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; number-list-defthms.lisp -- theorems about lists of numbers
; Copyright (C) 1997 Computational Logic, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by: Bill Bevier (bevier@cli.com)
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
(include-book "number-list-defuns")
(include-book "deflist")
(deflabel number-list-defthms-section)
(defmacro real/rational-listp (x)
#+:non-standard-analysis
`(real-listp ,x)
#-:non-standard-analysis
`(rational-listp ,x))
; ------------------------------------------------------------
; Theorems
; ------------------------------------------------------------
(deflist rational-listp (l) rationalp
(:options :omit-defun))
#+:non-standard-analysis
(deflist real-listp (l) realp
(:options :omit-defun))
(deflist integer-listp (l) integerp
(:options :omit-defun))
(deflist natural-listp (l) (lambda (x) (naturalp x))
(:options :omit-defun))
; ---------- forwardchaining rules ----------
(defthm natural-listp-forward-to-integer-listp
(implies (natural-listp l)
(integer-listp l))
:rule-classes :forward-chaining)
(defthm integer-listp-forward-to-eqlable-listp
(implies (integer-listp l)
(eqlable-listp l)))
; ---------- LINEAR ----------
(defthm <=maxlist
(implies (member-equal x l)
(<= x (maxlist l)))
:rule-classes (:rewrite :linear :forward-chaining))
(defthm minlist<=
(implies (member-equal x l)
(<= (minlist l) x))
:rule-classes (:rewrite :linear :forward-chaining))
; ---------- MEMBER-EQUAL ----------
(defthm member-equal-maxlist
(implies (and (real/rational-listp l)
(consp l))
(member-equal (maxlist l) l)))
(defthm member-equal-minlist
(implies (and (real/rational-listp l)
(consp l))
(member-equal (minlist l) l)))
(defthm member-equal-excess-natural
(implies (natural-listp l)
(not (member-equal (excess-natural l) l)))
:hints (("Goal" :do-not-induct t
:use ((:instance <=maxlist (x (excess-natural l)) (l l)))
:in-theory (disable <=maxlist))))
(defthm numerically-sorted-numeric-insert
(implies (and (real/rationalp x)
(real/rational-listp l)
(numerically-sorted l))
(numerically-sorted (numeric-insert x l)))
:hints (("Goal" :induct (numeric-insert x l))))
(defthm numerically-sorted-numeric-sort
(implies (real/rational-listp l)
(numerically-sorted (numeric-sort l))))
(deftheory number-list-defthms
(set-difference-theories
(current-theory :here)
(current-theory 'number-list-defthms-section)))
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