lean/OAI/Computability/VertexCover/PCP/PCPIteration.lean

import Mathlib
import OAI.Computability.VertexCover.PCP.AlphabetRetraction
import OAI.Computability.VertexCover.PCP.FiniteGraph
import OAI.Computability.VertexCover.PCP.AmplificationIteration
import OAI.Computability.VertexCover.PCP.RoundGap
import OAI.Computability.VertexCover.PCP.RoundSize
import OAI.Computability.VertexCover.Fourier.SourceNonempty

namespace OAI

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noncomputable section

namespace UniqueGames.Foundations.PCP.PCPIteration

open FiniteGraph

abbrev Label := AlphabetRetraction.Label64
abbrev Graph := Bundle Label
abbrev Addresses := AmplificationRound.Addresses

def step (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) : Graph := by
  classical
  exact Bundle.ofGraph (AmplificationRound.graph addresses complete G.graph)

def run (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) : Nat → Graph → Graph :=
  AmplificationIteration.run (step addresses complete)

def initial (F : Target.Formula) : Graph := Bundle.ofGraph (AlphabetRetraction.initial64 F)

def iterationCount (F : Target.Formula) : Nat := AmplificationIteration.rounds (initial F).size

def output (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) : Graph := run addresses complete (iterationCount F) (initial F)

def polynomialDegree : Nat := AmplificationIteration.rounds RoundSize.sizeFactor

def finalClauseGap : ℚ := 1 / (40960 * (FinalConstants.walkLength : ℚ))

theorem finalClauseGap_positive : 0 < finalClauseGap := by
  apply one_div_pos.mpr
  exact mul_pos (by norm_num) (by exact_mod_cast FinalConstants.walkLength_positive)

theorem sizeFactor_le_power : RoundSize.sizeFactor ≤ 2 ^ polynomialDegree :=
  (AmplificationIteration.rounds_large RoundSize.sizeFactor).le

theorem initial_satisfiable_iff (F : Target.Formula) :
    (initial F).Satisfiable ↔ F.Satisfiable :=
  AlphabetRetraction.initial64_satisfiable_iff F

theorem initial_size_positive (F : Target.Formula) : 0 < (initial F).size :=
  (initial F).size_positive

theorem initial_size_bound (F : Target.Formula) :
    (initial F).size ≤ 10 * F.clauses.length + 2 := by
  change Fintype.card (InitialGraph.CompactVertex F) +
    Fintype.card (InitialGraph.CompactDart F) ≤ _
  rw [InitialGraph.build_dart_count]
  have h := InitialGraph.build_vertex_bound F
  omega

theorem step_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (G : Graph) (satisfied : G.Satisfiable) : (step addresses complete G).Satisfiable :=
  AmplificationRound.completeness addresses complete G.graph satisfied

theorem step_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) :
    min (2 * G.gap) FinalConstants.cap ≤ (step addresses complete G).gap :=
  RoundGap.amplifies addresses complete G.graph

theorem step_size (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) :
    (step addresses complete G).size ≤ RoundSize.sizeFactor * G.size :=
  RoundSize.output_total_le G.graph

theorem run_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (n : Nat) (G : Graph) (satisfied : G.Satisfiable) :
    (run addresses complete n G).Satisfiable :=
  AmplificationIteration.run_preserves (step addresses complete) Bundle.Satisfiable
    (step_completeness addresses complete) n G satisfied

theorem run_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (n : Nat) (G : Graph) :
    min (2 ^ n * G.gap) FinalConstants.cap ≤ (run addresses complete n G).gap :=
  AmplificationIteration.run_gap (step addresses complete) Bundle.gap
    FinalConstants.cap FinalConstants.cap_positive.le (step_gap addresses complete) n G

theorem run_size (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (n : Nat) (G : Graph) :
    (run addresses complete n G).size ≤ RoundSize.sizeFactor ^ n * G.size :=
  AmplificationIteration.run_size (step addresses complete) Bundle.size RoundSize.sizeFactor
    (step_size addresses complete) n G

theorem output_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) (satisfied : F.Satisfiable) :
    (output addresses complete F).Satisfiable :=
  run_completeness addresses complete (iterationCount F) (initial F)
    ((initial_satisfiable_iff F).mpr satisfied)

theorem output_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) (unsat : ¬ F.Satisfiable) :
    FinalConstants.cap ≤ (output addresses complete F).gap := by
  have hinitial : ¬ (initial F).Satisfiable :=
    fun h => unsat ((initial_satisfiable_iff F).mp h)
  exact AmplificationIteration.run_reaches_cap (step addresses complete) Bundle.gap
    FinalConstants.cap FinalConstants.cap_positive.le FinalConstants.cap_le_one
    (step_gap addresses complete) (initial F).size (initial F)
    (initial F).gap_nonnegative ((initial F).one_le_size_mul_gap hinitial)

theorem output_satisfiable_iff (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) : (output addresses complete F).Satisfiable ↔ F.Satisfiable := by
  constructor
  · intro satisfied
    by_contra unsat
    have hgap := output_gap addresses complete F unsat
    rw [(output addresses complete F).gap_eq_zero_iff.mpr satisfied] at hgap
    exact (not_le_of_gt FinalConstants.cap_positive) hgap
  · exact output_completeness addresses complete F

theorem output_size_polynomial (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) :
    (output addresses complete F).size ≤
      (2 * (initial F).size) ^ polynomialDegree * (initial F).size :=
  AmplificationIteration.run_size_polynomial (step addresses complete) Bundle.size
    (initial_size_positive F) sizeFactor_le_power (step_size addresses complete) (initial F)

theorem output_size_clause_bound (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) :
    (output addresses complete F).size ≤
      (2 * (10 * F.clauses.length + 2)) ^ polynomialDegree * (10 * F.clauses.length + 2) := by
  have h := initial_size_bound F
  exact (output_size_polynomial addresses complete F).trans
    (Nat.mul_le_mul (Nat.pow_le_pow_left (Nat.mul_le_mul_left 2 h) _) h)

theorem output_count_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (F : Target.Formula) (unsat : ¬ F.Satisfiable)
    (labeling : (output addresses complete F).Vertex → Label) :
    Fintype.card (output addresses complete F).Dart ≤
      FinalConstants.walkLength * (output addresses complete F).rejectionCount labeling := by
  have h := ((output addresses complete F).le_gap_iff FinalConstants.cap).mp
    (output_gap addresses complete F unsat) labeling
  have ht : (0 : ℝ) < FinalConstants.walkLength := by
    exact_mod_cast FinalConstants.walkLength_positive
  have h' : (Fintype.card (output addresses complete F).Dart : ℝ) /
      FinalConstants.walkLength ≤
      ((output addresses complete F).rejectionCount labeling : ℝ) := by
    simpa only [FinalConstants.cap, one_div_mul_eq_div] using h
  have h'' := (div_le_iff₀ ht).mp h'
  exact_mod_cast (show (Fintype.card (output addresses complete F).Dart : ℝ) ≤
    (FinalConstants.walkLength : ℝ) *
      ((output addresses complete F).rejectionCount labeling : ℝ) by
        simpa only [mul_comm] using h'')

section FinalCNF

variable (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
  (F : Target.Formula) {n m : Nat}
  (vertices : (output addresses complete F).Vertex ≃ Fin n)
  (darts : (output addresses complete F).Dart ≃ Fin m)

def finalCNF : Target.Formula :=
  FinalBooleanVerifier.cnf (output addresses complete F).graph vertices darts

theorem finalCNF_satisfiable_iff :
    (finalCNF addresses complete F vertices darts).Satisfiable ↔ F.Satisfiable :=
  (FinalBooleanVerifier.cnf_satisfiable_iff _ vertices darts).trans
    (output_satisfiable_iff addresses complete F)

