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

Information

name	list-take-drop
version	1.13
description	Definitions and theorems about the list take and drop functions
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List Number.Natural

Files

Package tarball list-take-drop-1.13.tgz
Theory file list-take-drop.thy (included in the package tarball)

Defined Constants

Data
- List
  - drop
  - take

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

→
bool
Data
- List
  - list
Number
- Natural
  - natural

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - F
  - T
- List
  - ::
  - @
  - []
  - head
  - length
  - nth
  - tail
Number
- Natural
  - +
  - -
  - <
  - ≤
  - suc
  - zero

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)