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
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((for-down (a,i,n,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
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((for-down (a,i,n,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
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let a, c be Int_position ; for i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let i be Integer; for n being Element of NAT st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
let n be Element of NAT ; ( s . (DataLoc ((s . a),i)) <= 0 implies IExec ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS)) )
set d1 = DataLoc ((s . a),i);
set FOR = for-down (a,i,n,I);
set pFOR = stop (for-down (a,i,n,I));
set s3 = Initialize s;
set P3 = P +* (stop (for-down (a,i,n,I)));
set s4 = Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1);
set P4 = P +* (stop (for-down (a,i,n,I)));
set i1 = (a,i) <=0_goto ((card I) + 3);
set i2 = AddTo (a,i,(- n));
set i3 = goto (- ((card I) + 2));
set SAl = Start-At (((card I) + 3),SCMPDS);
A2:
IC (Initialize s) = 0
by MEMSTR_0:def 8;
A3:
not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
A4:
stop (for-down (a,i,n,I)) c= P +* (stop (for-down (a,i,n,I)))
by FUNCT_4:25;
not a in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),i)) =
(Initialize s) . (DataLoc ((s . a),i))
by FUNCT_4:11
.=
s . (DataLoc ((s . a),i))
by A3, FUNCT_4:11
;
A6:
for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
by Th15;
A7: Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),(0 + 1)) =
Following ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (for-down (a,i,n,I)))),(Initialize s))
by EXTPRO_1:2
.=
Exec (((a,i) <=0_goto ((card I) + 3)),(Initialize s))
by A6, SCMPDS_6:11
;
assume
s . (DataLoc ((s . a),i)) <= 0
; IExec ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
then A8: IC (Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1)) =
ICplusConst ((Initialize s),((card I) + 3))
by A7, A5, SCMPDS_2:56
.=
0 + ((card I) + 3)
by A2, SCMPDS_6:12
;
A11:
card (for-down (a,i,n,I)) = (card I) + 3
by Th60;
then
(card I) + 3 in dom (stop (for-down (a,i,n,I)))
by COMPOS_1:64;
then (P +* (stop (for-down (a,i,n,I)))) . ((card I) + 3) =
(stop (for-down (a,i,n,I))) . ((card I) + 3)
by A4, GRFUNC_1:2
.=
halt SCMPDS
by A11, COMPOS_1:64
;
then A12:
CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1))) = halt SCMPDS
by A8, PBOOLE:143;
then A13:
P +* (stop (for-down (a,i,n,I))) halts_on Initialize s
by EXTPRO_1:29;
A14:
CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Initialize s)) = (a,i) <=0_goto ((card I) + 3)
by A6, SCMPDS_6:11;
now let l be
Element of
NAT ;
( l < 0 + 1 implies CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),l))) <> halt SCMPDS )assume
l < 0 + 1
;
CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),l))) <> halt SCMPDSthen
l = 0
by NAT_1:13;
hence
CurInstr (
(P +* (stop (for-down (a,i,n,I)))),
(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),l)))
<> halt SCMPDS
by A14, EXTPRO_1:2;
verum end;
then
for l being Element of NAT st CurInstr ((P +* (stop (for-down (a,i,n,I)))),(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l
;
then B17:
LifeSpan ((P +* (stop (for-down (a,i,n,I)))),(Initialize s)) = 1
by A12, A13, EXTPRO_1:def 15;
A19:
now let x be
set ;
( x in dom (IExec ((for-down (a,i,n,I)),P,(Initialize s))) implies (IExec ((for-down (a,i,n,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1 )A20:
dom (Start-At (((card I) + 3),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
assume A21:
x in dom (IExec ((for-down (a,i,n,I)),P,(Initialize s)))
;
(IExec ((for-down (a,i,n,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A21, SCMPDS_4:6;
suppose A22:
x is
Int_position
;
(IExec ((for-down (a,i,n,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A23:
not
x in dom (Start-At (((card I) + 3),SCMPDS))
by A20, TARSKI:def 1;
TT:
not
x in dom (Start-At (0,SCMPDS))
by A22, SCMPDS_4:18;
thus (IExec ((for-down (a,i,n,I)),P,(Initialize s))) . x =
(Comput ((P +* (stop (for-down (a,i,n,I)))),(Initialize s),1)) . x
by B17, A13, EXTPRO_1:23
.=
(Initialize s) . x
by A7, A22, SCMPDS_2:56
.=
s . x
by TT, FUNCT_4:11
.=
(s +* (Start-At (((card I) + 3),SCMPDS))) . x
by A23, FUNCT_4:11
;
verum end; suppose A24:
x = IC
;
(IExec ((for-down (a,i,n,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 3),SCMPDS))) . b1thus (IExec ((for-down (a,i,n,I)),P,(Initialize s))) . x =
(card I) + 3
by A8, B17, A24, A13, EXTPRO_1:23
.=
(s +* (Start-At (((card I) + 3),SCMPDS))) . x
by A24, FUNCT_4:113
;
verum end; end;
dom (IExec ((for-down (a,i,n,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 ((for-down (a,i,n,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 3),SCMPDS))
by A19, FUNCT_1:2; verum