lean/OAI/Probability/GaussianReplacement/MixedGeometry.lean

import OAI.Probability.GaussianReplacement.Geometry

namespace OAI

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace CurrentProjection

section MixedGeometry
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E]


def AmbientGrowth (μ : Measure E) (a : ℕ) (A : ℝ) : Prop :=
  ∀ z : E, ∀ t : ℝ, 0 < t → μ (Metric.closedBall z t) ≤ ENNReal.ofReal (A * t ^ a)

omit [MeasurableSpace E] [BorelSpace E] in
lemma infDist_eq_norm_sub_projection (V : Submodule ℝ E) (v : E) :
    Metric.infDist v (V : Set E) = ‖v - V.starProjection v‖ := by
  rw [Submodule.starProjection_minimal, Metric.infDist_eq_iInf]
  simp only [dist_eq_norm]
  rfl

omit [MeasurableSpace E] [BorelSpace E] in
lemma infDist_submodule_smul (V : Submodule ℝ E) (v : E) (c : ℝ) :
    Metric.infDist (c • v) (V : Set E) = |c| * Metric.infDist v (V : Set E) := by
  simp only [infDist_eq_norm_sub_projection, map_smul, ← smul_sub, norm_smul, Real.norm_eq_abs]


theorem exists_radial_tube_cover (V : Submodule ℝ E) (s : E)
    {R η : ℝ} (hR : 0 < R) (hη : 0 < η) (hη1 : η ≤ 1) :
    ∃ t : Finset V, (t.card : ℝ) ≤ (3 / η) ^ Module.finrank ℝ V ∧
      ∀ u : E, ‖u - s‖ ≤ R → Metric.infDist (u - s) (V : Set E) ≤ η * R →
        ∃ y ∈ t, dist u (s + R • (y : E)) ≤ 2 * (η * R) := by
  classical
  obtain ⟨t, _, htcard, hnet⟩ := exists_unit_ball_net (E := V) hη hη1
  refine ⟨t, htcard, ?_⟩
  intro u hu huV
  let z : E := R⁻¹ • (u - s)
  have hz : ‖z‖ ≤ 1 := by
    dsimp [z]
    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hR)]
    exact (inv_mul_le_iff₀ hR).mpr (by simpa using hu)
  obtain ⟨y, hyt, hzy⟩ := hnet (V.orthogonalProjectionOnto z)
    ((V.norm_orthogonalProjectionOnto_apply_le z).trans hz)
  have hzV : ‖z - V.starProjection z‖ ≤ η := by
    rw [← infDist_eq_norm_sub_projection]
    dsimp [z]
    rw [infDist_submodule_smul, abs_of_pos (inv_pos.mpr hR)]
    exact (inv_mul_le_iff₀ hR).mpr (by simpa [mul_comm] using huV)
  have hdist : dist z (y : E) ≤ 2 * η := by
    calc
      dist z (y : E) ≤ dist z (V.orthogonalProjectionOnto z : E) +
          dist (V.orthogonalProjectionOnto z : E) (y : E) := dist_triangle _ _ _
      _ ≤ η + η := add_le_add (by simpa [dist_eq_norm] using hzV) hzy
      _ = 2 * η := by ring
  refine ⟨y, hyt, ?_⟩
  have he : u = s + R • z := by
    dsimp [z]
    rw [smul_inv_smul₀ hR.ne']
    abel
  rw [he, dist_add_left, dist_smul₀, Real.norm_eq_abs, abs_of_pos hR]
  nlinarith


