| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.Pushforward |
| import OAI.Geometry.CAT0Fillings.Currents.PushMajorant |
| import OAI.Geometry.CAT0Fillings.Prism.Differential |
| import OAI.Geometry.CAT0Fillings.Swept.RadialChart |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric Matrix |
| open scoped Topology NNReal BigOperators |
|
|
| namespace CAT0Fillings |
|
|
| local instance {k : ℕ} : MeasurableSpace (Matrix (Fin k) (Fin k) ℝ) := borel _ |
| local instance {k : ℕ} : BorelSpace (Matrix (Fin k) (Fin k) ℝ) := ⟨rfl⟩ |
|
|
| lemma measurable_matrix_inverse {E : Type*} [MeasurableSpace E] {k : ℕ} |
| (P : E → Matrix (Fin k) (Fin k) ℝ) (hm : Measurable P) : |
| Measurable (fun z => (P z)⁻¹) := by |
| have hd : Measurable (fun z => (P z).det) := continuous_id.matrix_det.measurable.comp hm |
| have ha : Measurable (fun z => (P z).adjugate) := continuous_id.matrix_adjugate.measurable.comp hm |
| simp only [Matrix.inv_def,Ring.inverse_eq_inv] |
| convert hd.inv.smul ha using 1 |
|
|
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] [nonemptyX : Nonempty X] {k : ℕ} (C : IntegerChart X k) |
|
|
| include nonemptyX |
|
|
| omit nonemptyX in |
| lemma aestronglyMeasurable_sweptWeight [Nonempty X] |
| (p : Euc k → Seminorm ℝ (Euc k)) (hpm : ∀ v, Measurable (fun z => p z v)) |
| (o : X) {g : X → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : |
| AEStronglyMeasurable (C.sweptWeight p o g) (volume.restrict C.domain) := by |
| obtain ⟨L,U,hL,_⟩ := C.bilipschitz |
| obtain ⟨R,hR,heR⟩ := (C.scalar_lipschitzOn hL (LipschitzWith.dist_right o)).extend_real |
| let P (z : Euc k) := polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis |
| have hPm : Measurable P := by |
| have hh : Measurable (fun z i j => P z i j) := |
| measurable_pi_iff.mpr fun i => measurable_pi_iff.mpr fun j => |
| measurable_polarizationMatrix p (EuclideanSpace.basisFun (Fin k) ℝ).toBasis hpm i j |
| have heq : _root_.borel (Fin k → Fin k → ℝ) = MeasurableSpace.pi := |
| BorelSpace.measurable_eq.symm |
| change @Measurable (Euc k) (Fin k → Fin k → ℝ) _ (_root_.borel _) P |
| rw [heq] |
| exact hh |
| have hRd : Measurable (fun z => differentialRow (fderiv ℝ R z)) := by |
| apply measurable_pi_iff.mpr |
| intro i |
| simp only [differentialRow] |
| fun_prop |
| have hPi := measurable_matrix_inverse P hPm |
| let J (z : Euc k) := C.scalar (dist o) z*C.scalar g z* |
| Real.sqrt (1-differentialRow (fderiv ℝ R z) ⬝ᵥ |
| (P z)⁻¹.mulVec (differentialRow (fderiv ℝ R z)))*Real.sqrt (P z).det |
| have hJ : Measurable J := by |
| have hprod : Measurable (fun z => differentialRow (fderiv ℝ R z) ⬝ᵥ |
| (P z)⁻¹.mulVec (differentialRow (fderiv ℝ R z))) := by |
| have hc : Continuous (fun q : (Fin k → ℝ) × Matrix (Fin k) (Fin k) ℝ => |
| q.1 ⬝ᵥ q.2.mulVec q.1) := by fun_prop |
| exact hc.measurable.comp (hRd.prodMk hPi) |
| exact (((C.measurable_scalar (LipschitzWith.dist_right o)).mul (C.measurable_scalar hg)).mul |
| ((measurable_const.sub hprod).sqrt)).mul |
