Entry Value
Name natural-prime_6
Conclusion primes = sunfold next_sieve init_sieve
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
  • !h1 h2 t1 t2. CONS h1 t1 = CONS h2 t2 <=> h1 = h2 /\ t1 = t2
  • !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. snth (scons h t) (SUC n) = snth t n
  • !b f. ?fn. fn [] = b /\ (!h t. fn (CONS h t) = f h t (fn t))
  • !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2
  • !t1 t2. ~(t1 ==> t2) <=> t1 /\ ~t2
  • !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2
  • !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)
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f n x. funpow f (SUC n) x = funpow f n (f x)
  • !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
  • !f b. sunfold f b = (let a,b' = f b in scons a (sunfold f b'))
  • !p. (!b. p b) <=> p T /\ p F
  • !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)
  • !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 + 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
  • !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
  • !n a b. ~(n = 0) ==> (SUC a MOD n = SUC b MOD n <=> a MOD n = b MOD n)
  • !n a b. ~(n = 0) ==> (a MOD n + b MOD n) MOD n = (a + b) MOD n
  • !m n p. ~(n * p = 0) ==> m MOD (n * p) MOD n = m MOD n
  • !m n p. m <= n /\ n <= p ==> m <= p
  • !a b c. divides a b /\ divides b c ==> divides a c
  • !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. 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 = 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
  • !a b. ~(a = 0) ==> (divides a b <=> b MOD a = 0)
  • !m n. ~(m = 0) ==> (m * n) DIV m = n
  • !n m. ~(n = 0) ==> m MOD n MOD n = m MOD n
  • !a b. ~(b = 0) /\ divides a b ==> a <= b
  • !m n. m < n ==> m <= n
  • !m n. m <= n \/ n <= m
  • !n. ~(n < n)
  • !n. ~(SUC n = _0)
  • !n. 0 < n <=> ~(n = 0)
  • !a. divides a 2 <=> a = 1 \/ a = 2
  • !n. BIT0 (SUC n) = SUC (SUC (BIT0 n))
  • !m. SUC m = m + 1
  • !m. m * 1 = m
  • !m. 1 * m = m
  • !n. SUC n - 1 = n
  • !n. ~(n = 0) ==> n MOD n = 0
  • !n. ~(n = 0) ==> 0 MOD n = 0
  • !m. m = 0 \/ (?n. m = SUC n)
  • !a. divides a a
  • !x. FST x,SND x = x
  • !s n. shd (sdrop s n) = snth s n
  • !s n. snth s (SUC n) = snth (stl s) n
  • !s n. stake s (SUC n) = APPEND (stake s n) [snth s n]
  • BIT0 0 = 0
  • (minimal n. T) = 0
  • primes = sunfold next_sieve init_sieve
  • Constructive Lemmas
  • T
  • !x y z. x = y /\ y = z ==> x = z
  • !h1 h2 t1 t2. CONS h1 t1 = CONS h2 t2 <=> h1 = h2 /\ t1 = t2
  • !x h t. MEM x (CONS h t) <=> x = h \/ MEM x t
  • !x y s. x IN y INSERT s <=> x = y \/ x IN s
  • !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. set_of_list (CONS h t) = h INSERT set_of_list t
  • !x s t. x INSERT s UNION t = x INSERT (s UNION t)
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x s. {x} UNION s = x INSERT s
  • !h t n. snth (scons h t) (SUC n) = snth t n
  • !h t. snth (scons h t) 0 = h
  • !a. ?x. x = a
  • !a. ?!x. x = a
  • !x. ~(x IN {})
  • !x. ~MEM x []
  • !x. x = x
  • !x. (@y. y = x) = x
  • !x. I x = x
  • !b f. ?fn. fn [] = b /\ (!h t. fn (CONS h t) = f h t (fn t))
  • !y s f. y IN IMAGE f s <=> (?x. y = f x /\ x IN s)
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 /\ q1 ==> p2 /\ q2
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 \/ q1 ==> p2 \/ q2
  • !p q r. p /\ (q \/ r) <=> p /\ q \/ p /\ r
  • !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
  • !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)
  • !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 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 x s. IMAGE f (x INSERT s) = f x INSERT IMAGE f s
  • !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. set_of_list (MAP f l) = IMAGE f (set_of_list l)
  • !f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}
  • !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 [] = []
  • !f. IMAGE f {} = {}
