let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be State of SCMPDS; for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be Program of SCMPDS; for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be halt-free shiftable Program of SCMPDS; for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position ; for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P implies IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A3: if<0 (a,k1,I,J) =
(((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J
by SCMPDS_4:50
.=
((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J)
by SCMPDS_4:50
;
A4: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),(0 + 1)) =
Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),0)))
by EXTPRO_1:4
.=
Following ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))
by EXTPRO_1:3
.=
Exec (((a,k1) >=0_goto ((card I) + 2)),(Initialize s))
by A3, Th22
;
A5:
not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS))
by SCMPDS_4:59;
not a in dom (Start-At (0,SCMPDS))
by SCMPDS_4:59;
then A6: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) =
(Initialize s) . (DataLoc ((s . a),k1))
by FUNCT_4:12
.=
s . (DataLoc ((s . a),k1))
by A5, FUNCT_4:12
;
A7:
IC (Initialize s) = 0
by COMPOS_1:223;
A8:
dom (ProgramPart s) = NAT
by COMPOS_1:34;
assume
s . (DataLoc ((s . a),k1)) >= 0
; ( not J is_closed_on s,P or not J is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A9: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) =
ICplusConst ((Initialize s),((card I) + 2))
by A4, A6, SCMPDS_2:69
.=
0 + ((card I) + 2)
by A7, Th23
;
now let a be
Int_position ;
(Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) . athus (Initialize s) . a =
(Initialize s) . a
.=
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) . a
by A4, SCMPDS_2:69
;
verum end;
then A11:
DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1))
by SCMPDS_4:23;
card ((((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) =
(card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1)))
by AFINSQ_1:20
.=
(card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + 1
by SCMPDS_5:6
.=
((card I) + 1) + 1
by Th15
.=
(card I) + (1 + 1)
;
then A12:
Shift ((stop J),((card I) + 2)) c= stop (if<0 (a,k1,I,J))
by Th24;
stop (if<0 (a,k1,I,J)) c= P +* (stop (if<0 (a,k1,I,J)))
by FUNCT_4:26;
then
Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J)))
by A12, XBOOLE_1:1;
then A13:
Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J)))
;
assume A14:
J is_closed_on s,P
; ( not J is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A15:
( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) )
by Th38, FUNCT_4:26;
UU:
stop J c= P +* (stop J)
by FUNCT_4:26;
assume A16:
J is_halting_on s,P
; IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then A17:
P +* (stop J) halts_on Initialize s
by Def3;
A19:
Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))
by EXTPRO_1:5;
A22: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) =
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))))
by A19
.=
CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s))))))
by A15, A13, A9, A11, Th45, UU
.=
halt SCMPDS
by A17, EXTPRO_1:def 14
;
then A23:
P +* (stop (if<0 (a,k1,I,J))) halts_on Initialize s
by EXTPRO_1:30;
A24:
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (a,k1) >=0_goto ((card I) + 2)
by A3, Th22;
now let l be
Element of
NAT ;
( l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS )assume A25:
l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1
;
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDSA26:
Comput (
(P +* (stop (if<0 (a,k1,I,J)))),
(Initialize s),
0)
= Initialize s
by EXTPRO_1:3;
per cases
( l = 0 or l <> 0 )
;
suppose
l = 0
;
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDSthen
CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l)))
= CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Initialize s))
by A26;
hence
CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l)))
<> halt SCMPDS
by A24;
verum end; suppose
l <> 0
;
not CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) = halt SCMPDSthen consider n being
Nat such that A27:
l = n + 1
by NAT_1:6;
reconsider n =
n as
Element of
NAT by ORDINAL1:def 13;
A28:
n < LifeSpan (
(P +* (stop J)),
(Initialize s))
by A25, A27, XREAL_1:8;
assume A29:
CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l)))
= halt SCMPDS
;
contradictionA31:
Comput (
(P +* (stop (if<0 (a,k1,I,J)))),
(Initialize s),
(n + 1))
= Comput (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),
n)
by EXTPRO_1:5;
CurInstr (
(P +* (stop J)),
(Comput ((P +* (stop J)),(Initialize s),n))) =
CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),n)))
by A15, A13, A9, A11, Th45, UU
.=
halt SCMPDS
by A27, A29, A31
;
hence
contradiction
by A17, A28, EXTPRO_1:def 14;
verum end; end;
then
for l being Element of NAT st CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop J)),(Initialize s))) + 1 <= l
;
then A34:
LifeSpan ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (LifeSpan ((P +* (stop J)),(Initialize s))) + 1
by A22, A23, EXTPRO_1:def 14;
A36: DataPart (Result ((P +* (stop J)),(Initialize s))) =
DataPart (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))
by A17, EXTPRO_1:23
.=
DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))
by A15, A13, A9, A11, Th45, UU
.=
DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))
by EXTPRO_1:5
.=
DataPart (Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)))
by A23, A34, EXTPRO_1:23
;
A37:
now let x be
set ;
( x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) implies (IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )A38:
dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )}
by FUNCOP_1:19;
assume A39:
x in dom (IExec ((if<0 (a,k1,I,J)),P,s))
;
(IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),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 ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:52;
then A41:
not
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A38, TARSKI:def 1;
A42:
not
x in dom (s | NAT)
by A8, A40, SCMPDS_2:53;
hence (IExec ((if<0 (a,k1,I,J)),P,s)) . x =
(Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
by FUNCT_4:12
.=
(Result ((P +* (stop J)),(Initialize s))) . x
by A36, A40, SCMPDS_4:23
.=
(IExec (J,P,s)) . x
by A42, FUNCT_4:12
.=
((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A41, FUNCT_4:12
;
verum end; suppose A43:
x = IC
;
(IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1A44:
not
x in dom (s | NAT)
by A8, A43, COMPOS_1:3;
then A45:
IC (Result ((P +* (stop J)),(Initialize s))) =
IC (IExec (J,P,s))
by A43, FUNCT_4:12
.=
card J
by A14, A16, Th48
;
A46:
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A38, A43, TARSKI:def 1;
thus (IExec ((if<0 (a,k1,I,J)),P,s)) . x =
(Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
by A44, FUNCT_4:12
.=
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) . x
by A23, A34, EXTPRO_1:23
.=
IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))
by A43, EXTPRO_1:5
.=
(IC (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) + ((card I) + 2)
by A15, A13, A9, A11, Th45, UU
.=
(IC (Result ((P +* (stop J)),(Initialize s)))) + ((card I) + 2)
by A17, EXTPRO_1:23
.=
IC (Start-At (((card J) + ((card I) + 2)),SCMPDS))
by A45, FUNCOP_1:87
.=
((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A43, A46, FUNCT_4:14
;
verum end; end;
dom (IExec ((if<0 (a,k1,I,J)),P,s)) =
the carrier of SCMPDS
by PARTFUN1:def 4
.=
dom ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)))
by PARTFUN1:def 4
;
hence
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A37, FUNCT_1:9; verum