let s be State of SCMPDS ; :: thesis:
let I be Program of SCMPDS ; :: thesis:
let a, c be Int_position ; :: thesis:
let i be Integer; :: thesis:
set b = DataLoc (s . a),i;
assume A1: s . (DataLoc (s . a),i) = 0 ; :: thesis:
set WHL = while<>0 a,i,I;
set pWH = stop (while<>0 a,i,I);
set iWH = Initialized (stop (while<>0 a,i,I));
set s3 = s +* (Initialized (stop (while<>0 a,i,I)));
set s4 = Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1;
set s5 = Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2;
set i1 = a,i <>0_goto 2;
set i2 = goto ((card I) + 2);
set i3 = goto (- ((card I) + 2));
set SAl = Start-At (inspos ((card I) + 3));
A2: IC (s +* (Initialized (stop (while<>0 a,i,I)))) = inspos 0 by SCMPDS_6:21;
A3: while<>0 a,i,I = (a,i <>0_goto 2) ';' (((goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 2)))) by Lm10;
A4: Computation (s +* (Initialized (stop (while<>0 a,i,I)))),(0 + 1) = Following (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),0 ) by AMI_1:14
.= Following (s +* (Initialized (stop (while<>0 a,i,I)))) by AMI_1:13
.= Exec (a,i <>0_goto 2),(s +* (Initialized (stop (while<>0 a,i,I)))) by A3, SCMPDS_6:22 ;
A5: ( not DataLoc (s . a),i in dom (Initialized (stop (while<>0 a,i,I))) & DataLoc (s . a),i in dom s ) by SCMPDS_2:49, SCMPDS_4:31;
( not a in dom (Initialized (stop (while<>0 a,i,I))) & a in dom s ) by SCMPDS_2:49, SCMPDS_4:31;
then A6: (s +* (Initialized (stop (while<>0 a,i,I)))) . (DataLoc ((s +* (Initialized (stop (while<>0 a,i,I)))) . a),i) = (s +* (Initialized (stop (while<>0 a,i,I)))) . (DataLoc (s . a),i) by FUNCT_4:12
.= 0 by A1, A5, FUNCT_4:12 ;
A7: Initialized (stop (while<>0 a,i,I)) c= s +* (Initialized (stop (while<>0 a,i,I))) by FUNCT_4:26;
then A8: stop (while<>0 a,i,I) c= Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1 by AMI_1:81, SCMPDS_4:57;
A9: inspos 1 in dom (while<>0 a,i,I) by Th14;
then A10: inspos 1 in dom (stop (while<>0 a,i,I)) by SCMPDS_6:18;
A11: IC (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1) = Next (IC (s +* (Initialized (stop (while<>0 a,i,I))))) by A4, A6, SCMPDS_2:67
.= inspos (0 + 1) by A2 ;
A12: (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1) . (inspos 1) = (stop (while<>0 a,i,I)) . (inspos 1) by A8, A10, GRFUNC_1:8
.= (while<>0 a,i,I) . (inspos 1) by A9, SCMPDS_6:19
.= goto ((card I) + 2) by Th15 ;
A13: Computation (s +* (Initialized (stop (while<>0 a,i,I)))),(1 + 1) = Following (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1) by AMI_1:14
.= Exec (goto ((card I) + 2)),(Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1) by A11, A12 ;
A14: stop (while<>0 a,i,I) c= Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2 by A7, AMI_1:81, SCMPDS_4:57;
A15: card (while<>0 a,i,I) = (card I) + 3 by Th12;
then A16: inspos ((card I) + 3) in dom (stop (while<>0 a,i,I)) by SCMPDS_6:25;
A17: IC (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2) = ICplusConst (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1),((card I) + 2) by A13, SCMPDS_2:66
.= inspos (((card I) + 2) + 1) by A11, SCMPDS_6:23
.= inspos ((card I) + (2 + 1)) ;
(Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2) . (inspos ((card I) + 3)) = (stop (while<>0 a,i,I)) . (inspos ((card I) + 3)) by A14, A16, GRFUNC_1:8
.= halt SCMPDS by A15, SCMPDS_6:25 ;
then A18: CurInstr (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2) = halt SCMPDS by A17;
then s +* (Initialized (stop (while<>0 a,i,I))) is halting by AMI_1:def 20;
then A19: Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2 = Result (s +* (Initialized (stop (while<>0 a,i,I)))) by A18, AMI_1:def 22;
A20: IExec (while<>0 a,i,I),s = (Result (s +* (Initialized (stop (while<>0 a,i,I))))) +* (s | NAT ) by SCMPDS_4:def 8;
A21: dom (s | NAT ) = NAT by SCMPDS_6:1;

A22: now

let x be set ; :: thesis:
assume A23: x in dom (IExec (while<>0 a,i,I),s) ; :: thesis:
A24: dom (Start-At (inspos ((card I) + 3))) = {(IC SCMPDS )} by FUNCOP_1:19;

per cases by A23, SCMPDS_4:20;

suppose A25: x is Int_position ; :: thesis:

then x <> IC SCMPDS by SCMPDS_2:52;
then A26: not x in dom (Start-At (inspos ((card I) + 3))) by A24, TARSKI:def 1;

now

assume x in dom (s | NAT ) ; :: thesis:
then reconsider l = x as Instruction-Location of SCMPDS by A21, AMI_1:def 4;
l = x ;
hence contradiction by A25, SCMPDS_2:53; :: thesis:

end;

hence (IExec (while<>0 a,i,I),s) . x = (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),2) . x by A19, A20, FUNCT_4:12
.= (Computation (s +* (Initialized (stop (while<>0 a,i,I)))),1) . x by A13, A25, SCMPDS_2:66
.= (s +* (Initialized (stop (while<>0 a,i,I)))) . x by A4, A25, SCMPDS_2:67
.= s . x by A25, SCMPDS_5:19
.= (s +* (Start-At (inspos ((card I) + 3)))) . x by A26, FUNCT_4:12 ;
:: thesis:

end;

suppose A27: x = IC SCMPDS ; :: thesis:

now

assume x in dom (s | NAT ) ; :: thesis:
then reconsider l = x as Instruction-Location of SCMPDS by A21, AMI_1:def 4;
l = x ;
hence contradiction by A27, AMI_1:48; :: thesis:

end;

hence (IExec (while<>0 a,i,I),s) . x = inspos ((card I) + 3) by A19, A20, A17, A27, FUNCT_4:12
.= (s +* (Start-At (inspos ((card I) + 3)))) . x by A27, SCMPDS_7:12 ;
:: thesis:

end;

suppose x is Instruction-Location of SCMPDS ; :: thesis:

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

end;

dom (IExec (while<>0 a,i,I),s) = the carrier of SCMPDS by AMI_1:79
.= dom (s +* (Start-At (inspos ((card I) + 3)))) by AMI_1:79 ;
hence IExec (while<>0 a,i,I),s = s +* (Start-At (inspos ((card I) + 3))) by A22, FUNCT_1:9; :: thesis: