Datasets:

roseking
/

openAI-math-dataset

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import OAI.Computability.UniqueGames.PCP.AlphabetRetractionLemmas
import OAI.Computability.UniqueGames.PCP.CayleySampling
import OAI.Computability.UniqueGames.PCP.CayleyTailCount
import OAI.Computability.UniqueGames.PCP.InitialGraphLemmas
import OAI.Computability.UniqueGames.PCP.InputLemmas

namespace OAI

noncomputable section

/-!
# Finite existence of the Cayley expander base

The sample space is the actual finite type of indexed Boolean-cube generators.
For each nonzero character, the checked sign sampling equivalence supplies its
uniform word law. A finite union bound then selects a generator family whose
nonconstant character biases are small. There is no random-graph hypothesis.

The fixed numbers below are manipulated symbolically; no enumeration of the
generator sample space is executed. The subsequent square-and-zigzag family
is a separate construction.
-/

namespace UniqueGamesTheorem.Foundations.PCP.Expanders

open scoped BigOperators
open UniqueGamesTheorem.Foundations.Hastad
open CayleySpectral

abbrev Generators (n m : Nat) := Fin (4 * m) → Cube (Fin n)

/-- The zero character is excluded from the bad-event count. -/
def badFrequency {n m : Nat} (g : Generators n m) (s : Cube (Fin n)) : ℝ :=
  if s = zeroFrequency then 0 else
    if CayleyTailCount.badWord m (fun d => AssignmentTester.parity s (g d)) then 1 else 0

def badMass {n m : Nat} (g : Generators n m) : ℝ :=
  ∑ s : Cube (Fin n), badFrequency g s

theorem badFrequency_nonnegative {n m : Nat}
    (g : Generators n m) (s : Cube (Fin n)) : 0 ≤ badFrequency g s := by
  unfold badFrequency
  split_ifs <;> norm_num

theorem mean_badFrequency_le {n m : Nat} (s : Cube (Fin n)) :
    (𝔼 g : Generators n m, badFrequency g s) ≤ 2 * (2 / 3 : ℝ) ^ m := by
  classical
  by_cases hs : s = zeroFrequency
  · simp only [badFrequency, ite_eq_left hs, Finset.expect_const_zero]
    positivity
  · have hs' : s ≠ (fun _ => false) := hs
    simp only [badFrequency, ite_eq_right hs]
    rw [CayleySampling.expect_parity_samples s hs'
      (fun w : Fin (4 * m) → Bool => if CayleyTailCount.badWord m w then (1 : ℝ) else 0)]
    exact CayleyTailCount.twoTail_probability m (by simp)

theorem mean_badMass_le (n m : Nat) (hm : 3 * (n + 2) ≤ m) :
    (𝔼 g : Generators n m, badMass g) ≤ 1 / 2 := by
  calc
    _ = ∑ s : Cube (Fin n), 𝔼 g : Generators n m, badFrequency g s := by
      unfold badMass
      rw [Finset.expect_sum_comm]
    _ ≤ ∑ _s : Cube (Fin n), 2 * (2 / 3 : ℝ) ^ m :=
      Finset.sum_le_sum (fun s _ => mean_badFrequency_le s)
    _ = (2 : ℝ) ^ n * (2 * (2 / 3 : ℝ) ^ m) := by
      simp [Cube]
    _ ≤ 1 / 2 := CayleyTailCount.union_bound_le_half n m hm

/-- Actual finite generators with a simultaneous bound for every nonconstant
Walsh character. The proof selects a point with bad-event count less than one.
-/
theorem exists_generators (n m : Nat) (hm : 3 * (n + 2) ≤ m) :
    ∃ g : Generators n m, ∀ s : Cube (Fin n),
      s ≠ zeroFrequency → |eigenvalue g s| ≤ (1 / 2 : ℝ) := by
  classical
  have hmpos : 0 < m := by omega
  let : Nonempty (Fin (4 * m)) := ⟨⟨0, by omega⟩⟩
  have havg : (𝔼 g : Generators n m, badMass g) < (1 : ℝ) :=
    lt_of_le_of_lt (mean_badMass_le n m hm) (by norm_num)
  obtain ⟨g, _, hg⟩ := Finset.exists_lt_of_expect_lt Finset.univ_nonempty havg
  refine ⟨g, fun s hs => ?_⟩
  have hgood : ¬ CayleyTailCount.badWord m
      (fun d => AssignmentTester.parity s (g d)) := by
    intro hbad
    have hsingle : badFrequency g s ≤ badMass g :=
      Finset.single_le_sum (fun t _ => badFrequency_nonnegative g t) (Finset.mem_univ s)
    simp only [badFrequency, ite_eq_right hs, ite_eq_left hbad] at hsingle
    exact (not_le_of_gt hg) hsingle
  unfold eigenvalue
  simp_rw [← AssignmentTester.bitSign_parity]
  exact CayleyTailCount.not_badWord_bias m (by simp) hmpos _ hgood

