| import Mathlib |
| import OAI.Geometry.CAT0Fillings.BV.JointPoincare |
| import OAI.Geometry.CAT0Fillings.Reconstruction.BorelAtoms |
| import OAI.Geometry.CAT0Fillings.Reconstruction.AtomCarrier |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients Slicing |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| abbrev FullGraphIndex := JointPieceIndex × AtomCode |
|
|
| theorem fullSlice_compact_atom_graphs {k : ℕ} {T : Functional X (k+1)} |
| (h : NormalApprox (k+1) T) (hX : IsCAT0 X) |
| (π : Fin (k+1) → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) : |
| ∃ (D : FullGraphIndex → Set (Euc (k+1))) |
| (L U : FullGraphIndex → ℝ≥0) |
| (γ : ∀ i : FullGraphIndex, D i → Fin i.2.size → X), |
| (∀ i, IsCompact (D i)) ∧ (∀ᵐ z : Euc (k+1), z ∈ ⋃ i, D i) ∧ |
| (∀ i j, LipschitzWith (L i) (fun t => γ i t j)) ∧ |
| (∀ i j, AntilipschitzWith (U i) (fun t => γ i t j)) ∧ |
| (∀ i (t : D i) j, coordinateMap π (γ i t j) = t) ∧ |
| (∀ i (t : D i), UnitRepresentation (fullSlice h π t) i.2.weights (γ i t)) ∧ |
| (∀ i (t : D i), Function.Injective (γ i t)) := by |
| classical |
| have hπb (i) := boundedLip_of_lipschitz (hπ i) |
| obtain ⟨D,L,γ,hD,hcover,hL,hI,hrep,hinj,hweights⟩ := joint_integer_atom_graphs |
| (μ := volume) (fullSlice h π) (Eventually.of_forall (fullSlice_integral h π)) |
| (fun b hb hb1 => (fullSlice_eval_integrable h hX π hπb ⟨⟨1,hb⟩,1,hb1⟩ |
| (fun j => Fin.elim0 j)).locallyIntegrable) |
| (fullSliceControl h π K) (C := 4*(2:ℝ)^(k+1)*(k+1)) (κ := 2) |
| (by positivity) (by norm_num) |
| (fun b hb hb1 a _r hr => fullSlice_ball_poincare h hX π hπ hb hb1 a hr) |
| obtain ⟨A,hA⟩ := coordinateMap_lipschitz π hπ |
| let (i : FullGraphIndex) : CompactSpace (D i) := isCompact_iff_compactSpace.mp (hD i) |
| let F (i : FullGraphIndex) := {t : D i | ∀ j, coordinateMap π (γ i t j) = t} |
| have hF (i) : IsCompact (F i) := by |
| apply IsClosed.isCompact |
| simpa only [F,ofPred_forall, Function.comp_def] using isClosed_iInter (fun j => isClosed_eq (hA.continuous.comp (hL i j).continuous) continuous_subtype_val) |
| let D' (i : FullGraphIndex) : Set (Euc (k+1)) := Subtype.val '' F i |
| have hsub (i) : D' i ⊆ D i := by |
| rintro z ⟨t,ht,rfl⟩ |
| exact t.property |
| let γ' (i : FullGraphIndex) (t : D' i) := γ i ⟨t,hsub i t.property⟩ |
| have hbase (i) (t : D' i) (j) : coordinateMap π (γ' i t j) = t := by |
| obtain ⟨s,hs,he⟩ := t.property |
| have ht : (⟨t,hsub i t.property⟩ : D i) = s := Subtype.ext he.symm |
| change coordinateMap π (γ i ⟨t,hsub i t.property⟩ j) = t |
| rw [ht,hs j,he] |
| refine ⟨D',L,fun _ => A,γ',?_,?_,?_,?_,hbase,?_,?_⟩ |
| · intro i |
| exact (hF i).image continuous_subtype_val |
| · filter_upwards [hcover,ae_fullSlice_mass_fiber h hX π hπb] with z hz hmass |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hz |
| apply mem_iUnion.mpr |
| refine ⟨i,⟨⟨z,hi⟩,?_,rfl⟩⟩ |
| intro j |
| exact (hrep i ⟨z,hi⟩).atom_in_carrier (fullSlice_approx h π z).metric |
| (hweights i ⟨z,hi⟩) (hinj i ⟨z,hi⟩) |
| (measurableSet_eq_fun hA.continuous.measurable measurable_const) hmass j |
| · intro i j |
| apply LipschitzWith.of_dist_le_mul |
| intro s t |
| exact (hL i j).dist_le_mul ⟨s,hsub i s.property⟩ ⟨t,hsub i t.property⟩ |
| · intro i j |
| apply AntilipschitzWith.of_le_mul_dist |
| intro s t |
