let ss be State of SCM+FSA; :: thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st ss . (intloc 0) = 1 & ss . a <> 0 & Directed J is_pseudo-closed_on ss holds
DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss))
set D = Int-Locations \/ FinSeq-Locations;
set A = NAT ;
let I, J be Program of SCM+FSA; :: thesis: for a being read-write Int-Location st ss . (intloc 0) = 1 & ss . a <> 0 & Directed J is_pseudo-closed_on ss holds
DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss))
let a be read-write Int-Location ; :: thesis: ( ss . (intloc 0) = 1 & ss . a <> 0 & Directed J is_pseudo-closed_on ss implies DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss)) )
set J0 = Directed J;
set s = Initialized ss;
set s0 = Initialized (Initialized ss);
set J9 = J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)));
set s00 = (Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)));
set s3 = (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)));
set s4 = Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1);
set s5 = Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2);
set i = a =0_goto ((card J) + 3);
assume A1: ss . (intloc 0) = 1 ; :: thesis: ( not ss . a <> 0 or not Directed J is_pseudo-closed_on ss or DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss)) )
set s1 = (Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)));
assume ss . a <> 0 ; :: thesis: ( not Directed J is_pseudo-closed_on ss or DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss)) )
then (Initialized ss) . a <> 0 by SCMFSA6C:3;
then A2: (Initialized (Initialized ss)) . a <> 0 by SCMFSA6C:3;
A3: 0 in dom (if=0 (a,I,J)) by Th54;
if=0 (a,I,J) c= (if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then dom (if=0 (a,I,J)) c= dom ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
then A4: ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 0 = ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) . 0 by A3, FUNCT_4:14
.= (if=0 (a,I,J)) . 0 by A3, SCMFSA6B:7
.= a =0_goto ((card J) + 3) by Th55 ;
(Initialized ss) +* (Initialized (if=0 (a,I,J))) = (Initialized (Initialized ss)) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
then A5: (Initialized ss) +* (Initialized (if=0 (a,I,J))) = (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by Th15;
A6: if=0 (a,I,J) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26, SCMFSA6B:5;
if=0 (a,I,J) c= (if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A7: dom (if=0 (a,I,J)) c= dom ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
A8: (Directed J) ';' (Stop SCM+FSA) = J ';' (Stop SCM+FSA) by SCMFSA8A:41;
IC SCM+FSA in dom ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
then A9: IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = IC ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:14
.= 0 by COMPOS_1:142 ;
if=0 (a,I,J) = ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) ';' (Stop SCM+FSA) by SCMFSA8B:def 1;
then if=0 (a,I,J) = (((Macro (a =0_goto ((card J) + 3))) ';' J) ';' (Goto ((card I) + 1))) ';' (I ';' (Stop SCM+FSA)) by SCMFSA6A:67;
then if=0 (a,I,J) = ((Macro (a =0_goto ((card J) + 3))) ';' J) ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA))) by SCMFSA6A:67;
then A10: if=0 (a,I,J) = (Macro (a =0_goto ((card J) + 3))) ';' (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))) by SCMFSA6A:67;
card (Macro (a =0_goto ((card J) + 3))) = 2 by COMPOS_1:150;
then A11: ProgramPart (Relocated ((J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))),2)) c= if=0 (a,I,J) by A10, FUNCT_4:26;
A12: if=0 (a,I,J) c= (if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
(if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
then if=0 (a,I,J) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by A12, XBOOLE_1:1;
then ProgramPart (Relocated ((J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))),2)) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by A11, XBOOLE_1:1;
then ProgramPart (ProgramPart (Relocated ((J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))),2))) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2) by AMI_1:99;
then A13: ProgramPart (Relocated ((J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))),2)) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2) by RELAT_1:209;
ProgramPart (Relocated ((Directed J),2)) c= ProgramPart (Relocated ((J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA)))),2)) by Th12, SCMFSA6A:55;
then A14: ProgramPart (Relocated ((Directed J),2)) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2) by A13, XBOOLE_1:1;
Y: (ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) /. (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) by COMPOS_1:38;
A15: Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(0 + 1)) = Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) by EXTPRO_1:3
.= Exec ((a =0_goto ((card J) + 3)),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) by A9, A4, Y ;
assume Directed J is_pseudo-closed_on ss ; :: thesis: DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((J ';' (Stop SCM+FSA)),ss))
then A16: Directed J is_pseudo-closed_on Initialized ss by A1, Th53;
then J ';' (Stop SCM+FSA) is_halting_on Initialized ss by Th58;
then A17: ProgramPart ((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) halts_on (Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by SCMFSA7B:def 8;
DataPart (Initialized ss) = DataPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) by SCMFSA8A:11;
then A18: Directed J is_pseudo-closed_on (Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))) by A16, Th52;
(Initialized ss) +* (Initialized (J ';' (Stop SCM+FSA))) = (Initialized (Initialized ss)) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
then A19: (Initialized ss) +* (Initialized (J ';' (Stop SCM+FSA))) = (Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by Th15;
A20: Directed (Directed J) = Directed J by SCMFSA6A:63;
card (if=0 (a,I,J)) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:14
.= ((card J) + 2) + ((card I) + 2) ;
then ((card J) + 2) + 0 < card (if=0 (a,I,J)) by XREAL_1:10;
then A21: (card J) + 2 in dom (if=0 (a,I,J)) by AFINSQ_1:70;
card (if=0 (a,I,J)) = ((card I) + (card J)) + (3 + 1) by SCMFSA8B:14
.= (((card I) + (card J)) + 3) + 1 ;
then ((card I) + (card J)) + 3 < card (if=0 (a,I,J)) by NAT_1:13;
then A22: ((card I) + (card J)) + 3 in dom (if=0 (a,I,J)) by AFINSQ_1:70;
A23: if=0 (a,I,J) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26, SCMFSA6B:5;
set s9 = Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2));
A24: (Directed J) +* (Start-At (0,SCM+FSA)) c= (Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
LifeSpan ((ProgramPart ((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))) = pseudo-LifeSpan ((Initialized ss),(Directed J)) by A16, Th58;
then A25: DataPart (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed J))))) = DataPart (Comput ((ProgramPart ((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))))) by A16, A20, A8, Th58;
A26: 1 in dom (if=0 (a,I,J)) by Th54;
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a = (Initialized ss) . a by Th28
.= (Initialized (Initialized ss)) . a by SCMFSA6C:3 ;
then A27: IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) = succ (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) by A2, A15, SCMFSA_2:96
.= 0 + 1 by A9 ;
A28: (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . 1 = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 1 by AMI_1:54
.= ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) . 1 by A26, A7, FUNCT_4:14
.= (if=0 (a,I,J)) . 1 by A26, SCMFSA6B:7
.= goto 2 by Th55 ;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))) by COMPOS_1:38;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) by AMI_1:123;
A29: Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(1 + 1)) = Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))) by EXTPRO_1:4
.= Exec ((goto 2),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))) by A27, A28, Y, T ;
then A30: IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) = 2 by SCMFSA_2:95;
A31: DataPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) = DataPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) by SCMFSA6A:39, SCMFSA8A:14;

