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|
; Copyright (C) 2007 by Shant Harutunian
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
(include-book "arith-nsa4")
(defthm abs-realp
(implies
(realp x)
(realp (abs x)))
:rule-classes :type-prescription)
(defthm abs-non-neg-thm
(implies
(and
(realp x)
(<= 0 x))
(equal (abs x)
x)))
(defthm abs-neg-thm
(implies
(and
(realp x)
(< x 0))
(equal (abs x)
(- x))))
(defthm abs-pos-*-left-thm
(implies
(and
(realp x)
(realp y)
(<= 0 x))
(equal (abs (* x y))
(* x (abs y)))))
(defthm abs-pos-*-right-thm
(implies
(and
(realp x)
(realp y)
(<= 0 x))
(equal (abs (* y x))
(* x (abs y)))))
(defthm abs-neg-*-left-thm
(implies
(and
(realp x)
(realp y)
(< x 0))
(equal (abs (* x y))
(* (- x) (abs y)))))
(defthm abs-neg-*-right-thm
(implies
(and
(realp x)
(realp y)
(< x 0))
(equal (abs (* y x))
(* (- x) (abs y)))))
(defthm abs-triangular-inequality-thm
(implies
(and
(realp x)
(realp y))
(<= (abs (+ x y))
(+ (abs x) (abs y))))
:rule-classes :linear)
(defthm abs-triangular-inequality-3way-thm
(implies
(and
(realp x)
(realp y)
(realp z))
(<= (abs (+ x y z))
(+ (abs x) (abs y) (abs z)))))
(defthm abs-is-non-neg-thm
(implies
(realp x)
(<= 0 (abs x)))
:rule-classes :type-prescription)
(defthm abs0-thm
(implies
(equal (abs x) 0)
(equal x 0))
:rule-classes :forward-chaining)
(defthm abs-*-thm
(implies
(and
(realp x)
(realp y))
(equal (abs (* x y)) (* (abs x) (abs y)))))
(defthm abs-<-*-thm
(implies
(and
(realp x)
(realp y)
(realp a)
(<= (abs x) (abs y)))
(<= (abs (* a x)) (abs (* a y))))
:rule-classes :linear
:hints (("Goal" :in-theory (disable abs <-*-LEFT-CANCEL)
:use ((:instance pos-factor-<=-thm (x (abs x))
(y (abs y))
(a (abs a)))
(:instance abs-*-thm (x a) (y x))
(:instance abs-*-thm (x a) (y y))))))
(defthm abs-limited-thm
(implies
(and
(realp x)
(i-limited x))
(i-limited (abs x)))
:rule-classes ((:rewrite) (:type-prescription)))
(defthm abs-standard-numberp
(implies
(and
(realp x)
(standard-numberp x))
(standard-numberp (abs x))))
|
