PrimeNumberTheoremAnd

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition. -/ lemma swap_diff_le {a b : ℝ} (P : TaggedPartition a b) (j : Fin (P.n + 1)) : (swapTaggedPartition P).x j.succ - (swapTaggedPartition P).x j.castSucc ≤ 2 * P.mesh := by cases j using Fin.inductionOn; · unfold swapTaggedPartition; unfold swap_x; rcases P with ⟨ ⟨ n, x, hx ⟩, xi, hxi ⟩; rcases n with ( _ | n ) <;> norm_num [ Fin.cons ] at *; · simp_all +decide [ Fin.snoc ]; unfold IPart.IPartition.mesh; aesop; · have := hxi 0; norm_num at *; linarith! [ hx ( show 0 ≤ 1 from Nat.zero_le _ ), IPart.mesh_ge_diff ⟨ n + 1, x, hx, by tauto, by tauto ⟩ 0 ] ; · unfold swapTaggedPartition; unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.ext_iff ]; · rename_i i hi; have h_diff : P.xi (i.succ.castLT hi) ≤ P.x (i.succ.castLT hi).succ ∧ P.x (i.succ.castLT hi).succ ≤ P.x (i.succ.castLT hi).castSucc + P.mesh := by have h_diff : P.x (i.succ.castLT hi).succ - P.x (i.succ.castLT hi).castSucc ≤ P.mesh := by exact IPart.mesh_ge_diff P.toIPartition (i.succ.castLT hi); exact ⟨ P.h_xi _ |>.2, by linarith ⟩; have h_diff : P.x (i.succ.castLT hi).castSucc - P.x i.castSucc ≤ P.mesh := by have h_diff : ∀ j : Fin P.n, P.x (j.succ) - P.x j.castSucc ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; convert h_diff i using 1; linarith [ P.h_xi i ]; · rename_i i hi; -- Since $P.xi i \leq b$, we have $b - P.xi i \leq b - P.xi i$. have h_le : b - P.xi i ≤ P.mesh := by have h_le : b - P.xi i ≤ P.x (Fin.last P.n) - P.x i.castSucc := by have := P.h_xi i; linarith [ P.right, P.monotone ( show i.castSucc ≤ Fin.last P.n from Fin.le_last _ ), P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ) ]; refine le_trans h_le ?_; have h_le : ∀ j : Fin P.n, P.x (Fin.succ j) - P.x (Fin.castSucc j) ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; cases hi.eq_or_lt <;> simp_all +decide [ Fin.eq_last_of_not_lt ]; · convert h_le i using 1; congr ; aesop; · linarith [ Fin.is_lt i ]; linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg P.toIPartition ]

/- The mesh of the swapped partition is at most twice the mesh of the original partition. -/ lemma mesh_swap_le {a b : ℝ} (P : TaggedPartition a b) : (swapTaggedPartition P).mesh ≤ 2 * P.mesh := by convert Basic.max_getD_le _ _; · exact mul_nonneg zero_le_two ( IPart.mesh_nonneg _ ); · simp +zetaDelta at *; intro i; linarith! [ swap_diff_le P i ] ;

/- Given a partition P and a function g differentiable on [a,b] with derivative g', there exists a tagged partition T with the same points as P such that the increments of g satisfy the Mean Value Theorem equation on each subinterval.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition. -/ lemma swap_diff_le {a b : ℝ} (P : TaggedPartition a b) (j : Fin (P.n + 1)) : (swapTaggedPartition P).x j.succ - (swapTaggedPartition P).x j.castSucc ≤ 2 * P.mesh := by cases j using Fin.inductionOn; · unfold swapTaggedPartition; unfold swap_x; rcases P with ⟨ ⟨ n, x, hx ⟩, xi, hxi ⟩; rcases n with ( _ | n ) <;> norm_num [ Fin.cons ] at *; · simp_all +decide [ Fin.snoc ]; unfold IPart.IPartition.mesh; aesop; · have := hxi 0; norm_num at *; linarith! [ hx ( show 0 ≤ 1 from Nat.zero_le _ ), IPart.mesh_ge_diff ⟨ n + 1, x, hx, by tauto, by tauto ⟩ 0 ] ; · unfold swapTaggedPartition; unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.ext_iff ]; · rename_i i hi; have h_diff : P.xi (i.succ.castLT hi) ≤ P.x (i.succ.castLT hi).succ ∧ P.x (i.succ.castLT hi).succ ≤ P.x (i.succ.castLT hi).castSucc + P.mesh := by have h_diff : P.x (i.succ.castLT hi).succ - P.x (i.succ.castLT hi).castSucc ≤ P.mesh := by exact IPart.mesh_ge_diff P.toIPartition (i.succ.castLT hi); exact ⟨ P.h_xi _ |>.2, by linarith ⟩; have h_diff : P.x (i.succ.castLT hi).castSucc - P.x i.castSucc ≤ P.mesh := by have h_diff : ∀ j : Fin P.n, P.x (j.succ) - P.x j.castSucc ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; convert h_diff i using 1; linarith [ P.h_xi i ]; · rename_i i hi; -- Since $P.xi i \leq b$, we have $b - P.xi i \leq b - P.xi i$. have h_le : b - P.xi i ≤ P.mesh := by have h_le : b - P.xi i ≤ P.x (Fin.last P.n) - P.x i.castSucc := by have := P.h_xi i; linarith [ P.right, P.monotone ( show i.castSucc ≤ Fin.last P.n from Fin.le_last _ ), P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ) ]; refine le_trans h_le ?_; have h_le : ∀ j : Fin P.n, P.x (Fin.succ j) - P.x (Fin.castSucc j) ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; cases hi.eq_or_lt <;> simp_all +decide [ Fin.eq_last_of_not_lt ]; · convert h_le i using 1; congr ; aesop; · linarith [ Fin.is_lt i ]; linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg P.toIPartition ]

/- The mesh of the swapped partition is at most twice the mesh of the original partition. -/ lemma mesh_swap_le {a b : ℝ} (P : TaggedPartition a b) : (swapTaggedPartition P).mesh ≤ 2 * P.mesh := by convert Basic.max_getD_le _ _; · exact mul_nonneg zero_le_two ( IPart.mesh_nonneg _ ); · simp +zetaDelta at *; intro i; linarith! [ swap_diff_le P i ] ;

/- Given a partition P and a function g differentiable on [a,b] with derivative g', there exists a tagged partition T with the same points as P such that the increments of g satisfy the Mean Value Theorem equation on each subinterval. -/

lemma mvt_tags_for_partition {a b : ℝ} (P : IPart.IPartition a b) {g : ℝ → ℝ} {g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by have h_mean_value : ∀ i : Fin P.n, ∃ ξ ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), g (P.x i.succ) - g (P.x i.castSucc) = g' ξ * (P.x i.succ - P.x i.castSucc) := by intro i by_cases h_eq : P.x i.castSucc = P.x i.succ; · aesop; · have h_mvt : ∃ ξ ∈ Set.Ioo (P.x i.castSucc) (P.x i.succ), deriv g ξ = (g (P.x i.succ) - g (P.x i.castSucc)) / (P.x i.succ - P.x i.castSucc) := by have := exists_deriv_eq_slope g ( lt_of_le_of_ne ( P.monotone ( Nat.le_succ _ ) ) h_eq ); exact this ( continuousOn_of_forall_continuousAt fun x hx => HasDerivAt.continuousAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ) ) ( fun x hx => DifferentiableAt.differentiableWithinAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.differentiableAt ) ); obtain ⟨ ξ, hξ₁, hξ₂ ⟩ := h_mvt; exact ⟨ ξ, ⟨ hξ₁.1.le, hξ₁.2.le ⟩, by rw [ ← hg ξ ⟨ by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.deriv, hξ₂, div_mul_cancel₀ _ ( sub_ne_zero_of_ne <| Ne.symm h_eq ) ] ⟩ ; choose ξ hξ using h_mean_value; refine' ⟨ ⟨ P, ξ, fun i => hξ i |>.1 ⟩, rfl, fun i => hξ i |>.2 ⟩

/- The sum of absolute differences of g over P equals the Riemann-Stieltjes sum of |g'| with respect to id over a tagged partition T associated with P. -/ lemma sumAbsDiff_eq_RSSum_abs_deriv {a b : ℝ} (P : IPart.IPartition a b) {g g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ SumDiff.sumAbsDiff g P = RSSum (fun x => |g' x|) (fun x => x) T := by obtain ⟨ T, rfl, hT ⟩ := Integ.mvt_tags_for_partition P hg; refine' ⟨ T, rfl, Finset.sum_congr rfl fun i _ => _ ⟩ ; rw [ hT i ] ; rw [ abs_mul, abs_of_nonneg ( sub_nonneg.2 <| T.monotone <| Nat.le_succ _ ) ]

/- The identity function has bounded variation on [a, b].

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition. -/ lemma swap_diff_le {a b : ℝ} (P : TaggedPartition a b) (j : Fin (P.n + 1)) : (swapTaggedPartition P).x j.succ - (swapTaggedPartition P).x j.castSucc ≤ 2 * P.mesh := by cases j using Fin.inductionOn; · unfold swapTaggedPartition; unfold swap_x; rcases P with ⟨ ⟨ n, x, hx ⟩, xi, hxi ⟩; rcases n with ( _ | n ) <;> norm_num [ Fin.cons ] at *; · simp_all +decide [ Fin.snoc ]; unfold IPart.IPartition.mesh; aesop; · have := hxi 0; norm_num at *; linarith! [ hx ( show 0 ≤ 1 from Nat.zero_le _ ), IPart.mesh_ge_diff ⟨ n + 1, x, hx, by tauto, by tauto ⟩ 0 ] ; · unfold swapTaggedPartition; unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.ext_iff ]; · rename_i i hi; have h_diff : P.xi (i.succ.castLT hi) ≤ P.x (i.succ.castLT hi).succ ∧ P.x (i.succ.castLT hi).succ ≤ P.x (i.succ.castLT hi).castSucc + P.mesh := by have h_diff : P.x (i.succ.castLT hi).succ - P.x (i.succ.castLT hi).castSucc ≤ P.mesh := by exact IPart.mesh_ge_diff P.toIPartition (i.succ.castLT hi); exact ⟨ P.h_xi _ |>.2, by linarith ⟩; have h_diff : P.x (i.succ.castLT hi).castSucc - P.x i.castSucc ≤ P.mesh := by have h_diff : ∀ j : Fin P.n, P.x (j.succ) - P.x j.castSucc ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; convert h_diff i using 1; linarith [ P.h_xi i ]; · rename_i i hi; -- Since $P.xi i \leq b$, we have $b - P.xi i \leq b - P.xi i$. have h_le : b - P.xi i ≤ P.mesh := by have h_le : b - P.xi i ≤ P.x (Fin.last P.n) - P.x i.castSucc := by have := P.h_xi i; linarith [ P.right, P.monotone ( show i.castSucc ≤ Fin.last P.n from Fin.le_last _ ), P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ) ]; refine le_trans h_le ?_; have h_le : ∀ j : Fin P.n, P.x (Fin.succ j) - P.x (Fin.castSucc j) ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; cases hi.eq_or_lt <;> simp_all +decide [ Fin.eq_last_of_not_lt ]; · convert h_le i using 1; congr ; aesop; · linarith [ Fin.is_lt i ]; linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg P.toIPartition ]

/- The mesh of the swapped partition is at most twice the mesh of the original partition. -/ lemma mesh_swap_le {a b : ℝ} (P : TaggedPartition a b) : (swapTaggedPartition P).mesh ≤ 2 * P.mesh := by convert Basic.max_getD_le _ _; · exact mul_nonneg zero_le_two ( IPart.mesh_nonneg _ ); · simp +zetaDelta at *; intro i; linarith! [ swap_diff_le P i ] ;

/- Given a partition P and a function g differentiable on [a,b] with derivative g', there exists a tagged partition T with the same points as P such that the increments of g satisfy the Mean Value Theorem equation on each subinterval. -/

lemma mvt_tags_for_partition {a b : ℝ} (P : IPart.IPartition a b) {g : ℝ → ℝ} {g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by have h_mean_value : ∀ i : Fin P.n, ∃ ξ ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), g (P.x i.succ) - g (P.x i.castSucc) = g' ξ * (P.x i.succ - P.x i.castSucc) := by intro i by_cases h_eq : P.x i.castSucc = P.x i.succ; · aesop; · have h_mvt : ∃ ξ ∈ Set.Ioo (P.x i.castSucc) (P.x i.succ), deriv g ξ = (g (P.x i.succ) - g (P.x i.castSucc)) / (P.x i.succ - P.x i.castSucc) := by have := exists_deriv_eq_slope g ( lt_of_le_of_ne ( P.monotone ( Nat.le_succ _ ) ) h_eq ); exact this ( continuousOn_of_forall_continuousAt fun x hx => HasDerivAt.continuousAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ) ) ( fun x hx => DifferentiableAt.differentiableWithinAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.differentiableAt ) ); obtain ⟨ ξ, hξ₁, hξ₂ ⟩ := h_mvt; exact ⟨ ξ, ⟨ hξ₁.1.le, hξ₁.2.le ⟩, by rw [ ← hg ξ ⟨ by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.deriv, hξ₂, div_mul_cancel₀ _ ( sub_ne_zero_of_ne <| Ne.symm h_eq ) ] ⟩ ; choose ξ hξ using h_mean_value; refine' ⟨ ⟨ P, ξ, fun i => hξ i |>.1 ⟩, rfl, fun i => hξ i |>.2 ⟩

/- The sum of absolute differences of g over P equals the Riemann-Stieltjes sum of |g'| with respect to id over a tagged partition T associated with P. -/ lemma sumAbsDiff_eq_RSSum_abs_deriv {a b : ℝ} (P : IPart.IPartition a b) {g g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ SumDiff.sumAbsDiff g P = RSSum (fun x => |g' x|) (fun x => x) T := by obtain ⟨ T, rfl, hT ⟩ := Integ.mvt_tags_for_partition P hg; refine' ⟨ T, rfl, Finset.sum_congr rfl fun i _ => _ ⟩ ; rw [ hT i ] ; rw [ abs_mul, abs_of_nonneg ( sub_nonneg.2 <| T.monotone <| Nat.le_succ _ ) ]

/- The identity function has bounded variation on [a, b]. -/ lemma boundedVariationOn_id {a b : ℝ} : BoundedVariationOn (fun x => x) (Set.Icc a b) := by unfold BoundedVariationOn; simp +decide [ eVariationOn ]; norm_num [ edist_dist ]; refine' ne_of_lt ( lt_of_le_of_lt ( iSup_le _ ) ENNReal.coe_lt_top ); swap; exact ⟨ |b - a|, abs_nonneg _ ⟩; norm_num [ dist_eq_norm ]; intro n f hf h; rw [ ← ENNReal.ofReal_sum_of_nonneg ] <;> norm_num; -- The sum of the absolute differences of f over the range n is bounded by the total variation of f over [a, b], which is |b - a|. have h_sum_abs_diff : ∑ i ∈ Finset.range n, |f (i + 1) - f i| ≤ |b - a| := by rw [ Finset.sum_congr rfl fun i hi => abs_of_nonneg <| sub_nonneg.mpr <| hf <| Nat.le_succ _ ]; rw [ Finset.sum_range_sub ( fun i => f i ) ]; cases abs_cases ( b - a ) <;> linarith [ h 0, h n ]; exact ENNReal.ofReal_le_of_le_toReal ( by simpa using h_sum_abs_diff )

/- The Riemann-Stieltjes integral of |g'| is less than or equal to the total variation of g.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition. -/ lemma swap_diff_le {a b : ℝ} (P : TaggedPartition a b) (j : Fin (P.n + 1)) : (swapTaggedPartition P).x j.succ - (swapTaggedPartition P).x j.castSucc ≤ 2 * P.mesh := by cases j using Fin.inductionOn; · unfold swapTaggedPartition; unfold swap_x; rcases P with ⟨ ⟨ n, x, hx ⟩, xi, hxi ⟩; rcases n with ( _ | n ) <;> norm_num [ Fin.cons ] at *; · simp_all +decide [ Fin.snoc ]; unfold IPart.IPartition.mesh; aesop; · have := hxi 0; norm_num at *; linarith! [ hx ( show 0 ≤ 1 from Nat.zero_le _ ), IPart.mesh_ge_diff ⟨ n + 1, x, hx, by tauto, by tauto ⟩ 0 ] ; · unfold swapTaggedPartition; unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.ext_iff ]; · rename_i i hi; have h_diff : P.xi (i.succ.castLT hi) ≤ P.x (i.succ.castLT hi).succ ∧ P.x (i.succ.castLT hi).succ ≤ P.x (i.succ.castLT hi).castSucc + P.mesh := by have h_diff : P.x (i.succ.castLT hi).succ - P.x (i.succ.castLT hi).castSucc ≤ P.mesh := by exact IPart.mesh_ge_diff P.toIPartition (i.succ.castLT hi); exact ⟨ P.h_xi _ |>.2, by linarith ⟩; have h_diff : P.x (i.succ.castLT hi).castSucc - P.x i.castSucc ≤ P.mesh := by have h_diff : ∀ j : Fin P.n, P.x (j.succ) - P.x j.castSucc ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; convert h_diff i using 1; linarith [ P.h_xi i ]; · rename_i i hi; -- Since $P.xi i \leq b$, we have $b - P.xi i \leq b - P.xi i$. have h_le : b - P.xi i ≤ P.mesh := by have h_le : b - P.xi i ≤ P.x (Fin.last P.n) - P.x i.castSucc := by have := P.h_xi i; linarith [ P.right, P.monotone ( show i.castSucc ≤ Fin.last P.n from Fin.le_last _ ), P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ) ]; refine le_trans h_le ?_; have h_le : ∀ j : Fin P.n, P.x (Fin.succ j) - P.x (Fin.castSucc j) ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; cases hi.eq_or_lt <;> simp_all +decide [ Fin.eq_last_of_not_lt ]; · convert h_le i using 1; congr ; aesop; · linarith [ Fin.is_lt i ]; linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg P.toIPartition ]

/- The mesh of the swapped partition is at most twice the mesh of the original partition. -/ lemma mesh_swap_le {a b : ℝ} (P : TaggedPartition a b) : (swapTaggedPartition P).mesh ≤ 2 * P.mesh := by convert Basic.max_getD_le _ _; · exact mul_nonneg zero_le_two ( IPart.mesh_nonneg _ ); · simp +zetaDelta at *; intro i; linarith! [ swap_diff_le P i ] ;

/- Given a partition P and a function g differentiable on [a,b] with derivative g', there exists a tagged partition T with the same points as P such that the increments of g satisfy the Mean Value Theorem equation on each subinterval. -/

lemma mvt_tags_for_partition {a b : ℝ} (P : IPart.IPartition a b) {g : ℝ → ℝ} {g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by have h_mean_value : ∀ i : Fin P.n, ∃ ξ ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), g (P.x i.succ) - g (P.x i.castSucc) = g' ξ * (P.x i.succ - P.x i.castSucc) := by intro i by_cases h_eq : P.x i.castSucc = P.x i.succ; · aesop; · have h_mvt : ∃ ξ ∈ Set.Ioo (P.x i.castSucc) (P.x i.succ), deriv g ξ = (g (P.x i.succ) - g (P.x i.castSucc)) / (P.x i.succ - P.x i.castSucc) := by have := exists_deriv_eq_slope g ( lt_of_le_of_ne ( P.monotone ( Nat.le_succ _ ) ) h_eq ); exact this ( continuousOn_of_forall_continuousAt fun x hx => HasDerivAt.continuousAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ) ) ( fun x hx => DifferentiableAt.differentiableWithinAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.differentiableAt ) ); obtain ⟨ ξ, hξ₁, hξ₂ ⟩ := h_mvt; exact ⟨ ξ, ⟨ hξ₁.1.le, hξ₁.2.le ⟩, by rw [ ← hg ξ ⟨ by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.deriv, hξ₂, div_mul_cancel₀ _ ( sub_ne_zero_of_ne <| Ne.symm h_eq ) ] ⟩ ; choose ξ hξ using h_mean_value; refine' ⟨ ⟨ P, ξ, fun i => hξ i |>.1 ⟩, rfl, fun i => hξ i |>.2 ⟩

/- The sum of absolute differences of g over P equals the Riemann-Stieltjes sum of |g'| with respect to id over a tagged partition T associated with P. -/ lemma sumAbsDiff_eq_RSSum_abs_deriv {a b : ℝ} (P : IPart.IPartition a b) {g g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ SumDiff.sumAbsDiff g P = RSSum (fun x => |g' x|) (fun x => x) T := by obtain ⟨ T, rfl, hT ⟩ := Integ.mvt_tags_for_partition P hg; refine' ⟨ T, rfl, Finset.sum_congr rfl fun i _ => _ ⟩ ; rw [ hT i ] ; rw [ abs_mul, abs_of_nonneg ( sub_nonneg.2 <| T.monotone <| Nat.le_succ _ ) ]

/- The identity function has bounded variation on [a, b]. -/ lemma boundedVariationOn_id {a b : ℝ} : BoundedVariationOn (fun x => x) (Set.Icc a b) := by unfold BoundedVariationOn; simp +decide [ eVariationOn ]; norm_num [ edist_dist ]; refine' ne_of_lt ( lt_of_le_of_lt ( iSup_le _ ) ENNReal.coe_lt_top ); swap; exact ⟨ |b - a|, abs_nonneg _ ⟩; norm_num [ dist_eq_norm ]; intro n f hf h; rw [ ← ENNReal.ofReal_sum_of_nonneg ] <;> norm_num; -- The sum of the absolute differences of f over the range n is bounded by the total variation of f over [a, b], which is |b - a|. have h_sum_abs_diff : ∑ i ∈ Finset.range n, |f (i + 1) - f i| ≤ |b - a| := by rw [ Finset.sum_congr rfl fun i hi => abs_of_nonneg <| sub_nonneg.mpr <| hf <| Nat.le_succ _ ]; rw [ Finset.sum_range_sub ( fun i => f i ) ]; cases abs_cases ( b - a ) <;> linarith [ h 0, h n ]; exact ENNReal.ofReal_le_of_le_toReal ( by simpa using h_sum_abs_diff )

/- The Riemann-Stieltjes integral of |g'| is less than or equal to the total variation of g. -/ lemma integral_le_variation {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (g' : ℝ → ℝ) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) (hg_bv : BoundedVariationOn g (Set.Icc a b)) (I : ℝ) (hI : RS.Integ.HasRSIntegral (fun x => |g' x|) id a b I) : I ≤ (eVariationOn g (Set.Icc a b)).toReal := by -- Fix ε > 0. have h_eps : ∀ ε > 0, I ≤ (eVariationOn g (Set.Icc a b)).toReal + ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ (T : Integ.TaggedPartition a b), T.mesh ≤ δ → |Integ.RSSum (fun x => |g' x|) id T - I| < ε := by exact hI ε hε_pos; -- Choose a uniform partition P with mesh ≤ δ. obtain ⟨P, hP⟩ : ∃ P : RS.IPart.IPartition a b, P.mesh ≤ δ := by -- Choose n such that (b - a) / (n + 1) ≤ δ. obtain ⟨n, hn⟩ : ∃ n : ℕ, (b - a) / (n + 1) ≤ δ := by exact ⟨ ⌊ ( b - a ) / δ⌋₊, by rw [ div_le_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / δ ), mul_div_cancel₀ ( b - a ) hδ_pos.ne' ] ⟩; exact ⟨ RS.IPart.uniformPartition n hab, by simpa only [ RS.IPart.mesh_uniformPartition ] using hn ⟩; -- Let T be the tagged partition obtained from the Mean Value Theorem applied to g on each subinterval of P. obtain ⟨T, hT⟩ : ∃ T : Integ.TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by exact RS.Integ.mvt_tags_for_partition P hg; -- By sum_abs_diff_le_variation, we have sumAbsDiff g P ≤ (eVariationOn g (Set.Icc a b)).toReal. have h_sum_abs_diff_le_variation : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by exact Integ.sum_abs_diff_le_variation P hg_bv; -- By sumAbsDiff_eq_RSSum_abs_deriv, we have RSSum (fun x => |g' x|) id T = sumAbsDiff g P. have h_RSSum_eq_sumAbsDiff : Integ.RSSum (fun x => |g' x|) id T = ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| := by have h_RSSum_eq_sumAbsDiff : Integ.RSSum (fun x => |g' x|) id T = ∑ i : Fin T.n, |g' (T.xi i)| * (T.x i.succ - T.x i.castSucc) := by unfold Integ.RSSum; aesop; rw [ h_RSSum_eq_sumAbsDiff, ← hT.1 ]; exact Finset.sum_congr rfl fun i _ => by rw [ hT.2 i, abs_mul, abs_of_nonneg ( sub_nonneg.mpr <| T.monotone <| Nat.le_succ _ ) ] ; linarith [ abs_lt.mp ( hδ T ( by simpa [ hT.1 ] using hP ) ) ]; exact le_of_forall_pos_le_add h_eps

def mvtTaggedPartition {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : TaggedPartition a b := Classical.choose (mvt_tags_for_partition P hg)

lemma mvtTaggedPartition_spec {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : (mvtTaggedPartition g g' P hg).toIPartition = P ∧ ∀ i : Fin (mvtTaggedPartition g g' P hg).n, g ((mvtTaggedPartition g g' P hg).x i.succ) - g ((mvtTaggedPartition g g' P hg).x i.castSucc) = g' ((mvtTaggedPartition g g' P hg).xi i) * ((mvtTaggedPartition g g' P hg).x i.succ - (mvtTaggedPartition g g' P hg).x i.castSucc) := Classical.choose_spec (mvt_tags_for_partition P hg)

/- The mesh of the MVT tagged partition equals the mesh of the original partition. -/ lemma mesh_mvtTaggedPartition {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : (mvtTaggedPartition g g' P hg).mesh = P.mesh := by have h := (mvtTaggedPartition_spec g g' P hg).1 convert congr_arg _ h using 1;

/- For any partition P, the sum of absolute differences of g over P is bounded by I + ε.

Sum of absolute increments of g over an untagged partition. -/ def sumAbsDiff {a b : ℝ} (g : ℝ → ℝ) (P : IPart.IPartition a b) : ℝ := ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)|

/- The absolute difference of g over a subinterval of P is bounded by the sum of absolute differences of g over the corresponding subintervals of R. -/ lemma sumAbsDiff_block_le {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (i : Fin P.n) : |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by rw [← hk_eq i.succ, ← hk_eq i.castSucc] rw [← IPart.sum_telescope_block g R (k i.castSucc) (k i.succ) (hk (Nat.le_succ _))] apply Finset.abs_sum_le_sum_abs

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_mono_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by -- Use refinement_index_map to get k. obtain ⟨k, , hk_eq⟩ : ∃ k : Fin (P.n + 1) → Fin (R.n + 1), Monotone k ∧ ∀ i, R.x (k i) = P.x i := by exact IPart.refinement_index_map P R hPR; -- Apply sumAbsDiff_block_le to bound the sum over P by the double sum over blocks of R. have h_sum_le_double_sum : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| := by exact Finset.sum_le_sum fun i _ => sumAbsDiff_block_le k ‹› ‹_› i; -- Use sum_partition_blocks to collapse the double sum into a single sum over Ico (k 0) (k P.n). have h_double_sum_to_single_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| := by have h_double_sum_to_single_sum : ∀ (n : ℕ) (k : Fin (n + 1) → ℕ) (hk : Monotone k), ∑ i : Fin n, ∑ j ∈ Finset.Ico (k i.castSucc) (k i.succ), |IPart.term g R j| = ∑ j ∈ Finset.Ico (k 0) (k (Fin.last n)), |IPart.term g R j| := by exact fun n k hk ↦ IPart.sum_partition_blocks n k hk fun j ↦ |IPart.term g R j|; convert h_double_sum_to_single_sum P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using ‹Monotone k› hij ) using 1; -- Bound this sum by the sum over range R.n (since terms are non-negative and the interval is a subset). have h_single_sum_le_range : ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last P.n)).val, |IPart.term g R j| ≤ ∑ j ∈ Finset.Ico 0 R.n, |IPart.term g R j| := by refine' Finset.sum_le_sum_of_subset_of_nonneg _ fun _ _ _ => abs_nonneg _; refine' Finset.Ico_subset_Ico _ _ <;> norm_num; exact Nat.le_of_lt_succ ( Fin.is_lt _ ); convert h_sum_le_double_sum.trans ( h_double_sum_to_single_sum.le.trans h_single_sum_le_range ) using 1; simp +decide [ sumAbsDiff, Finset.sum_range, Fin.cast_val_eq_self ]; refine' Finset.sum_congr rfl fun i hi => _ ; unfold IPart.term ; aesop

/- sumAbsDiff is monotonic with respect to partition refinement. -/ lemma sumAbsDiff_le_sumAbsDiff_of_refinement {a b : ℝ} {g : ℝ → ℝ} {P R : IPart.IPartition a b} (hPR : IPart.IsRefinement R P) : sumAbsDiff g P ≤ sumAbsDiff g R := by convert sumAbsDiff_mono_of_refinement hPR using 1

/- If x is a monotone sequence of points contained in partition P, then the sum of absolute differences of g along x is bounded by sumAbsDiff g P. -/ lemma sum_le_sumAbsDiff_of_points_subset {a b : ℝ} (g : ℝ → ℝ) (P : RS.IPart.IPartition a b) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (h_subset : ∀ i, x i ∈ P.points) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ RS.SumDiff.sumAbsDiff g P := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$. obtain ⟨k, hk⟩ : ∃ k : Fin (p + 1) → Fin (P.n + 1), Monotone k ∧ ∀ i : Fin (p + 1), P.x (k i) = x i := by -- By definition of $P.points$, there exists a monotone map $k : Fin (p + 1) → Fin (P.n + 1)$ such that $P.x (k i) = x i$ for all $i$ by the properties of the points of $P$. have h_k_exists : ∀ y ∈ P.points, ∃ i : Fin (P.n + 1), P.x i = y := by exact fun y hy => by rw [ IPart.IPartition.points ] at hy; aesop; choose k hk using h_k_exists; use fun i => k ( x i ) ( h_subset i ); have h_k_mono : ∀ i j : Fin (p + 1), x i ≤ x j → k (x i) (h_subset i) ≤ k (x j) (h_subset j) := by intros i j hij have h_k_mono : P.x (k (x i) (h_subset i)) ≤ P.x (k (x j) (h_subset j)) := by aesop; contrapose! h_k_mono; apply_rules [ P.monotone, lt_of_le_of_ne ]; · exact le_of_lt h_k_mono; · grind; exact ⟨ fun i j hij => h_k_mono i j <| hx hij, fun i => hk _ _ ⟩; -- Apply the lemma sum_telescope_abs_le to each term in the sum. have h_telescope : ∀ i : Fin p, |g (x (i.succ)) - g (x (i.castSucc))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by intros i have h_telescope : |g (P.x (k (i.succ))) - g (P.x (k (i.castSucc)))| ≤ ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| := by convert IPart.sum_telescope_abs_le g P ( k ( Fin.castSucc i ) ) ( k ( Fin.succ i ) ) _ using 1; exact hk.1 ( Nat.le_succ _ ); aesop; -- Apply the lemma sum_partition_blocks to combine the sums over the intervals [k i, k (i+1)). have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| ≤ ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by have h_sum_partition_blocks : ∑ i : Fin p, ∑ j ∈ Finset.Ico (k (i.castSucc)).val (k (i.succ)).val, |RS.IPart.term g P j| = ∑ j ∈ Finset.Ico (k 0).val (k (Fin.last p)).val, |RS.IPart.term g P j| := by convert RS.IPart.sum_partition_blocks p ( fun i => ( k i : ℕ ) ) hk.1 _ using 1; grind; refine le_trans ( Finset.sum_le_sum fun i _ => h_telescope i ) ( h_sum_partition_blocks.trans ?_ ); refine' le_trans ( Finset.sum_le_sum_of_subset_of_nonneg ( Finset.subset_iff.mpr _ ) fun _ _ _ => abs_nonneg _ ) _; exact Finset.range P.n; · simp +zetaDelta at *; intro i hi₁ hi₂; exact lt_of_lt_of_le hi₂ ( Nat.le_of_lt_succ <| by simp [ Fin.is_lt ] ) ; · simp +decide [ Finset.sum_range, IPart.term ]; rfl

lemma sum_le_K_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, sumAbsDiff g P ≤ K) (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b) : ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by -- Let s be the set of values of x union {a, b}. set s : Finset ℝ := Finset.image x Finset.univ ∪ {a, b} with hs_def; -- Construct a partition P from s using RS.IPart.fromPoints. obtain ⟨P, hP⟩ : ∃ P : IPart.IPartition a b, P.points = s := by refine' ⟨ IPart.fromPoints s _ _ _, _ ⟩ <;> norm_num [ hs_def ]; grind; convert IPart.points_fromPoints _ _ _ _; convert sum_le_sumAbsDiff_of_points_subset g P p x hx _ |> le_trans <| hK P; grind

/- If the sum of absolute differences of g over any partition is bounded by K, then the total variation of g is bounded by K. -/ lemma variation_le_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : IPart.IPartition a b, sumAbsDiff g P ≤ K) : (eVariationOn g (Set.Icc a b)).toReal ≤ K := by -- We know that for any monotone sequence x, the sum of absolute differences is ≤ K. have h_sum_le : ∀ (p : ℕ) (x : Fin (p + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin p, |g (x i.succ) - g (x i.castSucc)| ≤ K := by intro p x hx hx_mem exact sum_le_K_of_forall_sumAbsDiff_le hab g K hK p x hx hx_mem

-- Now we relate this to eVariationOn. -- eVariationOn is the supremum of these sums (converted to ENNReal). -- We show eVariationOn ≤ ENNReal.ofReal K. have h_var_le : eVariationOn g (Set.Icc a b) ≤ ENNReal.ofReal K := by -- Unfold eVariationOn to its definition as a supremum -- Since eVariationOn is defined as a supr, we use iSup_le -- Note: The definition might be wrapped. -- We can try to use eVariationOn properties or just unfold it. -- Assuming standard definition: -- ref: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Variation.html#eVariationOn -- It is iSup over p, x. refine' iSup_le _; rintro ⟨ p, ⟨ x, hx₁, hx₂ ⟩ ⟩; simp_all +decide [ edist_dist, Real.dist_eq ]; convert ENNReal.ofReal_le_ofReal ( h_sum_le p ( fun i => x i ) ( fun i j hij => hx₁ hij ) ( fun i => hx₂ i ) ) using 1; rw [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _, Finset.sum_range ]; exact rfl

/- If the sum of absolute differences of g over any partition is bounded by K, then the sum of absolute differences along any monotone sequence in [a, b] is bounded by K. -/

-- Finally, convert to Real. apply ENNReal.toReal_le_of_le_ofReal · -- K ≥ 0 have P := RS.IPart.uniformPartition 1 hab have := hK P calc 0 ≤ RS.SumDiff.sumAbsDiff g P := by apply Finset.sum_nonneg intro i _ apply abs_nonneg _ ≤ K := this · exact h_var_le

end SumDiff

/- First, a definition of tagged partitions, followed by some basic lemmas -/ /- A tagged Riemann-Stieltjes partition includes points xi in each subinterval [x_i, x_{i+1}]. -/

namespace Integ

structure TaggedPartition (a b : ℝ) extends IPart.IPartition a b where xi : Fin n → ℝ h_xi : ∀ i : Fin n, x i.castSucc ≤ xi i ∧ xi i ≤ x i.succ

/- The filter of tagged partitions with mesh tending to 0. -/ def TaggedPartitionFilter (a b : ℝ) : Filter (TaggedPartition a b) := Filter.comap (fun P => P.mesh) (nhds 0)

/- A uniform partition with left tags. We need that simply because our filter is a filter of tagged partitions. -/ def taggedUniformPartition {a b : ℝ} (n : ℕ) (hab : a < b) : TaggedPartition a b := let P := IPart.uniformPartition n hab { toIPartition := P xi := fun i => P.x i.castSucc h_xi := by intro i constructor · exact le_refl _ · apply P.monotone apply Nat.le_succ }

/- The partition filter is proper, that is, there exist partitions of [a,b] of arbitrarily small mesh. -/ lemma TaggedPartitionFilter_neBot {a b : ℝ} (hab : a < b) : Filter.NeBot (TaggedPartitionFilter a b) := by refine' Filter.neBot_iff.mpr _; rw [ Ne.eq_def, Filter.ext_iff ]; simp +zetaDelta at *; use ∅; intro h; obtain ⟨ U, hU₁, hU₂ ⟩ := Filter.mem_comap.mp h; simp_all +decide [ Set.subset_def ]; contrapose! hU₂; rcases Metric.mem_nhds_iff.mp hU₁ with ⟨ ε, εpos, hε ⟩; -- Choose $n$ such that $(b - a) / (n + 1) < \epsilon$. obtain ⟨ n, hn ⟩ : ∃ n : ℕ, (b - a) / (n + 1) < ε := by exact ⟨ ⌊ ( b - a ) / ε⌋₊, by rw [ div_lt_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / ε ), mul_div_cancel₀ ( b - a ) εpos.ne' ] ⟩; refine' ⟨ taggedUniformPartition n hab, hε _ ⟩; simp +zetaDelta at *; rw [ abs_of_nonneg ( IPart.mesh_nonneg _ ) ]; convert hn using 1; convert IPart.mesh_uniformPartition n hab using 1

/- A tagged partition where the tags are the left endpoints of the intervals. -/ def tagLeft {a b : ℝ} (P : IPart.IPartition a b) : TaggedPartition a b := { P with xi := fun i => P.x i.castSucc h_xi := fun i => ⟨le_refl _, P.monotone (Nat.le_succ _)⟩ }

/- We finally come to Riemann-Stieltjes sums and to the definition of Riemann-Stieltjes integrals!-/ /- The Riemann-Stieltjes sum S(P, f, g) is the sum of f(xi_i) * (g(x_{i+1}) - g(x_i)). -/ def RSSum {a b : ℝ} (f g : ℝ → ℝ) (P : TaggedPartition a b) : ℝ := ∑ i : Fin P.n, f (P.xi i) * (g (P.x i.succ) - g (P.x i.castSucc))

/- The Riemann-Stieltjes integral of f with respect to g on [a, b] has value I. -/ def HasRSIntegral (f g : ℝ → ℝ) (a b I : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P : TaggedPartition a b), P.mesh ≤ δ → |RSSum f g P - I| < ε

/- HasRSIntegral is equivalent to convergence along the tagged partition filter. -/ lemma hasRSIntegral_iff_tendsto {f g : ℝ → ℝ} {a b I : ℝ} : HasRSIntegral f g a b I ↔ Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by constructor <;> intro h; · refine' Metric.tendsto_nhds.mpr _; intro ε hε; obtain ⟨ δ, hδ, H ⟩ := h ε hε; filter_upwards [ Filter.preimage_mem_comap ( Iio_mem_nhds hδ ) ] with P hP using H P <| le_of_lt hP; · intro ε hε; obtain ⟨ δ, hδ ⟩ := Metric.tendsto_nhds.mp h ε hε; -- Since δ is a neighborhood of 0, there exists a δ' > 0 such that (0, δ') is contained in δ. obtain ⟨δ', hδ'⟩ : ∃ δ' > 0, Metric.ball 0 δ' ⊆ δ := by exact Metric.mem_nhds_iff.mp hδ.1; use δ' / 2; exact ⟨ half_pos hδ'.1, fun P hP => hδ.2 <| hδ'.2 <| mem_ball_zero_iff.mpr <| abs_lt.mpr ⟨ by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ], by linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg _ ] ⟩ ⟩

