let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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 i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let I be Program of SCMPDS; :: thesis: for a being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let a be Int_position ; :: thesis: for i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let i be Integer; :: thesis: for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let n be Element of NAT ; :: thesis: ( s . (DataLoc ((s . a),i)) <= 0 implies ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
set d1 = DataLoc ((s . a),i);
assume A1: s . (DataLoc ((s . a),i)) <= 0 ; :: thesis: ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
set i1 = (a,i) <=0_goto ((card I) + 3);
set i2 = AddTo (a,i,(- n));
set i3 = goto (- ((card I) + 2));
set FOR = for-down (a,i,n,I);
set pFOR = stop (for-down (a,i,n,I));
set s3 = Initialize s;
set P3 = P +* (stop (for-down (a,i,n,I)));
set s4 = Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1);
set P4 = P +* (stop (for-down (a,i,n,I)));
A3: IC (Initialize s) = 0 by MEMSTR_0:def 8;
A4: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),i)) = (Initialize s) . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by A4, FUNCT_4:11 ;
A6: for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2)))) by Th15;
Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (for-down (a,i,n,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,i) <=0_goto ((card I) + 3)),(Initialize s)) by A6, SCMPDS_6:11 ;
then A7: IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 3)) by A1, A5, SCMPDS_2:56
.= 0 + ((card I) + 3) by A3, SCMPDS_6:12 ;
A8: card (for-down (a,i,n,I)) = (card I) + 3 by Th60;
then A9: (card I) + 3 in dom (stop (for-down (a,i,n,I))) by COMPOS_1:64;
stop (for-down (a,i,n,I)) c= P +* (stop (for-down (a,i,n,I))) by FUNCT_4:25;
then (P +* (stop (for-down (a,i,n,I)))) . ((card I) + 3) = (stop (for-down (a,i,n,I))) . ((card I) + 3) by A9, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A11: CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1))) = halt SCMPDS by A7, PBOOLE:143;

now

let k be Element of NAT ; :: thesis: IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),b₁)) in dom (stop (for-down (a,i,n,I)))

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

suppose 0 < k ; :: thesis: IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),b₁)) in dom (stop (for-down (a,i,n,I)))

then 1 + 0 <= k by INT_1:7;
hence IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),k)) in dom (stop (for-down (a,i,n,I))) by A9, A7, A11, EXTPRO_1:5; :: thesis: verum

end;

suppose k = 0 ; :: thesis: IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),b₁)) in dom (stop (for-down (a,i,n,I)))

then Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),k)) in dom (stop (for-down (a,i,n,I))) by A3, COMPOS_1:36; :: thesis: verum

end;

end;

end;

hence for-down (a,i,n,I) is_closed_on s,P by SCMPDS_6:def 2; :: thesis: for-down (a,i,n,I) is_halting_on s,P
P +* (stop (for-down (a,i,n,I))) halts_on Initialize s by A11, EXTPRO_1:29;
hence for-down (a,i,n,I) is_halting_on s,P by SCMPDS_6:def 3; :: thesis: verum