Package modular-thm: Properties of modular arithmetic
Information
| name | modular-thm |
| version | 1.24 |
| description | Properties of modular arithmetic |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2012-03-08 |
| requires | bool natural modular-def |
| show | Data.Bool Number.Modular Number.Natural |
Files
- Package tarball modular-thm-1.24.tgz
- Theory file modular-thm.thy (included in the package tarball)
Theorems
⊦ fromNatural modulus = 0
⊦ ∀x. toNatural x < modulus
⊦ ~0 = 0
⊦ ∀x. ~~x = x
⊦ ∀x. x + 0 = x
⊦ ∀x. x ^ 1 = 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 y. x * y = y * x
⊦ ∀x y. x + y = y + x
⊦ ∀x. fromNatural x = 0 ⇔ divides modulus x
⊦ ∀x. ~x = 0 ⇔ x = 0
⊦ ∀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 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
⊦ ∀n. 0 ^ n = if n = 0 then 1 else 0
⊦ ∀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
⊦ ∀x m n. x ^ m * x ^ n = x ^ (m + n)
Input Type Operators
- →
- bool
- Number
- Modular
- modular
- Natural
- natural
- Modular
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ¬
- cond
- ⊤
- Bool
- Number
- Modular
- *
- +
- <
- ≤
- ^
- ~
- fromNatural
- modulus
- toNatural
- Natural
- *
- +
- -
- <
- ≤
- ^
- bit0
- bit1
- div
- divides
- mod
- suc
- zero
- Modular
Assumptions
⊦ ⊤
⊦ bit0 0 = 0
⊦ modulus mod modulus = 0
⊦ 0 mod modulus = 0
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀x. fromNatural (toNatural x) = x
⊦ ∀n. 0 * n = 0
⊦ ∀n. 0 + n = n
⊦ ∀n. n mod modulus < modulus
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀m. m ^ 0 = 1
⊦ ∀m. 1 * m = m
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀x. x ^ 0 = 1
⊦ ∀x. toNatural (fromNatural x) = x mod modulus
⊦ ∀x. ~x = fromNatural (modulus - toNatural x)
⊦ ∀m n. m * n = n * m
⊦ ∀m n. m + n = n + m
⊦ ∀m n. m < n ⇒ m ≤ n
⊦ ∀n. divides modulus n ⇔ n mod modulus = 0
⊦ ∀n. n mod modulus mod modulus = n mod modulus
⊦ ∀x y. x < y ⇔ ¬(y ≤ x)
⊦ ∀m n. ¬(m ≤ n) ⇔ n < m
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀x y. x ≤ y ⇔ toNatural x ≤ toNatural y
⊦ ∀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
⊦ ∀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
⊦ ∀m n. m ^ suc n = m * m ^ n
⊦ ∀m n. n ≤ m ⇒ m - n + n = m
⊦ ∀m n p. m * (n * p) = m * n * p
⊦ ∀m n p. m + (n + p) = m + n + p
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
⊦ ∀m n p. m * (n + p) = m * n + m * p
⊦ ∀m n p. (m + n) * p = m * p + n * p
⊦ ∀b f x y. f (if b then x else y) = if b then f x else f y
⊦ ∀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