Package natural-order-def: natural-order-def

Information

name	natural-order-def
version	1.0
description	natural-order-def
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-02-19
show	Data.Bool

Files

Package tarball natural-order-def-1.0.tgz
Theory file natural-order-def.thy (included in the package tarball)

Defined Constants

Number
- Natural
  - Number.Natural.<
  - Number.Natural.≤
  - Number.Natural.>
  - Number.Natural.≥

Theorems

⊦ ∀n m. Number.Natural.> m n ⇔ Number.Natural.< n m

⊦ ∀n m. Number.Natural.≥ m n ⇔ Number.Natural.≤ n m

⊦ (∀m. Number.Natural.< m Number.Numeral.zero ⇔ F) ∧
  ∀m n.
    Number.Natural.< m (Number.Natural.suc n) ⇔
    m = n ∨ Number.Natural.< m n

⊦ (∀m. Number.Natural.≤ m Number.Numeral.zero ⇔ m = Number.Numeral.zero) ∧
  ∀m n.
    Number.Natural.≤ m (Number.Natural.suc n) ⇔
    m = Number.Natural.suc n ∨ Number.Natural.≤ m n

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - select
  - F
  - T
Number
- Natural
  - Number.Natural.suc
- Numeral
  - Number.Numeral.zero

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ (∀) = λP. P = λx. T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀e f.
    ∃fn.
      fn Number.Numeral.zero = e ∧
      ∀n. fn (Number.Natural.suc n) = f (fn n) n