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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.

(* Monotonicity property *)
Lemma BC_to_TRS_qf_monotonicity_main bc st n s lx ly:
let trs := snd (BC_to_TRS bc st) in
arities bc = ok_arities n s →
Forall2 le lx ly →
asg_main trs lx ≤ asg_main trs ly.