proposition: Lemma TM_integral_error_0 (x0 : R) : contains iX0 (Xreal x0) -> i_validTM x0 iX Mf xF -> contains (I.convert TM_integral_error) (Xreal 0).
proof: Proof. move => Hx0X0 [_ _ ErrMf0 HX0X HPol]. case: HPol => [p Hcontains _]. replace 0 with ((x0 - x0) * 0) by ring. apply: J.mul_correct ErrMf0. apply: J.sub_correct (Hx0X0). exact: HX0X. Qed.
imports: Coq(ZArith Psatz Reals) Flocq(Raux) Coquelicot(Coquelicot) mathcomp.ssreflect(ssreflect ssrfun ssrbool eqtype ssrnat seq fintype bigop) Stdlib MathComp Coquelicot Xreal Interval Basic Xreal_derive Interval_compl Datatypes Taylor_poly Basic_rec Bound