def baseDimension : Nat := 448
def initialDegree : Nat := 2 ^ 16
def initialQuarterDegree : Nat := 2 ^ 14
def basePower : Nat := 7

theorem base_arithmetic :
    4 * initialQuarterDegree = initialDegree ∧
    3 * (baseDimension + 2) ≤ initialQuarterDegree ∧
    (2 : Nat) ^ baseDimension = (initialDegree ^ basePower) ^ 4 ∧
    (1 / 2 : ℝ) ^ basePower ≤ 1 / 100 := by
  refine ⟨?_, ?_, ?_, ?_⟩
  · norm_num [initialDegree, initialQuarterDegree]
  · norm_num [baseDimension, initialQuarterDegree]
  · unfold baseDimension initialDegree basePower
    rw [← pow_mul, ← pow_mul]
  · norm_num [basePower]

theorem exists_initial_generators :
    ∃ g : Generators baseDimension initialQuarterDegree,
      ∀ s : Cube (Fin baseDimension), s ≠ zeroFrequency →
        |eigenvalue g s| ≤ (1 / 2 : ℝ) :=
  exists_generators _ _ base_arithmetic.2.1

abbrev BaseVertex := Cube (Fin baseDimension)
abbrev BasePort := Fin basePower → Fin (4 * initialQuarterDegree)
def baseDegree : Nat := initialDegree ^ basePower

theorem card_basePort : Fintype.card BasePort = baseDegree := by
  simp only [BasePort, Fintype.card_fun, Fintype.card_fin]
  rw [base_arithmetic.1]
  rfl

theorem card_baseVertex : Fintype.card BaseVertex = baseDegree ^ 4 := by
  simpa only [BaseVertex, Cube, Fintype.card_fun, Fintype.card_bool,
    Fintype.card_fin, baseDegree] using base_arithmetic.2.2.1

/-- A genuine finite regular graph with the exact degree/size relation needed
for zigzag. Its spectral certificate follows from the actual generator witness.
-/
theorem exists_baseGraph :
    ∃ H : PoweringWalks.PortGraph BaseVertex BasePort,
      SpectralReturn.SpectralCertificate H (1 / 100 : ℝ) := by
  obtain ⟨g, hg⟩ := exists_initial_generators
  refine ⟨cayleyGraph (powerGenerators g basePower), ?_⟩
  apply cayley_spectralCertificate
  · norm_num
  · norm_num
  · intro s hs
    exact (power_eigenvalue_bound g (1 / 2) (by norm_num) hg basePower s hs).trans
      base_arithmetic.2.2.2

end UniqueGamesTheorem.Foundations.PCP.Expanders

/-! Exact changes of finite vertex and port coordinates. The graph rotation and
its normalized averaging operator are transported by the same equivalences. -/

namespace UniqueGamesTheorem.Foundations.PCP.GraphTransport

open PoweringWalks SpectralReturn

variable {V W D E : Type*}

def reindex (G : PortGraph V D) (vertices : V ≃ W) (ports : D ≃ E) :
    PortGraph W E where
  rot := (Equiv.prodCongr vertices ports).symm.trans
    (G.rot.trans (Equiv.prodCongr vertices ports))
  rot_involutive := by
    intro x
    change (Equiv.prodCongr vertices ports)
        (G.rot ((Equiv.prodCongr vertices ports).symm
          ((Equiv.prodCongr vertices ports)
            (G.rot ((Equiv.prodCongr vertices ports).symm x))))) = x
    rw [Equiv.symm_apply_apply, G.rot_involutive, Equiv.apply_symm_apply]

@[simp] theorem reindex_rot (G : PortGraph V D) (vertices : V ≃ W)
    (ports : D ≃ E) (v : V) (d : D) :
    (reindex G vertices ports).rot (vertices v, ports d) =
      (vertices (G.rot (v, d)).1, ports (G.rot (v, d)).2) := by
  simp [reindex, Equiv.trans_apply, Prod.map]

variable [Fintype V] [Fintype W] [Fintype D] [Fintype E]

omit [Fintype V] [Fintype W] in
theorem operator_reindex (G : PortGraph V D) (vertices : V ≃ W)
    (ports : D ≃ E) (f : W → ℝ) (w : W) :
    averagingOperator (reindex G vertices ports) f w =
      averagingOperator G (fun v => f (vertices v)) (vertices.symm w) := by
  unfold averagingOperator
  apply Fintype.expect_equiv ports.symm
  intro e
  rfl

theorem energy_reindex (vertices : V ≃ W) (f : W → ℝ) :
    energy (fun v => f (vertices v)) = energy f :=
  mean_equiv vertices (fun w => f w ^ 2)

