import Mathlib
import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated

namespace OAI

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namespace VertexCover.LabelCover

theorem norm_sum_le (Φ : LabelCover) {d : ℕ} {ι : Type*}
    (T : Finset ι) (f : ι → Φ.Coordinate d → ℝ) :
    Φ.compatibilityNorm (∑ i ∈ T, f i) ≤ ∑ i ∈ T, Φ.compatibilityNorm (f i) := by
  classical
  induction T using Finset.induction_on with
  | empty =>
    simp only [Finset.sum_empty]
    exact (Φ.norm_zero d).le
  | @insert i T hi ih =>
    simp only [Finset.sum_insert hi]
    exact (Φ.norm_add_le _ _).trans (add_le_add_right ih _)

theorem increment_sub (Φ : LabelCover) {d : ℕ} (i : Φ.Query d)
    (s t : Fin (Φ.WeightDimension d) → ℝ) :
    Φ.increment i (s-t) = Φ.increment i s - Φ.increment i t := by
  have hn : Φ.increment i (-t) = -Φ.increment i t := by
    ext p
    change Φ.increment i (fun k => -t k) p = -Φ.increment i t p
    have h := congrFun (Φ.increment_mul i (-1) t) p
    simpa only [neg_one_mul] using h
  rw [sub_eq_add_neg, Φ.increment_add, hn, sub_eq_add_neg]

noncomputable def continuousSum (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Φ.Coordinate d → ℝ :=
  ∑ j, Φ.increment (Φ.query seed j) (s j)

theorem sumVector_eq_continuousSum (Φ : LabelCover) {d : ℕ} (v : Φ.Vertex d) :
    Φ.sumVector v = Φ.continuousSum v.1 (fun j k => (gridRational d (v.2 j k) : ℝ)) := rfl

theorem continuousSum_neg (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
    Φ.continuousSum seed (-s) = -Φ.continuousSum seed s := by
  unfold continuousSum
  rw [← Finset.sum_neg_distrib]
  apply Finset.sum_congr rfl
  intro j _
  ext p
  change Φ.increment (Φ.query seed j) (fun k => -s j k) p =
    -Φ.increment (Φ.query seed j) (s j) p
  have h := congrFun (Φ.increment_mul (Φ.query seed j) (-1) (s j)) p
  simpa only [neg_one_mul] using h

theorem continuousSum_sub_same_seed (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
    Φ.continuousSum seed s - Φ.continuousSum seed t = Φ.continuousSum seed (s-t) := by
  simp only [continuousSum, ← Finset.sum_sub_distrib, Pi.sub_apply, increment_sub]

theorem continuousSum_close (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) {b : ℝ} (hb : 0 ≤ b)
    (hclose : ∀ j k, |s j k - t j k| ≤ b) :
    Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed t) ≤ d*b := by
  rw [Φ.continuousSum_sub_same_seed]
  apply (Φ.norm_sum_le _ _).trans
  calc
    (∑ j, Φ.compatibilityNorm (Φ.increment (Φ.query seed j) ((s-t) j))) ≤
        ∑ _j : Fin d, b := by
      apply Finset.sum_le_sum
      intro j _
      exact Φ.increment_norm_le _ _ hb (hclose j)
    _ = d*b := by simp

def InWeightCube (Φ : LabelCover) {d : ℕ}
    (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Prop :=
  ∀ j k, |s j k| ≤ 1

theorem continuousSum_change_blocks (Φ : LabelCover) {d : ℕ}
    (seed seed' : Φ.Seeds d) (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ)
    (hs : Φ.InWeightCube s) (ht : Φ.InWeightCube t) (B : Finset (Fin d))
    (heq : ∀ j ∉ B, Φ.query seed j = Φ.query seed' j ∧ s j = t j) :
    Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed' t) ≤ 2*B.card := by
  classical
  unfold continuousSum
  rw [← Finset.sum_sub_distrib]
  apply (Φ.norm_sum_le _ _).trans
  calc
    (∑ j, Φ.compatibilityNorm
        (Φ.increment (Φ.query seed j) (s j) - Φ.increment (Φ.query seed' j) (t j))) ≤
        ∑ j : Fin d, if j ∈ B then (2 : ℝ) else 0 := by
      apply Finset.sum_le_sum
      intro j _
      by_cases hj : j ∈ B
      · rw [ite_eq_left hj]
        have hb := Φ.norm_sub_triangle (Φ.increment (Φ.query seed j) (s j)) 0
          (Φ.increment (Φ.query seed' j) (t j))
        have hh1 := Φ.increment_norm_le (Φ.query seed j) (s j) (by norm_num : (0:ℝ) ≤ 1) (hs j)
        have hh2 := Φ.increment_norm_le (Φ.query seed' j) (t j) (by norm_num : (0:ℝ) ≤ 1) (ht j)
        simp only [sub_zero, zero_sub, Φ.norm_neg] at hb
        linarith
      · obtain ⟨hq, hw⟩ := heq j hj
        rw [hq, hw, sub_self, ite_eq_right hj]
        exact (Φ.norm_zero d).le
    _ = 2*B.card := by simp [Finset.sum_ite_mem, mul_comm]

theorem separator_continuousSum_odd (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) :
    Φ.separator A hA (Φ.continuousSum seed (-s)) =
      -Φ.separator A hA (Φ.continuousSum seed s) := by
  rw [Φ.continuousSum_neg, Φ.separator_odd]

end VertexCover.LabelCover

end
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end OAI