| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Reconstruction.Uniqueness |
| import OAI.Geometry.CAT0Fillings.Reconstruction.GraphCharts |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.Slicing |
| open Foundations MassMeasure BorelCoefficients |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| lemma IntegralSliceTree.isMetricCurrent {X : Type*} [MetricSpace X] [MeasurableSpace X] |
| [BorelSpace X] [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X k} |
| (h : IntegralSliceTree k T) : IsMetricCurrent T := by |
| cases k with |
| | zero => exact h.1 |
| | succ k => exact h.choose |
|
|
| lemma ae_superlevelSlice_metric_of_bounded_weak_limit |
| {k : ℕ} {Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)} |
| (hTs : ∀ j, IsIntegral (k+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) : |
| ∀ᵐ t : ℝ, IsMetricCurrent (superlevelSlice hT hB u t) := by |
| obtain ⟨hT',hB',hS⟩ := integral_slice_tree_of_bounded_weak_limit hTs M N hM hN hlim hX |
| exact (hS u Ku hu).mono fun _ ht => ht.isMetricCurrent |
|
|
| theorem superlevelSlice_borel_integral_of_bounded_weak_limit |
| {k : ℕ} {Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)} |
| (hTs : ∀ j, IsIntegral (k+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) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) |
| {f : X → ℝ} (hf : Measurable f) (B : ℝ≥0) (hBf : ∀ x, |f x| ≤ B) : |
| ActionIntegralAgreement volume T (Matrix.vecCons u π) (superlevelSlice hT hB u) π f := by |
| have hρ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (Matrix.vecCons u π i) := by |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact ⟨Ku,hu⟩ |
| · exact hπ j |
| have hS := ae_superlevelSlice_metric_of_bounded_weak_limit hTs hT hB M N hM hN hlim hX hu |
| obtain ⟨g,hg,hg0,hgb,_⟩ := |
| superlevelSlice_mass_bound_of_bounded_weak_limit hTs hT hB M N hM hN hlim hX hu |
| refine ActionIntegralAgreement.bounded_measurable hρ hπ hS hg |
| (Eventually.of_forall hg0) hgb ?_ hf B hBf |
| intro b hb |
| have hab : Admissible b π := ⟨hb,hπ⟩ |
| have ha := superlevelSlice_integral_of_bounded_weak_limit hTs hT hB M N hM hN hlim hX hu hab |
| have he : (fun t : ℝ => superlevelSlice hT hB u t b π) =ᵐ[volume] |
| (fun t => currentBorelAction (superlevelSlice hT hB u t) b π) := |
| hS.mono fun _ ht => (currentBorelAction_eq_action ht hab).symm |
| exact ⟨ha.1.congr he, (currentBorelAction_eq_action hT ⟨hb,hρ⟩).trans |
| (ha.2.trans (integral_congr_ae he))⟩ |
|
|
| end CAT0Fillings.Slicing |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients BorelRestriction Slicing |
|
|
| attribute [local instance] Classical.propDecidable |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| lemma integral_zero_unitRepresentation {S : Functional X 0} (hS : IsIntegral 0 S) : |
| ∃ (m : ℕ) (a : Fin m → ℤ) (x : Fin m → X), |
| (∀ i, a i ≠ 0) ∧ Function.Injective x ∧ UnitRepresentation S a x := by |
| obtain ⟨m,a,x,ha,hx,he⟩ := integerRectifiable_zero_distinct_representation hS.1 hS.2 |
| refine ⟨m,a,x,ha,hx,?_⟩ |
| intro b hb hb1 |
| rw [he] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [atomCurrent,ite_eq_left ⟨⟨⟨1,hb⟩,1,hb1⟩,fun j => Fin.elim0 j⟩] |
|
|
