proposition: Lemma middle_is_in_the_middle : forall x y, x < y -> x < middle x y < y.
proof: Proof. intros x y x_lt_y ; split. apply Rle_lt_trans with (middle x x). right ; symmetry ; apply middle_identity. unfold middle, Rdiv ; apply Rmult_lt_compat_r ; [lra |] ; apply Rplus_lt_compat_l ; assumption. apply Rlt_le_trans with (middle y y). unfold middle, Rdiv ; apply Rmult_lt_compat_r ; [lra |] ; apply Rplus_lt_compat_r ; assumption. right ; apply middle_identity. Qed.
imports: Rbase Rfunctions Rfunction_def Rtopology Lra MyRIneq MyNeq PSeries_reg Tactics