let P be Instruction-Sequence of SCMPDS; for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of
for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; for I being halt-free shiftable Program of
for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of ; for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of ; for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,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 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1:
s . (DataLoc ((s . a),k1)) = 0
; ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if=0 (a,k1,I,J);
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s2 = s;
set s3 = s;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
A2:
Initialize s = s
by MEMSTR_0:44;
then A3:
IC s = 0
by MEMSTR_0:47;
A4: 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
;
then A5:
Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if=0 (a,k1,I,J)))
by Lm6;
A6: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(0 + 1)) =
Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (if=0 (a,k1,I,J)))),s)
by EXTPRO_1:2
.=
Exec (((a,k1) <>0_goto ((card I) + 2)),s)
by A4, Th3, A2
;
s . (DataLoc ((s . a),k1)) =
s . (DataLoc ((s . a),k1))
.=
0
by A1
;
then A7: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) =
(IC s) + 1
by A6, SCMPDS_2:55
.=
0 + 1
by A3
;
for a being Int_position holds s . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) . a
by A6, SCMPDS_2:55;
then A8:
DataPart s = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1))
by SCMPDS_4:8;
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
assume A9:
I is_closed_on s,P
; ( not I is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A10:
I is_halting_on s,P
; IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then
(I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P
by A9, Th21;
then A11:
P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s
by A2;
(I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P
by A9, A10, Th21;
then A12:
( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) )
by A2, FUNCT_4:25;
A13:
stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))
by FUNCT_4:25;
A14:
Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))
by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),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)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))))
by A14
.=
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))))
by A12, A5, A7, A8, Th22, A13
.=
halt SCMPDS
by A11, EXTPRO_1:def 15
;
then A16:
P +* (stop (if=0 (a,k1,I,J))) halts_on s
by EXTPRO_1:29;
A17:
CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),s) = (a,k1) <>0_goto ((card I) + 2)
by A4, Th3, A2;
now for l being Nat st l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 holds
CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) <> halt SCMPDSlet l be
Nat;
( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS )assume A18:
l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1
;
CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) <> halt SCMPDSA19:
Comput (
(P +* (stop (if=0 (a,k1,I,J)))),
s,
0)
= 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)))),s,b1))) <> halt SCMPDSthen
CurInstr (
(P +* (stop (if=0 (a,k1,I,J)))),
(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l)))
= CurInstr (
(P +* (stop (if=0 (a,k1,I,J)))),
s)
by A19;
hence
CurInstr (
(P +* (stop (if=0 (a,k1,I,J)))),
(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l)))
<> halt SCMPDS
by A17;
verum end; suppose
l <> 0
;
not CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) = halt SCMPDSthen consider n being
Nat such that A20:
l = n + 1
by NAT_1:6;
reconsider n =
n as
Nat ;
A21:
n < LifeSpan (
(P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),
s)
by A18, A20, XREAL_1:6;
assume A22:
CurInstr (
(P +* (stop (if=0 (a,k1,I,J)))),
(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l)))
= halt SCMPDS
;
contradictionA23:
Comput (
(P +* (stop (if=0 (a,k1,I,J)))),
s,
(n + 1))
= Comput (
(P +* (stop (if=0 (a,k1,I,J)))),
(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),
n)
by EXTPRO_1:4;
CurInstr (
(P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),
(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),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)))),s,1)),n)))
by A12, A5, A7, A8, Th22, A13
.=
halt SCMPDS
by A20, A22, A23
;
hence
contradiction
by A11, A21, 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)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l
;
then A24:
LifeSpan ((P +* (stop (if=0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1
by A15, A16, EXTPRO_1:def 15;
A25: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) =
DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))
by A11, EXTPRO_1:23
.=
DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))
by A12, A5, A7, A8, Th22, A13
.=
DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))
by EXTPRO_1:4
.=
DataPart (Result ((P +* (stop (if=0 (a,k1,I,J)))),s))
by A16, A24, EXTPRO_1:23
;
A26:
now for x being object st x in dom (IExec ((if=0 (a,k1,I,J)),P,s)) holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . x = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . xlet x be
object ;
( x in dom (IExec ((if=0 (a,k1,I,J)),P,s)) implies (IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )A27:
dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )}
by FUNCOP_1:13;
assume A28:
x in dom (IExec ((if=0 (a,k1,I,J)),P,s))
;
(IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1per cases
( x is Int_position or x = IC )
by A28, SCMPDS_4:6;
suppose A29:
x is
Int_position
;
(IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1then
x <> IC
by SCMPDS_2:43;
then A30:
not
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A27, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x =
(Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.=
(Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x
by A25, A29, SCMPDS_4:8
.=
(IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.=
((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A30, FUNCT_4:11
;
verum end; suppose A31:
x = IC
;
(IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1A32:
IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) =
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.=
((card I) + (card J)) + 1
by A9, A10, Th23
;
A33:
x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A27, A31, TARSKI:def 1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x =
(Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.=
(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x
by A16, A24, EXTPRO_1:23
.=
IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))
by A31, EXTPRO_1:4
.=
(IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1
by A12, A5, A7, A8, Th22, A13
.=
(IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1
by A11, EXTPRO_1:23
.=
IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS))
by A32, FUNCOP_1:72
.=
((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x
by A31, A33, FUNCT_4:13
;
verum end; end; end;
dom (IExec ((if=0 (a,k1,I,J)),P,s)) =
the carrier of SCMPDS
by PARTFUN1:def 2
.=
dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)))
by PARTFUN1:def 2
;
hence IExec ((if=0 (a,k1,I,J)),P,s) =
(IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A26, FUNCT_1:2
.=
((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by A9, A10, Th24
.=
(IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
by MEMSTR_0:36
;
verum