let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
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)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be 0 -started 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)),P,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)),P,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)),P,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)),P,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 P3 = P +* (stop (while<0 (a,i,I)));
set s4 = Comput ((P +* (stop (while<0 (a,i,I)))),s,1);
set P4 = P +* (stop (while<0 (a,i,I)));
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set SAl = Start-At (((card I) + 2),SCMPDS);
I: Initialize s = s by MEMSTR_0:44;
A2: IC s = 0 by MEMSTR_0:def 8;
A3: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A15: CurInstr ((P +* (stop (while<0 (a,i,I)))),s) = (a,i) >=0_goto ((card I) + 2) by A3, SCMPDS_6:11, I;
A6: Comput ((P +* (stop (while<0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while<0 (a,i,I)))),s) by EXTPRO_1:2
.= Exec (((a,i) >=0_goto ((card I) + 2)),s) by A3, SCMPDS_6:11, I ;
A7: stop (while<0 (a,i,I)) c= P +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
A8: IExec ((while<0 (a,i,I)),P,s) = Result ((P +* (stop (while<0 (a,i,I)))),s) by SCMPDS_4:def 5;
assume s . (DataLoc ((s . a),i)) >= 0 ; :: thesis: IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) = ICplusConst (s,((card I) + 2)) by A6, SCMPDS_2:57
.= 0 + ((card I) + 2) by A2, SCMPDS_6:12 ;
A12: card (while<0 (a,i,I)) = (card I) + 2 by Th6;
then (card I) + 2 in dom (stop (while<0 (a,i,I))) by COMPOS_1:64;
then (P +* (stop (while<0 (a,i,I)))) . ((card I) + 2) = (stop (while<0 (a,i,I))) . ((card I) + 2) by A7, GRFUNC_1:2
.= halt SCMPDS by A12, COMPOS_1:64 ;
then A13: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1))) = halt SCMPDS by A9, PBOOLE:143;
then A14: P +* (stop (while<0 (a,i,I))) halts_on s by EXTPRO_1:29;

now

let l be Element of NAT ; :: thesis: ( l < 0 + 1 implies CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS )
A16: Comput ((P +* (stop (while<0 (a,i,I)))),s,0) = s by EXTPRO_1:2;
assume l < 0 + 1 ; :: thesis: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS
then l = 0 by NAT_1:13;
hence CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS by A15, A16; :: thesis: verum

end;

then for l being Element of NAT st CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) = halt SCMPDS holds
1 <= l ;
then B17: LifeSpan ((P +* (stop (while<0 (a,i,I)))),s) = 1 by A13, A14, EXTPRO_1:def 15;

A19: now

let x be set ; :: thesis: ( x in dom (IExec ((while<0 (a,i,I)),P,s)) implies (IExec ((while<0 (a,i,I)),P,s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁ )
A20: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A21: x in dom (IExec ((while<0 (a,i,I)),P,s)) ; :: thesis: (IExec ((while<0 (a,i,I)),P,s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A21, SCMPDS_4:6;

suppose A22: x is Int_position ; :: thesis: (IExec ((while<0 (a,i,I)),P,s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A23: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A20, TARSKI:def 1;
thus (IExec ((while<0 (a,i,I)),P,s)) . x = (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) . x by B17, A8, A14, EXTPRO_1:23
.= s . x by A6, A22, SCMPDS_2:57
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A23, FUNCT_4:11 ; :: thesis: verum

end;

suppose A24: x = IC ; :: thesis: (IExec ((while<0 (a,i,I)),P,s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

hence (IExec ((while<0 (a,i,I)),P,s)) . x = (card I) + 2 by A9, B17, A8, A14, EXTPRO_1:23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A24, FUNCT_4:113 ;
:: thesis: verum

end;

end;

end;

dom (IExec ((while<0 (a,i,I)),P,s)) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by A19, FUNCT_1:2; :: thesis: verum