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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")
(include-book "abs")
(deflabel nsa-theory-start)
(defthm standard-part-abs-0-thm
(iff (= (standard-part (abs x)) 0)
(= (standard-part x) 0)))
(defthm standard-numberp-times-type-thm
(implies
(and
(standard-numberp x)
(standard-numberp y))
(standard-numberp (* x y)))
:rule-classes ((:type-prescription)))
(defthm standard-numberp-plus-type-thm
(implies
(and
(standard-numberp x)
(standard-numberp y))
(standard-numberp (+ x y)))
:rule-classes ((:type-prescription)))
(encapsulate
()
(local (defthm arith-4
(implies
(and
(realp x)
(realp y)
(not (equal x 0))
(not (equal y 0))
(<= (abs x)
(abs y)))
(<= (abs (/ y))
(abs (/ x))))
:rule-classes nil))
(defthm standard-bound-x-implies-limited-x-thm
(implies
(and
(standard-numberp y)
(realp y)
(realp x)
(not (equal y 0))
(<= (abs x)
(abs y)))
(i-limited x))
:rule-classes nil
:hints (("Goal" :cases ((equal y 0) (not (equal y 0)))
:in-theory (disable abs))
("Goal'" :use ((:instance arith-4)
(:instance standard-part-<= (x (abs (/ y)))
(y (abs (/ x))))))))
(defthm limited-bound-x-implies-limited-x-thm
(implies
(and
(i-limited y)
(realp y)
(realp x)
(not (equal y 0))
(<= (abs x)
(abs y)))
(i-limited x))
:rule-classes nil
:hints (("Goal" :cases ((i-small y) (not (i-small y)))
:in-theory (disable abs))
("Subgoal 2" :use ((:instance arith-4)
(:instance standard-part-<= (x (abs (/ y)))
(y (abs (/ x))))))
("Subgoal 1" :use ((:instance arith-4)
(:instance standard-part-<= (x (abs (/ y)))
(y (abs (/ x)))))))))
(defthm plus-limited
(implies
(and
(realp x)
(realp y)
(i-limited x)
(i-limited y))
(i-limited (+ x y)))
:rule-classes :rewrite
:hints (("Goal" :use ((:instance standard+small->i-limited
(x (standard-part (+ x y)))
(eps (- (+ x y) (standard-part (+ x y)))))))))
(defthm times-limited
(implies
(and
(realp x)
(realp y)
(i-limited x)
(i-limited y))
(i-limited (* x y)))
:rule-classes :rewrite
:hints (("Goal" :use ((:instance standard+small->i-limited
(x (standard-part (* x y)))
(eps (- (* x y) (standard-part (* x
y)))))))))
(defthm divide-limited
(implies
(and
(realp x)
(realp y)
(i-limited x)
(i-limited y)
(not (i-small y)))
(i-limited (/ x y)))
:rule-classes :rewrite
:hints (("Goal" :use ((:instance standard+small->i-limited
(x (standard-part (/ x y)))
(eps (- (/ x y) (standard-part (/ x
y)))))))))
(defthm btwn-0-and-1-limited-thm
(implies
(and
(realp a)
(< 0 a)
(< a 1))
(i-limited a))
:rule-classes nil
:hints (("Goal" :cases ((i-small a) (not (i-small a))))
("Subgoal 1" :use ((:instance (:theorem (implies
(and
(realp x)
(realp y)
(not (equal x 0))
(not (equal y 0))
(< 0 x)
(< x 1))
(< 1 (/ x)))))
(:instance standard-part-<= (x 1)
(y (/ a)))))))
(defthm sandwich-limited-thm-hint-1
(implies
(and
(realp u)
(realp v)
(realp a)
(< 0 a)
(< a 1)
(i-limited u)
(i-limited v))
(i-limited (+ v (* a (- u v)))))
:rule-classes nil
:hints (("Goal" :use ((:instance btwn-0-and-1-limited-thm)))))
(defthm sandwich-limited-thm
(implies
(and
(realp u)
(realp v)
(realp x)
(< u x)
(< x v)
(i-limited u)
(i-limited v))
(i-limited x))
:rule-classes nil
:hints (("Goal" :in-theory (disable i-large
DISTRIBUTIVITY
distributivity-left)
:use ((:instance sandwich-limited-thm-hint-1
(a (/ (- x v) (- u v))))))))
(defthm /-large-integer-is-ismall-thm
(i-small (/ (i-large-integer)))
:hints (("Goal" :in-theory (disable i-large-integer-is-large)
:use ((:instance i-large-integer-is-large)))))
(deftheory nsa-theory
(set-difference-theories
(universal-theory :here)
(universal-theory 'nsa-theory-start)))
(defthm standard-part-limited-thm
(implies
(and
(realp x)
(i-limited x))
(i-limited (standard-part x)))
:hints (("Goal" :use ((:instance standards-are-limited
(x (standard-part x)))))))
(defthm /-large-integer-standard-part-thm
(equal (STANDARD-PART (/ (I-LARGE-INTEGER)))
0)
:hints (("Goal" :in-theory (disable /-LARGE-INTEGER-IS-ISMALL-THM)
:use ((:instance /-LARGE-INTEGER-IS-ISMALL-THM)))))
(defthm /-large-integer-limited-thm
(i-limited (/ (I-LARGE-INTEGER)))
:rule-classes ((:type-prescription) (:rewrite))
:hints (("Goal" :in-theory (disable /-LARGE-INTEGER-IS-ISMALL-THM)
:use ((:instance /-LARGE-INTEGER-IS-ISMALL-THM)))))
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