let P be Instruction-Sequence of SCMPDS; for s being 0 -started 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,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be 0 -started 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,s) = s +* (Start-At (((card I) + 2),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,s) = s +* (Start-At (((card I) + 2),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,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let i be Integer; ( s . (DataLoc ((s . a),i)) <= 0 implies IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set d1 = DataLoc ((s . a),i);
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,I));
set P3 = P +* (stop (while>0 (a,i,I)));
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),s,1);
set P4 = P +* (stop (while>0 (a,i,I)));
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set SAl = Start-At (((card I) + 2),SCMPDS);
A1:
Initialize s = s
by MEMSTR_0:44;
A2:
IC s = 0
by MEMSTR_0:def 11;
A3:
while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
by SCMPDS_4:15;
A4: Comput ((P +* (stop (while>0 (a,i,I)))),s,(0 + 1)) =
Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (while>0 (a,i,I)))),s)
.=
Exec (((a,i) <=0_goto ((card I) + 2)),s)
by A3, A1, SCMPDS_6:11
;
A5:
stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I)))
by FUNCT_4:25;
A6:
IExec ((while>0 (a,i,I)),P,s) = Result ((P +* (stop (while>0 (a,i,I)))),s)
by SCMPDS_4:def 5;
assume
s . (DataLoc ((s . a),i)) <= 0
; IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
then A7: IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) =
ICplusConst (s,((card I) + 2))
by A4, SCMPDS_2:56
.=
0 + ((card I) + 2)
by A2, SCMPDS_6:12
;
A8:
card (while>0 (a,i,I)) = (card I) + 2
by Th15;
then
(card I) + 2 in dom (stop (while>0 (a,i,I)))
by COMPOS_1:64;
then (P +* (stop (while>0 (a,i,I)))) . ((card I) + 2) =
(stop (while>0 (a,i,I))) . ((card I) + 2)
by A5, GRFUNC_1:2
.=
halt SCMPDS
by A8, COMPOS_1:64
;
then A9:
CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1))) = halt SCMPDS
by A7, PBOOLE:143;
then A10:
P +* (stop (while>0 (a,i,I))) halts_on s
by EXTPRO_1:29;
A11:
CurInstr ((P +* (stop (while>0 (a,i,I)))),s) = (a,i) <=0_goto ((card I) + 2)
by A3, A1, SCMPDS_6:11;
now for l being Nat st l < 0 + 1 holds
CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDSlet l be
Nat;
( l < 0 + 1 implies CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDS )assume
l < 0 + 1
;
CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDSthen
l = 0
by NAT_1:13;
hence
CurInstr (
(P +* (stop (while>0 (a,i,I)))),
(Comput ((P +* (stop (while>0 (a,i,I)))),s,l)))
<> halt SCMPDS
by A11;
verum end;
then
for l being Nat st CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) = halt SCMPDS holds
1 <= l
;
then A12:
LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) = 1
by A9, A10, EXTPRO_1:def 15;
A13:
now for x being object st x in dom (IExec ((while>0 (a,i,I)),P,s)) holds
(IExec ((while>0 (a,i,I)),P,s)) . x = (s +* (Start-At (((card I) + 2),SCMPDS))) . xlet x be
object ;
( x in dom (IExec ((while>0 (a,i,I)),P,s)) implies (IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )A14:
dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
assume A15:
x in dom (IExec ((while>0 (a,i,I)),P,s))
;
(IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A15, SCMPDS_4:6;
suppose A16:
x is
Int_position
;
(IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A17:
not
x in dom (Start-At (((card I) + 2),SCMPDS))
by A14, TARSKI:def 1;
thus (IExec ((while>0 (a,i,I)),P,s)) . x =
(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) . x
by A6, A12, A10, EXTPRO_1:23
.=
s . x
by A4, A16, SCMPDS_2:56
.=
(s +* (Start-At (((card I) + 2),SCMPDS))) . x
by A17, FUNCT_4:11
;
verum end; suppose A18:
x = IC
;
(IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1hence (IExec ((while>0 (a,i,I)),P,s)) . x =
(card I) + 2
by A6, A7, A12, A10, EXTPRO_1:23
.=
(s +* (Start-At (((card I) + 2),SCMPDS))) . x
by A18, FUNCT_4:113
;
verum end; end; end;
dom (IExec ((while>0 (a,i,I)),P,s)) =
the carrier of SCMPDS
by PARTFUN1:def 2
.=
dom (s +* (Start-At (((card I) + 2),SCMPDS)))
by PARTFUN1:def 2
;
hence
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
by A13, FUNCT_1:2; verum