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|
#|==========================================
Cayley-Dickson Construction
cayley1d.lisp
31 March 2017
Cons pairs of Octonions form a (16 dimensional) algebra.
A Real Vector Algebra with Identity:
A Real Vector Space with Multiplication
and a Multiplicative Identity.
The Vector Algebra also has a real valued Norm
and a real valued Dot (or Inner) Product.
This algebra has a NON-trivial conjugate and
its multiplication is NOT commutative and
also NOT associative. The norm does NOT satisfy
the Composition Law: Norm(xy) = Norm(x)Norm(y),
because the algebra has nontrivial zero divisors.
All nonzero vectors have multiplicative inverses.
In fact, this algebra is (isomorphic to) the Sedenions.
References:
J.H. Conway and D.A. Smith, On Quaternions and Octonions: Their Geometry,
Arithmetic, and Symmetry, A K Peters, 2003, pages 67--73.
H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer,
J. Neukirch, A. Prestel, and R. Remmert, Numbers, Springer, 1991, pp 256--261,
265--274
ACL2 Version 7.4(r) built March 30, 2017 08:51:54.
System books directory "/home/acl2/acl2-7.4r/acl2-7.4/books/".
ACL2 Version 7.4 built March 29, 2017 10:38:07.
System books directory "/home/acl2/acl2-7.4/acl2-7.4/books/".
===============================|#
#|====================================
To certify:
(certify-book "cayley1d"
0 ;; world with no commands
)
===============================
To use:
(include-book
"cayley1d"
:uncertified-okp nil
:defaxioms-okp nil
:skip-proofs-okp nil
)
=====================================|#
#|===============================================
:set-gag-mode t ; enable gag-mode, suppressing most proof commentary
(set-gag-mode t) ; same as above
:set-gag-mode :goals ; same as above, but print names of goals when produced
:set-gag-mode nil ; disable gag-mode
==============|#
#|==============================
(LD
"cayley1d.lisp") ; read and evaluate each form in file
==================|#
(in-package "ACL2")
(local
(include-book "arithmetic/top" :dir :system
:uncertified-okp nil
:defaxioms-okp nil
:skip-proofs-okp nil))
(include-book "cayley1c"
:uncertified-okp nil
:defaxioms-okp nil
:skip-proofs-okp nil)
;;===============================
;;==Real Vector Space Operations:
;; Predicate for set of vectors
(defun
16p (x)
(and (consp x)
(8p (car x))
(8p (cdr x))))
;; Zero vector
(defun
16_0 ()
(cons (8_0)(8_0)))
;; Vector addition
(defun
16_+ (x y)
(cons (8_+ (car x)(car y))
(8_+ (cdr x)(cdr y))))
;; Vector minus
(defun
16_- (x)
(cons (8_- (car x))
(8_- (cdr x))))
;; Scalar multiplication
(defun
S16_* (a x)
(cons (S8_* a (car x))
(S8_* a (cdr x))))
;;================================================
;;==Vector Multiplication and Identity Operations:
;; Vector multiplication
(defun
16_* (x y)
(cons (8_+ (8_* (car x)
(car y))
(8_- (8_* (8_conjugate (cdr y))
(cdr x))))
(8_+ (8_* (cdr y)
(car x))
(8_* (cdr x)
(8_conjugate (car y))))))
;; One vector
(defun
16_1 ()
(cons (8_1)(8_0)))
;;=================
;;==Norm operation:
;; Vector Norm
(defun
16_norm (x)
(+ (8_norm (car x))
(8_norm (cdr x))))
;;===================================
;;==Dot (or Inner) Product Operation:
;; Vector Dot Product
(defun
16_dot (x y)
(+ (8_dot (car x)(car y))
(8_dot (cdr x)(cdr y))))
;;========================
;;==Conjugation Operation:
;; Vector conjugate
(defun
16_conjugate (x)
(cons (8_conjugate (car x))
(8_- (cdr x))))
;;==========================
;; Real Vector Space Axioms:
(defthmD
16-Vector-closure
(and (16p (16_0))
(implies (and (16p x)
(16p y))
(16p (16_+ x y)))
(implies (16p x)
