lean/OAI/Analysis/DiskMaximal/Potentials.lean

import OAI.Analysis.DiskMaximal.Envelopes

namespace OAI

open MeasureTheory Set
open scoped ENNReal NNReal Topology ContDiff
open scoped RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology ENNReal
open scoped Real RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology RealInnerProductSpace ContDiff
open MeasureTheory Set Filter
open scoped Topology Interval NNReal
open MeasureTheory Set Filter
open scoped Topology NNReal Real
open MeasureTheory Set Metric
open scoped ENNReal NNReal Topology
open MeasureTheory Set
open scoped ENNReal NNReal
open MeasureTheory Set
open scoped ENNReal NNReal
open MeasureTheory Set
open scoped ENNReal NNReal Topology
open MeasureTheory Set Metric
open scoped ENNReal NNReal
open MeasureTheory Set
open scoped ENNReal NNReal
open Finset
open scoped RealInnerProductSpace
open MeasureTheory Set Metric
open scoped ENNReal NNReal Topology
open MeasureTheory Set
open scoped ENNReal NNReal RealInnerProductSpace
open MeasureTheory Set
open scoped ENNReal NNReal Topology
open MeasureTheory Set Metric
open scoped ENNReal NNReal RealInnerProductSpace
open scoped RealInnerProductSpace
open scoped RealInnerProductSpace
open scoped ContDiff
open Finset
open MeasureTheory Set Metric Filter
open scoped ENNReal NNReal RealInnerProductSpace ContDiff Topology
open MeasureTheory MeasureTheory.Measure Set Filter
open scoped ENNReal NNReal Topology RealInnerProductSpace ContDiff
open MeasureTheory Set Metric Filter Finset
open scoped ENNReal NNReal RealInnerProductSpace ContDiff Topology
open MeasureTheory Set
open scoped RealInnerProductSpace

noncomputable section

namespace PlanePotential
open LocalCancellation

lemma green_restrict_density {ψ : ℂ → ℝ} {R X : ℝ}
    (hψs : ∀ z, ψ z ≠ 0 → ‖z‖ ≤ R) (hRX : R ≤ X)
    (β : ℂ → ℂ) {x : ℂ} (hx : ‖x‖ ≤ X) (y : ℂ) :
    ⟪green ψ x y, ({y : ℂ | ‖y‖ ≤ X}.indicator β) y⟫ = ⟪green ψ x y,β y⟫ := by
  by_cases hy : ‖y‖ ≤ X
  · simp [hy]
  · rw [green_eq_zero_of_outside hψs hRX hx (lt_of_not_ge hy)]
    simp

lemma green_root_lp_local {ψ : ℂ → ℝ} {β : ℂ → ℂ}
    (hψm : Measurable ψ) {R X B : ℝ}
    (hψs : ∀ z, ψ z ≠ 0 → ‖z‖ ≤ R) (hψB : ∀ z, |ψ z| ≤ B)
    (hRX : R ≤ X) (hM : 0 < R+X) (hB : 0 ≤ B) (hX : 0 < X)
    (hβ : Integrable β) :
    (∫⁻ x in {x : ℂ | ‖x‖ ≤ X}, (ENNReal.ofReal |∫ y, ⟪green ψ x y,β y⟫|)^(3/2:ℝ)) ≤
    ENNReal.ofReal ((2*B*(R+X)^2)^(3/2:ℝ)*(4*Real.pi*(4*X)^(1/2:ℝ))) *
      (ENNReal.ofReal (∫ y, ‖β y‖))^(3/2:ℝ) := by
  let β' := {y : ℂ | ‖y‖ ≤ X}.indicator β
  have hi : Integrable β' := hβ.indicator (measurableSet_le measurable_norm measurable_const)
  have hs : ∀ y, β' y ≠ 0 → ‖y‖ ≤ X := by
    intro y hy
    by_contra hn
    exact hy (by simp [β',hn])
  have h := green_root_lp hψm hψs hψB hM hB (show 0 < X+X by linarith) hi hs
  have he : (fun x => (ENNReal.ofReal |∫ y, ⟪green ψ x y,β' y⟫|)^(3/2:ℝ)) =ᵐ[volume.restrict {x : ℂ | ‖x‖ ≤ X}]
      (fun x => (ENNReal.ofReal |∫ y, ⟪green ψ x y,β y⟫|)^(3/2:ℝ)) := by
    filter_upwards [ae_restrict_mem (measurableSet_le measurable_norm measurable_const)] with x hx
    congr 3
    exact integral_congr_ae (ae_of_all _ (green_restrict_density hψs hRX β hx))
  rw [lintegral_congr_ae he] at h
  have hm : (∫ y, ‖β' y‖) ≤ ∫ y, ‖β y‖ :=
    integral_mono hi.norm hβ.norm (fun y => norm_indicator_le_norm_self β y)
  have hm' := ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal hm) (by norm_num : (0:ℝ) ≤ 3/2)
  have h' := h.trans (mul_le_mul_right hm' _)
  simpa only [show 2*(X+X)=4*X by ring] using h'

lemma green_zero_piece_lp_ae {ψ : ℂ → ℝ} {β : ℂ → ℂ}
    (hψm : Measurable ψ) {R X B L a : ℝ}
    (hψs : ∀ z, ψ z ≠ 0 → ‖z‖ ≤ R) (hψB : ∀ z, |ψ z| ≤ B)
    (hψL : ∀ u v, |ψ u-ψ v| ≤ L*‖u-v‖)
    (hM : 0 < R+X) (hB : 0 ≤ B) (hL : 0 ≤ L) (ha : 0 < a)
    (z : ℂ) (hβ : Integrable β) (hzero : ∫ y, β y = 0)
    (hβs : ∀ᵐ y, β y ≠ 0 → ‖y-z‖ ≤ a) :
    (∫⁻ x in {x : ℂ | ‖x‖ ≤ X},
      (ENNReal.ofReal |∫ y, ⟪green ψ x y,β y⟫|)^(3/2:ℝ)) ≤
    ENNReal.ofReal ((8*Real.pi*(2*B*(R+X)^2)^(3/2:ℝ)+
      Real.pi*(8*B*(R+X)^2+(64/3:ℝ)*L*(R+X)^3)^(3/2:ℝ))*a^(1/2:ℝ)) *
      (ENNReal.ofReal (∫ y, ‖β y‖))^(3/2:ℝ) := by
  let β' := {y : ℂ | ‖y-z‖ ≤ a}.indicator β
  have he : β' =ᵐ[volume] β := by
    filter_upwards [hβs] with y hy
    by_cases hs : ‖y-z‖ ≤ a
    · simp [β',hs]
    · have hz : β y = 0 := by by_contra hn; exact hs (hy hn)
      simp [β',hs,hz]
  have hi : Integrable β' := hβ.congr he.symm
  have hz : ∫ y, β' y = 0 := (integral_congr_ae he).trans hzero
  have hs : ∀ y, β' y ≠ 0 → ‖y-z‖ ≤ a := by
    intro y hy
    by_contra hn
    exact hy (by simp [β',hn])
  have hp (x : ℂ) : (∫ y, ⟪green ψ x y,β' y⟫) = ∫ y, ⟪green ψ x y,β y⟫ :=
    integral_congr_ae (he.mono (fun y hy => by simpa only using congrArg (fun z => ⟪green ψ x y,z⟫) hy))
  have hm : (∫ y, ‖β' y‖) = ∫ y, ‖β y‖ := integral_congr_ae (he.mono (fun y hy => by simpa only using congrArg norm hy))
  simpa only [hp,hm] using green_zero_piece_lp hψm hψs hψB hψL hM hB hL ha z hi hz hs

/-- The local decomposition theorem does not demand artificial bounded support
of the original densities: the actual Green kernel vanishes outside the
observation ball. -/
theorem green_decomposition_norm_local {ι : Type*} [Fintype ι]
    {ψ g : ℂ → ℝ} (hψ : Continuous ψ) (hψc : HasCompactSupport ψ)
    (hg : ContDiff ℝ 1 g) {D R X B : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (hψ1 : ∫ z, ψ z = 1)
    (hψs : ∀ z, ψ z ≠ 0 → ‖z‖ ≤ R) (hψB : ∀ z, |ψ z| ≤ B)
    (hRX : R ≤ X) (hM : 0 < R+X) (hB : 0 ≤ B) (hX : 0 < X)
    (β : ι → ℂ → ℂ) (hβ : ∀ i, Integrable (β i))
    (hdecomp : ∀ᵐ y, ‖y‖ ≤ X → gradient g y = ∑ i, β i y) :
    eLpNorm (fun x => g x-∫ z, ψ z*g z) (3/2) (volume.restrict {x : ℂ | ‖x‖ ≤ X}) ≤
      ∑ i, eLpNorm (fun x => ∫ y, ⟪green ψ x y,β i y⟫) (3/2)
        (volume.restrict {x : ℂ | ‖x‖ ≤ X}) := by
  let β' := fun i => {y : ℂ | ‖y‖ ≤ X}.indicator (β i)
  have hi (i : ι) : Integrable (β' i) := (hβ i).indicator (measurableSet_le measurable_norm measurable_const)
  have hs (i : ι) (y : ℂ) (hy : β' i y ≠ 0) : ‖y‖ ≤ X := by
    by_contra hn
    exact hy (by simp [β',hn])
  have hd : ∀ᵐ y, ‖y‖ ≤ X → gradient g y = ∑ i, β' i y := by
    filter_upwards [hdecomp] with y hy hY
    simpa [β',hY] using hy hY
  have h := green_decomposition_norm hψ hψc hg hD hψ1 hψs hψB hRX hM hB
    (show 0 < X+X by linarith) β' hi hs hd
  apply h.trans_eq
  apply Finset.sum_congr rfl
  intro i _
  apply eLpNorm_congr_ae
  filter_upwards [ae_restrict_mem (measurableSet_le measurable_norm measurable_const)] with x hx
  exact integral_congr_ae (ae_of_all _ (green_restrict_density hψs hRX (β i) hx))

/-- Uniform fractional small-support gain expressed in the common price
exponent 1/100, including the unrestricted unit-size terms. -/
theorem exists_green_price : ∃ (ψ : ℂ → ℝ) (C : ℝ), 0 < C ∧
    ContDiff ℝ ∞ ψ ∧ HasCompactSupport ψ ∧ (∫ z, ψ z) = 1 ∧
    (∀ z, ψ z ≠ 0 → ‖z‖ ≤ 1) ∧
    ∀ b : ErrorPiece 2,
      eLpNorm (fun x => ∫ y, ⟪green ψ x y,b.density y⟫) (3/2)
        (volume.restrict {x : ℂ | ‖x‖ ≤ 8}) ≤
        ENNReal.ofReal (C*b.size^(1/100:ℝ)*b.mass) := by
  obtain ⟨ψ,B,L,hB,hL,hψ,hψc,hψ1,hψs,hψB,hψL⟩ := exists_kernel_cutoff
  let A : ℝ := (2*B*(1+8)^2)^(3/2:ℝ)*(4*Real.pi*(4*8)^(1/2:ℝ))
  let Z : ℝ := 8*Real.pi*(2*B*(1+8)^2)^(3/2:ℝ)+
    Real.pi*(8*B*(1+8)^2+(64/3:ℝ)*L*(1+8)^3)^(3/2:ℝ)
  have hA : 0 ≤ A := by dsimp [A]; positivity
  have hZ : 0 ≤ Z := by dsimp [Z]; positivity
  let C : ℝ := A^(2/3:ℝ)+Z^(2/3:ℝ)*2^(1/3:ℝ)+1
  have hC : 0 < C := by dsimp [C]; positivity
  refine ⟨ψ,C,hC,hψ,hψc,hψ1,hψs,?_⟩
  intro b
  have hb0 := b.size_pos.le
  have hm0 := b.mass_nonneg
  rw [eLpNorm_eq_eLpNorm' (by norm_num : (3/2:ℝ≥0∞) ≠ 0)
    (ENNReal.div_ne_top (by norm_num) (by norm_num))
    (potential_aestronglyMeasurable _ (green_measurable ψ hψ.continuous.measurable) b.integrable)]
  norm_num only [ENNReal.toReal_div, ENNReal.toReal_ofNat]
  by_cases hb : b.size < 1
  · have ha : 0 < 2*b.size := mul_pos (by norm_num) b.size_pos
    have hp := green_zero_piece_lp_ae hψ.continuous.measurable hψs hψB hψL (by norm_num : (0:ℝ) < 1+8)
      hB hL ha b.center b.integrable (b.zero_mass hb) (b.support hb)
    have hn := eLpNorm_three_halves_bound _ hp
    change _ ≤ (ENNReal.ofReal (Z*(2*b.size)^(1/2:ℝ)))^(2/3:ℝ)*ENNReal.ofReal b.mass at hn
    have he : (ENNReal.ofReal (Z*(2*b.size)^(1/2:ℝ)))^(2/3:ℝ)*ENNReal.ofReal b.mass =
        ENNReal.ofReal (Z^(2/3:ℝ)*2^(1/3:ℝ)*b.size^(1/3:ℝ)*b.mass) := by
      rw [ENNReal.ofReal_rpow_of_nonneg (by positivity) (by norm_num),← ENNReal.ofReal_mul (by positivity)]
      congr 1
      rw [Real.mul_rpow hZ (by positivity),← Real.rpow_mul (by positivity),show (1/2:ℝ)*(2/3)=1/3 by norm_num,
        Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) b.size_pos.le]
      ring
    rw [he] at hn
    apply hn.trans (ENNReal.ofReal_le_ofReal _)
    have hbpow : b.size^(1/3:ℝ) ≤ b.size^(1/100:ℝ) :=
      Real.rpow_le_rpow_of_exponent_ge b.size_pos b.size_le_one (by norm_num)
    have hZC : Z^(2/3:ℝ)*2^(1/3:ℝ) ≤ C := by dsimp [C]; linarith [Real.rpow_nonneg hA (2/3)]
    exact mul_le_mul_of_nonneg_right
      (le_trans (mul_le_mul_of_nonneg_left hbpow (by positivity))
        (mul_le_mul_of_nonneg_right hZC (by positivity))) b.mass_nonneg
  · have hb1 : b.size = 1 := le_antisymm b.size_le_one (le_of_not_gt hb)
    have hp := green_root_lp_local hψ.continuous.measurable hψs hψB (by norm_num : (1:ℝ) ≤ 8)
      (by norm_num : (0:ℝ) < 1+8) hB (by norm_num : (0:ℝ) < 8) b.integrable
    have hn := eLpNorm_three_halves_bound _ hp
    change _ ≤ (ENNReal.ofReal A)^(2/3:ℝ)*ENNReal.ofReal b.mass at hn
    rw [ENNReal.ofReal_rpow_of_nonneg hA (by norm_num),← ENNReal.ofReal_mul (by positivity)] at hn
    simp only [hb1,Real.one_rpow,mul_one]
    apply hn.trans (ENNReal.ofReal_le_ofReal _)
    apply mul_le_mul_of_nonneg_right _ b.mass_nonneg
    dsimp [C]
    linarith [mul_nonneg (Real.rpow_nonneg hZ (2/3)) (Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) (1/3))]
end PlanePotential

namespace DiskMaximalTools

lemma strong_three_halves_norm {f : ℂ → ℝ} (hf : Measurable f)
    (hf0 : ∀ x, 0 ≤ f x) [IsFiniteMeasure (volume.withDensity (fun x => ENNReal.ofReal (f x)))] :
    eLpNorm (fun x => (rationalMaximal (volume.withDensity (fun x => ENNReal.ofReal (f x))) x).toReal)
      (3/2) volume ≤ ENNReal.ofReal ((150*2^(1/2:ℝ))^(2/3:ℝ))*eLpNorm f (3/2) volume := by
  have h := strong_rationalMaximal hf hf0 (by norm_num : (1:ℝ) < 3/2)
  have hs := ENNReal.rpow_le_rpow h (by norm_num : (0:ℝ) ≤ 2/3)
  rw [ENNReal.mul_rpow_of_nonneg _ _ (by norm_num : (0:ℝ) ≤ 2/3)] at hs
  have hp : (50*(3/2:ℝ)*2^((3/2:ℝ)-1)/((3/2:ℝ)-1)) = 150*2^(1/2:ℝ) := by
    norm_num
    ring
  rw [hp,ENNReal.ofReal_rpow_of_nonneg (by positivity) (by norm_num)] at hs
  rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num : (3/2:ℝ≥0∞) ≠ 0)
    (ENNReal.div_ne_top (by norm_num) (by norm_num))
    (measurable_rationalMaximal _).ennreal_toReal.aestronglyMeasurable,
    eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num : (3/2:ℝ≥0∞) ≠ 0)
    (ENNReal.div_ne_top (by norm_num) (by norm_num)) hf.aestronglyMeasurable]
  norm_num only [ENNReal.toReal_div,ENNReal.toReal_ofNat,one_div_div]
  simp only [← ofReal_norm,Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg,
    abs_of_nonneg (hf0 _),ENNReal.ofReal_rpow_of_nonneg (hf0 _) (by norm_num : (0:ℝ) ≤ 3/2),
    ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg (by norm_num : (0:ℝ) ≤ 3/2)] at hs ⊢
  exact hs

lemma lintegral_le_three_halves_norm {f : ℂ → ℝ} (hf : AEStronglyMeasurable f volume)
    (hf0 : ∀ x, 0 ≤ f x) {Q : Set ℂ} (hQ : volume Q ≤ 1) :
    (∫⁻ x in Q, ENNReal.ofReal (f x)) ≤ eLpNorm f (3/2) volume := by
  have h := eLpNorm_le_eLpNorm_mul_rpow_measure_univ
    (show (1:ℝ≥0∞) ≤ 3/2 by rw [ENNReal.le_div_iff_mul_le (by norm_num) (by norm_num)]; norm_num) hf.restrict (μ := volume.restrict Q)
  have he : (1 / ENNReal.toReal 1 - 1 / ENNReal.toReal (3/2)) = (1/3:ℝ) := by norm_num
  rw [he,Measure.restrict_apply_univ] at h
  have hpow : (volume Q)^(1/3:ℝ) ≤ 1 := by
    simpa using ENNReal.rpow_le_rpow hQ (by norm_num : (0:ℝ) ≤ 1/3)
  calc
    _ = eLpNorm f 1 (volume.restrict Q) := by
      rw [eLpNorm_one_eq_lintegral_enorm hf.restrict]
      simp only [← ofReal_norm,Real.norm_eq_abs,abs_of_nonneg (hf0 _)]
    _ ≤ eLpNorm f (3/2) (volume.restrict Q)*(volume Q)^(1/3:ℝ) := h
    _ ≤ eLpNorm f (3/2) (volume.restrict Q) := by simpa using mul_le_mul_right hpow _
    _ ≤ _ := eLpNorm_mono_measure _ Measure.restrict_le_self
end DiskMaximalTools

namespace PlanePotential
open LocalCancellation

/-- The complete local L^(3/2) value estimate for a genuine finite vector
cancellation decomposition, with the common 1/100 price. -/
theorem local_value_lp : ∃ C : ℝ, 0 < C ∧
    ∀ {ι : Type} [Fintype ι] (b : ι → ErrorPiece 2) {g : ℂ → ℝ},
    ContDiff ℝ 1 g → (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) →
    (∀ᵐ y, ‖y‖ ≤ 8 → gradient g y = ∑ i, (b i).density y) →
    ∃ c : ℝ, eLpNorm (fun x => g x-c) (3/2) (volume.restrict {x : ℂ | ‖x‖ ≤ 8}) ≤
      ENNReal.ofReal (C * errorPrice b (1/100)) := by
  obtain ⟨ψ,C,hC,hψ,hψc,hψ1,hψs,hprice⟩ := exists_green_price
  obtain ⟨B,hB⟩ := hψc.exists_bound_of_continuous hψ.continuous
  have hB0 : 0 ≤ B := (norm_nonneg (ψ 0)).trans (hB 0)
  have hψB (z : ℂ) : |ψ z| ≤ B := by simpa only [Real.norm_eq_abs] using hB z
  refine ⟨C,hC,?_⟩
  intro ι _ b g hg hD hdecomp
  obtain ⟨D,hD⟩ := hD
  refine ⟨∫ z, ψ z*g z,?_⟩
  apply (green_decomposition_norm_local hψ.continuous hψc hg hD hψ1 hψs hψB
    (by norm_num : (1:ℝ) ≤ 8) (by norm_num : (0:ℝ) < 1+8) hB0 (by norm_num : (0:ℝ) < 8)
    (fun i => (b i).density) (fun i => (b i).integrable) hdecomp).trans
  apply (Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hprice (b i))).trans_eq
  rw [← ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (mul_nonneg hC.le (Real.rpow_nonneg (b i).size_pos.le _)) (b i).mass_nonneg)]
  congr 1
  rw [errorPrice,Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro i _
  ring
end PlanePotential

namespace DiskMoments
open DiskMaximalTools

lemma signedAverage_sub_const {g : ℂ → ℝ} (hg : Continuous g) (c : ℝ) (x : ℂ) {t : ℝ} (ht : 0 < t) :
    signedAverage (fun y => g y-c) x t = signedAverage g x t-c := by
  have hi : IntegrableOn g (Metric.ball x t) :=
    (hg.continuousOn.integrableOn_compact (isCompact_closedBall x t)).mono_set Metric.ball_subset_closedBall
  have hc : IntegrableOn (fun _ : ℂ => c) (Metric.ball x t) := integrableOn_const measure_ball_lt_top.ne
  unfold signedAverage
  rw [integral_sub hi hc,integral_const]
  have hv : (volume (Metric.ball x t)).toReal = Real.pi*t^2 := by
    rw [volume_disk_real x ht.le,ENNReal.toReal_ofReal (by positivity)]
  simp only [Measure.real,Measure.restrict_apply_univ,hv,smul_eq_mul]
  field_simp

/-- A true disk average of a continuous function is controlled by the local
absolute-density maximal function. No maximizing-radius selection is used. -/
lemma local_average_le {g : ℂ → ℝ} (hg : Continuous g) (c : ℝ)
    {x : ℂ} (hx : ‖x‖ ≤ 2) {t : ℝ} (ht : 0 < t) (ht4 : t ≤ 4) :
    ENNReal.ofReal |signedAverage g x t-c| ≤
      rationalMaximal (volume.withDensity (fun y => ENNReal.ofReal
        (({y : ℂ | ‖y‖ ≤ 8}.indicator (fun y => |g y-c|)) y))) x := by
  let f : ℂ → ℝ := {y : ℂ | ‖y‖ ≤ 8}.indicator (fun y => |g y-c|)
  have hf : Measurable f := ((hg.measurable.sub measurable_const).abs).indicator
    (measurableSet_le measurable_norm measurable_const)
  have hfi : Integrable f := by
    apply IntegrableOn.integrable_indicator _ (measurableSet_le measurable_norm measurable_const)
    apply ContinuousOn.integrableOn_compact
      (show IsCompact {y : ℂ | ‖y‖ ≤ 8} by simpa only [Metric.closedBall,dist_zero_right] using isCompact_closedBall (0:ℂ) 8)
    exact (hg.sub continuous_const).abs.continuousOn
  have hsub (y : ℂ) (hy : y ∈ Metric.ball x t) : ‖y‖ ≤ 8 := by
    have hy' : ‖y-x‖ < t := by simpa only [Metric.mem_ball,dist_eq_norm] using hy
    have hh := norm_add_le (y-x) x
    rw [sub_add_cancel] at hh
    linarith
  have he : (∫ y in Metric.ball x t, f y) = ∫ y in Metric.ball x t, |g y-c| := by
    apply setIntegral_congr_fun measurableSet_ball
    intro y hy
    simp [f,hsub y hy]
  have hav : |signedAverage g x t-c| ≤ (∫ y in Metric.ball x t, f y)/(Real.pi*t^2) := by
    rw [← signedAverage_sub_const hg c x ht,signedAverage,abs_div,abs_of_pos (by positivity : 0 < Real.pi*t^2),he]
    apply div_le_div_of_nonneg_right _ (by positivity)
    simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun y => g y-c) (μ := volume.restrict (Metric.ball x t))
  have hν : (volume.withDensity (fun y => ENNReal.ofReal (f y))) (Metric.ball x t) =
      ENNReal.ofReal (∫ y in Metric.ball x t, f y) := by
    rw [withDensity_apply _ measurableSet_ball]
    exact (ofReal_integral_eq_lintegral_ofReal hfi.integrableOn
      (ae_of_all _ (fun y => by dsimp [f]; exact Set.indicator_nonneg (fun _ _ => abs_nonneg _ ) y))).symm
  apply (ENNReal.ofReal_le_ofReal hav).trans
  rw [ENNReal.ofReal_div_of_pos (by positivity : 0 < Real.pi*t^2),← hν,← volume_disk_real x ht.le]
  exact disk_average_le_rationalMaximal_exact _ x ht
end DiskMoments

namespace PlanePotential
open LocalCancellation DiskMaximalTools DiskMoments

/-- Full local value-oscillation price, for every pair of positive radii at
most four. The measurable majorant is an actual centered Euclidean disk
maximal function, not an assumed analytic input. -/
theorem local_value_majorant : ∃ C : ℝ, 0 < C ∧
    ∀ {ι : Type} [Fintype ι] (b : ι → ErrorPiece 2) {g : ℂ → ℝ},
    ContDiff ℝ 1 g → (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) →
    (∀ᵐ y, ‖y‖ ≤ 8 → gradient g y = ∑ i, (b i).density y) →
    ∀ Q : Set ℂ, volume Q ≤ ENNReal.ofReal 1 →
    ∃ F : ℂ → ℝ≥0∞, Measurable F ∧
      (∫⁻ x in Q, F x) ≤ ENNReal.ofReal (C*errorPrice b (1/100)) ∧
      ∀ x, ‖x‖ ≤ 2 → ∀ t t', 0 < t → t ≤ 4 → 0 < t' → t' ≤ 4 →
        ENNReal.ofReal |signedAverage g x t-signedAverage g x t'| ≤ F x := by
  obtain ⟨C,hC,hlocal⟩ := local_value_lp
  let K : ℝ := (150*2^(1/2:ℝ))^(2/3:ℝ)
  have hK : 0 < K := by dsimp [K]; positivity
  refine ⟨2*K*C,by positivity,?_⟩
  intro ι _ b g hg hD hdecomp Q hQ
  obtain ⟨c,hc⟩ := hlocal b hg hD hdecomp
  let f : ℂ → ℝ := {y : ℂ | ‖y‖ ≤ 8}.indicator (fun y => |g y-c|)
  have hf : Measurable f := ((hg.continuous.measurable.sub measurable_const).abs).indicator
    (measurableSet_le measurable_norm measurable_const)
  have hf0 (y : ℂ) : 0 ≤ f y := Set.indicator_nonneg (fun _ _ => abs_nonneg _) y
  have hfi : Integrable f := by
    apply IntegrableOn.integrable_indicator _ (measurableSet_le measurable_norm measurable_const)
    apply ContinuousOn.integrableOn_compact
      (show IsCompact {y : ℂ | ‖y‖ ≤ 8} by simpa only [Metric.closedBall,dist_zero_right] using isCompact_closedBall (0:ℂ) 8)
    exact (hg.continuous.sub continuous_const).abs.continuousOn
  let ν := volume.withDensity (fun y => ENNReal.ofReal (f y))
  have hν : ν univ = ENNReal.ofReal (∫ y, f y) := by
    rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
    exact (ofReal_integral_eq_lintegral_ofReal hfi (ae_of_all _ hf0)).symm
  have : IsFiniteMeasure ν := ⟨by rw [hν]; exact ENNReal.ofReal_lt_top⟩
  let M : ℂ → ℝ := fun x => (rationalMaximal ν x).toReal
  have hM : Measurable M := (measurable_rationalMaximal ν).ennreal_toReal
  have hMf : ∀ x, rationalMaximal ν x ≠ ⊤ := by
    obtain ⟨B,hB⟩ := (isCompact_closedBall (0:ℂ) 8).exists_bound_of_continuousOn
      (hg.continuous.sub (continuous_const (y := c))).continuousOn
    have hbnd (y : ℂ) : f y ≤ max B 0 := by
      by_cases hy : ‖y‖ ≤ 8
      · have hby := hB y (by simpa only [Metric.mem_closedBall,dist_zero_right] using hy)
        simpa [f, hy, Real.norm_eq_abs] using hby.trans (le_max_left B 0)
      · simp [f, hy]
    intro x
    exact ne_of_lt ((rationalMaximal_le_of_density_le (fun y => ENNReal.ofReal_le_ofReal (hbnd y)) x).trans_lt ENNReal.ofReal_lt_top)
  have hfnorm : eLpNorm f (3/2) volume = eLpNorm (fun x => g x-c) (3/2) (volume.restrict {x : ℂ | ‖x‖ ≤ 8}) := by
    rw [eLpNorm_indicator_eq_eLpNorm_restrict (measurableSet_le measurable_norm measurable_const)]
    simpa only [Real.norm_eq_abs] using eLpNorm_norm (fun x => g x-c) (p := (3/2:ℝ≥0∞)) (μ := volume.restrict {x : ℂ | ‖x‖ ≤ 8})
      (hg.continuous.measurable.sub measurable_const).aestronglyMeasurable
  have hn : eLpNorm M (3/2) volume ≤ ENNReal.ofReal K*ENNReal.ofReal (C*errorPrice b (1/100)) := by
    apply (strong_three_halves_norm hf hf0).trans
    apply mul_le_mul_right _ _
    simpa only [hfnorm] using hc
  refine ⟨fun x => ENNReal.ofReal 2*ENNReal.ofReal (M x),measurable_const.mul hM.ennreal_ofReal,?_,?_⟩
  · rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
    apply (mul_le_mul_right (lintegral_le_three_halves_norm hM.aestronglyMeasurable
      (fun _ => ENNReal.toReal_nonneg) (by simpa using hQ)) _).trans
    apply (mul_le_mul_right hn _).trans_eq
    rw [← ENNReal.ofReal_mul hK.le,← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)]
    congr 1
    ring
  · intro x hx t t' ht ht4 ht' ht4'
    have ha : |signedAverage g x t-c| ≤ M x :=
      (ENNReal.ofReal_le_iff_le_toReal (hMf x)).mp (local_average_le hg.continuous c hx ht ht4)
    have hb : |signedAverage g x t'-c| ≤ M x :=
      (ENNReal.ofReal_le_iff_le_toReal (hMf x)).mp (local_average_le hg.continuous c hx ht' ht4')
    change ENNReal.ofReal _ ≤ ENNReal.ofReal 2 * ENNReal.ofReal (M x)
    rw [← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)]
    apply ENNReal.ofReal_le_ofReal
    have hh := abs_sub_le (signedAverage g x t) c (signedAverage g x t')
    rw [abs_sub_comm c] at hh
    linarith
end PlanePotential

namespace AffineDensities
lemma fderiv_affine_bound {g : ℂ → ℝ} (hg : Differentiable ℝ g) (p : ℂ) (r : ℝ)
    {D : ℝ} (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) :
    ∃ B : ℝ, ∀ X, ‖fderiv ℝ (fun Y => g (p+r • Y)) X‖ ≤ B := by
  refine ⟨D*|r|,?_⟩
  intro X
  have hA : HasFDerivAt (fun Y : ℂ => p+r • Y)
      (r • ContinuousLinearMap.id ℝ ℂ) X := by
    simpa using ((hasFDerivAt_id X).const_smul r).const_add p
  have hd := (hg (p+r • X)).hasFDerivAt.comp X hA
  simp only [Function.comp_def] at hd
  rw [hd.fderiv]
  apply (ContinuousLinearMap.opNorm_comp_le _ _).trans
  simp only [norm_smul,Real.norm_eq_abs,ContinuousLinearMap.norm_id, mul_one]
  exact mul_le_mul_of_nonneg_right (hD _) (abs_nonneg _)
end AffineDensities

namespace CommonPrices
open DyadicHats ConeCoordinates LocalCancellation AffineDensities DirectionNeighbors Finset DiskMoments PricePieces

