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; Copyright (C) 1997 Computational Logic, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by:
; Matt Kaufmann, Bishop Brock, and John Cowles, with help from Art Flatau
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
#-:non-standard-analysis
(defmacro real-listp (l)
`(rational-listp ,l))
(include-book "cowles/acl2-crg" :dir :system)
(defsection fc
:parents (arithmetic-1)
:short "Identity macro — does nothing, you can safely ignore this."
:long "<p>@(call fc) just expands to @('x'). This macro is a historic
artifact that was originally used in the @('arithmetic') library as a way to
experiment with using @(see force).</p>
@(def fc)"
#|
(defmacro fc (x)
(list 'force x))
|#
(defmacro fc (x)
x))
(defsection basic-sum-normalization
:parents (arithmetic-1)
:short "Trivial normalization and cancellation rules for sums."
(defthm commutativity-2-of-+
(equal (+ x (+ y z))
(+ y (+ x z))))
(defthm functional-self-inversion-of-minus
(equal (- (- x))
(fix x)))
(defthm distributivity-of-minus-over-+
(equal (- (+ x y))
(+ (- x) (- y))))
(defthm minus-cancellation-on-right
(equal (+ x y (- x))
(fix y)))
(defthm minus-cancellation-on-left
(equal (+ x (- x) y)
(fix y)))
; Note that the cancellation rules below (and similarly for *) aren't
; complete, in the sense that the element to cancel could be on the
; left side of one expression and the right side of the other. But
; perhaps those situations rarely arise in practice. (?)
(defthm right-cancellation-for-+
(equal (equal (+ x z)
(+ y z))
(equal (fix x) (fix y))))
(defthm left-cancellation-for-+
(equal (equal (+ x y)
(+ x z))
(equal (fix y) (fix z))))
(defthm equal-minus-0
(equal (equal 0 (- x))
(equal 0 (fix x))))
(defthm inverse-of-+-as=0
(equal (equal (- a b) 0)
(equal (fix a) (fix b))))
(defthm equal-minus-minus
(equal (equal (- a) (- b))
(equal (fix a) (fix b))))
(defthm fold-consts-in-+
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (+ x (+ y z))
(+ (+ x y) z)))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; Facts about * (and /)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The same as Inverse-of-*, from axioms.lisp, but with force.
#|
(defaxiom right-inverse-of-*
(implies (and (acl2-numberp x)
(not (equal x 0)))
(equal (* x (/ x)) 1)))
|#
#| The following proof of commutativity-2-of-* was originally obtained by using
John Cowles's macro acl2-asg::add-commutativity-2 as follows, and
then editing out package references in the statement of the final
theorem.
(acl2-asg::add-commutativity-2 equal
acl2-numberp
*
commutativity-of-*
commutativity-2-of-*)
|#
(defsection basic-product-normalization
:parents (arithmetic-1)
:short "Trivial normalization and cancellation rules for products."
(defthm commutativity-2-of-*
(equal (* x (* y z))
(* y (* x z)))
:hints (("Goal" :use (:functional-instance acl2-asg::commutativity-2-of-op
(acl2-asg::equiv equal)
(acl2-asg::pred (lambda (x) t))
(acl2-asg::op binary-*)))))
(defthm functional-self-inversion-of-/
(equal (/ (/ x)) (fix x))
:hints (("Goal" :use (:functional-instance
