Package relation-well-founded-thm: relation-well-founded-thm

Information

name	relation-well-founded-thm
version	1.13
description	relation-well-founded-thm
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-09-21
show	Data.Bool

Files

Package tarball relation-well-founded-thm-1.13.tgz
Theory file relation-well-founded-thm.thy (included in the package tarball)

Theorems

⊦ Relation.wellFounded Number.Natural.<

⊦ Relation.wellFounded (λx y. F)

⊦ ∀<< x. Relation.wellFounded << ⇒ ¬<< x x

⊦ ∀<< m.
Relation.wellFounded << ⇒ Relation.wellFounded (λx x'. << (m x) (m x'))

⊦ ∀<<.
Relation.wellFounded << ⇔ ¬∃s. ∀n. << (s (Number.Natural.suc n)) (s n)

⊦ ∀<< <<<.
(∀x y. << x y ⇒ <<< x y) ∧ Relation.wellFounded <<< ⇒
Relation.wellFounded <<

⊦ ∀<<.
Relation.wellFounded << ⇔ ∀P. (∀x. (∀y. << y x ⇒ P y) ⇒ P x) ⇒ ∀x. P x

⊦ ∀<<.
Relation.wellFounded << ⇔ ∀P. (∃x. P x) ⇔ ∃x. P x ∧ ∀y. << y x ⇒ ¬P y

⊦ ∀H.
(∀f g n. (∀m. Number.Natural.< m n ⇒ f m = g m) ⇒ H f n = H g n) ⇒
∃f. ∀n. f n = H f n

⊦ ∀<< <<<.
    Relation.wellFounded << ∧ Relation.wellFounded <<< ⇒
    Relation.wellFounded
      (λ(Data.Pair., x1 y1) (Data.Pair., x2 y2). << x1 x2 ∧ <<< y1 y2)

⊦ ∀<<.
    Relation.wellFounded << ⇒
    ∀H.
      (∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
      ∃f. ∀x. f x = H f x

⊦ ∀<<.
    Relation.wellFounded << ⇒
    ∀H.
      (∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
      ∃!f. ∀x. f x = H f x

⊦ ∀<<.
    (∀H.
       (∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
       ∃f. ∀x. f x = H f x) ⇒ Relation.wellFounded <<

⊦ ∀R S.
    Relation.wellFounded R ∧ Relation.wellFounded S ⇒
    Relation.wellFounded
      (λ(Data.Pair., r1 s1) (Data.Pair., r2 s2).
         R r1 r2 ∨ r1 = r2 ∧ S s1 s2)

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

⊦ ∀R S.
    Relation.wellFounded R ∧ (∀a. Relation.wellFounded (S a)) ⇒
    Relation.wellFounded
      (λ(Data.Pair., r1 s1) (Data.Pair., r2 s2).
         R r1 r2 ∨ r1 = r2 ∧ S r1 s1 s2)

⊦ ∀<<.
    Relation.wellFounded << ⇒
    ∀H S.
      (∀f g x.
         (∀z. << z x ⇒ f z = g z ∧ S z (f z)) ⇒
         H f x = H g x ∧ S x (H f x)) ⇒ ∃f. ∀x. f x = H f x

⊦ ∀<<.
    Relation.wellFounded << ⇒
    ∀H.
      (∀f g x. (∀z. << z x ⇒ f z = g z) ⇒ H f x = H g x) ⇒
      ∀f g. (∀x. f x = H f x) ∧ (∀x. g x = H g x) ⇒ f = g

⊦ ∀<<.
    (∀H.
       (∀f g x. (∀z. << z x ⇒ (f z ⇔ g z)) ⇒ (H f x ⇔ H g x)) ⇒
       ∀f g. (∀x. f x ⇔ H f x) ∧ (∀x. g x ⇔ H g x) ⇒ f = g) ⇒
    Relation.wellFounded <<

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - F
  - T
- Pair
  - Data.Pair.,
Function
- Function.id
Number
- Natural
  - Number.Natural.<
  - Number.Natural.suc
  - Number.Natural.zero
Relation
- Relation.wellFounded
Set
- Set.finite
- Set.fromPredicate
- Set.image
- Set.infinite
- Set.insert
- Set.∈
- Set.⊆
- Set.universe

Assumptions

⊦ T

⊦ Set.infinite Set.universe

⊦ ∀x. Set.∈ x Set.universe

⊦ F ⇔ ∀p. p

⊦ ∀x. Function.id x = x

⊦ ∀n. Number.Natural.< 0 (Number.Natural.suc n)

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀s. Set.infinite s ⇔ ¬Set.finite s

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

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

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀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

⊦ ∀s t. Set.finite t ∧ Set.⊆ s t ⇒ Set.finite s

⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (Data.Pair., p1 p2)

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

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

⊦ ∀m n. Number.Natural.< m n ∨ Number.Natural.< n m ∨ m = n

⊦ ∀PAIR'. ∃fn. ∀a0 a1. fn (Data.Pair., a0 a1) = PAIR' a0 a1

⊦ ∀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. (∀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

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

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

⊦ ∀s t. Set.⊆ s t ⇔ ∀x. Set.∈ x s ⇒ Set.∈ x t

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

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀p x. Set.∈ x { y. y | p y } ⇔ p x

⊦ ∀x y s. Set.∈ x (Set.insert y s) ⇔ x = y ∨ Set.∈ x s

⊦ (∃!) = λP. (∃) P ∧ ∀x y. P x ∧ P y ⇒ x = y

⊦ ∀P. (∀n. (∀m. Number.Natural.< 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 (Number.Natural.suc n) = f (fn n) n

⊦ ∀P. (∃n. P n) ⇔ ∃n. P n ∧ ∀m. Number.Natural.< m n ⇒ ¬P m

⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∧ C ⇒ B ∧ D

⊦ ∀A B C D. (B ⇒ A) ∧ (C ⇒ D) ⇒ (A ⇒ C) ⇒ B ⇒ D

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

⊦ (∀m. Number.Natural.< m 0 ⇔ F) ∧
  ∀m n.
    Number.Natural.< m (Number.Natural.suc n) ⇔
    m = n ∨ Number.Natural.< m n

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

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

⊦ ∀<<.
Relation.wellFounded << ⇔ ∀P. (∃x. P x) ⇒ ∃x. P x ∧ ∀y. << y x ⇒ ¬P y

⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)

⊦ ∀R.
(∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (Number.Natural.suc n)) ⇒
∀m n. Number.Natural.< m n ⇒ R m n

⊦ ∀p q r.
(p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
(p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)