Package relation: Relation operators

Information

name	relation
version	1.55
description	Relation operators
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
requires	bool function natural pair set
show	Data.Bool Data.Pair Function Number.Natural Relation Set

Files

Package tarball relation-1.55.tgz
Theory source file relation.thy (included in the package tarball)

Defined Constants

Number
- Natural
  - isSuc
Relation
- bigIntersect
- bigUnion
- empty
- fromSet
- intersect
- irreflexive
- measure
- reflexive
- subrelation
- toSet
- transitive
- transitiveClosure
- union
- universe
- wellFounded

Theorems

⊦ irreflexive empty

⊦ irreflexive isSuc

⊦ reflexive universe

⊦ transitive empty

⊦ transitive universe

⊦ transitive (<)

⊦ wellFounded empty

⊦ wellFounded (<)

⊦ wellFounded isSuc

⊦ subrelation isSuc (<)

⊦ empty = fromSet ∅

⊦ universe = fromSet universe

⊦ transitiveClosure isSuc = (<)

⊦ ∀m. wellFounded (measure m)

⊦ ∀r. transitive (transitiveClosure r)

⊦ ∀r. subrelation r r

⊦ ∀r. subrelation r (transitiveClosure r)

⊦ ∀x y. universe x y

⊦ ∀s. toSet (fromSet s) = s

⊦ ∀r. wellFounded r ⇒ irreflexive r

⊦ ∀r. fromSet (toSet r) = r

⊦ ∀x y. ¬empty x y

⊦ ∀r x. reflexive r ⇒ r x x

⊦ ∀r. reflexive r ⇔ ∀x. r x x

⊦ ∀s. bigIntersect s = fromSet (bigIntersect (image toSet s))

⊦ ∀s. bigUnion s = fromSet (bigUnion (image toSet s))

⊦ ∀r x. irreflexive r ⇒ ¬r x x

⊦ ∀r. irreflexive r ⇔ ∀x. ¬r x x

⊦ ∀m n. isSuc m n ⇔ suc m = n

⊦ ∀r. subrelation isSuc r ∧ transitive r ⇒ subrelation (<) r

⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r

⊦ ∀r s. subrelation r s ⇔ toSet r ⊆ toSet s

⊦ ∀r s. toSet r = toSet s ⇒ r = s

⊦ ∀r s. intersect r s = fromSet (toSet r ∩ toSet s)

⊦ ∀r s. union r s = fromSet (toSet r ∪ toSet s)

⊦ ∀r m. wellFounded r ⇒ wellFounded (λx y. r (m x) (m y))

⊦ ∀r s.
subrelation r s ∧ transitive s ⇒ subrelation (transitiveClosure r) s

⊦ ∀r s. subrelation r s ∧ subrelation s r ⇒ r = s

⊦ ∀m x y. measure m x y ⇔ m x < m y

⊦ ∀s x y. fromSet s x y ⇔ (x, y) ∈ s

⊦ ∀r. wellFounded r ⇔ ¬∃f. ∀n. r (f (suc n)) (f n)

⊦ ∀r x y. (x, y) ∈ toSet r ⇔ r x y

⊦ ∀r s t. subrelation r s ∧ subrelation s t ⇒ subrelation r t

⊦ ∀r s. subrelation r (bigIntersect s) ⇔ ∀t. t ∈ s ⇒ subrelation r t

⊦ ∀r s. subrelation r s ⇔ ∀x y. r x y ⇒ s x y

⊦ ∀r s. (∀x y. r x y ⇔ s x y) ⇒ r = s

⊦ ∀r. toSet r = { x y. (x, y) | r x y }

⊦ ∀s x y. bigIntersect s x y ⇔ ∀r. r ∈ s ⇒ r x y

⊦ ∀s x y. bigUnion s x y ⇔ ∃r. r ∈ s ∧ r x y

⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h x

⊦ ∀r.
transitiveClosure r =
bigIntersect { s. s | subrelation r s ∧ transitive s }

⊦ ∀r s x y. intersect r s x y ⇔ r x y ∧ s x y

⊦ ∀r s x y. union r s x y ⇔ r x y ∨ s x y

⊦ ∀r. transitive r ⇔ ∀x y z. r x y ∧ r y z ⇒ r x z

⊦ ∀m a b. (∀y. measure m y a ⇒ measure m y b) ⇔ m a ≤ m b

⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀r. wellFounded r ⇔ ∀p. (∃x. p x) ⇔ ∃x. p x ∧ ∀y. r y x ⇒ ¬p y

