| import Mathlib | |
| import OAI.Geometry.CAT0Fillings.Slicing.WeakSlices | |
| import OAI.Geometry.CAT0Fillings.Slicing.IntegralBoundary | |
| 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 scoped ENNReal NNReal Topology | |
| open Set MeasureTheory Filter | |
| open scoped Topology NNReal ENNReal | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| 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.Slicing | |
| open Set MeasureTheory Filter MassMeasure Foundations | |
| open scoped Topology NNReal | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [CompactSpace X] [Nonempty X] | |
| def IntegralSliceTree : (k : ℕ) → Functional X k → Prop | |
| | 0, T => IsIntegral 0 T | |
| | k+1, T => ∃ hT : IsMetricCurrent T, ∃ hB : IsMetricCurrent (boundarySucc T), | |
| ∀ (u : X → ℝ) (K : ℝ≥0), LipschitzWith K u → | |
| ∀ᵐ t : ℝ, IntegralSliceTree k (superlevelSlice hT hB u t) | |
| theorem integral_slice_tree_of_bounded_weak_limit {k : ℕ} | |
| {Ts : ℕ → Functional X k} {T : Functional X k} | |
| (hTs : ∀ j, IsIntegral k (Ts j)) (M N : ℝ≥0) | |
| (hM : ∀ j, mass (Ts j) ≤ M) (hN : ∀ j, mass (boundary (Ts j)) ≤ N) | |
| (hlim : ∀ b π, Tendsto (fun j => Ts j b π) atTop (𝓝 (T b π))) | |
| (hX : IsCAT0 X) : IntegralSliceTree k T := by | |
| induction k generalizing M N with | |
| | zero => exact integral_zero_weak_closed hTs M hM hlim | |
| | succ k ih => | |
| cases k with | |
| | zero => | |
| obtain ⟨hT,hB,_,hS⟩ := ae_integral_zero_slice_of_bounded_weak_limit hTs M N hM hN hlim hX | |
| exact ⟨hT,hB,hS⟩ | |
| | succ k => | |
| obtain ⟨hT,hB⟩ := normal_of_bounded_integral_weak_limit hTs M N hM hN hlim | |
| refine ⟨hT,hB,?_⟩ | |
| intro u K hK | |
| have hu : BoundedLip u := boundedLip_of_lipschitz hK | |
| obtain ⟨ψ,hψ,μs,νs,μ,ν,hμs,hμ,hνs,hν,hμlim,hνlim⟩ := | |
| exists_joint_weak_controlling_subsequence (fun j => (hTs j).1) | |
| (fun j => (hTs j).2.2.1) M N hM hN hlim (boundarySucc_weak_limit hlim) | |
| have hweak := ae_superlevelSlice_weak_tendsto (fun j => (hTs (ψ j)).1) hT | |
| (fun j => (hTs (ψ j)).2.2.1) hB μs νs μ ν hμs hμ hνs hν hμlim hνlim | |
| (fun b π => (hlim b π).comp hψ.tendsto_atTop) hu | |
| choose G H hG hH hG0 hH0 hGI hHI hS using fun j => | |
| superlevelSlice_normal_coarea_bound (hTs (ψ j)) hX hK | |
| have hsum j : Integrable (fun t => G j t + H j t) volume := (hG j).add (hH j) | |
| have hsum0 j : 0 ≤ᵐ[volume] (fun t => G j t + H j t) := | |
| Eventually.of_forall fun t => add_nonneg (hG0 j t) (hH0 j t) | |
| have hsumI j : (∫ t : ℝ, G j t + H j t) ≤ (K : ℝ)*(M+N) := by | |
| rw [integral_add (hG j) (hH j),mul_add] | |
| exact add_le_add ((hGI j).trans (mul_le_mul_of_nonneg_left (hM (ψ j)) K.coe_nonneg)) | |
| ((hHI j).trans (mul_le_mul_of_nonneg_left (hN (ψ j)) K.coe_nonneg)) | |
| have hfat := ae_exists_bounded_subsequence_of_integral_bound volume hsum hsum0 | |
| ((K : ℝ)*(M+N)) hsumI | |
| filter_upwards [hweak,ae_all_iff.mpr hS,hfat] with t ht hs hf | |
| obtain ⟨C,χ,hχ,hC⟩ := hf | |
| let D : ℝ≥0 := ⟨max C 0,le_max_right _ _⟩ | |
| let Ss : ℕ → Functional X (k+1) := fun j => | |
| superlevelSlice (hTs (ψ (χ j))).1 (hTs (ψ (χ j))).2.2.1 u t | |
| have hSs j : IsIntegral (k+1) (Ss j) := (hs (χ j)).1 | |
| have hSC j : mass (Ss j) ≤ D := | |
| (((hs (χ j)).2.1.trans (le_add_of_nonneg_right (hH0 (χ j) t))).trans (hC j)).trans (le_max_left _ _) | |
| have hBC j : mass (boundary (Ss j)) ≤ D := | |
| (((hs (χ j)).2.2.trans (le_add_of_nonneg_left (hG0 (χ j) t))).trans (hC j)).trans (le_max_left _ _) | |
| exact ih hSs D D hSC hBC (fun b π => (ht b π).comp hχ.tendsto_atTop) | |
| end CAT0Fillings.Slicing | |
| end | |
| end OAI | |