/
derive.v
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/
derive.v
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(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *)
From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice.
From mathcomp Require Import ssralg ssrnum fintype bigop order matrix interval.
Require Import boolp reals.
Require Import classical_sets posnum topology prodnormedzmodule normedtype landau forms.
(******************************************************************************)
(* This file provides a theory of differentiation. It includes the standard *)
(* rules of differentiation (differential of a sum, of a product, of *)
(* exponentiation, of the inverse, etc.) as well standard theorems (the *)
(* Extreme Value Theorem, Rolle's theorem, the Mean Value Theorem). *)
(* *)
(* Parsable notations: *)
(* 'd f x == the differential of a function f at a point x *)
(* differentiable f x == the function f is differentiable at a point x *)
(* 'J f x == the Jacobian of f at a point x *)
(* 'D_v f == the directional derivative of f along v *)
(* f^`() == the derivative of f of domain R *)
(* f^`(n) == the nth derivative of f of domain R *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Theory.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Reserved Notation "''d' f x" (at level 0, f at level 0, x at level 0,
format "''d' f x").
Reserved Notation "'is_diff' F" (at level 0, F at level 0,
format "'is_diff' F").
Reserved Notation "''J' f p" (at level 10, p, f at next level,
format "''J' f p").
Reserved Notation "''D_' v f" (at level 10, v, f at next level,
format "''D_' v f").
Reserved Notation "''D_' v f c" (at level 10, v, f at next level,
format "''D_' v f c"). (* printing *)
Reserved Notation "f ^` ()" (at level 8, format "f ^` ()").
Reserved Notation "f ^` ( n )" (at level 8, format "f ^` ( n )").
Section Differential.
Context {K : numDomainType} {V W : normedModType K}.
Definition diff (F : filter_on V) (_ : phantom (set (set V)) F) (f : V -> W) :=
(get (fun (df : {linear V -> W}) => continuous df /\ forall x,
f x = f (lim F) + df (x - lim F) +o_(x \near F) (x - lim F))).
Local Notation "''d' f x" := (@diff _ (Phantom _ [filter of x]) f).
Fact diff_key : forall T, T -> unit. Proof. by constructor. Qed.
CoInductive differentiable_def (f : V -> W) (x : filter_on V)
(phF : phantom (set (set V)) x) : Prop := DifferentiableDef of
(continuous ('d f x) /\
f = cst (f (lim x)) + 'd f x \o center (lim x) +o_x (center (lim x))).
Local Notation differentiable f F := (@differentiable_def f _ (Phantom _ [filter of F])).
Class is_diff_def (x : filter_on V) (Fph : phantom (set (set V)) x) (f : V -> W)
(df : V -> W) := DiffDef {
ex_diff : differentiable f x ;
diff_val : 'd f x = df :> (V -> W)
}.
Hint Mode is_diff_def - - ! - : typeclass_instances.
Lemma diffP (F : filter_on V) (f : V -> W) :
differentiable f F <->
continuous ('d f F) /\
(forall x, f x = f (lim F) + 'd f F (x - lim F) +o_(x \near F) (x - lim F)).
Proof. by split=> [[] |]; last constructor; rewrite funeqE. Qed.
Lemma diff_continuous (x : filter_on V) (f : V -> W) :
differentiable f x -> continuous ('d f x).
Proof. by move=> /diffP []. Qed.
(* We should have a continuous class or structure *)
Hint Extern 0 (continuous _) => exact: diff_continuous : core.
Lemma diffE (F : filter_on V) (f : V -> W) :
differentiable f F ->
forall x, f x = f (lim F) + 'd f F (x - lim F) +o_(x \near F) (x - lim F).
Proof. by move=> /diffP []. Qed.
Lemma littleo_center0 (x : V) (f : V -> W) (e : V -> V) :
[o_x e of f] = [o_ (0 : V) (e \o shift x) of f \o shift x] \o center x.
Proof.
rewrite /the_littleo /insubd /=; have [g /= _ <-{f}|/asboolP Nfe] /= := insubP.
rewrite insubT //= ?comp_shiftK //; apply/asboolP => _/posnumP[eps].
rewrite [\forall x \near _, _ <= _](near_shift x) sub0r; near=> y.
by rewrite /= subrK; near: y; have /eqoP := littleo_eqo g; apply.
rewrite insubF //; apply/asboolP => fe; apply: Nfe => _/posnumP[eps].
by rewrite [\forall x \near _, _ <= _](near_shift 0) subr0; apply: fe.
Grab Existential Variables. end_near. Qed.
End Differential.
Section Differential_numFieldType.
Context {K : numFieldType (*TODO: to numDomainType?*)} {V W : normedModType K}.
(* duplicate from Section Differential *)
Local Notation differentiable f F := (@differentiable_def _ _ _ f _ (Phantom _ [filter of F])).
Local Notation "''d' f x" := (@diff _ _ _ _ (Phantom _ [filter of x]) f).
Hint Extern 0 (continuous _) => exact: diff_continuous : core.
Lemma diff_locallyxP (x : V) (f : V -> W) :
differentiable f x <-> continuous ('d f x) /\
forall h, f (h + x) = f x + 'd f x h +o_(h \near 0 : V) h.
Proof.
split=> [dxf|[dfc dxf]].
split => //; apply: eqaddoEx => h; have /diffE -> := dxf.
rewrite lim_id // addrK; congr (_ + _); rewrite littleo_center0 /= addrK.
by congr ('o); rewrite funeqE => k /=; rewrite addrK.
apply/diffP; split=> //; apply: eqaddoEx; move=> y.
rewrite lim_id // -[in LHS](subrK x y) dxf; congr (_ + _).
rewrite -(comp_centerK x id) -[X in the_littleo _ _ _ X](comp_centerK x).
by rewrite -[_ (y - x)]/((_ \o (center x)) y) -littleo_center0.
Qed.
Lemma diff_locallyx (x : V) (f : V -> W) : differentiable f x ->
forall h, f (h + x) = f x + 'd f x h +o_(h \near 0 : V) h.
Proof. by move=> /diff_locallyxP []. Qed.
Lemma diff_locallyxC (x : V) (f : V -> W) : differentiable f x ->
forall h, f (x + h) = f x + 'd f x h +o_(h \near 0 : V) h.
Proof. by move=> ?; apply/eqaddoEx => h; rewrite [x + h]addrC diff_locallyx. Qed.
Lemma diff_locallyP (x : V) (f : V -> W) :
differentiable f x <->
continuous ('d f x) /\ (f \o shift x = cst (f x) + 'd f x +o_ (0 : V) id).
Proof. by apply: iff_trans (diff_locallyxP _ _) _; rewrite funeqE. Qed.
Lemma diff_locally (x : V) (f : V -> W) : differentiable f x ->
(f \o shift x = cst (f x) + 'd f x +o_ (0 : V) id).
Proof. by move=> /diff_locallyP []. Qed.
End Differential_numFieldType.
Notation "''d' f F" := (@diff _ _ _ _ (Phantom _ [filter of F]) f).
