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

Information

namelist-take-drop-thm
version1.52
descriptionProperties of the list take and drop functions
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2012-11-10
requiresbool
list-append
list-def
list-length
list-nth
list-take-drop-def
list-thm
natural
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 n + length (drop n l) = length l

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

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

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

n l i.
    n length l i < length l
    nth l i = if i < n then nth (take n l) i else nth (drop n l) (i - n)

Input Type Operators

Input Constants

Assumptions

length [] = 0

n. n n

p. p

(¬) = λp. p

t. (x. t) t

() = λp. p = λx.

t. t t

t. t t

t. t

t. t t

t. t

n. ¬(suc n = 0)

n. n - n = 0

l. [] @ l = l

l. l @ [] = l

l. drop 0 l = l

l. take 0 l = []

t. (t ) ¬t

m n. m m + n

() = λp q. p q p

m. suc m = m + 1

m. m 0 m = 0

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

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

l. length l = 0 l = []

x y. x = y y = x

m n. m + n - m = n

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

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

m n. suc m suc n m n

l1 l2. length (l1 @ l2) = length l1 + length l2

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

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 [] (h t. p t p (h :: t)) l. p l

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

l1 l2 k.
    k < length l1 + length l2
    nth (l1 @ l2) k =
    if k < length l1 then nth l1 k else nth l2 (k - length l1)