Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.NameCompaction

namespace OAI


namespace DFVSGames.BinaryNameMachine

open Turing
open DFVSGames.Foundations.Complexity
open MachineComposition
open DFVSGames.Reduction.MachineTransfer

def frame : List Bool → List Bool
  | [] => [false]
  | bit :: bits => true :: bit :: frame bits

def finalDigit : List Bool → Option Bool → Option Bool
  | [], previous => previous
  | bit :: bits, _ => finalDigit bits (some bit)

def canonical (bits : List Bool) : Bool := (finalDigit bits none).getD true

structure Result where
  accepted : Bool
  remaining : List Bool
  reversedPayload : List Bool
  deriving DecidableEq

def scanSpec : List Bool → List Bool → Option Bool → Result
  | [], acc, _ => ⟨false, [], acc⟩
  | false :: rest, acc, last => ⟨last.getD true, rest, acc⟩
  | [true], acc, _ => ⟨false, [], acc⟩
  | true :: bit :: rest, acc, _ => scanSpec rest (bit :: acc) (some bit)

def steps : List Bool → Nat
  | true :: _ :: rest => steps rest + 1
  | _ => 1

theorem steps_le (input : List Bool) : steps input ≤ input.length + 1 := by
  induction input using List.twoStepInduction with
  | nil => simp [steps]
  | singleton bit => cases bit <;> simp [steps]
  | cons_cons flag bit rest ih _ =>
      cases flag <;> simp only [steps, List.length_cons]
      · omega
      · omega

@[simp] theorem frame_length (bits : List Bool) :
    (frame bits).length = 2 * bits.length + 1 := by
  induction bits with
  | nil => simp [frame]
  | cons bit bits ih => simp only [frame, List.length_cons, ih]; omega

theorem scanSpec_frame (bits suffix acc : List Bool) (last : Option Bool) :
    scanSpec (frame bits ++ suffix) acc last =
      ⟨(finalDigit bits last).getD true, suffix, bits.reverse ++ acc⟩ := by
  induction bits generalizing acc last with
  | nil => simp [frame, scanSpec, finalDigit]
  | cons bit bits ih =>
      simpa [frame, scanSpec, finalDigit, List.reverse_cons, List.append_assoc] using
        ih (bit :: acc) (some bit)

@[simp] theorem steps_frame (bits suffix : List Bool) :
    steps (frame bits ++ suffix) = bits.length + 1 := by
  induction bits with
  | nil => simp [frame, steps]
  | cons bit bits ih => simp [frame, steps, ih]

section Program

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

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := A × Option Bool × Option Bool

def clean (ambient : A) : State A := (ambient, none, none)

def exitAt (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .load (fun state => clean state.1) (exitAt exit)

def scan (source destination : K) (again : Λ) (accepted rejected : Option Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .pop source (fun state head => (state.1, state.2.1, head))
    (.branch (fun state => state.2.2.isNone)
      (finish rejected)
      (.branch (fun state => state.2.2.getD false)
        (.pop source (fun state head => (state.1, state.2.1, head))
          (.branch (fun state => state.2.2.isSome)
            (.push destination (fun state => state.2.2.getD false)
              (.load (fun state => (state.1, state.2.2, none))
                (.goto fun _ => again)))
            (finish rejected)))
        (.branch (fun state => state.2.1.getD true)
          (finish accepted) (finish rejected))))

@[simp] theorem stepAux_finish (exit : Option Λ) (state : State A)
    (base : K → List Bool) :
    TM2.stepAux (finish exit) state base = ⟨exit, clean state.1, base⟩ := by
  cases exit <;> rfl

