| import Mathlib |
| import OAI.Geometry.Borsuk.Support.ActiveChambers |
| import OAI.Geometry.Borsuk.Support.NegativeFibre |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator Topology |
| open Set |
| section PurityChart |
| variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι] |
| local notation "E" => EuclideanSpace ℝ ν |
| local notation "M" => Matrix ν ν ℝ |
| local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ) |
|
|
| noncomputable def purityChart (f : E → ι → ℝ) (I : Finset ι) (w c : E) : Set M := |
| {A | IsUnit A ∧ HasPositiveLine A ∧ op A⁺ w ≠ 0 ∧ 0 < Matrix.trace A⁻ ∧ |
| lineRepresentative w A ∈ commonPositiveCone f I ∧ |
| lineRepresentative w A ∈ interior (signChamber (activeVectors f I) c)} |
|
|
| omit [Nonempty ν] in |
| lemma Admissible.purityChart_isOpen {f : E → ι → ℝ} (hf : Admissible f) |
| (I : Finset ι) (w c : E) {X : Type*} [TopologicalSpace X] |
| (A : X → M) (hA : Continuous A) (hSA : ∀ x, IsSelfAdjoint (A x)) : |
| IsOpen (A ⁻¹' purityChart f I w c) := by |
| have hop : Continuous (fun x => op (A x)⁺ w) := |
| (((op).toAlgEquiv.toLinearEquiv.toLinearMap.continuous_of_finiteDimensional).comp |
| (matrix_posPart_continuousOn.comp_continuous hA hSA)).clm_apply continuous_const |
| have hopen : IsOpen {x | op (A x)⁺ w ≠ 0} := (isClosed_eq hop continuous_const).isOpen_compl |
| have hrep := lineRepresentative_continuousOn A hA hSA w |
| have hpre : IsOpen ({x | op (A x)⁺ w ≠ 0} ∩ |
| (fun x => lineRepresentative w (A x)) ⁻¹' |
| (commonPositiveCone f I ∩ interior (signChamber (activeVectors f I) c))) := |
| hrep.isOpen_inter_preimage hopen ((hf.commonPositiveCone_isOpen I).inter isOpen_interior) |
| have hn : IsOpen {x | 0 < Matrix.trace (A x)⁻} := isOpen_lt continuous_const |
| (matrix_trace_continuous.comp (matrix_negPart_continuousOn.comp_continuous hA hSA)) |
| have he : A ⁻¹' purityChart f I w c = |
| ({x | IsUnit (A x) ∧ HasPositiveLine (A x)} ∩ {x | 0 < Matrix.trace (A x)⁻}) ∩ |
| ({x | op (A x)⁺ w ≠ 0} ∩ (fun x => lineRepresentative w (A x)) ⁻¹' |
| (commonPositiveCone f I ∩ interior (signChamber (activeVectors f I) c))) := by |
| ext x |
| simp only [purityChart,mem_preimage,mem_ofPred_eq,mem_inter_iff] |
| tauto |
| rw [he] |
| exact ((positive_line_isOpen A hA hSA).inter hn).inter hpre |
|
|
| omit [Fintype ι] in |
| omit [Nonempty ν] in |
| lemma purityChart_unit {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M} |
| (hA : A ∈ purityChart f I w c) : ‖lineRepresentative w A‖ = 1 := |
| lineRepresentative_norm hA.2.2.1 |
|
|
| omit [Fintype ι] in |
| lemma purityChart_fibre {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M} |
| (hA : A ∈ purityChart f I w c) : negativeFibre A ∈ matrixFibre (lineRepresentative w A) := |
| negativeFibre_mem hA.2.1 hA.2.2.1 hA.2.2.2.1 |
|
|
| omit [Fintype ι] in |
| lemma purityChart_fibre_unit {f : E → ι → ℝ} {I : Finset ι} {w c : E} {A : M} |
| (hSA : IsSelfAdjoint A) (hA : A ∈ purityChart f I w c) : |
| IsUnit (rankOne (lineRepresentative w A) + negativeFibre A) := by |
| have hf := purityChart_fibre hA |
| apply fibre_unit_of_split_unit hf.1 hf.2.2 (hasPositiveLine_trace_pos hA.2.1) hA.2.2.2.1 |
| rw [← negativeFibre_splitting hSA hA.2.1 hA.2.2.1 hA.2.2.2.1] |
| exact hA.1 |
|
|
| end PurityChart |
| end BorsukNine |
|
|
| end OAI |
|
|
