Package list-set: Viewing lists as finite sets

Information

name	list-set
version	1.12
description	Viewing lists as finite sets
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List

Files

Package tarball list-set-1.12.tgz
Theory file list-set.thy (included in the package tarball)

Defined Constants

Data
- List
  - fromSet
  - toSet

Theorems

⊦ ∀l. Set.finite (toSet l)

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

⊦ ∀l. toSet l = Set.∅ ⇔ l = []

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

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

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

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

Input Type Operators

→
bool
Data
- List
  - list
Number
- Natural
  - Number.Natural.natural
Set
- Set.set

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ¬
  - cond
  - F
  - T
- List
  - ::
  - []
  - length
Number
- Natural
  - Number.Natural.≤
  - Number.Natural.suc
  - Number.Natural.zero
Set
- Set.∅
- Set.finite
- Set.insert
- Set.∈
- 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 ((select) P)

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀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

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

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

⊦ ∀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)

⊦ 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)

⊦ ∀P.
    P Set.∅ ∧
    (∀x s. P s ∧ ¬Set.∈ x s ∧ Set.finite s ⇒ P (Set.insert x s)) ⇒
    ∀s. Set.finite s ⇒ P s

⊦ ∀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)