Package natural-order-def: Definition of natural number orderings
Information
| name | natural-order-def |
| version | 1.23 |
| description | Definition of natural number orderings |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-06-16 |
| requires | bool natural-thm |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-order-def-1.23.tgz
- Theory file natural-order-def.thy (included in the package tarball)
Defined Constants
- Number
- Natural
- <
- ≤
- >
- ≥
- Natural
Theorems
⊦ ∀m. ¬(m < 0)
⊦ ∀m. m ≤ 0 ⇔ m = 0
⊦ ∀m n. m > n ⇔ n < m
⊦ ∀m n. m ≥ n ⇔ n ≤ m
⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- ⊥
- ⊤
- Bool
- Number
- Natural
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ (∃) = λp. p ((select) p)
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y 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