lean/OAI/Computability/MinUncut/Machines/Machine.lean

import Mathlib
import OAI.Computability.MinUncut.Machines.Runtime

namespace OAI

section
namespace MinUncutGames.Foundations.Hastad.SourceMachine

open Turing
open MinUncutGames.Foundations.Complexity

variable {K Λ σ : Type} [DecidableEq K]

abbrev Alphabet (_ : K) := Bool

def fieldLoop (source destination : K) (loopLabel : Λ) (exit : Option Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) :=
  .pop source (fun state head => (state.1, head))
    (.branch (fun state => state.2.getD false)
      (.push destination (fun _ => true) (.goto fun _ => loopLabel))
      (.load (fun state => (state.1, none))
        (Reduction.MachineTransfer.exitAt destination exit)))

def fieldStart (destination : K) (loopLabel : Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) :=
  .push destination (fun _ => false) (.goto fun _ => loopLabel)

def fieldTapes (source destination : K) (base : K → List Bool)
    (input output : List Bool) : K → List Bool :=
  Function.update (Function.update base source input) destination output

@[simp] theorem fieldTapes_source (source destination : K) (hne : source ≠ destination)
    (base : K → List Bool) (input output : List Bool) :
    fieldTapes source destination base input output source = input := by
  simp [fieldTapes, hne]

@[simp] theorem fieldTapes_destination (source destination : K)
    (base : K → List Bool) (input output : List Bool) :
    fieldTapes source destination base input output destination = output := by
  simp [fieldTapes]

@[simp] theorem fieldTapes_self (source destination : K) (base : K → List Bool) :
    fieldTapes source destination base (base source) (base destination) = base := by
  simp [fieldTapes]

theorem fieldTapes_other (source destination p : K)
    (hpS : p ≠ source) (hpD : p ≠ destination)
    (base : K → List Bool) (input output : List Bool) :
    fieldTapes source destination base input output p = base p := by
  simp [fieldTapes, hpS, hpD]

private theorem update_field_source (source destination : K) (hne : source ≠ destination)
    (base : K → List Bool) (input output replacement : List Bool) :
    Function.update (fieldTapes source destination base input output) source replacement =
      fieldTapes source destination base replacement output := by
  funext p
  by_cases hs : p = source
  · subst p; simp [fieldTapes, hne]
  · by_cases hd : p = destination
    · subst p; simp [fieldTapes, Ne.symm hne]
    · simp [fieldTapes, hs, hd]

private theorem update_field_destination (source destination : K)
    (base : K → List Bool) (input output replacement : List Bool) :
    Function.update (fieldTapes source destination base input output) destination replacement =
      fieldTapes source destination base input replacement := by
  simp [fieldTapes]

theorem fieldStep_delimiter (source destination : K) (hne : source ≠ destination)
    (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLoop : program loopLabel = fieldLoop source destination loopLabel exit)
    (base : K → List Bool) (suffix output : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, register), fieldTapes source destination base (false :: suffix) output⟩ =
      some ⟨exit, (ambient, none), fieldTapes source destination base suffix output⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (fieldTapes source destination base (false :: suffix) output)) = _
  rw [atLoop]
  cases exit <;>
    simp [fieldLoop, Reduction.MachineTransfer.exitAt, TM2.stepAux, hne, update_field_source]

theorem fieldStep_true (source destination : K) (hne : source ≠ destination)
    (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLoop : program loopLabel = fieldLoop source destination loopLabel exit)
    (base : K → List Bool) (input output : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, register), fieldTapes source destination base (true :: input) output⟩ =
      some ⟨some loopLabel, (ambient, some true),
        fieldTapes source destination base input (true :: output)⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, register)
    (fieldTapes source destination base (true :: input) output)) = _
  rw [atLoop]
  simp [fieldLoop, TM2.stepAux, hne, update_field_source, update_field_destination]

theorem fieldLoopTrace (source destination : K) (hne : source ≠ destination)
    (loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLoop : program loopLabel = fieldLoop source destination loopLabel exit)
    (base : K → List Bool) (n : Nat) (suffix output : List Bool)
    (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[n + 1]
      (some ⟨some loopLabel, (ambient, register),
        fieldTapes source destination base (encodeWord n ++ suffix) output⟩) =
      some ⟨exit, (ambient, none),
        fieldTapes source destination base suffix (List.replicate n true ++ output)⟩ := by
  induction n generalizing output register with
  | zero =>
    simpa only [Nat.zero_add, Function.iterate_one, MachineComposition.advance_some,
      encodeWord, List.replicate_zero, List.nil_append, List.singleton_append] using
      fieldStep_delimiter source destination hne loopLabel exit program atLoop
        base suffix output ambient register
  | succ n ih =>
    rw [Function.iterate_succ_apply]
    simp only [encodeWord, List.replicate_succ, List.cons_append]
    change (MachineComposition.advance (TM2.step program))^[n + 1]
      (TM2.step program ⟨some loopLabel, (ambient, register),
        fieldTapes source destination base (true :: (encodeWord n ++ suffix)) output⟩) = _
    rw [fieldStep_true source destination hne loopLabel exit program atLoop, ih]
    have hrotate : List.replicate n true ++ true :: output =
        true :: (List.replicate n true ++ output) := by
      calc
        _ = (List.replicate n true ++ [true]) ++ output := by simp
        _ = List.replicate (n + 1) true ++ output := by rw [List.replicate_succ']
        _ = _ := by rw [List.replicate_succ, List.cons_append]
    rw [hrotate]

theorem fieldStartStep (source destination : K)
    (startLabel loopLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atStart : program startLabel = fieldStart destination loopLabel)
    (base : K → List Bool) (input output : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some startLabel, (ambient, register), fieldTapes source destination base input output⟩ =
      some ⟨some loopLabel, (ambient, register),
        fieldTapes source destination base input (false :: output)⟩ := by
  change some (TM2.stepAux (program startLabel) (ambient, register)
    (fieldTapes source destination base input output)) = _
  rw [atStart]
  simp [fieldStart, TM2.stepAux, update_field_destination]

def fieldInTime (source destination : K) (hne : source ≠ destination)
    (startLabel loopLabel : Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atStart : program startLabel = fieldStart destination loopLabel)
    (atLoop : program loopLabel = fieldLoop source destination loopLabel exit)
    (base : K → List Bool) (n : Nat) (suffix : List Bool)
    (hinput : base source = encodeWord n ++ suffix)
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some startLabel, (ambient, register), base⟩
      (some ⟨exit, (ambient, none),
        fieldTapes source destination base suffix (encodeWord n ++ base destination)⟩)
      (n + 2) where
  steps := n + 2
  evals_in_steps := by
    have hbase : fieldTapes source destination base (encodeWord n ++ suffix)
        (base destination) = base := by rw [← hinput, fieldTapes_self]
    conv_lhs => rw [← hbase]
    change (MachineComposition.advance (TM2.step program))^[(n + 1) + 1]
      (some ⟨some startLabel, (ambient, register),
        fieldTapes source destination base (encodeWord n ++ suffix) (base destination)⟩) = _
    rw [Function.iterate_succ_apply]
    change (MachineComposition.advance (TM2.step program))^[n + 1]
      (TM2.step program ⟨some startLabel, (ambient, register),
        fieldTapes source destination base (encodeWord n ++ suffix) (base destination)⟩) = _
    rw [fieldStartStep source destination startLabel loopLabel program atStart,
      fieldLoopTrace source destination hne loopLabel exit program atLoop]
    simp [encodeWord, List.append_assoc]
  steps_le_m := Nat.le_refl _

def afterField (source destination : K) (base : K → List Bool)
    (n : Nat) (suffix : List Bool) : K → List Bool :=
  fieldTapes source destination base suffix (encodeWord n ++ base destination)

def fieldNext (start : Fin 4 → Λ) (exit : Option Λ) (j : Fin 4) : Option Λ :=
  match j.val with
  | 0 => some (start 1)
  | 1 => some (start 2)
  | 2 => some (start 3)
  | _ => exit

def fourTapes (source : K) (destination : Fin 4 → K) (base : K → List Bool)
    (a b c d : Nat) (suffix : List Bool) : K → List Bool :=
  let t₀ := afterField source (destination 0) base a (encodeWords [b, c, d] ++ suffix)
  let t₁ := afterField source (destination 1) t₀ b (encodeWords [c, d] ++ suffix)
  let t₂ := afterField source (destination 2) t₁ c (encodeWords [d] ++ suffix)
  afterField source (destination 3) t₂ d suffix

theorem fourTapes_source (source : K) (destination : Fin 4 → K)
    (hsep : ∀ j, source ≠ destination j) (base : K → List Bool)
    (a b c d : Nat) (suffix : List Bool) :
    fourTapes source destination base a b c d suffix source = suffix := by
  simp [fourTapes, afterField, hsep]

theorem fourTapes_other (source : K) (destination : Fin 4 → K)
    (p : K) (hpS : p ≠ source) (hpD : ∀ j, p ≠ destination j)
    (base : K → List Bool) (a b c d : Nat) (suffix : List Bool) :
    fourTapes source destination base a b c d suffix p = base p := by
  simp [fourTapes, afterField, fieldTapes_other, hpS, hpD]

theorem fourField_time_eq_length (a b c d : Nat) :
    a + b + c + d + 8 = (encodeWords [a, b, c, d]).length + 4 := by
  simp [encodeWords, encodeWord]
  omega

def fourFieldsInTime (source : K) (destination : Fin 4 → K)
    (hsep : ∀ j, source ≠ destination j)
    (start loopLabel : Fin 4 → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atStart : ∀ j, program (start j) = fieldStart (destination j) (loopLabel j))
    (atLoop : ∀ j, program (loopLabel j) =
      fieldLoop source (destination j) (loopLabel j) (fieldNext start exit j))
    (base : K → List Bool) (a b c d : Nat) (suffix : List Bool)
    (hinput : base source = encodeWords [a, b, c, d] ++ suffix)
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (start 0), (ambient, register), base⟩
      (some ⟨exit, (ambient, none), fourTapes source destination base a b c d suffix⟩)
      (a + b + c + d + 8) := by
  let t₀ := afterField source (destination 0) base a (encodeWords [b, c, d] ++ suffix)
  let t₁ := afterField source (destination 1) t₀ b (encodeWords [c, d] ++ suffix)
  let t₂ := afterField source (destination 2) t₁ c (encodeWords [d] ++ suffix)
  have h₀ : base source = encodeWord a ++ (encodeWords [b, c, d] ++ suffix) := by
    simpa only [encodeWords, List.append_assoc] using hinput
  have h₁ : t₀ source = encodeWord b ++ (encodeWords [c, d] ++ suffix) := by
    simp [t₀, afterField, hsep, encodeWords, List.append_assoc]
  have h₂ : t₁ source = encodeWord c ++ (encodeWords [d] ++ suffix) := by
    simp [t₁, afterField, hsep, encodeWords, List.append_assoc]
  have h₃ : t₂ source = encodeWord d ++ suffix := by
    simp [t₂, afterField, hsep, encodeWords]
  have p₀ := fieldInTime source (destination 0) (hsep 0) (start 0) (loopLabel 0)
    (some (start 1)) program (atStart 0) (atLoop 0) base a
    (encodeWords [b, c, d] ++ suffix) h₀ ambient register
  have p₁ := fieldInTime source (destination 1) (hsep 1) (start 1) (loopLabel 1)
    (some (start 2)) program (atStart 1) (atLoop 1) t₀ b
    (encodeWords [c, d] ++ suffix) h₁ ambient none
  have p₂ := fieldInTime source (destination 2) (hsep 2) (start 2) (loopLabel 2)
    (some (start 3)) program (atStart 2) (atLoop 2) t₁ c
    (encodeWords [d] ++ suffix) h₂ ambient none
  have p₃ := fieldInTime source (destination 3) (hsep 3) (start 3) (loopLabel 3)
    exit program (atStart 3) (atLoop 3) t₂ d suffix h₃ ambient none
  let p₀₁ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀ p₁
  let p₀₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁ p₂
  let p := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁₂ p₃
  exact {
    toEvalsTo := p.toEvalsTo
    steps_le_m := by have h := p.steps_le_m; omega
  }

def fieldDestination (j : Fin 4) : Fin 5 := ⟨j.val + 1, by omega⟩

theorem fourTapes_concrete (base : Fin 5 → List Bool) (a b c d : Nat) (suffix : List Bool) :
    fourTapes 0 fieldDestination base a b c d suffix =
      fun p : Fin 5 => if p = 0 then suffix
        else if p = 1 then encodeWord a ++ base 1
        else if p = 2 then encodeWord b ++ base 2
        else if p = 3 then encodeWord c ++ base 3
        else encodeWord d ++ base 4 := by
  funext p
  have hp : p = 0 ∨ p = 1 ∨ p = 2 ∨ p = 3 ∨ p = 4 := by omega
  rcases hp with rfl | rfl | rfl | rfl | rfl <;>
    simp [fourTapes, afterField, fieldTapes, fieldDestination]

def groupingProgram (label : Fin 4 × Bool) :
    TM2.Stmt (fun _ : Fin 5 => Bool) (Fin 4 × Bool) (Unit × Option Bool) :=
  if label.2 then
    fieldLoop 0 (fieldDestination label.1) (label.1, true)
      (fieldNext (fun j => (j, false)) none label.1)
  else fieldStart (fieldDestination label.1) (label.1, true)

def groupingMachine : FinTM2 where
  K := Fin 5
  k₀ := 0
  k₁ := 1
  Γ _ := Bool
  Λ := Fin 4 × Bool
  main := (0, false)
  σ := Unit × Option Bool
  initialState := ((), none)
  m := groupingProgram

def groupingMachineInTime (base : Fin 5 → List Bool) (a b c d : Nat) (suffix : List Bool)
    (hinput : base 0 = encodeWords [a, b, c, d] ++ suffix) (register : Option Bool) :
    StateTransition.EvalsToInTime groupingMachine.step
      ⟨some (0, false), ((), register), base⟩
      (some ⟨none, ((), none), fourTapes (0 : Fin 5) fieldDestination base a b c d suffix⟩)
      (a + b + c + d + 8) :=
  fourFieldsInTime (0 : Fin 5) fieldDestination (by
    intro j h
    have hh := congrArg Fin.val h
    change 0 = j.val + 1 at hh
    omega)
    (fun j => (j, false)) (fun j => (j, true)) none groupingProgram
    (by intro j; simp [groupingProgram]) (by intro j; simp [groupingProgram])
    base a b c d suffix hinput () register

end MinUncutGames.Foundations.Hastad.SourceMachine

end

end OAI