let s be State of SCM+FSA; for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_on s & I is_halting_on s holds
( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s )
let I, J be Program of SCM+FSA; for a being read-write Int-Location st s . a > 0 & I is_closed_on s & I is_halting_on s holds
( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s )
let a be read-write Int-Location ; ( s . a > 0 & I is_closed_on s & I is_halting_on s implies ( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s ) )
set I1 = I ';' (Stop SCM+FSA);
set s1 = s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)));
set s3 = s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)));
set s4 = Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1);
set i = a >0_goto ((card J) + 3);
A1:
not a in dom ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by SCMFSA6B:12;
A2:
0 in dom (if>0 (a,I,J))
by Lm2;
if>0 (a,I,J) c= (if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))
by SCMFSA8A:9;
then
dom (if>0 (a,I,J)) c= dom ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by GRFUNC_1:8;
then A3: (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 0 =
((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))) . 0
by A2, FUNCT_4:14
.=
(if>0 (a,I,J)) . 0
by A2, SCMFSA6B:7
.=
a >0_goto ((card J) + 3)
by Lm3
;
IC SCM+FSA in dom ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by COMPOS_1:141;
then A4: IC (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) =
IC ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:14
.=
0
by COMPOS_1:142
;
A5:
if>0 (a,I,J) c= (if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))
by SCMFSA8A:9;
(if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)) c= s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:26;
then A6:
if>0 (a,I,J) c= s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by A5, XBOOLE_1:1;
A7:
if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' (I ';' (Stop SCM+FSA))
by SCMFSA6A:67;
card (((a >0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) =
card (((Macro (a >0_goto ((card J) + 3))) ';' J) ';' (Goto ((card I) + 1)))
by SCMFSA6A:def 6
.=
(card ((Macro (a >0_goto ((card J) + 3))) ';' J)) + (card (Goto ((card I) + 1)))
by SCMFSA6A:61
.=
(card ((Macro (a >0_goto ((card J) + 3))) ';' J)) + 1
by SCMFSA8A:29
.=
((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1
by SCMFSA6A:61
.=
((card J) + 2) + 1
by COMPOS_1:150
.=
(card J) + (2 + 1)
;
then
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= if>0 (a,I,J)
by A7, Lm1;
then
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by A6, XBOOLE_1:1;
then
ProgramPart (ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3)))) c= Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)
by AMI_1:99;
then A8:
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)
by RELAT_1:209;
Y:
(ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) /. (IC (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by COMPOS_1:38;
A9: Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(0 + 1)) =
Following ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0)))
by EXTPRO_1:4
.=
Following ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by EXTPRO_1:3
.=
Exec ((a >0_goto ((card J) + 3)),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by A4, A3, Y
;
A10:
DataPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) = DataPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))
by SCMFSA6A:39, SCMFSA8A:14;
A11:
now let f be
FinSeq-Location ;
(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . f = (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . fthus (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . f =
(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . f
by A10, SCMFSA6A:38
.=
(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . f
by A9, SCMFSA_2:97
;
verum end;
now let a be
Int-Location ;
(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . a = (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . athus (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . a =
(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a
by A10, SCMFSA6A:38
.=
(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . a
by A9, SCMFSA_2:97
;
verum end;
then A12:
DataPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) = DataPart (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by A11, SCMFSA6A:38;
assume
s . a > 0
; ( not I is_closed_on s or not I is_halting_on s or ( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s ) )
then
(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a > 0
by A1, FUNCT_4:12;
then A13:
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) = (card J) + 3
by A9, SCMFSA_2:97;
assume A14:
I is_closed_on s
; ( not I is_halting_on s or ( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s ) )
assume A15:
I is_halting_on s
; ( if>0 (a,I,J) is_closed_on s & if>0 (a,I,J) is_halting_on s )
then A16:
I ';' (Stop SCM+FSA) is_closed_on s
by A14, SCMFSA8A:46;
I ';' (Stop SCM+FSA) is_halting_on s
by A14, A15, SCMFSA8A:46;
then A17:
ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) halts_on s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by SCMFSA7B:def 8;
DataPart s = DataPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))
by SCMFSA8A:11;
then A18:
I ';' (Stop SCM+FSA) is_closed_on s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by A16, Th6;
A19:
(I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)) c= s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:26;
now let k be
Element of
NAT ;
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),b1)) in dom (if>0 (a,I,J))per cases
( 0 < k or k = 0 )
;
suppose
0 < k
;
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),b1)) in dom (if>0 (a,I,J))then consider k1 being
Nat such that A20:
k1 + 1
= k
by NAT_1:6;
reconsider k1 =
k1 as
Element of
NAT by ORDINAL1:def 13;
reconsider m =
IC (Comput ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k1)) as
Element of
NAT ;
m in dom (I ';' (Stop SCM+FSA))
by A16, SCMFSA7B:def 7;
then A21:
m < card (I ';' (Stop SCM+FSA))
by AFINSQ_1:70;
card (Stop SCM+FSA) = 1
by COMPOS_1:46;
then A22:
card (I ';' (Stop SCM+FSA)) = (card I) + 1
by SCMFSA6A:61;
card (if>0 (a,I,J)) =
((card I) + (card J)) + 4
by Th15
.=
((card J) + 3) + (card (I ';' (Stop SCM+FSA)))
by A22
;
then A23:
m + ((card J) + 3) < card (if>0 (a,I,J))
by A21, XREAL_1:8;
T:
ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by AMI_1:123;
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) =
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),k1))
by A20, EXTPRO_1:5
.=
m + ((card J) + 3)
by A19, A18, A8, A13, A12, Th11, T
;
hence
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) in dom (if>0 (a,I,J))
by A23, AFINSQ_1:70;
verum end; suppose
k = 0
;
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),b1)) in dom (if>0 (a,I,J))hence
IC (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) in dom (if>0 (a,I,J))
by A2, A4, EXTPRO_1:3;
verum end; end;
hence
if>0 (a,I,J) is_closed_on s
by SCMFSA7B:def 7; if>0 (a,I,J) is_halting_on s
T:
ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by AMI_1:123;
x:
Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) + 1)) = Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),(LifeSpan ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))))
by EXTPRO_1:5;
CurInstr ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) + 1)))) =
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))),(Comput ((ProgramPart (Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))),(Comput ((ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),(LifeSpan ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))))))
by x, T
.=
IncAddr ((CurInstr ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))))))),((card J) + 3))
by A19, A18, A8, A13, A12, Th11
.=
IncAddr ((halt SCM+FSA),((card J) + 3))
by A17, EXTPRO_1:def 14
.=
halt SCM+FSA
by SCMFSA_4:8
;
then
ProgramPart (s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) halts_on s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by EXTPRO_1:30;
hence
if>0 (a,I,J) is_halting_on s
by SCMFSA7B:def 8; verum