Package list-quant-thm: Properties of list quantifiers

Information

name	list-quant-thm
version	1.33
description	Properties of list quantifiers
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2012-02-10
requires	bool function set list-def list-set list-append list-map list-quant-def
show	Data.Bool Data.List Function Set

Files

Package tarball list-quant-thm-1.33.tgz
Theory file list-quant-thm.thy (included in the package tarball)

Theorems

⊦ ∀l. all (λx. ⊤) l

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

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

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

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

⊦ ∀P f l. all P (map f l) ⇔ all (P ∘ f) l

⊦ ∀P f l. exists P (map f l) ⇔ exists (P ∘ f) l

⊦ ∀P l. (∀x. all (P x) l) ⇔ all (λs. ∀x. P x s) l

⊦ ∀P l. (∃x. exists (P x) l) ⇔ exists (λs. ∃x. P x s) l

⊦ ∀P l1 l2. all P (l1 @ l2) ⇔ all P l1 ∧ all P l2

⊦ ∀P Q l. (∀x. P x ⇒ Q x) ⇒ all P l ⇒ all Q l

⊦ ∀f g l. all (λx. f x = g x) l ⇒ map f l = map g l

⊦ ∀P Q l. all P l ∧ all Q l ⇔ all (λx. P x ∧ Q x) l

⊦ ∀P Q l. all (λx. P x ⇒ Q x) l ∧ all P l ⇒ all Q l

Input Type Operators

→
bool
Data
- List
  - list
Set
- set

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - all
  - exists
  - map
  - toSet
Function
- id
- ∘
Set
- ∅
- insert
- ∈

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ toSet [] = ∅

⊦ ∀p. all p []

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀x. id x = x

⊦ ∀p. ¬exists p []

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀a. ∃x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀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. [] @ l = l

⊦ ∀f. map f [] = []

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ (∧) = 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

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

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t1 t2. ¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2

⊦ ∀t1 t2. ¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀p q. (∀x. p x ⇒ q) ⇔ (∃x. p x) ⇒ q

⊦ ∀p q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

⊦ ∀t1 t2 t3. (t1 ∧ t2) ∧ t3 ⇔ t1 ∧ t2 ∧ t3

⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

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

⊦ ∀f h t. map f (h :: t) = f h :: map f t

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

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

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

⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x

⊦ ∀p q. (∃x. p x ∨ q x) ⇔ (∃x. p x) ∨ ∃x. q x

⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x

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