Module mathcomp.analysis.realfun

From mathcomp Require Import all_ssreflect ssralg ssrint ssrnum archimedean.
From mathcomp Require Import matrix interval zmodp vector fieldext falgebra.
From mathcomp Require Import finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality ereal reals signed.
From mathcomp Require Import topology prodnormedzmodule normedtype derive.
From mathcomp Require Import sequences real_interval.
From HB Require Import structures.

# Real-valued functions over reals This file provides properties of standard real-valued functions over real numbers (e.g., the continuity of the inverse of a continuous function). ``` nondecreasing_fun f == the function f is non-decreasing nonincreasing_fun f == the function f is non-increasing increasing_fun f == the function f is (strictly) increasing decreasing_fun f == the function f is (strictly) decreasing derivable_oo_continuous_bnd f x y == f is derivable in `]x, y[ and continuous up to the boundary, i.e., f @ x^'+ --> f x and f @ y^'- --> f y itv_partition a b s == s is a partition of the interval `[a, b] itv_partitionL s c == the left side of splitting a partition at c itv_partitionR s c == the right side of splitting a partition at c variation a b f s == the sum of f at all points in the partition s variations a b f == the set of all variations of f between a and b bounded_variation a b f == all variations of f are bounded total_variation a b f == the sup over all variations of f from a to b neg_tv a f x == the decreasing component of f pos_tv a f x == the increasing component of f ``` * Limit superior and inferior for functions: ``` lime_sup f a/lime_inf f a == limit sup/inferior of the extended real- valued function f at point a ```

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldTopology.Exports.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Import numFieldNormedType.Exports.

Notation "'nondecreasing_fun' f" := ({homo f : n m / (n <= m)%O >-> (n <= m)%O})
 (at level 10).
Notation "'nonincreasing_fun' f" := ({homo f : n m / (n <= m)%O >-> (n >= m)%O})
 (at level 10).
Notation "'increasing_fun' f" := ({mono f : n m / (n <= m)%O >-> (n <= m)%O})
 (at level 10).
Notation "'decreasing_fun' f" := ({mono f : n m / (n <= m)%O >-> (n >= m)%O})
 (at level 10).

Lemma nondecreasing_funN {R : realType} a b (f : R -> R) :
 {in `[a, b] &, nondecreasing_fun f} <->
 {in `[a, b] &, nonincreasing_fun (\- f)}.

Proof.

split=> [h m n mab nab mn|h m n mab nab mn]; first by rewrite lerNr opprK h.
by rewrite -(opprK (f n)) -lerNr h.
Qed.

Lemma nonincreasing_funN {R : realType} a b (f : R -> R) :
{in `[a, b] &, nonincreasing_fun f} <->
{in `[a, b] &, nondecreasing_fun (\- f)}.

Proof.

apply: iff_sym; apply: (iff_trans (nondecreasing_funN a b (\- f))).
rewrite [in X in _ <-> X](_ : f = \- (\- f))//.
by apply/funext => x /=; rewrite opprK.
Qed.

Section fun_cvg.

Section fun_cvg_realFieldType.
Context {R : realFieldType}.

Lemma cvg_addrl (M : R) : M + r @[r --> +oo] --> +oo.

Proof.

move=> P [r [rreal rP]]; exists (r - M); split.
by rewrite realB// num_real.
by move=> m; rewrite ltrBlDl => /rP.
Qed.

Lemma cvg_addrr (M : R) : (r + M) @[r --> +oo] --> +oo.

Proof.

by under [X in X @ _]funext => n do rewrite addrC; exact: cvg_addrl. Qed.

Lemma cvg_centerr (M : R) (T : topologicalType) (f : R -> T) (l : T) :
(f (n - M) @[n --> +oo] --> l) = (f r @[r --> +oo] --> l).

Proof.

rewrite propeqE; split; last by apply: cvg_comp; exact: cvg_addrr.
gen have cD : f l / f r @[r --> +oo] --> l -> f (n + M) @[n --> +oo] --> l.
by apply: cvg_comp; exact: cvg_addrr.
move=> /cD /=.
by under [X in X @ _ --> l]funext => n do rewrite addrK.
Qed.

Lemma cvg_shiftr (M : R) (T : topologicalType) (f : R -> T) (l : T) :
(f (n + M) @[n --> +oo]--> l) = (f r @[r --> +oo] --> l).

Proof.

rewrite propeqE; split; last by apply: cvg_comp; exact: cvg_addrr.
rewrite -[X in X -> _](cvg_centerr M); apply: cvg_trans => /=.
apply: near_eq_cvg; near do rewrite subrK; exists M.
by rewrite num_real.
Unshelve. all: by end_near. Qed.

Lemma left_right_continuousP {T : topologicalType} (f : R -> T) x :
f @ x^'- --> f x /\ f @ x^'+ --> f x <-> f @ x --> f x.

Proof.

split; last by move=> cts; split; exact: cvg_within_filter.
move=> [+ +] U /= Uz => /(_ U Uz) + /(_ U Uz); near_simpl.
rewrite !near_withinE => lf rf; apply: filter_app lf; apply: filter_app rf.
near=> t => xlt xgt; have := @real_leVge R x t; rewrite !num_real.
move=> /(_ isT isT) /orP; rewrite !le_eqVlt => -[|] /predU1P[|//].
- by move=> <-; exact: nbhs_singleton.
- by move=> ->; exact: nbhs_singleton.
Unshelve. all: by end_near. Qed.

Lemma cvg_at_right_left_dnbhs (f : R -> R) (p : R) (l : R) :
f x @[x --> p^'+] --> l -> f x @[x --> p^'-] --> l ->
f x @[x --> p^'] --> l.

Proof.

move=> /cvgrPdist_le fppl /cvgrPdist_le fpnl; apply/cvgrPdist_le => e e0.
have {fppl}[a /= a0 fppl] := fppl _ e0; have {fpnl}[b /= b0 fpnl] := fpnl _ e0.
near=> t.
have : t != p by near: t; exact: nbhs_dnbhs_neq.
rewrite neq_lt => /orP[tp|pt].
- apply: fpnl => //=; near: t.
  exists (b / 2) => //=; first by rewrite divr_gt0.
  move=> z/= + _ => /lt_le_trans; apply.
  by rewrite ler_pdivrMr// ler_pMr// ler1n.
- apply: fppl =>//=; near: t.
  exists (a / 2) => //=; first by rewrite divr_gt0.
  move=> z/= + _ => /lt_le_trans; apply.
  by rewrite ler_pdivrMr// ler_pMr// ler1n.
Unshelve. all: by end_near. Qed.

End fun_cvg_realFieldType.

Section cvgr_fun_cvg_seq.
Context {R : realType}.

Lemma cvg_at_rightP (f : R -> R) (p l : R) :
 f x @[x --> p^'+] --> l <->
 (forall u : R^nat, (forall n, u n > p) /\ (u n @[n --> \oo] --> p) ->
 f (u n) @[n --> \oo] --> l).

Proof.

split=> [/cvgrPdist_le fpl u [up /cvgrPdist_lt ucvg]|pfl].
 apply/cvgrPdist_le => e e0.
 have [r /= r0 {}fpl] := fpl _ e0; have [s /= s0 {}ucvg] := ucvg _ r0.
 near=> t; apply: fpl => //=; apply: ucvg => /=.
 by near: t; exists s.
apply: contrapT => fpl; move: pfl; apply/existsNP.
suff: exists2 x : R ^nat,
 (forall k, x k > p) /\ x n @[n --> \oo] --> p & ~ f (x n) @[n --> \oo] --> l.
 by move=> [x_] h; exists x_; exact/not_implyP.
have [e He] : exists e : {posnum R}, forall d : {posnum R},
 exists xn : R, [/\ xn > p, `|xn - p| < d%:num & `|f xn - l| >= e%:num].
 apply: contrapT; apply: contra_not fpl => /forallNP h.
 apply/cvgrPdist_le => e e0; have /existsNP[d] := h (PosNum e0).
 move/forallNP => {}h; near=> t.
 have /not_and3P[abs|abs|/negP] := h t.
 - by exfalso; apply: abs; near: t; exact: nbhs_right_gt.
 - exfalso; apply: abs.
 by near: t; by exists d%:num => //= z/=; rewrite distrC.
 - by rewrite -ltNge distrC => /ltW.
have invn n : 0 < n.+1%:R^-1 :> R by rewrite invr_gt0.
exists (fun n => sval (cid (He (PosNum (invn n))))).
 split => [k|]; first by rewrite /sval/=; case: cid => x [].
 apply/cvgrPdist_lt => r r0; near=> t.
 rewrite /sval/=; case: cid => x [px xpt _].
 rewrite distrC (lt_le_trans xpt)// -(@invrK _ r) lef_pV2 ?posrE ?invr_gt0//.
 near: t; exists `|ceil r^-1|%N => // s /=.
 rewrite -ltnS -(@ltr_nat R) => /ltW; apply: le_trans.
 by rewrite natr_absz gtr0_norm -?ceil_gt0 ?invr_gt0// ceil_ge.
move=> /cvgrPdist_lt/(_ e%:num (ltac:(by [])))[] n _ /(_ _ (leqnn _)).
rewrite /sval/=; case: cid => // x [px xpn].
by rewrite leNgt distrC => /negP.
Unshelve. all: by end_near. Qed.

Lemma cvg_at_leftP (f : R -> R) (p l : R) :
 f x @[x --> p^'-] --> l <->
 (forall u : R^nat, (forall n, u n < p) /\ u n @[n --> \oo] --> p ->
 f (u n) @[n --> \oo] --> l).

Proof.

apply: (iff_trans (cvg_at_leftNP f p l)).
apply: (iff_trans (cvg_at_rightP _ _ _)).
split=> [pfl u [pu up]|pfl u [pu up]].
rewrite -(opprK u); apply: pfl.
by split; [move=> k; rewrite ltrNr opprK//|exact/cvgNP].
apply: pfl.
by split; [move=> k; rewrite ltrNl//|apply/cvgNP => /=; rewrite opprK].
Qed.

End cvgr_fun_cvg_seq.

Section cvge_fun_cvg_seq.
Context {R : realType}.

Lemma cvge_at_rightP (f : R -> \bar R) (p l : R) :
 f x @[x --> p^'+] --> l%:E <->
 (forall u : R^nat, (forall n, u n > p) /\ u n @[n --> \oo] --> p ->
 f (u n) @[n --> \oo] --> l%:E).

Proof.

split=> [/fine_cvgP [ffin_num fpl] u [pu up]|h].
 apply/fine_cvgP; split; last by move/cvg_at_rightP : fpl; exact.
 have [e /= e0 {}ffin_num] := ffin_num.
 move/cvgrPdist_lt : up => /(_ _ e0)[s /= s0 {}up]; near=> t.
 by apply: ffin_num => //=; apply: up => /=; near: t; exists s.
suff H : \forall F \near p^'+, f F \is a fin_num.
 by apply/fine_cvgP; split => //; apply/cvg_at_rightP => u /h /fine_cvgP[].
apply: contrapT => /not_near_at_rightP abs.
have invn n : 0 < n.+1%:R^-1 :> R by rewrite invr_gt0.
pose y_ n := sval (cid2 (abs (PosNum (invn n)))).
have py_ k : p < y_ k by rewrite /y_ /sval/=; case: cid2 => //= x /andP[].
have y_p : y_ n @[n --> \oo] --> p.
 apply/cvgrPdist_lt => e e0; near=> t.
 rewrite ltr0_norm// ?subr_lt0// opprB.
 rewrite /y_ /sval/=; case: cid2 => //= x /andP[_ + _].
 rewrite ltrBlDr => /lt_le_trans; apply.
 rewrite addrC lerD2r -(invrK e) lef_pV2// ?posrE ?invr_gt0//.
 near: t.
 exists `|ceil e^-1|%N => // k /= ek.
 rewrite (le_trans (ceil_ge _))// (@le_trans _ _ `|ceil e^-1|%:~R)//.
 by rewrite ger0_norm -?ceil_ge0// (lt_le_trans (ltrN10 _))// invr_ge0// ltW.
 by move: ek;rewrite -(leq_add2r 1) !addn1 -(ltr_nat R) => /ltW.
have /fine_cvgP[[m _ mfy_] /= _] := h _ (conj py_ y_p).
near \oo => n.
have mn : (m <= n)%N by near: n; exists m.
have {mn} := mfy_ _ mn.
by rewrite /y_ /sval; case: cid2 => /= x _.
Unshelve. all: by end_near. Qed.

Lemma cvge_at_leftP (f : R -> \bar R) (p l : R) :
 f x @[x --> p^'-] --> l%:E <->
 (forall u : R^nat, (forall n, u n < p) /\ u n @[n --> \oo] --> p ->
 f (u n) @[n --> \oo] --> l%:E).

Proof.

apply: (iff_trans (cvg_at_leftNP f p l%:E)).
apply: (iff_trans (cvge_at_rightP _ _ l)); split=> h u [up pu].
- rewrite (_ : u = \- (\- u))%R; last by apply/funext => ?/=; rewrite opprK.
by apply: h; split; [by move=> n; rewrite ltrNl opprK|exact: cvgN].
- by apply: h; split => [n|]; [rewrite ltrNl|move/cvgN : pu; rewrite opprK].
Qed.

End cvge_fun_cvg_seq.

Section cvgr_fun_dvg_seq.
Context {R : realType}.

Lemma cvg_pinftyP (f : R -> R) (l : R) :
f x @[x --> +oo] --> l <->
forall u : R^nat, (u n @[n --> \oo] --> +oo) -> f (u n) @[n --> \oo] --> l.

Proof.

split; first by move=> ? ? /cvg_comp; exact.
apply: contraPP => noncvg_f; apply/existsNP.
under eq_exists do rewrite not_implyE; apply/exists2P.
suff [e e_sep] : exists e : {posnum R},
 forall A, exists2 un, A < un & e%:num <= `|f un - l|.
 exists (fun n => sval (cid2 (e_sep n%:R))).
 apply/cvgryPgt => A; near=> n; case: cid2 => //= r nr _.
 by rewrite (le_lt_trans _ nr)//; near: n; exact: nbhs_infty_ger.
 apply/cvgrPdistC_lt => /= /(_ e%:num ltac:(by []))[N _ /(_ _ (leqnn N))].
 by case: cid2 => uN/= _ /le_lt_trans /[apply]; rewrite ltxx.
move: noncvg_f => /cvgrPdistC_lt /=; rewrite -existsPNP => -[eps eps_gt0].
move=> /not_near_inftyP eps_sep; exists (PosNum eps_gt0) => A /=.
have [x x_ltA /negP fxleps] := eps_sep _ (num_real A).
by exists x => //; rewrite leNgt.
Unshelve. all: by end_near. Qed.

Lemma cvg_ninftyP (f : R -> R) (l : R) :
f x @[x --> -oo] --> l <->
forall u : R^nat, (u n @[n --> \oo] --> -oo) -> f (u n) @[n --> \oo] --> l.

Proof.

rewrite cvgyNP cvg_pinftyP/= (@bij_forall R^nat _ -%R)//.
have u_opp (u : R^nat) :
    ((- u) n @[n --> \oo] --> +oo) = (u n @[n --> \oo] --> -oo).
  by rewrite propeqE cvgNry.
by under eq_forall do
  (rewrite u_opp; under (eq_cvg _ (nbhs l)) do rewrite opprK).
Qed.

End cvgr_fun_dvg_seq.

Section fun_cvg_realType.
Context {R : realType}.
Implicit Types f : R -> R.

Lemma nondecreasing_cvgr f : nondecreasing_fun f -> has_ubound (range f) ->
f r @[r --> +oo] --> sup (range f).

Proof.

move=> ndf ubf; set M := sup (range f).
have supf : has_sup (range f) by split => //; exists (f 0), 0.
apply/cvgrPdist_le => _/posnumP[e].
have [p Mefp] : exists p, M - e%:num <= f p.
have [_ -[p _] <- /ltW efp] := sup_adherent (gt0 e) supf.
by exists p; rewrite efp.
near=> n; have pn : p <= n by near: n; apply: nbhs_pinfty_ge; rewrite num_real.
rewrite ler_distlC (le_trans Mefp (ndf _ _ _))//= (@le_trans _ _ M) ?lerDl//.
by have /ubP := sup_upper_bound supf; apply; exists n.
Unshelve. all: by end_near. Qed.

Lemma nonincreasing_at_right_cvgr f a (b : itv_bound R) : (BRight a < b)%O ->
 {in Interval (BRight a) b &, nonincreasing_fun f} ->
 has_ubound (f @` [set` Interval (BRight a) b]) ->
 f x @[x --> a ^'+] --> sup (f @` [set` Interval (BRight a) b]).

Proof.

move=> ab lef ubf; set M := sup _.
have supf : has_sup [set f x | x in [set` Interval (BRight a) b]].
 split => //; case: b ab {lef ubf M} => [[|] t ta|[]] //=.
 - exists (f ((a + t) / 2)), ((a + t) / 2) => //=.
 by rewrite in_itv/= !midf_lt.
 - exists (f ((a + t) / 2)), ((a + t) / 2) => //=.
 by rewrite in_itv/= midf_lt// midf_le// ltW.
 - by exists (f (a + 1)), (a + 1) => //=; rewrite in_itv/= ltrDl andbT.
apply/cvgrPdist_le => _/posnumP[e].
have {supf} [p [ap pb]] :
 exists p, [/\ a < p, (BLeft p < b)%O & M - e%:num <= f p].
 have [_ -[p apb] <- /ltW efp] := sup_adherent (gt0 e) supf.
 move: apb; rewrite /= in_itv/= -[X in _ && X]/(BLeft p < b)%O => /andP[ap pb].
 by exists p; split.
rewrite lerBlDr {}/M.
move: b ab pb lef ubf => [[|] b|[//|]] ab pb lef ubf; set M := sup _ => Mefp.
- near=> r; rewrite ler_distl; apply/andP; split.
 + suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl.
 apply: sup_ubound => //=; exists r => //; rewrite in_itv/=.
 by apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_lt].
 + rewrite (le_trans Mefp)// lerD2r lef//=; last 2 first.
 by rewrite in_itv/= ap.
 by near: r; exact: nbhs_right_le.
 apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_lt].
- near=> r; rewrite ler_distl; apply/andP; split.
 + suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl.
 apply: sup_ubound => //=; exists r => //; rewrite in_itv/=.
 by apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_le].
 + rewrite (le_trans Mefp)// lerD2r lef//=; last 2 first.
 by rewrite in_itv/= ap.
 by near: r; exact: nbhs_right_le.
 by apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_le].
- near=> r; rewrite ler_distl; apply/andP; split.
 suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl.
 apply: sup_ubound => //=; exists r => //; rewrite in_itv/= andbT.
 by near: r; apply: nbhs_right_gt.
 rewrite (le_trans Mefp)// lerD2r lef//.
 - by rewrite in_itv/= andbT; near: r; exact: nbhs_right_gt.
 - by rewrite in_itv/= ap.
 - by near: r; exact: nbhs_right_le.
Unshelve. all: by end_near. Qed.

Lemma nonincreasing_at_right_is_cvgr f a :
    (\forall x \near a^'+, {in `]a, x[ &, nonincreasing_fun f}) ->
    (\forall x \near a^'+, has_ubound (f @` `]a, x[)) ->
  cvg (f x @[x --> a ^'+]).

Proof.

move=> nif ubf; apply/cvg_ex; near a^'+ => b.
by eexists; apply: (@nonincreasing_at_right_cvgr _ _ (BLeft b));
[rewrite bnd_simp|near: b..].
Unshelve. all: by end_near. Qed.

Lemma nondecreasing_at_right_cvgr f a (b : itv_bound R) : (BRight a < b)%O ->
 {in Interval (BRight a) b &, nondecreasing_fun f} ->
 has_lbound (f @` [set` Interval (BRight a) b]) ->
 f x @[x --> a ^'+] --> inf (f @` [set` Interval (BRight a) b]).

Proof.

move=> ab nif hlb; set M := inf _.
have ndNf : {in Interval (BRight a) b &, nonincreasing_fun (\- f)}.
  by move=> r s rab sab /nif; rewrite lerN2; exact.
have hub : has_ubound [set (\- f) x | x in [set` Interval (BRight a) b]].
  apply/has_ub_lbN; rewrite image_comp/=.
  rewrite [X in has_lbound X](_ : _ = f @` [set` Interval (BRight a) b])//.
  by apply: eq_imagel => y _ /=; rewrite opprK.
have /cvgN := nonincreasing_at_right_cvgr ab ndNf hub.
rewrite opprK [X in _ --> X -> _](_ : _ =
    inf (f @` [set` Interval (BRight a) b]))//.
by rewrite /inf; congr (- sup _); rewrite image_comp/=; exact: eq_imagel.
Qed.

Lemma nondecreasing_at_right_is_cvgr f a :
    (\forall x \near a^'+, {in `]a, x[ &, nondecreasing_fun f}) ->
    (\forall x \near a^'+, has_lbound (f @` `]a, x[)) ->
  cvg (f x @[x --> a ^'+]).

Proof.

move=> ndf lbf; apply/cvg_ex; near a^'+ => b.
by eexists; apply: (@nondecreasing_at_right_cvgr _ _ (BLeft b));
[rewrite bnd_simp|near: b..].
Unshelve. all: by end_near. Qed.

Lemma nondecreasing_at_left_is_cvgr f a :
 (\forall x \near a^'-, {in `]x, a[ &, {homo f : n m / n <= m}}) ->
 (\forall x \near a^'-, has_ubound [set f y | y in `]x, a[]) ->
 cvg (f x @[x --> a^'-]).

Proof.

move=> ndf ubf; suff: cvg ((f \o -%R) x @[x --> (- a)^'+]).
 move=> /cvg_ex[/= l fal].
 by apply/cvg_ex; exists l; exact/cvg_at_leftNP.
apply: @nonincreasing_at_right_is_cvgr.
- rewrite at_rightN near_simpl; apply: filterS ndf => x ndf y z.
 by rewrite -2!oppr_itvoo => yxa zxa yz; rewrite ndf// lerNr opprK.
- rewrite at_rightN near_simpl; apply: filterS ubf => x [r ubf].
 exists r => _/= [s sax <-]; rewrite ubf//=; exists (- s) => //.
 by rewrite oppr_itvoo.
Qed.

Let nondecreasing_at_right_is_cvgrW f a b r : a < r -> r 
{in `[a, b] &, nondecreasing_fun f} -> cvg (f x @[x --> r^'+]).

Proof.

move=> ar rb ndf H; apply: nondecreasing_at_right_is_cvgr.
- near=> s => x y xrs yrs xy; rewrite ndf//.
 + by apply: subset_itvW xrs; exact/ltW.
 + by apply: subset_itvW yrs; exact/ltW.
- near=> x; exists (f r) => _ /= [s srx <-]; rewrite ndf//.
 + by apply: subset_itv_oo_cc; rewrite in_itv/= ar.
 + by apply: subset_itvW srx; exact/ltW.
 + by move: srx; rewrite in_itv/= => /andP[/ltW].
Unshelve. all: by end_near. Qed.

Let nondecreasing_at_left_is_cvgrW f a b r : a < r -> r 
{in `[a, b] &, nondecreasing_fun f} -> cvg (f x @[x --> r^'-]).

Proof.

move=> ar rb ndf H; apply: nondecreasing_at_left_is_cvgr.
- near=> s => x y xrs yrs xy; rewrite ndf//.
 + by apply: subset_itvW xrs; exact/ltW.
 + by apply: subset_itvW yrs; exact/ltW.
- near=> x; exists (f r) => _ /= [s srx <-]; rewrite ndf//.
 + by apply: subset_itvW srx; exact/ltW.
 + by apply: subset_itv_oo_cc; rewrite in_itv/= ar.
 + by move: srx; rewrite in_itv/= => /andP[_ /ltW].
Unshelve. all: by end_near. Qed.

Lemma nondecreasing_at_left_at_right f a b :
{in `[a, b] &, nondecreasing_fun f} ->
{in `]a, b[, forall r, lim (f x @[x --> r^'-]) <= lim (f x @[x --> r^'+])}.

Proof.

move=> ndf r; rewrite in_itv/= => /andP[ar rb]; apply: limr_ge.
exact: nondecreasing_at_right_is_cvgrW ndf.
near=> x; apply: limr_le; first exact: nondecreasing_at_left_is_cvgrW ndf.
near=> y; rewrite ndf// ?in_itv/=.
- apply/andP; split; first by near: y; apply: nbhs_left_ge.
exact: (le_trans _ (ltW rb)).
- by rewrite (le_trans (ltW ar))/= ltW.
- exact: (@le_trans _ _ r).
Unshelve. all: by end_near. Qed.

Lemma nonincreasing_at_left_at_right f a b :
{in `[a, b] &, nonincreasing_fun f} ->
{in `]a, b[, forall r, lim (f x @[x --> r^'-]) >= lim (f x @[x --> r^'+])}.

Proof.

move=> nif; have ndNf : {in `[a, b] &, nondecreasing_fun (-%R \o f)}.
  by move=> x y xab yab xy /=; rewrite lerNl opprK nif.
move/nondecreasing_at_left_at_right : (ndNf) => H x.
rewrite in_itv/= => /andP[ax xb]; rewrite -[leLHS]opprK lerNl -!limN//.
- by apply: H; rewrite !in_itv/= ax.
- rewrite -(opprK f); apply: is_cvgN.
  exact: nondecreasing_at_right_is_cvgrW ndNf.
- rewrite -(opprK f);apply: is_cvgN.
  exact: nondecreasing_at_left_is_cvgrW ndNf.
Qed.

End fun_cvg_realType.
Arguments nondecreasing_at_right_cvgr {R f a} b.

Section fun_cvg_ereal.
Context {R : realType}.
Local Open Scope ereal_scope.

Lemma nondecreasing_cvge (f : R -> \bar R) :
nondecreasing_fun f -> f r @[r --> +oo%R] --> ereal_sup (range f).

Proof.

move=> ndf; set S := range f; set l := ereal_sup S.
have [Spoo|Spoo] := pselect (S +oo).
 have [N Nf] : exists N, forall n, (n >= N)%R -> f n = +oo.
 case: Spoo => N _ uNoo; exists N => n Nn.
 by have := ndf _ _ Nn; rewrite uNoo leye_eq => /eqP.
 have -> : l = +oo by rewrite /l /ereal_sup; exact: supremum_pinfty.
 rewrite -(cvg_shiftr `|N|); apply: cvg_near_cst.
 exists N; split; first by rewrite num_real.
 by move=> x /ltW Nx; rewrite Nf// ler_wpDr.
have [lpoo|lpoo] := eqVneq l +oo.
 rewrite lpoo; apply/cvgeyPge => M.
 have /ereal_sup_gt[_ [n _] <- Mun] : M%:E < l by rewrite lpoo// ltry.
 exists n; split; first by rewrite num_real.
 by move=> m /= nm; rewrite (le_trans (ltW Mun))// ndf// ltW.
have [fnoo|fnoo] := pselect (f = cst -oo).
 rewrite /l (_ : S = [set -oo]).
 by rewrite ereal_sup1 fnoo; exact: cvg_cst.
 apply/seteqP; split => [_ [n _] <- /[!fnoo]//|_ ->].
 by rewrite /S fnoo; exists 0%R.
have [/ereal_sup_ninfty lnoo|lnoo] := eqVneq l -oo.
 by exfalso; apply/fnoo/funext => n; rewrite (lnoo (f n))//; exists n.
have l_fin_num : l \is a fin_num by rewrite fin_numE lpoo lnoo.
set A := [set n | f n = -oo]; set B := [set n | f n != -oo].
have f_fin_num n : B n -> f n \is a fin_num.
 move=> Bn; rewrite fin_numE Bn/=.
 by apply: contra_notN Spoo => /eqP unpoo; exists n.
have [x Bx] : B !=set0.
 apply/set0P/negP => /eqP B0; apply/fnoo/funext => n.
 apply/eqP/negPn/negP => unnoo.
 by move/seteqP : B0 => [+ _] => /(_ n); apply.
have xB r : (x <= r)%R -> B r.
 move=> /ndf xr; apply/negP => /eqP urnoo.
 by move: xr; rewrite urnoo leeNy_eq; exact/negP.
rewrite -(@fineK _ l)//; apply/fine_cvgP; split.
 exists x; split; first by rewrite num_real.
 by move=> r A1r; rewrite f_fin_num //; exact/xB/ltW.
set g := fun n => if (n < x)%R then fine (f x) else fine (f n).
have <- : sup (range g) = fine l.
 apply: EFin_inj; rewrite -ereal_sup_EFin//; last 2 first.
 - exists (fine l) => /= _ [m _ <-]; rewrite /g /=.
 have [mx|xm] := ltP m x.
 by rewrite fine_le// ?f_fin_num//; apply: ereal_sup_ubound; exists x.
 rewrite fine_le// ?f_fin_num//; first exact/xB.
 by apply: ereal_sup_ubound; exists m.
 - by exists (g 0%R), 0%R.
 rewrite fineK//; apply/eqP; rewrite eq_le; apply/andP; split.
 apply: le_ereal_sup => _ /= [_ [m _] <-] <-.
 rewrite /g; have [_|xm] := ltP m x.
 by rewrite fineK// ?f_fin_num//; exists x.
 by rewrite fineK// ?f_fin_num//; [exists m|exact/xB].
 apply: ub_ereal_sup => /= _ [m _] <-.
 have [mx|xm] := ltP m x.
 rewrite (le_trans (ndf _ _ (ltW mx)))//.
 apply: ereal_sup_ubound => /=; exists (fine (f x)); last first.
 by rewrite fineK// f_fin_num.
 by exists m => //; rewrite /g mx.
 apply: ereal_sup_ubound => /=; exists (fine (f m)) => //.
 by exists m => //; rewrite /g ltNge xm.
 by rewrite fineK ?f_fin_num//; exact: xB.
suff: g x @[x --> +oo%R] --> sup (range g).
 apply: cvg_trans; apply: near_eq_cvg; near=> n.
 rewrite /g ifF//; apply/negbTE; rewrite -leNgt.
 by near: n; apply: nbhs_pinfty_ge; rewrite num_real.
apply: nondecreasing_cvgr.
- move=> m n mn; rewrite /g /=; have [_|xm] := ltP m x.
 + have [nx|nx] := ltP n x; first by rewrite fine_le// f_fin_num.
 by rewrite fine_le// ?f_fin_num//; [exact: xB|exact: ndf].
 + rewrite ltNge (le_trans xm mn)//= fine_le ?f_fin_num//.
 * exact: xB.
 * by apply: xB; rewrite (le_trans xm).
 * exact/ndf.
- exists (fine l) => /= _ [m _ <-]; rewrite /g /=.
 rewrite -lee_fin (fineK l_fin_num); apply: ereal_sup_ubound.
 have [_|xm] := ltP m x; first by rewrite fineK// ?f_fin_num//; eexists.
 by rewrite fineK// ?f_fin_num//; [exists m|exact/xB].
Unshelve. all: by end_near. Qed.

Lemma nondecreasing_is_cvge (f : R -> \bar R) :
nondecreasing_fun f -> (cvg (f r @[r --> +oo]))%R.

Proof.

by move=> u_nd u_ub; apply: cvgP; exact: nondecreasing_cvge. Qed.

Lemma nondecreasing_at_right_cvge (f : R -> \bar R) a (b : itv_bound R) :
 (BRight a < b)%O ->
 {in Interval (BRight a) b &, nondecreasing_fun f} ->
 f x @[x --> a ^'+] --> ereal_inf (f @` [set` Interval (BRight a) b]).

Proof.

