Package list-take-drop: Definitions and theorems about the list take and drop functions

Information

namelist-take-drop
version1.13
descriptionDefinitions and theorems about the list take and drop functions
authorJoe Hurd <joe@gilith.com>
licenseMIT
showData.Bool
Data.List
Number.Natural

Files

Defined Constants

Theorems

l. drop 0 l = l

l. take 0 l = []

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

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

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

(l. drop 0 l = l)
  n h t. n length t drop (suc n) (h :: t) = drop n t

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

(l. take 0 l = [])
  n h t. n length t take (suc n) (h :: t) = h :: take n t

Input Type Operators

Input Constants

Assumptions

T

n. n n

F p. p

(¬) = λp. p F

() = λP. P ((select) P)

t. (x. t) t

() = λp. p = λx. T

x. x = x T

n. ¬(suc n = 0)

n. n - n = 0

l. l @ [] = l

() = λp q. p q p

h t. head (h :: t) = h

h t. tail (h :: t) = t

l. length l = 0 l = []

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

m n. m + n - m = n

() = λp q. (λf. f p q) = λf. f T T

() = λP. q. (x. P x q) q

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. m n d. n = m + d

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

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

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

length [] = 0 h t. length (h :: t) = suc (length t)

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

e f. fn. fn 0 = e n. fn (suc n) = f (fn n) 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. 0 + n = n) m n. suc m + n = suc (m + n)

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

(m. m < 0 F) m n. m < suc n m = n m < n

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

(m. m 0 m = 0) m n. m suc n m = suc n m n

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 t T) (t F ¬t)