Package gfp: Parametric theory of GF(p) finite fields

Information

namegfp
version1.91
descriptionParametric theory of GF(p) finite fields
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
checksuma4b679d0cf3345a6ab344a6c333c9a175d422221
requiresbase
gfp-witness
natural-bits
natural-divides
natural-fibonacci
natural-prime
showData.Bool
Data.List
Data.Pair
Number.GF(p)
Number.Natural
Number.Natural.Fibonacci
Probability.Random

Files

Defined Type Operator

Defined Constants

Theorems

¬(oddprime = 0)

1 < oddprime

x. x x

¬(oddprime = 1)

¬divides oddprime 1

fromNatural oddprime = 0

oddprime mod oddprime = 0

0 mod oddprime = 0

2 < oddprime

x. ¬(x < x)

x. toNatural x < oddprime

¬(oddprime = 2)

¬divides oddprime 2

¬divides 2 oddprime

~0 = 0

x. ~~x = x

x. fromNatural (toNatural x) = x

n. n mod oddprime < oddprime

n. n mod oddprime n

¬(1 = 0)

1 mod oddprime = 1

x. x + 0 = x

x. x 1 = x

x. 0 + x = x

x. toNatural x div oddprime = 0

¬(2 = 0)

inv 1 = 1

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. x / 1 = x

x. 1 * x = x

n. toNatural (fromNatural n) = n mod oddprime

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

2 mod oddprime = 2

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. ¬(x = 0) inv (inv x) = x

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. ¬(x = 0) ¬(inv x = 0)

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

x. ¬(x = 0) x * inv x = 1

x. ¬(x = 0) inv x * x = 1

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

x y. ¬(x = 0) x * (y / x) = y

x y. ¬(x = 0) (y / x) * x = y

x y. ¬(x = 0) x * y / x = y

x y. ¬(x = 0) y * x / x = y

m n. divides oddprime (m * n) divides oddprime m divides oddprime n

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

x y. ¬(x = 0) y / x = y * inv x

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)

x n. ¬(x = 0) inv x n = inv (x 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 * y = x x = 0 y = 1

x y. y * x = x x = 0 y = 1

x y. x * y = 0 x = 0 y = 0

x n. x n = 0 x = 0 ¬(n = 0)

x. ¬(x = 0) inv x = 1 x = 1

x y. ¬(y = 0) x / y = divGcd (toNatural y) oddprime x 0

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

x y z. x * y = x * z x = 0 y = z

x y z. y * x = z * x x = 0 y = z

x y. x * y = if y = 0 then 0 else x / (1 / y)

x y z. ¬(x = 0) x * y = x * z y = z

x y z. ¬(x = 0) y * x = z * x y = z

x y. ¬(x = 0) ¬(y = 0) ¬(y / x = 0)

x y. ¬(x = 0) ¬(y = 0) inv (y / x) = x / y

b n d f p. expDiv b n d f p [] = if b then n / d else d / n

x y. ¬(x = 0) ¬(y = 0) inv x = inv y x = y

x y. ¬(x = 0) ¬(y = 0) inv x * inv y = inv (x * y)

x y z. ¬(y = 0) ¬(z = 0) x / (y / z) = x * z / y

x n.
    x n =
    if n = 0 then 1 else if x = 0 then 0 else expDiv 1 1 x 1 (encode n)

b n d f p h t.
    expDiv b n d f p (h :: t) =
    let s p / f in expDiv (¬b) d (if h then n / s else n) s f t

u v x1 x2.
    gcd u v = 1 fromNatural u * x2 = fromNatural v * x1
    fromNatural u * divGcd u v x1 x2 = x1
    fromNatural v * divGcd u v x1 x2 = x2

x n d f p l.
    ¬(x = 0) ¬(n = 0) ¬(d = 0)
    expDiv n d (x f) (inv (x p)) l =
    (n / d) * x decode.dest f p l
    expDiv n d (inv (x f)) (x p) l = (d / n) * x decode.dest f p l

u v x1 x2.
    divGcd u v x1 x2 =
    if u = 1 then x1
    else if v = 1 then x2
    else if even u then divGcd (u div 2) v (x1 / 2) x2
    else if even v then divGcd u (v div 2) x1 (x2 / 2)
    else if v u then divGcd (u - v) v (x1 - x2) x2
    else divGcd u (v - u) x1 (x2 - x1)

p.
    (v. p 1 v) (u. ¬(u = 1) p u 1)
    (u v. gcd (2 * u) v = 1 ¬(v = 1) p u v p (2 * u) v)
    (u v. gcd u (2 * v) = 1 ¬(u = 1) odd u p u v p u (2 * v))
    (u v. gcd u v = 1 even u ¬(v = 1) odd v p u v p (v + u) v)
    (u v. gcd u v = 1 ¬(u = 1) odd u even v p u v p u (u + v))
    u v. gcd u v = 1 p u v

p.
    (v x1 x2. p 1 v x1 x2 x1) (u x1 x2. p u 1 x1 x2 x2)
    (u v x1 x2 g.
       gcd (2 * u) v = 1 p u v x1 x2 g p (2 * u) v (2 * x1) x2 g)
    (u v x1 x2 g.
       gcd u (2 * v) = 1 p u v x1 x2 g p u (2 * v) x1 (2 * x2) g)
    (u v x1 x2 g.
       gcd u v = 1 p u v x1 x2 g p (v + u) v (x2 + x1) x2 g)
    (u v x1 x2 g.
       gcd u v = 1 p u v x1 x2 g p u (u + v) x1 (x1 + x2) g)
    u v x1 x2. gcd u v = 1 p u v x1 x2 (divGcd u v x1 x2)

External Type Operators

External Constants

Assumptions

odd oddprime

prime oddprime

¬odd 0

¬prime 0

¬prime 1

¬

¬

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. t t

t. t

t. t t

t. t

t. t t t

n. ¬(suc n = 0)

n. decode (encode n) = n

n. 0 * n = 0

n. 0 + n = n

m. m + 0 = m

a. gcd 0 a = a

a. gcd a 0 = a

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. even (2 * n)

n. bit1 n = suc (bit0 n)

n. ¬even n odd n

n. ¬odd n even n

m. m 0 = 1

m. 1 * m = m

m n. m m + n

() = λp q. p q p

t. (t ) (t )

n. odd (suc n) ¬odd n

m. m 0 m = 0

t1 t2. (if then t1 else t2) = t2

t1 t2. (if then t1 else t2) = t1

f p. decode.dest f p [] = 0

p x. p x p ((select) p)

n. 0 < n ¬(n = 0)

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

a. divides 2 a even a

l. decode l = decode.dest 1 0 l

x y. x = y y = x

x y. x = y y = x

t1 t2. t1 t2 t2 t1

m n. m * n = n * m

m n. m + n = n + m

a b. gcd a b = gcd b a

m n. m < n m n

m n. m n n m

m n. m + n - m = n

a. divides a 1 a = 1

m n. ¬(m < n) n m

m n. ¬(m n) n < m

m n. suc m n m < n

m. m = 0 n. m = suc n

p. (b. p b) p p

() = λp q. (λf. f p q) = λf. f

n. ¬(n = 0) n mod n = 0

p. ¬(x. p x) x. ¬p x

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

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

m n. n < m + n 0 < m

a b. gcd a (a + b) = gcd a b

a b. gcd (b + a) b = gcd a b

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

t1 t2. ¬(t1 t2) ¬t1 ¬t2

t1 t2. ¬(t1 t2) ¬t1 ¬t2

m n. even (m + n) even m even n

m n. m suc n = m * m n

m n. ¬(n = 0) m mod n < n

m n. ¬(n = 0) m mod n m

n. even n m. n = 2 * m

p. (x. p x) a b. p (a, b)

m n. m n d. n = m + d

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

m n. m n n m m = n

f. fn. a b. fn (a, b) = f a b

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

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

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

m n. m < n m n ¬(m = n)

m n. ¬(m = 0) m * n div m = n

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

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

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

p q r. p q r p q r

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 < m + p n < p

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

m n p. m + n m + p 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)

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

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

p n. prime p ¬divides p n gcd p n = 1

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

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

a. divides a 2 a = 1 a = 2

(∃!) = 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

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

p g h. f. x. f x = if p x then f (g x) else h x

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

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

p. p [] (h t. p t p (h :: t)) l. p l

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

m n p. m * p < n * p m < n ¬(p = 0)

a b. ¬(a = 0) s t. t * b + gcd b a = s * a

p c x y. p (if c then x else y) (c p x) (¬c p y)

b f. fn. fn [] = b h t. fn (h :: t) = f h t (fn t)

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

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

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

f p h t.
    decode.dest f p (h :: t) =
    let s f + p in let n decode.dest s f t in if h then s + n else n