Package natural-div-def: Definition of natural number division

Information

namenatural-div-def
version1.31
descriptionDefinition of natural number division
authorJoe Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2012-06-16
requiresbool
natural-add
natural-mult
natural-order
natural-thm
showData.Bool
Number.Natural

Files

Defined Constants

Theorems

even 0

odd 0

n. even (suc n) ¬even n

n. odd (suc n) ¬odd n

m n. ¬(n = 0) m mod n < n

m n. ¬(n = 0) (m div n) * n + m mod n = m

Input Type Operators

Input Constants

Assumptions

¬

¬

t. t t

p. p

t. t ¬t

(¬) = λp. p

() = λp. p ((select) p)

a. x. x = a

t. (λx. t x) = t

() = λp. p = λx.

t. ¬¬t t

t. ( t) t

t. t

t. t t

t. t t

t. t

t. t t

t. t

t. t t

t. t

t. t t

t. t

n. 0 * n = 0

n. 0 + n = n

t. ( t) ¬t

t. t ¬t

m. 1 * m = m

() = λp q. p q p

t. (t ) (t )

m. m 0 m = 0

t1 t2. (if then t1 else t2) = t2

t1 t2. (if then t1 else t2) = t1

t1 t2. t1 t2 t2 t1

m n. ¬(m < n) n m

m n. ¬(m n) n < m

() = λp q. (λf. f p q) = λf. f

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

t1 t2. ¬(t1 t2) t1 ¬t2

m n. n < m + n 0 < m

m n. m n d. n = m + d

() = λp q. r. (p r) (q r) r

p q. (x. p q x) p x. q x

p q. p (x. q x) x. p q x

p q. (x. p x) q x. p x q

t1 t2 t3. (t1 t2) t3 t1 t2 t3

m n p. m + (n + p) = m + n + p

p. (x. y. p x y) y. x. p x (y x)

m n p. (m + n) * p = m * p + n * p

(∃!) = λp. () p x y. p x p y x = y

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

p. (n. p n) n. p n m. m < n ¬p m