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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")
(include-book "../primitives")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
;; Extended base functions.
;;
;; We also introduce some extended arithmetic support. This isn't strictly
;; necessary since we can define these functions in terms of the base
;; functions. But, by making these base functions, we can give them far more
;; efficient Common Lisp implementations.
(definlined * (a b)
;; Multiply a and b.
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::* (nfix a) (nfix b)))
(definlined floor (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(let ((afix (nfix a))
(bfix (nfix b)))
(if (equal bfix 0)
0
(COMMON-LISP::floor afix bfix))))
(definlined mod (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(let ((afix (nfix a))
(bfix (nfix b)))
(if (equal bfix 0)
afix
(COMMON-LISP::mod afix bfix))))
(definlined expt (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::expt (nfix a) (nfix b)))
(definlined bitwise-shl (a n)
;; Shift a left by n bits.
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::ash (nfix a) (nfix n)))
(definlined bitwise-shr (a n)
;; Shift a right by n bits.
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::ash (nfix a) (COMMON-LISP::- (nfix n))))
(definlined bitwise-and (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::logand (nfix a) (nfix b)))
(definlined bitwise-or (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::logior (nfix a) (nfix b)))
(definlined bitwise-xor (a b)
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::logxor (nfix a) (nfix b)))
(definlined bitwise-nth (n a)
;; Is the nth bit of a set?
(declare (xargs :guard t
:guard-hints (("Goal" :in-theory (enable natp nfix)))
:export nil))
(COMMON-LISP::logbitp (nfix n) (nfix a)))
;; Common lisp also provides some other bitwise functions which we do not
;; support because they are not natural-valued. For example:
;;
;; (lognand 0 0) = -1
;; (lognor 0 0) = -1
;; (lognot 0) = -1
;; (logorc1 0 0) = -1
;; (logorc2 0 0) = -1
;; (logeqv 0 0) = -1
;;
;; I didn't find an immediate example where logandc1 or logandc2 would produce
;; a negative value, but I'm not going to add them until someone wants them,
;; because I think they are weird.
(defun dec-floor2-induction (n a)
(declare (xargs :guard t :measure (nfix n)))
(if (zp n)
a
(dec-floor2-induction (- n 1) (floor a 2))))
;; Axioms for the extended base functions.
;;
;; We introduce recursive formulations of each of our extended arithmetic
;; functions. This way, we only need one symbolic axiom per added base
;; function, and everything else is proven from the recursive definition. It's
;; vital that we get these right, so we've used ACL2 to prove the equivalence
;; with our ACL2 definitions.
(encapsulate
()
(local (in-theory (enable natp nfix zp < + - * expt floor mod
bitwise-shl bitwise-shr bitwise-and
bitwise-or bitwise-xor bitwise-nth)))
(local (in-theory (disable ACL2::expt ACL2::floor ACL2::mod
ACL2::ash ACL2::logand
ACL2::logior ACL2::logxor ACL2::logbitp)))
(local (include-book "arithmetic-3/floor-mod/floor-mod" :dir :system))
(local (include-book "bitwise-support"))
(defthmd definition-of-*
(equal (* a b)
(if (zp a)
0
(+ b (* (- a 1) b))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (* a b)
:scheme (dec-induction a))))
(defthmd definition-of-floor
(equal (floor a b)
(cond ((zp b) 0)
((< a b) 0)
(t (+ 1 (floor (- a b) b)))))
:hints(("Goal" :in-theory (disable acl2::prefer-positive-addends-equal)))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (floor a b)
:scheme (sub-induction a b))))
(defthmd definition-of-mod
(equal (mod a b)
(cond ((zp b) (nfix a))
((< a b) (nfix a))
(t (mod (- a b) b))))
:hints(("Goal" :in-theory (disable acl2::prefer-positive-addends-equal)))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (mod a b)
:scheme (sub-induction a b))))
(defthmd definition-of-expt
(equal (expt a b)
(if (zp b)
1
(* a (expt a (- b 1)))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (expt a b)
:scheme (dec-induction b))))
(defthmd definition-of-bitwise-shl
(equal (bitwise-shl a n)
(if (zp n)
(nfix a)
(* 2 (bitwise-shl a (- n 1)))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-shl a n)
:scheme (dec-induction n)))
:hints(("Goal" :in-theory (enable acl2::ash))))
(defthmd definition-of-bitwise-shr
(equal (bitwise-shr a n)
(if (zp n)
(nfix a)
(floor (bitwise-shr a (- n 1)) 2)))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-shr a n)
:scheme (dec-induction n)))
:hints(("Goal" :in-theory (enable acl2::ash))))
(encapsulate
()
;; We introduce the floor2-floor2 induction scheme without relying on ACL2
;; arithmetic, so we should be able to recreate this in Milawa.
(local (in-theory (disable floor < natp + -)))
(local (defthm termination-lemma
(implies (not (zp a))
(equal (< (floor a 2) a)
t))
:hints(("Goal"
:in-theory (e/d (definition-of-floor)
(floor < natp + -))))))
(defun floor2-floor2-induction (a b)
(declare (xargs :guard t :measure (nfix a)))
(if (or (zp a)
(zp b))
nil
(floor2-floor2-induction (floor a 2) (floor b 2)))))
(defthm definition-of-bitwise-and
(equal (bitwise-and a b)
(cond ((zp a) 0)
((zp b) 0)
(t (+ (if (or (equal (mod a 2) 0)
(equal (mod b 2) 0))
0
1)
(* 2 (bitwise-and (floor a 2) (floor b 2)))))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-and a b)
:scheme (floor2-floor2-induction a b))))
(defthm definition-of-bitwise-or
(equal (bitwise-or a b)
(cond ((zp a) (nfix b))
((zp b) (nfix a))
(t (+ (if (or (equal (mod a 2) 1)
(equal (mod b 2) 1))
1
0)
(* 2 (bitwise-or (floor a 2) (floor b 2)))))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-or a b)
:scheme (floor2-floor2-induction a b))))
(defthm definition-of-bitwise-xor
(equal (bitwise-xor a b)
(cond ((zp a) (nfix b))
((zp b) (nfix a))
(t (+ (if (equal (mod a 2) (mod b 2))
0
1)
(* 2 (bitwise-xor (floor a 2) (floor b 2)))))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-xor a b)
:scheme (floor2-floor2-induction a b))))
(defthm definition-of-bitwise-nth
(equal (bitwise-nth n a)
(if (zp n)
(equal (mod a 2) 1)
(bitwise-nth (- n 1) (floor a 2))))
:rule-classes ((:definition)
(:induction :corollary t
:pattern (bitwise-nth n a)
:scheme (dec-floor2-induction n a))))
)
;; From this point forward, all reasoning about our extended operations should
;; be done without referring to their under-the-hood implementations.
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition *))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition expt))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition floor))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition mod))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-shl))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-shr))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-and))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-or))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-xor))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition bitwise-nth))))
|
