lean/OAI/Computability/VertexCover/Analysis/BatchFunctionOwnLipschitz.lean

import Mathlib
import OAI.Computability.VertexCover.Reduction.SeparatorContinuousSumOdd
import OAI.Computability.VertexCover.Analysis.CanonicalGradientInvariant
import OAI.Computability.VertexCover.Analysis.BlockLawAc

namespace OAI

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namespace VertexCover.Cube
open MeasureTheory ProbabilityTheory
open scoped ENNReal

theorem integrable_continuous {ι : Type*} [Fintype ι] {f : (ι → ℝ) → ℝ}
    (hf : Continuous f) : Integrable f (law ι) := by
  rw [law_eq_restrict_volume]
  exact (hf.continuousOn.integrableOn_compact
    (isCompact_univ_pi (fun _ => isCompact_Icc))).smul_measure (by finiteness)

theorem integrable_continuous_block {ι κ : Type*} [Fintype ι] [Fintype κ]
    {f : (ι → κ → ℝ) → ℝ} (hf : Continuous f) : Integrable f (blockLaw ι κ) := by
  rw [blockLaw_eq_restrict_volume]
  exact (hf.continuousOn.integrableOn_compact
    (isCompact_univ_pi (fun _ => isCompact_univ_pi (fun _ => isCompact_Icc)))).smul_measure
      (by finiteness)

end VertexCover.Cube

namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory

noncomputable def batchLaw (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) :
    Measure (Φ.BatchWeights J) := VertexCover.Cube.blockLaw J (Fin (Φ.WeightDimension d))

instance batchLaw_probability (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) :
    IsProbabilityMeasure (Φ.batchLaw J) := by
  unfold batchLaw
  infer_instance

theorem canonicalGradient_ae_cube (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) :
    ∀ᵐ s ∂Φ.batchLaw J, 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
  let : (volume : Measure (Φ.BatchWeights J)).IsAddHaarMeasure := by
    change (Measure.pi (fun _ : J => (volume : Measure (Fin (Φ.WeightDimension d) → ℝ)))).IsAddHaarMeasure
    apply Measure.pi.isAddHaarMeasure
  exact (VertexCover.Cube.blockLaw_ac J (Fin (Φ.WeightDimension d))).ae_le
    (Φ.canonicalGradient_ae_derivative seed J c0 A hA volume)

theorem batchFunction_integrable (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) :
    Integrable (Φ.batchFunction seed J c0 A hA) (Φ.batchLaw J) :=
  VertexCover.Cube.integrable_continuous_block (Φ.batchFunction_continuous seed J c0 A hA)

theorem batchSum_update_sub (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (s : Φ.BatchWeights J) (j : J)
    (u v : Fin (Φ.WeightDimension d) → ℝ) :
    Φ.batchSum seed J (Function.update s j u) -
      Φ.batchSum seed J (Function.update s j v) = Φ.increment (Φ.query seed j) (u-v) := by
  classical
  simp only [batchSum, ← Finset.sum_sub_distrib]
  rw [Finset.sum_eq_single j]
  · simp only [Function.update_self, ← Φ.increment_sub]
  · intro k _ hkj
    rw [Function.update_of_ne hkj, Function.update_of_ne hkj, sub_self]
  · simp

theorem batchFunction_own_lipschitz (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) :
    LipschitzWith 1 (fun u => Φ.batchFunction seed J c0 A hA (Function.update s j u)) := by
  classical
  apply LipschitzWith.of_dist_le_mul
  intro u v
  simp only [NNReal.coe_one, one_mul, dist_eq_norm]
  apply (Φ.separator_lipschitz A hA _ _).trans
  rw [add_sub_add_left_eq_sub, Φ.batchSum_update_sub]
  apply Φ.increment_norm_le _ _ (_root_.norm_nonneg _)
  intro k
  simpa only [Pi.sub_apply, Real.norm_eq_abs] using norm_le_pi_norm (u-v) k

end VertexCover.LabelCover


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end OAI