import Mathlib
import OAI.Computability.VertexCover.Analysis.CanonicalGradientValid

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory

theorem canonicalGradient_measurable (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    Measurable (Φ.canonicalGradient seed J c0 A hA) := by
  classical
  apply Measurable.ite (VertexCover.AffineSelector.regular_measurableSet _ _)
  · apply Measurable.of_eval; intro batchIndex
    apply Measurable.of_eval; intro weightIndex
    exact (ContinuousLinearMap.apply ℝ ℝ
      (Pi.single batchIndex (Pi.single weightIndex 1))).measurable.comp
      (measurable_fderiv ℝ (Φ.batchFunction seed J c0 A hA))
  · exact measurable_const

theorem canonicalGradient_ae_derivative (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (μ : Measure (Φ.BatchWeights J)) [μ.IsAddHaarMeasure] :
    ∀ᵐ s ∂μ, DifferentiableAt ℝ (Φ.batchFunction seed J c0 A hA) s ∧
      Φ.canonicalGradient seed J c0 A hA s =
        Φ.batchGradient J (fderiv ℝ (Φ.batchFunction seed J c0 A hA) s) := by
  classical
  filter_upwards [VertexCover.AffineSelector.ae_regular μ (Φ.affineSlope seed J)
    (Φ.affineOffset c0 (A := A))] with s hs
  obtain ⟨im, ip, hder⟩ := Φ.batchFunction_hasFDerivAt seed J c0 A hA s hs
  exact ⟨hder.differentiableAt, by simp [canonicalGradient, hs]⟩

theorem canonicalGradient_block_bound (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) (j : J) :
    ∑ k, |Φ.canonicalGradient seed J c0 A hA s j k| ≤ 1 := by
  obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s
  simp_rw [heq]
  exact Φ.hull_block_bound seed hG j

theorem canonicalGradient_unused (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J)
    (j : J) (k : Fin (Φ.WeightDimension d))
    (hk : ∀ a : (Φ.query seed j).LocalLabel, (Φ.query seed j).slot a ≠ k) :
    Φ.canonicalGradient seed J c0 A hA s j k = 0 := by
  obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s
  rw [heq]
  exact Φ.hull_unused seed hG j k hk

theorem canonicalGradient_listMass (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J)
    (L : Φ.PrivateLists d) :
    (∑ j : J, ∑ a ∈ (L (Φ.query seed j)).toFinset,
      |Φ.canonicalGradient seed J c0 A hA s j ((Φ.query seed j).slot a)|) ≤
        Φ.listHitMaximum L seed J := by
  classical
  obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s
  simp_rw [heq]
  exact (Finset.sum_coe_sort J (fun j =>
    ∑ a ∈ (L (Φ.query seed j)).toFinset, |G j ((Φ.query seed j).slot a)|)).trans_le
      (Φ.listMass_le_hitMaximum L seed J hG)

theorem canonicalGradient_query_invariant (Φ : LabelCover) {d : ℕ}
    (seed seed' : Φ.Seeds d) (J : Finset (Fin d))
    (hq : ∀ j : J, Φ.query seed j = Φ.query seed' j)
    (c0 : Φ.Coordinate d → ℝ) (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    Φ.canonicalGradient seed J c0 A hA = Φ.canonicalGradient seed' J c0 A hA := by
  have hS : Φ.batchSum seed J = Φ.batchSum seed' J := by
    funext s
    unfold batchSum
    apply Finset.sum_congr rfl
    intro j hj
    rw [hq j]
  have hC : Φ.batchSumCLM seed J = Φ.batchSumCLM seed' J := by
    ext s p
    exact congrFun (congrFun hS s) p
  have hA' : (Φ.affineSlope seed J (A := A)) = Φ.affineSlope seed' J := by
    funext i
    simp only [affineSlope, hC]
  have hf : Φ.batchFunction seed J c0 A hA = Φ.batchFunction seed' J c0 A hA := by
    funext s
    simp only [batchFunction, hS]
  funext s
  unfold canonicalGradient
  have hr : VertexCover.AffineSelector.Regular (Φ.affineSlope seed J)
      (Φ.affineOffset c0 (A := A)) s ↔
      VertexCover.AffineSelector.Regular (Φ.affineSlope seed' J)
      (Φ.affineOffset c0 (A := A)) s := by rw [hA']
  simp only [hf]
  split_ifs <;> first | rfl | (exfalso; tauto)

end VertexCover.LabelCover


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end OAI