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 COMPOS_1:125;
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 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 ((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, COMPOS_1:132;
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, EXTPRO_1:30;
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 EXTPRO_1:3;
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; :: 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, EXTPRO_1:def 14;
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, EXTPRO_1:23;
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, 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) + 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