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

Information

Files

• Package tarball list-filter-1.0.tgz
• Theory file list-filter.thy (included in the package tarball)

Defined Constant

• Data
  • List
    • filter

Theorems

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

Input Type Operators

• →
• bool
• Data
  • List
    • list

Input Constants

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

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ (¬) = 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

⊦ ∀f g x. (f o g) x = f (g x)

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

⊦ ∀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) ∧ ∀h t l. (h :: t) @ l = h :: t @ l

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