import Mathlib
import OAI.Computability.VertexCover.Analysis.SeparatorBoundCube
import OAI.Computability.VertexCover.Reduction.PrivateProjectionGroupLower

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators

theorem spliceSeeds_at_inside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (e : ↥(internalPairs J)) :
    Φ.spliceSeeds J frozen hidden e = hidden e := by
  simp only [spliceSeeds, dite_eq_left e.2]

theorem spliceSeeds_at_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (e : PositionPair d)
    (he : e ∉ internalPairs J) : Φ.spliceSeeds J frozen hidden e = frozen e := by
  simp only [spliceSeeds, dite_eq_right he]

noncomputable def privateFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (j : Fin d) (seed : Φ.Seeds d) : Finset (Φ.Seeds d) := by
  classical
  exact Finset.univ.filter (fun t => Φ.privateSeedKey J j t = Φ.privateSeedKey J j seed)

@[simp] theorem mem_privateFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (j : Fin d) (seed t : Φ.Seeds d) : t ∈ Φ.privateFiber J j seed ↔
      Φ.privateSeedKey J j t = Φ.privateSeedKey J j seed := by
  classical
  simp only [privateFiber, Finset.mem_filter, Finset.mem_univ, true_and]

theorem spliceSeeds_restrict_of_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen seed : Φ.Seeds d) (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e) :
    Φ.spliceSeeds J frozen (fun e : ↥(internalPairs J) => seed e) = seed := by
  funext e
  by_cases he : e ∈ internalPairs J
  · simp only [spliceSeeds, dite_eq_left he]
  · simpa only [spliceSeeds, dite_eq_right he] using ho e he

noncomputable def privateFiberEquiv (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
    (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e) :
    Φ.ownFiber J frozen j (Φ.query seed j) ≃ Φ.privateFiber J j seed := by
  classical
  have reconstruct : ∀ t : Φ.privateFiber J j seed,
      Φ.spliceSeeds J frozen (fun e : ↥(internalPairs J) => t.1 e) = t.1 := by
    intro t
    apply Φ.spliceSeeds_restrict_of_outside
    intro e he
    have ht := (Φ.privateSeedKey_eq_iff J j t.1 seed).mp ((Φ.mem_privateFiber J j seed t.1).mp t.2)
    exact (ho e he).trans (ht.1 e he).symm
  exact {
    toFun := fun hidden => ⟨Φ.spliceSeeds J frozen hidden.1, by
      apply (Φ.mem_privateFiber J j seed _).mpr
      apply (Φ.privateSeedKey_eq_iff J j _ _).mpr
      exact ⟨fun e he => (Φ.spliceSeeds_at_outside J frozen hidden.1 e he).trans (ho e he),
        (Φ.mem_ownFiber J frozen j (Φ.query seed j) hidden.1).mp hidden.2⟩⟩
    invFun := fun t => ⟨fun e => t.1 e, by
      apply (Φ.mem_ownFiber J frozen j (Φ.query seed j) _).mpr
      rw [reconstruct t]
      exact ((Φ.privateSeedKey_eq_iff J j t.1 seed).mp
        ((Φ.mem_privateFiber J j seed t.1).mp t.2)).2⟩
    left_inv := fun hidden => by
      apply Subtype.ext
      funext e
      exact Φ.spliceSeeds_at_inside J frozen hidden.1 e
    right_inv := fun t => Subtype.ext (reconstruct t) }

theorem fiberAverage_eq_finiteMean {S T : Type*} [Fintype S]
    (q : S → T) (f : S → ℝ) (i : T) (ts : Finset S)
    (ht : ∀ s, s ∈ ts ↔ q s = i) :
    VertexCover.Average.fiberAverage q (fun _ => f) i =
      VertexCover.finiteMean (fun s : ts => f s) := by
  classical
  have he : Finset.univ.filter (fun s => q s = i) = ts := by
    ext s
    simpa only [Finset.mem_filter, Finset.mem_univ, true_and] using (ht s).symm
  unfold VertexCover.Average.fiberAverage
  rw [he, VertexCover.Average.finiteMean_finset ts f]

theorem privateSeedProjection_fiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
    (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
    (f : Φ.RestrictionSpace d) (s : Φ.RestrictionWeights d) :
    Φ.privateSeedProjection J j f seed s =
      VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j (Φ.query seed j) =>
        f (Φ.spliceSeeds J frozen hidden.1) s) := by
  classical
  change conditionMap _ f seed s = _
  rw [conditionMap_apply_function]
  calc
    _ = VertexCover.finiteMean (fun t : Φ.privateFiber J j seed => f t.1 s) := by
      exact fiberAverage_eq_finiteMean (Φ.privateSeedKey J j) (fun t => f t s)
        (Φ.privateSeedKey J j seed) (Φ.privateFiber J j seed)
        (fun t => Φ.mem_privateFiber J j seed t)
    _ = _ := (VertexCover.finiteMean_equiv (Φ.privateFiberEquiv J frozen j seed ho)
      (fun t : Φ.privateFiber J j seed => f t.1 s)).symm

theorem otherWeightProjection_apply (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (j : J) (f : Φ.RestrictionSpace d)
    (seed : Φ.Seeds d) (s : Φ.RestrictionWeights d) :
    Φ.otherWeightProjection J j f seed s =
      ∫ t, f seed (splice J s (Function.update t j (s j)))
        ∂CompactCube.blockLaw J (Fin (Φ.WeightDimension d)) := by
  change productMap (fun i => (average (CompactCube.law (Fin (Φ.WeightDimension d))) i).lift)
    (J.erase j).toList f seed s = _
  rw [productMap_lift, product_average_apply _ _ (J.erase j).nodup_toList,
    Finset.toList_toFinset]
  exact integral_plug_erase _ J j (f seed) s

theorem realWeights_splice (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (s : Φ.RestrictionWeights d)
    (t : J → Fin (Φ.WeightDimension d) → CompactCube.Interval) :
    CompactCube.realWeights (splice J s t) =
      Φ.spliceWeights J (CompactCube.realWeights s) (CompactCube.realWeights t) := by
  classical
  funext i k
  by_cases hi : i ∈ J
  · simp only [splice, spliceWeights, CompactCube.realWeights, dite_eq_left hi]
  · simp only [splice, spliceWeights, CompactCube.realWeights, dite_eq_right hi]

theorem otherWeightProjection_separator_splice (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (hidden : Φ.HiddenSeeds J) (s : Φ.RestrictionWeights d) :
    Φ.otherWeightProjection J j (Φ.restrictionFunction A hA)
      (Φ.spliceSeeds J frozen hidden) s =
      ∫ other, Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J
        (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
        (Function.update other j (CompactCube.realWeights s j)) ∂Φ.batchLaw J := by
  classical
  rw [Φ.otherWeightProjection_apply]
  simp only [restrictionFunction_apply, Φ.realWeights_splice, CompactCube.realWeights_update]
  simp_rw [← Φ.batchFunction_is_restriction J frozen (CompactCube.realWeights s) A hA]
  apply CompactCube.integral_realWeights (f := fun other : Φ.BatchWeights J =>
    Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J
      (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
      (Function.update other j (CompactCube.realWeights s j)))
  exact ((Φ.batchFunction_continuous _ J _ A hA).comp
    (continuous_id.update j continuous_const)).measurable.aestronglyMeasurable

theorem privateAverageProjection_separator (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
    (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
    Φ.privateAverageProjection J j (Φ.restrictionFunction A hA) seed s =
      Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
        j (Φ.query seed j) (CompactCube.realWeights s j) := by
  change Φ.privateSeedProjection J j
    (Φ.otherWeightProjection J j (Φ.restrictionFunction A hA)) seed s = _
  rw [Φ.privateSeedProjection_fiber J frozen j seed ho]
  unfold ownMean
  congr 1
  funext hidden
  exact Φ.otherWeightProjection_separator_splice J frozen j A hA hidden.1 s

theorem outsideSum_update (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) (j : J)
    (x : Fin (Φ.WeightDimension d) → ℝ) :
    Φ.outsideSum J frozen (Function.update s j x) = Φ.outsideSum J frozen s := by
  classical
  apply Finset.sum_congr rfl
  intro k hk
  have hkj : k ≠ j := by
    intro h
    exact (Finset.mem_compl.mp hk) (h ▸ j.2)
  rw [Function.update_of_ne hkj]

theorem privateProjection_separator (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
    (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
    Φ.privateProjection J j (Φ.restrictionFunction A hA) seed s =
      Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
        j (Φ.query seed j) (CompactCube.realWeights s j) -
      ∫ x, Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA
        j (Φ.query seed j) x ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)) := by
  classical
  change Φ.privateAverageProjection J j (Φ.restrictionFunction A hA) seed s -
    Φ.weightProjection j (Φ.privateAverageProjection J j (Φ.restrictionFunction A hA)) seed s = _
  rw [Φ.privateAverageProjection_separator J frozen j seed ho,
    Φ.weightProjection_apply]
  simp_rw [Φ.privateAverageProjection_separator J frozen j seed ho,
    CompactCube.realWeights_update, Φ.outsideSum_update J frozen _ j,
    Function.update_self]
  congr 1
  exact CompactCube.integral_coe
    (Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.measurable.aestronglyMeasurable

theorem privateProjection_coordinate_variance (Φ : LabelCover) {d : ℕ}
    (J : Finset (Fin d)) (frozen : Φ.Seeds d) (j : J) (seed : Φ.Seeds d)
    (ho : ∀ e, e ∉ internalPairs J → frozen e = seed e)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.RestrictionWeights d) :
    (∫ x, (Φ.privateProjection J j (Φ.restrictionFunction A hA) seed
      (Function.update s j x))^2 ∂CompactCube.law (Fin (Φ.WeightDimension d))) =
      ProbabilityTheory.variance (Φ.ownMean J frozen
        (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j))
        (VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := by
  classical
  simp_rw [Φ.privateProjection_separator J frozen j seed ho,
    CompactCube.realWeights_update, Φ.outsideSum_update J frozen _ j,
    Function.update_self]
  rw [ProbabilityTheory.variance_eq_integral
    (Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.measurable.aemeasurable]
  apply CompactCube.integral_coe (f := fun x =>
    (Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j) x -
      ∫ y, Φ.ownMean J frozen (Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA j (Φ.query seed j) y
        ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)))^2)
  exact (((Φ.ownMean_lipschitz J frozen _ A hA j (Φ.query seed j)).continuous.sub
    continuous_const).pow 2).measurable.aestronglyMeasurable

end VertexCover.LabelCover


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end OAI