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|
; Arithmetic-5 Library
; Written by Robert Krug
; Copyright/License:
; See the LICENSE file at the top level of the arithmetic-5 library.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; simplify.lisp
;;;
;;;
;;; This book contains three types of rules:
;;; 1. simplify-terms-such-as-ax+bx-rel-0
;;;
;;; assuming a, b, and x are acl2-numberp
;;; (equal (+ x (* b x)) 0) ==> (or (equal x 0) (equal (+ 1 b) 0))
;;; (equal (+ (* a x) (* b x)) 0) ==> (or (equal x 0) (equal (+ a b) 0))
;;;
;;; 2. Cancel "like" terms on either side of an equality or inequality.
;;;
;;; A couple of simple example of cancelling like terms:
;;;
;;; (equal (+ a (* 2 c) d)
;;; (+ b c d))
;;; ===>
;;; (equal (+ a c)
;;; b)
;;;
;;; Note that just as for normailze.liso, there are two distinct
;;; behaviours for cancelling like factors:
;;;
;;; Under the default theory, gather-exponents:
;;;
;;; (equal (* a (expt b n))
;;; (* c (expt b m)))
;;; ===>
;;; (equal (* a (expt b (- n m)))
;;; c)
;;;
;;; Under the other theory, scatter-exponents
;;;
;;; (equal (* a (expt b n))
;;; (* c (expt b m)))
;;; ===>
;;; no change
;;;
;;; Under both theories:
;;;
;;; (equal (* a (expt b (* 2 n)))
;;; (* c (expt b n)))
;;; ===>
;;; (equal (* a (expt b n))
;;; (* c))
;;;
;;; (equal (* a (expt c 2) d)
;;; (* b c d))
;;; ===>
;;; (equal (* a c)
;;; b)
;;;
;;; 3. Move "negative" terms form one side of an equality or
;;; inequality to the other.
;;;
;;; A couple of simple example of moving a negative term to the
;;; other side:
;;;
;;; (< (+ a (- b) c)
;;; (+ d e))
;;; ===>
;;; (< (+ a c)
;;; (+ b d e))
;;;
;;; Under the default theory, gather-exponents, we do not move
;;; ``negative'' exponents to the other side. It has proved too
;;; dificult to prevent loops in the general case, and so we avoid
;;; the issue entirely. We could certainly catch the ``simple''
;;; cases, but have not done so.
;;;
;;; Under the other theory, scatter-exponents:
;;;
;;; (equal (* a (/ b) c)
;;; (* d e))
;;; ===>
;;; (equal (* a c)
;;; (* b d e))
;;;
;;; (equal (* a (expt b (- n)) c)
;;; (* d (expt b m)))
;;; ===>
;;; (equal (* a c)
;;; (* d (expt b m) (expt b n)))
;;;
;;; Note that for multiplication, division or exponentiation to a negative
;;; power is considered to be negative.
;;;
;;; See common.lisp for a short description of the general strategy
;;; used in the last two of these types.
;;;
;;; The certification of this book could probably be sped up a good
;;; bit. It is rather slow.
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(include-book "common")
(local
(include-book "simplify-helper"))
(local
(include-book "basic"))
(table acl2-defaults-table :state-ok t)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(local
(defthm rewrite-equal-<-to-iff-<
(equal (equal (< a b)
(< c d))
(iff (< a b)
(< c d)))))
(local
(encapsulate
()
(local (include-book "../../support/top"))
(defthm equal-*-/-1
(equal (equal (* (/ x) y) z)
(if (equal (fix x) 0)
(equal z 0)
(and (acl2-numberp z)
(equal (fix y) (* x z))))))
(defthm equal-*-/-2
(equal (equal (* y (/ x)) z)
(if (equal (fix x) 0)
(equal z 0)
(and (acl2-numberp z)
(equal (fix y) (* z x))))))
))
(local
(encapsulate
()
(local (include-book "../../support/top"))
(defthm equal-*-1
(implies (not (equal (fix x) 0))
(equal (equal (* x y) (* x z))
(equal (fix y) (fix z)))))
(defthm equal-*-2
(implies (not (equal (fix x) 0))
(equal (equal (* y x) (* z x))
(equal (fix y) (fix z)))))
))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; simplify-terms-such-as-ax+bx-rel-0
;;; assuming a, b, and x are acl2-numberp
;;; (equal (+ x (* b x)) 0) ==> (or (equal x 0) (equal (+ 1 b) 0))
;;; (equal (+ (* a x) (* b x)) 0) ==> (or (equal x 0) (equal (+ a b) 0))
;;; We are, in affect, undoing distributivity when the term is being
;;; compared to zero.
;;; In the example
;;; (thm
;;; (implies (and (acl2-numberp a)
;;; (acl2-numberp b)
;;; (acl2-numberp x))
;;; (equal (+ (* a x) (* b x))
;;; 0))
;;; :otf-flg t)
;;; simplify-terms-such-as-ax+bx-rel-0-fn returns
;;; ((COMMON . X)
;;; (REMAINDER BINARY-+ A B))
(defun simplify-terms-such-as-ax+bx-rel-0-fn (sum)
(declare (xargs :guard t))
;; We look for any factors common to each addend of sum. If we
;; find any, we return a binding list with common bound to the
;; product of these factors, and remainder bound to the original
;; sum but with the common factors removed from each addend.
(if (eq (fn-symb sum) 'BINARY-+)
(let ((common-factors (common-factors (factors (arg1 sum))
(arg2 sum))))
(if common-factors
(let ((common (make-product common-factors))
(remainder (remainder common-factors sum)))
(list (cons 'common common)
(cons 'remainder remainder)))
nil))
nil))
;;; assuming a, b, and x are acl2-numberp
;;; (equal (+ x (* b x)) 0) ==> (or (equal x 0) (equal (+ 1 b) 0))
;;; (equal (+ (* a x) (* b x)) 0) ==> (or (equal x 0) (equal (+ a b) 0))
(defthm simplify-terms-such-as-ax+bx-=-0
(implies (and (bind-free
(simplify-terms-such-as-ax+bx-rel-0-fn sum)
(common remainder))
(acl2-numberp common)
(acl2-numberp remainder)
(equal sum
(* common remainder)))
(equal (equal sum 0)
(or (equal common 0)
(equal remainder 0)))))
(defthm simplify-terms-such-as-ax+bx-<-0-rational-remainder
(implies (and (bind-free
(simplify-terms-such-as-ax+bx-rel-0-fn sum)
(common remainder))
(acl2-numberp common)
(real/rationalp remainder)
(equal sum
(* common remainder)))
(equal (< sum 0)
(cond ((< common 0)
(< 0 remainder))
((< 0 common)
(< remainder 0))
(t
nil)))))
(defthm simplify-terms-such-as-ax+bx-<-0-rational-common
(implies (and (bind-free
(simplify-terms-such-as-ax+bx-rel-0-fn sum)
(common remainder))
(real/rationalp common)
(acl2-numberp remainder)
(equal sum
(* common remainder)))
(equal (< sum 0)
(cond ((< common 0)
(< 0 remainder))
((< 0 common)
(< remainder 0))
(t
nil))))
:hints (("Goal" :use (:instance simplify-terms-such-as-ax+bx-<-0-rational-remainder
(common remainder)
(remainder common)))))
(defthm simplify-terms-such-as-0-<-ax+bx-rational-remainder
(implies (and (bind-free
(simplify-terms-such-as-ax+bx-rel-0-fn sum)
(common remainder))
(acl2-numberp common)
(real/rationalp remainder)
(equal sum
(* common remainder)))
(equal (< 0 sum)
(cond ((< 0 common)
(< 0 remainder))
((< common 0)
(< remainder 0))
(t
nil)))))
(defthm simplify-terms-such-as-0-<-ax+bx-rational-common
(implies (and (bind-free
(simplify-terms-such-as-ax+bx-rel-0-fn sum)
(common remainder))
(real/rationalp common)
(acl2-numberp remainder)
(equal sum
(* common remainder)))
(equal (< 0 sum)
(cond ((< 0 common)
(< 0 remainder))
((< common 0)
(< remainder 0))
(t
nil))))
:hints (("Goal" :use (:instance simplify-terms-such-as-0-<-ax+bx-rational-remainder
(common remainder)
(remainder common)))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; simplify-sums-equal and simplify-sums-<
;;; We wish to cancel like addends from both sides of af an equality
;;; or inequality. We scan the sums on either side of the relation,
;;; and construct a pair of addend-info-lists. We then find the first
;;; match in these lists and cancel it.