/- The Cauchy condition for Riemann-Stieltjes sums. -/ def RSCauchyCondition (f g : ℝ → ℝ) (a b : ℝ) : Prop := ∀ ε > 0, ∃ δ > 0, ∀ (P Q : TaggedPartition a b), P.mesh ≤ δ → Q.mesh ≤ δ → |RSSum f g P - RSSum f g Q| < ε

/- The image of the partition filter under the Riemann sum map is a Cauchy filter. -/ lemma cauchy_map_RSSum {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (h : RSCauchyCondition f g a b) : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by have h_cauchy : ∀ ε > 0, ∃ S ∈ TaggedPartitionFilter a b, ∀ P Q : TaggedPartition a b, P ∈ S → Q ∈ S → |RSSum f g P - RSSum f g Q| < ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ := h ε hε_pos use {P : TaggedPartition a b | P.mesh ≤ δ}; exact ⟨ Filter.mem_comap.mpr ⟨ Set.Iic δ, Iic_mem_nhds hδ_pos, by aesop ⟩, hδ ⟩; rw [ Metric.cauchy_iff ]; constructor; · apply_rules [ Filter.map_neBot ]; exact TaggedPartitionFilter_neBot hab; · exact fun ε ε_pos => by obtain ⟨ S, hS₁, hS₂ ⟩ := h_cauchy ε ε_pos; exact ⟨ RSSum f g '' S, Filter.image_mem_map hS₁, by aesop ⟩ ;

/- The Cauchy condition implies the existence of the Riemann-Stieltjes integral. -/ theorem RSCauchy_implies_exists (f g : ℝ → ℝ) (a b : ℝ) (hab : a < b) (h : RSCauchyCondition f g a b) : ∃ I, HasRSIntegral f g a b I := by -- By cauchy_map_RSSum, the filter map (RSSum f g) (TaggedPartitionFilter a b) is Cauchy. have h_cauchy : Cauchy (Filter.map (RSSum f g) (TaggedPartitionFilter a b)) := by exact cauchy_map_RSSum hab h -- Since ℝ is a complete metric space, every Cauchy filter converges. obtain ⟨I, hI⟩ : ∃ I : ℝ, Filter.Tendsto (RSSum f g) (TaggedPartitionFilter a b) (nhds I) := by exact CompleteSpace.complete h_cauchy exact ⟨ I, by simpa only [ hasRSIntegral_iff_tendsto ] using hI ⟩

/- Rewrite S(P) as a double sum using the refinement index map. -/ lemma RSSum_P_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g P = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, f (P.xi i) * IPart.term g R.toIPartition j := by apply Finset.sum_congr rfl fun i _ => ?_; have h_telescope : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, IPart.term g R.toIPartition j = g (R.x (k i.succ)) - g (R.x (k i.castSucc)) := by convert IPart.sum_telescope_block g R.toIPartition ( k i.castSucc ) ( k i.succ ) ( hk ( Nat.le_succ _ ) ) using 1; simp_all +decide [ ← Finset.mul_sum _ _ _, ← Finset.sum_mul ]

/- Term in Riemann sum of R at index j. -/ def term_f_g {a b : ℝ} (f g : ℝ → ℝ) (R : TaggedPartition a b) (j : ℕ) : ℝ := if h : j < R.n then f (R.xi ⟨j, h⟩) * IPart.term g R.toIPartition j else 0

/- Rewrite the Riemann sum of R as a double sum over blocks defined by P, using the index map k. -/ lemma RSSum_R_rewrite {a b : ℝ} (f g : ℝ → ℝ) (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_0 : k 0 = 0) (hk_eq : ∀ i, R.x (k i) = P.x i) : RSSum f g R = ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, term_f_g f g R j := by have h_rewrite : ∑ j ∈ Finset.range R.n, term_f_g f g R j = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, term_f_g f g R j := by -- Since $R.x m = b$ and $R.x$ is monotone, for $j \geq m$, $R.x j = b$. have h_monotone : ∀ j : Fin (R.n + 1), (k (Fin.last P.n)).val ≤ j.val → R.x j = b := by intros j hj have h_monotone : ∀ i j : Fin (R.n + 1), i ≤ j → R.x i ≤ R.x j := by exact R.monotone; have h_monotone : R.x (k (Fin.last P.n)) = b := by exact hk_eq _ ▸ P.right; exact le_antisymm ( by linarith [ R.monotone ( show j ≤ Fin.last _ from Fin.le_last _ ), R.right ] ) ( by linarith [ ‹∀ i j : Fin ( R.n + 1 ), i ≤ j → R.x i ≤ R.x j› _ _ ( show k ( Fin.last P.n ) ≤ j from hj ) ] ); -- Since $R.x j = b$ for $j \geq m$, the term_f_g $f$ $g$ $R$ $j$ is zero for $j \geq m$. have h_zero : ∀ j : ℕ, (k (Fin.last P.n)).val ≤ j → j < R.n → term_f_g f g R j = 0 := by intros j hj₁ hj₂ have h_x_eq_b : R.x ⟨j + 1, by linarith⟩ = b := by exact h_monotone _ ( Nat.le_succ_of_le hj₁ ); have h_x_eq_b : R.x ⟨j, by linarith⟩ = b := by exact h_monotone ⟨ j, by linarith ⟩ hj₁; unfold term_f_g IPart.term; aesop; rw [ ← Finset.sum_range_add_sum_Ico _ ( show ( k ( Fin.last P.n ) |> Fin.val ) ≤ R.n from Nat.le_of_lt_succ <| Fin.is_lt _ ) ]; simp +zetaDelta at *; exact Finset.sum_eq_zero fun x hx => h_zero x ( Finset.mem_Ico.mp hx |>.1 ) ( Finset.mem_Ico.mp hx |>.2 ); convert h_rewrite using 1; · unfold RSSum term_f_g; unfold IPart.term; simp +decide [ Finset.sum_range, Fin.sum_univ_castSucc ] ; rfl; · convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => by simpa using hk hij ) ( fun j => term_f_g f g R j ) using 1; aesop

/- Now we start proving meaningful bounds.-/

/- The sum of absolute differences of g over a partition is bounded by the total variation of g. -/ lemma sum_abs_diff_le_variation {g : ℝ → ℝ} {a b : ℝ} (P : IPart.IPartition a b) (hg : BoundedVariationOn g (Set.Icc a b)) : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by have h_sum_le_eVariationOn : ∑ i : Fin P.n, ENNReal.ofReal (abs ((g (P.x i.succ)) - (g (P.x i.castSucc)))) ≤ eVariationOn g (Set.Icc a b) := by refine' le_trans _ ( eVariationOn.mono g _ ); swap; exact Set.range P.x; · refine' le_trans _ ( le_ciSup _ ⟨ P.n, _ ⟩ ); rotate_right; use fun i => if hi : i ≤ P.n then P.x ⟨ i, by linarith ⟩ else P.x ⟨ P.n, by linarith ⟩; refine' ⟨ _, _ ⟩; intro i j hij; all_goals norm_num [ edist_dist ]; · split_ifs <;> try linarith [ P.monotone ( show ⟨ i, by linarith ⟩ ≤ ⟨ j, by linarith ⟩ from Nat.le_trans hij ( Nat.le_refl _ ) ) ]; · exact P.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by linarith ) ) ( Nat.le_refl _ ) ); · norm_num; · aesop; · rw [ Finset.sum_range ]; refine' Finset.sum_le_sum fun i _ => _; split_ifs <;> norm_num [ Nat.succ_le_iff ]; · rfl; · linarith [ Fin.is_lt i ]; · exact False.elim <| ‹¬ ( i : ℕ ) + 1 ≤ P.n› <| Nat.succ_le_of_lt <| Fin.is_lt i; · linarith [ Fin.is_lt i ]; · rintro _ ⟨ i, rfl ⟩ ; exact ⟨ by linarith [ P.monotone ( show 0 ≤ i from Nat.zero_le _ ), P.left ], by linarith [ P.monotone ( show i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ; convert ENNReal.toReal_mono _ h_sum_le_eVariationOn using 1; · rw [ ENNReal.toReal_sum ] ; aesop; exact fun _ _ => ENNReal.ofReal_ne_top; · exact hg

/- For a fixed interval of the coarser partition, the sum of the weighted differences is bounded by epsilon times the sum of the absolute differences of g. -/ lemma inner_sum_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (i : Fin P.n) : ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, (if h : j < R.n then |(f (P.xi i) - f (R.xi ⟨j, h⟩)) * IPart.term g R.toIPartition j| else 0) ≤ ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, |IPart.term g R.toIPartition j| := by rw [ Finset.mul_sum _ _ _ ]; refine' Finset.sum_le_sum fun j hj => _; split_ifs <;> simp_all +decide [ abs_mul ]; · refine' mul_le_mul_of_nonneg_right ( h_osc i _ _ _ ) ( abs_nonneg _ ); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( show k i.castSucc ≤ ⟨ j, by linarith ⟩ from hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans this.2 ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · positivity

