set WHL = while<0 F4(),F5(),F3();
set pWHL = stop (while<0 F4(),F5(),F3());
set s1 = (Initialize F2()) +* (stop (while<0 F4(),F5(),F3()));
set ps = ProgramPart F2();
set sI = (Initialize F2()) +* (stop F3());
set m1 = (LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2;
set s2 = (Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()));
set m2 = LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())));
I1:
(Initialize F2()) +* (stop (while<0 F4(),F5(),F3())) = F2() +* (Initialize (stop (while<0 F4(),F5(),F3())))
by SCMPDS_4:5;
I2:
(Initialize F2()) +* (stop F3()) = F2() +* (Initialize (stop F3()))
by SCMPDS_4:5;
I3:
(Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())) = (IExec F3(),F2()) +* (Initialize (stop (while<0 F4(),F5(),F3())))
by SCMPDS_4:5;
A6:
P1[ Dstate F2()]
by A4;
A7:
Initialize (stop F3()) c= (Initialize F2()) +* (stop F3())
by I2, FUNCT_4:26;
F3() is_closed_on F2()
by A2, A4, A5;
then A8:
F3() is_closed_on (Initialize F2()) +* (stop F3())
by SCMPDS_6:38;
F3() is_halting_on F2()
by A2, A4, A5;
then A9:
ProgramPart ((Initialize F2()) +* (stop F3())) halts_on (Initialize F2()) +* (stop F3())
by SCMPDS_6:def 3;
I4:
(Initialize ((Initialize F2()) +* (stop F3()))) +* (stop F3()) = ((Initialize F2()) +* (stop F3())) +* (Initialize (stop F3()))
by SCMPDS_4:5;
(Initialize F2()) +* (stop F3()) = (Initialize ((Initialize F2()) +* (stop F3()))) +* (stop F3())
by A7, I4, FUNCT_4:79;
then
ProgramPart ((Initialize ((Initialize F2()) +* (stop F3()))) +* (stop F3())) halts_on (Initialize ((Initialize F2()) +* (stop F3()))) +* (stop F3())
by A9;
then A10:
F3() is_halting_on (Initialize F2()) +* (stop F3())
by SCMPDS_6:def 3;
set Es = IExec F3(),F2();
set bj = DataLoc ((IExec F3(),F2()) . F4()),F5();
deffunc H1( State of SCMPDS ) -> Element of NAT = F1($1);
A11:
for t being State of SCMPDS st P1[ Dstate t] & t . F4() = F2() . F4() & t . (DataLoc (F2() . F4()),F5()) < 0 holds
( (IExec F3(),t) . F4() = t . F4() & F3() is_closed_on t & F3() is_halting_on t & H1( Dstate (IExec F3(),t)) < H1( Dstate t) & P1[ Dstate (IExec F3(),t)] )
by A5;
A12:
for t being State of SCMPDS st P1[ Dstate t] & H1( Dstate t) = 0 holds
t . (DataLoc (F2() . F4()),F5()) >= 0
by A3;
( ( H1(F2()) = H1(F2()) or P1[F2()] ) & while<0 F4(),F5(),F3() is_closed_on F2() & while<0 F4(),F5(),F3() is_halting_on F2() )
from SCMPDS_8:sch 1(A1, A12, A6, A11);
then A13:
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) halts_on (Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))
by SCMPDS_6:def 3;
deffunc H2( State of SCMPDS ) -> Element of NAT = F1($1);
A14:
(IExec F3(),F2()) . F4() = F2() . F4()
by A2, A4, A5;
then A15:
for t being State of SCMPDS st P1[ Dstate t] & H2( Dstate t) = 0 holds
t . (DataLoc ((IExec F3(),F2()) . F4()),F5()) >= 0
by A3;
A16:
for t being State of SCMPDS st P1[ Dstate t] & t . F4() = (IExec F3(),F2()) . F4() & t . (DataLoc ((IExec F3(),F2()) . F4()),F5()) < 0 holds
( (IExec F3(),t) . F4() = t . F4() & F3() is_closed_on t & F3() is_halting_on t & H2( Dstate (IExec F3(),t)) < H2( Dstate t) & P1[ Dstate (IExec F3(),t)] )
by A5, A14;
A17:
P1[ Dstate (IExec F3(),F2())]
by A2, A4, A5;
( ( H2( IExec F3(),F2()) = H2( IExec F3(),F2()) or P1[ IExec F3(),F2()] ) & while<0 F4(),F5(),F3() is_closed_on IExec F3(),F2() & while<0 F4(),F5(),F3() is_halting_on IExec F3(),F2() )
from SCMPDS_8:sch 1(A1, A15, A17, A16);
then A18:
ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) halts_on (Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))
by SCMPDS_6:def 3;
set s4 = Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1;
set i1 = F4(),F5() >=0_goto ((card F3()) + 2);
set i2 = goto (- ((card F3()) + 1));
set b = DataLoc (F2() . F4()),F5();
A19:
while<0 F4(),F5(),F3() = (F4(),F5() >=0_goto ((card F3()) + 2)) ';' (F3() ';' (goto (- ((card F3()) + 1))))
by SCMPDS_4:51;
set mI = LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()));
set s5 = Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())));
set l1 = (card F3()) + 1;
A20:
IC ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = 0
by SCMPDS_6:21;
A21: Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(0 + 1) =
Following (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),0 )
by AMI_1:14
.=
Following (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))
by AMI_1:13
.=
Exec (F4(),F5() >=0_goto ((card F3()) + 2)),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))
by A19, I1, SCMPDS_6:22
;
A22:
DataPart ((Initialize F2()) +* (stop F3())) = DataPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))
by FUNCT_7:134, SCMPDS_4:24;
now let a be
Int_position ;
