let ss be State of SCM+FSA; for I, J being Program of SCM+FSA
for a being read-write Int-Location st ss . (intloc 0) = 1 & ss . a = 0 & Directed I is_pseudo-closed_on ss holds
DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss))
set A = NAT ;
set D = Int-Locations \/ FinSeq-Locations;
let I, J be Program of SCM+FSA; for a being read-write Int-Location st ss . (intloc 0) = 1 & ss . a = 0 & Directed I is_pseudo-closed_on ss holds
DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss))
let a be read-write Int-Location ; ( ss . (intloc 0) = 1 & ss . a = 0 & Directed I is_pseudo-closed_on ss implies DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss)) )
set I0 = Directed I;
set s = Initialized ss;
set I1 = I ';' (Stop SCM+FSA);
set s00 = (Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)));
set s3 = (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)));
set s4 = Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1);
set i = a =0_goto ((card J) + 3);
assume A1:
ss . (intloc 0) = 1
; ( not ss . a = 0 or not Directed I is_pseudo-closed_on ss or DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss)) )
set s1 = (Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)));
assume
ss . a = 0
; ( not Directed I is_pseudo-closed_on ss or DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss)) )
then A2:
(Initialized ss) . a = 0
by SCMFSA6C:3;
IC SCM+FSA in dom ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by COMPOS_1:141;
then A3: IC ((Initialized ss) +* ((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
;
A4:
0 in dom (if=0 (a,I,J))
by Th54;
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 A5: ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 0 =
((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))) . 0
by A4, FUNCT_4:14
.=
(if=0 (a,I,J)) . 0
by A4, SCMFSA6B:7
.=
a =0_goto ((card J) + 3)
by Th55
;
Y:
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) /. (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by COMPOS_1:38;
A6: Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(0 + 1)) =
Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0)))
by EXTPRO_1:4
.=
Following ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by EXTPRO_1:3
.=
Exec ((a =0_goto ((card J) + 3)),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by A3, A5, Y
;
not a in dom ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by SCMFSA6B:12;
then
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a = 0
by A2, FUNCT_4:12;
then A7:
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) = (card J) + 3
by A6, SCMFSA_2:96;
assume
Directed I is_pseudo-closed_on ss
; DataPart (IExec ((if=0 (a,I,J)),ss)) = DataPart (IExec ((I ';' (Stop SCM+FSA)),ss))
then A8:
Directed I is_pseudo-closed_on Initialized ss
by A1, Th53;
then A9:
LifeSpan ((ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))) = pseudo-LifeSpan ((Initialized ss),(Directed I))
by Th58;
DataPart (Initialized ss) = DataPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))
by SCMFSA8A:11;
then A10:
Directed I is_pseudo-closed_on (Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))
by A8, Th52;
A11:
DataPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) = DataPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))
by SCMFSA6A:39, SCMFSA8A:14;
A12:
now let f be
FinSeq-Location ;
((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) . f = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . fthus ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) . f =
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . f
by A11, SCMFSA6A:38
.=
(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . f
by A6, SCMFSA_2:96
;
verum end;
now let a be
Int-Location ;
((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) . a = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . athus ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) . a =
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . a
by A11, SCMFSA6A:38
.=
(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)) . a
by A6, SCMFSA_2:96
;
verum end;
then A13:
DataPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))) = DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by A12, SCMFSA6A:38;
card (if=0 (a,I,J)) =
((card I) + (card J)) + (3 + 1)
by SCMFSA8B:14
.=
(((card I) + (card J)) + 3) + 1
;
then
((card I) + (card J)) + 3 < card (if=0 (a,I,J))
by NAT_1:13;
then A14:
((card I) + (card J)) + 3 in dom (if=0 (a,I,J))
by AFINSQ_1:70;
A15: card (((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) =
(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)
;
(Initialized ss) +* (Initialized (if=0 (a,I,J))) = (Initialized (Initialized ss)) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by SCMFSA8A:13;
then A16:
(Initialized ss) +* (Initialized (if=0 (a,I,J))) = (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by Th15;
A17:
if=0 (a,I,J) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:26, SCMFSA6B:5;
