Entry Value
Name natural-bits_127
Conclusion !n i. i < n ==> (?r. random_uniform n r = i)
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d) 
(\a. a = (\b. (\c. c) = (\c. c)))
(\d. (\e. e = (\f. (\c. c) = (\c. c)))
     (\g. (\h i.
               (\j k.
                    (\l. l j k) =
                    (\m. m ((\c. c) = (\c. c)) ((\c. c) = (\c. c))))
               h
               i <=>
               h)
          (d g)
          (d ((@) d)))) 
(\a. (\b. b = (\c. (\d. d) = (\d. d)))
     (\e. (\f g.
               (\h i.
                    (\j. j h i) =
                    (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
               f
               g <=>
               f)
          ((\l. l = (\m. (\d. d) = (\d. d)))
          (\n. (\f g.
                    (\h i.
                         (\j. j h i) =
                         (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
                    f
                    g <=>
                    f)
               (a n)
               e))
          e))
(\p. (\h i. (\j. j h i) = (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
     ((\q. q = (\r. (\d. d) = (\d. d)))
     (\s. (\q. q = (\r. (\d. d) = (\d. d)))
          (\t. (\f g.
                    (\h i.
                         (\j. j h i) =
                         (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
                    f
                    g <=>
                    f)
               (p s = p t)
               (s = t))))
     ((\u. (\f g.
                (\h i.
                     (\j. j h i) =
                     (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
                f
                g <=>
                f)
           u
           ((\b. b = (\c. (\d. d) = (\d. d))) (\d. d)))
     ((\q. q = (\r. (\d. d) = (\d. d)))
     (\v. (\w. (\b. b = (\c. (\d. d) = (\d. d)))
               (\x. (\f g.
                         (\h i.
                              (\j. j h i) =
                              (\k. k ((\d. d) = (\d. d)) ((\d. d) = (\d. d))))
                         f
                         g <=>
                         f)
                    ((\q. q = (\r. (\d. d) = (\d. d)))
                    (\y. (\f g.
                              (\h i.
                                   (\j. j h i) =
                                   (\k. k ((\d. d) = (\d. d))
                                        ((\d. d) = (\d. d))))
                              f
                              g <=>
                              f)
                         (w y)
                         x))
                    x))
          (\z. v = p z)))))
Classical Lemmas
  • !a b a' b'. a,b = a',b' <=> a = a' /\ b = b'
  • !a b. FST (a,b) = a
  • !a b. SND (a,b) = b
  • !e f. ?!fn. fn _0 = e /\ (!n. fn (SUC n) = f (fn n) n)
  • !a n. snth (sreplicate a) n = a
  • !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2
  • !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2
  • !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2
  • !t1 t2. ~t1 ==> ~t2 <=> t2 ==> t1
  • !b. ?r. random_bit r <=> b
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f n x. funpow f (SUC n) x = funpow f n (f x)
  • !f m n. funpow f (m + n) = funpow f m o funpow f n
  • !f n. funpow f (SUC n) = funpow f n o f
  • !f. funpow f 1 = f
  • !p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)
  • !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. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !f. ?fn. !a b. fn (a,b) = f a b
  • !p n. p n /\ (!m. m < n ==> ~p m) ==> (minimal) p = n
  • !p. (?n. p n) <=> (?n. p n /\ (!m. m < n ==> ~p m))
  • !p. (?n. p n) <=> p ((minimal) p) /\ (!m. m < (minimal) p ==> ~p m)
  • !p. (!n. (!m. m < n ==> p m) ==> p n) ==> (!n. p n)
  • !k p. 1 < k /\ p 0 /\ (!n. ~(n = 0) /\ p (n DIV k) ==> p n) ==> (!n. p n)
  • !n l. LENGTH l = n ==> (?r. random_bits n r = l)
  • !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
  • !m n p q. m = n + q * p ==> m MOD p = n MOD p
  • !m n p. m * p < n * p <=> m < n /\ ~(p = 0)
  • !m n p. m * n < m * p <=> ~(m = 0) /\ n < p
  • !m n p. m + n < m + p <=> n < p
  • !x m n. x EXP m < x EXP n <=> 2 <= x /\ m < n \/ x = 0 /\ ~(m = 0) /\ n = 0
  • !m n p. m * p <= n * p <=> m <= n \/ p = 0
  • !m n p. m * n <= m * p <=> m = 0 \/ n <= p
  • !m n p. n + m <= p + m <=> n <= p
  • !m n p. m + n <= m + p <=> n <= p
  • !x m n. x EXP m <= x EXP n <=> (if x = 0 then m = 0 ==> n = 0 else x = 1 \/ m <= n)
  • !m n p. m * p = n * p <=> m = n \/ p = 0
  • !m n p. m * n = m * p <=> m = 0 \/ n = p
  • !p m n. m + p = n + p <=> m = n
  • !m n p. m + n = m + p <=> 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 + m * p
  • !m n p. (m + n) * p = m * p + n * p
  • !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)
  • !a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b)
  • !n a b. ~(n = 0) ==> (a MOD n + b MOD n) MOD n = (a + b) MOD n
  • !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. m < n /\ n <= p ==> m < p
  • !k n m. 1 < k /\ ~(n = 0) ==> (log k n = m <=> k EXP m <= n /\ n < k EXP (m + 1))
  • !k n1 n2. 1 < k /\ ~(n1 = 0) /\ n1 <= n2 ==> log k n1 <= log k n2
  • !k m n. 1 < k /\ k EXP m <= n ==> m <= log k n
  • !m n p. m <= n /\ n < p ==> m < p
  • !m n p. m <= n /\ n <= p ==> m <= p
  • !k n m. k EXP m <= n /\ n < k EXP (m + 1) ==> log k n = m
  • !n k. bit_width (bit_bound n k) <= k
  • !m n. n * m DIV n <= m
  • !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. ODD (m + n) <=> ~(ODD m <=> ODD n)
  • !n1 n2. bit_hd (n1 * n2) <=> bit_hd n1 /\ bit_hd n2
  • !m n. ~(m < n) <=> n <= m
  • !m n. ~(m <= n) <=> n < m
  • !m n. m <= n /\ n <= m <=> m = n
  • !m n. m < n <=> (?d. n = m + SUC d)
  • !m n. m < m + n <=> 0 < n
  • !n x. 0 < x EXP n <=> ~(x = 0) \/ n = 0
  • !m n. SUC m < SUC n <=> m < n
  • !m n. m <= n <=> (?d. n = m + d)
  • !m n. m <= n <=> m < n \/ m = n
  • !m n. SUC m <= n <=> m < n
  • !m n. SUC m <= SUC n <=> m <= n
  • !m n. SUC m = SUC n <=> m = n
  • !m n. m * n = 0 <=> m = 0 \/ n = 0
  • !m n. m * n = 1 <=> m = 1 /\ n = 1
  • !m n. m + n = n <=> m = 0
  • !m n. m + n = m <=> n = 0
  • !m n. m + n = 0 <=> m = 0 /\ n = 0
  • !m n. m EXP n = 0 <=> m = 0 /\ ~(n = 0)
  • !x n. x EXP n = 1 <=> x = 1 \/ n = 0
  • !n i. bit_nth n (SUC i) <=> bit_nth (bit_tl n) i
  • !m n. LENGTH (interval m n) = n
  • !n k. bits_to_num (num_to_bitvec n k) = bit_bound n k
  • !m n. m * n = n * m
  • !m n. ~(n = 0) ==> m MOD n < n
  • !m n. ~(n = 0) ==> (m DIV n = 0 <=> m < n)
  • !m n. ~(n = 0) ==> (m MOD n = 0 <=> (?q. m = q * n))
  • !m n. ~(n = 0) ==> m DIV n * n + m MOD n = m
  • !m n. ~(m = 0) ==> (m * n) DIV m = n
  • !m n. ~(m = 0) ==> (m * n) MOD m = 0
  • !n m. ~(n = 0) ==> m MOD n MOD n = m MOD n
  • !k n. 1 < k /\ ~(n = 0) ==> n < k EXP (log k n + 1)
  • !k n. 1 < k /\ ~(n = 0) ==> k EXP log k n <= n
  • !k n. 1 < k /\ ~(n = 0) ==> (log k n = 0 <=> n < k)
  • !k n. 1 < k /\ ~(n = 0) ==> log k n = (if n < k then 0 else log k (n DIV k) + 1)
  • !k n. 1 < k /\ ~(n = 0) /\ n < k ==> log k n = 0
  • !n i. i < n ==> (?r. random_uniform n r = i)
  • !m n. m < n ==> m DIV n = 0
  • !k n. 1 < k ==> (?m. n <= k EXP m)
  • !n i. bit_nth n i ==> i < bit_width n
  • !m n. m <= n \/ n <= m
  • !n. EVEN (2 * n)
  • !n. ODD (SUC (2 * n))
  • !n. ~(n < n)
  • !n. ~(SUC n = _0)
  • !n. n <= SUC n
  • !n. EVEN n <=> (?m. n = 2 * m)
  • !n. EVEN n <=> n MOD 2 = 0
  • !n. ODD n <=> n MOD 2 = 1
  • !n. 0 < n <=> ~(n = 0)
  • !n. bit_nth (bit_tl n) = bit_nth n o SUC
  • !n. BIT0 (SUC n) = SUC (SUC (BIT0 n))
  • !m. SUC m = m + 1
  • !n. bits_to_num (num_to_bits n) = n
  • !m. m * 1 = m
  • !n. 2 * n = n + n
  • !m. 1 * m = m
  • !n. SUC n - 1 = n
  • !n. n DIV 1 = n
  • !n. n EXP 1 = n
  • !n. 0 EXP n = (if n = 0 then 1 else 0)
  • !n. 1 EXP n = 1
  • !n. ~(n = 0) ==> 0 DIV n = 0
  • !k. 1 < k ==> log k 1 = 0
  • !m. m = 0 \/ (?n. m = SUC n)
  • !r1 r2. ?r. split_random r = r1,r2
  • !s1 s2. ssplit (sinterleave s1 s2) = s1,s2
  • BIT0 0 = 0
    		•
    Constructive Lemmas
  • T
  • !x y z. x = y /\ y = z ==> x = z
  • !t1 t2. (if F then t1 else t2) = t2
  • !t1 t2. (if T then t1 else t2) = t1
  • !x y. x = y <=> y = x
  • !x y. x = y ==> y = x
  • !a b a' b'. a,b = a',b' <=> a = a' /\ b = b'
  • !a b. FST (a,b) = a
  • !a b. SND (a,b) = b
  • !e f. ?!fn. fn _0 = e /\ (!n. fn (SUC n) = f (fn n) n)
  • !h t n. n < LENGTH t ==> nth (CONS h t) (SUC n) = nth t n
  • !h t. HD (CONS h t) = h
  • !h t. nth (CONS h t) 0 = h
  • !h t. TL (CONS h t) = t
  • !h t. LENGTH (CONS h t) = SUC (LENGTH t)
  • !a n. snth (sreplicate a) n = a
  • !a. ?x. x = a
  • !a. ?!x. x = a
  • !x. x = x
  • !x. (@y. y = x) = x
  • !x. I x = x
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 /\ q1 ==> p2 /\ q2
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 \/ q1 ==> p2 \/ q2
  • !t1 t2 t3. (t1 /\ t2) /\ t3 <=> t1 /\ t2 /\ t3
  • !t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3
  • !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2
