| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.Chart |
| import OAI.Geometry.CAT0Fillings.Gradient.HilbertAtlas |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator |
|
|
| namespace CAT0Fillings |
|
|
| noncomputable def rawCovector {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) (v : Euc n) : Fin n → ℝ := |
| (CFC.sqrt P⁻¹)⁻¹.mulVec (WithLp.ofLp v) |
|
|
| lemma sqrt_inverse_matrix_det_ne_zero {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) |
| (hP : P.PosDef) : (CFC.sqrt P⁻¹).det ≠ 0 := by |
| have hh := hP.inv.posSemidef.det_sqrt |
| rw [hh,RCLike.sqrt_real] |
| exact (Real.sqrt_pos.mpr hP.inv.det_pos).ne' |
|
|
| lemma normalize_rawCovector {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) |
| (hP : P.PosDef) (v : Euc n) : |
| (CFC.sqrt P⁻¹).mulVec (rawCovector P v) = WithLp.ofLp v := by |
| rw [rawCovector,Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv _ |
| (isUnit_iff_ne_zero.mpr (sqrt_inverse_matrix_det_ne_zero P hP)),Matrix.one_mulVec] |
|
|
| lemma raw_normalizedCovector {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) |
| (hP : P.PosDef) (a : Fin n → ℝ) : |
| rawCovector P (WithLp.toLp 2 ((CFC.sqrt P⁻¹).mulVec a)) = a := by |
| rw [rawCovector,WithLp.ofLp_toLp,Matrix.mulVec_mulVec,Matrix.nonsing_inv_mul _ |
| (isUnit_iff_ne_zero.mpr (sqrt_inverse_matrix_det_ne_zero P hP)),Matrix.one_mulVec] |
|
|
| lemma rawCovector_norm {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) |
| (hP : P.PosDef) (v : Euc n) : |
| Real.sqrt (rawCovector P v ⬝ᵥ P⁻¹.mulVec (rawCovector P v)) = ‖v‖ := by |
| rw [dot_mulVec_sqrt P⁻¹ hP.inv.posSemidef,normalize_rawCovector P hP] |
| have he : (WithLp.ofLp v) ⬝ᵥ (WithLp.ofLp v) = ‖v‖^2 := by |
| rw [←real_inner_self_eq_norm_sq] |
| simp only [EuclideanSpace.inner_eq_star_dotProduct,star_trivial] |
| rw [he,Real.sqrt_sq (norm_nonneg _)] |
|
|
| noncomputable def normalizedDet {n : ℕ} (P : Matrix (Fin (n+1)) (Fin (n+1)) ℝ) |
| (R : Fin n → Fin (n+1) → ℝ) (v : Euc (n+1)) : ℝ := |
| Matrix.det (Matrix.vecCons (rawCovector P v) R) / Real.sqrt P.det |
|
|
| lemma normalizedDet_bound {n : ℕ} (P : Matrix (Fin (n+1)) (Fin (n+1)) ℝ) |
| (hP : P.PosDef) (R : Fin n → Fin (n+1) → ℝ) |
| (hR : ∀ i, Real.sqrt (R i ⬝ᵥ P⁻¹.mulVec (R i)) ≤ 1) (v : Euc (n+1)) : |
| |normalizedDet P R v| ≤ ‖v‖ := by |
| have h := abs_det_le_first_dual_norm (Matrix.vecCons (rawCovector P v) R) P hP hR |
| change |Matrix.det (Matrix.vecCons (rawCovector P v) R)| ≤ |
| Real.sqrt (rawCovector P v ⬝ᵥ P⁻¹.mulVec (rawCovector P v))*Real.sqrt P.det at h |
| rw [rawCovector_norm P hP] at h |
| rw [normalizedDet,abs_div,abs_of_pos (Real.sqrt_pos.mpr hP.det_pos)] |
| exact (div_le_iff₀ (Real.sqrt_pos.mpr hP.det_pos)).mpr h |
|
|
| lemma normalizedDet_bound_prod {n : ℕ} (P : Matrix (Fin (n+1)) (Fin (n+1)) ℝ) |
| (hP : P.PosDef) (R : Fin n → Fin (n+1) → ℝ) (K : Fin n → ℝ≥0) |
| (hR : ∀ i, Real.sqrt (R i ⬝ᵥ P⁻¹.mulVec (R i)) ≤ K i) (v : Euc (n+1)) : |