/-- A coordinate change preserves the actual spectral certificate, including
the uniform measure on the port alphabet. -/
theorem reindex_spectralCertificate (G : PortGraph V D) (vertices : V ≃ W)
    (ports : D ≃ E) (lambda : ℝ) (certificate : SpectralCertificate G lambda) :
    SpectralCertificate (reindex G vertices ports) lambda where
  nonnegative := certificate.nonnegative
  lt_one := certificate.lt_one
  contraction := by
    intro f hf
    have hf' : mean (fun v => f (vertices v)) = 0 :=
      (mean_equiv vertices f).trans hf
    have h := certificate.contraction (fun v => f (vertices v)) hf'
    have hout : energy (averagingOperator (reindex G vertices ports) f) =
        energy (averagingOperator G (fun v => f (vertices v))) := by
      rw [← energy_reindex vertices]
      congr 1
      funext v
      rw [operator_reindex, vertices.symm_apply_apply]
    rw [hout, ← energy_reindex vertices f]
    exact h

end UniqueGamesTheorem.Foundations.PCP.GraphTransport

section

/-! The actual half-lazy port graph retains a Boolean stay/step flag and an
ordinary port. This supplies the concrete graph underlying Boolean-tape
length smoothing. -/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringWalks

variable {V D : Type*}

def lazyRotate (G : PortGraph V D) : Edge V (Bool × D) → Edge V (Bool × D)
  | (v, (false, d)) => (v, (false, d))
  | (v, (true, d)) => ((G.rot (v, d)).1, (true, (G.rot (v, d)).2))

theorem lazyRotate_involutive (G : PortGraph V D) :
    Function.Involutive (lazyRotate G) := by
  rintro ⟨v, b, d⟩
  cases b with
  | false => rfl
  | true =>
    exact congrArg (fun e : V × D => (e.1, (true, e.2))) (rot_rot G (v, d))

def lazyGraph (G : PortGraph V D) : PortGraph V (Bool × D) where
  rot := {
    toFun := lazyRotate G
    invFun := lazyRotate G
    left_inv := lazyRotate_involutive G
    right_inv := lazyRotate_involutive G }
  rot_involutive := lazyRotate_involutive G

@[simp] theorem lazyGraph_rot_false (G : PortGraph V D) (v : V) (d : D) :
    (lazyGraph G).rot (v, (false, d)) = (v, (false, d)) := rfl

@[simp] theorem lazyGraph_rot_true (G : PortGraph V D) (v : V) (d : D) :
    (lazyGraph G).rot (v, (true, d)) = ((G.rot (v, d)).1, (true, (G.rot (v, d)).2)) := rfl

end UniqueGamesTheorem.Foundations.PCP.PoweringWalks

namespace UniqueGamesTheorem.Foundations.PCP.PoweringLazy

open PoweringWalks SpectralReturn

variable {V D : Type*}

theorem operator_binomialMean [Fintype D] (G : PortGraph V D)
    (n : Nat) (g : Nat → V → ℝ) (v : V) :
    averagingOperator G (fun w => LazySmoothing.binomialMean n (fun k => g k w)) v =
      LazySmoothing.binomialMean n (fun k => averagingOperator G (g k) v) := by
  unfold averagingOperator LazySmoothing.binomialMean
  exact Finset.expect_comm Finset.univ Finset.univ
    (fun d tape => g (LazySmoothing.bitCount tape) (G.rot (v, d)).1)

theorem averagingOperator_lazy [Fintype D] [Nonempty D]
    (G : PortGraph V D) (f : V → ℝ) (v : V) :
    averagingOperator (lazyGraph G) f v = (f v + averagingOperator G f v) / 2 := by
  change mean (fun bd : Bool × D => f ((lazyGraph G).rot (v, bd)).1) = _
  rw [mean_prod]
  change Finset.univ.expect (fun b : Bool =>
    mean (fun d : D => f ((lazyGraph G).rot (v, (b, d))).1)) = _
  rw [LazySmoothing.expect_bool]
  simp only [lazyGraph_rot_false, lazyGraph_rot_true]
  rw [mean_const]
  rfl

/-- The exact Boolean-tape mixture for the actual lazy transition operator. -/
theorem iterate_lazy_eq_binomialMean [Fintype D] [Nonempty D]
    (G : PortGraph V D) (n : Nat) (f : V → ℝ) :
    iterateOperator (lazyGraph G) n f =
      fun v => LazySmoothing.binomialMean n (fun k => iterateOperator G k f v) := by
  induction n with
  | zero =>
    funext v
    simp [iterateOperator]
  | succ n ih =>
    funext v
    change averagingOperator (lazyGraph G) (iterateOperator (lazyGraph G) n f) v = _
    rw [ih, averagingOperator_lazy, operator_binomialMean, LazySmoothing.mean_succ]
    rfl

theorem iterate_mem_Icc [Fintype D] [Nonempty D]
    (G : PortGraph V D) (f : V → ℝ) (hf : ∀ w, f w ∈ Set.Icc (0 : ℝ) 1)
    (n : Nat) (v : V) : iterateOperator G n f v ∈ Set.Icc (0 : ℝ) 1 := by
  induction n generalizing v with
  | zero => exact hf v
  | succ n ih =>
    change mean (fun d : D => iterateOperator G n f (G.rot (v, d)).1) ∈ Set.Icc (0 : ℝ) 1
    constructor
    · exact mean_nonnegative _ (fun d => (ih ((G.rot (v, d)).1)).1)
    · calc
        mean (fun d : D => iterateOperator G n f (G.rot (v, d)).1) ≤
            mean (fun _ : D => (1 : ℝ)) :=
          mean_mono (fun d => (ih ((G.rot (v, d)).1)).2)
        _ = 1 := mean_const _

