let s be State of SCM+FSA ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
let I, J be Program of SCM+FSA ; :: thesis:
let a be read-write Int-Location ; :: thesis:
set J0 = Directed J;
set s0 = Initialize s;
set J9 = J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )));
set s00 = s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ));
set s3 = s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ));
set s4 = Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1;
set s5 = Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2;
set i = a >0_goto ((card J) + 3);
A1: (Directed J) +* (Start-At 0 ,SCM+FSA ) c= s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
if>0 a,I,J = ((((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) ';' (Stop SCM+FSA ) by SCMFSA8B:def 2;
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 A2: 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 SCMFSA7B:6;
then A3: ProgramPart (Relocated (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )))),2) c= if>0 a,I,J by A2, FUNCT_4:26;
A4: 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 A5: (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . 0 = ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) . 0 by A4, FUNCT_4:14
.= (if>0 a,I,J) . 0 by A4, SCMFSA6B:7
.= a >0_goto ((card J) + 3) by Th55 ;
card (if>0 a,I,J) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:15
.= ((card J) + 2) + ((card I) + 2) ;
then A6: ((card J) + 2) + 0 < card (if>0 a,I,J) by XREAL_1:10;
then A7: (card J) + 2 in dom (if>0 a,I,J) by SCMFSA6A:15;
IC SCM+FSA in dom ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
then A8: IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by FUNCT_4:14
.= 0 by SF_MASTR:66 ;
set ss = Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2);
A9: if>0 a,I,J c= (if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
A10: if>0 a,I,J c= s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26, SCMFSA6B:5;
(if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ) c= s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then if>0 a,I,J c= s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by A9, XBOOLE_1:1;
then ProgramPart (Relocated (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )))),2) c= s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by A3, XBOOLE_1:1;
then ProgramPart (ProgramPart (Relocated (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )))),2)) c= Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2 by AMI_1:99;
then A11: ProgramPart (Relocated (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )))),2) c= Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2 by AMI_1:105;
ProgramPart (Relocated (Directed J),2) c= ProgramPart (Relocated (J ';' ((Goto ((card I) + 1)) ';' (I ';' (Stop SCM+FSA )))),2) by Th12, SCMFSA6A:55;
then A12: ProgramPart (Relocated (Directed J),2) c= Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2 by A11, XBOOLE_1:1;
card (if>0 a,I,J) = ((card I) + (card J)) + (3 + 1) by SCMFSA8B:15
.= (((card I) + (card J)) + 3) + 1 ;
then ((card I) + (card J)) + 3 < card (if>0 a,I,J) by NAT_1:13;
then A13: ((card I) + (card J)) + 3 in dom (if>0 a,I,J) by SCMFSA6A:15;
assume s . a <= 0 ; :: thesis:
then A14: (Initialize s) . a <= 0 by SCMFSA6C:3;
A15: 1 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 A16: dom (if>0 a,I,J) c= dom ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
A17: if>0 a,I,J c= s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26, SCMFSA6B:5;
assume A18: Directed J is_pseudo-closed_on s ; :: thesis:
then A19: pseudo-LifeSpan s,(Directed J) = LifeSpan (s +* ((J ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) by Th58;
A20: (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) . 1 = (s +* ((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 A15, A16, FUNCT_4:14
.= (if>0 a,I,J) . 1 by A15, SCMFSA6B:7
.= goto 2 by Th55 ;
DataPart s = DataPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))) by SCMFSA8A:11;
then A21: Directed J is_pseudo-closed_on s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )) by A18, Th52;
Y: (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) /. (IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . (IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) by AMI_1:150;
A22: Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(0 + 1) = Following (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),0 ) by AMI_1:14
.= Following (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= Exec (a >0_goto ((card J) + 3)),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) by A8, A5, Y ;
(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . a = s . a by Th28
.= (Initialize s) . a by SCMFSA6C:3 ;
then A23: IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) = succ (IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) by A14, A22, SCMFSA_2:97
.= 0 + 1 by A8 ;
Y: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1)) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1)) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1)) by AMI_1:150;
T: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) by AMI_1:144;
A24: Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(1 + 1) = Following (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) by AMI_1:14
.= Exec (goto 2),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) by A23, A20, Y, T ;
then A25: IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) = 2 by SCMFSA_2:95;
A26: DataPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))) = DataPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) by SCMFSA6A:39, SCMFSA8A:14;

