Package list-filter-thm: list-filter-thm

Information

name	list-filter-thm
version	1.12
description	list-filter-thm
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-09-21
show	Data.Bool

Files

Package tarball list-filter-thm-1.12.tgz
Theory file list-filter-thm.thy (included in the package tarball)

Theorems

⊦ ∀p l.
Number.Natural.≤ (Data.List.length (Data.List.filter p l))
(Data.List.length l)

⊦ ∀p l. Set.⊆ (Data.List.toSet (Data.List.filter p l)) (Data.List.toSet l)

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

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

Input Type Operators

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

Input Constants

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

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ n n

⊦ ∀s. Set.⊆ s s

⊦ F ⇔ ∀p. p

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

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀x s. Set.∪ (Set.insert x Set.∅) s = Set.insert x s

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

⊦ ∀f g x. Function.∘ f g x = f (g x)

⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

⊦ ∀s t u. Set.⊆ s t ∧ Set.⊆ t u ⇒ Set.⊆ s u

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

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

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

⊦ (∀s t. Set.⊆ s (Set.∪ s t)) ∧ ∀s t. Set.⊆ s (Set.∪ t s)

⊦ ∀s t u. Set.⊆ (Set.∪ s t) u ⇔ Set.⊆ s u ∧ Set.⊆ t u

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

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

⊦ (∀l. Data.List.@ Data.List.[] l = l) ∧
∀l h t.
Data.List.@ (Data.List.:: h t) l = Data.List.:: h (Data.List.@ t l)

⊦ (∀f. Data.List.map f Data.List.[] = Data.List.[]) ∧
  ∀f h t.
    Data.List.map f (Data.List.:: h t) =
    Data.List.:: (f h) (Data.List.map f t)

⊦ (∀P. Data.List.filter P Data.List.[] = Data.List.[]) ∧
  ∀P h t.
    Data.List.filter P (Data.List.:: h t) =
    if P h then Data.List.:: h (Data.List.filter P t)
    else Data.List.filter P t

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