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(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(local (include-book "../arithmetic/top"))
(defund shft (x s l)
(declare (xargs :guard (and (integerp s) (rationalp x))))
(mod (fl (* (expt 2 s) x)) (expt 2 (nfix l))))
(defthm shft-nonnegative-integer-type
(and (integerp (shft x s l))
(<= 0 (shft x s l)))
:rule-classes (:type-prescription))
;(:type-prescription shft) is no better than shft-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription shft)))
(defthm shft-natp
(natp (shft x s n)))
(defthm shft-bvecp-simple
(bvecp (shft x s n) n)
:hints (("Goal" :in-theory (enable bvecp shft))))
(local (include-book "bvecp"))
(defthm shft-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (shft x s n) k))
:hints (("Goal" :in-theory (disable shft-bvecp-simple)
:use shft-bvecp-simple)))
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