Package natural-def: Constructing the natural numbers
Information
| name | natural-def |
| version | 1.11 |
| description | Constructing the natural numbers |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2011-11-27 |
| requires | bool function axiom-infinity |
| show | Data.Bool Function Number.Natural |
Files
- Package tarball natural-def-1.11.tgz
- Theory file natural-def.thy (included in the package tarball)
Defined Type Operator
- Number
- Natural
- natural
- Natural
Defined Constants
- Number
- Natural
- suc
- zero
- Natural
Theorems
⊦ ∀n. ¬(suc n = 0)
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
Input Type Operators
- →
- bool
- ind
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- Bool
- Function
- injective
- surjective
Assumptions
⊦ T
⊦ ¬F ⇔ T
⊦ ¬T ⇔ F
⊦ F ⇔ ∀p. p
⊦ ∀t. t ∨ ¬t
⊦ (¬) = λp. p ⇒ F
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (T ⇔ t) ⇔ t
⊦ ∀t. F ∧ t ⇔ F
⊦ ∀t. T ∧ t ⇔ t
⊦ ∀t. t ∧ T ⇔ t
⊦ ∀t. F ⇒ t ⇔ T
⊦ ∀t. T ⇒ t ⇔ t
⊦ ∀t. t ⇒ T ⇔ T
⊦ ∀t. F ∨ t ⇔ t
⊦ ∀t. T ∨ t ⇔ T
⊦ ∀t. t ∨ F ⇔ t
⊦ ∀t. t ∨ T ⇔ T
⊦ ∀t. (F ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ F ⇔ ¬t
⊦ ∃f. injective f ∧ ¬surjective f
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x
⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x
⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x
⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀P Q. (∃x. P ∧ Q x) ⇔ P ∧ ∃x. Q x
⊦ ∀P Q. P ∧ (∃x. Q x) ⇔ ∃x. P ∧ Q x
⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x
⊦ ∀P Q. (∃x. P x) ∨ Q ⇔ ∃x. P x ∨ Q
⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3
⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)
⊦ ∀f. injective f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
⊦ ∀P Q. (∀x. P x ∧ Q x) ⇔ (∀x. P x) ∧ ∀x. Q x
⊦ ∀P Q. (∃x. P x ∨ Q x) ⇔ (∃x. P x) ∨ ∃x. Q x
⊦ ∀P Q. (∀x. P x ⇒ Q x) ⇒ (∃x. P x) ⇒ ∃x. Q x
⊦ ∀P Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x
⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∧ C ⇒ B ∧ D
⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∨ C ⇒ B ∨ D