(*<*) ―‹ ******************************************************************** * Project : HOL-CSP - A Shallow Embedding of CSP in Isabelle/HOL * Version : 2.0 * * Author : Burkhart Wolff, Safouan Taha. * (Based on HOL-CSP 1.0 by Haykal Tej and Burkhart Wolff) * * This file : A Combined CSP Theory * * Copyright (c) 2009 Université Paris-Sud, France * * 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 list of conditions and the following disclaimer. * * * 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. * * * Neither the name of the copyright holders nor the names of its * contributors may be used to endorse or promote products derived * from this software without specific prior written permission. * * 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 * OWNER 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. ******************************************************************************› (*>*) chapter‹ The CSP Operators › section‹The Undefined Process› theory Bot imports Process begin lift_definition BOT :: ‹'α process› is ‹({(s,X). front_tickFree s}, {d. front_tickFree d})› unfolding is_process_def FAILURES_def DIVERGENCES_def by (auto simp: tickFree_implies_front_tickFree elim: front_tickFree_dw_closed front_tickFree_append) lemma F_BOT: "ℱ BOT = {(s,X). front_tickFree s}" by (simp add: BOT.rep_eq FAILURES_def Failures.rep_eq) lemma D_BOT: "𝒟 BOT = {d. front_tickFree d}" by (simp add: BOT.rep_eq DIVERGENCES_def Divergences.rep_eq) lemma T_BOT: "𝒯 BOT = {s. front_tickFree s}" by (simp add: Traces.rep_eq TRACES_def Failures.rep_eq[symmetric] F_BOT) text‹ This is the key result: @{term "⊥"} --- which we know to exist from the process instantiation --- is equal \<^const>‹BOT› . › lemma BOT_is_UU[simp]: "BOT = ⊥" apply(auto simp: Pcpo.eq_bottom_iff Process.le_approx_def Ra_def min_elems_Collect_ftF_is_Nil Process.Nil_elem_T F_BOT D_BOT T_BOT elim: D_imp_front_tickFree) apply(metis Process.is_processT2) done lemma F_UU: "ℱ ⊥ = {(s,X). front_tickFree s}" using F_BOT by auto lemma D_UU: "𝒟 ⊥ = {d. front_tickFree d}" using D_BOT by auto lemma T_UU: "𝒯 ⊥ = {s. front_tickFree s}" using T_BOT by auto lemma BOT_iff_D: ‹P = ⊥ ⟷ [] ∈ 𝒟 P› apply (intro iffI, simp add: D_UU) apply (subst Process_eq_spec, safe) by (simp_all add: F_UU D_UU is_processT2 D_imp_front_tickFree) (metis append_Nil is_processT tickFree_Nil)+ end