Package list-take-drop: The list take and drop functions

Information

name	list-take-drop
version	1.57
description	The list take and drop functions
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
requires	bool list-append list-def list-dest list-length list-nth list-replicate list-thm natural
show	Data.Bool Data.List Number.Natural

Files

Package tarball list-take-drop-1.57.tgz
Theory source 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

⊦ ∀l₁ l₂. drop (length l₁) (l₁ @ l₂) = l₂

⊦ ∀l₁ l₂. take (length l₁) (l₁ @ l₂) = l₁

⊦ ∀h t. take 1 (h :: t) = h :: []

⊦ ∀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

⊦ ∀m n x. m ≤ n ⇒ take m (replicate x n) = replicate x m

⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t

⊦ ∀m n x. m ≤ n ⇒ drop m (replicate x n) = replicate x (n - m)

⊦ ∀n h t. n ≤ length t ⇒ take (suc n) (h :: t) = h :: take n t

⊦ ∀m n l. m + n ≤ length l ⇒ drop (m + n) l = drop m (drop n 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

⊦ ∀m n l. m + n ≤ length l ⇒ take (m + n) l = take m l @ take n (drop m l)

⊦ ∀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)

External Type Operators

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

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - head
  - length
  - nth
  - replicate
  - tail
Number
- Natural
  - +
  - -
  - <
  - ≤
  - bit0
  - bit1
  - suc
  - zero

Assumptions

⊦ ⊤

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) 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)

⊦ ∀m. m + 0 = m

⊦ ∀n. n - n = 0

⊦ ∀l. [] @ l = l

⊦ ∀l. l @ [] = l

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m n. m ≤ m + n

⊦ ∀m n. n ≤ 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

⊦ ∀x n. length (replicate x n) = n

⊦ ∀h t. head (h :: t) = h

⊦ ∀h t. tail (h :: t) = t

⊦ ∀l. length l = 0 ⇔ l = []

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀m n. m + n = n + m

⊦ ∀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. m + suc n = suc (m + n)

⊦ ∀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 ≤ m + p ⇔ 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

⊦ ∀l₁ l₂ l₃. l₁ @ l₂ @ l₃ = (l₁ @ l₂) @ l₃

⊦ ∀l l₁ l₂. l₁ @ l = l₂ @ l ⇔ l₁ = l₂

⊦ ∀l l₁ l₂. l₁ @ l = l₂ @ l ⇒ l₁ = l₂

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀h₁ h₂ t₁ t₂. h₁ :: t₁ = h₂ :: t₂ ⇔ h₁ = h₂ ∧ t₁ = t₂

⊦ ∀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)

⊦ ∀x n l₁ l₂.
    l₁ @ l₂ = replicate x n ⇔
    replicate x (length l₁) = l₁ ∧ replicate x (length l₂) = l₂ ∧
    length l₁ + length l₂ = n