lean/OAI/Geometry/Borsuk/SphereGeometry/IntegralLift.lean

import Mathlib
import OAI.Geometry.Borsuk.SphereGeometry.HomogeneousLift
import OAI.Geometry.Borsuk.SphereGeometry.SphereMeasure

namespace OAI

namespace BorsukNine
open scoped RealInnerProductSpace ContDiff
open scoped Manifold
open MeasureTheory Set Metric
open scoped ENNReal Pointwise Matrix
section IntegralLift
variable {ν ι : Type*} [Fintype ν] [Nonempty ν] [Fintype ι]
local notation "E" => EuclideanSpace ℝ ν

noncomputable def integralLift (f : E → ι → ℝ) (T : E →L[ℝ] E) : ι → ℝ :=
  fun i => (Fintype.card ν : ℝ) *
    ∫ x : UnitSphere E, homogeneousLift f (T x) i ∂sphereProbability E

omit [Nonempty ν] in
lemma Admissible.integrand_continuous {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) :
    Continuous (fun x : UnitSphere E => homogeneousLift f (T x) i) :=
  (continuous_apply i).comp (hf.lift_continuous.comp (T.continuous.comp continuous_subtype_val))

lemma Admissible.integrand_integrable {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) :
    Integrable (fun x : UnitSphere E => homogeneousLift f (T x) i) (sphereProbability E) :=
  (hf.integrand_continuous T i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)

omit [Nonempty ν] in
lemma Admissible.integralLift_nonneg {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) : 0 ≤ integralLift f T i :=
  mul_nonneg (Nat.cast_nonneg _) (integral_nonneg (fun _ => hf.lift_nonneg _ i))

lemma Admissible.integralLift_sum {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) : ∑ i, integralLift f T i =
      (Fintype.card ν : ℝ) * ∫ x : UnitSphere E, ‖T x‖ ^ 2 ∂sphereProbability E := by
  classical
  simp only [integralLift, ← Finset.mul_sum]
  rw [← integral_finsetSum _ (fun i _ => hf.integrand_integrable T i)]
  simp only [hf.lift_sum]

lemma Admissible.integralLift_pos_iff_sphere {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) :
    0 < integralLift f T i ↔ ∃ x : UnitSphere E, 0 < homogeneousLift f (T x) i := by
  have hc : (0 : ℝ) < Fintype.card ν := Nat.cast_pos.mpr Fintype.card_pos
  rw [integralLift, mul_pos_iff_of_pos_left hc]
  constructor
  · intro h
    by_contra hn
    push Not at hn
    have he : (fun x : UnitSphere E => homogeneousLift f (T x) i) = 0 := by
      funext x
      exact le_antisymm (hn x) (hf.lift_nonneg _ i)
    rw [he] at h
    change 0 < ∫ _ : UnitSphere E, (0 : ℝ) ∂sphereProbability E at h
    simp only [integral_zero, lt_self_iff_false] at h
  · rintro ⟨x, hx⟩
    exact integral_pos_of_integrable_nonneg_nonzero (hf.integrand_continuous T i)
      (hf.integrand_integrable T i) (fun _ => hf.lift_nonneg _ i) hx.ne'

lemma Admissible.integralLift_continuous {f : E → ι → ℝ} (hf : Admissible f) :
    Continuous (integralLift f) := by
  apply continuous_pi
  intro i
  apply continuous_const.mul
  have hc : Continuous (fun p : (E →L[ℝ] E) × UnitSphere E =>
      homogeneousLift f (p.1 p.2) i) :=
    (continuous_apply i).comp (hf.lift_continuous.comp
      (continuous_fst.clm_apply (continuous_subtype_val.comp continuous_snd)))
  have hi := continuous_parametric_integral_of_continuous
    (μ := sphereProbability E)
    (f := fun (T : E →L[ℝ] E) (x : UnitSphere E) => homogeneousLift f (T x) i)
    hc isCompact_univ
  simpa only [setIntegral_univ] using hi

lemma Admissible.integralLift_pos_iff {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) :
    0 < integralLift f T i ↔ ∃ x : E, 0 < homogeneousLift f (T x) i := by
  rw [hf.integralLift_pos_iff_sphere]
  constructor
  · rintro ⟨x, hx⟩
    exact ⟨x, hx⟩
  · rintro ⟨x, hx⟩
    have hx0 : x ≠ 0 := by
      rintro rfl
      simp at hx
    have hn : 0 < ‖x‖⁻¹ := inv_pos.mpr (norm_pos_iff.mpr hx0)
    refine ⟨⟨‖x‖⁻¹ • x, mem_sphere_zero_iff_norm.mpr (norm_smul_inv_norm (𝕜 := ℝ) hx0)⟩, ?_⟩
    simp only [map_smul, homogeneousLift_smul_pos f _ hn, Pi.smul_apply, smul_eq_mul]
    exact mul_pos (sq_pos_of_pos hn) hx

lemma Admissible.integralLift_support {f : E → ι → ℝ} (hf : Admissible f)
    (T : E →L[ℝ] E) (i : ι) :
    0 < integralLift f T i ↔ ∃ v : E,
      v ∈ LinearMap.range T.toLinearMap ∧ ‖v‖ = 1 ∧ 0 < f v i := by
  rw [hf.integralLift_pos_iff]
  constructor
  · rintro ⟨x, hx⟩
    have hx0 : T x ≠ 0 := by
      intro hz
      simp only [hz, homogeneousLift_zero, Pi.zero_apply, lt_self_iff_false] at hx
    refine ⟨‖T x‖⁻¹ • T x, ⟨‖T x‖⁻¹ • x, by simp⟩,
      norm_smul_inv_norm (𝕜 := ℝ) hx0, ?_⟩
    exact (mul_pos_iff_of_pos_left (sq_pos_of_pos (norm_pos_iff.mpr hx0))).mp hx
  · rintro ⟨v, ⟨x, rfl⟩, hv, hx⟩
    refine ⟨x, ?_⟩
    rw [homogeneousLift_unit f (show ‖T x‖ = 1 from hv)]
    exact hx

lemma Admissible.integralLift_orthogonal {f : E → ι → ℝ} (hf : Admissible f)
    (T S : E →L[ℝ] E) (hTS : ∀ x y, ⟪T x, S y⟫ = 0) (i : ι) :
    integralLift f T i * integralLift f S i = 0 := by
  apply mul_eq_zero.mpr
  by_contra hn
  push Not at hn
  have ht := lt_of_le_of_ne (hf.integralLift_nonneg T i) (Ne.symm hn.1)
  have hs := lt_of_le_of_ne (hf.integralLift_nonneg S i) (Ne.symm hn.2)
  obtain ⟨x, hx⟩ := (hf.integralLift_pos_iff T i).mp ht
  obtain ⟨y, hy⟩ := (hf.integralLift_pos_iff S i).mp hs
  have hp := hf.lift_orthogonal (hTS x y) i
  exact (ne_of_gt (mul_pos hx hy)) hp

omit [Nonempty ν] [Fintype ι] in
lemma integralLift_smul_nonneg (f : E → ι → ℝ) (T : E →L[ℝ] E)
    {t : ℝ} (ht : 0 ≤ t) : integralLift f (t • T) = t ^ 2 • integralLift f T := by
  ext i
  simp only [integralLift, smul_apply, homogeneousLift_smul_nonneg f _ ht,
    Pi.smul_apply, smul_eq_mul, integral_const_mul]
  ring

end IntegralLift

end BorsukNine

end OAI