theorem AmbientGrowth.radial_tube {μ : Measure E} {a : ℕ} {A : ℝ}
    (hμ : AmbientGrowth μ a A) (V : Submodule ℝ E) (s : E)
    {R η : ℝ} (hR : 0 < R) (hη : 0 < η) (hη1 : η ≤ 1) :
    μ {u | ‖u - s‖ ≤ R ∧ Metric.infDist (u - s) (V : Set E) ≤ η * R} ≤
      ENNReal.ofReal ((3 / η) ^ Module.finrank ℝ V * (A * (2 * (η * R)) ^ a)) := by
  classical
  obtain ⟨t, htcard, htcover⟩ := exists_radial_tube_cover V s hR hη hη1
  calc
    μ {u | ‖u - s‖ ≤ R ∧ Metric.infDist (u - s) (V : Set E) ≤ η * R} ≤
        μ (⋃ y ∈ t, Metric.closedBall (s + R • (y : E)) (2 * (η * R))) := by
      apply measure_mono
      intro u hu
      obtain ⟨y, hy, huy⟩ := htcover u hu.1 hu.2
      exact mem_iUnion.2 ⟨y, mem_iUnion.2 ⟨hy, huy⟩⟩
    _ ≤ ∑ y ∈ t, μ (Metric.closedBall (s + R • (y : E)) (2 * (η * R))) :=
      measure_biUnion_finset_le _ _
    _ ≤ ∑ _y ∈ t, ENNReal.ofReal (A * (2 * (η * R)) ^ a) := by
      exact Finset.sum_le_sum fun y _ => hμ _ _ (by positivity)
    _ = (t.card : ℝ≥0∞) * ENNReal.ofReal (A * (2 * (η * R)) ^ a) := by simp
    _ ≤ ENNReal.ofReal ((3 / η) ^ Module.finrank ℝ V) *
        ENNReal.ofReal (A * (2 * (η * R)) ^ a) := by
      gcongr
      exact_mod_cast ENNReal.ofReal_le_ofReal htcard
    _ = _ := (ENNReal.ofReal_mul (by positivity)).symm

end MixedGeometry


section MixedRadial
variable {E : Type*} [NormedAddCommGroup E] [innerProductSpaceRealE : InnerProductSpace ℝ E]
  [finiteDimensionalRealE : FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

lemma tube_radius_margin (h a k : ℕ) (A R η : ℝ)
    (hA : 0 ≤ A) (hR : 0 ≤ R) (hη : 0 < η) (hη1 : η ≤ 1) (hmargin : h+k ≤ a) :
    (3 / η) ^ h * (A * (2 * (η * R)) ^ a) ≤
      (3 ^ h * 2 ^ a * A * R ^ a) * η ^ k := by
  have he : η ^ a = η ^ h * η ^ (a-h) := by
    rw [← pow_add, Nat.add_sub_of_le (by omega : h ≤ a)]
  calc
    (3 / η) ^ h * (A * (2 * (η * R)) ^ a) =
        (3 ^ h * 2 ^ a * A * R ^ a) * η ^ (a-h) := by
      simp only [div_pow, mul_pow]
      rw [he]
      field_simp
    _ ≤ _ := mul_le_mul_of_nonneg_left
      (pow_le_pow_of_le_one hη.le hη1 (by omega)) (by positivity)

lemma AmbientGrowth.radial_tube_pow {μ : Measure E} {a : ℕ} {A : ℝ}
    (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (V : Submodule ℝ E) (s : E)
    (k : ℕ) (hmargin : Module.finrank ℝ V + k ≤ a)
    {R η : ℝ} (hR : 0 < R) (hη : 0 < η) (hη1 : η ≤ 1) :
    μ {u | ‖u - s‖ ≤ R ∧ Metric.infDist (u - s) (V : Set E) ≤ η * R} ≤
      ENNReal.ofReal (3 ^ Module.finrank ℝ V * 2 ^ a * A * R ^ a) *
        ENNReal.ofReal (η ^ k) := by
  refine (hμ.radial_tube V s hR hη hη1).trans ?_
  rw [← ENNReal.ofReal_mul (by positivity)]
  exact ENNReal.ofReal_le_ofReal (tube_radius_margin _ _ k A R η hA hR.le hη hη1 hmargin)

def radialShell (s : E) (R : ℝ) : Set E :=
  {u | R / 2 < ‖u - s‖ ∧ ‖u - s‖ ≤ R}

omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
lemma measurableSet_radialShell (s : E) (R : ℝ) : MeasurableSet (radialShell s R) := by
  have hf : Measurable (fun u : E => ‖u - s‖) := by fun_prop
  exact (measurableSet_lt measurable_const hf).inter (measurableSet_le hf measurable_const)


