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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.
;; Extended by Ruben Gamboa to include the reals when ACL2(r) is used. This is
;; nothing more than a trivial change of rationalp to realp.
;; Modified by Jared Davis in January 2014 to add xdoc, simplify hints, and
;; generally organize into sections.
(in-package "ACL2")
(include-book "equalities")
(local (defthm equal-of-booleans-rewrite
;; [Jared] this rule avoids needing many corollaries and ugly uses of
;; the previous rule iff-equal
(implies (and (booleanp x)
(booleanp y))
(equal (equal x y)
(iff x y)))))
(defsection inequalities-of-sums
:parents (arithmetic-1)
:short "Basic normalization to move negative terms to the other side of an
inequality."
(defthm <-0-minus
(equal (< 0 (- x))
(< x 0)))
(defthm <-minus-zero
(equal (< (- x) 0)
(< 0 x)))
(defthm <-0-+-negative-1
(equal (< 0 (+ (- y) x))
(< y x)))
(defthm <-0-+-negative-2
(equal (< 0 (+ x (- y)))
(< y x)))
(defthm <-+-negative-0-1
(equal (< (+ (- y) x) 0)
(< x y)))
(defthm <-+-negative-0-2
(equal (< (+ x (- y)) 0)
(< x y))))
; ??? I'm not convinced we should be apply FC to REAL/RATIONALP hypotheses,
; but for now I'll go ahead and do so at times.
(defsection inequalities-of-products
:parents (arithmetic-1)
:short "Rules for reducing inequalities between products, canceling like
factors, and comparing products against 0."
; The following two lemmas could be extended by adding two more such
; lemmas, i.e. for (< (* x z) (* z y)) and (< (* z x) (* y z)), but
; rather than incur that overhead here and in any other such cases
; (and besides, how about for example (< (* x z a) (* z a y))?), I'll
; wait for metalemmas to handle such things.
(defthm <-*-right-cancel
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp z)))
(equal (< (* x z) (* y z))
(cond ((< 0 z) (< x y))
((equal z 0) nil)
(t (< y x)))))
:hints (("Goal" :use ((:instance (:theorem
(implies (and (real/rationalp a)
(< 0 a)
(real/rationalp b)
(< 0 b))
(< 0 (* a b))))
(a (abs (- y x)))
(b (abs z)))))))
(defthm <-*-left-cancel
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp z)))
(equal (< (* z x) (* z y))
(cond ((< 0 z) (< x y))
((equal z 0) nil)
(t (< y x))))))
(defthm <-*-0
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (* x y) 0)
(and (not (equal x 0))
(not (equal y 0))
(iff (< x 0) (< 0 y))))))
(defthm 0-<-*
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< 0 (* x y))
(and (not (equal x 0))
(not (equal y 0))
(iff (< 0 x) (< 0 y))))))
(defthm <-*-x-y-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (* x y) y)
(cond ((equal y 0) nil)
((< 0 y) (< x 1))
(t (< 1 x)))))
:hints (("Goal" :use ((:instance <-*-right-cancel
(z y)
(x x)
(y 1))))))
(defthm <-*-y-x-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (* y x) y)
(cond ((equal y 0) nil)
((< 0 y) (< x 1))
(t (< 1 x))))))
(defthm <-y-*-x-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (* x y))
(cond ((equal y 0) nil)
((< 0 y) (< 1 x))
(t (< x 1)))))
:hints (("Goal" :use ((:instance <-*-right-cancel
(z y)
(x 1)
(y x))))))
(defthm <-y-*-y-x
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (* y x))
(cond ((equal y 0) nil)
((< 0 y) (< 1 x))
(t (< x 1)))))))
(defsection inequalities-of-reciprocals
:parents (arithmetic-1)
:short "Basic rules for moving reciprocals across inequalities, comparing
reciprocals, and canceling reciprocals by multiplying across an inequality."
