(* similar remark apply here… *)

Lemma BC_to_TRS_infos_subst_rec_third g h0 h1 st n s f:
f ∈ all_lhs_funcs (snd (BC_to_TRS h1 (fst (BC_to_TRS h0 (fst (BC_to_TRS g st)))))) →
get_info (snd ((BC_to_TRS g >>+ BC_to_TRS h0 >>++ BC_to_TRS h1
>>+ Mnew_fun >>== Mmr (S n) s) st)) f =
get_info (snd (BC_to_TRS h1 (fst (BC_to_TRS h0 (fst (BC_to_TRS g st)))))) f.