A32: now

let f be FinSeq-Location ; :: thesis: ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) . f = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) . f
thus ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) . f = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . f by A31, SCMFSA6A:38
.= (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . f by A15, SCMFSA_2:96
.= (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) . f by A29, SCMFSA_2:95 ; :: thesis: verum

end;

now

let a be Int-Location ; :: thesis: ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) . a = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) . a
thus ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) . a = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a by A31, SCMFSA6A:38
.= (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . a by A15, SCMFSA_2:96
.= (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) . a by A29, SCMFSA_2:95 ; :: thesis: verum

end;

then A33: DataPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))) = DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) by A32, SCMFSA6A:38;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) by COMPOS_1:38;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) by AMI_1:123;
A34: IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) = IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)),(pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))))) by EXTPRO_1:5
.= (IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J)))))) + 2 by A24, A18, A14, A30, A33, T, Th51
.= (IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed J)))))) + 2 by A16, Th50
.= (card (ProgramPart (Directed J))) + 2 by A16, SCMFSA8A:def 5
.= (card (Directed J)) + 2 by RELAT_1:209
.= (card J) + 2 by SCMFSA8A:34 ;
then A35: CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . ((card J) + 2) by Y, AMI_1:54
.= (if=0 (a,I,J)) . ((card J) + 2) by A23, A21, GRFUNC_1:8
.= goto (((card I) + (card J)) + 3) by Th63 ;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1)))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1)))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1)))) by COMPOS_1:38;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) by AMI_1:123;
TX3: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1))) by AMI_1:123;
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1))) = IC (Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))))) by EXTPRO_1:4
.= ((card I) + (card J)) + 3 by A35, T, SCMFSA_2:95 ;
then A36: CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1)))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (((card I) + (card J)) + 3) by Y, TX3, AMI_1:54
.= (if=0 (a,I,J)) . (((card I) + (card J)) + 3) by A6, A22, GRFUNC_1:8
.= halt SCM+FSA by Th61 ;
then A37: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) halts_on (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by EXTPRO_1:30;
Y: (ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) /. (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) by COMPOS_1:38;
A38: CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = a =0_goto ((card J) + 3) by A9, A4, Y;

now

A39: 0 + 2 < ((card I) + (card J)) + 3 by XREAL_1:10;
then A40: 2 in dom (if=0 (a,I,J)) by Th56;
A41: (if=0 (a,I,J)) . 2 <> halt SCM+FSA by A39, Th56;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2))) by COMPOS_1:38;
TX2: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) by AMI_1:123;
A42: CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 2 by A30, Y, TX2, AMI_1:54
.= (if=0 (a,I,J)) . 2 by A40, Th26 ;
let k be Element of NAT ; :: thesis: ( CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = halt SCM+FSA implies (pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + (1 + 2) <= k )
TX3: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) by AMI_1:123;
assume A43: CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = halt SCM+FSA ; :: thesis: (pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + (1 + 2) <= k
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) by COMPOS_1:38;
u: Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0) = (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by EXTPRO_1:3;
CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0))) = a =0_goto ((card J) + 3) by A38, u;
then A44: k <> 0 by A43;
xx: k <> 1 by A27, A28, A43, Y, TX3;
2 <> k by A43, A41, A42;
then 2 < k by A44, xx, NAT_1:27;
then consider k2 being Nat such that
A45: 2 + k2 = k by NAT_1:10;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),k2)) as Element of NAT ;
assume not (pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + (1 + 2) <= k ; :: thesis: contradiction
then k < ((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 1) + 2 ;
then k2 < (pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 1 by A45, XREAL_1:8;
then A46: k2 <= pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J)) by NAT_1:13;
then A47: k2 <= pseudo-LifeSpan ((Initialized ss),(Directed J)) by A16, Th50;

