lean/OAI/Computability/MinUncut/PCP/Generator.lean

import Mathlib
import OAI.Computability.MinUncut.PCP.TupleBound
import OAI.Computability.MinUncut.Machines.RuntimeSpace

namespace OAI

namespace MinUncutGames.Foundations.Hastad.SourceGenerator

open Turing Complexity Target
open SourceGeneratorModel SourceGeneratorProgram SourceGeneratorContract

noncomputable section

variable {u D : Nat}
local instance generatorExtraDecidableEq : DecidableEq (Extra u D) := Classical.decEq _

def prefixPolynomial (u D : Nat) : Polynomial Nat :=
  SourceStartup.timePolynomial u D + SourceGeneratorTraversal.timePolynomial u D + 1

def timePolynomial (u D : Nat) (hD : 0 < D) : Polynomial Nat :=
  SourceRuntimeSpace.completedTime (SourceGeneratorProgram.machine u D hD)
    (clearKeys u D).length (prefixPolynomial u D)

def beforeFinish (F : Formula) (hD : 0 < D) : (SourceGeneratorProgram.machine u D hD).Cfg :=
  ⟨SourceRuntimeFinish.entry (clearKeys u D) SourceGeneratorProgram.Label.finish,
    (initialAmbient u D hD, none), SourceGeneratorTraversal.traversalOutput F hD⟩

def enterFinishInTime (F : Formula) (hD : 0 < D) :
    StateTransition.EvalsToInTime (SourceGeneratorProgram.machine u D hD).step
      ⟨some .finishEnter, (initialAmbient u D hD, none),
        SourceGeneratorTraversal.traversalOutput F hD⟩
      (some (beforeFinish F hD)) 1 where
  steps := 1
  evals_in_steps := by
    change some (TM2.stepAux
      (Reduction.MachineTransfer.exitAt (accumulatorTape u D)
        (SourceRuntimeFinish.entry (clearKeys u D) SourceGeneratorProgram.Label.finish))
      _ _) = _
    cases h : SourceRuntimeFinish.entry (clearKeys u D) SourceGeneratorProgram.Label.finish <;>
      simp only [Reduction.MachineTransfer.exitAt, beforeFinish, TM2.stepAux] <;>
      erw [h] <;> rfl
  steps_le_m := Nat.le_refl _

def prefixInTime (F : Formula) (hm : 0 < F.clauses.length) (hD : 0 < D) :
    StateTransition.EvalsToInTime (SourceGeneratorProgram.machine u D hD).step
      (initList (SourceGeneratorProgram.machine u D hD) (formulaBits F))
      (some (beforeFinish F hD)) ((prefixPolynomial u D).eval (formulaBits F).length) := by
  have start := startupInTime (u := u) F hm hD
  let traversal := Classical.choice (SourceGeneratorTraversal.traversalInPolynomialTime
    (u := u) F hm hD)
  have first := StateTransition.EvalsToInTime.trans
    (SourceGeneratorProgram.machine u D hD).step _ _ _ _ _ start traversal
  have joined := StateTransition.EvalsToInTime.trans
    (SourceGeneratorProgram.machine u D hD).step _ _ _ _ _ first (enterFinishInTime F hD)
  refine { steps := joined.steps, evals_in_steps := joined.evals_in_steps, steps_le_m := ?_ }
  apply joined.steps_le_m.trans
  simp only [prefixPolynomial, Polynomial.eval_add, Polynomial.eval_one]
  omega

theorem output_empty (F : Formula) (hD : 0 < D) :
    SourceGeneratorTraversal.traversalOutput F hD (outputTape u D) = [] := by
  rw [SourceGeneratorTraversal.traversalOutput_frame F hD _
    (Ne.symm (accumulator_ne_output u D))]
  exact SourceStartup.ready_work_blank F (.extra (.inr ()))
    (by simp) (by simp) (by simp)