(defun addend-val (addend)
(declare (xargs :guard t))
;; We wish to subtract the ``smaller'' of two matching addends.
;;
(cond ((variablep addend)
(list 0 1 0))
((constant-p addend)
(let ((val (unquote addend)))
(if (rationalp val) ; OK, no constant reals
(list 0 0 (abs val))
(list 0 0 1))))
((eq (ffn-symb addend) 'UNARY--)
(addend-val (arg1 addend)))
((and (eq (ffn-symb addend) 'BINARY-*)
(constant-p (arg1 addend)))
(let ((val (unquote (arg1 addend))))
(if (rationalp val)
(list (abs val) 0 0)
(list 1 0 0))))
(t
(list 0 1 0))))
(defun addend-info-entry (x)
(declare (xargs :guard t))
(list (addend-pattern x) (addend-val x) x))
(defun addend-info-list (x)
(declare (xargs :guard t))
(if (eq (fn-symb x) 'BINARY-+)
(cons (addend-info-entry (arg1 x))
(addend-info-list (arg2 x)))
(list (addend-info-entry x))))
(local
(encapsulate
()
(local
(defthm temp-1
(good-val-triple-p (addend-val x))))
(defthm addend-info-list-thm
(info-list-p (addend-info-list x)))
))
(defun assoc-addend (x info-list)
(declare (xargs :guard (info-list-p info-list)))
(cond ((endp info-list)
nil)
((matching-addend-patterns-p x (caar info-list))
(car info-list))
(t
(assoc-addend x (cdr info-list)))))
(defun first-match-in-addend-info-lists (info-list1 info-list2 mfc state)
(declare (xargs :guard (and (info-list-p info-list1)
(info-list-p info-list2))
:guard-hints (("Goal" :in-theory (disable good-val-triple-p
negate-match
val-<)))))
(if (endp info-list1)
nil
(let ((temp (assoc-addend (car (car info-list1)) info-list2)))
(if temp
(cond ((and ;; We want the ``smaller'' match
(val-< (cadr (car info-list1))
(cadr temp))
;; prevent various odd loops
(stable-under-rewriting-sums (negate-match
(caddr (car info-list1)))
mfc state))
(list (cons 'x (negate-match (caddr (car info-list1))))))
((stable-under-rewriting-sums (negate-match
(caddr temp))
mfc state)
(list (cons 'x (negate-match (caddr temp)))))
(t
(first-match-in-addend-info-lists (cdr info-list1) info-list2
mfc state)))
(first-match-in-addend-info-lists (cdr info-list1) info-list2
mfc state)))))
(defun find-matching-addends (lhs rhs mfc state)
(declare (xargs :guard t
:verify-guards nil))
(let* ((info-list1 (addend-info-list lhs))
(info-list2 (if info-list1
(addend-info-list rhs)
nil)))
(if info-list2
(first-match-in-addend-info-lists info-list1 info-list2
mfc state)
nil)))
(verify-guards find-matching-addends)
(defthm simplify-sums-equal
(implies (and (syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-+ lhs mfc state))
(syntaxp (in-term-order-+ rhs mfc state))
(acl2-numberp lhs)
(acl2-numberp rhs)
(bind-free
(find-matching-addends lhs rhs mfc state)
(x)))
(equal (equal lhs rhs)
(equal (+ x lhs) (+ x rhs)))))
(local
(in-theory (disable simplify-sums-equal)))
(defthm simplify-sums-<
(implies (and (syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-+ lhs mfc state))
(syntaxp (in-term-order-+ rhs mfc state))
(acl2-numberp lhs)
(acl2-numberp rhs)
(bind-free
(find-matching-addends lhs rhs mfc state)
(x)))
(equal (< lhs rhs)
(< (+ x lhs) (+ x rhs)))))
(local
(in-theory (disable simplify-sums-<)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun negative-addend-p (x)
(declare (xargs :guard t))
(or (and (eq (fn-symb x) 'UNARY--)
(or (variablep (arg1 x))
(not (equal (ffn-symb (arg1 x)) 'UNARY--))))
(and (eq (fn-symb x) 'BINARY-*)
(rational-constant-p (arg1 x))
(< (unquote (arg1 x)) 0))))
(defun find-negative-addend1 (x mfc state)
(declare (xargs :guard t))
(cond ((not (eq (fn-symb x) 'BINARY-+))
(if (and (negative-addend-p x)
;; prevent various odd loops
(stable-under-rewriting-sums (negate-match x)
mfc state))
(list (cons 'x (negate-match x)))
nil))
((and (negative-addend-p (arg1 x))
(stable-under-rewriting-sums (negate-match (arg1 x))
mfc state))
(list (cons 'x (negate-match (arg1 x)))))
((eq (fn-symb (arg2 x)) 'BINARY-+)
(find-negative-addend1 (arg2 x) mfc state))
((and (negative-addend-p (arg2 x))
(stable-under-rewriting-sums (negate-match (arg2 x))
mfc state))
(list (cons 'x (negate-match (arg2 x)))))
(t
nil)))
(defun find-negative-addend (lhs rhs mfc state)
(declare (xargs :guard t))
(let ((temp1 (find-negative-addend1 lhs mfc state)))
(if temp1
temp1
(let ((temp2 (find-negative-addend1 rhs mfc state)))
(if temp2
temp2
nil)))))
(defthm prefer-positive-addends-equal
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (in-term-order-+ lhs mfc state))
(syntaxp (in-term-order-+ rhs mfc state))
(syntaxp (or (equal (fn-symb lhs) 'BINARY-+)
(equal (fn-symb rhs) 'BINARY-+)))
(bind-free
(find-negative-addend lhs rhs mfc state)
(x)))
(equal (equal lhs rhs)
(equal (+ x lhs) (+ x rhs)))))
(local
(in-theory (disable prefer-positive-addends-equal)))
(defthm prefer-positive-addends-<
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (in-term-order-+ lhs mfc state))
(syntaxp (in-term-order-+ rhs mfc state))
(syntaxp (or (equal (fn-symb lhs) 'BINARY-+)
(equal (fn-symb rhs) 'BINARY-+)))
(bind-free
(find-negative-addend lhs rhs mfc state)
(x)))
(equal (< lhs rhs)
(< (+ x lhs) (+ x rhs)))))
(local
(in-theory (disable prefer-positive-addends-<)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm simplify-products-gather-exponents-equal
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-matching-factors-gather-exponents lhs rhs mfc state)
(x))
;; Something is not right if x = +/-1. This will
;; presumably be rewritten away later. We abort
;; for now.
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (acl2-numberp x))
(case-split (not (equal x 0))))
(equal (equal lhs rhs)
(equal (* x lhs) (* x rhs)))))
(local
(in-theory (disable simplify-products-gather-exponents-equal)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm simplify-products-gather-exponents-<
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-rational-matching-factors-gather-exponents lhs rhs
mfc state)
(x))
;; Something is not right if x = +/-1. This will
;; presumably be rewritten away later. We abort
;; for now.
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (real/rationalp x))
(case-split (not (equal x 0))))
(equal (< lhs rhs)
(cond ((< 0 x)
(< (* x lhs) (* x rhs)))
(t
(< (* x rhs) (* x lhs)))))))
(local
(in-theory (disable simplify-products-gather-exponents-<)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm simplify-products-scatter-exponents-equal
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-matching-factors-scatter-exponents lhs rhs mfc state)
(x))
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (acl2-numberp x))
(case-split (not (equal x 0))))
(equal (equal lhs rhs)
(equal (* x lhs) (* x rhs)))))
(local
(in-theory (disable simplify-products-scatter-exponents-equal)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm simplify-products-scatter-exponents-<
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (not (quotep lhs)))
(syntaxp (not (quotep rhs)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-rational-matching-factors-scatter-exponents lhs rhs
mfc state)
(x))
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (real/rationalp x))
(case-split (not (equal x 0))))
(equal (< lhs rhs)
(cond ((< 0 x)
(< (* x lhs) (* x rhs)))
(t
(< (* x rhs) (* x lhs)))))))
(local
(in-theory (disable simplify-products-scatter-exponents-<)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
#|
;;; We do not include theorems such as the below
;;; prefer-positive-exponents-gather-exponents-equal because it is too
;;; difficult to prevent looping.
