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(in-package "ACL2")
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defund bits (x i j)
(declare (xargs :guard (and (natp x)
(natp i)
(natp j))
:verify-guards nil))
(mbe :logic (if (or (not (integerp i))
(not (integerp j)))
0
(fl (/ (mod x (expt 2 (1+ i))) (expt 2 j))))
:exec (if (< i j)
0
(logand (ash x (- j)) (1- (ash 1 (1+ (- i j))))))))
(defund bitn (x n)
(declare (xargs :guard (and (natp x)
(natp n))
:verify-guards nil))
(mbe :logic (bits x n n)
:exec (if (evenp (ash x (- n))) 0 1)))
(local (include-book "../arithmetic/top"))
(local (include-book "bvecp"))
(include-book "cat-def")
(local (include-book "bits"))
(local (include-book "bitn"))
(local (include-book "cat"))
(defund mulcat (l n x)
; We introduce mbe not because we want particularly fast execution, but because
; the existing logic definition does not satisfy the guard of cat, which can't
; be changed because of the guard of bits.
(declare (xargs :guard (and (integerp l)
(< 0 l)
(acl2-numberp n)
(natp x))
:verify-guards nil))
(mbe :logic (if (and (integerp n) (> n 0))
(cat (mulcat l (1- n) x)
(* l (1- n))
x
l)
0)
:exec (cond
((eql n 1)
(bits x (1- l) 0))
((and (integerp n) (> n 0))
(cat (mulcat l (1- n) x)
(* l (1- n))
x
l))
(t 0))))
;(defthm rationalp-mulcat
; (and (rationalp (mulcat l n x))
; (<= 0 (mulcat l n x)))
; :rule-classes :type-prescription)
;(verify-guards mulcat)
(local (in-theory (disable a15)))
(defthm mulcat-nonnegative-integer-type
(and (integerp (mulcat l n x))
(<= 0 (mulcat l n x)))
:hints (("Goal" :in-theory (enable mulcat)))
:rule-classes (:type-prescription))
;this rule is no better than mulcat-nonnegative-integer-type and might be worse:
(in-theory (disable (:type-prescription mulcat)))
(defthm mulcat-1
(implies (natp l)
(equal (mulcat l 1 x)
(bits x (1- l) 0)))
:hints (("Goal" :in-theory (enable mulcat bits-tail)
:expand ((mulcat l 1 x)))))
(defthm mulcat-bvecp-simple
(implies (and (= p (* l n))
(case-split (natp l)))
(bvecp (mulcat l n x) p))
:hints (("Goal" :in-theory (enable mulcat)))
:rule-classes ())
(defthm mulcat-bvecp
(implies (and (>= p (* l n))
(case-split (integerp p))
(case-split (natp l)))
(bvecp (mulcat l n x) p))
:hints (("Goal" :in-theory (disable bvecp-longer)
:use ((:instance mulcat-bvecp-simple (p (* l n)))
(:instance bvecp-longer (x (mulcat l n x)) (k2 p) (k1 (* l n)))))))
(defthm mulcat-0
(equal (mulcat l n 0) 0)
:hints (("Goal" :in-theory (enable mulcat))))
(defthm mulcat-0-two
(equal (mulcat l 0 x) 0)
:hints (("Goal" :in-theory (enable mulcat))))
(defthm bvecp-mulcat-1
(implies (natp n)
(bvecp (mulcat 1 n 1) n))
:rule-classes ())
(local (defthm mulcat-n-1-1
(implies (and (natp n)
(> n 0))
(equal (mulcat 1 n 1)
(1+ (* 2 (mulcat 1 (1- n) 1)))))
:hints (("Goal" :in-theory (enable mulcat cat bits-tail)))))
(defthm mulcat-n-1
(implies (case-split (<= 0 n))
(equal (mulcat 1 n 1)
(1- (expt 2 n))))
:hints (("Goal" :in-theory (enable mulcat expt-split))))
(defun mulcat-induct (n n2)
(IF (AND (INTEGERP N) (> N 0) (INTEGERP N2) (> N2 0))
(mulcat-induct (1- n) (1- n2))
0))
(local (include-book "merge")) ;yuck
;BOZO prove a bits-mulcat? could be used to prove-bitn-mulcat
;BOZO generalize to bits of mulcat when x is larger than 1?
;not general (note the 1 for the l parameter)
;and to when (<= n m)
;add to lib/ ?
(defthm bitn-mulcat-1
(implies (and (< m n)
(case-split (bvecp x 1))
(case-split (natp m))
(case-split (integerp n)) ;(case-split (natp n))
)
(equal (bitn (mulcat 1 n x) m)
x))
:hints (("Goal" :induct (mulcat-induct n m)
:do-not '(generalize)
:expand ((mulcat 1 n x))
:in-theory (enable mulcat))))
(defthm bitn-mulcat-2
(implies (and (<= (* l n) n2)
(natp n)
(natp l)
(natp n2)
(case-split (bvecp x l))
)
(equal (bitn (mulcat l n x) n2)
0))
:hints (("Goal" :use ((:instance bvecp-bitn-0 (x (mulcat l n x)) (n n2))))))
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