((Initialize F2()) +* (stop F3())) . a = (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1) . athus ((Initialize F2()) +* (stop F3())) . a =
((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) . a
by A22, SCMPDS_4:23
.=
(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1) . a
by A21, SCMPDS_2:69
;
verum end;
then A23:
DataPart ((Initialize F2()) +* (stop F3())) = DataPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)
by SCMPDS_4:23;
set m3 = (LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1;
set s6 = Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1);
(card F3()) + 1 < (card F3()) + 2
by XREAL_1:8;
then A24:
(card F3()) + 1 in dom (while<0 F4(),F5(),F3())
by Th7;
thus
( F1(F2()) = F1(F2()) or P1[F2()] )
; IExec (while<0 F4(),F5(),F3()),F2() = IExec (while<0 F4(),F5(),F3()),(IExec F3(),F2())
set m0 = LifeSpan (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())));
A25: dom (ProgramPart F2()) =
(dom F2()) /\ NAT
by RELAT_1:90
.=
(({(IC SCMPDS )} \/ SCM-Data-Loc ) \/ NAT ) /\ NAT
by SCMPDS_4:19
.=
NAT
by XBOOLE_1:21
;
A26: (IExec F3(),F2()) | NAT =
((Result (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) +* (ProgramPart F2())) | NAT
by SCMPDS_4:def 8
.=
ProgramPart F2()
by A25, FUNCT_4:24
;
set s7 = Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1);
A27:
IExec F3(),F2() = (Result (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) +* (F2() | NAT )
by SCMPDS_4:def 8;
A28:
dom (ProgramPart F2()) = NAT
by COMPOS_1:34;
( while<0 F4(),F5(),F3() c= Initialize (stop (while<0 F4(),F5(),F3())) & Initialize (stop (while<0 F4(),F5(),F3())) c= (Initialize F2()) +* (stop (while<0 F4(),F5(),F3())) )
by I1, FUNCT_4:26, SCMPDS_6:17;
then A29:
while<0 F4(),F5(),F3() c= (Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))
by XBOOLE_1:1;
Shift F3(),1 c= while<0 F4(),F5(),F3()
by Lm2;
then
Shift F3(),1 c= (Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))
by A29, XBOOLE_1:1;
then A30:
Shift F3(),1 c= Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1
by AMI_1:81;
((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) . (DataLoc (((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) . F4()),F5()) =
((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) . (DataLoc (F2() . F4()),F5())
by SCMPDS_5:19
.=
F2() . (DataLoc (F2() . F4()),F5())
by SCMPDS_5:19
;
then A31: IC (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1) =
succ (IC ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))))
by A2, A21, SCMPDS_2:69
.=
0 + 1
by A20
;
then A32:
IC (Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))) = (card F3()) + 1
by A1, A7, A10, A8, A23, A30, SCMPDS_7:36;
Y:
(ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))) /. (IC (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))) = (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1)) . (IC (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1)))
by COMPOS_1:38;
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)
by AMI_1:123;
then A33:
Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) = Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))
by AMI_1:51;
then A34: CurInstr (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1)) =
(Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))) . ((card F3()) + 1)
by A1, A7, A10, A8, A31, A23, A30, Y, SCMPDS_7:36
.=
(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1) . ((card F3()) + 1)
by AMI_1:54
.=
((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) . ((card F3()) + 1)
by AMI_1:54
.=
(while<0 F4(),F5(),F3()) . ((card F3()) + 1)
by A24, A29, GRFUNC_1:8
.=
goto (- ((card F3()) + 1))
by Th8
;
T:
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))
by AMI_1:123;
A35: Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1) =
Following (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))
by AMI_1:14
.=
Exec (goto (- ((card F3()) + 1))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1))
by A34, T
;
then IC (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1)) =
ICplusConst (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1)),(0 - ((card F3()) + 1))
by SCMPDS_2:66
.=
0
by A32, A33, SCMPDS_7:1
;
then A36:
IC ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) = IC (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2))
by SCMPDS_6:21;
A37:
DataPart (Comput (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))) = DataPart (Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))))
by A1, A7, A10, A8, A31, A23, A30, SCMPDS_7:36;
now let x be
Int_position ;
(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1)) . x = ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) . xA38:
not
