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;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; numerator-and-denominator.lisp
;;;
;;;
;;; Some simple facts about numerator and denominator.
;;;
;;; This book should be expanded some day.
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
(local (include-book "../pass1/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-positive
(implies (rationalp x)
(equal (< 0 (numerator x))
(< 0 x)))
:rule-classes ((:rewrite)
(:type-prescription
:corollary
(implies (and (rationalp x)
(< 0 x))
(< 0 (numerator x))))
(:type-prescription
:corollary
(implies (and (rationalp x)
(<= 0 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))
(< (numerator x) 0)))
(:type-prescription
:corollary
(implies (and (rationalp x)
(<= x 0))
(<= (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
(equal (* r (denominator r))
(if (rationalp r)
(numerator r)
(fix r))))
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