move=> ab ndf; set S := (X in ereal_inf X); set l := ereal_inf S.
have [Snoo|Snoo] := pselect (S -oo).
 case: (Snoo) => N/=.
 rewrite in_itv/= -[X in _ && X]/(BLeft N < b)%O => /andP[aN Nb] fNpoo.
 have Nf n : (a < n <= N)%R -> f n = -oo.
 move=> /andP[an nN]; apply/eqP.
 rewrite eq_le leNye andbT -fNpoo ndf//.
 by rewrite in_itv/= -[X in _ && X]/(BLeft n < b)%O an (le_lt_trans _ Nb).
 by rewrite in_itv/= -[X in _ && X]/(BLeft N < b)%O (lt_le_trans an nN).
 have -> : l = -oo.
 by rewrite /l /ereal_inf /ereal_sup supremum_pinfty//=; exists -oo.
 apply: cvg_near_cst; exists (N - a)%R => /=; first by rewrite subr_gt0.
 move=> y /= + ay; rewrite ltr0_norm ?subr_lt0// opprB => ayNa.
 by rewrite Nf// ay/= -(subrK a y) -lerBrDr ltW.
have [lnoo|lnoo] := eqVneq l -oo.
 rewrite lnoo; apply/cvgeNyPle => M.
 have /ereal_inf_lt[x [y]]/= : M%:E > l by rewrite lnoo ltNyr.
 rewrite in_itv/= -[X in _ && X]/(BLeft y < b)%O/= => /andP[ay yb] <- fyM.
 exists (y - a)%R => /=; first by rewrite subr_gt0.
 move=> z /= + az.
 rewrite ltr0_norm ?subr_lt0// opprB ltrBlDr subrK => zy.
 rewrite (le_trans _ (ltW fyM))// ndf ?ltW//.
 by rewrite in_itv/= -[X in _ && X]/(BLeft z < b)%O/= az/= (lt_trans _ yb).
 by rewrite in_itv/= -[X in _ && X]/(BLeft y < b)%O/= (lt_trans az zy).
have [fpoo|fpoo] := pselect {in Interval (BRight a) b, forall x, f x = +oo}.
 rewrite {}/l in lnoo *; rewrite {}/S in Snoo lnoo *.
 rewrite [X in ereal_inf X](_ : _ = [set +oo]).
 rewrite ereal_inf1; apply/cvgeyPgey; near=> M.
 move: b ab {ndf lnoo Snoo} fpoo => [[|] b|[//|]] ab fpoo.
 - near=> x; rewrite fpoo ?leey// in_itv/=.
 by apply/andP; split; near: x; [exact: nbhs_right_gt|exact: nbhs_right_lt].
 - near=> x; rewrite fpoo ?leey// in_itv/=.
 by apply/andP; split; near: x; [exact: nbhs_right_gt|exact: nbhs_right_le].
 - near=> x; rewrite fpoo ?leey// in_itv/= andbT.
 by near: x; exact: nbhs_right_gt.
 apply/seteqP; split => [_ [n _] <- /[!fpoo]//|_ ->].
 move: b ab ndf lnoo Snoo fpoo => [[|] s|[//|]] ab ndf lnoo Snoo fpoo /=.
 - by exists ((a + s) / 2)%R; rewrite ?fpoo// in_itv/= !midf_lt.
 - by exists ((a + s) / 2)%R; rewrite ?fpoo// in_itv/= !(midf_lt, midf_le)// ltW.
 - by exists (a + 1)%R; rewrite ?fpoo// in_itv/= andbT ltrDl.
have [/ereal_inf_pinfty lpoo|lpoo] := eqVneq l +oo.
 by exfalso; apply/fpoo => r rab; rewrite (lpoo (f r))//; exists r.
have l_fin_num : l \is a fin_num by rewrite fin_numE lpoo lnoo.
set A := [set r | [/\ (a < r)%R, (BLeft r < b)%O & f r != +oo]].
have f_fin_num r : r \in A -> f r \is a fin_num.
 rewrite inE /A/= => -[ar rb] frnoo; rewrite fin_numE frnoo andbT.
 apply: contra_notN Snoo => /eqP frpoo.
 by exists r => //=; rewrite in_itv/= -[X in _ && X]/(BLeft r < b)%O ar rb.
have [x [ax xb fxpoo]] : A !=set0.
 apply/set0P/negP => /eqP A0; apply/fpoo => x.
 rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O => /andP[ax xb].
 apply/eqP/negPn/negP => unnoo.
 by move/seteqP : A0 => [+ _] => /(_ x); apply; rewrite /A/= ax.
have axA r : (a < r <= x)%R -> r \in A.
 move=> /andP[ar rx]; move: (rx) => /ndf rafx; rewrite /A /= inE; split => //.
 by rewrite (le_lt_trans _ xb).
 apply/negP => /eqP urnoo.
 move: rafx; rewrite urnoo.
 rewrite in_itv/= -[X in _ && X]/(BLeft r < b)%O ar/=.
 rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O ax/=.
 by rewrite leye_eq (negbTE fxpoo) -falseE; apply; rewrite (le_lt_trans _ xb).
rewrite -(@fineK _ l)//; apply/fine_cvgP; split.
 exists (x - a)%R => /=; first by rewrite subr_gt0.
 move=> z /= + az.
 rewrite ltr0_norm ?subr_lt0// opprB ltrBlDr subrK// => zx.
 by rewrite f_fin_num// axA// az/= ltW.
set g := fun n => if (a < n < x)%R then fine (f n) else fine (f x).
have <- : inf [set g x | x in [set` Interval (BRight a) b]] = fine l.
 apply: EFin_inj; rewrite -ereal_inf_EFin//; last 2 first.
 - exists (fine l) => /= _ [m _ <-]; rewrite /g /=.
 case: ifPn => [/andP[am mx]|].
 rewrite fine_le// ?f_fin_num//; first by rewrite axA// am (ltW mx).
 apply: ereal_inf_lbound; exists m => //=.
 rewrite in_itv/= -[X in _ && X]/(BLeft m < b)%O am/=.
 by rewrite (le_lt_trans _ xb) ?ltW.
 rewrite negb_and -!leNgt => /orP[ma|xm].
 rewrite fine_le// ?f_fin_num ?inE//.
 apply: ereal_inf_lbound; exists x => //=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O ax xb.
 rewrite fine_le// ?f_fin_num ?inE//.
 apply: ereal_inf_lbound; exists x => //=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O ax xb.
 - rewrite {}/l in lnoo lpoo l_fin_num *.
 rewrite {}/S in Snoo lnoo lpoo l_fin_num *.
 rewrite {}/A in f_fin_num axA *.
 move: b ab {xb ndf lnoo lpoo l_fin_num f_fin_num Snoo fpoo axA} =>
 [[|] s|[//|]] ab /=.
 + exists (g ((a + s) / 2))%R, ((a + s) / 2)%R => //=.
 by rewrite /= in_itv/= !midf_lt.
 + exists (g ((a + s) / 2))%R, ((a + s) / 2)%R => //=.
 by rewrite /= in_itv/= !(midf_lt, midf_le)// ltW.
 + exists (g (a + 1)%R), (a + 1)%R => //=.
 by rewrite in_itv/= andbT ltrDl.
 rewrite fineK//; apply/eqP; rewrite eq_le; apply/andP; split; last first.
 apply: le_ereal_inf => _ /= [_ [m _] <-] <-.
 rewrite /g; case: ifPn => [/andP[am mx]|].
 rewrite fineK// ?f_fin_num//; last by rewrite axA// am ltW.
 exists m => //=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft m < b)%O am/= (lt_trans _ xb).
 rewrite negb_and -!leNgt => /orP[ma|xm].
 rewrite fineK//; last by rewrite f_fin_num ?inE.
 exists x => //=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O ax xb.
 exists x => /=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft x < b)%O ax xb.
 by rewrite fineK// f_fin_num ?inE.
 apply: lb_ereal_inf => /= y [m] /=.
 rewrite in_itv/= -[X in _ && X]/(BLeft m < b)%O => /andP[am mb] <-{y}.
 have [mx|xm] := ltP m x.
 apply: ereal_inf_lbound => /=; exists (fine (f m)); last first.
 by rewrite fineK// f_fin_num// axA// am (ltW mx).
 by exists m; [rewrite in_itv/= am|rewrite /g am mx].
 rewrite (@le_trans _ _ (f x))//; last first.
 by apply: ndf => //; rewrite in_itv//= ?ax ?am.
 apply: ereal_inf_lbound => /=; exists (fine (f x)); last first.
 by rewrite fineK// f_fin_num ?inE.
 by exists x; [rewrite in_itv/= ax|rewrite /g ltxx andbF].
suff: g x @[x --> a^'+] --> inf [set g x | x in [set` Interval (BRight a) b]].
 apply: cvg_trans; apply: near_eq_cvg; near=> n.
 rewrite /g /=; case: ifPn => [//|].
 rewrite negb_and -!leNgt => /orP[na|xn].
 exfalso.
 move: na; rewrite leNgt => /negP; apply.
 by near: n; exact: nbhs_right_gt.
 suff nx : (n < x)%R by rewrite ltNge xn in nx.
 near: n; exists ((x - a) / 2)%R; first by rewrite /= divr_gt0// subr_gt0.
 move=> y /= /[swap] ay.
 rewrite ltr0_norm// ?subr_lt0// opprB ltrBlDr => /lt_le_trans; apply.
 by rewrite -lerBrDr ler_pdivrMr// ler_pMr// ?ler1n// subr_gt0.
apply: nondecreasing_at_right_cvgr => //.
- move=> m n; rewrite !in_itv/= -[X in _ && X]/(BLeft m < b)%O.
 rewrite -[X in _ -> _ && X -> _]/(BLeft n < b)%O.
 move=> /andP[am mb] /andP[an nb] mn.
 rewrite /g /=; case: ifPn => [/andP[_ mx]|].
 rewrite (lt_le_trans am mn) /=; have [nx|nn0] := ltP n x.
 rewrite fine_le ?f_fin_num ?ndf//; first by rewrite axA// am (ltW mx).
 by rewrite axA// (ltW nx) andbT (lt_le_trans am).
 by rewrite in_itv/= am.
 by rewrite in_itv/= an.
 rewrite fine_le ?f_fin_num//.
 + by rewrite axA// am (ltW (lt_le_trans mx _)).
 + by rewrite inE.
 + rewrite ndf//; last exact/ltW.
 by rewrite !in_itv/= am.
 by rewrite !in_itv/= ax.
 rewrite negb_and -!leNgt => /orP[|xm]; first by rewrite leNgt am.
 by rewrite (lt_le_trans am mn)/= ltNge (le_trans xm mn).
- exists (fine l) => /= _ [m _ <-]; rewrite /g /=.
 rewrite -lee_fin (fineK l_fin_num); apply: ereal_inf_lbound.
 case: ifPn => [/andP[am mn0]|].
 rewrite fineK//; last by rewrite f_fin_num// axA// am (ltW mn0).
 exists m => //=.
 by rewrite in_itv/= -[X in _ && X]/(BLeft m < b)%O am (lt_trans _ xb).
 rewrite negb_and -!leNgt => /orP[ma|xm].
 rewrite fineK//; first by exists x => //=; rewrite in_itv/= ax.
 by rewrite f_fin_num ?inE.
 by rewrite fineK// ?f_fin_num ?inE//; exists x => //=; rewrite in_itv/= ax.
Unshelve. all: by end_near. Qed.

Lemma nondecreasing_at_right_is_cvge (f : R -> \bar R) (a : R) :
(\forall x \near a^'+, {in `]a, x[ &, nondecreasing_fun f}) ->
cvg (f x @[x --> a ^'+]).

Proof.

move=> ndf; apply/cvg_ex; near a^'+ => b.
by eexists; apply: (@nondecreasing_at_right_cvge _ _ (BLeft b));
[rewrite bnd_simp|near: b..].
Unshelve. all: by end_near. Qed.

Lemma nonincreasing_at_right_cvge (f : R -> \bar R) a (b : itv_bound R) :
(BRight a < b)%O -> {in Interval (BRight a) b &, nonincreasing_fun f} ->
f x @[x --> a ^'+] --> ereal_sup (f @` [set` Interval (BRight a) b]).

Proof.

move=> ab nif; have ndNf : {in Interval (BRight a) b &,
{homo (\- f) : n m / (n <= m)%R >-> n <= m}}.
by move=> r s rab sab /nif; rewrite leeN2; exact.
have /cvgeN := nondecreasing_at_right_cvge ab ndNf.
under eq_fun do rewrite oppeK.
set lhs := (X in _ --> X -> _); set rhs := (X in _ -> _ --> X).
suff : lhs = rhs by move=> ->.
rewrite {}/rhs {}/lhs; rewrite /ereal_inf oppeK; congr ereal_sup.
by rewrite image_comp/=; apply: eq_imagel => x _ /=; rewrite oppeK.
Qed.

Lemma nonincreasing_at_right_is_cvge (f : R -> \bar R) a :
(\forall x \near a^'+, {in `]a, x[ &, nonincreasing_fun f}) ->
cvg (f x @[x --> a ^'+]).

Proof.

move=> nif; apply/cvg_ex; near a^'+ => b.
by eexists; apply: (@nonincreasing_at_right_cvge _ _ (BLeft b));
[rewrite bnd_simp|near: b..].
Unshelve. all: by end_near. Qed.

End fun_cvg_ereal.

End fun_cvg.
Arguments nondecreasing_at_right_cvge {R f a} b.
Arguments nondecreasing_at_right_is_cvge {R f a}.
Arguments nonincreasing_at_right_cvge {R f a} b.
Arguments nonincreasing_at_right_is_cvge {R f a}.

Section lime_sup_inf.
Variable R : realType.
Local Open Scope ereal_scope.
Implicit Types (f g : R -> \bar R) (a r s l : R).

Definition lime_sup f a : \bar R := limf_esup f a^'.

Definition lime_inf f a : \bar R := - lime_sup (\- f) a.

Let sup_ball f a r := ereal_sup [set f x | x in ball a r `\ a].

Let sup_ball_le f a r s : (r <= s)%R -> sup_ball f a r <= sup_ball f a s.

Proof.

move=> rs; apply: ub_ereal_sup => /= _ /= [t [rt ta] <-].
by apply: ereal_sup_ubound => /=; exists t => //; split => //; exact: le_ball rt.
Qed.

Let sup_ball_is_cvg f a : cvg (sup_ball f a e @[e --> 0^'+]).

Proof.

apply: nondecreasing_at_right_is_cvge; near=> e.
by move=> x y; rewrite !in_itv/= => /andP[x0 xe] /andP[y0 ye] /sup_ball_le.
Unshelve. all: by end_near. Qed.

Let inf_ball f a r := - sup_ball (\- f) a r.

Let inf_ballE f a r : inf_ball f a r = ereal_inf [set f x | x in ball a r `\ a].

Proof.

by rewrite /inf_ball /ereal_inf; congr (- _); rewrite /sup_ball -image_comp.
Qed.

Let inf_ball_le f a r s : (s <= r)%R -> inf_ball f a r <= inf_ball f a s.

Proof.

by move=> sr; rewrite /inf_ball leeNl oppeK sup_ball_le. Qed.

Let inf_ball_is_cvg f a : cvg (inf_ball f a e @[e --> 0^'+]).

Proof.

apply: nonincreasing_at_right_is_cvge; near=> e.
by move=> x y; rewrite !in_itv/= => /andP[x0 xe] /andP[y0 ye] /inf_ball_le.
Unshelve. all: by end_near. Qed.

Let le_sup_ball f g a :
(\forall r \near 0^'+, forall y : R, y != a -> ball a r y -> f y <= g y) ->
\forall r \near 0^'+, sup_ball f a r <= sup_ball g a r.

Proof.

move=> [e/= e0 fg].
near=> r; apply: ub_ereal_sup => /= _ [s [pas /= /eqP ps]] <-.
rewrite (@le_trans _ _ (g s))//.
by rewrite (fg r)//= sub0r normrN gtr0_norm.
by apply: ereal_sup_ubound => /=; exists s => //; split => //; exact/eqP.
Unshelve. all: by end_near. Qed.

Lemma lime_sup_lim f a : lime_sup f a = lim (sup_ball f a e @[e --> 0^'+]).

Proof.

apply/eqP; rewrite eq_le; apply/andP; split.
apply: lime_ge => //; near=> e; apply: ereal_inf_lbound => /=.
by exists (ball a e `\ a) => //=; exact: dnbhs_ball.
apply: lb_ereal_inf => /= _ [A [r /= r0 arA] <-].
apply: lime_le => //; near=> e.
apply: le_ereal_sup => _ [s [ase /eqP sa] <- /=].
exists s => //; apply: arA => //=; apply: (lt_le_trans ase).
by near: e; exact: nbhs_right_le.
Unshelve. all: by end_near. Qed.

Lemma lime_inf_lim f a : lime_inf f a = lim (inf_ball f a e @[e --> 0^'+]).

Proof.

rewrite /lime_inf lime_sup_lim -limeN; last exact: sup_ball_is_cvg.
by rewrite /sup_ball; under eq_fun do rewrite -image_comp.
Qed.

Lemma lime_supE f a :
lime_sup f a = ereal_inf [set sup_ball f a e | e in `]0, +oo[ ]%R.

Proof.

rewrite lime_sup_lim; apply/cvg_lim => //.
apply: nondecreasing_at_right_cvge => //.
by move=> x y; rewrite !in_itv/= !andbT => x0 y0; exact: sup_ball_le.
Qed.

Lemma lime_infE f a :
lime_inf f a = ereal_sup [set inf_ball f a e | e in `]0, +oo[ ]%R.

Proof.

by rewrite /lime_inf lime_supE /ereal_inf oppeK image_comp. Qed.

Lemma lime_infN f a : lime_inf (\- f) a = - lime_sup f a.

Proof.

by rewrite /lime_sup -limf_einfN. Qed.

Lemma lime_supN f a : lime_sup (\- f) a = - lime_inf f a.

Proof.

by rewrite /lime_inf oppeK. Qed.

Lemma __deprecated__lime_sup_ge0 f a : (forall x, 0 <= f x) -> 0 <= lime_sup f a.

Proof.

by move=> f0; exact: limf_esup_ge0. Qed.

Lemma lime_inf_ge0 f a : (forall x, 0 <= f x) -> 0 <= lime_inf f a.

Proof.

move=> f0; rewrite lime_inf_lim; apply: lime_ge; first exact: inf_ball_is_cvg.
near=> b; rewrite inf_ballE.
by apply: lb_ereal_inf => /= _ [r [abr/= ra]] <-; exact: f0.
Unshelve. all: by end_near. Qed.

Lemma lime_supD f g a : lime_sup f a +? lime_sup g a ->
lime_sup (f \+ g)%E a <= lime_sup f a + lime_sup g a.

Proof.

move=> fg; rewrite !lime_sup_lim -limeD//; last first.
 by rewrite -!lime_sup_lim.
apply: lee_lim => //.
- apply: nondecreasing_at_right_is_cvge; near=> e => x y; rewrite !in_itv/=.
 by move=> /andP[? ?] /andP[? ?] xy; apply: leeD => //; exact: sup_ball_le.
- near=> a0; apply: ub_ereal_sup => _ /= [a1 [a1ae a1a]] <-.
 by apply: leeD; apply: ereal_sup_ubound => /=; exists a1.
Unshelve. all: by end_near. Qed.

Lemma lime_sup_le f g a :
(\forall r \near 0^'+, forall y, y != a -> ball a r y -> f y <= g y) ->
lime_sup f a <= lime_sup g a.

Proof.

by move=> fg; rewrite !lime_sup_lim; apply: lee_lim => //; exact: le_sup_ball.
Qed.

Lemma lime_inf_sup f a : lime_inf f a <= lime_sup f a.

Proof.

rewrite lime_inf_lim lime_sup_lim; apply: lee_lim => //.
near=> r; rewrite ereal_sup_le//.
have ? : exists2 x, ball a r x /\ x <> a & f x = f (a + r / 2)%R.
 exists (a + r / 2)%R => //; split.
 rewrite /ball/= opprD addrA subrr sub0r normrN gtr0_norm ?divr_gt0//.
 by rewrite ltr_pdivrMr// ltr_pMr// ltr1n.
 by apply/eqP; rewrite gt_eqF// ltr_pwDr// divr_gt0.
by exists (f (a + r / 2)) => //=; rewrite inf_ballE ereal_inf_lbound.
Unshelve. all: by end_near. Qed.

Local Lemma lim_lime_sup' f a l :
f r @[r --> a] --> l%:E -> lime_sup f a <= l%:E.

Proof.

move=> fpA; apply/lee_addgt0Pr => e e0; rewrite lime_sup_lim.
apply: lime_le => //.
move/fine_cvg : (fpA) => /cvgrPdist_le fpA1.
move/fcvg_is_fine : (fpA); rewrite near_map => -[d d0] fpA2.
have := fpA1 _ e0 => -[q /= q0] H.
near=> x; apply: ub_ereal_sup => //= _ [y [pry /= yp <-]].
have ? : f y \is a fin_num.
 apply: fpA2.
 rewrite /ball_ /= (lt_le_trans pry)//.
 by near: x; exact: nbhs_right_le.
rewrite -lee_subel_addl// -(@fineK _ (f y)) // -EFinB lee_fin.
rewrite (le_trans (ler_norm _))// distrC H// /ball_/= ltr_distlC.
move: pry; rewrite /ball/= ltr_distlC => /andP[pay ypa].
have xq : (x <= q)%R by near: x; exact: nbhs_right_le.
apply/andP; split.
 by rewrite (le_lt_trans _ pay)// lerB.
by rewrite (lt_le_trans ypa)// lerD2l.
Unshelve. all: by end_near.
Qed.

Local Lemma lim_lime_inf' f a (l : R) :
f r @[r --> a] --> l%:E -> l%:E <= lime_inf f a.

