proposition: Lemma derive_pt_recip_interv : forall (f g:R->R) (lb ub x:R) (lb_lt_ub:lb < ub) (x_encad:f lb < x < f ub) (f_incr:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) (g_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) (Prf:forall a : R, lb <= a <= ub -> derivable_pt f a) (f_eq_g:forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) (Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad g_wf Prf) <> 0), derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr Prf Df_neq) = 1 / (derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))).
proof: Proof. intros. assert(g_incr := (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g)). apply derive_pt_recip_interv_prelim0 with (lb:=f lb) (ub:=f ub) ; [intuition |assumption | intuition |]. intro Hfalse ; apply Df_neq. rewrite pr_nu_var2_interv with (g:=f) (lb:=lb) (ub:=ub) (pr2:= (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g))) ; [intuition | intuition | | intuition]. exact (derive_pt_recip_interv_prelim1_0 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g). Qed.
imports: Rbase Ranalysis_reg Rfunctions Rseries RiemannInt SeqProp Lia Lra