  • !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2
  • !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2
  • !t1 t2. t1 /\ t2 <=> t2 /\ t1
  • !t1 t2. ~t1 ==> ~t2 <=> t2 ==> t1
  • !t1 t2. t1 \/ t2 <=> t2 \/ t1
  • !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)
  • !b. ?r. random_bit r <=> b
  • !t. (!x. t) <=> t
  • !t. (?x. t) <=> t
  • !t. ~ ~t <=> t
  • !t. F /\ t <=> F
  • !t. T /\ t <=> t
  • !t. t /\ F <=> F
  • !t. t /\ T <=> t
  • !t. t /\ t <=> t
  • !t. (F <=> t) <=> ~t
  • !t. (T <=> t) <=> t
  • !t. (t <=> F) <=> ~t
  • !t. (t <=> T) <=> t
  • !t. F ==> t <=> T
  • !t. T ==> t <=> t
  • !t. t ==> F <=> ~t
  • !t. t ==> T <=> T
  • !t. F \/ t <=> t
  • !t. T \/ t <=> T
  • !t. t \/ F <=> t
  • !t. t \/ T <=> T
  • !t. t \/ t <=> t
  • !t. t ==> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f n x. funpow f (SUC n) x = funpow f n (f x)
  • !f m n. funpow f (m + n) = funpow f m o funpow f n
  • !f n. funpow f (SUC n) = f o funpow f n
  • !f n. funpow f (SUC n) = funpow f n o f
  • !f. funpow f 0 = I
  • !f. funpow f 1 = f
  • !f h t. MAP f (CONS h t) = CONS (f h) (MAP f t)
  • !f y. (\x. f x) y = f y
  • !f g. (!x. f x = g x) <=> f = g
  • !f g. (!x. f x = g x) ==> f = g
  • !f l. LENGTH (MAP f l) = LENGTH l
  • !f. ONE_ONE f <=> (!x1 x2. f x1 = f x2 ==> x1 = x2)
  • !f. ONTO f <=> (!y. ?x. y = f x)
  • !f. f o I = f
  • !f. I o f = f
  • !t. (\x. t x) = t
  • !f. MAP f [] = []
  • !p a. (?x. a = x /\ p x) <=> p a
  • !p x. (!y. p y <=> y = x) ==> (@) p = x
  • !p x. p x ==> p ((@) p)
  • !p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)
  • !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) <=> (!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)
  • !p q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x)
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. (?!x. p x) <=> (?x. !y. p y <=> x = y)
  • !p. (?!x. p x) <=> (?x. p x) /\ (!x x'. p x /\ p x' ==> x = x')
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !f b h t. foldr f b (CONS h t) = f h (foldr f b t)
  • !f b. foldr f b [] = b
  • !f. ?fn. !a b. fn (a,b) = f a b
  • !p. (!x y. p x y) <=> (!y x. p x y)
  • !r. (!x. ?y. r x y) <=> (?f. !x. r x (f x))
  • !p. (!x. ?!y. p x y) <=> (?f. !x y. p x y <=> f x = y)
  • !f g x. (f o g) x = f (g x)
  • !f g l. MAP (f o g) l = MAP f (MAP g l)
  • !f g h. (f o g) o h = f o g o h
  • !p. p [] /\ (!h t. p t ==> p (CONS h t)) ==> (!l. p l)
  • !p n. p n /\ (!m. m < n ==> ~p m) ==> (minimal) p = n
  • !p. (?n. p n) <=> (?n. p n /\ (!m. m < n ==> ~p m))
  • !p. (?n. p n) <=> p ((minimal) p) /\ (!m. m < (minimal) p ==> ~p m)
  • !p. (!n. (!m. m < n ==> p m) ==> p n) ==> (!n. p n)
  • !p. p _0 /\ (!n. p n ==> p (SUC n)) ==> (!n. p n)
  • !k p. 1 < k /\ p 0 /\ (!n. ~(n = 0) /\ p (n DIV k) ==> p n) ==> (!n. p n)
  • !n l. LENGTH l = n ==> (?r. random_bits n r = l)
  • !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
  • !m n p q. m = n + q * p ==> m MOD p = n MOD p