theorem finalCNF_clause_count :
    (finalCNF addresses complete F vertices darts).clauses.length = m * 40960 :=
  FinalBooleanVerifier.cnf_clause_count _ vertices darts

theorem finalCNF_variable_count :
    (finalCNF addresses complete F vertices darts).«variables» = n * 6 + m * 36864 :=
  FinalBooleanVerifier.cnf_variable_count _ vertices darts

theorem finalCNF_nonempty : (finalCNF addresses complete F vertices darts).clauses ≠ [] := by
  have hm : Fintype.card (output addresses complete F).Dart = m := by
    simpa only [Fintype.card_fin] using Fintype.card_congr darts
  have hpositive : 0 < m := hm ▸ (Fintype.card_pos :
    0 < Fintype.card (output addresses complete F).Dart)
  exact FinalBooleanVerifier.cnf_nonempty _ vertices darts hpositive

theorem finalCNF_gap (unsat : ¬ F.Satisfiable)
    (assignment : Fin (finalCNF addresses complete F vertices darts).«variables» → Bool) :
    (finalCNF addresses complete F vertices darts).clauses.length ≤
      (FinalConstants.walkLength * 40960) *
        NameCompaction.failedCount (finalCNF addresses complete F vertices darts) assignment := by
  have source (labeling : (output addresses complete F).Vertex → Label) :
      1 * Fintype.card (output addresses complete F).Dart ≤
        FinalConstants.walkLength * (output addresses complete F).graph.rejectionCount labeling := by
    simpa only [one_mul, Bundle.rejectionCount] using output_count_gap addresses complete F unsat labeling
  simpa only [one_mul, finalCNF] using FinalBooleanVerifier.cnf_gap
    (output addresses complete F).graph vertices darts 1 FinalConstants.walkLength source assignment

theorem finalCNF_clauseGap (unsat : ¬ F.Satisfiable) :
    Hastad.SourceGap.ClauseGap (finalCNF addresses complete F vertices darts) finalClauseGap :=
  Hastad.SourceNonempty.cnf_clauseGap_unit (output addresses complete F).graph
    vertices darts FinalConstants.walkLength FinalConstants.walkLength_positive
    (output_count_gap addresses complete F unsat)

theorem finalCNF_total_size :
    (finalCNF addresses complete F vertices darts).«variables» +
        (finalCNF addresses complete F vertices darts).clauses.length ≤
      77824 * (output addresses complete F).size := by
  have hn : Fintype.card (output addresses complete F).Vertex = n := by
    simpa only [Fintype.card_fin] using Fintype.card_congr vertices
  have hm : Fintype.card (output addresses complete F).Dart = m := by
    simpa only [Fintype.card_fin] using Fintype.card_congr darts
  have hsize : (output addresses complete F).size = n + m := congrArg₂ Nat.add hn hm
  calc
    _ = n * 6 + m * 36864 + m * 40960 :=
      congrArg₂ Nat.add (finalCNF_variable_count addresses complete F vertices darts)
        (finalCNF_clause_count addresses complete F vertices darts)
    _ ≤ 77824 * (n + m) := by omega
    _ = 77824 * (output addresses complete F).size :=
      congrArg (fun k => 77824 * k) hsize.symm

theorem finalCNF_polynomial_size :
    (finalCNF addresses complete F vertices darts).«variables» +
        (finalCNF addresses complete F vertices darts).clauses.length ≤
      77824 * ((2 * (10 * F.clauses.length + 2)) ^ polynomialDegree *
        (10 * F.clauses.length + 2)) :=
  (finalCNF_total_size addresses complete F vertices darts).trans
    (Nat.mul_le_mul_left _ (output_size_clause_bound addresses complete F))

end FinalCNF

end UniqueGames.Foundations.PCP.PCPIteration

end


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end OAI