Package list-nub-thm: list-nub-thm

Information

Files

• Package tarball list-nub-thm-1.13.tgz
• Theory file list-nub-thm.thy (included in the package tarball)

Theorems

⊦ ∀l. Data.List.nub (Data.List.nub l) = Data.List.nub l

⊦ ∀l.
    Data.List.nubReverse (Data.List.nubReverse l) = Data.List.nubReverse l

⊦ ∀l. Data.List.toSet (Data.List.nub l) = Data.List.toSet l

⊦ ∀l. Data.List.toSet (Data.List.nubReverse l) = Data.List.toSet l

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

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

⊦ ∀l x. Data.List.member x (Data.List.nub l) ⇔ Data.List.member x l

⊦ ∀l x. Data.List.member x (Data.List.nubReverse l) ⇔ Data.List.member x 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.length
    • Data.List.member
    • Data.List.nub
    • Data.List.nubReverse
    • Data.List.reverse
    • Data.List.toSet
• Number
  • Natural
    • Number.Natural.≤
    • Number.Natural.suc
    • Number.Natural.zero
• Set
  • Set.∈

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (~) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀l. Data.List.reverse (Data.List.reverse l) = l

⊦ ∀l. Data.List.length (Data.List.reverse l) = Data.List.length l

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀l.
    Data.List.nub l =
    Data.List.reverse (Data.List.nubReverse (Data.List.reverse l))

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀x l. Data.List.member x l ⇔ Set.∈ x (Data.List.toSet l)

⊦ ∀l x. Data.List.member x (Data.List.reverse l) ⇔ Data.List.member x l

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

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

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀s t. s = t ⇔ ∀x. Set.∈ x s ⇔ Set.∈ x t

⊦ ∀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)

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

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

⊦ (∀x. Data.List.member x Data.List.[] ⇔ F) ∧
  ∀x h t.
    Data.List.member x (Data.List.:: h t) ⇔ x = h ∨ Data.List.member x t

⊦ (∀m. Number.Natural.≤ m 0 ⇔ m = 0) ∧
  ∀m n.
    Number.Natural.≤ m (Number.Natural.suc n) ⇔
    m = Number.Natural.suc n ∨ Number.Natural.≤ m n

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

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