Package list-nth: Definitions and theorems about the list nth function

Information

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

Files

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

Defined Constant

Data
- List
  - nth

Theorems

⊦ ∀h t. nth 0 (h :: t) = h

⊦ ∀l i. i < length l ⇒ Set.∈ (nth i l) (toSet l)

⊦ ∀l. ¬(l = []) ⇒ last l = nth (length l - 1) l

⊦ ∀P l. all P l ⇔ ∀i. i < length l ⇒ P (nth i l)

⊦ ∀P l. exists P l ⇔ ∃i. i < length l ∧ P (nth i l)

⊦ ∀h t n. n < length t ⇒ nth (suc n) (h :: t) = nth n t

⊦ ∀x l. Set.∈ x (toSet l) ⇔ ∃i. i < length l ∧ x = nth i l

⊦ ∀f l i. i < length l ⇒ nth i (map f l) = f (nth i l)

⊦ ∀l m.
length l = length m ∧ (∀i. i < length l ⇒ nth i l = nth i m) ⇒ l = m

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

⊦ ∀k l m.
k < length l + length m ⇒
nth k (l @ m) = if k < length l then nth k l else nth (k - length l) m

Input Type Operators

→
bool
Data
- List
  - list
Number
- Natural
  - natural
Set
- Set.set

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - F
  - T
- List
  - ::
  - @
  - []
  - all
  - exists
  - head
  - last
  - length
  - map
  - tail
  - toSet
Number
- Natural
  - +
  - -
  - <
  - ≤
  - bit1
  - suc
  - zero
Set
- Set.∅
- Set.insert
- Set.∈

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ 1 = suc 0

⊦ ∀x. ¬Set.∈ x Set.∅

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(suc n = 0)

⊦ ∀m. m - 0 = m

⊦ ∀n. n - n = 0

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀n. suc n - 1 = n

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀h t. head (h :: t) = h

⊦ ∀h t. tail (h :: t) = t

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀l. length l = 0 ⇔ l = []

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀l f. length (map f l) = length l

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀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

⊦ ∀l m. length (l @ m) = length l + length m

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

⊦ ∀P. (∀x y. P x y) ⇔ ∀y x. P x y

⊦ ∀h t. last (h :: t) = if t = [] then h else last t

⊦ ∀P l. all P l ⇔ ∀x. Set.∈ x (toSet l) ⇒ P x

⊦ ∀P l. exists P l ⇔ ∃x. Set.∈ x (toSet l) ∧ P x

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀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

⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet t)

⊦ ∀x y s. Set.∈ x (Set.insert y s) ⇔ x = y ∨ Set.∈ x s

⊦ ∀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)

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

⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

⊦ (∀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

⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f 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 ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)