let ss be State of SCM+FSA ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
set A = NAT ;
let I, J be Program of SCM+FSA ; :: thesis:
let a be read-write Int-Location ; :: thesis:
set J0 = Directed J;
set s = Initialize ss;
set s0 = Initialize (Initialize ss);
set J9 = J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )));
set s00 = (Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )));
set s3 = (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )));
set s4 = Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1;
set s5 = Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2;
set i = a =0_goto (insloc ((card J) + 3));
assume A1: ss . (intloc 0 ) = 1 ; :: thesis:
set s1 = (Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )));
assume ss . a <> 0 ; :: thesis:
then (Initialize ss) . a <> 0 by SCMFSA6C:3;
then A2: (Initialize (Initialize ss)) . a <> 0 by SCMFSA6C:3;
A3: insloc 0 in dom (if=0 a,I,J) by Th54;
if=0 a,I,J c= (if=0 a,I,J) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then dom (if=0 a,I,J) c= dom ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
then A4: ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 0 ) = ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (insloc 0 ) by A3, FUNCT_4:14
.= (if=0 a,I,J) . (insloc 0 ) by A3, SCMFSA6B:7
.= a =0_goto (insloc ((card J) + 3)) by Th55 ;
(Initialize ss) +* (Initialized (if=0 a,I,J)) = (Initialize (Initialize ss)) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
then A5: (Initialize ss) +* (Initialized (if=0 a,I,J)) = (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by Th15;
A6: if=0 a,I,J c= (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26, SCMFSA6B:5;
if=0 a,I,J c= (if=0 a,I,J) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A7: dom (if=0 a,I,J) c= dom ((if=0 a,I,J) +* (Start-At (insloc 0 ))) 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 (insloc 0 ))) by SF_MASTR:65;
then A9: IC ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by FUNCT_4:14
.= insloc 0 by SF_MASTR:66 ;
if=0 a,I,J = ((((a =0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I) ';' (Stop SCM+FSA ) by SCMFSA8B:def 1;
then if=0 a,I,J = (((Macro (a =0_goto (insloc ((card J) + 3)))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' (I ';' (Stop SCM+FSA )) by SCMFSA6A:67;
then if=0 a,I,J = ((Macro (a =0_goto (insloc ((card J) + 3)))) ';' J) ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA ))) by SCMFSA6A:67;
then A10: if=0 a,I,J = (Macro (a =0_goto (insloc ((card J) + 3)))) ';' (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))) by SCMFSA6A:67;
card (Macro (a =0_goto (insloc ((card J) + 3)))) = 2 by SCMFSA7B:6;
then A11: ProgramPart (Relocated (J ';' ((Goto (insloc ((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 (insloc 0 )) by SCMFSA8A:9;
(if=0 a,I,J) +* (Start-At (insloc 0 )) c= (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then if=0 a,I,J c= (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by A12, XBOOLE_1:1;
then ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) c= (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by A11, XBOOLE_1:1;
then ProgramPart [(ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2))] c= Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by AMI_1:99;
then A13: ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) c= Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by AMI_1:105;
ProgramPart (Relocated (Directed J),2) c= ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) by Th12, SCMFSA6A:55;
then A14: ProgramPart (Relocated (Directed J),2) c= Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by A13, XBOOLE_1:1;
A15: Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(0 + 1) = Following (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),0 ) by AMI_1:14
.= Following ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by AMI_1:13
.= Exec (a =0_goto (insloc ((card J) + 3))),((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by A9, A4 ;
assume Directed J is_pseudo-closed_on ss ; :: thesis:
then A16: Directed J is_pseudo-closed_on Initialize ss by A1, Th53;
then J ';' (Stop SCM+FSA ) is_halting_on Initialize ss by Th58;
then A17: (Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) is halting by SCMFSA7B:def 8;
DataPart (Initialize ss) = DataPart ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))) by SCMFSA8A:11;
then A18: Directed J is_pseudo-closed_on (Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 ))) by A16, Th52;
(Initialize ss) +* (Initialized (J ';' (Stop SCM+FSA ))) = (Initialize (Initialize ss)) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
then A19: (Initialize ss) +* (Initialized (J ';' (Stop SCM+FSA ))) = (Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))) 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: insloc ((card J) + 2) in dom (if=0 a,I,J) by SCMFSA6A:15;
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: insloc (((card I) + (card J)) + 3) in dom (if=0 a,I,J) by SCMFSA6A:15;
A23: if=0 a,I,J c= (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26, SCMFSA6B:5;
set s9 = Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2);
A24: (Directed J) +* (Start-At (insloc 0 )) c= (Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
LifeSpan ((Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) = pseudo-LifeSpan (Initialize ss),(Directed J) by A16, Th58;
then A25: DataPart (Computation ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan (Initialize ss),(Directed J))) = DataPart (Computation ((Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize ss) +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))))) by A16, A20, A8, Th58;
A26: insloc 1 in dom (if=0 a,I,J) by Th54;
((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . a = (Initialize ss) . a by Th28
.= (Initialize (Initialize ss)) . a by SCMFSA6C:3 ;
then A27: IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) = Next (IC ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))))) by A2, A15, SCMFSA_2:96
.= insloc (0 + 1) by A9 ;
A28: (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) . (insloc 1) = ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 1) by AMI_1:54
.= ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (insloc 1) by A26, A7, FUNCT_4:14
.= (if=0 a,I,J) . (insloc 1) by A26, SCMFSA6B:7
.= goto (insloc 2) by Th55 ;
A29: Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(1 + 1) = Following (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) by AMI_1:14
.= Exec (goto (insloc 2)),(Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) by A27, A28 ;
then A30: IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) = insloc 2 by SCMFSA_2:95;
A31: DataPart ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))) = DataPart ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by SCMFSA6A:39, SCMFSA8A:14;

