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(*                     Translation produces a QI                             *)
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Definition  valid_QI_prog trs := valid_QI variable function constructor
trs.(rules) mcs qic (qif trs) cs.
Definition subterm_QI_prog trs := subterm function (qif trs).
Definition monotonicity_QI_prog trs := monotonicity_qif function (qif trs).
Definition compatible_QI_prog trs := compatible_QI variable function constructor
trs.(rules) qic (qif trs).
Definition noqif:=λ (f:function) (l:list nat), 0.

Lemma map_BC_to_TRS_arity_norm bcl st trs n s:
let trsl := snd (lm_to_ml (map BC_to_TRS bcl) st) in
(∀ bc, bc ∈ bcl → arities bc = ok_arities n s) →
trs ∈ trsl →
norm (get_info_l trsl trs.(main)) = n.