⊦ ∀r. wellFounded r ⇔ ∀p. (∃x. p x) ⇒ ∃x. p x ∧ ∀y. r y x ⇒ ¬p y

⊦ ∀r s.
wellFounded r ∧ wellFounded s ⇒
wellFounded (λ(x₁, y₁) (x₂, y₂). r x₁ x₂ ∧ s y₁ y₂)

⊦ ∀r.
    wellFounded r ⇒
    ∀h.
      (∀f g x. (∀z. r z x ⇒ f z = g z) ⇒ h f x = h g x) ⇒
      ∃!f. ∀x. f x = h f x

⊦ ∀r s.
wellFounded r ∧ wellFounded s ⇒
wellFounded (λ(x₁, y₁) (x₂, y₂). r x₁ x₂ ∨ x₁ = x₂ ∧ s y₁ y₂)

⊦ ∀r.
(∀x. ¬r x x) ∧ (∀x y z. r x y ∧ r y z ⇒ r x z) ∧
(∀x. finite { y. y | r y x }) ⇒ wellFounded r

⊦ ∀r s.
wellFounded r ∧ (∀a. wellFounded (s a)) ⇒
wellFounded (λ(x₁, y₁) (x₂, y₂). r x₁ x₂ ∨ x₁ = x₂ ∧ s x₁ y₁ y₂)

External Type Operators

→
bool
Data
- Pair
  - ×
Number
- Natural
  - natural
Set
- set

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- Pair
  - ,
Function
- id
Number
- Natural
  - +
  - <
  - ≤
  - suc
  - zero
Set
- ∅
- bigIntersect
- bigUnion
- finite
- fromPredicate
- image
- infinite
- insert
- ∩
- ∈
- ⊆
- ∪
- universe

Assumptions

⊦ ⊤

⊦ infinite universe

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ ∀x. x ∈ universe

⊦ ∀t. t ⇒ t

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀x. id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀m. ¬(m < 0)

⊦ ∀n. ¬(n < n)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀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

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀m. m + 0 = m

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀s. infinite s ⇔ ¬finite s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀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

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

⊦ ∀p x. x ∈ fromPredicate p ⇔ p x

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = 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

⊦ ∀t₁ t₂. ¬t₁ ⇒ ¬t₂ ⇔ t₂ ⇒ t₁

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m < suc n ⇔ m < n

⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀m n. m < n ∨ n < m ∨ m = n

⊦ ∀s t. s ⊆ t ∧ t ⊆ s ⇒ s = t

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d

⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x

⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x

⊦ ∀p q. p ∧ (∀x. q x) ⇔ ∀x. p ∧ q x

⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x

⊦ ∀p q. p ⇒ (∀x. q x) ⇔ ∀x. p ⇒ q x

⊦ ∀p q. p ∨ (∀x. q x) ⇔ ∀x. p ∨ q x

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n

⊦ ∀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

⊦ ∀p q. (∀x. p x) ∨ q ⇔ ∀x. p x ∨ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

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

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

⊦ ∀m n p. m < n ∧ n < p ⇒ m < p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀s t u. s ⊆ t ∧ t ⊆ u ⇒ s ⊆ u

⊦ ∀s t. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t

⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇒ s = t

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀t u. t ⊆ bigIntersect u ⇔ ∀s. s ∈ u ⇒ t ⊆ s

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀s x. x ∈ bigIntersect s ⇔ ∀t. t ∈ s ⇒ x ∈ t

⊦ ∀s x. x ∈ bigUnion s ⇔ ∃t. t ∈ s ∧ x ∈ t

⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x

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

⊦ ∀s t x. x ∈ s ∩ t ⇔ x ∈ s ∧ x ∈ t

⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n

⊦ ∀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 ⇒ 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

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

⊦ ∀p₁ p₂ q₁ q₂. (p₁ ⇒ p₂) ∧ (q₁ ⇒ q₂) ⇒ p₁ ∧ q₁ ⇒ p₂ ∧ q₂

⊦ ∀p₁ p₂ q₁ q₂. (p₂ ⇒ p₁) ∧ (q₁ ⇒ q₂) ⇒ (p₁ ⇒ q₁) ⇒ p₂ ⇒ q₂

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀f s. (∀x y. f x = f y ⇒ x = y) ∧ infinite s ⇒ infinite (image f s)

⊦ ∀p f s. (∀y. y ∈ image f s ⇒ p y) ⇔ ∀x. x ∈ s ⇒ p (f x)