/- The difference between the Riemann sums of a partition and its refinement is bounded by ε times the total variation of g, provided the oscillation of f on the coarser partition is bounded by ε. -/ lemma RSSum_diff_bound {f g : ℝ → ℝ} {a b : ℝ} (P R : TaggedPartition a b) (k : Fin (P.n + 1) → Fin (R.n + 1)) (hk : Monotone k) (hk_eq : ∀ i, R.x (k i) = P.x i) (hk_0 : k 0 = 0) (ε : ℝ) (hε : 0 ≤ ε) (h_osc : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε) (hg : BoundedVariationOn g (Set.Icc a b)) : |RSSum f g P - RSSum f g R| ≤ ε * (eVariationOn g (Set.Icc a b)).toReal := by -- Applying the triangle inequality and the bounds from inner_sum_bound, we get: have h_triangle : abs (RSSum f g P - RSSum f g R) ≤ ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) := by convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun i _ => ?_ using 2; any_goals try infer_instance; rw [ RSSum_P_rewrite f g P R k hk hk_eq, RSSum_R_rewrite f g P R k hk hk_0 hk_eq ]; rw [ ← Finset.sum_sub_distrib ]; rw [ Finset.mul_sum _ _ _ ]; convert Finset.abs_sum_le_sum_abs _ _ |> le_trans <| Finset.sum_le_sum fun j hj => ?_ using 2; rw [ ← Finset.sum_sub_distrib ]; · infer_instance; · infer_instance; · unfold term_f_g; split_ifs <;> simp_all +decide [ abs_mul ]; · rw [ ← sub_mul ]; rw [ abs_mul ]; gcongr; apply h_osc i (R.xi ⟨j, by linarith⟩); · have := R.h_xi ⟨ j, by linarith ⟩; exact le_trans ( hk_eq _ ▸ R.monotone ( Nat.le_trans ( Nat.le_of_lt_succ ( by aesop ) ) hj.1 ) ) this.1; · have := R.h_xi ⟨ j, by linarith ⟩; exact this.2.trans ( hk_eq ( Fin.succ i ) ▸ R.monotone ( Nat.le_of_lt_succ ( by aesop ) ) ); · linarith [ Fin.is_lt ( k i.castSucc ), Fin.is_lt ( k i.succ ) ]; -- Summing over $i$, we get: have h_sum : ∑ i : Fin P.n, ε * ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) ≤ ε * ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by have h_sum : ∑ i : Fin P.n, ∑ j ∈ Finset.Ico (k i.castSucc).val (k i.succ).val, abs (IPart.term g R.toIPartition j) = ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) := by convert IPart.sum_partition_blocks P.n ( fun i => ( k i : ℕ ) ) ( fun i j hij => hk hij ) ( fun j => |IPart.term g R.toIPartition j| ) using 1; aesop; rw [ ← h_sum, Finset.mul_sum _ _ _ ]; -- Since $k (Fin.last P.n) \leq R.n$, we have: have h_last : ∑ j ∈ Finset.Ico 0 (k (Fin.last P.n)).val, abs (IPart.term g R.toIPartition j) ≤ ∑ j ∈ Finset.range R.n, abs (IPart.term g R.toIPartition j) := by exact Finset.sum_le_sum_of_subset_of_nonneg ( fun x hx => Finset.mem_range.mpr ( by linarith [ Finset.mem_Ico.mp hx, Fin.is_lt ( k ( Fin.last P.n ) ) ] ) ) fun _ _ _ => abs_nonneg ; refine le_trans h_triangle <| h_sum.trans <| mul_le_mul_of_nonneg_left ( h_last.trans ? ) hε; convert sum_abs_diff_le_variation R.toIPartition hg using 1; simp +decide [ Finset.sum_range, IPart.term ]; rfl

/- If the mesh of a partition is small enough such that any two points within distance mesh satisfy the uniform continuity condition, then the oscillation of f on each subinterval is bounded. -/ lemma oscillation_bound_of_mesh {f : ℝ → ℝ} {a b : ℝ} {ε δ : ℝ} (P : TaggedPartition a b) (h_mesh : P.mesh ≤ δ) (h_unif : ∀ x y, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε) : ∀ i : Fin P.n, ∀ x ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), |f (P.xi i) - f x| ≤ ε := by intros i x hx have h_dist : |P.xi i - x| ≤ P.mesh := by -- By definition of mesh, we know that for any $i$, $|P.x i - P.x (i + 1)| \leq P.mesh$. have h_mesh_le : ∀ i : Fin P.n, |P.x (i.castSucc) - P.x (i.succ)| ≤ P.mesh := by intro i have h_diff_le_mesh : ∀ i : Fin P.n, P.x i.succ - P.x i.castSucc ≤ P.mesh := by intro i have h_diff_le_mesh : P.x i.succ - P.x i.castSucc ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)) := by exact Finset.mem_image_of_mem _ ( Finset.mem_univ _ ); have h_diff_le_mesh : ∀ y ∈ Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n)), y ≤ P.mesh := by intros y hy have h_diff_le_mesh : y ≤ Finset.max (Finset.image (fun i => P.x i.succ - P.x i.castSucc) (Finset.univ : Finset (Fin P.n))) := by exact Finset.le_max hy; rw [ show P.mesh = ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ).max.getD 0 from ?_ ]; · cases h : Finset.max ( Finset.image ( fun i : Fin P.n => P.x i.succ - P.x i.castSucc ) Finset.univ ) <;> aesop; · exact rfl exact h_diff_le_mesh _ ‹›; rw [ abs_sub_comm, abs_of_nonneg ] <;> linarith [ h_diff_le_mesh i, P.monotone ( show i.castSucc ≤ i.succ from Nat.le_succ _ ) ]; exact abs_le.mpr ⟨ by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ], by linarith [ abs_le.mp ( h_mesh_le i ), P.h_xi i, hx.1, hx.2 ] ⟩; apply h_unif _ _ ? ?_ <| h_dist.trans h_mesh; · exact ⟨ by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ) ], by linarith [ P.h_xi i, P.left, P.right, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ) ] ⟩; · exact ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩

/- If f is continuous and g has bounded variation, the Riemann-Stieltjes sums satisfy the Cauchy condition. -/ lemma RS_cauchy_of_continuous_bv {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : RSCauchyCondition f g a b := by intro ε hε let V := (eVariationOn g (Set.Icc a b)).toReal let ε' := ε / (3 * (V + 1)); -- Choose δ such that |x - y| ≤ δ implies |f x - f y| ≤ ε'. obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ x y : ℝ, x ∈ Set.Icc a b → y ∈ Set.Icc a b → |x - y| ≤ δ → |f x - f y| ≤ ε' := by have := Metric.uniformContinuousOn_iff.mp ( isCompact_Icc.uniformContinuousOn_of_continuous hf ) ε' ( by exact div_pos hε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] ) ); exact ⟨ this.choose / 2, half_pos this.choose_spec.1, fun x y hx hy hxy => le_of_lt ( this.choose_spec.2 x hx y hy ( by rw [ Real.dist_eq ] ; exact lt_of_le_of_lt hxy ( half_lt_self this.choose_spec.1 ) ) ) ⟩; refine' ⟨ δ, hδ_pos, fun P Q hP hQ => _ ⟩; -- Let R_untagged = IPart.union P.toIPart.IPartition Q.toIPart.IPartition. set R_untagged := IPart.union P.toIPartition Q.toIPartition; -- Apply RSSum_diff_bound to (P, R) and (Q, R) to get |S(P) - S(R)| ≤ ε' * V and |S(Q) - S(R)| ≤ ε' * V. have h_diff_P_R : |RSSum f g P - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · apply IPart.index_map_zero; · exact IPart.union_strictMono P.toIPartition Q.toIPartition · exact Classical.choose_spec ( IPart.refinement_index_map P.toIPartition R_untagged ( IPart.union_refines_left P.toIPartition Q.toIPartition ) ) |>.2; · exact div_nonneg hε.le ( mul_nonneg zero_le_three ( add_nonneg ( ENNReal.toReal_nonneg ) zero_le_one ) ); · apply oscillation_bound_of_mesh P hP hδ; · assumption have h_diff_Q_R : |RSSum f g Q - RSSum f g (tagLeft R_untagged)| ≤ ε' * V := by apply RSSum_diff_bound; any_goals assumption; case k => exact fun i => ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |> Classical.choose |> fun k => k i; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.1; · exact Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2; · exact IPart.index_map_zero ( IPart.union_strictMono _ _ ) _ ( Classical.choose_spec ( IPart.refinement_index_map _ _ ( IPart.union_refines_right _ _ ) ) |>.2 ); · positivity; · apply oscillation_bound_of_mesh Q hQ hδ; exact abs_lt.mpr ⟨ by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ], by nlinarith [ abs_le.mp h_diff_P_R, abs_le.mp h_diff_Q_R, show 0 ≤ ε' by positivity, show 0 ≤ V by exact ENNReal.toReal_nonneg, mul_div_cancel₀ ε ( by linarith [ show 0 ≤ V by exact ENNReal.toReal_nonneg ] : ( 3 * ( V + 1 ) ) ≠ 0 ) ] ⟩

/- Auxiliary definitions for the swapped partition points and tags. -/ def swap_x {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 2) → ℝ := Fin.cons a (Fin.snoc P.xi b)

def swap_xi {a b : ℝ} (P : TaggedPartition a b) : Fin (P.n + 1) → ℝ := P.x

/- Given a tagged partition P, construct a 'swapped' partition Q where the points of Q are the tags of P (interleaved with endpoints) and the tags of Q are the points of P. -/ def swapTaggedPartition {a b : ℝ} (P : TaggedPartition a b) : TaggedPartition a b := { n := P.n + 1 x := swap_x P monotone := by intro i j hij; induction' i using Fin.cases with i ih_i ; induction' j using Fin.cases with j ih_j ; aesop; · unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.castSucc_lt_last ]; · have := P.h_xi ( Fin.castLT j ‹› ); linarith [ P.monotone ( show 0 ≤ Fin.castSucc ( Fin.castLT j ‹› ) from Nat.zero_le _ ), P.left ]; · linarith [ P.left, P.right, P.monotone ( show 0 ≤ Fin.last P.n from Nat.zero_le _ ) ]; · induction' j using Fin.inductionOn with j ih_j; · tauto; · cases hij.eq_or_lt <;> simp_all +decide [ swap_x ]; simp +decide [ Fin.snoc, * ]; split_ifs <;> simp_all +decide [ Fin.castLT ]; · -- Since $P$ is a partition, its points are monotone. have h_monotone : Monotone P.x := by exact P.monotone; exact h_monotone ( Nat.succ_le_of_lt ‹› ) |> le_trans ( P.h_xi _ |>.2 ) |> le_trans <| P.h_xi _ |>.1; · exact P.h_xi _ |>.2.trans ( P.monotone ( Fin.le_last _ ) ) |> le_trans <| by linarith [ P.right ] ; · exact absurd ‹› ( not_le_of_gt ( Nat.lt_of_lt_of_le ‹_› ( Nat.le_of_lt_succ ( by linarith [ Fin.is_lt i, Fin.is_lt j ] ) ) ) ) left := by rfl right := by -- By definition of swap_x, the last element of the list is b. simp [swap_x] xi := swap_xi P h_xi := by intro i unfold swap_x swap_xi refine' ⟨ _, _ ⟩; · induction' i using Fin.inductionOn with i IH; · exact P.left.ge; · simp +zetaDelta at *; exact P.h_xi i |>.2; · induction i using Fin.lastCases <;> simp_all +decide [ Fin.cons, Fin.snoc ]; · simp +decide [ Fin.last, Fin.snoc ]; exact P.right.le; · exact P.h_xi _ |>.1 }