  • !p h t. ALL p (CONS h t) <=> p h /\ ALL p t
  • !p a. (?x. a = x /\ p x) <=> p a
  • !p x. x IN GSPEC p <=> p x
  • !p x. x IN {y | p y} <=> p x
  • !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 q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x)
  • !p l. (!x. MEM x l ==> p x) <=> ALL p l
  • !p l. ALL p l <=> (!x. x IN set_of_list l ==> p 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)
  • !p. ALL p []
  • !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)
  • !p f l. ALL p (MAP f l) <=> ALL (p o f) l
  • !p f s. (!y. y IN IMAGE f s ==> p y) <=> (!x. x IN s ==> p (f x))
  • !f b. sunfold f b = stream (\n. FST (f (funpow (SND o f) n b)))
  • !f b. sunfold f b = (let a,b' = f b in scons a (sunfold f b'))
  • !f g h. (f o g) o h = f o g o h
  • !p. (!b. p b) <=> p T /\ p F
  • !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)
  • !l h t. APPEND (CONS h t) l = CONS h (APPEND t l)
  • !l x. MEM x l <=> x IN set_of_list l
  • !l1 l2 x. MEM x (APPEND l1 l2) <=> MEM x l1 \/ MEM x l2
  • !l1 l2. set_of_list (APPEND l1 l2) = set_of_list l1 UNION set_of_list l2
  • !l. APPEND [] l = l
  • !l. APPEND l [] = 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 + 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
  • !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
  • !n a b. ~(n = 0) ==> (SUC a MOD n = SUC b MOD n <=> a MOD n = b MOD n)
  • !n a b. ~(n = 0) ==> (a MOD n + b MOD n) MOD n = (a + b) MOD n
  • !m n p. ~(n * p = 0) ==> m MOD (n * p) MOD n = m MOD n
  • !m n p. m <= n /\ n <= p ==> m <= p
  • !a b c. divides a b /\ divides b c ==> divides a c
  • !m n. m <= m + 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. 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 = m <=> n = 0
  • !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. ~(n = 0) ==> m MOD n < n
  • !m n. ~(n = 0) ==> m DIV n * n + m MOD n = m
  • !a b. ~(a = 0) ==> (divides a b <=> b MOD a = 0)
  • !m n. ~(m = 0) ==> (m * n) DIV m = n
  • !n m. ~(n = 0) ==> m MOD n MOD n = m MOD n
  • !a b. ~(b = 0) /\ divides a b ==> a <= b
  • !m n. m < n ==> m <= n
  • !m n. m = n ==> m <= n
  • !m n. m <= n \/ n <= m
  • !n. ~(n < n)
  • !m. ~(m < 0)
  • !n. ~(SUC n = _0)
  • !n. n < SUC n
  • !n. 0 < SUC n
  • !n. n <= n
  • !n. 0 <= n
  • !n. 0 < n <=> ~(n = 0)
  • !m. m <= 0 <=> m = 0
  • !a. divides a 2 <=> a = 1 \/ a = 2
  • !a. divides 0 a <=> a = 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
  • !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. SUC n - 1 = n
  • !n. ~(n = 0) ==> n MOD n = 0
  • !n. ~(n = 0) ==> 0 MOD n = 0
  • !m. m = 0 \/ (?n. m = SUC n)
  • !a. divides a a
  • !a. divides a 0
  • !x. ?a b. x = a,b
  • !x. FST x,SND x = x
  • !s t x. x IN s UNION t <=> x IN s \/ x IN t
  • !s t u. (s UNION t) UNION u = s UNION t UNION u
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. s UNION t = {x | x IN s \/ x IN t}
  • !s t. s UNION t = t UNION s
  • !s t. (!x. x IN s <=> x IN t) ==> s = t
  • !s. {} UNION s = s
  • !s. s UNION {} = s
  • !s n. shd (sdrop s n) = snth s n
  • !s n. snth s (SUC n) = snth (stl s) n
  • !s n. stake s (SUC n) = APPEND (stake s n) [snth s n]
  • !s n. stake s (SUC n) = CONS (shd s) (stake (stl s) n)
  • !s n. sdrop s n = stream (\m. snth s (m + n))
  • !s1 s2. (!n. snth s1 n = snth s2 n) ==> s1 = s2
  • !s. stake s 0 = []
  • !s. stream (snth s) = s
  • ?f. ONE_ONE f /\ ~ONTO f
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • I = (\x. x)
  • LET_END = (\t. t)
  • (~) = (\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)
  • LET = (\f. f)
  • (!) = (\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
  • (minimal n. T) = 0
  • {} = {x | F}
  • set_of_list [] = {}
  • primes = sunfold next_sieve init_sieve
  • Contained Package natural-prime
    Comment Natural-prime package from OpenTheory.
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