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(in-package "ACL2")
(local (include-book "integerp"))
(local (include-book "predicate"))
(local (include-book "fp2"))
;a funny little rule:
;can be expensive!
;perhaps should export disabled
;make a forward-chaining rule?
(defthmd even-int-implies-int
(implies (and (integerp (* 1/2 x))
(rationalp x) ;gen?
)
(integerp x))
:rule-classes ((:rewrite :backchain-limit-lst (1 nil)))
:hints (("Goal" :in-theory (disable integerp-prod)
:use (:instance integerp-prod (x (* 1/2 x)) (y 2)))))
;this book is currently a mess.
;normal forms: leave evenp and oddp enabled
;from basic
(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)))))
;is this sort of thing elsewhere? integerp.lisp?
(defthm integerp-+
(implies (and (integerp x)
(integerp y))
(integerp (+ x y))))
(defthm x-or-x/2
(implies (integerp x)
(or (integerp (/ x 2)) (integerp (/ (1+ x) 2))))
:rule-classes ()
:hints (("Goal" :in-theory (disable integerp-+)
:use ((:instance integerp-+ (x (+ 1/2 (* -1/2 X))) (y x))
(:instance x-or-x/2-4)
(:instance x-or-x/2-4 (x (- x)))))))
;end stuff from basic
(encapsulate
()
(local (defthm hack-int
(implies (and (integerp x)
(integerp y))
(integerp (+ x y)))))
(defthm integerp-sum-of-odds-over-2
(implies (and (rationalp x)
(rationalp y)
(integerp (* 2 x)) ;these two hyps say x is of the form (odd)/2
(not (integerp x)) ;
)
(equal (integerp (+ x y))
(and (integerp (* 2 y))
(not (integerp y)) ; (oddp (* 2 y)) ;rephrase the oddp hyps?
)))
:hints (("Goal" :in-theory (disable even-int-implies-int)
:use ( (:instance even-int-implies-int (x (+ (* 2 x) (* 2 y))))
(:instance hack-int (x (+ 1/2 X)) (y (+ 1/2 y)))
(:instance x-or-x/2 (x (* 2 x)))
(:instance x-or-x/2 (x (* 2 y)))))))
)
;derive the results below from the above (or eliminate them)
(in-theory (disable integerp-sum-of-odds-over-2))
;make this a rewrite?
;special case
(defthm integerp-sum-of-odds-over-2-leading-constant
(implies (and (syntaxp (and (quotep x)
(integerp (* 2 x)) ;;these two hyps say x is of the form (odd)/2
(not (integerp x)) ;;
))
(rationalp x)
(rationalp y)
(integerp (* 2 x)) ;;these two hyps say x is of the form (odd)/2
(not (integerp x)) ;;
(integerp (* 2 y))
(not (integerp y)) ; (oddp (* 2 y)) ;rephrase the oddp hyps?
)
(integerp (+ x y)))
:hints (("Goal" :use integerp-sum-of-odds-over-2)))
;do we need all this stuff?
;(defthm even-or-odd
; (implies (integerp x)
; (or (evenp x) (oddp x)))
;:rule-classes nil)
;should be a rewrite? - general rewrites for evenp/oddp of sum/diff?
;(defthm odd-+-1
; (implies (and (integerp x)
; (oddp x))
; (evenp (+ 1 x)))
;:hints (("Goal" :use (:instance x-or-x/2 ))))
;(defthm odd/2-plus-1/2
; (implies (and (integerp x)
; (oddp x))
; (integerp (+ 1/2 (/ x 2))))
;:hints (("Goal" :use (:instance odd-+-1))))
;hack, don't leave enabled ;rewrite?
;(defthm integerp-next
; (implies (and (rationalp x)
; (integerp (+ x 1)))
; (integerp x)))
;(defthm odd/2-minus-1/2
; (implies (and (integerp x)
; (oddp x))
; (integerp (+ -1/2 (/ x 2))))
;:hints (("Goal" :use (:instance odd/2-plus-1/2))))
;(in-theory (disable integerp-next))
;(defthm odd/2-minus-1/2-alt
; (implies (and (integerp x)
; (oddp x))
; (integerp (+ -1/2 (* 1/2 x))))
; :hints (("Goal" :in-theory (disable odd/2-minus-1/2)
; :use (:instance odd/2-minus-1/2))))
;floor of odd/2 is odd/2 -1/2
;; :hints (("Goal" :use ( (:instance fl-unique
; (x (* 1/2 x))
; (n (- (* 1/2 x) 1/2)))))))
;needed?
(defthm even-and-odd-alternate-eric
(implies (and (rationalp x)
(integerp (* 2 x)))
(equal (integerp (+ 1/2 x))
(not (integerp x)))))
;should this go in type.lisp?
;needed? ;lets change any leading constant of -1/2 to 1/2 and elim this rule
(defthm even-and-odd-alternate-3
(implies (and (integerp x))
(equal (integerp (+ -1/2 (* -1/2 x)))
(not (integerp (* 1/2 x)))))
:hints (("Goal" :in-theory (disable integerp-minus)
:use (:instance integerp-minus (x (+ -1/2 (* -1/2 x)))))))
#| never finished this
(defthm integerp-+-odd-over-2-reduce
(implies (and (rationalp x)
(integerp (* 2 x)) ;these two hyps say x is of the form (odd)/2
(not (integerp x))
(rationalp y))
(implies (integerp (+ x y))
(and (integerp (* 2 y)) ;these two hyps say x is of the form (odd)/2
(not (integerp y))))) ;
:otf-flg t
:hints (("Goal" :use integerp-sum-of-odds-over-2)))
|#
(defthm even-and-odd-alternate-eric-2-bk
(implies (rationalp x)
(implies (and (integerp (* 2 x))
(not (integerp x)))
(integerp (+ 1/2 x)))))
;if s is even, then s-1 is odd
(defthm even-odd-5
(implies (and (rationalp x)
(integerp (* 1/2 x)))
(and (integerp (- x 1))
(not (integerp (* 1/2 (- x 1))))))
:hints (("Goal" :in-theory (enable even-int-implies-int)))
)
(defthm even-and-odd-alternate-eric-2-fw
(implies (rationalp x)
(implies (integerp (+ 1/2 x))
(and (integerp (* 2 x))
(not (integerp x)))))
:hints (("Goal"
:in-theory (disable even-odd-5)
:use (:instance even-odd-5 (x (+ 1 (* 2 x)))))))
;replace the 1/2 rules above and similarly generalize the rules at the top to be equal rules
(defthm even-and-odd-alternate-eric-2
(implies (rationalp x)
(equal (integerp (+ 1/2 x))
(and (integerp (* 2 x))
(not (integerp x))))))
(in-theory (disable even-and-odd-alternate-eric-2-fw even-and-odd-alternate-eric-2-bk))
|
