let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let s be State of SCMPDS; for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let I be Program of SCMPDS; for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let a, c be Int_position; for i being Integer st s . (DataLoc ((s . a),i)) = 0 holds
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let i be Integer; ( s . (DataLoc ((s . a),i)) = 0 implies IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS)) )
set b = DataLoc ((s . a),i);
assume A1:
s . (DataLoc ((s . a),i)) = 0
; IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
set i1 = (a,i) <>0_goto 2;
set i2 = goto ((card I) + 2);
set i3 = goto (- ((card I) + 2));
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:
IC (Initialize s) = 0
by MEMSTR_0:47;
A3:
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
by FUNCT_4:25;
then A4:
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
;
A5:
stop (while<>0 (a,i,I)) c= P +* (stop (while<>0 (a,i,I)))
by A3;
A6:
not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
A7:
while<>0 (a,i,I) = ((a,i) <>0_goto 2) ';' (((goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 2))))
by Lm8;
A8: 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 A7, SCMPDS_6:11
;
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 A4, GRFUNC_1:2
.=
(while<>0 (a,i,I)) . 1
by A9, COMPOS_1:63
.=
goto ((card I) + 2)
by Th11
;
set SAl = Start-At (((card I) + 3),SCMPDS);
not a in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),i)) =
(Initialize s) . (DataLoc ((s . a),i))
by FUNCT_4:11
.=
0
by A1, A6, FUNCT_4:11
;
then A11: IC (Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)) =
(IC (Initialize s)) + 1
by A8, SCMPDS_2:55
.=
0 + 1
by A2
;
A12:
(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;
A13: 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 A11, A10, A12
;
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 A11, 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;
A16:
card (while<>0 (a,i,I)) = (card I) + 3
by Th8;
then
(card I) + 3 in dom (stop (while<>0 (a,i,I)))
by COMPOS_1:64;
then (P +* (stop (while<>0 (a,i,I)))) . ((card I) + 3) =
(stop (while<>0 (a,i,I))) . ((card I) + 3)
by A5, GRFUNC_1:2
.=
halt SCMPDS
by A16, COMPOS_1:64
;
then A17:
CurInstr ((P +* (stop (while<>0 (a,i,I)))),(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2))) = halt SCMPDS
by A14, A15;
then
P +* (stop (while<>0 (a,i,I))) halts_on Initialize s
by EXTPRO_1:29;
then A18:
Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2) = Result ((P +* (stop (while<>0 (a,i,I)))),(Initialize s))
by A17, EXTPRO_1:def 9;
A19:
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = Result ((P +* (stop (while<>0 (a,i,I)))),(Initialize s))
by SCMPDS_4:def 5;
A20:
now for x being object st x in dom (IExec ((while<>0 (a,i,I)),P,(Initialize s))) holds
(IExec ((while<>0 (a,i,I)),P,(Initialize s))) . x = (s +* (Start-At (((card I) + 3),SCMPDS))) . xlet x be
object ;
( x in dom (IExec ((while<>0 (a,i,I)),P,(Initialize s))) implies (IExec ((while<>0 (a,i,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1 )A21:
dom (Start-At (((card I) + 3),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
assume A22:
x in dom (IExec ((while<>0 (a,i,I)),P,(Initialize s)))
;
(IExec ((while<>0 (a,i,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A22, SCMPDS_4:6;
suppose A23:
x is
Int_position
;
(IExec ((while<>0 (a,i,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A24:
not
x in dom (Start-At (((card I) + 3),SCMPDS))
by A21, TARSKI:def 1;
A25:
not
x in dom (Start-At (0,SCMPDS))
by A23, SCMPDS_4:18;
thus (IExec ((while<>0 (a,i,I)),P,(Initialize s))) . x =
(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),2)) . x
by A18, A19
.=
(Comput ((P +* (stop (while<>0 (a,i,I)))),(Initialize s),1)) . x
by A13, A23, SCMPDS_2:54
.=
(Initialize s) . x
by A8, A23, SCMPDS_2:55
.=
s . x
by A25, FUNCT_4:11
.=
(s +* (Start-At (((card I) + 3),SCMPDS))) . x
by A24, FUNCT_4:11
;
verum end; end;
dom (IExec ((while<>0 (a,i,I)),P,(Initialize s))) =
the carrier of SCMPDS
by PARTFUN1:def 2
.=
dom (s +* (Start-At (((card I) + 3),SCMPDS)))
by PARTFUN1:def 2
;
hence
IExec ((while<>0 (a,i,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
by A20, FUNCT_1:2; verum