let P be Instruction-Sequence of SCMPDS; :: thesis:
let s be 0 -started State of SCMPDS; :: thesis:
let I be halt-free shiftable Program of ; :: thesis:
let J be shiftable Program of ; :: thesis:
let a be Int_position; :: thesis:
let k1 be Integer; :: thesis:
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; :: thesis:
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if=0 (a,k1,I,J);
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s2 = s;
set s3 = s;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
A2: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A4: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if=0 (a,k1,I,J))) by Lm6;
A6: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 2)),s) by A4, Th3, A2 ;
s . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . a),k1))
.= 0 by A1 ;
then A7: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) = (IC s) + 1 by A6, SCMPDS_2:55
.= 0 + 1 by A3 ;
for a being Int_position holds s . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) . a by A6, SCMPDS_2:55;
then A8: DataPart s = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
assume A9: I is_closed_on s,P ; :: thesis:
assume A10: I is_halting_on s,P ; :: thesis:
then (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A9, Th21;
then A11: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by A2;
(I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A9, A10, Th21;
then A12: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by A2, FUNCT_4:25;
A13: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A14: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A14
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A12, A5, A7, A8, Th22, A13
.= halt SCMPDS by A11, EXTPRO_1:def 15 ;
then A16: P +* (stop (if=0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),s) = (a,k1) <>0_goto ((card I) + 2) by A4, Th3, A2;

now :: thesis:

let l be Nat; :: thesis:
assume A18: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; :: thesis:
A19: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;

per cases ;

suppose l = 0 ; :: thesis:

then CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),s) by A19;
hence CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A17; :: thesis:

end;

suppose l <> 0 ; :: thesis:

then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Nat ;
A21: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; :: thesis:
A23: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,n))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),n))) by A12, A5, A7, A8, Th22, A13
.= halt SCMPDS by A20, A22, A23 ;
hence contradiction by A11, A21, EXTPRO_1:def 15; :: thesis:

end;

then for l being Nat st CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if=0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A15, A16, EXTPRO_1:def 15;
A25: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A11, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A12, A5, A7, A8, Th22, A13
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) by A16, A24, EXTPRO_1:23 ;

A26: now :: thesis:

let x be object ; :: thesis:
A27: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if=0 (a,k1,I,J)),P,s)) ; :: thesis:

per cases by A28, SCMPDS_4:6;

suppose A29: x is Int_position ; :: thesis:

then x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x by A25, A29, SCMPDS_4:8
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; :: thesis:

end;

suppose A31: x = IC ; :: thesis:

A32: IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.= ((card I) + (card J)) + 1 by A9, A10, Th23 ;
A33: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, A31, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1 by A12, A5, A7, A8, Th22, A13
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A11, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; :: thesis:

end;

dom (IExec ((if=0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A9, A10, Th24
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
:: thesis: