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;;;***************************************************************
;;;An ACL2 Library of Floating Point Arithmetic
;;;David M. Russinoff
;;;Advanced Micro Devices, Inc.
;;;February, 1998
;;;***************************************************************
(in-package "ACL2")
(local (include-book "arithmetic/top" :dir :system))
(local (defun kmin (x k)
(if (integerp k)
(if (> k 0)
(if (evenp (* (expt 2 k) x))
(kmin x (1- k))
k)
1)
1)))
(local (defthm kmin-pos
(implies (and (integerp k)
(> k 0))
(and (integerp (kmin x k))
(> (kmin x k) 0)))
:rule-classes ()))
(local (defthm kmin-integerp
(implies (and (rationalp x)
(not (integerp x))
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x)))
(integerp (* (expt 2 (kmin x k)) x)))
:rule-classes ()))
(local (defthm kmin-oddp
(implies (and (rationalp x)
(not (integerp x))
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x)))
(not (evenp (* (expt 2 (kmin x k)) x))))
:rule-classes ()))
(local (in-theory (disable kmin)))
(local (defun revenp (n)
(if (and (integerp n) (>= n 0))
(if (= n 0)
t
(if (= n 1)
()
(revenp (- n 2))))
())))
(local (defthm half-lemma
(implies (and (integerp n)
(>= n 0))
(iff (revenp n) (evenp n)))
:rule-classes ()))
(local (defthm int-*-closed
(implies (and (integerp x) (integerp y))
(integerp (* x y)))))
(local (defthm evenp-evenp
(implies (and (integerp x)
(>= x 0)
(integerp y)
(>= y 0)
(evenp x))
(evenp (* x y)))
:hints (("Goal" :use ((:instance int-*-closed (x (/ x 2))))))))
(local (defthm evenp-oddp
(implies (and (integerp n)
(>= n 0))
(iff (revenp n)
(not (revenp (1+ n)))))
:rule-classes ()))
(local (defthm evenp-oddp-2
(implies (and (integerp n)
(>= n 0))
(iff (evenp n)
(not (evenp (1+ n)))))
:rule-classes ()
:hints (("Goal" :use ((:instance evenp-oddp)
(:instance half-lemma)
(:instance half-lemma (n (1+ n))))))))
(local (defthm evenp-oddp-3
(implies (and (integerp n)
(>= n 0)
(not (evenp n)))
(and (>= (1- n) 0)
(evenp (1- n))))
:rule-classes ()
:hints (("Goal" :use ((:instance evenp-oddp-2 (n (1- n))))))))
(local (defthm evenp-plus
(implies (and (integerp n)
(integerp m)
(>= n 0)
(>= m 0)
(evenp n)
(evenp m))
(evenp (+ n m)))
:rule-classes ()))
(local (defthm evenp-x2+2x
(implies (and (integerp x)
(>= x 0)
(evenp x))
(evenp (+ (* x x) (* 2 x))))
:rule-classes ()
:hints (("Goal" :use ((:instance evenp-plus (n (* x x)) (m (* 2 x)))
(:instance evenp-evenp (y x)))))))
(local (defthm hack1
(implies (integerp x)
(= (+ 1 (* 2 X) (* X X))
(+ 1 x x (* X X))))
:rule-classes ()))
(local (defthm oddp-oddp
(implies (and (integerp x)
(>= x 0)
(evenp x))
(not (evenp (* (1+ x) (1+ x)))))
:hints (("Goal" :in-theory (disable evenp)
:use ((:instance evenp-x2+2x)
(:instance hack1)
(:instance evenp-oddp-2 (n (+ (* x x) (* 2 x)))))))))
(local (defthm oddp-oddp-2
(implies (and (integerp x)
(>= x 0)
(not (evenp x)))
(not (evenp (* x x))))
:hints (("Goal" :in-theory (disable evenp)
:use ((:instance oddp-oddp (x (1- x)))
(:instance evenp-oddp-3 (n x)))))))
(local (defthm kmin-oddp-square
(implies (and (rationalp x)
(not (integerp x))
(>= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x)))
(not (evenp (* (* (expt 2 (kmin x k)) x) (* (expt 2 (kmin x k)) x)))))
:rule-classes ()
:hints (("Goal" :in-theory (disable evenp)
:use ((:instance oddp-oddp-2 (x (* (expt 2 (kmin x k)) x)))
(:instance kmin-oddp)
(:instance kmin-pos)
(:instance kmin-integerp))))))
(local (defthm hack2
(implies (and (rationalp x)
(integerp m))
(= (* (* (expt 2 m) x) (* (expt 2 m) x))
(* (* 2 x x) (expt 2 (1- (* 2 m))))))
:rule-classes ()))
(local (defthm kmin-oddp-corollary
(implies (and (rationalp x)
(not (integerp x))
(>= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x)))
(not (evenp (* (* 2 x x) (expt 2 (1- (* 2 (kmin x k))))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance kmin-oddp-square)
(:instance hack2 (m (kmin x k)))
(:instance kmin-pos))))))
(local (defthm evenp-expt
(implies (and (integerp n)
(> n 0))
(evenp (expt 2 n)))))
(local (defthm evenp-expt-2
(implies (and (integerp n)
(> n 0))
(and (integerp (expt 2 n))
(> (expt 2 n) 0)
(evenp (expt 2 n))))
:rule-classes ()))
(local (defthm evenp-expt-3
(implies (and (integerp n)
(> n 0))
(and (integerp (expt 2 (1- (* 2 n))))
(> (expt 2 (1- (* 2 n))) 0)
(evenp (expt 2 (1- (* 2 n))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance evenp-expt-2 (n (1- (* 2 n)))))))))
(local (defthm evenp-2-kmin
(implies (and (rationalp x)
(not (integerp x))
(>= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x)))
(and (integerp (expt 2 (1- (* 2 (kmin x k)))))
(> (expt 2 (1- (* 2 (kmin x k)))) 0)
(evenp (expt 2 (1- (* 2 (kmin x k)))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance evenp-expt-3 (n (kmin x k)))
(:instance kmin-pos)
(:instance kmin-integerp))))))
(local (defthm 2xx-lemma-1
(implies (and (rationalp x)
(not (integerp x))
(>= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x))
(integerp (* 2 x x)))
(evenp (* (* 2 x x) (expt 2 (1- (* 2 (kmin x k)))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance evenp-2-kmin)
(:instance evenp-evenp (x (expt 2 (1- (* 2 (kmin x k))))) (y (* 2 x x))))))))
(local (defthm x-2xx-1
(implies (and (rationalp x)
(>= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x))
(integerp (* 2 x x)))
(integerp x))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance 2xx-lemma-1)
(:instance kmin-oddp-corollary))))))
(local (defthm x-2xx-2
(implies (and (rationalp x)
(<= x 0)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x))
(integerp (* 2 x x)))
(integerp x))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt evenp)
:use ((:instance x-2xx-1 (x (- x))))))))
(local (defthm x-2xx-3
(implies (and (rationalp x)
(integerp k)
(> k 0)
(integerp (* (expt 2 k) x))
(integerp (* 2 x x)))
(integerp x))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt)
:use ((:instance x-2xx-2)
(:instance x-2xx-1))))))
; Added for Version_2.6.
(local (in-theory (enable exponents-add-unrestricted)))
(local (defthm x-2xx-4
(implies (and (rationalp x)
(integerp k)
(<= k 0)
(integerp (* (expt 2 k) x)))
(integerp x))
:rule-classes ()))
(defthm x-2xx
(implies (and (rationalp x)
(integerp k)
(integerp (* (expt 2 k) x))
(integerp (* 2 x x)))
(integerp x))
:rule-classes ()
:hints (("Goal" :in-theory (disable expt)
:use ((:instance x-2xx-3)
(:instance x-2xx-4)))))
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