lean/OAI/Computability/MaxCut/PCP/ExpanderFamily.lean

import Mathlib
import OAI.Computability.MaxCut.PCP.CayleySpectral

namespace OAI

/-!
# A concrete finite family from the Cayley base

The initial graph has one vertex and the fixed number of loop ports. Each
subsequent graph is the actual square-and-zigzag construction. Consequently
the vertex sizes are exact powers of one fixed integer, including size one.
The finite base is selected from the proved existence theorem.
-/

namespace MaxCutGames.Foundations.PCP.ExpanderFamily

open PoweringWalks SpectralReturn Expanders

abbrev Port := BasePort × BasePort
abbrev CloudPort := Port × Port
def growth : Nat := baseDegree ^ 4

theorem card_port : Fintype.card Port = baseDegree ^ 2 := by
  change Fintype.card (BasePort × BasePort) = baseDegree ^ 2
  rw [Fintype.card_prod, card_basePort, pow_two]

theorem card_cloudPort : Fintype.card CloudPort = growth := by
  change Fintype.card (Port × Port) = growth
  rw [Fintype.card_prod, card_port]
  unfold growth
  ring

theorem growth_gt_one : 1 < growth := by
  norm_num [growth, baseDegree, initialDegree, basePower]

theorem port_degree_ge_eight : 8 ≤ Fintype.card Port := by
  rw [card_port]
  norm_num [baseDegree, initialDegree, basePower]

noncomputable def baseVertexEquiv : BaseVertex ≃ CloudPort :=
  Fintype.equivOfCardEq (card_baseVertex.trans card_cloudPort.symm)

noncomputable def baseOnCloud : PortGraph CloudPort BasePort :=
  GraphTransport.reindex (Classical.choose exists_baseGraph) baseVertexEquiv (Equiv.refl _)

theorem baseOnCloud_certificate : SpectralCertificate baseOnCloud (1 / 100 : ℝ) :=
  GraphTransport.reindex_spectralCertificate _ _ _ _ (Classical.choose_spec exists_baseGraph)

def Vertex : Nat → Type
  | 0 => Unit
  | n + 1 => Vertex n × CloudPort

instance vertexFintype (n : Nat) : Fintype (Vertex n) := by
  induction n with
  | zero => exact inferInstanceAs (Fintype Unit)
  | succ n ih =>
    letI : Fintype (Vertex n) := ih
    change Fintype (Vertex n × CloudPort)
    infer_instance

instance basePortNonempty : Nonempty BasePort :=
  ⟨fun _ => ⟨0, by norm_num [initialQuarterDegree]⟩⟩

instance vertexNonempty (n : Nat) : Nonempty (Vertex n) := by
  induction n with
  | zero => exact inferInstanceAs (Nonempty Unit)
  | succ n ih =>
    let : Nonempty (Vertex n) := ih
    change Nonempty (Vertex n × CloudPort)
    infer_instance

/-- The family is a recursive rotation algorithm when its fixed base is given. -/
def graph (H : PortGraph CloudPort BasePort) : (n : Nat) → PortGraph (Vertex n) Port
  | 0 => { rot := Equiv.refl _, rot_involutive := fun _ => rfl }
  | n + 1 => ZigzagGraphs.zigzag (ZigzagGraphs.square (graph H n)) H

theorem graph_certificate (H : PortGraph CloudPort BasePort)
    (hH : SpectralCertificate H (1 / 100 : ℝ)) (n : Nat) :
    SpectralCertificate (graph H n) (1 / 2 : ℝ) := by
  induction n with
  | zero =>
    refine ⟨by norm_num, by norm_num, ?_⟩
    intro f hf
    have hm : mean f = f () := by
      change mean (fun x : Unit => f x) = f ()
      have hfun : (fun x : Unit => f x) = fun _ : Unit => f () := by
        funext x
        cases x
        rfl
      rw [hfun, mean_const]
    have hz : f = fun _ => 0 := by
      funext x
      cases x
      exact hm.symm.trans hf
    rw [hz]
    simp [energy, mean, averagingOperator]
  | succ n ih =>
    exact ZigzagSpectral.square_zigzag_halfCertificate (graph H n) H ih hH

noncomputable def family (n : Nat) : PortGraph (Vertex n) Port := graph baseOnCloud n

theorem family_certificate (n : Nat) : SpectralCertificate (family n) (1 / 2 : ℝ) :=
  graph_certificate baseOnCloud baseOnCloud_certificate n

theorem card_vertex (n : Nat) : Fintype.card (Vertex n) = growth ^ n := by
  induction n with
  | zero => rfl
  | succ n ih =>
    change Fintype.card (Vertex n × CloudPort) = growth ^ (n + 1)
    rw [Fintype.card_prod, ih, card_cloudPort, pow_succ]

/-- The least geometric level covering a requested size. `Nat.clog` is the
ordinary finite ceiling logarithm algorithm. -/
def level (k : Nat) : Nat := Nat.clog growth k

def size (k : Nat) : Nat := growth ^ level k

theorem le_size (k : Nat) : k ≤ size k := Nat.le_pow_clog growth_gt_one k

theorem size_le_mul {k : Nat} (hk : 0 < k) : size k ≤ growth * k := by
  by_cases hk1 : k = 1
  · subst k
    simpa [size, level] using growth_gt_one.le
  · have hk2 : 1 < k := by omega
    have hl : 0 < level k := Nat.clog_pos growth_gt_one hk2
    have hp : growth ^ (level k).pred < k :=
      Nat.pow_pred_clog_lt_self growth_gt_one hk2
    have he : (level k).pred + 1 = level k := Nat.succ_pred_eq_of_pos hl
    calc
      size k = growth ^ ((level k).pred + 1) := by rw [he]; rfl
      _ = growth * growth ^ (level k).pred := by rw [pow_succ, Nat.mul_comm]
      _ ≤ growth * k := Nat.mul_le_mul_left growth hp.le

theorem padded_family_size {k : Nat} (hk : 0 < k) :
    k ≤ Fintype.card (Vertex (level k)) ∧
      Fintype.card (Vertex (level k)) ≤ growth * k := by
  rw [card_vertex]
  exact ⟨le_size k, size_le_mul hk⟩

end MaxCutGames.Foundations.PCP.ExpanderFamily

/-!
# Padding actual tail fibers to the geometric graph-family sizes

Every nonempty cloud is padded to the next family size. Empty clouds stay
empty, so their old vertices do not create new darts. Explicit equivalences
identify padded tail fibers and partition all darts by their original tails.
These facts give the global size bound without an expansion assumption.
-/

namespace MaxCutGames.Foundations.PCP.CloudPadding

open scoped BigOperators
open DegreeReplacement

variable {V E A : Type*}

/-- Leave empty clouds empty; pad every positive size to a geometric level. -/
def paddedSize (k : Nat) : Nat := if k = 0 then 0 else ExpanderFamily.size k

@[simp] theorem paddedSize_zero : paddedSize 0 = 0 := rfl

theorem paddedSize_of_pos {k : Nat} (hk : 0 < k) :
    paddedSize k = ExpanderFamily.size k := by
  simp only [paddedSize, ite_eq_right (Nat.ne_of_gt hk)]

theorem le_paddedSize (k : Nat) : k ≤ paddedSize k := by
  by_cases hk : k = 0
  · subst k
    simp
  · rw [paddedSize, ite_eq_right hk]
    exact ExpanderFamily.le_size k

theorem paddedSize_le_mul (k : Nat) : paddedSize k ≤ ExpanderFamily.growth * k := by
  by_cases hk : k = 0
  · subst k
    simp
  · rw [paddedSize, ite_eq_right hk]
    exact ExpanderFamily.size_le_mul (Nat.pos_of_ne_zero hk)

/-- The actual padded tail fiber contains old darts and exactly the dummy
darts indexed by that tail, with transport along the sigma index equality. -/
def paddedCloudEquiv (G : ConstraintGraph V E A) (dummy : V → Type*) (v : V) :
    Cloud (paddedGraph G dummy) v ≃ Cloud G v ⊕ dummy v where
  toFun := by
    rintro ⟨e, he⟩
    rcases e with e | ⟨w, d⟩
    · exact Sum.inl ⟨e, he⟩
    · change w = v at he
      cases he
      exact Sum.inr d
  invFun := fun z => match z with
    | Sum.inl e => ⟨Sum.inl e.val, e.property⟩
    | Sum.inr d => ⟨Sum.inr ⟨v, d⟩, rfl⟩
  left_inv := by
    rintro ⟨e, he⟩
    rcases e with e | ⟨w, d⟩
    · rfl
    · change w = v at he
      cases he
      rfl
  right_inv := by
    intro z
    cases z <;> rfl

/-- Every original dart occurs in exactly one cloud. -/
def dartCloudEquiv (G : ConstraintGraph V E A) : E ≃ Σ v : V, Cloud G v where
  toFun e := ⟨G.tail e, ⟨e, rfl⟩⟩
  invFun z := z.2.val
  left_inv _ := rfl
  right_inv := by
    rintro ⟨v, ⟨e, he⟩⟩
    cases he
    rfl

variable [Fintype V] [DecidableEq V] [Fintype E]

theorem sum_card_cloud (G : ConstraintGraph V E A) :
    (∑ v, Fintype.card (Cloud G v)) = Fintype.card E := by
  simpa only [Fintype.card_sigma] using (Fintype.card_congr (dartCloudEquiv G)).symm

theorem card_padded_cloud (G : ConstraintGraph V E A) (dummy : V → Type*)
    [∀ v, Fintype (dummy v)] (v : V) :
    Fintype.card (Cloud (paddedGraph G dummy) v) =
      Fintype.card (Cloud G v) + Fintype.card (dummy v) := by
  simpa only [Fintype.card_sum] using Fintype.card_congr (paddedCloudEquiv G dummy v)

/-- The precise number of always-accepting loop darts to add at each vertex. -/
abbrev dummy (G : ConstraintGraph V E A) (v : V) :=
  Fin (paddedSize (Fintype.card (Cloud G v)) - Fintype.card (Cloud G v))

theorem card_cloud (G : ConstraintGraph V E A) (v : V) :
    Fintype.card (Cloud (paddedGraph G (dummy G)) v) =
      paddedSize (Fintype.card (Cloud G v)) := by
  rw [card_padded_cloud]
  simp only [dummy, Fintype.card_fin]
  exact Nat.add_sub_of_le (le_paddedSize _)

omit [Fintype V] in
theorem card_dummy_of_empty (G : ConstraintGraph V E A) (v : V)
    (hk : Fintype.card (Cloud G v) = 0) : Fintype.card (dummy G v) = 0 := by
  simp [dummy, hk]

theorem card_cloud_of_empty (G : ConstraintGraph V E A) (v : V)
    (hk : Fintype.card (Cloud G v) = 0) :
    Fintype.card (Cloud (paddedGraph G (dummy G)) v) = 0 := by
  rw [card_cloud, hk, paddedSize_zero]

theorem card_cloud_eq_family (G : ConstraintGraph V E A) (v : V)
    (hk : 0 < Fintype.card (Cloud G v)) :
    Fintype.card (Cloud (paddedGraph G (dummy G)) v) =
      Fintype.card (ExpanderFamily.Vertex
        (ExpanderFamily.level (Fintype.card (Cloud G v)))) := by
  rw [card_cloud, paddedSize_of_pos hk, ExpanderFamily.card_vertex]
  rfl

/-- Exact count over all padded tail fibers, including empty original fibers. -/
theorem card_paddedDart_eq_sum (G : ConstraintGraph V E A) :
    Fintype.card (PaddedDart G (dummy G)) =
      ∑ v, paddedSize (Fintype.card (Cloud G v)) := by
  calc
    Fintype.card (PaddedDart G (dummy G)) =
        ∑ v, Fintype.card (Cloud (paddedGraph G (dummy G)) v) :=
      (sum_card_cloud (paddedGraph G (dummy G))).symm
    _ = ∑ v, paddedSize (Fintype.card (Cloud G v)) := by
      apply Finset.sum_congr rfl
      intro v _
      exact card_cloud G v

/-- Padding increases the total number of darts by at most the fixed family
growth factor, independent of the number of empty vertices. -/
theorem card_paddedDart_le (G : ConstraintGraph V E A) :
    Fintype.card (PaddedDart G (dummy G)) ≤ ExpanderFamily.growth * Fintype.card E := by
  rw [card_paddedDart_eq_sum]
  calc
    (∑ v, paddedSize (Fintype.card (Cloud G v))) ≤
        ∑ v, ExpanderFamily.growth * Fintype.card (Cloud G v) := by
      apply Finset.sum_le_sum
      intro v _
      exact paddedSize_le_mul _
    _ = ExpanderFamily.growth * Fintype.card E := by
      rw [← Finset.mul_sum, sum_card_cloud]

end MaxCutGames.Foundations.PCP.CloudPadding

/-!
# Executable indexing of the actual outgoing-dart clouds

Clouds are enumerated by filtering the input table's increasing dart-index
list. Ranking searches that list, and selection reads its indexed entry.
The same functions work for empty clouds without selecting a default dart.
Padded cloud indices put all old darts first and the dummy indices afterward.
All data functions below are executable; their proofs establish the connection
to the semantic graph's actual tail fibers.
-/

namespace MaxCutGames.Foundations.PCP.PreprocessingCloudIndex

open scoped BigOperators
open GraphTables DegreeReplacement

/-- An explicit complete list supplies both directions of a finite encoding.
The forward direction searches the list; its inverse performs list indexing. -/
def listEquiv {α : Type*} [BEq α] [LawfulBEq α] (xs : List α)
    (nodup : xs.Nodup) (complete : ∀ a, a ∈ xs) : α ≃ Fin xs.length where
  toFun a := ⟨xs.idxOf a, List.idxOf_lt_length_of_mem (complete a)⟩
  invFun i := xs.get i
  left_inv a := List.idxOf_get _
  right_inv i := by
    apply Fin.ext
    exact List.get_idxOf nodup i

@[simp] theorem listEquiv_val {α : Type*} [BEq α] [LawfulBEq α]
    (xs : List α) (nodup : xs.Nodup) (complete : ∀ a, a ∈ xs) (a : α) :
    (listEquiv xs nodup complete a).val = xs.idxOf a := rfl

@[simp] theorem listEquiv_symm_apply {α : Type*} [BEq α] [LawfulBEq α]
    (xs : List α) (nodup : xs.Nodup) (complete : ∀ a, a ∈ xs)
    (i : Fin xs.length) : (listEquiv xs nodup complete).symm i = xs.get i := rfl

/-- Retain the input dart order and test only the stored tail index. -/
def cloudDarts (t : Table) (v : Fin t.vertices) : List (Fin t.darts) :=
  (List.finRange t.darts).filter (fun e => decide (t.rows[e].tail = v))

@[simp] theorem mem_cloudDarts (t : Table) (v : Fin t.vertices) (e : Fin t.darts) :
    e ∈ cloudDarts t v ↔ t.rows[e].tail = v := by
  simp [cloudDarts]

theorem cloudDarts_nodup (t : Table) (v : Fin t.vertices) :
    (cloudDarts t v).Nodup :=
  List.Nodup.filter _ (List.nodup_finRange t.darts)

/-- Filtering preserves the original ordering, including repeated row values
at different dart indices. -/
theorem cloudDarts_sublist (t : Table) (v : Fin t.vertices) :
    (cloudDarts t v).Sublist (List.finRange t.darts) := List.filter_sublist

def cloudSize (t : Table) (v : Fin t.vertices) : Nat := (cloudDarts t v).length

theorem cloudSize_le_darts (t : Table) (v : Fin t.vertices) :
    cloudSize t v ≤ t.darts := by
  simpa only [cloudSize, List.length_finRange] using (cloudDarts_sublist t v).length_le

/-- Read the original dart stored at a valid local cloud index. -/
def cloudSelect (t : Table) (v : Fin t.vertices) (i : Fin (cloudSize t v)) :
    Cloud (semantics t) v :=
  ⟨(cloudDarts t v).get i,
    (mem_cloudDarts t v _).1 (List.get_mem (cloudDarts t v) i)⟩

/-- Find the local index of an original dart belonging to this cloud. -/
def cloudRank (t : Table) (v : Fin t.vertices) (e : Cloud (semantics t) v) :
    Fin (cloudSize t v) :=
  ⟨(cloudDarts t v).idxOf e.val,
    List.idxOf_lt_length_of_mem ((mem_cloudDarts t v e.val).2 e.property)⟩

@[simp] theorem cloudSelect_val (t : Table) (v : Fin t.vertices)
    (i : Fin (cloudSize t v)) :
    (cloudSelect t v i).val = (cloudDarts t v).get i := rfl

@[simp] theorem cloudRank_val (t : Table) (v : Fin t.vertices)
    (e : Cloud (semantics t) v) :
    (cloudRank t v e).val = (cloudDarts t v).idxOf e.val := rfl

@[simp] theorem cloudSelect_cloudRank (t : Table) (v : Fin t.vertices)
    (e : Cloud (semantics t) v) : cloudSelect t v (cloudRank t v e) = e := by
  apply Subtype.ext
  exact List.idxOf_get (cloudRank t v e).isLt

@[simp] theorem cloudRank_cloudSelect (t : Table) (v : Fin t.vertices)
    (i : Fin (cloudSize t v)) : cloudRank t v (cloudSelect t v i) = i := by
  apply Fin.ext
  exact List.get_idxOf (cloudDarts_nodup t v) i

/-- This equivalence contains executable rank and select functions, including
when both its domain and codomain are empty. -/
def cloudEquiv (t : Table) (v : Fin t.vertices) :
    Cloud (semantics t) v ≃ Fin (cloudSize t v) where
  toFun := cloudRank t v
  invFun := cloudSelect t v
  left_inv := cloudSelect_cloudRank t v
  right_inv := cloudRank_cloudSelect t v

theorem cloudSize_eq_card_cloud (t : Table) (v : Fin t.vertices) :
    cloudSize t v = Fintype.card (Cloud (semantics t) v) := by
  simpa only [Fintype.card_fin] using (Fintype.card_congr (cloudEquiv t v)).symm

theorem cloudDarts_eq_nil_iff (t : Table) (v : Fin t.vertices) :
    cloudDarts t v = [] ↔ ∀ e : Fin t.darts, t.rows[e].tail ≠ v := by
  simp only [List.eq_nil_iff_forall_not_mem, mem_cloudDarts]

theorem cloudSize_eq_zero_iff (t : Table) (v : Fin t.vertices) :
    cloudSize t v = 0 ↔ ∀ e : Fin t.darts, t.rows[e].tail ≠ v := by
  rw [cloudSize, List.length_eq_zero_iff, cloudDarts_eq_nil_iff]

theorem sum_cloudSize (t : Table) : (∑ v, cloudSize t v) = t.darts := by
  calc
    _ = ∑ v, Fintype.card (Cloud (semantics t) v) := by
      apply Finset.sum_congr rfl
      intro v _
      exact cloudSize_eq_card_cloud t v
    _ = Fintype.card (Fin t.darts) := CloudPadding.sum_card_cloud (semantics t)
    _ = t.darts := Fintype.card_fin t.darts

/-- An original global dart carries its own owner and executable local rank. -/
def oldCloudIndex (t : Table) (e : Fin t.darts) :
    Fin (cloudSize t t.rows[e].tail) := cloudRank t t.rows[e].tail ⟨e, rfl⟩

@[simp] theorem cloudSelect_oldCloudIndex (t : Table) (e : Fin t.darts) :
    (cloudSelect t t.rows[e].tail (oldCloudIndex t e)).val = e :=
  congrArg Subtype.val (cloudSelect_cloudRank t t.rows[e].tail ⟨e, rfl⟩)

/-- Padding counts are supplied as executable natural-valued data, without
choosing enumerations of the semantic clouds by their cardinalities. -/
abbrev PaddedCloud (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) :=
  Cloud (paddedGraph (semantics t) (fun u => Fin (padding u))) v

def paddedOld (t : Table) (padding : Fin t.vertices → Nat) (v : Fin t.vertices)
    (e : Cloud (semantics t) v) : PaddedCloud t padding v :=
  ⟨Sum.inl e.val, e.property⟩

def paddedNew (t : Table) (padding : Fin t.vertices → Nat) (v : Fin t.vertices)
    (i : Fin (padding v)) : PaddedCloud t padding v :=
  ⟨Sum.inr ⟨v, i⟩, rfl⟩

/-- Local padded order: old cloud indices first, then increasing dummy index. -/
def paddedCloudEquiv (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) : PaddedCloud t padding v ≃ Fin (cloudSize t v + padding v) :=
  (CloudPadding.paddedCloudEquiv (semantics t) (fun u => Fin (padding u)) v).trans
    ((Equiv.sumCongr (cloudEquiv t v) (Equiv.refl (Fin (padding v)))).trans
      finSumFinEquiv)

def paddedCloudRank (t : Table) (padding : Fin t.vertices → Nat) (v : Fin t.vertices) :
    PaddedCloud t padding v → Fin (cloudSize t v + padding v) :=
  paddedCloudEquiv t padding v

def paddedCloudSelect (t : Table) (padding : Fin t.vertices → Nat) (v : Fin t.vertices) :
    Fin (cloudSize t v + padding v) → PaddedCloud t padding v :=
  (paddedCloudEquiv t padding v).symm

@[simp] theorem paddedCloudSelect_rank (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (e : PaddedCloud t padding v) :
    paddedCloudSelect t padding v (paddedCloudRank t padding v e) = e :=
  (paddedCloudEquiv t padding v).symm_apply_apply e

@[simp] theorem paddedCloudRank_select (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (cloudSize t v + padding v)) :
    paddedCloudRank t padding v (paddedCloudSelect t padding v i) = i :=
  (paddedCloudEquiv t padding v).apply_symm_apply i

@[simp] theorem paddedCloudRank_old (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (e : Cloud (semantics t) v) :
    paddedCloudRank t padding v (paddedOld t padding v e) =
      Fin.castAdd (padding v) (cloudRank t v e) := rfl

@[simp] theorem paddedCloudRank_new (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (padding v)) :
    paddedCloudRank t padding v (paddedNew t padding v i) =
      Fin.natAdd (cloudSize t v) i := rfl

@[simp] theorem paddedCloudRank_old_val (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (e : Cloud (semantics t) v) :
    (paddedCloudRank t padding v (paddedOld t padding v e)).val =
      (cloudRank t v e).val := rfl

@[simp] theorem paddedCloudRank_new_val (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (padding v)) :
    (paddedCloudRank t padding v (paddedNew t padding v i)).val =
      cloudSize t v + i.val := rfl

@[simp] theorem paddedCloudSelect_castAdd (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (cloudSize t v)) :
    paddedCloudSelect t padding v (Fin.castAdd (padding v) i) =
      paddedOld t padding v (cloudSelect t v i) := by
  apply (paddedCloudEquiv t padding v).injective
  change (paddedCloudEquiv t padding v)
      ((paddedCloudEquiv t padding v).symm (Fin.castAdd (padding v) i)) = _
  rw [Equiv.apply_symm_apply]
  change Fin.castAdd (padding v) i =
    Fin.castAdd (padding v) (cloudRank t v (cloudSelect t v i))
  rw [cloudRank_cloudSelect]

@[simp] theorem paddedCloudSelect_natAdd (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) (i : Fin (padding v)) :
    paddedCloudSelect t padding v (Fin.natAdd (cloudSize t v) i) =
      paddedNew t padding v i :=
  (paddedCloudEquiv t padding v).symm_apply_apply (paddedNew t padding v i)

theorem card_paddedCloud (t : Table) (padding : Fin t.vertices → Nat)
    (v : Fin t.vertices) :
    Fintype.card (PaddedCloud t padding v) = cloudSize t v + padding v := by
  simpa only [Fintype.card_fin] using Fintype.card_congr (paddedCloudEquiv t padding v)

end MaxCutGames.Foundations.PCP.PreprocessingCloudIndex

namespace MaxCutGames.Foundations.PCP.ExpanderTables

open PoweringWalks SpectralReturn

def rowIndex (v d : Nat) : Fin v × Fin d ≃ Fin (v * d) := finProdFinEquiv

theorem rowIndex_val (v d : Nat) (x : Fin v × Fin d) :
    (rowIndex v d x).val = x.2.val + d * x.1.val := rfl

/-- Stored reverse-dart indices, with a proof of the finite involution law. -/
structure Table (v d : Nat) where
  rows : Vector (Fin (v * d)) (v * d)
  involutive : Function.Involutive (fun i : Fin (v * d) => rows[i])

def reverseIndex {v d : Nat} (table : Table v d) (i : Fin (v * d)) : Fin (v * d) :=
  table.rows[i]

def lookup {v d : Nat} (table : Table v d) (x : Fin v × Fin d) : Fin v × Fin d :=
  (rowIndex v d).symm (reverseIndex table (rowIndex v d x))

theorem lookup_involutive {v d : Nat} (table : Table v d) :
    Function.Involutive (lookup table) := by
  intro x
  simp only [lookup, Equiv.apply_symm_apply]
  exact (congrArg (rowIndex v d).symm (table.involutive (rowIndex v d x))).trans
    ((rowIndex v d).symm_apply_apply x)

def graph {v d : Nat} (table : Table v d) : PortGraph (Fin v) (Fin d) where
  rot := ZigzagGraphs.involutionEquiv (lookup table) (lookup_involutive table)
  rot_involutive := lookup_involutive table

@[simp] theorem graph_rot {v d : Nat} (table : Table v d) (x : Fin v × Fin d) :
    (graph table).rot x = lookup table x := rfl

def ofGraph {v d : Nat} (G : PortGraph (Fin v) (Fin d)) : Table v d where
  rows := Vector.ofFn (fun i => rowIndex v d (G.rot ((rowIndex v d).symm i)))
  involutive := by
    intro i
    simp only [Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta, Equiv.symm_apply_apply]
    exact (congrArg (rowIndex v d) (G.rot_involutive ((rowIndex v d).symm i))).trans
      ((rowIndex v d).apply_symm_apply i)

@[simp] theorem lookup_ofGraph {v d : Nat} (G : PortGraph (Fin v) (Fin d))
    (x : Fin v × Fin d) : lookup (ofGraph G) x = G.rot x := by
  simp only [lookup, reverseIndex, ofGraph, Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta,
    Equiv.symm_apply_apply]

private theorem graph_ext_inline_ExpanderTables {V D : Type*} {G H : PortGraph V D}
    (h : ∀ x, G.rot x = H.rot x) : G = H := by
  have hr : G.rot = H.rot := Equiv.ext h
  cases G
  cases H
  cases hr
  rfl

@[simp] theorem graph_ofGraph {v d : Nat} (G : PortGraph (Fin v) (Fin d)) :
    graph (ofGraph G) = G := graph_ext_inline_ExpanderTables (lookup_ofGraph G)

theorem row_count {v d : Nat} (table : Table v d) : table.rows.toList.length = v * d := by
  simp

/-- Explicit mixed-radix vertex coordinates for one square/zigzag step. -/
def vertexIndex (v q : Nat) : Fin v × (Fin q × Fin q) ≃ Fin (v * (q * q)) :=
  (Equiv.prodCongr (Equiv.refl _) (rowIndex q q)).trans (rowIndex v (q * q))

/-- The actual square/zigzag graph, expressed in the table's integer coordinates. -/
def stepGraph {v d : Nat} (G : Table v (d * d))
    (H : Table ((d * d) * (d * d)) d) :
    PortGraph (Fin (v * ((d * d) * (d * d)))) (Fin (d * d)) :=
  GraphTransport.reindex
    (ZigzagGraphs.zigzag (ZigzagGraphs.square (graph G))
      (GraphTransport.reindex (graph H) (rowIndex (d * d) (d * d)).symm (Equiv.refl _)))
    (vertexIndex v (d * d)) (rowIndex d d)

/-- Materialize every row of the next table in the declared order. -/
def step {v d : Nat} (G : Table v (d * d))
    (H : Table ((d * d) * (d * d)) d) :
    Table (v * ((d * d) * (d * d))) (d * d) := ofGraph (stepGraph G H)

/-- A row uses exactly the displayed four lookups in already stored tables. -/
def stepLookup {v d : Nat} (G : Table v (d * d))
    (H : Table ((d * d) * (d * d)) d)
    (x : Fin (v * ((d * d) * (d * d))) × Fin (d * d)) :
    Fin (v * ((d * d) * (d * d))) × Fin (d * d) :=
  let vc := (rowIndex v ((d * d) * (d * d))).symm x.1
  let ports := (rowIndex d d).symm x.2
  let first := lookup H (vc.2, ports.1)
  let squarePorts := (rowIndex (d * d) (d * d)).symm first.1
  let outerFirst := lookup G (vc.1, squarePorts.1)
  let outerSecond := lookup G (outerFirst.1, squarePorts.2)
  let last := lookup H (rowIndex (d * d) (d * d) (outerSecond.2, outerFirst.2), ports.2)
  (rowIndex v ((d * d) * (d * d)) (outerSecond.1, last.1),
    rowIndex d d (last.2, first.2))

theorem lookup_step {v d : Nat} (G : Table v (d * d))
    (H : Table ((d * d) * (d * d)) d)
    (x : Fin (v * ((d * d) * (d * d))) × Fin (d * d)) :
    lookup (step G H) x = stepLookup G H x := by
  simp [step, stepGraph, GraphTransport.reindex,
    Equiv.trans_apply, Equiv.prodCongr_apply,
    Equiv.prodCongr_symm, Equiv.refl_apply, vertexIndex, Equiv.apply_symm_apply,
    ZigzagGraphs.zigzag, ZigzagGraphs.square, ZigzagGraphs.involutionEquiv,
    ZigzagGraphs.cloudFirst, ZigzagGraphs.crossCloud,
    graph_rot,
    stepLookup, Prod.map, Equiv.symm_symm]
  all_goals repeat' constructor

theorem step_certificate {v d : Nat} [NeZero v] [NeZero d]
    (G : Table v (d * d)) (H : Table ((d * d) * (d * d)) d)
    (hG : SpectralCertificate (graph G) (1 / 2 : ℝ))
    (hH : SpectralCertificate (graph H) (1 / 100 : ℝ)) :
    SpectralCertificate (graph (step G H)) (1 / 2 : ℝ) := by
  rw [step, graph_ofGraph]
  apply GraphTransport.reindex_spectralCertificate
  apply ZigzagSpectral.square_zigzag_halfCertificate
  · exact hG
  · exact GraphTransport.reindex_spectralCertificate _ _ _ _ hH

def initial (d : Nat) : Table 1 (d * d) :=
  ofGraph { rot := Equiv.refl _, rot_involutive := fun _ => rfl }

theorem initial_certificate (d : Nat) :
    SpectralCertificate (graph (initial d)) (1 / 2 : ℝ) := by
  refine ⟨by norm_num, by norm_num, ?_⟩
  intro f hf
  have hfun : f = fun _ => f 0 := funext (fun x => congrArg f (Subsingleton.elim x 0))
  have hm : mean f = f 0 := (congrArg mean hfun).trans (mean_const _)
  have hz : f 0 = 0 := hm.symm.trans hf
  rw [hfun, hz]
  simp [energy, mean, averagingOperator]

def vertexCount (q : Nat) : Nat → Nat
  | 0 => 1
  | n + 1 => vertexCount q n * (q * q)

theorem vertexCount_eq (q n : Nat) : vertexCount q n = (q * q) ^ n := by
  induction n with
  | zero => rfl
  | succ n ih => simp only [vertexCount, ih, pow_succ]

theorem vertexCount_positive {q : Nat} (hq : 0 < q) (n : Nat) :
    0 < vertexCount q n := by
  rw [vertexCount_eq]
  exact Nat.pow_pos (Nat.mul_pos hq hq)

/-- A bottom-up vector of rotation rows. The recursive result is passed once
to `step`, whose row formula reads only that already materialized table. -/
def family {d : Nat} (H : Table ((d * d) * (d * d)) d) :
    (n : Nat) → Table (vertexCount (d * d) n) (d * d)
  | 0 => initial d
  | n + 1 => step (family H n) H

theorem family_certificate {d : Nat} [NeZero d]
    (H : Table ((d * d) * (d * d)) d)
    (hH : SpectralCertificate (graph H) (1 / 100 : ℝ)) (n : Nat) :
    SpectralCertificate (graph (family H n)) (1 / 2 : ℝ) := by
  induction n with
  | zero => exact initial_certificate d
  | succ n ih =>
    let : NeZero (vertexCount (d * d) n) :=
      ⟨Nat.ne_of_gt (vertexCount_positive (Nat.mul_pos (NeZero.pos d) (NeZero.pos d)) n)⟩
    exact step_certificate _ _ ih hH

theorem family_row_count {d : Nat} (H : Table ((d * d) * (d * d)) d) (n : Nat) :
    (family H n).rows.toList.length = ((d * d) * (d * d)) ^ n * (d * d) := by
  rw [row_count, vertexCount_eq]

/-- The only selected data is one fixed finite table, independent of the input. -/
theorem exists_base_table : ∃ H : Table
    ((Expanders.baseDegree * Expanders.baseDegree) *
      (Expanders.baseDegree * Expanders.baseDegree)) Expanders.baseDegree,
    SpectralCertificate (graph H) (1 / 100 : ℝ) := by
  classical
  have hv : Fintype.card ExpanderFamily.CloudPort =
      (Expanders.baseDegree * Expanders.baseDegree) *
        (Expanders.baseDegree * Expanders.baseDegree) := by
    rw [ExpanderFamily.card_cloudPort]
    unfold ExpanderFamily.growth
    ring
  let vertices := Fintype.equivFinOfCardEq hv
  let ports := Fintype.equivFinOfCardEq Expanders.card_basePort
  let G := GraphTransport.reindex ExpanderFamily.baseOnCloud vertices ports
  refine ⟨ofGraph G, ?_⟩
  rw [graph_ofGraph]
  exact GraphTransport.reindex_spectralCertificate _ _ _ _
    ExpanderFamily.baseOnCloud_certificate

end MaxCutGames.Foundations.PCP.ExpanderTables

namespace MaxCutGames.Foundations.PCP.PreprocessingLevels

def levelLoop (g k : Nat) : Nat → Nat → Nat → Nat × Nat
  | 0, e, s => (e, s)
  | fuel + 1, e, s =>
      if k ≤ s then (e, s)
      else levelLoop g k fuel (e + 1) (s * g)

/-- At most `fuel` continuing iterations can increment the exponent. -/
theorem levelLoop_level_le (g k fuel e s : Nat) :
    (levelLoop g k fuel e s).1 ≤ e + fuel := by
  induction fuel generalizing e s with
  | zero => simp only [levelLoop, Nat.add_zero, le_refl]
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · omega
      · have h := ih (e + 1) (s * g)
        omega

/-- The carried size remains the exact power of the carried exponent. -/
theorem levelLoop_power (g k fuel e : Nat) :
    (levelLoop g k fuel e (g ^ e)).2 =
      g ^ (levelLoop g k fuel e (g ^ e)).1 := by
  induction fuel generalizing e with
  | zero => rfl
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · rfl
      · simpa only [pow_succ] using ih (e + 1)

theorem levelLoop_covers (g k fuel e s : Nat) (hg : 1 < g)
    (hs : 0 < s) (hbudget : k ≤ s + fuel) :
    k ≤ (levelLoop g k fuel e s).2 := by
  induction fuel generalizing e s with
  | zero => simpa only [levelLoop, Nat.add_zero] using hbudget
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · assumption
      · apply ih
        · exact Nat.mul_pos hs (by omega)
        · have htwo : 2 ≤ g := by omega
          have hmul : s * 2 ≤ s * g := Nat.mul_le_mul_left s htwo
          omega

/-- The search never passes an exponent whose power already covers the
request. This remains true even without a sufficient fuel budget. -/
theorem levelLoop_le_of_covering_power (g k fuel e j : Nat)
    (he : e ≤ j) (hj : k ≤ g ^ j) :
    (levelLoop g k fuel e (g ^ e)).1 ≤ j := by
  induction fuel generalizing e with
  | zero => exact he
  | succ fuel ih =>
      simp only [levelLoop]
      split
      · exact he
      · rename_i hnot
        have hne : e ≠ j := by
          intro h
          subst e
          exact hnot hj
        have hnext : e + 1 ≤ j := by omega
        simpa only [pow_succ] using ih (e + 1) hnext

def searchResult (k : Nat) : Nat × Nat :=
  levelLoop ExpanderFamily.growth k k 0 1

/-- An executable exponent, computed with at most `k` multiplications. -/
def boundedLevel (k : Nat) : Nat := (searchResult k).1

/-- The algorithm returns the power it has already computed. -/
def paddedSize (k : Nat) : Nat := (searchResult k).2

@[simp] theorem boundedLevel_zero : boundedLevel 0 = 0 := rfl

@[simp] theorem paddedSize_zero : paddedSize 0 = 1 := rfl

theorem boundedLevel_le_input (k : Nat) : boundedLevel k ≤ k := by
  simpa only [boundedLevel, searchResult, Nat.zero_add] using
    levelLoop_level_le ExpanderFamily.growth k k 0 1

theorem paddedSize_power (k : Nat) :
    paddedSize k = ExpanderFamily.growth ^ boundedLevel k := by
  simpa only [paddedSize, boundedLevel, searchResult, pow_zero] using
    levelLoop_power ExpanderFamily.growth k k 0

theorem le_paddedSize (k : Nat) : k ≤ paddedSize k := by
  exact levelLoop_covers ExpanderFamily.growth k k 0 1
    ExpanderFamily.growth_gt_one (by omega) (by omega)

/-- The bounded multiplication loop computes the existing ceiling logarithm. -/
theorem boundedLevel_eq_level (k : Nat) :
    boundedLevel k = ExpanderFamily.level k := by
  apply Nat.le_antisymm
  · simpa only [boundedLevel, searchResult, pow_zero] using
      levelLoop_le_of_covering_power ExpanderFamily.growth k k 0
        (ExpanderFamily.level k) (Nat.zero_le _) (ExpanderFamily.le_size k)
  · apply (Nat.clog_le_iff_le_pow ExpanderFamily.growth_gt_one).2
    rw [← paddedSize_power]
    exact le_paddedSize k

theorem paddedSize_eq_size (k : Nat) : paddedSize k = ExpanderFamily.size k := by
  rw [paddedSize_power, boundedLevel_eq_level]
  rfl

/-- The returned exponent is the least exponent whose power covers `k`. -/
theorem boundedLevel_le_of_le_power {k j : Nat}
    (h : k ≤ ExpanderFamily.growth ^ j) : boundedLevel k ≤ j := by
  rw [boundedLevel_eq_level]
  exact (Nat.clog_le_iff_le_pow ExpanderFamily.growth_gt_one).2 h

theorem paddedSize_positive (k : Nat) : 0 < paddedSize k := by
  rw [paddedSize_power]
  exact Nat.pow_pos (by have h := ExpanderFamily.growth_gt_one; omega)

/-- Positive cloud sizes and positive whole-graph vertex counts increase by
at most the single fixed growth factor. -/
theorem paddedSize_bounds {k : Nat} (hk : 0 < k) :
    k ≤ paddedSize k ∧ paddedSize k ≤ ExpanderFamily.growth * k := by
  rw [paddedSize_eq_size]
  exact ⟨ExpanderFamily.le_size k, ExpanderFamily.size_le_mul hk⟩

/-- Empty clouds remain empty; nonempty clouds use the geometric family. -/
def cloudPaddedSize (k : Nat) : Nat := if k = 0 then 0 else paddedSize k

@[simp] theorem cloudPaddedSize_zero : cloudPaddedSize 0 = 0 := rfl

theorem cloudPaddedSize_of_pos {k : Nat} (hk : 0 < k) :
    cloudPaddedSize k = paddedSize k := by
  simp only [cloudPaddedSize, Nat.ne_of_gt hk, ite_false]

theorem cloudPaddedSize_bounds (k : Nat) :
    k ≤ cloudPaddedSize k ∧ cloudPaddedSize k ≤ ExpanderFamily.growth * k := by
  by_cases hk : k = 0
  · subst k
    simp only [cloudPaddedSize_zero, Nat.mul_zero, le_refl, and_self]
  · rw [cloudPaddedSize_of_pos (Nat.pos_of_ne_zero hk)]
    exact paddedSize_bounds (Nat.pos_of_ne_zero hk)

/-- The executable table family has exactly the size returned by the loop. -/
theorem table_vertexCount_eq_paddedSize (k : Nat) :
    ExpanderTables.vertexCount (Expanders.baseDegree * Expanders.baseDegree)
        (boundedLevel k) = paddedSize k := by
  rw [ExpanderTables.vertexCount_eq]
  have hg : (Expanders.baseDegree * Expanders.baseDegree) *
      (Expanders.baseDegree * Expanders.baseDegree) = ExpanderFamily.growth := by
    unfold ExpanderFamily.growth
    ring
  rw [hg]
  exact (paddedSize_power k).symm

theorem family_row_count_at_size
    (H : ExpanderTables.Table
      ((Expanders.baseDegree * Expanders.baseDegree) *
        (Expanders.baseDegree * Expanders.baseDegree)) Expanders.baseDegree)
    (k : Nat) :
    (ExpanderTables.family H (boundedLevel k)).rows.toList.length =
      paddedSize k * (Expanders.baseDegree * Expanders.baseDegree) := by
  rw [ExpanderTables.row_count, table_vertexCount_eq_paddedSize]

end MaxCutGames.Foundations.PCP.PreprocessingLevels

end OAI