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

Information

Files

• Package tarball list-filter-thm-1.11.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.o 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.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

⊦ ∀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.o 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)