Notation differentiable f F := (@differentiable_def _ _ _ f _ (Phantom _ [filter of F])).
Notation "'is_diff' F" := (is_diff_def (Phantom _ [filter of F])).
Hint Extern 0 (differentiable _ _) => solve[apply: ex_diff] : core.
Hint Extern 0 ({for _, continuous _}) => exact: diff_continuous : core.
Lemma differentiableP (R : numDomainType) (V W : normedModType R) (f : V -> W) x :
differentiable f x -> is_diff x f ('d f x).
Proof. by move=> ?; apply: DiffDef. Qed.
Section jacobian.
Definition jacobian n m (R : numFieldType) (f : 'rV[R]_n.+1 -> 'rV[R]_m.+1) p :=
lin1_mx ('d f p).
End jacobian.
Notation "''J' f p" := (jacobian f p).
Section DifferentialR.
Context {R : numFieldType} {V W : normedModType R}.
(* split in multiple bits:
- a linear map which is locally bounded is a little o of 1
- the identity is a littleo of 1
*)
Lemma differentiable_continuous (x : V) (f : V -> W) :
differentiable f x -> {for x, continuous f}.
Proof.
move=> /diff_locallyP [dfc]; rewrite -addrA.
rewrite (littleo_bigO_eqo (cst (1 : R^o))); last first.
apply/eqOP; near=> k; rewrite /cst [`|1 : R^o|]normr1 mulr1.
near=> y; rewrite ltW //; near: y; apply/locally_normP.
by exists k; [near: k; exists 0; rewrite real0|move=> ? /=; rewrite sub0r normrN].
rewrite addfo; first by move=> /eqolim; rewrite cvg_shift add0r.
by apply/eqolim0P; apply: (cvg_trans (dfc 0)); rewrite linear0.
Grab Existential Variables. all: end_near. Qed.
Section littleo_lemmas.
Variables (X Y Z : normedModType R).
Lemma normm_littleo x (f : X -> Y) : `| [o_(x \near x) (1 : R^o) of f x]| = 0.
Proof.
rewrite /cst /=; have [e /(_ (`|e x|/2) _)/locally_singleton /=] := littleo.
rewrite pmulr_lgt0 // [`|1 : R^o|]normr1 mulr1 [X in X <= _]splitr.
rewrite ger_addr pmulr_lle0 // => /implyP.
case : real_ltgtP; rewrite ?realE ?normrE //=.
by apply/orP; left.
Qed.
Lemma littleo_lim0 (f : X -> Y) (h : _ -> Z) (x : X) :
f @ x --> (0 : Y) -> [o_x f of h] x = 0.
Proof.
move/eqolim0P => ->; have [k /(_ _ [gt0 of 1])/=] := littleo.
by move=> /locally_singleton; rewrite mul1r normm_littleo normr_le0 => /eqP.
Qed.
End littleo_lemmas.
Section diff_locally_converse_tentative.
(* if there exist A and B s.t. f(a + h) = A + B h + o(h) then
f is differentiable at a, A = f(a) and B = f'(a) *)
(* this is a consequence of diff_continuous and eqolim0 *)
(* indeed the differential being b *: idfun is locally bounded *)
(* and thus a littleo of 1, and so is id *)
(* This can be generalized to any dimension *)
Lemma diff_locally_converse_part1 (f : R^o -> R^o) (a : R^o) (b : R^o) (x : R^o) :
f \o shift x = cst a + b *: idfun +o_ (0 : [filteredType R^o of R^o]) id -> f x = a.
Proof.
rewrite funeqE => /(_ 0) /=; rewrite add0r => ->.
by rewrite -[LHS]/(_ 0 + _ 0 + _ 0) /cst [X in a + X]scaler0 littleo_lim0 ?addr0.
Qed.
End diff_locally_converse_tentative.
Definition derive (f : V -> W) a v :=
lim ((fun h => h^-1 *: ((f \o shift a) (h *: v) - f a)) @ locally' (0 : R^o)).
Local Notation "''D_' v f" := (derive f ^~ v).
Local Notation "''D_' v f c" := (derive f c v). (* printing *)
Definition derivable (f : V -> W) a v :=
cvg ((fun h => h^-1 *: ((f \o shift a) (h *: v) - f a)) @ locally' (0 : R^o)).
Class is_derive (a v : V) (f : V -> W) (df : W) := DeriveDef {
ex_derive : derivable f a v;
derive_val : 'D_v f a = df
}.
Lemma derivable_locally (f : V -> W) a v :
derivable f a v ->
(fun h => (f \o shift a) (h *: v)) = (cst (f a)) +
(fun h => h *: ('D_v f a)) +o_ (locally (0 : [filteredType R^o of R^o])) id.
Proof.
move=> df; apply/eqaddoP => _/posnumP[e].
rewrite -locally_nearE locally_simpl /= locallyE'; split; last first.
rewrite /at_point opprD -![(_ + _ : _ -> _) _]/(_ + _) scale0r add0r.
by rewrite addrA subrr add0r normrN scale0r !normr0 mulr0.
have /eqolimP := df; rewrite -[lim _]/(derive _ _ _).
move=> /eqaddoP /(_ e%:num) /(_ [gt0 of e%:num]).
apply: filter_app; rewrite /= !near_simpl near_withinE; near=> h => hN0.
rewrite /= opprD -![(_ + _ : _ -> _) _]/(_ + _) -![(- _ : _ -> _) _]/(- _).
rewrite /cst /= [`|1 : R^o|]normr1 mulr1 => dfv.
rewrite addrA -[X in X + _]scale1r -(@mulVf _ h) //.
rewrite mulrC -scalerA -scalerBr normmZ.
rewrite -ler_pdivl_mull; last by rewrite normr_gt0.
by rewrite mulrCA mulVf ?mulr1; last by rewrite normr_eq0.
Grab Existential Variables. all: end_near. Qed.
Lemma derivable_locallyP (f : V -> W) a v :
derivable f a v <->
(fun h => (f \o shift a) (h *: v)) = (cst (f a)) +
(fun h => h *: ('D_v f a)) +o_ (locally (0 : [filteredType R^o of R^o])) id.
Proof.
split; first exact: derivable_locally.
move=> df; apply/cvg_ex; exists ('D_v f a).
apply/(@eqolimP _ R^o _ (locally'_filter_on _))/eqaddoP => _/posnumP[e].
have /eqaddoP /(_ e%:num) /(_ [gt0 of e%:num]) := df.
rewrite /= !(near_simpl, near_withinE); apply: filter_app; near=> h.
rewrite /= opprD -![(_ + _ : _ -> _) _]/(_ + _) -![(- _ : _ -> _) _]/(- _).
rewrite /cst /= [`|1 : R^o|]normr1 mulr1 addrA => dfv hN0.
rewrite -[X in _ - X]scale1r -(@mulVf _ h) //.
rewrite -scalerA -scalerBr normmZ normfV ler_pdivr_mull ?normr_gt0 //.
by rewrite mulrC.
Grab Existential Variables. all: end_near. Qed.
