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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 ta_of_p_subst qif2 p qic s qif1:
term_assignment variable function constructor qic qif1 (subst s (term_from_pattern p)) =
term_assignment variable function constructor qic qif2 (subst s (term_from_pattern p)).