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 2;
set j = 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 s5 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<=0 (a,k1,I)));
set P5 = P +* (stop (if<=0 (a,k1,I)));
A2:
if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I)
by SCMPDS_4:16;
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
;
A5:
IC (Initialize s) = 0
by MEMSTR_0:47;
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 2),(Initialize s))
by A2, Th3
;
then A6: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) =
(IC (Initialize s)) + 1
by A1, A4, SCMPDS_2:56
.=
0 + 1
by A5
;
A7:
stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I)))
by FUNCT_4:25;
then A8:
stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I)))
;
A9:
1 in dom (if<=0 (a,k1,I))
by Lm9;
then
1 in dom (stop (if<=0 (a,k1,I)))
by COMPOS_1:62;
then A10: (P +* (stop (if<=0 (a,k1,I)))) . 1 =
(stop (if<=0 (a,k1,I))) . 1
by A8, GRFUNC_1:2
.=
(if<=0 (a,k1,I)) . 1
by A9, COMPOS_1:63
.=
goto ((card I) + 1)
by Lm10
;
A11:
card (if<=0 (a,k1,I)) = (card I) + 2
by Lm8;
then A12:
(card I) + 2 in dom (stop (if<=0 (a,k1,I)))
by COMPOS_1:64;
A13:
(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;
Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(1 + 1)) =
Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)))
by EXTPRO_1:3
.=
Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)))
by A6, A10, A13
;
then A14: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2)) =
ICplusConst ((Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),((card I) + 1))
by SCMPDS_2:54
.=
((card I) + 1) + 1
by A6, Th4
.=
(card I) + (1 + 1)
;
A15:
(P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2)))
by PBOOLE:143;
stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I)))
by A7;
then (P +* (stop (if<=0 (a,k1,I)))) . ((card I) + 2) =
(stop (if<=0 (a,k1,I))) . ((card I) + 2)
by A12, GRFUNC_1:2
.=
halt SCMPDS
by A11, COMPOS_1:64
;
then A16:
CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS
by A14, A15;
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)))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 (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 A5, COMPOS_1:36;
verum end; 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 A16, EXTPRO_1:29;
hence
if<=0 (a,k1,I) is_halting_on s,P
; verum