(16p (16_- x)))
(implies (and (real/rationalp a)
(16p x))
(16p (S16_* a x)))))
(defthmD
Associativity-of-16_+
(implies (and (16p x)
(16p y)
(16p z))
(equal (16_+ (16_+ x y) z)
(16_+ x (16_+ y z)))))
(defthmD
Commutativity-of-16_+
(implies (and (16p x)
(16p y))
(equal (16_+ x y)
(16_+ y x))))
(defthmD
Unicity-of-16_0
(implies (16p x)
(equal (16_+ (16_0) x)
x)))
(defthmD
Inverse-of-16_+
(implies (16p x)
(equal (16_+ x (16_- x))
(16_0))))
(defthmD
Associativity-of-S16_*
(implies (and (real/rationalp a)
(real/rationalp b)
(16p x))
(equal (S16_* a (S16_* b x))
(S16_* (* a b) x))))
(defthmD
Unicity-of-Scalar16-1
(implies (16p x)
(equal (S16_* 1 x) x)))
(defthmD
Distributivity-S16_*-scalar-+
(implies (and (real/rationalp a)
(real/rationalp b)
(16p x))
(equal (S16_* (+ a b) x)
(16_+ (S16_* a x)(S16_* b x)))))
(defthmD
Distributivity-S16_*-16_+
(implies (and (real/rationalp a)
(16p x)
(16p y))
(equal (S16_* a (16_+ x y))
(16_+ (S16_* a x)(S16_* a y)))))
;;=======================================
;; Additional Real Vector Algebra Axioms:
(defthmD
16_1&16_*-closure
(and (16p (16_1))
(implies (and (16p x)
(16p y))
(16p (16_* x y)))))
(defthmD
Not-16_1=16_0
(not (equal (16_1)(16_0))))
(defthmD
Left-Distributivity-16_*_16_+
(implies (and (real/rationalp a)
(real/rationalp b)
(16p x)
(16p y)
(16p z))
(equal (16_* x
(16_+ (S16_* a y)
(S16_* b z)))
(16_+ (S16_* a
(16_* x y))
(S16_* b
(16_* x z))))))
(defthmD
Right-Distributivity-16_*_16_+
(implies (and (real/rationalp a)
(real/rationalp b)
(16p x)
(16p y)
(16p z))
(equal (16_* (16_+ (S16_* a x)
(S16_* b y))
z)
(16_+ (S16_* a
(16_* x z))
(S16_* b
(16_* y z))))))
(defthmD
Unicity-of-16_1
(implies (16p x)
(and (equal (16_* (16_1) x) x)
(equal (16_* x (16_1)) x))))
;;===============================================
;; Additional Vector Norm and Dot Product Axioms:
(defthmD
Realp>=0-16_norm
(implies (16p x)
(and (real/rationalp (16_norm x))
(>= (16_norm x) 0))))
(defthmD
16_norm=0
(implies (16p x)
(equal (equal (16_norm x) 0)
(equal x (16_0)))))
#|===============================
This is false because of zero divisors.
See below for example.
(defthmD
16-Composition-Law
(implies (and (16p x)
(16p y))
(equal (16_norm (16_* x y))
(* (16_norm x)
(16_norm y)))))
===========================|#
(defthmD
16_dot-def
(equal (16_dot x y)
(* 1/2 (+ (16_norm (16_+ x y))
(- (16_norm x))
(- (16_norm y)))))
:rule-classes :definition)
(defthmD
Distributivity-16_dot-16_+
(implies (and (real/rationalp a)
(real/rationalp b)
(16p x)
(16p y)
(16p z))
(equal (16_dot (16_+ (S16_* a x)
(S16_* b y))
z)
(+ (* a (16_dot x z))
(* b (16_dot y z))))))
(defun-sk
Forall-u-16_dot-u-x=0 (x)
(forall (u)
(implies (16p u)
(equal (16_dot u x) 0)))
:rewrite :direct)
(defthmD
Forall-u-16_dot-u-x=0-def
(equal (Forall-u-16_dot-u-x=0 x)
(let ((u (Forall-u-16_dot-u-x=0-witness x)))
(implies (16p u)
(equal (16_dot u x) 0))))
:rule-classes :definition)
;; redundant
(defthm
Forall-u-16_dot-u-x=0-necc
(implies (Forall-u-16_dot-u-x=0 x)
(implies (16p u)
(equal (16_dot u x) 0))))
(local
(defthmD
16_dot=0
(implies (16p x)
(equal (equal (16_dot x x) 0)
(equal x (16_0))))))
(defthm ;;16_dot is nonsingular
Forall-u-16_dot-u-x=0->x=8_0
(implies (and (16p x)
(Forall-u-16_dot-u-x=0 x))
(equal x (16_0)))
:rule-classes nil
:hints (("Goal"
:in-theory (disable (:DEFINITION 16_DOT))
:use 16_dot=0)))
(in-theory (disable Forall-u-16_dot-u-x=0-necc))
(defthmD
16_conjugate-def-rewrite
(implies (16p x)
(equal (16_conjugate x)
(16_+ (S16_* (* 2 (16_dot x (16_1)))
(16_1))
(16_- x)))))
;;=================================
(defthmD
16_norm=8_norm
