Package gfp-def: Definition of GF(p) finite fields

Information

namegfp-def
version1.74
descriptionDefinition of GF(p) finite fields
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2014-11-01
checksum479aa0ae67f8d14bd60cddf123fbbc4db56d8ea3
requiresbase
gfp-witness
natural-bits
natural-divides
natural-prime
showData.Bool
Number.GF(p)
Number.Natural
Probability.Random

Files

Defined Type Operator

Defined Constants

Theorems

¬(oddprime = 0)

x. x x

fromNatural oddprime = 0

oddprime mod oddprime = 0

0 mod oddprime = 0

x. ¬(x < x)

x. toNatural x < oddprime

~0 = 0

x. ~~x = x

x. fromNatural (toNatural x) = x

n. n mod oddprime < oddprime

n. n mod oddprime n

x. x + 0 = x

x. x 1 = x

x. 0 + x = x

x. toNatural x div oddprime = 0

x. x 0 = 1

x. x * 0 = 0

x. x + ~x = 0

x. 0 * x = 0

x. ~x + x = 0

x. toNatural x mod oddprime = toNatural x

x. x * 1 = x

x. 1 * x = x

n. toNatural (fromNatural n) = n mod oddprime

r. random r = fromNatural (Uniform.random oddprime r)

x. ~x = fromNatural (oddprime - toNatural x)

x y. x * y = y * x

x y. x + y = y + x

n. divides oddprime n n mod oddprime = 0

n. n < oddprime toNatural (fromNatural n) = n

n. n < oddprime n mod oddprime = n

x. fromNatural x = 0 divides oddprime x

n. n mod oddprime mod oddprime = n mod oddprime

x y. x - y = x + ~y

x y. ¬(x < y) y x

x y. ¬(x y) y < x

x. ~x = 0 x = 0

x y. x < y toNatural x < toNatural y

x y. x y toNatural x toNatural y

x y. x * ~y = ~(x * y)

x y. ~x * y = ~(x * y)

x y. ~x = ~y x = y

x y. toNatural x = toNatural y x = y

m n. fromNatural (m n) = fromNatural m n

x y. x + y = x y = 0

x y. y + x = x y = 0

x y. ~x + ~y = ~(x + y)

x n. x suc n = x * x n

x1 y1. fromNatural (x1 * y1) = fromNatural x1 * fromNatural y1

x1 y1. fromNatural (x1 + y1) = fromNatural x1 + fromNatural y1

x y. toNatural (x * y) = toNatural x * toNatural y mod oddprime

x y. toNatural (x + y) = (toNatural x + toNatural y) mod oddprime

x y z. x * y * z = x * (y * z)

x y z. x + y + z = x + (y + z)

x y z. x + y = x + z y = z

x y z. y + x = z + x y = z

x1 x2 x3. x1 < x2 x2 < x3 x1 < x3

x1 x2 x3. x1 < x2 x2 x3 x1 < x3

x1 x2 x3. x1 x2 x2 < x3 x1 < x3

x1 x2 x3. x1 x2 x2 x3 x1 x3

n. 0 n = if n = 0 then 1 else 0

x y. fromNatural x = fromNatural y x mod oddprime = y mod oddprime

x y z. x * (y + z) = x * y + x * z

x y z. (y + z) * x = y * x + z * x

x m n. x m * x n = x (m + n)

m n.
    (m mod oddprime) * (n mod oddprime) mod oddprime = m * n mod oddprime

m n.
    (m mod oddprime + n mod oddprime) mod oddprime = (m + n) mod oddprime

x y. x < oddprime y < oddprime fromNatural x = fromNatural y x = y

External Type Operators

External Constants

Assumptions

prime oddprime

¬prime 0

¬

¬

bit0 0 = 0

t. t t

n. n n

p. p

t. t ¬t

(¬) = λp. p

() = λp. p ((select) 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

n. 0 * n = 0

n. 0 + n = n

t. ( t) ¬t

t. t ¬t

n. bit1 n = suc (bit0 n)

m. m 0 = 1

m. 1 * m = m

() = λp q. p q p

t. (t ) (t )

n. 0 < n ¬(n = 0)

x y. x = y y = x

t1 t2. t1 t2 t2 t1

m n. m * n = n * m

m n. m + n = n + m

m n. m < n m n

m n. ¬(m < n) n m

() = λp q. (λf. f p q) = λf. f

n. ¬(n = 0) n mod n = 0

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

m n. m < n m div n = 0

m n. m < n m mod n = m

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

n. 0 n = if n = 0 then 1 else 0

m n. m suc n = m * m n

m n. ¬(n = 0) m mod n < n

m n. ¬(n = 0) m mod n m

f g. (x. f x = g x) f = g

() = λp q. r. (p r) (q r) r

m n. n m m - n + n = m

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

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

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

m n p. m < n n < p m < p

m n p. m < n n p m < p

m n p. m n n < p m < p

m n p. m n n p m p

r. (x. y. r x y) f. x. r x (f x)

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

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

n m. ¬(n = 0) m mod n mod n = m mod n

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

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

(∃!) = λp. () p x y. p x p y x = y

b f x y. f (if b then x else y) = if b then f x else f y

e f. ∃!fn. fn 0 = e n. fn (suc n) = f (fn n) n

m n. ¬(n = 0) (m div n) * n + m mod n = m

n m p. ¬(n = 0) (m mod n) * (p mod n) mod n = m * p mod n

n a b. ¬(n = 0) (a mod n + b mod n) mod n = (a + b) mod n