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
for m being Element of NAT st m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) holds
CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)))
let I be Program of SCM+FSA; :: thesis: ( I is_closed_onInit s & I is_halting_onInit s implies for m being Element of NAT st m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) holds
CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) )
set s1 = s +* (Initialized I);
set s2 = s +* (Initialized (loop I));
assume that
A1: I is_closed_onInit s and
A2: I is_halting_onInit s ; :: thesis: for m being Element of NAT st m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) holds
CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)))
let m be Element of NAT ; :: thesis: ( m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) implies CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) )
A3: IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) in dom I by A1, Def4;
then A4: IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) in dom (loop I) by FUNCT_4:105;
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) by AMI_1:123;
T2: ProgramPart (s +* (Initialized (loop I))) = ProgramPart (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)) by AMI_1:123;
Y: (ProgramPart (s +* (Initialized I))) /. (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by TX, COMPOS_1:38;
I c= Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by Th67, AMI_1:81;
then A5: CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by A3, Y, GRFUNC_1:8;
A6: loop I c= Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) by Th67, AMI_1:81;
assume A7: m < LifeSpan ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) ; :: thesis: CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)))
Z: (ProgramPart (s +* (Initialized (loop I)))) /. (IC (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) = (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)) . (IC (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) by T2, COMPOS_1:38;
ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) by A2, Def5;
then I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) <> halt SCM+FSA by A7, A5, EXTPRO_1:def 14;
then A8: I . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = (loop I) . (IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) by FUNCT_4:111;
IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) = IC (Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)) by A1, A2, A7, Th68, COMPOS_1:24;
hence CurInstr ((ProgramPart (s +* (Initialized I))),(Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m))) = CurInstr ((ProgramPart (s +* (Initialized (loop I)))),(Comput ((ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m))) by A6, A4, A5, A8, Z, GRFUNC_1:8; :: thesis: verum