reserve x for set,
  k for Element of NAT;
reserve s for State of SCMPDS;
reserve d1,d2,d3,d4,d5 for Element of SCM-Data-Loc,
  k1,k2,k3,k4,k5,k6 for Integer;
reserve I for Instruction of SCMPDS;
reserve a,b,c for Int_position;

theorem
  k1 <> 0 implies goto k1 is non halting
proof
  assume
A1: k1<>0;
  set n=|.k1.|;
  reconsider loc=n+1 as Element of NAT;
  consider s be State of SCMPDS such that
A2: s.NAT=loc and
  for d being Int_position holds s.d = 0 by Th58;
  -n<=k1 by ABSVALUE:4;
  then 0-n<=k1;
  then
A3: (n+k1)*1>=0 by XREAL_1:20;
  ex m be Element of NAT st m=IC s & ICplusConst(s,k1)=|.m+k1.| by Def17;
  then
A4: Exec(goto k1, s).IC SCMPDS = |.n+k1+1.| by A2,Th1,Th51
    .= |.n+k1.|+|.1.| by A3,ABSVALUE:11
    .=|.(n+k1).|+1 by ABSVALUE:def 1
    .=(n+k1)+1 by A3,ABSVALUE:def 1
    .=n+1+k1;
  assume goto k1 is halting;
  then Exec(goto k1,s).IC SCMPDS = n+1 by A2,Th1;
  hence contradiction by A1,A4;
end;
