Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Geometry.CAT0Fillings.Currents.Basic
import OAI.Geometry.CAT0Fillings.Compactness.Ascoli
import OAI.Geometry.CAT0Fillings.Compactness.Convolution

namespace OAI

section

open Set Filter MeasureTheory Metric ContinuousLinearMap
open scoped Topology NNReal ENNReal Convolution

namespace CAT0Fillings.JointBV
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure]

theorem totallyBounded_L1_of_translation {ι : Type*}
    (f : ι → E → ℝ) (hf : ∀ i, Integrable (f i) μ)
    (M : ℝ≥0) (hM : ∀ i, (∫ x, |f i x| ∂μ) ≤ M)
    (R : ℝ) (hs : ∀ i, Function.support (f i) ⊆ closedBall (0:E) R)
    {C : ℝ} (hC : 0 ≤ C)
    (ht : ∀ i w, (∫ x, |f i (x+w)-f i x| ∂μ) ≤ C*‖w‖) :
    TotallyBounded (range fun i => (hf i).toL1 (f i)) := by
  apply totallyBounded_range_of_uniform_approx
  intro ε hε
  let δ := ε/(C+1)
  have hδ : 0 < δ := div_pos hε (by linarith)
  let η : ContDiffBump (0:E) := ⟨δ/2,δ,half_pos hδ,by linarith⟩
  let ρ := η.normed μ
  let g (i) := f i ⋆[lsmul ℝ ℝ, μ] ρ
  have hρc : HasCompactSupport ρ := η.hasCompactSupport_normed
  have hρ : Continuous ρ := η.continuous_normed
  have hρi : Integrable ρ μ := hρ.integrable_of_hasCompactSupport hρc
  have hg (i) : Integrable (g i) μ := (hf i).integrable_convolution (lsmul ℝ ℝ) hρi
  obtain ⟨K,hK⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hρc (η.contDiff_normed (n := (⊤ : ℕ∞))) (by simp)
  obtain ⟨B,hB⟩ := hρ.bounded_above_of_compact_support hρc
  have hB0 : 0 ≤ B := (norm_nonneg (ρ 0)).trans (hB 0)
  have hgs (i) : Function.support (g i) ⊆ closedBall (0:E) (R+δ) := by
    apply convolution_scalar_support (hs i)
    dsimp [ρ]
    rw [η.support_normed_eq]
    exact ball_subset_closedBall
  have hgb (i) (x) : |g i x| ≤ B*M :=
    convolution_scalar_bound (hf i) hB0 (fun y => hB y) (hM i) x
  have hgl (i) : LipschitzWith (K*M) (g i) :=
    convolution_scalar_lipschitz (hf i) hρc hK (hM i)
  refine ⟨fun i => (hg i).toL1 (g i),
    totallyBounded_L1_of_compact_lipschitz_family g hg (K*M) hgl (B*M) (R+δ) hgb hgs,?_⟩
  intro i
  rw [dist_toL1_eq_integral]
  have he := mollifier_error_of_translation (hf i) hC (ht i) η
  have hcomm : (ρ ⋆[lsmul ℝ ℝ, μ] f i) = g i := by
    have hflip : (lsmul ℝ ℝ).flip = lsmul ℝ ℝ := by ext; simp
    simpa only [hflip,g] using (convolution_flip (L := lsmul ℝ ℝ) (f := f i) (g := ρ) (μ := μ))
  rw [hcomm] at he
  apply he.trans_lt
  change C*δ < ε
  dsimp [δ]
  rw [←mul_div_assoc]
  apply (div_lt_iff₀ (by linarith : 0 < C+1)).mpr
  nlinarith

end CAT0Fillings.JointBV
end

section

open Set Filter MeasureTheory Metric
open scoped Topology ENNReal NNReal

namespace CAT0Fillings.JointBV
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure]

lemma integral_le_of_closedBall_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E}
    [μ.IsAddHaarMeasure] {f : E → ℝ} (hf : Integrable f μ) {C : ℝ}
    (hb : ∀ r : ℝ, (∫ x in closedBall (0:E) r, f x ∂μ) ≤ C) :
    (∫ x, f x ∂μ) ≤ C := by
  have hi : IntegrableOn f (⋃ n : ℕ, closedBall (0:E) n) μ := by
    simpa only [iUnion_closedBall_nat,integrableOn_univ] using hf
  have hl := tendsto_setIntegral_of_monotone (fun n : ℕ => measurableSet_closedBall)
    (fun i j hij => closedBall_subset_closedBall (Nat.cast_le.mpr hij)) hi
  simp only [iUnion_closedBall_nat,Measure.restrict_univ] at hl
  exact le_of_tendsto hl (Eventually.of_forall fun n : ℕ => hb n)

lemma full_segment_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {f : E → ℝ} {f' : E → E →L[ℝ] ℝ}
    (hf : ∀ x, HasFDerivAt f (f' x) x) (hf' : Continuous f') (x w : E) :
    |f (x+w)-f x| ≤ ‖w‖*∫ t in Icc (0:ℝ) 1, ‖f' (x+t • w)‖ := by
  let p (t : ℝ) := x+t • w
  have hp (t : ℝ) : HasDerivAt p w t := by
    simpa only [p,one_smul,id_eq] using ((hasDerivAt_id t).smul_const w).const_add x
  have hd (t : ℝ) := (hf (p t)).comp_hasDerivAt t (hp t)
  have hc : Continuous (fun t => f' (p t) w) := by dsimp [p]; fun_prop
  have hi := intervalIntegral.integral_eq_sub_of_hasDerivAt
    (a := (0:ℝ)) (b := 1) (fun t _ => hd t) (hc.intervalIntegrable _ _)
  change (∫ t in (0:ℝ)..1, f' (p t) w) = f (p 1)-f (p 0) at hi
  simp only [p,zero_smul,one_smul,add_zero] at hi
  rw [←hi]
  apply (intervalIntegral.abs_integral_le_integral_abs zero_le_one).trans
  have hh := intervalIntegral.integral_mono_on (μ := volume) zero_le_one
    (hc.abs.intervalIntegrable _ _)
    ((show Continuous (fun t : ℝ => ‖w‖*‖f' (p t)‖) by dsimp [p]; fun_prop).intervalIntegrable _ _)
    (fun t _ => by simpa only [Real.norm_eq_abs,mul_comm] using (f' (p t)).le_opNorm w)
  apply hh.trans_eq
  rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_of_le zero_le_one,
    ←integral_Icc_eq_integral_Ioc]
lemma smooth_translation_integral_bound {f : E → ℝ} {f' : E → E →L[ℝ] ℝ}
    (hf : ∀ x, HasFDerivAt f (f' x) x) (hf' : Continuous f')
    (hfi : Integrable f μ) (hgi : Integrable (fun x => ‖f' x‖) μ) (w : E) :
    (∫ x, |f (x+w)-f x| ∂μ) ≤ ‖w‖*∫ x, ‖f' x‖ ∂μ := by
  let m : Measure ℝ := volume.restrict (Icc 0 1)
  have : IsFiniteMeasure m := by dsimp [m]; infer_instance
  have he (t : ℝ) : (∫ x, ‖f' (x+t • w)‖ ∂μ) = ∫ x, ‖f' x‖ ∂μ :=
    integral_add_right_eq_self (fun x => ‖f' x‖) (t • w)
  have hp : Integrable (fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) (μ.prod m) := by
    apply (integrable_prod_iff' (show AEStronglyMeasurable
      (fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) (μ.prod m) from
        (show Continuous (fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) by fun_prop).aestronglyMeasurable)).mpr
    refine ⟨Eventually.of_forall fun t => hgi.comp_add_right (t • w),?_⟩
    simp only [norm_norm,he]
    exact integrable_const (μ := m) _
  have hh := integral_mono ((hfi.comp_add_right w).sub hfi).abs
    (hp.integral_prod_left.const_mul ‖w‖) (fun x => full_segment_bound hf hf' x w)
  rw [integral_const_mul,integral_integral_swap hp] at hh
  simp only [he,integral_const,smul_eq_mul] at hh
  have hm : m.real univ = 1 := by simp [m,measureReal_def,Real.volume_Icc]
  simpa only [←measureReal_def,hm,one_mul,Pi.sub_apply] using hh
end CAT0Fillings.JointBV
end
end OAI