A27: now

let f be FinSeq-Location ; :: thesis:
thus (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))) . f = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . f by A26, SCMFSA6A:38
.= (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) . f by A22, SCMFSA_2:97
.= (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) . f by A24, SCMFSA_2:95 ; :: thesis:

end;

now

let a be Int-Location ; :: thesis:
thus (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))) . a = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . a by A26, SCMFSA6A:38
.= (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),1) . a by A22, SCMFSA_2:97
.= (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) . a by A24, SCMFSA_2:95 ; :: thesis:

end;

then A28: DataPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))) = DataPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) by A27, SCMFSA6A:38;
Y: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2))) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2))) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2)) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2))) by AMI_1:150;
T: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) by AMI_1:144;
A29: IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2)) = IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2),(pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J))) by AMI_1:51
.= (IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)))) + 2 by A1, A21, A12, A25, A28, T, Th51
.= (IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(pseudo-LifeSpan s,(Directed J)))) + 2 by A18, Th50
.= (card (ProgramPart (Directed J))) + 2 by A18, SCMFSA8A:def 5
.= (card (Directed J)) + 2 by AMI_1:105
.= (card J) + 2 by SCMFSA8A:34 ;
then A30: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2)) = (s +* ((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 A10, A7, GRFUNC_1:8
.= goto (((card I) + (card J)) + 3) by Th64 ;
Y: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1))) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1))) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1)) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1))) by AMI_1:150;
T: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2)) by AMI_1:144;
TX: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1)) by AMI_1:144;
A31: IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1)) = IC (Following (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2))) by AMI_1:14
.= ((card I) + (card J)) + 3 by A30, T, SCMFSA_2:95 ;
then A32: CurInstr (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1)) = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . (((card I) + (card J)) + 3) by Y, TX, AMI_1:54
.= (if>0 a,I,J) . (((card I) + (card J)) + 3) by A17, A13, GRFUNC_1:8
.= halt SCM+FSA by Th62 ;
then A33: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) halts_on s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by TX, AMI_1:146;
hence if>0 a,I,J is_halting_on s by SCMFSA7B:def 8; :: thesis:

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

then Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k = s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by AMI_1:13;
then IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) = 0 by Th31;
hence IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) in dom (if>0 a,I,J) by Th54; :: thesis:

end;

suppose ( 0 < k & k = 1 ) ; :: thesis:

hence IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) in dom (if>0 a,I,J) by A23, Th54; :: thesis:

end;

suppose A34: ( 0 < k & k <> 1 & k < (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2 ) ; :: thesis:

then 0 + 1 <= k by INT_1:20;
then 1 < k by A34, XXREAL_0:1;
then 0 + (1 + 1) <= k by INT_1:20;
then consider k2 being Nat such that
A35: 2 + k2 = k by NAT_1:10;
card (if>0 a,I,J) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:15
.= ((card J) + 2) + ((card I) + 2) ;
then A36: (card J) + 2 <= card (if>0 a,I,J) by NAT_1:12;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),k2) as Element of NAT ;
A37: k2 < pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J) by A34, A35, XREAL_1:8;
then k2 < pseudo-LifeSpan s,(Directed J) by A18, Th50;
then n in dom (Directed J) by A18, SCMFSA8A:31;
then n < card (Directed J) by SCMFSA6A:15;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:8;
then n + 2 < (card J) + 2 by SCMFSA8A:34;
then A38: n + 2 < card (if>0 a,I,J) by A36, XXREAL_0:2;
T: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) by AMI_1:144;
IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) = IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2),k2) by A35, AMI_1:51
.= (IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),k2)) + 2 by A1, A21, A12, A25, A28, A37, Th51, T ;
hence IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) in dom (if>0 a,I,J) by A38, SCMFSA6A:15; :: thesis:

end;

suppose A39: ( 0 < k & k <> 1 & (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2 <= k ) ; :: thesis:

hereby :: thesis:

per cases ) + 2 = k or (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2 < k ) by A39, XXREAL_0:1;

suppose (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2 = k ; :: thesis:

hence IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) in dom (if>0 a,I,J) by A6, A29, SCMFSA6A:15; :: thesis:

end;

suppose (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2 < k ; :: thesis:

then ((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 2) + 1 <= k by INT_1:20;
hence IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) in dom (if>0 a,I,J) by A13, A31, A32, AMI_1:52; :: thesis:

end;

end;

end;

end;

end;

end;

hence if>0 a,I,J is_closed_on s by SCMFSA7B:def 7; :: thesis:
V: (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) /. (IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . (IC (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))) by AMI_1:150;
A40: CurInstr (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = a >0_goto ((card J) + 3) by A8, A5, V;

now

A41: 0 + 2 < ((card I) + (card J)) + 3 by XREAL_1:10;
then A42: 2 in dom (if>0 a,I,J) by Th57;
A43: (if>0 a,I,J) . 2 <> halt SCM+FSA by A41, Th57;
Y: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2)) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2)) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2)) by AMI_1:150;
A44: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2)),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . 2 by A25, Y, AMI_1:54
.= (if>0 a,I,J) . 2 by A42, Th26 ;
let k be Element of NAT ; :: thesis:
assume A45: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) = halt SCM+FSA ; :: thesis:
u: Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),0 = s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )) by AMI_1:13;
U: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
CurInstr (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),0 )),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),0 ) = a >0_goto ((card J) + 3) by A40, u;
then A46: k <> 0 by A45, SCMFSA_2:49, SCMFSA_2:124;
xx: k <> 1 by A23, A20, A45, U, SCMFSA_2:47, SCMFSA_2:124;
2 <> k by A45, A43, A44;
then 2 < k by A46, xx, NAT_1:27;
then consider k2 being Nat such that
A47: 2 + k2 = k by NAT_1:10;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),k2) as Element of NAT ;
assume not (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + (1 + 2) <= k ; :: thesis:
then k < ((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 1) + 2 ;
then k2 < (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 1 by A47, XREAL_1:8;
then A48: k2 <= pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J) by NAT_1:13;
then A49: k2 <= pseudo-LifeSpan s,(Directed J) by A18, Th50;

A50: now

per cases by A49, XXREAL_0:1;

suppose A51: k2 = pseudo-LifeSpan s,(Directed J) ; :: thesis:

((card I) + (card J)) + (2 + 1) = (((card J) + 2) + 1) + (card I) ;
then A52: ((card J) + 2) + 1 <= ((card I) + (card J)) + 3 by NAT_1:11;
IC (Comput (ProgramPart (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),k2) = card (ProgramPart (Directed J)) by A18, A51, SCMFSA8A:def 5
.= card (Directed J) by AMI_1:105 ;
then n = card J by SCMFSA8A:34;
hence n + 2 < ((card I) + (card J)) + 3 by A52, NAT_1:13; :: thesis:

end;

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

then n in dom (Directed J) by A18, SCMFSA8A:31;
then n < card (Directed J) by SCMFSA6A:15;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:8;
then A53: 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 A53, XXREAL_0:2; :: thesis:

end;

end;

end;

then A54: n + 2 in dom (if>0 a,I,J) by Th57;
T: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2) by AMI_1:144;
A55: IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) = IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),2),k2) by A47, AMI_1:51
.= n + 2 by A1, A21, A12, A25, A28, A48, Th51, T ;
Y: (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) by AMI_1:150;
CurInstr (ProgramPart (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k) = (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) by Y, AMI_1:54
.= (if>0 a,I,J) . (IC (Comput (ProgramPart (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))),k)) by A55, A54, Th26 ;
hence contradiction by A45, A55, A50, Th57; :: thesis:

end;

then LifeSpan (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At 0 ,SCM+FSA ))),(Directed J)) + 3 by A32, A33, TX, AMI_1:def 46;
hence LifeSpan (s +* ((if>0 a,I,J) +* (Start-At 0 ,SCM+FSA ))) = (LifeSpan (s +* ((J ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))) + 3 by A18, A19, Th50; :: thesis: