| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Reconstruction.Localization |
| import OAI.Geometry.CAT0Fillings.Reconstruction.ScalarCharts |
| import OAI.Geometry.CAT0Fillings.Reconstruction.ScalarKernel |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients BorelRestriction Slicing |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| lemma chart_mass_compl_image_zero {k : ℕ} (C : IntegerChart X k) |
| (hC : IsMetricCurrent C.action) : currentMassMeasure hC C.imageᶜ = 0 := by |
| have he := C.restrictCurrent_image_eq hC C.measurableSet_image (subset_refl _) |
| have hs := massMeasure_restrict_compl_zero hC C.measurableSet_image |
| simpa only [he] using hs |
|
|
| theorem scalar_graph_restriction_eq_chart |
| {Ts : ℕ → Functional X 1} {T : Functional X 1} |
| (hTs : ∀ j, IsIntegral 1 (Ts j)) |
| (hT : IsMetricCurrent T) (hB : IsMetricCurrent (boundarySucc T)) |
| (M N : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M) |
| (hN : ∀ j, mass (boundarySucc (Ts j)) ≤ N) |
| (hlim : ∀ b π, Tendsto (fun j => Ts j b π) atTop (𝓝 (T b π))) |
| (hX : IsCAT0 X) {u : X → ℝ} {Ku : ℝ≥0} (hu : LipschitzWith Ku u) |
| {D : Set ℝ} (hD : IsCompact D) {m : ℕ} (γ : D → Fin m → X) |
| (hγ : ∀ j, ∃ L U : ℝ≥0, LipschitzWith L (fun t => γ t j) ∧ |
| AntilipschitzWith U (fun t => γ t j)) (a : Fin m → ℤ) |
| (hbase : ∀ t j, u (γ t j) = t) (hinj : ∀ t, Function.Injective (γ t)) |
| (hrep : ∀ t : D, UnitRepresentation (superlevelSlice hT hB u t) a (γ t)) |
| (j : Fin m) : |
| restrictCurrent hT (Set.range (fun t => γ t j)) = |
| (scalarGraphChart hD (fun t => γ t j) (hγ j) (a j)).action := by |
| classical |
| let : CompactSpace D := isCompact_iff_compactSpace.mp hD |
| let C := scalarGraphChart hD (fun t => γ t j) (hγ j) (a j) |
| let E := Set.range (fun t => γ t j) |
| have hγc (i : Fin m) : Continuous (fun t => γ t i) := (hγ i).choose_spec.choose_spec.1.continuous |
| have hE : MeasurableSet E := (isCompact_range (hγc j)).measurableSet |
| have hR := restrictCurrent_isMetricCurrent hT hE |
| have hC : IsMetricCurrent C.action := C.action_isMetricCurrent |
| obtain ⟨ht,hb,hI⟩ := integral_slice_tree_of_bounded_weak_limit hTs M N hM hN hlim hX |
| have hI' : ∀ᵐ t : ℝ, IsIntegral 0 (superlevelSlice hT hB u t) := hI u Ku hu |
| apply current_eq_on_scalar_graph (currentMassMeasure hR) hR hC |
| (currentMassMeasure_controls hR) (currentMassMeasure hC) (currentMassMeasure_controls hC) |
| hD (fun t => γ t j) (hγ j).choose_spec.choose_spec.1 hu (fun t => hbase t j) |
| (massMeasure_restrict_compl_zero hT hE) |
| · have he : C.image = E := scalarGraphChart_image hD _ (hγ j) (a j) |
| simpa only [he] using chart_mass_compl_image_zero C hC |
| intro b hb |
| let f := E.indicator b |
| have hf : Measurable f := hb.continuous.measurable.indicator hE |
| obtain ⟨B,hBb⟩ := hb.2 |
| let B' : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ |
| have hfB (x : X) : |f x| ≤ B' := by |
| by_cases hx : x ∈ E |
| · exact (by simpa only [f,indicator_of_mem hx] using (show |b x| ≤ B' from (hBb x).trans (le_max_left B 0))) |
| · simp only [f,indicator_of_notMem hx,abs_zero] |
| exact B'.coe_nonneg |
| have ha := superlevelSlice_borel_integral_of_bounded_weak_limit hTs hT hB M N hM hN hlim hX hu |