;;; Add an example here!!!
(defthm prefer-positive-exponents-gather-exponents-equal
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-rational-negative-factor-gather-exponents lhs rhs
mfc state)
(x))
(case-split (rationalp x))
(case-split (not (equal x 0))))
(equal (equal lhs rhs)
(equal (* (/ x) lhs) (* (/ x) rhs)))))
|#
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; This should probably be refactored and cleaned up. Do I have
;;; divisive-factor-p defined anywhere already? Am I consistent in
;;; my notions? Is it consistent with invert-match?
(defun find-divisive-factor-scatter-exponents2 (x mfc state)
(declare (xargs :guard t))
(cond ((or (variablep x)
(fquotep x))
nil)
;; Is this correct?
((eq (ffn-symb x) 'UNARY--)
(find-divisive-factor-scatter-exponents2 (arg1 x) mfc state))
((eq (ffn-symb x) 'UNARY-/)
(if (stable-under-rewriting-products (invert-match x) mfc state)
(list (cons 'x (invert-match x)))
nil))
((eq (ffn-symb x) 'EXPT)
(cond ((eq (fn-symb (arg1 x)) 'UNARY-/)
(if (stable-under-rewriting-products (invert-match x) mfc state)
(list (cons 'x (invert-match x)))
nil))
((and (quotep (arg1 x))
(consp (cdr (arg1 x)))
(not (integerp (cadr (arg1 x))))
(rationalp (cadr (arg1 x))) ; OK, no realp constants
(eql (numerator (cadr (arg1 x))) 1))
(if (stable-under-rewriting-products (invert-match x) mfc state)
(list (cons 'x (invert-match x)))
nil))
((eq (fn-symb (arg2 x)) 'UNARY--)
(if (stable-under-rewriting-products (invert-match x) mfc state)
(list (cons 'x (invert-match x)))
nil))
((and (eq (fn-symb (arg2 x)) 'BINARY-*)
(rational-constant-p (arg1 (arg2 x)))
(< (unquote (arg1 (arg2 x))) 0))
(if (stable-under-rewriting-products (invert-match x) mfc state)
(list (cons 'x (invert-match x)))
nil))
(t
nil)))
((eq (ffn-symb x) 'BINARY-*)
(let ((temp (find-divisive-factor-scatter-exponents2 (arg1 x)
mfc state)))
(if temp
temp
(find-divisive-factor-scatter-exponents2 (arg2 x)
mfc state))))
(t
nil)))
(defun find-divisive-factor-scatter-exponents1 (x mfc state)
(declare (xargs :guard t))
(cond ((or (variablep x)
(fquotep x))
nil)
((eq (ffn-symb x) 'BINARY-+)
(let ((temp (find-divisive-factor-scatter-exponents2 (arg1 x)
mfc state)))
(if temp
temp
(find-divisive-factor-scatter-exponents1 (arg2 x)
mfc state))))
(t
(find-divisive-factor-scatter-exponents2 x mfc state))))
(defun find-divisive-factor-scatter-exponents (lhs rhs mfc state)
(declare (xargs :guard t))
(let ((temp1 (find-divisive-factor-scatter-exponents1 lhs mfc state)))
(if temp1
temp1
(let ((temp2 (find-divisive-factor-scatter-exponents1 rhs mfc state)))
(if temp2
temp2
nil)))))
(defthm prefer-positive-exponents-scatter-exponents-equal
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
; (syntaxp (not (equal rhs ''0)))
; (syntaxp (not (equal lhs ''0)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-divisive-factor-scatter-exponents lhs rhs
mfc state)
(x))
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (acl2-numberp x))
(case-split (not (equal x 0))))
(equal (equal lhs rhs)
(equal (* x lhs) (* x rhs)))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; This should probably be refactored and cleaned up. Do I have
;;; divisive-factor-p defined anywhere already?
