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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 max_match x y z:
x ≤ match max x (max y z) with
| 0 => 1
| S n => S n
end.