Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.MaxCut.PCP.CayleySampling

namespace OAI

/-!
# Walsh diagonalization of actual Boolean-cube Cayley graphs

Every port is an indexed generator. The rotation keeps the port and adds its
generator by pointwise xor, so loops and repeated generators keep their full
multiplicity. The Fourier multiplier and energy estimate are proved for this
actual averaging operator; no Fourier/operator compatibility is assumed.
-/

noncomputable section

namespace MaxCutGames.Foundations.PCP.CayleySpectral

open scoped BigOperators
open Finset
open MaxCutGames.Foundations.Hastad
open PoweringWalks

variable {I D : Type*}

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

theorem cubeXor_cancel_right (x y : Cube I) :
    cubeXor (cubeXor x y) y = x := by
  funext i
  cases hx : x i <;> cases hy : y i <;> simp [cubeXor, hx, hy]

theorem cubeXor_assoc (x y z : Cube I) :
    cubeXor (cubeXor x y) z = cubeXor x (cubeXor y z) := by
  funext i
  cases hx : x i <;> cases hy : y i <;> cases hz : z i <;>
    simp [cubeXor, hx, hy, hz]

@[simp] theorem cubeXor_zeroFrequency (x : Cube I) :
    cubeXor x zeroFrequency = x := by
  funext i
  cases hx : x i <;> simp [cubeXor, zeroFrequency, hx]

/-- Uniform cube measure is invariant under this explicit xor translation. -/
def xorTranslation (y : Cube I) : Cube I ≃ Cube I where
  toFun x := cubeXor x y
  invFun x := cubeXor x y
  left_inv x := cubeXor_cancel_right x y
  right_inv x := cubeXor_cancel_right x y

def rotate (g : D → Cube I) (p : Cube I × D) : Cube I × D :=
  (cubeXor p.1 (g p.2), p.2)

theorem rotate_involutive (g : D → Cube I) : Function.Involutive (rotate g) := by
  rintro ⟨x, d⟩
  change (cubeXor (cubeXor x (g d)) (g d), d) = (x, d)
  rw [cubeXor_cancel_right]

/-- The undirected regular port graph generated by the indexed cube vectors. -/
def cayleyGraph (g : D → Cube I) : PortGraph (Cube I) D where
  rot :=
    { toFun := rotate g
      invFun := rotate g
      left_inv := rotate_involutive g
      right_inv := rotate_involutive g }
  rot_involutive := rotate_involutive g

@[simp] theorem cayleyGraph_rot (g : D → Cube I) (x : Cube I) (d : D) :
    (cayleyGraph g).rot (x, d) = (cubeXor x (g d), d) := rfl

/-- Expose the first actual port in a finite port word. -/
def splitPortWord (n : Nat) : (Fin (n + 1) → D) ≃ D × (Fin n → D) where
  toFun p := (p 0, fun j => p j.succ)
  invFun z := Fin.cases z.1 z.2
  left_inv p := by
    funext j
    exact Fin.cases rfl (fun _ => rfl) j
  right_inv z := rfl

/-- The generator of a powered port is the xor of its original port word. -/
def powerGenerators (g : D → Cube I) : (n : Nat) → (Fin n → D) → Cube I
  | 0, _ => zeroFrequency
  | n + 1, p => cubeXor (g (p 0)) (powerGenerators g n (fun j => p j.succ))

/-- These are actual walk displacements, including repeated vertices and ports. -/
theorem wordEnd_eq_powerGenerators (g : D → Cube I) (n : Nat)
    (x : Cube I) (p : Fin n → D) :
    wordEnd (cayleyGraph g) n x p = cubeXor x (powerGenerators g n p) := by
  induction n generalizing x with
  | zero => simp [wordEnd, powerGenerators]
  | succ n ih =>
    change wordEnd (cayleyGraph g) n (cubeXor x (g (p 0))) (fun j => p j.succ) =
      cubeXor x (cubeXor (g (p 0)) (powerGenerators g n (fun j => p j.succ)))
    rw [ih, cubeXor_assoc]

variable [Fintype I] [DecidableEq I] [Fintype D]

/-- Average along the actual outgoing ports of the Cayley graph. -/
def cayleyAverage (g : D → Cube I) (f : Cube I → ℝ) (x : Cube I) : ℝ :=
  𝔼 d, f ((cayleyGraph g).rot (x, d)).1

/-- The eigenvalue is the empirical bias of the generator characters. -/
def eigenvalue (g : D → Cube I) (s : Cube I) : ℝ :=
  𝔼 d, walsh s (g d)

omit [Fintype I] [DecidableEq I] in
@[simp] theorem averagingOperator_cayleyGraph (g : D → Cube I) (f : Cube I → ℝ) :
    SpectralReturn.averagingOperator (cayleyGraph g) f = cayleyAverage g f := rfl