(defthm /-preserves-positive
(implies (fc (real/rationalp x))
(equal (< 0 (/ x))
(< 0 x))))
(defthm /-preserves-negative
(implies (fc (real/rationalp x))
(equal (< (/ x) 0)
(< x 0))))
(defthm /-inverts-order-1
(implies (and (< 0 x)
(< x y)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(< (/ y) (/ x)))
:hints (("Goal" :use ((:instance <-*-right-cancel (z (/ (* x y))))))))
(defthm /-inverts-order-2
(implies (and (< y 0)
(< x y)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(< (/ y) (/ x)))
:hints (("Goal"
:use ((:instance <-*-right-cancel (z (/ (* x y)))))
:in-theory (disable <-*-right-cancel))))
(defthm /-inverts-weak-order
(implies (and (< 0 x)
(<= x y)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(not (< (/ x) (/ y))))
:hints (("Goal"
:use /-inverts-order-1
:in-theory (disable /-inverts-order-1))))
(defthm <-unary-/-negative-left
(implies (and (< x 0)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (/ x) y)
(< (* x y) 1)))
:hints (("Goal"
:use ((:instance <-*-left-cancel (z x) (x (/ x)) (y y)))
:in-theory (e/d (Uniqueness-of-*-inverses)
(equal-/)))))
(defthm <-unary-/-negative-right
(implies (and (< x 0)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (/ x))
(< 1 (* x y))))
:hints (("Goal" :use ((:instance <-*-right-cancel (z x) (x (/ x)) (y y))))))
(defthm <-unary-/-positive-left
(implies (and (< 0 x)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (/ x) y)
(< 1 (* x y))))
:hints (("Goal" :use ((:instance <-*-left-cancel (z x) (x (/ x)) (y y))))))
(defthm <-unary-/-positive-right
(implies (and (< 0 x)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (/ x))
(< (* x y) 1)))
:hints (("Goal"
:use ((:instance <-*-left-cancel (z x) (x (/ x)) (y y)))
:in-theory (e/d (Uniqueness-of-*-inverses)
(<-unary-/-positive-left
equal-/
<-*-left-cancel)))))
(defthm <-*-/-right
(implies (and (< 0 y)
(fc (real/rationalp a))
(fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< a (* x (/ y)))
(< (* a y) x)))
:hints (("Goal" :use ((:instance <-*-right-cancel
(x a)
(y (* x (/ y)))
(z y))))))
(defthm <-*-/-right-commuted
(implies (and (< 0 y)
(fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp a)))
(equal (< x (* (/ y) a))
(< (* x y) a))))
(defthm <-*-/-left
(implies (and (< 0 y)
(fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp a)))
(equal (< (* x (/ y)) a)
(< x (* a y))))
:hints (("Goal" :use ((:instance <-*-right-cancel
(y a)
(x (* x (/ y)))
(z y))))))
(defthm <-*-/-left-commuted
(implies (and (< 0 y)
(fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp a)))
(equal (< (* (/ y) x) a)
(< x (* y a))))))
#| Already covered by EQUAL-*-/-2 from rationals-equalities.lisp
(defthm equal-binary-quotient
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y))
(fc (real/rationalp z))
(fc (not (equal z 0))))
(equal (equal x (/ y z))
(equal (* x z) y)))
:hints (("Goal" :use
((:instance
(:theorem
(implies (and (real/rationalp x)
(real/rationalp y)
(real/rationalp z)
(not (equal z 0))
(equal x y))
(equal (* x z) (* y z))))
(x (* x z))
(y y)
(z (/ z)))))))
|#
(defsection inequalities-of-products-2
:extension inequalities-of-products
;; Jared: I found I couldn't move *-preserves->=-for-nonnegatives up into the
;; rule up ordinary inequalities-of-products section without breaking proofs
;; like coi/super-ihs/iter-sqrt/sqrt-epsilon-delta-aux-6. Apparently these
;; proofs are sensitive to the order of rules here. Ugh.