/-- The modal lower bound survives every allowed nearby actual lazy length. -/
theorem lazy_endpoint_modal_transfer [Fintype D] [Nonempty D]
    (G : PortGraph V D) (q M m : Nat) (hq : 1 ≤ q) (hM : 1 ≤ M)
    (hlo : (4 * q * M) ^ 2 - M ≤ m) (hhi : m ≤ (4 * q * M) ^ 2 + M)
    (f : V → ℝ) (hf : ∀ w, f w ∈ Set.Icc (0 : ℝ) 1) (v : V)
    (hmodal : 1 / (q : ℝ) ≤ iterateOperator (lazyGraph G) ((4 * q * M) ^ 2) f v) :
    1 / (2 * (q : ℝ)) ≤ iterateOperator (lazyGraph G) m f v := by
  rw [iterate_lazy_eq_binomialMean] at hmodal ⊢
  exact LazySmoothing.modal_transfer q M m hq hM hlo hhi
    (fun k => iterateOperator G k f v) (fun k => iterate_mem_Icc G f hf k v) hmodal

end UniqueGamesTheorem.Foundations.PCP.PoweringLazy

end

/-! Spectral contraction for the actual Boolean-flag lazy port graph. The
operator identity is proved in `PoweringLazy`; the estimate here averages the
elementary two-term square inequality. -/

namespace UniqueGamesTheorem.Foundations.PCP.LazySpectral

open PoweringWalks SpectralReturn

variable {V D : Type*}

theorem half_sum_sq_le (a b : ℝ) : ((a + b) / 2) ^ 2 ≤ (a ^ 2 + b ^ 2) / 2 := by
  nlinarith [sq_nonneg (a - b)]

/-- Lazification averages the input and one actual graph step, so its energy
is at most the corresponding average of the two energies. -/
theorem lazy_energy_le_average [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (f : V → ℝ) :
    energy (averagingOperator (lazyGraph G) f) ≤
      (energy f + energy (averagingOperator G f)) / 2 := by
  calc
    energy (averagingOperator (lazyGraph G) f) =
        mean (fun v => ((f v + averagingOperator G f v) / 2) ^ 2) := by
      unfold energy
      congr 1
      funext v
      rw [PoweringLazy.averagingOperator_lazy]
    _ ≤ mean (fun v => (f v ^ 2 + averagingOperator G f v ^ 2) / 2) :=
      mean_mono (fun v => half_sum_sq_le _ _)
    _ = mean (fun v => (1 / 2 : ℝ) * (f v ^ 2 + averagingOperator G f v ^ 2)) := by
      congr 1
      funext v
      ring
    _ = (1 / 2 : ℝ) * (energy f + energy (averagingOperator G f)) := by
      rw [mean_mul_left, mean_add]
      rfl
    _ = (energy f + energy (averagingOperator G f)) / 2 := by ring

/-- The concrete lazy graph inherits a squared-energy bound from an actual
certificate for its underlying graph. -/
theorem lazy_energy_bound [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (lambda : ℝ) (hG : SpectralCertificate G lambda)
    (f : V → ℝ) (hf : mean f = 0) :
    energy (averagingOperator (lazyGraph G) f) ≤
      ((1 + lambda ^ 2) / 2) * energy f := by
  calc
    energy (averagingOperator (lazyGraph G) f) ≤
        (energy f + energy (averagingOperator G f)) / 2 := lazy_energy_le_average G f
    _ ≤ (energy f + lambda ^ 2 * energy f) / 2 := by
      have h := hG.contraction f hf
      linarith
    _ = ((1 + lambda ^ 2) / 2) * energy f := by ring

/-- An actual `7/8` certificate gives the stated `31/32` certificate after
lazification: its intermediate squared-energy factor is `113/128`. -/
theorem lazy_certificate_31_32 [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (hG : SpectralCertificate G (7 / 8)) :
    SpectralCertificate (lazyGraph G) (31 / 32) where
  nonnegative := by norm_num
  lt_one := by norm_num
  contraction f hf := by
    have h := lazy_energy_bound G (7 / 8) hG f hf
    have hcoef : ((1 + (7 / 8 : ℝ) ^ 2) / 2) ≤ (31 / 32 : ℝ) ^ 2 := by norm_num
    exact h.trans (mul_le_mul_of_nonneg_right hcoef (energy_nonnegative f))

end UniqueGamesTheorem.Foundations.PCP.LazySpectral

/-! Actual finite walk averages and their operator representations.
All changes of probability coordinates come from explicit equivalences. -/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringReturn

open PoweringWalks SpectralReturn
open PoweringMoment (bit)

