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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
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