/-- The actual common-price value estimate, including every terminal error and
all roots. This is the third local analytic input to signed run assembly. -/
theorem value_cell_price : ∃ C : ℝ, 0 < C ∧
    ∀ {N : ℕ} {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) (hN : R/r+1 ≤ (N:ℝ))
    {g : ℂ → ℝ}, ContDiff ℝ 1 g → Integrable (gradient g) →
    (∀ y, gradient g y ≠ 0 → ‖y‖ ≤ R) →
    (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) → ∀ (k : ℕ) (p : ℤ × ℤ)
    {n : ℂ}, n ∈ fixedPalette.labels → ∀ Q : Set ℂ, volume Q ≤ ENNReal.ofReal 1 →
    ∃ F : ℂ → ℝ≥0∞, Measurable F ∧
      (∫⁻ X in Q, F X) ≤ ENNReal.ofReal (C*(scale r k)⁻¹*
        (rootMass (gradient g) r k (cellRoots N p)+cellPrice N r (gradient g) k (cellRoots N p))) ∧
      ∀ X, ‖X‖ ≤ 2 → ∀ t t', 0 < t → t ≤ 4 → 0 < t' → t' ≤ 4 →
        ENNReal.ofReal |signedAverage (fun Y => g (point (scale r k) p+scale r k • Y)) X t-
          signedAverage (fun Y => g (point (scale r k) p+scale r k • Y)) X t'| ≤ F X := by
  obtain ⟨C,hC,hmajor⟩ := PlanePotential.local_value_majorant
  refine ⟨4*C,by positivity,?_⟩
  intro N r R hr hR hN g hg hh hsupp hD k p n hn Q hQ
  let b := sizeErrors hr hh k (point (scale r k) p) (cellRoots N p) n
  have hg' : ContDiff ℝ 1 (fun Y => g (point (scale r k) p+scale r k • Y)) := hg.comp (by fun_prop)
  obtain ⟨D,hD⟩ := hD
  have hD' := fderiv_affine_bound (hg.differentiable one_ne_zero) (point (scale r k) p) (scale r k) hD
  have hd : ∀ᵐ X, ‖X‖ ≤ 8 →
      gradient (fun Y => g (point (scale r k) p+scale r k • Y)) X = ∑ i, (b i).density X := by
    filter_upwards with X hX
    exact (sizeErrors_gradient hr hR hN (hg.differentiable one_ne_zero) hh hsupp k p
      (fixedPalette.unit n hn) X (by linarith)).symm
  obtain ⟨F,hFm,hFi,hFb⟩ := hmajor b hg' hD' hd Q hQ
  refine ⟨F,hFm,?_,hFb⟩
  apply hFi.trans
  apply ENNReal.ofReal_le_ofReal
  have hb := sizeErrors_price hr hR hN hh hsupp k (point (scale r k) p) (cellRoots N p) hn
  have hDp : 0 ≤ cellPrice N r (gradient g) k (cellRoots N p) :=
    sum_nonneg (fun q _ => rootPrice_nonneg hr hR hN hh hsupp k q)
  change errorPrice b (1/100) ≤ _ at hb
  have hrk := scale_pos hr k
  nlinarith [mul_le_mul_of_nonneg_left hb hC.le,inv_nonneg.mpr hrk.le,
    mul_nonneg (inv_nonneg.mpr hrk.le) hDp]
end CommonPrices

namespace PhysicalCells
open MeasureTheory Set

def unitCell : Set ℂ := {X | X.re ∈ Ico (0:ℝ) 1 ∧ X.im ∈ Ico (0:ℝ) 1}
def toUnit (p : ℂ) (r : ℝ) (y : ℂ) : ℂ := r⁻¹ • (y-p)
def cell (p : ℂ) (r : ℝ) : Set ℂ := toUnit p r ⁻¹' unitCell

lemma unitCell_measurable : MeasurableSet unitCell :=
  (Complex.measurable_re measurableSet_Ico).inter (Complex.measurable_im measurableSet_Ico)
lemma toUnit_continuous (p : ℂ) (r : ℝ) : Continuous (toUnit p r) := by unfold toUnit; fun_prop
lemma cell_measurable (p : ℂ) (r : ℝ) : MeasurableSet (cell p r) :=
  (toUnit_continuous p r).measurable unitCell_measurable
lemma volume_unitCell : volume unitCell = 1 := by
  have he := Complex.volume_preserving_equiv_real_prod.measure_preimage
    (s := Ico (0:ℝ) 1 ×ˢ Ico (0:ℝ) 1) (measurableSet_Ico.prod measurableSet_Ico).nullMeasurableSet
  change volume unitCell = _ at he
  rw [he]
  change (volume.prod volume) (Ico (0:ℝ) 1 ×ˢ Ico (0:ℝ) 1) = 1
  rw [Measure.prod_prod]
  norm_num
lemma norm_unitCell {X : ℂ} (hX : X ∈ unitCell) : ‖X‖ ≤ 2 := by
  have h := Complex.norm_le_abs_re_add_abs_im X
  rw [abs_of_nonneg hX.1.1,abs_of_nonneg hX.2.1] at h
  linarith [hX.1.2,hX.2.2]
lemma toUnit_affine (p : ℂ) {r : ℝ} (hr : r ≠ 0) (X : ℂ) : toUnit p r (p+r • X) = X := by
  simp [toUnit,hr]
lemma affine_toUnit (p : ℂ) {r : ℝ} (hr : r ≠ 0) (y : ℂ) : p+r • toUnit p r y = y := by
  simp [toUnit,hr]

lemma lintegral_smul (F : ℂ → ℝ≥0∞) {r : ℝ} (hr : r ≠ 0) :
    (∫⁻ X : ℂ, F (r • X)) = ENNReal.ofReal ((r^2)⁻¹)*(∫⁻ y, F y) := by
  have he := (lintegral_map_equiv (μ := volume) F
    (Homeomorph.smul (isUnit_iff_ne_zero.mpr hr).unit).toMeasurableEquiv).symm
  change (∫⁻ X : ℂ, F (r • X)) = ∫⁻ y, F y ∂Measure.map (fun X : ℂ => r • X) volume at he
  rw [he,Measure.map_addHaar_smul volume hr,lintegral_smul_measure]
  simp only [Complex.finrank_real_complex,abs_of_nonneg (inv_nonneg.mpr (sq_nonneg r)),smul_eq_mul]

lemma lintegral_toUnit (F : ℂ → ℝ≥0∞) (p : ℂ) {r : ℝ} (hr : r ≠ 0) :
    (∫⁻ y : ℂ, F (toUnit p r y)) = ENNReal.ofReal (r^2)*(∫⁻ X, F X) := by
  calc
    (∫⁻ y : ℂ, F (r⁻¹ • (y-p))) = ∫⁻ y : ℂ, F (r⁻¹ • y) :=
      lintegral_sub_right_eq_self (μ := (volume : Measure ℂ)) (fun y : ℂ => F (r⁻¹ • y)) p
    _ = _ := by rw [lintegral_smul _ (inv_ne_zero hr),inv_pow,inv_inv]

/-- Exact physical-space change of variables, with the two-dimensional volume
factor and no integrability or measurability assumptions on the majorant. -/
lemma lintegral_cell (F : ℂ → ℝ≥0∞) (p : ℂ) {r : ℝ} (hr : r ≠ 0) :
    (∫⁻ y in cell p r, F y) = ENNReal.ofReal (r^2)*∫⁻ X in unitCell, F (p+r • X) := by
  rw [← lintegral_indicator (cell_measurable p r),← lintegral_indicator unitCell_measurable]
  have he (y : ℂ) : (cell p r).indicator F y =
      unitCell.indicator (fun X => F (p+r • X)) (toUnit p r y) := by
    by_cases hy : y ∈ cell p r
    · rw [Set.indicator_of_mem hy,Set.indicator_of_mem (show toUnit p r y ∈ unitCell from hy),affine_toUnit p hr]
    · rw [Set.indicator_of_notMem hy,Set.indicator_of_notMem (show toUnit p r y ∉ unitCell from hy)]
  simp_rw [he]
  exact lintegral_toUnit _ p hr

lemma cell_majorant {F : ℂ → ℝ≥0∞} (p : ℂ) {r : ℝ} (hr : r ≠ 0) :
    (∫⁻ y in cell p r, F (toUnit p r y)) = ENNReal.ofReal (r^2)*∫⁻ X in unitCell, F X := by
  rw [lintegral_cell _ p hr]
  simp only [toUnit_affine p hr]
end PhysicalCells

namespace DiskMoments
lemma signedAverage_affine (g : ℂ → ℝ) (p X : ℂ) {r t : ℝ} (hr : 0 < r) (ht : 0 < t) :
    signedAverage (fun Y => g (p+r • Y)) X t = signedAverage g (p+r • X) (r*t) := by
  rw [signedAverage_fixedDomain _ _ ht,signedAverage_fixedDomain _ _ (mul_pos hr ht)]
  congr 1
  apply setIntegral_congr_fun measurableSet_ball
  intro Z _
  change g (p+r • (X+t • Z)) = g (p+r • X+(r*t) • Z)
  congr 1
  rw [smul_add,smul_smul]
  abel

lemma gradient_signedAverage {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (x : ℂ) {t : ℝ} (ht : 0 < t) :
    gradient (fun y => signedAverage g y t) x =
      (Real.pi*t^2)⁻¹ • ∫ y in Metric.ball x t, gradient g y := by
  have hi : IntegrableOn (fun z : ℂ => fderiv ℝ g (x+t • z)) (Metric.ball 0 1) :=
    (((hg.continuous_fderiv one_ne_zero).comp (by fun_prop)).continuousOn.integrableOn_compact
      (isCompact_closedBall (0:ℂ) 1)).mono_set Metric.ball_subset_closedBall
  rw [gradient,(hasFDerivAt_signedAverage hg hD x ht).fderiv,map_smul]
  have he := (InnerProductSpace.toDual ℝ ℂ).symm.toContinuousLinearMap.integral_comp_comm hi
  change (∫ z in Metric.ball (0:ℂ) 1, (InnerProductSpace.toDual ℝ ℂ).symm (fderiv ℝ g (x+t • z))) =
    (InnerProductSpace.toDual ℝ ℂ).symm (∫ z in Metric.ball (0:ℂ) 1, fderiv ℝ g (x+t • z)) at he
  rw [← he]
  change (1/Real.pi) • (∫ z in Metric.ball (0:ℂ) 1, gradient g (x+t • z)) = _
  rw [integral_scale_ball (fun z => gradient g (x+z)) ht,integral_translate_ball,smul_smul]
  congr 1
  ring

lemma gradient_average_affine {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (p X : ℂ) {r t : ℝ} (hr : 0 < r) (ht : 0 < t) :
    gradient (fun Y => signedAverage (fun Z => g (p+r • Z)) Y t) X =
      r • gradient (fun y => signedAverage g y (r*t)) (p+r • X) := by
  have he : (fun Y => signedAverage (fun Z => g (p+r • Z)) Y t) =
      (fun Y => signedAverage g (p+r • Y) (r*t)) := by
    funext Y
    exact signedAverage_affine g p Y hr ht
  rw [he]
  exact AffineDensities.gradient_affine (fun y =>
    (hasFDerivAt_signedAverage hg hD y (mul_pos hr ht)).differentiableAt) p r X

def PhysicalNeighborhoodRadius (g : ℂ → ℝ) (r : ℝ) (x : ℂ) (t : ℝ) : Prop :=
  t ∈ Icc r (2*r) ∧ ∀ s, 0 < s → |s-t| ≤ r/4 → signedAverage g x s ≤ signedAverage g x t

def PhysicalEndpointRadius (g : ℂ → ℝ) (r : ℝ) (x : ℂ) (t : ℝ) : Prop :=
  t ∈ Icc r (2*r) ∧ (t = r ∨ t = 2*r ∨ ∀ s ∈ Icc r (2*r), signedAverage g x s ≤ signedAverage g x t)

lemma neighborhoodRadius_affine {g : ℂ → ℝ} (p X : ℂ) {r t : ℝ} (hr : 0 < r)
    (ht : PhysicalNeighborhoodRadius g r (p+r • X) (r*t)) :
    NeighborhoodRadius (fun Y => g (p+r • Y)) X t := by
  have ht1 : 1 ≤ t := (le_of_mul_le_mul_left (by simpa only [mul_one] using ht.1.1) hr)
  have ht2 : t ≤ 2 := (le_of_mul_le_mul_left (by simpa only [mul_comm 2 r] using ht.1.2) hr)
  refine ⟨⟨ht1,ht2⟩,?_⟩
  intro s hs habs
  rw [signedAverage_affine _ _ _ hr hs,signedAverage_affine _ _ _ hr (by linarith)]
  apply ht.2 (r*s) (mul_pos hr hs)
  rw [← mul_sub,abs_mul,abs_of_pos hr]
  nlinarith

lemma endpointRadius_affine {g : ℂ → ℝ} (p X : ℂ) {r t : ℝ} (hr : 0 < r)
    (ht : PhysicalEndpointRadius g r (p+r • X) (r*t)) :
    EndpointRadius (fun Y => g (p+r • Y)) X t := by
  have ht1 : 1 ≤ t := (le_of_mul_le_mul_left (by simpa only [mul_one] using ht.1.1) hr)
  have ht2 : t ≤ 2 := (le_of_mul_le_mul_left (by simpa only [mul_comm 2 r] using ht.1.2) hr)
  refine ⟨⟨ht1,ht2⟩,?_⟩
  rcases ht.2 with he | he | he
  · left
    nlinarith
  · right; left
    nlinarith
  · right; right
    intro s hs
    rw [signedAverage_affine _ _ _ hr (by linarith [hs.1]),signedAverage_affine _ _ _ hr (by linarith)]
    exact he (r*s) ⟨by nlinarith [hs.1],by nlinarith [hs.2]⟩
end DiskMoments

namespace PhysicalCells
lemma derivative_majorant_integral {F : ℂ → ℝ≥0∞} {C P : ℝ} (p : ℂ) {r : ℝ}
    (hr : 0 < r) (hF : (∫⁻ X in unitCell, F X) ≤ ENNReal.ofReal (C*r⁻¹*P)) :
    (∫⁻ y in cell p r, ENNReal.ofReal r⁻¹*F (toUnit p r y)) ≤ ENNReal.ofReal (C*P) := by
  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,cell_majorant p hr.ne']
  apply (mul_le_mul' le_rfl (mul_le_mul' le_rfl hF)).trans_eq
  rw [← ENNReal.ofReal_mul (sq_nonneg r),← ENNReal.ofReal_mul (inv_nonneg.mpr hr.le)]
  congr 1
  field_simp

lemma value_majorant_integral {F : ℂ → ℝ≥0∞} {C P : ℝ} (p : ℂ) {r : ℝ}
    (hr : 0 < r) (hF : (∫⁻ X in unitCell, F X) ≤ ENNReal.ofReal (C*r⁻¹*P)) :
    (∫⁻ y in cell p r, F (toUnit p r y)) ≤ ENNReal.ofReal (C*r*P) := by
  rw [cell_majorant p hr.ne']
  apply (mul_le_mul' le_rfl hF).trans_eq
  rw [← ENNReal.ofReal_mul (sq_nonneg r)]
  congr 1
  field_simp
end PhysicalCells

namespace DiskMoments
lemma average_gradient_norm_le {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (x : ℂ) {t : ℝ} (ht : 1 ≤ t) :
    ‖gradient (fun y => signedAverage g y t) x‖ ≤ ‖∫ y in Metric.ball x t, gradient g y‖ := by
  have hp : 0 < Real.pi*t^2 := mul_pos Real.pi_pos (sq_pos_of_pos (by linarith))
  have h1 : 1 ≤ Real.pi*t^2 := by nlinarith [Real.pi_gt_three,sq_nonneg (t-1)]
  rw [gradient_signedAverage hg hD x (by linarith),norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hp)]
  exact mul_le_of_le_one_left (norm_nonneg _) ((inv_le_one₀ hp).mpr h1)

