import Mathlib
import OAI.Combinatorics.IndependentSets.Expansion.PoweringTest
import OAI.Combinatorics.IndependentSets.PCP.ClauseVerifier
import OAI.Combinatorics.IndependentSets.Expansion.PoweringWitness

namespace OAI

namespace IndependentSetsGames.Foundations.PCP.PoweringNumeric

theorem min_density_ratio (ε C k t : ℝ)
    (hε : 0 < ε) (hC : 0 < C) (hk : 0 ≤ k)
    (ht : 0 < t) (hkt : k ≤ t) :
    min ε (1 / t) / (C + 1) ≤ ε / (C + k * ε) := by
  have hx : 0 ≤ min ε (1 / t) :=
    le_min hε.le (one_div_nonneg.mpr ht.le)
  have hxt : min ε (1 / t) * t ≤ 1 :=
    (le_div_iff₀ ht).mp (min_le_right ε (1 / t))
  have hkx : k * min ε (1 / t) ≤ 1 := by
    calc
      k * min ε (1 / t) ≤ t * min ε (1 / t) :=
        mul_le_mul_of_nonneg_right hkt hx
      _ ≤ 1 := by simpa only [mul_comm] using hxt
  have hC1 : 0 < C + 1 := by positivity
  have hden : 0 < C + k * ε :=
    add_pos_of_pos_of_nonneg hC (mul_nonneg hk hε.le)
  apply (div_le_div_iff₀ hC1 hden).2
  calc
    min ε (1 / t) * (C + k * ε) =
        C * min ε (1 / t) + ε * (k * min ε (1 / t)) := by ring
    _ ≤ C * ε + ε * 1 :=
      add_le_add
        (mul_le_mul_of_nonneg_left (min_le_left ε (1 / t)) hC.le)
        (mul_le_mul_of_nonneg_left hkx hε.le)
    _ = ε * (C + 1) := by ring

theorem count_moment_ratio_lower (α ε C t : ℝ) (m : ℕ)
    (hm : 0 < m) (hε : 0 < ε) (hC : 0 < C) (ht : 0 < t)
    (hmt : (m : ℝ) - 1 ≤ t) :
    (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / t) ≤
      (α ^ 2 * (m : ℝ) * ε) ^ 2 /
        ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε)) := by
  have hm0 : (0 : ℝ) < (m : ℝ) := Nat.cast_pos.mpr hm
  have hm1 : (1 : ℝ) ≤ (m : ℝ) := by
    exact_mod_cast (Nat.succ_le_iff.mpr hm)
  have hk : 0 ≤ (m : ℝ) - 1 := sub_nonneg.mpr hm1
  have hden : 0 < C + ((m : ℝ) - 1) * ε :=
    add_pos_of_pos_of_nonneg hC (mul_nonneg hk hε.le)
  have hαm : 0 ≤ α ^ 4 * (m : ℝ) := by positivity
  have hratio := min_density_ratio ε C ((m : ℝ) - 1) t hε hC hk ht hmt
  calc
    (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / t) =
        (α ^ 4 * (m : ℝ)) * (min ε (1 / t) / (C + 1)) := by ring
    _ ≤ (α ^ 4 * (m : ℝ)) * (ε / (C + ((m : ℝ) - 1) * ε)) :=
      mul_le_mul_of_nonneg_left hratio hαm
    _ = (α ^ 2 * (m : ℝ) * ε) ^ 2 /
        ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε)) := by
      field_simp [ne_of_gt hm0, ne_of_gt hε, ne_of_gt hden]

end IndependentSetsGames.Foundations.PCP.PoweringNumeric
noncomputable section

namespace IndependentSetsGames.Foundations.PCP.PoweringSoundness

open scoped BigOperators
open PoweringWalks PoweringLabels PoweringOpinions PoweringTest
open SpectralReturn PoweringWitness PoweringMomentBound
open PoweringMoment (bit hits)

variable {V D A : Type*}

def center (q M : Nat) : Nat := (4 * q * M) ^ 2

theorem le_center (q M : Nat) (hq : 1 ≤ q) (hM : 1 ≤ M) : M ≤ center q M := by
  have hfour : 1 ≤ 4 * q := by omega
  have hx : M ≤ 4 * q * M := by simpa only [Nat.one_mul] using Nat.mul_le_mul_right M hfour
  have hone : 1 ≤ 4 * q * M := hM.trans hx
  have hsquare : 4 * q * M ≤ (4 * q * M) * (4 * q * M) := by
    simpa only [Nat.mul_one] using Nat.mul_le_mul_left (4 * q * M) hone
  exact hx.trans (by simpa only [center, pow_two] using hsquare)