Lemma derivable_locallyx (f : V -> W) a v :
derivable f a v -> forall h, f (a + h *: v) = f a + h *: 'D_v f a
+o_(h \near (locally (0 : [filteredType R^o of R^o]))) h.
Proof.
move=> /derivable_locally; rewrite funeqE => df.
by apply: eqaddoEx => h; have /= := (df h); rewrite addrC => ->.
Qed.
Lemma derivable_locallyxP (f : V -> W) a v :
derivable f a v <-> forall h, f (a + h *: v) = f a + h *: 'D_v f a
+o_(h \near (locally (0 : [filteredType R^o of R^o]))) h.
Proof.
split; first exact: derivable_locallyx.
move=> df; apply/derivable_locallyP; apply/eqaddoE; rewrite funeqE => h.
by rewrite /= addrC df.
Qed.
End DifferentialR.
Notation "''D_' v f" := (derive f ^~ v).
Notation "''D_' v f c" := (derive f c v). (* printing *)
Hint Extern 0 (derivable _ _ _) => solve[apply: ex_derive] : core.
Section DifferentialR_numFieldType.
Context {R : numFieldType} {V W : normedModType R}.
Lemma deriveE (f : V -> W) (a v : V) :
differentiable f a -> 'D_v f a = 'd f a v.
Proof.
rewrite /derive => /diff_locally -> /=; set k := 'o _.
evar (g : R -> W); rewrite [X in X @ _](_ : _ = g) /=; last first.
rewrite funeqE=> h; rewrite !scalerDr scalerN /cst /=.
by rewrite addrC !addrA addNr add0r linearZ /= scalerA /g.
apply: cvg_map_lim => //.
pose g1 : R -> W := fun h => (h^-1 * h) *: 'd f a v.
pose g2 : R -> W := fun h : R => h^-1 *: k (h *: v ).
rewrite (_ : g = g1 + g2) ?funeqE // -(addr0 (_ _ v)); apply: (@cvgD _ _ [topologicalType of R^o]).
rewrite -(scale1r (_ _ v)); apply: cvgZl => /= X [e e0].
rewrite /ball_ /= => eX.
apply/locallyP; rewrite locally_E.
by exists e => //= x _ x0; apply eX; rewrite mulVr // ?unitfE // subrr normr0.
rewrite /g2.
have [/eqP ->|v0] := boolP (v == 0).
rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst.
by rewrite funeqE => ?; rewrite scaler0 /k littleo_lim0 // scaler0.
apply/cvg_distP => e e0.
rewrite nearE /=; apply/locallyP; rewrite locally_E.
have /(littleoP [littleo of k]) /locallyP[i i0 Hi] : 0 < e / (2 * `|v|).
by rewrite divr_gt0 // pmulr_rgt0 // normr_gt0.
exists (i / `|v|); first by rewrite divr_gt0 // normr_gt0.
move=> /= j; rewrite /ball /= /ball_ add0r normrN.
rewrite ltr_pdivl_mulr ?normr_gt0 // => jvi j0.
rewrite add0r normrN normmZ -ltr_pdivl_mull ?normr_gt0 ?invr_neq0 //.
have /Hi/le_lt_trans -> // : ball 0 i (j *: v).
by rewrite -ball_normE /ball_ add0r normrN (le_lt_trans _ jvi) // normmZ.
rewrite -(mulrC e) -mulrA -ltr_pdivl_mull // mulrA mulVr ?unitfE ?gt_eqF //.
rewrite normrV ?unitfE // div1r invrK ltr_pdivr_mull; last first.
by rewrite pmulr_rgt0 // normr_gt0.
rewrite normmZ mulrC -mulrA.
by rewrite ltr_pmull ?ltr1n // pmulr_rgt0 ?normm_gt0 // normr_gt0.
Qed.
End DifferentialR_numFieldType.
Section DifferentialR2.
Variable R : numFieldType.
Implicit Type (V : normedModType R).
Lemma derivemxE m n (f : 'rV[R]_m.+1 -> 'rV[R]_n.+1) (a v : 'rV[R^o]_m.+1) :
differentiable f a -> 'D_ v f a = v *m jacobian f a.
Proof. by move=> /deriveE->; rewrite /jacobian mul_rV_lin1. Qed.
Definition derive1 V (f : R -> V) (a : R) :=
lim ((fun h => h^-1 *: (f (h + a) - f a)) @ locally' (0 : R^o)).
Local Notation "f ^` ()" := (derive1 f).
Lemma derive1E V (f : R -> V) a : f^`() a = 'D_1 (f : R^o -> _) a.
Proof.
rewrite /derive1 /derive; set d := (fun _ : R => _); set d' := (fun _ : R => _).
by suff -> : d = d' by []; rewrite funeqE=> h; rewrite /d /d' /= [h%:A](mulr1).
Qed.
(* Is it necessary? *)
Lemma derive1E' V f a : differentiable (f : R^o -> V) a ->
f^`() a = 'd f a 1.
Proof. by move=> ?; rewrite derive1E deriveE. Qed.
Definition derive1n V n (f : R -> V) := iter n (@derive1 V) f.
Local Notation "f ^` ( n )" := (derive1n n f).
Lemma derive1n0 V (f : R -> V) : f^`(0) = f.
Proof. by []. Qed.
Lemma derive1n1 V (f : R -> V) : f^`(1) = f^`().
Proof. by []. Qed.
Lemma derive1nS V (f : R -> V) n : f^`(n.+1) = f^`(n)^`().
Proof. by []. Qed.
Lemma derive1Sn V (f : R -> V) n : f^`(n.+1) = f^`()^`(n).
Proof. exact: iterSr. Qed.
End DifferentialR2.
Notation "f ^` ()" := (derive1 f).
Notation "f ^` ( n )" := (derive1n n f).
Section DifferentialR3.
Variable R : numFieldType.
Fact dcst (V W : normedModType R) (a : W) (x : V) : continuous (0 : V -> W) /\
cst a \o shift x = cst (cst a x) + \0 +o_ (0 : V) id.
Proof.
split; first exact: continuous_cst.
apply/eqaddoE; rewrite addr0 funeqE => ? /=; rewrite -[LHS]addr0; congr (_ + _).
by rewrite littleoE; last exact: littleo0_subproof.
Qed.
Variables (V W : normedModType R).
Fact dadd (f g : V -> W) x :
differentiable f x -> differentiable g x ->
continuous ('d f x \+ 'd g x) /\
(f + g) \o shift x = cst ((f + g) x) + ('d f x \+ 'd g x) +o_ (0 : V) id.
Proof.
move=> df dg; split => [?|]; do ?exact: continuousD.
apply/(@eqaddoE R); rewrite funeqE => y /=; rewrite -[(f + g) _]/(_ + _).
by rewrite ![_ (_ + x)]diff_locallyx// addrACA addox addrACA.
Qed.
Fact dopp (f : V -> W) x :
differentiable f x -> continuous (- ('d f x : V -> W)) /\
(- f) \o shift x = cst (- f x) \- 'd f x +o_ (0 : V) id.
Proof.
move=> df; split; first by move=> ?; apply: continuousN.
apply/eqaddoE; rewrite funeqE => y /=.
by rewrite -[(- f) _]/(- (_ _)) diff_locallyx// !opprD oppox.
