let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; :: thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
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)));
set i = (a,k1) <=0_goto ((card I) + 2);
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
I: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A7: 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 A8: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if>0 (a,k1,I,J))) by Lm6;
A9: 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 A7, Th22, I ;
assume s . (DataLoc ((s . a),k1)) > 0 ; :: thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A10: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) = succ (IC s) by A9, SCMPDS_2:56
.= 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 A9, SCMPDS_2:56;
then A12: DataPart s = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume A13: I is_closed_on s,P ; :: thesis: ( not I is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A14: I is_halting_on s,P ; :: thesis: IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A13, Th44;
then A15: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, I;
(I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A13, A14, Th44;
then A16: ( 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 Th38, FUNCT_4:25, I;
UU: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A18: 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;
A21: 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 A18
.= 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 A16, A8, A10, A12, Th45, UU
.= halt SCMPDS by A15, EXTPRO_1:def 15 ;
then A22: P +* (stop (if>0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A23: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),s) = (a,k1) <=0_goto ((card I) + 2) by A7, Th22, I;

now

let l be Element of NAT ; :: thesis: ( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b₁))) <> halt SCMPDS )
assume A24: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; :: thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b₁))) <> halt SCMPDS
A25: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;

per cases ( l = 0 or l <> 0 ) ;

suppose l = 0 ; :: thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b₁))) <> halt SCMPDS

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 A25;
hence CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A23; :: thesis: verum

end;

suppose l <> 0 ; :: thesis: not CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b₁))) = halt SCMPDS

then consider n being Nat such that
A26: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def 12;
A27: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A24, A26, XREAL_1:6;
assume A28: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; :: thesis: contradiction
A32: 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 A16, A8, A10, A12, Th45, UU
.= halt SCMPDS by A26, A28, A32 ;
hence contradiction by A15, A27, EXTPRO_1:def 15; :: thesis: verum

end;

end;

end;

then for l being Element of 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 A33: LifeSpan ((P +* (stop (if>0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A21, A22, EXTPRO_1:def 15;
A35: 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 A15, 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 A16, A8, A10, A12, Th45, UU
.= 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 A22, A33, EXTPRO_1:23 ;

A36: now

let x be set ; :: thesis: ( x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁ )
A37: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A38: x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) ; :: thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A38, SCMPDS_4:6;

suppose A39: x is Int_position ; :: thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A40: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A37, 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 A35, A39, 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 A40, FUNCT_4:11 ; :: thesis: verum

end;

suppose A42: x = IC ; :: thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

A44: 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 A13, A14, Th46 ;
A45: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A37, A42, 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 A22, A33, 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 A42, 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 A16, A8, A10, A12, Th45, UU
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A15, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A44, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A42, A45, FUNCT_4:13 ; :: thesis: verum

end;

end;

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 A36, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A13, A14, Th47
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
:: thesis: verum