let s be State of ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
let I, J be Program of ; :: thesis:
let a be read-write Int-Location ; :: thesis:
set I0 = Directed I;
set I1 = I ';' (Stop SCM+FSA );
set s00 = s +* ((Directed I) +* (Start-At (insloc 0 )));
set s3 = s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )));
set s4 = Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1;
set i = a >0_goto (insloc ((card J) + 3));
A1: (Directed I) +* (Start-At (insloc 0 )) c= s +* ((Directed I) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
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 A2: insloc (((card I) + (card J)) + 3) in dom (if>0 a,I,J) by SCMFSA6A:15;
IC SCM+FSA in dom ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by SF_MASTR:65;
then A3: IC (s +* ((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 ;
A4: 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 A5: (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 0 ) = ((if>0 a,I,J) +* (Start-At (insloc 0 ))) . (insloc 0 ) by A4, FUNCT_4:14
.= (if>0 a,I,J) . (insloc 0 ) by A4, SCMFSA6B:7
.= a >0_goto (insloc ((card J) + 3)) by Th55 ;
A6: card (((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) = (card ((Macro (a >0_goto (insloc ((card J) + 3)))) ';' J)) + (card (Goto (insloc ((card I) + 1)))) by SCMFSA6A:61
.= (card ((Macro (a >0_goto (insloc ((card J) + 3)))) ';' J)) + 1 by SCMFSA8A:29
.= ((card (Macro (a >0_goto (insloc ((card J) + 3))))) + (card J)) + 1 by SCMFSA6A:61
.= ((card J) + 2) + 1 by SCMFSA7B:6
.= (card J) + (2 + 1) ;
A7: Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),(0 + 1) = Following (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),0 ) by AMI_1:14
.= Following (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) by AMI_1:13
.= Exec (a >0_goto (insloc ((card J) + 3))),(s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) by A3, A5 ;
A8: 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= s +* ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then A9: if>0 a,I,J c= s +* ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by A8, XBOOLE_1:1;
A10: if>0 a,I,J c= s +* ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26, SCMFSA6B:5;
A11: 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 2;
then if>0 a,I,J = (((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' (I ';' (Stop SCM+FSA )) by SCMFSA6A:67;
then ProgramPart (Relocated (I ';' (Stop SCM+FSA )),((card J) + 3)) c= if>0 a,I,J by A6, FUNCT_4:26;
then ProgramPart (Relocated (I ';' (Stop SCM+FSA )),((card J) + 3)) c= s +* ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by A9, XBOOLE_1:1;
then ProgramPart [(ProgramPart (Relocated (I ';' (Stop SCM+FSA )),((card J) + 3)))] c= Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1 by AMI_1:99;
then A12: ProgramPart (Relocated (I ';' (Stop SCM+FSA )),((card J) + 3)) c= Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1 by AMI_1:105;
ProgramPart (Relocated (Directed I),((card J) + 3)) c= ProgramPart (Relocated (I ';' (Stop SCM+FSA )),((card J) + 3)) by Th12, SCMFSA6A:55;
then A13: ProgramPart (Relocated (Directed I),((card J) + 3)) c= Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1 by A12, XBOOLE_1:1;
A14: DataPart (s +* ((Directed I) +* (Start-At (insloc 0 )))) = DataPart (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) by SCMFSA6A:39, SCMFSA8A:14;

A15: now

let f be FinSeq-Location ; :: thesis:
thus (s +* ((Directed I) +* (Start-At (insloc 0 )))) . f = (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . f by A14, SCMFSA6A:38
.= (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1) . f by A7, SCMFSA_2:97 ; :: thesis:

end;

let a be Int-Location ; :: thesis:
thus (s +* ((Directed I) +* (Start-At (insloc 0 )))) . a = (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . a by A14, SCMFSA6A:38
.= (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1) . a by A7, SCMFSA_2:97 ; :: thesis:

end;

then A16: DataPart (s +* ((Directed I) +* (Start-At (insloc 0 )))) = DataPart (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1) by A15, SCMFSA6A:38;
not a in dom ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by SCMFSA6B:12;
then A17: (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . a = s . a by FUNCT_4:12;
assume s . a > 0 ; :: thesis:
then A18: IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1) = insloc ((card J) + 3) by A7, A17, SCMFSA_2:97;
assume A19: Directed I is_pseudo-closed_on s ; :: thesis:
then A20: pseudo-LifeSpan s,(Directed I) = LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) by Th58;
DataPart s = DataPart (s +* ((Directed I) +* (Start-At (insloc 0 )))) by SCMFSA8A:11;
then A21: Directed I is_pseudo-closed_on s +* ((Directed I) +* (Start-At (insloc 0 ))) by A19, Th52;
A22: IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1)) = IC (Computation (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1),(pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I))) by AMI_1:51
.= (IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)))) + ((card J) + 3) by A1, A21, A13, A18, A16, Th51
.= (IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan s,(Directed I)))) + ((card J) + 3) by A19, Th50
.= (insloc (card (ProgramPart (Directed I)))) + ((card J) + 3) by A19, SCMFSA8A:def 5
.= (insloc (card (Directed I))) + ((card J) + 3) by AMI_1:105
.= insloc ((card I) + ((card J) + 3)) by SCMFSA8A:34
.= insloc (((card I) + (card J)) + 3) ;
then A23: CurInstr (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1)) = (s +* ((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 A10, A2, GRFUNC_1:8
.= halt SCM+FSA by Th62 ;
then A24: ProgramPart (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) halts_on s +* ((if>0 a,I,J) +* (Start-At (insloc 0 ))) by AMI_1:146;
hence if>0 a,I,J is_halting_on s by SCMFSA7B:def 8; :: thesis:

let k be Element of NAT ; :: thesis:

suppose k = 0 ; :: thesis:

