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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.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; building-blocks-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"))
(local
(defun rationalp-guard-fn (args)
(if (endp (cdr args))
`((rationalp ,(car args)))
(cons `(rationalp ,(car args))
(rationalp-guard-fn (cdr args))))))
(local
(defmacro rationalp-guard (&rest args)
(if (endp (cdr args))
`(rationalp ,(car args))
(cons 'and
(rationalp-guard-fn args)))))
(local
(defun real/rationalp-guard-fn (args)
(if (endp (cdr args))
`((real/rationalp ,(car args)))
(cons `(real/rationalp ,(car args))
(real/rationalp-guard-fn (cdr args))))))
(local
(defmacro real/rationalp-guard (&rest args)
(if (endp (cdr args))
`(real/rationalp ,(car args))
(cons 'and
(real/rationalp-guard-fn args)))))
(local
(defthm niq-bounds
(implies (and (integerp i)
(<= 0 i)
(integerp j)
(< 0 j))
(and (<= (nonnegative-integer-quotient i j)
(/ i j))
(< (+ (/ i j) -1)
(nonnegative-integer-quotient i j))))
:hints (("Subgoal *1/1''" :use (:instance NORMALIZE-<-/-TO-*-3-3
(X 1)
(Z J)
(Y I))))
:rule-classes ((:linear
:trigger-terms ((nonnegative-integer-quotient i j))))))
(local
(defthm floor-bounds-1
(implies (real/rationalp-guard x y)
(and (< (+ (/ x y) -1)
(floor x y))
(<= (floor x y)
(/ x y))))
:rule-classes ((:generalize)
(:linear :trigger-terms ((floor x y))))))
(local
(defthm floor-bounds-2
(implies (and (real/rationalp-guard x y)
(integerp (/ x y)))
(equal (floor x y)
(/ x y)))
:rule-classes ((:generalize)
(:linear :trigger-terms ((floor x y))))))
(local
(defthm floor-bounds-3
(implies (and (real/rationalp-guard x y)
(not (integerp (/ x y))))
(< (floor x y)
(/ x y)))
:rule-classes ((:generalize)
(:linear :trigger-terms ((floor x y))))))
(local
(in-theory (disable floor)))
(defthm one
(IMPLIES (AND (INTEGERP x)
(integerp y)
(<= 0 y))
(<= 0
(LOGAND x y)))
:rule-classes :linear)
(defthm two
(IMPLIES (AND (INTEGERP x)
(integerp y)
(<= 0 y))
(<= (LOGAND x y)
y))
:rule-classes :linear)
(defthm rewrite-floor-x*y-z-left
(implies (and (real/rationalp x)
(real/rationalp y)
(real/rationalp z)
(not (equal z 0)))
(equal (floor (* x y) z)
(floor y (/ z x)))))
(defun power-of-2-measure (x)
(declare (xargs :guard (and (real/rationalp x) (not (equal x 0)))))
(cond ((or (not (real/rationalp x))
(<= x 0)) 0)
((< x 1) (cons (cons 1 1) (floor (/ x) 1)))
(t (floor x 1))))
(defun power-of-2-helper (x)
(declare (xargs :guard t
:measure (power-of-2-measure x)
:hints (("Subgoal 2.2'" :use (:instance
(:theorem
(IMPLIES (AND (REAL/RATIONALP X)
(< 2 X))
(< (FLOOR X 2) (FLOOR X 1))))
(x (/ x)))
:in-theory (enable NORMALIZE-<-/-TO-*-1)))))
(cond ((or (not (real/rationalp x))
(<= x 0))
0)
((< x 1) (+ -1 (power-of-2-helper (* 2 x))))
((<= 2 x) (+ 1 (power-of-2-helper (* 1/2 x))))
((equal x 1) 0)
(t 0) ;got a number in the doubly-open interval (1,2)
))
|
