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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.

(* Swapping QIs *)
(* in the following Lemmas, qif2 is first for easier partial application *)
Lemma term_assignment_of_value qif2 v qic qif1:
term_assignment variable function constructor qic qif1 (term_from_value v) =
term_assignment variable function constructor qic qif2 (term_from_value v).