Package natural-divides-lcm-thm: Properties of natural number least common multiple

Information

namenatural-divides-lcm-thm
version1.0
descriptionProperties of natural number least common multiple
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2014-11-01
checksum492b9462ef1667dfab2de59249d4df44d8eca454
requiresbase
natural-divides-def
natural-divides-gcd
natural-divides-lcm-def
natural-divides-thm
showData.Bool
Number.Natural

Files

Theorems

a. lcm a 0 = 0

a. lcm a a = a

a. lcm a 1 = a

a. lcm 1 a = a

a b. divides a (lcm a b)

a b. divides b (lcm a b)

a b. lcm a b = lcm b a

a b. lcm a b = a divides b a

a b. lcm b a = a divides b a

a b c. divides a b divides a (lcm b c)

a b c. divides a b divides a (lcm c b)

a b c. lcm (lcm a b) c = lcm a (lcm b c)

a b. lcm a b = 0 a = 0 b = 0

a b c. divides (lcm a b) c divides a c divides b c

a b c. gcd a (lcm b c) = lcm (gcd a b) (gcd a c)

a b c. lcm a (gcd b c) = gcd (lcm a b) (lcm a c)

a b c. gcd (lcm b c) a = lcm (gcd b a) (gcd c a)

a b c. lcm (gcd b c) a = gcd (lcm b a) (lcm c a)

a b c. lcm (a * b) (a * c) = a * lcm b c

a b c. lcm (b * a) (c * a) = lcm b c * a

a b c. divides a c divides b c divides (lcm a b) c

a b. lcm a b = 1 a = 1 b = 1

a b l.
    divides a l divides b l
    (c. divides a c divides b c divides l c) lcm a b = l

External Type Operators

External Constants

Assumptions

¬

¬

bit0 0 = 0

t. t t

a. divides a 0

a. divides a a

p. p

t. t ¬t

a. divides 1 a

(¬) = λp. p

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx.

t. ¬¬t t

t. t

t. t t

t. t t t

t. t t

t. t

t. t t

n. 0 * n = 0

m. m * 0 = 0

n. 0 + n = n

a. gcd a 0 = a

a. gcd a a = a

a. lcm 0 a = 0

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. bit1 n = suc (bit0 n)

m. m * 1 = m

a b. divides (gcd a b) a

a b. divides (gcd a b) b

() = λp q. p q p

t. (t ) (t )

n. bit0 (suc n) = suc (suc (bit0 n))

x y. x = y y = x

m n. m * n = n * m

m n. m + n = n + m

a b. gcd a b = gcd b a

a. divides a 1 a = 1

m n. ¬(m n) n < m

() = λp q. (λf. f p q) = λf. f

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

m n. m + suc n = suc (m + n)

m n. suc m + n = suc (m + n)

t1 t2. ¬(t1 t2) ¬t1 ¬t2

a b c. divides a b divides a (b * c)

a b c. divides b a divides (gcd c b) a

a b c. divides (a * b) c divides a c

a b. divides a b c. c * a = b

() = λp q. r. (p r) (q r) r

m n. m n n m m = n

a b. lcm a b * gcd a b = a * b

a b. divides a b divides b a a = b

m n. m < n d. n = m + suc d

p q r. p q r p q r

m n p. m * (n * p) = m * n * p

a b c. divides a b divides b c divides a c

m n. m * n = 0 m = 0 n = 0

a b. gcd a b = 0 a = 0 b = 0

p. p 0 (n. p n p (suc n)) n. p n

a b c. divides c (gcd a b) divides c a divides c b

a b c. divides (gcd a (b * c)) (gcd a b * gcd a c)

a b c. gcd (a * b) (a * c) = a * gcd b c

m n p. m * n = m * p m = 0 n = p

m n p. m * p = n * p m = n p = 0

a b c. divides (a * b) (a * c) a = 0 divides b c

a b c. divides (b * a) (c * a) a = 0 divides b c

a b c d. divides a c divides b d divides (a * b) (c * d)

a b c. gcd b c = 1 divides b a divides c a divides (b * c) a

p.
    p 0 0 (a b. gcd a b = 1 p a b)
    (c a b. ¬(c = 0) p a b p (c * a) (c * b)) a b. p a b