Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Geometry.CAT0Fillings.Slicing.NormalRestriction

namespace OAI

section
open Set MeasureTheory Measure Filter Module
open Set Filter MeasureTheory Measure ContinuousLinearMap
open scoped Topology Convolution NNReal
open Set Filter MeasureTheory Measure Metric
open scoped Topology ContDiff
open Set Filter Metric
open Filter Set
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open scoped ENNReal NNReal Topology
open Set MeasureTheory Filter
open scoped Topology NNReal
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings

attribute [local instance] Classical.propDecidable

universe u

end CAT0Fillings

open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder

namespace CAT0Fillings
attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable
open BorelCoefficients BorelRestriction MassMeasure

end CAT0Fillings

open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings

attribute [local instance] Classical.propDecidable

universe u

end CAT0Fillings

open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal

namespace CAT0Fillings
open Set MeasureTheory Filter

attribute [local instance] Classical.propDecidable
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] {k : ℕ}

end CAT0Fillings

open scoped Topology NNReal
open Set MeasureTheory Filter
open scoped ENNReal NNReal Topology

namespace CAT0Fillings.Slicing
open Set MeasureTheory Filter SmoothCutoff BorelCoefficients BorelRestriction MassMeasure Foundations
open scoped Topology NNReal

variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X]

noncomputable def levelBarrier (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) : ℝ :=
  1-profile a 0 |u x-t|

omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
lemma levelBarrier_bounds (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) :
    0 ≤ levelBarrier a t u x ∧ levelBarrier a t u x ≤ 1 := by
  have hp := profile_bounds a 0 |u x-t|
  dsimp only [levelBarrier]
  constructor <;> linarith [hp.1,hp.2]

omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
lemma levelBarrier_boundedLip {u : X → ℝ} (hu : BoundedLip u) (a : ℝ≥0) (t : ℝ) :
    BoundedLip (levelBarrier a t u) := by
  obtain ⟨K,hK⟩ := hu.1
  have hv : BoundedLip (fun x => |u x-t|) := by
    obtain ⟨M,hM⟩ := hu.2
    refine ⟨⟨K,?_⟩, M+|t|, fun x => ?_⟩
    · apply LipschitzWith.of_dist_le_mul
      intro x y
      rw [Real.dist_eq]
      calc
        _ ≤ |(u x-t)-(u y-t)| := abs_abs_sub_abs_le_abs_sub _ _
        _ = dist (u x) (u y) := by rw [Real.dist_eq]; congr 1; ring
        _ ≤ K*dist x y := hK.dist_le_mul x y
    · rw [abs_abs]
      exact (abs_sub (u x) t).trans (by linarith [hM x])
  have h := (BoundedLip.const (X := X) 1).add
    ((profile_comp_boundedLip hv a 0).const_mul (-1))
  change BoundedLip (fun x => 1-profile a 0 |u x-t|)
  simpa only [Function.comp_def,neg_one_mul,sub_eq_add_neg] using h

omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
lemma profile_error_le_levelBarrier (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) :
    |profile a t (u x)-({x | t < u x}.indicator (fun _ => (1:ℝ)) x)| ≤
      levelBarrier a t u x := by
  by_cases hx : t < u x
  · rw [indicator_of_mem (show x ∈ {x | t < u x} from hx),abs_of_nonpos (by linarith [(profile_bounds a t (u x)).2])]
    dsimp only [levelBarrier,profile]
    rw [abs_of_pos (sub_pos.mpr hx),sub_zero]
    linarith
  · rw [indicator_of_notMem (show x ∉ {x | t < u x} from hx),profile_zero a (le_of_not_gt hx),sub_self,abs_zero]
    exact (levelBarrier_bounds a t u x).1

omit [CompactSpace X] in
lemma levelBarrier_integral_tendsto (μ : Measure X) [IsFiniteMeasure μ]
    {u : X → ℝ} (hu : BoundedLip u) (t : ℝ) (ht : μ {x | u x = t} = 0) :
    Tendsto (fun n : ℕ => ∫ x, levelBarrier ((n:ℝ≥0)+1) t u x ∂μ) atTop (𝓝 0) := by
  have hne : ∀ᵐ x ∂μ, u x ≠ t := by simpa only [ae_iff,not_not] using ht
  have hpt : ∀ᵐ x ∂μ, Tendsto (fun n : ℕ => levelBarrier ((n:ℝ≥0)+1) t u x) atTop (𝓝 0) := by
    filter_upwards [hne] with x hx
    have hpos : 0 < |u x-t| := abs_pos.mpr (sub_ne_zero.mpr hx)
    have hp : Tendsto (fun n : ℕ => profile ((n:ℝ≥0)+1) 0 |u x-t|) atTop (𝓝 1) := by
      simpa only [indicator_of_mem (show |u x-t| ∈ Ioi 0 from hpos)] using profile_sequence_tendsto 0 |u x-t|
    simpa only [levelBarrier,sub_self] using (tendsto_const_nhds (x := (1:ℝ))).sub hp
  have hdom (n : ℕ) : ∀ᵐ x ∂μ, ‖levelBarrier ((n:ℝ≥0)+1) t u x‖ ≤ (1:ℝ) := by
    exact Eventually.of_forall fun x => by
      rw [Real.norm_eq_abs,abs_of_nonneg (levelBarrier_bounds _ _ _ _).1]
      exact (levelBarrier_bounds _ _ _ _).2
  have hh := tendsto_integral_of_dominated_convergence (μ := μ)
    (fun _ => (1:ℝ)) (fun n => (levelBarrier_boundedLip hu _ t).continuous.aestronglyMeasurable)
    (integrable_const _) hdom hpt
  simpa only [integral_zero] using hh

lemma restrictCurrent_apply_of_controls {k : ℕ} {T : Functional X k}
    (hT : IsMetricCurrent T) (μ : Measure X) [IsFiniteMeasure μ] (hμ : Controls T μ)
    {E : Set X} (hE : MeasurableSet E) {b : X → ℝ} {π : Fin k → X → ℝ}
    (hab : Admissible b π) :
    restrictCurrent hT E b π = borelAction μ hT (E.indicator b) π := by
  rw [restrictCurrent_apply hT E hab]
  exact borelAction_independent _ hT (currentMassMeasure_controls hT) hμ
    ((integrable_boundedLip _ hab.1).indicator hE)
    ((integrable_boundedLip _ hab.1).indicator hE) π hab.2

lemma restriction_profile_error_bound {k : ℕ} {T : Functional X k}
    (hT : IsMetricCurrent T) (μ : Measure X) [IsFiniteMeasure μ] (hμ : Controls T μ)
    {u : X → ℝ} (hu : BoundedLip u) (a : ℝ≥0) (t : ℝ)
    {b : X → ℝ} (hb : BoundedLip b) (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0)
    (hK : ∀ i, LipschitzWith (K i) (π i)) (M : ℝ) (hM : ∀ x, |b x| ≤ M) :
    |restrictCurrent hT {x | t < u x} b π - T (fun x => profile a t (u x)*b x) π| ≤
      ((∏ i, (K i : ℝ))*M) * ∫ x, levelBarrier a t u x ∂μ := by
  have hab : Admissible b π := ⟨hb,fun i => ⟨K i,hK i⟩⟩
  have hE : MeasurableSet {x | t < u x} := measurableSet_lt measurable_const hu.continuous.measurable
  have hpb : BoundedLip (fun x => profile a t (u x)*b x) := (profile_comp_boundedLip hu a t).mul hb
  rw [restrictCurrent_apply_of_controls hT μ hμ hE hab,
    ←borelAction_eq μ hT hμ ⟨hpb,hab.2⟩]
  apply (borelAction_difference_bound μ hT hμ ((integrable_boundedLip μ hb).indicator hE)
    (integrable_boundedLip μ hpb) π K hK).trans
  have hi : ∫ x, |({x | t < u x}.indicator b x)-profile a t (u x)*b x| ∂μ ≤
      M*∫ x, levelBarrier a t u x ∂μ := by
    rw [←integral_const_mul]
    apply integral_mono ((integrable_boundedLip μ hb).indicator hE |>.sub
      (integrable_boundedLip μ hpb) |>.abs)
      ((integrable_boundedLip μ (levelBarrier_boundedLip hu a t)).const_mul M)
    intro x
    have he : ({x | t < u x}.indicator b x)-profile a t (u x)*b x =
        (profile a t (u x)-({x | t < u x}.indicator (fun _ => (1:ℝ)) x)) * (-b x) := by
      by_cases hx : t < u x <;> simp [hx]
      ring
    change |({x | t < u x}.indicator b x)-profile a t (u x)*b x| ≤ M*levelBarrier a t u x
    rw [he,abs_mul,abs_neg,mul_comm]
    exact mul_le_mul (hM x) (profile_error_le_levelBarrier a t u x) (abs_nonneg _)
      (le_trans (abs_nonneg _) (hM x))
  exact (mul_le_mul_of_nonneg_left hi (by positivity)).trans_eq (by ring)

