lean/OAI/Geometry/CubicTorus/Compactness/GraphMollifier.lean

import Mathlib
import OAI.Geometry.CubicTorus.Regularity.CoverEssentialBoundary

namespace OAI

section
section
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal NNReal Topology RealInnerProductSpace symmDiff
namespace CubicTorus

theorem isMinimizerAt_primary_cone_C1_offvertex {V : ℝ} (hV : V ∈ Ioo 0 1)
    {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E)
    (hcenter : quotientMap cubeCenter ∈ frontier (openRepresentative E))
    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (hrlim : Tendsto r atTop (𝓝 0)) :
    ∃ (A : Set Cover) (σ : FiniteMeasure BlowupNormalBundle) (θ : ℝ) (φ : ℕ → ℕ),
      MeasurableSet A ∧ 0 < θ ∧ StrictMono φ ∧
      Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 2,
        |blowupCharacteristic E cubeCenter (r (φ i)) x-A.indicator (fun _ ↦ (1:ℝ)) x|)
        atTop (𝓝 0) ∧
      (∀ b : ℝ, 0 < b → b ≤ 1 →
        coverLocalPerimeter A (Metric.closedBall (0 : Cover) b) = ENNReal.ofReal (θ*b^2)) ∧
      (∀ B : Set Cover, MeasurableSet B → ∀ a b : ℝ, 0 < a → a < b → b ≤ 1 →
        A ∆ B ⊆ Metric.ball (0 : Cover) a →
        coverLocalPerimeter A (Metric.closedBall (0 : Cover) b) ≤
          coverLocalPerimeter B (Metric.closedBall (0 : Cover) b)) ∧
      (∀ t : ℝ, 1 ≤ t → ∀ᵐ x : Cover ∂volume, x ∈ Metric.ball (0 : Cover) 1 →
        (t⁻¹ • x ∈ A ↔ x ∈ A)) ∧
      (∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F →
        tsupport F ⊆ Metric.closedBall (0 : Cover) 1 →
        (∫ z, tangentialDivergence F z.1 z.2 ∂(σ : Measure BlowupNormalBundle)) = 0 ∧
        (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(σ : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) ∧
      ∀ p : Cover, ‖p‖ < 1 → p ≠ 0 → p ∈ coverEssentialBoundary A → CoverC1BoundaryAt A p := by
  obtain ⟨μ,hμ⟩ := exists_normalMeasure hE hmin.1.2.ne
  obtain ⟨H,hH⟩ := isMinimizerAt_centered_multiplier hV hE hmin hcenter hμ
  obtain ⟨A,σ,θ,φ,hA,hθ,hφ,hconv,hper,hcomp,hcone,hstat⟩ :=
    isMinimizerAt_minimizing_cone_blowup hV hE hmin hμ hH hcenter r hr hrlim
  refine ⟨A,σ,θ,φ,hA,hθ,hφ,hconv,hper,hcomp,hcone,hstat,?_⟩
  intro p hp hp0 hboundary
  exact actual_primary_offvertex_C1 hV hE hmin hcenter hμ hH
    (fun i ↦ r (φ i)) (fun i ↦ hr _) (hrlim.comp hφ.tendsto_atTop)
    hA hconv hcone hstat p hp hp0 hboundary

end CubicTorus

end
end

section

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

def graphMollifier (i : ℕ) : ContDiffBump (0 : Plane) where
  rIn := ((i:ℝ)+1)⁻¹/2
  rOut := ((i:ℝ)+1)⁻¹
  rIn_pos := by positivity
  rIn_lt_rOut := by
    have hh : 0 < ((i:ℝ)+1)⁻¹ := by positivity
    linarith

lemma graphMollifier_radius_tendsto :
    Tendsto (fun i ↦ (graphMollifier i).rOut) atTop (𝓝 0) := by
  simpa only [graphMollifier,one_div] using
    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))

lemma graphSmooth_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (y : Plane) :
    Tendsto (fun i ↦ graphSmooth (graphMollifier i) g y) atTop (𝓝 (g y)) := by
  apply tendsto_iff_norm_sub_tendsto_zero.mpr
  apply squeeze_zero (fun _ ↦ norm_nonneg _) (fun i ↦ ?_)
    (by simpa only [mul_zero] using graphMollifier_radius_tendsto.const_mul (K:ℝ))
  exact graphSmooth_error _ hg y

theorem graphSmooth_fderiv_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {y : Plane} (hc : ContinuousAt (fderiv ℝ g) y) :
    Tendsto (fun i ↦ fderiv ℝ (graphSmooth (graphMollifier i) g) y)
      atTop (𝓝 (fderiv ℝ g y)) := by
  rw [Metric.tendsto_nhds]
  intro ε hε
  obtain ⟨δ,hδ,hclose⟩ := Metric.continuousAt_iff.mp hc (ε/2) (by positivity)
  filter_upwards [graphMollifier_radius_tendsto.eventually (Iio_mem_nhds hδ)] with i hi
  have ha := lipschitz_probability_average_hasFDerivAt (graphSmoothingMeasure_ac (graphMollifier i))
    hg (graphMollifier i).rOut_pos.le (graphSmoothingMeasure_radius (graphMollifier i)) y
  change dist (fderiv ℝ (fun z ↦ ∫ w, g (z-w) ∂graphSmoothingMeasure (graphMollifier i)) y) _ < ε
  rw [ha.2.fderiv,dist_eq_norm]
  have he : (∫ _w : Plane, fderiv ℝ g y ∂graphSmoothingMeasure (graphMollifier i)) = fderiv ℝ g y := by simp
  rw [← he,← integral_sub ha.1 (integrable_const _)]
  apply lt_of_le_of_lt (norm_integral_le_integral_norm _) ?_
  have hb : (∫ w, ‖fderiv ℝ g (y-w)-fderiv ℝ g y‖ ∂graphSmoothingMeasure (graphMollifier i)) ≤ ε/2 := by
    calc
      _ ≤ ∫ _w : Plane, ε/2 ∂graphSmoothingMeasure (graphMollifier i) := by
        apply integral_mono_ae (ha.1.sub (integrable_const _)).norm (integrable_const _)
        filter_upwards [graphSmoothingMeasure_radius (graphMollifier i)] with w hw
        have hh : dist (y-w) y < δ := by simpa only [dist_eq_norm,sub_sub_cancel_left,norm_neg] using hw.trans_lt hi
        simpa only [dist_eq_norm,Pi.sub_apply] using (hclose hh).le
      _ = _ := by simp
  exact hb.trans_lt (by linarith)