(local (defthm *-preserves->=-1
(implies (and (real/rationalp x1)
(real/rationalp x2)
(real/rationalp y1)
(>= x1 x2)
(>= x2 0)
(>= y1 0))
(>= (* x1 y1) (* x2 y1)))
:rule-classes nil))
(local (defthm *-preserves->=-2
(implies (and (real/rationalp x2)
(real/rationalp y1)
(real/rationalp y2)
(>= y1 y2)
(>= x2 0)
(>= y2 0))
(>= (* x2 y1) (* x2 y2)))
:rule-classes nil))
(defthm *-preserves->=-for-nonnegatives
(implies (and (>= x1 x2)
(>= y1 y2)
(>= x2 0)
(>= y2 0)
(fc (real/rationalp x1))
(fc (real/rationalp x2))
(fc (real/rationalp y1))
(fc (real/rationalp y2)))
(>= (* x1 y1) (* x2 y2)))
:hints (("Goal" :use
(*-preserves->=-1 *-preserves->=-2)
:in-theory (disable <-*-left-cancel <-*-right-cancel))))
(local (defthm *-preserves->-1
(implies (and (real/rationalp x1)
(real/rationalp x2)
(real/rationalp y1)
(> x1 x2)
(>= x2 0)
(> y1 0))
(>= (* x1 y1) (* x2 y1)))
:rule-classes nil))
(local (defthm *-preserves->-2
(implies (and (real/rationalp x2)
(real/rationalp y1)
(real/rationalp y2)
(>= y1 y2)
(>= x2 0)
(>= y2 0))
(>= (* x2 y1) (* x2 y2)))
:rule-classes nil))
(defthm *-preserves->-for-nonnegatives-1
(implies (and (> x1 x2)
(>= y1 y2)
(>= x2 0)
(> y2 0)
(fc (real/rationalp x1))
(fc (real/rationalp x2))
(fc (real/rationalp y1))
(fc (real/rationalp y2)))
(> (* x1 y1) (* x2 y2)))
:hints (("Goal" :use (*-preserves->-1 *-preserves->-2)
:in-theory (disable <-*-left-cancel <-*-right-cancel
*-preserves->=-for-nonnegatives))))
(defthm x*y>1-positive-lemma
(implies (and (> x i)
(> y j)
(real/rationalp i)
(real/rationalp j)
(< 0 i)
(< 0 j)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(> (* x y) (* i j)))
:hints (("Goal" :use ((:instance 0-<-*
(x (- x i))
(y (- y j)))))))
(defthm x*y>1-positive
(implies (and (> x 1)
(> y 1)
(fc (real/rationalp x))
(fc (real/rationalp y)))
(> (* x y) 1))
:rule-classes (:linear :rewrite)
:hints (("Goal" :use ((:instance x*y>1-positive-lemma
(i 1) (j 1)))))))
(defsection inequalities-of-exponents
:parents (arithmetic-1)
:short "Rules for resolving inequalities between exponents."
(defthm expt->-1
(implies (and (< 1 r)
(< 0 i)
(fc (real/rationalp r))
(fc (integerp i)))
(< 1 (expt r i)))
:rule-classes :linear)
(local (defun Math-induction-start-at-k (k n)
(declare (xargs :guard (and (integerp k)
(integerp n)
(not (< n k)))
:measure (if (and (integerp k)
(integerp n)
(< k n))
(- n k)
0)))
(if (and (integerp k)
(integerp n)
(< k n))
(Math-induction-start-at-k k (+ -1 n))
t)))
(local (defthm hack
(implies (and (< (* x y) z)
(real/rationalp x)
(real/rationalp y)
(real/rationalp z)
(< 1 x)
(< 0 y))
(< y z))
:hints
(("Goal" :use ((:instance (:theorem
(implies (and (real/rationalp x)
(real/rationalp y)
(real/rationalp z)
(< 0 y)
;; w = (* x1 y)
(real/rationalp w)
(< 0 w)
(< (+ y w) z))
(< y z)))
(w (* (- x 1) y))))))))
(defthm expt-is-increasing-for-base>1
(implies (and (< 1 r)
(< i j)
(fc (real/rationalp r))
(fc (integerp i))
(fc (integerp j)))
(< (expt r i)
(expt r j)))
:rule-classes (:rewrite :linear)
:hints (("Goal"
:in-theory (enable Exponents-add-unrestricted)
:induct (Math-induction-start-at-k (+ 1 i) j))))
(defthm Expt-is-decreasing-for-pos-base<1
(implies (and (< 0 r)
(< r 1)
(< i j)
(fc (real/rationalp r))
(fc (integerp i))
(fc (integerp j)))
(< (expt r j)
(expt r i)))
:hints (("Goal" :use ((:instance expt-is-increasing-for-base>1 (r (/ r)))))))
(defthm Expt-is-weakly-increasing-for-base>1
(implies (and (< 1 r)
(<= i j)
(fc (real/rationalp r))
(fc (integerp i))
(fc (integerp j)))
(<= (expt r i)
(expt r j)))
:hints (("Goal" :use expt-is-increasing-for-base>1
:in-theory (disable expt-is-increasing-for-base>1))))
(defthm Expt-is-weakly-decreasing-for-pos-base<1
(implies (and (< 0 r)
(< r 1)
(<= i j)
(fc (real/rationalp r))
(fc (integerp i))
(fc (integerp j)))
(<= (expt r j)
(expt r i)))
:hints (("Goal" :use expt-is-decreasing-for-pos-base<1
:in-theory (disable expt-is-decreasing-for-pos-base<1)))))
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