Proof.

move=> fpA; apply/lee_subgt0Pr => e e0; rewrite lime_inf_lim.
apply: lime_ge => //.
move/fine_cvg : (fpA) => /cvgrPdist_le fpA1.
move/fcvg_is_fine : (fpA); rewrite near_map => -[d d0] fpA2.
have := fpA1 _ e0 => -[q /= q0] H.
near=> x.
rewrite inf_ballE.
apply: lb_ereal_inf => //= _ [y [pry /= yp <-]].
have ? : f y \is a fin_num.
 apply: fpA2.
 rewrite /ball_ /= (lt_le_trans pry)//.
 by near: x; exact: nbhs_right_le.
rewrite -(@fineK _ (f y)) // -EFinB lee_fin lerBlDr -lerBlDl.
rewrite (le_trans (ler_norm _))// H// /ball_/= ltr_distlC.
move: pry; rewrite /ball/= ltr_distlC => /andP[pay ypa].
have xq : (x <= q)%R by near: x; exact: nbhs_right_le.
apply/andP; split.
 by rewrite (le_lt_trans _ pay)// lerB.
by rewrite (lt_le_trans ypa)// lerD2l.
Unshelve. all: by end_near.
Qed.

Lemma lim_lime_inf f a (l : R) :
f r @[r --> a] --> l%:E -> lime_inf f a = l%:E.

Proof.

move=> h; apply/eqP; rewrite eq_le.
by rewrite lim_lime_inf'// andbT (le_trans (lime_inf_sup _ _))// lim_lime_sup'.
Qed.

Lemma lim_lime_sup f a (l : R) :
f r @[r --> a] --> l%:E -> lime_sup f a = l%:E.

Proof.

move=> h; apply/eqP; rewrite eq_le.
by rewrite lim_lime_sup'//= (le_trans _ (lime_inf_sup _ _))// lim_lime_inf'.
Qed.

Local Lemma lime_supP f a l :
lime_sup f a = l%:E -> forall e : {posnum R}, exists d : {posnum R},
forall x, (ball a d%:num `\ a) x -> f x < l%:E + e%:num%:E.

Proof.

rewrite lime_supE => fal.
have H (e : {posnum R}) :
 exists d : {posnum R}, l%:E <= sup_ball f a d%:num < l%:E + e%:num%:E.
 apply: contrapT => /forallNP H.
 have : ereal_inf [set sup_ball f a r | r in `]0%R, +oo[] \is a fin_num.
 by rewrite fal.
 move=> /lb_ereal_inf_adherent-/(_ e%:num ltac:(by []))[y] /=.
 case=> r; rewrite in_itv/= andbT => r0 <-{y}.
 rewrite ltNge => /negP; apply.
 have /negP := H (PosNum r0).
 rewrite negb_and => /orP[|].
 rewrite -ltNge => farl.
 have : ereal_inf [set sup_ball f a r | r in `]0%R, +oo[] < l%:E.
 rewrite (le_lt_trans _ farl)//; apply: ereal_inf_lbound => /=; exists r => //.
 by rewrite in_itv/= r0.
 by rewrite fal ltxx.
 by rewrite -leNgt; apply: le_trans; rewrite leeD2r// fal.
move=> e; have [d /andP[lfp fpe]] := H e.
exists d => r /= [] prd rp.
by rewrite (le_lt_trans _ fpe)//; apply: ereal_sup_ubound => /=; exists r.
Qed.

Local Lemma lime_infP f a l :
lime_inf f a = l%:E -> forall e : {posnum R}, exists d : {posnum R},
forall x, (ball a d%:num `\ a) x -> l%:E - e%:num%:E < f x.

Proof.

move=> /(congr1 oppe); rewrite -lime_supN => /lime_supP => H e.
have [d {}H] := H e.
by exists d => r /H; rewrite lteNl oppeD// EFinN oppeK.
Qed.

Lemma lime_sup_inf_at_right f a l :
lime_sup f a = l%:E -> lime_inf f a = l%:E -> f x @[x --> a^'+] --> l%:E.

Proof.

move=> supfpl inffpl; apply/cvge_at_rightP => u [pu up].
have fu : \forall n \near \oo, f (u n) \is a fin_num.
 have [dsup Hdsup] := lime_supP supfpl (PosNum ltr01).
 have [dinf Hdinf] := lime_infP inffpl (PosNum ltr01).
 near=> n; rewrite fin_numE; apply/andP; split.
 apply/eqP => fxnoo.
 suff : (ball a dinf%:num `\ a) (u n) by move=> /Hdinf; rewrite fxnoo.
 split; last by apply/eqP; rewrite gt_eqF.
 by near: n; move/cvgrPdist_lt : up; exact.
 apply/eqP => fxnoo.
 suff : (ball a dsup%:num `\ a) (u n) by move=> /Hdsup; rewrite fxnoo.
 split; last by apply/eqP; rewrite gt_eqF.
 by near: n; move/cvgrPdist_lt : up; exact.
apply/fine_cvgP; split => /=; first exact: fu.
apply/cvgrPdist_le => _/posnumP[e].
have [d1 Hd1] : exists d1 : {posnum R},
 l%:E - e%:num%:E <= ereal_inf [set f x | x in ball a d1%:num `\ a].
 have : l%:E - e%:num%:E < lime_inf f a.
 by rewrite inffpl lteBlDr// lteDl.
 rewrite lime_infE => /ereal_sup_gt[x /= [r]]; rewrite in_itv/= andbT.
 move=> r0 <-{x} H; exists (PosNum r0); rewrite ltW//.
 by rewrite -inf_ballE.
have [d2 Hd2] : exists d2 : {posnum R},
 ereal_sup [set f x | x in ball a d2%:num `\ a] <= l%:E + e%:num%:E.
 have : lime_sup f a < l%:E + e%:num%:E by rewrite supfpl lteDl.
 rewrite lime_supE => /ereal_inf_lt[x /= [r]]; rewrite in_itv/= andbT.
 by move=> r0 <-{x} H; exists (PosNum r0); rewrite ltW.
pose d := minr d1%:num d2%:num.
have d0 : (0 < d)%R by rewrite lt_min; apply/andP; split => //=.
move/cvgrPdist_lt : up => /(_ _ d0)[m _] {}ucvg.
near=> n.
rewrite /= ler_distlC; apply/andP; split.
 rewrite -lee_fin EFinB (le_trans Hd1)//.
 rewrite (@le_trans _ _ (ereal_inf [set f x | x in ball a d `\ a]))//.
 apply: le_ereal_inf => _/= [r [adr ra] <-]; exists r => //; split => //.
 by rewrite /ball/= (lt_le_trans adr)// /d ge_min lexx.
 apply: ereal_inf_lbound => /=; exists (u n).
 split; last by apply/eqP; rewrite eq_sym lt_eqF.
 by apply: ucvg => //=; near: n; exists m.
 by rewrite fineK//; by near: n.
rewrite -lee_fin EFinD (le_trans _ Hd2)//.
rewrite (@le_trans _ _ (ereal_sup [set f x | x in ball a d `\ a]))//; last first.
 apply: le_ereal_sup => z/= [r [adr rp] <-{z}]; exists r => //; split => //.
 by rewrite /ball/= (lt_le_trans adr)// /d ge_min lexx orbT.
apply: ereal_sup_ubound => /=; exists (u n).
 split; last by apply/eqP; rewrite eq_sym lt_eqF.
 by apply: ucvg => //=; near: n; exists m.
by rewrite fineK//; near: n.
Unshelve. all: by end_near. Qed.

Lemma lime_sup_inf_at_left f a l :
lime_sup f a = l%:E -> lime_inf f a = l%:E -> f x @[x --> a^'-] --> l%:E.

Proof.

move=> supfal inffal; apply/cvg_at_leftNP/lime_sup_inf_at_right.
- by rewrite /lime_sup -limf_esup_dnbhsN.
- by rewrite /lime_inf /lime_sup -(limf_esup_dnbhsN (-%E \o f)) limf_esupN oppeK.
Qed.

End lime_sup_inf.
#[deprecated(since="mathcomp-analysis 1.3.0", note="use `limf_esup_ge0` instead")]
Notation lime_sup_ge0 := __deprecated__lime_sup_ge0 (only parsing).

Section derivable_oo_continuous_bnd.
Context {R : numFieldType} {V : normedModType R}.

Definition derivable_oo_continuous_bnd (f : R -> V) (x y : R) :=
  [/\ {in `]x, y[, forall x, derivable f x 1},
      f @ x^'+ --> f x & f @ y^'- --> f y].

Lemma derivable_oo_continuous_bnd_within (f : R -> V) (x y : R) :
  derivable_oo_continuous_bnd f x y -> {within `[x, y], continuous f}.

Proof.

move=> [fxy fxr fyl]; apply/subspace_continuousP => z /=.
rewrite in_itv/= => /andP[]; rewrite le_eqVlt => /predU1P[<-{z} xy|].
 have := cvg_at_right_within fxr; apply: cvg_trans; apply: cvg_app.
 by apply: within_subset => z/=; rewrite in_itv/= => /andP[].
move=> /[swap].
rewrite le_eqVlt => /predU1P[->{z} xy|zy xz].
 have := cvg_at_left_within fyl; apply: cvg_trans; apply: cvg_app.
 by apply: within_subset => z/=; rewrite in_itv/= => /andP[].
apply: cvg_within_filter.
apply/differentiable_continuous; rewrite -derivable1_diffP.
by apply: fxy; rewrite in_itv/= xz zy.
Qed.

End derivable_oo_continuous_bnd.

Section real_inverse_functions.
Variable R : realType.
Implicit Types (a b : R) (f g : R -> R).

This lemma should be used with caution. Generally `{within I, continuous f}` is what one would intend. So having `{in I, continuous f}` as a condition may indicate potential issues at the endpoints of the interval.

Lemma continuous_subspace_itv (I : interval R) (f : R -> R) :
{in I, continuous f} -> {within [set` I], continuous f}.

Proof.

move=> ctsf; apply: continuous_in_subspaceT => x Ix; apply: ctsf.
by move: Ix; rewrite inE.
Qed.

Lemma itv_continuous_inj_le f (I : interval R) :
 (exists x y, [/\ x \in I, y \in I, x < y & f x <= f y]) ->
 {within [set` I], continuous f} -> {in I &, injective f} ->
 {in I &, {mono f : x y / x <= y}}.

Proof.

gen have fxy : f / {in I &, injective f} ->
 {in I &, forall x y, x < y -> f x != f y}.
 move=> fI x y xI yI xLy; apply/negP => /eqP /fI => /(_ xI yI) xy.
 by move: xLy; rewrite xy ltxx.
gen have main : f / forall c, {within [set` I], continuous f} ->
 {in I &, injective f} ->
 {in I &, forall a b, f a < f b -> a < c -> c f a < f c /\ f c < f b}.
 move=> c fC fI a b aI bI faLfb aLc cLb.
 have intP := interval_is_interval aI bI.
 have cI : c \in I by rewrite intP// (ltW aLc) ltW.
 have ctsACf : {within `[a, c], continuous f}.
 apply: (continuous_subspaceW _ fC) => x; rewrite /= inE => /itvP axc.
 by rewrite intP// axc/= (le_trans _ (ltW cLb))// axc.
 have ctsCBf : {within `[c, b], continuous f}.
 apply: (continuous_subspaceW _ fC) => x /=; rewrite inE => /itvP axc.
 by rewrite intP// axc andbT (le_trans (ltW aLc)) ?axc.
 have [aLb alb'] : a fbLfc.
 have := fbLfc; suff /eqP -> : c == b by rewrite ltxx.
 rewrite eq_le (ltW cLb) /=.
 have [d /andP[ad dc] fdEfb] : exists2 d, a <= d <= c & f d = f b.
 have aLc' : a <= c by rewrite ltW.
 apply: IVT => //; last first.
 by case: ltrgtP faLfc; rewrite // (ltW faLfb) // ltW.
 rewrite -(fI _ _ _ _ fdEfb) //.
 move: ad dc; rewrite le_eqVlt =>/predU1P[<-//| /ltW L] dc.
 by rewrite intP// L (le_trans _ (ltW cLb)).
 - have [fbLfc | fcLfb | fbEfc] /= := ltrgtP (f b) (f c).
 + by have := lt_trans fbLfc fcLfa; rewrite ltNge (ltW faLfb).
 + have [d /andP[cLd dLb] /eqP] : exists2 d, c <= d <= b & f d = f a.
 have cLb' : c <= b by rewrite ltW.
 apply: IVT => //; last by case: ltrgtP fcLfb; rewrite // !ltW.
 have /(fxy f fI) : a < d by rewrite (lt_le_trans aLc).
 suff dI' : d \in I by rewrite eq_sym=> /(_ aI dI') => /negbTE ->.
 move: dLb; rewrite le_eqVlt => /predU1P[->//|/ltW db].
 by rewrite intP// db (le_trans (ltW aLc)).
 + by move: fcLfa; rewrite -fbEfc ltNge (ltW faLfb).
 by move/(fxy _ fI) : aLc=> /(_ aI cI); rewrite faEfc.
move=> [u [v [uI vI ulv +]]] fC fI; rewrite le_eqVlt => /predU1P[fufv|fuLfv].
 by move/fI: fufv => /(_ uI vI) uv; move: ulv; rewrite uv ltxx.
have aux a c b : a \in I -> b \in I -> a < c -> c 
 (f a < f c -> f a < f b /\ f c < f b) /\
 (f c < f b -> f a < f b /\ f a < f c).
 move=> aI bI aLc cLb; have aLb := lt_trans aLc cLb.
 have cI : c \in I by rewrite (interval_is_interval aI bI)// (ltW aLc)/= ltW.
 have fanfb : f a != f b by apply: (fxy f fI).
 have decr : f b < f a -> f b < f c /\ f c < f a.
 have ofC : {within [set` I], continuous (-f)}.
 move=> ?; apply: continuous_comp; [exact: fC | exact: continuousN].
 have ofI : {in I &, injective (-f)} by move=>> ? ? /oppr_inj/fI ->.
 rewrite -[X in X < _ -> _](opprK (f b)) ltrNl => ofaLofb.
 have := main _ c ofC ofI a b aI bI ofaLofb aLc cLb.
 by (do 2 rewrite ltrNl opprK); rewrite and_comm.
 split=> [faLfc|fcLfb].
 suff L : f a < f b by have [] := main f c fC fI a b aI bI L aLc cLb.
 by case: ltgtP decr fanfb => // fbfa []//; case: ltgtP faLfc.
 suff L : f a < f b by have [] := main f c fC fI a b aI bI L aLc cLb.
 by case: ltgtP decr fanfb => // fbfa []//; case: ltgtP fcLfb.
have{main fC} whole a c b := main f c fC fI a b.
have low a c b : f a < f c -> a \in I -> b \in I ->
 a < c -> c f a < f b /\ f c < f b.
 by move=> L aI bI ac cb; case: (aux a c b aI bI ac cb)=> [/(_ L)].
have high a c b : f c < f b -> a \in I -> b \in I ->
 a < c -> c f a < f b /\ f a < f c.
 by move=> L aI bI ac cb; case: (aux a c b aI bI ac cb)=> [_ /(_ L)].
apply: le_mono_in => x y xI yI xLy.
have [uLx | xLu | xu] := ltrgtP u x.
- suff fuLfx : f u < f x by have [] := low u x y fuLfx uI yI uLx xLy.
 have [xLv | vLx | -> //] := ltrgtP x v; first by case: (whole u x v).
 by case: (low u v x).
- have fxLfu : f x < f u by have [] := high x u v fuLfv xI vI xLu ulv.
 have [yLu | uLy | -> //] := ltrgtP y u; first by case: (whole x y u).
 by case: (low x u y).
move: xLy; rewrite -xu => uLy.
have [yLv | vLy | -> //] := ltrgtP y v; first by case: (whole u y v).
by case: (low u v y).
Qed.

Lemma itv_continuous_inj_ge f (I : interval R) :
 (exists x y, [/\ x \in I, y \in I, x < y & f y <= f x]) ->
 {within [set` I], continuous f} -> {in I &, injective f} ->
 {in I &, {mono f : x y /~ x <= y}}.

Proof.

move=> [a [b [aI bI ab fbfa]]] fC fI x y xI yI.
suff : (- f) y <= (- f) x = (y <= x) by rewrite lerNl opprK.
apply: itv_continuous_inj_le xI => // [|x1 x1I | x1 x2 x1I x2I].
- by exists a, b; split => //; rewrite lerNl opprK.
- by apply/continuousN/fC.
by move/oppr_inj; apply/fI.
Qed.

Lemma itv_continuous_inj_mono f (I : interval R) :
{within [set` I], continuous f} -> {in I &, injective f} -> monotonous I f.

Proof.

move=> fC fI.
case: (pselect (exists a b, [/\ a \in I , b \in I & a < b])); last first.
 move=> N2I; left => x y xI yI; suff -> : x = y by rewrite ?lexx.
 by apply: contra_notP N2I => /eqP; case: ltgtP; [exists x, y|exists y, x|].
move=> [a [b [aI bI lt_ab]]].
have /orP[faLfb|fbLfa] := le_total (f a) (f b).
 by left; apply: itv_continuous_inj_le => //; exists a, b; rewrite ?faLfb.
by right; apply: itv_continuous_inj_ge => //; exists a, b; rewrite ?fbLfa.
Qed.

Lemma segment_continuous_inj_le f a b :
f a <= f b -> {within `[a, b], continuous f} -> {in `[a, b] &, injective f} ->
{in `[a, b] &, {mono f : x y / x <= y}}.

Proof.

move=> fafb fct finj; have [//|] := itv_continuous_inj_mono fct finj.
have [aLb|bLa|<-] := ltrgtP a b; first 1 last.
- by move=> _ x ?; rewrite itv_ge// -ltNge.
- by move=> _ x y /itvxxP-> /itvxxP->; rewrite !lexx.
move=> /(_ a b); rewrite !bound_itvE fafb.
by move=> /(_ (ltW aLb) (ltW aLb)); rewrite lt_geF.
Qed.

Lemma segment_continuous_inj_ge f a b :
f a >= f b -> {within `[a, b], continuous f} -> {in `[a, b] &, injective f} ->
{in `[a, b] &, {mono f : x y /~ x <= y}}.