theorem restrictCurrent_weak_tendsto {ι : Type*} {l : Filter ι} {k : ℕ}
    {Ts : ι → Functional X k} {T : Functional X k}
    (hTs : ∀ j, IsMetricCurrent (Ts j)) (hT : IsMetricCurrent T)
    (μs : ι → FiniteMeasure X) (μ : FiniteMeasure X)
    (hμs : ∀ j, Controls (Ts j) (μs j)) (hμ : Controls T μ)
    (hμlim : Tendsto μs l (𝓝 μ))
    (hlim : ∀ b π, Tendsto (fun j => Ts j b π) l (𝓝 (T b π)))
    {u : X → ℝ} (hu : BoundedLip u) (t : ℝ) (ht : (μ : Measure X) {x | u x = t} = 0)
    (b : X → ℝ) (π : Fin k → X → ℝ) :
    Tendsto (fun j => restrictCurrent (hTs j) {x | t < u x} b π) l
      (𝓝 (restrictCurrent hT {x | t < u x} b π)) := by
  by_cases hab : Admissible b π
  swap
  · simp only [restrictCurrent,ite_eq_right hab]
    exact tendsto_const_nhds
  choose K hK using hab.2
  obtain ⟨M,hM⟩ := hab.1.2
  let C := (∏ i, (K i : ℝ)) * max M 0
  have hC : 0 ≤ C := by dsimp [C]; positivity
  have hbound (x : X) : |b x| ≤ max M 0 := (hM x).trans (le_max_left _ _)
  let P (n : ℕ) (U : Functional X k) := U (fun x => profile ((n:ℝ≥0)+1) t (u x)*b x) π
  let A (n : ℕ) (ν : FiniteMeasure X) := C*∫ x, levelBarrier ((n:ℝ≥0)+1) t u x ∂(ν : Measure X)
  have hA : Tendsto (fun n => A n μ) atTop (𝓝 0) := by
    simpa only [A,mul_zero] using (levelBarrier_integral_tendsto (μ : Measure X) hu t ht).const_mul C
  have hAlim (n : ℕ) : Tendsto (fun j => A n (μs j)) l (𝓝 (A n μ)) := by
    let f : C(X,ℝ) := ⟨levelBarrier ((n:ℝ≥0)+1) t u,(levelBarrier_boundedLip hu _ t).continuous⟩
    exact ((FiniteMeasure.continuous_integral_continuousMap f).tendsto μ |>.comp hμlim).const_mul C
  have hbT (n : ℕ) : dist (restrictCurrent hT {x | t < u x} b π) (P n T) ≤ A n μ :=
    restriction_profile_error_bound hT μ hμ hu _ t hab.1 π K hK _ hbound
  have hbTs (n : ℕ) (j : ι) : dist (restrictCurrent (hTs j) {x | t < u x} b π) (P n (Ts j)) ≤ A n (μs j) :=
    restriction_profile_error_bound (hTs j) (μs j) (hμs j) hu _ t hab.1 π K hK _ hbound
  apply Metric.tendsto_nhds.mpr
  intro ε hε
  obtain ⟨n,hn⟩ := ((hA.eventually (gt_mem_nhds (show (0:ℝ) < ε/4 by positivity))).exists)
  have he1 : ∀ᶠ j in l, A n (μs j) < ε/3 :=
    (hAlim n).eventually (gt_mem_nhds (show A n μ < ε/3 by linarith))
  have he2 : ∀ᶠ j in l, dist (P n (Ts j)) (P n T) < ε/3 :=
    Metric.tendsto_nhds.mp (hlim _ _) (ε/3) (by positivity)
  filter_upwards [he1,he2] with j hj1 hj2
  have htri := dist_triangle (restrictCurrent (hTs j) {x | t < u x} b π) (P n (Ts j)) (P n T)
  have htri2 := dist_triangle (restrictCurrent (hTs j) {x | t < u x} b π) (P n T)
    (restrictCurrent hT {x | t < u x} b π)
  have hb := hbT n
  rw [dist_comm] at hb
  linarith [hbTs n j]

end CAT0Fillings.Slicing

open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings

attribute [local instance] Classical.propDecidable

universe u

end CAT0Fillings

open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal

end

end OAI

	import Mathlib
	import OAI.Geometry.CAT0Fillings.Slicing.NormalRestriction

	namespace OAI

	section
	open Set MeasureTheory Measure Filter Module
	open Set Filter MeasureTheory Measure ContinuousLinearMap
	open scoped Topology Convolution NNReal
	open Set Filter MeasureTheory Measure Metric
	open scoped Topology ContDiff
	open Set Filter Metric
	open Filter Set
	open Set Filter MeasureTheory TopologicalSpace
	open scoped Topology ENNReal
	open Set MeasureTheory
	open scoped RealInnerProductSpace
	open Matrix
	open scoped RealInnerProductSpace MatrixOrder
	open MeasureTheory Filter Set Metric
	open scoped Topology Pointwise NNReal
	open scoped ENNReal NNReal Topology
	open Set MeasureTheory Filter
	open scoped Topology NNReal
	open Set Filter MeasureTheory
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings

	attribute [local instance] Classical.propDecidable

	universe u

	end CAT0Fillings

	open Filter Set
	open scoped Topology NNReal
	open Set Filter MeasureTheory TopologicalSpace
	open scoped Topology ENNReal
	open MeasureTheory Filter Set Metric
	open scoped Topology Pointwise NNReal
	open Set MeasureTheory
	open scoped RealInnerProductSpace
	open Matrix
	open scoped RealInnerProductSpace MatrixOrder

	namespace CAT0Fillings
	attribute [local instance] Classical.propDecidable

	attribute [local instance] Classical.propDecidable

	attribute [local instance] Classical.propDecidable

	attribute [local instance] Classical.propDecidable

	attribute [local instance] Classical.propDecidable

	attribute [local instance] Classical.propDecidable
	open BorelCoefficients BorelRestriction MassMeasure

	end CAT0Fillings

	open Set Filter MeasureTheory
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings

	attribute [local instance] Classical.propDecidable

	universe u

	end CAT0Fillings

	open MeasureTheory Filter Set Metric
	open scoped Topology Pointwise NNReal

	namespace CAT0Fillings
	open Set MeasureTheory Filter

	attribute [local instance] Classical.propDecidable
	variable {X : Type*} [MetricSpace X] [MeasurableSpace X] {k : ℕ}

	end CAT0Fillings

	open scoped Topology NNReal
	open Set MeasureTheory Filter
	open scoped ENNReal NNReal Topology

	namespace CAT0Fillings.Slicing
	open Set MeasureTheory Filter SmoothCutoff BorelCoefficients BorelRestriction MassMeasure Foundations
	open scoped Topology NNReal

	variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X]

	noncomputable def levelBarrier (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) : ℝ :=
	1-profile a 0 \|u x-t\|

	omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
	lemma levelBarrier_bounds (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) :
	0 ≤ levelBarrier a t u x ∧ levelBarrier a t u x ≤ 1 := by
	have hp := profile_bounds a 0 \|u x-t\|
	dsimp only [levelBarrier]
	constructor <;> linarith [hp.1,hp.2]

	omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
	lemma levelBarrier_boundedLip {u : X → ℝ} (hu : BoundedLip u) (a : ℝ≥0) (t : ℝ) :
	BoundedLip (levelBarrier a t u) := by
	obtain ⟨K,hK⟩ := hu.1
	have hv : BoundedLip (fun x => \|u x-t\|) := by
	obtain ⟨M,hM⟩ := hu.2
	refine ⟨⟨K,?_⟩, M+\|t\|, fun x => ?_⟩
	· apply LipschitzWith.of_dist_le_mul
	intro x y
	rw [Real.dist_eq]
	calc
	_ ≤ \|(u x-t)-(u y-t)\| := abs_abs_sub_abs_le_abs_sub _ _
	_ = dist (u x) (u y) := by rw [Real.dist_eq]; congr 1; ring
	_ ≤ K*dist x y := hK.dist_le_mul x y
	· rw [abs_abs]
	exact (abs_sub (u x) t).trans (by linarith [hM x])
	have h := (BoundedLip.const (X := X) 1).add
	((profile_comp_boundedLip hv a 0).const_mul (-1))
	change BoundedLip (fun x => 1-profile a 0 \|u x-t\|)
	simpa only [Function.comp_def,neg_one_mul,sub_eq_add_neg] using h

	omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in
	lemma profile_error_le_levelBarrier (a : ℝ≥0) (t : ℝ) (u : X → ℝ) (x : X) :
	\|profile a t (u x)-({x \| t < u x}.indicator (fun _ => (1:ℝ)) x)\| ≤
	levelBarrier a t u x := by
	by_cases hx : t < u x
	· rw [indicator_of_mem (show x ∈ {x \| t < u x} from hx),abs_of_nonpos (by linarith [(profile_bounds a t (u x)).2])]
	dsimp only [levelBarrier,profile]
	rw [abs_of_pos (sub_pos.mpr hx),sub_zero]
	linarith
	· rw [indicator_of_notMem (show x ∉ {x \| t < u x} from hx),profile_zero a (le_of_not_gt hx),sub_self,abs_zero]
	exact (levelBarrier_bounds a t u x).1

	omit [CompactSpace X] in
	lemma levelBarrier_integral_tendsto (μ : Measure X) [IsFiniteMeasure μ]
	{u : X → ℝ} (hu : BoundedLip u) (t : ℝ) (ht : μ {x \| u x = t} = 0) :
	Tendsto (fun n : ℕ => ∫ x, levelBarrier ((n:ℝ≥0)+1) t u x ∂μ) atTop (𝓝 0) := by
	have hne : ∀ᵐ x ∂μ, u x ≠ t := by simpa only [ae_iff,not_not] using ht
	have hpt : ∀ᵐ x ∂μ, Tendsto (fun n : ℕ => levelBarrier ((n:ℝ≥0)+1) t u x) atTop (𝓝 0) := by
	filter_upwards [hne] with x hx
	have hpos : 0 < \|u x-t\| := abs_pos.mpr (sub_ne_zero.mpr hx)
	have hp : Tendsto (fun n : ℕ => profile ((n:ℝ≥0)+1) 0 \|u x-t\|) atTop (𝓝 1) := by
	simpa only [indicator_of_mem (show \|u x-t\| ∈ Ioi 0 from hpos)] using profile_sequence_tendsto 0 \|u x-t\|
	simpa only [levelBarrier,sub_self] using (tendsto_const_nhds (x := (1:ℝ))).sub hp
	have hdom (n : ℕ) : ∀ᵐ x ∂μ, ‖levelBarrier ((n:ℝ≥0)+1) t u x‖ ≤ (1:ℝ) := by
	exact Eventually.of_forall fun x => by
	rw [Real.norm_eq_abs,abs_of_nonneg (levelBarrier_bounds _ _ _ _).1]
	exact (levelBarrier_bounds _ _ _ _).2
	have hh := tendsto_integral_of_dominated_convergence (μ := μ)
	(fun _ => (1:ℝ)) (fun n => (levelBarrier_boundedLip hu _ t).continuous.aestronglyMeasurable)
	(integrable_const _) hdom hpt
	simpa only [integral_zero] using hh

