lean/OAI/Computability/MinUncut/Encoding/Decoder.lean

import Mathlib
import OAI.Computability.MinUncut.PCP.Folding

namespace OAI

section
noncomputable section

namespace MinUncutGames.Foundations.Hastad

open scoped BigOperators
open Finset

variable {I J : Type*} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]

def parityProjection (π : J → I) (s : Cube J) : Cube I := fun i =>
  decide (Odd ((support s).filter fun y => π y = i).card)

theorem walsh_support (s x : Cube I) :
    walsh s x = ∏ i ∈ support s, bitSign (x i) := by
  unfold walsh support
  rw [Finset.prod_filter]
  apply Finset.prod_congr rfl
  intro i _
  cases s i <;> simp [bitSign]

theorem sign_power_parity (b : Bool) (n : Nat) :
    bitSign b ^ n = bitSign (decide (Odd n) && b) := by
  cases b
  · simp [bitSign]
  · simp only [Bool.and_true, bitSign, ite_true, decide_eq_true_eq]
    rw [neg_one_pow_eq_ite]
    by_cases h : Even n
    · simp [h, Nat.not_odd_iff_even.mpr h]
    · simp [h, Nat.not_even_iff_odd.mp h]

theorem walsh_pullback (π : J → I) (s : Cube J) (f : Cube I) :
    walsh s (fun y => f (π y)) = walsh (parityProjection π s) f := by
  rw [walsh_support]
  rw [← Finset.prod_fiberwise' (support s) π (fun i => bitSign (f i))]
  unfold walsh parityProjection
  apply Finset.prod_congr rfl
  intro i _
  simp only [Finset.prod_const]
  exact sign_power_parity (f i) _

theorem projectedCoefficient_eq (π : J → I) (A : Cube I → ℝ) (s : Cube J) :
    projectedCoefficient π A s = coefficient A (parityProjection π s) := by
  simp only [projectedCoefficient, coefficient, walsh_pullback]

omit [DecidableEq J] in
theorem projection_has_preimage (π : J → I) (s : Cube J) (i : I)
    (hi : i ∈ support (parityProjection π s)) :
    ∃ y ∈ support s, π y = i := by
  have hodd : Odd ((support s).filter fun y => π y = i).card := by
    have hbit := (Finset.mem_filter.mp hi).2
    simpa only [parityProjection, decide_eq_true_eq] using hbit
  have hp : 0 < ((support s).filter fun y => π y = i).card := by
    obtain ⟨n, hn⟩ := hodd
    omega
  obtain ⟨y, hy⟩ := Finset.card_pos.mp hp
  exact ⟨y, (Finset.mem_filter.mp hy).1, (Finset.mem_filter.mp hy).2⟩

theorem scaled_geometric_power_le_one {t : ℝ} (ht : 0 ≤ t) (ht' : t ≤ 1) (n : Nat) :
    (n : ℝ) * (1 - t) * t ^ n ≤ 1 := by
  have hsum : (n : ℝ) * t ^ n ≤ ∑ j ∈ Finset.range n, t ^ j := by
    calc
      (n : ℝ) * t ^ n = ∑ _j ∈ Finset.range n, t ^ n := by simp
      _ ≤ ∑ j ∈ Finset.range n, t ^ j := by
        apply Finset.sum_le_sum
        intro j hj
        exact pow_le_pow_of_le_one ht ht' (Nat.le_of_lt (Finset.mem_range.mp hj))
  have hm := mul_le_mul_of_nonneg_right hsum (sub_nonneg.mpr ht')
  rw [geom_sum_mul_neg] at hm
  have hp : 0 ≤ t ^ n := pow_nonneg ht _
  nlinarith

