Package list-member-thm: Properties of the list member function

Information

name	list-member-thm
version	1.39
description	Properties of the list member function
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2012-02-10
requires	bool function natural set list-def list-length list-set list-append list-map list-quant list-filter list-reverse list-nth list-member-def
show	Data.Bool Data.List Function Number.Natural Set

Files

Package tarball list-member-thm-1.39.tgz
Theory file list-member-thm.thy (included in the package tarball)

Theorems

⊦ ∀x l. member x l ⇔ x ∈ toSet l

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

⊦ ∀l n. n < length l ⇒ member (nth n l) l

⊦ ∀s. finite s ⇒ ∀x. member x (fromSet s) ⇔ x ∈ s

⊦ ∀P l. (∀x. member x l ⇒ P x) ⇔ all P l

⊦ ∀P l. (∃x. P x ∧ member x l) ⇔ exists P l

⊦ ∀P l x. member x (filter P l) ⇔ P x ∧ member x l

⊦ ∀x l1 l2. member x (l1 @ l2) ⇔ member x l1 ∨ member x l2

⊦ ∀l x. member x l ⇔ ∃i. i < length l ∧ x = nth i l

⊦ ∀f y l. member y (map f l) ⇔ ∃x. member x l ∧ y = f x

⊦ ∀P Q l. (∀x. member x l ∧ P x ⇒ Q x) ∧ all P l ⇒ all Q l

⊦ ∀P Q l. (∀x. member x l ∧ P x ⇒ Q x) ∧ exists P l ⇒ exists Q l

Input Type Operators

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

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - all
  - exists
  - filter
  - fromSet
  - length
  - map
  - member
  - nth
  - reverse
  - toSet
Function
- id
Number
- Natural
  - <
  - suc
  - zero
Set
- ∅
- finite
- insert
- ∈

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ reverse [] = []

⊦ toSet [] = ∅

⊦ ∀p. all p []

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬member x []

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀x. id x = x

⊦ ∀n. 0 < suc n

⊦ ∀p. ¬exists p []

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. ⊥ ∧ t ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊥ ⇔ ⊥

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀m. m < 0 ⇔ ⊥

⊦ ∀l. [] @ l = l

⊦ ∀f. map f [] = []

⊦ ∀P. filter P [] = []

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀s. finite s ⇒ toSet (fromSet s) = s

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀h t. nth 0 (h :: t) = h

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀h t. toSet (h :: t) = insert h (toSet t)

⊦ ∀m n. suc m < suc n ⇔ m < n

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

⊦ ∀x l. reverse (x :: l) = reverse l @ x :: []

⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀p q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀p h t. all p (h :: t) ⇔ p h ∧ all p t

⊦ ∀p h t. exists p (h :: t) ⇔ p h ∨ exists p t

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

⊦ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t

⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s

⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x

⊦ ∀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

⊦ ∀P h t. filter P (h :: t) = if P h then h :: filter P t else filter P t

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