	lemma restrictCurrent_apply_of_controls {k : ℕ} {T : Functional X k}
	(hT : IsMetricCurrent T) (μ : Measure X) [IsFiniteMeasure μ] (hμ : Controls T μ)
	{E : Set X} (hE : MeasurableSet E) {b : X → ℝ} {π : Fin k → X → ℝ}
	(hab : Admissible b π) :
	restrictCurrent hT E b π = borelAction μ hT (E.indicator b) π := by
	rw [restrictCurrent_apply hT E hab]
	exact borelAction_independent _ hT (currentMassMeasure_controls hT) hμ
	((integrable_boundedLip _ hab.1).indicator hE)
	((integrable_boundedLip _ hab.1).indicator hE) π hab.2

	lemma restriction_profile_error_bound {k : ℕ} {T : Functional X k}
	(hT : IsMetricCurrent T) (μ : Measure X) [IsFiniteMeasure μ] (hμ : Controls T μ)
	{u : X → ℝ} (hu : BoundedLip u) (a : ℝ≥0) (t : ℝ)
	{b : X → ℝ} (hb : BoundedLip b) (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0)
	(hK : ∀ i, LipschitzWith (K i) (π i)) (M : ℝ) (hM : ∀ x, \|b x\| ≤ M) :
	\|restrictCurrent hT {x \| t < u x} b π - T (fun x => profile a t (u x)*b x) π\| ≤
	((∏ i, (K i : ℝ))M) ∫ x, levelBarrier a t u x ∂μ := by
	have hab : Admissible b π := ⟨hb,fun i => ⟨K i,hK i⟩⟩
	have hE : MeasurableSet {x \| t < u x} := measurableSet_lt measurable_const hu.continuous.measurable
	have hpb : BoundedLip (fun x => profile a t (u x)*b x) := (profile_comp_boundedLip hu a t).mul hb
	rw [restrictCurrent_apply_of_controls hT μ hμ hE hab,
	←borelAction_eq μ hT hμ ⟨hpb,hab.2⟩]
	apply (borelAction_difference_bound μ hT hμ ((integrable_boundedLip μ hb).indicator hE)
	(integrable_boundedLip μ hpb) π K hK).trans
	have hi : ∫ x, \|({x \| t < u x}.indicator b x)-profile a t (u x)*b x\| ∂μ ≤
	M*∫ x, levelBarrier a t u x ∂μ := by
	rw [←integral_const_mul]
	apply integral_mono ((integrable_boundedLip μ hb).indicator hE \|>.sub
	(integrable_boundedLip μ hpb) \|>.abs)
	((integrable_boundedLip μ (levelBarrier_boundedLip hu a t)).const_mul M)
	intro x
	have he : ({x \| t < u x}.indicator b x)-profile a t (u x)*b x =
	(profile a t (u x)-({x \| t < u x}.indicator (fun _ => (1:ℝ)) x)) * (-b x) := by
	by_cases hx : t < u x <;> simp [hx]
	ring
	change \|({x \| t < u x}.indicator b x)-profile a t (u x)b x\| ≤ MlevelBarrier a t u x
	rw [he,abs_mul,abs_neg,mul_comm]
	exact mul_le_mul (hM x) (profile_error_le_levelBarrier a t u x) (abs_nonneg _)
	(le_trans (abs_nonneg _) (hM x))
	exact (mul_le_mul_of_nonneg_left hi (by positivity)).trans_eq (by ring)

