lean/OAI/Computability/UniqueGames/PCP/MachineRawInitialTable.lean

import OAI.Computability.UniqueGames.Machines.MachineLemmas
import OAI.Computability.UniqueGames.PCP.RawInitialMachineLoop

namespace OAI

section

/-! The actual finite machine computing the initial graph's full serialized
table. The runtime bound counts every parse, row, cleanup and output reversal. -/

namespace UniqueGamesTheorem.Foundations.PCP.MachineRawInitialTable

open Turing Target Complexity RawInitialMachineModel RawInitialMachineLoopData
open RawInitialMachineBudget RawInitialMachineLoop

private theorem trace_trans {α : Type*} (f : α → α) {a b : Nat} {x y z : α}
    (first : f^[a] x = y) (second : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

def prefixBits (F : Formula) : List Bool :=
  encodeWords ([F.variables + F.clauses.length + 1, 6 * F.clauses.length + 1] ++
    clauseOutput F.variables 0 F.clauses)

theorem outputBits (F : Formula) :
    GraphTables.tableBits (RawInitialTables.table F) = prefixBits F ++
      encodeWords (RawInitialRows.dummyWords F.variables F.clauses.length) := by
  change encodeWords (GraphTables.tableWords (RawInitialTables.table F)) = _
  rw [tableWords_decomposition, encodeWords_append]
  rfl

theorem machineTrace (F : Formula) :
    (MachineComposition.advance (TM2.step program))^[fullBudget F]
      (some (initList machine (formulaBits F))) =
      some (haltList machine (GraphTables.tableBits (RawInitialTables.table F))) := by
  have startup := RawInitialMachineStart.formulaStartTrace F
  rw [startTapes_eq] at startup
  have loop := loopTrace 0 F.clauses
    (encodeWords [F.variables + F.clauses.length + 1, 6 * F.clauses.length + 1]).reverse
    initialState
  have hprefix : (encodeWords (clauseOutput F.variables 0 F.clauses)).reverse ++
      (encodeWords [F.variables + F.clauses.length + 1, 6 * F.clauses.length + 1]).reverse =
      (prefixBits F).reverse := by
    simp only [prefixBits, encodeWords_append, List.reverse_append]
  rw [hprefix, Nat.zero_add] at loop
  have finish := RawInitialMachineFinish.finishTrace F.variables F.clauses.length
    (prefixBits F).reverse (finalSigns initialState.1 F.clauses, none)
  rw [List.reverse_reverse, ← outputBits F] at finish
  have total := trace_trans _ (trace_trans _ startup loop) finish
  have time : (3 * F.variables + 5 * F.clauses.length + 13 + loopTime 0 F.clauses) +
      (3 * F.variables + 5 * F.clauses.length + 17 +
        (encodeWords (RawInitialRows.dummyWords F.variables F.clauses.length)).length +
        (prefixBits F).reverse.length) = fullBudget F := by
    rw [fullBudget, outputBits, List.length_append, List.length_reverse]
    omega
  simpa only [time] using total

def outputsInTime (F : Formula) :
    TM2OutputsInTime machine (formulaBits F)
      (some (GraphTables.tableBits (RawInitialTables.table F)))
      (timePolynomial.eval (formulaBits F).length) where
  steps := fullBudget F
  evals_in_steps := machineTrace F
  steps_le_m := fullBudget_le F

noncomputable def computableInPolyTime :
    TM2ComputableInPolyTime formulaEncoding.encode GraphTables.encoding.encode
      RawInitialTables.table where
  tm := machine
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := timePolynomial
  outputsFun F := by
    change TM2OutputsInTime machine ((formulaBits F).map id)
      (some ((GraphTables.tableBits (RawInitialTables.table F)).map id))
      (timePolynomial.eval (formulaBits F).length)
    have hi := @List.map_id (machine.Γ machine.k₀) (formulaBits F)
    have ho := @List.map_id (machine.Γ machine.k₁)
      (GraphTables.tableBits (RawInitialTables.table F))
    rw [hi, ho]
    exact outputsInTime F

end UniqueGamesTheorem.Foundations.PCP.MachineRawInitialTable

end

end OAI