| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.MatrixKernel |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace ContDiff |
| open scoped Manifold |
| open MeasureTheory Set Metric |
| open scoped ENNReal Pointwise Matrix |
| section SignedExtension |
| open scoped MatrixOrder Matrix.Norms.L2Operator |
| variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι] |
| local notation "E" => EuclideanSpace ℝ ν |
| local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ) |
|
|
| omit [Nonempty ν] in |
| lemma matrix_images_orthogonal {P N : Matrix ν ν ℝ} (hP : 0 ≤ P) (hPN : P * N = 0) |
| (x y : E) : ⟪op P x, op N y⟫ = 0 := by |
| rw [← ContinuousLinearMap.adjoint_inner_right, ← ContinuousLinearMap.star_eq_adjoint] |
| have hs : star (op P) = op P := by rw [← map_star, hP.star_eq] |
| rw [hs] |
| change ⟪x, (op P * op N) y⟫ = 0 |
| rw [← map_mul, hPN, map_zero] |
| simp |
|
|
| lemma Admissible.positiveExtension_orthogonal {f : E → ι → ℝ} (hf : Admissible f) |
| {P N : Matrix ν ν ℝ} (hP : 0 ≤ P) (hN : 0 ≤ N) (hPN : P * N = 0) (i : ι) : |
| positiveExtension f P i * positiveExtension f N i = 0 := by |
| apply mul_eq_zero.mpr |
| by_contra hn |
| push Not at hn |
| have hp := lt_of_le_of_ne (hf.positiveExtension_nonneg P i) (Ne.symm hn.1) |
| have hn' := lt_of_le_of_ne (hf.positiveExtension_nonneg N i) (Ne.symm hn.2) |
| obtain ⟨v, ⟨x, rfl⟩, hv, hx⟩ := (hf.positiveExtension_support hP i).mp hp |
| obtain ⟨w, ⟨y, rfl⟩, hw, hy⟩ := (hf.positiveExtension_support hN i).mp hn' |
| exact hf.orthogonal _ _ hv hw (matrix_images_orthogonal hP hPN x y) i ⟨hx, hy⟩ |
|
|
| noncomputable def signedExtension (f : E → ι → ℝ) (A : Matrix ν ν ℝ) : ι → ℝ := |
| positiveExtension f A⁺ - positiveExtension f A⁻ |
|
|
| omit [Nonempty ν] [Fintype ι] in |
| lemma signedExtension_odd (f : E → ι → ℝ) (A : Matrix ν ν ℝ) : |
| signedExtension f (-A) = -signedExtension f A := by |
| simp [signedExtension, CFC.posPart_neg, CFC.negPart_neg] |
|
|
| omit [Nonempty ν] [Fintype ι] in |
| lemma signedExtension_smul_nonneg (f : E → ι → ℝ) (A : Matrix ν ν ℝ) |
| {t : ℝ} (ht : 0 ≤ t) : signedExtension f (t • A) = t • signedExtension f A := by |
| rw [signedExtension, CFC.posPart_smul_of_nonneg ht, CFC.negPart_smul_of_nonneg ht, |
| positiveExtension_smul_nonneg f (CFC.posPart_nonneg A) ht, |
| positiveExtension_smul_nonneg f (CFC.negPart_nonneg A) ht, ← smul_sub] |
| rfl |
|
|
| lemma Admissible.signedExtension_sum {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) : |
| ∑ i, signedExtension f A i = Matrix.trace A := by |
| classical |
| simp only [signedExtension, Pi.sub_apply, Finset.sum_sub_distrib, |
| hf.positiveExtension_sum (CFC.posPart_nonneg A), hf.positiveExtension_sum (CFC.negPart_nonneg A)] |
| rw [← Matrix.trace_sub, CFC.posPart_sub_negPart A hA] |
|
|
| lemma Admissible.signedExtension_abs_sum {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) : |
| ∑ i, |signedExtension f A i| = Matrix.trace (CFC.abs A) := by |
| classical |
| have hcoord (i : ι) : |signedExtension f A i| = |
| positiveExtension f A⁺ i + positiveExtension f A⁻ i := by |
| have hp := hf.positiveExtension_nonneg A⁺ i |
| have hn := hf.positiveExtension_nonneg A⁻ i |
| have hprod := hf.positiveExtension_orthogonal (CFC.posPart_nonneg A) |
| (CFC.negPart_nonneg A) (CFC.posPart_mul_negPart A) i |
| rcases mul_eq_zero.mp hprod with hz | hz |
| · simp [signedExtension, hz, abs_of_nonneg hn] |
| · simp [signedExtension, hz, abs_of_nonneg hp] |
| simp_rw [hcoord] |
| rw [Finset.sum_add_distrib, hf.positiveExtension_sum (CFC.posPart_nonneg A), |
| hf.positiveExtension_sum (CFC.negPart_nonneg A), ← Matrix.trace_add, |
| CFC.posPart_add_negPart A hA] |
|
|
| omit [Nonempty ν] in |
| lemma matrix_posPart_continuousOn : |
| ContinuousOn (fun A : Matrix ν ν ℝ => A⁺) {A | IsSelfAdjoint A} := by |
| have hc : Continuous (fun A : Matrix ν ν ℝ => (1 / 2 : ℝ) • (CFC.abs A + A)) := |
| (continuous_const (y := (1 / 2 : ℝ))).smul (CFC.continuous_abs.add continuous_id) |
| apply hc.continuousOn.congr |
| intro A hA |
| change A⁺ = (1 / 2 : ℝ) • (CFC.abs A + A) |
| rw [CFC.abs_add_self A hA, two_smul, smul_add] |
| module |
|
|
| omit [Nonempty ν] in |
| lemma matrix_negPart_continuousOn : |
| ContinuousOn (fun A : Matrix ν ν ℝ => A⁻) {A | IsSelfAdjoint A} := by |
| have hc : Continuous (fun A : Matrix ν ν ℝ => (1 / 2 : ℝ) • (CFC.abs A - A)) := |
| (continuous_const (y := (1 / 2 : ℝ))).smul (CFC.continuous_abs.sub continuous_id) |
| apply hc.continuousOn.congr |
| intro A hA |
| change A⁻ = (1 / 2 : ℝ) • (CFC.abs A - A) |
| rw [CFC.abs_sub_self A hA, two_smul, smul_add] |
| module |
|
|
| lemma Admissible.signedExtension_continuousOn {f : E → ι → ℝ} (hf : Admissible f) : |
| ContinuousOn (signedExtension f) {A : Matrix ν ν ℝ | IsSelfAdjoint A} := by |
| apply ContinuousOn.sub |
| · exact hf.positiveExtension_continuousOn.comp matrix_posPart_continuousOn |
| (fun A _ => CFC.posPart_nonneg A) |
| · exact hf.positiveExtension_continuousOn.comp matrix_negPart_continuousOn |
| (fun A _ => CFC.negPart_nonneg A) |
|
|
| end SignedExtension |
|
|
| end BorsukNine |
|
|
| end OAI |
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|