	theorem restrictCurrent_weak_tendsto {ι : Type*} {l : Filter ι} {k : ℕ}
	{Ts : ι → Functional X k} {T : Functional X k}
	(hTs : ∀ j, IsMetricCurrent (Ts j)) (hT : IsMetricCurrent T)
	(μs : ι → FiniteMeasure X) (μ : FiniteMeasure X)
	(hμs : ∀ j, Controls (Ts j) (μs j)) (hμ : Controls T μ)
	(hμlim : Tendsto μs l (𝓝 μ))
	(hlim : ∀ b π, Tendsto (fun j => Ts j b π) l (𝓝 (T b π)))
	{u : X → ℝ} (hu : BoundedLip u) (t : ℝ) (ht : (μ : Measure X) {x \| u x = t} = 0)
	(b : X → ℝ) (π : Fin k → X → ℝ) :
	Tendsto (fun j => restrictCurrent (hTs j) {x \| t < u x} b π) l
	(𝓝 (restrictCurrent hT {x \| t < u x} b π)) := by
	by_cases hab : Admissible b π
	swap
	· simp only [restrictCurrent,ite_eq_right hab]
	exact tendsto_const_nhds
	choose K hK using hab.2
	obtain ⟨M,hM⟩ := hab.1.2
	let C := (∏ i, (K i : ℝ)) * max M 0
	have hC : 0 ≤ C := by dsimp [C]; positivity
	have hbound (x : X) : \|b x\| ≤ max M 0 := (hM x).trans (le_max_left _ _)
	let P (n : ℕ) (U : Functional X k) := U (fun x => profile ((n:ℝ≥0)+1) t (u x)*b x) π
	let A (n : ℕ) (ν : FiniteMeasure X) := C*∫ x, levelBarrier ((n:ℝ≥0)+1) t u x ∂(ν : Measure X)
	have hA : Tendsto (fun n => A n μ) atTop (𝓝 0) := by
	simpa only [A,mul_zero] using (levelBarrier_integral_tendsto (μ : Measure X) hu t ht).const_mul C
	have hAlim (n : ℕ) : Tendsto (fun j => A n (μs j)) l (𝓝 (A n μ)) := by
	let f : C(X,ℝ) := ⟨levelBarrier ((n:ℝ≥0)+1) t u,(levelBarrier_boundedLip hu _ t).continuous⟩
	exact ((FiniteMeasure.continuous_integral_continuousMap f).tendsto μ \|>.comp hμlim).const_mul C
	have hbT (n : ℕ) : dist (restrictCurrent hT {x \| t < u x} b π) (P n T) ≤ A n μ :=
	restriction_profile_error_bound hT μ hμ hu _ t hab.1 π K hK _ hbound
	have hbTs (n : ℕ) (j : ι) : dist (restrictCurrent (hTs j) {x \| t < u x} b π) (P n (Ts j)) ≤ A n (μs j) :=
	restriction_profile_error_bound (hTs j) (μs j) (hμs j) hu _ t hab.1 π K hK _ hbound
	apply Metric.tendsto_nhds.mpr
	intro ε hε
	obtain ⟨n,hn⟩ := ((hA.eventually (gt_mem_nhds (show (0:ℝ) < ε/4 by positivity))).exists)
	have he1 : ∀ᶠ j in l, A n (μs j) < ε/3 :=
	(hAlim n).eventually (gt_mem_nhds (show A n μ < ε/3 by linarith))
	have he2 : ∀ᶠ j in l, dist (P n (Ts j)) (P n T) < ε/3 :=
	Metric.tendsto_nhds.mp (hlim _ _) (ε/3) (by positivity)
	filter_upwards [he1,he2] with j hj1 hj2
	have htri := dist_triangle (restrictCurrent (hTs j) {x \| t < u x} b π) (P n (Ts j)) (P n T)
	have htri2 := dist_triangle (restrictCurrent (hTs j) {x \| t < u x} b π) (P n T)
	(restrictCurrent hT {x \| t < u x} b π)
	have hb := hbT n
	rw [dist_comm] at hb
	linarith [hbTs n j]

	end CAT0Fillings.Slicing

	open Set Filter MeasureTheory
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings

	attribute [local instance] Classical.propDecidable

	universe u

	end CAT0Fillings

	open MeasureTheory Filter Set Metric
	open scoped Topology Pointwise NNReal

	end

	end OAI