∀P. (∃x y. P x y) ⇔ ∃y x. P x y
⊦ bit0 0 = 0 ∧ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀h t. last (h :: t) = if t = [] then h else last t
⊦ ∀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 (exp m n) ⇔ even m ∧ ¬(n = 0)
⊦ ∀m n. ¬(m = 0) ⇒ m * n div m = n
⊦ ∀m n. ¬(m = 0) ⇒ m * n mod m = 0
⊦ ∀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
⊦ ∀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 = ∅
⊦ ∀x y z. x = y ∧ y = z ⇒ x = z
⊦ ∀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)
⊦ ∀t1 t2 t3. t1 ∧ t2 ∧ t3 ⇔ (t1 ∧ t2) ∧ t3
⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r
⊦ ∀t1 t2 t3. t1 ∨ t2 ∨ t3 ⇔ (t1 ∨ t2) ∨ t3
⊦ ∀p q r. p ∧ q ⇒ r ⇔ p ⇒ q ⇒ r
⊦ ∀p q r. p ∧ q ⇒ r ⇔ q ⇒ p ⇒ r
⊦ ∀n x i. i < n ⇒ nth i (replicate n x) = x
⊦ ∀x y n. x ≤ y ⇒ exp x n ≤ exp y n
⊦ ∀m n p. distance m p ≤ distance m n + distance n p
⊦ ∀m n p. m * (n * p) = m * n * p
⊦ ∀m n p. m + (n + p) = m + n + p
⊦ ∀m n p. exp m (n * p) = exp (exp m n) p
⊦ ∀m n p. m + n = m + p ⇔ n = p
⊦ ∀m n p. 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. m + p ≤ n + p ⇔ m ≤ n
⊦ ∀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. Real.≤ y z ⇒ Real.≤ (Real.+ x y) (Real.+ x z)
⊦ ∀x y z. Real.* x (Real.* y z) = Real.* (Real.* x y) z
⊦ ∀x y z. Real.+ x (Real.+ y z) = Real.+ (Real.+ x y) z
⊦ ∀x y z. Real.≤ x y ∧ Real.≤ y z ⇒ Real.≤ x z
⊦ ∀x. ¬(x = 0) ⇒ Real.* (Real.inv x) x = 1
⊦ ∀l m n. l @ m @ n = (l @ m) @ n
⊦ ∀l. ¬(l = []) ⇒ length (tail l) = length l - 1
⊦ ∀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. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ 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 l. map (g o f) l = map g (map f l)
⊦ ∀P x. (∀y. P y ⇔ y = x) ⇒ (select) P = x
⊦ ∀f g s. image (f o g) s = image f (image g s)
⊦ ∀P f l. all P (map f l) ⇔ all (P o f) l
⊦ ∀P f l. exists P (map f l) ⇔ exists (P o f) l
⊦ ∀f g h. (f o (g o h)) = (f o g o h)
⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)
⊦ ∀P. (∀x. ∃!y. P x y) ⇔ ∃!f. ∀x. P x (f x)
⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2
⊦ ∀b f g. (λx. if b then f x else g x) = if b then f else g
⊦ ∀n x. toSet (replicate n x) = if n = 0 then ∅ else insert x ∅
⊦ ∀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. 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
⊦ ∀l m. head (l @ m) = if l = [] then head m else head l
⊦ ∀p q. last (p @ q) = if q = [] then last p else last q
⊦ ∀l m. l @ m = [] ⇔ l = [] ∧ 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
⊦ ∀P. (∀a. P (left a)) ∧ (∀a. P (right a)) ⇒ ∀x. P x
⊦ length [] = 0 ∧ ∀h t. length (h :: t) = suc (length t)
⊦ ∀n. hasSize { m. m | m ≤ n } (n + 1)
⊦ ∀l. ¬(l = []) ⇒ last l = nth (length l - 1) l
⊦ ∀s t. disjoint s t ⇔ ¬∃x. x ∈ s ∧ x ∈ t
⊦ ∀s t. disjoint s (bigUnion t) ⇔ ∀x. x ∈ t ⇒ disjoint s x
⊦ ∀t f. t ⊆ bigIntersect f ⇔ ∀s. s ∈ f ⇒ t ⊆ s