variable {V D : Type*}

def portConsEquiv (D : Type*) (n : Nat) :
    (D × (Fin n → D)) ≃ (Fin (n + 1) → D) where
  toFun z := Fin.cases z.1 z.2
  invFun r := (r 0, fun j => r j.succ)
  left_inv z := by
    apply Prod.ext
    · rfl
    · funext j
      rfl
  right_inv r := by
    funext j
    exact Fin.cases rfl (fun _ => rfl) j

theorem mean_word_cons_head [Fintype D] (n : Nat)
    (F : (Fin (n + 1) → D) → ℝ) :
    mean F = mean (fun d : D => mean (fun r : Fin n → D => F (Fin.cases d r))) := by
  calc
    mean F = mean (fun z : D × (Fin n → D) => F (Fin.cases z.1 z.2)) :=
      (mean_equiv (portConsEquiv D n) F).symm
    _ = _ := mean_prod _

theorem mean_word_cons_tail [Fintype D] (n : Nat)
    (F : (Fin (n + 1) → D) → ℝ) :
    mean F = mean (fun r : Fin n → D => mean (fun d : D => F (Fin.cases d r))) := by
  calc
    mean F = mean (fun z : (Fin n → D) × D => F (Fin.cases z.2 z.1)) :=
      (mean_equiv ((Equiv.prodComm (Fin n → D) D).trans (portConsEquiv D n)) F).symm
    _ = _ := mean_prod _

theorem iterateOperator_succ_right [Fintype D] (G : PortGraph V D)
    (n : Nat) (f : V → ℝ) :
    iterateOperator G n (averagingOperator G f) = iterateOperator G (n + 1) f := by
  induction n with
  | zero => rfl
  | succ n ih => exact congrArg (averagingOperator G) ih

theorem mean_wordEnd [Fintype D] [Nonempty D] (G : PortGraph V D) :
    ∀ (n : Nat) (v : V) (f : V → ℝ),
      mean (fun r : Fin n → D => f (wordEnd G n v r)) = iterateOperator G n f v := by
  intro n
  induction n with
  | zero =>
    intro v f
    change mean (fun _ : Fin 0 → D => f v) = f v
    exact mean_const _
  | succ n ih =>
    intro v f
    rw [mean_word_cons_head]
    change mean (fun d : D => mean (fun r : Fin n → D =>
      f (wordEnd G n (next G v d) r))) = _
    calc
      _ = mean (fun d : D => iterateOperator G n f (next G v d)) := by
        congr 1
        funext d
        exact ih (next G v d) f
      _ = iterateOperator G (n + 1) f v := rfl

/-- Conditional endpoint factorization at an arbitrary fixed pivot dart. -/
theorem mean_pivot_endpoints [Fintype D] [Nonempty D] (G : PortGraph V D) :
    ∀ (n : Nat) (k : Fin (n + 1)) (e : Edge V D) (φ ψ : V → ℝ),
      mean (fun r : Fin n → D =>
        φ (leftFromPivot G n k e.1 r) * ψ (rightFromPivot G n k e r)) =
        iterateOperator G k.val φ e.1 *
          iterateOperator G (n - k.val) ψ (next G e.1 e.2) := by
  intro n
  induction n with
  | zero =>
    intro k e φ ψ
    have hk : k = 0 := Fin.eq_zero k
    subst k
    change mean (fun _ : Fin 0 → D => φ e.1 * ψ (next G e.1 e.2)) =
      φ e.1 * ψ (next G e.1 e.2)
    exact mean_const _
  | succ n ih =>
    intro k e φ ψ
    refine Fin.cases ?_ (fun j => ?_) k
    · change mean (fun r : Fin (n + 1) → D =>
        φ e.1 * ψ (wordEnd G (n + 1) (next G e.1 e.2) r)) =
          φ e.1 * iterateOperator G (n + 1) ψ (next G e.1 e.2)
      rw [mean_mul_left, mean_wordEnd]
    · simp only [Fin.val_succ, Nat.add_sub_add_right]
      rw [mean_word_cons_tail]
      change mean (fun r : Fin n → D => mean (fun d : D =>
        φ (next G (leftFromPivot G n j e.1 r) d) *
          ψ (rightFromPivot G n j e r))) =
        iterateOperator G (j.val + 1) φ e.1 *
          iterateOperator G (n - j.val) ψ (next G e.1 e.2)
      calc
        _ = mean (fun r : Fin n → D =>
            averagingOperator G φ (leftFromPivot G n j e.1 r) *
              ψ (rightFromPivot G n j e r)) := by
          congr 1
          funext r
          rw [mean_mul_right]
          rfl
        _ = iterateOperator G j.val (averagingOperator G φ) e.1 *
            iterateOperator G (n - j.val) ψ (next G e.1 e.2) :=
          ih j e (averagingOperator G φ) ψ
        _ = _ := by rw [iterateOperator_succ_right]

