Record C1_fun : Type := mkC1
  {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.Record C1_fun : Type := mkC1
  {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.Record C1_fun : Type := mkC1
  {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.Record C1_fun : Type := mkC1
  {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.Record C1_fun : Type := mkC1
  {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.

(* Specifications: *)
Lemma RiemannInt_P31 : forall (f:C1_fun) (a b:R), a <= b -> antiderivative (derive f (diff0 f)) f a b.
Lemma RiemannInt_P32 : forall (f:C1_fun) (a b:R), Riemann_integrable (derive f (diff0 f)) a b.
Lemma RiemannInt_P33 : forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b), a <= b -> RiemannInt pr = f b - f a.
Lemma FTC_Riemann : forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b), RiemannInt pr = f b - f a.
Lemma RiemannInt_const_bound : forall f a b l u (h : Riemann_integrable f a b), a <= b -> (forall x, a < x < b -> l <= f x <= u) -> l * (b - a) <= RiemannInt h <= u * (b - a).
Lemma Riemann_integrable_scal : forall f a b k, Riemann_integrable f a b -> Riemann_integrable (fun x => k * f x) a b.
Lemma Riemann_integrable_Ropp : forall f a b, Riemann_integrable f a b -> Riemann_integrable (fun x => - f x) a b.