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; See the top-level arithmetic-2 LICENSE file for authorship,
; copyright, and license information.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; integerp.lisp
;;;
;;;
;;; This book contains several lemmatta about when a sum or
;;; product is or is not an integer.
;;;
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
(local (include-book "../pass1/top"))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The first two cannot be type-prescriptions.
(defthm integerp-1a
(implies (and (integerp (+ x y))
(acl2-numberp x)
(integerp y))
(integerp x)))
(defthm integerp-1b
(implies (and (integerp (+ x y))
(acl2-numberp y)
(integerp x))
(integerp y)))
; The next eight rules are rather weak, and do not cover all the cases.
; Perhaps I should do something about it.
(defthm nintegerp-1a
(implies (and (real/rationalp x)
(real/rationalp y)
(< 0 x)
(< x y))
(not (integerp (* x (/ y)))))
:hints (("Goal" :use (:theorem
(implies (and (real/rationalp x)
(real/rationalp y)
(< 0 x)
(< x y))
(and (< 0 (* x (/ y)))
(< (* x (/ y)) 1))))
:in-theory (enable prefer-*-to-/)))
:rule-classes :type-prescription)
(defthm nintegerp-1b
(implies (and (real/rationalp x)
(real/rationalp y)
(< 0 x)
(< x y))
(not (integerp (* (/ y) x))))
:rule-classes :type-prescription)
(defthm nintegerp-2a
(implies (and (real/rationalp x)
(real/rationalp y)
(< x 0)
(< y x))
(not (integerp (* x (/ y)))))
:hints (("Goal" :use (:instance nintegerp-1a
(x (- x))
(y (- y)))))
:rule-classes :type-prescription)
(defthm nintegerp-2b
(implies (and (real/rationalp x)
(real/rationalp y)
(< x 0)
(< y x))
(not (integerp (* (/ y) x))))
:rule-classes :type-prescription)
(defthm nintegerp-3a
(implies (and (real/rationalp x)
(real/rationalp y)
(< 0 x)
(< y (- x)))
(not (integerp (* x (/ y)))))
:hints (("Goal" :use (:instance nintegerp-1a
(y (- y)))))
:rule-classes :type-prescription)
(defthm nintegerp-3b
(implies (and (real/rationalp x)
(real/rationalp y)
(< 0 x)
(< y (- x)))
(not (integerp (* (/ y) x))))
:rule-classes :type-prescription)
(defthm nintegerp-4a
(implies (and (real/rationalp x)
(real/rationalp y)
(< x 0)
(< (- x) y))
(not (integerp (* x (/ y)))))
:hints (("Goal" :use (:instance nintegerp-1a
(x (- x)))))
:rule-classes :type-prescription)
(defthm nintegerp-4b
(implies (and (real/rationalp x)
(real/rationalp y)
(< x 0)
(< (- x) y))
(not (integerp (* (/ y) x))))
:rule-classes :type-prescription)
(encapsulate
()
(local
(defthm nintegerp-expt-helper
(implies (and (< 1 x)
(real/rationalp x)
(< n 0)
(integerp n))
(and (< 0 (expt x n))
(< (expt x n) 1)))
:rule-classes nil))
(defthm nintegerp-expt
(implies (and (real/rationalp x)
(< 1 x)
(integerp n)
(< n 0))
(not (integerp (expt x n))))
:hints (("Goal" :use nintegerp-expt-helper))
:rule-classes :type-prescription)
(defthm nintegerp-/
(implies (and (real/rationalp x)
(< 1 x))
(not (integerp (/ x))))
:hints (("Goal" :use ((:instance nintegerp-expt
(n -1)))
:in-theory (disable nintegerp-expt)))
:rule-classes :type-prescription)
)
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