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; The ACL2 Matrix Algebra Book. Summary of definitions and algebra in matrix.lisp.
; Copyright (C) 2002 Ruben Gamboa and 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:
; Ruben Gamboa and John Cowles
; Department of Computer Science
; University of Wyoming
; Laramie, WY 82071-3682 U.S.A.
; Summer and Fall 2002.
; Last modified 1 November 2002.
#|
To certify in ACL2 Version 2.6
(ld ;; Newline to fool dependency scanner
"defpkg.lsp")
(certify-book "kalman-demo" 1)
|#
(in-package "KALMAN")
(include-book "kalman-proof")
(defun kalman-loop-body (xhatmin-prev pminus-prev k)
(let* ((gain (m-* pminus-prev
(m-* (m-trans (h k))
(m-inv (m-+ (m-* (h k)
(m-* pminus-prev
(m-trans (h k))))
(r k))))))
(xhat (m-+ xhatmin-prev
(m-* gain
(m-- (z k)
(m-* (h k) xhatmin-prev)))))
(pplus (m-* (m-- (m-id (n))
(m-* gain (h k)))
pminus-prev))
(xhatmin (m-* (phi k) xhat))
(pminus (m-+ (m-* (phi k)
(m-* pplus
(m-trans (phi k))))
(q k))))
(mv gain xhat pplus xhatmin pminus)))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(declare (ignore xhat pplus xhatmin pminus))
(equal gain (gain k))))
:hints (("Goal"
:in-theory (disable acl2::*-+-right acl2::*-+-left
acl2::right-distributivity-of-m-*-over-m-+
acl2::left-distributivity-of-m-*-over-m-+)))))
(local
(defthm lemma-2
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(declare (ignore gain pplus xhatmin pminus))
(equal xhat (xhat k))))
:hints (("Goal"
:in-theory (disable acl2::*-+-right acl2::*-+-left
acl2::right-distributivity-of-m-*-over-m-+
acl2::left-distributivity-of-m-*-over-m-+)))))
(local
(defthm lemma-3
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(declare (ignore gain xhat xhatmin pminus))
(equal pplus (pplus k))))
:hints (("Goal"
:in-theory (disable pplus-as-mean
acl2::*-+-right acl2::*-+-left
acl2::right-distributivity-of-m-*-over-m-+
acl2::left-distributivity-of-m-*-over-m-+)))))
(local
(defthm lemma-4
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(declare (ignore gain xhat pplus pminus))
(equal xhatmin (xhatmin (1+ k)))))
:hints (("Goal"
:in-theory (disable acl2::*-+-right acl2::*-+-left
acl2::right-distributivity-of-m-*-over-m-+
acl2::left-distributivity-of-m-*-over-m-+)))))
(local
(defthm lemma-5
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(declare (ignore gain xhat pplus xhatmin))
(equal pminus (pminus (1+ k)))))
:hints (("Goal"
:in-theory (disable pplus-as-mean
pminus-as-mean
pminus-as-mean-almost
acl2::*-+-right acl2::*-+-left
acl2::right-distributivity-of-m-*-over-m-+
acl2::left-distributivity-of-m-*-over-m-+)))))
(defthm kalman-loop-invariant
(implies (and (integerp k)
(<= 0 k)
(equal xhatmin-prev (xhatmin k))
(equal pminus-prev (pminus k)))
(mv-let (gain xhat pplus xhatmin pminus)
(kalman-loop-body xhatmin-prev pminus-prev k)
(and (equal gain (gain k))
(equal xhat (xhat k))
(equal pplus (pplus k))
(equal xhatmin (xhatmin (1+ k)))
(equal pminus (pminus (1+ k))))))
:hints (("Goal"
:use ((:instance lemma-1)
(:instance lemma-2)
(:instance lemma-3)
(:instance lemma-4)
(:instance lemma-5)))))
)
|
