II: Initialize F₂() = F₂() by MEMSTR_0:44;
set WHL = while<0 (F₅(),F₆(),F₄());
set pWHL = stop (while<0 (F₅(),F₆(),F₄()));
set P1 = F₃() +* (stop (while<0 (F₅(),F₆(),F₄())));
set PI = F₃() +* (stop F₄());
set m1 = (LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2;
set s2 = Initialize (IExec (F₄(),F₃(),F₂()));
set P2 = F₃() +* (stop (while<0 (F₅(),F₆(),F₄())));
set m2 = LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))));
A9: P₁[F₂()] by A4;
A10: stop F₄() c= F₃() +* (stop F₄()) by FUNCT_4:25;
A11: F₄() is_closed_on F₂(),F₃() +* (stop F₄()) by A2, A4, A5;
F₄() is_halting_on F₂(),F₃() by A2, A4, A5;
then A12: F₃() +* (stop F₄()) halts_on F₂() by SCMPDS_6:def 3, II;
(F₃() +* (stop F₄())) +* (stop F₄()) halts_on F₂() by A12, FUNCT_4:25, FUNCT_4:98;
then A14: F₄() is_halting_on F₂(),F₃() +* (stop F₄()) by SCMPDS_6:def 3, II;
set Es = IExec (F₄(),F₃(),F₂());
set bj = DataLoc (((Initialize (IExec (F₄(),F₃(),F₂()))) . F₅()),F₆());
set EP = F₃();
A15: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P₁[t] & t . F₅() = F₂() . F₅() & t . (DataLoc ((F₂() . F₅()),F₆())) < 0 holds
( (IExec (F₄(),Q,t)) . F₅() = t . F₅() & F₄() is_closed_on t,Q & F₄() is_halting_on t,Q & F₁((Initialize (IExec (F₄(),Q,t)))) < F₁(t) & P₁[ Initialize (IExec (F₄(),Q,t))] ) by A5;
A16: for t being 0 -started State of SCMPDS st P₁[t] & F₁(t) = 0 holds
t . (DataLoc ((F₂() . F₅()),F₆())) >= 0 by A3;
( while<0 (F₅(),F₆(),F₄()) is_closed_on F₂(),F₃() & while<0 (F₅(),F₆(),F₄()) is_halting_on F₂(),F₃() ) from SCMPDS_8:sch 1(A16, A9, A15);
then A17: F₃() +* (stop (while<0 (F₅(),F₆(),F₄()))) halts_on F₂() by SCMPDS_6:def 3, II;
B18: (IExec (F₄(),F₃(),F₂())) . F₅() = (Initialize (IExec (F₄(),F₃(),F₂()))) . F₅() by SCMPDS_5:15;
A18: (IExec (F₄(),F₃(),F₂())) . F₅() = F₂() . F₅() by A2, A4, A5;
then A19: for t being 0 -started State of SCMPDS st P₁[t] & F₁(t) = 0 holds
t . (DataLoc (((Initialize (IExec (F₄(),F₃(),F₂()))) . F₅()),F₆())) >= 0 by A3, B18;
A20: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P₁[t] & t . F₅() = (Initialize (IExec (F₄(),F₃(),F₂()))) . F₅() & t . (DataLoc (((Initialize (IExec (F₄(),F₃(),F₂()))) . F₅()),F₆())) < 0 holds
( (IExec (F₄(),Q,t)) . F₅() = t . F₅() & F₄() is_closed_on t,Q & F₄() is_halting_on t,Q & F₁((Initialize (IExec (F₄(),Q,t)))) < F₁(t) & P₁[ Initialize (IExec (F₄(),Q,t))] ) by A5, A18, B18;
A21: P₁[ Initialize (IExec (F₄(),F₃(),F₂()))] by A2, A4, A5;
( while<0 (F₅(),F₆(),F₄()) is_closed_on Initialize (IExec (F₄(),F₃(),F₂())),F₃() & while<0 (F₅(),F₆(),F₄()) is_halting_on Initialize (IExec (F₄(),F₃(),F₂())),F₃() ) from SCMPDS_8:sch 1(A19, A21, A20);
then A22: F₃() +* (stop (while<0 (F₅(),F₆(),F₄()))) halts_on Initialize (Initialize (IExec (F₄(),F₃(),F₂()))) by SCMPDS_6:def 3;
set s4 = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1);
set P4 = F₃() +* (stop (while<0 (F₅(),F₆(),F₄())));
set i1 = (F₅(),F₆()) >=0_goto ((card F₄()) + 2);
set i2 = goto (- ((card F₄()) + 1));
set b = DataLoc ((F₂() . F₅()),F₆());
A23: while<0 (F₅(),F₆(),F₄()) = ((F₅(),F₆()) >=0_goto ((card F₄()) + 2)) ';' (F₄() ';' (goto (- ((card F₄()) + 1)))) by SCMPDS_4:15;
set mI = LifeSpan ((F₃() +* (stop F₄())),F₂());
set s5 = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)),(LifeSpan ((F₃() +* (stop F₄())),F₂())));
set P5 = F₃() +* (stop (while<0 (F₅(),F₆(),F₄())));
set l1 = (card F₄()) + 1;
A24: IC F₂() = 0 by MEMSTR_0:def 8;
A25: Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(0 + 1)) = Following ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),0))) by EXTPRO_1:3
.= Following ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()) by EXTPRO_1:2
.= Exec (((F₅(),F₆()) >=0_goto ((card F₄()) + 2)),F₂()) by A23, SCMPDS_6:11, II ;
for a being Int_position holds F₂() . a = (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)) . a by A25, SCMPDS_2:57;
then A27: DataPart F₂() = DataPart (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)) by SCMPDS_4:8;
set m3 = (LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1;
set s6 = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1));
set P6 = F₃() +* (stop (while<0 (F₅(),F₆(),F₄())));
(card F₄()) + 1 < (card F₄()) + 2 by XREAL_1:6;
then A28: (card F₄()) + 1 in dom (while<0 (F₅(),F₆(),F₄())) by Th7;
set m0 = LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂());
set s7 = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1));
XX: while<0 (F₅(),F₆(),F₄()) c= stop (while<0 (F₅(),F₆(),F₄())) by AFINSQ_1:74;
stop (while<0 (F₅(),F₆(),F₄())) c= F₃() +* (stop (while<0 (F₅(),F₆(),F₄()))) by FUNCT_4:25;
then A33: while<0 (F₅(),F₆(),F₄()) c= F₃() +* (stop (while<0 (F₅(),F₆(),F₄()))) by XX, XBOOLE_1:1;
Shift (F₄(),1) c= while<0 (F₅(),F₆(),F₄()) by Lm2;
then A34: Shift (F₄(),1) c= F₃() +* (stop (while<0 (F₅(),F₆(),F₄()))) by A33, XBOOLE_1:1;
A35: IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)) = succ (IC F₂()) by A2, A25, SCMPDS_2:57
.= 0 + 1 by A24 ;
then A36: IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)),(LifeSpan ((F₃() +* (stop F₄())),F₂())))) = (card F₄()) + 1 by A10, A14, A11, A27, A34, SCMPDS_7:18;
A37: (F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))) /. (IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)))) = (F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))) . (IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)))) by PBOOLE:143;
A38: Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)) = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)),(LifeSpan ((F₃() +* (stop F₄())),F₂()))) by EXTPRO_1:4;
then A39: CurInstr ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)))) = (F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))) . ((card F₄()) + 1) by A10, A14, A11, A35, A27, A34, A37, SCMPDS_7:18
.= (while<0 (F₅(),F₆(),F₄())) . ((card F₄()) + 1) by A28, A33, GRFUNC_1:2
.= goto (- ((card F₄()) + 1)) by Th8 ;
A41: Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1)) = Following ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F₄()) + 1))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1)))) by A39 ;
then IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1))) = ICplusConst ((Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1))),(0 - ((card F₄()) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: IC (Initialize (IExec (F₄(),F₃(),F₂()))) = IC (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2))) by MEMSTR_0:def 8;
A43: DataPart (Comput ((F₃() +* (stop F₄())),F₂(),(LifeSpan ((F₃() +* (stop F₄())),F₂())))) = DataPart (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)),(LifeSpan ((F₃() +* (stop F₄())),F₂())))) by A10, A14, A11, A35, A27, A34, SCMPDS_7:18;

