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|
; Lemmas about explode-nonnegative-integer
; Copyright (C) 2005-2006 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>
;
; explode-nonnegative-integer.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "decimal")
(local (include-book "ihs/quotient-remainder-lemmas" :dir :system))
(local (include-book "std/lists/revappend" :dir :system))
(local (include-book "std/lists/append" :dir :system))
; 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 (in-theory (disable floor mod)))
(local (defun simpler-explode-nonnegative-integer (n base ans)
(declare (xargs :guard (and (integerp n)
(<= 0 n)
(print-base-p base))))
(if (or (zp n)
(not (print-base-p base)))
ans
(simpler-explode-nonnegative-integer
(floor n base)
base
(cons (digit-to-char (mod n base)) 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 base ans)
(or (simpler-explode-nonnegative-integer n base 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 base)
(declare (xargs :guard (and (natp n)
(print-base-p base))))
(if (or (zp n)
(not (print-base-p base)))
nil
(cons (digit-to-char (mod n base))
(basic-eni-core (floor n base) base)))))
(local (defun basic-eni-induction (n m base)
(declare (xargs :guard (and (natp n)
(natp m)
(print-base-p base))))
(cond ((zp n)
nil)
((zp m)
nil)
((not (print-base-p base))
nil)
(t
(basic-eni-induction (floor n base) (floor m base) base)))))
(local (defthm basic-eni-core-under-iff
(iff (basic-eni-core n base)
(and (not (zp n))
(print-base-p base)))))
(local (defthm consp-of-basic-eni-core
(equal (consp (basic-eni-core n base))
(and (not (zp n))
(if (print-base-p base)
t
nil)))
:hints(("Goal" :expand (basic-eni-core n base)))))
(local (defthm equal-of-basic-eni-cores
(implies (force (print-base-p base))
(equal (equal (basic-eni-core n base)
(basic-eni-core m base))
(equal (nfix n)
(nfix m))))
:hints(("Goal"
:in-theory (disable basic-eni-core)
:induct (basic-eni-induction n m base)
:expand ((:free (base) (basic-eni-core n base))
(:free (base) (basic-eni-core m base)))
:do-not '(generalize fertilize)))))
(local (defthm equal-of-basic-eni-core-with-list-zero
(not (equal (basic-eni-core n base) '(#\0)))
:hints(("Goal" :in-theory (enable digit-to-char)))))
(local (defthm basic/simpler-equivalence
(equal (simpler-explode-nonnegative-integer n base acc)
(revappend (basic-eni-core n base) acc))))
(local (defthm equal-of-simpler-explode-nonnegative-integers
(implies (force (print-base-p base))
(equal (equal (simpler-explode-nonnegative-integer n base acc)
(simpler-explode-nonnegative-integer m base acc))
(equal (nfix n) (nfix m))))))
(local (defthm simpler-eni-when-nonzero
(implies (and (not (zp n))
(print-base-p base))
(simpler-explode-nonnegative-integer n base acc))))
(local (defthm simpler-eni-degenerate-lemma
(equal (equal (simpler-explode-nonnegative-integer n base acc) '(#\0))
(and (equal acc '(#\0))
(or (zp n)
(not (print-base-p base)))))
:hints(("Goal"
:induct (simpler-explode-nonnegative-integer n base acc)
:expand ((:free (base) (basic-eni-core n base)))
:in-theory (e/d (digit-to-char)
(basic-eni-core))))))
(local (defthm not-of-simpler-explode-nonnegative-integer
(equal (not (simpler-explode-nonnegative-integer n base acc))
(and (equal acc nil)
(or (zp n)
(not (print-base-p base)))))))
(local (defthm true-listp-of-simpler-explode-nonnegative-integer
(equal (true-listp (simpler-explode-nonnegative-integer n base acc))
(true-listp acc))))
(local (defthm equal-of-explode-nonnegative-integers-lemma
(implies (and (natp n)
(natp m)
(not (equal n m))
(not (simpler-explode-nonnegative-integer n base acc))
(force (print-base-p base)))
(simpler-explode-nonnegative-integer m base acc))))
(defthm equal-of-explode-nonnegative-integers
(implies (and (natp n)
(natp m)
(force (print-base-p base)))
(equal (equal (explode-nonnegative-integer n base acc)
(explode-nonnegative-integer m base 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 base acc))
(true-listp acc)))
(defthm true-listp-of-explode-nonnegative-integer-type
(implies (true-listp acc)
(true-listp (explode-nonnegative-integer n base acc)))
:rule-classes :type-prescription)
(defthm character-listp-of-explode-nonnegative-integer
(equal (character-listp (explode-nonnegative-integer n base acc))
(character-listp acc)))
(local (defthm digit-listp-of-basic-eni-core
(str::digit-listp (basic-eni-core n 10))))
(local (defthm digit-listp-of-simpler-eni
(implies (str::digit-listp acc)
(str::digit-listp (simpler-explode-nonnegative-integer n 10 acc)))))
(defthm digit-listp-of-explode-nonnegative-integer
(implies (str::digit-listp acc)
(str::digit-listp (explode-nonnegative-integer n 10 acc))))
(encapsulate
()
(local (in-theory (disable revappend-removal)))
(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))))))
(local (defthm lemma2
(implies (and (not (zp n))
(print-base-p base))
(consp (basic-eni-core n base)))))
(defthm nonzeroness-of-explode-nonnegative-integer-when-nonzero
(implies (and (not (zp n))
(force (print-base-p base)))
(not (equal (explode-nonnegative-integer n base nil)
'(#\0))))
:hints(("Goal"
:in-theory (e/d (digit-to-char)
(basic-eni-core))
:expand ((:free (base) (basic-eni-core n base)))))))
(defthm digit-val-of-digit-to-char
(implies (and (force (natp n))
(force (<= 0 n))
(force (<= n 9)))
(equal (str::digit-val (digit-to-char n))
n))
:hints(("Goal" :in-theory (enable str::digit-val
digit-to-char))))
(defsection digit-to-char-of-digit-val
(local (defun test (n)
(declare (xargs :ruler-extenders :all))
(and (let ((char (code-char n)))
(or (not (str::digitp char))
(equal (digit-to-char (str::digit-val char))
char)))
(if (zp n)
t
(test (- n 1))))))
(local (defthm l0
(implies (and (test n)
(natp i)
(natp n)
(<= i n))
(let ((char (code-char i)))
(implies (str::digitp char)
(equal (digit-to-char (str::digit-val char))
char))))))
(local (defthm l1
(implies (and (natp i)
(<= i 255))
(let ((char (code-char i)))
(implies (str::digitp char)
(equal (digit-to-char (str::digit-val char))
char))))
:hints(("Goal" :use ((:instance l0 (n 255)))))))
(defthm digit-to-char-of-digit-val
(implies (str::digitp char)
(equal (digit-to-char (str::digit-val char))
char))
:hints(("Goal" :use ((:instance l1 (i (char-code char))))))))
(defund basic-unexplode-core (x)
(declare (xargs :guard (and (character-listp x)
(str::digit-listp x))))
(if (consp x)
(+ (str::digit-val (car x))
(* 10 (basic-unexplode-core (cdr x))))
0))
(local (defthm basic-unexplode-core-of-basic-eni-core
(implies (force (natp n))
(equal (basic-unexplode-core (basic-eni-core n 10))
n))
:hints(("Goal" :in-theory (enable basic-eni-core
basic-unexplode-core)))))
(defund unexplode-nonnegative-integer (x)
(declare (xargs :guard (and (character-listp x)
(str::digit-listp x))))
(basic-unexplode-core (revappend x nil)))
(encapsulate
()
(local (include-book "std/lists/rev" :dir :system))
(defthm unexplode-nonnegative-integer-of-explode-nonnegative-integer
(implies (force (natp n))
(equal (unexplode-nonnegative-integer (explode-nonnegative-integer n 10 nil))
n))
:hints(("Goal" :in-theory (e/d (unexplode-nonnegative-integer)
(basic-eni-core))))))
|
