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

Information

Files

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

Theorems

⊦ ∀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.o P f) l)

Input Type Operators

• →
• bool
• Data
  • List
    • Data.List.list

Input Constants

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

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ (¬) = 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

⊦ ∀f g x. Function.o f g x = f (g x)

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

⊦ ∀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) ∧
  ∀h t l.
    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.[]) ∧
  ∀h P 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)