now

let x be Int_position ; :: thesis: (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1))) . x = (Initialize (IExec (F₄(),F₃(),F₂()))) . x
A44: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),1)),(LifeSpan ((F₃() +* (stop F₄())),F₂())))) . x = (Comput ((F₃() +* (stop F₄())),F₂(),(LifeSpan ((F₃() +* (stop F₄())),F₂())))) . x by A43, SCMPDS_4:8
.= (Result ((F₃() +* (stop F₄())),F₂())) . x by A12, EXTPRO_1:23
.= (IExec (F₄(),F₃(),F₂())) . x by SCMPDS_4:def 5 ;
hence (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1))) . x = (IExec (F₄(),F₃(),F₂())) . x by A38, A41, SCMPDS_2:54
.= (Initialize (IExec (F₄(),F₃(),F₂()))) . x by A44, FUNCT_4:11 ;
:: thesis: verum

end;

then A46: DataPart (Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 1) + 1))) = DataPart (Initialize (IExec (F₄(),F₃(),F₂()))) by SCMPDS_4:8;
A48: Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2)) = Initialize (IExec (F₄(),F₃(),F₂())) by A46, A42, MEMSTR_0:78;
then CurInstr ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2)))) = (F₅(),F₆()) >=0_goto ((card F₄()) + 2) by A23, SCMPDS_6:11;
then LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()) > (LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2 by A17, EXTPRO_1:36, SCMPDS_6:18;
then consider nn being Nat such that
A49: LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()) = ((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 12;
Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2) + (LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))))))) = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))),(LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))))) by A48, EXTPRO_1:4;
then CurInstr ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2) + (LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))))))))) = halt SCMPDS by A22, EXTPRO_1:def 15;
then ((LifeSpan ((F₃() +* (stop F₄())),F₂())) + 2) + (LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))))) >= LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()) by A17, EXTPRO_1:def 15;
then A51: LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))) >= nn by A49, XREAL_1:6;
A52: Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂(),(LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()))) = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))),nn) by A48, A49, EXTPRO_1:4;
then CurInstr ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))),nn))) = halt SCMPDS by A17, EXTPRO_1:def 15;
then nn >= LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))) by A22, EXTPRO_1:def 15;
then nn = LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))) by A51, XXREAL_0:1;
then Result ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),F₂()) = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))),(LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))))) by A17, A52, EXTPRO_1:23;
hence IExec ((while<0 (F₅(),F₆(),F₄())),F₃(),F₂()) = Comput ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂()))),(LifeSpan ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))))) by SCMPDS_4:def 5
.= Result ((F₃() +* (stop (while<0 (F₅(),F₆(),F₄())))),(Initialize (IExec (F₄(),F₃(),F₂())))) by A22, EXTPRO_1:23
.= IExec ((while<0 (F₅(),F₆(),F₄())),F₃(),(Initialize (IExec (F₄(),F₃(),F₂())))) by SCMPDS_4:def 5 ;
:: thesis: verum