theorem noise_decay_bound {ε : ℝ} (hε : 0 < ε) (hε' : ε ≤ 1 / 2)
    {k : Nat} (hk : 0 < k) :
    4 * ε * ((1 - 2 * ε) ^ k) ^ 2 ≤ 1 / (k : ℝ) := by
  have h := scaled_geometric_power_le_one (t := 1 - 2 * ε)
    (by linarith) (by linarith) (2 * k)
  have hkp : (0 : ℝ) < k := Nat.cast_pos.mpr hk
  apply (le_div_iff₀ hkp).mpr
  have hpow : (1 - 2 * ε) ^ (2 * k) = ((1 - 2 * ε) ^ k) ^ 2 := by
    rw [Nat.mul_comm, pow_mul]
  rw [hpow] at h
  norm_num only [Nat.cast_mul, Nat.cast_ofNat] at h
  nlinarith

def zeroMask : Cube I := fun _ => false

omit [DecidableEq I] in
theorem support_eq_empty_iff (s : Cube I) : support s = ∅ ↔ s = zeroMask := by
  constructor
  · intro h
    funext i
    have hi : i ∉ support s := by rw [h]; simp
    have : s i ≠ true := by simpa [support] using hi
    cases hs : s i <;> simp_all [zeroMask]
  · intro h
    subst s
    simp [zeroMask, support]

omit [Fintype I] [DecidableEq J] in
theorem parityProjection_zero (π : J → I) :
    parityProjection π (zeroMask : Cube J) = zeroMask := by
  funext i
  simp [parityProjection, support, zeroMask]

def agreementProbability (π : J → I) (a : Cube I) (b : Cube J) : ℝ :=
  (∑ i ∈ support a, (((support b).filter fun y => π y = i).card : ℝ)) /
    ((support a).card * (support b).card : ℝ)

omit [DecidableEq J] in
theorem agreementProbability_nonneg (π : J → I) (a : Cube I) (b : Cube J) :
    0 ≤ agreementProbability π a b := by
  unfold agreementProbability
  apply div_nonneg
  · exact Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _)
  · exact mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)

omit [DecidableEq J] in
theorem agreementProbability_projection (π : J → I) (b : Cube J)
    (ha : (support (parityProjection π b)).Nonempty) :
    1 / ((support b).card : ℝ) ≤ agreementProbability π (parityProjection π b) b := by
  have hca : (0 : ℝ) < (support (parityProjection π b)).card :=
    Nat.cast_pos.mpr (Finset.card_pos.mpr ha)
  have hb : (support b).Nonempty := by
    obtain ⟨i, hi⟩ := ha
    obtain ⟨y, hy, _⟩ := projection_has_preimage π b i hi
    exact ⟨y, hy⟩
  have hcb : (0 : ℝ) < (support b).card := Nat.cast_pos.mpr (Finset.card_pos.mpr hb)
  have hsum : ((support (parityProjection π b)).card : ℝ) ≤
      ∑ i ∈ support (parityProjection π b),
        (((support b).filter fun y => π y = i).card : ℝ) := by
    calc
      _ = ∑ _i ∈ support (parityProjection π b), (1 : ℝ) := by simp
      _ ≤ _ := by
        apply Finset.sum_le_sum
        intro i hi
        obtain ⟨y, hy, hπ⟩ := projection_has_preimage π b i hi
        have hp : 0 < ((support b).filter fun y => π y = i).card :=
          Finset.card_pos.mpr ⟨y, Finset.mem_filter.mpr ⟨hy, hπ⟩⟩
        exact_mod_cast hp
  unfold agreementProbability
  apply (div_le_div_iff₀ hcb (mul_pos hca hcb)).mpr
  nlinarith

def decoderSuccess (π : J → I) (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  ∑ b, coefficient (fun g => bitSign (B g)) b ^ 2 *
    (∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b)

def decodingEnergy (π : J → I) (A : Cube I → ℝ) (B : Cube J → ℝ) : ℝ :=
  ∑ b, projectedCoefficient π A b ^ 2 * coefficient B b ^ 2 / (support b).card

theorem decodingEnergy_le_success (π : J → I) (A : Cube I → Bool) (B : Cube J → Bool)
    (hzero : coefficient (fun f => bitSign (A f)) zeroMask = 0) :
    decodingEnergy π (fun f => bitSign (A f)) (fun g => bitSign (B g)) ≤
      decoderSuccess π A B := by
  unfold decodingEnergy decoderSuccess
  apply Finset.sum_le_sum
  intro b _
  rw [projectedCoefficient_eq]
  have hsum :
      coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        agreementProbability π (parityProjection π b) b ≤
      ∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b := by
    apply Finset.single_le_sum (s := Finset.univ)
      (f := fun a => coefficient (fun f => bitSign (A f)) a ^ 2 * agreementProbability π a b)
    · intro a _
      exact mul_nonneg (sq_nonneg _) (agreementProbability_nonneg π a b)
    · exact Finset.mem_univ _
  have hm := mul_le_mul_of_nonneg_left hsum
    (sq_nonneg (coefficient (fun g => bitSign (B g)) b))
  by_cases ha : (support (parityProjection π b)).Nonempty
  · have hagree := agreementProbability_projection π b ha
    have ht := mul_le_mul_of_nonneg_left hagree
      (mul_nonneg (sq_nonneg (coefficient (fun f => bitSign (A f))
        (parityProjection π b))) (sq_nonneg (coefficient (fun g => bitSign (B g)) b)))
    have heq : coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        coefficient (fun g => bitSign (B g)) b ^ 2 / (support b).card =
      (coefficient (fun f => bitSign (A f)) (parityProjection π b) ^ 2 *
        coefficient (fun g => bitSign (B g)) b ^ 2) * (1 / (support b).card) := by ring
    rw [heq]
    exact ht.trans (by convert hm using 1; ring)
  · have hempty : support (parityProjection π b) = ∅ := Finset.not_nonempty_iff_eq_empty.mp ha
    have hz := (support_eq_empty_iff _).mp hempty
    simpa [hz, hzero] using hm

theorem bias_sq_le_weighted_energy (ε : ℝ) (π : J → I)
    (A : Cube I → ℝ) (B : Cube J → Bool) :
    testBias ε π A (fun g => bitSign (B g)) ^ 2 ≤
      ∑ b, (projectedCoefficient π A b * coefficient (fun g => bitSign (B g)) b *
        (1 - 2 * ε) ^ (support b).card) ^ 2 := by
  rw [testBias_fourier]
  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
    (fun b => projectedCoefficient π A b * coefficient (fun g => bitSign (B g)) b *
      (1 - 2 * ε) ^ (support b).card)
    (fun b => coefficient (fun g => bitSign (B g)) b)
  rw [sign_parseval, mul_one] at h
  convert h using 1
  congr 1
  apply Finset.sum_congr rfl
  intro b _
  ring

theorem decoder_four_epsilon_bias_sq (ε : ℝ) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2)
    (hzero : coefficient (fun f => bitSign (A f)) zeroMask = 0) :
    4 * ε * testBias ε π (fun f => bitSign (A f)) (fun g => bitSign (B g)) ^ 2 ≤
      decoderSuccess π A B := by
  apply le_trans _ (decodingEnergy_le_success π A B hzero)
  have hcs := mul_le_mul_of_nonneg_left
    (bias_sq_le_weighted_energy ε π (fun f => bitSign (A f)) B)
    (by linarith : 0 ≤ 4 * ε)
  apply hcs.trans
  rw [Finset.mul_sum]
  unfold decodingEnergy
  apply Finset.sum_le_sum
  intro b _
  by_cases hk : 0 < (support b).card
  · have hd := noise_decay_bound hε hε' hk
    have hm := mul_le_mul_of_nonneg_left hd
      (mul_nonneg (sq_nonneg (projectedCoefficient π (fun f => bitSign (A f)) b))
        (sq_nonneg (coefficient (fun g => bitSign (B g)) b)))
    convert hm using 1 <;> ring
  · have hb : support b = ∅ := Finset.card_eq_zero.mp (by omega)
    have hbzero := (support_eq_empty_iff b).mp hb
    subst b
    simp [projectedCoefficient_eq, parityProjection_zero, hzero, support, zeroMask]

theorem folded_decoder_bound (ε : ℝ) (π : J → I)
    (i₀ : I) (table : HalfCube i₀ → Bool) (B : Cube J → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ table f))
      (fun g => bitSign (B g)) ^ 2 ≤ decoderSuccess π (foldedAnswer i₀ table) B := by
  exact decoder_four_epsilon_bias_sq ε π (foldedAnswer i₀ table) B hε hε'
    (foldedAnswer_zero_coefficient i₀ table)

def validAgreementProbability (valid : J → Bool) (π : J → I)
    (a : Cube I) (b : Cube J) : ℝ :=
  (∑ i ∈ support a,
    (((support b).filter fun y => π y = i ∧ valid y = true).card : ℝ)) /
    ((support a).card * (support b).card : ℝ)

omit [DecidableEq J] in
theorem validAgreementProbability_eq (valid : J → Bool) (π : J → I)
    (a : Cube I) (b : Cube J) (hb : ∀ y ∈ support b, valid y = true) :
    validAgreementProbability valid π a b = agreementProbability π a b := by
  unfold validAgreementProbability agreementProbability
  congr 1
  apply Finset.sum_congr rfl
  intro i _
  congr 1
  apply congrArg Finset.card
  apply Finset.filter_congr
  intro y hy
  simp [hb y hy]

def validDecoderSuccess (valid : J → Bool) (π : J → I)
    (A : Cube I → Bool) (B : Cube J → Bool) : ℝ :=
  ∑ b, coefficient (fun g => bitSign (B g)) b ^ 2 *
    (∑ a, coefficient (fun f => bitSign (A f)) a ^ 2 *
      validAgreementProbability valid π a b)

theorem conditioned_validDecoderSuccess (valid : J → Bool) (π : J → I)
    (A : Cube I → Bool) (j₀ : {j : J // valid j = true})
    (table : HalfCube j₀ → Bool) :
    validDecoderSuccess valid π A (conditionedFoldedAnswer valid j₀ table) =
      decoderSuccess π A (conditionedFoldedAnswer valid j₀ table) := by
  unfold validDecoderSuccess decoderSuccess
  apply Finset.sum_congr rfl
  intro b _
  by_cases hc : coefficient (fun g => bitSign (conditionedFoldedAnswer valid j₀ table g)) b = 0
  · simp [hc]
  · have hb : ∀ y ∈ support b, valid y = true := by
      apply conditioned_coefficient_support valid (fun g => bitSign (foldedAnswer j₀ table g)) b
      exact hc
    congr 1
    apply Finset.sum_congr rfl
    intro a _
    rw [validAgreementProbability_eq valid π a b hb]

theorem conditioned_folded_decoder_bound (ε : ℝ) (π : J → I) (valid : J → Bool)
    (i₀ : I) (tableA : HalfCube i₀ → Bool)
    (j₀ : {j : J // valid j = true}) (tableB : HalfCube j₀ → Bool)
    (hε : 0 < ε) (hε' : ε ≤ 1 / 2) :
    4 * ε * testBias ε π (fun f => bitSign (foldedAnswer i₀ tableA f))
      (fun g => bitSign (conditionedFoldedAnswer valid j₀ tableB g)) ^ 2 ≤
      validDecoderSuccess valid π (foldedAnswer i₀ tableA)
        (conditionedFoldedAnswer valid j₀ tableB) := by
  rw [conditioned_validDecoderSuccess]
  exact folded_decoder_bound ε π i₀ tableA
    (conditionedFoldedAnswer valid j₀ tableB) hε hε'

end MinUncutGames.Foundations.Hastad

end
end

end OAI