let s be State of SCMPDS; for P being the Instructions of SCMPDS -valued ManySortedSet of NAT
for I being halt-free Program of SCMPDS
for J being shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(IExec (I,P,s)))),(card I))
let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; for I being halt-free Program of SCMPDS
for J being shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(IExec (I,P,s)))),(card I))
let I be halt-free Program of SCMPDS; for J being shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(IExec (I,P,s)))),(card I))
let J be shiftable Program of SCMPDS; ( I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(IExec (I,P,s)))),(card I)) )
set ps = s | NAT;
set IJ = I ';' J;
set s1 = Initialize s;
set P1 = P +* (stop I);
set m1 = LifeSpan ((P +* (stop I)),(Initialize s));
set s2 = Initialize s;
set P2 = P +* (stop (I ';' J));
set s3 = Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))));
set P3 = (P +* (stop I)) +* (stop J);
set m3 = LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))));
set sE = IExec (I,P,s);
assume that
A2:
I is_closed_on s,P
and
A3:
I is_halting_on s,P
and
A4:
J is_closed_on IExec (I,P,s),P
and
A5:
J is_halting_on IExec (I,P,s),P
; IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(IExec (I,P,s)))),(card I))
A6:
DataPart (Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = DataPart (Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))
by A2, A3, A4, A5, Lm1;
J is_closed_on Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))),P +* (stop I)
by A3, A4, A5, Th42;
then A7:
J is_closed_on Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))),(P +* (stop I)) +* (stop J)
by SCMPDS_6:38;
A8:
stop J c= (P +* (stop I)) +* (stop J)
by FUNCT_4:26;
B8:
Start-At (0,SCMPDS) c= Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))
by FUNCT_4:26;
A9:
Shift ((stop J),(card I)) c= P +* (stop (I ';' J))
by A2, A3, A4, A5, Lm1;
A10:
IC (Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I
by A2, A3, A4, A5, Lm1;
then A11:
IC (Comput ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))))) = (IC (Comput (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))))) + (card I)
by A8, B8, A6, A9, A7, SCMPDS_6:45;
A12:
DataPart (Comput ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))))) = DataPart (Comput (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))))
by A8, B8, A10, A6, A9, A7, SCMPDS_6:45;
set R1 = Initialize (IExec (I,P,s));
set R2 = Initialize (Result ((P +* (stop I)),(Initialize s)));
A17: dom (s | NAT) =
(dom s) /\ NAT
by RELAT_1:90
.=
(({(IC )} \/ SCM-Data-Loc) \/ NAT) /\ NAT
by SCMPDS_4:19
.=
NAT
by XBOOLE_1:21
;
B18:
stop J c= P +* (stop J)
by FUNCT_4:26;
B19:
stop J c= (P +* (stop I)) +* (stop J)
by FUNCT_4:26;
NPP (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = NPP ((Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) +* (s | NAT))
by COMPOS_1:236;
then A20:
NPP (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = NPP ((Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) +* (s | NAT))
;
A22:
P +* (stop I) halts_on Initialize s
by A3, SCMPDS_6:def 3;
then A23:
Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = Initialize (Result ((P +* (stop I)),(Initialize s)))
by EXTPRO_1:23;
A24:
DataPart (IExec (I,P,s)) = DataPart (Initialize (IExec (I,P,s)))
by COMPOS_1:80;
then A25:
J is_closed_on Initialize (IExec (I,P,s)),P +* (stop J)
