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 (s +* (Initialized (loop 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 (s +* (Initialized (loop I)))),(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 (s +* (Initialized (loop 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 (s +* (Initialized (loop I)))),(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 (s +* (Initialized (loop 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 (s +* (Initialized (loop I)))),(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;
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;
Y: (ProgramPart (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 TX, COMPOS_1:38;
A8: CurInstr ((ProgramPart (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 ;
T2: ProgramPart (s +* (Initialized (loop I))) = ProgramPart (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),(LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))))) by AMI_1:123;
Y: (ProgramPart (s +* (Initialized (loop 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 T2, COMPOS_1:38;
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 EXTPRO_1:def 14;
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 (s +* (Initialized (loop 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, 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 (s +* (Initialized (loop I)))),(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 (s +* (Initialized (loop I)))),(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 (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) <> halt SCM+FSA

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 (s +* (Initialized (loop I)))),(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, EXTPRO_1:def 14;
hence CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) <> halt SCM+FSA by A2, A3, A13, Th69; :: thesis: verum

end;

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

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

end;

end;