let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being Program of
for a, c being Int_position
for i being Integer
for n being Nat st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))

let s be State of SCMPDS; :: thesis: for I being Program of
for a, c being Int_position
for i being Integer
for n being Nat st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))

let I be Program of ; :: thesis: for a, c being Int_position
for i being Integer
for n being Nat st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))

let a, c be Int_position; :: thesis: for i being Integer
for n being Nat st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))

let i be Integer; :: thesis: for n being Nat st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))

let n be Nat; :: thesis: ( s . (DataLoc ((s . a),i)) >= 0 implies IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS)) )
set d1 = DataLoc ((s . a),i);
set FOR = for-up (a,i,n,I);
set pFOR = stop (for-up (a,i,n,I));
set s3 = Initialize s;
set P3 = P +* (stop (for-up (a,i,n,I)));
set s4 = Comput ((P +* (stop (for-up (a,i,n,I)))),(),1);
set P4 = P +* (stop (for-up (a,i,n,I)));
set i1 = (a,i) >=0_goto ((card I) + 3);
set i3 = goto (- ((card I) + 2));
set SAl = Start-At (((card I) + 3),SCMPDS);
A1: IC () = 0 by MEMSTR_0:def 11;
A2: not DataLoc ((s . a),i) in dom () by SCMPDS_4:18;
A3: stop (for-up (a,i,n,I)) c= P +* (stop (for-up (a,i,n,I))) by FUNCT_4:25;
not a in dom () by SCMPDS_4:18;
then A4: () . (DataLoc ((() . a),i)) = () . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by ;
A5: for-up (a,i,n,I) = ((a,i) >=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,n))) ';' (goto (- ((card I) + 2)))) by Th2;
A6: Comput ((P +* (stop (for-up (a,i,n,I)))),(),(0 + 1)) = Following ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),0))) by EXTPRO_1:3
.= Exec (((a,i) >=0_goto ((card I) + 3)),()) by ;
assume s . (DataLoc ((s . a),i)) >= 0 ; :: thesis: IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS))
then A7: IC (Comput ((P +* (stop (for-up (a,i,n,I)))),(),1)) = ICplusConst ((),((card I) + 3)) by
.= 0 + ((card I) + 3) by ;
A8: card (for-up (a,i,n,I)) = (card I) + 3 by Th30;
then (card I) + 3 in dom (stop (for-up (a,i,n,I))) by COMPOS_1:64;
then (P +* (stop (for-up (a,i,n,I)))) . ((card I) + 3) = (stop (for-up (a,i,n,I))) . ((card I) + 3) by
.= halt SCMPDS by ;
then A9: CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),1))) = halt SCMPDS by ;
then A10: P +* (stop (for-up (a,i,n,I))) halts_on Initialize s by EXTPRO_1:29;
A11: CurInstr ((P +* (stop (for-up (a,i,n,I)))),()) = (a,i) >=0_goto ((card I) + 3) by ;
now :: thesis: for l being Nat st l < 0 + 1 holds
CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),l))) <> halt SCMPDS
let l be Nat; :: thesis: ( l < 0 + 1 implies CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),l))) <> halt SCMPDS )
assume l < 0 + 1 ; :: thesis: CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),l))) <> halt SCMPDS
then l = 0 by NAT_1:13;
hence CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),l))) <> halt SCMPDS by A11; :: thesis: verum
end;
then for l being Nat st CurInstr ((P +* (stop (for-up (a,i,n,I)))),(Comput ((P +* (stop (for-up (a,i,n,I)))),(),l))) = halt SCMPDS holds
1 <= l ;
then A12: LifeSpan ((P +* (stop (for-up (a,i,n,I)))),()) = 1 by ;
A13: now :: thesis: for x being object st x in dom (IExec ((for-up (a,i,n,I)),P,())) holds
(IExec ((for-up (a,i,n,I)),P,())) . x = (s +* (Start-At (((card I) + 3),SCMPDS))) . x
let x be object ; :: thesis: ( x in dom (IExec ((for-up (a,i,n,I)),P,())) implies (IExec ((for-up (a,i,n,I)),P,())) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1 )
A14: dom (Start-At (((card I) + 3),SCMPDS)) = {()} by FUNCOP_1:13;
assume A15: x in dom (IExec ((for-up (a,i,n,I)),P,())) ; :: thesis: (IExec ((for-up (a,i,n,I)),P,())) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1
per cases by ;
suppose A16: x is Int_position ; :: thesis: (IExec ((for-up (a,i,n,I)),P,())) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1
then x <> IC by SCMPDS_2:43;
then A17: not x in dom (Start-At (((card I) + 3),SCMPDS)) by ;
A18: not x in dom () by ;
thus (IExec ((for-up (a,i,n,I)),P,())) . x = (Comput ((P +* (stop (for-up (a,i,n,I)))),(),1)) . x by
.= () . x by
.= s . x by
.= (s +* (Start-At (((card I) + 3),SCMPDS))) . x by ; :: thesis: verum
end;
suppose A19: x = IC ; :: thesis: (IExec ((for-up (a,i,n,I)),P,())) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1
thus (IExec ((for-up (a,i,n,I)),P,())) . x = (card I) + 3 by
.= (s +* (Start-At (((card I) + 3),SCMPDS))) . x by ; :: thesis: verum
end;
end;
end;
dom (IExec ((for-up (a,i,n,I)),P,())) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom (s +* (Start-At (((card I) + 3),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((for-up (a,i,n,I)),P,()) = s +* (Start-At (((card I) + 3),SCMPDS)) by ; :: thesis: verum