Package modular-equiv-thm: modular-equiv-thm

Information

name	modular-equiv-thm
version	1.0
description	modular-equiv-thm
author	Joe Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2011-02-19
show	Data.Bool

Files

Package tarball modular-equiv-thm-1.0.tgz
Theory file modular-equiv-thm.thy (included in the package tarball)

Theorems

⊦ ∀x. Number.Modular.equivalent x x

⊦ ∀x y z.
Number.Modular.equivalent x y ∧ Number.Modular.equivalent y z ⇒
Number.Modular.equivalent x z

⊦ ∀x y.
    Number.Modular.equivalent x = Number.Modular.equivalent y ⇔
    Number.Natural.mod x Number.Modular.modulus =
    Number.Natural.mod y Number.Modular.modulus

⊦ ∀x y.
    Number.Natural.< x Number.Modular.modulus ∧
    Number.Natural.< y Number.Modular.modulus ∧
    Number.Modular.equivalent x = Number.Modular.equivalent y ⇒ x = y

⊦ ∀x1 x2 y1 y2.
    Number.Modular.equivalent x1 x2 ∧ Number.Modular.equivalent y1 y2 ⇒
    Number.Modular.equivalent (Number.Natural.* x1 y1)
      (Number.Natural.* x2 y2)

⊦ ∀x1 x2 y1 y2.
    Number.Modular.equivalent x1 x2 ∧ Number.Modular.equivalent y1 y2 ⇒
    Number.Modular.equivalent (Number.Natural.+ x1 y1)
      (Number.Natural.+ x2 y2)

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ¬
  - F
  - T
Number
- Modular
  - Number.Modular.equivalent
  - Number.Modular.modulus
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.<
  - Number.Natural.mod

Assumptions

⊦ T

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀n.
Number.Natural.< n Number.Modular.modulus ⇒
Number.Natural.mod n Number.Modular.modulus = n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

⊦ ∀x y.
    Number.Modular.equivalent x y ⇔
    Number.Natural.mod x Number.Modular.modulus =
    Number.Natural.mod y Number.Modular.modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.* (Number.Natural.mod m Number.Modular.modulus)
         (Number.Natural.mod n Number.Modular.modulus))
      Number.Modular.modulus =
    Number.Natural.mod (Number.Natural.* m n) Number.Modular.modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.mod m Number.Modular.modulus)
         (Number.Natural.mod n Number.Modular.modulus))
      Number.Modular.modulus =
    Number.Natural.mod (Number.Natural.+ m n) Number.Modular.modulus

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)