Package gfp-div-thm: Properties of GF(p) field division
Information
name | gfp-div-thm |
version | 1.23 |
description | Properties of GF(p) field division |
author | Joe Hurd <joe@gilith.com> |
license | MIT |
provenance | HOL Light theory extracted on 2012-03-08 |
requires | bool natural gfp-def gfp-thm gfp-div-def |
show | Data.Bool Number.GF(p) Number.Natural |
Files
- Package tarball gfp-div-thm-1.23.tgz
- Theory file gfp-div-thm.thy (included in the package tarball)
Theorems
⊦ inv 1 = 1
⊦ ∀x. x / 1 = x
⊦ ∀x. ¬(x = 0) ⇒ inv (inv x) = x
⊦ ∀x. ¬(x = 0) ⇒ ¬(inv x = 0)
⊦ ∀x. ¬(x = 0) ⇒ x * inv x = 1
⊦ ∀x y. ¬(x = 0) ⇒ x * (y / x) = y
⊦ ∀x y. ¬(x = 0) ⇒ (y / x) * x = y
⊦ ∀x y. ¬(x = 0) ⇒ y * x / x = y
⊦ ∀x y. ¬(x = 0) ⇒ y / x = y * inv x
⊦ ∀x n. ¬(x = 0) ⇒ inv x ^ n = inv (x ^ n)
⊦ ∀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. ¬(x = 0) ∧ inv x = 1 ⇒ x = 1
⊦ ∀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
⊦ ∀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
Input Type Operators
- →
- bool
- Number
- GF(p)
- gfp
- Natural
- natural
- GF(p)
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- cond
- ⊥
- ⊤
- Bool
- Number
- GF(p)
- *
- /
- ^
- fromNatural
- inv
- Natural
- bit1
- suc
- zero
- GF(p)
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ⊥ ⇔ ∀p. p
⊦ ∀t. t ∨ ¬t
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ⊥ ∧ t ⇔ ⊥
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ¬(1 = 0)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀x. x ^ 0 = 1
⊦ ∀x. x * 0 = 0
⊦ ∀x. 0 * x = 0
⊦ ∀x. x * 1 = x
⊦ ∀x. 1 * x = x
⊦ ∀x y. x = y ⇒ y = x
⊦ ∀x y. x * y = y * x
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀x n. x ^ suc n = x * x ^ n
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀x y z. x = y ∧ y = z ⇒ x = z
⊦ ∀x y z. x * y * z = x * (y * z)
⊦ ∀x. ¬(x = 0) ⇒ inv x * x = 1
⊦ ∀x y. ¬(x = 0) ⇒ x * y / x = y
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
⊦ ∀x y. x * y = 0 ⇔ x = 0 ∨ y = 0
⊦ ∀x n. x ^ n = 0 ⇔ x = 0 ∧ ¬(n = 0)
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)