by A4, A5, SCMPDS_6:37;
A26:
J is_halting_on Initialize (IExec (I,P,s)),P +* (stop J)
by A4, A5, A24, SCMPDS_6:37;
X1:
Start-At (0,SCMPDS) c= Initialize (IExec (I,P,s))
by FUNCT_4:26;
X2:
Start-At (0,SCMPDS) c= Initialize (Result ((P +* (stop I)),(Initialize s)))
by FUNCT_4:26;
Result ((P +* (stop I)),(Initialize s)) = Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))
by A22, EXTPRO_1:23;
then
NPP ((Result ((P +* (stop I)),(Initialize s))) +* (s | NAT)) = NPP (Result ((P +* (stop I)),(Initialize s)))
by A20;
then
NPP (IExec (I,P,s)) = NPP (Result ((P +* (stop I)),(Initialize s)))
;
then
NPP (Initialize (IExec (I,P,s))) = NPP (Initialize (Result ((P +* (stop I)),(Initialize s))))
by COMPOS_1:81;
then
NPP (Result ((P +* (stop J)),(Initialize (IExec (I,P,s))))) = NPP (Result (((P +* (stop I)) +* (stop J)),(Initialize (Result ((P +* (stop I)),(Initialize s))))))
by X1, X2, B18, B19, A25, A26, Th28;
then
NPP (Result (((P +* (stop I)) +* (stop J)),(Initialize (Result ((P +* (stop I)),(Initialize s)))))) = NPP (Result ((P +* (stop J)),(Initialize (IExec (I,P,s)))))
;
then A27:
IC (Result (((P +* (stop I)) +* (stop J)),(Initialize (Result ((P +* (stop I)),(Initialize s)))))) = IC (Result ((P +* (stop J)),(Initialize (IExec (I,P,s)))))
by COMPOS_1:230;
A29:
(IExec (I,P,s)) | NAT = s | NAT
by PBOOLE:157;
I ';' J is_halting_on s,P
by A2, A3, A4, A5, Th43;
then A30:
P +* (stop (I ';' J)) halts_on Initialize s
by SCMPDS_6:def 3;
J is_halting_on Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))),P +* (stop I)
by A3, A4, A5, Th42;
then A31:
(P +* (stop I)) +* (stop J) halts_on Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))
by SCMPDS_6:def 3;
SS:
Start-At (0,SCMPDS) c= Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))
by FUNCT_4:26;
IExec (I,P,s) = (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) +* (s | NAT)
by A22, EXTPRO_1:23;
then
NPP (IExec (I,P,s)) = NPP (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))
by A20;
then TT:
NPP (Initialize (IExec (I,P,s))) = NPP (Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))
by COMPOS_1:81;
NPP (Result ((P +* (stop J)),(Initialize (IExec (I,P,s))))) = NPP (Result (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))
by X1, B18, B19, A25, A26, Th28, SS, TT;
then
NPP (Result ((P +* (stop J)),(Initialize (IExec (I,P,s))))) = NPP (Result (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))
;
then
(Result ((P +* (stop J)),(Initialize (IExec (I,P,s))))) +* (s | NAT) = (Result (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))) +* (s | NAT)
by A17, COMPOS_1:238;
then A32:
IExec (J,P,(IExec (I,P,s))) = (Comput (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))))) +* (s | NAT)
by A31, A29, EXTPRO_1:23;
A34:
Result ((P +* (stop I)),(Initialize s)) = Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))
by A22, EXTPRO_1:23;
A35: IC (IExec ((I ';' J),P,s)) =
IC (Result ((P +* (stop (I ';' J))),(Initialize s)))
by SCMPDS_5:22
.=
IC (Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop (I ';' J))),(Initialize s)))))
by A30, EXTPRO_1:23
.=
IC (Comput ((P +* (stop (I ';' J))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + (LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))))))
by A2, A3, A4, A5, A23, Lm1
.=
(IC (Comput (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))))) + (card I)
by A11, EXTPRO_1:5
.=