Proof.

move=> fafb fct finj; have [|//] := itv_continuous_inj_mono fct finj.
have [aLb|bLa|<-] := ltrgtP a b; first 1 last.
- by move=> _ x ?; rewrite itv_ge// -ltNge.
- by move=> _ x y /itvxxP-> /itvxxP->; rewrite !lexx.
move=> /(_ b a); rewrite !bound_itvE fafb.
by move=> /(_ (ltW aLb) (ltW aLb)); rewrite lt_geF.
Qed.

The condition "f a <= f b" is unnecessary because the last interval condition is vacuously true otherwise.

Lemma segment_can_le a b f g : a <= b ->
 {within `[a, b], continuous f} ->
 {in `[a, b], cancel f g} ->
 {in `[f a, f b] &, {mono g : x y / x <= y}}.

Proof.

move=> aLb ctf fK; have [fbLfa | faLfb] := ltrP (f b) (f a).
by move=> x y; rewrite itv_ge// -ltNge.
have [aab bab] : a \in `[a, b] /\ b \in `[a, b] by rewrite !bound_itvE.
case: ltgtP faLfb => // [faLfb _|-> _ _ _ /itvxxP-> /itvxxP->]; rewrite ?lexx//.
have lt_ab : a // ->; rewrite ltxx.
have w : exists x y, [/\ x \in `[a, b], y \in `[a, b], x < y & f x <= f y].
by exists a, b; rewrite !bound_itvE (ltW faLfb).
have fle := itv_continuous_inj_le w ctf (can_in_inj fK).
move=> x y xin yin; have := IVT aLb ctf.
case: (ltrgtP (f a) (f b)) faLfb => // _ _ ivt.
by have [[u uin <-] [v vin <-]] := (ivt _ xin, ivt _ yin); rewrite !fK// !fle.
Qed.

Section negation_itv.
Local Definition itvN_oppr a b := @GRing.opp R.
Local Lemma itv_oppr_is_fun a b :
isFun _ _ `[- b, - a]%classic `[a, b]%classic (itvN_oppr a b).

Proof.

by split=> x /=; rewrite oppr_itvcc. Qed.

HB.instance Definition _ a b := itv_oppr_is_fun a b.
End negation_itv.

The condition "f b <= f a" is unnecessary---see seg...increasing above

Lemma segment_can_ge a b f g : a <= b ->
 {within `[a, b], continuous f} ->
 {in `[a, b], cancel f g} ->
 {in `[f b, f a] &, {mono g : x y /~ x <= y}}.

Proof.

move=> aLb fC fK x y xfbfa yfbfa; rewrite -lerN2.
apply: (@segment_can_le (- b) (- a) (f \o -%R) (- g));
    rewrite /= ?lerN2 ?opprK //.
  pose fun_neg : subspace `[-b,-a] -> subspace `[a,b] := itvN_oppr a b.
  move=> z; apply: (@continuous_comp _ _ _ [fun of fun_neg]); last exact: fC.
  exact/subspaceT_continuous/continuous_subspaceT/opp_continuous.
by move=> z zab; rewrite -[- g]/(@GRing.opp _ \o g)/= fK ?opprK// oppr_itvcc.
Qed.

Lemma segment_can_mono a b f g : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b], cancel f g} ->
monotonous (f @`[a, b]) g.

Proof.

move=> le_ab fct fK; rewrite /monotonous/=; case: ltrgtP => fab; [left|right..];
do ?by [apply: segment_can_le|apply: segment_can_ge].
by move=> x y /itvxxP<- /itvxxP<-; rewrite !lexx.
Qed.

Lemma segment_continuous_surjective a b f : a <= b ->
{within `[a, b], continuous f} -> set_surj `[a, b] (f @`[a, b]) f.

Proof.

by move=> ? fct y/= /IVT[]// x; exists x. Qed.

Lemma segment_continuous_le_surjective a b f : a <= b -> f a <= f b ->
{within `[a, b], continuous f} -> set_surj `[a, b] `[f a, f b] f.

Proof.

move=> le_ab f_ab /(segment_continuous_surjective le_ab).
by rewrite (min_idPl _)// (max_idPr _).
Qed.

Lemma segment_continuous_ge_surjective a b f : a <= b -> f b <= f a ->
{within `[a, b], continuous f} -> set_surj `[a, b] `[f b, f a] f.

Proof.

move=> le_ab f_ab /(segment_continuous_surjective le_ab).
by rewrite (min_idPr _)// (max_idPl _).
Qed.

Lemma continuous_inj_image_segment a b f : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b] &, injective f} ->
f @` `[a, b] = f @`[a, b]%classic.

Proof.

move=> leab fct finj; apply: mono_surj_image_segment => //.
exact: itv_continuous_inj_mono.
exact: segment_continuous_surjective.
Qed.

Lemma continuous_inj_image_segmentP a b f : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b] &, injective f} ->
forall y, reflect (exists2 x, x \in `[a, b] & f x = y) (y \in f @`[a, b]).

Proof.

move=> /continuous_inj_image_segment/[apply]/[apply]/predeqP + y => /(_ y) faby.
by apply/(equivP idP); symmetry.
Qed.

Lemma segment_continuous_can_sym a b f g : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b], cancel f g} ->
{in f @`[a, b], cancel g f}.

Proof.

move=> aLb ctf fK; have g_mono := segment_can_mono aLb ctf fK.
have f_mono := itv_continuous_inj_mono ctf (can_in_inj fK).
have f_surj := segment_continuous_surjective aLb ctf.
have fIP := mono_surj_image_segmentP aLb f_mono f_surj.
suff: {in f @`[a, b], {on `[a, b], cancel g & f}}.
by move=> gK _ /fIP[x xab <-]; rewrite fK.
have: {in f @`[a, b] &, {on `[a, b] &, injective g}}.
by move=> _ _ /fIP [x xab <-] /fIP[y yab <-]; rewrite !fK// => _ _ ->.
by apply/ssrbool.inj_can_sym_in_on => x xab; rewrite ?fK ?mono_mem_image_segment.
Qed.

Lemma segment_continuous_le_can_sym a b f g : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b], cancel f g} ->
{in `[f a, f b], cancel g f}.

Proof.

move=> aLb fct fK x xfafb; apply: (segment_continuous_can_sym aLb fct fK).
have : f a <= f b by rewrite (itvP xfafb).
by case: ltrgtP xfafb => // ->.
Qed.

Lemma segment_continuous_ge_can_sym a b f g : a <= b ->
{within `[a, b], continuous f} -> {in `[a, b], cancel f g} ->
{in `[f b, f a], cancel g f}.

Proof.

move=> aLb fct fK x xfafb; apply: (segment_continuous_can_sym aLb fct fK).
have : f a >= f b by rewrite (itvP xfafb).
by case: ltrgtP xfafb => // ->.
Qed.

Lemma segment_inc_surj_continuous a b f :
{in `[a, b] &, {mono f : x y / x <= y}} -> set_surj `[a, b] `[f a, f b] f ->
{within `[a, b], continuous f}.

Proof.

move=> fle f_surj; have [f_inj flt] := (inc_inj_in fle, leW_mono_in fle).
have [aLb|bLa|] := ltgtP a b; first last.
- by move=> ->; rewrite set_itv1; exact: continuous_subspace1.
- rewrite continuous_subspace_in => z /set_mem /=; rewrite in_itv /=.
 by move=> /andP[/le_trans] /[apply]; rewrite leNgt bLa.
have le_ab : a <= b by rewrite ltW.
have [aab bab] : a \in `[a, b] /\ b \in `[a, b] by rewrite !bound_itvE ltW.
have fab : f @` `[a, b] = `[f a, f b]%classic by exact:inc_surj_image_segment.
pose g := pinv `[a, b] f; have fK : {in `[a, b], cancel f g}.
 by rewrite -[mem _]mem_setE; apply: pinvKV; rewrite !mem_setE.
have gK : {in `[f a, f b], cancel g f} by move=> z zab; rewrite pinvK// fab inE.
have gle : {in `[f a, f b] &, {mono g : x y / x <= y}}.
 apply: can_mono_in (fle); first by move=> *; rewrite gK.
 move=> z zfab; have {zfab} : `[f a, f b]%classic z by [].
 by rewrite -fab => -[x xab <-]; rewrite fK.
have glt := leW_mono_in gle.
rewrite continuous_subspace_in => x xab.
have xabcc : x \in `[a, b] by move: xab; rewrite mem_setE.
have fxab : f x \in `[f a, f b] by rewrite in_itv/= !fle.
have := xabcc; rewrite in_itv //= => /andP [ax xb].
apply/cvgrPdist_lt => _ /posnumP[e]; rewrite !near_simpl; near=> y.
rewrite (@le_lt_trans _ _ (e%:num / 2%:R))//; last first.
 by rewrite ltr_pdivrMr// ltr_pMr// ltr1n.
rewrite ler_distlC; near: y.
pose u := minr (f x + e%:num / 2) (f b).
pose l := maxr (f x - e%:num / 2) (f a).
have ufab : u \in `[f a, f b].
 rewrite !in_itv /= ge_min ?le_min lexx ?fle // le_ab orbT ?andbT.
 by rewrite ler_wpDr // fle.
have lfab : l \in `[f a, f b].
 rewrite !in_itv/= ge_max ?le_max lexx ?fle// le_ab orbT ?andbT.
 by rewrite lerBlDr ler_wpDr// fle // lexx.
have guab : g u \in `[a, b].
 rewrite !in_itv; apply/andP; split; have := ufab; rewrite in_itv => /andP.
 by case; rewrite /= -[f _ <= _]gle // ?fK // bound_itvE fle.
 by case => _; rewrite /= -[_ <= f _]gle // ?fK // bound_itvE fle.
have glab : g l \in `[a, b].
 rewrite !in_itv; apply/andP; split; have := lfab; rewrite in_itv /= => /andP.
 by case; rewrite -[f _ <= _]gle // ?fK // bound_itvE fle.
 by case => _; rewrite -[_ <= f _]gle // ?fK // bound_itvE fle.
have faltu : f a < u.
 rewrite /u comparable_lt_min ?real_comparable ?num_real// flt// aLb andbT.
 by rewrite (@le_lt_trans _ _ (f x)) ?fle// ltrDl.
have lltfb : l < f b.
 rewrite /u comparable_gt_max ?real_comparable ?num_real// flt// aLb andbT.
 by rewrite (@lt_le_trans _ _ (f x)) ?fle// ltrBlDr ltrDl.
case: pselect => // _; rewrite near_withinE; near_simpl.
have Fnbhs : Filter (nbhs x) by apply: nbhs_filter.
have := ax; rewrite le_eqVlt => /orP[/eqP|] {}ax.
 near=> y => /[dup] yab; rewrite /= in_itv => /andP[ay yb]; apply/andP; split.
 by rewrite (@le_trans _ _ (f a)) ?fle// lerBlDr ax ler_wpDr.
 apply: ltW; suff : f y /andP[->].
 rewrite -?[f y < _]glt// ?fK//; last by rewrite in_itv /= !fle.
 by near: y; near_simpl; apply: open_lt; rewrite /= -flt ?gK// -ax.
have := xb; rewrite le_eqVlt => /orP[/eqP {}xb {ax}|{}xb].
 near=> y => /[dup] yab; rewrite /= in_itv /= => /andP[ay yb].
 apply/andP; split; last by rewrite (@le_trans _ _ (f b)) ?fle// xb ler_wpDr.
 apply: ltW; suff : l < f y by rewrite gt_max => /andP[->].
 rewrite -?[_ < f y]glt// ?fK//; last by rewrite in_itv /= !fle.
 by near: y; near_simpl; apply: open_gt; rewrite /= -flt// gK// xb.
have xoab : x \in `]a, b[ by rewrite in_itv /=; apply/andP; split.
near=> y; suff: l <= f y <= u.
 by rewrite ge_max le_min -!andbA => /and4P[-> _ ->].
have ? : y \in `[a, b] by apply: subset_itv_oo_cc; near: y; apply: near_in_itv.
have fyab : f y \in `[f a, f b] by rewrite in_itv/= !fle// ?ltW.
rewrite -[l <= _]gle -?[_ <= u]gle// ?fK //.
apply: subset_itv_oo_cc; near: y; apply: near_in_itv; rewrite in_itv /=.
rewrite -[x]fK // !glt//= lt_min gt_max ?andbT ltrBlDr ltr_pwDr //.
by apply/and3P; split; rewrite // flt.
Unshelve. all: by end_near. Qed.

Lemma segment_dec_surj_continuous a b f :
 {in `[a, b] &, {mono f : x y /~ x <= y}} ->
 set_surj `[a, b] `[f b, f a] f ->
 {within `[a, b], continuous f}.

Proof.

move=> fge f_surj; suff: {within `[a, b], continuous (- f)}.
 move=> contNf x xab; rewrite -[f]opprK.
 exact/continuous_comp/opp_continuous/contNf.
apply: segment_inc_surj_continuous.
 by move=> x y xab yab; rewrite lerN2 fge.
by move=> y /=; rewrite -oppr_itvcc => /f_surj[x ? /(canLR opprK)<-]; exists x.
Qed.

Lemma segment_mono_surj_continuous a b f :
monotonous `[a, b] f -> set_surj `[a, b] (f @`[a, b]) f ->
{within `[a, b], continuous f}.

Proof.

rewrite continuous_subspace_in => -[fle|fge] f_surj x /set_mem /= xab.
 have leab : a <= b by rewrite (itvP xab).
 have fafb : f a <= f b by rewrite fle // ?bound_itvE.
 by apply: segment_inc_surj_continuous => //; case: ltrP f_surj fafb.
have leab : a <= b by rewrite (itvP xab).
have fafb : f b <= f a by rewrite fge // ?bound_itvE.
by apply: segment_dec_surj_continuous => //; case: ltrP f_surj fafb.
Qed.

Lemma segment_can_le_continuous a b f g : a <= b ->
 {within `[a, b], continuous f} ->
 {in `[a, b], cancel f g} ->
 {within `[(f a), (f b)], continuous g}.

Proof.

move=> aLb ctf; rewrite continuous_subspace_in => fK x /set_mem /= xab.
have faLfb : f a <= f b by rewrite (itvP xab).
apply: segment_inc_surj_continuous; first exact: segment_can_le.
rewrite !fK ?bound_itvE//=; apply: (@can_surj _ _ f); first by rewrite mem_setE.
exact/image_subP/mem_inc_segment/segment_continuous_inj_le/(can_in_inj fK).
Qed.

Lemma segment_can_ge_continuous a b f g : a <= b ->
 {within `[a, b], continuous f} ->
 {in `[a, b], cancel f g} ->
 {within `[(f b), (f a)], continuous g}.

Proof.

move=> aLb ctf; rewrite continuous_subspace_in => fK x /set_mem /= xab.
have fbLfa : f b <= f a by rewrite (itvP xab).
apply: segment_dec_surj_continuous; first exact: segment_can_ge.
rewrite !fK ?bound_itvE//=; apply: (@can_surj _ _ f); first by rewrite mem_setE.
exact/image_subP/mem_dec_segment/segment_continuous_inj_ge/(can_in_inj fK).
Qed.

Lemma segment_can_continuous a b f g : a <= b ->
 {within `[a, b], continuous f} ->
 {in `[a, b], cancel f g} ->
 {within f @`[a, b], continuous g}.

Proof.

move=> aLb crf fK x; case: lerP => // _;
by [apply: segment_can_ge_continuous|apply: segment_can_le_continuous].
Qed.

Lemma near_can_continuousAcan_sym f g (x : R) :
{near x, cancel f g} -> {near x, continuous f} ->
{near f x, continuous g} /\ {near f x, cancel g f}.

Proof.

move=> fK fct; near (0 : R)^'+ => e; have e_gt0 : 0 < e by [].
have xBeLxDe : x - e <= x + e by rewrite lerD2l gt0_cp.
have fcte : {in `[x - e, x + e], continuous f}.
 by near: e; apply/at_right_in_segment.
have fwcte : {within `[x - e, x + e], continuous f}.
 apply: continuous_in_subspaceT => y yI.
 by apply: fcte; move/set_mem: yI.
have fKe : {in `[x - e, x + e], cancel f g}
 by near: e; apply/at_right_in_segment.
have nearfx : \forall y \near f x, y \in f @`](x - e), (x + e)[.
 apply: near_in_itv; apply: mono_mem_image_itvoo; last first.
 by rewrite in_itv/= -ltr_distlC subrr normr0.
 apply: itv_continuous_inj_mono => //.
 by apply: (@can_in_inj _ _ _ _ g); near: e; apply/at_right_in_segment.
have fxI : f x \in f @`]x - e, x + e[ by exact: (nbhs_singleton nearfx).
split; near=> y; first last.
 rewrite (@segment_continuous_can_sym (x - e) (x + e))//.
 by apply: subset_itv_oo_cc; near: y.
near: y; apply: (filter_app _ _ nearfx); near_simpl; near=> y => yfe.
have : {within f @`]x - e, (x + e)[, continuous g}.
 apply: continuous_subspaceW; last exact: (segment_can_continuous _ fwcte _).
 exact: subset_itv_oo_cc.
rewrite continuous_open_subspace; first by apply; exact: mem_set.
exact: interval_open.
Unshelve. all: by end_near. Qed.

Lemma near_can_continuous f g (x : R) :
{near x, cancel f g} -> {near x, continuous f} -> {near f x, continuous g}.

Proof.

by move=> fK fct; have [] := near_can_continuousAcan_sym fK fct. Qed.

Lemma near_continuous_can_sym f g (x : R) :
{near x, continuous f} -> {near x, cancel f g} -> {near f x, cancel g f}.

Proof.

by move=> fct fK; have [] := near_can_continuousAcan_sym fK fct. Qed.

End real_inverse_functions.

Section real_inverse_function_instances.
Variable R : realType.

Lemma exprn_continuous n : continuous (@GRing.exp R ^~ n).

Proof.

move=> x; elim: n=> [|n /(continuousM cvg_id) ih]; first exact: cst_continuous.
by rewrite /continuous_at exprS; under eq_fun do rewrite exprS; exact: ih.
Qed.