/-- The pivot and both endpoint opinions are averaged over actual walks. -/
theorem mean_edge_endpoints [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (n : Nat) (k : Fin (n + 1)) (φ ψ : Edge V D → V → ℝ) :
    mean (fun w : Walk V D (n + 1) =>
      φ (edgeAt G n w k) w.1 * ψ (edgeAt G n w k) (endpoint G w)) =
      mean (fun e : Edge V D =>
        iterateOperator G k.val (φ e) e.1 *
          iterateOperator G (n - k.val) (ψ e) (next G e.1 e.2)) := by
  let F : Edge V D × (Fin n → D) → ℝ := fun z =>
    φ z.1 (leftFromPivot G n k z.1.1 z.2) *
      ψ z.1 (rightFromPivot G n k z.1 z.2)
  calc
    _ = mean (fun w => F (pivotEquiv G n k w)) := by
      congr 1
      funext w
      dsimp [F]
      rw [leftFromPivot_actual, rightFromPivot_actual, pivotEquiv_fst]
    _ = mean F := mean_equiv (pivotEquiv G n k) F
    _ = mean (fun e : Edge V D => mean (fun r : Fin n → D =>
        φ e (leftFromPivot G n k e.1 r) * ψ e (rightFromPivot G n k e r))) := mean_prod F
    _ = _ := by
      congr 1
      funext e
      exact mean_pivot_endpoints G n k e (φ e) (ψ e)

theorem bit_bool (b : Bool) : bit (b = true) = (if b then 1 else 0 : ℝ) := by
  cases b <;> simp [bit]

theorem mean_edge_bit [Fintype D] (bad : V × D → Bool) (v : V) :
    mean (fun d : D => bit (bad (v, d) = true)) = edgeProfile bad v := by
  simp only [bit_bool, mean, edgeProfile]

theorem mean_marked_bit [Fintype D] (G : PortGraph V D)
    (bad : V × D → Bool) (f : V → ℝ) (v : V) :
    mean (fun d : D => bit (bad (v, d) = true) * f (next G v d)) =
      markedStep G bad f v := by
  change mean (fun d : D => bit (bad (v, d) = true) * f (next G v d)) =
    mean (fun d : D => if bad (v, d) then f (next G v d) else 0)
  congr 1
  funext d
  cases bad (v, d) <;> simp [bit]

theorem edgeAt_zero (G : PortGraph V D) (n : Nat) (w : Walk V D (n + 1)) :
    edgeAt G n w 0 = (w.1, w.2 0) := by
  cases n <;> rfl

theorem word_first_hit_mean [Fintype D] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) (n : Nat) (v : V) :
    mean (fun p : Fin (n + 1) → D => bit (bad (edgeAt G n (v, p) 0) = true)) =
      edgeProfile bad v := by
  rw [mean_word_cons_head]
  simp_rw [edgeAt_zero]
  change mean (fun d : D => mean (fun _ : Fin n → D => bit (bad (v, d) = true))) = _
  calc
    _ = mean (fun d : D => bit (bad (v, d) = true)) := by simp only [mean_const]
    _ = edgeProfile bad v := mean_edge_bit bad v

theorem word_hit_mean [Fintype D] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) :
    ∀ (n : Nat) (v : V) (k : Fin (n + 1)),
      mean (fun p : Fin (n + 1) → D => bit (bad (edgeAt G n (v, p) k) = true)) =
        iterateOperator G k.val (edgeProfile bad) v := by
  intro n
  induction n with
  | zero =>
    intro v k
    have hk : k = 0 := Fin.eq_zero k
    subst k
    exact word_first_hit_mean G bad 0 v
  | succ n ih =>
    intro v k
    refine Fin.cases ?_ (fun j => ?_) k
    · exact word_first_hit_mean G bad (n + 1) v
    · rw [mean_word_cons_head]
      change mean (fun d : D => mean (fun p : Fin (n + 1) → D =>
        bit (bad (edgeAt G n (next G v d, p) j) = true))) =
          averagingOperator G (iterateOperator G j.val (edgeProfile bad)) v
      calc
        _ = mean (fun d : D => iterateOperator G j.val (edgeProfile bad) (next G v d)) := by
          congr 1
          funext d
          exact ih (next G v d) j
        _ = _ := rfl

