let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) >= 0 ; :: thesis: ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) >=0_goto ((card I) + 1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
A3: IC (Initialize s) = 0 by COMPOS_1:223;
A4: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:59;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:59;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A4, FUNCT_4:12 ;
Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:4
.= Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:3
.= Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s)) by Th22 ;
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A1, A5, SCMPDS_2:69
.= 0 + ((card I) + 1) by A3, Th23 ;
A7: card (if<0 (a,k1,I)) = (card I) + 1 by Th15;
then A8: (card I) + 1 in dom (stop (if<0 (a,k1,I))) by Th25;
A9: (P +* (stop (if<0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:158;
stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) by FUNCT_4:26;
then stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) ;
then (P +* (stop (if<0 (a,k1,I)))) . ((card I) + 1) = (stop (if<0 (a,k1,I))) . ((card I) + 1) by A8, GRFUNC_1:8
.= halt SCMPDS by A7, Th25 ;
then A11: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A6, A9;

now

let k be Element of NAT ; :: thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b₁)) in dom (stop (if<0 (a,k1,I)))

per cases ( 0 < k or k = 0 ) ;

suppose 0 < k ; :: thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b₁)) in dom (stop (if<0 (a,k1,I)))

then 1 + 0 <= k by INT_1:20;
hence IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A8, A6, A11, EXTPRO_1:6; :: thesis: verum

end;

suppose k = 0 ; :: thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b₁)) in dom (stop (if<0 (a,k1,I)))

then Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:3;
hence IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A3, COMPOS_1:135; :: thesis: verum

end;

end;

end;

hence if<0 (a,k1,I) is_closed_on s,P by Def2; :: thesis: if<0 (a,k1,I) is_halting_on s,P
P +* (stop (if<0 (a,k1,I))) halts_on Initialize s by A11, EXTPRO_1:30;
hence if<0 (a,k1,I) is_halting_on s,P by Def3; :: thesis: verum