	import Mathlib
	import OAI.Geometry.CAT0Fillings.Currents.Basic
	import OAI.Geometry.CAT0Fillings.Compactness.Ascoli
	import OAI.Geometry.CAT0Fillings.Compactness.Convolution

	namespace OAI

	section

	open Set Filter MeasureTheory Metric ContinuousLinearMap
	open scoped Topology NNReal ENNReal Convolution

	namespace CAT0Fillings.JointBV
	variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
	[MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure]

	theorem totallyBounded_L1_of_translation {ι : Type*}
	(f : ι → E → ℝ) (hf : ∀ i, Integrable (f i) μ)
	(M : ℝ≥0) (hM : ∀ i, (∫ x, \|f i x\| ∂μ) ≤ M)
	(R : ℝ) (hs : ∀ i, Function.support (f i) ⊆ closedBall (0:E) R)
	{C : ℝ} (hC : 0 ≤ C)
	(ht : ∀ i w, (∫ x, \|f i (x+w)-f i x\| ∂μ) ≤ C*‖w‖) :
	TotallyBounded (range fun i => (hf i).toL1 (f i)) := by
	apply totallyBounded_range_of_uniform_approx
	intro ε hε
	let δ := ε/(C+1)
	have hδ : 0 < δ := div_pos hε (by linarith)
	let η : ContDiffBump (0:E) := ⟨δ/2,δ,half_pos hδ,by linarith⟩
	let ρ := η.normed μ
	let g (i) := f i ⋆[lsmul ℝ ℝ, μ] ρ
	have hρc : HasCompactSupport ρ := η.hasCompactSupport_normed
	have hρ : Continuous ρ := η.continuous_normed
	have hρi : Integrable ρ μ := hρ.integrable_of_hasCompactSupport hρc
	have hg (i) : Integrable (g i) μ := (hf i).integrable_convolution (lsmul ℝ ℝ) hρi
	obtain ⟨K,hK⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hρc (η.contDiff_normed (n := (⊤ : ℕ∞))) (by simp)
	obtain ⟨B,hB⟩ := hρ.bounded_above_of_compact_support hρc
	have hB0 : 0 ≤ B := (norm_nonneg (ρ 0)).trans (hB 0)
	have hgs (i) : Function.support (g i) ⊆ closedBall (0:E) (R+δ) := by
	apply convolution_scalar_support (hs i)
	dsimp [ρ]
	rw [η.support_normed_eq]
	exact ball_subset_closedBall
	have hgb (i) (x) : \|g i x\| ≤ B*M :=
	convolution_scalar_bound (hf i) hB0 (fun y => hB y) (hM i) x
	have hgl (i) : LipschitzWith (K*M) (g i) :=
	convolution_scalar_lipschitz (hf i) hρc hK (hM i)
	refine ⟨fun i => (hg i).toL1 (g i),
	totallyBounded_L1_of_compact_lipschitz_family g hg (KM) hgl (BM) (R+δ) hgb hgs,?_⟩
	intro i
	rw [dist_toL1_eq_integral]
	have he := mollifier_error_of_translation (hf i) hC (ht i) η
	have hcomm : (ρ ⋆[lsmul ℝ ℝ, μ] f i) = g i := by
	have hflip : (lsmul ℝ ℝ).flip = lsmul ℝ ℝ := by ext; simp
	simpa only [hflip,g] using (convolution_flip (L := lsmul ℝ ℝ) (f := f i) (g := ρ) (μ := μ))
	rw [hcomm] at he
	apply he.trans_lt
	change C*δ < ε
	dsimp [δ]
	rw [←mul_div_assoc]
	apply (div_lt_iff₀ (by linarith : 0 < C+1)).mpr
	nlinarith

	end CAT0Fillings.JointBV
	end

	section

	open Set Filter MeasureTheory Metric
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings.JointBV
	variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
	[MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure]

