Package natural-factorial: Definitions and theorems about natural number factorial

Information

Files

• Package tarball natural-factorial-1.5.tgz
• Theory file natural-factorial.thy (included in the package tarball)

Defined Constant

• Number
  • Natural
    • factorial

Theorems

⊦ ∀n. 0 < factorial n

⊦ ∀n. ¬(factorial n = 0)

⊦ ∀n. 1 ≤ factorial n

⊦ ∀m n. m ≤ n ⇒ factorial m ≤ factorial n

⊦ factorial 0 = 1 ∧ ∀n. factorial (suc n) = suc n * factorial n

Input Type Operators

• →
• bool
• Number
  • Natural
    • natural

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ~
    • F
    • T
• Number
  • Natural
    • *
    • +
    • <
    • ≤
    • bit1
    • suc
    • zero

Assumptions

⊦ T

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ 1 = suc 0

⊦ ∀n. 0 < suc n

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ (⇒) = 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 T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀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

⊦ ∀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

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ (∀n. 0 + n = n) ∧ (∀m. m + 0 = m) ∧ (∀m n. suc m + n = suc (m + n)) ∧
  ∀m n. m + suc n = suc (m + n)

⊦ (∀n. 0 * n = 0) ∧ (∀m. m * 0 = 0) ∧ (∀n. 1 * n = n) ∧ (∀m. m * 1 = m) ∧
  (∀m n. suc m * n = m * n + n) ∧ ∀m n. m * suc n = m + m * n