| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.FibreContinuity |
| import OAI.Geometry.Borsuk.Projectors.FiniteOpenGlue |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open Set |
| lemma compact_target_hits {P X Y : Type*} [TopologicalSpace P] [CompactSpace P] |
| [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] |
| (φ : P → X) (hφ : Continuous φ) (K : ℝ × P → Y) |
| (hK : ContinuousOn K (Icc (0:ℝ) 1 ×ˢ univ)) (y : Y) : |
| IsCompact {x : X | ∃ p : P, φ p = x ∧ ∃ t ∈ Icc (0:ℝ) 1, K (t,p) = y} := by |
| have hC : IsCompact ((Icc (0:ℝ) 1 ×ˢ (univ : Set P)) ∩ K ⁻¹' {y}) := |
| ((isCompact_Icc.prod isCompact_univ) : IsCompact (Icc (0:ℝ) 1 ×ˢ (univ : Set P))).of_isClosed_subset |
| (hK.preimage_isClosed_of_isClosed (isClosed_Icc.prod isClosed_univ) isClosed_singleton) inter_subset_left |
| have he : {x : X | ∃ p : P, φ p = x ∧ ∃ t ∈ Icc (0:ℝ) 1, K (t,p) = y} = |
| (fun q : ℝ × P => φ q.2) '' ((Icc (0:ℝ) 1 ×ˢ (univ : Set P)) ∩ K ⁻¹' {y}) := by |
| ext x |
| constructor |
| · rintro ⟨p,hp,t,ht,hk⟩ |
| exact ⟨(t,p),⟨⟨ht,mem_univ _⟩,hk⟩,hp⟩ |
| · rintro ⟨⟨t,p⟩,⟨⟨ht,-⟩,hk⟩,hp⟩ |
| exact ⟨p,hp,t,ht,hk⟩ |
| rw [he] |
| exact hC.image (hφ.comp continuous_snd) |
| end BorsukNine |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| open Set |
| section LevelModel |
| variable {ν ι κ : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι] [DecidableEq ι] [Fintype κ] |
| local notation "E" => EuclideanSpace ℝ ν |
| local notation "M" => Matrix ν ν ℝ |
|
|
| noncomputable def balancedModel {f : E → ι → ℝ} {w : E} {r : ι → ℝ} |
| (p : levelFibre f w r) : traceSphere ν := |
| ⟨(1/2:ℝ) • rankOne p.val.1 - (1/2:ℝ) • p.val.2, |
| balancedFibre_mem p.property.1.1 p.property.2.1 p.property.2.2.1 p.property.2.2.2⟩ |
|
|
| omit [Fintype ι] [DecidableEq ι] in |
| omit [Nonempty ν] in |
| lemma balancedModel_continuous (f : E → ι → ℝ) (w : E) (r : ι → ℝ) : |
| Continuous (balancedModel (f := f) (w := w) (r := r)) := by |
| apply Continuous.subtype_mk |
| change Continuous (fun p : levelFibre f w r => (1/2:ℝ) • rankOne p.val.1 - (1/2:ℝ) • p.val.2) |
| exact ((continuous_const : Continuous (fun _ : levelFibre f w r => (1/2:ℝ))).smul |
| (rankOne_continuous.comp continuous_subtype_val.fst)).sub |
| ((continuous_const : Continuous (fun _ : levelFibre f w r => (1/2:ℝ))).smul continuous_subtype_val.snd) |
|
|
| noncomputable def modelChambers (f : E → ι → ℝ) (I : Finset ι) (w : E) (r : ι → ℝ) |
| (c : κ → E) (j : κ) : Set (levelFibre f w r) := |
| {p | p.val.1 ∈ interior (signChamber (activeVectors f I) (c j))} |
|
|
| omit [Nonempty ν] [Fintype ι] [DecidableEq ι] [Fintype κ] in |
| lemma modelChambers_open (f : E → ι → ℝ) (I : Finset ι) (w : E) (r : ι → ℝ) (c : κ → E) (j : κ) : |
| IsOpen (modelChambers f I w r c j) := |
| isOpen_interior.preimage continuous_subtype_val.fst |
|
|
| omit [Nonempty ν] [Fintype ι] [DecidableEq ι] [Fintype κ] in |
| lemma modelChambers_disjoint {f : E → ι → ℝ} {I : Finset ι} {w : E} {r : ι → ℝ} {c : κ → E} |
| (hd : Pairwise (fun i j => Disjoint (interior (signChamber (activeVectors f I) (c i))) |
| (interior (signChamber (activeVectors f I) (c j))))) : |
| Pairwise (fun i j => Disjoint (modelChambers f I w r c i) (modelChambers f I w r c j)) := by |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun p hp hq => Set.disjoint_left.mp (hd hij) hp hq) |
|
|
| omit [Nonempty ν] [Fintype ι] [DecidableEq ι] [Fintype κ] in |
| lemma modelChambers_cover {f : E → ι → ℝ} {I : Finset ι} {w : E} {r : ι → ℝ} {c : κ → E} |
| (hc : orientedLevel f w r ⊆ ⋃ j, interior (signChamber (activeVectors f I) (c j))) : |
| (⋃ j, modelChambers f I w r c j) = univ := by |
| apply eq_univ_of_forall |
| intro p |
| obtain ⟨j,hj⟩ := mem_iUnion.mp (hc p.property.1) |
| exact mem_iUnion.mpr ⟨j,hj⟩ |