/- The Riemann-Stieltjes sum of g with respect to f on P is equal to the boundary terms minus the Riemann-Stieltjes sum of f with respect to g on the swapped partition Q. -/ lemma RSSum_swap_identity {a b : ℝ} {f g : ℝ → ℝ} (P : TaggedPartition a b) : RSSum g f P = f b * g b - f a * g a - RSSum f g (swapTaggedPartition P) := by unfold swapTaggedPartition Integ.RSSum; -- By definition of swap_x and swap_xi, we can expand the sums. have h_expand : ∑ x : Fin (P.n + 1), f (swap_xi P x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) = ∑ x : Fin (P.n + 1), f (P.x x) * (g (swap_x P x.succ) - g (swap_x P x.castSucc)) := by rfl; have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.succ) = f b * g b + ∑ x : Fin P.n, f (P.x x.castSucc) * g (P.xi x) := by rw [ Fin.sum_univ_castSucc ]; simp +decide [ add_comm, swap_x ]; exact Or.inl ( congr_arg f ( P.right ) ); have h_telescope : ∑ x : Fin (P.n + 1), f (P.x x) * g (swap_x P x.castSucc) = f a * g a + ∑ x : Fin P.n, f (P.x x.succ) * g (P.xi x) := by convert Fin.sum_univ_succ _ using 1; congr! 2; · exact P.left.symm ▸ rfl; · unfold swap_x; aesop; simp_all +decide [ mul_sub, sub_mul, mul_comm, Finset.sum_add_distrib, Finset.mul_sum _ _ _ ] ; ring

/- The length of any interval in the swapped partition is at most twice the mesh of the original partition. -/ lemma swap_diff_le {a b : ℝ} (P : TaggedPartition a b) (j : Fin (P.n + 1)) : (swapTaggedPartition P).x j.succ - (swapTaggedPartition P).x j.castSucc ≤ 2 * P.mesh := by cases j using Fin.inductionOn; · unfold swapTaggedPartition; unfold swap_x; rcases P with ⟨ ⟨ n, x, hx ⟩, xi, hxi ⟩; rcases n with ( _ | n ) <;> norm_num [ Fin.cons ] at *; · simp_all +decide [ Fin.snoc ]; unfold IPart.IPartition.mesh; aesop; · have := hxi 0; norm_num at *; linarith! [ hx ( show 0 ≤ 1 from Nat.zero_le _ ), IPart.mesh_ge_diff ⟨ n + 1, x, hx, by tauto, by tauto ⟩ 0 ] ; · unfold swapTaggedPartition; unfold swap_x; simp +decide [ Fin.cons, Fin.snoc ]; split_ifs <;> simp_all +decide [ Fin.ext_iff ]; · rename_i i hi; have h_diff : P.xi (i.succ.castLT hi) ≤ P.x (i.succ.castLT hi).succ ∧ P.x (i.succ.castLT hi).succ ≤ P.x (i.succ.castLT hi).castSucc + P.mesh := by have h_diff : P.x (i.succ.castLT hi).succ - P.x (i.succ.castLT hi).castSucc ≤ P.mesh := by exact IPart.mesh_ge_diff P.toIPartition (i.succ.castLT hi); exact ⟨ P.h_xi _ |>.2, by linarith ⟩; have h_diff : P.x (i.succ.castLT hi).castSucc - P.x i.castSucc ≤ P.mesh := by have h_diff : ∀ j : Fin P.n, P.x (j.succ) - P.x j.castSucc ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; convert h_diff i using 1; linarith [ P.h_xi i ]; · rename_i i hi; -- Since $P.xi i \leq b$, we have $b - P.xi i \leq b - P.xi i$. have h_le : b - P.xi i ≤ P.mesh := by have h_le : b - P.xi i ≤ P.x (Fin.last P.n) - P.x i.castSucc := by have := P.h_xi i; linarith [ P.right, P.monotone ( show i.castSucc ≤ Fin.last P.n from Fin.le_last _ ), P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ) ]; refine le_trans h_le ?_; have h_le : ∀ j : Fin P.n, P.x (Fin.succ j) - P.x (Fin.castSucc j) ≤ P.mesh := by exact fun j ↦ IPart.mesh_ge_diff P.toIPartition j; cases hi.eq_or_lt <;> simp_all +decide [ Fin.eq_last_of_not_lt ]; · convert h_le i using 1; congr ; aesop; · linarith [ Fin.is_lt i ]; linarith [ show 0 ≤ P.mesh from IPart.mesh_nonneg P.toIPartition ]

/- The mesh of the swapped partition is at most twice the mesh of the original partition. -/ lemma mesh_swap_le {a b : ℝ} (P : TaggedPartition a b) : (swapTaggedPartition P).mesh ≤ 2 * P.mesh := by convert Basic.max_getD_le _ _; · exact mul_nonneg zero_le_two ( IPart.mesh_nonneg _ ); · simp +zetaDelta at *; intro i; linarith! [ swap_diff_le P i ] ;

/- Given a partition P and a function g differentiable on [a,b] with derivative g', there exists a tagged partition T with the same points as P such that the increments of g satisfy the Mean Value Theorem equation on each subinterval. -/

lemma mvt_tags_for_partition {a b : ℝ} (P : IPart.IPartition a b) {g : ℝ → ℝ} {g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by have h_mean_value : ∀ i : Fin P.n, ∃ ξ ∈ Set.Icc (P.x i.castSucc) (P.x i.succ), g (P.x i.succ) - g (P.x i.castSucc) = g' ξ * (P.x i.succ - P.x i.castSucc) := by intro i by_cases h_eq : P.x i.castSucc = P.x i.succ; · aesop; · have h_mvt : ∃ ξ ∈ Set.Ioo (P.x i.castSucc) (P.x i.succ), deriv g ξ = (g (P.x i.succ) - g (P.x i.castSucc)) / (P.x i.succ - P.x i.castSucc) := by have := exists_deriv_eq_slope g ( lt_of_le_of_ne ( P.monotone ( Nat.le_succ _ ) ) h_eq ); exact this ( continuousOn_of_forall_continuousAt fun x hx => HasDerivAt.continuousAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ ) ) ( fun x hx => DifferentiableAt.differentiableWithinAt ( hg x ⟨ by linarith [ hx.1, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hx.2, P.monotone ( show Fin.succ i ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.differentiableAt ) ); obtain ⟨ ξ, hξ₁, hξ₂ ⟩ := h_mvt; exact ⟨ ξ, ⟨ hξ₁.1.le, hξ₁.2.le ⟩, by rw [ ← hg ξ ⟨ by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show 0 ≤ Fin.castSucc i from Nat.zero_le _ ), P.left ], by linarith [ hξ₁.1, hξ₁.2, P.monotone ( show i.succ ≤ Fin.last P.n from Fin.le_last _ ), P.right ] ⟩ |> HasDerivAt.deriv, hξ₂, div_mul_cancel₀ _ ( sub_ne_zero_of_ne <| Ne.symm h_eq ) ] ⟩ ; choose ξ hξ using h_mean_value; refine' ⟨ ⟨ P, ξ, fun i => hξ i |>.1 ⟩, rfl, fun i => hξ i |>.2 ⟩

/- The sum of absolute differences of g over P equals the Riemann-Stieltjes sum of |g'| with respect to id over a tagged partition T associated with P. -/ lemma sumAbsDiff_eq_RSSum_abs_deriv {a b : ℝ} (P : IPart.IPartition a b) {g g' : ℝ → ℝ} (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : ∃ T : TaggedPartition a b, T.toIPartition = P ∧ SumDiff.sumAbsDiff g P = RSSum (fun x => |g' x|) (fun x => x) T := by obtain ⟨ T, rfl, hT ⟩ := Integ.mvt_tags_for_partition P hg; refine' ⟨ T, rfl, Finset.sum_congr rfl fun i _ => _ ⟩ ; rw [ hT i ] ; rw [ abs_mul, abs_of_nonneg ( sub_nonneg.2 <| T.monotone <| Nat.le_succ _ ) ]

/- The identity function has bounded variation on [a, b]. -/ lemma boundedVariationOn_id {a b : ℝ} : BoundedVariationOn (fun x => x) (Set.Icc a b) := by unfold BoundedVariationOn; simp +decide [ eVariationOn ]; norm_num [ edist_dist ]; refine' ne_of_lt ( lt_of_le_of_lt ( iSup_le _ ) ENNReal.coe_lt_top ); swap; exact ⟨ |b - a|, abs_nonneg _ ⟩; norm_num [ dist_eq_norm ]; intro n f hf h; rw [ ← ENNReal.ofReal_sum_of_nonneg ] <;> norm_num; -- The sum of the absolute differences of f over the range n is bounded by the total variation of f over [a, b], which is |b - a|. have h_sum_abs_diff : ∑ i ∈ Finset.range n, |f (i + 1) - f i| ≤ |b - a| := by rw [ Finset.sum_congr rfl fun i hi => abs_of_nonneg <| sub_nonneg.mpr <| hf <| Nat.le_succ _ ]; rw [ Finset.sum_range_sub ( fun i => f i ) ]; cases abs_cases ( b - a ) <;> linarith [ h 0, h n ]; exact ENNReal.ofReal_le_of_le_toReal ( by simpa using h_sum_abs_diff )

