Package list-set: List to set conversions

Information

name	list-set
version	1.30
description	List to set conversions
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool natural set list-def list-thm list-length
show	Data.Bool Data.List Number.Natural Set

Files

Package tarball list-set-1.30.tgz
Theory file list-set.thy (included in the package tarball)

Defined Constants

Data
- List
  - fromSet
  - toSet

Theorems

⊦ toSet [] = ∅

⊦ ∀l. finite (toSet l)

⊦ ∀l. size (toSet l) ≤ length l

⊦ ∀l. toSet l = ∅ ⇔ l = []

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

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

⊦ ∀h t. toSet (h :: t) = insert h (toSet t)

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

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - []
  - length
Number
- Natural
  - ≤
  - suc
  - zero
Set
- ∅
- finite
- insert
- ∈
- size

Assumptions

⊦ ⊤

⊦ finite ∅

⊦ ¬⊥ ⇔ ⊤

⊦ length [] = 0

⊦ size ∅ = 0

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀x s. ¬(insert x s = ∅)

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

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

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

⊦ ∀x s.
finite s ⇒ size (insert x s) = if x ∈ s then size s else suc (size s)

⊦ ∀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)

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