let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
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,P & if=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; 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,P & if=0 (a,k1,I) is_halting_on s,P )
let I be 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,P & if=0 (a,k1,I) is_halting_on s,P )
let a be 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 k1 be Integer; ( 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
; ( 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)));
A2:
IC (Initialize s) = 0
by MEMSTR_0:47;
A3:
not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
not a in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) =
(Initialize s) . (DataLoc ((s . a),k1))
by FUNCT_4:11
.=
s . (DataLoc ((s . a),k1))
by A3, FUNCT_4:11
;
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_1:3
.=
Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))
by EXTPRO_1:2
.=
Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s))
by Th3
;
then A5: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) =
ICplusConst ((Initialize s),((card I) + 1))
by A1, A4, SCMPDS_2:55
.=
0 + ((card I) + 1)
by A2, Th4
;
A6:
card (if=0 (a,k1,I)) = (card I) + 1
by Th1;
then A7:
(card I) + 1 in dom (stop (if=0 (a,k1,I)))
by COMPOS_1:64;
A8:
(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:143;
stop (if=0 (a,k1,I)) c= P +* (stop (if=0 (a,k1,I)))
by FUNCT_4:25;
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 A7, GRFUNC_1:2
.=
halt SCMPDS
by A6, COMPOS_1:64
;
then A9:
CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS
by A5, A8;
now for k being Nat holds IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I)))let k be
Nat;
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
k = 0
;
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_1:2;
hence
IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I)))
by A2, COMPOS_1:36;
verum end; end;
hence
if=0 (a,k1,I) is_closed_on s,P
; if=0 (a,k1,I) is_halting_on s,P
P +* (stop (if=0 (a,k1,I))) halts_on Initialize s
by A9, EXTPRO_1:29;
hence
if=0 (a,k1,I) is_halting_on s,P
; verum