def gain (q M : Nat) (lambda : ℝ) : ℝ :=
  (1 / (2 * (q : ℝ))) ^ 4 * ((2 * M + 1 : Nat) : ℝ) /
    ((1 + 2 / (1 - lambda)) + 1)

variable [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
  [Fintype A] [Nonempty A]

def decodedError (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n N : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) (fallback : A) : ℝ :=
  edgeDensity (decodedBad G accepts (decoded G (n + 1) N selectors labels fallback))

def poweredRejection (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) : ℝ :=
  mean (fun d : Dart V D n => bit ((poweredGraph G accepts n selectors).edgeSatisfied labels d = false))

omit [Fintype V] [Nonempty V] [Nonempty D] in
theorem witness_bit_eq (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n N : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) (fallback : A)
    (w : Walk V D (n + 1)) (k : Fin (n + 1)) :
    bit (witness G accepts n selectors labels fallback
      (decoded G (n + 1) N selectors labels fallback) w k) =
      vertexWitness G
        (decodedBad G accepts (decoded G (n + 1) N selectors labels fallback))
        (matchFn G (n + 1) N selectors labels fallback) n k w := by
  simp only [witness, vertexWitness, endpointWitness, matchFn,
    PoweringMoment.bit_mul, next, and_assoc]

omit [Nonempty V] in
theorem modal_witness_mean (G : PortGraph V D) (M : Nat) (hM : 1 ≤ M)
    (accepts : Edge V (Bool × D) → A → A → Bool)
    (selectors : ∀ v, AddressSelector (lazyGraph G) (2 * center (Fintype.card A) M + 1) v)
    (labels : V → PaddedLabel (Bool × D) (2 * center (Fintype.card A) M + 1) A)
    (fallback : A) (k : Fin (2 * center (Fintype.card A) M + 1))
    (hlo : center (Fintype.card A) M - M ≤ k.val)
    (hhi : k.val ≤ center (Fintype.card A) M + M) :
    decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
      (center (Fintype.card A) M) selectors labels fallback / (4 * (Fintype.card A : ℝ) ^ 2) ≤
      mean (fun w => bit (witness (lazyGraph G) accepts (2 * center (Fintype.card A) M)
        selectors labels fallback (decoded (lazyGraph G) (2 * center (Fintype.card A) M + 1)
          (center (Fintype.card A) M) selectors labels fallback) w k)) := by
  have h := lazy_middle_witness_mean G (Fintype.card A) M Fintype.card_pos hM k hlo hhi
    (decodedBad (lazyGraph G) accepts
      (decoded (lazyGraph G) (2 * center (Fintype.card A) M + 1)
        (center (Fintype.card A) M) selectors labels fallback))
    (matchFn (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
    (matchFn_mem_Icc (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
    (decoded_modal_baseline (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
  simp_rw [witness_bit_eq]
  convert h using 1 <;> rfl

