Package set-fold-thm: Properties of a fold operation on finite sets

Information

name	set-fold-thm
version	1.25
description	Properties of a fold operation on finite sets
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2012-03-18
requires	bool set-finite set-fold-def set-thm
show	Data.Bool Set

Files

Package tarball set-fold-thm-1.25.tgz
Theory file set-fold-thm.thy (included in the package tarball)

Theorems

⊦ ∀f b.
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ⇒
    fold f b ∅ = b ∧
    ∀x s.
      finite s ⇒
      fold f b s =
      if x ∈ s then f x (fold f b (delete s x)) else fold f b (delete s x)

⊦ ∀f g b s.
    finite s ∧ (∀x. x ∈ s ⇒ f x = g x) ∧
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ∧
    (∀x y s. ¬(x = y) ⇒ g x (g y s) = g y (g x s)) ⇒
    fold f b s = fold g b s

Input Type Operators

→
bool
Set
- set

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
Set
- ∅
- delete
- finite
- fold
- insert
- ∈

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ ∀t. t ⇒ t

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (∀x. t) ⇔ 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

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀s x. finite s ⇒ finite (delete s x)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀x s. delete s x = s ⇔ ¬(x ∈ s)

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

⊦ ∀x s. x ∈ s ⇒ insert x (delete s x) = s

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀p q. (∀x. p ⇒ q x) ⇔ p ⇒ ∀x. q x

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

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

⊦ ∀s x y. x ∈ delete s y ⇔ x ∈ s ∧ ¬(x = y)

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

⊦ ∀f b.
    (∀x y s. ¬(x = y) ⇒ f x (f y s) = f y (f x s)) ⇒
    fold f b ∅ = b ∧
    ∀x s.
      finite s ⇒
      fold f b (insert x s) =
      if x ∈ s then fold f b s else f x (fold f b s)