let s be State of SCM+FSA; for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s holds
( IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) = card I & DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) )
let I be Program of SCM+FSA; ( I is_closed_on s & I is_halting_on s implies ( IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) = card I & DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) ) )
set s1 = s +* (I +* (Start-At (0,SCM+FSA)));
set s2 = s +* ((Directed I) +* (Start-At (0,SCM+FSA)));
set m1 = LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))));
assume that
A1:
I is_closed_on s
and
A2:
I is_halting_on s
; ( IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) = card I & DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) )
A3:
ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA)))
by A2, SCMFSA7B:def 8;
set l1 = IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))));
A4:
IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom I
by A1, SCMFSA7B:def 7;
then
IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom I
by A1, A2, Th35, COMPOS_1:24;
then A5:
IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom (Directed I)
by FUNCT_4:105;
Directed I c= (Directed I) +* (Start-At (0,SCM+FSA))
by Th9;
then A6:
dom (Directed I) c= dom ((Directed I) +* (Start-At (0,SCM+FSA)))
by GRFUNC_1:8;
A7:
dom I = dom (Directed I)
by FUNCT_4:105;
Y:
(ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by COMPOS_1:38;
TX:
ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))
by AMI_1:123;
I c= I +* (Start-At (0,SCM+FSA))
by Th9;
then
dom I c= dom (I +* (Start-At (0,SCM+FSA)))
by GRFUNC_1:8;
then
(s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (I +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A4, FUNCT_4:14;
then A8: I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) =
(s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A4, COMPOS_1:145
.=
CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by Y, TX, AMI_1:54
.=
halt SCM+FSA
by A3, EXTPRO_1:def 14
;
IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))
by A1, A2, Th35, COMPOS_1:24;
then A9: (s +* ((Directed I) +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) =
((Directed I) +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A5, A6, FUNCT_4:14
.=
(Directed I) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A4, A7, COMPOS_1:145
.=
goto (card I)
by A4, A8, FUNCT_4:112
;
Y:
(ProgramPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) /. (IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by COMPOS_1:38;
A10: CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))),(Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) =
(Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A1, A2, Th35, Y, COMPOS_1:24
.=
goto (card I)
by A9, AMI_1:54
;
T:
ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))
by AMI_1:123;
A11: Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1)) =
Following ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by EXTPRO_1:4
.=
Exec ((goto (card I)),(Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))
by A10, T
;
hence
IC (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) = card I
by SCMFSA_2:95; DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1)))
A12:
( ( for a being Int-Location holds (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) . a = (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . a ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1))) . f = (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . f ) )
by A11, SCMFSA_2:95;
DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))
by A1, A2, Th35, SCMFSA6A:39;
hence
DataPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = DataPart (Comput ((ProgramPart (s +* ((Directed I) +* (Start-At (0,SCM+FSA))))),(s +* ((Directed I) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))) + 1)))
by A12, SCMFSA6A:38; verum