@[simp] theorem coefficient_zeroFrequency (f : Cube I → ℝ) :
    coefficient f zeroFrequency = 𝔼 x, f x := by
  simp [coefficient, zeroFrequency, walsh, bitSign]

/-- Change «variables» through xor translation in the actual Fourier average. -/
theorem coefficient_xorTranslation (f : Cube I → ℝ) (s v : Cube I) :
    coefficient (fun x => f (cubeXor x v)) s = walsh s v * coefficient f s := by
  calc
    coefficient (fun x => f (cubeXor x v)) s =
        𝔼 x, f x * walsh s (cubeXor x v) := by
      unfold coefficient
      apply Fintype.expect_equiv (xorTranslation v)
      intro x
      change f (cubeXor x v) * walsh s x =
        f (cubeXor x v) * walsh s (cubeXor (cubeXor x v) v)
      rw [cubeXor_cancel_right]
    _ = 𝔼 x, (f x * walsh s x) * walsh s v := by
      apply Finset.expect_congr rfl
      intro x _
      rw [walsh_xor]
      ring
    _ = coefficient f s * walsh s v := by
      rw [← Finset.expect_mul]
      rfl
    _ = walsh s v * coefficient f s := mul_comm _ _

/-- Each Walsh character is an eigenvector of the actual Cayley averaging. -/
theorem cayleyAverage_walsh (g : D → Cube I) (s x : Cube I) :
    cayleyAverage g (walsh s) x = eigenvalue g s * walsh s x := by
  unfold cayleyAverage
  simp only [cayleyGraph_rot, walsh_xor]
  rw [← Finset.mul_expect]
  rw [mul_comm]
  rfl

/-- Full Walsh diagonalization, obtained by swapping the two finite averages. -/
theorem coefficient_cayleyAverage (g : D → Cube I) (f : Cube I → ℝ) (s : Cube I) :
    coefficient (cayleyAverage g f) s = eigenvalue g s * coefficient f s := by
  calc
    coefficient (cayleyAverage g f) s =
        𝔼 x, 𝔼 d, f (cubeXor x (g d)) * walsh s x := by
      unfold coefficient cayleyAverage
      simp only [cayleyGraph_rot, Finset.expect_mul]
    _ = 𝔼 d, 𝔼 x, f (cubeXor x (g d)) * walsh s x := by
      rw [Finset.expect_comm]
    _ = 𝔼 d, walsh s (g d) * coefficient f s := by
      apply Finset.expect_congr rfl
      intro d _
      exact coefficient_xorTranslation f s (g d)
    _ = eigenvalue g s * coefficient f s := by
      rw [← Finset.expect_mul]
      rfl

/-- Uniform finite port-word splitting multiplies the actual eigenvalues. -/
theorem eigenvalue_powerGenerators (g : D → Cube I) (n : Nat) (s : Cube I) :
    eigenvalue (powerGenerators g n) s = eigenvalue g s ^ n := by
  induction n with
  | zero =>
    let : Nonempty (Fin 0 → D) := ⟨fun i => Fin.elim0 i⟩
    simp [eigenvalue, powerGenerators, zeroFrequency, walsh, bitSign]
  | succ n ih =>
    calc
      eigenvalue (powerGenerators g (n + 1)) s =
          𝔼 z : D × (Fin n → D),
            walsh s (cubeXor (g z.1) (powerGenerators g n z.2)) := by
        unfold eigenvalue
        apply Fintype.expect_equiv (splitPortWord n)
        intro p
        rfl
      _ = 𝔼 d, 𝔼 p : Fin n → D,
          walsh s (cubeXor (g d) (powerGenerators g n p)) := by
        exact SpectralReturn.mean_prod _
      _ = eigenvalue g s * eigenvalue (powerGenerators g n) s := by
        simp_rw [walsh_xor, ← Finset.mul_expect]
        rw [← Finset.expect_mul]
        rfl
      _ = eigenvalue g s ^ (n + 1) := by
        rw [ih, pow_succ, mul_comm]

/-- The port degree of the powered graph is the exact number of port words. -/
theorem power_port_card (n : Nat) :
    Fintype.card (Fin n → D) = Fintype.card D ^ n := by
  simp only [Fintype.card_fun, Fintype.card_fin]

theorem seven_power_port_card :
    Fintype.card (Fin 7 → Fin (2 ^ 16)) = (2 ^ 16) ^ 7 := by
  simp only [Fintype.card_fun, Fintype.card_fin]

