
theorem
  for PTN being Petri_net, Sd being non empty Subset of the carrier of
PTN holds Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0.:Sd
= {FALSE} for t being transition of PTN st t is_firable_on M0 holds Firing(t,M0
  ).:Sd = {FALSE}
proof
  let PTN be Petri_net, Sd be non empty Subset of the carrier of PTN;
  thus Sd is Deadlock-like implies for M0 being Boolean_marking of PTN st M0.:
Sd = {FALSE} for t being transition of PTN st t is_firable_on M0 holds Firing(t
  ,M0).:Sd = {FALSE}
  proof
    assume Sd is Deadlock-like;
    then
A1: *'Sd is Subset of Sd*';
    let M0 be Boolean_marking of PTN such that
A2: M0.:Sd = {FALSE};
    let t be transition of PTN;
    assume t is_firable_on M0;
    then M0.:*'{t} c= {TRUE};
    then
A3: M0.:*'{t} misses {FALSE} by XBOOLE_1:63,ZFMISC_1:11;
    then *'{t} misses Sd by A2,Th2;
    then not t in *'Sd by A1,PETRI:19;
    then {t}*' misses Sd by PETRI:20;
    hence Firing(t,M0).:Sd = (M0 +* (*'{t}-->FALSE)).:Sd by Th3
      .= {FALSE} by A2,A3,Th2,Th3;
  end;
  thus thesis by Lm1;
end;