(implies (8p x)
(and (equal (16_norm (cons x (8_0)))
(8_norm x))
(equal (16_norm (cons (8_0) x))
(8_norm x)))))
(defthmD
8p-8_0-orthogonal-8_0-8p
(implies (and (8p x)
(8p y))
(equal (16_dot (cons x (8_0))
(cons (8_0) y))
0)))
(defthmD
16_1-def
(equal (16_1)
(cons (8_1)(8_0)))
:rule-classes :definition)
(defthmD
16_*-cons=cons-8_*
(implies (and (8p x)
(8p y))
(equal (16_* (cons x (8_0))
(cons y (8_0)))
(cons (8_* x y)(8_0)))))
(defthmD
8p*i=cons-8_0-8p
(implies (8p x)
(equal (16_* (cons x (8_0))
(cons (8_0)(8_1)))
(cons (8_0) x))))
;;=========================
(defun
16_/ (x)
(S16_* (/ (16_norm x))
(16_conjugate x)))
(defthmD
16p-16_/
(implies (and (16p x)
(not (equal x (16_0))))
(16p (16_/ x))))
(defthmD
16_*-16_conjugate-right
(implies (and (real/rationalp a)
(not (equal a 0))
(16p x))
(equal (16_* x (S16_* (/ a)
(16_conjugate x)))
(S16_* (* (/ a)
(16_norm x))
(16_1)))))
(defthmD
16_*-16_conjugate-left
(implies (and (real/rationalp a)
(not (equal a 0))
(16p x))
(equal (16_* (S16_* (/ a)
(16_conjugate x))
x)
(S16_* (* (/ a)
(16_norm x))
(16_1)))))
(defthmD
16_/=16_*-inverse-right
(implies (and (16p x)
(not (equal x (16_0))))
(equal (16_* x (16_/ x))
(16_1)))
:hints (("Goal"
:in-theory (disable (:DEFINITION |16P|)
(:DEFINITION 16_*)
(:DEFINITION S16_*)
(:DEFINITION 16_NORM)
(:DEFINITION 16_CONJUGATE)
(:DEFINITION |16_1|)
(:DEFINITION |16_0|)
(:EXECUTABLE-COUNTERPART |16_1|)
(:EXECUTABLE-COUNTERPART |16_0|))
:use (Realp>=0-16_norm
16_norm=0
(:instance
Unicity-of-Scalar16-1
(x (16_1)))
(:instance
16_*-16_conjugate-right
(a (16_norm x)))))))
(defthmD
16_/=16_*-inverse-left
(implies (and (16p x)
(not (equal x (16_0))))
(equal (16_* (16_/ x) x)
(16_1)))
:hints (("Goal"
:in-theory (disable (:DEFINITION |16P|)
(:DEFINITION 16_*)
(:DEFINITION S16_*)
(:DEFINITION 16_NORM)
(:DEFINITION 16_CONJUGATE)
(:DEFINITION |16_1|)
(:DEFINITION |16_0|)
(:EXECUTABLE-COUNTERPART |16_1|)
(:EXECUTABLE-COUNTERPART |16_0|))
:use (Realp>=0-16_norm
16_norm=0
(:instance
Unicity-of-Scalar16-1
(x (16_1)))
(:instance
16_*-16_conjugate-left
(a (16_norm x)))))))
;;=============================
(defthmD
16_conjugate-is-NOT-trivial
(and (16p (cons (8_0)(8_1)))
(not (equal (16_conjugate (cons (8_0)(8_1)))
(identity (cons (8_0)(8_1)))))))
(defthmD
16_*-is-NOT-commutative
(let ((i (cons (cons (cons (cons 0 1)(2_0))(4_0))(8_0)))
(j (cons (cons (cons (2_0)(2_1))(4_0))(8_0)))
(k (cons (cons (cons (2_0)(cons 0 1))(4_0))(8_0))))
(and (16p i)
(16p j)
(16p k)
(equal (16_- k)
(cons (cons (cons (2_0)(cons 0 -1))(4_0))(8_0)))
(not (equal (16_- k)
k))
(equal (16_* i j) k)
(equal (16_* j i)(16_- k))
(not (equal (16_* i j)
(16_* j i))))))
(defthmD
16_*-is-NOT-associative
(let ((e5 (cons (cons (4_0)(4_1))(8_0)))
(e6 (cons (cons (4_0)(cons (cons 0 1)(2_0)))(8_0)))
(e7 (cons (cons (4_0)(cons (2_0)(2_1)))(8_0)))
(e8 (cons (cons (4_0)(cons (2_0)(cons 0 1)))(8_0))))
(and (16p e5)
(16p e6)
(16p e7)
(16p e8)
(equal (16_- e8)
(cons (cons (4_0)(cons (2_0)(cons 0 -1)))(8_0)))
(not (equal (16_- e8)
e8))
(equal (16_* (16_* e5 e6) e7)
(16_- e8))
(equal (16_* e5 (16_* e6 e7))
e8)
(not (equal (16_* (16_* e5 e6) e7)
(16_* e5 (16_* e6 e7)))))))
;; The 16_* product of nonzero values could be zero.
;; Thus the Composition Law: Norm(xy) = Norm(x)Norm(y),
;; does not hold
(defthmD
16_*-zero-divisors
(let ((x (cons (cons (cons (2_0)(cons 0 1))(4_0))
(cons (cons (2_0)(2_1))(4_0))))
(y (cons (cons (4_0)(cons (2_0)(2_1)))
(cons (4_0)(cons (2_0)(cons 0 -1))))))
(and (16p x)
(16p y)
(equal (16_* x y)(16_0))
(not (equal (16_norm (16_* x y))
(* (16_norm x)(16_norm y)))))))
|
