(* similar remark apply here… *)

(* functions are within the bounds *)
Lemma funcs_bounds_aux bcl st st' f f_up:
fmono f_up →
(∀ bc, bc ∈ bcl → ∀ s f', f' ∈ all_lhs_funcs (snd (BC_to_TRS bc s)) →
s+1 <= f' <= f_up (fst (BC_to_TRS bc s))) →
st <= st' →
f ∈ map lhs_func (get_rules (snd (lm_to_ml (map BC_to_TRS bcl) st'))) →
st+1 <= f <= f_up (fst (lm_to_ml (map BC_to_TRS bcl) st')).