/-- A bias bound on all nonconstant generator characters contracts the actual
mean-zero function energy. The constant coefficient is eliminated by its exact
identification with the mean, and every other coefficient is bounded termwise.
-/
theorem cayley_energy_contraction (g : D → Cube I) (lambda : ℝ) (_hlambda : 0 ≤ lambda)
    (hbias : ∀ s : Cube I, s ≠ zeroFrequency → |eigenvalue g s| ≤ lambda)
    (f : Cube I → ℝ) (hmean : (𝔼 x, f x) = 0) :
    (𝔼 x, cayleyAverage g f x ^ 2) ≤ lambda ^ 2 * (𝔼 x, f x ^ 2) := by
  classical
  have hzero : coefficient f zeroFrequency = 0 := by
    rw [coefficient_zeroFrequency]
    exact hmean
  calc
    (𝔼 x, cayleyAverage g f x ^ 2) =
        ∑ s, coefficient (cayleyAverage g f) s ^ 2 := (walsh_parseval _).symm
    _ = ∑ s, (eigenvalue g s * coefficient f s) ^ 2 := by
      simp only [coefficient_cayleyAverage]
    _ ≤ ∑ s, lambda ^ 2 * coefficient f s ^ 2 := by
      apply Finset.sum_le_sum
      intro s _
      by_cases hs : s = zeroFrequency
      · subst s
        simp [hzero]
      · have hb := hbias s hs
        have hlo : -lambda ≤ eigenvalue g s := (abs_le.mp hb).1
        have hhi : eigenvalue g s ≤ lambda := (abs_le.mp hb).2
        have hprod : 0 ≤ (lambda - eigenvalue g s) * (lambda + eigenvalue g s) :=
          mul_nonneg (sub_nonneg.mpr hhi) (by linarith)
        have hsq : eigenvalue g s ^ 2 ≤ lambda ^ 2 := by nlinarith
        calc
          (eigenvalue g s * coefficient f s) ^ 2 =
              eigenvalue g s ^ 2 * coefficient f s ^ 2 := by rw [mul_pow]
          _ ≤ lambda ^ 2 * coefficient f s ^ 2 :=
            mul_le_mul_of_nonneg_right hsq (sq_nonneg _)
    _ = lambda ^ 2 * (𝔼 x, f x ^ 2) := by
      rw [← Finset.mul_sum, walsh_parseval]

/-- The Fourier proof supplies the graph certificate used in return bounds. -/
theorem cayley_spectralCertificate (g : D → Cube I) (lambda : ℝ)
    (hlambda : 0 ≤ lambda) (hlambdaone : lambda < 1)
    (hbias : ∀ s : Cube I, s ≠ zeroFrequency → |eigenvalue g s| ≤ lambda) :
    SpectralReturn.SpectralCertificate (cayleyGraph g) lambda where
  nonnegative := hlambda
  lt_one := hlambdaone
  contraction := by
    intro f hf
    change (𝔼 x, cayleyAverage g f x ^ 2) ≤ lambda ^ 2 * (𝔼 x, f x ^ 2)
    exact cayley_energy_contraction g lambda hlambda hbias f hf

theorem power_eigenvalue_bound (g : D → Cube I) (lambda : ℝ) (hlambda : 0 ≤ lambda)
    (hbias : ∀ s : Cube I, s ≠ zeroFrequency → |eigenvalue g s| ≤ lambda)
    (n : Nat) (s : Cube I) (hs : s ≠ zeroFrequency) :
    |eigenvalue (powerGenerators g n) s| ≤ lambda ^ n := by
  rw [eigenvalue_powerGenerators, abs_pow]
  induction n with
  | zero => simp
  | succ n ih =>
    rw [pow_succ, pow_succ]
    exact mul_le_mul ih (hbias s hs) (abs_nonneg _) (pow_nonneg hlambda _)

/-- Seven actual generator steps turn one-half character bias into `2⁻⁷`. -/
theorem sevenStep_halfCertificate (g : D → Cube I)
    (hbias : ∀ s : Cube I, s ≠ zeroFrequency → |eigenvalue g s| ≤ (1 / 2 : ℝ)) :
    SpectralReturn.SpectralCertificate (cayleyGraph (powerGenerators g 7))
      ((1 / 2 : ℝ) ^ 7) := by
  apply cayley_spectralCertificate
  · norm_num
  · norm_num
  · intro s hs
    exact power_eigenvalue_bound g (1 / 2) (by norm_num) hbias 7 s hs

end MaxCutGames.Foundations.PCP.CayleySpectral
end

/-!
# 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.
-/

noncomputable section

namespace MaxCutGames.Foundations.PCP.Expanders

open scoped BigOperators
open MaxCutGames.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]
  · dsimp only [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 MaxCutGames.Foundations.PCP.Expanders
end

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

namespace MaxCutGames.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 MaxCutGames.Foundations.PCP.GraphTransport

/-!
# Actual graph squares and zigzag products

Rotations operate on actual vertex-port pairs. All returned ports are retained,
including ports belonging to loops and parallel edges. Each construction is a
palindrome of involutions and hence defines a reversible regular port graph.
-/

namespace MaxCutGames.Foundations.PCP.ZigzagGraphs

open PoweringWalks

variable {V D E : Type*}

/-- Regard a checked involution as a permutation with the same inverse. -/
def involutionEquiv {A : Type*} (f : A → A) (hf : Function.Involutive f) : A ≃ A where
  toFun := f
  invFun := f
  left_inv := hf
  right_inv := hf

/-- A palindrome of two involutions is itself an involution. -/
theorem palindrome_involutive {A : Type*} (B P : A ≃ A)
    (hB : Function.Involutive B) (hP : Function.Involutive P) :
    Function.Involutive (B.trans (P.trans B)) := by
  intro x
  change B (P (B (B (P (B x))))) = x
  rw [hB, hP, hB]

/-- Cross an original edge using the first port; retain its return port. -/
def squareFirst (G : PortGraph V D) (s : V × (D × D)) : V × (D × D) :=
  let step := G.rot (s.1, s.2.1)
  (step.1, (step.2, s.2.2))

theorem squareFirst_involutive (G : PortGraph V D) : Function.Involutive (squareFirst G) := by
  rintro ⟨v, d₁, d₂⟩
  change ((G.rot (G.rot (v, d₁))).1, ((G.rot (G.rot (v, d₁))).2, d₂)) =
    (v, (d₁, d₂))
  rw [G.rot_involutive]

def squareSwap (s : V × (D × D)) : V × (D × D) := (s.1, (s.2.2, s.2.1))

theorem squareSwap_involutive : Function.Involutive (squareSwap (V := V) (D := D)) := by
  rintro ⟨v, d₁, d₂⟩
  rfl

/-- The actual graph square. The two return ports appear in reverse order. -/
def square (G : PortGraph V D) : PortGraph V (D × D) := by
  let B := involutionEquiv (squareFirst G) (squareFirst_involutive G)
  let P := involutionEquiv (squareSwap (V := V) (D := D)) squareSwap_involutive
  exact
    { rot := B.trans (P.trans B)
      rot_involutive := palindrome_involutive B P
        (squareFirst_involutive G) squareSwap_involutive }

@[simp] theorem square_rot_apply (G : PortGraph V D) (v : V) (d₁ d₂ : D) :
    (square G).rot (v, (d₁, d₂)) =
      let first := G.rot (v, d₁)
      let second := G.rot (first.1, d₂)
      (second.1, (second.2, first.2)) := rfl

@[simp] theorem square_next (G : PortGraph V D) (v : V) (d₁ d₂ : D) :
    next (square G) v (d₁, d₂) = next G (next G v d₁) d₂ := rfl

/-- One inner step in a cloud, retaining its return port and the unused port. -/
def cloudFirst (H : PortGraph D E) (s : (V × D) × (E × E)) :
    (V × D) × (E × E) :=
  let step := H.rot (s.1.2, s.2.1)
  ((s.1.1, step.1), (step.2, s.2.2))

theorem cloudFirst_involutive (H : PortGraph D E) :
    Function.Involutive (cloudFirst (V := V) H) := by
  rintro ⟨⟨v, d⟩, ⟨e₁, e₂⟩⟩
  change ((v, (H.rot (H.rot (d, e₁))).1), ((H.rot (H.rot (d, e₁))).2, e₂)) =
    ((v, d), (e₁, e₂))
  rw [H.rot_involutive]

/-- Cross the outer graph edge and swap the two inner ports. -/
def crossCloud (G : PortGraph V D) (s : (V × D) × (E × E)) :
    (V × D) × (E × E) := (G.rot s.1, (s.2.2, s.2.1))

theorem crossCloud_involutive (G : PortGraph V D) :
    Function.Involutive (crossCloud (E := E) G) := by
  rintro ⟨x, e₁, e₂⟩
  change (G.rot (G.rot x), (e₁, e₂)) = (x, (e₁, e₂))
  rw [G.rot_involutive]

/-- The actual zigzag product: inner step, outer edge, inner step. -/
def zigzag (G : PortGraph V D) (H : PortGraph D E) : PortGraph (V × D) (E × E) := by
  let B := involutionEquiv (cloudFirst (V := V) H) (cloudFirst_involutive H)
  let P := involutionEquiv (crossCloud (E := E) G) (crossCloud_involutive G)
  exact
    { rot := B.trans (P.trans B)
      rot_involutive := palindrome_involutive B P
        (cloudFirst_involutive H) (crossCloud_involutive G) }