| ((continuous_id.matrix_det.measurable.comp hPm).sqrt) |
| apply hJ.aestronglyMeasurable.congr |
| filter_upwards [ae_fderivWithin_eq_fderiv_extension volume C.borel hR heR] with z hz |
| simp only [J,sweptWeight,hz,P] |
|
|
| omit nonemptyX in |
| lemma ae_sweptWeight_nonneg [Nonempty X] |
| (p : Euc k → Seminorm ℝ (Euc k)) (o : X) {g : X → ℝ} |
| (hg0 : ∀ x, 0 ≤ g x) : 0 ≤ᵐ[volume.restrict C.domain] C.sweptWeight p o g := by |
| filter_upwards [ae_restrict_mem C.borel] with z hz |
| simp only [sweptWeight,C.scalar_eq hz] |
| exact mul_nonneg (mul_nonneg (mul_nonneg dist_nonneg (hg0 _)) (Real.sqrt_nonneg _)) |
| (Real.sqrt_nonneg _) |
|
|
| omit nonemptyX in |
| lemma ae_sweptWeight_bound [Nonempty X] |
| (p : Euc k → Seminorm ℝ (Euc k)) |
| (hp : ∀ᵐ z ∂volume.restrict C.domain, |
| (∀ hz : z ∈ C.domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| C.domain C.param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) |
| (o : X) {g : X → ℝ} (hg0 : ∀ x, 0 ≤ g x) |
| {B : ℝ} (hB : ∀ x, dist o x*g x ≤ B) : |
| ∀ᵐ z ∂volume.restrict C.domain, |
| C.sweptWeight p o g z ≤ B*Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det := by |
| filter_upwards [hp,ae_restrict_mem C.borel] with z hpz hz |
| let G := polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis |
| let α := differentialRow (fderivWithin ℝ (C.scalar (dist o)) C.domain z) |
| have hG : G.PosDef := polarizationMatrix_posDef _ _ hpz.2.2 hpz.2.1 |
| have hq : 0 ≤ α ⬝ᵥ G⁻¹.mulVec α := hG.inv.posSemidef.dotProduct_mulVec_nonneg α |
| have hs : Real.sqrt (1-α ⬝ᵥ G⁻¹.mulVec α) ≤ 1 := by |
| apply (Real.sqrt_le_left zero_le_one).mpr |
| linarith |
| change C.scalar (dist o) z*C.scalar g z*Real.sqrt (1-α ⬝ᵥ G⁻¹.mulVec α)*Real.sqrt G.det ≤ _ |
| rw [C.scalar_eq hz,C.scalar_eq hz] |
| apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _) |
| exact ((mul_le_mul_of_nonneg_left hs (mul_nonneg dist_nonneg (hg0 _))).trans_eq |
| (mul_one _)).trans (hB _) |
|
|
| theorem integrable_sweptWeight |
| (p : Euc k → Seminorm ℝ (Euc k)) (hpm : ∀ v, Measurable (fun z => p z v)) |
| (hp : ∀ᵐ z ∂volume.restrict C.domain, |
| (∀ hz : z ∈ C.domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| C.domain C.param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) |
| (hρ : Integrable (fun z => |(C.multiplicity z : ℝ)| *Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det) |
| (volume.restrict C.domain)) |
| (o : X) {g : X → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| (hg0 : ∀ x, 0 ≤ g x) {B : ℝ} (hB : ∀ x, dist o x*g x ≤ B) : |
| Integrable (fun z => |(C.multiplicity z : ℝ)| * C.sweptWeight p o g z) |
| (volume.restrict C.domain) := by |
| apply (hρ.const_mul B).mono' |
| (C.integrable.abs.aestronglyMeasurable.mul (C.aestronglyMeasurable_sweptWeight p hpm o hg)) |
| filter_upwards [C.ae_sweptWeight_nonneg p o hg0,C.ae_sweptWeight_bound p hp o hg0 hB] with z hn hb |
| simp only [Pi.mul_apply] |
| rw [Real.norm_eq_abs,abs_mul,abs_abs,abs_of_nonneg hn] |