acl2-agp::Involution-of-inv
(acl2-agp::equiv equal)
(acl2-agp::pred (lambda (x)
(and (acl2-numberp x)
(not (equal x 0)))))
(acl2-agp::op binary-*)
(acl2-agp::id (lambda () 1))
(acl2-agp::inv unary-/)))))
(defthm distributivity-of-/-over-*
(equal (/ (* x y))
(* (/ x) (/ y)))
:hints (("Goal" :use (:functional-instance
acl2-agp::Distributivity-of-inv-over-op
(acl2-agp::equiv equal)
(acl2-agp::pred (lambda (x)
(and (acl2-numberp x)
(not (equal x 0)))))
(acl2-agp::op binary-*)
(acl2-agp::id (lambda () 1))
(acl2-agp::inv unary-/)))))
(defthm /-cancellation-on-right
(implies (and (fc (acl2-numberp x))
(fc (not (equal x 0))))
(equal (* x y (/ x))
(fix y)))
:hints (("Goal" :use (:functional-instance
acl2-agp::inv-cancellation-on-right
(acl2-agp::equiv equal)
(acl2-agp::pred (lambda (x)
(and (acl2-numberp x)
(not (equal x 0)))))
(acl2-agp::op binary-*)
(acl2-agp::id (lambda () 1))
(acl2-agp::inv unary-/)))))
(defthm /-cancellation-on-left
(implies (and (fc (acl2-numberp x))
(fc (not (equal 0 x))))
(equal (* x (/ x) y)
(fix y)))
:hints (("Goal" :use /-cancellation-on-right)))
(local
(defthm right-cancellation-for-*-lemma
(implies (and (equal (* x z) (* y z))
(acl2-numberp z)
(not (equal 0 z))
(acl2-numberp x)
(acl2-numberp y))
(equal (equal x y) t))
:hints (("Goal" :use (:functional-instance
acl2-agp::Right-cancellation-for-op
(acl2-agp::equiv equal)
(acl2-agp::pred (lambda (x)
(and (acl2-numberp x)
(not (equal x 0)))))
(acl2-agp::op binary-*)
(acl2-agp::id (lambda () 1))
(acl2-agp::inv unary-/))))))
(defthm right-cancellation-for-*
(equal (equal (* x z) (* y z))
(or (equal 0 (fix z))
(equal (fix x) (fix y)))))
(defthm left-cancellation-for-*
(equal (equal (* z x) (* z y))
(or (equal 0 (fix z))
(equal (fix x) (fix y)))))
(defthm Zero-is-only-zero-divisor
(equal (equal (* x y) 0)
(or (equal (fix x) 0)
(equal (fix y) 0))))
(defthm equal-*-x-y-x
(equal (equal (* x y) x)
(or (equal x 0)
(and (equal y 1)
(acl2-numberp x))))
:hints (("Goal" :use ((:instance right-cancellation-for-*
(z x)
(x y)
(y 1))))))
(defthm equal-*-x-y-y
(equal (equal (* x y) y)
(or (equal y 0)
(and (equal x 1)
(acl2-numberp y))))
:hints (("Goal" :use ((:instance right-cancellation-for-*
(z y)
(x x)
(y 1))))))
(local (defthm equal-/-lemma
(implies (and (acl2-numberp x)
(acl2-numberp y)
(equal (* x y) 1))
(equal y (/ x)))
:rule-classes nil
:hints (("Goal" :use (:functional-instance
acl2-agp::Uniqueness-of-op-inverses
(acl2-agp::equiv equal)
(acl2-agp::pred (lambda (x)
(and (acl2-numberp x)
(not (equal x 0)))))
(acl2-agp::op binary-*)
(acl2-agp::id (lambda () 1))
(acl2-agp::inv unary-/))))))
(defthm equal-/
(implies (and (fc (acl2-numberp x))
(fc (not (equal 0 x))))
(equal (equal (/ x) y)
(equal 1 (* x y))))
:hints (("Goal" :use equal-/-lemma)))
; The following hack helps in the application of equal-/ when forcing is
; turned off.
(defthm numerator-nonzero-forward
(implies (not (equal (numerator r) 0))
(and (not (equal r 0))
(acl2-numberp r)))
:rule-classes
((:forward-chaining :trigger-terms
((numerator r)))))
; The following loops with the lemma equal-/ just proved but is
; sometimes useful.
