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 COMPOS_1:125;
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, COMPOS_1:132;
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, COMPOS_1:132;
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 EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (while<>0 (a,i,I))))),((Initialize s) +* (stop (while<>0 (a,i,I))))) by EXTPRO_1:3
.= 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 EXTPRO_1:4
.= 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 EXTPRO_1:30;
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, EXTPRO_1:def 8;
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, FUNCT_4:121 ;
:: 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