lemma graphSmooth_gradient_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {y : Plane} (hc : ContinuousAt (fderiv ℝ g) y) :
    Tendsto (fun i ↦ gradient (graphSmooth (graphMollifier i) g) y) atTop (𝓝 (gradient g y)) := by
  exact ((InnerProductSpace.toDual ℝ Plane).symm.continuous.tendsto _).comp
    (graphSmooth_fderiv_tendsto hg hc)

end CubicTorus

end
end

section

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

lemma graph_trace_volume_zero (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} (hg : Continuous g) :
    volume {x : Cover | ⟪n,x-a⟫ = graphHeightExtension a n g x} = 0 := by
  have hz : volume {x : Cover | ⟪n,x-a⟫ = 0} = 0 := by
    simpa only [Set.preimage_ofPred_eq,sub_eq_add_neg] using
      (measurePreserving_add_right volume (-a)).quasiMeasurePreserving.preimage_null
        (hyperplane_volume_zero hn)
  have hh := (graphShear_measurePreserving a hn hg.neg.measurable).quasiMeasurePreserving.preimage_null hz
  convert hh using 2
  ext x
  simp only [Set.mem_ofPred_eq,Set.mem_preimage,graphShear,normalMove,graphHeightExtension,
    add_sub_right_comm,inner_add_right,inner_smul_right,real_inner_self_eq_norm_sq,hn,
    one_pow,mul_one,Pi.neg_apply]
  constructor <;> intro h <;> linarith

lemma measurable_graphSublevelIndicator (a n : Cover) {g : Plane → ℝ} (hg : Measurable g) :
    Measurable (graphSublevelIndicator a n g) := by
  exact measurable_const.indicator (measurableSet_lt (by fun_prop)
    (hg.comp ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const))))

lemma graphSublevelIndicator_bound (a n : Cover) (g : Plane → ℝ) (x : Cover) :
    |graphSublevelIndicator a n g x| ≤ 1 := by
  classical
  by_cases hx : ⟪n,x-a⟫ < graphHeightExtension a n g x <;>
    simp [graphSublevelIndicator,hx]

lemma graphSublevelIndicator_tendsto {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    (a n : Cover) {x : Cover} (hx : ⟪n,x-a⟫ ≠ graphHeightExtension a n g x) :
    Tendsto (fun i ↦ graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x)
      atTop (𝓝 (graphSublevelIndicator a n g x)) := by
  have ht := graphSmooth_tendsto hg (normalPlaneCoordinates n (x-a))
  have he : ∀ᶠ i in atTop,
      graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x = graphSublevelIndicator a n g x := by
    by_cases hl : ⟪n,x-a⟫ < graphHeightExtension a n g x
    · filter_upwards [ht.eventually (Ioi_mem_nhds hl)] with i hi
      change (if ⟪n,x-a⟫ < graphSmooth (graphMollifier i) g (normalPlaneCoordinates n (x-a)) then (1:ℝ) else 0) = _
      rw [ite_eq_left hi]
      exact (indicator_of_mem hl (fun _ ↦ (1:ℝ))).symm
    · have hh : graphHeightExtension a n g x < ⟪n,x-a⟫ := lt_of_le_of_ne (not_lt.mp hl) hx.symm
      filter_upwards [ht.eventually (Iio_mem_nhds hh)] with i hi
      have hi' : ¬ ⟪n,x-a⟫ < graphHeightExtension a n (graphSmooth (graphMollifier i) g) x := not_lt.mpr hi.le
      simp [graphSublevelIndicator,hl,hi']
  apply tendsto_const_nhds.congr'
  filter_upwards [he] with i hi
  exact hi.symm

lemma graphSublevel_flux_mollification_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hs : HasCompactSupport F) :
    Tendsto (fun i ↦ ∫ x : Cover, graphSublevelIndicator a n (graphSmooth (graphMollifier i) g) x * divergence F x)
      atTop (𝓝 (∫ x : Cover, graphSublevelIndicator a n g x * divergence F x)) := by
  have hi : Integrable (divergence F) := (continuous_divergence hF).integrable_of_hasCompactSupport
    (hs.of_isClosed_subset isClosed_closure (divergence_tsupport_subset F))
  apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖divergence F x‖)
    (fun i ↦ ((measurable_graphSublevelIndicator a n (graphSmooth_contDiff _ hg).continuous.measurable).mul
      (continuous_divergence hF).measurable).aestronglyMeasurable) hi.norm
  · intro i
    filter_upwards [] with x
    simp only [Pi.mul_apply,norm_mul]
    exact mul_le_of_le_one_left (norm_nonneg _) (graphSublevelIndicator_bound a n _ x)
  · have hae : ∀ᵐ x : Cover ∂volume, ⟪n,x-a⟫ ≠ graphHeightExtension a n g x := by
      simpa only [ae_iff,not_not] using graph_trace_volume_zero a hn hg.continuous
    filter_upwards [hae] with x hx
    exact (graphSublevelIndicator_tendsto hg a n hx).mul_const _

end CubicTorus

end
end

section

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

lemma graphAreaVector_norm_linear {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (y : Plane) :
    ‖graphAreaVector n g y‖ ≤ 1+(K:ℝ) := by
  calc
    _ ≤ ‖n‖+‖normalPlaneEmbedding n (gradient g y)‖ := norm_sub_le _ _
    _ = 1+‖fderiv ℝ g y‖ := by rw [hn,normalPlaneEmbedding_norm,plane_gradient_norm]
    _ ≤ _ := add_le_add le_rfl (norm_fderiv_le_of_lipschitz ℝ hg)

lemma graph_trace_zero_outside {X : Type*} [Zero X] {a n : Cover} (hn : ‖n‖ = 1)
    {F : Cover → X} {R : ℝ} (hs : tsupport F ⊆ Metric.closedBall a R)
    (g : Plane → ℝ) {y : Plane} (hy : y ∉ Metric.closedBall (0 : Plane) R) :
    F (a+normalGraph n g y) = 0 := by
  apply image_eq_zero_of_notMem_tsupport
  intro h
  have hh := hs h
  rw [Metric.mem_closedBall,dist_eq_norm,add_sub_cancel_left] at hh
  exact hy (by simpa only [Metric.mem_closedBall,dist_zero_right] using
    (normalGraph_norm_ge hn g y).trans hh)

theorem integral_C1graphSublevel_divergence (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F)
    (hs : tsupport F ⊆ Metric.closedBall a R) :
    (∫ x : Cover, graphSublevelIndicator a n g x * divergence F x) =
      ∫ y : Plane, ⟪graphAreaVector n g y, F (a+normalGraph n g y)⟫ := by
  have hFs : HasCompactSupport F := (isCompact_closedBall a R).of_isClosed_subset isClosed_closure hs
  obtain ⟨M,hM⟩ := hF.continuous.norm.bddAbove_range_of_hasCompactSupport hFs.norm
  have hMb (x : Cover) : ‖F x‖ ≤ M := hM (mem_range_self x)
  have hM0 : 0 ≤ M := (norm_nonneg (F 0)).trans (hMb 0)
  let B : Plane → ℝ := (Metric.closedBall (0 : Plane) R).indicator (fun _ ↦ (1+(K:ℝ))*M)
  have hBi : Integrable B := (integrableOn_const (isCompact_closedBall (0 : Plane) R).measure_lt_top.ne).integrable_indicator Metric.isClosed_closedBall.measurableSet
  have ht : Tendsto (fun i ↦ ∫ y : Plane,
      ⟪graphAreaVector n (graphSmooth (graphMollifier i) g) y,
        F (a+normalGraph n (graphSmooth (graphMollifier i) g) y)⟫) atTop
      (𝓝 (∫ y : Plane, ⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫)) := by
    apply tendsto_integral_of_dominated_convergence B
      (fun i ↦ ((graphAreaVector_continuous n (graphSmooth_contDiff _ hg)).inner
        (graph_trace_continuous a n (graphSmooth_contDiff _ hg).continuous hF.continuous)).aestronglyMeasurable) hBi
    · intro i
      filter_upwards [] with y
      by_cases hy : y ∈ Metric.closedBall (0 : Plane) R
      · rw [show B y = (1+(K:ℝ))*M from indicator_of_mem hy _]
        exact (norm_inner_le_norm _ _).trans
          (mul_le_mul (graphAreaVector_norm_linear hn (graphSmooth_lipschitz _ hg) y)
            (hMb _) (norm_nonneg _) (by positivity))
      · rw [graph_trace_zero_outside hn hs _ hy,inner_zero_right,norm_zero,
          show B y = 0 from indicator_of_notMem hy _]
    · filter_upwards [] with y
      by_cases hy : y ∈ Metric.closedBall (0 : Plane) R
      · have hd := graphSmooth_gradient_tendsto hg (hc y hy)
        have hnG : Tendsto (fun i ↦ normalGraph n (graphSmooth (graphMollifier i) g) y)
            atTop (𝓝 (normalGraph n g y)) :=
          tendsto_const_nhds.add ((graphSmooth_tendsto hg y).smul_const n)
        have hnA : Tendsto (fun i ↦ graphAreaVector n (graphSmooth (graphMollifier i) g) y)
            atTop (𝓝 (graphAreaVector n g y)) :=
          tendsto_const_nhds.sub (((normalPlaneLinearIsometry n).continuous.tendsto _).comp hd)
        exact hnA.inner ((hF.continuous.tendsto _).comp (tendsto_const_nhds.add hnG))
      · simp only [graph_trace_zero_outside hn hs _ hy,inner_zero_right]
        exact tendsto_const_nhds
  apply tendsto_nhds_unique (graphSublevel_flux_mollification_tendsto a hn hg hF hFs)
  simpa only [integral_graphSublevel_divergence a hn (graphSmooth_contDiff _ hg) hF hFs] using ht

end CubicTorus

end
end

section

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

lemma graphAreaVector_continuousAt (n : Cover) {g : Plane → ℝ} {y : Plane}
    (hc : ContinuousAt (fderiv ℝ g) y) : ContinuousAt (graphAreaVector n g) y := by
  exact continuousAt_const.sub ((normalPlaneLinearIsometry n).continuous.continuousAt.comp
    ((InnerProductSpace.toDual ℝ Plane).symm.continuous.continuousAt.comp hc))

lemma graph_weighted_continuous (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} (hg : Continuous g) {R : ℝ}
    {φ : Cover → ℝ} (hφ : Continuous φ) (hs : tsupport φ ⊆ Metric.closedBall a R)
    {ψ : Plane → ℝ} (hψ : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt ψ y) :
    Continuous (fun y : Plane ↦ φ (a+normalGraph n g y)*ψ y) := by
  rw [continuous_iff_continuousAt]
  intro y
  by_cases hy : y ∈ Metric.closedBall (0 : Plane) R
  · exact (graph_trace_continuous a n hg hφ).continuousAt.mul (hψ y hy)
  · have he : (fun z : Plane ↦ φ (a+normalGraph n g z)*ψ z) =ᶠ[𝓝 y] fun _ ↦ 0 := by
      filter_upwards [Metric.isClosed_closedBall.isOpen_compl.mem_nhds hy] with z hz
      rw [graph_trace_zero_outside hn hs _ hz,zero_mul]
    exact continuousAt_const.congr_of_eventuallyEq he

lemma graph_weighted_integrable (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} (hg : Continuous g) {R : ℝ}
    {φ : Cover → ℝ} (hφ : Continuous φ) (hsφ : HasCompactSupport φ)
    (hs : tsupport φ ⊆ Metric.closedBall a R)
    {ψ : Plane → ℝ} (hψ : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt ψ y) :
    Integrable (fun y : Plane ↦ φ (a+normalGraph n g y)*ψ y) :=
  (graph_weighted_continuous a hn hg hφ hs hψ).integrable_of_hasCompactSupport
    (graph_trace_compactSupport a hn hg hsφ).mul_right