(defun find-rational-divisive-factor-scatter-exponents2 (x mfc state)
(declare (xargs :guard t))
(cond ((or (variablep x)
(fquotep x))
nil)
((eq (ffn-symb x) 'UNARY--)
(find-rational-divisive-factor-scatter-exponents2 (arg1 x) mfc state))
((eq (ffn-symb x) 'UNARY-/)
(if (and (proveably-real/rational 'x `((x . ,x)) mfc state)
(stable-under-rewriting-products (invert-match x) mfc state))
(list (cons 'x (invert-match x)))
nil))
((eq (ffn-symb x) 'EXPT)
(cond ((eq (fn-symb (arg1 x)) 'UNARY-/)
(if (and (proveably-real/rational 'x `((x . ,x)) mfc state)
(stable-under-rewriting-products (invert-match x) mfc state))
(list (cons 'x (invert-match x)))
nil))
((and (quotep (arg1 x))
(consp (cdr (arg1 x)))
(not (integerp (cadr (arg1 x))))
(rationalp (cadr (arg1 x))) ; OK, no realp constants
(eql (numerator (cadr (arg1 x))) 1))
(if (and (proveably-real/rational 'x `((x . ,x)) mfc state)
(stable-under-rewriting-products (invert-match x) mfc state))
(list (cons 'x (invert-match x)))
nil))
((eq (fn-symb (arg2 x)) 'UNARY--)
(if (and (proveably-real/rational 'x `((x . ,x)) mfc state)
(stable-under-rewriting-products (invert-match x) mfc state))
(list (cons 'x (invert-match x)))
nil))
((and (eq (fn-symb (arg2 x)) 'BINARY-*)
(rational-constant-p (arg1 (arg2 x)))
(< (unquote (arg1 (arg2 x))) 0))
(if (and (proveably-real/rational 'x `((x . ,x)) mfc state)
(stable-under-rewriting-products (invert-match x) mfc state))
(list (cons 'x (invert-match x)))
nil))
(t
nil)))
((eq (ffn-symb x) 'BINARY-*)
(let ((temp (find-rational-divisive-factor-scatter-exponents2 (arg1 x) mfc state)))
(if temp
temp
(find-rational-divisive-factor-scatter-exponents2 (arg2 x) mfc state))))
(t
nil)))
(defun find-rational-divisive-factor-scatter-exponents1 (x mfc state)
(declare (xargs :guard t))
(cond ((or (variablep x)
(fquotep x))
nil)
((eq (ffn-symb x) 'BINARY-+)
(let ((temp (find-rational-divisive-factor-scatter-exponents2 (arg1 x) mfc state)))
(if temp
temp
(find-rational-divisive-factor-scatter-exponents1 (arg2 x) mfc state))))
(t
(find-rational-divisive-factor-scatter-exponents2 x mfc state))))
(defun find-rational-divisive-factor-scatter-exponents (lhs rhs mfc state)
(declare (xargs :guard t))
(let ((temp1 (find-rational-divisive-factor-scatter-exponents1 lhs mfc state)))
(if temp1
temp1
(let ((temp2 (find-rational-divisive-factor-scatter-exponents1 rhs mfc state)))
(if temp2
temp2
nil)))))
(defthm prefer-positive-exponents-scatter-exponents-<
(implies (and (acl2-numberp lhs)
(acl2-numberp rhs)
; (syntaxp (not (equal rhs ''0)))
; (syntaxp (not (equal lhs ''0)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-rational-divisive-factor-scatter-exponents lhs rhs
mfc state)
(x)) (syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (real/rationalp x))
(case-split (not (equal x 0))))
(equal (< lhs rhs)
(cond ((< 0 x)
(< (* x lhs) (* x rhs)))
(t
(< (* x rhs) (* x lhs)))))))
(defthm prefer-positive-exponents-scatter-exponents-<-2
(implies (and (real/rationalp lhs)
(real/rationalp rhs)
; (syntaxp (not (equal rhs ''0)))
; (syntaxp (not (equal lhs ''0)))
(syntaxp (in-term-order-* lhs mfc state))
(syntaxp (in-term-order-* rhs mfc state))
(bind-free
(find-divisive-factor-scatter-exponents lhs rhs
mfc state)
(x))
(syntaxp (not (equal x ''1)))
(syntaxp (not (equal x ''-1)))
(case-split (acl2-numberp x))
(case-split (not (equal x 0))))
(equal (< lhs rhs)
(cond ((< 0 x)
(< (* x lhs) (* x rhs)))
(t
(< (* x rhs) (* x lhs))))))
:hints (("Goal" :in-theory (enable big-helper-2))))
|
