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) + 2),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) + 2),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) + 2),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) + 2),SCMPDS ) )
set d1 = DataLoc (s . a),i;
set WHL = while<0 a,i,I;
set pWHL = 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 i1 = a,i >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
I1: s +* (Initialize (stop (while<0 a,i,I))) = (Initialize s) +* (stop (while<0 a,i,I)) by SCMPDS_4:5;
set SAl = Start-At ((card I) + 2),SCMPDS ;
A1: IC ((Initialize s) +* (stop (while<0 a,i,I))) = 0 by SCMPDS_6:21;
A2: while<0 a,i,I = (a,i >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:51;
A3: not DataLoc (s . a),i in dom (Initialize (stop (while<0 a,i,I))) by SCMPDS_4:31;
not a in dom (Initialize (stop (while<0 a,i,I))) by SCMPDS_4:31;
then A4: ((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
.= s . (DataLoc (s . a),i) by A3, I1, FUNCT_4:12 ;
A5: 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 ((card I) + 2)),((Initialize s) +* (stop (while<0 a,i,I))) by A2, I1, SCMPDS_6:22 ;
Initialize (stop (while<0 a,i,I)) c= (Initialize s) +* (stop (while<0 a,i,I)) by I1, FUNCT_4:26;
then A6: 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;
A7: 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;
assume s . (DataLoc (s . a),i) >= 0 ; :: thesis: IExec (while<0 a,i,I),s = s +* (Start-At ((card I) + 2),SCMPDS )
then A8: IC (Comput (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))),1) = ICplusConst ((Initialize s) +* (stop (while<0 a,i,I))),((card I) + 2) by A5, A4, SCMPDS_2:69
.= 0 + ((card I) + 2) by A1, SCMPDS_6:23 ;
Y: (ProgramPart (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)) = (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 COMPOS_1:38;
TX: ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))),1) = ProgramPart ((Initialize s) +* (stop (while<0 a,i,I))) by AMI_1:123;
A9: card (while<0 a,i,I) = (card I) + 2 by Th6;
then (card I) + 2 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))),1) . ((card I) + 2) = (stop (while<0 a,i,I)) . ((card I) + 2) by A6, GRFUNC_1:8
.= halt SCMPDS by A9, SCMPDS_6:25 ;
then A10: CurInstr (ProgramPart (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) = halt SCMPDS by A8, Y;
then A11: ProgramPart ((Initialize s) +* (stop (while<0 a,i,I))) halts_on (Initialize s) +* (stop (while<0 a,i,I)) by TX, AMI_1:146;
A12: CurInstr (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))) = a,i >=0_goto ((card I) + 2) by A2, I1, SCMPDS_6:22;

now

let l be Element of NAT ; :: thesis: ( l < 0 + 1 implies 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))),l) <> halt SCMPDS )
u: Comput (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))),0 = (Initialize s) +* (stop (while<0 a,i,I)) by AMI_1:13;
assume l < 0 + 1 ; :: thesis: 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))),l) <> halt SCMPDS
then l = 0 by NAT_1:13;
then 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))),l) = CurInstr (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))) by u;
hence 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))),l) <> halt SCMPDS by A12, SCMPDS_6:31; :: thesis: verum

end;

then for l being Element of NAT st 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))),l) = halt SCMPDS holds
1 <= l ;
then LifeSpan (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))) = 1 by A10, A11, TX, AMI_1:def 46;
then A13: Comput (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))),1 = Result (ProgramPart ((Initialize s) +* (stop (while<0 a,i,I)))),((Initialize s) +* (stop (while<0 a,i,I))) by A11, AMI_1:122;
A14: dom (ProgramPart s) = NAT by COMPOS_1:34;

A15: 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) + 2),SCMPDS )) . b₁ )
A16: dom (Start-At ((card I) + 2),SCMPDS ) = {(IC SCMPDS )} by FUNCOP_1:19;
assume A17: x in dom (IExec (while<0 a,i,I),s) ; :: thesis: (IExec (while<0 a,i,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 A17, SCMPDS_4:20;

suppose A18: x is Int_position ; :: thesis: (IExec (while<0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A19: not x in dom (Start-At ((card I) + 2),SCMPDS ) by A16, TARSKI:def 1;
not x in dom (s | NAT ) by A14, A18, 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))),1) . x by A13, A7, FUNCT_4:12
.= ((Initialize s) +* (stop (while<0 a,i,I))) . x by A5, A18, SCMPDS_2:69
.= s . x by A18, SCMPDS_5:19
.= (s +* (Start-At ((card I) + 2),SCMPDS )) . x by A19, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A20: x = IC SCMPDS ; :: thesis: (IExec (while<0 a,i,I),s) . b₁ = (s +* (Start-At ((card I) + 2),SCMPDS )) . b₁

not x in dom (s | NAT ) by A14, A20, COMPOS_1:3;
hence (IExec (while<0 a,i,I),s) . x = (card I) + 2 by A8, A13, A7, A20, FUNCT_4:12
.= (s +* (Start-At ((card I) + 2),SCMPDS )) . x by A20, 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) + 2),SCMPDS )) . b₁

hence (IExec (while<0 a,i,I),s) . x = (s +* (Start-At ((card I) + 2),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) + 2),SCMPDS )) by PARTFUN1:def 4 ;
hence IExec (while<0 a,i,I),s = s +* (Start-At ((card I) + 2),SCMPDS ) by A15, FUNCT_1:9; :: thesis: verum