| Entry |
Value |
|
Name |
natural-divides_21 |
|
Conclusion |
!a b. divides (FST (egcd a b)) b |
|
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)!t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2!t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2!t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2!p q. (!x. p \/ q x) <=> p \/ (!x. q x)!p q. p \/ (!x. q x) <=> (!x. p \/ q x)!t. ~ ~t <=> t!t. (t <=> T) \/ (t <=> 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)!r. WF r <=> (!p. (!x. (!y. r y x ==> p y) ==> p x) ==> (!x. p x))!f. ?fn. !a b. fn (a,b) = f a b!m. WF (MEASURE m)!r m. WF r ==> WF (\x y. r (m x) (m y))!p. (!n. (!m. m < n ==> p m) ==> p n) ==> (!n. p n)!p. (!x. p x) <=> (!a b. p (a,b))!p. (?x. p x) <=> (?a b. p (a,b))!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. 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!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 /\ n < p ==> m < p!m n p. p <= n ==> m * (n - p) = m * n - m * p!m n p. n <= m ==> (m - n) * p = m * p - n * p!m n. ~(m <= n) <=> n < m!m n. SUC m < SUC n <=> m < n!m n. m <= n <=> (?d. n = m + d)!m n. SUC m <= n <=> m < n!m n. SUC m <= SUC n <=> m <= n!m n. m = n * m <=> m = 0 \/ n = 1!m n. m = m * n <=> m = 0 \/ n = 1!m n. SUC m = SUC n <=> m = n!m n. n * m = m <=> m = 0 \/ n = 1!m n. m * n = 0 <=> m = 0 \/ n = 0!m n. m * n = 1 <=> m = 1 /\ n = 1!m n. m + n = m <=> n = 0!m n. m + n = 0 <=> m = 0 /\ n = 0!m n. m * n = n * m!m n. ~(n = 0) ==> m MOD n < 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 <= n \/ n <= m!a b. divides (FST (egcd a b)) b!n. ~(SUC n = _0)!n. 0 < n <=> ~(n = 0)!n. BIT0 (SUC n) = SUC (SUC (BIT0 n))!m. m * 1 = m!m. 1 * m = m!x. FST x,SND x = xWF (<)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)!a. ?x. x = a!a. ?!x. x = a!x. x = x!x. (@y. y = 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!p q r. p ==> q ==> r <=> p /\ q ==> r!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. (?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)!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 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. ONE_ONE f <=> (!x1 x2. f x1 = f x2 ==> x1 = x2)!f. ONTO f <=> (!y. ?x. y = f x)!t. (\x. t x) = t!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)!r. WF r <=> (!p. (!x. (!y. r y x ==> p y) ==> p x) ==> (!x. p x))!r. WF r <=> (!p. (?x. p x) ==> (?x. p x /\ (!y. r y x ==> ~p y)))!f. ?fn. !a b. fn (a,b) = f a b!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)!m x y. MEASURE m x y <=> m x < m y!m. WF (MEASURE m)!r m. WF r ==> WF (\x y. r (m x) (m y))!p. (!n. (!m. m < n ==> p m) ==> p n) ==> (!n. p n)!p. p _0 /\ (!n. p n ==> p (SUC n)) ==> (!n. p n)!p. (!x. p x) <=> (!a b. p (a,b))!p. (?x. p x) <=> (?a b. p (a,b))!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. 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!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!m n p. m < n /\ n < p ==> m < p!m n p. p <= n ==> m * (n - p) = m * n - m * p!m n p. n <= m ==> (m - n) * p = m * p - n * p!m n. m <= m + n!m n. ~(m <= n) <=> n < m!m n. m < SUC n <=> m <= n!m n. m < SUC n <=> m = n \/ m < n!m n. SUC m < SUC n <=> m < n!m n. m <= n <=> (?d. n = m + d)!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. m = n * m <=> m = 0 \/ n = 1!m n. m = m * n <=> m = 0 \/ n = 1!m n. SUC m = SUC n <=> m = n!m n. n * m = m <=> m = 0 \/ n = 1!m n. m * n = 0 <=> m = 0 \/ n = 0!m n. m * n = 1 <=> m = 1 /\ n = 1!m n. m + n = m <=> n = 0!m n. m + n = 0 <=> m = 0 /\ n = 0!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 + n) - m = n!m n. ~(n = 0) ==> m MOD n < 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 <= n \/ n <= m!a b. divides (FST (egcd a b)) b!a b. divides (gcd a b) b!a b. divides (gcd a b) a!m. ~(m < 0)!n. ~(SUC n = _0)!n. 0 < SUC n!n. n <= n!n. 0 <= n!n. 0 < n <=> ~(n = 0)!m. m <= 0 <=> m = 0!n. BIT0 (SUC n) = SUC (SUC (BIT0 n))!n. BIT1 n = SUC (BIT0 n)!n. PRE (SUC n) = n!m. m * 0 = 0!m. m * 1 = m!n. 0 * n = 0!m. 1 * m = m!m. m + 0 = m!n. 0 + n = n!n. n - n = 0!x. ?a b. x = a,b!x. FST x,SND x = x?f. ONE_ONE f /\ ~ONTO fWF (<)F <=> (!p. p)T <=> (\p. p) = (\p. p)~F <=> T~T <=> F(~) = (\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))NUMERAL = (\n. n)BIT0 0 = 0 |
|
Contained Package |
natural-divides |
|
Comment |
Natural-divides package from OpenTheory. |
