let P be Instruction-Sequence of SCM+FSA; for s being State of SCM+FSA
for I, J being MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . (intloc 0) = 1 & s . a = 0 & Directed I is_pseudo-closed_on s,P holds
DataPart (IExec ((if=0 (a,I,J)),P,s)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,s))
let ss be State of SCM+FSA; for I, J being MacroInstruction 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,P holds
DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss))
set D = Data-Locations ;
let I, J be MacroInstruction 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,P holds
DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss))
let a be read-write Int-Location; ( ss . (intloc 0) = 1 & ss . a = 0 & Directed I is_pseudo-closed_on ss,P implies DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss)) )
set I0 = Directed I;
set s = Initialized ss;
set I1 = I ";" (Stop SCM+FSA);
set s00 = Initialize (Initialized ss);
set P00 = P +* (Directed I);
set s3 = Initialize (Initialized ss);
set P3 = P +* (if=0 (a,I,J));
A1:
if=0 (a,I,J) c= P +* (if=0 (a,I,J))
by FUNCT_4:25;
set s4 = Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1);
set i = a =0_goto ((card J) + 3);
A2:
Directed I c= P +* (Directed I)
by FUNCT_4:25;
assume A3:
ss . (intloc 0) = 1
; ( not ss . a = 0 or not Directed I is_pseudo-closed_on ss,P or DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss)) )
set s1 = Initialize (Initialized ss);
set P1 = P +* (I ";" (Stop SCM+FSA));
assume
ss . a = 0
; ( not Directed I is_pseudo-closed_on ss,P or DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss)) )
then A4:
(Initialized ss) . a = 0
by SCMFSA_M:37;
A5:
IC in dom (Start-At (0,SCM+FSA))
by MEMSTR_0:15;
A6: IC (Initialize (Initialized ss)) =
IC (Initialize (Initialized ss))
.=
IC (Start-At (0,SCM+FSA))
by A5, FUNCT_4:13
.=
0
by FUNCOP_1:72
;
A7:
0 in dom (if=0 (a,I,J))
by AFINSQ_1:65;
A8: (P +* (if=0 (a,I,J))) . 0 =
(if=0 (a,I,J)) . 0
by A7, FUNCT_4:13
.=
a =0_goto ((card J) + 3)
by Th18
;
A9: Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),(0 + 1)) =
Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),0)))
by EXTPRO_1:3
.=
Following ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)))
.=
Exec ((a =0_goto ((card J) + 3)),(Initialize (Initialized ss)))
by A6, A8, PBOOLE:143
;
A10:
a <> IC
by SCMFSA_2:56;
not a in dom (Start-At (0,SCM+FSA))
by A10, TARSKI:def 1;
then A11:
not a in dom (Start-At (0,SCM+FSA))
;
(Initialize (Initialized ss)) . a = 0
by A11, A4, FUNCT_4:11;
then A12:
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)) = (card J) + 3
by A9, SCMFSA_2:70;
assume
Directed I is_pseudo-closed_on ss,P
; DataPart (IExec ((if=0 (a,I,J)),P,ss)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss))
then A13:
Directed I is_pseudo-closed_on Initialized ss,P
by A3, Th16;
then A14:
LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize (Initialized ss))) = pseudo-LifeSpan ((Initialized ss),P,(Directed I))
by Th21;
A15:
Directed I is_pseudo-closed_on Initialize (Initialized ss),P +* (Directed I)
by A13;
A16:
for f being FinSeq-Location holds (Initialize (Initialized ss)) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)) . f
by A9, SCMFSA_2:70;
for a being Int-Location holds (Initialize (Initialized ss)) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)) . a
by A9, SCMFSA_2:70;
then A17:
DataPart (Initialize (Initialized ss)) = DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1))
by A16, SCMFSA_M:2;
card (if=0 (a,I,J)) =
((card I) + (card J)) + (3 + 1)
by SCMFSA8B:11
.=
(((card I) + (card J)) + 3) + 1
;
then
((card I) + (card J)) + 3 < card (if=0 (a,I,J))
by NAT_1:13;
then A18:
((card I) + (card J)) + 3 in dom (if=0 (a,I,J))
by AFINSQ_1:66;
A19: 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:21
.=
(card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + 1
by SCMFSA8A:15
.=
((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1