lemma setIntegral_graphSublevel_divergence (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F)
    (hs : tsupport F ⊆ Metric.closedBall a R) :
    (∫ x in {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x}, divergence F x) =
      ∫ y : Plane, ⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫ := by
  rw [← integral_C1graphSublevel_divergence a hn hg hc hF hs]
  have hm : MeasurableSet {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x} :=
    measurableSet_lt (by fun_prop) (hg.continuous.measurable.comp
      ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const)))
  rw [← integral_indicator hm]
  congr 1
  funext x
  classical
  by_cases hx : ⟪n,x-a⟫ < graphHeightExtension a n g x <;>
    simp [graphSublevelIndicator,hx]

theorem blowup_graph_patch_weighted_flux {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) =
        ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1)
    {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) :
    (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*⟪(z.2 : Cover),F z.1⟫
      ∂(ν : Measure BlowupNormalBundle)) =
      ∫ y : Plane, φ (a+normalGraph n g y)*⟪graphAreaVector n g y,F (a+normalGraph n g y)⟫ := by
  let H : Cover → Cover := fun x ↦ φ x • F x
  have hH : ContDiff ℝ (⊤ : ℕ∞) H := hφ.smul hF
  have hsH : HasCompactSupport H := hsφ.smul_right
  have hHO : tsupport H ⊆ O := (tsupport_smul_subset_left _ _).trans hφO
  have hnH (x) : ‖H x‖ ≤ 1 := by
    rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (hφb x).1]
    exact (mul_le_mul (hφb x).2 (hnF x) (norm_nonneg _) zero_le_one).trans_eq (one_mul 1)
  have he : (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, ⟪(z.2 : Cover),H z.1⟫
      ∂(ν : Measure BlowupNormalBundle)) = ∫ z, ⟪(z.2 : Cover),H z.1⟫ ∂(ν : Measure BlowupNormalBundle) := by
    apply setIntegral_eq_integral_of_forall_compl_eq_zero
    intro z hz
    have hh : H z.1 = 0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hz (hHO h))
    simp only [hh,inner_zero_right]
  have hm : MeasurableSet {x : Cover | ⟪n,x-a⟫ < graphHeightExtension a n g x} :=
    measurableSet_lt (by fun_prop) (hg.continuous.measurable.comp
      ((normalPlaneCoordinates n).measurable.comp (measurable_id.sub measurable_const)))
  have hh := hflux H hH hsH hHO hnH
  rw [setIntegral_divergence_congr_local hA hm hgraph
      ((tsupport_smul_subset_left _ _).trans hφU),
    setIntegral_graphSublevel_divergence a hn hg hc hH
      ((tsupport_smul_subset_left _ _).trans hφR)] at hh
  rw [← he] at hh
  simpa only [H,inner_smul_right] using hh

end CubicTorus

end
end

section

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

lemma graphSlopeNormal_contDiff {n : Cover} (hn : ‖n‖ = 1) :
    ContDiff ℝ (⊤ : ℕ∞) (graphSlopeNormal n) := by
  have hv : ContDiff ℝ (⊤ : ℕ∞) (fun p : Plane ↦ n-normalPlaneEmbedding n p) :=
    contDiff_const.sub (normalPlaneLinearIsometry n).contDiff
  have hz (p : Plane) : n-normalPlaneEmbedding n p ≠ 0 := by
    intro h
    have hh := graphSlopeVector_norm_ge hn p
    rw [h,norm_zero] at hh
    norm_num at hh
  exact ((hv.norm ℝ hz).inv (fun p ↦ norm_ne_zero_iff.mpr (hz p))).smul hv

def graphUnitNormalExtension (a n : Cover) (g : Plane → ℝ) (x : Cover) : Cover :=
  graphSlopeNormal n (gradient g (normalPlaneCoordinates n (x-a)))

lemma graphUnitNormalExtension_unit (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    (g : Plane → ℝ) (x : Cover) : ‖graphUnitNormalExtension a n g x‖ = 1 :=
  graphSlopeNormal_unit hn _

lemma graphUnitNormalExtension_contDiff (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) :
    ContDiff ℝ (⊤ : ℕ∞) (graphUnitNormalExtension a n g) := by
  have hd := hg.fderiv_right (m := (⊤ : ℕ∞)) (by simp)
  have hgrad : ContDiff ℝ (⊤ : ℕ∞) (gradient g) :=
    (InnerProductSpace.toDual ℝ Plane).symm.contDiff.comp hd
  exact (graphSlopeNormal_contDiff hn).comp (hgrad.comp
    ((normalPlaneCoordinates n).contDiff.comp (contDiff_id.sub contDiff_const)))

lemma graphUnitNormalExtension_graph (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    (g f : Plane → ℝ) (y : Plane) :
    graphUnitNormalExtension a n f (a+normalGraph n g y) = graphSlopeNormal n (gradient f y) := by
  unfold graphUnitNormalExtension
  rw [add_sub_cancel_left,normalGraph, map_add,map_smul,
    normalPlaneCoordinates_embedding,normalPlaneCoordinates_normal hn,smul_zero,add_zero]

lemma graphUnitNormalExtension_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {x : Cover}
    (hc : ContinuousAt (fderiv ℝ g) (normalPlaneCoordinates n (x-a))) :
    Tendsto (fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) x)
      atTop (𝓝 (graphUnitNormalExtension a n g x)) :=
  ((graphSlopeNormal_contDiff hn).continuous.tendsto _).comp (graphSmooth_gradient_tendsto hg hc)

lemma graphAreaVector_inner_unit {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) (y : Plane) :
    ⟪graphAreaVector n g y,graphSlopeNormal n (gradient g y)⟫ = ‖graphAreaVector n g y‖ := by
  have hh : 0 < ‖graphAreaVector n g y‖ := lt_of_lt_of_le (by norm_num) (graphSlopeVector_norm_ge hn _)
  rw [graphSlopeNormal,inner_smul_right,show n-normalPlaneEmbedding n (gradient g y)=graphAreaVector n g y from rfl,
    real_inner_self_eq_norm_sq]
  field_simp

end CubicTorus

end
end

section

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

lemma graph_area_trace_mollified_normal_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    {φ : Cover → ℝ} (hφ : Continuous φ) (hsφ : HasCompactSupport φ)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφ0 : ∀ x, 0 ≤ φ x) :
    Tendsto (fun i ↦ ∫ y : Plane, φ (a+normalGraph n g y)*
      ⟪graphAreaVector n g y,graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g) (a+normalGraph n g y)⟫)
      atTop (𝓝 (∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)) := by
  have hi := graph_weighted_integrable a hn hg.continuous hφ hsφ hφR
    (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
  apply tendsto_integral_of_dominated_convergence
    (fun y ↦ φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) ?_ hi
  · intro i
    filter_upwards [] with y
    rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφ0 _)]
    apply mul_le_mul_of_nonneg_left _ (hφ0 _)
    exact (norm_inner_le_norm _ _).trans_eq (by rw [graphUnitNormalExtension_unit a hn,mul_one])
  · filter_upwards [] with y
    by_cases hy : y ∈ Metric.closedBall (0 : Plane) R
    · have ht : Tendsto (fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g)
          (a+normalGraph n g y)) atTop (𝓝 (graphSlopeNormal n (gradient g y))) := by
        simp only [graphUnitNormalExtension_graph a hn]
        exact ((graphSlopeNormal_contDiff hn).continuous.tendsto _).comp
          (graphSmooth_gradient_tendsto hg (hc y hy))
      simpa only [graphAreaVector_inner_unit hn] using ((show Tendsto (fun _ : ℕ ↦ graphAreaVector n g y) atTop
        (𝓝 (graphAreaVector n g y)) from tendsto_const_nhds).inner ht).const_mul (φ (a+normalGraph n g y))
    · simp only [graph_trace_zero_outside hn hφR _ hy,zero_mul]
      exact tendsto_const_nhds
  · intro i
    exact (graph_weighted_continuous a hn hg.continuous hφ hφR (fun y hy ↦
      (graphAreaVector_continuousAt n (hc y hy)).inner
        (graph_trace_continuous a n hg.continuous
          (graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg)).continuous).continuousAt)).aestronglyMeasurable

theorem blowup_graph_patch_weighted_mass {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O})
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) =
        ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) :
    (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1 ∂(ν : Measure BlowupNormalBundle)) =
      ∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by
  let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}
  have hiφ : Integrable (fun z : BlowupNormalBundle ↦ φ z.1) μ :=
    Integrable.of_bound (hφ.continuous.comp continuous_fst).aestronglyMeasurable 1
      (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hφb z.1).1]; exact (hφb z.1).2)
  have hiArea := graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR
    (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
  have hiF (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) :
      Integrable (fun z : BlowupNormalBundle ↦ φ z.1*⟪(z.2 : Cover),F z.1⟫) μ := by
    simpa only [mul_comm] using hiφ.bdd_mul
      (show Continuous (fun z : BlowupNormalBundle ↦ ⟪(z.2 : Cover),F z.1⟫) from by fun_prop).aestronglyMeasurable
      (ae_of_all μ fun z ↦ blowup_test_inner_bound hnF z)
  have hfield (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hnF : ∀ x, ‖F x‖ ≤ 1) :
      (∫ z, φ z.1*⟪(z.2 : Cover),F z.1⟫ ∂μ) ≤
        ∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by
    rw [show (∫ z, φ z.1*⟪(z.2 : Cover),F z.1⟫ ∂μ) = _ from
      blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb hF hnF]
    apply integral_mono (graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR
      (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).inner
        (graph_trace_continuous a n hg.continuous hF.continuous).continuousAt)) hiArea
    intro y
    apply mul_le_mul_of_nonneg_left _ (hφb _).1
    exact (real_inner_le_norm _ _).trans (mul_le_of_le_one_right (norm_nonneg _) (hnF _))
  refine le_antisymm ?_ ?_
  · exact blowupNormal_weighted_saturation heq hflux hiφ (by norm_num : (0 : ℝ)<1)
      (fun z ↦ (hφb z.1).2) (fun F hF _ _ hnF ↦ hfield F hF hnF)
  · apply le_of_tendsto (graph_area_trace_mollified_normal_tendsto a hn hg hc
      hφ.continuous hsφ hφR (fun x ↦ (hφb x).1))
    apply Eventually.of_forall
    intro i
    let F := graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g)
    have hF : ContDiff ℝ (⊤ : ℕ∞) F := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg)
    have hnF (x : Cover) : ‖F x‖ ≤ 1 := (graphUnitNormalExtension_unit a hn _ x).le
    rw [← blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb hF hnF]
    apply integral_mono (hiF F hF hnF) hiφ
    intro z
    apply mul_le_of_le_one_right (hφb _).1
    exact (real_inner_le_norm (z.2 : Cover) (F z.1)).trans (by rw [z.2.property,one_mul]; exact hnF _)

end CubicTorus

end
end

section

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

lemma normalPlaneCoordinates_norm_le (n x : Cover) : ‖normalPlaneCoordinates n x‖ ≤ ‖x‖ := by
  simpa only [map_zero,dist_zero_right,NNReal.coe_one,one_mul] using
    (normalPlaneCoordinates_lipschitz n).dist_le_mul x 0

lemma graph_weighted_normal_tendsto (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    {φ : Cover → ℝ} (hφR : tsupport φ ⊆ Metric.closedBall a R) (z : BlowupNormalBundle) :
    Tendsto (fun i ↦ φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n
      (graphSmooth (graphMollifier i) g) z.1⟫) atTop
      (𝓝 (φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫)) := by
  by_cases hz : z.1 ∈ tsupport φ
  · have hcoord : normalPlaneCoordinates n (z.1-a) ∈ Metric.closedBall (0 : Plane) R := by
      simp only [Metric.mem_closedBall,dist_zero_right]
      exact (normalPlaneCoordinates_norm_le n _).trans (by simpa [Metric.mem_closedBall,dist_eq_norm] using hφR hz)
    exact ((show Tendsto (fun _ : ℕ ↦ (z.2 : Cover)) atTop (𝓝 (z.2 : Cover)) from
      tendsto_const_nhds).inner (graphUnitNormalExtension_tendsto a hn hg (hc _ hcoord))).const_mul _
  · simp only [image_eq_zero_of_notMem_tsupport hz,zero_mul]
    exact tendsto_const_nhds

