import Mathlib
import OAI.Computability.VertexCover.Reduction.SeparatorStarOscillation
import OAI.Computability.VertexCover.Analysis.SeparatorRoundingErrorAe

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory

theorem query_splice_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (j : Fin d) (hj : j ∉ J) :
    Φ.query (Φ.spliceSeeds J frozen hidden) j = Φ.query frozen j := by
  classical
  apply Φ.query_eq_of_questions
  · intro k h
    have hn : (⟨(j,k),h⟩ : PositionPair d) ∉ internalPairs J := by
      simp [internalPairs, hj]
    simp [spliceSeeds, hn]
  · intro k h
    have hn : (⟨(k,j),h⟩ : PositionPair d) ∉ internalPairs J := by
      simp [internalPairs, hj]
    simp [spliceSeeds, hn]

noncomputable def spliceWeights (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Fin d → Fin (Φ.WeightDimension d) → ℝ) (s : Φ.BatchWeights J) :
      Fin d → Fin (Φ.WeightDimension d) → ℝ := by
  classical
  exact fun j => if h : j ∈ J then s ⟨j,h⟩ else frozen j

noncomputable def outsideSum (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
    Φ.Coordinate d → ℝ := ∑ j ∈ Jᶜ, Φ.increment (Φ.query frozen j) (weights j)

theorem continuousSum_splice (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ)
    (hidden : Φ.HiddenSeeds J) (s : Φ.BatchWeights J) :
    Φ.continuousSum (Φ.spliceSeeds J frozen hidden) (Φ.spliceWeights J weights s) =
      Φ.outsideSum J frozen weights + Φ.batchSum (Φ.spliceSeeds J frozen hidden) J s := by
  classical
  unfold continuousSum outsideSum batchSum
  rw [← Finset.sum_add_sum_compl J]
  rw [add_comm]
  congr 1
  · apply Finset.sum_congr rfl
    intro j hj
    have hnj : j ∉ J := Finset.mem_compl.mp hj
    simp only [spliceWeights, dite_eq_right hnj, Φ.query_splice_outside J frozen hidden j hnj]
  · rw [← Finset.sum_coe_sort]
    apply Finset.sum_congr rfl
    intro j _
    simp only [spliceWeights, dite_eq_left j.2]

theorem batchFunction_is_restriction (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (hidden : Φ.HiddenSeeds J) (s : Φ.BatchWeights J) :
    Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J (Φ.outsideSum J frozen weights) A hA s =
      Φ.separator A hA (Φ.continuousSum (Φ.spliceSeeds J frozen hidden)
        (Φ.spliceWeights J weights s)) := by
  rw [Φ.continuousSum_splice]
  rfl

theorem separator_bound_on_cube (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (seed : Φ.Seeds d)
    (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) (hs : Φ.InWeightCube s) :
    |Φ.separator A hA (Φ.continuousSum seed s)| ≤ d := by
  have hzero : Φ.continuousSum seed 0 = 0 := by
    change Φ.continuousSumCLM seed 0 = 0
    exact map_zero _
  have hl := Φ.separator_lipschitz A hA (Φ.continuousSum seed s) 0
  have hn := Φ.continuousSum_close seed s 0 (by norm_num : (0:ℝ)≤1)
    (by simpa only [InWeightCube, Pi.zero_apply, sub_zero] using hs)
  rw [hzero] at hn
  simpa only [Φ.separator_zero, sub_zero, mul_one] using hl.trans hn

end VertexCover.LabelCover


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end OAI