Qed.
Lemma is_diff_eq (V' W' : normedModType R) (f f' g : V' -> W') (x : V') :
is_diff x f f' -> f' = g -> is_diff x f g.
Proof. by move=> ? <-. Qed.
Fact dscale (f : V -> W) k x :
differentiable f x -> continuous (k \*: 'd f x) /\
(k *: f) \o shift x = cst ((k *: f) x) + k \*: 'd f x +o_ (0 : V) id.
Proof.
move=> df; split; first by move=> ?; apply: continuousZr.
apply/eqaddoE; rewrite funeqE => y /=.
by rewrite -[(k *: f) _]/(_ *: _) diff_locallyx // !scalerDr scaleox.
Qed.
(* NB: could be generalized with K : absRingType instead of R; DONE? *)
Fact dscalel (k : V -> R^o) (f : W) x :
differentiable k x ->
continuous (fun z : V => 'd k x z *: f) /\
(fun z => k z *: f) \o shift x =
cst (k x *: f) + (fun z => 'd k x z *: f) +o_ (0 : V) id.
Proof.
move=> df; split.
move=> ?; exact/continuousZl/(@diff_continuous _ _ [normedModType R of R^o]).
apply/eqaddoE; rewrite funeqE => y /=.
by rewrite diff_locallyx //= !scalerDl scaleolx.
Qed.
Fact dlin (V' W' : normedModType R) (f : {linear V' -> W'}) x :
continuous f -> f \o shift x = cst (f x) + f +o_ (0 : V') id.
Proof.
move=> df; apply: eqaddoE; rewrite funeqE => y /=.
rewrite linearD addrC -[LHS]addr0; congr (_ + _).
by rewrite littleoE; last exact: littleo0_subproof. (*fixme*)
Qed.
(* TODO: generalize *)
Lemma compoO_eqo (U V' W' : normedModType R) (f : U -> V')
(g : V' -> W') :
[o_ (0 : V') id of g] \o [O_ (0 : U) id of f] =o_ (0 : U) id.
Proof.
apply/eqoP => _ /posnumP[e].
have /bigO_exP [_ /posnumP[k]] := bigOP [bigO of [O_ (0 : U) id of f]].
have := littleoP [littleo of [o_ (0 : V') id of g]].
move=> /(_ (e%:num / k%:num)) /(_ _) /locallyP [//|_ /posnumP[d] hd].
apply: filter_app; near=> x => leOxkx; apply: le_trans (hd _ _) _; last first.
rewrite -ler_pdivl_mull //; apply: le_trans leOxkx _.
by rewrite invf_div mulrA -[_ / _ * _]mulrA mulVf // mulr1.
rewrite -ball_normE /= distrC subr0 (le_lt_trans leOxkx) //.
rewrite -ltr_pdivl_mull //; near: x; rewrite /= !locally_simpl.
apply/locallyP; exists (k%:num ^-1 * d%:num)=> // x.
by rewrite -ball_normE /= distrC subr0.
Grab Existential Variables. all: end_near. Qed.
Lemma compoO_eqox (U V' W' : normedModType R) (f : U -> V')
(g : V' -> W') :
forall x : U, [o_ (0 : V') id of g] ([O_ (0 : U) id of f] x) =o_(x \near 0 : U) x.
Proof. by move=> x; rewrite -[X in X = _]/((_ \o _) x) compoO_eqo. Qed.
(* TODO: generalize *)
Lemma compOo_eqo (U V' W' : normedModType R) (f : U -> V')
(g : V' -> W') :
[O_ (0 : V') id of g] \o [o_ (0 : U) id of f] =o_ (0 : U) id.
Proof.
apply/eqoP => _ /posnumP[e].
have /bigO_exP [_ /posnumP[k]] := bigOP [bigO of [O_ (0 : V') id of g]].
move=> /locallyP [_ /posnumP[d] hd].
have ekgt0 : e%:num / k%:num > 0 by [].
have /(_ _ ekgt0) := littleoP [littleo of [o_ (0 : U) id of f]].
apply: filter_app; near=> x => leoxekx; apply: le_trans (hd _ _) _; last first.
by rewrite -ler_pdivl_mull // mulrA [_^-1 * _]mulrC.
rewrite -ball_normE /= distrC subr0; apply: le_lt_trans leoxekx _.
rewrite -ltr_pdivl_mull //; near: x; rewrite /= locally_simpl.
apply/locallyP; exists ((e%:num / k%:num) ^-1 * d%:num)=> // x.
by rewrite -ball_normE /= distrC subr0.
Grab Existential Variables. all: end_near. Qed.
End DifferentialR3.
Section DifferentialR3_realFieldType.
Variable R : realFieldType.
Lemma littleo_linear0 (V W : normedModType R) (f : {linear V -> W}) :
(f : V -> W) =o_ (0 : V) id -> f = cst 0 :> (V -> W).
Proof.
move/eqoP => oid.
rewrite funeqE => x; apply/eqP; case: (ler0P `|x|) => [|xn0].
by rewrite normr_le0 => /eqP ->; rewrite linear0.
rewrite -normr_le0 -(mul0r `|x|) -ler_pdivr_mulr //.
apply/ler0_addgt0P => _ /posnumP[e]; rewrite ler_pdivr_mulr //.
have /oid /locallyP [_ /posnumP[d] dfe] := posnum_gt0 e.
set k := ((d%:num / 2) / (PosNum xn0)%:num)^-1.
rewrite -{1}(@scalerKV _ _ k _ x) // linearZZ normmZ.
rewrite -ler_pdivl_mull; last by rewrite gtr0_norm.
rewrite mulrCA (@le_trans _ _ (e%:num * `|k^-1 *: x|)) //; last first.
by rewrite ler_pmul // normmZ normfV.
apply dfe.
rewrite -ball_normE /ball_ sub0r normrN normmZ.
rewrite invrK -ltr_pdivl_mulr // ger0_norm // ltr_pdivr_mulr //.
by rewrite -mulrA mulVf ?lt0r_neq0 // mulr1 [X in _ < X]splitr ltr_addl.
Qed.
Lemma diff_unique (V W : normedModType R) (f : V -> W)
(df : {linear V -> W}) x :
continuous df -> f \o shift x = cst (f x) + df +o_ (0 : V) id ->
'd f x = df :> (V -> W).
