; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 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
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;   DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>

(in-package "MILAWA")
(include-book "hypboxp")
(%interactive)


;; BOZO move this to primitives
;; BOZO why bother, why not just use the two-nats-measure to admit it?

(defthm rank-of-fifth
  (equal (< (rank (fifth x)) (rank x))
         (consp x)))

(%autoprove rank-of-fifth)


(defthm rank-of-fifth-weak
  (equal (< (rank x) (rank (fifth x)))
         nil))

(%autoprove rank-of-fifth-weak)

(defthm rank-of-cdr-of-cdr-of-cdr-weak
  (equal (< (rank x)
            (rank (cdr (cdr (cdr x)))))
         nil))

(%autoprove rank-of-cdr-of-cdr-of-cdr-weak
            (%restrict default rank (memberp x '(x (cdr x) (cdr (cdr x))))))

(defthm rank-of-cdr-of-cdr-of-cdr
  (equal (< (rank (cdr (cdr (cdr x))))
            (rank x))
         (consp x)))

(%autoprove rank-of-cdr-of-cdr-of-cdr
            (%restrict default rank (memberp x '(x (cdr x) (cdr (cdr x))))))


(%autoadmit rw.flag-eqtracep)
(%autoadmit rw.eqtracep)
(%autoadmit rw.eqtrace-listp)

(%autoprove definition-of-rw.eqtracep
            (%restrict default rw.flag-eqtracep (equal x 'x))
            (%enable default rw.eqtracep rw.eqtrace-listp))

(%autoprove definition-of-rw.eqtrace-listp
            (%restrict default rw.flag-eqtracep (equal x 'x))
            (%enable default rw.eqtracep rw.eqtrace-listp))

(%autoprove rw.flag-eqtracep-of-trace
            (%enable default rw.eqtracep))

(%autoprove rw.flag-eqtracep-of-list
            (%enable default rw.eqtrace-listp))



(defmacro %rw.flag-eqtracep-induction-raw (flag x)
  ;; This is a lot better than autoinduct for controlling cases
  `(%induct (two-nats-measure (rank ,x) (if (equal ,flag 'trace) '1 '0))
            ((equal ,flag 'trace)
             (((,flag 'list) (,x (cdr (cdr (cdr ,x)))))))
            ((and (not (equal ,flag 'trace))
                  (not (consp ,x)))
             nil)
            ((and (not (equal ,flag 'trace))
                  (consp ,x))
             (((,flag 'trace) (,x (car ,x)))
              ((,flag 'list) (,x (cdr ,x)))))))

(%autoprove lemma-for-booleanp-of-rw.eqtracep
            (%rw.flag-eqtracep-induction-raw flag x)
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%restrict default definition-of-rw.eqtrace-listp (equal x 'x)))

(%autoprove booleanp-of-rw.eqtracep
            (%use (%instance (%thm lemma-for-booleanp-of-rw.eqtracep)
                             (flag 'trace))))

(%autoprove booleanp-of-rw.eqtrace-listp
            (%use (%instance (%thm lemma-for-booleanp-of-rw.eqtracep)
                             (flag 'list))))

(%autoprove rw.eqtrace-listp-when-not-consp
            (%restrict default definition-of-rw.eqtrace-listp (equal x 'x)))

(%autoprove rw.eqtrace-listp-of-cons
            (%restrict default definition-of-rw.eqtrace-listp (equal x '(cons a x))))

(%deflist rw.eqtrace-listp (x)
          (rw.eqtracep x))



(%autoadmit rw.eqtrace->method)
(%autoadmit rw.eqtrace->iffp)
(%autoadmit rw.eqtrace->lhs)
(%autoadmit rw.eqtrace->rhs)
(%autoadmit rw.eqtrace->subtraces)

(%autoprove forcing-symbolp-of-rw.eqtrace->method
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->method))

(%autoprove forcing-booleanp-of-rw.eqtrace->iffp
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->iffp))

(%autoprove forcing-logic.termp-of-rw.eqtrace->lhs
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->lhs))

(%autoprove forcing-logic.termp-of-rw.eqtrace->rhs
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->rhs))

(%autoprove forcing-rw.eqtrace-listp-of-rw.eqtrace->subtraces
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->subtraces))

(%autoprove forcing-logic.term-<-of-rw.eqtrace->lhs-and-rw.eqtrace->rhs
            (%restrict default definition-of-rw.eqtracep (equal x 'x))
            (%enable default rw.eqtrace->lhs rw.eqtrace->rhs))

(%autoprove forcing-logic.term-<-of-rw.eqtrace->lhs-and-rw.eqtrace->rhs-free)

;; BOZO move to primitives
(%autoprove |(< a (+ b c d e f a g))|)

;; BOZO really want this still?  Don't think so...
(%autoprove rank-of-rw.eqtrace->subtraces-weak
            (%enable default rw.eqtrace->subtraces))



(%autoadmit rw.flag-eqtrace-atblp)
(%autoadmit rw.eqtrace-atblp)
(%autoadmit rw.eqtrace-list-atblp)

(%autoprove definition-of-rw.eqtrace-atblp
            (%restrict default rw.flag-eqtrace-atblp (equal x 'x))
            (%enable default rw.eqtrace-atblp rw.eqtrace-list-atblp))

(%autoprove definition-of-rw.eqtrace-list-atblp
            (%restrict default rw.flag-eqtrace-atblp (equal x 'x))
            (%enable default rw.eqtrace-atblp rw.eqtrace-list-atblp))

(%autoprove rw.flag-eqtrace-atblp-of-trace
            (%enable default rw.eqtrace-atblp))

(%autoprove rw.flag-eqtrace-atblp-of-list
            (%enable default rw.eqtrace-list-atblp))