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

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

variable (source destination : K) (distinct : source ≠ destination)
variable (again : Λ) (accepted rejected : Option Λ)
variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
variable (atScan : program again = scan source destination again accepted rejected)
variable (base : K → List Bool) (ambient : A)

include distinct atScan

theorem step_empty (output : List Bool) (last register : Option Bool) :
    TM2.step program
      ⟨some again, (ambient, last, register), tapesAt source destination base [] output⟩ =
      some ⟨rejected, clean ambient, tapesAt source destination base [] output⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atScan]
  simp [scan, TM2.stepAux, distinct, update_source_inline_MachineBinaryNameMachine]

theorem step_end (rest output : List Bool) (last register : Option Bool) :
    TM2.step program
      ⟨some again, (ambient, last, register),
        tapesAt source destination base (false :: rest) output⟩ =
      some ⟨if last.getD true then accepted else rejected, clean ambient,
        tapesAt source destination base rest output⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atScan]
  cases h : last.getD true <;>
    simp [scan, TM2.stepAux, distinct, update_source_inline_MachineBinaryNameMachine, h]

theorem step_truncated (output : List Bool) (last register : Option Bool) :
    TM2.step program
      ⟨some again, (ambient, last, register), tapesAt source destination base [true] output⟩ =
      some ⟨rejected, clean ambient, tapesAt source destination base [] output⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atScan]
  simp [scan, TM2.stepAux, distinct, update_source_inline_MachineBinaryNameMachine]

theorem step_digit (bit : Bool) (rest output : List Bool) (last register : Option Bool) :
    TM2.step program
      ⟨some again, (ambient, last, register),
        tapesAt source destination base (true :: bit :: rest) output⟩ =
      some ⟨some again, (ambient, some bit, none),
        tapesAt source destination base rest (bit :: output)⟩ := by
  change some (TM2.stepAux (program again) _ _) = _
  rw [atScan]
  simp [scan, TM2.stepAux, distinct, update_source_inline_MachineBinaryNameMachine, update_destination_inline_MachineBinaryNameMachine]

theorem scanTrace (input output : List Bool) (last register : Option Bool) :
    (advance (TM2.step program))^[steps input]
      (some ⟨some again, (ambient, last, register),
        tapesAt source destination base input output⟩) =
      some ⟨if (scanSpec input output last).accepted then accepted else rejected,
        clean ambient, tapesAt source destination base
          (scanSpec input output last).remaining (scanSpec input output last).reversedPayload⟩ := by
  induction input using List.twoStepInduction generalizing output last register with
  | nil =>
      simpa only [steps, scanSpec, Bool.false_eq_true, ite_false,
        Function.iterate_one, advance_some] using
        step_empty source destination distinct again accepted rejected program atScan base ambient
          output last register
  | singleton flag =>
      cases flag
      · have h := step_end source destination distinct again accepted rejected program atScan
          base ambient [] output last register
        cases hlast : last.getD true <;>
          simpa [steps, scanSpec, hlast] using h
      · simpa only [steps, scanSpec, Bool.false_eq_true, ite_false,
          Function.iterate_one, advance_some] using
          step_truncated source destination distinct again accepted rejected program atScan base ambient
            output last register
  | cons_cons flag bit rest ih _ =>
      cases flag
      · have h := step_end source destination distinct again accepted rejected program atScan
          base ambient (bit :: rest) output last register
        cases hlast : last.getD true <;>
          simpa [steps, scanSpec, hlast] using h
      · rw [steps, Function.iterate_succ_apply]
        simp only [advance_some]
        rw [step_digit source destination distinct again accepted rejected program atScan]
        exact ih (bit :: output) (some bit) none

def scanInTime (input output : List Bool) (last register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some again, (ambient, last, register), tapesAt source destination base input output⟩
      (some ⟨if (scanSpec input output last).accepted then accepted else rejected,
        clean ambient, tapesAt source destination base
          (scanSpec input output last).remaining (scanSpec input output last).reversedPayload⟩)
      (input.length + 1) where
  steps := steps input
  evals_in_steps := scanTrace source destination distinct again accepted rejected program atScan
    base ambient input output last register
  steps_le_m := steps_le input

theorem framedTrace (bits suffix output : List Bool) (register : Option Bool) :
    (advance (TM2.step program))^[bits.length + 1]
      (some ⟨some again, (ambient, none, register),
        tapesAt source destination base (frame bits ++ suffix) output⟩) =
      some ⟨if canonical bits then accepted else rejected, clean ambient,
        tapesAt source destination base suffix (bits.reverse ++ output)⟩ := by
  have h := scanTrace source destination distinct again accepted rejected program atScan
    base ambient (frame bits ++ suffix) output none register
  cases hlast : (finalDigit bits none).getD true <;>
    simpa [steps_frame, scanSpec_frame, canonical, hlast] using h