(IC (Result (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))) + (card I)
by A31, EXTPRO_1:23
.=
(IC (Result (((P +* (stop I)) +* (stop J)),(Initialize (Result ((P +* (stop I)),(Initialize s))))))) + (card I)
by A34
.=
(IC (IExec (J,P,(IExec (I,P,s))))) + (card I)
by A27, SCMPDS_5:22
;
IExec ((I ';' J),P,s) =
(Comput ((P +* (stop (I ';' J))),(Initialize s),(LifeSpan ((P +* (stop (I ';' J))),(Initialize s))))) +* (s | NAT)
by A30, EXTPRO_1:23
.=
(Comput ((P +* (stop (I ';' J))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + (LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))))) +* (s | NAT)
by A2, A3, A4, A5, A23, Lm1
;
then A36: DataPart (IExec ((I ';' J),P,s)) =
DataPart (Comput ((P +* (stop (I ';' J))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + (LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))))))))
by A17, AMI_2:29, FUNCT_4:76, SCMPDS_2:100
.=
DataPart (Comput (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))),(LifeSpan (((P +* (stop I)) +* (stop J)),(Initialize (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))))))
by A12, EXTPRO_1:5
.=
DataPart (IExec (J,P,(IExec (I,P,s))))
by A17, A32, AMI_2:29, FUNCT_4:76, SCMPDS_2:100
;
hereby verum
reconsider l =
(IC (IExec (J,P,(IExec (I,P,s))))) + (card I) as
Element of
NAT ;
A37:
dom (Start-At (l,SCMPDS)) = {(IC )}
by FUNCOP_1:19;
A38:
now let x be
set ;
( x in dom (IExec ((I ';' J),P,s)) implies (IExec ((I ';' J),P,s)) . b1 = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . b1 )assume A39:
x in dom (IExec ((I ';' J),P,s))
;
(IExec ((I ';' J),P,s)) . b1 = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . b1per cases
( x is Int_position or x = IC or x is Element of NAT )
by A39, SCMPDS_4:20;
suppose A40:
x is
Int_position
;
(IExec ((I ';' J),P,s)) . b1 = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:52;
then A41:
not
x in dom (Start-At (l,SCMPDS))
by A37, TARSKI:def 1;
(IExec ((I ';' J),P,s)) . x = (IExec (J,P,(IExec (I,P,s)))) . x
by A36, A40, SCMPDS_4:23;
hence
(IExec ((I ';' J),P,s)) . x = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . x
by A41, FUNCT_4:12;
verum end; suppose A42:
x = IC
;
(IExec ((I ';' J),P,s)) . b1 = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . b1then
x in {(IC )}
by TARSKI:def 1;
then A43:
x in dom (Start-At (l,SCMPDS))
by FUNCOP_1:19;
thus (IExec ((I ';' J),P,s)) . x =
(Start-At (l,SCMPDS)) . (IC )
by A35, A42, FUNCOP_1:87
.=
((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . x
by A42, A43, FUNCT_4:14
;
verum end; suppose A44:
x is
Element of
NAT
;
(IExec ((I ';' J),P,s)) . b1 = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . b1then
x <> IC
by COMPOS_1:3;
then A45:
not
x in dom (Start-At (l,SCMPDS))
by A37, TARSKI:def 1;
(IExec ((I ';' J),P,s)) | NAT =
s | NAT
by PBOOLE:157
.=
(IExec (J,P,(IExec (I,P,s)))) | NAT
by A29, PBOOLE:157
;
then
(IExec ((I ';' J),P,s)) . x = (IExec (J,P,(IExec (I,P,s)))) . x
by A44, COMPOS_1:127;
hence
(IExec ((I ';' J),P,s)) . x = ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS))) . x
by A45, FUNCT_4:12;
verum end; end; dom (IExec ((I ';' J),P,s)) =
the
carrier of
SCMPDS
by PARTFUN1:def 4
.=
dom ((IExec (J,P,(IExec (I,P,s)))) +* (Start-At (((IC (IExec (J,P,(IExec (I,P,s))))) + (card I)),SCMPDS)))
by PARTFUN1:def 4
;
hence
IExec (
(I ';' J),
P,
s)
= IncIC (
(IExec (J,P,(IExec (I,P,s)))),
(card I))
by A38, FUNCT_1:9;
verum
end;