Lemma sqr_continuous : continuous (@exprz R ^~ 2).

Proof.

exact: (@exprn_continuous 2%N). Qed.

Lemma sqrt_continuous : continuous (@Num.sqrt R).

Proof.

move=> x; case: (ltrgtP x 0) => [xlt0 | xgt0 | ->].
- apply: (near_cst_continuous 0).
 by near do rewrite ltr0_sqrtr//; apply: (cvgr_lt x).
 pose I b : set R := [set` `]0 ^+ 2, b ^+ 2[].
 suff main b : 0 <= b -> {in I b, continuous (@Num.sqrt R)}.
 near +oo_R => M; apply: (main M); rewrite // /I !inE/= in_itv/= expr0n xgt0.
 by rewrite -ltr_sqrt ?exprn_gt0// sqrtr_sqr gtr0_norm/=.
 move=> b0; rewrite -continuous_open_subspace; last exact: interval_open.
 apply: continuous_subspaceW; first exact: subset_itv_oo_cc.
 apply: (@segment_can_le_continuous _ _ _ (@GRing.exp _^~ _)) => //.
 by apply: continuous_subspaceT; exact: exprn_continuous.
 by move=> y y0b; rewrite sqrtr_sqr ger0_norm// (itvP y0b).
- rewrite /continuous_at sqrtr0; apply/cvgr0Pnorm_lt => _ /posnumP[e]; near=> y.
 have [ylt0|yge0] := ltrP y 0; first by rewrite ltr0_sqrtr ?normr0.
 rewrite ger0_norm ?sqrtr_ge0//; have: `|y| < e%:num ^+ 2 by [].
 by rewrite -ltr_sqrt// ger0_norm// sqrtr_sqr ger0_norm.
Unshelve. all: by end_near. Qed.

End real_inverse_function_instances.

Section is_derive_inverse.
Variable R : realType.

Lemma is_derive1_caratheodory (f : R -> R) (x a : R) :
 is_derive x 1 f a <->
 exists g, [/\ forall z, f z - f x = g z * (z - x),
 {for x, continuous g} & g x = a].

Proof.

split => [Hd|[g [fxE Cg gxE]]].
 exists (fun z => if z == x then a else (f(z) - f(x)) / (z - x)); split.
 - move=> z; case: eqP => [->|/eqP]; first by rewrite !subrr mulr0.
 by rewrite -subr_eq0 => /divfK->.
 - apply/continuous_withinNshiftx; rewrite eqxx /=.
 pose g1 h := (h^-1 *: ((f \o shift x) h%:A - f x)).
 have F1 : g1 @ 0^' --> a by case: Hd => H1 <-.
 apply: cvg_trans F1; apply: near_eq_cvg; rewrite /g1 !fctE.
 near=> i.
 rewrite ifN; first by rewrite addrK mulrC /= [_%:A]mulr1.
 rewrite -subr_eq0 addrK.
 by near: i; rewrite near_withinE /= near_simpl; near=> x1.
 by rewrite eqxx.
suff Hf : h^-1 *: ((f \o shift x) h%:A - f x) @[h --> 0^'] --> a.
 have F1 : 'D_1 f x = a by apply: cvg_lim.
 rewrite -F1 in Hf.
 by constructor.
 have F1 : (g \o shift x) y @[y --> 0^'] --> a.
 by rewrite -gxE; apply/continuous_withinNshiftx.
apply: cvg_trans F1; apply: near_eq_cvg.
near=> y.
rewrite /= fxE /= addrK [_%:A]mulr1.
suff yNZ : y != 0 by rewrite [RHS]mulrC mulfK.
by near: y; rewrite near_withinE /= near_simpl; near=> x1.
Unshelve. all: by end_near. Qed.

Lemma is_derive_0_is_cst (f : R -> R) x y :
(forall x, is_derive x (1 : R) f 0) -> f x = f y.

Proof.

move=> Hd.
wlog xLy : x y / x <= y by move=> H; case: (leP x y) => [/H |/ltW /H].
rewrite -(subKr (f y) (f x)).
have [| _ _] := MVT_segment xLy; last by rewrite mul0r => ->; rewrite subr0.
apply/continuous_subspaceT=> r.
exact/differentiable_continuous/derivable1_diffP.
Qed.

Global Instance is_derive1_comp (f g : R -> R) (x a b : R) :
is_derive (g x) 1 f a -> is_derive x 1 g b ->
is_derive x 1 (f \o g) (a * b).

Proof.

move=> [fgxv <-{a}] [gv <-{b}]; apply: (@DeriveDef _ _ _ _ _ (f \o g)).
apply/derivable1_diffP/differentiable_comp; first exact/derivable1_diffP.
by move/derivable1_diffP in fgxv.
by rewrite -derive1E (derive1_comp gv fgxv) 2!derive1E.
Qed.

Lemma is_deriveV (f : R -> R) (x t v : R) :
f x != 0 -> is_derive x v f t ->
is_derive x v (fun y => (f y)^-1) (- (f x) ^- 2 *: t).

Proof.

move=> fxNZ Df.
constructor; first by apply: derivableV => //; case: Df.
by rewrite deriveV //; case: Df => _ ->.
Qed.

Lemma is_derive_inverse (f g : R -> R) l x :
  {near x, cancel f g} ->
  {near x, continuous f} ->
  is_derive x 1 f l -> l != 0 -> is_derive (f x) 1 g l^-1.

Proof.

move=> fgK fC fD lNZ.
have /is_derive1_caratheodory [h [fE hC hxE]] := fD.
(* There should be something simpler *)
have gfxE : g (f x) = x by have [d Hd]:= nbhs_ex fgK; apply: Hd.
pose g1 y := if y == f x then (h (g y))^-1
 else (g y - g (f x)) / (y - f x).
apply/is_derive1_caratheodory.
exists g1; split; first 2 last.
- by rewrite /g1 eqxx gfxE hxE.
- move=> z; rewrite /g1; case: eqP => [->|/eqP]; first by rewrite !subrr mulr0.
 by rewrite -subr_eq0 => /divfK.
have F1 : (h (g x))^-1 @[x --> f x] --> g1 (f x).
 rewrite /g1 eqxx; apply: continuousV; first by rewrite /= gfxE hxE.
 apply: continuous_comp; last by rewrite gfxE.
 by apply: nbhs_singleton (near_can_continuous _ _).
apply: cvg_sub0 F1.
apply/cvgrPdist_lt => eps eps_gt0 /=; rewrite !near_simpl /=.
near=> y; rewrite sub0r normrN !fctE.
have fgyE : f (g y) = y by near: y; apply: near_continuous_can_sym.
rewrite /g1; case: eqP => [_|/eqP x1Dfx]; first by rewrite subrr normr0.
have -> : y - f x = h (g y) * (g y - x) by rewrite -fE fgyE.
rewrite gfxE invfM mulrC divfK ?subrr ?normr0 // subr_eq0.
by apply: contra x1Dfx => /eqP<-; apply/eqP.
Unshelve. all: by end_near. Qed.

End is_derive_inverse.

#[global] Hint Extern 0 (is_derive _ _ (fun _ => (_ _)^-1) _) =>
(eapply is_deriveV; first by []) : typeclass_instances.

Section interval_partition.
Context {R : realType}.
Implicit Type (a b : R) (s : seq R).

a :: s is a partition of the interval a, b

Definition itv_partition a b s := [/\ path <%R a s & last a s == b].

Lemma itv_partition_nil a b : itv_partition a b [::] -> a = b.

Proof.

by move=> [_ /eqP <-]. Qed.

Lemma itv_partition_cons a b x s :
itv_partition a b (x :: s) -> itv_partition x b s.

Proof.

by rewrite /itv_partition/= => -[/andP[]]. Qed.

Lemma itv_partition1 a b : a itv_partition a b [:: b].

Proof.

by rewrite /itv_partition /= => ->. Qed.

Lemma itv_partition_size_neq0 a b s :
(size s > 0)%N -> itv_partition a b s -> a < b.

Proof.

elim: s a => // x [_ a _|h t ih a _]; rewrite /itv_partition /=.
by rewrite andbT => -[ax /eqP <-].
move=> [] /andP[ax /andP[xy] ht /eqP tb].
by rewrite (lt_trans ax)// ih// /itv_partition /= xy/= tb.
Qed.

Lemma itv_partitionxx a s : itv_partition a a s -> s = [::].

Proof.

case: s => //= h t [/= /andP[ah /lt_path_min/allP ht] /eqP hta].
suff : h < a by move/lt_trans => /(_ _ ah); rewrite ltxx.
apply/ht; rewrite -hta.
by have := mem_last h t; rewrite inE hta lt_eqF.
Qed.

Lemma itv_partition_le a b s : itv_partition a b s -> a <= b.

Proof.

case: s => [/itv_partition_nil ->//|h t /itv_partition_size_neq0 - /(_ _)/ltW].
exact.
Qed.

Lemma itv_partition_cat a b c s t :
itv_partition a b s -> itv_partition b c t -> itv_partition a c (s ++ t).

Proof.

rewrite /itv_partition => -[sa /eqP asb] [bt btc].
by rewrite cat_path// sa /= last_cat asb.
Qed.

Lemma itv_partition_nth_size def a b s : itv_partition a b s ->
nth def (a :: s) (size s) = b.

Proof.

by elim: s a => [a/= /itv_partition_nil//|y t ih a /= /itv_partition_cons/ih].
Qed.

Lemma itv_partition_nth_ge a b s m : (m < (size s).+1)%N ->
itv_partition a b s -> a <= nth b (a :: s) m.

Proof.

elim: m s a b => [s a b _//|n ih [//|h t] a b].
rewrite ltnS => nh [/= /andP[ah ht] lb].
by rewrite (le_trans (ltW ah))// ih.
Qed.

Lemma itv_partition_nth_le a b s m : (m < (size s).+1)%N ->
itv_partition a b s -> nth b (a :: s) m <= b.

Proof.

elim: m s a => [s a _|n ih]; first exact: itv_partition_le.
by move=> [//|a h t /= nt] H; rewrite ih//; exact: itv_partition_cons H.
Qed.

Lemma nondecreasing_fun_itv_partition a b f s :
 {in `[a, b] &, nondecreasing_fun f} -> itv_partition a b s ->
 let F : nat -> R := f \o nth b (a :: s) in
 forall k, (k < size s)%N -> F k <= F k.+1.

Proof.

move=> ndf abs F k ks.
have [_] := nondecreasing_seqP F; apply => m n mn; rewrite /F/=.
have [ms|ms] := ltnP m (size s).+1; last first.
  rewrite nth_default//.
  have [|ns] := ltnP n (size s).+1; last by rewrite nth_default.
  by move=> /(leq_ltn_trans mn); rewrite ltnS leqNgt ms.
have [ns|ns] := ltnP n (size s).+1; last first.
  rewrite [in leRHS]nth_default//=; apply/ndf/itv_partition_nth_le => //.
    by rewrite in_itv/= itv_partition_nth_le// andbT itv_partition_nth_ge.
  by rewrite in_itv/= lexx andbT; exact: (itv_partition_le abs).
move: abs; rewrite /itv_partition => -[] sa sab.
move: mn; rewrite leq_eqVlt => /predU1P[->//|mn].
apply/ndf/ltW/sorted_ltn_nth => //=; last exact: lt_trans.
  by rewrite in_itv/= itv_partition_nth_le// andbT itv_partition_nth_ge.
by rewrite in_itv/= itv_partition_nth_le// andbT itv_partition_nth_ge.
Qed.

Lemma nonincreasing_fun_itv_partition a b f s :
 {in `[a, b] &, nonincreasing_fun f} -> itv_partition a b s ->
 let F : nat -> R := f \o nth b (a :: s) in
 forall k, (k < size s)%N -> F k.+1 <= F k.

Proof.

move/nonincreasing_funN => ndNf abs F k ks; rewrite -(opprK (F k)) lerNr.
exact: (nondecreasing_fun_itv_partition ndNf abs).
Qed.

given a partition of a, b and c, returns a partition of a, c

Definition itv_partitionL s c := rcons [seq x <- s | x < c] c.

Lemma itv_partitionLP a b c s : a < c -> c itv_partition a b s ->
itv_partition a c (itv_partitionL s c).

Proof.

move=> ac bc [] al /eqP htb; split.
 rewrite /itv_partitionL rcons_path/=; apply/andP; split.
 by apply: path_filter => //; exact: lt_trans.
 exact: (last_filterP [pred x | x < c]).
by rewrite /itv_partitionL last_rcons.
Qed.

given a partition of a, b and c, returns a partition of c, b

Definition itv_partitionR s c := [seq x <- s | c < x].

Lemma itv_partitionRP a b c s : a < c -> c itv_partition a b s ->
itv_partition c b (itv_partitionR s c).

Proof.

move=> ac cb [] sa /eqP alb; rewrite /itv_partition; split.
  move: sa; rewrite lt_path_sortedE => /andP[allas ss].
  rewrite lt_path_sortedE filter_all/=.
  by apply: sorted_filter => //; exact: lt_trans.
exact/eqP/(path_lt_last_filter ac).
Qed.

Lemma in_itv_partition c s : sorted <%R s -> c \in s ->
s = itv_partitionL s c ++ itv_partitionR s c.

Proof.

elim: s c => // h t ih c /= ht.
rewrite inE => /predU1P[->{c}/=|ct].
  rewrite ltxx /itv_partitionL /= ltxx /itv_partitionR/= path_lt_filter0//=.
  by rewrite path_lt_filterT.
rewrite /itv_partitionL/=; case: ifPn => [hc|].
  by rewrite ltNge (ltW hc)/= /= [in LHS](ih _ _ ct)//; exact: path_sorted ht.
rewrite -leNgt le_eqVlt => /predU1P[ch|ch].
  by rewrite ch ltxx path_lt_filter0//= /itv_partitionR path_lt_filterT.
move: ht; rewrite lt_path_sortedE => /andP[/allP/(_ _ ct)].
by move=> /lt_trans-/(_ _ ch); rewrite ltxx.
Qed.

Lemma notin_itv_partition c s : sorted <%R s -> c \notin s ->
s = [seq x <- s | x < c] ++ itv_partitionR s c.

Proof.

elim: s c => // h t ih c /= ht.
rewrite inE negb_or => /andP[]; rewrite neq_lt => /orP[ch|ch] ct.
  rewrite ch ltNge (ltW ch)/= path_lt_filter0/= /itv_partitionR; last first.
    exact: path_lt_head ht.
  by rewrite path_lt_filterT//; exact: path_lt_head ht.
by rewrite ch/= ltNge (ltW ch)/= -ih//; exact: path_sorted ht.
Qed.

Lemma itv_partition_rev a b s : itv_partition a b s ->
itv_partition (- b) (- a) (rev (belast (- a) (map -%R s))).

Proof.

move=> [sa /eqP alb]; split.
 rewrite (_ : - b = last (- a) (map -%R s)); last by rewrite last_map alb.
 rewrite rev_path// path_map.
 by apply: sub_path sa => x y xy/=; rewrite ltrNr opprK.
case: s sa alb => [_ <-//|h t] /= /andP[ah ht] <-{b}.
by rewrite rev_cons last_rcons.
Qed.

End interval_partition.

Section variation.
Context {R : realType}.
Implicit Types (a b : R) (f g : R -> R).

Definition variation a b f s := let F := f \o nth b (a :: s) in
\sum_(0 <= n < size s) `|F n.+1 - F n|%R.

Lemma variation_zip a b f s : itv_partition a b s ->
variation a b f s = \sum_(x <- zip s (a :: s)) `|f x.1 - f x.2|.

Proof.

elim: s a b => // [a b|h t ih a b].
by rewrite /itv_partition /= => -[_ /eqP <-]; rewrite /variation/= !big_nil.
rewrite /itv_partition /variation => -[]/= /andP[ah ht] /eqP htb.
rewrite big_nat_recl//= big_cons/=; congr +%R.
have /ih : itv_partition h b t by split => //; exact/eqP.
by rewrite /variation => ->; rewrite !big_seq; apply/eq_bigr => r rt.
Qed.

Lemma variation_prev a b f s : itv_partition a b s ->
variation a b f s = \sum_(x <- s) `|f x - f (prev (locked (a :: s)) x)|.

Proof.

move=> [] sa /eqP asb; rewrite /variation [in LHS]/= (big_nth b) !big_nat.
apply: eq_bigr => i /andP[_ si]; congr (`| _ - f _ |).
rewrite -lock.
rewrite prev_nth inE gt_eqF; last first.
  rewrite -[a]/(nth b (a :: s) 0) -[ltRHS]/(nth b (a :: s) i.+1).
  exact: lt_sorted_ltn_nth.
rewrite orFb mem_nth// index_uniq//.
  by apply: set_nth_default => /=; rewrite ltnS ltnW.
by apply: (sorted_uniq lt_trans) => //; apply: path_sorted sa.
Qed.

Lemma variation_next a b f s : itv_partition a b s ->
variation a b f s =
\sum_(x <- belast a s) `|f (next (locked (a :: s)) x) - f x|.

Proof.

move=> [] sa /eqP asb; rewrite /variation [in LHS]/= (big_nth b) !big_nat.
rewrite size_belast; apply: eq_bigr => i /andP[_ si].
congr (`| f _ - f _ |); last first.
by rewrite lastI -cats1 nth_cat size_belast// si.
rewrite -lock next_nth.
rewrite {1}lastI mem_rcons inE mem_nth ?size_belast// orbT.
rewrite lastI -cats1 index_cat mem_nth ?size_belast//.
rewrite index_uniq ?size_belast//.
exact: set_nth_default.
have /lt_sorted_uniq : sorted <%R (a :: s) by [].
by rewrite lastI rcons_uniq => /andP[].
Qed.

Lemma variation_nil a b f : variation a b f [::] = 0.

Proof.

by rewrite /variation/= big_nil. Qed.

Lemma variation_ge0 a b f s : 0 <= variation a b f s.

Proof.

exact/sumr_ge0. Qed.