theorem blowup_graph_patch_normal_ae {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O})
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) :
    ∀ᵐ z ∂(ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O},
      0 < φ z.1 → (z.2 : Cover)=graphUnitNormalExtension a n g z.1 := by
  let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}
  let F : ℕ → Cover → Cover := fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g)
  let f : ℕ → BlowupNormalBundle → ℝ := fun i z ↦ φ z.1*⟪(z.2 : Cover),F i z.1⟫
  let f₀ : BlowupNormalBundle → ℝ := fun z ↦ φ z.1*⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫
  have hF (i) : ContDiff ℝ (⊤ : ℕ∞) (F i) := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg)
  have hFn (i) (x) : ‖F i x‖ ≤ 1 := (graphUnitNormalExtension_unit a hn _ x).le
  have hf (i) : AEStronglyMeasurable (f i) μ :=
    (show Continuous (f i) from (hφ.continuous.comp continuous_fst).mul
      (continuous_subtype_val.comp continuous_snd |>.inner ((hF i).continuous.comp continuous_fst))).aestronglyMeasurable
  have ht : ∀ᵐ z ∂μ, Tendsto (fun i ↦ f i z) atTop (𝓝 (f₀ z)) :=
    ae_of_all μ (graph_weighted_normal_tendsto a hn hg hc hφR)
  have hb (i) (z : BlowupNormalBundle) : ‖f i z‖ ≤ 1 := by
    dsimp only [f]
    rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1]
    exact (mul_le_mul (hφb _).2 (blowup_test_inner_bound (hFn i) z) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1)
  have hb₀ (z : BlowupNormalBundle) : ‖f₀ z‖ ≤ 1 := by
    dsimp only [f₀]
    rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1]
    exact (mul_le_mul (hφb _).2 (blowup_test_inner_bound
      (fun x ↦ (graphUnitNormalExtension_unit a hn _ x).le) z) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1)
  have hi₀ : Integrable f₀ μ := Integrable.of_bound
    (aestronglyMeasurable_of_tendsto_ae atTop hf ht) 1 (ae_of_all μ hb₀)
  have hilim : Tendsto (fun i ↦ ∫ z, f i z ∂μ) atTop (𝓝 (∫ z, f₀ z ∂μ)) :=
    tendsto_integral_of_dominated_convergence (fun _ ↦ (1:ℝ)) hf
      (integrable_const 1) (fun i ↦ ae_of_all μ (hb i)) ht
  have heint : (∫ z, f₀ z ∂μ) = ∫ z, φ z.1 ∂μ := by
    apply tendsto_nhds_unique hilim
    rw [show (∫ z, φ z.1 ∂μ) = _ from
      blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb]
    have hh := graph_area_trace_mollified_normal_tendsto a hn hg hc hφ.continuous hsφ hφR
      (fun x ↦ (hφb x).1)
    convert hh using 1
    funext i
    exact blowup_graph_patch_weighted_flux hA a hn hg hc hgraph hflux hφ hsφ hφU hφO hφR hφb (hF i) (hFn i)
  have hiφ : Integrable (fun z : BlowupNormalBundle ↦ φ z.1) μ :=
    Integrable.of_bound (hφ.continuous.comp continuous_fst).aestronglyMeasurable 1
      (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hφb _).1]; exact (hφb _).2)
  have hinon (z : BlowupNormalBundle) : 0 ≤ φ z.1-f₀ z := by
    have hh : ⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫ ≤ 1 := by
      exact (real_inner_le_norm _ _).trans_eq (by rw [z.2.property,graphUnitNormalExtension_unit a hn,one_mul])
    exact sub_nonneg.mpr (mul_le_of_le_one_right (hφb _).1 hh)
  have hezero : (∫ z, (φ z.1-f₀ z) ∂μ)=0 := by rw [integral_sub hiφ hi₀,heint,sub_self]
  have hae := (integral_eq_zero_iff_of_nonneg_ae (ae_of_all μ hinon) (hiφ.sub hi₀)).mp hezero
  filter_upwards [hae] with z hz
  intro hzφ
  have hi : ⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫ = 1 := by
    have hh : φ z.1*(1-⟪(z.2 : Cover),graphUnitNormalExtension a n g z.1⟫)=0 := by
      simpa only [mul_sub,mul_one,f₀,Pi.zero_apply] using hz
    exact (sub_eq_zero.mp ((mul_eq_zero.mp hh).resolve_left hzφ.ne')).symm
  exact (inner_eq_one_iff_of_norm_eq_one z.2.property (graphUnitNormalExtension_unit a hn g z.1)).mp hi

end CubicTorus

end
end

section

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

theorem blowup_graph_patch_signed_mass {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O})
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1)
    {f : Cover → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hfb : ∀ x, |f x| ≤ 1) :
    (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*f z.1 ∂(ν : Measure BlowupNormalBundle)) =
      ∫ y : Plane, φ (a+normalGraph n g y)*f (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by
  let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}
  let ψ : Cover → ℝ := fun x ↦ φ x*((1+f x)/2)
  have hψ : ContDiff ℝ (⊤ : ℕ∞) ψ := hφ.mul ((contDiff_const.add hf).div_const 2)
  have hsψ : tsupport ψ ⊆ tsupport φ := tsupport_mul_subset_left
  have hψb (x : Cover) : ψ x ∈ Icc 0 1 := by
    have hx := abs_le.mp (hfb x)
    have hhalf : (1+f x)/2 ∈ Icc 0 1 := ⟨by linarith,by linarith⟩
    exact ⟨mul_nonneg (hφb x).1 hhalf.1,
      (mul_le_mul (hφb x).2 hhalf.2 hhalf.1 zero_le_one).trans_eq (one_mul 1)⟩
  have hsψ' : HasCompactSupport ψ := hsφ.of_isClosed_subset (isClosed_tsupport _) hsψ
  have hmψ := blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hψ hsψ'
    (hsψ.trans hφU) (hsψ.trans hφO) (hsψ.trans hφR) hψb
  have hmφ := blowup_graph_patch_weighted_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb
  have hI (w : Cover → ℝ) (hw : Continuous w) (hb : ∀ x, w x ∈ Icc 0 1) :
      Integrable (fun z : BlowupNormalBundle ↦ w z.1) μ :=
    Integrable.of_bound (hw.comp continuous_fst).aestronglyMeasurable 1
      (ae_of_all μ fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (hb _).1]; exact (hb _).2)
  have hGφ := graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR
    (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
  have hGψ := graph_weighted_integrable a hn hg.continuous hψ.continuous hsψ' (hsψ.trans hφR)
    (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
  calc
    (∫ z, φ z.1*f z.1 ∂μ) = ∫ z, (2*ψ z.1-φ z.1) ∂μ := by
      congr 1; funext z; dsimp only [ψ]; ring
    _ = 2*(∫ z, ψ z.1 ∂μ)-(∫ z, φ z.1 ∂μ) := by
      rw [integral_sub ((hI ψ hψ.continuous hψb).const_mul 2) (hI φ hφ.continuous hφb),integral_const_mul]
    _ = 2*(∫ y : Plane, ψ (a+normalGraph n g y)*‖graphAreaVector n g y‖)-
        (∫ y : Plane, φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) := by rw [hmψ,hmφ]
    _ = ∫ y : Plane, (2*(ψ (a+normalGraph n g y)*‖graphAreaVector n g y‖)-
        φ (a+normalGraph n g y)*‖graphAreaVector n g y‖) := by
      rw [integral_sub (hGψ.const_mul 2) hGφ,integral_const_mul]
    _ = _ := by congr 1; funext y; dsimp only [ψ]; ring

end CubicTorus

end
end

section

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

lemma weighted_tendsto_on_tsupport {X : Type*} [TopologicalSpace X]
    {φ : X → ℝ} {f : ℕ → X → ℝ} {g : X → ℝ}
    (h : ∀ x ∈ tsupport φ, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) (x : X) :
    Tendsto (fun i ↦ φ x*f i x) atTop (𝓝 (φ x*g x)) := by
  by_cases hx : x ∈ tsupport φ
  · exact (h x hx).const_mul _
  · simp only [image_eq_zero_of_notMem_tsupport hx,zero_mul]
    exact tendsto_const_nhds

theorem blowup_graph_patch_limit_mass {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O})
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1)
    {f : ℕ → Cover → ℝ} {f₀ : Cover → ℝ}
    (hf : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (f i)) (hfb : ∀ i x, |f i x| ≤ 1)
    (hlim : ∀ x ∈ tsupport φ, Tendsto (fun i ↦ f i x) atTop (𝓝 (f₀ x))) :
    (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*f₀ z.1 ∂(ν : Measure BlowupNormalBundle)) =
      ∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by
  let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}
  have hleft : Tendsto (fun i ↦ ∫ z, φ z.1*f i z.1 ∂μ) atTop (𝓝 (∫ z, φ z.1*f₀ z.1 ∂μ)) := by
    apply tendsto_integral_of_dominated_convergence (fun _ ↦ (1:ℝ))
    · intro i
      exact ((hφ.continuous.mul (hf i).continuous).comp continuous_fst).aestronglyMeasurable
    · exact integrable_const 1
    · intro i
      filter_upwards [] with z
      rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1,Real.norm_eq_abs]
      exact (mul_le_mul (hφb _).2 (hfb i _) (abs_nonneg _) zero_le_one).trans_eq (one_mul 1)
    · exact ae_of_all μ fun z ↦ weighted_tendsto_on_tsupport hlim z.1
  have hright : Tendsto (fun i ↦ ∫ y : Plane,
      φ (a+normalGraph n g y)*f i (a+normalGraph n g y)*‖graphAreaVector n g y‖) atTop
      (𝓝 (∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖)) := by
    apply tendsto_integral_of_dominated_convergence (fun y ↦ φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)
    · intro i
      have hh := graph_weighted_continuous a hn hg.continuous hφ.continuous hφR
        (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
      have hmeas : AEStronglyMeasurable (fun y : Plane ↦
          (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)*f i (a+normalGraph n g y))
          (volume : Measure Plane) :=
        (hh.mul (graph_trace_continuous a n hg.continuous (hf i).continuous)).aestronglyMeasurable
      simpa only [mul_assoc,mul_comm,mul_left_comm] using hmeas
    · exact graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR
        (fun y hy ↦ (graphAreaVector_continuousAt n (hc y hy)).norm)
    · intro i
      filter_upwards [] with y
      rw [norm_mul,norm_mul,Real.norm_eq_abs,abs_of_nonneg (hφb _).1,Real.norm_eq_abs,
        norm_norm]
      exact mul_le_mul_of_nonneg_right (mul_le_of_le_one_right (hφb _).1 (hfb i _)) (norm_nonneg _)
    · filter_upwards [] with y
      exact (weighted_tendsto_on_tsupport hlim _).mul_const _
  apply tendsto_nhds_unique hleft
  convert hright using 1
  funext i
  exact blowup_graph_patch_signed_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb (hf i) (hfb i)

end CubicTorus

end
end

section

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

lemma contDiff_divergence {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) :
    ContDiff ℝ (⊤ : ℕ∞) (divergence F) := by
  have hd := hF.fderiv_right (m := (⊤ : ℕ∞)) (by simp)
  unfold divergence
  apply ContDiff.sum
  intro i _
  exact (EuclideanSpace.proj i).contDiff.comp (hd.clm_apply contDiff_const)

lemma contDiff_tangentialDivergence_normal {F N : Cover → Cover}
    (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hN : ContDiff ℝ (⊤ : ℕ∞) N) :
    ContDiff ℝ (⊤ : ℕ∞) (fun x ↦ tangentialDivergence F x (N x)) :=
  (contDiff_divergence hF).sub (hN.inner ℝ ((hF.fderiv_right (m := (⊤ : ℕ∞)) (by simp)).clm_apply hN))

lemma continuous_tangentialDivergence_normal (F : Cover → Cover) (x : Cover) :
    Continuous (fun v : Cover ↦ tangentialDivergence F x v) :=
  continuous_const.sub (continuous_id.inner ((fderiv ℝ F x).continuous.comp continuous_id))