Proof.
move=> dfc dxf; apply/subr0_eq; rewrite -[LHS]/(_ \- _).
apply/littleo_linear0/eqoP/eq_some_oP => /=; rewrite funeqE => y /=.
have hdf h :
(f \o shift x = cst (f x) + h +o_ (0 : V) id) ->
h = f \o shift x - cst (f x) +o_ (0 : V) id.
move=> hdf; apply: eqaddoE.
rewrite hdf addrAC (addrC _ h) addrK.
rewrite -[LHS]addr0 -addrA; congr (_ + _).
by apply/eqP; rewrite eq_sym addrC addr_eq0 oppo.
rewrite (hdf _ dxf).
suff /diff_locally /hdf -> : differentiable f x.
by rewrite opprD addrCA -(addrA (_ - _)) addKr oppox addox.
apply/diffP; apply: (@getPex _ (fun (df : {linear V -> W}) => continuous df /\
forall y, f y = f (lim x) + df (y - lim x) +o_(y \near x) (y - lim x))).
exists df; split=> //; apply: eqaddoEx => z.
rewrite (hdf _ dxf) !addrA lim_id // /(_ \o _) /= subrK [f _ + _]addrC addrK.
rewrite -addrA -[LHS]addr0; congr (_ + _).
apply/eqP; rewrite eq_sym addrC addr_eq0 oppox; apply/eqP.
by rewrite littleo_center0 (comp_centerK x id) -[- _ in RHS](comp_centerK x).
Qed.
Lemma diff_cst (V W : normedModType R) a x : ('d (cst a) x : V -> W) = 0.
Proof. by apply/diff_unique; have [] := dcst a x. Qed.
Variables (V W : normedModType R).
Lemma differentiable_cst (W' : normedModType R) (a : W') (x : V) :
differentiable (cst a) x.
Proof. by apply/diff_locallyP; rewrite diff_cst; have := dcst a x. Qed.
Global Instance is_diff_cst (a : W) (x : V) : is_diff x (cst a) 0.
Proof. exact: DiffDef (differentiable_cst _ _) (diff_cst _ _). Qed.
Lemma diffD (f g : V -> W) x :
differentiable f x -> differentiable g x ->
'd (f + g) x = 'd f x \+ 'd g x :> (V -> W).
Proof. by move=> df dg; apply/diff_unique; have [] := dadd df dg. Qed.
Lemma differentiableD (f g : V -> W) x :
differentiable f x -> differentiable g x -> differentiable (f + g) x.
Proof.
by move=> df dg; apply/diff_locallyP; rewrite diffD //; have := dadd df dg.
Qed.
Global Instance is_diffD (f g df dg : V -> W) x :
is_diff x f df -> is_diff x g dg -> is_diff x (f + g) (df + dg).
Proof.
move=> dfx dgx; apply: DiffDef; first exact: differentiableD.
by rewrite diffD // !diff_val.
Qed.
Lemma differentiable_sum n (f : 'I_n -> V -> W) (x : V) :
(forall i, differentiable (f i) x) -> differentiable (\sum_(i < n) f i) x.
Proof.
elim: n f => [f _| n IH f H]; first by rewrite big_ord0.
rewrite big_ord_recr /=; apply/differentiableD; [apply/IH => ? |]; exact: H.
Qed.
Lemma diffN (f : V -> W) x :
differentiable f x -> 'd (- f) x = - ('d f x : V -> W) :> (V -> W).
Proof.
move=> df; rewrite -[RHS]/(@GRing.opp _ \o _); apply/diff_unique;
by have [] := dopp df.
Qed.
Lemma differentiableN (f : V -> W) x :
differentiable f x -> differentiable (- f) x.
Proof.
by move=> df; apply/diff_locallyP; rewrite diffN //; have := dopp df.
Qed.
Global Instance is_diffN (f df : V -> W) x :
is_diff x f df -> is_diff x (- f) (- df).
Proof.
move=> dfx; apply: DiffDef; first exact: differentiableN.
by rewrite diffN // diff_val.
Qed.
Global Instance is_diffB (f g df dg : V -> W) x :
is_diff x f df -> is_diff x g dg -> is_diff x (f - g) (df - dg).
Proof. by move=> dfx dgx; apply: is_diff_eq. Qed.
Lemma diffB (f g : V -> W) x :
differentiable f x -> differentiable g x ->
'd (f - g) x = 'd f x \- 'd g x :> (V -> W).
Proof. by move=> /differentiableP df /differentiableP dg; rewrite diff_val. Qed.
Lemma differentiableB (f g : V -> W) x :
differentiable f x -> differentiable g x -> differentiable (f \- g) x.
Proof. by move=> /differentiableP df /differentiableP dg. Qed.
Lemma diffZ (f : V -> W) k x :
differentiable f x -> 'd (k *: f) x = k \*: 'd f x :> (V -> W).
Proof. by move=> df; apply/diff_unique; have [] := dscale k df. Qed.
Lemma differentiableZ (f : V -> W) k x :
differentiable f x -> differentiable (k *: f) x.
Proof.
by move=> df; apply/diff_locallyP; rewrite diffZ //; have := dscale k df.
Qed.
Global Instance is_diffZ (f df : V -> W) k x :
is_diff x f df -> is_diff x (k *: f) (k *: df).
Proof.
move=> dfx; apply: DiffDef; first exact: differentiableZ.
by rewrite diffZ // diff_val.
Qed.
Lemma diffZl (k : V -> R^o) (f : W) x : differentiable k x ->
'd (fun z => k z *: f) x = (fun z => 'd k x z *: f) :> (_ -> _).
Proof.
move=> df; set g := RHS; have glin : linear g.
by move=> a u v; rewrite /g linearP /= scalerDl -scalerA.
by apply:(@diff_unique _ _ _ (Linear glin)); have [] := dscalel f df.
Qed.
Lemma differentiableZl (k : V -> R^o) (f : W) x :
differentiable k x -> differentiable (fun z => k z *: f) x.
Proof.
by move=> df; apply/diff_locallyP; rewrite diffZl //; have [] := dscalel f df.
Qed.
Lemma diff_lin (V' W' : normedModType R) (f : {linear V' -> W'}) x :
continuous f -> 'd f x = f :> (V' -> W').
Proof. by move=> fcont; apply/diff_unique => //; apply: dlin. Qed.
Lemma linear_differentiable (V' W' : normedModType R) (f : {linear V' -> W'})
x : continuous f -> differentiable f x.
Proof.
by move=> fcont; apply/diff_locallyP; rewrite diff_lin //; have := dlin x fcont.
Qed.
Global Instance is_diff_id (x : V) : is_diff x id id.
Proof.
apply: DiffDef.
by apply: (@linear_differentiable _ _ [linear of idfun]) => ? //.
by rewrite (@diff_lin _ _ [linear of idfun]) // => ? //.
Qed.
Global Instance is_diff_scaler (k : R) (x : V) : is_diff x ( *:%R k) ( *:%R k).
Proof.
apply: DiffDef; first exact/linear_differentiable/scaler_continuous.
by rewrite diff_lin //; apply: scaler_continuous.
Qed.
Global Instance is_diff_scalel (x k : [filteredType R^o of R^o]) :
is_diff k ( *:%R ^~ x) ( *:%R ^~ x).
Proof.
have -> : *:%R ^~ x = GRing.scale_linear [lmodType R of R^o] x.
by rewrite funeqE => ? /=; rewrite [_ *: _]mulrC.
apply: DiffDef; first exact/linear_differentiable/scaler_continuous.
by rewrite diff_lin //; apply: scaler_continuous.
Qed.
Lemma differentiable_coord m n (M : 'M[R^o]_(m.+1, n.+1)) i j :
differentiable (fun N : 'M[R]_(m.+1, n.+1) => N i j : R^o) M.