theorem AmbientGrowth.shell_normalized_inverse {μ : Measure E} {a m : ℕ} {A : ℝ}
    (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (V : Submodule ℝ E) (s : E)
    (hmargin : Module.finrank ℝ V + (m+1) ≤ a) {R : ℝ} (hR : 0 < R) :
    ∫⁻ u in radialShell s R,
      ENNReal.ofReal (((Metric.infDist (u-s) (V : Set E) / R)⁻¹) ^ m) ∂μ ≤
      ENNReal.ofReal ((1 + 3 ^ Module.finrank ℝ V * 2 ^ (a+m+1)) * A * R ^ a) := by
  let f : E → ℝ := fun u => Metric.infDist (u-s) (V : Set E) / R
  have hf : Measurable f := by
    exact ((Metric.continuous_infDist_pt _).measurable.comp (by fun_prop)).div_const R
  have hnonneg : ∀ u, 0 ≤ f u := fun u => div_nonneg Metric.infDist_nonneg hR.le
  have htube (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) :
      (μ.restrict (radialShell s R)) {u | f u ≤ η} ≤
        ENNReal.ofReal (3 ^ Module.finrank ℝ V * 2 ^ a * A * R ^ a) *
          ENNReal.ofReal (η ^ (m+1)) := by
    rw [Measure.restrict_apply (measurableSet_le hf measurable_const)]
    refine (measure_mono ?_).trans (hμ.radial_tube_pow hA V s (m+1) hmargin hR hη hη1)
    intro u hu
    exact ⟨hu.2.2, (div_le_iff₀ hR).mp hu.1⟩
  have hmass : μ (radialShell s R) ≤ ENNReal.ofReal (A * R ^ a) := by
    refine (measure_mono ?_).trans (hμ s R hR)
    intro u hu
    exact (Metric.mem_closedBall.mpr (by simpa [dist_eq_norm] using hu.2))
  calc
    _ ≤ μ (radialShell s R) + ENNReal.ofReal (3 ^ Module.finrank ℝ V * 2 ^ a * A * R ^ a) *
        (2 : ℝ≥0∞) ^ (m+1) := by
      simpa only [f, Measure.restrict_apply_univ] using
        inverse_moment_of_tube_bound (μ.restrict (radialShell s R)) f m hf hnonneg _ htube
    _ ≤ ENNReal.ofReal (A * R ^ a) +
        ENNReal.ofReal (3 ^ Module.finrank ℝ V * 2 ^ a * A * R ^ a) * (2 : ℝ≥0∞)^(m+1) :=
      add_le_add hmass le_rfl
    _ = _ := by
      rw [← ENNReal.ofReal_ofNat (n := 2), ← ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 2),
        ← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_add (by positivity) (by positivity)]
      congr 1
      rw [show a+m+1 = a+(m+1) by omega, pow_add]
      ring



def mixedInverse (s : E) (V : Submodule ℝ E) (m r : ℕ) (u : E) : ℝ≥0∞ :=
  ENNReal.ofReal (‖u-s‖⁻¹ ^ r * (Metric.infDist (u-s) (V : Set E))⁻¹ ^ m)

omit [FiniteDimensional ℝ E] in
lemma measurable_mixedInverse (s : E) (V : Submodule ℝ E) (m r : ℕ) :
    Measurable (mixedInverse s V m r) := by
  have hf : Measurable (fun u : E => Metric.infDist (u-s) (V : Set E)) :=
    (Metric.continuous_infDist_pt _).measurable.comp (by fun_prop)
  exact Measurable.ennreal_ofReal ((by fun_prop : Measurable (fun u : E => ‖u-s‖⁻¹ ^ r)).mul
    (hf.inv.pow_const m))

def shellConstant (h a m r : ℕ) : ℝ := 2 ^ r * (1 + 3 ^ h * 2 ^ (a+m+1))

lemma inverse_div_factor (x R : ℝ) (m : ℕ) (hR : R ≠ 0) :
    x⁻¹ ^ m = R⁻¹ ^ m * (x / R)⁻¹ ^ m := by
  rw [← mul_pow, inv_div, div_eq_mul_inv]
  simp [hR]