|
|
| noncomputable def levelHomotopy (f : E → ι → ℝ) (I : Finset ι) (w : E) (r : ι → ℝ) |
| (c : κ → E) (g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) : |
| ℝ × levelFibre f w r → EuclideanSpace ℝ ↥(Iᶜ : Finset ι) := |
| finiteOpenGlue (fun j => {p | p.2 ∈ modelChambers f I w r c j}) |
| (fun j p => fibreHomotopy f I (c j) (g j) p.1 p.2.val.1 p.2.val.2) |
|
|
| omit [Nonempty ν] in |
| lemma levelHomotopy_eq {f : E → ι → ℝ} {I : Finset ι} {w : E} {r : ι → ℝ} {c : κ → E} |
| (hd : Pairwise (fun i j => Disjoint (interior (signChamber (activeVectors f I) (c i))) |
| (interior (signChamber (activeVectors f I) (c j))))) |
| (g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) {p : levelFibre f w r} {j : κ} |
| (hp : p.val.1 ∈ interior (signChamber (activeVectors f I) (c j))) (t : ℝ) : |
| levelHomotopy f I w r c g (t,p) = fibreHomotopy f I (c j) (g j) t p.val.1 p.val.2 := by |
| apply finiteOpenGlue_eq |
| · intro a b hab |
| exact Set.disjoint_left.mpr (fun q ha hb => Set.disjoint_left.mp (hd hab) ha hb) |
| · exact hp |
|
|
| lemma Admissible.levelHomotopy_continuousOn {f : E → ι → ℝ} (hf : Admissible f) |
| {I : Finset ι} (hI : I.Nonempty) {w : E} {r : ι → ℝ} (c : κ → E) |
| (hc : ∀ j, c j ∈ commonPositiveCone f I) (hcn : ∀ j, ‖c j‖ = 1) |
| (hd : Pairwise (fun i j => Disjoint (interior (signChamber (activeVectors f I) (c i))) |
| (interior (signChamber (activeVectors f I) (c j))))) |
| (hcover : orientedLevel f w r ⊆ ⋃ j, interior (signChamber (activeVectors f I) (c j))) |
| (g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hg : ∀ j, Continuous (g j)) : |
| ContinuousOn (levelHomotopy f I w r c g) (Icc (0:ℝ) 1 ×ˢ univ) := by |
| rw [← modelChambers_cover hcover] |
| apply finiteOpenGlue_continuousOn_prod (modelChambers_open f I w r c) (modelChambers_disjoint hd) |
| intro j |
| have hpar : Continuous (fun p : ℝ × levelFibre f w r => (p.1,p.2.val.1,p.2.val.2)) := |
| continuous_fst.prodMk ((continuous_subtype_val.comp continuous_snd).fst.prodMk |
| (continuous_subtype_val.comp continuous_snd).snd) |
| have hmap : MapsTo (fun p : ℝ × levelFibre f w r => (p.1,p.2.val.1,p.2.val.2)) |
| (Icc (0:ℝ) 1 ×ˢ modelChambers f I w r c j) |
| {p : ℝ × E × M | p.1 ∈ Icc 0 1 ∧ ‖p.2.1‖ = 1 ∧ |
| p.2.1 ∈ signChamber (activeVectors f I) (c j) ∧ p.2.2 ∈ matrixFibre p.2.1} := |
| fun p hp => ⟨hp.1,p.2.property.1.1,interior_subset hp.2,p.2.property.2⟩ |
| exact ContinuousOn.comp' (g := fun p : ℝ × E × M => fibreHomotopy f I (c j) (g j) p.1 p.2.1 p.2.2) |
| (f := fun p : ℝ × levelFibre f w r => (p.1,p.2.val.1,p.2.val.2)) |
| (hf.fibreHomotopy_continuousOn hI (hc j) (hcn j) (g j) (hg j)) hpar.continuousOn hmap |
|
|
| lemma Admissible.level_hits_compact {f : E → ι → ℝ} (hf : Admissible f) |
| {I : Finset ι} (hI : I.Nonempty) {w : E} {r : ι → ℝ} (c : κ → E) |
| (hc : ∀ j, c j ∈ commonPositiveCone f I) (hcn : ∀ j, ‖c j‖ = 1) |
| (hd : Pairwise (fun i j => Disjoint (interior (signChamber (activeVectors f I) (c i))) |
| (interior (signChamber (activeVectors f I) (c j))))) |
| (hcover : orientedLevel f w r ⊆ ⋃ j, interior (signChamber (activeVectors f I) (c j))) |
| (g : κ → M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hg : ∀ j, Continuous (g j)) |
| (z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) : |
| IsCompact {A : traceSphere ν | ∃ p : levelFibre f w r, balancedModel p = A ∧ |
| ∃ t ∈ Icc (0:ℝ) 1, levelHomotopy f I w r c g (t,p) = z} := by |
| let : CompactSpace (levelFibre f w r) := isCompact_iff_compactSpace.mp (hf.levelFibre_compact w r) |
| exact compact_target_hits balancedModel (balancedModel_continuous f w r) |
| (levelHomotopy f I w r c g) (hf.levelHomotopy_continuousOn hI c hc hcn hd hcover g hg) z |
|
|
| end LevelModel |
| end BorsukNine |
|
|
| end OAI |
|
|
