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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(**     Definition of the derivative,continuity          *)
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Local Open Scope R_scope.
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Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.
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Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in f (D_x D x0) (f x0) x0.
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Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0.

(* Specifications: *)
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Lemma cont_deriv :
forall (f d:R -> R) (D:R -> Prop) (x0:R),
D_in f d D x0 -> continue_in f D x0.
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Lemma Dconst :
forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0.
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Lemma Dx :
forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0.
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Lemma Dadd :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0.
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Lemma Dmult :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0.
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Lemma Dmult_const :
forall (D:R -> Prop) (f df:R -> R) (x0 a:R),
D_in f df D x0 -> D_in (fun x:R => a * f x) (fun x:R => a * df x) D x0.
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Lemma Dopp :
forall (D:R -> Prop) (f df:R -> R) (x0:R),
D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0.
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Lemma Dminus :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0.
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Lemma Dx_pow_n :
forall (n:nat) (D:R -> Prop) (x0:R),
D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0.
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Lemma Dcomp :
forall (Df Dg:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df Df x0 ->
D_in g dg Dg (f x0) ->
D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0.
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Lemma D_pow_n :
forall (n:nat) (D:R -> Prop) (x0:R) (expr dexpr:R -> R),
D_in expr dexpr D x0 ->
D_in (fun x:R => expr x ^ n)
(fun x:R => INR n * expr x ^ (n - 1) * dexpr x) (
Dgf D D expr) x0.