lean/OAI/Computability/UniqueGames/Machines/MachineExpanderFamilyLoop.lean

import OAI.Computability.UniqueGames.Machines.MachineExpanderFamilyBoundsLemmas
import OAI.Computability.UniqueGames.Machines.MachineExpanderFamilyTransferLemmas

namespace OAI

/-! Actual counted repetition of complete table-expansion cycles. The input
counter is physically decremented, and the invariant carries the exact table
at the completed family level. -/

namespace UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open MachineExpanderFamilyBounds

noncomputable def levelLoopCost {d : Nat} (H : Table (cloudSize d) d) : Nat → Nat → Nat
  | _, 0 => 1
  | start, remaining + 1 => 1 + resizeCost (family H start) H +
      levelLoopCost H (start + 1) remaining

theorem levelLoopCost_prefix {d : Nat} (H : Table (cloudSize d) d) (start remaining : Nat) :
    levelLoopCost H start remaining + resizeSum H start =
      resizeSum H (start + remaining) + remaining + 1 := by
  induction remaining generalizing start with
  | zero => simp [levelLoopCost, Nat.add_comm]
  | succ remaining ih =>
      have h := ih (start + 1)
      rw [resizeSum_succ] at h
      have index : start + 1 + remaining = start + (remaining + 1) := by omega
      rw [index] at h
      simp only [levelLoopCost]
      omega

theorem levelLoopCost_zero_start {d : Nat} (H : Table (cloudSize d) d) (remaining : Nat) :
    levelLoopCost H 0 remaining = resizeSum H remaining + remaining + 1 := by
  simpa only [resizeSum_zero, Nat.add_zero, Nat.zero_add] using
    levelLoopCost_prefix H 0 remaining

def levelLoopTapes {d : Nat} (H : Table (cloudSize d) d)
    (completed remaining : Nat) (suffix : List Bool) : ∀ tape, List (Alphabet tape) :=
  boundaryTapes remaining (vertexCount (degree d) completed)
    (encodeWords (rotationWords (family H completed))) suffix

structure LevelLoopRun {d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (completed remaining : Nat) (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
    ⟨some (.inr .levelGuard), state, levelLoopTapes H completed remaining suffix⟩
    (some ⟨some (.inr .done), finalState, familyTapes H (completed + remaining) suffix⟩)
    (levelLoopCost H completed remaining)

private def familySingleStep {S : Type*} (f : S → Option S) (a b : S)
    (h : f a = some b) : StateTransition.EvalsToInTime f a (some b) 1 where
  steps := 1
  evals_in_steps := by change f a = some b; exact h
  steps_le_m := Nat.le_refl _

/-- Every body is the already constructed actual resize run. There is no
whole-loop, body-trace, or computational-correctness premise. -/
noncomputable def levelLoopInTime {d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (completed remaining : Nat) (state : State ρ d) (suffix : List Bool) :
    LevelLoopRun positive H growth completed remaining state suffix := by
  induction remaining generalizing completed state with
  | zero =>
      refine ⟨clearRegister state, caller_clearRegister state, ?_⟩
      simpa only [levelLoopCost, levelLoopTapes, familyTapes, Nat.add_zero] using
        familySingleStep _ _ _ (levelGuard_boundary_zeroStep positive H growth
          (vertexCount (degree d) completed) (encodeWords (rotationWords (family H completed)))
          suffix state)
  | succ remaining ih =>
      let cycle := resizeInTime positive H growth (family H completed) remaining
        (clearRegister state) suffix
      let rest := ih (completed + 1) cycle.finalState
      let guard := familySingleStep _ _ _ (levelGuard_boundary_succStep positive H growth
        remaining (vertexCount (degree d) completed)
        (encodeWords (rotationWords (family H completed))) suffix state)
      have cycleRun : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
          ⟨some (.inr (.affine .copyCount .seed)), clearRegister state,
            levelLoopTapes H completed remaining suffix⟩
          (some ⟨some (.inr .levelGuard), cycle.finalState,
            levelLoopTapes H (completed + 1) remaining suffix⟩)
          (resizeCost (family H completed) H) := cycle.execution
      let first := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
        _ _ _ _ _ guard cycleRun
      let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
        _ _ _ _ _ first rest.execution
      have index : completed + 1 + remaining = completed + (remaining + 1) := by omega
      refine ⟨rest.finalState, ?_, ?_⟩
      · exact rest.caller_preserved.trans
          (cycle.caller_preserved.trans (caller_clearRegister state))
      · rw [index] at all
        refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
        have bound := all.steps_le_m
        simp only [levelLoopCost]
        omega

end UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily

end OAI