Proof.
have @f : {linear 'M[R]_(m.+1, n.+1) -> R^o}.
by exists (fun N : 'M[R]_(_, _) => N i j); eexists; move=> ? ?; rewrite !mxE.
rewrite (_ : (fun _ => _) = f) //; exact/linear_differentiable/coord_continuous.
Qed.
Lemma linear_lipschitz (V' W' : normedModType R) (f : {linear V' -> W'}) :
continuous f -> exists2 k, k > 0 & forall x, `|f x| <= k * `|x|.
Proof.
move=> /(_ 0); rewrite linear0 => /(_ _ (locally_ball 0 1%:pos)).
move=> /locallyP [_ /posnumP[e] he]; exists (2 / e%:num) => // x.
case: (lerP `|x| 0) => [|xn0].
by rewrite normr_le0 => /eqP->; rewrite linear0 !normr0 mulr0.
set k := 2 / e%:num * (PosNum xn0)%:num.
have kn0 : k != 0 by [].
have abskgt0 : `|k| > 0 by rewrite normr_gt0.
rewrite -[x in X in X <= _](scalerKV kn0) linearZZ normmZ -ler_pdivl_mull //.
suff /he : ball 0 e%:num (k^-1 *: x).
rewrite -ball_normE /= distrC subr0 => /ltW /le_trans; apply.
by rewrite ger0_norm // mulVf.
rewrite -ball_normE /= distrC subr0 normmZ.
rewrite normfV ger0_norm // invrM ?unitfE // mulrAC mulVf //.
by rewrite invf_div mul1r [X in _ < X]splitr; apply: ltr_spaddr.
Qed.
Lemma linear_eqO (V' W' : normedModType R) (f : {linear V' -> W'}) :
continuous f -> (f : V' -> W') =O_ (0 : V') id.
Proof.
move=> /linear_lipschitz [k kgt0 flip]; apply/eqO_exP; exists k => //.
exact: filterE.
Qed.
Lemma diff_eqO (V' W' : normedModType R) (x : filter_on V') (f : V' -> W') :
differentiable f x -> ('d f x : V' -> W') =O_ (0 : V') id.
Proof. by move=> /diff_continuous /linear_eqO; apply. Qed.
Lemma compOo_eqox (U V' W' : normedModType R) (f : U -> V')
(g : V' -> W') : forall x,
[O_ (0 : V') id of g] ([o_ (0 : U) id of f] x) =o_(x \near 0 : U) x.
Proof. by move=> x; rewrite -[X in X = _]/((_ \o _) x) compOo_eqo. Qed.
Fact dcomp (U V' W' : normedModType R) (f : U -> V') (g : V' -> W') x :
differentiable f x -> differentiable g (f x) ->
continuous ('d g (f x) \o 'd f x) /\ forall y,
g (f (y + x)) = g (f x) + ('d g (f x) \o 'd f x) y +o_(y \near 0 : U) y.
Proof.
move=> df dg; split; first by move=> ?; apply: continuous_comp.
apply: eqaddoEx => y; rewrite diff_locallyx// -addrA diff_locallyxC// linearD.
rewrite addrA -addrA; congr (_ + _ + _).
rewrite diff_eqO // ['d f x : _ -> _]diff_eqO //.
by rewrite {2}eqoO addOx compOo_eqox compoO_eqox addox.
Qed.
Lemma diff_comp (U V' W' : normedModType R) (f : U -> V') (g : V' -> W') x :
differentiable f x -> differentiable g (f x) ->
'd (g \o f) x = 'd g (f x) \o 'd f x :> (U -> W').
Proof. by move=> df dg; apply/diff_unique; have [? /funext] := dcomp df dg. Qed.
Lemma differentiable_comp (U V' W' : normedModType R) (f : U -> V')
(g : V' -> W') x : differentiable f x -> differentiable g (f x) ->
differentiable (g \o f) x.
Proof.
move=> df dg; apply/diff_locallyP; rewrite diff_comp //;
by have [? /funext]:= dcomp df dg.
Qed.
Global Instance is_diff_comp (U V' W' : normedModType R) (f df : U -> V')
(g dg : V' -> W') x : is_diff x f df -> is_diff (f x) g dg ->
is_diff x (g \o f) (dg \o df) | 99.
Proof.
move=> dfx dgfx; apply: DiffDef; first exact: differentiable_comp.
by rewrite diff_comp // !diff_val.
Qed.
Lemma bilinear_schwarz (U V' W' : normedModType R)
(f : {bilinear U -> V' -> W'}) : continuous (fun p => f p.1 p.2) ->
exists2 k, k > 0 & forall u v, `|f u v| <= k * `|u| * `|v|.
Proof.
move=> /(_ 0); rewrite linear0r => /(_ _ (locally_ball 0 1%:pos)).
move=> /locallyP [_ /posnumP[e] he]; exists ((2 / e%:num) ^+2) => // u v.
case: (lerP `|u| 0) => [|un0].
by rewrite normr_le0 => /eqP->; rewrite linear0l !normr0 mulr0 mul0r.
case: (lerP `|v| 0) => [|vn0].
by rewrite normr_le0 => /eqP->; rewrite linear0r !normr0 mulr0.
rewrite -[`|u|]/((PosNum un0)%:num) -[`|v|]/((PosNum vn0)%:num).
set ku := 2 / e%:num * (PosNum un0)%:num.
set kv := 2 / e%:num * (PosNum vn0)%:num.
rewrite -[X in f X](@scalerKV _ _ ku) // linearZl_LR normmZ.
rewrite gtr0_norm // -ler_pdivl_mull //.
rewrite -[X in f _ X](@scalerKV _ _ kv) // linearZr_LR normmZ.
rewrite gtr0_norm // -ler_pdivl_mull //.
suff /he : ball 0 e%:num (ku^-1 *: u, kv^-1 *: v).
rewrite -ball_normE /= distrC subr0 => /ltW /le_trans; apply.
rewrite ler_pdivl_mull // mulr1 ler_pdivl_mull //.
by rewrite mulrA [ku * _]mulrAC expr2.
rewrite -ball_normE /= distrC subr0.
have -> : (ku^-1 *: u, kv^-1 *: v) =
(e%:num / 2) *: ((PosNum un0)%:num ^-1 *: u, (PosNum vn0)%:num ^-1 *: v).
rewrite invrM ?unitfE // [kv ^-1]invrM ?unitfE //.
rewrite mulrC -[_ *: u]scalerA [X in X *: v]mulrC -[_ *: v]scalerA.
by rewrite invf_div.
rewrite normmZ ger0_norm // -mulrA gtr_pmulr // ltr_pdivr_mull // mulr1.
rewrite prod_normE /= (_ : _%:nng = 1%:nng); last first.
by apply/val_inj => /=; rewrite normmZ normrV ?unitfE ?gt_eqF // normr_id mulVf ?gt_eqF.
rewrite (_ : _%:nng = 1%:nng); last first.
by apply/val_inj => /=; rewrite normmZ normrV ?unitfE ?gt_eqF // normr_id mulVf ?gt_eqF.
by rewrite joinxx /= ltr1n.