theorem word_pair_hit_mean [Fintype D] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) :
    ∀ (n : Nat) (v : V) (i j : Fin (n + 1)), i < j →
      mean (fun p : Fin (n + 1) → D =>
        bit (bad (edgeAt G n (v, p) i) = true) *
          bit (bad (edgeAt G n (v, p) j) = true)) =
        iterateOperator G i.val (markedStep G bad
          (iterateOperator G (j.val - i.val - 1) (edgeProfile bad))) v := by
  intro n
  induction n with
  | zero =>
    intro v i j hij
    have hi : i = 0 := Fin.eq_zero i
    have hj : j = 0 := Fin.eq_zero j
    subst i
    subst j
    exact ((lt_irrefl _) hij).elim
  | succ n ih =>
    intro v i
    refine Fin.cases ?_ (fun a => ?_) i
    · intro j
      refine Fin.cases ?_ (fun b => ?_) j
      · intro hij
        exact ((lt_irrefl _) hij).elim
      · intro _hij
        simp only [Fin.val_zero, Fin.val_succ, Nat.sub_zero, Nat.add_sub_cancel]
        rw [mean_word_cons_head]
        change mean (fun d : D => mean (fun p : Fin (n + 1) → D =>
          bit (bad (v, d) = true) * bit (bad (edgeAt G n (next G v d, p) b) = true))) =
          markedStep G bad (iterateOperator G b.val (edgeProfile bad)) v
        calc
          _ = mean (fun d : D => bit (bad (v, d) = true) *
              iterateOperator G b.val (edgeProfile bad) (next G v d)) := by
            congr 1
            funext d
            rw [mean_mul_left, word_hit_mean]
          _ = _ := mean_marked_bit G bad _ v
    · intro j
      refine Fin.cases ?_ (fun b => ?_) j
      · intro hij
        exact (Nat.not_lt_zero _ hij).elim
      · intro hij
        have hab : a < b := Nat.lt_of_succ_lt_succ hij
        simp only [Fin.val_succ, Nat.add_sub_add_right]
        rw [mean_word_cons_head]
        change mean (fun d : D => mean (fun p : Fin (n + 1) → D =>
          bit (bad (edgeAt G n (next G v d, p) a) = true) *
            bit (bad (edgeAt G n (next G v d, p) b) = true))) =
          averagingOperator G (iterateOperator G a.val (markedStep G bad
            (iterateOperator G (b.val - a.val - 1) (edgeProfile bad)))) v
        calc
          _ = mean (fun d : D => iterateOperator G a.val (markedStep G bad
              (iterateOperator G (b.val - a.val - 1) (edgeProfile bad))) (next G v d)) := by
            congr 1
            funext d
            exact ih (next G v d) a b hab
          _ = _ := rfl

theorem hit_mean [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) (n : Nat) (k : Fin (n + 1)) :
    mean (fun w : Walk V D (n + 1) => bit (bad (edgeAt G n w k) = true)) =
      edgeDensity bad := by
  calc
    _ = mean (fun v : V => mean (fun p : Fin (n + 1) → D =>
        bit (bad (edgeAt G n (v, p) k) = true))) :=
      mean_prod (A := V) (B := Fin (n + 1) → D)
        (fun w => bit (bad (edgeAt G n w k) = true))
    _ = mean (iterateOperator G k.val (edgeProfile bad)) := by
      congr 1
      funext v
      exact word_hit_mean G bad n v k
    _ = mean (edgeProfile bad) := mean_iterate G k.val _
    _ = _ := rfl

theorem pair_hit_mean [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) (n : Nat)
    (i j : Fin (n + 1)) (hij : i < j) :
    mean (fun w : Walk V D (n + 1) => bit (bad (edgeAt G n w i) = true) *
      bit (bad (edgeAt G n w j) = true)) = returnMass G bad (j.val - i.val - 1) := by
  calc
    _ = mean (fun v : V => mean (fun p : Fin (n + 1) → D =>
        bit (bad (edgeAt G n (v, p) i) = true) *
          bit (bad (edgeAt G n (v, p) j) = true))) :=
      mean_prod (A := V) (B := Fin (n + 1) → D)
        (fun w => bit (bad (edgeAt G n w i) = true) *
          bit (bad (edgeAt G n w j) = true))
    _ = mean (iterateOperator G i.val (markedStep G bad
        (iterateOperator G (j.val - i.val - 1) (edgeProfile bad)))) := by
      congr 1
      funext v
      exact word_pair_hit_mean G bad n v i j hij
    _ = mean (markedStep G bad
        (iterateOperator G (j.val - i.val - 1) (edgeProfile bad))) := mean_iterate G i.val _
    _ = _ := rfl

theorem pair_event_mean [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
    (G : PortGraph V D) (bad : V × D → Bool) (n : Nat)
    (i j : Fin (n + 1)) (hij : i < j) :
    mean (fun w : Walk V D (n + 1) =>
      bit (bad (edgeAt G n w i) = true ∧ bad (edgeAt G n w j) = true)) =
        returnMass G bad (j.val - i.val - 1) := by
  simpa only [PoweringMoment.bit_mul] using pair_hit_mean G bad n i j hij

end UniqueGamesTheorem.Foundations.PCP.PoweringReturn

/-! The second moment of the actual marked-edge count in a consecutive walk
window. Pair events are reduced to the checked spectral return operator. -/

namespace UniqueGamesTheorem.Foundations.PCP.PoweringMomentBound

open scoped BigOperators
open PoweringWalks SpectralReturn PoweringReturn
open PoweringMoment (bit hits)

