let s be State of ; :: thesis: for I being Program of
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 ; :: 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 2;
set j = goto ((card I) + 1);
set IF = if<=0 a,k1,I;
set pIF = stop (if<=0 a,k1,I);
set IsIF = Initialized (stop (if<=0 a,k1,I));
set s3 = s +* (Initialized (stop (if<=0 a,k1,I)));
set s4 = Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1;
set s5 = Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2;
A2: if<=0 a,k1,I = (a,k1 <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A3: not DataLoc (s . a),k1 in dom (Initialized (stop (if<=0 a,k1,I))) by SCMPDS_4:31;
not a in dom (Initialized (stop (if<=0 a,k1,I))) by SCMPDS_4:31;
then A4: (s +* (Initialized (stop (if<=0 a,k1,I)))) . (DataLoc ((s +* (Initialized (stop (if<=0 a,k1,I)))) . a),k1) = (s +* (Initialized (stop (if<=0 a,k1,I)))) . (DataLoc (s . a),k1) by FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A3, FUNCT_4:12 ;
A5: IC (s +* (Initialized (stop (if<=0 a,k1,I)))) = inspos 0 by FUNCT_4:26, SCMPDS_5:18;
Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),(0 + 1) = Following (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),0 ) by AMI_1:14
.= Following (s +* (Initialized (stop (if<=0 a,k1,I)))) by AMI_1:13
.= Exec (a,k1 <=0_goto 2),(s +* (Initialized (stop (if<=0 a,k1,I)))) by A2, Th22 ;
then A6: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) = Next (IC (s +* (Initialized (stop (if<=0 a,k1,I))))) by A1, A4, SCMPDS_2:68
.= inspos (0 + 1) by A5 ;
A7: Initialized (stop (if<=0 a,k1,I)) c= s +* (Initialized (stop (if<=0 a,k1,I))) by FUNCT_4:26;
then A8: stop (if<=0 a,k1,I) c= Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1 by AMI_1:81, SCMPDS_4:57;
A9: inspos 1 in dom (if<=0 a,k1,I) by Lm9;
then inspos 1 in dom (stop (if<=0 a,k1,I)) by Th18;
then A10: (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) . (inspos 1) = (stop (if<=0 a,k1,I)) . (inspos 1) by A8, GRFUNC_1:8
.= (if<=0 a,k1,I) . (inspos 1) by A9, Th19
.= goto ((card I) + 1) by Lm10 ;
A11: card (if<=0 a,k1,I) = (card I) + 2 by Lm8;
then A12: inspos ((card I) + 2) in dom (stop (if<=0 a,k1,I)) by Th25;
Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),(1 + 1) = Following (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) by AMI_1:14
.= Exec (goto ((card I) + 1)),(Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) by A6, A10 ;
then A13: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) = ICplusConst (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1),((card I) + 1) by SCMPDS_2:66
.= inspos (((card I) + 1) + 1) by A6, Th23
.= inspos ((card I) + (1 + 1)) ;
stop (if<=0 a,k1,I) c= Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2 by A7, AMI_1:81, SCMPDS_4:57;
then (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) . (inspos ((card I) + 2)) = (stop (if<=0 a,k1,I)) . (inspos ((card I) + 2)) by A12, GRFUNC_1:8
.= halt SCMPDS by A11, Th25 ;
then A14: CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) = halt SCMPDS by A13;

now

let k be Element of NAT ; :: thesis: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),b₁) in dom (stop (if<=0 a,k1,I))
A15: ( k = 0 or 0 + 1 <= k ) by INT_1:20;

per cases ( k = 0 or k = 1 or 1 < k ) by A15, XXREAL_0:1;

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

then Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),k = s +* (Initialized (stop (if<=0 a,k1,I))) by AMI_1:13;
hence IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),k) in dom (stop (if<=0 a,k1,I)) by A5, SCMPDS_4:75; :: thesis: verum

end;

suppose k = 1 ; :: thesis: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),b₁) in dom (stop (if<=0 a,k1,I))

hence IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),k) in dom (stop (if<=0 a,k1,I)) by A9, A6, Th18; :: thesis: verum

end;

suppose 1 < k ; :: thesis: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),b₁) in dom (stop (if<=0 a,k1,I))

then 1 + 1 <= k by INT_1:20;
hence IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),k) in dom (stop (if<=0 a,k1,I)) by A12, A13, A14, AMI_1:52; :: 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 (s +* (Initialized (stop (if<=0 a,k1,I)))) halts_on s +* (Initialized (stop (if<=0 a,k1,I))) by A14, AMI_1:146;
hence if<=0 a,k1,I is_halting_on s by Def3; :: thesis: verum