Qed.
Lemma bilinear_eqo (U V' W' : normedModType R) (f : {bilinear U -> V' -> W'}) :
continuous (fun p => f p.1 p.2) -> (fun p => f p.1 p.2) =o_ (0 : U * V') id.
Proof.
move=> fc; have [_ /posnumP[k] fschwarz] := bilinear_schwarz fc.
apply/eqoP=> _ /posnumP[e]; near=> x; rewrite (le_trans (fschwarz _ _))//.
rewrite ler_pmul ?pmulr_rge0 //; last by rewrite nng_lexU /= lexx orbT.
rewrite -ler_pdivl_mull //.
suff : `|x| <= k%:num ^-1 * e%:num by apply: le_trans; rewrite nng_lexU /= lexx.
near: x; rewrite !near_simpl; apply/locally_le_locally_norm.
by exists (k%:num ^-1 * e%:num) => // ? /=; rewrite distrC subr0 => /ltW.
Grab Existential Variables. all: end_near. Qed.
Fact dbilin (U V' W' : normedModType R) (f : {bilinear U -> V' -> W'}) p :
continuous (fun p => f p.1 p.2) ->
continuous (fun q => (f p.1 q.2 + f q.1 p.2)) /\
(fun q => f q.1 q.2) \o shift p = cst (f p.1 p.2) +
(fun q => f p.1 q.2 + f q.1 p.2) +o_ (0 : U * V') id.
Proof.
move=> fc; split=> [q|].
by apply: (@continuousD _ _ _ (fun q => f p.1 q.2) (fun q => f q.1 p.2));
move=> A /(fc (_.1, _.2)) /= /locallyP [_ /posnumP[e] fpqe_A];
apply/locallyP; exists e%:num => // r [??]; exact: (fpqe_A (_.1, _.2)).
apply/eqaddoE; rewrite funeqE => q /=.
rewrite linearDl !linearDr addrA addrC.
rewrite -[f q.1 _ + _ + _]addrA [f q.1 _ + _]addrC addrA [f q.1 _ + _]addrC.
by congr (_ + _); rewrite -[LHS]/((fun p => f p.1 p.2) q) bilinear_eqo.
Qed.
Lemma diff_bilin (U V' W' : normedModType R) (f : {bilinear U -> V' -> W'}) p :
continuous (fun p => f p.1 p.2) -> 'd (fun q => f q.1 q.2) p =
(fun q => f p.1 q.2 + f q.1 p.2) :> (U * V' -> W').
Proof.
move=> fc; have lind : linear (fun q => f p.1 q.2 + f q.1 p.2).
by move=> ???; rewrite linearPr linearPl scalerDr addrACA.
have -> : (fun q => f p.1 q.2 + f q.1 p.2) = Linear lind by [].
by apply/diff_unique; have [] := dbilin p fc.
Qed.
Lemma differentiable_bilin (U V' W' : normedModType R)
(f : {bilinear U -> V' -> W'}) p :
continuous (fun p => f p.1 p.2) -> differentiable (fun p => f p.1 p.2) p.
Proof.
by move=> fc; apply/diff_locallyP; rewrite diff_bilin //; apply: dbilin p fc.
Qed.
Definition Rmult_rev (y x : R) := x * y.
Canonical rev_Rmult := @RevOp _ _ _ Rmult_rev (@GRing.mul [ringType of R])
(fun _ _ => erefl).
Lemma Rmult_is_linear x : linear (@GRing.mul [ringType of R] x : R^o -> R^o).
Proof. by move=> ???; rewrite mulrDr scalerAr. Qed.
Canonical Rmult_linear x := Linear (Rmult_is_linear x).
Lemma Rmult_rev_is_linear y : linear (Rmult_rev y : R^o -> R^o).
Proof. by move=> ???; rewrite /Rmult_rev mulrDl scalerAl. Qed.
Canonical Rmult_rev_linear y := Linear (Rmult_rev_is_linear y).
Canonical Rmult_bilinear :=
[bilinear of (@GRing.mul [ringType of [lmodType R of R^o]])].
Global Instance is_diff_Rmult (p : [filteredType R^o * R^o of R^o * R^o]) :
is_diff p (fun q => q.1 * q.2) (fun q => p.1 * q.2 + q.1 * p.2).
Proof.
apply: DiffDef; last by rewrite diff_bilin // => ?; apply: mul_continuous.
by apply: differentiable_bilin =>?; apply: mul_continuous.
Qed.
Lemma eqo_pair (U V' W' : normedModType R) (F : filter_on U)
(f : U -> V') (g : U -> W') :
(fun t => ([o_F id of f] t, [o_F id of g] t)) =o_F id.
Proof.
apply/eqoP => _/posnumP[e]; near=> x; rewrite nng_leUx /=.
by apply/andP; split; near: x; apply: littleoP.
Grab Existential Variables. all: end_near. Qed.
Fact dpair (U V' W' : normedModType R) (f : U -> V') (g : U -> W') x :
differentiable f x -> differentiable g x ->
continuous (fun y => ('d f x y, 'd g x y)) /\
(fun y => (f y, g y)) \o shift x = cst (f x, g x) +
(fun y => ('d f x y, 'd g x y)) +o_ (0 : U) id.
Proof.
move=> df dg; split=> [?|]; first by apply: cvg_pair; apply: diff_continuous.
apply/eqaddoE; rewrite funeqE => y /=.
rewrite ![_ (_ + x)]diff_locallyx//.
(* fixme *)
have -> : forall h e, (f x + 'd f x y + [o_ (0 : U) id of h] y,
g x + 'd g x y + [o_ (0 : U) id of e] y) =
(f x, g x) + ('d f x y, 'd g x y) +
([o_ (0 : U) id of h] y, [o_ (0 : U) id of e] y) by [].
by congr (_ + _); rewrite -[LHS]/((fun y => (_ y, _ y)) y) eqo_pair.
Qed.
Lemma diff_pair (U V' W' : normedModType R) (f : U -> V') (g : U -> W') x :
differentiable f x -> differentiable g x -> 'd (fun y => (f y, g y)) x =
(fun y => ('d f x y, 'd g x y)) :> (U -> V' * W').
Proof.
move=> df dg.
have lin_pair : linear (fun y => ('d f x y, 'd g x y)).
by move=> ???; rewrite !linearPZ.
have -> : (fun y => ('d f x y, 'd g x y)) = Linear lin_pair by [].
by apply: diff_unique; have [] := dpair df dg.
Qed.
Lemma differentiable_pair (U V' W' : normedModType R) (f : U -> V')
(g : U -> W') x : differentiable f x -> differentiable g x ->
differentiable (fun y => (f y, g y)) x.
Proof.
by move=> df dg; apply/diff_locallyP; rewrite diff_pair //; apply: dpair.
Qed.
Global Instance is_diff_pair (U V' W' : normedModType R) (f df : U -> V')
(g dg : U -> W') x : is_diff x f df -> is_diff x g dg ->
is_diff x (fun y => (f y, g y)) (fun y => (df y, dg y)).
