| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.FibreHomotopy |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| open Set |
| section FibreContinuity |
| variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι] [DecidableEq ι] |
| local notation "E" => EuclideanSpace ℝ ν |
| local notation "M" => Matrix ν ν ℝ |
| local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ) |
|
|
| lemma Admissible.negativeCoordinates_continuousOn {f : E → ι → ℝ} (hf : Admissible f) |
| (I : Finset ι) : ContinuousOn (negativeCoordinates f I) {Q : M | 0 ≤ Q} := by |
| apply (PiLp.continuous_toLp 2 _).comp_continuousOn |
| apply continuousOn_pi.mpr |
| intro i |
| exact (continuous_apply i.val).comp_continuousOn hf.positiveExtension_continuousOn |
|
|
| omit [Nonempty ν] in |
| lemma centeredFibre_continuousOn (c : E) : |
| ContinuousOn (fun p : E × M => centeredFibre c p.1 p.2) {p | c+p.1 ≠ 0} := |
| householder_matrix_joint_continuousOn c |
|
|
| omit [Nonempty ν] in |
| lemma fibreTransport_continuousOn_comp {X : Type*} [TopologicalSpace X] |
| {S : Set X} (c : E) {u v : X → E} {Q : X → M} |
| (hu : ContinuousOn u S) (hv : ContinuousOn v S) (hQ : ContinuousOn Q S) |
| (hc : ∀ x ∈ S, c+u x ≠ 0 ∧ c+v x ≠ 0) : |
| ContinuousOn (fun x => fibreTransport c (u x) (v x) (Q x)) S := by |
| exact ContinuousOn.comp' (g := fun p : E × E × M => fibreTransport c p.1 p.2.1 p.2.2) |
| (f := fun x : X => (u x,v x,Q x)) (s := S) |
| (fibreTransport_continuousOn c) (hu.prodMk (hv.prodMk hQ)) hc |
|
|
| omit [DecidableEq ν] [Nonempty ν] in |
| lemma chamberPath_matrix_continuousOn {Z : Set E} {c : E} (hcz : c ∈ Z) |
| (hcs : c ∈ safeNormals Z) : |
| ContinuousOn (fun p : ℝ × E × M => chamberPath c p.2.1 p.1) |
| {p | p.1 ∈ Icc 0 1 ∧ p.2.1 ∈ signChamber Z c} := by |
| have hp : Continuous (fun p : ℝ × E × M => (p.1,p.2.1)) := |
| continuous_fst.prodMk continuous_snd.fst |
| exact ContinuousOn.comp' (g := fun p : ℝ × E => chamberPath c p.2 p.1) |
| (f := fun p : ℝ × E × M => (p.1,p.2.1)) |
| (chamberPath_continuousOn hcz hcs) hp.continuousOn (fun _ h => h) |
|
|
| omit [DecidableEq ι] in |
| omit [Nonempty ν] in |
| lemma transportedFibre_continuousOn {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι} |
| (hI : I.Nonempty) {c : E} (hc : c ∈ commonPositiveCone f I) : |
| ContinuousOn (fun p : ℝ × E × M => transportedFibre c p.1 p.2.1 p.2.2) |
| {p | p.1 ∈ Icc 0 1 ∧ p.2.1 ∈ signChamber (activeVectors f I) c} := by |
| have hcs := hf.commonPositiveCone_safe I hc |
| have hcz := commonPositiveCone_active hI hc |
| apply fibreTransport_continuousOn_comp c continuous_snd.fst.continuousOn |
| (chamberPath_matrix_continuousOn hcz hcs) continuous_snd.snd.continuousOn |
| intro p hp |
| exact ⟨chamber_add_ne_zero hcz hcs hp.2, |
| chamber_add_ne_zero hcz hcs (chamberPath_mem hcz hcs hp.2 hp.1)⟩ |
|
|
| lemma Admissible.fibreHomotopy_continuousOn {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι} |
| (hI : I.Nonempty) {c : E} (hc : c ∈ commonPositiveCone f I) (hcn : ‖c‖ = 1) |
| (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hg : Continuous g) : |
| ContinuousOn (fun p : ℝ × E × M => fibreHomotopy f I c g p.1 p.2.1 p.2.2) |
| {p | p.1 ∈ Icc 0 1 ∧ ‖p.2.1‖ = 1 ∧ |
| p.2.1 ∈ signChamber (activeVectors f I) c ∧ p.2.2 ∈ matrixFibre p.2.1} := by |
| have hcs := hf.commonPositiveCone_safe I hc |
| have hcz := commonPositiveCone_active hI hc |
| let S : Set (ℝ × E × M) := {p | p.1 ∈ Icc 0 1 ∧ ‖p.2.1‖ = 1 ∧ |
| p.2.1 ∈ signChamber (activeVectors f I) c ∧ p.2.2 ∈ matrixFibre p.2.1} |
| have hT : ContinuousOn (fun p : ℝ × E × M => transportedFibre c p.1 p.2.1 p.2.2) S := |
| (transportedFibre_continuousOn hf hI hc).mono (fun _ hp => ⟨hp.1,hp.2.2.1⟩) |
| have hC : ContinuousOn (fun p : ℝ × E × M => centeredFibre c p.2.1 p.2.2) S := |
| (centeredFibre_continuousOn c).comp continuous_snd.continuousOn |
| (fun _ hp => chamber_add_ne_zero hcz hcs hp.2.2.1) |
| have hET := (hf.negativeCoordinates_continuousOn I).comp hT |
| (fun _ hp => (transportedFibre_mem hf hI hc hcn hp.2.1 hp.2.2.1 hp.1 hp.2.2.2).1) |
| have hEC := (hf.negativeCoordinates_continuousOn I).comp hC |
| (fun _ hp => (centeredFibre_mem hcn hp.2.1 hp.2.2.2).1) |
| exact hET.add (continuous_fst.continuousOn.smul ((hg.comp_continuousOn hC).sub hEC)) |
|
|
| end FibreContinuity |
| end BorsukNine |
|
|
| end OAI |
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|
