Package natural-even-odd: Definitions and theorems about natural number even and odd

Information

Files

• Package tarball natural-even-odd-1.0.tgz
• Theory file natural-even-odd.thy (included in the package tarball)

Defined Constants

• Number
  • Natural
    • even
    • odd

Theorems

⊦ ∀n. even n ∨ odd n

⊦ ∀n. ¬(even n ∧ odd n)

⊦ ∀n. even (2 * n)

⊦ ∀n. ¬even n ⇔ odd n

⊦ ∀n. ¬odd n ⇔ even n

⊦ ∀n. odd (suc (2 * n))

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀m n. odd (m * n) ⇔ odd m ∧ odd n

⊦ ∀n. even n ⇔ ∃m. n = 2 * m

⊦ (even 0 ⇔ T) ∧ ∀n. even (suc n) ⇔ ¬even n

⊦ (odd 0 ⇔ F) ∧ ∀n. odd (suc n) ⇔ ¬odd n

⊦ ∀m n. odd (m + n) ⇔ ¬(odd m ⇔ odd n)

⊦ ∀m n. odd (exp m n) ⇔ odd m ∨ n = 0

⊦ ∀n. odd n ⇔ ∃m. n = suc (2 * m)

⊦ ∀m n. even (exp m n) ⇔ even m ∧ ¬(n = 0)

⊦ ∀n. (∃k m. odd m ∧ n = exp 2 k * m) ⇔ ¬(n = 0)

⊦ ∀n. (even n ⇒ ∃m. n = 2 * m) ∧ (¬even n ⇒ ∃m. n = suc (2 * m))

Input Type Operators

• →
• bool
• Number
  • Natural
    • natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • select
    • F
    • T
• Number
  • Natural
    • *
    • +
    • <
    • exp
    • suc
  • Numeral
    • bit0
    • bit1
    • zero

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ 1 = suc 0

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(suc n = 0)

⊦ 2 = suc 1

⊦ ∀n. bit0 n = n + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. bit1 n = suc (n + n)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀n. 2 * n = n + n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

⊦ ∀P. (∃x y. P x y) ⇔ ∃y x. P x y

⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x

⊦ ∀P Q. (∃x. P x) ∧ Q ⇔ ∃x. P x ∧ Q

⊦ ∀P Q. (∃x. P x) ∨ Q ⇔ ∃x. P x ∨ Q

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n. exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀P. (∀n. (∀m. m < n ⇒ P m) ⇒ P n) ⇒ ∀n. P n

⊦ ∀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

⊦ ∀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)

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

⊦ (∀m. m < 0 ⇔ F) ∧ ∀m n. m < suc n ⇔ m = n ∨ m < n

⊦ ∀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)

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

⊦ ∀p q r.
    (p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
    (p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)

⊦ (∀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