Theory Bot

(*<*)
―‹ ********************************************************************
 * 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
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chapter‹ The CSP Operators ›

section‹The Undefined Process›

theory     Bot
imports    Process
begin 

lift_definition BOT ::  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: " 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 constBOT .
›

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: "  = {(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