lean/OAI/Computability/MinUncut/PCP/PoweringRowData.lean

import Mathlib
import OAI.Computability.MinUncut.Machines.PoweringMachineRow

namespace OAI

section
namespace MinUncutGames.Foundations.PCP.PoweringRowData

open MinUncutGames.Foundations.Complexity
open Turing
open PoweringMachineRow MachineFixedBlockMap
open PoweringWalks PoweringLabels PoweringAddresses PoweringReach
open PoweringOpinionTables PoweringEnumeration

variable {vertices d : Nat}

theorem firstMatch_of_first {S : Nat} (mask : Fin S → Bool) (i : Fin S)
    (hi : mask i = true) (hfirst : ∀ j : Fin S, j.val < i.val → mask j = false) :
    firstMatch mask = some i := by
  unfold firstMatch
  apply List.find?_ofFn_eq_some.mpr
  refine ⟨hi, i, rfl, ?_⟩
  intro j hj
  change ¬mask j = true
  simp [hfirst j hj]

theorem firstMatch_firstIndex {X : Type*} (p : X → Prop) [DecidablePred p]
    (xs : List X) (h : ∃ a ∈ xs, p a) :
    firstMatch (fun i : Fin xs.length => decide (p (xs.get i))) =
      some (firstIndex p xs h) := by
  apply firstMatch_of_first
  · exact decide_eq_true (firstIndex_matches p xs h)
  · intro j hj
    exact decide_eq_false (firstIndex_is_first p xs h j hj)

def endpointMask (G : PortGraph (Fin vertices) (Fin d)) (t : Nat)
    (center target : Fin vertices) (i : AddressIndex d t) : Bool :=
  decide ((wordToBall G t center ((allAddresses d t).get i)).val = target)

theorem firstMatch_endpointMask (G : PortGraph (Fin vertices) (Fin d)) (t : Nat)
    (center : Fin vertices) (u : Ball G t center) :
    firstMatch (endpointMask G t center u.val) = some (addressIndex G t center u) := by
  unfold endpointMask addressIndex
  exact firstMatch_firstIndex
    (fun w : PortWords (Fin d) t => (wordToBall G t center w).val = u.val)
    (allAddresses d t) (by
      obtain ⟨w, hw⟩ := wordToBall_surjective G t center u
      exact ⟨w, mem_allAddresses d t w, congrArg Subtype.val hw⟩)

def slotCount (d n : Nat) : Nat := (allAddresses d (n + 1)).length

abbrev RowBuffer (d n : Nat) := Buffer (inputSize (n + 1) (slotCount d n))

def fixedLabelAt (d n : Nat) (a : Fin (PoweringTables.labelCount d n))
    (i : Fin (slotCount d n)) : GraphTables.Label :=
  decodeLabel d (n + 1) 64 a ((allAddresses d (n + 1)).get i)

theorem fixedLabelAt_addressIndex (G : PortGraph (Fin vertices) (Fin d)) (n : Nat)
    (center : Fin vertices) (u : Ball G (n + 1) center)
    (a : Fin (PoweringTables.labelCount d n)) :
    fixedLabelAt d n a (addressIndex G (n + 1) center u) =
      decode (finitePortSelector G (n + 1) center) (decodeLabel d (n + 1) 64 a) u := by
  simpa only [fixedLabelAt, labelTable_get] using
    labelTable_addressIndex G (n + 1) center u (decodeLabel d (n + 1) 64 a)

def walkData (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1))
    (k : Fin (n + 1)) : Field (slotCount d n) → Bool
  | .inl (a, b) => PortTables.accepts input (edgeAt (PortTables.portGraph input) n w k) a b
  | .inr (.inl i) => endpointMask (PortTables.portGraph input) (n + 1) w.1
      (edgeAt (PortTables.portGraph input) n w k).1 i
  | .inr (.inr i) => endpointMask (PortTables.portGraph input) (n + 1)
      (endpoint (PortTables.portGraph input) w)
      ((PortTables.portGraph input).rot (edgeAt (PortTables.portGraph input) n w k)).1 i

def walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) : RowBuffer d n :=
  packData (walkData input n w)

@[simp] theorem basePredicate_walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1))
    (a b : GraphTables.Label) :
    basePredicate (walkBits input n w) k a b =
      PortTables.accepts input (edgeAt (PortTables.portGraph input) n w k) a b := by
  simp [basePredicate, walkBits, walkData]