@[simp] theorem zigzag_rot_apply (G : PortGraph V D) (H : PortGraph D E)
    (v : V) (d : D) (e₁ e₂ : E) :
    (zigzag G H).rot ((v, d), (e₁, e₂)) =
      let first := H.rot (d, e₁)
      let middle := G.rot (v, first.1)
      let last := H.rot (middle.2, e₂)
      ((middle.1, last.1), (last.2, first.2)) := rfl

theorem natCard_square_ports : Nat.card (D × D) = Nat.card D ^ 2 := by
  simp [Nat.card_prod, pow_two]

theorem natCard_square_edges :
    Nat.card (Edge V (D × D)) = Nat.card V * Nat.card D ^ 2 := by
  simp [Edge, Nat.card_prod, pow_two]

theorem natCard_zigzag_vertices :
    Nat.card (V × D) = Nat.card V * Nat.card D := Nat.card_prod V D

theorem natCard_zigzag_ports : Nat.card (E × E) = Nat.card E ^ 2 := by
  simp [Nat.card_prod, pow_two]

theorem natCard_zigzag_edges :
    Nat.card (Edge (V × D) (E × E)) =
      Nat.card V * Nat.card D * Nat.card E ^ 2 := by
  simp [Edge, Nat.card_prod, pow_two]

/-! ## Exact averaging operators -/

noncomputable section

/-- Average only the inner cloud coordinate, keeping the outer vertex fixed. -/
def cloudOperator [Fintype E] (H : PortGraph D E) (f : V × D → ℝ) (x : V × D) : ℝ :=
  SpectralReturn.mean (fun e => f (x.1, (H.rot (x.2, e)).1))

/-- The permutation on cloud vertices induced by the actual outer rotation. -/
def graphPermutation (G : PortGraph V D) (f : V × D → ℝ) (x : V × D) : ℝ :=
  f (G.rot x)

/-- Uniform averaging on the graph square is two original averaging steps. -/
theorem averagingOperator_square [Fintype D] (G : PortGraph V D) (f : V → ℝ) :
    SpectralReturn.averagingOperator (square G) f =
      SpectralReturn.averagingOperator G (SpectralReturn.averagingOperator G f) := by
  funext v
  let w : D × D → ℝ := fun ds => f (G.rot ((G.rot (v, ds.1)).1, ds.2)).1
  change SpectralReturn.mean w =
    SpectralReturn.mean (fun d₁ => SpectralReturn.mean (fun d₂ => w (d₁, d₂)))
  exact SpectralReturn.mean_prod w

/-- The zigzag operator is precisely inner average, outer permutation, inner
average. This factorization follows from the actual rotation formula. -/
theorem averagingOperator_zigzag [Fintype E] (G : PortGraph V D)
    (H : PortGraph D E) (f : V × D → ℝ) :
    SpectralReturn.averagingOperator (zigzag G H) f =
      cloudOperator H (graphPermutation G (cloudOperator H f)) := by
  funext x
  rcases x with ⟨v, d⟩
  let w : E × E → ℝ := fun es =>
    let first := H.rot (d, es.1)
    let middle := G.rot (v, first.1)
    let last := H.rot (middle.2, es.2)
    f (middle.1, last.1)
  change SpectralReturn.mean w =
    SpectralReturn.mean (fun e₁ => SpectralReturn.mean (fun e₂ => w (e₁, e₂)))
  exact SpectralReturn.mean_prod w

end

end MaxCutGames.Foundations.PCP.ZigzagGraphs

/-!
# Energy contraction for the actual zigzag graph

The proof splits a function into its cloud average and its cloud-centered
part. All operators are the actual graph averages from `ZigzagGraphs`.
-/

namespace MaxCutGames.Foundations.PCP.ZigzagSpectral

open PoweringWalks SpectralReturn ZigzagGraphs

noncomputable section

variable {V D E : Type*}

def cloudMean [Fintype D] (f : V × D → ℝ) (v : V) : ℝ :=
  mean (fun d => f (v, d))

def liftCloud (f : V → ℝ) (x : V × D) : ℝ := f x.1

def projection [Fintype D] (f : V × D → ℝ) : V × D → ℝ :=
  liftCloud (cloudMean f)

def centered [Fintype D] (f : V × D → ℝ) : V × D → ℝ := f - projection f

theorem mean_liftCloud [Fintype V] [Fintype D] [Nonempty D] (f : V → ℝ) :
    mean (liftCloud (D := D) f) = mean f := by
  change mean (fun x : V × D => f x.1) = mean f
  rw [mean_prod]
  simp only [mean_const]

theorem energy_liftCloud [Fintype V] [Fintype D] [Nonempty D] (f : V → ℝ) :
    energy (liftCloud (D := D) f) = energy f :=
  mean_liftCloud (D := D) (fun v => f v ^ 2)