x in dom (Initialize (stop (while<0 F4(),F5(),F3())))
by SCMPDS_4:31;
A39:
not
x in dom (F2() | NAT )
by A28, SCMPDS_2:53;
(Comput (ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1)),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),1),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))) . x =
(Comput (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3())))) . x
by A37, SCMPDS_4:23
.=
(Result (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) . x
by A9, AMI_1:122
.=
(IExec F3(),F2()) . x
by A27, A39, FUNCT_4:12
;
hence (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1)) . x =
(IExec F3(),F2()) . x
by A33, A35, SCMPDS_2:66
.=
((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) . x
by A38, I3, FUNCT_4:12
;
verum end;
then A40:
DataPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 1) + 1)) = DataPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))
by SCMPDS_4:23;
TX:
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2))
by AMI_1:123;
ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) =
ProgramPart (((Result (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) +* (ProgramPart F2())) +* (Initialize (stop (while<0 F4(),F5(),F3()))))
by I3, SCMPDS_4:def 8
.=
(ProgramPart ((Result (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) +* (ProgramPart F2()))) +* (ProgramPart (Initialize (stop (while<0 F4(),F5(),F3()))))
by FUNCT_4:75
.=
(ProgramPart F2()) +* (ProgramPart (Initialize (stop (while<0 F4(),F5(),F3()))))
by A25, FUNCT_4:24
.=
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))
by I1, FUNCT_4:75
.=
ProgramPart (Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2))
by AMI_1:123
;
then A41:
Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2) = (Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))
by A40, A36, SCMPDS_7:7;
then
CurInstr (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2)) = F4(),F5() >=0_goto ((card F3()) + 2)
by A19, TX, I3, SCMPDS_6:22;
then
LifeSpan (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) > (LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2
by A13, SCMPDS_6:2, SCMPDS_6:31;
then consider nn being Nat such that
A42:
LifeSpan (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = ((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2) + nn
by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
T:
ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))
by A41, AMI_1:123;
Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2) + (LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))))) = Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))),(LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))))
by A41, AMI_1:51;
then
CurInstr (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),(Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2) + (LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))))) = halt SCMPDS
by A18, T, AMI_1:def 46;
then
((LifeSpan (ProgramPart ((Initialize F2()) +* (stop F3()))),((Initialize F2()) +* (stop F3()))) + 2) + (LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))) >= LifeSpan (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))
by A13, AMI_1:def 46;
then A43:
LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))) >= nn
by A42, XREAL_1:8;
A44:
Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))),(LifeSpan (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))) = Comput (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))),nn
by A41, A42, AMI_1:51;
then
CurInstr (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),(Comput (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))),nn) = halt SCMPDS
by A13, T, AMI_1:def 46;
then
nn >= LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))
by A18, AMI_1:def 46;
then
nn = LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))
by A43, XXREAL_0:1;
then
Result (ProgramPart ((Initialize F2()) +* (stop (while<0 F4(),F5(),F3())))),((Initialize F2()) +* (stop (while<0 F4(),F5(),F3()))) = Comput (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))),(LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))))
by A13, A44, T, AMI_1:122;
hence IExec (while<0 F4(),F5(),F3()),F2() =
(Comput (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))),(LifeSpan (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3()))))) +* (ProgramPart F2())
by SCMPDS_4:def 8
.=
(Result (ProgramPart ((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))),((Initialize (IExec F3(),F2())) +* (stop (while<0 F4(),F5(),F3())))) +* ((IExec F3(),F2()) | NAT )
by A18, A26, AMI_1:122
.=
IExec (while<0 F4(),F5(),F3()),(IExec F3(),F2())
by SCMPDS_4:def 8
;
verum