| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Swept.Current |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal BigOperators Matrix.Norms.L2Operator |
|
|
| namespace CAT0Fillings |
|
|
| lemma inverse_quadratic_le {k : ℕ} (G : Matrix (Fin k) (Fin k) ℝ) |
| (α : Fin k → ℝ) (hG : G.PosDef) {K : ℝ} (hK : 0 ≤ K) |
| (hα : ∀ v, |α ⬝ᵥ v| ≤ K * Real.sqrt (v ⬝ᵥ G.mulVec v)) : |
| α ⬝ᵥ G⁻¹.mulVec α ≤ K^2 := by |
| have hi : G.mulVec (G⁻¹.mulVec α) = α := by |
| rw [Matrix.mulVec_mulVec, Matrix.mul_nonsing_inv G (isUnit_iff_ne_zero.mpr hG.det_pos.ne'), |
| Matrix.one_mulVec] |
| have hq : 0 ≤ α ⬝ᵥ G⁻¹.mulVec α := hG.inv.posSemidef.dotProduct_mulVec_nonneg α |
| have hh := hα (G⁻¹.mulVec α) |
| rw [hi,dotProduct_comm (G⁻¹.mulVec α),abs_of_nonneg hq] at hh |
| have hs := Real.sq_sqrt hq |
| have hb : Real.sqrt (α ⬝ᵥ G⁻¹.mulVec α) ≤ K := by |
| by_contra! hn |
| have hp := mul_pos (lt_of_le_of_lt hK hn) (sub_pos.mpr hn) |
| nlinarith |
| nlinarith [Real.sqrt_nonneg (α ⬝ᵥ G⁻¹.mulVec α)] |
|
|
| 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) |
|
|
| local instance : MeasurableSpace (Matrix (Fin k) (Fin k) ℝ) := borel _ |
| local instance : BorelSpace (Matrix (Fin k) (Fin k) ℝ) := ⟨rfl⟩ |
|
|
| noncomputable def covector (i : ℕ) (u : X → ℝ) (z : Euc k) : Fin k → ℝ := |
| differentialRow (fderivWithin ℝ ((q.chart i).scalar u) (q.chart i).domain z) |
|
|
| noncomputable def normSq (i : ℕ) (u : X → ℝ) (z : Euc k) : ℝ := |
| q.covector i u z ⬝ᵥ (q.gram i z)⁻¹.mulVec (q.covector i u z) |
|
|
| noncomputable def duNorm (i : ℕ) (u : X → ℝ) (z : Euc k) : ℝ := |
| Real.sqrt (q.normSq i u z) |
|
|
| lemma measurable_gram (i : ℕ) : Measurable (q.gram i) := by |
| have hh : Measurable (fun z a b => q.gram i z a b) := |
| measurable_pi_iff.mpr fun a => measurable_pi_iff.mpr fun b => |
| measurable_polarizationMatrix (q.field i) |
| (EuclideanSpace.basisFun (Fin k) ℝ).toBasis (q.measurable i) a b |
| 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 _) _ |
| rw [heq] |
| exact hh |
|
|
| lemma aestronglyMeasurable_covector (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : AEStronglyMeasurable (q.covector i u) |
| (volume.restrict (q.chart i).domain) := by |
| obtain ⟨L,U,hL,_⟩ := (q.chart i).bilipschitz |
| obtain ⟨R,hR,heR⟩ := ((q.chart i).scalar_lipschitzOn hL hu).extend_real |
| have hm : Measurable (fun z => differentialRow (fderiv ℝ R z)) := by |
| apply measurable_pi_iff.mpr |
| intro j |
| simp only [differentialRow] |
| fun_prop |
| apply hm.aestronglyMeasurable.congr |
| filter_upwards [ae_fderivWithin_eq_fderiv_extension volume (q.chart i).borel hR heR] with z hz |
| simp only [covector,hz] |
|
|
| lemma aestronglyMeasurable_normSq (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : AEStronglyMeasurable (q.normSq i u) |
| (volume.restrict (q.chart i).domain) := by |
| have hc : Continuous (fun r : (Fin k → ℝ) × Matrix (Fin k) (Fin k) ℝ => |
| r.1 ⬝ᵥ r.2.mulVec r.1) := by fun_prop |
| exact hc.comp_aestronglyMeasurable ((q.aestronglyMeasurable_covector i hu).prodMk |
| (measurable_matrix_inverse _ (q.measurable_gram i)).aestronglyMeasurable) |
|
|
| lemma aestronglyMeasurable_duNorm (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : AEStronglyMeasurable (q.duNorm i u) |
| (volume.restrict (q.chart i).domain) := |
| Real.continuous_sqrt.comp_aestronglyMeasurable (q.aestronglyMeasurable_normSq i hu) |
|
|
| lemma ae_normSq_bounds (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| 0 ≤ q.normSq i u z ∧ q.normSq i u z ≤ (K : ℝ)^2 := by |
| obtain ⟨L,U,hL,_⟩ := (q.chart i).bilipschitz |
| have hb := MetricDifferentiation.ae_scalarOn_derivative_bound volume (q.chart i).borel hL |
| ((q.differential i).mono fun z hz => hz.1) hu |
| filter_upwards [q.differential i,hb] with z hz hbz |
| have hG : (q.gram i z).PosDef := polarizationMatrix_posDef _ _ hz.2.2 hz.2.1 |
| refine ⟨hG.inv.posSemidef.dotProduct_mulVec_nonneg _,?_⟩ |
| apply inverse_quadratic_le _ _ hG K.coe_nonneg |
| intro v |
| let b := (EuclideanSpace.basisFun (Fin k) ℝ).toBasis |
| let w : Euc k := WithLp.toLp 2 v |
| have hr : b.repr w = v := by ext j; simp [b,w] |
| have hquad : v ⬝ᵥ (q.gram i z).mulVec v = (q.field i z w)^2 := by |
| simpa only [gram,hr] using polarizationMatrix_quadratic b (q.field i z) hz.2.2 hz.2.1 w |
| rw [hquad,Real.sqrt_sq (apply_nonneg (q.field i z) w)] |
| have he : (q.chart i).scalar u = MetricDifferentiation.scalarOn (q.chart i).param u := by |
| ext x |
| rfl |
| simpa only [covector,differentialRow_dot,he,w] using hbz w |
|
|
| lemma ae_duNorm_bounds (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) : ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| 0 ≤ q.duNorm i u z ∧ q.duNorm i u z ≤ (K : ℝ) ∧ |
| q.duNorm i u z ^2 = q.normSq i u z := by |
| filter_upwards [q.ae_normSq_bounds i hu] with z hz |
| exact ⟨Real.sqrt_nonneg _, (Real.sqrt_le_left K.coe_nonneg).mpr hz.2, Real.sq_sqrt hz.1⟩ |
|
|
| lemma integrable_gradient (i : ℕ) {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| Integrable (fun z => q.density i z * q.duNorm i u z) |
| (volume.restrict (q.chart i).domain) := by |
| apply ((q.density_integrable i).mul_const (K : ℝ)).mono' |
| ((q.density_integrable i).aestronglyMeasurable.mul (q.aestronglyMeasurable_duNorm i hu)) |
| filter_upwards [q.ae_duNorm_bounds i hu] with z hz |
| have density_nonneg : 0 ≤ q.density i z := |
| mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
| simp only [Pi.mul_apply] |
| rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg density_nonneg hz.1)] |
| exact mul_le_mul_of_nonneg_left hz.2.1 density_nonneg |
|
|
| lemma integrable_energy (i : ℕ) {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| Integrable (fun z => q.density i z * q.normSq i u z) |
| (volume.restrict (q.chart i).domain) := by |
| apply ((q.density_integrable i).mul_const ((K : ℝ)^2)).mono' |
| ((q.density_integrable i).aestronglyMeasurable.mul (q.aestronglyMeasurable_normSq i hu)) |
| filter_upwards [q.ae_normSq_bounds i hu] with z hz |
| have density_nonneg : 0 ≤ q.density i z := |
| mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
| simp only [Pi.mul_apply] |
| rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg density_nonneg hz.1)] |
| exact mul_le_mul_of_nonneg_left hz.2 density_nonneg |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal BigOperators MatrixOrder |
|
|
