import Mathlib
import OAI.Computability.VertexCover.Reduction.CompatibleSingleton

namespace OAI

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namespace VertexCover.LabelCover

noncomputable def compatibleSelections (Φ : LabelCover) (d : ℕ) :
    Finset (Finset (Φ.Coordinate d)) := by
  classical
  exact Finset.univ.filter Φ.Compatible

@[simp] theorem mem_compatibleSelections (Φ : LabelCover) {d : ℕ}
    (I : Finset (Φ.Coordinate d)) : I ∈ Φ.compatibleSelections d ↔ Φ.Compatible I := by
  classical
  simp [compatibleSelections]

theorem compatible_empty (Φ : LabelCover) (d : ℕ) :
    Φ.Compatible (∅ : Finset (Φ.Coordinate d)) := by
  simp [Compatible]

theorem compatibleSelections_nonempty (Φ : LabelCover) (d : ℕ) :
    (Φ.compatibleSelections d).Nonempty :=
  ⟨∅, (mem_compatibleSelections Φ ∅).mpr (Φ.compatible_empty d)⟩

noncomputable def compatibilityNorm (Φ : LabelCover) {d : ℕ}
    (z : Φ.Coordinate d → ℝ) : ℝ :=
  (Φ.compatibleSelections d).sup' (Φ.compatibleSelections_nonempty d)
    (fun I => |∑ p ∈ I, z p|)

theorem form_le_norm (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ)
    (I : Finset (Φ.Coordinate d)) (hI : Φ.Compatible I) :
    |∑ p ∈ I, z p| ≤ Φ.compatibilityNorm z := by
  exact Finset.le_sup' (fun J : Finset (Φ.Coordinate d) => |∑ p ∈ J, z p|)
    ((mem_compatibleSelections Φ I).mpr hI)

theorem norm_le_iff (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ) (b : ℝ) :
    Φ.compatibilityNorm z ≤ b ↔ ∀ I, Φ.Compatible I → |∑ p ∈ I, z p| ≤ b := by
  constructor
  · intro hb I hI
    exact (Φ.form_le_norm z I hI).trans hb
  · intro h
    exact Finset.sup'_le _ _ (fun I hI => h I ((mem_compatibleSelections Φ I).mp hI))

theorem norm_nonneg (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ) :
    0 ≤ Φ.compatibilityNorm z := by
  simpa using Φ.form_le_norm z ∅ (Φ.compatible_empty d)

theorem norm_zero (Φ : LabelCover) (d : ℕ) :
    Φ.compatibilityNorm (fun _ : Φ.Coordinate d => (0 : ℝ)) = 0 := by
  apply le_antisymm ((Φ.norm_le_iff _ _).mpr ?_) (Φ.norm_nonneg _)
  intro I _
  simp

theorem coordinate_le_norm (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ)
    (p : Φ.Coordinate d) : |z p| ≤ Φ.compatibilityNorm z := by
  classical
  simpa using Φ.form_le_norm z {p} (Φ.compatible_singleton p)

theorem norm_eq_zero_iff (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ) :
    Φ.compatibilityNorm z = 0 ↔ z = 0 := by
  constructor
  · intro h
    funext p
    have hh := Φ.coordinate_le_norm z p
    rw [h] at hh
    exact abs_nonpos_iff.mp hh
  · rintro rfl
    exact Φ.norm_zero d

theorem norm_add_le (Φ : LabelCover) {d : ℕ} (z w : Φ.Coordinate d → ℝ) :
    Φ.compatibilityNorm (z + w) ≤ Φ.compatibilityNorm z + Φ.compatibilityNorm w := by
  apply (Φ.norm_le_iff _ _).mpr
  intro I hI
  calc
    |∑ p ∈ I, (z + w) p| = |(∑ p ∈ I, z p) + ∑ p ∈ I, w p| := by
      simp only [Pi.add_apply, Finset.sum_add_distrib]
    _ ≤ |∑ p ∈ I, z p| + |∑ p ∈ I, w p| := abs_add_le _ _
    _ ≤ Φ.compatibilityNorm z + Φ.compatibilityNorm w :=
      add_le_add (Φ.form_le_norm z I hI) (Φ.form_le_norm w I hI)

theorem norm_mul (Φ : LabelCover) {d : ℕ} (a : ℝ) (z : Φ.Coordinate d → ℝ) :
    Φ.compatibilityNorm (fun p => a * z p) = |a| * Φ.compatibilityNorm z := by
  apply le_antisymm
  · apply (Φ.norm_le_iff _ _).mpr
    intro I hI
    rw [← Finset.mul_sum, abs_mul]
    exact mul_le_mul_of_nonneg_left (Φ.form_le_norm z I hI) (abs_nonneg a)
  · obtain ⟨I, hI, hmax⟩ := Finset.exists_mem_eq_sup'
      (Φ.compatibleSelections_nonempty d) (fun I => |∑ p ∈ I, z p|)
    change Φ.compatibilityNorm z = _ at hmax
    rw [hmax]
    have hh := Φ.form_le_norm (fun p => a * z p) I ((mem_compatibleSelections Φ I).mp hI)
    simpa only [← Finset.mul_sum, abs_mul] using hh

theorem norm_neg (Φ : LabelCover) {d : ℕ} (z : Φ.Coordinate d → ℝ) :
    Φ.compatibilityNorm (-z) = Φ.compatibilityNorm z := by
  change Φ.compatibilityNorm (fun p => -z p) = Φ.compatibilityNorm z
  simpa only [neg_one_mul, abs_neg, abs_one, one_mul] using Φ.norm_mul (-1) z

theorem norm_attained_on_support (Φ : LabelCover) {d : ℕ}
    (z : Φ.Coordinate d → ℝ) (B : Finset (Φ.Query d))
    (hz : ∀ p, p.1 ∉ B → z p = 0) :
    ∃ I : Finset (Φ.Coordinate d), Φ.Compatible I ∧
      (∀ p ∈ I, p.1 ∈ B) ∧ Φ.compatibilityNorm z = |∑ p ∈ I, z p| := by
  classical
  obtain ⟨J, hJ, heq⟩ := Finset.exists_mem_eq_sup'
    (Φ.compatibleSelections_nonempty d) (fun I => |∑ p ∈ I, z p|)
  let I := J.filter (fun p => p.1 ∈ B)
  have hIJ : I ⊆ J := Finset.filter_subset _ _
  refine ⟨I, Φ.compatible_mono ((mem_compatibleSelections Φ J).mp hJ) hIJ,
    fun p hp => (Finset.mem_filter.mp hp).2, ?_⟩
  change Φ.compatibilityNorm z = _ at heq
  rw [heq]
  congr 1
  symm
  apply Finset.sum_subset hIJ
  intro p hp hpI
  apply hz p
  intro hpB
  exact hpI (Finset.mem_filter.mpr ⟨hp, hpB⟩)

end VertexCover.LabelCover


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end OAI