Entry Value
Name natural-divides_57
Conclusion !a b. a <= b ==> gcd a (b - a) = gcd a 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
  • !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)
  • !p. (?x. ~p x) <=> ~(!x. p x)
  • !p. ~(!x. p x) <=> (?x. ~p x)
  • !p. ~(?x. p x) <=> (!x. ~p x)
  • !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 * n = m * p <=> m = 0 \/ n = p
  • !m n p. m + n = m + p <=> n = 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. 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 <=> (?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 = 0 <=> m = 0 /\ n = 0
  • !m n. m * n = n * m
  • !m n. n <= m ==> m - n + n = m
  • !a b. a <= b ==> gcd a (b - a) = gcd a b
  • !n. ~(SUC n = _0)
  • !n. BIT0 (SUC n) = SUC (SUC (BIT0 n))
  • !m. m * 1 = m
  • !m. 1 * m = m
  • BIT0 0 = 0
    		•
    Constructive Lemmas
  • T
  • !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)
  • !a. ?x. x = a
  • !a. ?!x. x = a
  • !x. 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
  • !a b. (a <=> b) ==> a ==> b
  • !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 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. p x ==> p ((@) p)
  • !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 ==> 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. (!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)
  • !p. p _0 /\ (!n. p n ==> p (SUC n)) ==> (!n. p n)
  • !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 * n = m * p <=> m = 0 \/ n = p
  • !m n p. m + n = m + p <=> n = 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. p <= n ==> m * (n - p) = m * n - m * p
  • !m n p. n <= m ==> (m - n) * p = m * p - n * p
  • !m n. m < SUC n <=> m <= n
  • !m n. m < SUC n <=> m = 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 = 0 <=> m = 0 /\ n = 0
  • !a b. divides a b <=> (?c. c * a = b)
  • !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 <= m ==> m - n + n = m
  • !a b. a <= b ==> gcd a (b - a) = gcd a b
  • !a b. divides (gcd a b) b
  • !a b. divides (gcd a b) a
  • !m. ~(m < 0)
  • !n. ~(SUC n = _0)
  • !n. n <= n
  • !n. 0 <= n
  • !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
  • ?f. ONE_ONE f /\ ~ONTO f
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • (~) = (\p. p ==> F)
  • (/\) = (\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.
    Back to main package pageBack to contained package page