Package natural-thm: Properties of natural numbers
Information
| name | natural-thm |
| version | 1.10 |
| description | Properties of natural numbers |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-01-04 |
| requires | bool natural-def |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-thm-1.10.tgz
- Theory file natural-thm.thy (included in the package tarball)
Theorems
⊦ ∀m. m = 0 ∨ ∃n. m = suc n
⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- F
- T
- Bool
- Number
- Natural
- suc
- zero
- Natural
Assumptions
⊦ T
⊦ ¬F ⇔ T
⊦ ¬T ⇔ F
⊦ F ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ F
⊦ ∀a. ∃!x. x = a
⊦ ∀t. (∃x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. F ∧ t ⇔ F
⊦ ∀t. T ∧ t ⇔ t
⊦ ∀t. F ∨ t ⇔ t
⊦ ∀t. T ∨ t ⇔ T
⊦ ∀t. t ∨ F ⇔ t
⊦ ∀n. ¬(suc n = 0)
⊦ ∀t. (F ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ F ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀x y. x = y ⇔ y = x
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x
⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∧ q1 ⇒ p2 ∧ q2
⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∨ q1 ⇒ p2 ∨ q2
⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y
⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'