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

end;

suppose A25: ( 0 < k & k < (pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1 ) ; :: thesis:

card (if>0 a,I,J) = ((card I) + (card J)) + (3 + 1) by SCMFSA8B:15
.= (((card I) + (card J)) + 3) + 1 ;
then A26: ((card I) + (card J)) + 3 < card (if>0 a,I,J) by XREAL_1:31;
0 + 1 <= k by A25, INT_1:20;
then consider k1 being Nat such that
A27: 1 + k1 = k by NAT_1:10;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k1) as Element of NAT by ORDINAL1:def 13;
A28: k1 < pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I) by A25, A27, XREAL_1:8;
then k1 < pseudo-LifeSpan s,(Directed I) by A19, Th50;
then insloc n in dom (Directed I) by A19, SCMFSA8A:31;
then n < card (Directed I) by SCMFSA6A:15;
then n + ((card J) + 3) < (card (Directed I)) + ((card J) + 3) by XREAL_1:8;
then n + ((card J) + 3) < (card I) + ((card J) + 3) by SCMFSA8A:34;
then A29: n + ((card J) + 3) < card (if>0 a,I,J) by A26, XXREAL_0:2;
IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) = IC (Computation (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1),k1) by A27, AMI_1:51
.= (IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k1)) + ((card J) + 3) by A1, A21, A13, A18, A16, A28, Th51 ;
hence IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if>0 a,I,J) by A29, SCMFSA6A:15; :: thesis:

end;

suppose ( 0 < k & (pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1 <= k ) ; :: thesis:

hence IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if>0 a,I,J) by A2, A22, A23, AMI_1:52; :: thesis:

end;

end;

end;

hence if>0 a,I,J is_closed_on s by SCMFSA7B:def 7; :: thesis:

set J1 = (((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I;
let k be Element of NAT ; :: thesis:
assume A30: CurInstr (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) = halt SCM+FSA ; :: thesis:
assume not (pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1 <= k ; :: thesis:
then A31: k <= pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I) by NAT_1:13;
A32: insloc 0 in dom (if>0 a,I,J) by Th54;
A33: InsCode (a >0_goto (insloc ((card J) + 3))) = 8 by SCMFSA_2:49;
CurInstr (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),0 ) = CurInstr (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) by AMI_1:13
.= (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 0 ) by Th31
.= (if>0 a,I,J) . (insloc 0 ) by A32, Th26
.= a >0_goto (insloc ((card J) + 3)) by Th55 ;
then consider k1 being Nat such that
A34: k1 + 1 = k by A30, A33, NAT_1:6, SCMFSA_2:124;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k1) as Element of NAT by ORDINAL1:def 13;
k1 < k by A34, XREAL_1:31;
then A35: k1 < pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I) by A31, XXREAL_0:2;
then k1 < pseudo-LifeSpan s,(Directed I) by A19, Th50;
then insloc n in dom (Directed I) by A19, SCMFSA8A:31;
then n < card (Directed I) by SCMFSA6A:15;
then n + ((card J) + 3) < (card (Directed I)) + ((card J) + 3) by XREAL_1:8;
then A36: n + ((card J) + 3) < (card I) + ((card J) + 3) by SCMFSA8A:34;
A37: IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) = IC (Computation (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),1),k1) by A34, AMI_1:51
.= (IC (Computation (s +* ((Directed I) +* (Start-At (insloc 0 )))),k1)) + ((card J) + 3) by A1, A21, A13, A18, A16, A35, Th51 ;
card ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I) = (card (((Macro (a >0_goto (insloc ((card J) + 3)))) ';' J) ';' (Goto (insloc ((card I) + 1))))) + (card I) by SCMFSA6A:61
.= ((card ((Macro (a >0_goto (insloc ((card J) + 3)))) ';' J)) + (card (Goto (insloc ((card I) + 1))))) + (card I) by SCMFSA6A:61
.= ((card ((Macro (a >0_goto (insloc ((card J) + 3)))) ';' J)) + 1) + (card I) by SCMFSA8A:29
.= (((card (Macro (a >0_goto (insloc ((card J) + 3))))) + (card J)) + 1) + (card I) by SCMFSA6A:61
.= ((2 + (card J)) + 1) + (card I) by SCMFSA7B:6
.= ((card I) + (card J)) + 3 ;
then IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I) by A37, A36, SCMFSA6A:15;
then A38: IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (Directed ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I)) by FUNCT_4:105;
then A39: (Directed ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I)) . (IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k)) in rng (Directed ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I)) by FUNCT_1:def 5;
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 XREAL_1:31;
then n + ((card J) + 3) < card (if>0 a,I,J) by A36, XXREAL_0:2;
then A40: IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if>0 a,I,J) by A37, SCMFSA6A:15;
A41: CurInstr (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k) = (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) . (IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k)) by AMI_1:54
.= (if>0 a,I,J) . (IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k)) by A40, Th26 ;
Directed ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I) c= if>0 a,I,J by A11, SCMFSA6A:55;
then (if>0 a,I,J) . (IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k)) = (Directed ((((a >0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I)) . (IC (Computation (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))),k)) by A38, GRFUNC_1:8;
hence contradiction by A30, A39, A41, AMI_1:def 53; :: thesis:

end;

then LifeSpan (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) = (pseudo-LifeSpan (s +* ((Directed I) +* (Start-At (insloc 0 )))),(Directed I)) + 1 by A23, A24, AMI_1:def 46;
hence LifeSpan (s +* ((if>0 a,I,J) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))))) + 1 by A19, A20, Th50; :: thesis: