Package list-take-drop-thm: Properties of the list take and drop functions

Information

namelist-take-drop-thm
version1.28
descriptionProperties of the list take and drop functions
authorJoe Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2011-11-27
requiresbool
natural
list-def
list-thm
list-length
list-append
list-nth
list-take-drop-def
showData.Bool
Data.List
Number.Natural

Files

Theorems

l. drop (length l) l = []

l. take (length l) l = l

n l. n length l length (take n l) = n

n l. n length l length (drop n l) = length l - n

n l. n length l take n l @ drop n l = l

n l i. n + i < length l nth i (drop n l) = nth (n + i) l

n l i. n length l i < n nth i (take n l) = nth i l

Input Type Operators

Input Constants

Assumptions

T

length [] = 0

n. n n

F p. p

(¬) = λp. p F

t. (x. t) t

() = λp. p = λx. T

t. F t F

t. T t t

t. t F F

t. t T t

t. F t T

t. T t t

t. t T T

n. ¬(suc n = 0)

m. m < 0 F

n. 0 + n = n

n. n - n = 0

l. [] @ l = l

l. l @ [] = l

l. drop 0 l = l

l. take 0 l = []

m n. m m + n

() = λp q. p q p

m. m 0 m = 0

l. length l = 0 l = []

h t. nth 0 (h :: t) = h

m n. m + n - m = n

h t. length (h :: t) = suc (length t)

m n. suc m n m < n

() = λp q. (λf. f p q) = λf. f T T

() = λP. q. (x. P x q) q

m n. m + suc n = suc (m + n)

m n. suc m + n = suc (m + n)

m n. suc m = suc n m = n

m n. suc m < suc n m < n

m n. suc m suc n m n

l m. length (l @ m) = length l + length m

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

m n p. m n n p m p

l h t. (h :: t) @ l = h :: t @ l

P. P 0 (n. P n P (suc n)) n. P n

P. P [] (a0 a1. P a1 P (a0 :: a1)) x. P x

h t n. n < length t nth (suc n) (h :: t) = nth n t

n h t. n length t drop (suc n) (h :: t) = drop n t

h1 h2 t1 t2. h1 :: t1 = h2 :: t2 h1 = h2 t1 = t2

n h t. n length t take (suc n) (h :: t) = h :: take n t