Package natural-sub-def: Definition of natural number subtraction
Information
| name | natural-sub-def |
| version | 1.15 |
| description | Definition of natural number subtraction |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| provenance | HOL Light theory extracted on 2011-12-18 |
| requires | bool natural-def natural-thm natural-dest natural-add |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-sub-def-1.15.tgz
- Theory file natural-sub-def.thy (included in the package tarball)
Defined Constant
- Number
- Natural
- -
- Natural
Theorem
⊦ ∀m n. m + n - n = m
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- T
- Bool
- Number
- Natural
- +
- pre
- suc
- zero
- Natural
Assumptions
⊦ T
⊦ (∃) = λp. p ((select) p)
⊦ (∀) = λp. p = λx. T
⊦ ∀n. pre (suc n) = n
⊦ ∀m. m + 0 = m
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y
⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n