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|
; 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
; 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>
(in-package "MILAWA")
(%interactive)
(include-book
;; Fooling dependency scanner with newline because of provisional
;; certification problems with loading other books. Generally we certify all
;; ACL2 books before doing the Milawa translation, so this isn't a problem.
;; Bug Jared to fix omake stuff if you need this to work.
"../../utilities/arithmetic/multiply")
;; This file gives a demo of using our highest-level proof checker.
;;
;; This is probably completely stupid, since there are 13 other directories
;; filled with examples of proofs, and the particular proof-checker being
;; used is utterly irrelevant except for proof sizes.
;;
;; On the other hand, at least we get to test that our interface is
;; creating proofs that are accepted by level11.proofp.
;;
;; What shall we prove? Well, multiplication is not one of our primitives.
;; Nor, in fact, is it used anywhere in Milawa. But once upon a time, I looked
;; at writing a more complete library for basic arithmetic, so if you look in
;; Sources/ACL2/utilities/arithmetic you will find some various lemmas about
;; multiplication, division, and so on. In particular, to see what theorems
;; we are proving, see
;;
;; Sources/ACL2/utilities/arithmetic/multiply.lisp
;;
;; I imagined adding multiplication as a new primitive. But for this simple
;; example file, I will just give it the recursive definition I had imagined
;; would be its defining axiom. There's no ACL2 analogue, so we use %defun
;; explicitly.
;;
;; In every other file you'll see %autoadmit being used instead, except
;; probably for something like prepare-for-bootstrapping.lisp in the utilities
;; directory. But if you look at the macros in the ACL2/interface directory,
;; you'll find that %defun and %admit can be used manually. There are similar
;; facilities called %prove and %qed, instead of %autoprove. If you are going
;; to use Milawa at all, you will probably need to read through the interface
;; files to see what commands are available.
(%building t) ;; Turn on proof building, for demo purposes
(%saving t) ;; Turn on proof saving, for demo purposes
(%checking t) ;; Turn on proof checking, for demo purposes
;; Introduce multiply manually, since the ACL2 definition in
;; extended-primitives is "under the hood" and not legitimate.
(encapsulate
()
(%defun * (a b)
(if (zp a)
0
(+ b (* (- a 1) b)))
:measure (nfix a))
(%auto)
(%admit))
(%autoprove natp-of-*
(%restrict default * (equal a 'a)))
(%autoprove *-when-not-natp-left-cheap
(%restrict default * (equal a 'a)))
(%autoprove *-when-not-natp-right-cheap
(%dec-induction a)
(%restrict default * (equal a 'a)))
(%autoprove *-when-zp-left-cheap
(%restrict default * (equal a 'a)))
(%autoprove *-when-zp-right-cheap
(%dec-induction a)
(%restrict default * (equal a 'a)))
(%autoprove |(* 0 a)|
(%restrict default * (equal a ''0)))
(%autoprove |(* a 0)|
(%dec-induction a)
(%restrict default * (equal a 'a)))
(%autoprove |(* (nfix a) b)|
(%enable default nfix))
(%autoprove |(* a (nfix b))|
(%enable default nfix))
(%autoprove |(* 1 a)|
(%restrict default * (equal a ''1)))
(%autoprove |(* a 1)|
(%dec-induction a)
(%restrict default * (equal a 'a)))
(%autoprove |(equal (* a b) 0)|
(%dec-induction a)
(%restrict default * (equal a 'a)))
(%autoprove |(* (+ a c) b)|
(%dec-induction a)
(%restrict default *
(or (equal a '(+ '1 c))
(equal a '(+ a c))
(equal a 'a)
(equal a 'c))))
(%autoprove |(* a (+ b c))|
(%dec-induction a)
(%restrict default *
(and (equal a 'a)
(or (equal b 'b)
(equal b '(+ b c))
(equal b 'c)))))
(%autoprove |(* (- a b) c)|
(%dec-dec-induction a b)
(%restrict default *
(and (equal b 'c)
(or (equal a '(- a b))
(equal a '(- a '1))
(equal a 'a)
(equal a 'b)))))
(%autoprove |(* a (- b c))|
(%dec-induction a)
(%restrict default *
(and (equal a 'a)
(or (equal b 'b)
(equal b 'c)
(equal b '(- b c)))))
(%disable default |(* (- a b) c)|))
(%autoprove |(< a (* a b))|
(%dec-induction a))
(%autoprove |(< b (* a b))|
(%dec-induction a)
(%restrict default * (equal a 'a))
(%disable default |(* (- a b) c)|))
(%autoprove |(< (* a b) a)|
(%dec-induction a))
(%autoprove |(< (* a b) b)|
(%dec-induction a))
(%autoprove |(< (* a c) (* b c))|
(%dec-dec-induction a b))
(%autoprove |(< (* a b) (* a c))|
(%dec-induction a))
(%autoprove associativity-of-*
(%dec-induction a))
(%autoprove commutativity-of-*
(%dec-induction a))
(%autoprove commutativity-of-*-two
(%use (%instance (%thm commutativity-of-*)
(a a) (b (* b c)))))
(%autoprove |(= a (* a b))|
(%restrict default *
(or (and (equal a 'a) (equal b 'b))
(and (equal a 'b) (equal b '(- a '1)))))
(%disable default |(* a (- b c))|))
(%autoprove |(= 1 (* a b))|
(%restrict default * (and (equal a 'a) (equal b 'b)))
(%use (%instance (%thm |(* a (- b c))|)
(a b) (b a) (c 1)))
(%disable default |(* a (- b c))|))
;; Keeping current with ACL2 file if any theorems are added:
(%ensure-exactly-these-rules-are-missing "../../utilities/arithmetic/multiply"
;; no rules are missing, but if we wanted
;; to exclude some, we'd list them here.
)
;; When you're done with a bunch of files, you can save an events file like
;; this. The %finish command inserts a finish command so that processing the
;; .events file gives you a new image with the events loaded. You should also
;; clear out the thmfiles table any time you run save-events, so you don't
;; process the same events again later. I typically do this once per
;; directory.
(%finish "user")
(%save-events "user.events")
(ACL2::table tactic-harness 'thmfiles nil)
|