⊦ ∀s. finite s ⇒ toSet (fromSet s) = s ∧ length (fromSet s) = size s
⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
⊦ ∀P l x. member x (filter P l) ⇔ P x ∧ member x l
⊦ ∀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
⊦ ∀P l. (∀x. all (P x) l) ⇔ all (λs. ∀x. P x s) l
⊦ ∀P l. (∃x. exists (P x) l) ⇔ exists (λs. ∃x. P x s) l
⊦ factorial 0 = 1 ∧ ∀n. factorial (suc n) = suc n * factorial n
⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀x s. x ∈ rest s ⇔ x ∈ s ∧ ¬(x = choice s)
⊦ ∀NONE' SOME'. ∃fn. fn none = NONE' ∧ ∀a. fn (some a) = SOME' 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)
⊦ ∀m n. ¬(n = 0) ⇒ (m div n = 0 ⇔ m < n)
⊦ ∀m n. ¬(n = 0) ⇒ m mod n mod n = m mod n
⊦ ∀n x. 0 < exp x n ⇔ ¬(x = 0) ∨ n = 0
⊦ ∀m n. exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)
⊦ ∀n l. n ≤ length l ⇒ length (drop n l) = length l - n
⊦ ∀n l. n ≤ length l ⇒ take n l @ drop n l = l
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ concat [] = [] ∧ ∀h t. concat (h :: t) = h @ concat t
⊦ toSet [] = ∅ ∧ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ (∀s t. s ⊆ s ∪ t) ∧ ∀s t. s ⊆ t ∪ s
⊦ (∀s t. s ∩ t ⊆ s) ∧ ∀s t. t ∩ s ⊆ s
⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s
⊦ ∀x l1 l2. member x (l1 @ l2) ⇔ member x l1 ∨ member x l2
⊦ ∀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 ∧ r ∨ q ∧ r
⊦ ∀p q r. p ∧ q ∨ r ⇔ (p ∨ r) ∧ (q ∨ r)
⊦ ∀m n i. i < n ⇒ nth i (interval m n) = m + i
⊦ ∀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. exp m (n + p) = exp m n * exp 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. exp (m * n) p = exp m p * exp n p
⊦ ∀n. size { m. m | m ≤ n } = n + 1
⊦ ∀x y z. Real.* x (Real.+ y z) = Real.+ (Real.* x y) (Real.* x z)
⊦ ∀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 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. (∀y. ∃x. f x = y) ⇔ ∃g. ∀y. f (g y) = y
⊦ ∀f. (∀m. ∃l. map f l = m) ⇔ ∀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. filter P (l1 @ l2) = filter P l1 @ filter P l2
⊦ ∀P f l. filter P (map f l) = map f (filter (P o f) l)
⊦ (∃!) = λP. (∃) P ∧ ∀x y. P x ∧ P y ⇒ x = y
⊦ ∀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. exp x n = 1 ⇔ x = 1 ∨ n = 0
⊦ ∀s P. { x. x | x ∈ s ∧ P x } ⊆ s
⊦ ∀f s. { x. f x | x ∈ s } = image f s
⊦ ∀P. (∃!x. P x) ⇔ ∃x. P x ∧ ∀y. P y ⇒ y = x
⊦ ∀P. (∀n. (∀m. m < n ⇒ P m) ⇒ P n) ⇒ ∀n. P n
⊦ ∀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 i 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 o h)
⊦ ∀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
⊦ ∀t. (λp. t p) = λ(x, y). t (x, y)
⊦ ∀e f. ∃fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
⊦ ∀e f. ∃fn. fn 0 = e ∧ ∀n. fn (suc n) = f n (fn 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. Real.≤ 0 x ∧ Real.≤ 0 y ⇒ Real.≤ 0 (Real.* 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 Q l. (∀x. P x ⇒ Q x) ⇒ all P l ⇒ all Q l
