Package natural-factorial: Natural number factorial
Information
| name | natural-factorial |
| version | 1.39 |
| description | Natural number factorial |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| checksum | 2d5aa3c8377c9f9089a75ebaf29913c94b5e91ef |
| requires | bool natural-add natural-def natural-mult natural-numeral natural-order natural-thm |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-factorial-1.39.tgz
- Theory source file natural-factorial.thy (included in the package tarball)
Defined Constant
- Number
- Natural
- factorial
- Natural
Theorems
⊦ factorial 0 = 1
⊦ ∀n. 0 < factorial n
⊦ ∀n. ¬(factorial n = 0)
⊦ ∀n. 1 ≤ factorial n
⊦ ∀n. factorial (suc n) = suc n * factorial n
⊦ ∀m n. m ≤ n ⇒ factorial m ≤ factorial n
External Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- ⊤
- Bool
- Number
- Natural
- *
- +
- <
- ≤
- bit0
- bit1
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ bit0 0 = 0
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ∀n. 0 < suc n
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. t ∧ ⊤ ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀m. m + 0 = m
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ ∀m. 1 * m = m
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀n. 0 < n ⇔ ¬(n = 0)
⊦ ∀m n. suc m ≤ n ⇔ m < n
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d
⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p
⊦ ∀m n. 0 < m * n ⇔ 0 < m ∧ 0 < n
⊦ ∀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
⊦ ∀m n p. m * p ≤ n * p ⇔ m ≤ n ∨ p = 0