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; An ACL2 Tarai Function book.
; Copyright (C) 2000 John R. Cowles, University of Wyoming
; This book 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 book 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 book; if not, write to the Free Software
; Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
; Written by:
; John Cowles
; Department of Computer Science
; University of Wyoming
; Laramie, WY 82071-3682 U.S.A.
;; 25 July 00
;; These are implementations of Knuth's (original and revised)
;; and Bailey's versions of the f function for Knuth's theorem 4.
;; For lists of integers, the revised Knuth and Bailey versions
;; compute the same values.
;; (certify-book "C:/acl2/tak/tarai11")
(in-package "ACL2")
(defmacro
last-el (lst)
"Return the last element in the list lst."
(list 'car (list 'last lst)))
(defun
Gb (lst)
"Bailey's g function for Knuth's Theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(cond ((consp (nthcdr 3 lst)) ;; (len lst) > 3
(if (or (equal (first lst)
(+ (second lst) 1))
(> (second lst)
(+ (third lst) 1)))
(Gb (rest lst))
(max (third lst)
(last-el lst))))
(t (last-el lst)))) ;; (len lst) <= 3
(defun
Gko (lst)
"Knuth's original g function for Knuth's Theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(cond ((consp (nthcdr 2 lst)) ;; (len lst) > 2
(cond ((equal (first lst)
(+ (second lst) 1))
(Gko (rest lst)))
((equal (second lst)
(+ (third lst) 1))
(max (third lst)
(last-el lst)))
(t (last-el lst))))
(t (last-el lst)))) ;; (len lst) <= 2
(defun
Gkr (lst)
"Knuth's revised g function for Knuth's Theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(cond ((consp (nthcdr 2 lst)) ;; (len lst) > 2
(cond ((and (> (first lst)
(+ (second lst) 1))
(equal (+ (second lst) 1)
(+ (third lst) 2)))
(max (third lst)
(last-el lst)))
(t (Gkr (rest lst)))))
(t (last-el lst)))) ;; (len lst) <= 2
(defun
decreasing-p (lst)
"Determine if the elements in lst are strictly decreasing."
(declare (xargs :guard (rational-listp lst)))
(if (consp (rest lst)) ;; (len lst) > 1
(and (> (first lst)(second lst))
(decreasing-p (rest lst)))
t))
(defun
first-non-decrease (lst)
"Return the front of lst up to and including the first
element that does not strictly decrease."
(declare (xargs :guard (and (rational-listp lst)
(not (decreasing-p lst)))))
(if (consp (rest lst)) ;; (len lst) > 1
(if (<= (first lst)(second lst))
(list (first lst)(second lst))
(cons (first lst)
(first-non-decrease (rest lst))))
lst))
(defun
Fb (lst)
"Bailey's f function for Knuth's theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(if (decreasing-p lst)
(first lst)
(Gb (first-non-decrease lst))))
(defun
Fko (lst)
"Knuth's original f function for Knuth's theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(if (decreasing-p lst)
(first lst)
(Gko (first-non-decrease lst))))
(defun
Fkr (lst)
"Knuth's revised f function for Knuth's theorem 4.
The input lst is intended to be a nonempty
list of integers."
(declare (xargs :guard (and (integer-listp lst)
(consp lst))))
(if (decreasing-p lst)
(first lst)
(Gkr (first-non-decrease lst))))
(defun
front (lst)
"Return lst with (last-el lst) removed."
(declare (xargs :guard (listp lst)))
(if (consp (rest lst)) ;; (len lst) > 1
(cons (first lst) (front (rest lst)))
nil))
(defthm
Gb=Gkr-1
(let ((lst (list first)))
(equal (Gb lst)(Gkr lst))))
(defthm
Gb=Gkr-2
(let ((lst (list first second)))
(equal (Gb lst)(Gkr lst))))
(defthm
Gb=Gkr<=2
(implies (and (true-listp lst)
(not (consp (nthcdr 2 lst)))) ;; (len lst) <= 2
(equal (Gb lst)(Gkr lst)))
:hints (("Goal"
:in-theory (disable Gb Gkr))))
(defthm
Gb=Gkr
(implies (and (integer-listp lst)
(decreasing-p (front lst)))
(equal (Gb lst)(Gkr lst))))
(defthm
integer-listp-first-non-decrease
(implies (integer-listp lst)
(integer-listp (first-non-decrease lst))))
(defthm
decreasing-p-front-first-non-decrease
(decreasing-p (front (first-non-decrease lst))))
(defthm
Fb=Fkr
(implies (integer-listp lst)
(equal (Fb lst)(Fkr lst))))
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