let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being halt-free 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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; :: thesis: for I being halt-free 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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let I be halt-free 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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1: 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)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
set pI = stop I;
set pIF = stop (if=0 (a,k1,I));
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
A4: IC (Initialize s) = 0 by MEMSTR_0:47;
A5: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A6: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th22 ;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A5, FUNCT_4:11 ;
then A7: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A6, SCMPDS_2:55
.= 0 + 1 by A4 ;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) . a by A6, SCMPDS_2:55;
then A9: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume A10: I is_closed_on s,P ; :: thesis: ( not I is_halting_on s,P or IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A11: I is_closed_on Initialize s,P +* (stop I) by Th38;
set SAl = Start-At (((card I) + 1),SCMPDS);
A13: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if=0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
UU: stop I c= P +* (stop I) by FUNCT_4:25;
assume A14: I is_halting_on s,P ; :: thesis: IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
then A15: P +* (stop I) halts_on Initialize s by Def3;
A17: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A20: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A11, A13, A7, A9, Th45, UU
.= halt SCMPDS by A15, EXTPRO_1:def 15 ;
then A21: P +* (stop (if=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A22: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto ((card I) + 1) by Th22;

now

let l be Element of NAT ; :: thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS )
assume A23: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; :: thesis: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b₁))) <> halt SCMPDS
A24: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;

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

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

then CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by A24;
hence CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A22; :: thesis: verum

end;

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

then consider n being Nat such that
A25: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def 12;
A26: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A23, A25, XREAL_1:6;
assume A27: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; :: thesis: contradiction
A31: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),n))) by A11, A13, A7, A9, Th45, UU
.= halt SCMPDS by A25, A27, A31 ;
hence contradiction by A15, A26, 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)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A32: LifeSpan ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A20, A21, EXTPRO_1:def 15;
A34: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A15, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A11, A13, A7, A9, Th45, UU
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) by A21, A32, EXTPRO_1:23 ;

A35: now

let x be set ; :: thesis: ( x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b₁ = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b₁ )
A36: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A37: x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) ; :: thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b₁ = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A37, SCMPDS_4:6;

suppose A38: x is Int_position ; :: thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b₁ = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A39: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A36, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A34, A38, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A39, FUNCT_4:11 ; :: thesis: verum

end;

suppose A41: x = IC ; :: thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b₁ = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b₁

A43: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A10, A14, Th48 ;
A44: x in dom (Start-At (((card I) + 1),SCMPDS)) by A36, A41, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A21, A32, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A41, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 1 by A11, A13, A7, A9, Th45, UU
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 1 by A15, EXTPRO_1:23
.= IC (Start-At (((card I) + 1),SCMPDS)) by A43, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A41, A44, FUNCT_4:13 ; :: thesis: verum

end;

end;

end;

dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A35, FUNCT_1:2; :: thesis: verum