Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.List.OfFn
import Mathlib.Tactic.FieldSimp
import OAI.Computability.BinPacking.Expanders.ExpanderTableEnumeration
import OAI.Computability.BinPacking.PCP.PoweringMomentBound
import OAI.Computability.BinPacking.PCP.RawInitialPredicate

namespace OAI

section

open scoped BigOperators

namespace BinPackingGames.Foundations.PCP.Overlay

open PoweringWalks SpectralReturn

variable {V D E A : Type*}

def originalPortGraph (G : ConstraintGraph V (V × D) A) : PortGraph V D where
  rot := G.reverse
  rot_involutive := G.reverse_involutive

def rotate (G : PortGraph V D) (H : PortGraph V E) :
    V × (D ⊕ E) → V × (D ⊕ E)
  | (v, Sum.inl d) => ((G.rot (v, d)).1, Sum.inl (G.rot (v, d)).2)
  | (v, Sum.inr e) => ((H.rot (v, e)).1, Sum.inr (H.rot (v, e)).2)

theorem rotate_involutive (G : PortGraph V D) (H : PortGraph V E) :
    Function.Involutive (rotate G H) := by
  rintro ⟨v, p⟩
  rcases p with d | e
  · exact congrArg (fun a : V × D => (a.1, (Sum.inl a.2 : D ⊕ E)))
      (G.rot_involutive (v, d))
  · exact congrArg (fun a : V × E => (a.1, (Sum.inr a.2 : D ⊕ E)))
      (H.rot_involutive (v, e))

def portGraph (G : PortGraph V D) (H : PortGraph V E) : PortGraph V (D ⊕ E) where
  rot :=
    { toFun := rotate G H
      invFun := rotate G H
      left_inv := rotate_involutive G H
      right_inv := rotate_involutive G H }
  rot_involutive := rotate_involutive G H

@[simp] theorem portGraph_rot_inl (G : PortGraph V D) (H : PortGraph V E)
    (v : V) (d : D) :
    (portGraph G H).rot (v, Sum.inl d) =
      ((G.rot (v, d)).1, Sum.inl (G.rot (v, d)).2) := rfl

@[simp] theorem portGraph_rot_inr (G : PortGraph V D) (H : PortGraph V E)
    (v : V) (e : E) :
    (portGraph G H).rot (v, Sum.inr e) =
      ((H.rot (v, e)).1, Sum.inr (H.rot (v, e)).2) := rfl

def constraintGraph (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) :
    ConstraintGraph V (V × (D ⊕ E)) A where
  reverse := (portGraph (originalPortGraph G) H).rot
  reverse_involutive := (portGraph (originalPortGraph G) H).rot_involutive
  tail := Prod.fst
  accepts p a b := match p.2 with
    | Sum.inl d => G.accepts (p.1, d) a b
    | Sum.inr _ => true
  reverse_accepts := by
    rintro ⟨v, p⟩ a b
    rcases p with d | e
    · exact G.reverse_accepts (v, d) a b
    · rfl

@[simp] theorem constraintGraph_tail (G : ConstraintGraph V (V × D) A)
    (H : PortGraph V E) : (constraintGraph G H).tail = Prod.fst := rfl

@[simp] theorem constraintGraph_portGraph (G : ConstraintGraph V (V × D) A)
    (H : PortGraph V E) :
    originalPortGraph (constraintGraph G H) = portGraph (originalPortGraph G) H := rfl

@[simp] theorem edgeSatisfied_inl (G : ConstraintGraph V (V × D) A)
    (H : PortGraph V E) (htail : G.tail = Prod.fst)
    (labeling : V → A) (v : V) (d : D) :
    (constraintGraph G H).edgeSatisfied labeling (v, Sum.inl d) =
      G.edgeSatisfied labeling (v, d) := by
  change G.accepts (v, d) (labeling v) (labeling (G.reverse (v, d)).1) =
    G.accepts (v, d) (labeling (G.tail (v, d)))
      (labeling (G.tail (G.reverse (v, d))))
  rw [htail]

@[simp] theorem edgeSatisfied_inr (G : ConstraintGraph V (V × D) A)
    (H : PortGraph V E) (labeling : V → A) (v : V) (e : E) :
    (constraintGraph G H).edgeSatisfied labeling (v, Sum.inr e) = true := rfl

theorem complete (G : ConstraintGraph V (V × D) A) (H : PortGraph V E)
    (htail : G.tail = Prod.fst) (labeling : V → A)
    (h : ∀ a, G.edgeSatisfied labeling a = true) :
    ∀ a, (constraintGraph G H).edgeSatisfied labeling a = true := by
  rintro ⟨v, p⟩
  rcases p with d | e
  · rw [edgeSatisfied_inl G H htail]
    exact h (v, d)
  · rfl

theorem satisfiable_iff (G : ConstraintGraph V (V × D) A) (H : PortGraph V E)
    (htail : G.tail = Prod.fst) :
    (constraintGraph G H).Satisfiable ↔ G.Satisfiable := by
  constructor
  · rintro ⟨labeling, h⟩
    refine ⟨labeling, ?_⟩
    rintro ⟨v, d⟩
    rw [← edgeSatisfied_inl G H htail]
    exact h (v, Sum.inl d)
  · rintro ⟨labeling, h⟩
    exact ⟨labeling, complete G H htail labeling h⟩

theorem rejectionCount_eq [Fintype V] [Fintype D] [Fintype E]
    (G : ConstraintGraph V (V × D) A) (H : PortGraph V E)
    (htail : G.tail = Prod.fst) (labeling : V → A) :
    (constraintGraph G H).rejectionCount labeling = G.rejectionCount labeling := by
  classical
  simp only [DegreeReplacement.rejectionCount_eq_sum, Fintype.sum_prod_type,
    Fintype.sum_sum_type]
  simp [edgeSatisfied_inl G H htail]

theorem card_ports [Fintype D] [Fintype E] :
    Fintype.card (D ⊕ E) = Fintype.card D + Fintype.card E := Fintype.card_sum

theorem card_darts [Fintype V] [Fintype D] [Fintype E] :
    Fintype.card (V × (D ⊕ E)) =
      Fintype.card (V × D) + Fintype.card (V × E) := by
  simp only [Fintype.card_prod, Fintype.card_sum, Nat.mul_add]

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

omit [Fintype V] in

theorem operator_mix [Nonempty D] [Nonempty E]
    (G : PortGraph V D) (H : PortGraph V E) (f : V → ℝ) (v : V) :
    averagingOperator (portGraph G H) f v =
      (Fintype.card D : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) *
        averagingOperator G f v +
      (Fintype.card E : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) *
        averagingOperator H f v := by
  have hd : (0 : ℝ) < Fintype.card D := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card D)
  have he : (0 : ℝ) < Fintype.card E := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card E)
  change mean (fun p : D ⊕ E => f ((portGraph G H).rot (v, p)).1) = _
  rw [mean_eq_sum_div_card]
  simp only [Fintype.sum_sum_type, Fintype.card_sum, Nat.cast_add,
    portGraph_rot_inl, portGraph_rot_inr]
  unfold averagingOperator
  rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card]
  field_simp [hd.ne', he.ne', (add_pos hd he).ne']

theorem energy_convex_mix (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1)
    (f g : V → ℝ) :
    energy (fun v => a * f v + b * g v) ≤ a * energy f + b * energy g := by
  calc
    energy (fun v => a * f v + b * g v) ≤
        mean (fun v => a * (f v) ^ 2 + b * (g v) ^ 2) := by
      apply mean_mono
      intro v
      have hnormalize := congrArg (fun t : ℝ => t * (a * (f v) ^ 2 + b * (g v) ^ 2)) hab
      have hnonneg := mul_nonneg (mul_nonneg ha hb) (sq_nonneg (f v - g v))
      nlinarith
    _ = a * energy f + b * energy g := by
      rw [mean_add, mean_mul_left, mean_mul_left]
      rfl

theorem energy_bound [Nonempty V] [Nonempty D] [Nonempty E]
    (G : PortGraph V D) (H : PortGraph V E) (lambda : ℝ)
    (certificate : SpectralCertificate H lambda) (f : V → ℝ) (hf : mean f = 0) :
    energy (averagingOperator (portGraph G H) f) ≤
      ((Fintype.card D : ℝ) + (Fintype.card E : ℝ) * lambda ^ 2) /
        ((Fintype.card D : ℝ) + Fintype.card E) * energy f := by
  let a : ℝ := (Fintype.card D : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E)
  let b : ℝ := (Fintype.card E : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E)
  have hd : (0 : ℝ) < Fintype.card D := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card D)
  have he : (0 : ℝ) < Fintype.card E := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card E)
  have hden : (0 : ℝ) < (Fintype.card D : ℝ) + Fintype.card E := add_pos hd he
  have ha : 0 ≤ a := div_nonneg hd.le hden.le
  have hb : 0 ≤ b := div_nonneg he.le hden.le
  have hab : a + b = 1 := by
    dsimp [a, b]
    rw [← add_div, div_self hden.ne']
  have hop : averagingOperator (portGraph G H) f =
      fun v => a * averagingOperator G f v + b * averagingOperator H f v := by
    funext v
    exact operator_mix G H f v
  calc
    energy (averagingOperator (portGraph G H) f) ≤
        a * energy (averagingOperator G f) + b * energy (averagingOperator H f) := by
      rw [hop]
      exact energy_convex_mix a b ha hb hab _ _
    _ ≤ a * energy f + b * (lambda ^ 2 * energy f) :=
      add_le_add
        (mul_le_mul_of_nonneg_left (ZigzagSpectral.averaging_energy_le G f) ha)
        (mul_le_mul_of_nonneg_left (certificate.contraction f hf) hb)
    _ = _ := by
      dsimp [a, b]
      ring

theorem energy_bound_three_quarters [Nonempty V]
    (G : PortGraph V D) (H : PortGraph V E)
    (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E)
    (certificate : SpectralCertificate H (1 / 2 : ℝ))
    (f : V → ℝ) (hf : mean f = 0) :
    energy (averagingOperator (portGraph G H) f) ≤ (3 / 4 : ℝ) * energy f := by
  let _ : Nonempty E := Fintype.card_pos_iff.mp (lt_of_lt_of_le Nat.zero_lt_one hpositive)
  let _ : Nonempty D := Fintype.card_pos_iff.mp (by omega)
  have he : (1 : ℝ) ≤ Fintype.card E := by exact_mod_cast hpositive
  have hd : (Fintype.card D : ℝ) = (Fintype.card E : ℝ) + 1 := by exact_mod_cast hdegree
  have hden : (0 : ℝ) < (Fintype.card D : ℝ) + Fintype.card E := by linarith
  have hcoefficient :
      ((Fintype.card D : ℝ) + (Fintype.card E : ℝ) * (1 / 2 : ℝ) ^ 2) /
        ((Fintype.card D : ℝ) + Fintype.card E) ≤ (3 / 4 : ℝ) := by
    apply (div_le_iff₀ hden).mpr
    nlinarith
  exact (energy_bound G H (1 / 2) certificate f hf).trans
    (mul_le_mul_of_nonneg_right hcoefficient (energy_nonnegative f))

theorem spectralCertificate_seven_eighths [Nonempty V]
    (G : PortGraph V D) (H : PortGraph V E)
    (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E)
    (certificate : SpectralCertificate H (1 / 2 : ℝ)) :
    SpectralCertificate (portGraph G H) (7 / 8 : ℝ) where
  nonnegative := by norm_num
  lt_one := by norm_num
  contraction := by
    intro f hf
    exact (energy_bound_three_quarters G H hdegree hpositive certificate f hf).trans
      (mul_le_mul_of_nonneg_right (by norm_num : (3 / 4 : ℝ) ≤ (7 / 8 : ℝ) ^ 2)
        (energy_nonnegative f))

theorem constraintGraph_spectralCertificate [Nonempty V]
    (G : ConstraintGraph V (V × D) A) (H : PortGraph V E)
    (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E)
    (certificate : SpectralCertificate H (1 / 2 : ℝ)) :
    SpectralCertificate (originalPortGraph (constraintGraph G H)) (7 / 8 : ℝ) := by
  rw [constraintGraph_portGraph]
  exact spectralCertificate_seven_eighths (originalPortGraph G) H
    hdegree hpositive certificate

end BinPackingGames.Foundations.PCP.Overlay

end

namespace BinPackingGames.Foundations.PCP.LazyConstraint

open scoped BigOperators
open PoweringWalks

variable {V D A : Type*}

def constraintGraph (G : ConstraintGraph V (V × D) A) :
    ConstraintGraph V (V × (Bool × D)) A where
  reverse := (lazyGraph (Overlay.originalPortGraph G)).rot
  reverse_involutive := (lazyGraph (Overlay.originalPortGraph G)).rot_involutive
  tail := Prod.fst
  accepts e a b := if e.2.1 then G.accepts (e.1,e.2.2) a b else true
  reverse_accepts := by
    rintro ⟨v,b,d⟩ a c
    cases b with
    | false => rfl
    | true => exact G.reverse_accepts (v,d) a c

@[simp] theorem constraintGraph_tail (G : ConstraintGraph V (V × D) A) :
    (constraintGraph G).tail = Prod.fst := rfl

@[simp] theorem constraintGraph_reverse (G : ConstraintGraph V (V × D) A) :
    (constraintGraph G).reverse = (lazyGraph (Overlay.originalPortGraph G)).rot := rfl

@[simp] theorem constraintGraph_portGraph (G : ConstraintGraph V (V × D) A) :
    Overlay.originalPortGraph (constraintGraph G) =
      lazyGraph (Overlay.originalPortGraph G) := rfl

@[simp] theorem edgeSatisfied_false (G : ConstraintGraph V (V × D) A)
    (labeling : V → A) (v : V) (d : D) :
    (constraintGraph G).edgeSatisfied labeling (v,(false,d)) = true := rfl

@[simp] theorem edgeSatisfied_true (G : ConstraintGraph V (V × D) A)
    (htail : G.tail = Prod.fst) (labeling : V → A) (v : V) (d : D) :
    (constraintGraph G).edgeSatisfied labeling (v,(true,d)) =
      G.edgeSatisfied labeling (v,d) := by
  change G.accepts (v,d) (labeling v) (labeling (G.reverse (v,d)).1) =
    G.accepts (v,d) (labeling (G.tail (v,d)))
      (labeling (G.tail (G.reverse (v,d))))
  rw [htail]

theorem complete (G : ConstraintGraph V (V × D) A) (htail : G.tail = Prod.fst)
    (labeling : V → A) (h : ∀ e, G.edgeSatisfied labeling e = true) :
    ∀ e, (constraintGraph G).edgeSatisfied labeling e = true := by
  rintro ⟨v,b,d⟩
  cases b with
  | false => rfl
  | true =>
    rw [edgeSatisfied_true G htail]
    exact h (v,d)

theorem complete_iff (G : ConstraintGraph V (V × D) A) (htail : G.tail = Prod.fst)
    (labeling : V → A) :
    (∀ e, (constraintGraph G).edgeSatisfied labeling e = true) ↔
      ∀ e, G.edgeSatisfied labeling e = true := by
  constructor
  · intro h
    rintro ⟨v,d⟩
    rw [← edgeSatisfied_true G htail]
    exact h (v,(true,d))
  · exact complete G htail labeling

theorem satisfiable_iff (G : ConstraintGraph V (V × D) A) (htail : G.tail = Prod.fst) :
    (constraintGraph G).Satisfiable ↔ G.Satisfiable := by
  constructor
  · rintro ⟨labeling,h⟩
    exact ⟨labeling, (complete_iff G htail labeling).mp h⟩
  · rintro ⟨labeling,h⟩
    exact ⟨labeling, complete G htail labeling h⟩

theorem rejectionCount_eq [Fintype V] [Fintype D]
    (G : ConstraintGraph V (V × D) A) (htail : G.tail = Prod.fst)
    (labeling : V → A) :
    (constraintGraph G).rejectionCount labeling = G.rejectionCount labeling := by
  classical
  simp only [DegreeReplacement.rejectionCount_eq_sum, Fintype.sum_prod_type,
    Fintype.sum_bool]
  simp [edgeSatisfied_true G htail]

theorem card_ports [Fintype D] :
    Fintype.card (Bool × D) = 2 * Fintype.card D := by
  simp

theorem card_darts [Fintype V] [Fintype D] :
    Fintype.card (V × (Bool × D)) = 2 * Fintype.card (V × D) := by
  simp [Fintype.card_prod, Nat.mul_left_comm]

theorem rejection_density_eq_half [Fintype V] [Fintype D]
    (G : ConstraintGraph V (V × D) A) (htail : G.tail = Prod.fst)
    (labeling : V → A) :
    ((constraintGraph G).rejectionCount labeling : ℝ) /
        (Fintype.card (V × (Bool × D)) : ℝ) =
      ((G.rejectionCount labeling : ℝ) / (Fintype.card (V × D) : ℝ)) / 2 := by
  rw [rejectionCount_eq G htail, card_darts]
  simp only [Nat.cast_mul, Nat.cast_ofNat, div_eq_mul_inv, mul_inv_rev]
  ring

end BinPackingGames.Foundations.PCP.LazyConstraint

namespace BinPackingGames.Foundations.PCP.PreprocessingOverlayTables

open PoweringWalks

def overlayPorts (d e : Nat) : (Fin d ⊕ Fin e) ≃ Fin (d + e) := finSumFinEquiv

@[simp] theorem overlayPorts_inl_val (d e : Nat) (i : Fin d) :
    (overlayPorts d e (Sum.inl i)).val = i.val := rfl

@[simp] theorem overlayPorts_inr_val (d e : Nat) (i : Fin e) :
    (overlayPorts d e (Sum.inr i)).val = d + i.val := rfl

def lazyPorts (d : Nat) : (Bool × Fin d) ≃ Fin (2 * d) :=
  (Equiv.prodCongr finTwoEquiv.symm (Equiv.refl (Fin d))).trans finProdFinEquiv

@[simp] theorem lazyPorts_false_val (d : Nat) (i : Fin d) :
    (lazyPorts d (false, i)).val = i.val := by
  simp [lazyPorts, finTwoEquiv, finProdFinEquiv]

@[simp] theorem lazyPorts_true_val (d : Nat) (i : Fin d) :
    (lazyPorts d (true, i)).val = i.val + d := by
  simp [lazyPorts, finTwoEquiv, finProdFinEquiv]

def materialize {n d : Nat} {D : Type*}
    (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label)
    (ports : D ≃ Fin d) : PortTables.Table n d :=
  PortTables.ofPortGraph
    (GraphTransport.reindex (Overlay.originalPortGraph G) (Equiv.refl _) ports)
    (fun x a b => G.accepts (x.1, ports.symm x.2) a b)
    (by
      intro x a b
      simp only [GraphTransport.reindex, Overlay.originalPortGraph,
        Equiv.trans_apply, Equiv.prodCongr_apply, Equiv.prodCongr_symm,
        Equiv.refl_apply, Equiv.symm_apply_apply, Prod.map]
      convert G.reverse_accepts (x.1, ports.symm x.2) a b using 1
      rfl)

theorem rotation_materialize {n d : Nat} {D : Type*}
    (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label)
    (ports : D ≃ Fin d) (x : Fin n × Fin d) :
    PortTables.rotation (materialize G ports) x =
      ((G.reverse (x.1, ports.symm x.2)).1,
        ports (G.reverse (x.1, ports.symm x.2)).2) := by
  simp only [materialize, PortTables.rotation_ofPortGraph, GraphTransport.reindex,
    Overlay.originalPortGraph, Equiv.trans_apply, Equiv.prodCongr_apply,
    Equiv.prodCongr_symm, Equiv.refl_apply, Prod.map]
  rfl

theorem rotation_materialize_image {n d : Nat} {D : Type*}
    (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label)
    (ports : D ≃ Fin d) (v : Fin n) (i : D) :
    PortTables.rotation (materialize G ports) (v, ports i) =
      ((G.reverse (v, i)).1, ports (G.reverse (v, i)).2) := by
  rw [rotation_materialize, Equiv.symm_apply_apply]

@[simp] theorem accepts_materialize {n d : Nat} {D : Type*}
    (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label)
    (ports : D ≃ Fin d) (x : Fin n × Fin d) (a b : PortTables.Label) :
    PortTables.accepts (materialize G ports) x a b =
      G.accepts (x.1, ports.symm x.2) a b := by
  exact PortTables.accepts_ofPortGraph _ _ _ x a b

private theorem constraintGraph_ext {V E A : Type*} {G H : ConstraintGraph V E A}
    (hreverse : ∀ e, G.reverse e = H.reverse e)
    (htail : ∀ e, G.tail e = H.tail e)
    (haccepts : ∀ e a b, G.accepts e a b = H.accepts e a b) : G = H := by
  rcases G with ⟨r, hi, t, p, hp⟩
  rcases H with ⟨r', hi', t', p', hp'⟩
  have hr : r = r' := Equiv.ext hreverse
  cases hr
  have ht : t = t' := funext htail
  cases ht
  have ha : p = p' := funext fun e => funext fun a => funext fun b => haccepts e a b
  cases ha
  rfl

theorem baseGraph_materialize {n d : Nat} {D : Type*}
    (G : ConstraintGraph (Fin n) (Fin n × D) PortTables.Label)
    (ports : D ≃ Fin d) (htail : G.tail = Prod.fst) :
    PortTables.baseGraph (materialize G ports) =
      G.reindex (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) ports) (Equiv.refl _) := by
  apply constraintGraph_ext
  · intro x
    rw [PortTables.baseGraph_reverse, rotation_materialize]
    rfl
  · intro x
    change x.1 = G.tail (x.1, ports.symm x.2)
    rw [htail]
  · intro x a b
    rw [PortTables.baseGraph_accepts, accepts_materialize]
    rfl

def overlay {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
    PortTables.Table n (d + e) :=
  materialize (Overlay.constraintGraph (PortTables.baseGraph G) (ExpanderTables.graph H))
    (overlayPorts d e)

theorem overlay_rotation_old {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin d) :
    PortTables.rotation (overlay G H) (v, overlayPorts d e (Sum.inl i)) =
      ((PortTables.rotation G (v, i)).1,
        overlayPorts d e (Sum.inl (PortTables.rotation G (v, i)).2)) := by
  exact rotation_materialize_image _ _ v (Sum.inl i)

theorem overlay_rotation_expander {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin e) :
    PortTables.rotation (overlay G H) (v, overlayPorts d e (Sum.inr i)) =
      ((ExpanderTables.lookup H (v, i)).1,
        overlayPorts d e (Sum.inr (ExpanderTables.lookup H (v, i)).2)) := by
  exact rotation_materialize_image _ _ v (Sum.inr i)

@[simp] theorem overlay_accepts_old {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin d) (a b : PortTables.Label) :
    PortTables.accepts (overlay G H) (v, overlayPorts d e (Sum.inl i)) a b =
      PortTables.accepts G (v, i) a b := by
  simp only [overlay, accepts_materialize, Equiv.symm_apply_apply]
  rfl

@[simp] theorem overlay_accepts_expander {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) (i : Fin e) (a b : PortTables.Label) :
    PortTables.accepts (overlay G H) (v, overlayPorts d e (Sum.inr i)) a b = true := by
  simp only [overlay, accepts_materialize, Equiv.symm_apply_apply]
  rfl

theorem overlay_semantics {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) :
    PortTables.baseGraph (overlay G H) =
      (Overlay.constraintGraph (PortTables.baseGraph G) (ExpanderTables.graph H)).reindex
        (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) (overlayPorts d e))
        (Equiv.refl _) :=
  baseGraph_materialize _ _ rfl

def lazy {n d : Nat} (G : PortTables.Table n d) : PortTables.Table n (2 * d) :=
  materialize (LazyConstraint.constraintGraph (PortTables.baseGraph G)) (lazyPorts d)

theorem lazy_rotation_false {n d : Nat} (G : PortTables.Table n d)
    (v : Fin n) (i : Fin d) :
    PortTables.rotation (lazy G) (v, lazyPorts d (false, i)) =
      (v, lazyPorts d (false, i)) := by
  exact rotation_materialize_image _ _ v (false, i)

theorem lazy_rotation_true {n d : Nat} (G : PortTables.Table n d)
    (v : Fin n) (i : Fin d) :
    PortTables.rotation (lazy G) (v, lazyPorts d (true, i)) =
      ((PortTables.rotation G (v, i)).1,
        lazyPorts d (true, (PortTables.rotation G (v, i)).2)) := by
  exact rotation_materialize_image _ _ v (true, i)

@[simp] theorem lazy_accepts_false {n d : Nat} (G : PortTables.Table n d)
    (v : Fin n) (i : Fin d) (a b : PortTables.Label) :
    PortTables.accepts (lazy G) (v, lazyPorts d (false, i)) a b = true := by
  simp only [lazy, accepts_materialize, Equiv.symm_apply_apply]
  rfl

@[simp] theorem lazy_accepts_true {n d : Nat} (G : PortTables.Table n d)
    (v : Fin n) (i : Fin d) (a b : PortTables.Label) :
    PortTables.accepts (lazy G) (v, lazyPorts d (true, i)) a b =
      PortTables.accepts G (v, i) a b := by
  simp only [lazy, accepts_materialize, Equiv.symm_apply_apply]
  rfl

theorem lazy_semantics {n d : Nat} (G : PortTables.Table n d) :
    PortTables.baseGraph (lazy G) =
      (LazyConstraint.constraintGraph (PortTables.baseGraph G)).reindex
        (Equiv.refl _) (Equiv.prodCongr (Equiv.refl _) (lazyPorts d)) (Equiv.refl _) :=
  baseGraph_materialize _ _ rfl

end BinPackingGames.Foundations.PCP.PreprocessingOverlayTables

namespace BinPackingGames.Foundations.PCP.PreprocessingLazyWords

open PortTables GraphTables PreprocessingOverlayTables
open BinPackingGames.Foundations.Complexity

variable {n d : Nat}

def row (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    DartRow n (n * d) :=
  ⟨v, table.reverseIndex[rowIndex n d (v, p)], table.relations[rowIndex n d (v, p)]⟩

def trueRelation : RelationTable := Vector.replicate 4096 true

def stayRow (d : Nat) (v : Fin n) (p : Fin d) : DartRow n (n * (2 * d)) :=
  ⟨v, rowIndex n (2 * d) (v, lazyPorts d (false, p)), trueRelation⟩

def moveRow (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    DartRow n (n * (2 * d)) :=
  ⟨v, rowIndex n (2 * d) ((rotation table (v, p)).1,
    lazyPorts d (true, (rotation table (v, p)).2)),
    table.relations[rowIndex n d (v, p)]⟩

def reverseMap (d j : Nat) : Nat := 2 * d * (j / d) + d + j % d

private theorem relation_ext {r s : RelationTable}
    (h : ∀ a b, relationAt r a b = relationAt s a b) : r = s := by
  apply Vector.ext
  intro i hi
  simpa only [relationAt, Prod.eta, Equiv.apply_symm_apply, Fin.getElem_fin] using
    h (relationIndex.symm ⟨i, hi⟩).1 (relationIndex.symm ⟨i, hi⟩).2

theorem row_lazy_stay (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    row (lazy table) v (lazyPorts d (false, p)) = stayRow d v p := by
  have hr : (lazy table).reverseIndex[rowIndex n (2 * d) (v, lazyPorts d (false, p))] =
      rowIndex n (2 * d) (v, lazyPorts d (false, p)) := by
    rw [← rowIndex_rotation, lazy_rotation_false]
  have hp : (lazy table).relations[rowIndex n (2 * d) (v, lazyPorts d (false, p))] =
      trueRelation := by
    apply relation_ext
    intro a b
    simpa only [PortTables.accepts, relationAt, trueRelation, Fin.getElem_fin,
      Vector.getElem_replicate] using lazy_accepts_false table v p a b
  exact congrArg₂ (DartRow.mk v) hr hp

theorem row_lazy_move (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    row (lazy table) v (lazyPorts d (true, p)) = moveRow table v p := by
  have hr : (lazy table).reverseIndex[rowIndex n (2 * d) (v, lazyPorts d (true, p))] =
      rowIndex n (2 * d) ((rotation table (v, p)).1,
        lazyPorts d (true, (rotation table (v, p)).2)) := by
    rw [← rowIndex_rotation, lazy_rotation_true]
  have hp : (lazy table).relations[rowIndex n (2 * d) (v, lazyPorts d (true, p))] =
      table.relations[rowIndex n d (v, p)] := by
    apply relation_ext
    intro a b
    exact lazy_accepts_true table v p a b
  exact congrArg₂ (DartRow.mk v) hr hp

theorem stayRow_reverse_val (v : Fin n) (p : Fin d) :
    (stayRow d v p).reverseIndex.val = 2 * d * v.val + p.val := by
  simp only [stayRow, rowIndex_val, lazyPorts_false_val]
  omega

theorem moveRow_reverse_val (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    (moveRow table v p).reverseIndex.val =
      reverseMap d (row table v p).reverseIndex.val := by
  simp only [moveRow, rowIndex_val, lazyPorts_true_val, reverseMap, row]
  change (table.reverseIndex[rowIndex n d (v, p)]).val % d + d +
      (2 * d) * ((table.reverseIndex[rowIndex n d (v, p)]).val / d) = _
  omega

theorem relationWords_true : relationWords trueRelation = List.replicate 4096 1 := by
  simp only [relationWords, trueRelation, Vector.toList_replicate,
    List.map_replicate, bitWord, ite_true]

theorem stayRow_words (v : Fin n) (p : Fin d) :
    rowWords (stayRow d v p) =
      [v.val, 2 * d * v.val + p.val] ++ List.replicate 4096 1 := by
  rw [rowWords, stayRow_reverse_val]
  exact congrArg (fun words => [v.val, 2 * d * v.val + p.val] ++ words) relationWords_true

theorem moveRow_words (table : PortTables.Table n d) (v : Fin n) (p : Fin d) :
    rowWords (moveRow table v p) =
      [v.val, reverseMap d (row table v p).reverseIndex.val] ++
        relationWords (row table v p).relation := by
  rw [rowWords, moveRow_reverse_val]
  rfl

