let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) = 0 holds
IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS )
let I be Program of SCMPDS ; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) = 0 holds
IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS )
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc (s . a),k1) = 0 holds
IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS )
let k1 be Integer; :: thesis: ( s . (DataLoc (s . a),k1) = 0 implies IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS ) )
set b = DataLoc (s . a),k1;
assume A1: s . (DataLoc (s . a),k1) = 0 ; :: thesis: IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS )
set SAl = Start-At ((card I) + 2),SCMPDS ;
set i = a,k1 <>0_goto 2;
set j = goto ((card I) + 1);
set IF = if<>0 a,k1,I;
set pIF = stop (if<>0 a,k1,I);
set s3 = (Initialize s) +* (stop (if<>0 a,k1,I));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1;
set s5 = Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2;
I2: s +* (Initialize (stop (if<>0 a,k1,I))) = (Initialize s) +* (stop (if<>0 a,k1,I)) by SCMPDS_4:5;
A2: if<>0 a,k1,I = (a,k1 <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A3: Initialize (stop (if<>0 a,k1,I)) c= (Initialize s) +* (stop (if<>0 a,k1,I)) by I2, FUNCT_4:26;
then A4: stop (if<>0 a,k1,I) c= Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1 by AMI_1:81, SCMPDS_4:57;
A5: stop (if<>0 a,k1,I) c= Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2 by A3, AMI_1:81, SCMPDS_4:57;
A6: not DataLoc (s . a),k1 in dom (Initialize (stop (if<>0 a,k1,I))) by SCMPDS_4:31;
A7: IC ((Initialize s) +* (stop (if<>0 a,k1,I))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A8: Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))) by AMI_1:13
.= Exec (a,k1 <>0_goto 2),((Initialize s) +* (stop (if<>0 a,k1,I))) by A2, Th22, I2 ;
not a in dom (Initialize (stop (if<>0 a,k1,I))) by SCMPDS_4:31;
then ((Initialize s) +* (stop (if<>0 a,k1,I))) . (DataLoc (((Initialize s) +* (stop (if<>0 a,k1,I))) . a),k1) = ((Initialize s) +* (stop (if<>0 a,k1,I))) . (DataLoc (s . a),k1) by I2, FUNCT_4:12
.= 0 by A1, A6, I2, FUNCT_4:12 ;
then A9: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) = succ (IC ((Initialize s) +* (stop (if<>0 a,k1,I)))) by A8, SCMPDS_2:67
.= 0 + 1 by A7 ;
A10: 1 in dom (if<>0 a,k1,I) by Lm9;
then 1 in dom (stop (if<>0 a,k1,I)) by Th18;
then A11: (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) . 1 = (stop (if<>0 a,k1,I)) . 1 by A4, GRFUNC_1:8
.= (if<>0 a,k1,I) . 1 by A10, Th19
.= goto ((card I) + 1) by Lm10 ;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1)) = (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1)) by COMPOS_1:38;
T: ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) by AMI_1:123;
A12: Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),(1 + 1) = Following (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) by AMI_1:14
.= Exec (goto ((card I) + 1)),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) by A9, A11, Y, T ;
then A13: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) = ICplusConst (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1),((card I) + 1) by SCMPDS_2:66
.= ((card I) + 1) + 1 by A9, Th23
.= (card I) + (1 + 1) ;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2)) = (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2)) by COMPOS_1:38;
TX: ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) by AMI_1:123;
A14: card (if<>0 a,k1,I) = (card I) + 2 by Lm8;
then (card I) + 2 in dom (stop (if<>0 a,k1,I)) by Th25;
then (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) . ((card I) + 2) = (stop (if<>0 a,k1,I)) . ((card I) + 2) by A5, GRFUNC_1:8
.= halt SCMPDS by A14, Th25 ;
then A15: CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) = halt SCMPDS by A13, Y, TX;
then A16: ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I))) halts_on (Initialize s) +* (stop (if<>0 a,k1,I)) by AMI_1:146;
A17: CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))) = a,k1 <>0_goto 2 by A2, Th22, I2;

now

let l be Element of NAT ; :: thesis: ( l < 1 + 1 implies CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),b₁) <> halt SCMPDS )
assume l < 1 + 1 ; :: thesis: CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),b₁) <> halt SCMPDS
then A18: l <= 1 by NAT_1:13;
u: Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),0 = (Initialize s) +* (stop (if<>0 a,k1,I)) by AMI_1:13;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l)) = (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l)) by COMPOS_1:38;
TX3: ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) by AMI_1:123;

per cases ( l = 0 or l = 1 ) by A18, NAT_1:26;

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

then CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) = CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))) by u;
hence CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) <> halt SCMPDS by A17, Th29; :: thesis: verum

end;

suppose l = 1 ; :: thesis: CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),b₁) <> halt SCMPDS

hence CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) <> halt SCMPDS by A9, A11, Y, TX3, SCMPDS_2:85; :: thesis: verum

end;

end;

end;

then for l being Element of NAT st CurInstr (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),l) = halt SCMPDS holds
2 <= l ;
then LifeSpan (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))) = 2 by A15, A16, AMI_1:def 46;
then A19: Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2 = Result (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))) by A16, AMI_1:122;
A20: dom (ProgramPart s) = NAT by COMPOS_1:34;

A21: now

A22: dom (Start-At ((card I) + 2),SCMPDS ) = {(IC SCMPDS )} by FUNCOP_1:19;
let x be set ; :: thesis: ( x in dom (IExec (if<>0 a,k1,I),s) implies (IExec (if<>0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁ )
assume A24: x in dom (IExec (if<>0 a,k1,I),s) ; :: thesis: (IExec (if<>0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A24, SCMPDS_4:20;

suppose A25: x is Int_position ; :: thesis: (IExec (if<>0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A26: not x in dom (Start-At ((card I) + 2),SCMPDS ) by A22, TARSKI:def 1;
not x in dom (s | NAT ) by A20, A25, SCMPDS_2:53;
hence (IExec (if<>0 a,k1,I),s) . x = (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),2) . x by A19, FUNCT_4:12
.= (Comput (ProgramPart ((Initialize s) +* (stop (if<>0 a,k1,I)))),((Initialize s) +* (stop (if<>0 a,k1,I))),1) . x by A12, A25, SCMPDS_2:66
.= ((Initialize s) +* (stop (if<>0 a,k1,I))) . x by A8, A25, SCMPDS_2:67
.= s . x by A25, SCMPDS_5:19
.= (s +* (Start-At ((card I) + 2),SCMPDS )) . x by A26, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A27: x = IC SCMPDS ; :: thesis: (IExec (if<>0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

not x in dom (s | NAT ) by A20, A27, COMPOS_1:3;
hence (IExec (if<>0 a,k1,I),s) . x = (card I) + 2 by A13, A19, A27, FUNCT_4:12
.= (s +* (Start-At ((card I) + 2),SCMPDS )) . x by A27, FUNCT_4:121 ;
:: thesis: verum

end;

suppose x is Element of NAT ; :: thesis: (IExec (if<>0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

hence (IExec (if<>0 a,k1,I),s) . x = (s +* (Start-At ((card I) + 2),SCMPDS )) . x by Th27; :: thesis: verum

end;

end;

end;

dom (IExec (if<>0 a,k1,I),s) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom (s +* (Start-At ((card I) + 2),SCMPDS )) by PARTFUN1:def 4 ;
hence IExec (if<>0 a,k1,I),s = s +* (Start-At ((card I) + 2),SCMPDS ) by A21, FUNCT_1:9; :: thesis: verum