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
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
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
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let I be Program of ; for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) <= 0 implies IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
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)));
set i = (a,k1) <=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1:
IC (Initialize s) = 0
by MEMSTR_0:47;
A2:
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 A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) =
(Initialize s) . (DataLoc ((s . a),k1))
by FUNCT_4:11
.=
s . (DataLoc ((s . a),k1))
by A2, FUNCT_4:11
;
A4: 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
;
assume
s . (DataLoc ((s . a),k1)) <= 0
; IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) =
ICplusConst ((Initialize s),((card I) + 1))
by A4, A3, SCMPDS_2:56
.=
0 + ((card I) + 1)
by A1, Th4
;
stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I)))
by FUNCT_4:25;
then A6:
stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I)))
;
A7:
(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;
A8:
card (if>0 (a,k1,I)) = (card I) + 1
by Th1;
then
(card I) + 1 in dom (stop (if>0 (a,k1,I)))
by COMPOS_1:64;
then (P +* (stop (if>0 (a,k1,I)))) . ((card I) + 1) =
(stop (if>0 (a,k1,I))) . ((card I) + 1)
by A6, GRFUNC_1:2
.=
halt SCMPDS
by A8, 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, A7;
then A10:
P +* (stop (if>0 (a,k1,I))) halts_on Initialize s
by EXTPRO_1:29;
A11:
CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto ((card I) + 1)
by Th3;
now for l being Nat st l < 1 holds
CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDSlet l be
Nat;
( l < 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS )A12:
Comput (
(P +* (stop (if>0 (a,k1,I)))),
(Initialize s),
0)
= Initialize s
by EXTPRO_1:2;
assume
l < 1
;
CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDSthen
l < 1
+ 0
;
then
l = 0
by NAT_1:13;
then
CurInstr (
(P +* (stop (if>0 (a,k1,I)))),
(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l)))
= CurInstr (
(P +* (stop (if>0 (a,k1,I)))),
(Initialize s))
by A12;
hence
CurInstr (
(P +* (stop (if>0 (a,k1,I)))),
(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l)))
<> halt SCMPDS
by A11;
verum end;
then
for l being Nat st CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l
;
then
LifeSpan ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = 1
by A9, A10, EXTPRO_1:def 15;
then A13:
Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1) = Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))
by A10, EXTPRO_1:23;
A14:
now for x being object st x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (s +* (Start-At (((card I) + 1),SCMPDS))) . xA15:
dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
let x be
object ;
( x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1 )assume A16:
x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s)))
;
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A16, SCMPDS_4:6;
suppose A17:
x is
Int_position
;
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A18:
not
x in dom (Start-At (((card I) + 1),SCMPDS))
by A15, TARSKI:def 1;
A19:
not
x in dom (Start-At (0,SCMPDS))
by A17, SCMPDS_4:18;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x =
(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . x
by A13
.=
(Initialize s) . x
by A4, A17, SCMPDS_2:56
.=
s . x
by A19, FUNCT_4:11
.=
(s +* (Start-At (((card I) + 1),SCMPDS))) . x
by A18, FUNCT_4:11
;
verum end; end; end;
dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) =
the carrier of SCMPDS
by PARTFUN1:def 2
.=
dom (s +* (Start-At (((card I) + 1),SCMPDS)))
by PARTFUN1:def 2
;
hence
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
by A14, FUNCT_1:2; verum