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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(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;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) /. (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 COMPOS_1:38;
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by AMI_1:123;
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, TX, 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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)
Z: (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)) /. (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 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, AMI_1:def 46;
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 (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m)),(Comput (ProgramPart (s +* (Initialized (loop I)))),(s +* (Initialized (loop I))),m) by A6, A4, A5, A8, Z, TX, GRFUNC_1:8; :: thesis: verum