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

Information

Files

• Package tarball list-take-drop-1.0.tgz
• Theory file list-take-drop.thy (included in the package tarball)

Defined Constants

• Data
  • List
    • drop
    • take

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

⊦ (∀l. drop 0 l = l) ∧ ∀n h t. drop (suc n) (h :: t) = drop n t

⊦ ∀n l i. n ≤ length l ∧ i < n ⇒ nth i (take n l) = nth i l

⊦ (∀l. take 0 l = []) ∧ ∀n h t. take (suc n) (h :: t) = h :: take n t

⊦ ∀n l i.
    n ≤ length l ∧ i < length l - n ⇒ nth i (drop n l) = nth (n + i) l

Input Type Operators

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

Input Constants

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

Assumptions

⊦ T

⊦ 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)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. tail (h :: t) = t

⊦ ∀t h. head (h :: t) = h

⊦ ∀m. m = 0 ∨ ∃n. m = suc 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

⊦ ∀m n. suc m - suc n = m - n

⊦ ∀x. x = [] ∨ ∃a0 a1. x = a0 :: a1

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ 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

⊦ (∀n. 0 + n = n) ∧ ∀m n. suc m + n = suc (m + n)

⊦ (∀m. m - 0 = m) ∧ ∀m n. m - suc n = pre (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) ∧ ∀h t l. (h :: t) @ l = h :: t @ l

⊦ (∀m. m ≤ 0 ⇔ m = 0) ∧ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ (∀h t. nth 0 (h :: t) = h) ∧ ∀h t n. nth (suc n) (h :: t) = nth n t

⊦ ∀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)