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

namespace OAI

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namespace VertexCover.LabelCover
namespace Decoding

variable (Φ : LabelCover) {d : ℕ}
variable (iU : Fin Φ.u → Φ.Query d) (iV : Fin Φ.v → Φ.Query d)
variable (pU : ∀ x, x ∈ (iU x).scopeU) (pV : ∀ y, y ∈ (iV y).scopeV)
variable (LU : ∀ x, List (iU x).LocalLabel) (LV : ∀ y, List (iV y).LocalLabel)

def leftAt (ρ : ℕ) (x : Fin Φ.u) : Fin Φ.qU :=
  if h : ρ < (LU x).length then ((LU x)[ρ]).1.1 ⟨x, pU x⟩ else ⟨0, Φ.qU_pos⟩

def rightAt (τ : ℕ) (y : Fin Φ.v) : Fin Φ.qV :=
  if h : τ < (LV y).length then ((LV y)[τ]).1.2 ⟨y, pV y⟩ else ⟨0, Φ.qV_pos⟩

def PairHit (c : Fin Φ.M) : Prop :=
  ∃ labU ∈ LU (Φ.left c), ∃ labV ∈ LV (Φ.right c),
    Φ.Compatible {⟨iU (Φ.left c), labU⟩, ⟨iV (Φ.right c), labV⟩}

noncomputable def pairHits : Finset (Fin Φ.M) := by
  classical
  exact Finset.univ.filter (PairHit Φ iU iV LU LV)

theorem pairHit_of_selection (c : Fin Φ.M) (I : Finset (Φ.Coordinate d))
    (hI : Φ.Compatible I)
    (labU : (iU (Φ.left c)).LocalLabel) (hlabU : labU ∈ LU (Φ.left c))
    (labV : (iV (Φ.right c)).LocalLabel) (hlabV : labV ∈ LV (Φ.right c))
    (hlabUI : ⟨iU (Φ.left c), labU⟩ ∈ I) (hlabVI : ⟨iV (Φ.right c), labV⟩ ∈ I) :
    PairHit Φ iU iV LU LV c := by
  classical
  refine ⟨labU, hlabU, labV, hlabV, Φ.compatible_mono hI ?_⟩
  intro p hp
  simp only [Finset.mem_insert, Finset.mem_singleton] at hp
  rcases hp with rfl | rfl
  · exact hlabUI
  · exact hlabVI

theorem pairHit_decodes {ell : ℕ}
    (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell)
    (c : Fin Φ.M) (hc : PairHit Φ iU iV LU LV c) :
    ∃ ρ : Fin ell, ∃ τ : Fin ell,
      Φ.Satisfies (leftAt Φ iU pU LU ρ, rightAt Φ iV pV LV τ) c := by
  classical
  obtain ⟨labU, hlabU, labV, hlabV, hcompat⟩ := hc
  obtain ⟨ρ, hρ, eρ⟩ := List.mem_iff_getElem.mp hlabU
  obtain ⟨τ, hτ, eτ⟩ := List.mem_iff_getElem.mp hlabV
  refine ⟨⟨ρ, lt_of_lt_of_le hρ (hLU _)⟩, ⟨τ, lt_of_lt_of_le hτ (hLV _)⟩, ?_⟩
  have hs := hcompat.2.2.2 ⟨iU (Φ.left c), labU⟩ (by simp)
    ⟨iV (Φ.right c), labV⟩ (by simp) c (pU _) (pV _)
  simpa only [Satisfies, leftAt, rightAt, dite_eq_left hρ, dite_eq_left hτ, eρ, eτ] using hs

include pU pV

theorem pairHits_card_le {ell : ℕ} {σ : ℝ}
    (hval : Φ.value ≤ σ)
    (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell) :
    ((pairHits Φ iU iV LU LV).card : ℝ) ≤ (ell : ℝ)^2 * σ * Φ.M := by
  classical
  let good : (Fin ell × Fin ell) → Finset (Fin Φ.M) := fun a =>
    Finset.univ.filter (Φ.Satisfies (leftAt Φ iU pU LU a.1, rightAt Φ iV pV LV a.2))
  have hs : pairHits Φ iU iV LU LV ⊆ Finset.univ.biUnion good := by
    intro c hc
    have hh : PairHit Φ iU iV LU LV c := (Finset.mem_filter.mp hc).2
    obtain ⟨ρ, τ, ha⟩ := pairHit_decodes Φ iU iV pU pV LU LV hLU hLV c hh
    exact Finset.mem_biUnion.mpr ⟨(ρ, τ), Finset.mem_univ _, Finset.mem_filter.mpr
      ⟨Finset.mem_univ _, ha⟩⟩
  have hcard := (Finset.card_le_card hs).trans Finset.card_biUnion_le
  calc
    ((pairHits Φ iU iV LU LV).card : ℝ) ≤
        ((∑ a : Fin ell × Fin ell, (good a).card : ℕ) : ℝ) := Nat.cast_le.mpr hcard
    _ = ∑ a : Fin ell × Fin ell, ((good a).card : ℝ) := by simp
    _ ≤ ∑ _a : Fin ell × Fin ell, σ * Φ.M := by
      apply Finset.sum_le_sum
      intro a _
      exact Φ.count_le_of_value_le hval _
    _ = (ell : ℝ)^2 * σ * Φ.M := by
      simp [Fintype.card_prod, pow_two]
      ring

theorem pairHits_probability_le {ell : ℕ} {σ : ℝ}
    (hval : Φ.value ≤ σ)
    (hLU : ∀ x, (LU x).length ≤ ell) (hLV : ∀ y, (LV y).length ≤ ell) :
    ((pairHits Φ iU iV LU LV).card : ℝ) / Φ.M ≤ (ell : ℝ)^2 * σ := by
  apply (div_le_iff₀ (Nat.cast_pos.mpr Φ.M_pos)).mpr
  exact pairHits_card_le Φ iU iV pU pV LU LV hval hLU hLV

end Decoding
end VertexCover.LabelCover


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end OAI