  • !m n p. m * p < n * p <=> m < n /\ ~(p = 0)
  • !m n p. m * n < m * p <=> ~(m = 0) /\ n < p
  • !m n p. m + n < m + p <=> n < p
  • !x m n. x EXP m < x EXP n <=> 2 <= x /\ m < n \/ x = 0 /\ ~(m = 0) /\ n = 0
  • !m n p. m * p <= n * p <=> m <= n \/ p = 0
  • !m n p. m * n <= m * p <=> m = 0 \/ n <= p
  • !m n p. n + m <= p + m <=> n <= p
  • !m n p. m + n <= m + p <=> n <= p
  • !x m n. x EXP m <= x EXP n <=> (if x = 0 then m = 0 ==> n = 0 else x = 1 \/ m <= n)
  • !m n p. m * p = n * p <=> m = n \/ p = 0
  • !m n p. m * n = m * p <=> m = 0 \/ n = p
  • !p m n. m + p = n + p <=> m = n
  • !m n p. m + n = m + p <=> 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 + m * p
  • !m n p. (m + n) * p = m * p + n * p
  • !m n p. m + n + p = (m + n) + p
  • !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)
  • !a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b)
  • !n a b. ~(n = 0) ==> (a MOD n + b MOD n) MOD n = (a + b) MOD n
  • !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. m < n /\ n <= p ==> m < p
  • !k n m. 1 < k /\ ~(n = 0) ==> (log k n = m <=> k EXP m <= n /\ n < k EXP (m + 1))
  • !k n1 n2. 1 < k /\ ~(n1 = 0) /\ n1 <= n2 ==> log k n1 <= log k n2
  • !k m n. 1 < k /\ k EXP m <= n ==> m <= log k n
  • !m n p. m <= n /\ n < p ==> m < p
  • !m n p. m <= n /\ n <= p ==> m <= p
  • !k n m. k EXP m <= n /\ n < k EXP (m + 1) ==> log k n = m
  • !m n. m <= m + n
  • !n k. bit_width (bit_bound n k) <= k
  • !m n. n * m DIV n <= m
  • !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. ODD (m + n) <=> ~(ODD m <=> ODD n)
  • !n1 n2. bit_hd (n1 * n2) <=> bit_hd n1 /\ bit_hd n2
  • !m n. ~(m < n) <=> n <= m
  • !m n. ~(m <= n) <=> n < m
  • !m n. m <= n /\ n <= m <=> m = n
  • !m n. m < n <=> (?d. n = m + SUC d)
  • !m n. m < SUC n <=> m <= n
  • !m n. m < SUC n <=> m = n \/ m < n
  • !m n. m < m + n <=> 0 < n
  • !n x. 0 < x EXP n <=> ~(x = 0) \/ n = 0
  • !m n. SUC m < SUC n <=> m < n
  • !m n. m <= n <=> (?d. n = m + d)
  • !m n. m <= n <=> m < n \/ m = n
  • !m n. m <= SUC n <=> m = SUC n \/ m <= n
  • !m n. SUC m <= n <=> m < n
  • !m n. SUC m <= SUC n <=> m <= n
  • !m n. SUC m = SUC n <=> m = n
  • !m n. m * n = 0 <=> m = 0 \/ n = 0
  • !m n. m * n = 1 <=> m = 1 /\ n = 1
  • !m n. m + n = n <=> m = 0
  • !m n. m + n = m <=> n = 0
  • !m n. m + n = 0 <=> m = 0 /\ n = 0
  • !m n. m EXP n = 0 <=> m = 0 /\ ~(n = 0)
  • !x n. x EXP n = 1 <=> x = 1 \/ n = 0
  • !n i. bit_nth n i <=> bit_hd (bit_shr n i)
  • !n i. bit_nth n (SUC i) <=> bit_nth (bit_tl n) i
  • !m n. MAP SUC (interval m n) = interval (SUC m) n
  • !m n. interval m (SUC n) = CONS m (interval (SUC m) n)
  • !m n. LENGTH (interval m n) = n
  • !n k. bits_to_num (num_to_bitvec n k) = bit_bound n k
  • !m n. m * n = n * m
  • !m n. m * SUC n = m + m * n
  • !m n. SUC m * n = m * n + n
  • !m n. m + n = n + m
  • !m n. m + SUC n = SUC (m + n)
  • !m n. SUC m + n = SUC (m + n)
  • !m n. (m + n) - n = m
  • !m n. m EXP SUC n = m * m EXP n
