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);
A1: Initialize s = s by MEMSTR_0:44;
A2: IC s = 0 by MEMSTR_0:def 11;
A3: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A4: 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)
.= Exec (((a,i) <=0_goto ((card I) + 2)),s) by A3, A1, SCMPDS_6:11 ;
A5: stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
A6: 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 A7: IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) = ICplusConst (s,((card I) + 2)) by A4, SCMPDS_2:56
.= 0 + ((card I) + 2) by A2, SCMPDS_6:12 ;
A8: card (while>0 (a,i,I)) = (card I) + 2 by Th15;
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 A5, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1))) = halt SCMPDS by A7, PBOOLE:143;
then A10: P +* (stop (while>0 (a,i,I))) halts_on s by EXTPRO_1:29;
A11: CurInstr ((P +* (stop (while>0 (a,i,I)))),s) = (a,i) <=0_goto ((card I) + 2) by A3, A1, SCMPDS_6:11;

now :: thesis: for l being Nat st l < 0 + 1 holds
CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDS

let l be 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 )
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 A11; :: thesis: verum

end;

then for l being 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 A12: LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) = 1 by A9, A10, EXTPRO_1:def 15;

A13: now :: thesis: for x being object st x in dom (IExec ((while>0 (a,i,I)),P,s)) holds
(IExec ((while>0 (a,i,I)),P,s)) . x = (s +* (Start-At (((card I) + 2),SCMPDS))) . x

let x be object ; :: 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 )
A14: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A15: 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 A15, SCMPDS_4:6;

suppose A16: 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 A17: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A14, 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 A6, A12, A10, EXTPRO_1:23
.= s . x by A4, A16, SCMPDS_2:56
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A17, FUNCT_4:11 ; :: thesis: verum

end;

suppose A18: 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 A6, A7, A12, A10, EXTPRO_1:23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A18, 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 A13, FUNCT_1:2; :: thesis: verum