let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being Program of
for J being halt-free shiftable Program of
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,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be State of SCMPDS; for I being Program of
for J being halt-free shiftable Program of
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,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be Program of ; for J being halt-free shiftable Program of
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,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be halt-free shiftable Program of ; 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,(Initialize s)) = (IExec (J,P,(Initialize 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,(Initialize s)) = (IExec (J,P,(Initialize 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,(Initialize s)) = (IExec (J,P,(Initialize 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);
A1: if<0 (a,k1,I,J) =
(((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J
by SCMPDS_4:14
.=
((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J)
by SCMPDS_4:14
;
A2: 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_1:3
.=
Following ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))
by EXTPRO_1:2
.=
Exec (((a,k1) >=0_goto ((card I) + 2)),(Initialize s))
by A1, Th3
;
A3:
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 A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) =
(Initialize s) . (DataLoc ((s . a),k1))
by FUNCT_4:11
.=
s . (DataLoc ((s . a),k1))
by A3, FUNCT_4:11
;
A5:
IC (Initialize s) = 0
by MEMSTR_0:47;
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,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) =
ICplusConst ((Initialize s),((card I) + 2))
by A2, A4, SCMPDS_2:57
.=
0 + ((card I) + 2)
by A5, Th4
;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) . a
by A2, SCMPDS_2:57;
then A7:
DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1))
by SCMPDS_4:8;
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_1:17
.=
(card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + 1
by COMPOS_1:54
.=
((card I) + 1) + 1
by Th1
.=
(card I) + (1 + 1)
;
then A8:
Shift ((stop J),((card I) + 2)) c= stop (if<0 (a,k1,I,J))
by Th5;
stop (if<0 (a,k1,I,J)) c= P +* (stop (if<0 (a,k1,I,J)))
by FUNCT_4:25;
then
Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J)))
by A8, XBOOLE_1:1;
then A9:
Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J)))
;
assume A10:
J is_closed_on s,P
; ( not J is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A11:
( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) )
by FUNCT_4:25;
A12:
stop J c= P +* (stop J)
by FUNCT_4:25;
assume A13:
J is_halting_on s,P
; IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then A14:
P +* (stop J) halts_on Initialize s
;
A15:
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_1:4;
A16: 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 A15
.=
CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s))))))
by A11, A9, A6, A7, Th22, A12
.=
halt SCMPDS
by A14, EXTPRO_1:def 15
;
then A17:
P +* (stop (if<0 (a,k1,I,J))) halts_on Initialize s
by EXTPRO_1:29;
A18:
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (a,k1) >=0_goto ((card I) + 2)
by A1, Th3;
now for l being Nat st l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 holds
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) <> halt SCMPDSlet l be
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 A19:
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 SCMPDSA20:
Comput (
(P +* (stop (if<0 (a,k1,I,J)))),
(Initialize s),
0)
= Initialize s
by EXTPRO_1:2;
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 A20;
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 A18;
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 A21:
l = n + 1
by NAT_1:6;
reconsider n =
n as
Nat ;
A22:
n < LifeSpan (
(P +* (stop J)),
(Initialize s))
by A19, A21, XREAL_1:6;
assume A23:
CurInstr (
(P +* (stop (if<0 (a,k1,I,J)))),
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l)))
= halt SCMPDS
;
contradictionA24:
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_1:4;
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 A11, A9, A6, A7, Th22, A12
.=
halt SCMPDS
by A21, A23, A24
;
hence
contradiction
by A14, A22, EXTPRO_1:def 15;
verum end; end; end;
then
for l being 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 A25:
LifeSpan ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (LifeSpan ((P +* (stop J)),(Initialize s))) + 1
by A16, A17, EXTPRO_1:def 15;
A26: DataPart (Result ((P +* (stop J)),(Initialize s))) =
DataPart (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))
by A14, 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 A11, A9, A6, A7, Th22, A12
.=
DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))
by EXTPRO_1:4
.=
DataPart (Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)))
by A17, A25, EXTPRO_1:23
;
A27:
now for x being object st x in dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . x = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . xlet x be
object ;
( x in dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) implies (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )A28:
dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
assume A29:
x in dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s)))
;
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A29, SCMPDS_4:6;
suppose A30:
x is
Int_position
;
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A31:
not
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A28, TARSKI:def 1;
thus (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . x =
(Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
.=
(Result ((P +* (stop J)),(Initialize s))) . x
by A26, A30, SCMPDS_4:8
.=
(IExec (J,P,(Initialize s))) . x
.=
((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A31, FUNCT_4:11
;
verum end; suppose A32:
x = IC
;
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1A33:
IC (Result ((P +* (stop J)),(Initialize s))) =
IC (IExec (J,P,(Initialize s)))
.=
card J
by A10, A13, Th25
;
A34:
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A28, A32, TARSKI:def 1;
thus (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . x =
(Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
.=
(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) . x
by A17, A25, 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 A32, EXTPRO_1:4
.=
(IC (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) + ((card I) + 2)
by A11, A9, A6, A7, Th22, A12
.=
(IC (Result ((P +* (stop J)),(Initialize s)))) + ((card I) + 2)
by A14, EXTPRO_1:23
.=
IC (Start-At (((card J) + ((card I) + 2)),SCMPDS))
by A33, FUNCOP_1:72
.=
((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A32, A34, FUNCT_4:13
;
verum end; end; end;
dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) =
the carrier of SCMPDS
by PARTFUN1:def 2
.=
dom ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)))
by PARTFUN1:def 2
;
hence
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A27, FUNCT_1:2; verum