Package natural-exp-thm: natural-exp-thm

Information

name	natural-exp-thm
version	1.0
description	natural-exp-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-exp-thm-1.0.tgz
Theory file natural-exp-thm.thy (included in the package tarball)

Theorems

⊦ ∀n. Number.Natural.exp n (Number.Numeral.bit1 Number.Numeral.zero) = n

⊦ ∀n.
Number.Natural.exp (Number.Numeral.bit1 Number.Numeral.zero) n =
Number.Numeral.bit1 Number.Numeral.zero

⊦ ∀n.
    Number.Natural.exp n
      (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero)) =
    Number.Natural.* n n

⊦ ∀n.
    Number.Natural.exp Number.Numeral.zero n =
    (if n = Number.Numeral.zero
     then Number.Numeral.bit1 Number.Numeral.zero else Number.Numeral.zero)

⊦ ∀m n p.
Number.Natural.exp m (Number.Natural.* n p) =
Number.Natural.exp (Number.Natural.exp m n) p

⊦ ∀m n.
Number.Natural.exp m n = Number.Numeral.zero ⇔
m = Number.Numeral.zero ∧ ¬(n = Number.Numeral.zero)

⊦ ∀m n p.
Number.Natural.exp m (Number.Natural.+ n p) =
Number.Natural.* (Number.Natural.exp m n) (Number.Natural.exp m p)

⊦ ∀p m n.
Number.Natural.exp (Number.Natural.* m n) p =
Number.Natural.* (Number.Natural.exp m p) (Number.Natural.exp n p)

⊦ ∀x n.
Number.Natural.exp x n = Number.Numeral.bit1 Number.Numeral.zero ⇔
x = Number.Numeral.bit1 Number.Numeral.zero ∨ n = Number.Numeral.zero

Input Type Operators

→
bool
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∨
  - ¬
  - cond
  - F
  - T
Number
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.exp
  - Number.Natural.suc
- Numeral
  - Number.Numeral.bit0
  - Number.Numeral.bit1
  - Number.Numeral.zero

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

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

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

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

⊦ ∀n. Number.Numeral.bit0 n = Number.Natural.+ n n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. Number.Numeral.bit1 n = Number.Natural.suc (Number.Natural.+ n n)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n. Number.Natural.* m n = Number.Natural.* n m

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀m n.
Number.Natural.+ m n = Number.Numeral.zero ⇔
m = Number.Numeral.zero ∧ n = Number.Numeral.zero

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

⊦ ∀m n.
    Number.Natural.* m n = Number.Numeral.bit1 Number.Numeral.zero ⇔
    m = Number.Numeral.bit1 Number.Numeral.zero ∧
    n = Number.Numeral.bit1 Number.Numeral.zero

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

⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

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

⊦ ∀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)

⊦ ∀m n p.
    Number.Natural.* m n = Number.Natural.* n m ∧
    Number.Natural.* (Number.Natural.* m n) p =
    Number.Natural.* m (Number.Natural.* n p) ∧
    Number.Natural.* m (Number.Natural.* n p) =
    Number.Natural.* n (Number.Natural.* m p)

⊦ (∀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)