⊦ ∀P. (∃n. P n) ⇔ ∃n. P n ∧ ∀m. m < n ⇒ ¬P m
⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x
⊦ ∀<< <<<. (∀x y. << x y ⇒ <<< x y) ∧ wellFounded <<< ⇒ wellFounded <<
⊦ reverse [] = [] ∧ ∀x l. reverse (x :: l) = reverse l @ x :: []
⊦ ∀h t n. n < length t ⇒ nth (suc n) (h :: t) = nth n t
⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t
⊦ ∀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. exp x n = exp 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. exp x n ≤ exp y n ⇔ x ≤ y ∨ n = 0
⊦ ∀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 i (map f l) = f (nth i l)
⊦ ∀P. (∃n. P n) ⇔ P ((minimal) P) ∧ ∀m. m < (minimal) P ⇒ ¬P m
⊦ (∀n. 0 + n = n) ∧ ∀m n. suc m + n = suc (m + n)
⊦ ∀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 }
⊦ ∀f y l. member y (map f l) ⇔ ∃x. member x l ∧ y = f x
⊦ ∀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 P l ∧ all Q l ⇔ all (λx. P x ∧ Q x) 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 }
⊦ cond = λt t1 t2. select x. ((t ⇔ T) ⇒ x = t1) ∧ ((t ⇔ F) ⇒ x = t2)
⊦ ∀a b n. ¬(a = 0) ⇒ (n ≤ b div a ⇔ a * n ≤ b)
⊦ ∀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. exp x n < exp y n ⇔ x < y ∧ ¬(n = 0)
⊦ ∀x y n. x < y ∧ ¬(n = 0) ⇒ exp x n < exp 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
⊦ ∀m n. ¬(n = 0) ⇒ (m mod n = 0 ⇔ ∃q. m = q * n)
⊦ ∀s t. s ⊂ t ⇔ s ⊆ t ∧ ∃a. a ∈ t ∧ ¬(a ∈ s)
⊦ ∀INL' INR'. ∃fn. (∀a. fn (left a) = INL' a) ∧ ∀a. fn (right a) = INR' a
⊦ (∀x. replicate 0 x = []) ∧ ∀x n. replicate (suc n) x = x :: replicate n x
⊦ (∀n. 0 * n = 0) ∧ ∀m n. suc m * n = m * n + n
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2
⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = 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 }
⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∧ C ⇒ B ∧ D
⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∨ C ⇒ B ∨ D
⊦ ∀A B C D. (B ⇒ A) ∧ (C ⇒ D) ⇒ (A ⇒ C) ⇒ B ⇒ D
⊦ ∀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
⊦ ∀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)
⊦ ∀f s t. s ⊆ image f t ⇔ ∃u. u ⊆ t ∧ s = image f u
⊦ ∀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. bigIntersect s ∪ t = bigIntersect { x. x ∪ t | x ∈ s }
⊦ ∀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
⊦ ∀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 ≤ n + p ∧ n ≤ m + p
⊦ ∀n. (∃k m. odd m ∧ n = exp 2 k * m) ⇔ ¬(n = 0)
⊦ (∀m. interval m 0 = []) ∧
∀m n. interval m (suc n) = m :: interval (suc m) n
⊦ (∀m. exp m 0 = 1) ∧ ∀m n. exp m (suc n) = m * exp m n
⊦ ∀x s. finite s ⇒ size (delete s x) = if x ∈ s then size s - 1 else size s
⊦ ∀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 } = ∅
⊦ ∀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)
⊦ ∀NIL' CONS'.
∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)
⊦ ∀m n p. ¬(n = 0) ⇒ m * (p mod n) mod n = m * p mod n
⊦ ∀m n p. ¬(n = 0) ⇒ m mod n * p mod n = m * p mod n
⊦ ∀m n p. ¬(n = 0) ⇒ exp (m mod n) p mod n = exp m p mod 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. (∃q. m = n * q) ⇔ if n = 0 then m = 0 else m mod n = 0
⊦ ∀n l i. n ≤ length l ∧ i < n ⇒ nth i (take n l) = nth i l
⊦ ∀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 } = exp 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'
⊦ ∀<<. wellFounded << ⇔ ∀P. (∀x. (∀y. << y x ⇒ P y) ⇒ P x) ⇒ ∀x. P x
⊦ (∀m. m < 0 ⇔ F) ∧ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ (∀x. Real.exp x 0 = 1) ∧
∀x n. Real.exp x (suc n) = Real.* x (Real.exp x n)
⊦ (∀b f. case.none.some b f none = b) ∧
∀b f a. case.none.some b f (some a) = f a
⊦ ∀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
⊦ ∀s n. hasSize s n ⇒ hasSize { t. t | t ⊆ s } (exp 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
⊦ ∀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
⊦ ∀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) ∧ exists P l ⇒ exists Q l
⊦ ∀f l1 l2 n. length l1 = n ∧ length l2 = n ⇒ length (zipWith f l1 l2) = n
⊦ nubReverse [] = [] ∧
∀h t.
nubReverse (h :: t) =
if member h t then nubReverse t else h :: nubReverse t
⊦ (∀l. [] @ l = l) ∧ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀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 }
⊦ ∀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)
⊦ ∀<<. wellFounded << ⇔ ∀P. (∃x. P x) ⇔ ∃x. P x ∧ ∀y. << y x ⇒ ¬P y
⊦ ∀<<. wellFounded << ⇔ ∀P. (∃x. P x) ⇒ ∃x. P x ∧ ∀y. << y x ⇒ ¬P y
⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)
⊦ ∀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
⊦ ∀P. (∃B. ∀n. P n ≤ B) ⇔ ∃A B. ∀n. n * P n ≤ A * n + B
⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t
⊦ (∀f. image f ∅ = ∅) ∧
∀f x s. image f (insert x s) = insert (f x) (image f s)
⊦ (∀P. all P [] ⇔ T) ∧ ∀P h t. all P (h :: t) ⇔ P h ∧ all P t
⊦ (∀P. exists P [] ⇔ F) ∧ ∀P h t. exists P (h :: t) ⇔ P h ∨ exists P t
⊦ (∀h. last (h :: []) = h) ∧ ∀h k t. last (h :: k :: t) = last (k :: t)
⊦ ∀b A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ (if b then A else C) ⇒ if b then B else D
⊦ ∀m n p. ¬(n = 0) ⇒ m mod n * (p mod n) mod n = m * p mod n
⊦ ∀a b n. ¬(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
⊦ ∀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 o g)
⊦ ∀P f s. (∃t. t ⊆ image f s ∧ P t) ⇔ ∃t. t ⊆ s ∧ P (image f 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 }
⊦ ∀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 }
⊦ size ∅ = 0 ∧
∀x s.
finite s ⇒ size (insert x s) = if x ∈ s then size s else suc (size s)
⊦ (∀x. member x [] ⇔ F) ∧ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t
⊦ ∀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
⊦ (∀m. m ≤ 0 ⇔ m = 0) ∧ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n
⊦ ∀P a b. (a, b) ∈ { x y. x, y | P x y } ⇔ P a b
⊦ ∀P. { p. p | P p } = { a b. a, b | P (a, b) }
⊦ (∀b f. case.[].:: b f [] = b) ∧ ∀b f h t. case.[].:: b f (h :: t) = f h t
⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)
⊦ ∀l m.
length l = length m ∧ (∀i. i < length l ⇒ nth i l = nth i m) ⇒ l = m
⊦ ∀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'
⊦ (∀l. drop 0 l = l) ∧
∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t
⊦ ∀n l i.
n ≤ length l ∧ i < length l - n ⇒ nth i (drop n l) = nth (n + i) l
⊦ ∀n. (even n ⇒ ∃m. n = 2 * m) ∧ (¬even n ⇒ ∃m. n = suc (2 * m))
⊦ ∀s t. finite s ∧ finite t ⇒ finite { x y. x, y | x ∈ s ∧ y ∈ t }
⊦ ∀f s a.