| namespace CAT0Fillings |
|
|
| lemma abs_det_le_prod_dual_norm {n : ℕ} (L P : Matrix (Fin n) (Fin n) ℝ) (hP : P.PosDef) : |
| |L.det| ≤ (∏ i, Real.sqrt (L i ⬝ᵥ P⁻¹.mulVec (L i))) * Real.sqrt P.det := by |
| let Q := CFC.sqrt P⁻¹ |
| have hQsym : Q.transpose = Q := |
| (Matrix.isHermitian_iff_isSymm.mp (CFC.sqrt_nonneg P⁻¹).posSemidef.isHermitian).eq |
| have hQdet : Q.det = (Real.sqrt P.det)⁻¹ := by |
| have hh := hP.inv.posSemidef.det_sqrt |
| simpa only [Q,RCLike.sqrt_real,Matrix.det_nonsing_inv,Ring.inverse_eq_inv, |
| Real.sqrt_inv] using hh |
| have hJ : 0 < Real.sqrt P.det := Real.sqrt_pos.mpr hP.det_pos |
| have hrow (i : Fin n) : (L * Q) i = Q.mulVec (L i) := by |
| ext j |
| change (∑ t, L i t * Q t j) = ∑ t, Q j t * L i t |
| apply Finset.sum_congr rfl |
| intro t ht |
| have hs : Q t j = Q j t := congrFun (congrFun hQsym j) t |
| rw [hs,mul_comm] |
| have hnorm (i : Fin n) : |
| ‖(WithLp.toLp 2 ((L*Q) i) : EuclideanSpace ℝ (Fin n))‖ = |
| Real.sqrt (L i ⬝ᵥ P⁻¹.mulVec (L i)) := by |
| have hn : ‖(WithLp.toLp 2 ((L*Q) i) : EuclideanSpace ℝ (Fin n))‖^2 = |
| L i ⬝ᵥ P⁻¹.mulVec (L i) := by |
| rw [←real_inner_self_eq_norm_sq] |
| simp only [EuclideanSpace.inner_eq_star_dotProduct,star_trivial,hrow] |
| exact (dot_mulVec_sqrt P⁻¹ hP.inv.posSemidef (L i)).symm |
| rw [←hn,Real.sqrt_sq (norm_nonneg _)] |
| have heq : |L.det| = |(L*Q).det| * Real.sqrt P.det := by |
| rw [Matrix.det_mul,abs_mul,hQdet,abs_of_pos (inv_pos.mpr hJ)] |
| field_simp |
| rw [heq] |
| apply mul_le_mul_of_nonneg_right _ hJ.le |
| simpa only [hnorm] using abs_det_le_prod_row_norm (L*Q) |
|
|
| lemma abs_det_le_first_dual_norm {n : ℕ} (L P : Matrix (Fin (n+1)) (Fin (n+1)) ℝ) |
| (hP : P.PosDef) |
| (hL : ∀ i : Fin n, Real.sqrt (L i.succ ⬝ᵥ P⁻¹.mulVec (L i.succ)) ≤ 1) : |
| |L.det| ≤ Real.sqrt (L 0 ⬝ᵥ P⁻¹.mulVec (L 0)) * Real.sqrt P.det := by |
| apply (abs_det_le_prod_dual_norm L P hP).trans |
| apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _) |
| rw [Fin.prod_univ_succ] |
| exact mul_le_of_le_one_right (Real.sqrt_nonneg _) |
| (Finset.prod_le_one₀ (fun i _ => Real.sqrt_nonneg _) (fun i _ => hL i)) |
|
|
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| theorem ae_jacobian_le_duNorm (i : ℕ) {u : X → ℝ} {K : ℝ≥0} |
| (hu : LipschitzWith K u) (π : Fin k → X → ℝ) (hπ : ∀ j, LipschitzWith 1 (π j)) : |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| |(q.chart i).jacobian (Fin.cons u π) z| ≤ |
| q.duNorm i u z * Real.sqrt (q.gram i z).det := by |
| have hp : ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| ∀ j, q.duNorm i ((Fin.cons u π : Fin (k+1) → X → ℝ) j) z ≤ |
| (Fin.cons (K : ℝ) (fun _ => (1 : ℝ)) : Fin (k+1) → ℝ) j := by |
| rw [eventually_all] |
| intro coordinate |
| refine Fin.cases ?_ (fun tail => ?_) coordinate |
| · exact (q.ae_duNorm_bounds i hu).mono fun _ bounds => bounds.2.1 |
| · exact (q.ae_duNorm_bounds i (hπ tail)).mono fun _ bounds => by simpa using bounds.2.1 |
| filter_upwards [q.differential i,hp] with z hz hpz |
| apply abs_det_le_first_dual_norm _ _ |
| (polarizationMatrix_posDef _ _ hz.2.2 hz.2.1) |
| intro j |
| exact hpz j.succ |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| end OAI |
|
|
