Package natural-order-min-max-thm: Properties of natural number min and max functions

Information

namenatural-order-min-max-thm
version1.35
descriptionProperties of natural number min and max functions
authorJoe Leslie-Hurd <joe@gilith.com>
licenseHOLLight
provenanceHOL Light theory extracted on 2015-04-04
checksum7b4855f1fec6c1e01d3499f60b7ce3b519386272
requiresbool
natural-order-def
natural-order-min-max-def
natural-order-thm
showData.Bool
Number.Natural

Files

Theorems

(minimal n. ) = 0

n. max 0 n = n

n. max n 0 = n

n. max n n = n

n. min 0 n = 0

n. min n 0 = 0

n. min n n = n

m n. m max m n

m n. n max m n

m n. min m n m

m n. min m n n

m n. max m n = max n m

m n. min m n = min n m

m n. max (suc m) (suc n) = suc (max m n)

m n. min (suc m) (suc n) = suc (min m n)

m n. max m n = 0 m = 0 n = 0

m n p. m < max n p m < n m < p

m n p. m < min n p m < n m < p

m n p. m max n p m n m p

m n p. m min n p m n m p

m n p. max n p < m n < m p < m

m n p. min n p < m n < m p < m

m n p. max n p m n m p m

m n p. min n p m n m p m

p n. p n (m. m < n ¬p m) (minimal) p = n

External Type Operators

External Constants

Assumptions

¬

¬

t. t t

n. 0 n

n. n n

p. p

m. ¬(m < 0)

(¬) = λp. p

t. (x. t) t

() = λp. p = λx.

t. t t

t. t t

t. t

t. t t

t. t t

t. t ¬t

() = λp q. p q p

m. m 0 m = 0

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

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

m n. m < n m n

m n. m n n m

m n. ¬(m < n) n m

m n. ¬(m n) n < m

m n. suc m n m < n

() = λp q. (λf. f p q) = λf. f

() = λp. q. (x. p x q) q

m n. suc m suc n m n

t1 t2. ¬(t1 t2) ¬t1 ¬t2

m n. max m n = if m n then n else m

m n. min m n = if m n then m else n

m n. m n n m m = n

m n p. m n n p m p

m n. m suc n m = suc n m n

p. (n. p n) p ((minimal) p) m. m < (minimal) p ¬p m

p c x y. p (if c then x else y) (c p x) (¬c p y)