Package montgomery-def: Definition of Montgomery multiplication

Information

name	montgomery-def
version	1.6
description	Definition of Montgomery multiplication
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2014-01-13
requires	bool natural
show	Data.Bool Number.Natural

Files

Package tarball montgomery-def-1.6.tgz
Theory source file montgomery-def.thy (included in the package tarball)

Defined Constants

Number
- Natural
  - Montgomery
    - Montgomery.doubleExp
    - Montgomery.reduce

Theorems

⊦ ∀n r k a. Montgomery.doubleExp n r k a 0 = a

⊦ ∀n r k a. Montgomery.reduce n r k a = (a + (a * k mod r) * n) div r

⊦ ∀n r k a m.
    Montgomery.doubleExp n r k a (suc m) =
    let b ← Montgomery.reduce n r k (a * a) in
    let c ← if r ≤ b then b - n else b in
    Montgomery.doubleExp n r k c m

External Type Operators

→
bool
Number
- Natural
  - natural

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - cond
  - ⊤
Number
- Natural
  - *
  - +
  - -
  - ≤
  - div
  - mod
  - suc
  - zero

Assumptions

⊦ ⊤

⊦ (∃) = λp. p ((select) p)

⊦ (∀) = λp. p = λx. ⊤

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

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