lemma radial_power_cancel (R : ℝ) (a k : ℕ) (hR : R ≠ 0) (hk : k ≤ a) :
    R⁻¹ ^ k * R ^ a = R ^ (a-k) := by
  have he : R ^ a = R ^ k * R ^ (a-k) := by rw [← pow_add, Nat.add_sub_of_le hk]
  rw [he, ← mul_assoc, ← mul_pow]
  simp [hR]


theorem AmbientGrowth.shell_mixed_inverse {μ : Measure E} {a m r : ℕ} {A : ℝ}
    (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (V : Submodule ℝ E) (s : E)
    (hmargin : Module.finrank ℝ V + (m+1) ≤ a) (hdegree : m+r ≤ a)
    {R : ℝ} (hR : 0 < R) :
    ∫⁻ u in radialShell s R, mixedInverse s V m r u ∂μ ≤
      ENNReal.ofReal (shellConstant (Module.finrank ℝ V) a m r * A * R ^ (a-(m+r))) := by
  let f : E → ℝ≥0∞ := fun u =>
    ENNReal.ofReal (((Metric.infDist (u-s) (V : Set E) / R)⁻¹) ^ m)
  have hf : Measurable f := by
    have hh := (Metric.continuous_infDist_pt (V : Set E)).measurable.comp
      (show Measurable (fun u : E => u-s) by fun_prop)
    exact ((hh.div_const R).inv.pow_const m).ennreal_ofReal
  have hbound : ∀ u ∈ radialShell s R,
      mixedInverse s V m r u ≤ ENNReal.ofReal (2 ^ r * R⁻¹ ^ (m+r)) * f u := by
    intro u hu
    have hu0 : 0 < ‖u-s‖ := lt_trans (by positivity : 0 < R/2) hu.1
    have hinv : ‖u-s‖⁻¹ ≤ 2 / R := by
      have h := (inv_le_inv₀ hu0 (by positivity : (0 : ℝ) < R/2)).mpr hu.1.le
      simpa only [inv_div] using h
    have hpow : ‖u-s‖⁻¹ ^ r ≤ (2 / R) ^ r :=
      pow_le_pow_left₀ (by positivity) hinv r
    rw [← ENNReal.ofReal_mul (by positivity)]
    apply ENNReal.ofReal_le_ofReal
    calc
      ‖u-s‖⁻¹ ^ r * (Metric.infDist (u-s) (V : Set E))⁻¹ ^ m ≤
          (2/R)^r * (Metric.infDist (u-s) (V : Set E))⁻¹ ^ m :=
        mul_le_mul_of_nonneg_right hpow (pow_nonneg (inv_nonneg.mpr Metric.infDist_nonneg) _)
      _ = _ := by
        rw [inverse_div_factor (Metric.infDist (u-s) (V : Set E)) R m hR.ne']
        simp only [div_eq_mul_inv, inv_pow, pow_add]
        ring
  calc
    _ ≤ ∫⁻ u in radialShell s R, ENNReal.ofReal (2 ^ r * R⁻¹ ^ (m+r)) * f u ∂μ := by
      apply lintegral_mono_ae
      filter_upwards [ae_restrict_mem (measurableSet_radialShell s R)] with u hu
      exact hbound u hu
    _ = ENNReal.ofReal (2 ^ r * R⁻¹ ^ (m+r)) * ∫⁻ u in radialShell s R, f u ∂μ :=
      lintegral_const_mul _ hf
    _ ≤ ENNReal.ofReal (2 ^ r * R⁻¹ ^ (m+r)) *
        ENNReal.ofReal ((1 + 3 ^ Module.finrank ℝ V * 2 ^ (a+m+1)) * A * R ^ a) := by
      gcongr
      exact hμ.shell_normalized_inverse hA V s hmargin hR
    _ = _ := by
      rw [← ENNReal.ofReal_mul (by positivity)]
      congr 1
      calc
        _ = (2^r * (1 + 3 ^ Module.finrank ℝ V * 2 ^ (a+m+1)) * A) *
            (R⁻¹ ^ (m+r) * R ^ a) := by ring
        _ = _ := by rw [radial_power_cancel R a (m+r) hR.ne' hdegree]; rfl

end MixedRadial


lemma dyadic_min_sum_le (B D : ℝ) (hB : 0 < B) (hD : 0 < D) :
    (∑' i : ℕ, ENNReal.ofReal (min B (D * (1/2 : ℝ)^i))) ≤
      ENNReal.ofReal (3 * B * (1 + max (Real.log (D/B)) 0)) := by
  let L : ℝ := max (Real.log (D/B)) 0
  let N : ℕ := ⌈L / Real.log 2⌉₊
  have hL : 0 ≤ L := le_max_right _ _
  have hlog : (1/2 : ℝ) ≤ Real.log 2 := by
    have := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ) < 2)
    norm_num at this ⊢
    exact this
  have hlog0 : 0 < Real.log 2 := by linarith
  have hN : L ≤ (N:ℝ) * Real.log 2 := by
    exact (div_le_iff₀ hlog0).mp (Nat.le_ceil _)
  have hDB : D / B ≤ (2:ℝ)^N := by
    apply (Real.log_le_log_iff (div_pos hD hB) (by positivity)).mp
    rw [Real.log_pow]
    exact (le_max_left _ _).trans hN
  have hcut : D * (1/2:ℝ)^N ≤ B := by
    have hh : D ≤ (2:ℝ)^N * B := (div_le_iff₀ hB).mp hDB
    calc
      D * (1/2:ℝ)^N ≤ ((2:ℝ)^N * B) * (1/2:ℝ)^N :=
        mul_le_mul_of_nonneg_right hh (by positivity)
      _ = B := by rw [mul_comm ((2:ℝ)^N), mul_assoc, ← mul_pow]; norm_num
  have hNupper : (N:ℝ) ≤ 2 * L + 1 := by
    have hh := (Nat.ceil_lt_add_one (div_nonneg hL hlog0.le)).le
    have hdiv : L / Real.log 2 ≤ 2 * L := by
      apply (div_le_iff₀ hlog0).mpr
      nlinarith
    dsimp [N]
    linarith
  let f : ℕ → ℝ≥0∞ := fun i => ENNReal.ofReal (min B (D * (1/2:ℝ)^i))
  calc
    ∑' i, f i = (∑ i ∈ Finset.range N, f i) + ∑' i, f (i+N) := by
      exact ENNReal.summable.sum_add_tsum_nat_add'.symm
    _ ≤ (N : ℝ≥0∞) * ENNReal.ofReal B +
        ∑' i : ℕ, ENNReal.ofReal B * (1/2:ℝ≥0∞)^i := by
      apply add_le_add
      · calc
          ∑ i ∈ Finset.range N, f i ≤ ∑ _i ∈ Finset.range N, ENNReal.ofReal B := by
            exact Finset.sum_le_sum fun i _ => ENNReal.ofReal_le_ofReal (min_le_left _ _)
          _ = _ := by simp
      · apply ENNReal.tsum_le_tsum
        intro i
        dsimp [f]
        rw [← (show ENNReal.ofReal (1/2:ℝ) = (1/2:ℝ≥0∞) by
          rw [ENNReal.ofReal_div_of_pos (by norm_num)]; norm_num)]
        rw [← ENNReal.ofReal_pow (by norm_num : (0:ℝ) ≤ 1/2),
          ← ENNReal.ofReal_mul hB.le]
        apply ENNReal.ofReal_le_ofReal
        refine (min_le_right _ _).trans ?_
        calc
          D * (1/2:ℝ)^(i+N) = (D * (1/2:ℝ)^N) * (1/2:ℝ)^i := by rw [pow_add]; ring
          _ ≤ _ := mul_le_mul_of_nonneg_right hcut (by positivity)
    _ = ENNReal.ofReal ((N:ℝ)*B + 2*B) := by
      rw [ENNReal.tsum_mul_left, ENNReal.tsum_geometric]
      norm_num
      rw [← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (by positivity),
        ← ENNReal.ofReal_ofNat (n:=2), ← ENNReal.ofReal_mul hB.le,
        ← ENNReal.ofReal_add (by positivity) (by positivity)]
      congr 1; ring
    _ ≤ ENNReal.ofReal (3 * B * (1+L)) := by
      apply ENNReal.ofReal_le_ofReal
      nlinarith

lemma shellConstant_nonneg (h a m r : ℕ) : 0 ≤ shellConstant h a m r := by
  unfold shellConstant
  positivity

lemma shellConstant_mono (h m r : ℕ) {a b : ℕ} (hab : a ≤ b) :
    shellConstant h a m r ≤ shellConstant h b m r := by
  unfold shellConstant
  gcongr
  norm_num

section MixedRadialSum
variable {E : Type*} [NormedAddCommGroup E] [innerProductSpaceRealE : InnerProductSpace ℝ E]
  [finiteDimensionalRealE : FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E]

omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma mem_radialShell_dyadic {s u : E} (hu : 0 < ‖u-s‖) (hu2 : ‖u-s‖ ≤ 2) :
    ∃ i : ℕ, u ∈ radialShell s (2 * (1/2:ℝ)^i) := by
  obtain ⟨i, hi, hj⟩ := exists_nat_pow_near_of_lt_one (show 0 < ‖u-s‖/2 by positivity)
    (show ‖u-s‖/2 ≤ 1 by linarith) (by norm_num : (0:ℝ) < 1/2) (by norm_num : (1/2:ℝ) < 1)
  refine ⟨i, ?_, ?_⟩
  · rw [pow_succ] at hi
    linarith
  · linarith



theorem AmbientGrowth.mixed_inverse_le {μ : Measure E} {n m r : ℕ} {B D : ℝ}
    (hBμ : AmbientGrowth μ (m+r) B) (hDμ : AmbientGrowth μ n D)
    (hB : 0 < B) (hD : 0 < D) (hr : 1 ≤ r) (hn : m+r < n)
    (V : Submodule ℝ E) (s : E) (hmargin : Module.finrank ℝ V + (m+1) ≤ m+r)
    (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) :
    ∫⁻ u, mixedInverse s V m r u ∂μ ≤
      ENNReal.ofReal (4 * shellConstant (Module.finrank ℝ V) n m r *
        B * (1 + max (Real.log (D/B)) 0)) := by
  let K : ℝ := shellConstant (Module.finrank ℝ V) n m r
  have hK : 0 ≤ K := shellConstant_nonneg _ _ _ _
  let S : ℕ → Set E := fun i => radialShell s (2 * (1/2:ℝ)^i)
  have hS : MeasurableSet (⋃ i, S i) := MeasurableSet.iUnion fun i => measurableSet_radialShell _ _
  have heq : (fun u => mixedInverse s V m r u) =ᵐ[μ]
      (⋃ i, S i).indicator (mixedInverse s V m r) := by
    filter_upwards [hsupp] with u hu
    by_cases hz : u = s
    · subst u
      simp [mixedInverse, Set.indicator, Nat.ne_of_gt (show 0 < r by omega)]
    · have hpos : 0 < ‖u-s‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hz)
      obtain ⟨i, hi⟩ := mem_radialShell_dyadic hpos hu
      exact (Set.indicator_of_mem (mem_iUnion.mpr ⟨i, hi⟩) _).symm
  have hzero : ∫⁻ u in S 0, mixedInverse s V m r u ∂μ ≤ ENNReal.ofReal (K*B) := by
    have hh := hBμ.shell_mixed_inverse hB.le V s hmargin le_rfl (by norm_num : (0:ℝ) < 2)
    have hk := shellConstant_mono (Module.finrank ℝ V) m r hn.le
    dsimp [S]
    simp only [pow_zero, mul_one, Nat.sub_self] at hh ⊢
    refine hh.trans (ENNReal.ofReal_le_ofReal ?_)