/- The Riemann-Stieltjes integral of |g'| is less than or equal to the total variation of g. -/ lemma integral_le_variation {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (g' : ℝ → ℝ) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) (hg_bv : BoundedVariationOn g (Set.Icc a b)) (I : ℝ) (hI : RS.Integ.HasRSIntegral (fun x => |g' x|) id a b I) : I ≤ (eVariationOn g (Set.Icc a b)).toReal := by -- Fix ε > 0. have h_eps : ∀ ε > 0, I ≤ (eVariationOn g (Set.Icc a b)).toReal + ε := by intro ε hε_pos obtain ⟨δ, hδ_pos, hδ⟩ : ∃ δ > 0, ∀ (T : Integ.TaggedPartition a b), T.mesh ≤ δ → |Integ.RSSum (fun x => |g' x|) id T - I| < ε := by exact hI ε hε_pos; -- Choose a uniform partition P with mesh ≤ δ. obtain ⟨P, hP⟩ : ∃ P : RS.IPart.IPartition a b, P.mesh ≤ δ := by -- Choose n such that (b - a) / (n + 1) ≤ δ. obtain ⟨n, hn⟩ : ∃ n : ℕ, (b - a) / (n + 1) ≤ δ := by exact ⟨ ⌊ ( b - a ) / δ⌋₊, by rw [ div_le_iff₀ ] <;> nlinarith [ Nat.lt_floor_add_one ( ( b - a ) / δ ), mul_div_cancel₀ ( b - a ) hδ_pos.ne' ] ⟩; exact ⟨ RS.IPart.uniformPartition n hab, by simpa only [ RS.IPart.mesh_uniformPartition ] using hn ⟩; -- Let T be the tagged partition obtained from the Mean Value Theorem applied to g on each subinterval of P. obtain ⟨T, hT⟩ : ∃ T : Integ.TaggedPartition a b, T.toIPartition = P ∧ ∀ i : Fin T.n, g (T.x i.succ) - g (T.x i.castSucc) = g' (T.xi i) * (T.x i.succ - T.x i.castSucc) := by exact RS.Integ.mvt_tags_for_partition P hg; -- By sum_abs_diff_le_variation, we have sumAbsDiff g P ≤ (eVariationOn g (Set.Icc a b)).toReal. have h_sum_abs_diff_le_variation : ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| ≤ (eVariationOn g (Set.Icc a b)).toReal := by exact Integ.sum_abs_diff_le_variation P hg_bv; -- By sumAbsDiff_eq_RSSum_abs_deriv, we have RSSum (fun x => |g' x|) id T = sumAbsDiff g P. have h_RSSum_eq_sumAbsDiff : Integ.RSSum (fun x => |g' x|) id T = ∑ i : Fin P.n, |g (P.x i.succ) - g (P.x i.castSucc)| := by have h_RSSum_eq_sumAbsDiff : Integ.RSSum (fun x => |g' x|) id T = ∑ i : Fin T.n, |g' (T.xi i)| * (T.x i.succ - T.x i.castSucc) := by unfold Integ.RSSum; aesop; rw [ h_RSSum_eq_sumAbsDiff, ← hT.1 ]; exact Finset.sum_congr rfl fun i _ => by rw [ hT.2 i, abs_mul, abs_of_nonneg ( sub_nonneg.mpr <| T.monotone <| Nat.le_succ _ ) ] ; linarith [ abs_lt.mp ( hδ T ( by simpa [ hT.1 ] using hP ) ) ]; exact le_of_forall_pos_le_add h_eps

def mvtTaggedPartition {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : TaggedPartition a b := Classical.choose (mvt_tags_for_partition P hg)

lemma mvtTaggedPartition_spec {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : (mvtTaggedPartition g g' P hg).toIPartition = P ∧ ∀ i : Fin (mvtTaggedPartition g g' P hg).n, g ((mvtTaggedPartition g g' P hg).x i.succ) - g ((mvtTaggedPartition g g' P hg).x i.castSucc) = g' ((mvtTaggedPartition g g' P hg).xi i) * ((mvtTaggedPartition g g' P hg).x i.succ - (mvtTaggedPartition g g' P hg).x i.castSucc) := Classical.choose_spec (mvt_tags_for_partition P hg)

/- The mesh of the MVT tagged partition equals the mesh of the original partition. -/ lemma mesh_mvtTaggedPartition {a b : ℝ} (g g' : ℝ → ℝ) (P : IPart.IPartition a b) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) : (mvtTaggedPartition g g' P hg).mesh = P.mesh := by have h := (mvtTaggedPartition_spec g g' P hg).1 convert congr_arg _ h using 1;

/- For any partition P, the sum of absolute differences of g over P is bounded by I + ε. -/ lemma sumAbsDiff_le_integral_add_eps {a b : ℝ} (hab : a < b) (g g' : ℝ → ℝ) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) (I : ℝ) (hI : HasRSIntegral (fun x => |g' x|) id a b I) (ε : ℝ) (hε : 0 < ε) (P : IPart.IPartition a b) : RS.SumDiff.sumAbsDiff g P ≤ I + ε := by -- Use hI with ε to get η > 0 such that for any tagged partition T with T.mesh ≤ η, |RSSum - I| < ε. obtain ⟨η, hη_pos, hη⟩ : ∃ η > 0, ∀ (T : TaggedPartition a b), T.mesh ≤ η → |RSSum (fun x => |g' x|) (fun x => x) T - I| < ε := by exact hI ε hε; -- Use exists_refinement_mesh_le to find a refinement Q of P with Q.mesh ≤ η. obtain ⟨Q, hQ_ref, hQ_mesh⟩ : ∃ Q : IPart.IPartition a b, IPart.IsRefinement Q P ∧ Q.mesh ≤ η := by exact IPart.exists_refinement_mesh_le hab P hη_pos; -- Use sumAbsDiff_mono_of_refinement to show sumAbsDiff g P ≤ sumAbsDiff g Q. have h_sum_abs_diff_mono : SumDiff.sumAbsDiff g P ≤ SumDiff.sumAbsDiff g Q := by exact SumDiff.sumAbsDiff_le_sumAbsDiff_of_refinement hQ_ref; -- Construct a tagged partition T from Q using mvtTaggedPartition. obtain ⟨T, hT⟩ : ∃ T : TaggedPartition a b, T.toIPartition = Q ∧ SumDiff.sumAbsDiff g Q = RSSum (fun x => |g' x|) (fun x => x) T := by have := sumAbsDiff_eq_RSSum_abs_deriv Q hg; exact this; -- Show T.mesh = Q.mesh ≤ η. have hT_mesh : T.mesh = Q.mesh := by exact hT.1 ▸ rfl generalize_proofs at *; ( linarith [ abs_lt.mp ( hη T ( by linarith ) ) ])

/- The total variation of g on [a, b] is bounded by the Riemann-Stieltjes integral of |g'| with respect to id. -/ lemma variation_le_integral_abs_deriv {a b : ℝ} (hab : a < b) (g g' : ℝ → ℝ) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) (hg' : ContinuousOn g' (Set.Icc a b)) (I : ℝ) (hI : RS.Integ.HasRSIntegral (fun x => |g' x|) id a b I) : (eVariationOn g (Set.Icc a b)).toReal ≤ I := by apply le_of_forall_pos_le_add intro ε hε apply SumDiff.variation_le_of_forall_sumAbsDiff_le hab g (I + ε) intro P apply sumAbsDiff_le_integral_add_eps hab g g' hg I hI ε hε P

lemma boundedVariationOn_of_forall_sumAbsDiff_le {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) (K : ℝ) (hK : ∀ P : RS.IPart.IPartition a b, RS.SumDiff.sumAbsDiff g P ≤ K) : BoundedVariationOn g (Set.Icc a b) := by -- By definition of BoundedVariationOn, if for all partitions P of [a, b], the sum of absolute differences of g over P is bounded by K, then g has bounded variation on [a, b]. apply ne_of_lt; refine' lt_of_le_of_lt ( iSup_le _ ) _; exact ENNReal.ofReal K; · simp +decide [ edist_dist ]; -- By the properties of the partition, the sum of the absolute differences over the partition P is equal to the sum of the absolute differences of g over the partition P. have h_sum_eq : ∀ (n : ℕ) (x : Fin (n + 1) → ℝ) (hx : Monotone x) (hx_mem : ∀ i, x i ∈ Set.Icc a b), ∑ i : Fin n, |g (x i.succ) - g (x i.castSucc)| ≤ K := by exact fun n x hx hx_mem ↦ SumDiff.sum_le_K_of_forall_sumAbsDiff_le hab g K hK n x hx hx_mem; intro n x hx hx'; specialize h_sum_eq n ( fun i => x i ) ( fun i j hij => hx hij ) ( fun i => hx' i ) ; simp_all +decide [ dist_eq_norm, Finset.sum_range ] ; simpa only [ ENNReal.ofReal_sum_of_nonneg fun _ _ => abs_nonneg _ ] using ENNReal.ofReal_le_ofReal h_sum_eq; · exact ENNReal.ofReal_lt_top

lemma boundedVariationOn_of_hasRSIntegral_abs_deriv {a b : ℝ} (hab : a < b) (g g' : ℝ → ℝ) (hg : ∀ x ∈ Set.Icc a b, HasDerivAt g (g' x) x) (I : ℝ) (hI : HasRSIntegral (fun x => |g' x|) id a b I) : BoundedVariationOn g (Set.Icc a b) := by apply boundedVariationOn_of_forall_sumAbsDiff_le hab g I; intro P; -- By the properties of the Riemann-Stieltjes integral, we know that the sum of the absolute differences of g over P is less than or equal to I. have h_sum_le_I : ∀ ε > 0, RS.SumDiff.sumAbsDiff g P ≤ I + ε := by exact fun ε a_1 ↦ sumAbsDiff_le_integral_add_eps hab g g' hg I hI ε a_1 P; exact le_of_forall_pos_le_add h_sum_le_I

namespace MVThms

/- If f is continuous on [a, b] and g has bounded variation on [a, b], then the Riemann-Stieltjes integral of f with respect to g exists. -/ theorem Theorem_A1 {f g : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b)) (hg : BoundedVariationOn g (Set.Icc a b)) : ∃ I, HasRSIntegral f g a b I := by apply RSCauchy_implies_exists; · exact hab · apply RS_cauchy_of_continuous_bv hab hf hg

/- Integration by parts for Riemann-Stieltjes integrals: if f is integrable with respect to g, then g is integrable with respect to f, and the integral is given by the integration by parts formula.

It would perhaps (?) be better to refactor this entire file so that we're not using explicit constants but instead systematically using big Oh notation... The punchline would be: