diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 6d418d4f81..06fa683ac9 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -59,6 +59,51 @@ + lemma `continuous_uncurry` + lemma `curry_continuous` + lemma `uncurry_continuous` +- in file `normedtype.v`, + + new lemma `continuous_within_itvP`. + +- in `ereal.v`: + + lemma `ereal_supy` + +- in `mathcomp_extra.v`: + + lemmas `last_filterP`, + `path_lt_filter0`, `path_lt_filterT`, `path_lt_head`, `path_lt_last_filter`, + `path_lt_le_last` + +- in file `realfun.v`, + + new definitions `itv_partition`, `itv_partitionL`, `itv_partitionR`, + `variation`, `variations`, `bounded_variation`, `total_variation`, + `neg_tv`, and `pos_tv`. + + + new lemmas `left_right_continuousP`, + `nondecreasing_funN`, `nonincreasing_funN` + + + new lemmas `itv_partition_nil`, `itv_partition_cons`, `itv_partition1`, + `itv_partition_size_neq0`, `itv_partitionxx`, `itv_partition_le`, + `itv_partition_cat`, `itv_partition_nth_size`, + `itv_partition_nth_ge`, `itv_partition_nth_le`, + `nondecreasing_fun_itv_partition`, `nonincreasing_fun_itv_partition`, + `itv_partitionLP`, `itv_partitionRP`, `in_itv_partition`, + `notin_itv_partition`, `itv_partition_rev`, + + + new lemmas `variation_zip`, `variation_prev`, `variation_next`, `variation_nil`, + `variation_ge0`, `variationN`, `variation_le`, `nondecreasing_variation`, + `nonincreasing_variation`, `variationD`, `variation_itv_partitionLR`, + `le_variation`, `variation_opp_rev`, `variation_rev_opp` + + + new lemmas `variations_variation`, `variations_neq0`, `variationsN`, `variationsxx` + + + new lemmas `bounded_variationxx`, `bounded_variationD`, `bounded_variationN`, + `bounded_variationl`, `bounded_variationr`, `variations_opp`, + `nondecreasing_bounded_variation` + + + new lemmas `total_variationxx`, `total_variation_ge`, `total_variation_ge0`, + `bounded_variationP`, `nondecreasing_total_variation`, `total_variationN`, + `total_variation_le`, `total_variationD`, `neg_tv_nondecreasing`, + `total_variation_pos_neg_tvE`, `fine_neg_tv_nondecreasing`, + `neg_tv_bounded_variation`, `total_variation_right_continuous`, + `neg_tv_right_continuous`, `total_variation_opp`, + `total_variation_left_continuous`, `total_variation_continuous` ### Changed diff --git a/classical/mathcomp_extra.v b/classical/mathcomp_extra.v index 59c5481c80..7e4e7091bc 100644 --- a/classical/mathcomp_extra.v +++ b/classical/mathcomp_extra.v @@ -934,3 +934,54 @@ Qed. Definition monotonous d (T : porderType d) (pT : predType T) (A : pT) (f : T -> T) := {in A &, {mono f : x y / (x <= y)%O}} \/ {in A &, {mono f : x y /~ (x <= y)%O}}. + +(* NB: these lemmas have been introduced to develop the theory of bounded variation *) +Section path_lt. +Context d {T : orderType d}. +Implicit Types (a b c : T) (s : seq T). + +Lemma last_filterP a (P : pred T) s : + P a -> P (last a [seq x <- s | P x]). +Proof. +by elim: s a => //= t1 t2 ih a Pa; case: ifPn => //= Pt1; exact: ih. +Qed. + +Lemma path_lt_filter0 a s : path <%O a s -> [seq x <- s | (x < a)%O] = [::]. +Proof. +move=> /lt_path_min/allP sa; rewrite -(filter_pred0 s). +apply: eq_in_filter => x xs. +by apply/negbTE; have := sa _ xs; rewrite ltNge; apply: contra => /ltW. +Qed. + +Lemma path_lt_filterT a s : path <%O a s -> [seq x <- s | (a < x)%O] = s. +Proof. +move=> /lt_path_min/allP sa; rewrite -[RHS](filter_predT s). +by apply: eq_in_filter => x xs; exact: sa. +Qed. + +Lemma path_lt_head a b s : (a < b)%O -> path <%O b s -> path <%O a s. +Proof. +by elim: s b => // h t ih b /= ab /andP[bh ->]; rewrite andbT (lt_trans ab). +Qed. + +(* TODO: this lemma feels a bit too technical, generalize? *) +Lemma path_lt_last_filter a b c s : + (a < c)%O -> (c < b)%O -> path <%O a s -> last a s = b -> + last c [seq x <- s | (c < x)%O] = b. +Proof. +elim/last_ind : s a b c => /= [|h t ih a b c ac cb]. + move=> a b c ac cb _ ab. + by apply/eqP; rewrite eq_le (ltW cb) -ab (ltW ac). +rewrite rcons_path => /andP[ah ht]; rewrite last_rcons => tb. +by rewrite filter_rcons tb cb last_rcons. +Qed. + +Lemma path_lt_le_last a s : path <%O a s -> (a <= last a s)%O. +Proof. +elim: s a => // a [_ c /andP[/ltW//]|b t ih i/= /and3P[ia ab bt]] /=. +have /= := ih a; rewrite ab bt => /(_ erefl). +by apply: le_trans; exact/ltW. +Qed. + +End path_lt. +Arguments last_filterP {d T a} P s. diff --git a/theories/ereal.v b/theories/ereal.v index 896c4fc7e3..1201c4ea38 100644 --- a/theories/ereal.v +++ b/theories/ereal.v @@ -511,6 +511,11 @@ case: xgetP => /=; first by move=> _ -> -[] /ubP geS _; apply geS. by case: (ereal_supremums_neq0 S) => /= x0 Sx0; move/(_ x0). Qed. +Lemma ereal_supy S : S +oo -> ereal_sup S = +oo. +Proof. +by move=> Soo; apply/eqP; rewrite eq_le leey/=; exact: ereal_sup_ub. +Qed. + Lemma ereal_sup_le S x : (exists2 y, S y & x <= y) -> x <= ereal_sup S. Proof. by move=> [y Sy] /le_trans; apply; exact: ereal_sup_ub. Qed. diff --git a/theories/normedtype.v b/theories/normedtype.v index 7b55266357..01ea11f428 100644 --- a/theories/normedtype.v +++ b/theories/normedtype.v @@ -2075,6 +2075,51 @@ by apply: xe_A => //; rewrite eq_sym. Qed. Arguments cvg_at_leftE {R V} f x. +Lemma continuous_within_itvP {R : realType } a b (f : R -> R) : + a < b -> + {within `[a,b], continuous f} <-> + {in `]a,b[, continuous f} /\ f @ a^'+ --> f a /\ f @b^'- --> f b. +Proof. +move=> ab; split=> [abf|]. + split. + suff : {in `]a, b[%classic, continuous f}. + by move=> P c W; apply: P; rewrite inE. + rewrite -continuous_open_subspace; last exact: interval_open. + by move: abf; exact/continuous_subspaceW/subset_itvW. + have [aab bab] : a \in `[a, b] /\ b \in `[a, b]. + by rewrite !in_itv/= !lexx (ltW ab). + split; apply/cvgrPdist_lt => eps eps_gt0 /=. + + move/continuous_withinNx/cvgrPdist_lt/(_ _ eps_gt0) : (abf a). + rewrite /dnbhs/= near_withinE !near_simpl// /prop_near1 /nbhs/=. + rewrite -nbhs_subspace_in// /within/= near_simpl. + apply: filter_app; exists (b - a); rewrite /= ?subr_gt0// => c cba + ac. + apply=> //; rewrite ?gt_eqF// !in_itv/= (ltW ac)/=; move: cba => /=. + by rewrite ltr0_norm ?subr_lt0// opprB ltr_add2r => /ltW. + + move/continuous_withinNx/cvgrPdist_lt/(_ _ eps_gt0) : (abf b). + rewrite /dnbhs/= near_withinE !near_simpl /prop_near1 /nbhs/=. + rewrite -nbhs_subspace_in// /within/= near_simpl. + apply: filter_app; exists (b - a); rewrite /= ?subr_gt0// => c cba + ac. + apply=> //; rewrite ?lt_eqF// !in_itv/= (ltW ac)/= andbT; move: cba => /=. + by rewrite gtr0_norm ?subr_gt0// ltr_add2l ltr_oppr opprK => /ltW. +case=> ctsoo [ctsL ctsR]; apply/subspace_continuousP => x /andP[]. +rewrite !bnd_simp/= !le_eqVlt => /predU1P[<-{x}|ax] /predU1P[|]. +- by move/eqP; rewrite lt_eqF. +- move=> _; apply/cvgrPdist_lt => eps eps_gt0 /=. + move/cvgrPdist_lt/(_ _ eps_gt0): ctsL; rewrite /at_right !near_withinE. + apply: filter_app; exists (b - a); rewrite /= ?subr_gt0// => c cba + ac. + have : a <= c by move: ac => /andP[]. + by rewrite le_eqVlt => /predU1P[->|/[swap] /[apply]//]; rewrite subrr normr0. +- move=> ->; apply/cvgrPdist_lt => eps eps_gt0 /=. + move/cvgrPdist_lt/(_ _ eps_gt0): ctsR; rewrite /at_left !near_withinE. + apply: filter_app; exists (b - a); rewrite /= ?subr_gt0 // => c cba + ac. + have : c <= b by move: ac => /andP[]. + by rewrite le_eqVlt => /predU1P[->|/[swap] /[apply]//]; rewrite subrr normr0. +- move=> xb; have aboox : x \in `]a, b[ by rewrite !in_itv/= ax. + rewrite within_interior; first exact: ctsoo. + suff : `]a, b[ `<=` interior `[a, b] by exact. + by rewrite -open_subsetE; [exact: subset_itvW| exact: interval_open]. +Qed. + (* TODO: generalize to R : numFieldType *) Section hausdorff. diff --git a/theories/realfun.v b/theories/realfun.v index efdda22805..e4502e512c 100644 --- a/theories/realfun.v +++ b/theories/realfun.v @@ -14,16 +14,30 @@ From HB Require Import structures. (* 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 *) +(* 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 *) (* *) -(* lime_sup f a/lime_inf f a == limit sup/inferior of the extended *) -(* real-valued function f at point a *) +(* derivable_oo_continuous_bnd f x y == f is derivable on `]x, y[ and *) +(* continuous up to the boundary *) (* *) -(* derivable_oo_continuous_bnd f x y == f is derivable on `]x, y[ and *) -(* continuous up to the boundary *) +(* 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 *) (* ``` *) (* *) (******************************************************************************) @@ -49,6 +63,23 @@ Notation "'increasing_fun' f" := ({mono f : n m / (n <= m)%O >-> (n <= m)%O}) 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. @@ -87,6 +118,18 @@ 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. @@ -105,7 +148,6 @@ rewrite neq_lt => /orP[tp|pt]. 