    simpa [K] using mul_le_mul_of_nonneg_right hk hB.le
  have htail (i : ℕ) :
      ∫⁻ u in S (i+1), mixedInverse s V m r u ∂μ ≤
        ENNReal.ofReal K * ENNReal.ofReal (min B (D*(1/2:ℝ)^i)) := by
    have hR : 0 < (1/2:ℝ)^i := by positivity
    have hR1 : (1/2:ℝ)^i ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
    have hSi : S (i+1) = radialShell s ((1/2:ℝ)^i) := by
      dsimp [S]
      congr 1
      rw [pow_succ]
      ring
    rw [hSi, ← ENNReal.ofReal_mul hK]
    rw [mul_min_of_nonneg _ _ hK, ENNReal.ofReal_min]
    apply le_min
    · have hh := hBμ.shell_mixed_inverse hB.le V s hmargin le_rfl hR
      simp only [Nat.sub_self, pow_zero, mul_one] at hh
      refine hh.trans (ENNReal.ofReal_le_ofReal ?_)
      exact mul_le_mul_of_nonneg_right (shellConstant_mono _ _ _ hn.le) hB.le
    · have hh := hDμ.shell_mixed_inverse hD.le V s (hmargin.trans hn.le) hn.le hR
      refine hh.trans (ENNReal.ofReal_le_ofReal ?_)
      have hrpow : ((1/2:ℝ)^i)^(n-(m+r)) ≤ (1/2:ℝ)^i := by
        simpa using pow_le_pow_of_le_one hR.le hR1 (show 1 ≤ n-(m+r) by omega)
      simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hrpow (mul_nonneg hK hD.le)
  calc
    _ = ∫⁻ u in ⋃ i, S i, mixedInverse s V m r u ∂μ := by
      rw [← lintegral_indicator hS]
      exact lintegral_congr_ae heq
    _ ≤ ∑' i, ∫⁻ u in S i, mixedInverse s V m r u ∂μ := lintegral_iUnion_le _ _
    _ = (∫⁻ u in S 0, mixedInverse s V m r u ∂μ) +
        ∑' i, ∫⁻ u in S (i+1), mixedInverse s V m r u ∂μ := tsum_eq_zero_add' ENNReal.summable
    _ ≤ ENNReal.ofReal (K*B) + ENNReal.ofReal K *
        (∑' i : ℕ, ENNReal.ofReal (min B (D*(1/2:ℝ)^i))) := by
      rw [← ENNReal.tsum_mul_left]
      exact add_le_add hzero (ENNReal.tsum_le_tsum htail)
    _ ≤ ENNReal.ofReal (K*B) + ENNReal.ofReal K *
        ENNReal.ofReal (3 * B * (1+max (Real.log (D/B)) 0)) := by
      gcongr
      exact dyadic_min_sum_le B D hB hD
    _ ≤ _ := by
      rw [← ENNReal.ofReal_mul hK,
        ← ENNReal.ofReal_add (mul_nonneg hK hB.le) (by positivity)]
      apply ENNReal.ofReal_le_ofReal
      have hL : 0 ≤ max (Real.log (D/B)) 0 := le_max_right _ _
      have hh : 0 ≤ K * B * max (Real.log (D/B)) 0 := by positivity
      dsimp [K] at *
      nlinarith

end MixedRadialSum

lemma two_pow_le_exp (k : ℕ) : (2:ℝ)^k ≤ Real.exp (k:ℝ) := by
  calc
    (2:ℝ)^k ≤ (Real.exp 1)^k :=
      pow_le_pow_left₀ (by norm_num) (by have h := Real.add_one_le_exp (1:ℝ); linarith) k
    _ = _ := by rw [← Real.exp_nat_mul]; simp