(∀x. f x = f a ⇒ x = a) ⇒
image f (delete s a) = delete (image f s) (f a)
⊦ ∀P x y. P (distance x y) ⇔ ∀d. (x = y + d ⇒ P d) ∧ (y = x + d ⇒ P d)
⊦ ∀P. (∃x. P x) ∧ (∃M. ∀x. P x ⇒ x ≤ M) ⇔ ∃m. P m ∧ ∀x. P x ⇒ x ≤ m
⊦ ∀f. (∀l m. map f l = map f m ⇒ l = m) ⇔ ∀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
⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m div n = q ∧ m mod n = r
⊦ ∀a b c d. ¬(b = 0) ∧ b * c < (a + 1) * d ⇒ c div d ≤ a div b
⊦ (∀l. take 0 l = []) ∧
∀n h t. n ≤ length t ⇒ take (suc n) (h :: t) = h :: take n t
⊦ (∀P. filter P [] = []) ∧
∀P h t. filter P (h :: t) = if P h then h :: filter P t else filter P t
⊦ ∀P Q. (∃B. ∀i. P i ≤ Q i + B) ⇔ ∃B N. ∀i. N ≤ i ⇒ P i ≤ Q i + B
⊦ ∀P. (∀m n. P m n ⇔ P n m) ∧ (∀m n. m ≤ n ⇒ P m n) ⇒ ∀m n. P m n
⊦ ∀P.
P ∅ ∧ (∀x s. P s ∧ ¬(x ∈ s) ∧ finite s ⇒ P (insert x s)) ⇒
∀s. finite s ⇒ P s
⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)
⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)
⊦ ∀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 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
⊦ (∀h t. nth 0 (h :: t) = h) ∧
∀h t n. n < length t ⇒ nth (suc n) (h :: t) = nth n 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 }
⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)
⊦ ∀x m n.
exp x m = exp x n ⇔ if x = 0 then m = 0 ⇔ n = 0 else x = 1 ∨ m = n
⊦ ∀x m n.
exp x m ≤ exp 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 t. (∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ image f s = t
⊦ (∀a. { x. x | x = a } = insert a ∅) ∧ ∀a. { x. x | a = x } = insert a ∅
⊦ ∀k l m.
k < length l + length m ⇒
nth k (l @ m) = if k < length l then nth k l else nth (k - length l) m
⊦ ∀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.
(∀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
⊦ ∀x m n. exp x m < exp x n ⇔ 2 ≤ x ∧ m < n ∨ x = 0 ∧ ¬(m = 0) ∧ n = 0
⊦ ∀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
⊦ ∀R.
(∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
((∀m n. m < n ⇒ R m n) ⇔ ∀n. R n (suc n))
⊦ ∀R.
(∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (suc n)) ⇒
∀m n. m < n ⇒ R m n
⊦ ∀H.
(∀f g n. (∀m. m < n ⇒ f m = g m) ⇒ H f n = H g n) ⇒ ∃f. ∀n. f n = H f n
⊦ (∀s. hasSize s 0 ⇔ s = ∅) ∧
∀s n. hasSize s (suc n) ⇔ ∃a t. hasSize t n ∧ ¬(a ∈ t) ∧ s = insert a t
⊦ ∀a b c d. b * c < (a + 1) * d ∧ a * d < (c + 1) * b ⇒ a div b = c div d
⊦ ∀P.
(∀m. P m m) ∧ (∀m n. P m n ⇔ P n m) ∧ (∀m n. m < n ⇒ P m n) ⇒
∀m y. P m y
⊦ ∀d t.
{ f. f | (∀x. x ∈ ∅ ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ ∅) ⇒ f x = d } =
insert (λx. d) ∅
⊦ ∀m n p.
m * n = n * m ∧ m * n * p = m * (n * p) ∧ m * (n * p) = n * (m * p)
⊦ ∀m n p.
m + n = n + m ∧ m + n + p = m + (n + p) ∧ m + (n + p) = n + (m + p)
⊦ ∀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
⊦ ∀<< <<<.
wellFounded << ∧ wellFounded <<< ⇒
wellFounded (λ(x1, y1) (x2, y2). << x1 x2 ∧ <<< 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 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 }
⊦ ∀f s t.
