lean/OAI/Probability/GaussianPropeller/CapReduction.lean

import OAI.Probability.GaussianPropeller.CapNumeric

namespace OAI

open MeasureTheory ProbabilityTheory
open scoped ENNReal
open scoped RealInnerProductSpace
open scoped RealInnerProductSpace
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal RealInnerProductSpace
open Filter
open scoped Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
open scoped RealInnerProductSpace
open Set Filter
open scoped Topology RealInnerProductSpace
open scoped NNReal
open Set Filter
open scoped Topology RealInnerProductSpace NNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology BigOperators
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
open Metric
open MeasureTheory ProbabilityTheory Set
open scoped RealInnerProductSpace ENNReal

namespace GaussianPropeller.Reduction
open Coords
variable {d k n : ℕ} [NeZero k]

omit [NeZero k] in
lemma closedCell_cone [NeZero k] (z : Fin k → Space d) (i : Fin k) (r:ℝ) (hr:0<r)
    (x:Space d) : r•x ∈ closedCell z i ↔ x∈closedCell z i := by
  simp only [closedCell, mem_ofPred_eq,inner_smul_right]
  exact forall_congr' (fun j => mul_le_mul_iff_right₀ hr)

lemma coordinates_law {V : Submodule ℝ (Space d)}
    (b : OrthonormalBasis (Fin (n+1)) ℝ V) :
    (gaussian d).map (fun x => b.repr (V.orthogonalProjectionOnto x)) = gaussian (n+1) := by
  have ho : Orthonormal ℝ (fun j => (b j : Space d)) := by
    constructor
    · intro j; exact b.orthonormal.1 j
    · intro j l hjl; exact b.orthonormal.2 hjl
  have he : (fun x => b.repr (V.orthogonalProjectionOnto x)) =
      WithLp.toLp 2 ∘ (fun x => fun j => ⟪(b j:Space d),x⟫) := by
    funext x; ext j
    simp only [Function.comp_apply,WithLp.ofLp_toLp,OrthonormalBasis.repr_apply_apply,
      Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left]
  unfold gaussian
  rw [he,← Measure.map_map (PiLp.continuous_toLp 2 _).measurable (by fun_prop),
    Analytic.map_orthonormal_stdGaussian _ ho,map_pi_eq_stdGaussian]

lemma exists_span_cone {z : Fin k → Space d} (hsum : ∑ j,z j=0)
    (hneg : Pairwise (fun j l=>⟪z j,z l⟫<0)) (i:Fin k) (hzi:z i≠0)
    (hdim : k-1=n+1) (hcen : centroid (closedCell z i)=z i) :
    ∃ K : Set (Space (n+1)), MeasurableSet K ∧
      (∀ r:ℝ, 0<r → ∀ x, r•x∈K ↔ x∈K) ∧
      (gaussian (n+1)).real K=(gaussian d).real (closedCell z i) ∧
      (∫ x, K.indicator (fun x : Space (n+1)=> x 0) x ∂gaussian (n+1))=‖z i‖ := by
  let V := Submodule.span ℝ (range z)
  let zv : Fin k → V := fun j=>⟨z j,Submodule.subset_span (mem_range_self j)⟩
  have hnorm : 0<‖z i‖ := norm_pos_iff.mpr hzi
  let e : V := ‖z i‖⁻¹ • zv i
  have he : ‖e‖=1 := by simp [e,zv,norm_smul,hnorm.ne']
  obtain ⟨b,hb⟩ := exists_basis_zero
    (show Module.finrank ℝ V=n+1 by rw [Gram.finrank_span z hneg hsum i,Fintype.card_fin,hdim]) e he
  let T : Space d → Space (n+1) := fun x=> b.repr (V.orthogonalProjectionOnto x)
  let w : Fin k → Space (n+1) := fun j=>b.repr (zv j)
  let K := closedCell w i
  have hT : Continuous T := b.repr.continuous.comp V.orthogonalProjectionOnto.continuous
  have hm : (gaussian d).map T=gaussian (n+1) := coordinates_law b
  have hs : ∀ j x, ⟪w j,T x⟫=⟪z j,x⟫ := by
    intro j x
    dsimp [w,T]
    rw [b.repr.inner_map_map,Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left]
  have hp : T ⁻¹' K=closedCell z i := by
    ext x
    simp only [K,closedCell,mem_preimage,mem_ofPred_eq,hs]
  refine ⟨K,measurableSet_closedCell w i,closedCell_cone w i,?_,?_⟩
  · rw [← hm,measureReal_def,Measure.map_apply hT.measurable (measurableSet_closedCell w i),hp]
    rfl
  have ht0 : ∀ x, T x 0=‖z i‖⁻¹*⟪z i,x⟫ := by
    intro x
    dsimp [T]
    rw [b.repr_apply_apply,Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left,hb]
    simp only [e,Submodule.coe_smul,zv,real_inner_smul_left]
  have hi := integral_map (μ:=gaussian d) (φ:=T)
    (f:=K.indicator (fun x:Space (n+1)=>x 0)) hT.aemeasurable
    (((by fun_prop : Continuous (fun x:Space (n+1)=>x 0)).stronglyMeasurable.indicator
      (measurableSet_closedCell w i)).aestronglyMeasurable)
  rw [hm] at hi
  rw [hi]
  have heq : (fun x => K.indicator (fun x:Space (n+1)=>x 0) (T x)) =
      (closedCell z i).indicator (fun x=>‖z i‖⁻¹*⟪z i,x⟫) := by
    funext x
    have hh : T x∈K ↔ x∈closedCell z i := Set.ext_iff.mp hp x
    by_cases hx : x∈closedCell z i
    · simp only [indicator_of_mem hx,indicator_of_mem (hh.mpr hx),ht0]
    · simp only [indicator_of_notMem hx,indicator_of_notMem (mt hh.mp hx)]
  rw [heq,integral_indicator (measurableSet_closedCell z i),integral_const_mul]
  have hinner : (∫ x in closedCell z i, ⟪z i,x⟫ ∂gaussian d) =
      ⟪z i,centroid (closedCell z i)⟫ := Pair.integral_inner_eq_inner_integral _ _
  rw [hinner]
  change ‖z i‖⁻¹*⟪z i,centroid (closedCell z i)⟫=‖z i‖
  rw [hcen,real_inner_self_eq_norm_sq]
  field_simp

lemma minimal_span_cone {A:Fin k→Set (Space d)} (hA:MinimalOptimal A)
    (hpos:0<value A) (hactive:∀ j,gaussian d (A j)≠0) (i:Fin k) (hdim:k-1=n+1) :
    ∃ K : Set (Space (n+1)), MeasurableSet K ∧
      (∀ r:ℝ, 0<r → ∀ x, r•x∈K ↔ x∈K) ∧
      (gaussian (n+1)).real K=(gaussian d).real (A i) ∧
      (∫ x, K.indicator (fun x : Space (n+1)=> x 0) x ∂gaussian (n+1))=‖centroid (A i)‖ := by
  obtain ⟨K,hK,hcone,hP,hM⟩ := exists_span_cone (sum_centroid_eq_zero hA.1.1)
    (fun j l hjl => negative_inner_of_minimal hA hjl (hactive j) (hactive l))
    i (active_centroid_ne_zero hA hpos (hactive i)) hdim
    (centroid_closedCell hA hpos (hactive i))
  refine ⟨K,hK,hcone,hP.trans ?_,hM⟩
  rw [measureReal_def,measureReal_def,measure_congr (active_eq_closedCell_ae hA hpos (hactive i))]

end GaussianPropeller.Reduction

end OAI