theorem mean_cloudMean [Fintype V] [Fintype D] (f : V × D → ℝ) :
    mean (cloudMean f) = mean f := (mean_prod f).symm

theorem energy_prod [Fintype V] [Fintype D] (f : V × D → ℝ) :
    energy f = mean (fun v => energy (fun d => f (v, d))) :=
  mean_prod (fun x => f x ^ 2)

theorem cloudMean_centered_zero [Fintype D] [Nonempty D]
    (f : V × D → ℝ) (v : V) : cloudMean (centered f) v = 0 := by
  change mean (fun d => f (v, d) - cloudMean f v) = 0
  rw [mean_sub, mean_const]
  exact sub_self _

theorem energy_centered_eq_sub [Fintype V] [Fintype D] [Nonempty D]
    (f : V × D → ℝ) : energy (centered f) = energy f - energy (projection f) := by
  have hpoint (v : V) : energy (fun d => centered f (v, d)) =
      energy (fun d => f (v, d)) - (cloudMean f v) ^ 2 := by
    change energy (fun d => f (v, d) - cloudMean f v) = _
    rw [energy_sub_const]
    change energy (fun d => f (v, d)) - 2 * cloudMean f v * cloudMean f v +
      (cloudMean f v) ^ 2 = _
    ring
  have hproj : energy (projection f) = energy (cloudMean f) := energy_liftCloud _
  calc
    energy (centered f) = mean (fun v => energy (fun d => centered f (v, d))) :=
      energy_prod _
    _ = mean (fun v => energy (fun d => f (v, d)) - (cloudMean f v) ^ 2) := by
      congr 1
      funext v
      exact hpoint v
    _ = mean (fun v => energy (fun d => f (v, d))) -
        mean (fun v => (cloudMean f v) ^ 2) := mean_sub _ _
    _ = energy f - energy (projection f) := by
      rw [← energy_prod f, hproj]
      rfl

theorem projection_energy_decomposition [Fintype V] [Fintype D] [Nonempty D]
    (f : V × D → ℝ) : energy f = energy (projection f) + energy (centered f) := by
  rw [energy_centered_eq_sub]
  ring

theorem energy_projection_le [Fintype V] [Fintype D] [Nonempty D]
    (f : V × D → ℝ) : energy (projection f) ≤ energy f := by
  have h := projection_energy_decomposition f
  have hn := energy_nonnegative (centered f)
  linarith

theorem energy_centered_le [Fintype V] [Fintype D] [Nonempty D]
    (f : V × D → ℝ) : energy (centered f) ≤ energy f := by
  rw [energy_centered_eq_sub]
  exact sub_le_self _ (energy_nonnegative _)

/-- Squared normalized mean is bounded by normalized squared energy. -/
theorem mean_sq_le_energy {A : Type*} [Fintype A] [Nonempty A] (f : A → ℝ) :
    mean f ^ 2 ≤ energy f := by
  have h := correlation_sq_le f (fun _ => 1)
  simpa [correlation, energy] using h

/-- Every actual reversible regular graph average is an energy contraction. -/
theorem averaging_energy_le [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
    (G : PortGraph V D) (f : V → ℝ) : energy (averagingOperator G f) ≤ energy f := by
  calc
    energy (averagingOperator G f) ≤ mean (averagingOperator G (fun v => f v ^ 2)) := by
      apply mean_mono
      intro v
      exact mean_sq_le_energy (fun d => f (G.rot (v, d)).1)
    _ = energy f := mean_operator G _

theorem cloudOperator_add [Fintype E] (H : PortGraph D E) (f g : V × D → ℝ) :
    cloudOperator H (f + g) = cloudOperator H f + cloudOperator H g := by
  funext x
  exact mean_add _ _

theorem cloudOperator_sub [Fintype E] (H : PortGraph D E) (f g : V × D → ℝ) :
    cloudOperator H (f - g) = cloudOperator H f - cloudOperator H g := by
  funext x
  exact mean_sub _ _

theorem cloudOperator_projection [Fintype D] [Fintype E] [Nonempty E]
    (H : PortGraph D E) (f : V × D → ℝ) : cloudOperator H (projection f) = projection f := by
  funext x
  change mean (fun _ : E => cloudMean f x.1) = cloudMean f x.1
  exact mean_const _

theorem cloudOperator_centered [Fintype D] [Fintype E] [Nonempty E]
    (H : PortGraph D E) (f : V × D → ℝ) :
    cloudOperator H (centered f) = cloudOperator H f - projection f := by
  unfold centered
  rw [cloudOperator_sub, cloudOperator_projection]