(∀y. y ∈ t ⇒ ∃x. x ∈ s ∧ f x = y) ⇔
∃g. ∀y. y ∈ t ⇒ g y ∈ s ∧ f (g y) = y
⊦ (∀P. (∀x. x ∈ ∅ ⇒ P x) ⇔ T) ∧
∀P a s. (∀x. x ∈ insert a s ⇒ P x) ⇔ P a ∧ ∀x. x ∈ s ⇒ P x
⊦ (∀P. (∃x. x ∈ ∅ ∧ P x) ⇔ F) ∧
∀P a s. (∃x. x ∈ insert a s ∧ P x) ⇔ P a ∨ ∃x. x ∈ s ∧ P x
⊦ ∀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
⊦ ∀P.
(∃x. P x) ∧ (∃M. ∀x. P x ⇒ Real.≤ x M) ⇒
∃M. (∀x. P x ⇒ Real.≤ x M) ∧ ∀M'. (∀x. P x ⇒ Real.≤ x M') ⇒ Real.≤ M M'
⊦ ∀<<.
wellFounded << ⇒
∀H.
(∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
∃f. ∀x. f x = H f x
⊦ ∀<<.
wellFounded << ⇒
∀H.
(∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
∃!f. ∀x. f x = H f x
⊦ ∀<<.
(∀H.
(∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
∃f. ∀x. f x = H f x) ⇒ wellFounded <<
⊦ ∀P A B.
P 0 0 = 0 ∧ (∀m n. P m n ≤ A * (m + n) + B) ⇒
∃B. ∀m n. P m n ≤ B * (m + n)
⊦ (∀f. zipWith f [] [] = []) ∧
∀f h1 h2 t1 t2.
length t1 = length t2 ⇒
zipWith f (h1 :: t1) (h2 :: t2) = f h1 h2 :: zipWith f t1 t2
⊦ ∀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 }
⊦ ∀R S.
wellFounded R ∧ wellFounded S ⇒
wellFounded (λ(r1, s1) (r2, s2). R r1 r2 ∨ r1 = r2 ∧ S s1 s2)
⊦ ∀<<.
(∀x. ¬<< x x) ∧ (∀x y z. << x y ∧ << y z ⇒ << x z) ∧
(∀x. finite { y. y | << y x }) ⇒ wellFounded <<
⊦ ∀R.
(∀x. R x x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
((∀m n. m ≤ n ⇒ R m n) ⇔ ∀n. R n (suc n))
⊦ ∀R.
(∀x. R x x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (suc n)) ⇒
∀m n. m ≤ n ⇒ R m n
⊦ (∀n. 0 + n = n) ∧ (∀m. m + 0 = m) ∧ (∀m n. suc m + n = suc (m + n)) ∧
∀m n. m + suc n = suc (m + n)
⊦ ∀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)
⊦ ∀s.
¬(s = ∅) ∧ (∃m. ∀x. x ∈ s ⇒ Real.≤ x m) ⇒
(∀x. x ∈ s ⇒ Real.≤ x (Real.sup s)) ∧
∀m. (∀x. x ∈ s ⇒ Real.≤ x m) ⇒ Real.≤ (Real.sup s) m
⊦ ∀R S.
wellFounded R ∧ (∀a. wellFounded (S a)) ⇒
wellFounded (λ(r1, s1) (r2, s2). R r1 r2 ∨ r1 = r2 ∧ S r1 s1 s2)
⊦ ∀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 }
⊦ ∀d s t.
finite s ∧ finite t ⇒
size { f. f | (∀x. x ∈ s ⇒ f x ∈ t) ∧ ∀x. ¬(x ∈ s) ⇒ f x = d } =
exp (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
⊦ ∀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 }
(exp 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 ⇔ F) }
⊦ ∀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 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
⊦ ∀<<.
wellFounded << ⇒
∀H S.
(∀f g x.
