proposition: Lemma Is_refining_T_step_productive_q: forall n xi alpha beta, Is_refining_T xi -> step_productive_q n xi alpha beta.
proof: Proof. induction n; intros xi alpha beta H_refining. constructor; apply Is_refining_T_emits_q; trivial. apply (step_productive_q_S n _ _ _ (Is_refining_T_emits_q _ alpha beta H_refining)). apply IHn. apply (Is_refining_T_modulus_q _ alpha beta H_refining). Qed.
imports: Reals QArithSternBrocot(R_addenda) Fourier_solvable_ineqs Fourier digits ub LNP_Digit Refining_T Incl_T quadratic qcorrectness productivity_M