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|
; Arithmetic-5 Library
; Written by Robert Krug
; Copyright/License:
; See the LICENSE file at the top level of the arithmetic-5 library.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; numerator-and-denominator.lisp
;;;
;;; Some simple facts about numerator and denominator.
;;;
;;; Some of the theorems in this book could be generalized, perhaps by
;;; using bind-free. Other useful theorems could probably be added.
;;; However, I choose to leave this book as it is. This book is
;;; sufficient for what I need, and I do not have enough experience to
;;; know what else might be needed.
;;;
;;; If you are reasoning about numerator or denominator, you are
;;; probably going about it the wrong way. If you truly need to
;;; reason about numerator and denominator and discover some useful
;;; theorems to add, please send them to the ACL2 maintainers. They
;;; could be added to this book.
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(local
(include-book "../../support/top"))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Type-set already knows that (numerator x) and (denominator x)
;;; are integers, and that 0 < (denominator x).
(defthm Rational-implies2 ; Redundant, from axioms.lisp
(implies (rationalp x)
(equal (* (/ (denominator x)) (numerator x)) x)))
(local
(in-theory (enable rewrite-linear-equalities-to-iff)))
(defthm numerator-zero
(implies (rationalp x)
(equal (equal (numerator x) 0)
(equal x 0)))
:rule-classes ((:rewrite)
(:type-prescription
:corollary
(implies (equal x 0)
(equal (numerator x) 0)))))
(defthm numerator-positive
(implies (rationalp x)
(equal (< 0 (numerator x))
(< 0 x)))
:rule-classes ((:rewrite)
(:type-prescription
:corollary
(implies (and (rationalp x)
(< 0 x))
(and (integerp (numerator x))
(< 0 (numerator x)))))
(:type-prescription
:corollary
(implies (and (rationalp x)
(<= 0 x))
(and (integerp (numerator x))
(<= 0 (numerator x)))))))
(defthm numerator-negative
(implies (rationalp x)
(equal (< (numerator x) 0)
(< x 0)))
:rule-classes ((:rewrite)
(:type-prescription
:corollary
(implies (and (rationalp x)
(< x 0))
(and (integerp (numerator x))
(< (numerator x) 0))))
(:type-prescription
:corollary
(implies (and (rationalp x)
(<= x 0))
(and (integerp (numerator x))
(<= (numerator x) 0))))))
(defthm numerator-minus
(equal (numerator (- i))
(- (numerator i))))
(defthm denominator-minus
(implies (rationalp x)
(equal (denominator (- x))
(denominator x))))
(defthm integerp==>numerator-=-x
(implies (integerp x)
(equal (numerator x)
x)))
(defthm integerp==>denominator-=-1
(implies (integerp x)
(equal (denominator x)
1)))
(defthm equal-denominator-1
(equal (equal (denominator x) 1)
(or (integerp x)
(not (rationalp x)))))
(defthm |(* r (denominator r))|
(implies (rationalp r)
(equal (* r (denominator r))
(numerator r))))
;;; From arithmetic/equalities.lisp
;;; This next rule is too general. I disable it right away.
;;; I do, however, include a couple of rules similar to this
;;; below.
(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))))
(in-theory (disable /r-when-abs-numerator=1))
(defthm |(equal (/ r) (denominator r))|
(implies (rationalp r)
(equal (equal (/ r) (denominator r))
(equal (numerator r) 1))))
(defthm |(equal (/ r) (- (denominator r)))|
(implies (rationalp r)
(equal (equal (/ r) (- (denominator r)))
(equal (numerator r) -1))))
;;; From arithmetic/rationals.lisp
; We name this Numerator-goes-down-by-integer-division-linear rather than
; Numerator-goes-down-by-integer-division to avoid a conflict with system books
; arithmetic/rationals.lisp.
(defthm
Numerator-goes-down-by-integer-division-linear
(implies (and (integerp x)
(< 0 x)
(rationalp r))
(<= (abs (numerator (* (/ x) r)))
(abs (numerator r))))
:hints (("Goal" :use numerator-goes-down-by-integer-division-pass1))
:rule-classes ((:linear
:corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= 0 r))
(<= (numerator (* (/ x) r))
(numerator r))))
(:linear
:corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= 0 r))
(<= (numerator (* r (/ x)))
(numerator r))))
(:linear
:corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= r 0))
(<= (numerator r)
(numerator (* (/ x) r)))))
(:linear
:corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= r 0))
(<= (numerator r)
(numerator (* r (/ x))))))))
;;; From rtl/rel5/denominator.lisp
(defthm denominator-bound
(implies (and (integerp x)
(integerp y)
(< 0 y))
(<= (denominator (* x (/ y)))
y))
:rule-classes ((:linear
:corollary
(implies (and (integerp x)
(integerp y)
(< 0 y))
(<= (denominator (* x (/ y)))
y)))
(:linear
:corollary
(implies (and (integerp x)
(integerp y)
(< 0 y))
(<= (denominator (* (/ y) x))
y)))))
(defthm |(denominator (+ x r))|
(and
(implies (and (rationalp r)
(integerp x))
(equal (denominator (+ x r))
(denominator r)))
(implies (and (rationalp r)
(integerp x))
(equal (denominator (+ r x))
(denominator r)))))
|
