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;; Copyright (C) 2018, Regents of the University of Texas
;; Written by Cuong Chau
;; License: A 3-clause BSD license. See the LICENSE file distributed with
;; ACL2.
;; Cuong Chau <ckcuong@cs.utexas.edu>
;; March 2019
(in-package "ADE")
(include-book "../comparators/v-less")
(local (include-book "gcd-alg"))
(local (include-book "../adders/subtractor"))
(local (include-book "arithmetic-3/top" :dir :system))
;; ======================================================================
;; A greatest-common-divisor specification
(local
(defthm v-<-correct-instance
(implies (and (natp data-size)
(equal (len x) (* 2 data-size))
(bvp x)
(v-< nil t
(rev (take data-size x))
(rev (nthcdr data-size x))))
(< (v-to-nat (take data-size x))
(v-to-nat (nthcdr data-size x))))
:hints (("Goal"
:use (:instance v-<-correct-1
(a (take data-size x))
(b (nthcdr data-size x)))
:in-theory (disable v-<-correct-1)))
:rule-classes :linear))
(local
(defthm v-to-nat-of-v-zp
(equal (v-zp x)
(equal (v-to-nat x) 0))
:hints (("Goal" :in-theory (enable v-zp v-nzp v-to-nat)))))
(local
(defun my-count (x)
(nfix (+ (v-to-nat (take (/ (len x) 2) x))
(v-to-nat (nthcdr (/ (len x) 2) x))))))
(local
(defun gcd$op (x)
(declare (xargs :hints (("Goal"
:in-theory (e/d ()
(v-not-take
v-not-nthcdr))))
:measure (my-count x)))
(b* ((data-size (/ (len x) 2))
(a (take data-size x))
(b (nthcdr data-size x))
(a-b (v-adder-output t a (v-not b)))
(b-a (v-adder-output t b (v-not a)))
(a<b (v-< nil t (rev a) (rev b))))
(cond
((or (atom x)
(zp data-size)
(not (bvp x)))
x)
((v-zp a) b)
((v-zp b) a)
((equal a b) a)
(t (gcd$op
(v-if a<b
(append b-a a)
(append a-b b))))))))
(defun gcd$op (x)
(declare (xargs :measure (:? x)))
(b* ((data-size (/ (len x) 2))
(a (take data-size x))
(b (nthcdr data-size x))
(a-b (v-adder-output t a (v-not b)))
(b-a (v-adder-output t b (v-not a)))
(a<b (v-< nil t (rev a) (rev b))))
(cond
((or (atom x)
(zp data-size)
(not (bvp x)))
x)
((v-zp a) b)
((v-zp b) a)
((equal a b) a)
(t (gcd$op
(v-if a<b
(append b-a a)
(append a-b b)))))))
(defthm bvp-gcd$op
(implies (and (natp (/ (len x) 2))
(bvp x))
(bvp (gcd$op x))))
(defthm len-gcd$op
(implies (and (natp (/ (len x) 2))
(bvp x))
(equal (len (gcd$op x))
(/ (len x) 2))))
(local
(defthm gcd$op-lemma-aux
(implies (and (bv2p a b)
(not (v-< nil t (rev a) (rev b)))
(equal (v-to-nat a) 0))
(equal a b))
:hints (("Goal" :use (v-to-nat-equality
v-<-correct-2)))
:rule-classes nil))
(local
(defthm v-adder-output-lemma
(implies (atom a)
(not (v-adder-output c a b)))
:hints (("Goal" :in-theory (enable v-adder-output)))))
(defthm gcd$op-lemma
(b* ((a (take data-size x))
(b (nthcdr data-size x))
(a-b (v-adder-output t a (v-not b)))
(b-a (v-adder-output t b (v-not a)))
(a<b (v-< nil t (rev a) (rev b))))
(implies (and (natp data-size)
(equal data-size (/ (len x) 2))
(bvp x))
(equal (gcd$op (v-if a<b
(append b-a a)
(append a-b b)))
(gcd$op x))))
:hints (("Goal"
:induct (gcd$op x)
:in-theory (e/d ()
(v-to-nat-equality
v-not-take
v-not-nthcdr)))
("Subgoal *1/3"
:use (:instance
v-to-nat-equality
(a (v-adder-output t
(take data-size x)
(v-not (nthcdr data-size x))))
(b (take data-size x))))
("Subgoal *1/2"
:use ((:instance
v-to-nat-equality
(a (v-adder-output t
(nthcdr data-size x)
(v-not (take data-size x))))
(b (nthcdr data-size x)))
(:instance
gcd$op-lemma-aux
(a (take data-size x))
(b (nthcdr data-size x)))))))
;; Prove that gcd$op correctly computes the greatest common divisor
(local
(defthm v-to-nat-of-GCD$OP-is-GCD-ALG
(implies (and (equal data-size (/ (len x) 2))
(bvp x))
(equal (v-to-nat (gcd$op x))
(gcd-alg (v-to-nat (take data-size x))
(v-to-nat (nthcdr data-size x)))))
:hints (("Goal" :in-theory (e/d ()
(v-not-take
v-not-nthcdr))))))
(in-theory (disable gcd$op))
(defthmd gcd$op-commutative
(implies (bv2p a b)
(equal (gcd$op (append a b))
(gcd$op (append b a))))
:hints (("Goal"
:use (:instance v-to-nat-equality
(a (gcd$op (append a b)))
(b (gcd$op (append b a))))
:in-theory (e/d (gcd-alg-commutative)
(v-to-nat-equality))))
:rule-classes ((:rewrite :loop-stopper ((a b)))))
;; The operation of GCD over a data sequence
(defun gcd$op-map (x)
(if (atom x)
nil
(cons (gcd$op (car x))
(gcd$op-map (cdr x)))))
(defthm len-of-gcd$op-map
(equal (len (gcd$op-map x))
(len x)))
(defthm gcd$op-map-of-append
(equal (gcd$op-map (append x y))
(append (gcd$op-map x) (gcd$op-map y))))
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