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; See the top-level arithmetic-2 LICENSE file for authorship,
; copyright, and license information.
;;
;; inequalities.lisp
;;
(in-package "ACL2")
(local (include-book "basic-arithmetic"))
; ??? 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.
(defmacro fc (x)
x)
;; I need iff-equal for the next batch of theorems up till
;; <-*-right-cancel (which is in fact proved in
;; inequalities-helper.lisp).
(local
(defthm iff-equal
(equal (equal (< w x) (< y z))
(iff (< w x) (< y z)))))
(defthm /-preserves-positive
(implies (real/rationalp x)
(equal (< 0 (/ x))
(< 0 x))))
(defthm /-preserves-negative
(implies (real/rationalp x)
(equal (< (/ x) 0)
(< x 0))))
(defthm <-0-minus
(equal (< 0 (- x))
(< x 0)))
(defthm <-minus-zero
(equal (< (- x) 0)
(< 0 x)))
(defthm <-minus-minus
(equal (< (- x) (- y))
(< y 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)))
(defthm <-*-0
(implies (and (real/rationalp x)
(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 (real/rationalp x)
(real/rationalp y))
(equal (< 0 (* x y))
(and (not (equal x 0))
(not (equal y 0))
(iff (< 0 x)
(< 0 y))))))
; 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))
((< z 0) (< y x))
((equal z 0) nil)
(t nil))))
: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))))
:in-theory (disable |0-<-*|))))
(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))
((< z 0) (< y x))
((equal z 0) nil)
(t nil)))))
(defthm <-*-x-y-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (* x y) y)
(cond
((< 0 y) (< x 1))
((< y 0) (< 1 x))
((equal y 0) nil)
(t nil))))
:hints (("Goal" :use ((:instance <-*-right-cancel
(x x)
(y 1)
(z y))))))
(defthm <-*-y-x-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< (* y x) y)
(cond
((< 0 y) (< x 1))
((< y 0) (< 1 x))
((equal y 0) nil)
(t nil)))))
(defthm <-y-*-x-y
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (* x y))
(cond
((< 0 y) (< 1 x))
((< y 0) (< x 1))
((equal y 0) nil)
(t nil))))
:hints (("Goal" :use ((:instance <-*-right-cancel
(x 1)
(y x)
(z y))))))
(defthm <-y-*-y-x
(implies (and (fc (real/rationalp x))
(fc (real/rationalp y)))
(equal (< y (* y x))
(cond
((< 0 y) (< 1 x))
((< y 0) (< x 1))
((equal y 0) nil)
(t nil)))))
|