A32: now

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

end;

now

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

end;

then A33: DataPart ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))) = DataPart (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) by A32, SCMFSA6A:38;
A34: IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2)) = IC (Computation (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2),(pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J))) by AMI_1:51
.= (IC (Computation ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)))) + 2 by A24, A18, A14, A30, A33, Th51
.= (IC (Computation ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan (Initialize ss),(Directed J)))) + 2 by A16, Th50
.= (insloc (card (ProgramPart (Directed J)))) + 2 by A16, SCMFSA8A:def 5
.= (insloc (card (Directed J))) + 2 by AMI_1:105
.= insloc ((card J) + 2) by SCMFSA8A:34 ;
then A35: CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2)) = ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc ((card J) + 2)) by AMI_1:54
.= (if=0 a,I,J) . (insloc ((card J) + 2)) by A23, A21, GRFUNC_1:8
.= goto (insloc (((card I) + (card J)) + 3)) by Th63 ;
IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2) + 1)) = IC (Following (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2))) by AMI_1:14
.= insloc (((card I) + (card J)) + 3) by A35, SCMFSA_2:95 ;
then A36: CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2) + 1)) = ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc (((card I) + (card J)) + 3)) by AMI_1:54
.= (if=0 a,I,J) . (insloc (((card I) + (card J)) + 3)) by A6, A22, GRFUNC_1:8
.= halt SCM+FSA by Th61 ;
then A37: (Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) is halting by AMI_1:def 20;
A38: CurInstr ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = a =0_goto (insloc ((card J) + 3)) by A9, A4;

now

A39: 0 + 2 < ((card I) + (card J)) + 3 by XREAL_1:10;
then A40: insloc 2 in dom (if=0 a,I,J) by Th56;
A41: (if=0 a,I,J) . (insloc 2) <> halt SCM+FSA by A39, Th56;
A42: CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) = ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 2) by A30, AMI_1:54
.= (if=0 a,I,J) . (insloc 2) by A40, Th26 ;
let k be Element of NAT ; :: thesis:
assume A43: CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = halt SCM+FSA ; :: thesis:
CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),0 ) = a =0_goto (insloc ((card J) + 3)) by A38, AMI_1:13;
then A44: k <> 0 by A43, SCMFSA_2:48, SCMFSA_2:124;
k <> 1 by A27, A28, A43, SCMFSA_2:47, SCMFSA_2:124;
then 2 < k by A43, A44, A41, A42, 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 (Computation ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),k2) as Element of NAT by ORDINAL1:def 13;
assume not (pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + (1 + 2) <= k ; :: thesis:
then k < ((pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 1) + 2 ;
then k2 < (pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 1 by A45, XREAL_1:8;
then A46: k2 <= pseudo-LifeSpan ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J) by NAT_1:13;
then A47: k2 <= pseudo-LifeSpan (Initialize ss),(Directed J) by A16, Th50;

A48: now

per cases by A47, XXREAL_0:1;

suppose A49: k2 = pseudo-LifeSpan (Initialize ss),(Directed J) ; :: thesis:

((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 (Computation ((Initialize ss) +* ((Directed J) +* (Start-At (insloc 0 )))),k2) = insloc (card (ProgramPart (Directed J))) by A16, A49, SCMFSA8A:def 5
.= insloc (card (Directed J)) by AMI_1:105 ;
then n = card J by SCMFSA8A:34;
hence n + 2 < ((card I) + (card J)) + 3 by A50, NAT_1:13; :: thesis:

end;

suppose k2 < pseudo-LifeSpan (Initialize ss),(Directed J) ; :: thesis:

then insloc n in dom (Directed J) by A16, SCMFSA8A:31;
then n < card (Directed J) by SCMFSA6A:15;
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:

end;

then A52: insloc (n + 2) in dom (if=0 a,I,J) by Th56;
A53: IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = IC (Computation (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2),k2) by A45, AMI_1:51
.= insloc (n + 2) by A24, A18, A14, A30, A33, A46, Th51 ;
CurInstr (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k)) by AMI_1:54
.= (if=0 a,I,J) . (IC (Computation ((Initialize ss) +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k)) by A53, A52, Th26 ;
hence contradiction by A43, A53, A48, Th56; :: thesis:

end;