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

Information

name	list-nub
version	1.15
description	Definitions and theorems about the list nub function
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List

Files

Package tarball list-nub-1.15.tgz
Theory file list-nub.thy (included in the package tarball)

Defined Constants

Data
- List
  - nub
  - nubReverse

Theorems

⊦ ∀l. nub (nub l) = nub l

⊦ ∀l. nubReverse (nubReverse l) = nubReverse l

⊦ ∀l. toSet (nub l) = toSet l

⊦ ∀l. toSet (nubReverse l) = toSet l

⊦ ∀l. Number.Natural.≤ (length (nub l)) (length l)

⊦ ∀l. Number.Natural.≤ (length (nubReverse l)) (length l)

⊦ ∀l. nub l = reverse (nubReverse (reverse l))

⊦ ∀l x. member x (nub l) ⇔ member x l

⊦ ∀l x. member x (nubReverse l) ⇔ member x l

⊦ nubReverse [] = [] ∧
  ∀h t.
    nubReverse (h :: t) =
    if member h t then nubReverse t else h :: nubReverse t

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - F
  - T
- List
  - ::
  - []
  - length
  - member
  - reverse
  - toSet
Number
- Natural
  - Number.Natural.≤
  - Number.Natural.suc
  - Number.Natural.zero
Set
- Set.∈

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀l. reverse (reverse l) = l

⊦ ∀l. length (reverse l) = length l

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀x l. member x l ⇔ Set.∈ x (toSet l)

⊦ ∀l x. member x (reverse l) ⇔ member x l

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀m n.
Number.Natural.≤ (Number.Natural.suc m) (Number.Natural.suc n) ⇔
Number.Natural.≤ m n

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

⊦ ∀s t. s = t ⇔ ∀x. Set.∈ x s ⇔ Set.∈ x t

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

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

⊦ ∀NIL' CONS'.
∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)

⊦ (∀x. member x [] ⇔ F) ∧ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t

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

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

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)