(encapsulate
()
(local (defthm Uniqueness-of-*-inverses-lemma
(equal (equal (* x y) 1)
(and (not (equal x 0))
(acl2-numberp x)
(equal y (/ x))))))
(defthm Uniqueness-of-*-inverses
(equal (equal (* x y) 1)
(and (not (equal (fix x) 0))
(equal y (/ x))))
:hints (("Goal" :in-theory (disable equal-/)))))
(in-theory (disable Uniqueness-of-*-inverses))
(theory-invariant
(not (and (active-runep '(:rewrite Uniqueness-of-*-inverses))
(active-runep '(:rewrite equal-/)))))
(encapsulate
()
(local (defthm equal-/-/-lemma
(implies (and (fc (acl2-numberp a))
(fc (acl2-numberp b))
(fc (not (equal a 0)))
(fc (not (equal b 0))))
(equal (equal (/ a) (/ b))
(equal a b)))
:hints
(("Goal" :use ((:instance (:theorem (implies
(and (fc (acl2-numberp a))
(fc (acl2-numberp b))
(fc (not (equal a 0)))
(fc (not (equal b 0))))
(implies (equal a b)
(equal (/ a) (/ b)))))
(a (/ a)) (b (/ b))))))
:rule-classes nil))
(defthm equal-/-/
(equal (equal (/ a) (/ b))
(equal (fix a) (fix b)))
:hints (("Goal" :use equal-/-/-lemma
:in-theory (disable equal-/)))))
(defthm equal-*-/-1
(implies (and (acl2-numberp x)
(not (equal x 0)))
(equal (equal (* (/ x) y) z)
(and (acl2-numberp z)
(equal (fix y) (* x z))))))
(defthm equal-*-/-2
(implies (and (acl2-numberp x)
(not (equal x 0)))
(equal (equal (* y (/ x)) z)
(and (acl2-numberp z)
(equal (fix y) (* z x))))))
(defthm fold-consts-in-*
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (* x (* y z))
(* (* x y) z))))
(defthm times-zero
;; We could prove an analogous rule about non-numeric coefficients, but
;; this one has efficiency advantages: it doesn't match too often, it has
;; no hypothesis, and also we know that the 0 is the first argument so we
;; don't need two versions. Besides, we won't need this too often; it's
;; a type-reasoning fact. But it did seem useful in the proof of a meta
;; lemma about times cancellation, so we include it here.
(equal (* 0 x) 0)))
(defsection basic-products-with-negations
:parents (arithmetic-1)
:short "Rules for normalizing products with negative factors, and reciprocals
of negations."
(local (defthm functional-commutativity-of-minus-*-right-lemma
(implies (and (fc (acl2-numberp x))
(fc (acl2-numberp y)))
(equal (* x (- y))
(- (* x y))))
:hints (("Goal" :use (:functional-instance
acl2-crg::functional-commutativity-of-minus-times-right
(acl2-crg::equiv equal)
(acl2-crg::pred acl2-numberp)
(acl2-crg::plus binary-+)
(acl2-crg::times binary-*)
(acl2-crg::zero (lambda () 0))
(acl2-crg::minus unary--))))
:rule-classes nil))
(defthm functional-commutativity-of-minus-*-right
(equal (* x (- y))
(- (* x y)))
:hints (("Goal" :use functional-commutativity-of-minus-*-right-lemma)))
(defthm functional-commutativity-of-minus-*-left
(equal (* (- x) y)
(- (* x y))))
(defthm reciprocal-minus
(equal (/ (- x))
(- (/ x)))
:hints (("Goal" :cases
((and (fc (acl2-numberp x))
(fc (not (equal x 0)))))))))
(defsection basic-rational-identities
:parents (arithmetic-1 numerator denominator)
:short "Basic cancellation rules for @('numerator') and @('denominator') terms."
