let s be State of SCMPDS ; for I being parahalting Program of SCMPDS holds
( not Initialized I c= s or CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = halt SCMPDS or IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = card I )
let I be parahalting Program of SCMPDS ; ( not Initialized I c= s or CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = halt SCMPDS or IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = card I )
set IsI = Initialized (stop I);
set ss = s +* (Initialized (stop I));
set m = LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)));
set s1 = Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))));
set s2 = Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))));
set Ik = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))));
A1:
ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I))
by FUNCT_4:26, SCMPDS_4:63;
reconsider n = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))) as Element of NAT ;
A2:
Initialized (stop I) c= s +* (Initialized (stop I))
by FUNCT_4:26;
then A3:
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))) in dom (stop I)
by SCMPDS_4:def 9;
XX:
( (Initialized (stop I)) +* (s +* (Initialized (stop I))) = s +* (Initialized (stop I)) & (s +* (Initialized (stop I))) +* (Initialized (stop I)) = s +* (Initialized (stop I)) )
by A2, FUNCT_4:79;
xx:
( (Initialized (stop I)) +* (s +* (Initialized (stop I))) = s +* (Initialized (stop I)) & (s +* (Initialized (stop I))) +* (Initialized (stop I)) = s +* (Initialized (stop I)) )
by A2, FUNCT_4:79;
card (stop I) = (card I) + 1
by LL, SCMPDS_4:45;
then
n < (card I) + 1
by A3, AFINSQ_1:70;
then A4:
n <= card I
by INT_1:20;
stop I c= Initialized (stop I)
by SCMPDS_4:9;
then A5:
stop I c= s +* (Initialized (stop I))
by A2, XBOOLE_1:1;
assume A6:
Initialized I c= s
; ( CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = halt SCMPDS or IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = card I )
then A7: IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) =
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))
by Th29, AMI_1:121
.=
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))))
by XX
;
A8:
I c= s
by A6, SCMPDS_4:57;
now per cases
( n < card I or n = card I )
by A4, XXREAL_0:1;
case
n < card I
;
halt SCMPDS = CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))then A9:
n in dom I
by AFINSQ_1:70;
Z:
(ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))) /. (IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))) = (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) . (IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))))
by AMI_1:150;
V:
(ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) /. (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) = (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))))
by AMI_1:150;
TX:
ProgramPart (s +* (Initialized (stop I))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))
by AMI_1:123;
thus halt SCMPDS =
CurInstr (ProgramPart (s +* (Initialized (stop I)))),
(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))
by A1, AMI_1:def 46
.=
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))))),
(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I)))))
by xx, TX
.=
(s +* (Initialized (stop I))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))))
by V, AMI_1:54
.=
(stop I) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))))
by A3, A5, GRFUNC_1:8
.=
I . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (ProgramPart ((s +* (Initialized (stop I))) +* (Initialized (stop I)))),((s +* (Initialized (stop I))) +* (Initialized (stop I))))))
by A9, SCMPDS_4:37
.=
s . (IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))))
by A8, A7, A9, GRFUNC_1:8
.=
CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),
(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))
by Z, AMI_1:54
;
verum end;
hence
( CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I)))))),(Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = halt SCMPDS or IC (Comput (ProgramPart s),s,(LifeSpan (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))))) = card I )
; verum