let P be Instruction-Sequence of SCM+FSA; :: thesis:
let s be State of SCM+FSA; :: thesis:
set D = Data-Locations ;
let I, J be MacroInstruction of SCM+FSA ; :: thesis:
let a be read-write Int-Location; :: thesis:
set J0 = Directed J;
set s0 = Initialized s;
set J9 = J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)));
set s00 = Initialize s;
set P00 = P +* (Directed J);
set s3 = Initialize s;
set P3 = P +* (if>0 (a,I,J));
A1: if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),1);
set s5 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),2);
set i = a >0_goto ((card J) + 3);
A2: Directed J c= P +* (Directed J) by FUNCT_4:25;
if>0 (a,I,J) = (((Macro (a >0_goto ((card J) + 3))) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
then if>0 (a,I,J) = ((Macro (a >0_goto ((card J) + 3))) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25;
then A3: if>0 (a,I,J) = (Macro (a >0_goto ((card J) + 3))) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:25;
card (Macro (a >0_goto ((card J) + 3))) = 2 by COMPOS_1:56;
then A4: Reloc ((J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))),2) c= if>0 (a,I,J) by A3, SCMFSA6A:38;
A5: 0 in dom (if>0 (a,I,J)) by AFINSQ_1:65;
A6: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by A5, FUNCT_4:13
.= a >0_goto ((card J) + 3) by Th18 ;
card (if>0 (a,I,J)) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:12
.= ((card J) + 2) + ((card I) + 2) ;
then ((card J) + 2) + 0 < card (if>0 (a,I,J)) by XREAL_1:8;
then A7: (card J) + 2 in dom (if>0 (a,I,J)) by AFINSQ_1:66;
A8: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
A9: IC (Initialize s) = IC (Initialize s)
.= IC (Start-At (0,SCM+FSA)) by A8, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
set ss = Comput ((P +* (if>0 (a,I,J))),(Initialize s),((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2));
set PP = P +* (if>0 (a,I,J));
if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
then A10: Reloc ((J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))),2) c= P +* (if>0 (a,I,J)) by A4, XBOOLE_1:1;
Reloc ((Directed J),2) c= Reloc ((J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))),2) by COMPOS_1:44, SCMFSA6A:16;
then A11: Reloc ((Directed J),2) c= P +* (if>0 (a,I,J)) by A10, XBOOLE_1:1;
card (if>0 (a,I,J)) = ((card I) + (card J)) + (3 + 1) by SCMFSA8B:12
.= (((card I) + (card J)) + 3) + 1 ;
then ((card I) + (card J)) + 3 < card (if>0 (a,I,J)) by NAT_1:13;
then A12: ((card I) + (card J)) + 3 in dom (if>0 (a,I,J)) by AFINSQ_1:66;
assume s . a <= 0 ; :: thesis:
then A13: (Initialized s) . a <= 0 by SCMFSA_M:37;
A14: 1 in dom (if>0 (a,I,J)) by Th17;
assume A15: Directed J is_pseudo-closed_on s,P ; :: thesis:
then A16: pseudo-LifeSpan (s,P,(Directed J)) = LifeSpan ((P +* (J ";" (Stop SCM+FSA))),(Initialize s)) by Th21;
A17: (P +* (if>0 (a,I,J))) . 1 = (if>0 (a,I,J)) . 1 by A14, FUNCT_4:13
.= goto 2 by Th18 ;
A18: Directed J is_pseudo-closed_on Initialize s,P +* (Directed J) by A15;
A19: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialize s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialize s)) by A9, A6, PBOOLE:143 ;
A20: a <> IC by SCMFSA_2:56;
not a in dom (Start-At (0,SCM+FSA)) by A20, TARSKI:def 1;
then not a in dom (Start-At (0,SCM+FSA)) ;
then (Initialize s) . a = s . a by FUNCT_4:11
.= (Initialized s) . a by SCMFSA_M:37 ;
then A21: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) = (IC (Initialize s)) + 1 by A13, A19, SCMFSA_2:71
.= 0 + 1 by A9 ;
A22: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(1 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto 2),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1))) by A21, A17, PBOOLE:143 ;
then A23: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) = 2 by SCMFSA_2:69;