lemma average_gradient_negative_le {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (x n : ℂ) {t : ℝ} (ht : 1 ≤ t) :
    max 0 (-inner ℝ n (gradient (fun y => signedAverage g y t) x)) ≤
      max 0 (-inner ℝ n (∫ y in Metric.ball x t, gradient g y)) := by
  have hp : 0 < Real.pi*t^2 := mul_pos Real.pi_pos (sq_pos_of_pos (by linarith))
  have h1 : 1 ≤ Real.pi*t^2 := by nlinarith [Real.pi_gt_three,sq_nonneg (t-1)]
  rw [gradient_signedAverage hg hD x (by linarith),inner_smul_right,← mul_neg,
    ← mul_zero (Real.pi*t^2)⁻¹,← mul_max_of_nonneg _ _ (inv_nonneg.mpr hp.le)]
  simp only [mul_zero]
  exact mul_le_of_le_one_left (le_max_left _ _) ((inv_le_one₀ hp).mpr h1)

lemma physical_gradient_norm_majorant {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (p X : ℂ) {r t : ℝ} (hr : 0 < r) (ht : 0 < t)
    {F : ℝ≥0∞}
    (hF : ENNReal.ofReal ‖gradient (fun Y => signedAverage (fun Z => g (p+r • Z)) Y t) X‖ ≤ F) :
    ENNReal.ofReal ‖gradient (fun y => signedAverage g y (r*t)) (p+r • X)‖ ≤
      ENNReal.ofReal r⁻¹*F := by
  have h : ENNReal.ofReal r⁻¹*_ ≤ ENNReal.ofReal r⁻¹*F := mul_le_mul' le_rfl hF
  rw [gradient_average_affine hg hD p X hr ht,norm_smul,Real.norm_eq_abs,abs_of_pos hr,
    ← ENNReal.ofReal_mul (inv_nonneg.mpr hr.le),inv_mul_cancel_left₀ hr.ne'] at h
  exact h

lemma physical_gradient_negative_majorant {g : ℂ → ℝ} (hg : ContDiff ℝ 1 g) {D : ℝ}
    (hD : ∀ y, ‖fderiv ℝ g y‖ ≤ D) (p X n : ℂ) {r t : ℝ} (hr : 0 < r) (ht : 0 < t)
    {F : ℝ≥0∞}
    (hF : ENNReal.ofReal (max 0 (-inner ℝ n
      (gradient (fun Y => signedAverage (fun Z => g (p+r • Z)) Y t) X))) ≤ F) :
    ENNReal.ofReal (max 0 (-inner ℝ n (gradient (fun y => signedAverage g y (r*t)) (p+r • X)))) ≤
      ENNReal.ofReal r⁻¹*F := by
  have h : ENNReal.ofReal r⁻¹*_ ≤ ENNReal.ofReal r⁻¹*F := mul_le_mul' le_rfl hF
  rw [gradient_average_affine hg hD p X hr ht,inner_smul_right,← mul_neg,
    ← mul_zero r,← mul_max_of_nonneg _ _ hr.le,
    ← ENNReal.ofReal_mul (inv_nonneg.mpr hr.le),inv_mul_cancel_left₀ hr.ne'] at h
  simpa only [mul_zero] using h
end DiskMoments

namespace CommonPrices
open DyadicHats ConeCoordinates LocalCancellation AffineDensities DirectionNeighbors Finset DiskMoments PricePieces PhysicalCells

theorem physical_directional_price : ∃ C : ℝ, 0 < C ∧
    ∀ {N : ℕ} {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) (hN : R/r+1 ≤ (N:ℝ))
    {g : ℂ → ℝ}, ContDiff ℝ 1 g → Integrable (gradient g) →
    (∀ y, gradient g y ≠ 0 → ‖y‖ ≤ R) →
    (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) → ∀ (k : ℕ) (p : ℤ × ℤ)
    {n : ℂ}, n ∈ fixedPalette.labels → ∀ {s : ℝ}, (s = 1 ∨ s = -1) →
    ∃ F : ℂ → ℝ≥0∞, Measurable F ∧
      (∫⁻ y in cell (point (scale r k) p) (scale r k), F y) ≤ ENNReal.ofReal
        (C*(badPrice (gradient g) r k (cellRoots N p) n+cellPrice N r (gradient g) k (cellRoots N p))) ∧
      ∀ y ∈ cell (point (scale r k) p) (scale r k), ∀ t,
        PhysicalNeighborhoodRadius g (scale r k) y t →
        ENNReal.ofReal (max 0 (-inner ℝ (axis n s) (gradient (fun z => signedAverage g z t) y))) ≤ F y := by
  obtain ⟨C,hC,hmajor⟩ := directional_cell_price
  refine ⟨C,hC,?_⟩
  intro N r R hr hR hN g hg hh hsupp hD k p n hn s hs
  obtain ⟨D,hD⟩ := hD
  obtain ⟨D',hD'⟩ := fderiv_affine_bound (hg.differentiable one_ne_zero) (point (scale r k) p) (scale r k) hD
  obtain ⟨F,hFm,hFi,hFb⟩ := hmajor hr hR hN hg hh hsupp k p hn hs unitCell (by rw [volume_unitCell]; norm_num)
  refine ⟨fun y => ENNReal.ofReal (scale r k)⁻¹*F (toUnit (point (scale r k) p) (scale r k) y),
    measurable_const.mul (hFm.comp (toUnit_continuous _ _).measurable),
    derivative_majorant_integral _ (scale_pos hr k) hFi,?_⟩
  intro y hy t ht
  let X := toUnit (point (scale r k) p) (scale r k) y
  have hX : ‖X‖ ≤ 2 := norm_unitCell hy
  have he : point (scale r k) p+scale r k • X = y := affine_toUnit _ (scale_pos hr k).ne' y
  have ht' : PhysicalNeighborhoodRadius g (scale r k)
      (point (scale r k) p+scale r k • X) (scale r k*(t/scale r k)) := by
    simpa only [he,mul_div_cancel₀ _ (scale_pos hr k).ne'] using ht
  have htn := neighborhoodRadius_affine _ X (scale_pos hr k) ht'
  have hg' : ContDiff ℝ 1 (fun Z => g (point (scale r k) p+scale r k • Z)) := hg.comp (by fun_prop)
  have hb := (ENNReal.ofReal_le_ofReal (average_gradient_negative_le hg' hD' X (axis n s) htn.1.1)).trans
    (hFb X hX _ htn)
  have hf := physical_gradient_negative_majorant hg hD (point (scale r k) p) X (axis n s)
    (scale_pos hr k) (by linarith [htn.1.1]) hb
  simpa only [he,mul_div_cancel₀ _ (scale_pos hr k).ne'] using hf

theorem physical_size_price : ∃ C : ℝ, 0 < C ∧
    ∀ {N : ℕ} {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) (hN : R/r+1 ≤ (N:ℝ))
    {g : ℂ → ℝ}, ContDiff ℝ 1 g → Integrable (gradient g) →
    (∀ y, gradient g y ≠ 0 → ‖y‖ ≤ R) →
    (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) → ∀ (k : ℕ) (p : ℤ × ℤ)
    {n : ℂ}, n ∈ fixedPalette.labels →
    ∃ F : ℂ → ℝ≥0∞, Measurable F ∧
      (∫⁻ y in cell (point (scale r k) p) (scale r k), F y) ≤ ENNReal.ofReal
        (C*(rootMass (gradient g) r k (cellRoots N p)+cellPrice N r (gradient g) k (cellRoots N p))) ∧
      ∀ y ∈ cell (point (scale r k) p) (scale r k), ∀ t,
        PhysicalEndpointRadius g (scale r k) y t →
        ENNReal.ofReal ‖gradient (fun z => signedAverage g z t) y‖ ≤ F y := by
  obtain ⟨C,hC,hmajor⟩ := size_cell_price
  refine ⟨C,hC,?_⟩
  intro N r R hr hR hN g hg hh hsupp hD k p n hn
  obtain ⟨D,hD⟩ := hD
  obtain ⟨D',hD'⟩ := fderiv_affine_bound (hg.differentiable one_ne_zero) (point (scale r k) p) (scale r k) hD
  obtain ⟨F,hFm,hFi,hFb⟩ := hmajor hr hR hN hg hh hsupp k p hn unitCell (by rw [volume_unitCell]; norm_num)
  refine ⟨fun y => ENNReal.ofReal (scale r k)⁻¹*F (toUnit (point (scale r k) p) (scale r k) y),
    measurable_const.mul (hFm.comp (toUnit_continuous _ _).measurable),
    derivative_majorant_integral _ (scale_pos hr k) hFi,?_⟩
  intro y hy t ht
  let X := toUnit (point (scale r k) p) (scale r k) y
  have hX : ‖X‖ ≤ 2 := norm_unitCell hy
  have he : point (scale r k) p+scale r k • X = y := affine_toUnit _ (scale_pos hr k).ne' y
  have ht' : PhysicalEndpointRadius g (scale r k)
      (point (scale r k) p+scale r k • X) (scale r k*(t/scale r k)) := by
    simpa only [he,mul_div_cancel₀ _ (scale_pos hr k).ne'] using ht
  have htn := endpointRadius_affine _ X (scale_pos hr k) ht'
  have hg' : ContDiff ℝ 1 (fun Z => g (point (scale r k) p+scale r k • Z)) := hg.comp (by fun_prop)
  have hb := (ENNReal.ofReal_le_ofReal (average_gradient_norm_le hg' hD' X htn.1.1)).trans
    (hFb X hX _ htn)
  have hf := physical_gradient_norm_majorant hg hD (point (scale r k) p) X
    (scale_pos hr k) (by linarith [htn.1.1]) hb
  simpa only [he,mul_div_cancel₀ _ (scale_pos hr k).ne'] using hf