| exact (mul_le_mul_of_nonneg_left hb (abs_nonneg _)).trans_eq (by ring) |
| end IntegerChart |
| end CAT0Fillings |
| end |
| |
| section |
| |
| open Set Filter MeasureTheory Metric Matrix |
| open scoped Topology NNReal BigOperators |
| |
| namespace CAT0Fillings |
| open CurrentOperations |
| |
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] {k : ℕ} (C : IntegerChart X k) |
| |
| lemma integrable_prism_tail {f : Euc k → ℝ} |
| (hf : Integrable f (volume.restrict C.domain)) : |
| Integrable (fun z => f (Prism.tail k z)) (volume.restrict C.prism.domain) := by |
| have hi := hf.comp_snd (volume.restrict (Icc (0:ℝ) 1)) |
| rw [Measure.prod_restrict] at hi |
| exact ((Prism.split_measurePreserving k).restrict_preimage |
| (measurableSet_Icc.prod C.borel)).integrable_comp_emb |
| (Prism.split k).toHomeomorph.measurableEmbedding |>.mpr hi |
| |
| lemma integral_prism_tail {f : Euc k → ℝ} |
| (hf : Integrable f (volume.restrict C.domain)) : |
| (∫ z, f (Prism.tail k z) ∂volume.restrict C.prism.domain) = |
| ∫ z, f z ∂volume.restrict C.domain := by |
| rw [←C.prism_split_measurePreserving.integral_comp |
| (Prism.split k).symm.toHomeomorph.measurableEmbedding] |
| have heq (q : ℝ × Euc k) : Prism.tail k ((Prism.split k).symm q) = q.2 := by |
| change (Prism.split k ((Prism.split k).symm q)).2 = _ |
| rw [ContinuousLinearEquiv.apply_symm_apply] |
| simp_rw [heq] |
| rw [integral_prod _ (hf.comp_snd _)] |
| simp [integral_const,Measure.real,Real.volume_Icc] |
| |
| variable [Nonempty X] [MeasurableSpace X] [BorelSpace X] |
| |
| theorem swept_prism_mass_le |
| (seg : X → X → ℝ → X) |
| (hcomp : ∀ o x y a b, a ∈ Icc (0:ℝ) 1 → b ∈ Icc (0:ℝ) 1 → |
| dist (seg o x a) (seg o y b)^2 ≤ (a*dist o x-b*dist o y)^2+ |
| a*b*(dist x y^2-(dist o x-dist o y)^2)) |
| (o : X) (g : X → ℝ) {Kg KF : ℝ≥0} (hg : LipschitzWith Kg g) |
| (hg0 : ∀ y, 0 ≤ g y) (hg1 : ∀ y, g y ≤ 1) |
| (F : ℝ × X → X) (hF : LipschitzWith KF F) |
| (hFeq : ∀ t ∈ Icc (0:ℝ) 1, ∀ y, F (t,y) = seg o y (1-t*g y)) |
| (p : Euc k → Seminorm ℝ (Euc k)) |
| (hI : Integrable (fun z => |(C.multiplicity z : ℝ)| *C.sweptWeight p o g z) |
| (volume.restrict C.domain)) |
| (hp : ∀ᵐ z ∂volume.restrict C.domain, |
| (∀ hz : z ∈ C.domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| C.domain C.param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) : |
| mass (pushCurrent F C.prism.action) ≤ |
| ∫ z, |(C.multiplicity z : ℝ)| *C.sweptWeight p o g z ∂volume.restrict C.domain := by |
| have hi : Integrable (fun z => |(C.prism.multiplicity z : ℝ)| * |
| C.sweptWeight p o g (Prism.tail k z)) (volume.restrict C.prism.domain) := |
| C.integrable_prism_tail hI |
| have hm := C.prism.push_mass_le_majorant hF hi |
| (C.ae_prism_tail (C.ae_sweptWeight_nonneg p o hg0)) |
| (C.ae_swept_jacobian_bound seg hcomp o g hg hg0 hg1 F hF hFeq p hp) |
| exact hm.trans_eq (C.integral_prism_tail hI) |
| |
| end IntegerChart |
| end CAT0Fillings |
| end |
| |
| end OAI |
| |