end Program

def machine : FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ _ := Bool
  Λ := Fin 3
  main := 0
  σ := State Unit
  initialState := clean ()
  m label := if label = 0 then scan false true 0 (some 1) (some 2) else .halt

theorem machine_finiteAlphabet (k : machine.K) : Finite (machine.Γ k) := by
  change Finite Bool
  infer_instance

def machineInTime (input output : List Bool) :
    StateTransition.EvalsToInTime machine.step
      ⟨some (0 : Fin 3), clean (), tapesAt false true (fun _ : Bool => []) input output⟩
      (some ⟨if (scanSpec input output none).accepted then some (1 : Fin 3) else some (2 : Fin 3),
        clean (), tapesAt false true (fun _ : Bool => [])
          (scanSpec input output none).remaining (scanSpec input output none).reversedPayload⟩)
      (input.length + 1) :=
  scanInTime false true (by decide) (0 : Fin 3) (some 1) (some 2) machine.m
    (by simp [machine]) (fun _ : Bool => []) () input output none none

end DFVSGames.BinaryNameMachine


namespace DFVSGames.BinaryParsing

open BinaryEncoding BinaryFormula

theorem frame_eq (bits : List Bool) : BinaryNameMachine.frame bits = frame bits := by
  induction bits with
  | nil => rfl
  | cons bit bits ih => simp [BinaryNameMachine.frame, frame, ih]

theorem canonical_cons_cons (a b : Bool) (bits : List Bool) :
    BinaryNameMachine.canonical (a :: b :: bits) = BinaryNameMachine.canonical (b :: bits) := rfl

theorem canonical_iff (bits : List Bool) :
    BinaryNameMachine.canonical bits = true ↔ (bitsValue bits).bits = bits := by
  induction bits with
  | nil => simp [BinaryNameMachine.canonical, BinaryNameMachine.finalDigit, bitsValue]
  | cons bit bits ih =>
      cases bits with
      | nil => cases bit <;> decide
      | cons next bits =>
          rw [canonical_cons_cons, ih]
          constructor
          · intro htail
            have hn : bitsValue (next :: bits) ≠ 0 := by
              intro hz
              rw [hz, Nat.zero_bits] at htail
              contradiction
            rw [bitsValue, Nat.bits_append_bit _ bit (fun h => False.elim (hn h)), htail]
          · intro hfull
            calc
              (bitsValue (next :: bits)).bits =
                  (Nat.bit bit (bitsValue (next :: bits))).div2.bits := by rw [Nat.div2_bit]
              _ = (Nat.bit bit (bitsValue (next :: bits))).bits.tail :=
                Nat.div2_bits_eq_tail _
              _ = (bit :: next :: bits).tail := congrArg List.tail hfull
              _ = next :: bits := rfl

@[simp] theorem canonical_nat_bits (name : Nat) :
    BinaryNameMachine.canonical name.bits = true := by
  apply (canonical_iff _).mpr
  rw [bitsValue_bits]

theorem parseFrame_sound (input digits rest : List Bool)
    (parsed : parseFrame input = some (digits, rest)) : input = frame digits ++ rest := by
  induction input using List.twoStepInduction generalizing digits rest with
  | nil => simp [parseFrame] at parsed
  | singleton flag =>
      cases flag
      · simp only [parseFrame, Option.some.injEq, Prod.mk.injEq] at parsed
        rcases parsed with ⟨rfl, rfl⟩
        rfl
      · simp [parseFrame] at parsed
  | cons_cons flag bit input ih _ =>
      cases flag
      · simp only [parseFrame, Option.some.injEq, Prod.mk.injEq] at parsed
        rcases parsed with ⟨rfl, rfl⟩
        rfl
      · cases h : parseFrame input with
        | none => simp [parseFrame, h] at parsed
        | some pair =>
            rcases pair with ⟨ds, tail⟩
            have hs := ih ds tail h
            simp [parseFrame, h] at parsed
            rcases parsed with ⟨rfl, rfl⟩
            simp [frame, hs]

