natural-bits

Entry	Value
Name	natural-bits_45
Conclusion	!m n. n <= bit_or m n
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	!e f. ?!fn. fn _0 = e /\ (!n. fn (SUC n) = f (fn n) n) !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2 !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2 !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2 !t. ~ ~t <=> t !t. (t <=> T) \/ (t <=> F) !f n. funpow f (SUC n) = funpow f n o f !f. funpow f 1 = f !f l i. i < LENGTH l ==> nth (MAP f l) i = f (nth l i) !p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y) !p. (?x. ~p x) <=> ~(!x. p x) !p. ~(!x. p x) <=> (?x. ~p x) !p. ~(?x. p x) <=> (!x. ~p x) !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) !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. n + m < p + m <=> 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. MAX n p <= m <=> n <= m /\ p <= m !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)) !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 i. i < n ==> nth (interval m n) i = m + i !m n. n <= bit_or m n !m n. m <= bit_or m n !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) !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 < 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. 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 !m n. LENGTH (interval m n) = n !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 !m n. m < n ==> m <= n !m n. m < n ==> m DIV n = 0 !k n. 1 < k ==> (?m. n <= k EXP m) !m n. m <= n \/ n <= m !n. EVEN (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. 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. 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 !m. m = 0 \/ (?n. m = SUC n) 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 !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. ?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 !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) !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. 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 i. i < LENGTH l ==> nth (MAP f l) i = f (nth l i) !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. 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) <=> (!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 !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) !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. n + m < p + m <=> 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. MAX n p <= m <=> n <= m /\ p <= m !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)) !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 i. i < n ==> nth (interval m n) i = m + i !m n. n <= m + n !m n. n <= bit_or m n !m n. m <= m + n !m n. m <= bit_or m n !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) !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 < n <=> m <= n /\ ~(m = n) !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 !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 !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 EXP SUC n = m * m EXP n !m n. MAX m n = (if m <= n then n else 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 !m n. m < n ==> m <= n !m n. m < n ==> m DIV n = 0 !k n. 1 < k ==> (?m. n <= k EXP m) !m n. m <= n \/ n <= m !n. EVEN (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. ~EVEN n <=> ODD n !n. ~ODD n <=> EVEN n !n. 0 < n <=> ~(n = 0) !m. m <= 0 <=> m = 0 !m. interval m 0 = [] !n. BIT0 (SUC n) = SUC (SUC (BIT0 n)) !n. BIT1 n = SUC (BIT0 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. 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. EVEN n \/ ODD n !m. m = 0 \/ (?n. m = SUC n) ?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) (~) = (\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)) NUMERAL = (\n. n) BIT0 0 = 0 LENGTH [] = 0
Contained Package	natural-bits
Comment	Probability package from OpenTheory.

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