lean/OAI/Computability/BinPacking/PCP/RawInitialMachineLoop.lean

import OAI.Computability.BinPacking.PCP.RawInitialMachineBody
import OAI.Computability.BinPacking.PCP.RawInitialMachineFinish

namespace OAI

namespace BinPackingGames.Foundations.PCP.RawInitialMachineLoop

open Turing Target Complexity RawInitialMachineModel RawInitialMachineLoopData
open RawInitialMachineBudget RawInitialMachineBody RawInitialMachineFinish

def finalSigns {n : Nat} (incoming : RawInitialMachineModel.Signs) :
    List (Clause n) → RawInitialMachineModel.Signs
  | [] => incoming
  | c :: cs => finalSigns (RawInitialRows.clauseSigns c) cs

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]

theorem empty_loopTapes (n i : Nat) (rev : List Bool) :
    loopTapes n i 0 [] rev = finishTapes n i rev := by
  funext tape
  cases tape <;> rfl

theorem loopTrace {n : Nat} (i : Nat) (cs : List (Clause n))
    (rev : List Bool) (state : State) :
    (MachineComposition.advance (TM2.step program))^[loopTime i cs]
      (some ⟨some .guard, state, loopTapes n i cs.length (clauseInput cs) rev⟩) =
      some ⟨some (.scan .dummyVariables), (finalSigns state.1 cs, none),
        finishTapes n (i + cs.length) ((encodeWords (clauseOutput n i cs)).reverse ++ rev)⟩ := by
  induction cs generalizing i rev state with
  | nil =>
      have h := RawInitialMachineBody.guardTrace_zero n i [] rev state
      rw [empty_loopTapes] at h
      simpa only [loopTime, List.length_nil, clauseInput_nil, clauseOutput_nil,
        finalSigns, encodeWords, List.reverse_nil, List.nil_append, Nat.add_zero,
        empty_loopTapes] using h
  | cons c cs ih =>
      have first := bodyTrace i cs.length c (clauseInput cs) rev state
      have rest := ih (i + 1)
        ((encodeWords (RawInitialRows.clauseWords n i
          (RawInitialRows.clauseNames c) (RawInitialRows.clauseSigns c))).reverse ++ rev)
        (RawInitialRows.clauseSigns c, none)
      have total := trace_trans _ first rest
      have hi : i + 1 + cs.length = i + (cs.length + 1) := by omega
      simpa only [loopTime, List.length_cons, clauseInput_cons, clauseOutput_cons,
        encodeWords_append, List.reverse_append, List.append_assoc, finalSigns, hi] using total

def loopInTime {n : Nat} (i : Nat) (cs : List (Clause n))
    (rev : List Bool) (state : State) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .guard, state, loopTapes n i cs.length (clauseInput cs) rev⟩
      (some ⟨some (.scan .dummyVariables), (finalSigns state.1 cs, none),
        finishTapes n (i + cs.length) ((encodeWords (clauseOutput n i cs)).reverse ++ rev)⟩)
      (loopTime i cs) where
  steps := loopTime i cs
  evals_in_steps := loopTrace i cs rev state
  steps_le_m := Nat.le_refl _

end BinPackingGames.Foundations.PCP.RawInitialMachineLoop

end OAI