(∀z. << 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
⊦ ∀<<.
wellFounded << ⇒
∀H.
(∀f g x. (∀z. << 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
⊦ ∀<<.
(∀H.
(∀f g x. (∀z. << 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 <<
⊦ ∀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 (insert x s) b =
if x ∈ s then fold f s b else f x (fold f s b)
⊦ (∀n. 0 * n = 0) ∧ (∀m. m * 0 = 0) ∧ (∀n. 1 * n = n) ∧ (∀m. m * 1 = m) ∧
(∀m n. suc m * n = m * n + n) ∧ ∀m n. m * suc n = m + 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 s b =
if x ∈ s then f x (fold f (delete s x) b) else fold f (delete s x) b
⊦ ∀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
⊦ (∀n. ¬(n = 0) ⇒ 0 < n) ∧ (∀n. ¬(n = 0) ⇒ 1 ≤ n) ∧
(∀n. 0 < n ⇒ ¬(n = 0)) ∧ (∀n. 0 < n ⇒ 1 ≤ n) ∧ (∀n. 1 ≤ n ⇒ 0 < n) ∧
∀n. 1 ≤ n ⇒ ¬(n = 0)
⊦ ∀s f g b.
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 s b = fold g s b
⊦ ∀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 })
⊦ (∀P f Q. (∀z. z ∈ { x. f x | P x } ⇒ Q z) ⇔ ∀x. P x ⇒ Q (f 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 ∈ { 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 ∈ { x. f x | P x } ∧ Q z) ⇔ ∃x. P x ∧ Q (f 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 ∈ { 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. bigIntersect { x. f x | P x } = { a. a | ∀x. P x ⇒ a ∈ f x }) ∧
(∀P f.
bigIntersect { x y. f x y | P x y } =
{ a. a | ∀x y. P x y ⇒ a ∈ f x y }) ∧
∀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. f x | P x } = { a. a | ∃x. P x ∧ a ∈ f x }) ∧
(∀P f.
bigUnion { x y. f x y | P x y } =
{ a. a | ∃x y. P x y ∧ a ∈ f x y }) ∧
∀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 }
⊦ let a d ← (λe. d e) = d ∈ a = λb. (λc. c) = λc. c
⊦ let a d ←
let e g ←
(let h ← d g ∈
λi.
(let j ← h ∈
λk. (λl. l j k) = λm. m ((λc. c) = λc. c) ((λc. c) = λc. c))
i ⇔ h) (d ((select) d)) ∈
e = (λf. (λc. c) = λc. c) ∈
a = λb. (λc. c) = λc. c
⊦ let a o ←
(let h ←
let p r ←
let p s ←
(let f ← o r = o s ∈
λg.
(let h ← f ∈
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔ f)
(r = s) ∈
p = (λq. (λd. d) = λd. d) ∈
p = (λq. (λd. d) = λd. d) ∈
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d))
(let t ←
let p u ←
let v y ← u = o y ∈
let b w ←
(let f ←
let p x ←
(let f ← v x ∈
λg.
(let h ← f ∈
λi.
(λj. j h i) =
λk.
k ((λd. d) = λd. d)
((λd. d) = λd. d)) g ⇔ f) w ∈
p = (λq. (λd. d) = λd. d) ∈
λg.
(let h ← f ∈
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔
f) w ∈
b = (λc. (λd. d) = λd. d) ∈
p = (λq. (λd. d) = λd. d) ∈
(let f ← t ∈
λg.
(let h ← f ∈
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d))
g ⇔ f) (let b d ← d ∈ b = λc. (λd. d) = λd. d)) ∈
let b e ←
(let f ←
let l n ←
(let f ← a n ∈
λg.
(let h ← f ∈
λi.
(λj. j h i) =
λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔ f) e ∈
l = (λm. (λd. d) = λd. d) ∈
λg.
(let h ← f ∈
λi. (λj. j h i) = λk. k ((λd. d) = λd. d) ((λd. d) = λd. d)) g ⇔
f) e ∈
b = λc. (λd. d) = λd. d