lemma shellConstant_exp_bound (d h n m r : ℕ) (hd : 1 ≤ d)
    (hh : h ≤ d) (hn : n ≤ d) (hm : m ≤ d) (hr : r ≤ d) :
    4 * shellConstant h n m r ≤ Real.exp (9 * (d:ℝ)) := by
  have hA : 1 ≤ (3:ℝ)^h * 2^(n+m+1) :=
    one_le_mul_of_one_le_of_one_le (one_le_pow₀ (by norm_num)) (one_le_pow₀ (by norm_num))
  have hpow : (3:ℝ)^h ≤ 4^h := pow_le_pow_left₀ (by norm_num) (by norm_num) h
  calc
    4 * shellConstant h n m r ≤ 8 * (2:ℝ)^r * ((3:ℝ)^h * 2^(n+m+1)) := by
      unfold shellConstant
      have hp : 0 ≤ (2:ℝ)^r := by positivity
      nlinarith
    _ ≤ 8 * (2:ℝ)^r * ((4:ℝ)^h * 2^(n+m+1)) := by gcongr
    _ = (2:ℝ)^(r+2*h+n+m+4) := by
      norm_num [pow_add, pow_mul]
      ring
    _ ≤ (2:ℝ)^(9*d) := pow_le_pow_right₀ (by norm_num) (by omega)
    _ ≤ Real.exp (9 * (d:ℝ)) := by simpa using two_pow_le_exp (9*d)


def sphereLift {d : ℕ} (ρ : Measure (Sphere d)) : Measure (Vector d) :=
  ρ.map Subtype.val

lemma LocalGrowth.sphereLift {d a : ℕ} {ρ : Measure (Sphere d)} {A : ℝ}
    (hρ : LocalGrowth ρ a A) : AmbientGrowth (sphereLift ρ) a A := by
  intro z t ht
  rw [CurrentProjection.sphereLift, Measure.map_apply continuous_subtype_val.measurable Metric.isClosed_closedBall.measurableSet]
  exact hρ z t ht

lemma sphereLift_ae_norm_sub_le {d : ℕ} (ρ : Measure (Sphere d)) (s : Sphere d) :
    ∀ᵐ u ∂sphereLift ρ, ‖u-(s:Vector d)‖ ≤ 2 := by
  apply (MeasurableEmbedding.ae_map_iff
    (MeasurableEmbedding.subtype_coe Metric.isClosed_sphere.measurableSet)).mpr
  apply Filter.Eventually.of_forall
  intro u
  have hu : ‖(u:Vector d)‖ = 1 := by simpa only [Metric.mem_sphere, dist_zero_right] using u.property
  have hs : ‖(s:Vector d)‖ = 1 := by simpa only [Metric.mem_sphere, dist_zero_right] using s.property
  exact (norm_sub_le _ _).trans (by rw [hu, hs]; norm_num)


theorem localGrowth_mixed_inverse {d m r q : ℕ} (hdim : AdmissibleDimensions d m r q)
    {ρ : Measure (Sphere d)} {B D : ℝ} (hB : 0 < B) (hD : 0 < D)
    (hBρ : LocalGrowth ρ (m+r) B) (hDρ : LocalGrowth ρ (d-1) D)
    (s : Sphere d) (V : Submodule ℝ (Vector d)) (hV : Module.finrank ℝ V ≤ q-1) :
    ∫⁻ u, mixedInverse (s:Vector d) V m r (u:Vector d) ∂ρ ≤ momentConstant 9 d B D := by
  rcases hdim with ⟨hm, hr, hq, hmn, hmargin⟩
  have hdimV : Module.finrank ℝ V + (m+1) ≤ m+r := by omega
  have hh := hBρ.sphereLift.mixed_inverse_le hDρ.sphereLift hB hD hr hmn V (s:Vector d)
    hdimV (sphereLift_ae_norm_sub_le ρ s)
  rw [sphereLift, lintegral_map (measurable_mixedInverse _ _ _ _) continuous_subtype_val.measurable] at hh
  refine hh.trans (ENNReal.ofReal_le_ofReal ?_)
  change 4 * shellConstant (Module.finrank ℝ V) (d-1) m r * B *
    (1 + max (Real.log (D/B)) 0) ≤ Real.exp (9 * (d:ℝ)) * B * (1 + max (Real.log (D/B)) 0)
  gcongr
  exact shellConstant_exp_bound d _ (d-1) m r (by omega) (by omega) (by omega) (by omega) (by omega)




end CurrentProjection

end

end OAI