theorem rejection_lower_bound (G : PortGraph V D) (lambda : ℝ)
    (certificate : SpectralCertificate (lazyGraph G) lambda)
    (accepts : Edge V (Bool × D) → A → A → Bool)
    (reverse_accepts : ∀ e a b, accepts ((lazyGraph G).rot e) b a = accepts e a b)
    (M : Nat) (hM : 1 ≤ M)
    (selectors : ∀ v, AddressSelector (lazyGraph G) (2 * center (Fintype.card A) M + 1) v)
    (labels : V → PaddedLabel (Bool × D) (2 * center (Fintype.card A) M + 1) A)
    (fallback : A)
    (hε : 0 < decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
      (center (Fintype.card A) M) selectors labels fallback) :
    gain (Fintype.card A) M lambda *
        min (decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
          (center (Fintype.card A) M) selectors labels fallback)
          (1 / ((2 * center (Fintype.card A) M + 1 : Nat) : ℝ)) ≤
      poweredRejection (lazyGraph G) accepts (2 * center (Fintype.card A) M) selectors labels := by
  let q := Fintype.card A
  let N := center q M
  let n := 2 * N
  let m := 2 * M + 1
  let start := N - M
  let L := lazyGraph G
  let assignment := decoded L (n + 1) N selectors labels fallback
  let bad := decodedBad L accepts assignment
  let ε := edgeDensity bad
  let α : ℝ := 1 / (2 * (q : ℝ))
  let C : ℝ := 1 + 2 / (1 - lambda)
  have hNM : M ≤ N := le_center q M Fintype.card_pos hM
  have hw : start + m ≤ n + 1 := by dsimp [start, m, n]; omega
  let E := windowEvent L bad n start m hw
  let W : Fin m → Walk V (Bool × D) (n + 1) → Prop := fun i w =>
    witness L accepts n selectors labels fallback assignment w (windowIndex n start m hw i)
  let R : Walk V (Bool × D) (n + 1) → Prop := fun w =>
    pathAccepts L accepts n selectors w (labels w.1) (labels (endpoint L w)) = false
  have hWE : ∀ i w, W i w → E i w := by
    intro i w h
    exact h.1
  have hWR : ∀ i w, W i w → R w := by
    intro i w h
    exact witness_implies_path_rejection L accepts n selectors labels fallback assignment w
      (windowIndex n start m hw i) h
  have hpoint (i : Fin m) : ε / (4 * (q : ℝ) ^ 2) ≤ mean (fun w => bit (W i w)) := by
    have hlo : N - M ≤ (windowIndex n start m hw i).val := by
      dsimp [windowIndex, start]
      omega
    have hhi : (windowIndex n start m hw i).val ≤ N + M := by
      have hi := i.isLt
      dsimp [windowIndex, start]
      dsimp [m] at hi
      omega
    exact modal_witness_mean G M hM accepts selectors labels fallback
      (windowIndex n start m hw i) hlo hhi
  have hfirst : α ^ 2 * (m : ℝ) * ε ≤ mean (hits W) := by
    change α ^ 2 * (m : ℝ) * ε ≤ PoweringMoment.mean (hits W)
    rw [PoweringMoment.mean_hits]
    calc
      _ = ∑ _i : Fin m, ε / (4 * (q : ℝ) ^ 2) := by
        simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
        dsimp [α]
        ring
      _ ≤ _ := Finset.sum_le_sum (fun i _ => hpoint i)
  have hsecond : mean (fun w => hits E w ^ 2) ≤
      (m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε) :=
    window_second_moment_le L lambda certificate bad
      (decodedBad_rot L accepts reverse_accepts assignment) n start m hw
  have hε' : 0 < ε := hε
  have hm : 0 < m := by dsimp [m]; omega
  have hmReal : (0 : ℝ) < m := Nat.cast_pos.mpr hm
  have hqReal : (0 : ℝ) < q := Nat.cast_pos.mpr Fintype.card_pos
  have hden : 0 < 1 - lambda := sub_pos.mpr certificate.lt_one
  have hC : 0 < C := by dsimp [C]; positivity
  have hmOne : (1 : ℝ) ≤ m := by exact_mod_cast hm
  have hα : 0 < α := by dsimp [α]; positivity
  have ha : 0 < α ^ 2 * (m : ℝ) * ε := by positivity
  have hb : 0 < (m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε) := by
    have hdiff : 0 ≤ (m : ℝ) - 1 := sub_nonneg.mpr hmOne
    positivity
  have hCS := PoweringMoment.rejection_lower_bound E W R hWE hWR
    (α ^ 2 * (m : ℝ) * ε) ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε))
    ha hb hfirst hsecond
  have ht : (0 : ℝ) < ((n + 1 : Nat) : ℝ) := Nat.cast_pos.mpr (Nat.succ_pos n)
  have hmn : m ≤ n + 1 := by dsimp [m, n]; omega
  have hmt : (m : ℝ) - 1 ≤ ((n + 1 : Nat) : ℝ) := by
    have hmnReal : (m : ℝ) ≤ ((n + 1 : Nat) : ℝ) := by exact_mod_cast hmn
    linarith
  have hratio := PoweringNumeric.count_moment_ratio_lower α ε C ((n + 1 : Nat) : ℝ)
    m hm hε' hC ht hmt
  change (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / ((n + 1 : Nat) : ℝ)) ≤ _
  change _ ≤ mean (fun d : Dart V (Bool × D) n =>
    bit ((poweredGraph L accepts n selectors).edgeSatisfied labels d = false))
  rw [rejection_mean_eq_path_mean]
  exact hratio.trans hCS

end IndependentSetsGames.Foundations.PCP.PoweringSoundness
end

end OAI