Package modular: Parametric theory of modular arithmetic

Information

namemodular
version1.26
descriptionParametric theory of modular arithmetic
authorJoe Hurd <joe@gilith.com>
licenseMIT
requiresbool
natural
natural-divides
modular-witness
showData.Bool
Number.Modular
Number.Natural

Files

Defined Type Operator

Defined Constants

Theorems

fromNatural modulus = 0

modulus mod modulus = 0

0 mod modulus = 0

x. toNatural x < modulus

~0 = 0

x. ~~x = x

x. fromNatural (toNatural x) = x

n. n mod modulus < modulus

x. x + 0 = x

x. 0 + x = x

x. toNatural x div modulus = 0

x. x * 0 = 0

x. x + ~x = 0

x. 0 * x = 0

x. ~x + x = 0

x. toNatural x mod modulus = toNatural x

x. x * 1 = x

x. 1 * x = x

x. toNatural (fromNatural x) = x mod modulus

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

x y. x * y = y * x

x y. x + y = y + x

n. divides modulus n n mod modulus = 0

n. n < modulus n mod modulus = n

x. fromNatural x = 0 divides modulus x

n. n mod modulus mod modulus = n mod modulus

x y. x < y ¬(y x)

x y. x - y = x + ~y

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

x y. x + y = x y = 0

x y. y + x = x y = 0

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

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 modulus

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

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

x y. fromNatural x = fromNatural y x mod modulus = y mod modulus

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

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

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

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

x y. x < modulus y < modulus fromNatural x = fromNatural y x = y

Input Type Operators

Input Constants

Assumptions

T

¬(modulus = 0)

¬F T

¬T F

t. t t

F p. p

t. t ¬t

(¬) = λp. p F

() = 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. F t F

t. T t t

t. F t T

t. T t t

t. t T T

t. F t t

t. T t T

t. t F t

t. t T T

n. 0 * n = 0

n. 0 + n = n

t. (F t) ¬t

t. t F ¬t

m. 1 * m = m

() = λp q. p q p

t. (t T) (t F)

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 T T

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. ¬(n = 0) m mod n < n

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

P. (x. y. P x y) y. x. P x (y x)

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

m n. ¬(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

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

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

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