Package natural-sub-def: natural-sub-def

Information

name	natural-sub-def
version	1.3
description	natural-sub-def
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-07-20
show	Data.Bool

Files

Package tarball natural-sub-def-1.3.tgz
Theory file natural-sub-def.thy (included in the package tarball)

Defined Constant

Number
- Natural
  - Number.Natural.-

Theorem

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) n = m

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

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

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. Number.Natural.pre (Number.Natural.suc n) = n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀P.
P Number.Numeral.zero ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

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

⊦ (∀n. Number.Natural.+ Number.Numeral.zero n = n) ∧
  (∀m. Number.Natural.+ m Number.Numeral.zero = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)