| (fun z => Fin.elim0 z) (fun z => Fin.elim0 z) hf B' hfB |
| have hp : Matrix.vecCons u (fun z => Fin.elim0 z) = fun _ : Fin 1 => u := by |
| funext i |
| exact Fin.cases rfl (fun z => Fin.elim0 z) i |
| dsimp only [ActionIntegralAgreement] at ha |
| rw [hp] at ha |
| have hs := superlevelSlice_graph_action hT hB hu.continuous hI' hD γ hγc a hbase hinj hrep hb j |
| rw [restrictCurrent_apply hT E ⟨hb,fun _ => ⟨Ku,hu⟩⟩, |
| ←currentBorelAction_eq hT f (fun _ : Fin 1 => u),ha.2] |
| change (∫ t, currentBorelAction (superlevelSlice hT hB u t) f (fun z => Fin.elim0 z)) = C.action b _ |
| rw [scalarGraphChart_base_action hD _ (hγ j) (a j) hu (fun t => hbase t j) hb] |
| rw [←integral_indicator hD.measurableSet] |
| apply integral_congr_ae |
| filter_upwards [hs] with t ht |
| rw [ht] |
| by_cases htd : t ∈ D |
| · simp only [dite_eq_left htd,indicator_of_mem htd,scalarGraphFunction] |
| · simp only [dite_eq_right htd,indicator_of_notMem htd] |
| |
| end CAT0Fillings.SliceReconstruction |
| end |
| |
| section |
| |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
| |
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients BorelRestriction Slicing |
| |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
| |
| theorem joint_graph_restriction_eq_chart {k : ℕ} {T : Functional X k} |
| (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, BoundedLip (π i)) |
| {D : Set (Euc k)} (hD : IsCompact D) {m : ℕ} (γ : D → Fin m → X) |
| (hγ : ∀ j, ∃ L U : ℝ≥0, LipschitzWith L (fun t => γ t j) ∧ |
| AntilipschitzWith U (fun t => γ t j)) (a : Fin m → ℤ) |
| (hbase : ∀ t j, coordinateMap π (γ t j) = t) (hinj : ∀ t, Function.Injective (γ t)) |
| (hrep : ∀ t : D, UnitRepresentation (fullSlice h π t) a (γ t)) |
| (j : Fin m) : |
| restrictCurrent h.metric (Set.range (fun t => γ t j)) = |
| (jointGraphChart hD (fun t => γ t j) (hγ j) (a j)).action := by |
| classical |
| let : CompactSpace D := isCompact_iff_compactSpace.mp hD |
| let C := jointGraphChart hD (fun t => γ t j) (hγ j) (a j) |
| let E := Set.range (fun t => γ t j) |
| have hγc (i : Fin m) : Continuous (fun t => γ t i) := (hγ i).choose_spec.choose_spec.1.continuous |
| have hE : MeasurableSet E := (isCompact_range (hγc j)).measurableSet |
| have hR := restrictCurrent_isMetricCurrent h.metric hE |
| have hC : IsMetricCurrent C.action := C.action_isMetricCurrent |
| apply current_eq_on_joint_graph (currentMassMeasure hR) hR hC |
| (currentMassMeasure_controls hR) (currentMassMeasure hC) (currentMassMeasure_controls hC) |
| hD (fun t => γ t j) (hγ j).choose_spec.choose_spec.1 π (fun i => (hπ i).1) |
| (fun t => hbase t j) (massMeasure_restrict_compl_zero h.metric hE) |
| · have he : C.image = E := jointGraphChart_image hD _ (hγ j) (a j) |
| simpa only [he] using chart_mass_compl_image_zero C hC |
| intro ρ b hb |
| rw [restrictCurrent_reindex h.metric (fun i => (hπ i).1) ρ |
| (fun b hb => h.apply_reindex ⟨hb,fun i => (hπ i).1⟩ ρ) hE hb, |
| C.action_reindex ⟨hb,fun i => (hπ i).1⟩ ρ, |
| full_graph_restriction_base_action h hX π hπ hD γ hγ a hbase hinj hrep j hb] |
| |
| end CAT0Fillings.SliceReconstruction |
| end |
| |
| end OAI |
| |