Lemma variationN a b f s : variation a b (\- f) s = variation a b f s.

Proof.

by rewrite /variation; apply: eq_bigr => k _ /=; rewrite -opprD normrN.
Qed.

Lemma variation_le a b f g s :
variation a b (f \+ g)%R s <= variation a b f s + variation a b g s.

Proof.

rewrite [in leRHS]/variation -big_split/=.
apply: ler_sum => k _; apply: le_trans; last exact: ler_normD.
by rewrite /= addrACA addrA opprD addrA.
Qed.

Lemma nondecreasing_variation a b f s : {in `[a, b] &, nondecreasing_fun f} ->
itv_partition a b s -> variation a b f s = f b - f a.

Proof.

move=> ndf abs; rewrite /variation; set F : nat -> R := f \o nth _ (a :: s).
transitivity (\sum_(0 <= n < size s) (F n.+1 - F n)).
rewrite !big_nat; apply: eq_bigr => k; rewrite leq0n/= => ks.
by rewrite ger0_norm// subr_ge0; exact: nondecreasing_fun_itv_partition.
by rewrite telescope_sumr// /F/= (itv_partition_nth_size _ abs).
Qed.

Lemma nonincreasing_variation a b f s : {in `[a, b] &, nonincreasing_fun f} ->
itv_partition a b s -> variation a b f s = f a - f b.

Proof.

move=> /nonincreasing_funN ndNf abs; have := nondecreasing_variation ndNf abs.
by rewrite opprK addrC => <-; rewrite variationN.
Qed.

Lemma variationD a b c f s t : a <= c -> c <= b ->
itv_partition a c s -> itv_partition c b t ->
variation a c f s + variation c b f t = variation a b f (s ++ t).

Proof.

rewrite le_eqVlt => /predU1P[<-{c} cb|ac].
 by move=> /itv_partitionxx ->; rewrite variation_nil add0r.
rewrite le_eqVlt => /predU1P[<-{b}|cb].
 by move=> ? /itv_partitionxx ->; rewrite variation_nil addr0 cats0.
move=> acs cbt; rewrite /variation /= [in RHS]/index_iota subn0 size_cat.
rewrite iotaD add0n big_cat/= -[in X in _ = X + _](subn0 (size s)); congr +%R.
 rewrite -/(index_iota 0 (size s)) 2!big_nat.
 apply: eq_bigr => k /[!leq0n] /= ks.
 rewrite nth_cat ks -cat_cons nth_cat /= ltnS (ltnW ks).
 by rewrite !(set_nth_default b c)//= ltnS ltnW.
rewrite -[in RHS](addnK (size s) (size t)).
rewrite -/(index_iota (size s) (size t + size s)).
rewrite -{1}[in RHS](add0n (size s)) big_addn addnK 2!big_nat; apply: eq_bigr.
move=> k /[!leq0n]/= kt.
rewrite nth_cat {1}(addnC k) -ltn_subRL subnn ltn0 addnK.
case: k kt => [t0 /=|k kt].
 rewrite add0n -cat_cons nth_cat/= ltnS leqnn -last_nth.
 by case: acs => _ /eqP ->.
rewrite addSnnS (addnC k) -cat_cons nth_cat/= -ltn_subRL subnn ltn0.
by rewrite -(addnC k) addnK.
Qed.

Lemma variation_itv_partitionLR a b c f s : a < c -> c 
itv_partition a b s ->
variation a b f s <= variation a b f (itv_partitionL s c ++ itv_partitionR s c).

Proof.

move=> ac bc abs; have [cl|cl] := boolP (c \in s).
 by rewrite -in_itv_partition//; case: abs => /path_sorted.
rewrite /itv_partitionL [in leLHS](notin_itv_partition _ cl)//; last first.
 by apply: path_sorted; case: abs => + _; exact.
rewrite -notin_itv_partition//; last first.
 by apply: path_sorted; case: abs => /= + _; exact.
rewrite !variation_zip//; last first.
 by apply: itv_partition_cat;
 [exact: (itv_partitionLP _ bc)|exact: (itv_partitionRP ac)].
rewrite [in leLHS](notin_itv_partition _ cl); last first.
 by apply: path_sorted; case: abs => + _; exact.
set L := [seq x <- s | x < c].
rewrite -cats1 -catA.
move: L => L.
set B := itv_partitionR s c.
move: B => B.
elim/last_ind : L => [|L0 L1 _].
 rewrite !cat0s /=; case: B => [|B0 B1].
 by rewrite big_nil big_cons/= big_nil addr0.
 rewrite !big_cons/= addrA lerD// [leRHS]addrC.
 by rewrite (le_trans _ (ler_normD _ _))// addrA subrK.
rewrite -cats1.
rewrite (_ : a :: _ ++ B = (a :: L0) ++ [:: L1] ++ B)//; last first.
 by rewrite -!catA -cat_cons.
rewrite zip_cat; last by rewrite cats1 size_rcons.
rewrite (_ : a :: _ ++ _ ++ B = (a :: L0) ++ [:: L1] ++ [:: c] ++ B); last first.
 by rewrite -!catA -cat_cons.
rewrite zip_cat; last by rewrite cats1 size_rcons.
rewrite !big_cat lerD//.
case: B => [|B0 B1].
 by rewrite /= big_nil big_cons big_nil addr0.
rewrite -cat1s zip_cat// catA.
rewrite (_ : [:: L1] ++ _ ++ B1 = ([:: L1] ++ [:: c]) ++ [:: B0] ++ B1); last first.
 by rewrite catA.
rewrite zip_cat// !big_cat lerD//= !big_cons !big_nil !addr0/= [leRHS]addrC.
 by rewrite (le_trans _ (ler_normD _ _))// addrA subrK.
Qed.

Lemma le_variation a b f s x : variation a b f s <= variation a b f (x :: s).

Proof.

case: s => [|h t].
by rewrite variation_nil /variation/= big_nat_recl//= big_nil addr0.
rewrite /variation/= !big_nat_recl//= addrA lerD2r.
by rewrite (le_trans _ (ler_normD _ _))// (addrC (f x - _)) addrA subrK.
Qed.

Lemma variation_opp_rev a b f s : itv_partition a b s ->
variation a b f s =
variation (- b) (- a) (f \o -%R) (rev (belast (- a) (map -%R s))).

Proof.

move=> abl; rewrite belast_map /variation /= [LHS]big_nat_rev/= add0n.
rewrite size_rev size_map size_belast 2!big_nat.
apply: eq_bigr => k; rewrite leq0n /= => ks.
rewrite nth_rev ?size_map ?size_belast// [in RHS]distrC.
rewrite (nth_map a); last first.
  by rewrite size_belast ltn_subLR// addSn ltnS leq_addl.
rewrite opprK -rev_rcons nth_rev ?size_rcons ?size_map ?size_belast 1?ltnW//.
rewrite subSn// -map_rcons (nth_map b) ?size_rcons ?size_belast; last first.
  by rewrite ltnS ltn_subLR// addSn ltnS leq_addl.
rewrite opprK nth_rcons size_belast -subSn// subSS.
rewrite (ltn_subLR _ (ltnW ks)) if_same.
case: k => [|k] in ks *.
  rewrite add0n ltnn subn1 (_ : nth b s _ = b); last first.
    case: abl ks => _.
    elim/last_ind : s => // h t _; rewrite last_rcons => /eqP -> _.
    by rewrite nth_rcons size_rcons ltnn eqxx.
  rewrite (_ : nth b (a :: s) _ = nth a (belast a s) (size s).-1)//.
  case: abl ks => _.
  elim/last_ind : s => // h t _; rewrite last_rcons => /eqP -> _.
  rewrite belast_rcons size_rcons/= -rcons_cons nth_rcons/= ltnS leqnn.
  exact: set_nth_default.
rewrite addSn ltnS leq_addl//; congr (`| f _ - f _ |).
  elim/last_ind : s ks {abl} => // h t _; rewrite size_rcons ltnS => kh.
  rewrite belast_rcons nth_rcons subSS ltn_subLR//.
  by rewrite addSn ltnS leq_addl// subSn.
elim/last_ind : s ks {abl} => // h t _; rewrite size_rcons ltnS => kh.
rewrite belast_rcons subSS -rcons_cons nth_rcons /= ltn_subLR//.
rewrite addnS ltnS leq_addl; apply: set_nth_default => //.
by rewrite /= ltnS leq_subLR leq_addl.
Qed.

Lemma variation_rev_opp a b f s : itv_partition (- b) (- a) s ->
variation a b f (rev (belast b (map -%R s))) =
variation (- b) (- a) (f \o -%R) s.

Proof.

move=> abs; rewrite [in RHS]variation_opp_rev ?opprK//.
suff: (f \o -%R) \o -%R = f by move=> ->.
by apply/funext=> ? /=; rewrite opprK.
Qed.

Lemma variation_subseq a b f (s t : list R) :
 itv_partition a b s -> itv_partition a b t ->
 subseq s t ->
 variation a b f s <= variation a b f t.

Proof.

elim: t s a => [? ? ? /= _ /eqP ->//|a s IH [|x t] w].
 by rewrite variation_nil // variation_ge0.
move=> /[dup] /itv_partition_cons itvxb /[dup] /itv_partition_le wb itvxt.
move=> /[dup] /itv_partition_cons itvas itvws /=.
have ab : a <= b by exact: (itv_partition_le itvas).
have wa : w < a by case: itvws => /= /andP[].
have waW : w <= a := ltW wa.
case: ifPn => [|] nXA.
 move/eqP : nXA itvxt itvxb => -> itvat itvt /= ta.
 rewrite -[_ :: t]cat1s -[_ :: s]cat1s.
 rewrite -?(@variationD _ _ a)//; [|exact: itv_partition1..].
 by rewrite lerD// IH.
move=> xts; rewrite -[_ :: s]cat1s -(@variationD _ _ a) => //; last first.
 exact: itv_partition1.
have [y [s' s'E]] : exists y s', s = y :: s'.
 by case: {itvas itvws IH} s xts => // y s' ?; exists y, s'.
apply: (@le_trans _ _ (variation w b f s)).
 rewrite IH//.
 case: itvws => /= /andP[_]; rewrite s'E /= => /andP[ay ys' lyb].
 by split => //; rewrite (path_lt_head wa)//= ys' andbT.
by rewrite variationD //; [exact: le_variation | exact: itv_partition1].
Qed.

End variation.

Section bounded_variation.
Context {R : realType}.
Implicit Type (a b : R) (f : R -> R).

Definition variations a b f := [set variation a b f l | l in itv_partition a b].

Lemma variations_variation a b f s : itv_partition a b s ->
variations a b f (variation a b f s).

Proof.

by move=> abs; exists s. Qed.

Lemma variations_neq0 a b f : a variations a b f !=set0.

Proof.

move=> ab; exists (variation a b f [:: b]); exists [:: b] => //.
exact: itv_partition1.
Qed.

Lemma variationsN a b f : variations a b (\- f) = variations a b f.

Proof.

apply/seteqP; split => [_ [s abs] <-|r [s abs]].
by rewrite variationN; exact: variations_variation.
by rewrite -variationN => <-; exact: variations_variation.
Qed.

Lemma variationsxx a f : variations a a f = [set 0].

Proof.

apply/seteqP; split => [x [_ /itv_partitionxx ->]|x ->].
by rewrite /variation big_nil => <-.
by exists [::] => //=; rewrite /variation /= big_nil.
Qed.

Definition bounded_variation a b f := has_ubound (variations a b f).

Notation BV := bounded_variation.

Lemma bounded_variationxx a f : BV a a f.

Proof.

by exists 0 => r; rewrite variationsxx => ->. Qed.

Lemma bounded_variationD a b f g : a 
BV a b f -> BV a b g -> BV a b (f \+ g).

Proof.

move=> ab [r abfr] [s abgs]; exists (r + s) => _ [l abl] <-.
apply: le_trans; first exact: variation_le.
rewrite lerD//.
- by apply: abfr; exact: variations_variation.
- by apply: abgs; exact: variations_variation.
Qed.

Lemma bounded_variationN a b f : BV a b f -> BV a b (\- f).

Proof.

by rewrite /bounded_variation variationsN. Qed.

Lemma bounded_variationl a c b f : a <= c -> c <= b -> BV a b f -> BV a c f.

Proof.

rewrite le_eqVlt => /predU1P[<-{c} ? ?|ac]; first exact: bounded_variationxx.
rewrite le_eqVlt => /predU1P[<-{b}//|cb].
move=> [x Hx]; exists x => _ [s acs] <-.
rewrite (@le_trans _ _ (variation a b f (rcons s b)))//; last first.
apply/Hx/variations_variation; case: acs => sa /eqP asc.
by rewrite /itv_partition rcons_path last_rcons sa/= asc.
rewrite {2}/variation size_rcons -[leLHS]addr0 big_nat_recr//= lerD//.
rewrite /variation !big_nat ler_sum// => k; rewrite leq0n /= => ks.
rewrite nth_rcons// ks -cats1 -cat_cons nth_cat /= ltnS (ltnW ks).
by rewrite ![in leRHS](set_nth_default c)//= ltnS ltnW.
Qed.

Lemma bounded_variationr a c b f : a <= c -> c <= b -> BV a b f -> BV c b f.

Proof.

rewrite le_eqVlt => /predU1P[<-{c}//|ac].
rewrite le_eqVlt => /predU1P[<-{b} ?|cb]; first exact: bounded_variationxx.
move=> [x Hx]; exists x => _ [s cbs] <-.
rewrite (@le_trans _ _ (variation a b f (c :: s)))//; last first.
apply/Hx/variations_variation; case: cbs => cs csb.
by rewrite /itv_partition/= ac/= cs.
by rewrite {2}/variation/= -[leLHS]add0r big_nat_recl//= lerD.
Qed.

Lemma variations_opp a b f :
variations (- b) (- a) (f \o -%R) = variations a b f.

Proof.

rewrite eqEsubset; split=> [_ [s bas <-]| _ [s abs <-]].
eexists; last exact: variation_rev_opp.
by move/itv_partition_rev : bas; rewrite !opprK.
eexists; last by exact/esym/variation_opp_rev.
exact: itv_partition_rev abs.
Qed.

Lemma nondecreasing_bounded_variation a b f :
{in `[a, b] &, {homo f : x y / x <= y}} -> BV a b f.

Proof.

move=> incf; exists (f b - f a) => ? [l pabl <-]; rewrite le_eqVlt.
by rewrite nondecreasing_variation// eqxx.
Qed.

End bounded_variation.

Section total_variation.
Context {R : realType}.
Implicit Types (a b : R) (f : R -> R).

Definition total_variation a b f :=
ereal_sup [set x%:E | x in variations a b f].

Notation BV := bounded_variation.
Notation TV := total_variation.

Lemma total_variationxx a f : TV a a f = 0%E.

Proof.

by rewrite /total_variation variationsxx image_set1 ereal_sup1. Qed.

Lemma total_variation_ge a b f : a <= b -> (`|f b - f a|%:E <= TV a b f)%E.

Proof.

rewrite le_eqVlt => /predU1P[<-{b}|ab].
by rewrite total_variationxx subrr normr0.
apply: ereal_sup_ubound => /=; exists (variation a b f [:: b]).
exact/variations_variation/itv_partition1.
by rewrite /variation/= big_nat_recr//= big_nil add0r.
Qed.

Lemma total_variation_ge0 a b f : a <= b -> (0 <= TV a b f)%E.

Proof.

by move=> ab; rewrite (le_trans _ (total_variation_ge _ ab)). Qed.

Lemma bounded_variationP a b f : a <= b -> BV a b f <-> TV a b f \is a fin_num.

Proof.

rewrite le_eqVlt => /predU1P[<-{b}|ab].
by rewrite total_variationxx; split => // ?; exact: bounded_variationxx.
rewrite ge0_fin_numE; last exact/total_variation_ge0/ltW.
split=> [abf|].
by rewrite /total_variation ereal_sup_EFin ?ltry//; exact: variations_neq0.
rewrite /total_variation /bounded_variation ltey => /eqP; apply: contra_notP.
by move/hasNub_ereal_sup; apply; exact: variations_neq0.
Qed.

Lemma nondecreasing_total_variation a b f : a <= b ->
{in `[a, b] &, nondecreasing_fun f} -> TV a b f = (f b - f a)%:E.

Proof.

rewrite le_eqVlt => /predU1P[<-{b} ?|ab ndf].
 by rewrite total_variationxx subrr.
rewrite /total_variation [X in ereal_sup X](_ : _ = [set (f b - f a)%:E]).
 by rewrite ereal_sup1.
apply/seteqP; split => [x/= [s [t abt <-{s} <-{x}]]|x/= ->{x}].
 by rewrite nondecreasing_variation.
exists (variation a b f [:: b]) => //.
 exact/variations_variation/itv_partition1.
by rewrite nondecreasing_variation//; exact: itv_partition1.
Qed.

Lemma total_variationN a b f : TV a b (\- f) = TV a b f.

Proof.

by rewrite /TV; rewrite variationsN. Qed.

Lemma total_variation_le a b f g : a <= b ->
(TV a b (f \+ g)%R <= TV a b f + TV a b g)%E.

Proof.

rewrite le_eqVlt => /predU1P[<-{b}|ab].
 by rewrite !total_variationxx adde0.
have [abf|abf] := pselect (BV a b f); last first.
 rewrite {2}/total_variation hasNub_ereal_sup//; last first.
 exact: variations_neq0.
 rewrite addye ?leey// -ltNye (@lt_le_trans _ _ 0%E)//.
 exact/total_variation_ge0/ltW.
have [abg|abg] := pselect (BV a b g); last first.
 rewrite {3}/total_variation hasNub_ereal_sup//; last first.
 exact: variations_neq0.
 rewrite addey ?leey// -ltNye (@lt_le_trans _ _ 0%E)//.
 exact/total_variation_ge0/ltW.
move: abf abg => [r abfr] [s abgs].
have BVabfg : BV a b (f \+ g).
 by apply: bounded_variationD => //; [exists r|exists s].
apply: ub_ereal_sup => y /= [r' [s' abs <-{r'} <-{y}]].
apply: (@le_trans _ _ (variation a b f s' + variation a b g s')%:E).
 exact: variation_le.
by rewrite EFinD leeD// ereal_sup_le//;
 (eexists; last exact: lexx); (eexists; last reflexivity);
 exact: variations_variation.
Qed.