theorem parseName_sound (input : List Bool) (name : Nat) (rest : List Bool)
    (parsed : parseName input = some (name, rest)) : input = nameBits name ++ rest := by
  cases h : parseFrame input with
  | none => simp [parseName, h] at parsed
  | some pair =>
      rcases pair with ⟨digits, tail⟩
      by_cases canonical : digits = (bitsValue digits).bits
      · simp only [parseName, h, Option.bind_eq_bind, Option.bind_some] at parsed
        rw [ite_eq_left canonical] at parsed
        simp only [Option.some.injEq, Prod.mk.injEq] at parsed
        rcases parsed with ⟨rfl, rfl⟩
        have hs := parseFrame_sound input digits tail h
        exact hs.trans (congrArg (fun ds => frame ds ++ tail) canonical)
      · simp only [parseName, h, Option.bind_eq_bind, Option.bind_some, ite_eq_right canonical] at parsed
        cases parsed

theorem parseLiteral_sound (input : List Bool) (literal : Literal) (rest : List Bool)
    (parsed : parseLiteral input = some (literal, rest)) :
    input = literalBits literal ++ rest := by
  cases input with
  | nil => simp [parseLiteral] at parsed
  | cons sign input =>
      cases h : parseName input with
      | none => simp [parseLiteral, h] at parsed
      | some pair =>
          rcases pair with ⟨name, tail⟩
          have hs := parseName_sound input name tail h
          simp [parseLiteral, h] at parsed
          rcases parsed with ⟨rfl, rfl⟩
          simp [literalBits, hs]

theorem parseClause_sound (input : List Bool) (clause : Clause) (rest : List Bool)
    (parsed : parseClause input = some (clause, rest)) : input = clauseBits clause ++ rest := by
  cases ha : parseLiteral input with
  | none => simp [parseClause, ha] at parsed
  | some first =>
      rcases first with ⟨a, afterA⟩
      cases hb : parseLiteral afterA with
      | none => simp [parseClause, ha, hb] at parsed
      | some second =>
          rcases second with ⟨b, afterB⟩
          cases hc : parseLiteral afterB with
          | none => simp [parseClause, ha, hb, hc] at parsed
          | some third =>
              rcases third with ⟨c, afterC⟩
              have hA := parseLiteral_sound input a afterA ha
              have hB := parseLiteral_sound afterA b afterB hb
              have hC := parseLiteral_sound afterB c afterC hc
              have pairEq : (#v[a, b, c], afterC) = (clause, rest) := by
                apply Option.some.inj
                simpa only [parseClause, ha, hb, hc, Option.bind_eq_bind,
                  Option.bind_some, Option.pure_def] using parsed
              rcases Prod.mk.inj pairEq with ⟨rfl, rfl⟩
              simp [clauseBits, hA, hB, hC, List.append_assoc]

theorem parseClauses_sound (fuel : Nat) (input : List Bool)
    (clauses : List Clause) (rest : List Bool)
    (parsed : parseClauses fuel input = some (clauses, rest)) :
    input = clausesBits clauses ++ rest := by
  induction fuel generalizing input clauses rest with
  | zero => simp [parseClauses] at parsed
  | succ fuel ih =>
      cases input with
      | nil => simp [parseClauses] at parsed
      | cons flag input =>
          cases flag
          · simp only [parseClauses, Option.some.injEq, Prod.mk.injEq] at parsed
            rcases parsed with ⟨rfl, rfl⟩
            rfl
          · cases hc : parseClause input with
            | none => simp [parseClauses, hc] at parsed
            | some first =>
                rcases first with ⟨clause, afterClause⟩
                cases hs : parseClauses fuel afterClause with
                | none => simp [parseClauses, hc, hs] at parsed
                | some remaining =>
                    rcases remaining with ⟨cs, tail⟩
                    have hC := parseClause_sound input clause afterClause hc
                    have hS := ih afterClause cs tail hs
                    simp [parseClauses, hc, hs] at parsed
                    rcases parsed with ⟨rfl, rfl⟩
                    simp [clausesBits, hC, hS, List.append_assoc]