A48: now

per cases ( k2 = pseudo-LifeSpan ((Initialized ss),(Directed J)) or k2 < pseudo-LifeSpan ((Initialized ss),(Directed J)) ) by A47, XXREAL_0:1;

suppose A49: k2 = pseudo-LifeSpan ((Initialized ss),(Directed J)) ; :: thesis: n + 2 < ((card I) + (card J)) + 3

((card I) + (card J)) + (2 + 1) = (((card J) + 2) + 1) + (card I) ;
then A50: ((card J) + 2) + 1 <= ((card I) + (card J)) + 3 by NAT_1:11;
IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),k2)) = card (ProgramPart (Directed J)) by A16, A49, SCMFSA8A:def 5
.= card (Directed J) by RELAT_1:209 ;
then n = card J by SCMFSA8A:34;
hence n + 2 < ((card I) + (card J)) + 3 by A50, NAT_1:13; :: thesis: verum

end;

suppose k2 < pseudo-LifeSpan ((Initialized ss),(Directed J)) ; :: thesis: n + 2 < ((card I) + (card J)) + 3

then n in dom (Directed J) by A16, SCMFSA8A:31;
then n < card (Directed J) by AFINSQ_1:70;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:8;
then A51: n + 2 < (card J) + 2 by SCMFSA8A:34;
((card I) + (card J)) + (1 + 2) = ((card J) + 2) + ((card I) + 1) ;
then (card J) + 2 <= ((card I) + (card J)) + 3 by NAT_1:11;
hence n + 2 < ((card I) + (card J)) + 3 by A51, XXREAL_0:2; :: thesis: verum

end;

end;

end;

then A52: n + 2 in dom (if=0 (a,I,J)) by Th56;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) by AMI_1:123;
A53: IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)),k2)) by A45, EXTPRO_1:5
.= n + 2 by A24, A18, A14, A30, A33, A46, Th51, T ;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) by COMPOS_1:38;
TX3: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) by AMI_1:123;
CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) by Y, TX3, AMI_1:54
.= (if=0 (a,I,J)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) by A53, A52, Th26 ;
hence contradiction by A43, A53, A48, Th56; :: thesis: verum

end;

then A54: LifeSpan ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = ((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2) + 1 by A36, A37, EXTPRO_1:def 14;
Y: (ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) by COMPOS_1:38;
CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . ((card J) + 2) by A34, Y, AMI_1:54
.= (if=0 (a,I,J)) . ((card J) + 2) by A21, Th26
.= goto (((card I) + (card J)) + 3) by Th63 ;
then InsCode (CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))))) = 6 by SCMFSA_2:47;
then InsCode (CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))))) in {0,6,7,8} by ENUMSET1:def 2;
then A55: DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) = DataPart (Following ((ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2)))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))))) by Th32;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)) by AMI_1:123;
A56: DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) = DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),2)),(pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))))) by EXTPRO_1:5
.= DataPart (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))))) by A24, A18, A14, A30, T, A33, Th51 ;
T: ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))) by AMI_1:123;
thus DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((if=0 (a,I,J)),(Initialized ss))) by Th17
.= DataPart ((Result ((ProgramPart ((Initialized ss) +* (Initialized (if=0 (a,I,J))))),((Initialized ss) +* (Initialized (if=0 (a,I,J)))))) +* ((Initialized ss) | NAT)) by SCMFSA6B:def 1
.= DataPart (Result ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))) by A5, Th35
.= DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))))) by A37, EXTPRO_1:23
.= DataPart (Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(Directed J))) + 2))))) by A54, EXTPRO_1:4
.= DataPart (Comput ((ProgramPart ((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed J) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed J))))) by A16, A56, A55, Th50, T
.= DataPart (Result ((ProgramPart ((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((J ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) by A17, A25, EXTPRO_1:23
.= DataPart ((Result ((ProgramPart ((Initialized ss) +* (Initialized (J ';' (Stop SCM+FSA))))),((Initialized ss) +* (Initialized (J ';' (Stop SCM+FSA)))))) +* ((Initialized ss) | NAT)) by A19, Th35
.= DataPart (IExec ((J ';' (Stop SCM+FSA)),(Initialized ss))) by SCMFSA6B:def 1
.= DataPart (IExec ((J ';' (Stop SCM+FSA)),ss)) by Th17 ; :: thesis: verum