| |normalizedDet P R v| ≤ ‖v‖ * ∏ i, (K i : ℝ) := by |
| have h := abs_det_le_prod_dual_norm (Matrix.vecCons (rawCovector P v) R) P hP |
| rw [Fin.prod_univ_succ] at h |
| simp only [Matrix.cons_val_zero,Matrix.cons_val_succ] at h |
| rw [rawCovector_norm P hP] at h |
| have hprod := Finset.prod_le_prod₀ (fun i (_ : i ∈ Finset.univ) => Real.sqrt_nonneg |
| (R i ⬝ᵥ P⁻¹.mulVec (R i))) (fun i _ => hR i) |
| have hh := h.trans (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hprod (norm_nonneg v)) (Real.sqrt_nonneg _)) |
| rw [normalizedDet,abs_div,abs_of_pos (Real.sqrt_pos.mpr hP.det_pos)] |
| exact (div_le_iff₀ (Real.sqrt_pos.mpr hP.det_pos)).mpr hh |
|
|
| lemma normalizedDet_sub {n : ℕ} (P : Matrix (Fin (n+1)) (Fin (n+1)) ℝ) |
| (R : Fin n → Fin (n+1) → ℝ) (v w : Euc (n+1)) : |
| normalizedDet P R (v-w) = normalizedDet P R v-normalizedDet P R w := by |
| have hr : rawCovector P (v-w) = rawCovector P v-rawCovector P w := by |
| simp [rawCovector,Matrix.mulVec_sub] |
| unfold normalizedDet |
| rw [hr,←sub_div] |
| congr 1 |
| have he (a : Fin (n+1) → ℝ) : Matrix.vecCons a R = |
| Matrix.updateRow (Matrix.vecCons (0 : Fin (n+1) → ℝ) R) 0 a := by |
| ext i j |
| refine Fin.cases ?_ (fun i => ?_) i <;> simp [Matrix.updateRow] |
| have ha := Matrix.det_updateRow_add (Matrix.vecCons (0 : Fin (n+1) → ℝ) R) 0 |
| (rawCovector P v) ((-1:ℝ) • rawCovector P w) |
| rw [Matrix.det_updateRow_smul (Matrix.vecCons (0 : Fin (n+1) → ℝ) R) 0 (-1:ℝ) |
| (rawCovector P w)] at ha |
| simpa only [←he,neg_one_smul,sub_eq_add_neg,neg_one_mul] using ha |
|
|
| 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 aestronglyMeasurable_normalizedDet {α : Type*} [MeasurableSpace α] |
| (μ : Measure α) {n : ℕ} {P : α → Matrix (Fin (n+1)) (Fin (n+1)) ℝ} |
| {R : α → Fin n → Fin (n+1) → ℝ} {v : α → Euc (n+1)} |
| (hP : Measurable P) (hR : AEStronglyMeasurable R μ) (hv : AEStronglyMeasurable v μ) : |
| AEStronglyMeasurable (fun x => normalizedDet (P x) (R x) (v x)) μ := by |
| have hQ := measurable_matrix_inverse _ |
| (measurable_matrix_sqrt.comp (measurable_matrix_inverse _ hP)) |
| have hc : Continuous (fun w : Matrix (Fin (n+1)) (Fin (n+1)) ℝ × |
| (Fin n → Fin (n+1) → ℝ) × Euc (n+1) => |
| Matrix.det (Matrix.vecCons (w.1.mulVec (WithLp.ofLp w.2.2)) w.2.1)) := by |
| fun_prop |
| have hh := hc.comp_aestronglyMeasurable (hQ.aestronglyMeasurable.prodMk (hR.prodMk hv)) |
| have hJ : Measurable (fun x => Real.sqrt (P x).det) := by fun_prop |
| exact (hh.aemeasurable.div hJ.aemeasurable).aestronglyMeasurable |
|
|
| lemma orientation_bound (a : ℝ) : abs (a/abs a) ≤ 1 := by |
| by_cases ha : a = 0 |
| · simp [ha] |
| · rw [abs_div,abs_abs,div_self (abs_ne_zero.mpr ha)] |
|
|
| lemma orientation_density (a : ℝ) : abs a*(a/abs a) = a := by |
| by_cases ha : a = 0 |
| · simp [ha] |
| · field_simp |
|
|
| end CAT0Fillings |
| end |
|
|
| end OAI |
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|