theorem decodeFormula_sound (input : List Bool) (formula : Formula)
    (parsed : decodeFormula input = some formula) : input = formulaBits formula := by
  cases h : parseClauses (input.length + 1) input with
  | none => simp [decodeFormula, h] at parsed
  | some pair =>
      rcases pair with ⟨clauses, rest⟩
      by_cases empty : rest = []
      · subst rest
        simp [decodeFormula, h] at parsed
        subst formula
        simpa only [formulaBits, List.append_nil] using
          parseClauses_sound (input.length + 1) input clauses [] h
      · simp [decodeFormula, h, empty] at parsed

theorem decodeFormula_eq_some_iff (input : List Bool) (formula : Formula) :
    decodeFormula input = some formula ↔ input = formulaBits formula :=
  ⟨decodeFormula_sound input formula, fun h => h ▸ decodeFormula_encoded formula⟩

theorem scanSpec_nameBits (name : Nat) (suffix output : List Bool) :
    BinaryNameMachine.scanSpec (nameBits name ++ suffix) output none =
      ⟨true, suffix, name.bits.reverse ++ output⟩ := by
  unfold nameBits
  rw [← frame_eq, BinaryNameMachine.scanSpec_frame]
  have h := canonical_nat_bits name
  simpa only [BinaryNameMachine.canonical] using
    congrArg (fun accepted =>
      (⟨accepted, suffix, name.bits.reverse ++ output⟩ : BinaryNameMachine.Result)) h

end DFVSGames.BinaryParsing


namespace DFVSGames.BinaryOccurrenceRename

open BinaryFormula DFVSGames.Foundations

theorem literal_mem_names (F : Formula) (c : Clause) (hc : c ∈ F.clauses)
    (i : Fin 3) : (c)[i].name ∈ sourceNames F := by
  apply List.mem_flatMap.mpr
  refine ⟨c, hc, ?_⟩
  have hi : i = 0 ∨ i = 1 ∨ i = 2 := by omega
  rcases hi with rfl | rfl | rfl <;> simp [clauseNames]

def nameIndex (F : Formula) (name : Nat) (mem : name ∈ sourceNames F) :
    Fin (sourceNames F).length :=
  ⟨(sourceNames F).idxOf name, List.idxOf_lt_length_of_mem mem⟩

def literal (F : Formula) (l : Literal) (mem : l.name ∈ sourceNames F) :
    Target.Literal (sourceNames F).length :=
  ⟨nameIndex F l.name mem, l.positive⟩

def clause (F : Formula) (c : Clause) (mem : c ∈ F.clauses) :
    Target.Clause (sourceNames F).length :=
  #v[literal F (c)[0] (literal_mem_names F c mem 0),
    literal F (c)[1] (literal_mem_names F c mem 1),
    literal F (c)[2] (literal_mem_names F c mem 2)]

def renamed (F : Formula) : Target.Formula where
  «variables» := (sourceNames F).length
  clauses := F.clauses.attach.map (fun c => clause F c.val c.property)

def restrictAssignment (F : Formula) (A : Nat → Bool) :
    Fin (sourceNames F).length → Bool :=
  fun i => A (sourceNames F)[i.val]

def extendAssignment (F : Formula) (A : Fin (sourceNames F).length → Bool) : Nat → Bool :=
  fun name => if mem : name ∈ sourceNames F then A (nameIndex F name mem) else false

@[simp] theorem nameIndex_reads_name (F : Formula) (name : Nat)
    (mem : name ∈ sourceNames F) : (sourceNames F)[(nameIndex F name mem).val] = name :=
  List.getElem_idxOf _

@[simp] theorem clause_eval_restrict (F : Formula) (c : Clause) (mem : c ∈ F.clauses)
    (A : Nat → Bool) : (clause F c mem).eval (restrictAssignment F A) = c.eval A := by
  simp [clause, literal, Target.Clause.eval, Target.Literal.eval,
    BinaryFormula.Clause.eval, BinaryFormula.Literal.eval, restrictAssignment]

