Package list-filter: The list filter function

Information

name	list-filter
version	1.28
description	The list filter function
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool function natural set list-def list-length list-set list-append list-map
show	Data.Bool Data.List Function Number.Natural Set

Files

Package tarball list-filter-1.28.tgz
Theory file list-filter.thy (included in the package tarball)

Defined Constant

Data
- List
  - filter

Theorems

⊦ ∀P. filter P [] = []

⊦ ∀p l. length (filter p l) ≤ length l

⊦ ∀p l. toSet (filter p l) ⊆ toSet l

⊦ ∀P l1 l2. filter P (l1 @ l2) = filter P l1 @ filter P l2

⊦ ∀P f l. filter P (map f l) = map f (filter (P ∘ f) l)

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

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ¬
  - cond
  - F
  - T
- List
  - ::
  - @
  - []
  - length
  - map
  - toSet
Function
- ∘
Number
- Natural
  - ≤
  - suc
Set
- ∅
- insert
- ⊆
- ∪

Assumptions

⊦ T

⊦ ∀n. n ≤ n

⊦ ∀s. s ⊆ s

⊦ F ⇔ ∀p. p

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀t. T ∧ t ⇔ t

⊦ ∀l. [] @ l = l

⊦ ∀f. map f [] = []

⊦ ∀s t. s ⊆ s ∪ t

⊦ ∀s t. s ⊆ t ∪ s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t1 t2. (if F then t1 else t2) = t2

⊦ ∀t1 t2. (if T then t1 else t2) = t1

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

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

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

⊦ ∀x s. insert x ∅ ∪ s = insert x s

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀s t u. s ⊆ t ∧ t ⊆ u ⇒ s ⊆ u

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

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

⊦ ∀s t u. s ∪ t ⊆ u ⇔ s ⊆ u ∧ t ⊆ u

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

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

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