let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc (s . a),i) = 0 holds
IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS )
let I be Program of SCMPDS ; :: thesis: for a, c being Int_position
for i being Integer st s . (DataLoc (s . a),i) = 0 holds
IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS )
let a, c be Int_position ; :: thesis: for i being Integer st s . (DataLoc (s . a),i) = 0 holds
IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS )
let i be Integer; :: thesis: ( s . (DataLoc (s . a),i) = 0 implies IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS ) )
set b = DataLoc (s . a),i;
assume A1: s . (DataLoc (s . a),i) = 0 ; :: thesis: IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS )
set i1 = a,i <>0_goto 2;
set i2 = goto ((card I) + 2);
set i3 = goto (- ((card I) + 2));
set WHL = while<>0 a,i,I;
set pWH = stop (while<>0 a,i,I);
set iWH = Initialize (stop (while<>0 a,i,I));
set s3 = (Initialize s) +* (stop (while<>0 a,i,I));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1;
set s5 = Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2;
I1: (Initialize s) +* (stop (while<>0 a,i,I)) = s +* (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:5;
A2: IC ((Initialize s) +* (stop (while<>0 a,i,I))) = 0 by SCMPDS_6:21;
A3: Initialize (stop (while<>0 a,i,I)) c= (Initialize s) +* (stop (while<>0 a,i,I)) by I1, FUNCT_4:26;
then A4: stop (while<>0 a,i,I) c= Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1 by AMI_1:81, SCMPDS_4:57;
A5: stop (while<>0 a,i,I) c= Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2 by A3, AMI_1:81, SCMPDS_4:57;
A6: not DataLoc (s . a),i in dom (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:31;
A7: while<>0 a,i,I = (a,i <>0_goto 2) ';' (((goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 2)))) by Lm10;
A8: Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))) by AMI_1:13
.= Exec (a,i <>0_goto 2),((Initialize s) +* (stop (while<>0 a,i,I))) by A7, I1, SCMPDS_6:22 ;
A9: 1 in dom (while<>0 a,i,I) by Th14;
then 1 in dom (stop (while<>0 a,i,I)) by SCMPDS_6:18;
then A10: (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) . 1 = (stop (while<>0 a,i,I)) . 1 by A4, GRFUNC_1:8
.= (while<>0 a,i,I) . 1 by A9, SCMPDS_6:19
.= goto ((card I) + 2) by Th15 ;
set SAl = Start-At ((card I) + 3),SCMPDS ;
A11: dom (ProgramPart s) = NAT by COMPOS_1:34;
not a in dom (Initialize (stop (while<>0 a,i,I))) by SCMPDS_4:31;
then ((Initialize s) +* (stop (while<>0 a,i,I))) . (DataLoc (((Initialize s) +* (stop (while<>0 a,i,I))) . a),i) = ((Initialize s) +* (stop (while<>0 a,i,I))) . (DataLoc (s . a),i) by I1, FUNCT_4:12
.= 0 by A1, A6, I1, FUNCT_4:12 ;
then A12: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) = succ (IC ((Initialize s) +* (stop (while<>0 a,i,I)))) by A8, SCMPDS_2:67
.= 0 + 1 by A2 ;
T4: ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by AMI_1:123;
T5: ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) by AMI_1:123;
Y: (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1)) = (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1)) by T4, COMPOS_1:38;
A13: Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),(1 + 1) = Following (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by AMI_1:14
.= Exec (goto ((card I) + 2)),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) by A12, A10, Y ;
then A14: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) = ICplusConst (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1),((card I) + 2) by SCMPDS_2:66
.= ((card I) + 2) + 1 by A12, SCMPDS_6:23
.= (card I) + (2 + 1) ;
Y: (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)) = (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2)) by T5, COMPOS_1:38;
A15: card (while<>0 a,i,I) = (card I) + 3 by Th12;
then (card I) + 3 in dom (stop (while<>0 a,i,I)) by SCMPDS_6:25;
then (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) . ((card I) + 3) = (stop (while<>0 a,i,I)) . ((card I) + 3) by A5, GRFUNC_1:8
.= halt SCMPDS by A15, SCMPDS_6:25 ;
then A16: CurInstr (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) = halt SCMPDS by A14, Y;
then ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I))) halts_on (Initialize s) +* (stop (while<>0 a,i,I)) by AMI_1:146;
then A17: Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2 = Result (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))) by A16, AMI_1:def 22;
A18: IExec (while<>0 a,i,I),s = (Result (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I)))) +* (s | NAT ) by SCMPDS_4:def 8;

A19: now

let x be set ; :: thesis: ( x in dom (IExec (while<>0 a,i,I),s) implies (IExec (while<>0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 3),SCMPDS )) . b₁ )
A20: dom (Start-At ((card I) + 3),SCMPDS ) = {(IC SCMPDS )} by FUNCOP_1:19;
assume A21: x in dom (IExec (while<>0 a,i,I),s) ; :: thesis: (IExec (while<>0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 3),SCMPDS )) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A21, SCMPDS_4:20;

suppose A22: x is Int_position ; :: thesis: (IExec (while<>0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 3),SCMPDS )) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A23: not x in dom (Start-At ((card I) + 3),SCMPDS ) by A20, TARSKI:def 1;
not x in dom (s | NAT ) by A11, A22, SCMPDS_2:53;
hence (IExec (while<>0 a,i,I),s) . x = (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),2) . x by A17, A18, FUNCT_4:12
.= (Comput (ProgramPart ((Initialize s) +* (stop (while<>0 a,i,I)))),((Initialize s) +* (stop (while<>0 a,i,I))),1) . x by A13, A22, SCMPDS_2:66
.= ((Initialize s) +* (stop (while<>0 a,i,I))) . x by A8, A22, SCMPDS_2:67
.= s . x by A22, SCMPDS_5:19
.= (s +* (Start-At ((card I) + 3),SCMPDS )) . x by A23, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A24: x = IC SCMPDS ; :: thesis: (IExec (while<>0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 3),SCMPDS )) . b₁

not x in dom (s | NAT ) by A11, A24, COMPOS_1:3;
hence (IExec (while<>0 a,i,I),s) . x = (card I) + 3 by A14, A17, A18, A24, FUNCT_4:12
.= (s +* (Start-At ((card I) + 3),SCMPDS )) . x by A24, SCMPDS_7:12 ;
:: thesis: verum

end;

suppose x is Element of NAT ; :: thesis: (IExec (while<>0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 3),SCMPDS )) . b₁

hence (IExec (while<>0 a,i,I),s) . x = (s +* (Start-At ((card I) + 3),SCMPDS )) . x by SCMPDS_6:27; :: thesis: verum

end;

end;

end;

dom (IExec (while<>0 a,i,I),s) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom (s +* (Start-At ((card I) + 3),SCMPDS )) by PARTFUN1:def 4 ;
hence IExec (while<>0 a,i,I),s = s +* (Start-At ((card I) + 3),SCMPDS ) by A19, FUNCT_1:9; :: thesis: verum