A1: dom (id the Instructions of SCM+FSA ) = the Instructions of SCM+FSA by RELAT_1:71;
let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_onInit s & I is_halting_onInit s holds
( CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA ) )
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_onInit s & I is_halting_onInit s implies ( CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA ) ) )
set s1 = s +* (Initialized I);
set s2 = s +* (Initialized (loop I));
assume that
A2: I is_closed_onInit s and
A3: I is_halting_onInit s ; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA ) )
set k = LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I));
A4: rng I c= the Instructions of SCM+FSA by RELAT_1:def 19;
A5: IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) in dom I by A2, Def4;
then IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) in dom (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) by FUNCT_4:13;
then A6: IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) in dom (Initialized I) by FUNCT_4:15;
dom (loop I) = dom I by FUNCT_4:105;
then IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) in dom ((loop I) +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) by A5, FUNCT_4:13;
then A7: IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) in dom (Initialized (loop I)) by FUNCT_4:15;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) /. (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) = (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by COMPOS_1:38;
A8: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = (s +* (Initialized I)) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by Y, AMI_1:54
.= (Initialized I) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by A6, FUNCT_4:14
.= (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by FUNCT_4:15
.= I . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by Th70, FUNCT_4:12 ;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) /. (IC (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) = (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) . (IC (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by COMPOS_1:38;
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) by AMI_1:123;
A9: ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) by A3, Def5;
then A10: CurInstr (ProgramPart (s +* (Initialized I))),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = halt SCM+FSA by AMI_1:def 46;
IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = IC (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) by A2, A3, Th68, COMPOS_1:24;
hence A11: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) = (s +* (Initialized (loop I))) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by Y, AMI_1:54
.= (Initialized (loop I)) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by A7, FUNCT_4:14
.= ((loop I) +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by FUNCT_4:15
.= (loop I) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by Th70, FUNCT_4:12
.= (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto 0 )) * I) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))) by A4, FUNCT_7:118
.= ((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto 0 )) . (halt SCM+FSA ) by A10, A5, A8, TX, FUNCT_1:23
.= goto 0 by A1, FUNCT_7:33 ;
:: thesis: for m being Element of NAT st m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA
let m be Element of NAT ; :: thesis: ( m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) implies CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA )
assume A12: m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by AMI_1:123;

per cases ( m < LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) or m = LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) ) by A12, XXREAL_0:1;

suppose A13: m < LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA

then CurInstr (ProgramPart (s +* (Initialized I))),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) <> halt SCM+FSA by A9, AMI_1:def 46;
hence CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA by A2, A3, A13, Th69, TX; :: thesis: verum

end;

suppose m = LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA

hence CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) <> halt SCM+FSA by A11, SCMFSA_2:47, SCMFSA_2:124; :: thesis: verum

end;

end;