Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.FibreContinuity
import OAI.Geometry.Borsuk.Support.PurityChart
import OAI.Geometry.Borsuk.MatrixGeometry.BlockVector

namespace OAI

namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
open Set
section MixedHomotopy
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 "Y" => EuclideanSpace ℝ ι
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)

omit [DecidableEq ν] [Nonempty ν] [DecidableEq ι] in
lemma Admissible.cone_unit_positive {f : E → ι → ℝ} (_hf : Admissible f)
    {I : Finset ι} {u : E} (hu : ‖u‖ = 1) (hc : u ∈ commonPositiveCone f I) :
    ∀ i ∈ I, 0 < f u i := by
  intro i hi
  simpa only [homogeneousLift_unit f hu] using hc i hi

omit [DecidableEq ν] [Nonempty ν] [DecidableEq ι] in
lemma Admissible.maximal_cone_zero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
    (hmax : ∀ u : E, ‖u‖ = 1 → I ⊆ positiveSupport f u → positiveSupport f u = I)
    {u : E} (hu : ‖u‖ = 1) (hc : u ∈ commonPositiveCone f I) :
    ∀ i ∉ I, f u i = 0 := by
  have he : positiveSupport f u = I := hmax u hu (fun i hi =>
    (mem_positiveSupport f u i).mpr (hf.cone_unit_positive hu hc i hi))
  intro i hi
  apply le_antisymm _ (hf.nonneg u hu i)
  by_contra hp
  have hh := (mem_positiveSupport f u i).mpr (lt_of_not_ge hp)
  rw [he] at hh
  exact hi hh

omit [Nonempty ν] in
lemma traceSphere_weights {A : M} (hA : A ∈ traceSphere ν) :
    Matrix.trace A⁺ + Matrix.trace A⁻ = 1 := by
  rw [← Matrix.trace_add,CFC.posPart_add_negPart A hA.1]; exact hA.2

noncomputable def mixedHomotopy (f : E → ι → ℝ) (I : Finset ι) (w c : E)
    (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (t : ℝ) (A : M) : Y :=
  blockVector I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
    (fibreHomotopy f I c g t (lineRepresentative w A) (negativeFibre A))

lemma Admissible.mixedHomotopy_zero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
    (hmax : ∀ u : E, ‖u‖ = 1 → I ⊆ positiveSupport f u → positiveSupport f u = I)
    {w c : E} (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) {A : M}
    (hSA : IsSelfAdjoint A) (hA : A ∈ purityChart f I w c) :
    mixedHomotopy f I w c g 0 A = WithLp.toLp 2 (signedExtension f A) := by
  have hu := purityChart_unit hA
  have hQ := purityChart_fibre hA
  have hl := hasPositiveLine_trace_pos hA.2.1
  have hm := hA.2.2.2.1
  have hs := (hf.signedExtension_splitting hSA hu hl (positive_line_representative hA.2.1 hA.2.2.1)).2.2
  have hneg : positiveExtension f A⁻ = Matrix.trace A⁻ • positiveExtension f (negativeFibre A) := by
    calc
      positiveExtension f A⁻ = positiveExtension f (Matrix.trace A⁻ • negativeFibre A) :=
        congrArg (positiveExtension f) (negativeFibre_smul (ne_of_gt hm)).symm
      _ = _ := positiveExtension_smul_nonneg f hQ.1 hm.le
  have hpos := hf.cone_unit_positive hu hA.2.2.2.2.1
  have hzero := hf.maximal_cone_zero hmax hu hA.2.2.2.2.1
  have hsafe := hf.commonPositiveCone_safe I hA.2.2.2.2.1
  apply PiLp.ext
  intro i
  change blockVector I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
    (fibreHomotopy f I c g 0 (lineRepresentative w A) (negativeFibre A)) i = signedExtension f A i
  rw [fibreHomotopy_zero f I c g hu,hs,hneg]
  by_cases hi : i ∈ I
  · rw [blockVector_in _ _ _ _ _ hi]
    have hz := hf.positiveExtension_safe_zero hsafe hQ.1 hQ.2.2 hi
    simp only [Pi.sub_apply,Pi.smul_apply,smul_eq_mul,hz,mul_zero,sub_zero]
  · let j : ↥(Iᶜ : Finset ι) := ⟨i,Finset.mem_compl.mpr hi⟩
    have hj := blockVector_out I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
      (negativeCoordinates f I (negativeFibre A)) j
    rw [hj]
    simp only [Pi.sub_apply,Pi.smul_apply,smul_eq_mul,hzero i hi,mul_zero,zero_sub]
    exact neg_mul _ _

omit [Nonempty ν] in
lemma Admissible.mixedHomotopy_nonzero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
    (hI : I.Nonempty) {w c : E} (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι))
    (t : ℝ) {A : M} (hA : A ∈ purityChart f I w c) : mixedHomotopy f I w c g t A ≠ 0 :=
  blockVector_nonzero hI (hasPositiveLine_trace_pos hA.2.1)
    (hf.cone_unit_positive (purityChart_unit hA) hA.2.2.2.2.1) _

lemma Admissible.mixedHomotopy_continuousOn {f : E → ι → ℝ} (hf : Admissible f)
    {I : Finset ι} (hI : I.Nonempty) {w c : E} (hc : c ∈ commonPositiveCone f I) (hcn : ‖c‖ = 1)
    (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hg : Continuous g) :
    ContinuousOn (fun p : ℝ × traceSphere ν => mixedHomotopy f I w c g p.1 p.2.val)
      (Icc (0:ℝ) 1 ×ˢ {A : traceSphere ν | A.val ∈ purityChart f I w c}) := by
  let S : Set (ℝ × traceSphere ν) := Icc (0:ℝ) 1 ×ˢ {A : traceSphere ν | A.val ∈ purityChart f I w c}
  have hA : Continuous (fun p : ℝ × traceSphere ν => p.2.val) := continuous_subtype_val.comp continuous_snd
  have hSA : ∀ p : ℝ × traceSphere ν, IsSelfAdjoint p.2.val := fun p => p.2.property.1
  have hu : ContinuousOn (fun p : ℝ × traceSphere ν => lineRepresentative w p.2.val) S :=
    (lineRepresentative_continuousOn _ hA hSA w).mono (fun p hp => hp.2.2.2.1)
  have hQ : ContinuousOn (fun p : ℝ × traceSphere ν => negativeFibre p.2.val) S :=
    (negativeFibre_continuousOn _ hA hSA).mono (fun p hp => ne_of_gt hp.2.2.2.2.1)
  have htuple : ContinuousOn (fun p : ℝ × traceSphere ν =>
      (p.1,lineRepresentative w p.2.val,negativeFibre p.2.val)) S :=
    continuous_fst.continuousOn.prodMk (hu.prodMk hQ)
  have hF : ContinuousOn (fun p : ℝ × traceSphere ν =>
      fibreHomotopy f I c g p.1 (lineRepresentative w p.2.val) (negativeFibre p.2.val)) S :=
    ContinuousOn.comp' (g := fun p : ℝ × E × M => fibreHomotopy f I c g p.1 p.2.1 p.2.2)
      (f := fun p : ℝ × traceSphere ν => (p.1,lineRepresentative w p.2.val,negativeFibre p.2.val))
      (hf.fibreHomotopy_continuousOn hI hc hcn g hg) htuple
      (fun p hp => ⟨hp.1,purityChart_unit hp.2,interior_subset hp.2.2.2.2.2.2,purityChart_fibre hp.2⟩)
  have hl : Continuous (fun p : ℝ × traceSphere ν => Matrix.trace p.2.val⁺) :=
    matrix_trace_continuous.comp (matrix_posPart_continuousOn.comp_continuous hA hSA)
  have hm : Continuous (fun p : ℝ × traceSphere ν => Matrix.trace p.2.val⁻) :=
    matrix_trace_continuous.comp (matrix_negPart_continuousOn.comp_continuous hA hSA)
  have hfC : Continuous f := continuous_pi (fun i => (hf.smooth i).continuous)
  exact blockVector_continuousOn I hl.continuousOn (hfC.comp_continuousOn hu) hm.continuousOn hF
end MixedHomotopy
end BorsukNine
end OAI

	import Mathlib
	import OAI.Geometry.Borsuk.MatrixGeometry.FibreContinuity
	import OAI.Geometry.Borsuk.Support.PurityChart
	import OAI.Geometry.Borsuk.MatrixGeometry.BlockVector

	namespace OAI

	namespace BorsukNine
	open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
	open Set
	section MixedHomotopy
	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 "Y" => EuclideanSpace ℝ ι
	local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)

	omit [DecidableEq ν] [Nonempty ν] [DecidableEq ι] in
	lemma Admissible.cone_unit_positive {f : E → ι → ℝ} (_hf : Admissible f)
	{I : Finset ι} {u : E} (hu : ‖u‖ = 1) (hc : u ∈ commonPositiveCone f I) :
	∀ i ∈ I, 0 < f u i := by
	intro i hi
	simpa only [homogeneousLift_unit f hu] using hc i hi

	omit [DecidableEq ν] [Nonempty ν] [DecidableEq ι] in
	lemma Admissible.maximal_cone_zero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
	(hmax : ∀ u : E, ‖u‖ = 1 → I ⊆ positiveSupport f u → positiveSupport f u = I)
	{u : E} (hu : ‖u‖ = 1) (hc : u ∈ commonPositiveCone f I) :
	∀ i ∉ I, f u i = 0 := by
	have he : positiveSupport f u = I := hmax u hu (fun i hi =>
	(mem_positiveSupport f u i).mpr (hf.cone_unit_positive hu hc i hi))
	intro i hi
	apply le_antisymm _ (hf.nonneg u hu i)
	by_contra hp
	have hh := (mem_positiveSupport f u i).mpr (lt_of_not_ge hp)
	rw [he] at hh
	exact hi hh

	omit [Nonempty ν] in
	lemma traceSphere_weights {A : M} (hA : A ∈ traceSphere ν) :
	Matrix.trace A⁺ + Matrix.trace A⁻ = 1 := by
	rw [← Matrix.trace_add,CFC.posPart_add_negPart A hA.1]; exact hA.2

	noncomputable def mixedHomotopy (f : E → ι → ℝ) (I : Finset ι) (w c : E)
	(g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (t : ℝ) (A : M) : Y :=
	blockVector I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
	(fibreHomotopy f I c g t (lineRepresentative w A) (negativeFibre A))

	lemma Admissible.mixedHomotopy_zero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
	(hmax : ∀ u : E, ‖u‖ = 1 → I ⊆ positiveSupport f u → positiveSupport f u = I)
	{w c : E} (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) {A : M}
	(hSA : IsSelfAdjoint A) (hA : A ∈ purityChart f I w c) :
	mixedHomotopy f I w c g 0 A = WithLp.toLp 2 (signedExtension f A) := by
	have hu := purityChart_unit hA
	have hQ := purityChart_fibre hA
	have hl := hasPositiveLine_trace_pos hA.2.1
	have hm := hA.2.2.2.1
	have hs := (hf.signedExtension_splitting hSA hu hl (positive_line_representative hA.2.1 hA.2.2.1)).2.2
	have hneg : positiveExtension f A⁻ = Matrix.trace A⁻ • positiveExtension f (negativeFibre A) := by
	calc
	positiveExtension f A⁻ = positiveExtension f (Matrix.trace A⁻ • negativeFibre A) :=
	congrArg (positiveExtension f) (negativeFibre_smul (ne_of_gt hm)).symm
	_ = _ := positiveExtension_smul_nonneg f hQ.1 hm.le
	have hpos := hf.cone_unit_positive hu hA.2.2.2.2.1
	have hzero := hf.maximal_cone_zero hmax hu hA.2.2.2.2.1
	have hsafe := hf.commonPositiveCone_safe I hA.2.2.2.2.1
	apply PiLp.ext
	intro i
	change blockVector I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
	(fibreHomotopy f I c g 0 (lineRepresentative w A) (negativeFibre A)) i = signedExtension f A i
	rw [fibreHomotopy_zero f I c g hu,hs,hneg]
	by_cases hi : i ∈ I
	· rw [blockVector_in _ _ _ _ _ hi]
	have hz := hf.positiveExtension_safe_zero hsafe hQ.1 hQ.2.2 hi
	simp only [Pi.sub_apply,Pi.smul_apply,smul_eq_mul,hz,mul_zero,sub_zero]
	· let j : ↥(Iᶜ : Finset ι) := ⟨i,Finset.mem_compl.mpr hi⟩
	have hj := blockVector_out I (Matrix.trace A⁺) (f (lineRepresentative w A)) (Matrix.trace A⁻)
	(negativeCoordinates f I (negativeFibre A)) j
	rw [hj]
	simp only [Pi.sub_apply,Pi.smul_apply,smul_eq_mul,hzero i hi,mul_zero,zero_sub]
	exact neg_mul _ _

	omit [Nonempty ν] in
	lemma Admissible.mixedHomotopy_nonzero {f : E → ι → ℝ} (hf : Admissible f) {I : Finset ι}
	(hI : I.Nonempty) {w c : E} (g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι))
	(t : ℝ) {A : M} (hA : A ∈ purityChart f I w c) : mixedHomotopy f I w c g t A ≠ 0 :=
	blockVector_nonzero hI (hasPositiveLine_trace_pos hA.2.1)
	(hf.cone_unit_positive (purityChart_unit hA) hA.2.2.2.2.1) _

	lemma Admissible.mixedHomotopy_continuousOn {f : E → ι → ℝ} (hf : Admissible f)
	{I : Finset ι} (hI : I.Nonempty) {w c : E} (hc : c ∈ commonPositiveCone f I) (hcn : ‖c‖ = 1)
	(g : M → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (hg : Continuous g) :
	ContinuousOn (fun p : ℝ × traceSphere ν => mixedHomotopy f I w c g p.1 p.2.val)
	(Icc (0:ℝ) 1 ×ˢ {A : traceSphere ν \| A.val ∈ purityChart f I w c}) := by
	let S : Set (ℝ × traceSphere ν) := Icc (0:ℝ) 1 ×ˢ {A : traceSphere ν \| A.val ∈ purityChart f I w c}
	have hA : Continuous (fun p : ℝ × traceSphere ν => p.2.val) := continuous_subtype_val.comp continuous_snd
	have hSA : ∀ p : ℝ × traceSphere ν, IsSelfAdjoint p.2.val := fun p => p.2.property.1
	have hu : ContinuousOn (fun p : ℝ × traceSphere ν => lineRepresentative w p.2.val) S :=
	(lineRepresentative_continuousOn _ hA hSA w).mono (fun p hp => hp.2.2.2.1)
	have hQ : ContinuousOn (fun p : ℝ × traceSphere ν => negativeFibre p.2.val) S :=
	(negativeFibre_continuousOn _ hA hSA).mono (fun p hp => ne_of_gt hp.2.2.2.2.1)
	have htuple : ContinuousOn (fun p : ℝ × traceSphere ν =>
	(p.1,lineRepresentative w p.2.val,negativeFibre p.2.val)) S :=
	continuous_fst.continuousOn.prodMk (hu.prodMk hQ)
	have hF : ContinuousOn (fun p : ℝ × traceSphere ν =>
	fibreHomotopy f I c g p.1 (lineRepresentative w p.2.val) (negativeFibre p.2.val)) S :=
	ContinuousOn.comp' (g := fun p : ℝ × E × M => fibreHomotopy f I c g p.1 p.2.1 p.2.2)
	(f := fun p : ℝ × traceSphere ν => (p.1,lineRepresentative w p.2.val,negativeFibre p.2.val))
	(hf.fibreHomotopy_continuousOn hI hc hcn g hg) htuple
	(fun p hp => ⟨hp.1,purityChart_unit hp.2,interior_subset hp.2.2.2.2.2.2,purityChart_fibre hp.2⟩)
	have hl : Continuous (fun p : ℝ × traceSphere ν => Matrix.trace p.2.val⁺) :=
	matrix_trace_continuous.comp (matrix_posPart_continuousOn.comp_continuous hA hSA)
	have hm : Continuous (fun p : ℝ × traceSphere ν => Matrix.trace p.2.val⁻) :=
	matrix_trace_continuous.comp (matrix_negPart_continuousOn.comp_continuous hA hSA)
	have hfC : Continuous f := continuous_pi (fun i => (hf.smooth i).continuous)
	exact blockVector_continuousOn I hl.continuousOn (hfC.comp_continuousOn hu) hm.continuousOn hF

	end MixedHomotopy
	end BorsukNine

	end OAI