Let total_variationD1 a b c f : a <= c -> c <= b ->
(TV a b f >= TV a c f + TV c b f)%E.

Proof.

rewrite le_eqVlt=> /predU1P[<-{c}|ac]; first by rewrite total_variationxx add0e.
rewrite le_eqVlt=> /predU1P[<-{b}|cb]; first by rewrite total_variationxx adde0.
have [abf|abf] := pselect (BV a b f); last first.
 rewrite {3}/total_variation hasNub_ereal_sup ?leey//.
 by apply: variations_neq0 => //; rewrite (lt_trans ac).
have H s t : itv_partition a c s -> itv_partition c b t ->
 (TV a b f >= (variation a c f s)%:E + (variation c b f t)%:E)%E.
 move=> acs cbt; rewrite -EFinD; apply: ereal_sup_le.
 exists (variation a b f (s ++ t))%:E.
 eexists; last reflexivity.
 by exists (s ++ t) => //; exact: itv_partition_cat acs cbt.
 by rewrite variationD// ltW.
rewrite [leRHS]ereal_sup_EFin//; last first.
 by apply: variations_neq0; rewrite (lt_trans ac).
have acf : BV a c f := bounded_variationl (ltW ac) (ltW cb) abf.
have cbf : BV c b f := bounded_variationr (ltW ac) (ltW cb) abf.
rewrite {1 2}/total_variation ereal_sup_EFin//; last exact: variations_neq0.
rewrite ereal_sup_EFin//; last exact: variations_neq0.
rewrite -EFinD -sup_sumE; last 2 first.
 by split => //; exact: variations_neq0.
 by split => //; exact: variations_neq0.
apply: le_sup.
- move=> r/= [s [l' acl' <-{s}]] [t [l cbl] <-{t} <-{r}].
 exists (variation a b f (l' ++ l)); split; last by rewrite variationD// ltW.
 exact/variations_variation/(itv_partition_cat acl' cbl).
- have [r acfr] := variations_neq0 f ac.
 have [s cbfs] := variations_neq0 f cb.
 by exists (r + s); exists r => //; exists s.
- by split => //; apply: variations_neq0; rewrite (lt_trans ac).
Qed.

Let total_variationD2 a b c f : a <= c -> c <= b ->
(TV a b f <= TV a c f + TV c b f)%E.

Proof.

rewrite le_eqVlt => /predU1P[<-{c}|ac]; first by rewrite total_variationxx add0e.
rewrite le_eqVlt => /predU1P[<-{b}|cb]; first by rewrite total_variationxx adde0.
case : (pselect (bounded_variation a c f)); first last.
 move=> nbdac; have /eqP -> : TV a c f == +oo%E.
 have: (-oo < TV a c f)%E by apply: (lt_le_trans _ (total_variation_ge0 f (ltW ac))).
 by rewrite ltNye_eq => /orP [] => // /bounded_variationP => /(_ (ltW ac)).
 by rewrite addye ?leey // -ltNye (@lt_le_trans _ _ 0)%E // ?total_variation_ge0 // ltW.
case : (pselect (bounded_variation c b f)); first last.
 move=> nbdac; have /eqP -> : TV c b f == +oo%E.
 have: (-oo < TV c b f)%E.
 exact: (lt_le_trans _ (total_variation_ge0 f (ltW cb))).
 by rewrite ltNye_eq => /orP [] => // /bounded_variationP => /(_ (ltW cb)).
 rewrite addey ?leey // -ltNye (@lt_le_trans _ _ 0%E)//.
 exact/total_variation_ge0/ltW.
move=> bdAB bdAC.
rewrite /total_variation [x in (x + _)%E]ereal_sup_EFin //; last first.
 exact: variations_neq0.
rewrite [x in (_ + x)%E]ereal_sup_EFin //; last exact: variations_neq0.
rewrite -EFinD -sup_sumE /has_sup; [|(by split => //; exact: variations_neq0)..].
apply: ub_ereal_sup => ? [? [l pacl <- <-]]; rewrite lee_fin.
apply: (le_trans (variation_itv_partitionLR _ ac _ _)) => //.
apply: sup_ubound => /=.
 case: bdAB => M ubdM; case: bdAC => N ubdN; exists (N + M).
 move=> q [?] [i pabi <-] [? [j pbcj <-]] <-.
 by apply: lerD; [apply: ubdN;exists i|apply:ubdM;exists j].
exists (variation a c f (itv_partitionL l c)).
 by apply: variations_variation; exact: itv_partitionLP pacl.
exists (variation c b f (itv_partitionR l c)).
 by apply: variations_variation; exact: itv_partitionRP pacl.
by rewrite variationD// ?ltW//;
 [exact: itv_partitionLP pacl|exact: itv_partitionRP pacl].
Qed.

Lemma total_variationD a b c f : a <= c -> c <= b ->
(TV a b f = TV a c f + TV c b f)%E.

Proof.

by move=> ac cb; apply/eqP; rewrite eq_le; apply/andP; split;
[exact: total_variationD2|exact: total_variationD1].
Qed.

End total_variation.

Section variation_continuity.
Context {R : realType}.
Implicit Type f : R -> R.

Notation BV := bounded_variation.
Notation TV := total_variation.

Definition neg_tv a f (x : R) : \bar R := ((TV a x f - (f x)%:E) * 2^-1%:E)%E.

Definition pos_tv a f (x : R) : \bar R := neg_tv a (\- f) x.

Lemma total_variation_nondecreasing a b f :
{in `[a, b] &, nondecreasing_fun (TV a ^~ f)}.

Proof.

move=> x y; rewrite !in_itv/= => /andP[ax xb] /andP[ay yb] xy.
by rewrite (total_variationD f ax xy)// leeDl// total_variation_ge0.
Qed.

Lemma total_variation_bounded_variation a b (f : R -> R) : a <= b ->
BV a b f ->
BV a b (fine \o TV a ^~ f).

Proof.

move=> ab BVf; apply/bounded_variationP => //.
rewrite ge0_fin_numE; last exact: total_variation_ge0.
rewrite nondecreasing_total_variation/= ?ltry//.
move=> x y; rewrite !in_itv!/= => /andP[ax xb] /andP[ay yb] xy.
apply: fine_le.
- exact/(bounded_variationP _ ax)/(bounded_variationl ax xb).
- exact/(bounded_variationP _ ay)/(bounded_variationl ay yb).
- by apply: (@total_variation_nondecreasing _ b); rewrite ?in_itv //= ?ax ?ay.
Qed.

Lemma neg_tv_nondecreasing a b f :
{in `[a, b] &, nondecreasing_fun (neg_tv a f)}.

Proof.

move=> x y xab yab xy; have ax : a <= x.
by move: xab; rewrite in_itv //= => /andP [].
rewrite /neg_tv lee_pmul2r // leeBrDl // addeCA -EFinB.
rewrite [TV a y _](total_variationD _ ax xy) //.
apply: leeD => //; apply: le_trans; last exact: total_variation_ge.
by rewrite lee_fin ler_norm.
Qed.

Lemma bounded_variation_pos_neg_tvE a b f : BV a b f ->
{in `[a, b], f =1 (fine \o pos_tv a f) \- (fine \o neg_tv a f)}.

Proof.

move=> bdabf x; rewrite in_itv /= => /andP [ax xb].
have ffin: TV a x f \is a fin_num.
   apply/bounded_variationP => //.
   exact: (bounded_variationl _ xb).
have Nffin : TV a x (\- f) \is a fin_num.
  apply/bounded_variationP => //; apply/bounded_variationN.
  exact: (bounded_variationl ax xb).
rewrite /pos_tv /neg_tv /= total_variationN -fineB -?muleBl // ?fineM //.
- rewrite addeAC oppeD //= ?fin_num_adde_defl //.
  by rewrite addeA subee // add0e -EFinD //= opprK mulrDl -splitr.
- by rewrite fin_numB ?fin_numD ?ffin; apply/andP; split.
- by apply: fin_num_adde_defl; rewrite fin_numN fin_numD; apply/andP; split.
- by rewrite fin_numM // fin_numD; apply/andP; split.
- by rewrite fin_numM // fin_numD; apply/andP; split.
Qed.

Lemma fine_neg_tv_nondecreasing a b f : BV a b f ->
{in `[a, b] &, nondecreasing_fun (fine \o neg_tv a f)}.

Proof.

move=> bdv p q pab qab pq /=.
move: (pab) (qab); rewrite ?in_itv /= => /andP[ap pb] /andP[aq qb].
apply: fine_le; rewrite /neg_tv ?fin_numM // ?fin_numB /=.
- apply/andP; split => //; apply/bounded_variationP => //.
exact: (bounded_variationl _ pb).
- apply/andP; split => //; apply/bounded_variationP => //.
exact: (bounded_variationl _ qb).
exact: (neg_tv_nondecreasing _ pab).
Qed.

Lemma neg_tv_bounded_variation a b f : BV a b f -> BV a b (fine \o neg_tv a f).

Proof.

move=> ?; apply: nondecreasing_bounded_variation.
exact: fine_neg_tv_nondecreasing.
Qed.

Lemma total_variation_right_continuous a b x f : a <= x -> x 
 f @ x^'+ --> f x ->
 BV a b f ->
 fine \o TV a ^~ f @ x^'+ --> fine (TV a x f).

Proof.

move=> ax xb ctsf bvf; have ? : a <= b by apply:ltW; apply: (le_lt_trans ax).
apply/cvgrPdist_lt=> _/posnumP[eps].
have ? : Filter (nbhs x^'+) by exact: at_right_proper_filter.
have xbl := ltW xb.
have xbfin : TV x b f \is a fin_num.
 by apply/bounded_variationP => //; exact: (bounded_variationr _ _ bvf).
have [//|?] := @ub_ereal_sup_adherent R _ (eps%:num / 2) _ xbfin.
case=> ? [l + <- <-]; rewrite -/(total_variation x b f).
move: l => [|i j].
 by move=> /itv_partition_nil /eqP; rewrite lt_eqF.
move=> [/= /andP[xi ij /eqP ijb]] tv_eps.
apply: filter_app (nbhs_right_ge _).
apply: filter_app (nbhs_right_lt xi).
have e20 : 0 < eps%:num / 2 by [].
move/cvgrPdist_lt/(_ (eps%:num/2) e20) : ctsf; apply: filter_app.
near=> t => fxt ti xt; have ta : a <= t by exact: (le_trans ax).
have tb : t <= b by rewrite (le_trans (ltW ti))// -ijb path_lt_le_last.
rewrite -fineB; last 2 first.
 by apply/bounded_variationP => //; exact: bounded_variationl bvf.
 by apply/bounded_variationP => //; exact: bounded_variationl bvf.
rewrite (total_variationD _ ax xt).
have tbfin : TV t b f \is a fin_num.
 by apply/bounded_variationP => //; exact: (@bounded_variationr _ a).
have xtfin : TV x t f \is a fin_num.
 apply/bounded_variationP => //; apply: (@bounded_variationl _ _ _ b) => //.
 exact: (@bounded_variationr _ a).
rewrite oppeD ?fin_num_adde_defl// addeA subee //; first last.
 by apply/bounded_variationP => //; exact: (@bounded_variationl _ _ _ b).
rewrite sub0e fineN normrN ger0_norm; last first.
 by rewrite fine_ge0// total_variation_ge0.
move: (tv_eps); rewrite (total_variationD f _ tb) //.
move: xt; rewrite le_eqVlt => /predU1P[->|xt].
 by rewrite total_variationxx/=.
have : variation x b f (i :: j) <= variation x t f (t :: nil) +
 variation t b f (i :: j).
 rewrite variationD//; last 2 first.
 exact: itv_partition1.
 by rewrite /itv_partition/= ti ij ijb.
 exact: le_variation.
rewrite -lee_fin => /lt_le_trans /[apply].
rewrite {1}variation_prev; last exact: itv_partition1.
rewrite /= -addeA -lteBrDr; last by rewrite fin_numD; apply/andP.
rewrite EFinD -lte_fin ?fineK // oppeD //= ?fin_num_adde_defl // opprK addeA.
move/lt_trans; apply.
rewrite [in ltRHS](splitr (eps%:num)) EFinD lteD2rE// -addeA.
apply: (@le_lt_trans _ _ (variation x t f (t :: nil))%:E).
 rewrite [in leRHS]variation_prev; last exact: itv_partition1.
 rewrite geeDl// sube_le0; apply: ereal_sup_ubound => /=.
 exists (variation t b f (i :: j)) => //; apply: variations_variation.
 by rewrite /itv_partition/= ijb ij ti.
by rewrite /variation/= big_nat_recr//= big_nil add0r distrC lte_fin.
Unshelve. all: by end_near. Qed.

Lemma neg_tv_right_continuous a x b f : a <= x -> x 
 BV a b f ->
 f @ x^'+ --> f x ->
 fine \o neg_tv a f @ x^'+ --> fine (neg_tv a f x).

Proof.

move=> ax ? bvf fcts; have xb : x <= b by exact: ltW.
have xbfin : TV a x f \is a fin_num.
 by apply/bounded_variationP => //; exact: bounded_variationl bvf.
apply: fine_cvg; rewrite /neg_tv fineM // ?fin_numB ?xbfin //= EFinM.
under eq_fun => i do rewrite EFinN.
apply: (@cvg_trans _ (((TV a n f - (f n)%:E) * 2^-1%:E)%E @[n --> x^'+])).
 exact: cvg_id.
apply: cvgeMr; first by [].
rewrite fineD; [|by []..].
rewrite EFinB; apply: cvgeB; [by []| |].
 apply/ fine_cvgP; split; first exists (b - x).
 - by rewrite /= subr_gt0.
 - move=> t /= xtbx xt; have ? : a <= t.
 by apply: ltW; apply: (le_lt_trans ax).
 apply/bounded_variationP => //.
 apply: bounded_variationl bvf => //.
 move: xtbx; rewrite distrC ger0_norm ?subr_ge0; last by exact: ltW.
 by rewrite ltrBrDr -addrA [-_ + _]addrC subrr addr0 => /ltW.
 by apply: total_variation_right_continuous => //; last exact: bvf.
apply: cvg_comp; first exact: fcts.
apply/ fine_cvgP; split; first by near=> t => //.
by have -> : fine \o EFin = id by move=> ?; rewrite funeqE => ? /=.
Unshelve. all: by end_near. Qed.

Lemma total_variation_opp a b f : TV a b f = TV (- b) (- a) (f \o -%R).

Proof.

by rewrite /total_variation variations_opp. Qed.

Lemma total_variation_left_continuous a b x f : a < x -> x <= b ->
 f @ x^'- --> f x ->
 BV a b f ->
 fine \o TV a ^~ f @ x^'- --> fine (TV a x f).

Proof.

move=> ax xb fNcts bvf.
apply/cvg_at_leftNP; rewrite total_variation_opp.
have bvNf : BV (-b) (-a) (f \o -%R).
 by case: bvf => M; rewrite -variations_opp => ?; exists M.
have bx : - b <= - x by rewrite lerNl opprK.
have xa : - x < - a by rewrite ltrNl opprK.
have ? : - x <= - a by exact: ltW.
have ? : Filter (nbhs (-x)^'+) by exact: at_right_proper_filter.
have -> : fine (TV (-x) (-a) (f \o -%R)) =
 fine (TV (-b) (-a) (f \o -%R)) - fine (TV (-b) (-x) (f \o -%R)).
 apply/eqP; rewrite -subr_eq opprK addrC.
 rewrite -fineD; last 2 first.
 by apply/bounded_variationP => //; exact: bounded_variationl bvNf.
 by apply/bounded_variationP => //; exact: bounded_variationr bvNf.
 by rewrite -total_variationD.
have /near_eq_cvg/cvg_trans : {near (- x)^'+,
 (fun t => fine (TV (- b) (- a) (f \o -%R)) - fine (TV (- b) t (f \o -%R))) =1
 (fine \o (TV a)^~ f) \o -%R}.
 apply: filter_app (nbhs_right_lt xa).
 apply: filter_app (nbhs_right_ge _).
 near=> t => xt ta; have ? : -b <= t by exact: (le_trans bx).
 have ? : t <= -a by exact: ltW.
 apply/eqP; rewrite eq_sym -subr_eq opprK addrC.
 rewrite /= [TV a _ f]total_variation_opp opprK -fineD; last first.
 by apply/bounded_variationP => //; apply: bounded_variationr bvNf.
 by apply/bounded_variationP => //; apply: bounded_variationl bvNf.
 by rewrite -total_variationD.
apply.
apply: cvgB; first exact: cvg_cst.
apply: (total_variation_right_continuous _ _ _ bvNf).
- by rewrite lerNl opprK //.
- by rewrite ltrNl opprK //.
by apply/cvg_at_leftNP; rewrite /= opprK.
Unshelve. all: by end_near. Qed.

Lemma total_variation_continuous a b (f : R -> R) : a 
 {within `[a,b], continuous f} ->
 BV a b f ->
 {within `[a,b], continuous (fine \o TV a ^~ f)}.

Proof.

move=> ab /(@continuous_within_itvP _ _ _ _ ab) [int l r] bdf.
apply/continuous_within_itvP => //; split.
- move=> x /[dup] xab; rewrite in_itv /= => /andP [ax xb].
  apply/left_right_continuousP; split.
    apply: (total_variation_left_continuous _ (ltW xb)) => //.
    by have /left_right_continuousP[] := int x xab.
  apply: (total_variation_right_continuous _ xb) => //; first exact: ltW.
  by have /left_right_continuousP[] := int x xab.
- exact: (total_variation_right_continuous _ ab).
- exact: (total_variation_left_continuous ab).
Qed.

End variation_continuity.

Files

Module mathcomp.analysis.realfun