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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 BC_to_TRS_comp_rec_rec g h0 h1 st n s sub trs_h ps:
let myqif := qif (snd ((BC_to_TRS g >>+ BC_to_TRS h0 >>++ BC_to_TRS h1 >>+ Mnew_fun >>==
Mmr (S n) s) st)) in
arities h0 = ok_arities (S n) (S s) →
arities h1 = ok_arities (S n) (S s) →
trs_h=snd (BC_to_TRS h0 (fst (BC_to_TRS g st))) ∨
trs_h=snd (BC_to_TRS h1 (fst (BC_to_TRS h0 (fst (BC_to_TRS g st))))) →
ps=ps0 ∨ ps=ps1 →
term_assignment variable function constructor qic myqif
(subst sub (fapply trs_h.(main) (mk_args_h (S n) s (mk_call (S (n + s))
(S (fst (BC_to_TRS h1 (fst (BC_to_TRS h0 (fst (BC_to_TRS g st)))))))))))
≤ myqif (S (fst (BC_to_TRS h1 (fst (BC_to_TRS h0 (fst (BC_to_TRS g st)))))))
(map (term_assignment variable function constructor qic myqif)
(map (subst sub) (map term_from_pattern (ps px :: mk_pvl 1 (S (n + s)))))).