@[simp] theorem leftMatches_walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1)) :
    leftMatches (walkBits input n w) k = endpointMask (PortTables.portGraph input)
      (n + 1) w.1 (edgeAt (PortTables.portGraph input) n w k).1 := by
  funext i
  simp [leftMatches, walkBits, walkData]

@[simp] theorem rightMatches_walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1)) :
    rightMatches (walkBits input n w) k = endpointMask (PortTables.portGraph input)
      (n + 1) (endpoint (PortTables.portGraph input) w)
      ((PortTables.portGraph input).rot (edgeAt (PortTables.portGraph input) n w k)).1 := by
  funext i
  simp [rightMatches, walkBits, walkData]

theorem firstMatch_walkBits_left (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1)) :
    firstMatch (leftMatches (walkBits input n w) k) = some
      (addressIndex (PortTables.portGraph input) (n + 1) w.1
        (tailFromStart (PortTables.portGraph input) n w k)) := by
  rw [leftMatches_walkBits]
  simpa only [tailFromStart_val, slotCount] using
    firstMatch_endpointMask (PortTables.portGraph input) (n + 1) w.1
      (tailFromStart (PortTables.portGraph input) n w k)

theorem firstMatch_walkBits_right (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1)) :
    firstMatch (rightMatches (walkBits input n w) k) = some
      (addressIndex (PortTables.portGraph input) (n + 1)
        (endpoint (PortTables.portGraph input) w)
        (headFromEnd (PortTables.portGraph input) n w k)) := by
  rw [rightMatches_walkBits]
  simpa only [headFromEnd_val, slotCount] using
    firstMatch_endpointMask (PortTables.portGraph input) (n + 1)
      (endpoint (PortTables.portGraph input) w) (headFromEnd (PortTables.portGraph input) n w k)

theorem edgeAccept_walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1)) (k : Fin (n + 1))
    (a b : Fin (PoweringTables.labelCount d n)) :
    edgeAccept (fixedLabelAt d n) (walkBits input n w) k a b =
      PortTables.accepts input (edgeAt (PortTables.portGraph input) n w k)
        (decode (finitePortSelector (PortTables.portGraph input) (n + 1) w.1)
          (decodeLabel d (n + 1) 64 a) (tailFromStart (PortTables.portGraph input) n w k))
        (decode (finitePortSelector (PortTables.portGraph input) (n + 1)
          (endpoint (PortTables.portGraph input) w))
          (decodeLabel d (n + 1) 64 b) (headFromEnd (PortTables.portGraph input) n w k)) := by
  simp only [edgeAccept, firstMatch_walkBits_left, firstMatch_walkBits_right,
    basePredicate_walkBits, fixedLabelAt_addressIndex]

theorem rowAccept_walkBits (input : PortTables.Table vertices d) (n : Nat)
    (w : Walk (Fin vertices) (Fin d) (n + 1))
    (a b : Fin (PoweringTables.labelCount d n)) :
    rowAccept (fixedLabelAt d n) (walkBits input n w) a b =
      rowsAccepts (PortTables.accepts input) (walkRows (PortTables.portGraph input) n w)
        (labelTable (decodeLabel d (n + 1) 64 a))
        (labelTable (decodeLabel d (n + 1) 64 b)) := by
  rw [rowsAccepts_walkRows]
  apply Bool.eq_iff_iff.mpr
  simp only [rowAccept_eq_true, PoweringTest.pathAccepts_eq_true_iff, edgeAccept_walkBits]

def transposeField {S : Nat} : Field S → Field S
  | .inl (a, b) => .inl (b, a)
  | .inr (.inl i) => .inr (.inr i)
  | .inr (.inr i) => .inr (.inl i)

def transposeBits {t S : Nat} (bits : Buffer (inputSize t S)) : Buffer (inputSize t S) :=
  packData fun k field => bits (inputEquiv t S (k, transposeField field))

@[simp] theorem basePredicate_transposeBits {t S : Nat}
    (bits : Buffer (inputSize t S)) (k : Fin t) (a b : GraphTables.Label) :
    basePredicate (transposeBits bits) k a b = basePredicate bits k b a := by
  simp [basePredicate, transposeBits, transposeField]

