Package natural-prime-thm: Properties of prime natural numbers

Information

namenatural-prime-thm
version1.52
descriptionProperties of prime natural numbers
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2014-11-01
checksumc5aed1ace2f77d1cadd247f42960ee5b01f81736
requiresbool
natural
natural-divides
natural-gcd
natural-prime-def
showData.Bool
Number.Natural

Files

Theorems

¬prime 0

¬prime 1

prime 2

prime 3

n. p. n p prime p

p. prime p even p p = 2

n. ¬(n = 1) p. prime p divides p n

p1 p2. prime p1 prime p2 divides p1 p2 p1 = p2

p n. prime p (gcd p n = 1 ¬divides p n)

p n. prime p ¬divides p n gcd p n = 1

p m n. prime p (divides p (m * n) divides p m divides p n)

p m n. prime p ¬divides p m ¬divides p n ¬divides p (m * n)

n. prime n ¬(n = 0) ¬(n = 1) p. prime p p < n ¬divides p n

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

(¬) = 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

t. t t

t. t

t. t t

t. t

t. t t

t. t

t. t t t

n. ¬(factorial n = 0)

n. ¬(suc n = 0)

n. 0 + n = n

m. m + 0 = m

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. bit1 n = suc (bit0 n)

m. m * 1 = m

m. 1 * m = m

() = λp q. p q p

t. (t ) (t )

a. divides 0 a a = 0

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

a. divides 2 a even a

x y. x = y y = x

t1 t2. t1 t2 t2 t1

m n. m + n = n + m

m n. m = n m n

m n. m n n m

m n. distance m (m + n) = n

a. divides a 1 a = 1

m n. ¬(m < n) n m

m n. ¬(m n) n < m

p. (b. p b) p p

() = λp q. (λf. f p q) = λf. f

p. ¬(x. p x) x. ¬p x

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

t1 t2. ¬(t1 t2) t1 ¬t2

t1 t2. ¬t1 ¬t2 t2 t1

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

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

m n. suc m = suc n m = n

m n. m + n = n m = 0

a b. gcd a b = a divides a b

t1 t2. ¬(t1 t2) ¬t1 ¬t2

t1 t2. ¬(t1 t2) ¬t1 ¬t2

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

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

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

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

m n. m < n m n ¬(m = n)

a b. ¬(b = 0) divides a b a b

p q. (x. p x) q x. p x q

p q. (x. p x) q x. p x q

t1 t2 t3. (t1 t2) t3 t1 t2 t3

m n p. m + (n + p) = m + n + p

m n p. m n n p m p

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

a b. ¬(b = 0) b a divides b (factorial a)

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

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

a. divides a 2 a = 1 a = 2

a. divides a 3 a = 1 a = 3

p. (n. (m. m < n p m) p n) n. p n

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

a b s t. distance (s * a) (t * b) = 1 gcd a b = 1

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

p. prime p ¬(p = 1) n. divides n p n = 1 n = p