theorem graphPermutation_energy [Fintype V] [Fintype D] (G : PortGraph V D)
    (f : V × D → ℝ) : energy (graphPermutation G f) = energy f :=
  mean_equiv G.rot (fun x => f x ^ 2)

theorem cloud_energy_le [Fintype V] [Fintype D] [Fintype E]
    [Nonempty D] [Nonempty E] (H : PortGraph D E) (f : V × D → ℝ) :
    energy (cloudOperator H f) ≤ energy f := by
  calc
    energy (cloudOperator H f) =
        mean (fun v => energy (averagingOperator H (fun d => f (v, d)))) := energy_prod _
    _ ≤ mean (fun v => energy (fun d => f (v, d))) :=
      mean_mono (fun v => averaging_energy_le H _)
    _ = energy f := (energy_prod f).symm

theorem cloud_centered_contraction [Fintype V] [Fintype D] [Fintype E]
    (H : PortGraph D E) (lambda : ℝ) (hH : SpectralCertificate H lambda)
    (f : V × D → ℝ) (hzero : ∀ v, cloudMean f v = 0) :
    energy (cloudOperator H f) ≤ lambda ^ 2 * energy f := by
  calc
    energy (cloudOperator H f) =
        mean (fun v => energy (averagingOperator H (fun d => f (v, d)))) := energy_prod _
    _ ≤ mean (fun v => lambda ^ 2 * energy (fun d => f (v, d))) :=
      mean_mono (fun v => hH.contraction _ (hzero v))
    _ = lambda ^ 2 * mean (fun v => energy (fun d => f (v, d))) := mean_mul_left _ _
    _ = lambda ^ 2 * energy f := by rw [← energy_prod f]

theorem cloud_residual_bound [Fintype V] [Fintype D] [Fintype E]
    [Nonempty D] [Nonempty E] (H : PortGraph D E) (lambda : ℝ)
    (hH : SpectralCertificate H lambda) (f : V × D → ℝ) :
    energy (cloudOperator H f - projection f) ≤ lambda ^ 2 * energy f := by
  rw [← cloudOperator_centered H f]
  exact (cloud_centered_contraction H lambda hH (centered f)
    (cloudMean_centered_zero f)).trans
      (mul_le_mul_of_nonneg_left (energy_centered_le f) (sq_nonneg lambda))

/-- Cloud projection of the outer permutation is the genuine outer graph
average when its input is constant on each cloud. -/
theorem projection_graphPermutation_projection [Fintype D] (G : PortGraph V D)
    (f : V × D → ℝ) :
    projection (graphPermutation G (projection f)) =
      liftCloud (averagingOperator G (cloudMean f)) := rfl

theorem projected_outer_bound [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (lambda : ℝ) (hG : SpectralCertificate G lambda)
    (f : V × D → ℝ) (hf : mean f = 0) :
    energy (projection (graphPermutation G (projection f))) ≤
      lambda ^ 2 * energy (projection f) := by
  have hzero : mean (cloudMean f) = 0 := (mean_cloudMean f).trans hf
  have hleft : energy (projection (graphPermutation G (projection f))) =
      energy (averagingOperator G (cloudMean f)) := by
    rw [projection_graphPermutation_projection]
    exact energy_liftCloud _
  have hright : energy (projection f) = energy (cloudMean f) := energy_liftCloud _
  rw [hleft, hright]
  exact hG.contraction _ hzero

theorem energy_add_three_le {A : Type*} [Fintype A] (a b c : A → ℝ) :
    energy (a + b + c) ≤ 3 * (energy a + energy b + energy c) := by
  calc
    energy (a + b + c) ≤ mean (fun x => 3 * (a x ^ 2 + b x ^ 2 + c x ^ 2)) := by
      apply mean_mono
      intro x
      change (a x + b x + c x) ^ 2 ≤ 3 * (a x ^ 2 + b x ^ 2 + c x ^ 2)
      nlinarith [sq_nonneg (a x - b x), sq_nonneg (a x - c x), sq_nonneg (b x - c x)]
    _ = 3 * (energy a + energy b + energy c) := by
      rw [mean_mul_left, mean_add, mean_add]
      rfl

