| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.SignedExtension |
| import OAI.Geometry.Borsuk.MatrixGeometry.Rank |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| section SpectralSupport |
| 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) |
|
|
| lemma Admissible.signedExtension_pos_iff {f : E → ι → ℝ} (hf : Admissible f) |
| (A : Matrix ν ν ℝ) (i : ι) : |
| 0 < signedExtension f A i ↔ 0 < positiveExtension f A⁺ i := by |
| have hp := hf.positiveExtension_nonneg A⁺ i |
| have hn := hf.positiveExtension_nonneg A⁻ i |
| have hpn := hf.positiveExtension_orthogonal (CFC.posPart_nonneg A) |
| (CFC.negPart_nonneg A) (CFC.posPart_mul_negPart A) i |
| change 0 < positiveExtension f A⁺ i - positiveExtension f A⁻ i ↔ _ |
| constructor |
| · intro h; linarith |
| · intro h |
| have hz := (mul_eq_zero.mp hpn).resolve_left (ne_of_gt h) |
| rwa [hz,sub_zero] |
|
|
| lemma Admissible.signedExtension_neg_iff {f : E → ι → ℝ} (hf : Admissible f) |
| (A : Matrix ν ν ℝ) (i : ι) : |
| signedExtension f A i < 0 ↔ 0 < positiveExtension f A⁻ i := by |
| have hp := hf.positiveExtension_nonneg A⁺ i |
| have hn := hf.positiveExtension_nonneg A⁻ i |
| have hpn := hf.positiveExtension_orthogonal (CFC.posPart_nonneg A) |
| (CFC.negPart_nonneg A) (CFC.posPart_mul_negPart A) i |
| change positiveExtension f A⁺ i - positiveExtension f A⁻ i < 0 ↔ _ |
| constructor |
| · intro h; linarith |
| · intro h |
| have hz := (mul_eq_zero.mp hpn).resolve_right (ne_of_gt h) |
| rw [hz,zero_sub] |
| exact neg_neg_of_pos h |
|
|
| lemma Admissible.signedExtension_positive_support {f : E → ι → ℝ} (hf : Admissible f) |
| (A : Matrix ν ν ℝ) (i : ι) : |
| 0 < signedExtension f A i ↔ ∃ v : E, |
| v ∈ LinearMap.range (op A⁺).toLinearMap ∧ ‖v‖ = 1 ∧ 0 < f v i := by |
| rw [hf.signedExtension_pos_iff,hf.positiveExtension_support (CFC.posPart_nonneg A)] |
|
|
| lemma Admissible.signedExtension_negative_support {f : E → ι → ℝ} (hf : Admissible f) |
| (A : Matrix ν ν ℝ) (i : ι) : |
| signedExtension f A i < 0 ↔ ∃ v : E, |
| v ∈ LinearMap.range (op A⁻).toLinearMap ∧ ‖v‖ = 1 ∧ 0 < f v i := by |
| rw [hf.signedExtension_neg_iff,hf.positiveExtension_support (CFC.negPart_nonneg A)] |
|
|
| lemma Admissible.positiveExtension_range_mono {f : E → ι → ℝ} (hf : Admissible f) |
| {P Q : Matrix ν ν ℝ} (hP : 0 ≤ P) (hQ : 0 ≤ Q) |
| (hPQ : LinearMap.range (op P).toLinearMap ≤ LinearMap.range (op Q).toLinearMap) |
| {i : ι} (hi : 0 < positiveExtension f P i) : 0 < positiveExtension f Q i := by |
| obtain ⟨v,hv,hvn,hvi⟩ := (hf.positiveExtension_support hP i).mp hi |
| exact (hf.positiveExtension_support hQ i).mpr ⟨v,hPQ hv,hvn,hvi⟩ |
|
|
| omit [Nonempty ν] in |
| lemma rankOne_range (u : E) (hu : ‖u‖ = 1) : |
| LinearMap.range (op (rankOne u)).toLinearMap = Submodule.span ℝ {u} := by |
| apply le_antisymm |
| · rintro v ⟨w,rfl⟩ |
| rw [ContinuousLinearMap.coe_coe,rankOne_action] |
| exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton u)) |
| · apply Submodule.span_le.mpr |
| intro v hv |
| have hvu : v = u := Set.mem_singleton_iff.mp hv |
| subst v |
| refine ⟨u,?_⟩ |
| simp [rankOne_action,hu] |
|
|
| lemma Admissible.signedExtension_range_labels {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} {I : Finset ι} |
| (hI : ∀ i ∉ I, signedExtension f A i ≤ 0) |
| {v : E} (hv : v ∈ LinearMap.range (op A⁺).toLinearMap) (hvn : ‖v‖ = 1) |
| {i : ι} (hi : i ∉ I) : f v i = 0 := by |
| apply le_antisymm _ (hf.nonneg v hvn i) |
| by_contra hn |
| have hp : 0 < f v i := lt_of_not_ge hn |
| exact (not_lt_of_ge (hI i hi)) ((hf.signedExtension_positive_support A i).mpr ⟨v,hv,hvn,hp⟩) |
|
|
| omit [DecidableEq ν] [Nonempty ν] in |
|
|
| lemma subspace_eq_span_of_inner_separates (W : Submodule ℝ E) (w u : E) |
| (hu : u ∈ W) (huw : ⟪w,u⟫ ≠ 0) |
| (hsep : ∀ v ∈ W, ⟪w,v⟫ = 0 → v = 0) : W = Submodule.span ℝ {u} := by |
| apply le_antisymm |
| · intro v hv |
| have hz : v - (⟪w,v⟫ / ⟪w,u⟫) • u = 0 := hsep _ |
| (W.sub_mem hv (W.smul_mem _ hu)) (by rw [inner_sub_right,inner_smul_right]; field_simp; ring) |
| rw [sub_eq_zero] at hz |
| rw [hz] |
| exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton u)) |
| · exact Submodule.span_le.mpr (by simpa using hu) |
| |
| lemma Admissible.positive_range_separated {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} {I : Finset ι} {w : E} (hw : ‖w‖ = 1) |
| (hwi : ∀ i ∈ I, 0 < f w i) |
| (hI : ∀ i ∉ I, signedExtension f A i ≤ 0) |
| {v : E} (hv : v ∈ LinearMap.range (op A⁺).toLinearMap) (hwv : ⟪w,v⟫ = 0) : v = 0 := by |
| by_contra hn |
| let u : E := ‖v‖⁻¹ • v |
| have hu : ‖u‖ = 1 := by simp [u,norm_smul,norm_inv,hn] |
| have hur : u ∈ LinearMap.range (op A⁺).toLinearMap := Submodule.smul_mem _ _ hv |
| obtain ⟨i,hi⟩ := hf.support_nonempty hu |
| have hip : 0 < f u i := by simpa [positiveSupport] using hi |
| by_cases hiI : i ∈ I |
| · exact hf.orthogonal w u hw hu (by simp [u,inner_smul_right,hwv]) i ⟨hwi i hiI,hip⟩ |
| · have hz := hf.signedExtension_range_labels hI hur hu hiI |
| rw [hz] at hip |
| exact (lt_irrefl 0) hip |
|
|
| end SpectralSupport |
| end BorsukNine |
|
|
| end OAI |
|
|
