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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 "deflist")
(include-book "nat-listp")
(include-book "all-equalp")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
;; This was previously the following, but now we have :negatedp.
;; (deflist all-at-leastp (n x)
;; (<= n x)
;; :guard (and (natp n) (nat-listp x)))
(deflist all-at-leastp (n x)
(< x n)
:negatedp t
:guard (and (natp n)
(nat-listp x)))
;; We previously had this, but with negatedp we don't need it.
;; (defthm <-of-car-when-all-at-leastp
;; (implies (all-at-leastp n x)
;; (equal (< (car x) n)
;; (and (not (consp x))
;; (< 0 n)))))
;; (in-theory (disable <=-of-car-when-all-at-leastp))
(defthm all-at-leastp-when-all-equalp
(implies (all-equalp n x)
(equal (all-at-leastp m x)
(if (consp x)
(not (< n m))
t)))
:hints(("Goal" :induct (cdr-induction x))))
(defthm all-at-leastp-of-zero
(equal (all-at-leastp 0 x)
t)
:hints(("Goal" :induct (cdr-induction x))))
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