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 & if>0 (a,k1,I) is_halting_on s )
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 & if>0 (a,k1,I) is_halting_on s )
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 & if>0 (a,k1,I) is_halting_on s )
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies ( if>0 (a,k1,I) is_closed_on s & if>0 (a,k1,I) is_halting_on s ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) <= 0 ; :: thesis: ( if>0 (a,k1,I) is_closed_on s & if>0 (a,k1,I) is_halting_on s )
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) +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1);
I2: s +* (Initialize (stop (if>0 (a,k1,I)))) = (Initialize s) +* (stop (if>0 (a,k1,I))) by COMPOS_1:125;
A2: IC ((Initialize s) +* (stop (if>0 (a,k1,I)))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A3: not DataLoc ((s . a),k1) in dom (Initialize (stop (if>0 (a,k1,I)))) by SCMPDS_4:31;
not a in dom (Initialize (stop (if>0 (a,k1,I)))) by SCMPDS_4:31;
then A4: ((Initialize s) +* (stop (if>0 (a,k1,I)))) . (DataLoc ((((Initialize s) +* (stop (if>0 (a,k1,I)))) . a),k1)) = ((Initialize s) +* (stop (if>0 (a,k1,I)))) . (DataLoc ((s . a),k1)) by I2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A3, I2, FUNCT_4:12 ;
Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),(0 + 1)) = Following ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I))))) by EXTPRO_1:3
.= Exec (((a,k1) <=0_goto ((card I) + 1)),((Initialize s) +* (stop (if>0 (a,k1,I))))) by Th22, I2 ;
then A5: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1)) = ICplusConst (((Initialize s) +* (stop (if>0 (a,k1,I)))),((card I) + 1)) by A1, A4, SCMPDS_2:68
.= 0 + ((card I) + 1) by A2, Th23 ;
A6: card (if>0 (a,k1,I)) = (card I) + 1 by Th15;
then A7: (card I) + 1 in dom (stop (if>0 (a,k1,I))) by Th25;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1))) /. (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1))) = (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1)) . (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1))) by COMPOS_1:38;
Initialize (stop (if>0 (a,k1,I))) c= (Initialize s) +* (stop (if>0 (a,k1,I))) by I2, FUNCT_4:26;
then stop (if>0 (a,k1,I)) c= Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1) by AMI_1:81, COMPOS_1:132;
then (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1)) . ((card I) + 1) = (stop (if>0 (a,k1,I))) . ((card I) + 1) by A7, GRFUNC_1:8
.= halt SCMPDS by A6, Th25 ;
then A8: CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1))),(Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1))) = halt SCMPDS by A5, Y;
TX: ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),1)) by AMI_1:123;

now

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

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

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

then 1 + 0 <= k by INT_1:20;
hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),k)) in dom (stop (if>0 (a,k1,I))) by A7, A5, A8, TX, EXTPRO_1:6; :: thesis: verum

end;

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

then Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),k) = (Initialize s) +* (stop (if>0 (a,k1,I))) by EXTPRO_1:3;
hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I))))),((Initialize s) +* (stop (if>0 (a,k1,I)))),k)) in dom (stop (if>0 (a,k1,I))) by A2, COMPOS_1:135; :: thesis: verum

end;

end;

end;

hence if>0 (a,k1,I) is_closed_on s by Def2; :: thesis: if>0 (a,k1,I) is_halting_on s
ProgramPart ((Initialize s) +* (stop (if>0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if>0 (a,k1,I))) by A8, TX, EXTPRO_1:30;
hence if>0 (a,k1,I) is_halting_on s by Def3; :: thesis: verum