  • !m n. ~(n = 0) ==> m MOD n < n
  • !m n. ~(n = 0) ==> (m DIV n = 0 <=> m < n)
  • !m n. ~(n = 0) ==> (m MOD n = 0 <=> (?q. m = q * n))
  • !m n. ~(n = 0) ==> m DIV n * n + m MOD n = m
  • !m n. ~(m = 0) ==> (m * n) DIV m = n
  • !m n. ~(m = 0) ==> (m * n) MOD m = 0
  • !n m. ~(n = 0) ==> m MOD n MOD n = m MOD n
  • !k n. 1 < k /\ ~(n = 0) ==> n < k EXP (log k n + 1)
  • !k n. 1 < k /\ ~(n = 0) ==> k EXP log k n <= n
  • !k n. 1 < k /\ ~(n = 0) ==> (log k n = 0 <=> n < k)
  • !k n. 1 < k /\ ~(n = 0) ==> log k n = (if n < k then 0 else log k (n DIV k) + 1)
  • !k n. 1 < k /\ ~(n = 0) /\ n < k ==> log k n = 0
  • !n i. i < n ==> (?r. random_uniform n r = i)
  • !m n. m < n ==> m DIV n = 0
  • !k n. 1 < k ==> (?m. n <= k EXP m)
  • !n i. bit_nth n i ==> i < bit_width n
  • !m n. m <= n \/ n <= m
  • !n. EVEN (2 * n)
  • !n. ODD (SUC (2 * n))
  • !n. ~(n < n)
  • !m. ~(m < 0)
  • !n. ~(SUC n = _0)
  • !n. n < SUC n
  • !n. 0 < SUC n
  • !n. n <= n
  • !n. n <= SUC n
  • !n. 0 <= n
  • !n. EVEN n <=> (?m. n = 2 * m)
  • !n. EVEN n <=> n MOD 2 = 0
  • !n. EVEN (SUC n) <=> ~EVEN n
  • !n. ODD n <=> n MOD 2 = 1
  • !n. ODD (SUC n) <=> ~ODD n
  • !n. bit_hd n <=> ODD n
  • !n. bit_hd (SUC n) <=> ~bit_hd n
  • !n. ~EVEN n <=> ODD n
  • !n. ~ODD n <=> EVEN n
  • !n. 0 < n <=> ~(n = 0)
  • !m. m <= 0 <=> m = 0
  • !n. bit_nth (bit_tl n) = bit_nth n o SUC
  • !n. num_to_bits n = num_to_bitvec n (bit_width n)
  • !m. interval m 0 = []
  • !n. BIT0 (SUC n) = SUC (SUC (BIT0 n))
  • !n. BIT1 n = SUC (BIT0 n)
  • !n. PRE (SUC n) = n
  • !m. SUC m = m + 1
  • !n. bits_to_num (num_to_bits n) = n
  • !m. m * 0 = 0
  • !m. m * 1 = m
  • !n. 0 * n = 0
  • !n. 2 * n = n + n
  • !m. 1 * m = m
  • !m. m + 0 = m
  • !n. 0 + n = n
  • !n. SUC n - 1 = n
  • !n. n DIV 1 = n
  • !m. m EXP 0 = 1
  • !n. n EXP 1 = n
  • !n. 0 EXP n = (if n = 0 then 1 else 0)
  • !n. 1 EXP n = 1
  • !n. ~(n = 0) ==> 0 DIV n = 0
  • !k. 1 < k ==> log k 1 = 0
  • !n. EVEN n \/ ODD n
  • !m. m = 0 \/ (?n. m = SUC n)
  • !x. ?a b. x = a,b
  • !r1 r2. ?r. split_random r = r1,r2
  • !s1 s2. ssplit (sinterleave s1 s2) = s1,s2
  • !s. shd s = snth s 0
  • !s. stream (snth s) = s
  • !s. dest_random (mk_random s) = s
  • ?f. ONE_ONE f /\ ~ONTO f
  • EVEN 0
  • ~ODD 0
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • I = (\x. x)
  • sreplicate = siterate I
  • (~) = (\p. p ==> F)
  • COND = (\t t1 t2. @x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2))
  • (/\) = (\p q. (\f. f p q) = (\f. f T T))
  • (==>) = (\p q. p /\ q <=> p)
  • (\/) = (\p q. !r. (p ==> r) ==> (q ==> r) ==> r)
  • (!) = (\p. p = (\x. T))
  • (?) = (\p. !q. (!x. p x ==> q) ==> q)
  • (?) = (\p. p ((@) p))
  • (?!) = (\p. (?) p /\ (!x y. p x /\ p y ==> x = y))
  • (o) = (\f g x. f (g x))
  • random_bits = random_vector random_bit
  • NUMERAL = (\n. n)
  • BIT0 0 = 0
  • LENGTH [] = 0
    		•
    Contained Package natural-bits
    Comment Probability package from OpenTheory.
    Back to main package pageBack to contained package page