	lemma integral_le_of_closedBall_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
	[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E}
	[μ.IsAddHaarMeasure] {f : E → ℝ} (hf : Integrable f μ) {C : ℝ}
	(hb : ∀ r : ℝ, (∫ x in closedBall (0:E) r, f x ∂μ) ≤ C) :
	(∫ x, f x ∂μ) ≤ C := by
	have hi : IntegrableOn f (⋃ n : ℕ, closedBall (0:E) n) μ := by
	simpa only [iUnion_closedBall_nat,integrableOn_univ] using hf
	have hl := tendsto_setIntegral_of_monotone (fun n : ℕ => measurableSet_closedBall)
	(fun i j hij => closedBall_subset_closedBall (Nat.cast_le.mpr hij)) hi
	simp only [iUnion_closedBall_nat,Measure.restrict_univ] at hl
	exact le_of_tendsto hl (Eventually.of_forall fun n : ℕ => hb n)

	lemma full_segment_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
	[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
	{f : E → ℝ} {f' : E → E →L[ℝ] ℝ}
	(hf : ∀ x, HasFDerivAt f (f' x) x) (hf' : Continuous f') (x w : E) :
	\|f (x+w)-f x\| ≤ ‖w‖*∫ t in Icc (0:ℝ) 1, ‖f' (x+t • w)‖ := by
	let p (t : ℝ) := x+t • w
	have hp (t : ℝ) : HasDerivAt p w t := by
	simpa only [p,one_smul,id_eq] using ((hasDerivAt_id t).smul_const w).const_add x
	have hd (t : ℝ) := (hf (p t)).comp_hasDerivAt t (hp t)
	have hc : Continuous (fun t => f' (p t) w) := by dsimp [p]; fun_prop
	have hi := intervalIntegral.integral_eq_sub_of_hasDerivAt
	(a := (0:ℝ)) (b := 1) (fun t _ => hd t) (hc.intervalIntegrable _ _)
	change (∫ t in (0:ℝ)..1, f' (p t) w) = f (p 1)-f (p 0) at hi
	simp only [p,zero_smul,one_smul,add_zero] at hi
	rw [←hi]
	apply (intervalIntegral.abs_integral_le_integral_abs zero_le_one).trans
	have hh := intervalIntegral.integral_mono_on (μ := volume) zero_le_one
	(hc.abs.intervalIntegrable _ _)
	((show Continuous (fun t : ℝ => ‖w‖*‖f' (p t)‖) by dsimp [p]; fun_prop).intervalIntegrable _ _)
	(fun t _ => by simpa only [Real.norm_eq_abs,mul_comm] using (f' (p t)).le_opNorm w)
	apply hh.trans_eq
	rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_of_le zero_le_one,
	←integral_Icc_eq_integral_Ioc]

	lemma smooth_translation_integral_bound {f : E → ℝ} {f' : E → E →L[ℝ] ℝ}
	(hf : ∀ x, HasFDerivAt f (f' x) x) (hf' : Continuous f')
	(hfi : Integrable f μ) (hgi : Integrable (fun x => ‖f' x‖) μ) (w : E) :
	(∫ x, \|f (x+w)-f x\| ∂μ) ≤ ‖w‖*∫ x, ‖f' x‖ ∂μ := by
	let m : Measure ℝ := volume.restrict (Icc 0 1)
	have : IsFiniteMeasure m := by dsimp [m]; infer_instance
	have he (t : ℝ) : (∫ x, ‖f' (x+t • w)‖ ∂μ) = ∫ x, ‖f' x‖ ∂μ :=
	integral_add_right_eq_self (fun x => ‖f' x‖) (t • w)
	have hp : Integrable (fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) (μ.prod m) := by
	apply (integrable_prod_iff' (show AEStronglyMeasurable
	(fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) (μ.prod m) from
	(show Continuous (fun p : E × ℝ => ‖f' (p.1+p.2 • w)‖) by fun_prop).aestronglyMeasurable)).mpr
	refine ⟨Eventually.of_forall fun t => hgi.comp_add_right (t • w),?_⟩
	simp only [norm_norm,he]
	exact integrable_const (μ := m) _
	have hh := integral_mono ((hfi.comp_add_right w).sub hfi).abs
	(hp.integral_prod_left.const_mul ‖w‖) (fun x => full_segment_bound hf hf' x w)
	rw [integral_const_mul,integral_integral_swap hp] at hh
	simp only [he,integral_const,smul_eq_mul] at hh
	have hm : m.real univ = 1 := by simp [m,measureReal_def,Real.volume_Icc]
	simpa only [←measureReal_def,hm,one_mul,Pi.sub_apply] using hh

	end CAT0Fillings.JointBV
	end

	end OAI