proposition: Lemma derive_pt_recip_interv_prelim1_1_decr : forall (f g:R->R) (lb ub x:R), lb < ub -> f ub < x < f lb -> (forall x y : R, lb <= x -> x < y -> y <= ub -> f y < f x) -> (forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) -> (forall x, f ub <= x -> x <= f lb -> (comp f g) x = id x) -> lb <= g x <= ub.
proof: Proof. intros f g lb ub x lb_lt_ub x_encad f_decr g_wf f_eq_g. enough (-ub <= - g x <= - lb) by lra. unshelve eapply (derive_pt_recip_interv_prelim1_1 (mirr_fct f) (opp_fct g) (-ub) (-lb) (x)). - lra. - unfold mirr_fct; repeat rewrite Ropp_involutive; lra. - intros x0 y0 H1 H2 H3. unfold mirr_fct. apply f_decr; lra. - intros x0 H1 H2. unfold mirr_fct in H1,H2; unfold opp_fct. rewrite Ropp_involutive in H1,H2. pose proof g_wf x0 as g_wfs; lra. - intros x0 H1 H2. unfold mirr_fct in H1,H2 |-*; unfold opp_fct, comp. rewrite Ropp_involutive in H1,H2 |-*. apply f_eq_g; lra. Qed.
imports: Rbase Ranalysis_reg Rfunctions Rseries RiemannInt SeqProp Lia Lra