A18:
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= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:26;
then A19:
if=0 (a,I,J) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by A18, XBOOLE_1:1;
A20:
(Directed I) +* (Start-At (0,SCM+FSA)) c= (Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:26;
A21:
if=0 (a,I,J) = ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) ';' (Stop SCM+FSA)
by SCMFSA8B:def 1;
then
if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' (I ';' (Stop SCM+FSA))
by SCMFSA6A:67;
then
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= if=0 (a,I,J)
by A15, FUNCT_4:26;
then
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by A19, XBOOLE_1:1;
then
ProgramPart (ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3)))) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)
by AMI_1:99;
then A22:
ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3))) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)
by RELAT_1:209;
ProgramPart (Relocated ((Directed I),((card J) + 3))) c= ProgramPart (Relocated ((I ';' (Stop SCM+FSA)),((card J) + 3)))
by Th12, SCMFSA6A:55;
then A23:
ProgramPart (Relocated ((Directed I),((card J) + 3))) c= Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)
by A22, XBOOLE_1:1;
Y:
(ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1)))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1)))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1))))
by COMPOS_1:38;
T:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by AMI_1:123;
TX3:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1)))
by AMI_1:123;
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1))) =
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I)))))
by EXTPRO_1:5
.=
(IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I)))))) + ((card J) + 3)
by A20, A10, A23, A7, A13, Th51, T
.=
(IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed I)))))) + ((card J) + 3)
by A8, Th50
.=
(card (ProgramPart (Directed I))) + ((card J) + 3)
by A8, SCMFSA8A:def 5
.=
(card (Directed I)) + ((card J) + 3)
by RELAT_1:209
.=
(card I) + ((card J) + 3)
by SCMFSA8A:34
.=
((card I) + (card J)) + 3
;
then A24: CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),((pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1)))) =
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (((card I) + (card J)) + 3)
by Y, TX3, AMI_1:54
.=
(if=0 (a,I,J)) . (((card I) + (card J)) + 3)
by A17, A14, GRFUNC_1:8
.=
halt SCM+FSA
by Th61
;
then A25:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) halts_on (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by EXTPRO_1:30;
now set J1 =
(((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I;
let k be
Element of
NAT ;
( CurInstr ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = halt SCM+FSA implies (pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1 <= k )assume A26:
CurInstr (
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),
(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)))
= halt SCM+FSA
;
(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1 <= kassume
not
(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1
<= k
;
contradictionthen A27:
k <= pseudo-LifeSpan (
((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),
(Directed I))
by NAT_1:13;
A28:
0 in dom (if=0 (a,I,J))
by Th54;
u:
Comput (
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),
0)
= (Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))
by EXTPRO_1:3;
Y:
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) /. (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by COMPOS_1:38;
CurInstr (
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),
(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),0))) =
CurInstr (
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))))
by u
.=
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . 0
by Th31, Y
.=
(if=0 (a,I,J)) . 0
by A28, Th26
.=
a =0_goto ((card J) + 3)
by Th55
;
then
k <> 0
by A26;
then consider k1 being
Nat such that A30:
k1 + 1
= k
by NAT_1:6;
reconsider k1 =
k1 as
Element of
NAT by ORDINAL1:def 13;
reconsider n =
IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),k1)) as
Element of
NAT ;
k1 < k
by A30, XREAL_1:31;
then A31:
k1 < pseudo-LifeSpan (
((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),
(Directed I))
by A27, XXREAL_0:2;
then
k1 < pseudo-LifeSpan (
(Initialized ss),
(Directed I))
by A8, Th50;
then
n in dom (Directed I)
by A8, SCMFSA8A:31;
then
n < card (Directed I)
by AFINSQ_1:70;
then
n + ((card J) + 3) < (card (Directed I)) + ((card J) + 3)
by XREAL_1:8;
then A32:
n + ((card J) + 3) < (card I) + ((card J) + 3)
by SCMFSA8A:34;
T:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by AMI_1:123;
A33:
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) =
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),k1))
by A30, EXTPRO_1:5
.=
(IC (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),k1))) + ((card J) + 3)