@[simp] theorem leftMatches_transposeBits {t S : Nat}
    (bits : Buffer (inputSize t S)) (k : Fin t) :
    leftMatches (transposeBits bits) k = rightMatches bits k := by
  funext i
  simp [leftMatches, rightMatches, transposeBits, transposeField]

@[simp] theorem rightMatches_transposeBits {t S : Nat}
    (bits : Buffer (inputSize t S)) (k : Fin t) :
    rightMatches (transposeBits bits) k = leftMatches bits k := by
  funext i
  simp [leftMatches, rightMatches, transposeBits, transposeField]

theorem edgeAccept_transposeBits {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (k : Fin t) (a b : Fin q) :
    edgeAccept labelAt (transposeBits bits) k a b = edgeAccept labelAt bits k b a := by
  simp only [edgeAccept, leftMatches_transposeBits, rightMatches_transposeBits]
  cases firstMatch (leftMatches bits k) <;>
    cases firstMatch (rightMatches bits k) <;>
    simp

theorem rowAccept_transposeBits {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (a b : Fin q) :
    rowAccept labelAt (transposeBits bits) a b = rowAccept labelAt bits b a := by
  apply Bool.eq_iff_iff.mpr
  simp only [rowAccept_eq_true, edgeAccept_transposeBits]

def rowBits (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) : RowBuffer d n :=
  if e.1 then transposeBits (walkBits input n e.2) else walkBits input n e.2

def rowData (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n)
    (k : Fin (n + 1)) (field : Field (slotCount d n)) : Bool :=
  walkData input n e.2 k (if e.1 then transposeField field else field)

theorem rowBits_packData (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    rowBits input n e = packData (rowData input n e) := by
  rcases e with ⟨direction, w⟩
  cases direction <;> funext i <;>
    simp [rowBits, rowData, transposeBits, walkBits, packData]

@[simp] theorem rowBits_field (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n)
    (k : Fin (n + 1)) (field : Field (slotCount d n)) :
    rowBits input n e (inputEquiv (n + 1) (slotCount d n) (k, field)) =
      rowData input n e k field := by
  rw [rowBits_packData, packData_inputEquiv]

theorem rowAccept_rowBits (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n)
    (a b : Fin (PoweringTables.labelCount d n)) :
    rowAccept (fixedLabelAt d n) (rowBits input n e) a b =
      PoweringTables.rowAccepts input n e a b := by
  rcases e with ⟨direction, w⟩
  cases direction <;>
    simp only [rowBits, Bool.false_eq_true, ite_false, ite_true,
      rowAccept_transposeBits, rowAccept_walkBits, PoweringTables.rowAccepts]

theorem rowBlock_at_rowBits (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n)
    (a b : Fin (PoweringTables.labelCount d n)) :
    rowBlock (fixedLabelAt d n) (rowBits input n e)
        (GenericGraphTables.relationIndex (PoweringTables.labelCount d n) (a, b)) =
      PoweringTables.rowAccepts input n e a b := by
  rw [rowBlock_at, rowAccept_rowBits]

def dataWords (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) : List Nat :=
  (List.ofFn (rowBits input n e)).map GraphTables.bitWord

def dataTape (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) : List Bool :=
  encodeWords (dataWords input n e)

theorem dataTape_eq (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    dataTape input n e = encodeBits (List.ofFn (rowBits input n e)) :=
  (encodeBits_graphWords _).symm

theorem dataWords_length (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    (dataWords input n e).length = (n + 1) * (4096 + (slotCount d n + slotCount d n)) := by
  rw [dataWords, List.length_map]
  simpa only [inputSize] using (List.length_ofFn (f := rowBits input n e))

theorem dataTape_length_le (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    (dataTape input n e).length ≤ 2 * inputSize (n + 1) (slotCount d n) := by
  rw [dataTape_eq]
  simpa only [List.length_ofFn] using encodeBits_length_le (List.ofFn (rowBits input n e))

def emittedRelation (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    GenericGraphTables.RelationTable (PoweringTables.labelCount d n) :=
  Vector.ofFn (rowBlock (fixedLabelAt d n) (rowBits input n e))

theorem emittedRelation_eq (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    emittedRelation input n e = GenericGraphTables.relationOf (PoweringTables.rowAccepts input n e) := by
  unfold emittedRelation GenericGraphTables.relationOf
  apply congrArg Vector.ofFn
  funext i
  exact rowAccept_rowBits input n e _ _

theorem rowBlock_unary (input : PortTables.Table vertices d) (n : Nat)
    (e : PoweringTest.Dart (Fin vertices) (Fin d) n) :
    encodeBits (List.ofFn (rowBlock (fixedLabelAt d n) (rowBits input n e))) =
      encodeWords (GenericGraphTables.relationWords (emittedRelation input n e)) := by
  have hbit : GraphTables.bitWord = GenericGraphTables.bitWord := by
    funext b
    cases b <;> rfl
  have h := encodeBits_graphWords
    (List.ofFn (rowBlock (fixedLabelAt d n) (rowBits input n e)))
  rw [hbit] at h
  simpa only [emittedRelation, GenericGraphTables.relationWords, Vector.toList_ofFn] using h

theorem emittedRelation_table (input : PortTables.Table vertices d) (n : Nat)
    (i : Fin (PoweringTables.dartCount vertices d n)) :
    emittedRelation input n (decodeDart vertices d n i) =
      (PoweringTables.table input n).rows[i].relation := by
  apply Vector.ext
  intro j hj
  have h := PoweringTables.table_accepts input n i
    ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).1
    ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).2
  have hindex : GenericGraphTables.relationIndex (PoweringTables.labelCount d n)
      (((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).1,
        ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).2) =
        ⟨j, hj⟩ := by
    change GenericGraphTables.relationIndex (PoweringTables.labelCount d n)
      ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩) = _
    exact Equiv.apply_symm_apply _ _
  simp only [GenericGraphTables.acceptsAt, GenericGraphTables.relationAt, hindex,
    Fin.getElem_fin] at h
  simp only [emittedRelation, Vector.getElem_ofFn]
  change rowBlock (fixedLabelAt d n) (rowBits input n (decodeDart vertices d n i)) ⟨j, hj⟩ =
    (PoweringTables.table input n).rows[i].relation[j]
  change rowAccept (fixedLabelAt d n) (rowBits input n (decodeDart vertices d n i))
    ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).1
    ((GenericGraphTables.relationIndex (PoweringTables.labelCount d n)).symm ⟨j, hj⟩).2 = _
  rw [rowAccept_rowBits]
  exact h.symm

theorem rowBlock_unary_table (input : PortTables.Table vertices d) (n : Nat)
    (i : Fin (PoweringTables.dartCount vertices d n)) :
    encodeBits (List.ofFn (rowBlock (fixedLabelAt d n)
      (rowBits input n (decodeDart vertices d n i)))) =
      encodeWords (GenericGraphTables.relationWords (PoweringTables.table input n).rows[i].relation) := by
  rw [rowBlock_unary, emittedRelation_table]

def tableRowMachineInTime (input : PortTables.Table vertices d) (n : Nat)
    (i : Fin (PoweringTables.dartCount vertices d n)) (suffix : List Bool)
    (register : RowBuffer d n) (tapes : Bool → List Bool)
    (hinput : tapes false = dataTape input n (decodeDart vertices d n i) ++ suffix) :
    StateTransition.EvalsToInTime (PoweringMachineRow.machine (t := n + 1) (fixedLabelAt d n)).step
      ⟨some (), ((), register), tapes⟩
      (some ⟨none, ((), emptyBuffer (inputSize (n + 1) (slotCount d n))),
        Function.update (Function.update tapes false suffix) true
          (encodeWords (GenericGraphTables.relationWords
            (PoweringTables.table input n).rows[i].relation) ++ tapes true)⟩) 1 := by
  have hinput' : tapes false =
      encodeBits (List.ofFn (rowBits input n (decodeDart vertices d n i))) ++ suffix := by
    simpa only [dataTape_eq] using hinput
  have h := PoweringMachineRow.machineInTime (fixedLabelAt d n)
    (rowBits input n (decodeDart vertices d n i)) suffix register tapes hinput'
  simpa only [rowBlock_unary_table] using h

end MinUncutGames.Foundations.PCP.PoweringRowData

end
section
namespace MinUncutGames.Foundations.PCP.PoweringFieldPlan

open PoweringWalks PoweringLabels PoweringAddresses PoweringReach PoweringRowData
open MinUncutGames.Foundations.Complexity PoweringMachineRow

variable {V D : Type*}

def tailPorts : (n : Nat) → (Fin (n + 1) → D) → Fin (n + 1) → List D
  | 0, _, _ => []
  | n + 1, p, k => Fin.cases [] (fun j => p 0 :: tailPorts n (fun i => p i.succ) j) k

theorem tailPorts_length : ∀ (n : Nat) (p : Fin (n + 1) → D) (k : Fin (n + 1)),
    (tailPorts n p k).length = k.val := by
  intro n
  induction n with
  | zero => intro p k; have hk := Fin.eq_zero k; subst k; rfl
  | succ n ih =>
      intro p k
      refine Fin.cases rfl (fun j => ?_) k
      simp only [tailPorts, Fin.cases_succ, List.length_cons, ih, Fin.val_succ]

theorem walkEnd_tailPorts (G : PortGraph V D) :
    ∀ (n : Nat) (v : V) (p : Fin (n + 1) → D) (k : Fin (n + 1)),
      walkEnd G v (tailPorts n p k) = (edgeAt G n (v, p) k).1 := by
  intro n
  induction n with
  | zero => intro v p k; rfl
  | succ n ih =>
      intro v p k
      refine Fin.cases rfl (fun j => ?_) k
      exact ih (next G v (p 0)) (fun i => p i.succ) j

def headPorts (n : Nat) (p : Fin (n + 1) → D) (k : Fin (n + 1)) : List D :=
  tailPorts n p k ++ [p k]

theorem walkEnd_headPorts (G : PortGraph V D) (n : Nat) (v : V)
    (p : Fin (n + 1) → D) (k : Fin (n + 1)) :
    walkEnd G v (headPorts n p k) = (G.rot (edgeAt G n (v, p) k)).1 := by
  rw [headPorts, walkEnd_append, walkEnd_tailPorts]
  change (G.rot ((edgeAt G n (v, p) k).1, p k)).1 = _
  have hp : (edgeAt G n (v, p) k).2 = p k := edgeAt_port G n (v, p) k
  rw [← hp]

def addressPorts {d t : Nat} (i : PoweringOpinionTables.AddressIndex d t) : List (Fin d) :=
  List.ofFn ((allAddresses d t).get i).2

theorem walkEnd_addressPorts {d t : Nat} (G : PortGraph V (Fin d)) (v : V)
    (i : PoweringOpinionTables.AddressIndex d t) :
    walkEnd G v (addressPorts i) = (wordToBall G t v ((allAddresses d t).get i)).val :=
  (wordEnd_eq_walkEnd_ofFn G _ _ _).symm

inductive Instruction (d : Nat) where
  | relation (ports : List (Fin d)) (port : Fin d) (left right : Fin 64)
  | equal (left right : List (Fin d))

def Instruction.radius {d : Nat} : Instruction d → Nat
  | .relation ports _ _ _ => ports.length
  | .equal left right => max left.length right.length

theorem addressPorts_length_le {d t : Nat} (i : PoweringOpinionTables.AddressIndex d t) :
    (addressPorts i).length ≤ t := by
  simp only [addressPorts, List.length_ofFn]
  exact Nat.le_of_lt_succ ((allAddresses d t).get i).1.isLt

def evaluate {vertices d : Nat} (input : PortTables.Table vertices d)
    (start : Fin vertices) : Instruction d → Bool
  | .relation ports port a b => PortTables.accepts input
      (walkEnd (PortTables.portGraph input) start ports, port) a b
  | .equal left right => decide
      (walkEnd (PortTables.portGraph input) start left =
        walkEnd (PortTables.portGraph input) start right)

def fieldPlan {d : Nat} (n : Nat) (ports : Fin (n + 1) → Fin d)
    (k : Fin (n + 1)) : Field (slotCount d n) → Instruction d
  | .inl (a, b) => .relation (tailPorts n ports k) (ports k) a b
  | .inr (.inl i) => .equal (addressPorts i) (tailPorts n ports k)
  | .inr (.inr i) => .equal (List.ofFn ports ++ addressPorts i) (headPorts n ports k)

theorem evaluate_fieldPlan {vertices d : Nat} (input : PortTables.Table vertices d)
    (n : Nat) (start : Fin vertices) (ports : Fin (n + 1) → Fin d)
    (k : Fin (n + 1)) (field : Field (slotCount d n)) :
    evaluate input start (fieldPlan n ports k field) = walkData input n (start, ports) k field := by
  rcases field with ⟨a,b⟩ | i | i
  · simp only [evaluate, fieldPlan, walkData, walkEnd_tailPorts]
    have hp : (edgeAt (PortTables.portGraph input) n (start, ports) k).2 = ports k :=
      edgeAt_port (PortTables.portGraph input) n (start, ports) k
    rw [← hp]
  · simp only [evaluate, fieldPlan, walkData, endpointMask, walkEnd_tailPorts]
    have ha := walkEnd_addressPorts (t := n + 1) (PortTables.portGraph input) start i
    rw [ha]
  · simp only [evaluate, fieldPlan, walkData, endpointMask, walkEnd_append,
      walkEnd_headPorts]
    rw [← wordEnd_eq_walkEnd_ofFn]
    have ha := walkEnd_addressPorts (t := n + 1) (PortTables.portGraph input)
      (wordEnd (PortTables.portGraph input) (n + 1) start ports) i
    rw [ha]
    rfl

def directedFieldPlan {d : Nat} (n : Nat) (ports : Fin (n + 1) → Fin d)
    (direction : Bool) (k : Fin (n + 1)) (field : Field (slotCount d n)) : Instruction d :=
  fieldPlan n ports k (if direction then transposeField field else field)

theorem fieldPlan_radius {d : Nat} (n : Nat) (ports : Fin (n + 1) → Fin d)
    (k : Fin (n + 1)) (field : Field (slotCount d n)) :
    (fieldPlan n ports k field).radius ≤ 2 * (n + 1) := by
  have hk := k.isLt
  rcases field with ⟨a,b⟩ | i | i
  · simp only [fieldPlan, Instruction.radius, tailPorts_length]
    omega
  · have hi := addressPorts_length_le i
    simp only [fieldPlan, Instruction.radius, tailPorts_length]
    exact max_le (by omega) (by omega)
  · have hi := addressPorts_length_le i
    simp only [fieldPlan, Instruction.radius, headPorts, List.length_append,
      List.length_ofFn, List.length_singleton, tailPorts_length]
    exact max_le (by omega) (by omega)

theorem directedFieldPlan_radius {d : Nat} (n : Nat) (ports : Fin (n + 1) → Fin d)
    (direction : Bool) (k : Fin (n + 1)) (field : Field (slotCount d n)) :
    (directedFieldPlan n ports direction k field).radius ≤ 2 * (n + 1) :=
  fieldPlan_radius n ports k _

theorem evaluate_directedFieldPlan {vertices d : Nat} (input : PortTables.Table vertices d)
    (n : Nat) (start : Fin vertices) (ports : Fin (n + 1) → Fin d)
    (direction : Bool) (k : Fin (n + 1)) (field : Field (slotCount d n)) :
    evaluate input start (directedFieldPlan n ports direction k field) =
      rowData input n (direction, start, ports) k field :=
  evaluate_fieldPlan input n start ports k _

def rowPlan {d : Nat} (n : Nat) (ports : Fin (n + 1) → Fin d) (direction : Bool) :
    List (Instruction d) :=
  List.ofFn fun i : Fin (inputSize (n + 1) (slotCount d n)) =>
    let pair := (inputEquiv (n + 1) (slotCount d n)).symm i
    directedFieldPlan n ports direction pair.1 pair.2

theorem evaluate_rowPlan {vertices d : Nat} (input : PortTables.Table vertices d)
    (n : Nat) (start : Fin vertices) (ports : Fin (n + 1) → Fin d) (direction : Bool) :
    (rowPlan n ports direction).map (evaluate input start) =
      List.ofFn (rowBits input n (direction, start, ports)) := by
  rw [rowPlan, List.map_ofFn]
  apply congrArg List.ofFn
  funext i
  dsimp only [Function.comp_apply]
  rw [evaluate_directedFieldPlan, rowBits_packData]
  rfl

theorem dataTape_eq_plan {vertices d : Nat} (input : PortTables.Table vertices d)
    (n : Nat) (start : Fin vertices) (ports : Fin (n + 1) → Fin d) (direction : Bool) :
    dataTape input n (direction, start, ports) =
      encodeBits ((rowPlan n ports direction).map (evaluate input start)) := by
  rw [evaluate_rowPlan, dataTape_eq]

end MinUncutGames.Foundations.PCP.PoweringFieldPlan

end

end OAI