:long "<p>See also @(see more-rational-identities) for additional reductions
involving @('numerator') and @('denominator') terms.</p>"
(local (defthm numerator-integerp-lemma-1
(implies (rationalp x)
(equal (* (* (numerator x) (/ (denominator x))) (denominator x))
(numerator x)))
:rule-classes nil
:hints (("Goal" :in-theory (disable rational-implies2)))))
(local (defthm numerator-integerp-lemma
(implies (and (rationalp x)
(equal (* (numerator x) (/ (denominator x)))
x))
(equal (numerator x)
(* x (denominator x))))
:rule-classes nil
:hints (("Goal" :use (numerator-integerp-lemma-1)
:in-theory (disable rational-implies2)))))
(defthm numerator-when-integerp
(implies (integerp x)
(equal (numerator x)
x))
:hints (("Goal" :in-theory (disable rational-implies2)
:use ((:instance lowest-terms (r x)
(q 1)
(n (denominator x)))
rational-implies2
numerator-integerp-lemma))))
(defthm integerp==>denominator=1
(implies (integerp x)
(equal (denominator x) 1))
:hints (("Goal" :use (rational-implies2 numerator-when-integerp)
:in-theory (disable rational-implies2))))
(defthm equal-denominator-1
(equal (equal (denominator x) 1)
(or (integerp x)
(not (rationalp x))))
:hints (("Goal" :use (rational-implies2 completion-of-denominator)
:in-theory (disable rational-implies2))))
(defthm *-r-denominator-r
(equal (* r (denominator r))
(if (rationalp r)
(numerator r)
(fix r)))
:hints (("Goal" :use ((:instance rational-implies2 (x r)))
:in-theory (disable rational-implies2))))
(defthm /r-when-abs-numerator=1
(and (implies (equal (numerator r) 1)
(equal (/ r) (denominator r)))
(implies (equal (numerator r) -1)
(equal (/ r) (- (denominator r)))))
:hints (("Goal" :use ((:instance rational-implies2 (x r)))
:in-theory (disable rational-implies2)))))
;; Much of this is adapted from John Cowles's @('acl2-exp.lisp') book. There
;; are various modifications, however, including some by Ruben Gamboa to
;; support non-standard analysis in the non-standard version of ACL2, ACL2(r);
;; see :doc real.
(defsection basic-expt-type-rules
:parents (arithmetic-1 expt)
:short "Rules about when @('expt') produces integers, positive numbers, etc."
#+:non-standard-analysis
(defthm expt-type-prescription-realp
(implies (realp r)
(realp (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-rationalp
(implies (rationalp r)
(rationalp (expt r i)))
:rule-classes (:type-prescription :generalize))
;; This theorem was strengthened to allow all real numbers (but reduces to
;; the version with a rationalp hypothesis in for ACL2, as opposed to
;; ACL2(r)).
(defthm expt-type-prescription-positive
(implies (and (< 0 r)
(real/rationalp r))
(< 0 (expt r i)))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-nonzero
(implies (and (fc (acl2-numberp r))
(not (equal r 0)))
(not (equal 0 (expt r i))))
:rule-classes (:type-prescription :generalize))
(defthm expt-type-prescription-integerp
(implies (and (<= 0 i)
(integerp r))
(integerp (expt r i)))
:rule-classes (:type-prescription :generalize))
(in-theory
;; [Jared] Some of these type-prescription rules for expt, above, are
;; duplicates of built-in ACL2 rules:
;;
;; new rule duplicates
;; ---------------------------------------------------------------------------------
;; EXPT-TYPE-PRESCRIPTION-RATIONALP RATIONALP-EXPT-TYPE-PRESCRIPTION
;; EXPT-TYPE-PRESCRIPTION-NONZERO EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE
;;
;; Since the new rules above have :generalize rule-classes as well, I'm going to
;; disable the built-in ACL2 rules.
(disable RATIONALP-EXPT-TYPE-PRESCRIPTION
EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE)))
(defsection basic-expt-normalization
:parents (arithmetic-1 expt)
:short "Basic rules for normalizing and simplifying exponents."
;; [Jared] removing since it is redundant with expt-1, below
;; (defthm Left-nullity-of-1-for-expt
;; (equal (expt 1 i) 1))
(defthm Right-unicity-of-1-for-expt
(equal (expt r 1)
(fix r))
:hints (("Goal" :expand (expt r 1))))
(defthm expt-minus
(equal (expt r (- i))
(/ (expt r i))))
;; The following is superseded by exponents-add below, except for the case
;; that r = 0. But I'll leave it here; in fact it's quite natural to have
;; (roughly speaking) two versions of each rule about expt, based on the
;; disjunction in the guard for expt.
(defthm Exponents-add-for-nonneg-exponents
;; We don't need that r is non-zero for this one.
(implies (and (<= 0 i)
(<= 0 j)
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(encapsulate
()
(local (defthm Exponents-add-negative-negative
(implies (and (integerp i)
(integerp j)
(< i 0)
(< j 0))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j))))
:rule-classes nil))
(local (defthm Exponents-add-positive-negative
(implies (and (integerp i)
(integerp j)
(acl2-numberp r)
(not (equal r 0))
(< 0 i)
(< j 0))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j))))
:hints (("Goal" :expand (expt r (+ i j))))
:rule-classes nil))
(defthm Exponents-add
; The first two (syntaxp) hypotheses below are new for Version_2.6. Without
; this change there can be looping with the definition of expt, for example on
; the following (thanks to Eric Smith for reporting the problem from which this
; example was culled). (By the way, this example is probably not a theorem;
; the point here is to avoid looping.) But see also
; Exponents-add-unrestricted.
