| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Swept.Current |
| import OAI.Geometry.CAT0Fillings.Gradient.Chart |
| import OAI.Geometry.CAT0Fillings.Fillings.WeightedCone |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| namespace CAT0Fillings |
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X k} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| noncomputable def weightedMeasure (v : ℕ → Euc k → ℝ) : Measure X := |
| Measure.sum (fun i => densityPush (volume.restrict (q.chart i).domain) |
| (q.chart i).paramExtended (fun z => q.density i z * v i z)) |
|
|
| lemma weightedMeasure_le {v : ℕ → Euc k → ℝ} {c : ℝ≥0} |
| (hv : ∀ i, ∀ᵐ z ∂volume.restrict (q.chart i).domain, v i z ≤ c) : |
| q.weightedMeasure v ≤ c • MassMeasure.currentMassMeasure hT := by |
| apply Measure.le_iff.mpr |
| intro s hs |
| rw [weightedMeasure,Measure.sum_apply _ hs,Measure.smul_apply,q.measure_eq, |
| Measure.sum_apply _ hs] |
| change _ ≤ (c : ℝ≥0∞) * _ |
| rw [←ENNReal.tsum_mul_left] |
| apply ENNReal.tsum_le_tsum |
| intro i |
| have hi := densityPush_mono_ae (q.chart i).measurable_paramExtended |
| ((hv i).mono fun z hz => (mul_le_mul_of_nonneg_left hz (q.density_nonneg i z)).trans_eq |
| (mul_comm _ _)) |
| rw [densityPush_const_mul _ _ _ _ (q.chart i).measurable_paramExtended] at hi |
| exact hi s |
|
|
| lemma weightedMeasure_finite {v : ℕ → Euc k → ℝ} {c : ℝ≥0} |
| (hv : ∀ i, ∀ᵐ z ∂volume.restrict (q.chart i).domain, v i z ≤ c) : |
| IsFiniteMeasure (q.weightedMeasure v) := |
| isFiniteMeasure_of_le (c • MassMeasure.currentMassMeasure hT) (q.weightedMeasure_le hv) |
|
|
| noncomputable def gradientMeasure (u : X → ℝ) : Measure X := |
| q.weightedMeasure (fun i => q.duNorm i u) |
| noncomputable def energyMeasure (u : X → ℝ) : Measure X := |
| q.weightedMeasure (fun i => q.normSq i u) |
|
|
| lemma gradientMeasure_le {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| q.gradientMeasure u ≤ K • MassMeasure.currentMassMeasure hT := |
| q.weightedMeasure_le (fun i => (q.ae_duNorm_bounds i hu).mono fun _ hz => hz.2.1) |
|
|
| lemma energyMeasure_le {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| q.energyMeasure u ≤ K^2 • MassMeasure.currentMassMeasure hT := by |
| apply q.weightedMeasure_le |
| intro i |
| simpa only [NNReal.coe_pow] using (q.ae_normSq_bounds i hu).mono (fun _ hz => hz.2) |
|
|
| lemma gradientMeasure_finite {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| IsFiniteMeasure (q.gradientMeasure u) := |
| isFiniteMeasure_of_le _ (q.gradientMeasure_le hu) |
|
|
| lemma energyMeasure_finite {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| IsFiniteMeasure (q.energyMeasure u) := |
| isFiniteMeasure_of_le _ (q.energyMeasure_le hu) |
|
|
| lemma integral_gradientMeasure {u w : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) |
| (hw : Continuous w) : |
| (∫ x, w x ∂q.gradientMeasure u) = ∑' i, ∫ z, |
| (q.density i z * q.duNorm i u z)*w ((q.chart i).paramExtended z) |
| ∂volume.restrict (q.chart i).domain := by |
| let := q.gradientMeasure_finite hu |
| have hi := hw.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace w) |
| (μ := q.gradientMeasure u) |
| rw [gradientMeasure,weightedMeasure] at hi ⊢ |
| rw [integral_sum_measure hi] |
| apply tsum_congr |
| intro i |
| exact integral_densityPush _ (q.chart i).measurable_paramExtended (q.integrable_gradient i hu) |
| (Eventually.of_forall fun z => mul_nonneg (q.density_nonneg i z) (Real.sqrt_nonneg _)) hw |
|
|
| lemma gradient_summable {u w : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) |
| (hw : Continuous w) : |
| Summable (fun i => ∫ z, |
| (q.density i z * q.duNorm i u z)*w ((q.chart i).paramExtended z) |
| ∂volume.restrict (q.chart i).domain) := by |
| let := q.gradientMeasure_finite hu |
| have hi := hw.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace w) |
| (μ := q.gradientMeasure u) |
| apply (hasSum_integral_measure hi).summable.congr |
| intro i |
| exact integral_densityPush _ (q.chart i).measurable_paramExtended (q.integrable_gradient i hu) |
| (Eventually.of_forall fun z => mul_nonneg (q.density_nonneg i z) (Real.sqrt_nonneg _)) hw |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator |
|
|
| namespace CAT0Fillings |
|
|
| local instance {n : ℕ} : MeasurableSpace (Matrix (Fin n) (Fin n) ℝ) := borel _ |
| local instance {n : ℕ} : BorelSpace (Matrix (Fin n) (Fin n) ℝ) := ⟨rfl⟩ |
|
|
| lemma matrix_posSemidef_closed {n : ℕ} : |
| IsClosed {P : Matrix (Fin n) (Fin n) ℝ | P.PosSemidef} := by |
| simp only [Matrix.posSemidef_iff_dotProduct_mulVec] |
| apply IsClosed.inter |
| · change IsClosed {P : Matrix (Fin n) (Fin n) ℝ | P.conjTranspose = P} |
| exact isClosed_eq (by fun_prop) continuous_id |
| · have hh (v : Fin n → ℝ) : IsClosed {P : Matrix (Fin n) (Fin n) ℝ | |
| 0 ≤ star v ⬝ᵥ P.mulVec v} := isClosed_le continuous_const (by fun_prop) |
| have hs : {P : Matrix (Fin n) (Fin n) ℝ | ∀ v, 0 ≤ star v ⬝ᵥ P.mulVec v} = |
| ⋂ v, {P : Matrix (Fin n) (Fin n) ℝ | 0 ≤ star v ⬝ᵥ P.mulVec v} := by |
| ext P |
| simp only [mem_iInter,mem_ofPred_eq] |
| change IsClosed {P : Matrix (Fin n) (Fin n) ℝ | ∀ v, 0 ≤ star v ⬝ᵥ P.mulVec v} |
| rw [hs] |
| exact isClosed_iInter hh |
|
|
| lemma measurable_matrix_sqrt {n : ℕ} : |
| Measurable (CFC.sqrt : Matrix (Fin n) (Fin n) ℝ → Matrix (Fin n) (Fin n) ℝ) := by |
| classical |
| have hm : MeasurableSet {P : Matrix (Fin n) (Fin n) ℝ | 0 ≤ P} := by |
| simpa only [Matrix.nonneg_iff_posSemidef] using (matrix_posSemidef_closed (n := n)).measurableSet |
| have h := (CFC.continuousOn_sqrt (A := Matrix (Fin n) (Fin n) ℝ)).measurable_piecewise |
| (g := fun _ => 0) continuousOn_const hm |
| convert h using 1 |
| funext P |
| by_cases hp : 0 ≤ P |
| · simp only [Set.piecewise,mem_ofPred_eq,hp,ite_true] |
| · simp only [Set.piecewise,mem_ofPred_eq,hp,ite_false,CFC.sqrt_of_not_nonneg hp] |
|
|
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| noncomputable def normalizedCovector (i : ℕ) (u : X → ℝ) (z : Euc n) : Euc n := |
| WithLp.toLp 2 ((CFC.sqrt (q.gram i z)⁻¹).mulVec (q.covector i u z)) |
|
|
| lemma aestronglyMeasurable_normalizedCovector (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : AEStronglyMeasurable (q.normalizedCovector i u) |
| (volume.restrict (q.chart i).domain) := by |
| have hM := measurable_matrix_sqrt.comp (measurable_matrix_inverse _ (q.measurable_gram i)) |
| have hc : Continuous (fun r : Matrix (Fin n) (Fin n) ℝ × (Fin n → ℝ) => |
| (WithLp.toLp 2 (r.1.mulVec r.2) : Euc n)) := by fun_prop |
| exact hc.comp_aestronglyMeasurable |
| (hM.aestronglyMeasurable.prodMk (q.aestronglyMeasurable_covector i hu)) |
|
|
| lemma ae_normalizedCovector_norm_sq (i : ℕ) (u : X → ℝ) : |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| ‖q.normalizedCovector i u z‖^2 = q.normSq i u z := by |
| filter_upwards [q.differential i] with z hz |
| have hG : (q.gram i z).PosDef := polarizationMatrix_posDef _ _ hz.2.2 hz.2.1 |
| rw [←real_inner_self_eq_norm_sq] |
| simp only [normalizedCovector,EuclideanSpace.inner_eq_star_dotProduct,star_trivial] |
| exact (dot_mulVec_sqrt _ hG.inv.posSemidef _).symm |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| noncomputable def coordinateMeasure (i : ℕ) : Measure (Euc n) := |
| (volume.restrict (q.chart i).domain).withDensity (fun z => ENNReal.ofReal (q.density i z)) |
| noncomputable def atlasMeasure : Measure (ℕ × Euc n) := |
| Measure.sum (fun i => (q.coordinateMeasure i).map (Prod.mk i)) |
|
|
| instance coordinateMeasure_finite (i : ℕ) : IsFiniteMeasure (q.coordinateMeasure i) := |
| isFiniteMeasure_withDensity_ofReal (q.density_integrable i).hasFiniteIntegral |
|
|
| lemma coordinateMeasure_total (i : ℕ) : (q.coordinateMeasure i).real univ = |