(defmacro %rw.flag-eqtracep-induction (flag x)
  `(%induct (two-nats-measure (rank ,x) (if (equal ,flag 'trace) '1 '0))
            ((equal ,flag 'trace)
             (((,flag 'list) (,x (rw.eqtrace->subtraces ,x)))))
            ((and (not (equal ,flag 'trace))
                  (not (consp ,x)))
             nil)
            ((and (not (equal ,flag 'trace))
                  (consp ,x))
             (((,flag 'trace) (,x (car ,x)))
              ((,flag 'list) (,x (cdr ,x)))))))

(%autoprove lemma-for-booleanp-of-rw.eqtrace-atblp
            (%rw.flag-eqtracep-induction flag x)
            (%restrict default definition-of-rw.eqtrace-atblp (equal x 'x))
            (%restrict default definition-of-rw.eqtrace-list-atblp (equal x 'x)))

(%autoprove booleanp-of-rw.eqtrace-atblp
            (%use (%instance (%thm lemma-for-booleanp-of-rw.eqtrace-atblp)
                             (flag 'trace))))

(%autoprove booleanp-of-rw.eqtrace-list-atblp
            (%use (%instance (%thm lemma-for-booleanp-of-rw.eqtrace-atblp)
                             (flag 'list))))

(%autoprove rw.eqtrace-list-atblp-when-not-consp
            (%restrict default definition-of-rw.eqtrace-list-atblp (equal x 'x)))

(%autoprove rw.eqtrace-list-atblp-of-cons
            (%restrict default definition-of-rw.eqtrace-list-atblp (equal x '(cons a x))))

(%autoprove rw.eqtrace-atblp-of-nil
            (%restrict default definition-of-rw.eqtrace-atblp (equal x ''nil)))

(%deflist rw.eqtrace-list-atblp (x atbl)
          (rw.eqtrace-atblp x atbl))

(%autoprove forcing-logic.term-atblp-of-rw.eqtrace->lhs
            (%restrict default definition-of-rw.eqtrace-atblp (equal x 'x)))

(%autoprove forcing-logic.term-atblp-of-rw.eqtrace->rhs
            (%restrict default definition-of-rw.eqtrace-atblp (equal x 'x)))

(%autoprove forcing-rw.eqtrace-list-atblp-of-rw.eqtrace->subtraces
            (%restrict default definition-of-rw.eqtrace-atblp (equal x 'x)))




(%autoadmit rw.eqtrace)
(%noexec rw.eqtrace)

(%autoprove rw.eqtrace->method-of-rw.eqtrace
            (%enable default rw.eqtrace rw.eqtrace->method))

(%autoprove rw.eqtrace->iffp-of-rw.eqtrace
            (%enable default rw.eqtrace rw.eqtrace->iffp))

(%autoprove rw.eqtrace->lhs-of-rw.eqtrace
            (%enable default rw.eqtrace rw.eqtrace->lhs))

(%autoprove rw.eqtrace->rhs-of-rw.eqtrace
            (%enable default rw.eqtrace rw.eqtrace->rhs))

(%autoprove rw.eqtrace->subtraces-of-rw.eqtrace
            (%enable default rw.eqtrace rw.eqtrace->subtraces))

(%autoprove forcing-rw.eqtracep-of-rw.eqtrace
            (%restrict default definition-of-rw.eqtracep (equal x '(CONS (CONS LHS RHS) (CONS IFFP (CONS METHOD SUBTRACES)))))
            (%enable default rw.eqtrace))

(%autoprove forcing-rw.eqtrace-atblp-of-rw.eqtrace
            (%restrict default definition-of-rw.eqtrace-atblp
                       (equal x '(rw.eqtrace method iffp lhs rhs subtraces))))

(%autoprove rw.eqtrace-under-iff
            (%enable default rw.eqtrace))


(defsection rw.trace-list-iffps
  (local (%disable default forcing-booleanp-of-rw.eqtrace->iffp))
  (%defprojection :list (rw.eqtrace-list-iffps x)
                  :element (rw.eqtrace->iffp x)
                  :nil-preservingp t))

(defsection rw.eqtrace-list-lhses
  (local (%disable default forcing-booleanp-of-rw.eqtrace->iffp))
  (%defprojection :list (rw.eqtrace-list-lhses x)
                  :element (rw.eqtrace->lhs x)
                  :nil-preservingp t))

(defsection rw.eqtrace-list-rhses
  (local (%disable default forcing-booleanp-of-rw.eqtrace->iffp))
  (%defprojection :list (rw.eqtrace-list-rhses x)
                  :element (rw.eqtrace->rhs x)
                  :nil-preservingp t))

(%autoprove forcing-logic.term-listp-of-rw.eqtrace-list-lhses (%cdr-induction x))
(%autoprove forcing-logic.term-listp-of-rw.eqtrace-list-rhses (%cdr-induction x))
(%autoprove forcing-logic.term-list-atblp-of-rw.eqtrace-list-lhses (%cdr-induction x))
(%autoprove forcing-logic.term-list-atblp-of-rw.eqtrace-list-rhses (%cdr-induction x))
(%autoprove rw.eqtrace->iffp-when-all-equalp-of-rw.eqtrace-list-iffps (%cdr-induction row))
(%autoprove rw.eqtrace->iffp-when-all-equalp-of-rw.eqtrace-list-iffps-alt)


(%autoprove rw.eqtrace->lhs-when-all-equalp-of-rw.eqtrace-list-lhses
            (%cdr-induction x))

(%autoprove rw.eqtrace->lhs-when-all-equalp-of-rw.eqtrace-list-lhses-alt)

(%ensure-exactly-these-rules-are-missing "../../rewrite/assms/eqtracep")