by A20, A10, A23, A7, A13, A31, Th51, T
;
card ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) =
(card (((Macro (a =0_goto ((card J) + 3))) ';' J) ';' (Goto ((card I) + 1)))) + (card I)
by SCMFSA6A:61
.=
((card ((Macro (a =0_goto ((card J) + 3))) ';' J)) + (card (Goto ((card I) + 1)))) + (card I)
by SCMFSA6A:61
.=
((card ((Macro (a =0_goto ((card J) + 3))) ';' J)) + 1) + (card I)
by SCMFSA8A:29
.=
(((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1) + (card I)
by SCMFSA6A:61
.=
((2 + (card J)) + 1) + (card I)
by COMPOS_1:150
.=
((card I) + (card J)) + 3
;
then
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) in dom ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)
by A33, A32, AFINSQ_1:70;
then A34:
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) in dom (Directed ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I))
by FUNCT_4:105;
then A35:
(Directed ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) in rng (Directed ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I))
by FUNCT_1:def 5;
card (if=0 (a,I,J)) =
((card I) + (card J)) + (3 + 1)
by SCMFSA8B:14
.=
(((card I) + (card J)) + 3) + 1
;
then
((card I) + (card J)) + 3
< card (if=0 (a,I,J))
by XREAL_1:31;
then
n + ((card J) + 3) < card (if=0 (a,I,J))
by A32, XXREAL_0:2;
then A36:
IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) in dom (if=0 (a,I,J))
by A33, AFINSQ_1:70;
Y:
(ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)))
by COMPOS_1:38;
TX3:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))
by AMI_1:123;
A37:
CurInstr (
(ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),
(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) =
((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)))
by Y, TX3, AMI_1:54
.=
(if=0 (a,I,J)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)))
by A36, Th26
;
Directed ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I) c= if=0 (
a,
I,
J)
by A21, SCMFSA6A:55;
then
(if=0 (a,I,J)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k))) = (Directed ((((a =0_goto ((card J) + 3)) ';' J) ';' (Goto ((card I) + 1))) ';' I)) . (IC (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),k)))
by A34, GRFUNC_1:8;
hence
contradiction
by A26, A35, A37, COMPOS_1:def 7;
verum end;
then A38:
LifeSpan ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))) = (pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I))) + 1
by A24, A25, EXTPRO_1:def 14;
(Initialized ss) +* (Initialized (I ';' (Stop SCM+FSA))) = (Initialized (Initialized ss)) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by SCMFSA8A:13;
then A39:
(Initialized ss) +* (Initialized (I ';' (Stop SCM+FSA))) = (Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by Th15;
A40:
(Directed I) ';' (Stop SCM+FSA) = I ';' (Stop SCM+FSA)
by SCMFSA8A:41;
Directed (Directed I) = Directed I
by SCMFSA6A:63;
then A41:
DataPart (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed I))))) = DataPart (Comput ((ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan ((Initialized ss),(Directed I)))))
by A8, A40, Th58;
I ';' (Stop SCM+FSA) is_halting_on Initialized ss
by A8, Th58;
then A42:
ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) halts_on (Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))
by SCMFSA7B:def 8;
T:
ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1))
by AMI_1:123;
thus DataPart (IExec ((if=0 (a,I,J)),ss)) =
DataPart (IExec ((if=0 (a,I,J)),(Initialized ss)))
by Th17
.=
DataPart ((Result ((ProgramPart ((Initialized ss) +* (Initialized (if=0 (a,I,J))))),((Initialized ss) +* (Initialized (if=0 (a,I,J)))))) +* ((Initialized ss) | NAT))
by SCMFSA6B:def 1
.=
DataPart (Result ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))))
by A16, Th35
.=
DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))))))
by A25, EXTPRO_1:23
.=
DataPart (Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart ((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((if=0 (a,I,J)) +* (Start-At (0,SCM+FSA)))),1)),(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I)))))
by A38, EXTPRO_1:5
.=
DataPart (Comput ((ProgramPart ((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(pseudo-LifeSpan (((Initialized ss) +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(Directed I)))))
by A20, A10, A23, A7, A13, T, Th51
.=
DataPart (Comput ((ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))))))
by A8, A9, A41, Th50
.=
DataPart (Result ((ProgramPart ((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized ss) +* ((I ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))))
by A42, EXTPRO_1:23
.=
DataPart ((Result ((ProgramPart ((Initialized ss) +* (Initialized (I ';' (Stop SCM+FSA))))),((Initialized ss) +* (Initialized (I ';' (Stop SCM+FSA)))))) +* ((Initialized ss) | NAT))
by A39, Th35
.=
DataPart (IExec ((I ';' (Stop SCM+FSA)),(Initialized ss)))
by SCMFSA6B:def 1
.=
DataPart (IExec ((I ';' (Stop SCM+FSA)),ss))
by Th17
; verum