/-- A deliberately loose bound sufficient for the expander iteration. -/
theorem zigzag_energy_bound [Fintype V] [Fintype D] [Fintype E]
    [Nonempty D] [Nonempty E]
    (G : PortGraph V D) (H : PortGraph D E) (lambdaG lambdaH : ℝ)
    (hG : SpectralCertificate G lambdaG) (hH : SpectralCertificate H lambdaH)
    (f : V × D → ℝ) (hf : mean f = 0) :
    energy (averagingOperator (zigzag G H) f) ≤
      3 * (lambdaG ^ 2 + lambdaH ^ 2) * energy f := by
  let u := projection f
  let v := centered f
  let a := projection (graphPermutation G u)
  let b := cloudOperator H (graphPermutation G u) - a
  let c := cloudOperator H (graphPermutation G (cloudOperator H v))
  have hfactor : averagingOperator (zigzag G H) f = a + b + c := by
    rw [averagingOperator_zigzag]
    have hf' : f = u + v := by
      funext x
      change f x = projection f x + (f x - projection f x)
      ring
    have hbu : cloudOperator H u = u := cloudOperator_projection H f
    have hbf : cloudOperator H f = u + cloudOperator H v := by
      rw [hf', cloudOperator_add, hbu]
    rw [hbf]
    have hp : graphPermutation G (u + cloudOperator H v) =
        graphPermutation G u + graphPermutation G (cloudOperator H v) := rfl
    rw [hp, cloudOperator_add]
    change cloudOperator H (graphPermutation G u) + c =
      a + (cloudOperator H (graphPermutation G u) - a) + c
    funext x
    change cloudOperator H (graphPermutation G u) x + c x =
      a x + (cloudOperator H (graphPermutation G u) x - a x) + c x
    ring
  have ha : energy a ≤ lambdaG ^ 2 * energy u := projected_outer_bound G lambdaG hG f hf
  have hb : energy b ≤ lambdaH ^ 2 * energy u := by
    have h := cloud_residual_bound H lambdaH hH (graphPermutation G u)
    simpa only [graphPermutation_energy] using h
  have hc : energy c ≤ lambdaH ^ 2 * energy v := by
    calc
      energy c ≤ energy (graphPermutation G (cloudOperator H v)) := cloud_energy_le H _
      _ = energy (cloudOperator H v) := graphPermutation_energy G _
      _ ≤ lambdaH ^ 2 * energy v :=
        cloud_centered_contraction H lambdaH hH v (cloudMean_centered_zero f)
  rw [hfactor]
  calc
    energy (a + b + c) ≤ 3 * (energy a + energy b + energy c) := energy_add_three_le _ _ _
    _ ≤ 3 * (lambdaG ^ 2 * energy u + lambdaH ^ 2 * energy u + lambdaH ^ 2 * energy v) :=
      mul_le_mul_of_nonneg_left (add_le_add (add_le_add ha hb) hc) (by norm_num)
    _ ≤ 3 * (lambdaG ^ 2 + lambdaH ^ 2) * energy f := by
      have hdecomp : energy f = energy u + energy v := projection_energy_decomposition f
      rw [hdecomp]
      have hn := mul_nonneg (sq_nonneg lambdaG) (energy_nonnegative v)
      nlinarith

theorem square_certificate [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
    (G : PortGraph V D) (lambda : ℝ) (hG : SpectralCertificate G lambda) :
    SpectralCertificate (square G) (lambda ^ 2) where
  nonnegative := sq_nonneg lambda
  lt_one := by
    have hprod := mul_pos (sub_pos.mpr hG.lt_one)
      (show 0 < 1 + lambda by linarith [hG.nonnegative])
    nlinarith
  contraction f hf := by
    rw [averagingOperator_square]
    calc
      energy (averagingOperator G (averagingOperator G f)) ≤
          lambda ^ 2 * energy (averagingOperator G f) :=
        hG.contraction _ ((mean_operator G f).trans hf)
      _ ≤ lambda ^ 2 * (lambda ^ 2 * energy f) :=
        mul_le_mul_of_nonneg_left (hG.contraction f hf) (sq_nonneg lambda)
      _ = (lambda ^ 2) ^ 2 * energy f := by ring

/-- Squaring an outer graph with certificate `1/2`, then zigzagging with an
inner graph with certificate `1/100`, again has certificate `1/2`. -/
theorem square_zigzag_halfCertificate [Fintype V] [Fintype D] [Fintype E]
    [Nonempty V] [Nonempty D] [Nonempty E]
    (G : PortGraph V D) (H : PortGraph (D × D) E)
    (hG : SpectralCertificate G (1 / 2)) (hH : SpectralCertificate H (1 / 100)) :
    SpectralCertificate (zigzag (square G) H) (1 / 2) where
  nonnegative := by norm_num
  lt_one := by norm_num
  contraction f hf := by
    have h := zigzag_energy_bound (square G) H ((1 / 2 : ℝ) ^ 2) (1 / 100)
      (square_certificate G (1 / 2) hG) hH f hf
    have hcoef : 3 * (((1 / 2 : ℝ) ^ 2) ^ 2 + (1 / 100) ^ 2) ≤ (1 / 2 : ℝ) ^ 2 := by
      norm_num
    exact h.trans (mul_le_mul_of_nonneg_right hcoef (energy_nonnegative f))

end

end MaxCutGames.Foundations.PCP.ZigzagSpectral

end OAI