theorem physical_value_price : ∃ C : ℝ, 0 < C ∧
    ∀ {N : ℕ} {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) (hN : R/r+1 ≤ (N:ℝ))
    {g : ℂ → ℝ}, ContDiff ℝ 1 g → Integrable (gradient g) →
    (∀ y, gradient g y ≠ 0 → ‖y‖ ≤ R) →
    (∃ D : ℝ, ∀ y, ‖fderiv ℝ g y‖ ≤ D) → ∀ (k : ℕ) (p : ℤ × ℤ)
    {n : ℂ}, n ∈ fixedPalette.labels →
    ∃ F : ℂ → ℝ≥0∞, Measurable F ∧
      (∫⁻ y in cell (point (scale r k) p) (scale r k), F y) ≤ ENNReal.ofReal
        (C*scale r k*(rootMass (gradient g) r k (cellRoots N p)+cellPrice N r (gradient g) k (cellRoots N p))) ∧
      ∀ y ∈ cell (point (scale r k) p) (scale r k), ∀ t t',
        0 < t → t ≤ 4*scale r k → 0 < t' → t' ≤ 4*scale r k →
        ENNReal.ofReal |signedAverage g y t-signedAverage g y t'| ≤ F y := by
  obtain ⟨C,hC,hmajor⟩ := value_cell_price
  refine ⟨C,hC,?_⟩
  intro N r R hr hR hN g hg hh hsupp hD k p n hn
  obtain ⟨F,hFm,hFi,hFb⟩ := hmajor hr hR hN hg hh hsupp hD k p hn unitCell (by rw [volume_unitCell]; norm_num)
  refine ⟨fun y => F (toUnit (point (scale r k) p) (scale r k) y),
    hFm.comp (toUnit_continuous _ _).measurable,
    value_majorant_integral _ (scale_pos hr k) hFi,?_⟩
  intro y hy t t' ht hb ht' hb'
  let X := toUnit (point (scale r k) p) (scale r k) y
  have he : point (scale r k) p+scale r k • X = y := affine_toUnit _ (scale_pos hr k).ne' y
  have hrk := scale_pos hr k
  have hf := hFb X (norm_unitCell hy) (t/scale r k) (t'/scale r k)
    (div_pos ht hrk) ((div_le_iff₀ hrk).mpr hb) (div_pos ht' hrk) ((div_le_iff₀ hrk).mpr hb')
  simpa only [signedAverage_affine _ _ _ hrk (div_pos ht hrk),
    signedAverage_affine _ _ _ hrk (div_pos ht' hrk),he,mul_div_cancel₀ _ hrk.ne'] using hf
end CommonPrices

namespace DiskMoments
lemma signedAverage_abs_le {g : ℂ → ℝ} {M : ℝ} (hM : ∀ y, |g y| ≤ M)
    (x : ℂ) {t : ℝ} (ht : 0 < t) : |signedAverage g x t| ≤ M := by
  rw [signedAverage_fixedDomain g x ht,abs_div,abs_of_pos Real.pi_pos]
  have : IsFiniteMeasure (volume.restrict (Metric.ball (0:ℂ) 1)) := ⟨by simp⟩
  have hi := norm_integral_le_of_norm_le_const (μ := volume.restrict (Metric.ball (0:ℂ) 1))
    (C := M) (f := fun z : ℂ => g (x+t • z)) (ae_of_all _ (fun z => by simpa only [Real.norm_eq_abs] using hM (x+t • z)))
  have hv : (volume.restrict (Metric.ball (0:ℂ) 1)).real univ = Real.pi := by
    simp [measureReal_def,Complex.volume_ball]
  rw [hv,Real.norm_eq_abs] at hi
  exact (div_le_iff₀ Real.pi_pos).mpr hi

def oscillationSet (g : ℂ → ℝ) (r : ℝ) (x : ℂ) : Set ℝ :=
  {a | ∃ t t', 0 < t ∧ t ≤ 4*r ∧ 0 < t' ∧ t' ≤ 4*r ∧ a = |signedAverage g x t-signedAverage g x t'|}
def oscillation (g : ℂ → ℝ) (r : ℝ) (x : ℂ) : ℝ := sSup (oscillationSet g r x)

lemma oscillationSet_zero (g : ℂ → ℝ) {r : ℝ} (hr : 0 < r) (x : ℂ) :
    0 ∈ oscillationSet g r x := ⟨r,r,hr,by linarith,hr,by linarith,by simp⟩
lemma oscillationSet_bdd {g : ℂ → ℝ} (hg : Continuous g) (hc : HasCompactSupport g) (r : ℝ) (x : ℂ) :
    BddAbove (oscillationSet g r x) := by
  obtain ⟨M,hM⟩ := hc.exists_bound_of_continuous hg
  refine ⟨2*M,?_⟩
  rintro a ⟨t,t',ht,_,ht',_,rfl⟩
  have hb : ∀ y, |g y| ≤ M := by simpa only [Real.norm_eq_abs] using hM
  have h1 := signedAverage_abs_le hb x ht
  have h2 := signedAverage_abs_le hb x ht'
  linarith [abs_sub (signedAverage g x t) (signedAverage g x t')]
lemma oscillation_nonneg {g : ℂ → ℝ} (hg : Continuous g) (hc : HasCompactSupport g) {r : ℝ}
    (hr : 0 < r) (x : ℂ) : 0 ≤ oscillation g r x :=
  le_csSup (oscillationSet_bdd hg hc r x) (oscillationSet_zero g hr x)
lemma le_oscillation {g : ℂ → ℝ} (hg : Continuous g) (hc : HasCompactSupport g) {r : ℝ}
    (x : ℂ) {t t' : ℝ} (ht : 0 < t) (hb : t ≤ 4*r) (ht' : 0 < t') (hb' : t' ≤ 4*r) :
    |signedAverage g x t-signedAverage g x t'| ≤ oscillation g r x :=
  le_csSup (oscillationSet_bdd hg hc r x) ⟨t,t',ht,hb,ht',hb',rfl⟩
lemma oscillation_majorant {g : ℂ → ℝ} {r : ℝ} (hr : 0 < r) (x : ℂ) {F : ℝ≥0∞}
    (hF : ∀ t t', 0 < t → t ≤ 4*r → 0 < t' → t' ≤ 4*r →
      ENNReal.ofReal |signedAverage g x t-signedAverage g x t'| ≤ F) :
    ENNReal.ofReal (oscillation g r x) ≤ F := by
  by_cases htop : F = ⊤
  · simp only [htop,le_top]
  · apply (ENNReal.ofReal_le_iff_le_toReal htop).mpr
    apply csSup_le ⟨0,oscillationSet_zero g hr x⟩
    rintro a ⟨t,t',ht,hb,ht',hb',rfl⟩
    exact (ENNReal.ofReal_le_iff_le_toReal htop).mp (hF t t' ht hb ht' hb')
end DiskMoments

end

end OAI