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;; Processing Unicode Files with ACL2
;; Copyright (C) 2005-2006 by Jared Davis <jared@cs.utexas.edu>
;;
;; This program is free software; you can redistribute it and/or modify it
;; under the terms of the GNU General Public License as published by the Free
;; Software Foundation; either version 2 of the License, or (at your option)
;; any later version.
;;
;; This program is distributed in the hope that it will be useful but WITHOUT
;; ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
;; FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
;; more details.
;;
;; You should have received a copy of the GNU General Public License along with
;; this program; if not, write to the Free Software Foundation, Inc., 59 Temple
;; Place - Suite 330, Boston, MA 02111-1307, USA.
(in-package "ACL2")
(local (include-book "arithmetic-3/floor-mod/floor-mod" :dir :system))
(local (include-book "revappend"))
(include-book "base10-digit-charp")
;; Explode-nonnegative-integer is a particularly nasty function to try to
;; reason about because it is tail recursive and has a very nasty base case.
;; Instead of reasoning about it directly, we will split it up into the
;; following, simpler definition.
(local (defun simpler-explode-nonnegative-integer (n ans)
(declare (xargs :guard (and (integerp n)
(<= 0 n))))
(if (zp n)
ans
(simpler-explode-nonnegative-integer (floor n 10)
(cons (digit-to-char
(mod n 10))
ans)))))
;; We can now redefine explode-nonnegative-integer to be a simple nonrecursive
;; function that uses our simpler-explode-nonnegative-integer as its core, but
;; wraps it in a simple "or" statement. We will then disable the previous
;; definition of explode-nonnegative-integer, so that only our new definition
;; will be used.
(local (defthm explode-nonnegative-integer-redefinition
(equal (explode-nonnegative-integer n 10 ans)
(or (simpler-explode-nonnegative-integer n ans)
'(#\0)))
:rule-classes ((:definition :install-body nil))))
(local (in-theory (disable (:definition explode-nonnegative-integer))))
;; Sadly, even simpler-explode-nonnegative-integer is hard to reason about as
;; it is tail recursive. So, we will introduce a non tail-recursive function
;; in its place that does almost the same thing. We'll call this the "basic
;; explode-nonnegative-integer core", or the basic-eni-core for short.
(local (defun basic-eni-core (n)
(declare (xargs :guard (natp n)))
(if (zp n)
nil
(cons (digit-to-char (mod n 10))
(basic-eni-core (floor n 10))))))
(local (defun basic-eni-induction (n m)
(declare (xargs :guard (and (natp n)
(natp m))))
(if (zp n)
nil
(if (zp m)
nil
(basic-eni-induction (floor n 10) (floor m 10))))))
(local (defthm equal-of-basic-eni-cores
(equal (equal (basic-eni-core n)
(basic-eni-core m))
(equal (nfix n) (nfix m)))
:hints(("Goal" :induct (basic-eni-induction n m)))))
(local (defthm basic-eni-core-under-iff
(iff (basic-eni-core n)
(not (zp n)))))
(local (defthm equal-of-basic-eni-core-with-list-zero
(not (equal (basic-eni-core n) '(#\0)))
:hints(("Goal" :in-theory (enable digit-to-char)))))
(local (defthm basic/simpler-equivalence
(equal (simpler-explode-nonnegative-integer n acc)
(revappend (basic-eni-core n) acc))))
(local (defthm equal-of-simpler-explode-nonnegative-integers
(equal (equal (simpler-explode-nonnegative-integer n acc)
(simpler-explode-nonnegative-integer m acc))
(equal (nfix n) (nfix m)))))
(local (defthm simpler-eni-when-nonzero
(implies (not (zp n))
(simpler-explode-nonnegative-integer n acc))))
(local (defthm simpler-eni-degenerate-lemma
(equal (equal (simpler-explode-nonnegative-integer n acc) '(#\0))
(and (equal acc '(#\0))
(zp n)))
:hints(("Goal" :in-theory (enable digit-to-char)))))
(local (defthm not-of-simpler-explode-nonnegative-integer
(equal (not (simpler-explode-nonnegative-integer n acc))
(and (equal acc nil)
(zp n)))))
(local (defthm true-listp-of-simpler-explode-nonnegative-integer
(equal (true-listp (simpler-explode-nonnegative-integer n acc))
(true-listp acc))))
(local (defthm equal-of-explode-nonnegative-integers-lemma
(implies (and (integerp n) (<= 0 n)
(integerp m) (<= 0 m)
(not (equal n m))
(not (simpler-explode-nonnegative-integer n acc)))
(simpler-explode-nonnegative-integer m acc))))
(defthm equal-of-explode-nonnegative-integers
(implies (and (natp n)
(natp m))
(equal (equal (explode-nonnegative-integer n 10 acc)
(explode-nonnegative-integer m 10 acc))
(equal n m)))
:hints(("Goal" :in-theory (disable simpler-explode-nonnegative-integer
basic/simpler-equivalence))))
(defthm true-listp-of-explode-nonnegative-integer
(equal (true-listp (explode-nonnegative-integer n 10 acc))
(true-listp acc)))
(defthm true-listp-of-explode-nonnegative-integer-type
(implies (true-listp acc)
(true-listp (explode-nonnegative-integer n 10 acc)))
:rule-classes :type-prescription)
(local (defthm base10-digit-char-listp-of-basic-eni-core
(base10-digit-char-listp (basic-eni-core n))
:hints(("Goal" :in-theory (enable base10-digit-char-listp)))))
(local (defthm base10-digit-char-listp-of-simpler-eni
(implies (base10-digit-char-listp acc)
(base10-digit-char-listp
(simpler-explode-nonnegative-integer n acc)))))
(defthm base10-digit-char-listp-of-explode-nonnegative-integer
(implies (base10-digit-char-listp acc)
(base10-digit-char-listp (explode-nonnegative-integer n 10 acc))))
(encapsulate
()
(local (defthm lemma
(equal (equal (revappend x acc) '(#\0))
(or (and (equal acc nil)
(consp x)
(equal (car x) #\0)
(atom (cdr x)))
(and (equal acc '(#\0))
(atom x))))))
(defthm nonzeroness-of-explode-nonnegative-integer-when-nonzero
(implies (not (zp n))
(not (equal (explode-nonnegative-integer n 10 nil)
'(#\0))))
:hints(("Goal" :in-theory (enable digit-to-char)))))
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