#|
(thm (IMPLIES (AND (NOT (ZIP P))
(< 0 P)
(< (* 2 (+ P -1) (/ (EXPT 2 (+ P -1))))
1)
(INTEGERP P)
(< 1 P)
(INTEGERP Q)
(< 0 Q))
(< (* 2 P (/ (EXPT 2 P))) 1)))
|#
(implies (and (syntaxp (not (and (quotep i) (integerp (cadr i))
(or (equal (cadr i) 1)
(equal (cadr i) -1)))))
(syntaxp (not (and (quotep j) (integerp (cadr j))
(or (equal (cadr j) 1)
(equal (cadr j) -1)))))
(not (equal 0 r))
(fc (acl2-numberp r))
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j))))
:hints (("Goal" :use
(Exponents-add-negative-negative
Exponents-add-positive-negative
(:instance Exponents-add-positive-negative
(i j) (j i)))))))
(defthm Exponents-add-unrestricted
; The comment above in Exponents-add explains why we do not leave this rule
; enabled. But we include it in case it is of use. For example, Exponents-add
; is not sufficient for the proof of expt-is-increasing-for-base>1 in
; inequalities.lisp.
(implies (and (not (equal 0 r))
(fc (acl2-numberp r))
(fc (integerp i))
(fc (integerp j)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(in-theory (disable Exponents-add-unrestricted))
(defthm Distributivity-of-expt-over-*
(equal (expt (* a b) i)
(* (expt a i)
(expt b i))))
;; It's not clear to me whether the following rule belongs this way or the
;; other way around, but I'll leave it this way -- mk.
(defthm expt-1
(equal (expt 1 x) 1))
(defthm Exponents-multiply
(implies (and (fc (integerp i))
(fc (integerp j)))
(equal (expt (expt r i) j)
(expt r (* i j))))
:hints (("Goal" :cases
((not (acl2-numberp r))
(equal r 0)))))
(defthm Functional-commutativity-of-expt-/-base
(equal (expt (/ r) i)
(/ (expt r i))))
;; Added 6/01 by Matt Kaufmann in response to an example sent by John Cowles
;; that cannot be proved without it, shown below. Actually this rule was
;; suggested by J Moore.
(defthm equal-constant-+
(implies (syntaxp (and (quotep c1)
(quotep c2)))
(equal (equal (+ c1 x) c2)
(if (acl2-numberp c2)
(if (acl2-numberp x)
(equal x (- c2 c1))
(equal (fix c1) c2))
nil)))))
#| John Cowles's example (see rule above); without the rule above the following
hint is needed for the thm form below:
; :hints (("Goal"
; :use (:theorem
; (implies (equal (+ -1 x) 3)
; (equal x 4)))))
(include-book
"/meru1/cowles/acl2/ver2.5/acl2-sources/books/arithmetic/top-with-meta")
(defun ;; compute 2^n ; ; ;
pow (n)
(if (zp n)
1
(* 2 (pow (- n 1)))))
(defun e (x) ;; product from i=1 to x of 2^i - 1 ; ; ;
(if (zp x)
1
(* (- (pow x) 1)(e (- x 1)))))
(defun
e1 (x)
(if (zp x)
1
(* (pow x)(e1 (- x 1)))))
(thm
;; some complicated hyps removed ; ; ;
(IMPLIES (EQUAL (+ -1 X) 3)
(EQUAL (+ (* 384 (POW (+ -4 X)))
(- (* 768 (POW (+ -4 X)) (POW (+ -4 X))))
(* 3456
(/ (+ (- (* 2 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))))
(* 4 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X)))))))
(+ (- (* 64 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))))
(* 128 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X)))
(* 256 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X)))
(* 512 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X)))
(- (* 512 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))))
(- (* 1024 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))))
(- (* 2048 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))))
(* 4096 (E (+ -4 X))
(E1 (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X))
(POW (+ -4 X)))))))
|#
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