Package list-set-thm: list-set-thm

Information

Files

• Package tarball list-set-thm-1.9.tgz
• Theory file list-set-thm.thy (included in the package tarball)

Theorems

⊦ ∀l. Set.finite (Data.List.toSet l)

⊦ ∀l. Number.Natural.≤ (Set.size (Data.List.toSet l)) (Data.List.length l)

⊦ ∀l. Data.List.toSet l = Set.∅ ⇔ l = Data.List.[]

⊦ ∀s. Set.finite s ⇒ Data.List.toSet (Data.List.fromSet s) = s

⊦ ∀s. Set.finite s ⇒ Data.List.length (Data.List.fromSet s) = Set.size s

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.fromSet
    • Data.List.length
    • Data.List.toSet
• Number
  • Natural
    • Number.Natural.≤
    • Number.Natural.suc
    • Number.Natural.zero
• Set
  • Set.∅
  • Set.finite
  • Set.∈
  • Set.insert
  • Set.size

Assumptions

⊦ T

⊦ Set.finite Set.∅

⊦ Set.size Set.∅ = 0

⊦ ∀n. Number.Natural.≤ 0 n

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

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

⊦ (~) = λp. p ⇒ F

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. ¬(Data.List.:: h t = Data.List.[])

⊦ ∀x s. ¬(Set.insert x s = Set.∅)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀s x. Set.finite (Set.insert x s) ⇔ Set.finite s

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

⊦ ∀m n p.
    Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

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

⊦ ∀s.
    Set.finite s ⇒
    Data.List.toSet (Data.List.fromSet s) = s ∧
    Data.List.length (Data.List.fromSet s) = Set.size s

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

⊦ ∀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)

⊦ Set.size Set.∅ = 0 ∧
  ∀x s.
    Set.finite s ⇒
    Set.size (Set.insert x s) =
    if Set.∈ x s then Set.size s else Number.Natural.suc (Set.size s)

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

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

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