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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.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; integerp-helper.lisp
;;;
;;;
;;; This book contains some messy proofs which I want to hide.
;;; There is probably nothing to be gained by looking at it.
;;;
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
(local
(include-book "../../support/top"))
(include-book "building-blocks")
(include-book "default-hint")
(set-default-hints '((nonlinearp-default-hint stable-under-simplificationp
hist pspv)))
(table acl2-defaults-table :state-ok t)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(encapsulate
()
(local
(defun INDUCT-NAT (x)
(if (and (integerp x)
(> x 0))
(induct-nat (1- x))
())))
(local
(defthm x-or-x/2-4
(implies (and (integerp x) (>= x 0))
(or (integerp (/ x 2)) (integerp (/ (1+ x) 2))))
:rule-classes ()
:hints (("Goal" :induct (induct-nat x))
("Subgoal *1/1''" :use (:instance (:theorem
(implies (integerp a)
(integerp (+ 1 a))))
(a (+ -1/2 (* 1/2 X))))))))
(local
(defthm x-or-x/2-5
(IMPLIES (integerp x)
(integerp (- x)))
:rule-classes ()))
(local
(defthm foo
(implies (and (integerp x)
(integerp y))
(integerp (+ x y)))
:rule-classes ()))
(local
(defthm bar
(equal (+ X (- (* 1/2 X)))
(* 1/2 x))))
(local
(defthm x-or-x/2-11
(implies (and (integerp x) (<= x 0))
(or (integerp (/ x 2)) (integerp (/ (1+ x) 2))))
:rule-classes ()
:hints (("Goal" :in-theory (disable FUNCTIONAL-SELF-INVERSION-OF-MINUS)
:use ((:instance x-or-x/2-4 (x (- x)))
(:instance foo (x (+ 1/2 (- (* 1/2 X)))) (y x)))))))
(local
(defthm X-OR-X/2
(implies (integerp x)
(or (integerp (/ x 2)) (integerp (/ (1+ x) 2))))
:rule-classes ()
:hints (("Goal" :in-theory (disable FUNCTIONAL-SELF-INVERSION-OF-MINUS)
:use ((:instance x-or-x/2-4)
(:instance x-or-x/2-11))))))
;;; Expressions like (integerp (+ 1/2 x)) show up when one is reasoning
;;; about odd and even.
;;; Note1: We do not have to worry about expressions such as
;;; (integerp (+ -1/2 x)) or (integerp (+ 3/2 x)) because of
;;; integerp-+-reduce-leading-constant.
;;; Note 2: We could write a similar rule --- probably a meta rule
;;; --- for expressions such as (integerp (+ 1/3 x)) and
;;; (integerp (+ 3/10 x)). For (integerp (+ c/d x)), (* n x) is not
;;; an integer for all 0 < n < d. But this would probably be a messy
;;; proof to do --- it would depend on c/d being in lowest terms ---
;;; but I have not thought about it yet.
(defthm even-and-odd-alternate
(implies (acl2-numberp x)
(equal (integerp (+ 1/2 x))
(and (integerp (* 2 x))
(not (integerp x)))))
:hints (("Subgoal 3'"
:use ((:instance
(:theorem (implies (and (integerp x)
(integerp y))
(integerp (* x y))))
(x (+ 1/2 x))
(y 2))))
("Subgoal 2'"
:use ((:instance X-OR-X/2
(x (* 2 x)))))))
(local
(defun ind (x)
(cond ((not (integerp x))
t)
((< x -1)
(ind (+ 2 x)))
((< 1 x)
(ind (+ -2 x)))
((equal x -1)
t)
((equal x 0)
t)
((equal x 1)
t)
(t
t))))
(local
(defthm reduce-integerp
(implies (and (integerp x)
(acl2-numberp y))
(iff (integerp (+ x y))
(integerp y)))))
(defthm sum-is-even-helper
(implies (and (integerp x)
(integerp y))
(equal (integerp (+ (* 1/2 x) (* 1/2 y)))
(if (integerp (* 1/2 x))
(integerp (* 1/2 y))
(not (integerp (* 1/2 y))))))
:hints (("Goal" :induct (ind x))
("Subgoal *1/3.1" :use (:instance X-OR-X/2
(x y))
:in-theory (disable even-and-odd-alternate))))
)
|
