proposition: Lemma derivable_pt_recip_interv_decr (f g:R->R) (lb ub x : R) (lb_lt_ub:lb < ub) (x_encad:f ub < x < f lb) (f_eq_g:forall x : R, f ub <= x -> x <= f lb -> comp f g x = id x) (g_wf:forall x : R, f ub <= x -> x <= f lb -> lb <= g x <= ub) (f_decr:forall x y : R, lb <= x -> x < y -> y <= ub -> f y < f x) (f_derivable:forall a : R, lb <= a <= ub -> derivable_pt f a) : derive_pt f (g x) (derivable_pt_recip_interv_prelim1_decr f g lb ub x lb_lt_ub x_encad g_wf f_derivable) <> 0 -> derivable_pt g x.
proof: Proof. intros. apply derivable_pt_opp_rev. unshelve eapply (derivable_pt_recip_interv (mirr_fct f) (opp_fct g) (-ub) (-lb) (x)). - lra. - unfold mirr_fct; repeat rewrite Ropp_involutive; 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. apply derivable_pt_mirr, f_derivable; 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. - intros x0 y0 H1 H2 H3. unfold mirr_fct. apply f_decr; lra. - unshelve erewrite (pr_nu _ _ _). + apply derivable_pt_mirr. unfold opp_fct; rewrite Ropp_involutive. apply f_derivable; apply g_wf; lra. + rewrite derive_pt_mirr. unfold opp_fct; rewrite Ropp_involutive. match goal with H:context[derive_pt _ _ ?pr] |- _ => rewrite (pr_nu f (g x) _ pr) end. apply Ropp_neq_0_compat. assumption. Qed.
imports: Rbase Ranalysis_reg Rfunctions Rseries RiemannInt SeqProp Lia Lra