A24: now :: thesis:

let f be FinSeq-Location ; :: thesis:
thus (Initialize s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . f by A19, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . f by A22, SCMFSA_2:69 ; :: thesis:

end;

now :: thesis:

let a be Int-Location; :: thesis:
thus (Initialize s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . a by A19, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . a by A22, SCMFSA_2:69 ; :: thesis:

end;

then A25: DataPart (Initialize s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) by A24, SCMFSA_M:2;
IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2))) = IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)),(pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))))) by EXTPRO_1:4
.= (IC (Comput ((P +* (Directed J)),(Initialize s),(pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J)))))) + 2 by A18, A11, A23, A25, Th14, A2
.= (IC (Comput ((P +* (Directed J)),(Initialize s),(pseudo-LifeSpan (s,P,(Directed J)))))) + 2 by A15, Th13
.= (card (Directed J)) + 2 by A15, SCMFSA8A:def 4
.= (card J) + 2 by SCMFSA6A:36 ;
then A26: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2)))) = (P +* (if>0 (a,I,J))) . ((card J) + 2) by PBOOLE:143
.= (if>0 (a,I,J)) . ((card J) + 2) by A7, A1, GRFUNC_1:2
.= goto (((card I) + (card J)) + 3) by Th27 ;
IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),(((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2) + 1))) = IC (Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2))))) by EXTPRO_1:3
.= ((card I) + (card J)) + 3 by A26, SCMFSA_2:69 ;
then A27: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),(((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 2) + 1)))) = (P +* (if>0 (a,I,J))) . (((card I) + (card J)) + 3) by PBOOLE:143
.= (if>0 (a,I,J)) . (((card I) + (card J)) + 3) by A12, A1, GRFUNC_1:2
.= halt SCM+FSA by Th25 ;
then A28: P +* (if>0 (a,I,J)) halts_on Initialize s by EXTPRO_1:29;
hence if>0 (a,I,J) is_halting_on s,P ; :: thesis:
A29: CurInstr ((P +* (if>0 (a,I,J))),(Initialize s)) = a >0_goto ((card J) + 3) by A9, A6, PBOOLE:143;

now :: thesis:

A30: 0 + 2 < ((card I) + (card J)) + 3 by XREAL_1:8;
then A31: 2 in dom (if>0 (a,I,J)) by Th20;
A32: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),2))) = (P +* (if>0 (a,I,J))) . 2 by A23, PBOOLE:143
.= (if>0 (a,I,J)) . 2 by A31, A1, GRFUNC_1:2 ;
let k be Nat; :: thesis:
assume A33: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),k))) = halt SCM+FSA ; :: thesis:
A34: k <> 0 by A33, A29;
A35: k <> 1 by A21, A33, A17, PBOOLE:143;
2 <> k by A33, A30, Th20, A32;
then k <> 0 & ... & k <> 2 by A34, A35;
then 2 < k ;
then consider k2 being Nat such that
A36: 2 + k2 = k by NAT_1:10;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 12;
reconsider n = IC (Comput ((P +* (Directed J)),(Initialize s),k2)) as Element of NAT ;
assume not (pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + (1 + 2) <= k ; :: thesis:
then k < ((pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 1) + 2 ;
then k2 < (pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 1 by A36, XREAL_1:6;
then A37: k2 <= pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J)) by NAT_1:13;
then A38: k2 <= pseudo-LifeSpan (s,P,(Directed J)) by A15, Th13;

A39: now :: thesis:

per cases by A38, XXREAL_0:1;

suppose A40: k2 = pseudo-LifeSpan (s,P,(Directed J)) ; :: thesis:

((card I) + (card J)) + (2 + 1) = (((card J) + 2) + 1) + (card I) ;
then A41: ((card J) + 2) + 1 <= ((card I) + (card J)) + 3 by NAT_1:11;
IC (Comput ((P +* (Directed J)),(Initialize s),k2)) = card (Directed J) by A15, A40, SCMFSA8A:def 4;
then n = card J by SCMFSA6A:36;
hence n + 2 < ((card I) + (card J)) + 3 by A41, NAT_1:13; :: thesis:

end;

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

then n in dom (Directed J) by A15, SCMFSA8A:17;
then n < card (Directed J) by AFINSQ_1:66;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:6;
then A42: n + 2 < (card J) + 2 by SCMFSA6A:36;
((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 A42, XXREAL_0:2; :: thesis:

end;

end;

end;

then A43: n + 2 in dom (if>0 (a,I,J)) by Th20;
A44: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) = IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)),k2)) by A36, EXTPRO_1:4
.= n + 2 by A18, A11, A23, A25, A37, Th14, A2 ;
CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),k))) = (P +* (if>0 (a,I,J))) . (IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k))) by PBOOLE:143
.= (if>0 (a,I,J)) . (IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k))) by A44, A43, A1, GRFUNC_1:2 ;
hence contradiction by A33, A44, A39, Th20; :: thesis:

end;

then LifeSpan ((P +* (if>0 (a,I,J))),(Initialize s)) = (pseudo-LifeSpan ((Initialize s),(P +* (Directed J)),(Directed J))) + 3 by A27, A28, EXTPRO_1:def 15;
hence LifeSpan ((P +* (if>0 (a,I,J))),(s +* (Start-At (0,SCM+FSA)))) = (LifeSpan ((P +* (J ";" (Stop SCM+FSA))),(Initialize s))) + 3 by A15, A16, Th13; :: thesis: