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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.
;; The function Fb satisfies the tarai recursion and the
;; restricted tarai recursion for lists of length 6.
;; (certify-book "C:/acl2/tak/tarai2")
(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
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
rotate (lst)
"Return the result of shifting the first element
of the list lst onto the tail end of lst."
(declare (xargs :guard (true-listp lst)))
(if (consp lst)
(append (rest lst)(list (first lst)))
nil))
(defun
minus-1 (lst)
"Return the result of replacing (first lst)
with (first lst) - 1."
(declare (xargs :guard (and (rational-listp lst)
(consp lst))))
(if (consp lst)
(cons (- (first lst) 1)
(rest lst))
lst))
(defun
lst-rotates-with-minus-1 (n lst)
"Return the list of n+1 successive rotates of the
list lst with the first of each rotate replaced
by the first minus 1."
(declare (xargs :guard (and (rational-listp lst)
(consp lst)
(integerp n)
(>= n 0))))
(if (zp n)
(list (minus-1 lst))
(cons (minus-1 lst)
(lst-rotates-with-minus-1 (- n 1)
(rotate lst)))))
(defun
Fb-lst (lst)
(if (consp lst)
(cons (Fb (first lst))
(Fb-lst (rest lst)))
nil))
(defun
dec-front-len (lst)
"Return the number of strictly decreasing elements
at the front of the list lst."
(declare (xargs :guard (rational-listp lst)))
(cond ((consp (rest lst)) ;; (len lst) > 1
(if (<= (first lst)(second lst))
1
(+ 1 (dec-front-len (rest lst)))))
((consp lst) 1) ;; (len lst) = 1
(t 0))) ;; (len lst) = 0
;;----------------------------------------------------------
;; Fb satisfies the tarai recursion equation when lst is a
;; true list of integers of length 5.
(local
(defthm
lst-rotates-with-minus-1-6a
(let ((lst (list first second third forth fifth sixth)))
(equal (lst-rotates-with-minus-1 1 lst)
(list (list (- first 1) second third forth fifth sixth)
(list (- second 1) third forth fifth sixth first))))
:hints (("Goal"
:expand ((lst-rotates-with-minus-1
1
(list first second third forth fifth sixth))))))
)
(local
(defthm
lst-rotates-with-minus-1-6b
(let ((lst (list first second third forth fifth sixth)))
(equal (lst-rotates-with-minus-1 2 lst)
(list (list (- first 1) second third forth fifth sixth)
(list (- second 1) third forth fifth sixth first)
(list (- third 1) forth fifth sixth first second)))))
)
(local
(defthm
lst-rotates-with-minus-1-6c
(let ((lst (list first second third forth fifth sixth)))
(equal (lst-rotates-with-minus-1 3 lst)
(list (list (- first 1) second third forth fifth sixth)
(list (- second 1) third forth fifth sixth first)
(list (- third 1) forth fifth sixth first second)
(list (- forth 1) fifth sixth first second third)))))
)
(local
(defthm
lst-rotates-with-minus-1-6d
(let ((lst (list first second third forth fifth sixth)))
(equal (lst-rotates-with-minus-1 4 lst)
(list (list (- first 1) second third forth fifth sixth)
(list (- second 1) third forth fifth sixth first)
(list (- third 1) forth fifth sixth first second)
(list (- forth 1) fifth sixth first second third)
(list (- fifth 1) sixth first second third forth)
))))
)
(local
(defthm
lst-rotates-with-minus-1-6e
(let ((lst (list first second third forth fifth sixth)))
(equal
(lst-rotates-with-minus-1 5 lst)
(list (list (- first 1) second third forth fifth sixth)
(list (- second 1) third forth fifth sixth first)
(list (- third 1) forth fifth sixth first second)
(list (- forth 1) fifth sixth first second third)
(list (- fifth 1) sixth first second third forth)
(list (- sixth 1) first second third forth fifth)
))))
)
(defthm
;;Time: 703.87 seconds (prove: 327.38, print: 376.44, other: 0.05)
Fb-sat-tarai-def-6
(implies (and (integerp first)
(integerp second)
(integerp third)
(integerp forth)
(integerp fifth)
(integerp sixth))
(let ((lst (list first second third forth fifth sixth)))
(equal (Fb lst)
(if (<= (first lst)
(second lst))
(second lst)
(Fb (Fb-lst (lst-rotates-with-minus-1
(- (len lst) 1)
lst)))))))
:rule-classes nil)
(defthm
;;Time: 253.60 seconds (prove: 122.80, print: 130.74, other: 0.06)
Fb-sat-tarai-def-6a
(implies (and (integerp first)
(integerp second)
(integerp third)
(integerp forth)
(integerp fifth)
(integerp sixth))
(let ((lst (list first second third forth fifth sixth)))
(equal (Fb lst)
(if (<= (first lst)
(second lst))
(second lst)
(Fb (Fb-lst (lst-rotates-with-minus-1
(- (dec-front-len lst) 1)
lst)))))))
:rule-classes nil)
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