| ∫ z, q.density i z ∂volume.restrict (q.chart i).domain := by |
| rw [show (q.coordinateMeasure i).real univ = ∫ _ : Euc n, (1:ℝ) ∂q.coordinateMeasure i by simp] |
| rw [coordinateMeasure,integral_withDensity_eq_integral_toReal_smul₀ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| simp only [ENNReal.toReal_ofReal (q.density_nonneg i _),smul_eq_mul,mul_one] |
|
|
| instance atlasMeasure_finite : IsFiniteMeasure q.atlasMeasure := by |
| apply finite_sum_measure_of_summable_total |
| apply q.density_summable.congr |
| intro i |
| rw [←q.coordinateMeasure_total] |
| simp only [measureReal_def,Measure.map_apply measurable_prodMk_left MeasurableSet.univ,preimage_univ] |
|
|
| noncomputable def gradientFunction (u : X → ℝ) (w : ℕ × Euc n) : Euc n := |
| q.normalizedCovector w.1 u w.2 |
|
|
| lemma aestronglyMeasurable_gradientFunction {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : AEStronglyMeasurable (q.gradientFunction u) q.atlasMeasure := by |
| apply aestronglyMeasurable_sum_measure_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).aestronglyMeasurable_map_iff.mpr |
| exact (q.aestronglyMeasurable_normalizedCovector i hu).mono_ac |
| (withDensity_absolutelyContinuous _ _) |
|
|
| lemma ae_gradientFunction_bound {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| ∀ᵐ w ∂q.atlasMeasure, ‖q.gradientFunction u w‖ ≤ K := by |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| apply (withDensity_absolutelyContinuous _ _).ae_le |
| filter_upwards [q.ae_normalizedCovector_norm_sq i u,q.ae_normSq_bounds i hu] with z hz hb |
| change ‖q.normalizedCovector i u z‖ ≤ K |
| nlinarith [norm_nonneg (q.normalizedCovector i u z),K.coe_nonneg] |
|
|
| lemma memLp_gradientFunction {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| MemLp (q.gradientFunction u) 2 q.atlasMeasure := |
| MemLp.of_bound (q.aestronglyMeasurable_gradientFunction hu) K (q.ae_gradientFunction_bound hu) |
|
|
| noncomputable def gradient {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| Lp (Euc n) 2 q.atlasMeasure := (q.memLp_gradientFunction hu).toLp _ |
|
|
| lemma gradient_norm_sq {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| ‖q.gradient hu‖^2 = (q.energyMeasure u).real univ := by |
| have hi : Integrable (fun w => ‖q.gradientFunction u w‖^2) q.atlasMeasure := by |
| exact (q.memLp_gradientFunction hu).integrable_norm_pow (by norm_num) |
| have hleft : ‖q.gradient hu‖^2 = ∫ w, ‖q.gradientFunction u w‖^2 ∂q.atlasMeasure := by |
| rw [←real_inner_self_eq_norm_sq,L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [(q.memLp_gradientFunction hu).coeFn_toLp] with w hw |
| rw [gradient,hw,real_inner_self_eq_norm_sq] |
| have he : (q.energyMeasure u).real univ = ∑' i, ∫ z, |
| q.density i z*q.normSq i u z ∂volume.restrict (q.chart i).domain := by |
| let := q.energyMeasure_finite hu |
| rw [show (q.energyMeasure u).real univ = ∫ _ : X, (1:ℝ) ∂q.energyMeasure u by simp] |
| have hI : Integrable (fun _ : X => (1:ℝ)) (q.energyMeasure u) := integrable_const _ |
| rw [energyMeasure,weightedMeasure] at hI ⊢ |
| rw [integral_sum_measure hI] |
| apply tsum_congr |
| intro i |
| rw [integral_densityPush _ (q.chart i).measurable_paramExtended (q.integrable_energy i hu) |
| ((q.ae_normSq_bounds i hu).mono fun z hz => mul_nonneg (q.density_nonneg i z) hz.1) continuous_const] |
| simp only [mul_one] |
| rw [hleft,atlasMeasure,integral_sum_measure hi,he] |
| apply tsum_congr |
| intro i |
| rw [(measurableEmbedding_prodMk_left i).integral_map] |
| rw [coordinateMeasure,integral_withDensity_eq_integral_toReal_smul₀ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| apply integral_congr_ae |
| filter_upwards [q.ae_normalizedCovector_norm_sq i u] with z hz |
| simp only [ENNReal.toReal_ofReal (q.density_nonneg i z),smul_eq_mul,gradientFunction, hz] |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