theorem symmetric_gap_matrix_sum (ε : ℝ) (r : Nat → ℝ) :
    ∀ (m : Nat) (f : Fin m → Fin m → ℝ),
      (∀ i j, f i j = f j i) → (∀ i, f i i = ε) →
      (∀ i j, i.val < j.val → f i j = r (j.val - i.val - 1)) →
      (∑ i, ∑ j, f i j) = (m : ℝ) * ε +
        2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap := by
  intro m
  induction m with
  | zero => intro f _hSym _hDiag _hUpper; simp
  | succ m ih =>
    intro f hSym hDiag hUpper
    have hOld : (∑ i : Fin m, ∑ j : Fin m, f i.castSucc j.castSucc) =
        (m : ℝ) * ε + 2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap := by
      apply ih (fun i j => f i.castSucc j.castSucc)
      · intro i j; exact hSym i.castSucc j.castSucc
      · intro i; exact hDiag i.castSucc
      · intro i j hij; exact hUpper i.castSucc j.castSucc hij
    have hLast : (∑ i : Fin m, f i.castSucc (Fin.last m)) =
        ∑ gap ∈ Finset.range m, r gap := by
      calc
        _ = ∑ i : Fin m, r (m - 1 - i.val) := by
          apply Finset.sum_congr rfl
          intro i _
          rw [hUpper i.castSucc (Fin.last m) i.isLt]
          simp only [Fin.val_last, Fin.val_castSucc]
          congr 1
          omega
        _ = ∑ i ∈ Finset.range m, r (m - 1 - i) :=
          Fin.sum_univ_eq_sum_range (fun i => r (m - 1 - i)) m
        _ = _ := Finset.sum_range_reflect r m
    have hLastRow : (∑ j : Fin m, f (Fin.last m) j.castSucc) =
        ∑ gap ∈ Finset.range m, r gap := by
      calc
        _ = ∑ j : Fin m, f j.castSucc (Fin.last m) := by
          apply Finset.sum_congr rfl
          intro j _
          exact hSym (Fin.last m) j.castSucc
        _ = _ := hLast
    calc
      (∑ i, ∑ j, f i j) =
          (∑ i : Fin m, ∑ j : Fin m, f i.castSucc j.castSucc) +
            (∑ i : Fin m, f i.castSucc (Fin.last m)) +
            (∑ j : Fin m, f (Fin.last m) j.castSucc) + f (Fin.last m) (Fin.last m) := by
        simp only [Fin.sum_univ_castSucc, Finset.sum_add_distrib]
        ring
      _ = (m : ℝ) * ε +
          2 * (∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap) +
          (∑ gap ∈ Finset.range m, r gap) + (∑ gap ∈ Finset.range m, r gap) + ε := by
        rw [hOld, hLast, hLastRow, hDiag]
      _ = ((m + 1 : Nat) : ℝ) * ε +
          2 * ∑ j ∈ Finset.range (m + 1), ∑ gap ∈ Finset.range j, r gap := by
        rw [Finset.sum_range_succ]
        simp only [Nat.cast_add, Nat.cast_one]
        ring

def windowIndex (n start m : Nat) (h : start + m ≤ n + 1) (i : Fin m) : Fin (n + 1) :=
  ⟨start + i.val, (Nat.add_lt_add_left i.isLt start).trans_le h⟩

variable {V D : Type*} [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]

def windowEvent (G : PortGraph V D) (bad : V × D → Bool)
    (n start m : Nat) (h : start + m ≤ n + 1) (i : Fin m) (w : Walk V D (n + 1)) : Prop :=
  bad (edgeAt G n w (windowIndex n start m h i)) = true

theorem window_second_moment_eq (G : PortGraph V D) (bad : V × D → Bool)
    (n start m : Nat) (h : start + m ≤ n + 1) :
    mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) =
      (m : ℝ) * edgeDensity bad +
        2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, returnMass G bad gap := by
  rw [show mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) =
      ∑ i, ∑ j, mean (fun w => bit (windowEvent G bad n start m h i w ∧
        windowEvent G bad n start m h j w)) from
    PoweringMoment.second_moment_eq (windowEvent G bad n start m h)]
  apply symmetric_gap_matrix_sum
  · intro i j
    congr 1
    funext w
    simp only [and_comm]
  · intro i
    simp only [and_self]
    exact hit_mean G bad n (windowIndex n start m h i)
  · intro i j hij
    have hwin : windowIndex n start m h i < windowIndex n start m h j :=
      Nat.add_lt_add_left hij start
    have hp := pair_event_mean G bad n (windowIndex n start m h i)
      (windowIndex n start m h j) hwin
    simpa only [windowEvent, windowIndex, Nat.add_sub_add_left] using hp

/-- All moment terms come from actual walk events; the only analytic input
is the stated spectral contraction certificate of the given graph. -/
theorem window_second_moment_le (G : PortGraph V D) (lambda : ℝ)
    (certificate : SpectralCertificate G lambda) (bad : V × D → Bool)
    (reversal : ∀ e, bad (G.rot e) = bad e)
    (n start m : Nat) (h : start + m ≤ n + 1) :
    mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) ≤
      (m : ℝ) * edgeDensity bad *
        (1 + 2 / (1 - lambda) + ((m : ℝ) - 1) * edgeDensity bad) := by
  rw [window_second_moment_eq]
  exact secondMoment_return_envelope G lambda certificate bad reversal m

end UniqueGamesTheorem.Foundations.PCP.PoweringMomentBound

end

end OAI