Proof.
move=> dfx dgx; apply: DiffDef; first exact: differentiable_pair.
by rewrite diff_pair // !diff_val.
Qed.
Global Instance is_diffM (f g df dg : V -> R^o) x :
is_diff x f df -> is_diff x g dg -> is_diff x (f * g) (f x *: dg + g x *: df).
Proof.
move=> dfx dgx.
have -> : f * g = (fun p => p.1 * p.2) \o (fun y => (f y, g y)) by [].
(* TODO: type class inference should succeed or fail, not leave an evar *)
apply: is_diff_eq; do ?exact: is_diff_comp.
by rewrite funeqE => ?; rewrite /= [_ * g _]mulrC.
Qed.
Lemma diffM (f g : V -> R^o) x :
differentiable f x -> differentiable g x ->
'd (f * g) x = f x \*: 'd g x + g x \*: 'd f x :> (V -> R).
Proof. by move=> /differentiableP df /differentiableP dg; rewrite diff_val. Qed.
Lemma differentiableM (f g : V -> R^o) x :
differentiable f x -> differentiable g x -> differentiable (f * g) x.
Proof. by move=> /differentiableP df /differentiableP dg. Qed.
(* fixme using *)
(* (1 / (h + x) - 1 / x) / h = - 1 / (h + x) x = -1/x^2 + o(1) *)
Fact dinv (x : R) :
x != 0 -> continuous (fun h : R^o => - x ^- 2 *: h) /\
(fun x => x^-1 : R^o) \o shift x = cst (x^-1) +
(fun h : R^o => - x ^- 2 *: h) +o_ (0 : [filteredType R^o of R^o]) id.
Proof.
move=> xn0; suff: continuous (fun h : R^o => - (1 / x) ^+ 2 *: h) /\
(fun x => 1 / x : R^o) \o shift x = cst (1 / x) +
(fun h : R^o => - (1 / x) ^+ 2 *: h) +o_ (0 : [filteredType R^o of R^o]) id.
rewrite !mul1r !GRing.exprVn.
rewrite (_ : (fun x => x^-1 : R^o) = (fun x => 1 / x : R^o))//.
by rewrite funeqE => y; rewrite mul1r.
split; first by move=> ?; apply: continuousZr.
apply/eqaddoP => _ /posnumP[e]; near=> h.
rewrite -[(_ + _ : R -> R) h]/(_ + _) -[(- _ : R -> R) h]/(- _) /=.
rewrite opprD scaleNr opprK /cst /=.
rewrite -[- _]mulr1 -[X in - _ * X](mulfVK xn0) mulrA mulNr -expr2 mulNr.
rewrite [- _ + _]addrC -mulrBr.
rewrite -[X in X + _]mulr1 -[X in 1 / _ * X](@mulfVK _ (x ^+ 2)); last first.
by rewrite sqrf_eq0.
rewrite mulrA mulf_div mulr1.
have hDx_neq0 : h + x != 0.
near: h; rewrite !locally_simpl; apply/locally_normP.
exists `|x|; first by rewrite normr_gt0.
move=> h /=; rewrite distrC subr0 -subr_gt0 => lthx.
rewrite -(normr_gt0 (h + x : R^o)) addrC -[h]opprK.
apply: lt_le_trans (ler_dist_dist _ _).
by rewrite ger0_norm normrN //; apply: ltW.
rewrite addrC -[X in X * _]mulr1 -{2}[1](@mulfVK _ (h + x)) //.
rewrite mulrA expr_div_n expr1n mulf_div mulr1 [_ ^+ 2 * _]mulrC -mulrA.
rewrite -mulrDr mulrBr [1 / _ * _]mulrC normrM.
rewrite mulrDl mulrDl opprD addrACA addrA [x * _]mulrC expr2.
do 2 ?[rewrite -addrA [- _ + _]addrC subrr addr0].
rewrite div1r normfV [X in _ / X]normrM invfM [X in _ * X]mulrC.
rewrite mulrA mulrAC ler_pdivr_mulr ?normr_gt0 ?mulf_neq0 //.
rewrite mulrAC ler_pdivr_mulr ?normr_gt0 //.
have : `|h * h| <= `|x / 2| * (e%:num * `|x * x| * `|h : R^o|).
rewrite !mulrA; near: h; exists (`|x / 2| * e%:num * `|x * x|).
by rewrite !pmulr_rgt0 // normr_gt0 mulf_neq0.
by move=> h /ltW; rewrite distrC subr0 [`|h * _|]normrM => /ler_pmul; apply.
move=> /le_trans-> //; rewrite [X in X <= _]mulrC ler_pmul ?mulr_ge0 //.
near: h; exists (`|x| / 2); first by rewrite divr_gt0 ?normr_gt0.
move=> h; rewrite /= distrC subr0 => lthhx; rewrite addrC -[h]opprK.
apply: le_trans (@ler_dist_dist _ [normedModType R of R^o] _ _).
rewrite normrN [X in _ <= X]ger0_norm; last first.
rewrite subr_ge0; apply: ltW; apply: lt_le_trans lthhx _.
by rewrite ler_pdivr_mulr // -{1}(mulr1 `|x|) ler_pmul // ler1n.
rewrite ler_subr_addr -ler_subr_addl (splitr `|x : R^o|).
by rewrite normrM normfV (@ger0_norm _ 2) // -addrA subrr addr0; apply: ltW.
Grab Existential Variables. all: end_near. Qed.
Lemma diff_Rinv (x : [filteredType R^o of R^o]) : x != 0 ->
'd GRing.inv x = (fun h => - x ^- 2 *: h) :> (R^o -> R^o).
Proof.
move=> xn0; have -> : (fun h : R^o => - x ^- 2 *: h) =
GRing.scale_linear _ (- x ^- 2) by [].
by apply: diff_unique; have [] := dinv xn0.
Qed.
Lemma differentiable_Rinv (x : R^o) : x != 0 ->
differentiable (GRing.inv : R^o -> R^o) x.
Proof.
by move=> xn0; apply/diff_locallyP; rewrite diff_Rinv //; apply: dinv.
Qed.
Lemma diffV (f : V -> R^o) x : differentiable f x -> f x != 0 ->
'd (fun y => (f y)^-1) x = - (f x) ^- 2 \*: 'd f x :> (V -> R).
Proof.
move=> df fxn0.
by rewrite [LHS](diff_comp df (differentiable_Rinv fxn0)) diff_Rinv.
Qed.
Lemma differentiableV (f : V -> R^o) x :
differentiable f x -> f x != 0 -> differentiable (fun y => (f y)^-1) x.
Proof.
by move=> df fxn0; apply: differentiable_comp _ (differentiable_Rinv fxn0).
Qed.
Lemma exprfunE (T : pointedType) (K : ringType) (f : T -> K) n :
f ^+ n = (fun x => f x ^+ n).
Proof.
by elim: n => [|n ihn]; rewrite funeqE=> ?; [rewrite !expr0|rewrite !exprS ihn].
Qed.
Global Instance is_diffX (f df : V -> R^o) n x :