let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P )
let s be State of SCMPDS; for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P )
let I be Program of SCMPDS; for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P )
let a be Int_position; for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P )
let i be Integer; ( s . (DataLoc ((s . a),i)) = 0 implies ( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
assume A1:
s . (DataLoc ((s . a),i)) = 0
; ( while<>0 (a,i,I) is_closed_on s,P & while<>0 (a,i,I) is_halting_on s,P )
set WHL = while<>0 (a,i,I);
set pWH = stop (while<>0 (a,i,I));
set iWH = stop (while<>0 (a,i,I));
set s3 = Initialize s;
set P3 = P +* (stop (while<>0 (a,i,I)));
set s4 = Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2);
set P4 = P +* (stop (while<>0 (a,i,I)));
set P5 = P +* (stop (while<>0 (a,i,I)));
A2:
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 A3: (Initialize s) . (DataLoc (((Initialize s) . a),i)) =
(Initialize s) . (DataLoc ((s . a),i))
by FUNCT_4:11
.=
0
by A1, A2, FUNCT_4:11
;
set i1 = (a,i) <>0_goto 2;
set i2 = goto ((card I) + 2);
set i3 = goto (- ((card I) + 2));
A4:
IC (Initialize s) = 0
by MEMSTR_0:47;
A5:
while<>0 (a,i,I) = ((a,i) <>0_goto 2) ';' (((goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 2))))
by Lm8;
Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),(0 + 1)) =
Following ((P +* (stop (while<>0 (a,i,I)))),(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (while<>0 (a,i,I)))),(Initialize s))
.=
Exec (((a,i) <>0_goto 2),(Initialize s))
by A5, SCMPDS_6:11
;
then A6: IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)) =
(IC (Initialize s)) + 1
by A3, SCMPDS_2:55
.=
0 + 1
by A4
;
A7:
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
by FUNCT_4:25;
then A8:
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
;
A9:
1 in dom (while<>0 (a,i,I))
by Th10;
then
1 in dom (stop (while<>0 (a,i,I)))
by COMPOS_1:62;
then A10: (P +* (stop (while<>0 (a,i,I)))) . 1 =
(stop (while<>0 (a,i,I))) . 1
by A8, GRFUNC_1:2
.=
(while<>0 (a,i,I)) . 1
by A9, COMPOS_1:63
.=
goto ((card I) + 2)
by Th11
;
A11:
card (while<>0 (a,i,I)) = (card I) + 3
by Th8;
then A12:
(card I) + 3 in dom (stop (while<>0 (a,i,I)))
by COMPOS_1:64;
A13:
(P +* (stop (while<>0 (a,i,I)))) /. (IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1))) = (P +* (stop (while<>0 (a,i,I)))) . (IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)))
by PBOOLE:143;
Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),(1 + 1)) =
Following ((P +* (stop (while<>0 (a,i,I)))),(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)))
by EXTPRO_1:3
.=
Exec ((goto ((card I) + 2)),(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)))
by A6, A10, A13
;
then A14: IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2)) =
ICplusConst ((Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)),((card I) + 2))
by SCMPDS_2:54
.=
((card I) + 2) + 1
by A6, SCMPDS_6:12
.=
(card I) + (2 + 1)
;
A15:
(P +* (stop (while<>0 (a,i,I)))) /. (IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2))) = (P +* (stop (while<>0 (a,i,I)))) . (IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2)))
by PBOOLE:143;
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
by A7;
then (P +* (stop (while<>0 (a,i,I)))) . ((card I) + 3) =
(stop (while<>0 (a,i,I))) . ((card I) + 3)
by A12, GRFUNC_1:2
.=
halt SCMPDS
by A11, COMPOS_1:64
;
then A16:
CurInstr ((P +* (stop (while<>0 (a,i,I)))),(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2))) = halt SCMPDS
by A14, A15;
now for k being Nat holds IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),k)) in dom (stop (while<>0 (a,i,I)))let k be
Nat;
IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while<>0 (a,i,I)))
(
k = 0 or
0 < k )
;
then A17:
(
k = 0 or
0 + 1
<= k )
by INT_1:7;
per cases
( k = 0 or k = 1 or 1 < k )
by A17, XXREAL_0:1;
suppose
k = 0
;
IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while<>0 (a,i,I)))then
Comput (
(P +* (stop (while<>0 (a,i,I)))),
(Initialize s),
k)
= Initialize s
;
hence
IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),k)) in dom (stop (while<>0 (a,i,I)))
by A4, COMPOS_1:36;
verum end; end;
hence
while<>0 (a,i,I) is_closed_on s,P
by SCMPDS_6:def 2; while<>0 (a,i,I) is_halting_on s,P
P +* (stop (while<>0 (a,i,I))) halts_on Initialize s
by A16, EXTPRO_1:29;
hence
while<>0 (a,i,I) is_halting_on s,P
by SCMPDS_6:def 3; verum