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. @@ -250,36 +292,36 @@ have supf : has_sup [set f x | x in [set` Interval (BRight a) b]]. - exists (f ((a + t) / 2)), ((a + t) / 2) => //=. by rewrite in_itv/= midf_lt// midf_le// ltW. - by exists (f (a + 1)), (a + 1). - - by exists (f (a + 1)), (a + 1) => //=; rewrite in_itv/= ltr_addl andbT. + - 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 ler_subl_addr {}/M. +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 ler_subl_addr ler_addl. + + suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl. apply: sup_ub => //=; exists r => //; rewrite in_itv/=. by apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_lt]. - + rewrite (le_trans Mefp)// ler_add2r lef//=; last 2 first. + + 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 ler_subl_addr ler_addl. + + suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl. apply: sup_ub => //=; exists r => //; rewrite in_itv/=. by apply/andP; split; near: r; [exact: nbhs_right_gt|exact: nbhs_right_le]. - + rewrite (le_trans Mefp)// ler_add2r lef//=; last 2 first. + + 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 ler_subl_addr ler_addl. + suff: f r <= M by apply: le_trans; rewrite lerBlDr lerDl. apply: sup_ub => //=; exists r => //; rewrite in_itv/= andbT. by near: r; apply: nbhs_right_gt. - rewrite (le_trans Mefp)// ler_add2r lef//. + 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. @@ -302,7 +344,7 @@ Lemma nondecreasing_at_right_cvgr f a (b : itv_bound R) : (BRight a < b)%O -> 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 ler_opp2; exact. + 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])//. @@ -442,7 +484,7 @@ have [lnoo|lnoo] := eqVneq l -oo. 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 ltr_subl_addr subrK => zy. + 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). @@ -461,7 +503,7 @@ have [fpoo|fpoo] := pselect {in Interval (BRight a) b, forall x, f x = +oo}. 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 ltr_addl. + - 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. @@ -514,7 +556,7 @@ have <- : inf [set g x | x in [set` Interval (BRight a) b]] = fine l. + 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 ltr_addl. + 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]|]. @@ -549,8 +591,8 @@ suff: g x @[x --> a^'+] --> inf [set g x | x in [set` Interval (BRight a) b]]. 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 ltr_subl_addr => /lt_le_trans; apply. - by rewrite -ler_subr_addr ler_pdivr_mulr// ler_pmulr// ?ler1n// subr_gt0. + 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. @@ -1210,6 +1252,7 @@ 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}. @@ -1518,3 +1561,866 @@ 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 < b -> 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)) ler_oppr. +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 < b -> 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 < b -> 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 ltr_oppr 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. + +(* NB: not used yet but should allow for "term-by-term" comparisons *) +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_norm_add. +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. + +(* NB: this is the only lemma that uses variation_zip *) +Lemma variation_itv_partitionLR a b c f s : a < c -> c < b -> + 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 < b -> 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 < b -> + 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_ub => /=; 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 lee_add// 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_ub => /=. + 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 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 // lee_subr_addl // addeCA -EFinB. +rewrite [TV a y _](total_variationD _ ax xy) //. +apply: lee_add => //; 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 -Num.Theory.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 < b -> + 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 -lte_subr_addr; last by rewrite fin_numD; apply/andP. +rewrite EFinD -lte_fin ?fineK // oppeD //= ?fin_num_adde_defl // opprK addeA. +move/lt_trans; apply. +rewrite [x in (_ < x%:E)%E]Num.Theory.splitr EFinD addeC lte_add2lE //. +rewrite -addeA. +apply: (@le_lt_trans _ _ (variation x t f (t :: nil))%:E). + rewrite [in leRHS]variation_prev; last exact: itv_partition1. + rewrite gee_addl // sube_le0; apply: ereal_sup_ub => /=. + 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 < b -> + 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 ler_oppl opprK //. +- by rewrite ltr_oppl opprK //. +by apply/cvg_at_leftNP; rewrite /= opprK. +Unshelve. all: by end_near. Qed. + +Lemma total_variation_continuous a b (f : R -> R) : a < b -> + {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; (repeat 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.