| lemma graph_atom_mem_iff {X : Type*} [MetricSpace X] [MeasurableSpace X] |
| [BorelSpace X] [CompactSpace X] [Nonempty X] {D : Set ℝ} {m : ℕ} |
| (γ : D → Fin m → X) |
| {u : X → ℝ} (hbase : ∀ t j, u (γ t j) = t) |
| (hinj : ∀ t, Function.Injective (γ t)) (t : D) (i j : Fin m) : |
| γ t i ∈ Set.range (fun s => γ s j) ↔ i = j := by |
| constructor |
| · rintro ⟨s,he⟩ |
| have ht : s = t := Subtype.ext (by simpa only [hbase] using congrArg u he) |
| subst s |
| exact (hinj t he).symm |
| · rintro rfl |
| exact ⟨t,rfl⟩ |
|
|
| lemma superlevelSlice_graph_action {T : Functional X 1} |
| (hT : IsMetricCurrent T) (hB : IsMetricCurrent (boundarySucc T)) |
| {u : X → ℝ} (hu : Continuous u) |
| (hI : ∀ᵐ t : ℝ, IsIntegral 0 (superlevelSlice hT hB u t)) |
| {D : Set ℝ} (hD : IsCompact D) {m : ℕ} (γ : D → Fin m → X) |
| (hγ : ∀ j, Continuous (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)) |
| {b : X → ℝ} (hb : BoundedLip b) (j : Fin m) : |
| ∀ᵐ t : ℝ, currentBorelAction (superlevelSlice hT hB u t) |
| ((Set.range (fun s => γ s j)).indicator b) (fun z => Fin.elim0 z) = |
| if ht : t ∈ D then (a j : ℝ)*b (γ ⟨t,ht⟩ j) else 0 := by |
| classical |
| let : CompactSpace D := isCompact_iff_compactSpace.mp hD |
| let E := Set.range (fun s => γ s j) |
| have hE : MeasurableSet E := (isCompact_range (hγ j)).measurableSet |
| have hf : Measurable (E.indicator b) := hb.continuous.measurable.indicator hE |
| obtain ⟨B,hBb⟩ := hb.2 |
| let B' : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ |
| have hfB (x : X) : |E.indicator b x| ≤ B' := by |
| by_cases hx : x ∈ E |
| · rw [indicator_of_mem hx] |
| exact (hBb x).trans (le_max_left _ _) |
| · rw [indicator_of_notMem hx,abs_zero] |
| exact B'.coe_nonneg |
| filter_upwards [hI] with t htI |
| by_cases ht : t ∈ D |
| · rw [dite_eq_left ht,(hrep ⟨t,ht⟩).borelAction htI.1 hf hfB] |
| rw [Finset.sum_eq_single j] |
| · rw [indicator_of_mem (show γ ⟨t,ht⟩ j ∈ E from ⟨⟨t,ht⟩,rfl⟩)] |
| · intro i _ hij |
| rw [indicator_of_notMem ((graph_atom_mem_iff γ hbase hinj ⟨t,ht⟩ i j).not.mpr hij),mul_zero] |
| · simp |
| · rw [dite_eq_right ht] |
| obtain ⟨q,c,x,hc,hx,hrep'⟩ := integral_zero_unitRepresentation htI |
| apply hrep'.borelAction_zero htI.1 hf |
| intro i |
| apply indicator_of_notMem |
| rintro ⟨s,he⟩ |
| have hlevel := unitRepresentation_atoms_on_level hT hB hu t hc hx hrep' i |
| rw [←he,hbase] at hlevel |
| exact ht (hlevel ▸ s.property) |
|
|
| end CAT0Fillings.SliceReconstruction |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients BorelRestriction Slicing |
|
|
| attribute [local instance] Classical.propDecidable |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| lemma joint_graph_atom_mem_iff {X : Type*} [MetricSpace X] [MeasurableSpace X] |
| [BorelSpace X] [CompactSpace X] [Nonempty X] {k : ℕ} {D : Set (Euc k)} {m : ℕ} |
| (γ : D → Fin m → X) |
| {π : Fin k → X → ℝ} (hbase : ∀ t j, coordinateMap π (γ t j) = t) |
| (hinj : ∀ t, Function.Injective (γ t)) (t : D) (i j : Fin m) : |
| γ t i ∈ Set.range (fun s => γ s j) ↔ i = j := by |
| constructor |
| · rintro ⟨s,he⟩ |
| have ht : s = t := Subtype.ext (by simpa only [hbase] using congrArg (coordinateMap π) he) |
| subst s |
| exact (hinj t he).symm |
| · rintro rfl |
| exact ⟨t,rfl⟩ |
|
|
| lemma fullSlice_graph_action {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, Continuous (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)) |
| {b : X → ℝ} (hb : BoundedLip b) (j : Fin m) : |
| ∀ᵐ z : Euc k, currentBorelAction (fullSlice h π z) |
| ((Set.range (fun s => γ s j)).indicator b) (fun i => Fin.elim0 i) = |
| if hz : z ∈ D then (a j : ℝ)*b (γ ⟨z,hz⟩ j) else 0 := by |
| classical |
| let : CompactSpace D := isCompact_iff_compactSpace.mp hD |
| let E := Set.range (fun s => γ s j) |
| have hE : MeasurableSet E := (isCompact_range (hγ j)).measurableSet |
| have hf : Measurable (E.indicator b) := hb.continuous.measurable.indicator hE |
| obtain ⟨B,hBb⟩ := hb.2 |
| let B' : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ |
| have hfB (x : X) : |E.indicator b x| ≤ B' := by |
| by_cases hx : x ∈ E |
| · rw [indicator_of_mem hx] |
| exact (hBb x).trans (le_max_left _ _) |
| · rw [indicator_of_notMem hx,abs_zero] |
| exact B'.coe_nonneg |
| have hA : Continuous (coordinateMap π) := by |
| exact (parameterEquiv k).continuous.comp (continuous_pi fun i => (hπ i).continuous) |
| filter_upwards [ae_fullSlice_mass_fiber h hX π hπ] with z hzM |
| have hI := fullSlice_integral h π z |
| by_cases hz : z ∈ D |
| · rw [dite_eq_left hz,(hrep ⟨z,hz⟩).borelAction hI.1 hf hfB] |
| rw [Finset.sum_eq_single j] |
| · rw [indicator_of_mem (show γ ⟨z,hz⟩ j ∈ E from ⟨⟨z,hz⟩,rfl⟩)] |
| · intro i _ hij |
| rw [indicator_of_notMem ((joint_graph_atom_mem_iff γ hbase hinj ⟨z,hz⟩ i j).not.mpr hij),mul_zero] |
| · simp |
| · rw [dite_eq_right hz] |
| obtain ⟨q,c,x,hc,hx,hrep'⟩ := integral_zero_unitRepresentation hI |
| apply hrep'.borelAction_zero hI.1 hf |
| intro i |
| apply indicator_of_notMem |
| rintro ⟨s,he⟩ |
| have hlevel := hrep'.atom_in_carrier hI.1 hc hx |
| (measurableSet_eq_fun hA.measurable measurable_const) hzM i |
| change coordinateMap π (x i) = z at hlevel |
| rw [←he,hbase] at hlevel |
| exact hz (hlevel ▸ s.property) |
|
|
| lemma full_graph_restriction_base_action {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) {b : X → ℝ} (hb : BoundedLip b) : |
| restrictCurrent h.metric (Set.range (fun t => γ t j)) b π = |
| (jointGraphChart hD (fun t => γ t j) (hγ j) (a j)).action b π := by |
| classical |
| let : CompactSpace D := isCompact_iff_compactSpace.mp hD |
| 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 |
| 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 := fullSlice_borel_integral h hX π hπ hf B' hfB |
| have hs := fullSlice_graph_action h hX π hπ hD γ hγc a hbase hinj hrep hb j |
| rw [restrictCurrent_apply h.metric E ⟨hb,fun i => (hπ i).1⟩, |
| ←currentBorelAction_eq h.metric f π,ha.2] |
| rw [jointGraphChart_base_action hD _ (hγ j) (a j) π (fun i => (hπ i).1) (by |
| intro t i |
| exact congrArg (fun z : Euc k => z i) (hbase t j)) hb] |
| rw [←integral_indicator hD.measurableSet] |
| apply integral_congr_ae |
| filter_upwards [hs] with z hz |
| rw [hz] |
| by_cases hzD : z ∈ D |
| · simp only [dite_eq_left hzD,indicator_of_mem hzD,jointGraphFunction] |
| · simp only [dite_eq_right hzD,indicator_of_notMem hzD] |
|
|
| end CAT0Fillings.SliceReconstruction |
| end |
|
|
| end OAI |
|
|
