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

Information

Files

• Package tarball list-filter-1.11.tgz
• Theory file list-filter.thy (included in the package tarball)

Defined Constant

• Data
  • List
    • filter

Theorems

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

⊦ ∀p l. Set.⊆ (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 o f) l)

⊦ (∀P. filter P [] = []) ∧
  ∀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
    • Number.Natural.natural
• Set
  • Set.set

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ~
    • cond
    • F
    • T
  • List
    • ::
    • @
    • []
    • length
    • map
    • toSet
• Function
  • o
• 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

⊦ (∃) = λP. P ((select) P)

⊦ ∀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. (f o 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

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

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

⊦ (∀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)