theorem halted_configuration (F : Formula) (hD : 0 < D) :
    (⟨none, (initialAmbient u D hD, none),
      SourceRuntimeFinish.canonicalTapes (outputTape u D)
        (SourceGeneratorTraversal.traversalOutput F hD (accumulatorTape u D)).reverse⟩ :
      (SourceGeneratorProgram.machine u D hD).Cfg) =
      haltList (SourceGeneratorProgram.machine u D hD)
        (MinUncutGames.Reduction.SourceEncoding.inputBits (SourceOccurrences.sourceInput F u D hD)) := by
  rw [SourceGeneratorTraversal.traversalOutput_accumulator, List.reverse_reverse]
  simp only [haltList, SourceGeneratorProgram.machine]
  congr 1
  funext k
  by_cases hk : k = outputTape u D
  · subst k
    simp [SourceRuntimeFinish.canonicalTapes]
    erw [Function.update_self]
    rfl
  · simp [SourceRuntimeFinish.canonicalTapes, hk]
    erw [Function.update_of_ne hk]

def outputInTime (F : NonemptyFormula) (hD : 0 < D) :
    TM2OutputsInTime (SourceGeneratorProgram.machine u D hD) (inputEncoding F)
      (some (MinUncutGames.Reduction.SourceEncoding.inputBits (sourceMap u D hD F)))
      ((timePolynomial u D hD).eval (inputEncoding F).length) := by
  have hm : 0 < F.val.clauses.length := List.length_pos_iff.mpr F.property
  have prefixRun := prefixInTime (u := u) F.val hm hD
  have keepAcc : accumulatorTape u D ∉ clearKeys u D := by
    simp [mem_clearKeys]
  have keepOut : outputTape u D ∉ clearKeys u D := by
    simp [mem_clearKeys]
  have covers : ∀ k, k ≠ accumulatorTape u D → k ≠ outputTape u D → k ∈ clearKeys u D :=
    fun k ha ho => (mem_clearKeys u D k).mpr ⟨ha, ho⟩
  have finish := SourceRuntimeFinish.finishInTime (clearKeys u D)
    (accumulatorTape u D) (outputTape u D) (accumulator_ne_output u D)
    keepAcc keepOut covers (initialAmbient u D hD) SourceGeneratorProgram.Label.finish none
    (SourceGeneratorProgram.program u D hD) (fun _ => rfl)
    (SourceGeneratorTraversal.traversalOutput F.val hD) (output_empty F.val hD)
    (initialAmbient u D hD) none
  erw [halted_configuration F.val hD] at finish
  have joined := SourceRuntimeSpace.prefixAndFinishInTime
    (SourceGeneratorProgram.machine u D hD) (formulaBits F.val) (prefixPolynomial u D)
    prefixRun (clearKeys u D) (accumulatorTape u D) (outputTape u D)
    keepAcc keepOut covers (SourceGeneratorTraversal.traversalOutput F.val hD)
    (output_empty F.val hD) (fun _ => rfl) finish
  exact joined

def computableInPolyTime (u D : Nat) (hD : 0 < D) :
    TM2ComputableInPolyTime inputEncoding MinUncutGames.Reduction.SourceEncoding.inputBits
      (sourceMap u D hD) where
  tm := SourceGeneratorProgram.machine u D hD
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := timePolynomial u D hD
  outputsFun F := by
    change TM2OutputsInTime (machine u D hD) ((inputEncoding F).map id)
      (some ((MinUncutGames.Reduction.SourceEncoding.inputBits (sourceMap u D hD F)).map id)) _
    have input_eq : @List.map ((machine u D hD).Γ (machine u D hD).k₀)
        ((machine u D hD).Γ (machine u D hD).k₀) id (inputEncoding F) =
        inputEncoding F := List.map_id _
    have output_eq : @List.map ((machine u D hD).Γ (machine u D hD).k₁)
        ((machine u D hD).Γ (machine u D hD).k₁) id
        (MinUncutGames.Reduction.SourceEncoding.inputBits (sourceMap u D hD F)) =
        MinUncutGames.Reduction.SourceEncoding.inputBits (sourceMap u D hD F) := List.map_id _
    simpa only [input_eq, output_eq] using outputInTime (u := u) F hD

theorem finite_work_alphabets (u D : Nat) (hD : 0 < D)
    (k : (computableInPolyTime u D hD).tm.K) :
    Finite ((computableInPolyTime u D hD).tm.Γ k) := by
  change Finite Bool
  infer_instance

def forErrorComputableInPolyTime (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ) :
    TM2ComputableInPolyTime inputEncoding MinUncutGames.Reduction.SourceEncoding.inputBits
      (errorMap η ξ hη hη1 hξ) := by
  let D := SourceNoiseParameter.noiseDenominator ξ
  let hD := SourceNoiseParameter.noiseDenominator_pos ξ
  let u := SourceGap.repetitionCount η ξ hη hη1 hξ D hD
  exact computableInPolyTime u D hD

theorem forError_finite_work_alphabets (η ξ : ℚ) (hη : 0 < η) (hη1 : η ≤ 1) (hξ : 0 < ξ)
    (k : (forErrorComputableInPolyTime η ξ hη hη1 hξ).tm.K) :
    Finite ((forErrorComputableInPolyTime η ξ hη hη1 hξ).tm.Γ k) := by
  change Finite Bool
  infer_instance

end

end MinUncutGames.Foundations.Hastad.SourceGenerator

namespace MinUncutGames.Foundations.PCP

structure ConstraintGraph (V E A : Type*) where
  reverse : E ≃ E
  reverse_involutive : Function.Involutive reverse
  tail : E → V
  accepts : E → A → A → Bool
  reverse_accepts : ∀ e a b, accepts (reverse e) b a = accepts e a b

namespace ConstraintGraph

variable {V E A : Type*}

def head (G : ConstraintGraph V E A) (e : E) : V := G.tail (G.reverse e)

@[simp] theorem head_reverse (G : ConstraintGraph V E A) (e : E) :
    G.head (G.reverse e) = G.tail e := by
  simp only [head, G.reverse_involutive e]

def edgeSatisfied (G : ConstraintGraph V E A) (labeling : V → A) (e : E) : Bool :=
  G.accepts e (labeling (G.tail e)) (labeling (G.head e))

@[simp] theorem edgeSatisfied_reverse (G : ConstraintGraph V E A)
    (labeling : V → A) (e : E) :
    G.edgeSatisfied labeling (G.reverse e) = G.edgeSatisfied labeling e := by
  simpa only [edgeSatisfied, head, G.reverse_involutive e] using
    G.reverse_accepts e (labeling (G.tail e)) (labeling (G.tail (G.reverse e)))

def Satisfiable (G : ConstraintGraph V E A) : Prop :=
  ∃ labeling : V → A, ∀ e, G.edgeSatisfied labeling e = true

def rejectedDarts [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) :
    Finset E := Finset.univ.filter (fun e => G.edgeSatisfied labeling e = false)

def rejectionCount [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) : Nat :=
  (G.rejectedDarts labeling).card

theorem mem_rejectedDarts [Fintype E] (G : ConstraintGraph V E A)
    (labeling : V → A) (e : E) :
    e ∈ G.rejectedDarts labeling ↔ G.edgeSatisfied labeling e = false := by
  simp [rejectedDarts]

theorem rejectionCount_le [Fintype E] (G : ConstraintGraph V E A)
    (labeling : V → A) : G.rejectionCount labeling ≤ Fintype.card E := by
  exact Finset.card_le_card (Finset.filter_subset _ _)

theorem rejected_exists_of_not_satisfiable (G : ConstraintGraph V E A)
    (unsat : ¬ G.Satisfiable) (labeling : V → A) :
    ∃ e, G.edgeSatisfied labeling e = false := by
  classical
  by_contra none
  apply unsat
  refine ⟨labeling, fun e => ?_⟩
  have notFalse : G.edgeSatisfied labeling e ≠ false := by
    intro h
    exact none ⟨e, h⟩
  cases h : G.edgeSatisfied labeling e <;> simp_all

theorem rejectionCount_positive [Fintype E] (G : ConstraintGraph V E A)
    (unsat : ¬ G.Satisfiable) (labeling : V → A) :
    1 ≤ G.rejectionCount labeling := by
  obtain ⟨e, he⟩ := G.rejected_exists_of_not_satisfiable unsat labeling
  exact Finset.one_le_card.mpr ⟨e, (G.mem_rejectedDarts labeling e).mpr he⟩

end ConstraintGraph

end MinUncutGames.Foundations.PCP

end OAI