theorem ofFn_rowIndex {α : Type*} (f : Fin (n * d) → α) :
    List.ofFn f = (List.ofFn fun v : Fin n =>
      List.ofFn fun p : Fin d => f (rowIndex n d (v, p))).flatten := by
  rw [List.ofFn_mul]
  apply congrArg List.flatten
  congr 1
  funext v
  congr 1
  funext p
  apply congrArg f
  apply Fin.ext
  simp only [rowIndex_val]
  ac_rfl

theorem flatRows_list (table : PortTables.Table n d) :
    (flatRows table).toList =
      (List.ofFn fun v : Fin n => List.ofFn (row table v)).flatten := by
  rw [flatRows, Vector.toList_ofFn, ofFn_rowIndex]
  simp only [Equiv.symm_apply_apply]
  rfl

theorem ofFn_lazyPorts {α : Type*} (f : Fin (2 * d) → α) :
    List.ofFn f = List.ofFn (fun p => f (lazyPorts d (false, p))) ++
      List.ofFn (fun p => f (lazyPorts d (true, p))) := by
  have hf (p : Fin d) : rowIndex 2 d (0, p) = lazyPorts d (false, p) := by
    apply Fin.ext
    simp only [rowIndex_val, Fin.val_zero, Nat.mul_zero, Nat.add_zero, lazyPorts_false_val]
  have ht (p : Fin d) : rowIndex 2 d ((0 : Fin 1).succ, p) = lazyPorts d (true, p) := by
    apply Fin.ext
    simp only [rowIndex_val, Fin.val_succ, Fin.val_zero, Nat.zero_add,
      Nat.mul_one, lazyPorts_true_val]
  simpa only [List.ofFn_succ, List.ofFn_zero, List.flatten_cons, List.flatten_nil,
    List.append_nil, hf, ht] using (ofFn_rowIndex (n := 2) (d := d) f)

def vertexRows (table : PortTables.Table n d) (v : Fin n) :
    List (DartRow n (n * (2 * d))) :=
  List.ofFn (stayRow d v) ++ List.ofFn (moveRow table v)

theorem vertexRows_length (table : PortTables.Table n d) (v : Fin n) :
    (vertexRows table v).length = 2 * d := by
  simp only [vertexRows, List.length_append, List.length_ofFn]
  omega

theorem flatRows_lazy (table : PortTables.Table n d) :
    (flatRows (lazy table)).toList = (List.ofFn (vertexRows table)).flatten := by
  rw [flatRows_list]
  apply congrArg List.flatten
  congr 1
  funext v
  rw [ofFn_lazyPorts]
  simp only [row_lazy_stay, row_lazy_move]
  rfl

theorem vertexRows_words (table : PortTables.Table n d) (v : Fin n) :
    (vertexRows table v).flatMap rowWords =
      (List.ofFn fun p : Fin d =>
        [v.val, 2 * d * v.val + p.val] ++ List.replicate 4096 1).flatten ++
      (List.ofFn fun p : Fin d =>
        [v.val, reverseMap d (row table v p).reverseIndex.val] ++
          relationWords (row table v p).relation).flatten := by
  have hs : rowWords ∘ stayRow d v = fun p : Fin d =>
      [v.val, 2 * d * v.val + p.val] ++ List.replicate 4096 1 := by
    funext p
    exact stayRow_words v p
  have hm : rowWords ∘ moveRow table v = fun p : Fin d =>
      [v.val, reverseMap d (row table v p).reverseIndex.val] ++
        relationWords (row table v p).relation := by
    funext p
    exact moveRow_words table v p
  simp only [vertexRows, List.flatMap_def, List.map_append, List.map_ofFn,
    List.flatten_append]
  rw [hs, hm]

theorem tableWords_lazy (table : PortTables.Table n d) :
    PortTables.tableWords (lazy table) =
      [n, n * (2 * d)] ++ (List.ofFn (vertexRows table)).flatten.flatMap rowWords := by
  rw [PortTables.tableWords_eq, flatRows_lazy]

theorem tableBits_lazy (table : PortTables.Table n d) :
    PortTables.tableBits (lazy table) =
      encodeWords ([n, n * (2 * d)] ++
        (List.ofFn (vertexRows table)).flatten.flatMap rowWords) := by
  change encodeWords (PortTables.tableWords (lazy table)) = _
  rw [tableWords_lazy]

end BinPackingGames.Foundations.PCP.PreprocessingLazyWords

end OAI