theorem blowup_C1graph_stationary {A O U : Set Cover} (hA : MeasurableSet A)
    (a : Cover) {n : Cover} (hn : ‖n‖ = 1)
    {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g)
    {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y)
    (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x))
    {ν : FiniteMeasure BlowupNormalBundle}
    (heq : coverLocalPerimeter A O = (ν : Measure BlowupNormalBundle) {z | z.1 ∈ O})
    (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F →
      tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) →
      (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x)
    {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ)
    (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O)
    (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1)
    {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hsF : HasCompactSupport F)
    (hφF : ∀ x ∈ tsupport F, φ x = 1)
    (hstat : (∫ z, tangentialDivergence F z.1 z.2 ∂(ν : Measure BlowupNormalBundle))=0) :
    (∫ y : Plane, tangentialDivergence F (a+normalGraph n g y)
      (graphSlopeNormal n (gradient g y))*‖graphAreaVector n g y‖)=0 := by
  let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}
  let M : ℝ := ‖blowupTangentTest hF hsF‖+1
  have hM : 0 < M := by dsimp only [M]; positivity
  let N : ℕ → Cover → Cover := fun i ↦ graphUnitNormalExtension a n (graphSmooth (graphMollifier i) g)
  let N₀ := graphUnitNormalExtension a n g
  let f : ℕ → Cover → ℝ := fun i x ↦ M⁻¹*tangentialDivergence F x (N i x)
  let f₀ : Cover → ℝ := fun x ↦ M⁻¹*tangentialDivergence F x (N₀ x)
  have hN (i) : ContDiff ℝ (⊤ : ℕ∞) (N i) := graphUnitNormalExtension_contDiff a hn (graphSmooth_contDiff _ hg)
  have hf (i) : ContDiff ℝ (⊤ : ℕ∞) (f i) :=
    contDiff_const.mul (contDiff_tangentialDivergence_normal hF (hN i))
  have hfb (i) (x) : |f i x| ≤ 1 := by
    have hh := (blowupTangentTest hF hsF).norm_coe_le_norm
      (x,⟨N i x,graphUnitNormalExtension_unit a hn _ x⟩)
    change ‖tangentialDivergence F x (N i x)‖ ≤ ‖blowupTangentTest hF hsF‖ at hh
    dsimp only [f]
    rw [abs_mul,abs_of_pos (inv_pos.mpr hM)]
    apply (mul_le_mul_of_nonneg_left (hh.trans (le_add_of_nonneg_right zero_le_one)) (inv_nonneg.mpr hM.le)).trans_eq
    exact inv_mul_cancel₀ hM.ne'
  have hlim (x : Cover) (hx : x ∈ tsupport φ) : Tendsto (fun i ↦ f i x) atTop (𝓝 (f₀ x)) := by
    have hcoord : normalPlaneCoordinates n (x-a) ∈ Metric.closedBall (0 : Plane) R := by
      simp only [Metric.mem_closedBall,dist_zero_right]
      exact (normalPlaneCoordinates_norm_le n _).trans (by simpa [Metric.mem_closedBall,dist_eq_norm] using hφR hx)
    exact (((continuous_tangentialDivergence_normal F x).tendsto _).comp
      (graphUnitNormalExtension_tendsto a hn hg (hc _ hcoord))).const_mul M⁻¹
  have he := blowup_graph_patch_limit_mass hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb hf hfb hlim
  have hae := blowup_graph_patch_normal_ae hA a hn hg hc hgraph heq hflux hφ hsφ hφU hφO hφR hφb
  have hw (x v) : φ x*tangentialDivergence F x v=tangentialDivergence F x v := by
    by_cases hx : x ∈ tsupport F
    · rw [hφF x hx,one_mul]
    · rw [tangentialDivergence_zero_of_notMem_tsupport hx,mul_zero]
  have hFO : tsupport F ⊆ O := by
    intro x hx
    apply hφO
    exact subset_tsupport _ (by rw [Function.mem_support,hφF x hx]; exact one_ne_zero)
  have hleft : (∫ z, φ z.1*f₀ z.1 ∂μ)=0 := by
    calc
      (∫ z, φ z.1*f₀ z.1 ∂μ) = ∫ z, M⁻¹*tangentialDivergence F z.1 z.2 ∂μ := by
        apply integral_congr_ae
        filter_upwards [hae] with z hz
        by_cases hzφ : 0 < φ z.1
        · dsimp only [f₀,N₀]
          rw [← hz hzφ,← mul_assoc,mul_comm (φ z.1) M⁻¹,mul_assoc,hw]
        · have hφz : φ z.1=0 := le_antisymm (le_of_not_gt hzφ) (hφb _).1
          dsimp only [f₀]
          rw [hφz,zero_mul,← hw z.1 (z.2 : Cover),hφz,zero_mul,mul_zero]
      _ = M⁻¹*(∫ z, tangentialDivergence F z.1 z.2 ∂μ) := integral_const_mul _ _
      _ = M⁻¹*(∫ z, tangentialDivergence F z.1 z.2 ∂(ν : Measure BlowupNormalBundle)) := by
        congr 1
        apply setIntegral_eq_integral_of_forall_compl_eq_zero
        intro z hz
        exact tangentialDivergence_zero_of_notMem_tsupport (fun h ↦ hz (hFO h)) _
      _ = 0 := by rw [hstat,mul_zero]
  have hright : (∫ y : Plane, φ (a+normalGraph n g y)*f₀ (a+normalGraph n g y)*‖graphAreaVector n g y‖) =
      M⁻¹*(∫ y : Plane, tangentialDivergence F (a+normalGraph n g y)
        (graphSlopeNormal n (gradient g y))*‖graphAreaVector n g y‖) := by
    rw [← integral_const_mul]
    congr 1
    funext y
    dsimp only [f₀,N₀]
    rw [graphUnitNormalExtension_graph a hn,← mul_assoc (φ _) M⁻¹,mul_comm (φ _) M⁻¹,mul_assoc M⁻¹,hw]
    ring
  rw [hright,hleft] at he
  exact (mul_eq_zero.mp he.symm).resolve_left (inv_ne_zero hM.ne')

end CubicTorus

end
end

section

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

end CubicTorus
end
end
end

end OAI