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|
; Arithmetic-5 Library
; Written by Robert Krug
; Copyright/License:
; See the LICENSE file at the top level of the arithmetic-5 library.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; natp-posp.lisp
;;;
;;; This book is a modified copy of that in arithmetic/natp-posp. I
;;; discourage the use of natp and posp, except as abbreviations in
;;; a functions definition. I have no good examples to justify this,
;;; however, so use your own judgement.
;;;
;;; I include this book for those who want to use natp and posp.
;;; Use (enable-natp-posp-theory) and (disable-natp-posp-theory)
;;; to toggle the rules in this book. See top.lisp for their
;;; definition.
;;;
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(include-book "building-blocks")
(local
(include-book "../../support/top"))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; A few simple forward-chaining and rewrite rules to allow
;;; basic reasoning about natp and posp.
(defthm natp-fc-1
(implies (natp x)
(<= 0 x))
:rule-classes :forward-chaining)
(defthm natp-fc-2
(implies (natp x)
(integerp x))
:rule-classes :forward-chaining)
(defthm posp-fc-1
(implies (posp x)
(< 0 x))
:rule-classes :forward-chaining)
(defthm posp-fc-2
(implies (posp x)
(integerp x))
:rule-classes :forward-chaining)
(defthm natp-rw
(implies (and (integerp x)
(<= 0 x))
(natp x)))
(defthm posp-rw
(implies (and (integerp x)
(< 0 x))
(posp x)))
(defthm natp-posp
(implies (and (natp a)
(not (equal a 0)))
(posp a)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; We include a few simple rules for reducing natp or
;;; posp expressions to simpler ones.
;;; We generalize |(natp a) <=> (posp a+1)| and natp-posp--1
(defthm |(posp (+ c x))|
(implies (and (syntaxp (quotep c))
(< 0 c)
(integerp a))
(equal (posp (+ c a))
(natp (+ (- c 1) a)))))
(defthm |(natp (+ c x))|
(implies (and (syntaxp (quotep c))
(< c 0)
(integerp a))
(equal (natp (+ c x))
(posp (+ (+ c 1) x)))))
;;; We break natp-* into two rules. One which induces a case-split
;;; similar to that for |(< 0 (* x y))| to be used while rewriting a
;;; goal literal, the other duplicates natp-* and is used during
;;; back-chaining.
(defthm |(natp (* x y))|
(implies (and (syntaxp (rewriting-goal-literal x mfc state))
(integerp x)
(integerp y))
(equal (natp (* x y))
(or (and (natp x)
(natp y))
(and (natp (- x))
(natp (- y)))))))
(defthm |(natp (* x y)) backchaining|
(implies (and (syntaxp (not (rewriting-goal-literal x mfc state)))
(natp x)
(natp y))
(natp (* x y))))
;;; We do the same for posp. This was missing from
;;; arithmetic/natp-posp.lisp.
(defthm |(posp (* x y))|
(implies (and (syntaxp (rewriting-goal-literal x mfc state))
(integerp x)
(integerp y))
(equal (posp (* x y))
(or (and (posp x)
(posp y))
(and (posp (- x))
(posp (- y)))))))
(defthm |(posp (* x y)) backchaining|
(implies (and (syntaxp (not (rewriting-goal-literal x mfc state)))
(posp x)
(posp y))
(posp (* x y))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; A couple of simple type-prescription rules for natp and posp
;;; of differences.
(defthm |x < y => 0 < -x+y|
(implies (and (integerp x) (integerp y) (< x y))
(posp (+ (- x) y)))
:rule-classes ((:type-prescription)))
(defthm |x < y => 0 < y-x|
(implies (and (integerp x) (integerp y) (< x y))
(posp (+ y (- x))))
:rule-classes ((:type-prescription)))
;;; We do the same for posp. This was missing from
;;; arithmetic/natp-posp.lisp.
(defthm |x < y => 0 <= -x+y|
(implies (and (integerp x) (integerp y) (<= x y))
(natp (+ (- x) y)))
:rule-classes ((:type-prescription)))
(defthm |x < y => 0 <= y-x|
(implies (and (integerp x) (integerp y) (<= x y))
(natp (+ y (- x))))
:rule-classes ((:type-prescription)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; We do not include the following:
#|
(defthm posp-natp
(implies (posp (+ -1 x))
(natp (+ -2 x))))
|#
;;; It is claimed that it was needed for a couple of proofs,
;;; but we did not find this to be the case in ACL2 v-3.0.1. The hint
#|
("Subgoal *1/3.3" :use (:instance map-lemma-3.7
(i 0)
(j (CAR V))))
|#
;;; did, however, have to be added to map-lemma-4 in
;;; books/workshops/2003/sustik/support/dickson,lisp
;;; (This test was done with the arithmetic/natp-posp book,
;;; not this one which contains a generalization of the above:
;;; |(posp (+ c x))|
(in-theory (disable natp posp))
|
