(*********************************************************************************** * Copyright (c) University of Exeter, UK * * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions are met: * * * Redistributions of source code must retain the above copyright notice, this * * * Redistributions in binary form must reproduce the above copyright notice, * this list of conditions and the following disclaimer in the documentation * and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. * * SPDX-License-Identifier: BSD-3-Clause ***********************************************************************************) chapter‹Extended Division on Multi-Intervals› theory Extended_Multi_Interval_Division_Core imports Interval_Division_Non_Zero Multi_Interval begin section‹Division over List of Intervals› text‹ In this theory, we define an extended division operation on intervals. This is a formalization of the interval division given in~\<^cite>‹"moore.ea:introduction:2009"›. › definition inverse_interval :: "('a::{linorder,minus_mono,zero,one,inverse,infinity,uminus}) interval ⇒ ('a interval) list" where "inverse_interval a = ( if (¬ 0 ∈⇩i a) then [mk_interval ( 1 / (upper a), 1 / (lower a))] else if lower a = 0 ∧ 0 < upper a then [mk_interval (1/ upper a, ∞)] else if lower a < 0 ∧ 0 < upper a then [mk_interval (-∞, 1/lower a), mk_interval (1/upper a, ∞)] else if lower a < upper a ∧ upper a = 0 then [mk_interval(-∞, 1 / lower a)] else undefined )" definition ‹minverse = concat o (map inverse_interval)› section‹Multi-Interval Division› end