by SCMFSA6A:21
.=
((card J) + 2) + 1
by COMPOS_1:56
.=
(card J) + (2 + 1)
;
A20:
(Initialized ss) +* (Initialize ((intloc 0) .--> 1)) = Initialize (Initialized (Initialized ss))
by MEMSTR_0:44;
A21:
if=0 (a,I,J) c= P +* (if=0 (a,I,J))
by FUNCT_4:25;
if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))
by SCMFSA6A:25;
then
Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if=0 (a,I,J)
by A19, SCMFSA6A:38;
then A22:
Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if=0 (a,I,J))
by A21, XBOOLE_1:1;
Reloc ((Directed I),((card J) + 3)) c= Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3))
by COMPOS_1:44, SCMFSA6A:16;
then A23:
Reloc ((Directed I),((card J) + 3)) c= P +* (if=0 (a,I,J))
by A22, XBOOLE_1:1;
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),((pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1))) =
IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)),(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I)))))
by EXTPRO_1:4
.=
(IC (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I)))))) + ((card J) + 3)
by A15, A23, A12, A17, Th14, A2
.=
(IC (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),(pseudo-LifeSpan ((Initialized ss),P,(Directed I)))))) + ((card J) + 3)
by A13, Th13
.=
(card (Directed I)) + ((card J) + 3)
by A13, SCMFSA8A:def 4
.=
(card I) + ((card J) + 3)
by SCMFSA6A:36
.=
((card I) + (card J)) + 3
;
then A24: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),((pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1)))) =
(P +* (if=0 (a,I,J))) . (((card I) + (card J)) + 3)
by PBOOLE:143
.=
(if=0 (a,I,J)) . (((card I) + (card J)) + 3)
by A18, A1, GRFUNC_1:2
.=
halt SCM+FSA
by Th24
;
then A25:
P +* (if=0 (a,I,J)) halts_on Initialize (Initialized ss)
by EXTPRO_1:29;
now for k being Nat st CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k))) = halt SCM+FSA holds
(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1 <= kset J1 =
(((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I;
let k be
Nat;
( CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k))) = halt SCM+FSA implies (pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1 <= k )assume A26:
CurInstr (
(P +* (if=0 (a,I,J))),
(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)))
= halt SCM+FSA
;
(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1 <= kassume
not
(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1
<= k
;
contradictionthen A27:
k <= pseudo-LifeSpan (
(Initialize (Initialized ss)),
(P +* (Directed I)),
(Directed I))
by NAT_1:13;
A28:
0 in dom (if=0 (a,I,J))
by AFINSQ_1:65;
A29:
(P +* (if=0 (a,I,J))) /. (IC (Initialize (Initialized ss))) = (P +* (if=0 (a,I,J))) . (IC (Initialize (Initialized ss)))
by PBOOLE:143;
CurInstr (
(P +* (if=0 (a,I,J))),
(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),0))) =
(P +* (if=0 (a,I,J))) . 0
by A29, MEMSTR_0:16
.=
(if=0 (a,I,J)) . 0
by A28, A1, GRFUNC_1:2
.=
a =0_goto ((card J) + 3)
by Th18
;
then consider k1 being
Nat such that A30:
k1 + 1
= k
by A26, NAT_1:6;
reconsider k1 =
k1 as
Element of
NAT by ORDINAL1:def 12;
reconsider n =
IC (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),k1)) as
Element of
NAT ;
k1 < k
by A30, XREAL_1:29;
then A31:
k1 < pseudo-LifeSpan (
(Initialize (Initialized ss)),
(P +* (Directed I)),
(Directed I))
by A27, XXREAL_0:2;
then
k1 < pseudo-LifeSpan (
(Initialized ss),
P,
(Directed I))
by A13, Th13;
then
n in dom (Directed I)
by A13, SCMFSA8A:17;
then
n < card (Directed I)
by AFINSQ_1:66;
then
n + ((card J) + 3) < (card (Directed I)) + ((card J) + 3)
by XREAL_1:6;
then A32:
n + ((card J) + 3) < (card I) + ((card J) + 3)
by SCMFSA6A:36;
A33:
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)) =
IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)),k1))
by A30, EXTPRO_1:4
.=
(IC (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),k1))) + ((card J) + 3)
by A15, A23, A12, A17, A31, Th14, A2
;
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:21
.=
((card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1)))) + (card I)
by SCMFSA6A:21
.=
((card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + 1) + (card I)
by SCMFSA8A:15
.=
(((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1) + (card I)
by SCMFSA6A:21
.=
((2 + (card J)) + 1) + (card I)
by COMPOS_1:56
.=
((card I) + (card J)) + 3
;
then
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)) in dom ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I)
by A33, A32, AFINSQ_1:66;
then A34:
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)) in dom (Directed ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I))
by FUNCT_4:99;
then A35:
(Directed ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I)) . (IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k))) in rng (Directed ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I))
by FUNCT_1:def 3;
card (if=0 (a,I,J)) =
((card I) + (card J)) + (3 + 1)
by SCMFSA8B:11
.=
(((card I) + (card J)) + 3) + 1
;
then
((card I) + (card J)) + 3
< card (if=0 (a,I,J))
by XREAL_1:29;
then
n + ((card J) + 3) < card (if=0 (a,I,J))
by A32, XXREAL_0:2;
then A36:
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)) in dom (if=0 (a,I,J))
by A33, AFINSQ_1:66;
A37:
CurInstr (
(P +* (if=0 (a,I,J))),
(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k))) =
(P +* (if=0 (a,I,J))) . (IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)))
by PBOOLE:143
.=
(if=0 (a,I,J)) . (IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)))
by A36, A1, GRFUNC_1:2
;
Directed ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) c= if=0 (
a,
I,
J)
by SCMFSA6A:16;
then
(if=0 (a,I,J)) . (IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k))) = (Directed ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I)) . (IC (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),k)))
by A34, GRFUNC_1:2;
hence
contradiction
by A26, A35, A37, COMPOS_1:def 11;
verum end;
then A38:
LifeSpan ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss))) = (pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I))) + 1
by A24, A25, EXTPRO_1:def 15;
A39:
(Initialized ss) +* (Initialize ((intloc 0) .--> 1)) = Initialize (Initialized (Initialized ss))
by MEMSTR_0:44;
A40:
(Directed I) ";" (Stop SCM+FSA) = I ";" (Stop SCM+FSA)
;
A41:
DataPart (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),(pseudo-LifeSpan ((Initialized ss),P,(Directed I))))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize (Initialized ss)),(pseudo-LifeSpan ((Initialized ss),P,(Directed I)))))
by A13, A40, Th21;
I ";" (Stop SCM+FSA) is_halting_on Initialized ss,P
by A13, Th21;
then A42:
P +* (I ";" (Stop SCM+FSA)) halts_on Initialize (Initialized ss)
;
thus DataPart (IExec ((if=0 (a,I,J)),P,ss)) =
DataPart (IExec ((if=0 (a,I,J)),P,(Initialized ss)))
.=
DataPart (Result ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss))))
by A20
.=
DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),(LifeSpan ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss))))))
by A25, EXTPRO_1:23
.=
DataPart (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize (Initialized ss)),1)),(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I)))))
by A38, EXTPRO_1:4
.=
DataPart (Comput ((P +* (Directed I)),(Initialize (Initialized ss)),(pseudo-LifeSpan ((Initialize (Initialized ss)),(P +* (Directed I)),(Directed I)))))
by A15, A23, A12, A17, Th14, A2
.=
DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize (Initialized ss)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize (Initialized ss))))))
by A13, A14, A41, Th13
.=
DataPart (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialize (Initialized ss))))
by A42, EXTPRO_1:23
.=
DataPart (IExec ((I ";" (Stop SCM+FSA)),P,(Initialized ss)))
by A39
.=
DataPart (IExec ((I ";" (Stop SCM+FSA)),P,ss))
; verum