Package natural-factorial-thm: natural-factorial-thm

Information

name	natural-factorial-thm
version	1.0
description	natural-factorial-thm
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-02-19
show	Data.Bool

Files

Package tarball natural-factorial-thm-1.0.tgz
Theory file natural-factorial-thm.thy (included in the package tarball)

Theorems

⊦ ∀n. Number.Natural.< Number.Numeral.zero (Number.Natural.factorial n)

⊦ ∀n. ¬(Number.Natural.factorial n = Number.Numeral.zero)

⊦ ∀n.
Number.Natural.≤ (Number.Numeral.bit1 Number.Numeral.zero)
(Number.Natural.factorial n)

⊦ ∀m n.
    Number.Natural.≤ m n ⇒
    Number.Natural.≤ (Number.Natural.factorial m)
      (Number.Natural.factorial n)

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - F
  - T
Number
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.<
  - Number.Natural.≤
  - Number.Natural.factorial
  - Number.Natural.suc
- Numeral
  - Number.Numeral.bit1
  - Number.Numeral.zero

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ Number.Numeral.zero n

⊦ ∀n. Number.Natural.≤ n n

⊦ Number.Numeral.bit1 Number.Numeral.zero =
Number.Natural.suc Number.Numeral.zero

⊦ ∀n. Number.Natural.< Number.Numeral.zero (Number.Natural.suc n)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀n. Number.Natural.< Number.Numeral.zero n ⇔ ¬(n = Number.Numeral.zero)

⊦ ∀m n. Number.Natural.≤ (Number.Natural.suc m) n ⇔ Number.Natural.< m n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀m n.
Number.Natural.≤ (Number.Natural.suc m) (Number.Natural.suc n) ⇔
Number.Natural.≤ m n

⊦ ∀m n. Number.Natural.≤ m n ⇔ ∃d. n = Number.Natural.+ m d

⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

⊦ ∀m n.
    Number.Natural.< Number.Numeral.zero (Number.Natural.* m n) ⇔
    Number.Natural.< Number.Numeral.zero m ∧
    Number.Natural.< Number.Numeral.zero n

⊦ ∀P.
P Number.Numeral.zero ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ Number.Natural.factorial Number.Numeral.zero =
  Number.Numeral.bit1 Number.Numeral.zero ∧
  ∀n.
    Number.Natural.factorial (Number.Natural.suc n) =
    Number.Natural.* (Number.Natural.suc n) (Number.Natural.factorial n)

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.* m p) (Number.Natural.* n p) ⇔
Number.Natural.≤ m n ∨ p = Number.Numeral.zero

⊦ ∀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. Number.Natural.+ Number.Numeral.zero n = n) ∧
  (∀m. Number.Natural.+ m Number.Numeral.zero = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)

⊦ (∀n. Number.Natural.* Number.Numeral.zero n = Number.Numeral.zero) ∧
  (∀m. Number.Natural.* m Number.Numeral.zero = Number.Numeral.zero) ∧
  (∀n. Number.Natural.* (Number.Numeral.bit1 Number.Numeral.zero) n = n) ∧
  (∀m. Number.Natural.* m (Number.Numeral.bit1 Number.Numeral.zero) = m) ∧
  (∀m n.
     Number.Natural.* (Number.Natural.suc m) n =
     Number.Natural.+ (Number.Natural.* m n) n) ∧
  ∀m n.
    Number.Natural.* m (Number.Natural.suc n) =
    Number.Natural.+ m (Number.Natural.* m n)