| have hh := hA.dist_le_mul (γ' i s j) (γ' i t j) |
| rwa [hbase i s j,hbase i t j] at hh |
| · intro i t |
| exact hrep i ⟨t,hsub i t.property⟩ |
| · intro i t |
| exact hinj i ⟨t,hsub i t.property⟩ |
|
|
| end CAT0Fillings.SliceReconstruction |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.SliceReconstruction |
| open Foundations MassMeasure BorelCoefficients BorelRestriction |
|
|
| attribute [local instance] Classical.propDecidable |
| variable {X : Type*} [MetricSpace X] {k : ℕ} |
|
|
| noncomputable def jointGraphChart {D : Set (Euc k)} (hD : IsCompact D) (γ : D → X) |
| (hγ : ∃ L U : ℝ≥0, LipschitzWith L γ ∧ AntilipschitzWith U γ) (a : ℤ) : IntegerChart X k where |
| domain := D |
| borel := hD.measurableSet |
| bounded := hD.isBounded |
| param := γ |
| bilipschitz := hγ |
| multiplicity := fun _ => a |
| integrable := integrableOn_const (C := (a : ℝ)) (hD.measure_lt_top (μ := volume)).ne |
|
|
| lemma jointGraphChart_image {D : Set (Euc k)} (hD : IsCompact D) (γ : D → X) |
| (hγ : ∃ L U : ℝ≥0, LipschitzWith L γ ∧ AntilipschitzWith U γ) (a : ℤ) : |
| (jointGraphChart hD γ hγ a).image = Set.range γ := rfl |
|
|
| lemma jointGraphChart_base_jacobian {D : Set (Euc k)} (hD : IsCompact D) (γ : D → X) |
| (hγ : ∃ L U : ℝ≥0, LipschitzWith L γ ∧ AntilipschitzWith U γ) (a : ℤ) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K, LipschitzWith K (π i)) |
| (hbase : ∀ t : D, ∀ i, π i (γ t) = t.val i) : |
| ∀ᵐ z ∂volume.restrict D, (jointGraphChart hD γ hγ a).jacobian π z = 1 := by |
| let C := jointGraphChart hD γ hγ a |
| have he (i : Fin k) := C.ae_fderivWithin_scalar_eq_linear C.borel (subset_refl _) |
| (hπ i).choose_spec (EuclideanSpace.proj i) (by |
| intro z hz |
| rw [C.scalar_eq hz] |
| exact hbase ⟨z,hz⟩ i) |
| filter_upwards [ae_all_iff.mpr he] with z hz |
| change (Matrix.of fun i j => fderivWithin ℝ (C.scalar (π i)) D z (EuclideanSpace.single j 1)).det = 1 |
| change ∀ i, fderivWithin ℝ (C.scalar (π i)) D z = EuclideanSpace.proj i at hz |
| have hm : (Matrix.of fun i j => fderivWithin ℝ (C.scalar (π i)) D z (EuclideanSpace.single j 1)) = |
| (1 : Matrix (Fin k) (Fin k) ℝ) := by |
| ext i j |
| rw [Matrix.of_apply,hz i] |
| simp [Matrix.one_apply] |
| rw [hm,Matrix.det_one] |
|
|
| noncomputable def jointGraphFunction {D : Set (Euc k)} (γ : D → X) (b : X → ℝ) (z : Euc k) : ℝ := |
| if hz : z ∈ D then b (γ ⟨z,hz⟩) else 0 |
|
|
| lemma jointGraphChart_base_action [MeasurableSpace X] [BorelSpace X] |
| {D : Set (Euc k)} (hD : IsCompact D) (γ : D → X) |
| (hγ : ∃ L U : ℝ≥0, LipschitzWith L γ ∧ AntilipschitzWith U γ) (a : ℤ) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K, LipschitzWith K (π i)) |
| (hbase : ∀ t : D, ∀ i, π i (γ t) = t.val i) |
| {b : X → ℝ} (hb : BoundedLip b) : |
| (jointGraphChart hD γ hγ a).action b π = ∫ z in D, (a : ℝ)*jointGraphFunction γ b z := by |
| let C := jointGraphChart hD γ hγ a |
| rw [IntegerChart.action,ite_eq_left ⟨hb,hπ⟩] |
| apply integral_congr_ae |
| filter_upwards [jointGraphChart_base_jacobian hD γ hγ a π hπ hbase,ae_restrict_mem hD.measurableSet] with z hj hz |
| rw [hj,mul_one,C.scalar_eq hz] |
| change z ∈ D at hz |
| simp only [jointGraphFunction,dite_eq_left hz] |
| rfl |
|
|
| end CAT0Fillings.SliceReconstruction |
| end |
|
|
| end OAI |
|
|
