Package list-set: List to set conversions

Information

name	list-set
version	1.36
description	List to set conversions
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool list-def list-dest list-length natural set
show	Data.Bool Data.List Number.Natural Set

Files

Package tarball list-set-1.36.tgz
Theory file list-set.thy (included in the package tarball)

Defined Constants

Data
- List
  - all
  - exists
  - fromSet
  - member
  - toSet

Theorems

⊦ toSet [] = ∅

⊦ ∀l. finite (toSet l)

⊦ ∀p. all p []

⊦ ∀l. all (λx. ⊤) l

⊦ ∀x. ¬member x []

⊦ ∀p. ¬exists p []

⊦ ∀l. size (toSet l) ≤ length l

⊦ ∀l. all (λx. ⊥) l ⇔ null l

⊦ ∀l. toSet l = ∅ ⇔ null l

⊦ ∀l. toSet l = ∅ ⇔ l = []

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

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

⊦ ∀l x. member x l ⇔ x ∈ toSet l

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

⊦ ∀p l. ¬all p l ⇔ exists (λx. ¬p x) l

⊦ ∀p l. ¬exists p l ⇔ all (λx. ¬p x) l

⊦ ∀p l. ¬all (λx. ¬p x) l ⇔ exists p l

⊦ ∀p l. ¬exists (λx. ¬p x) l ⇔ all p l

⊦ ∀s. finite s ⇒ ∀x. member x (fromSet s) ⇔ x ∈ s

⊦ ∀p l. (∀x. member x l ⇒ p x) ⇔ all p l

⊦ ∀p l. (∃x. member x l ∧ p x) ⇔ exists p l

⊦ ∀p l. all p l ⇔ ∀x. x ∈ toSet l ⇒ p x

⊦ ∀p l. exists p l ⇔ ∃x. x ∈ toSet l ∧ p x

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

⊦ ∀p h t. all p (h :: t) ⇔ p h ∧ all p t

⊦ ∀p h t. exists p (h :: t) ⇔ p h ∨ exists p t

⊦ ∀p l. (∀x. all (p x) l) ⇔ all (λy. ∀x. p x y) l

⊦ ∀p l. (∃x. exists (p x) l) ⇔ exists (λy. ∃x. p x y) l

⊦ ∀x h t. member x (h :: t) ⇔ x = h ∨ member x t

⊦ ∀p q l. all (λx. p x ∧ q x) l ⇔ all p l ∧ all q l

⊦ ∀p q l. all (λx. p x ⇒ q x) l ∧ all p l ⇒ all q l

⊦ ∀p q l. (∀x. member x l ∧ p x ⇒ q x) ∧ all p l ⇒ all q l

⊦ ∀p q l. (∀x. member x l ∧ p x ⇒ q x) ∧ exists p l ⇒ exists q l

Input Type Operators

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

Input Constants

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

Assumptions

⊦ ⊤

⊦ null []

⊦ finite ∅

⊦ ¬⊥ ⇔ ⊤

⊦ length [] = 0

⊦ size ∅ = 0

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. (t ⇔ ⊤) ⇔ t

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀l. null l ⇔ l = []

⊦ ∀h t. ¬null (h :: t)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀x s. ¬(insert x s = ∅)

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

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

⊦ ∀s. (∃x. x ∈ s) ⇔ ¬(s = ∅)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀t₁ t₂. ¬(t₁ ⇒ t₂) ⇔ t₁ ∧ ¬t₂

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀t₁ t₂. ¬(t₁ ∨ t₂) ⇔ ¬t₁ ∧ ¬t₂

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

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀p. (∃x y. p x y) ⇔ ∃y x. p x y

⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x

⊦ ∀p q. p ⇒ (∀x. q x) ⇔ ∀x. p ⇒ q x

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

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

⊦ ∀x y s. x ∈ insert y s ⇔ x = y ∨ x ∈ s

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀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)

⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)

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