@[simp] theorem clause_eval_extend (F : Formula) (c : Clause) (mem : c ∈ F.clauses)
    (A : Fin (sourceNames F).length → Bool) :
    (clause F c mem).eval A = c.eval (extendAssignment F A) := by
  have evalLiteral (l : Literal) (hl : l.name ∈ sourceNames F) :
      (literal F l hl).eval A = l.eval (extendAssignment F A) := by
    simp only [literal, Target.Literal.eval, BinaryFormula.Literal.eval,
      extendAssignment, dite_eq_left hl]
    rfl
  simp [clause, Target.Clause.eval, BinaryFormula.Clause.eval, evalLiteral]

@[simp] theorem renamed_clause_count (F : Formula) :
    (renamed F).clauses.length = F.clauses.length := by simp [renamed]

@[simp] theorem renamed_variable_count (F : Formula) :
    (renamed F).«variables» = 3 * F.clauses.length := by
  exact clauseNames_flat_length F.clauses

theorem renamed_satisfiable_iff (F : Formula) : (renamed F).Satisfiable ↔ F.Satisfiable := by
  constructor
  · rintro ⟨A, hA⟩
    refine ⟨extendAssignment F A, ?_⟩
    intro c mem
    have hc : clause F c mem ∈ (renamed F).clauses :=
      List.mem_map.mpr ⟨⟨c, mem⟩, by simp, rfl⟩
    exact (clause_eval_extend F c mem A).symm.trans (hA _ hc)
  · rintro ⟨A, hA⟩
    refine ⟨restrictAssignment F A, ?_⟩
    intro c mem
    obtain ⟨original, _, rfl⟩ := List.mem_map.mp mem
    exact (clause_eval_restrict F original.val original.property A).trans
      (hA original.val original.property)

theorem nat_bits_injective : Function.Injective Nat.bits := by
  intro a b same
  have decoded := congrArg BinaryEncoding.bitsValue same
  simpa only [BinaryEncoding.bitsValue_bits] using decoded

theorem idxOf_binary_payloads (names : List Nat) (name : Nat) :
    (names.map Nat.bits).idxOf name.bits = names.idxOf name := by
  induction names with
  | nil => rfl
  | cons first names ih =>
      by_cases same : first = name
      · subst first
        simp
      · have different : first.bits ≠ name.bits := fun h => same (nat_bits_injective h)
        simp only [List.map_cons, List.idxOf_cons, beq_false_of_ne same,
          beq_false_of_ne different, ih]

theorem renamed_encoding_bound (F : Formula) :
    (Complexity.formulaBits (renamed F)).length ≤
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
  have h := Complexity.formulaBits_length_le (renamed F)
  simp only [renamed_clause_count, renamed_variable_count] at h
  nlinarith

theorem renamed_encoding_bound_bits (F : Formula) :
    (Complexity.formulaBits (renamed F)).length ≤
      9 * (BinaryEncoding.formulaBits F).length * (BinaryEncoding.formulaBits F).length +
      10 * (BinaryEncoding.formulaBits F).length + 2 := by
  have h := renamed_encoding_bound F
  have hm : F.clauses.length ≤ (BinaryEncoding.formulaBits F).length :=
    (BinaryEncoding.clauses_length_lt_bits F.clauses).le
  have hsq := Nat.mul_self_le_mul_self hm
  nlinarith

end DFVSGames.BinaryOccurrenceRename


namespace DFVSGames.BinaryLanguage

open DFVSGames.Foundations

def rejectBinary : BinaryFormula.Formula where
  clauses := [#v[⟨0, true⟩, ⟨0, true⟩, ⟨0, true⟩],
    #v[⟨0, false⟩, ⟨0, false⟩, ⟨0, false⟩]]

theorem rejectBinary_unsatisfiable : ¬rejectBinary.Satisfiable := by
  rintro ⟨assignment, satisfied⟩
  have positive := satisfied #v[⟨0, true⟩, ⟨0, true⟩, ⟨0, true⟩]
    (by simp [rejectBinary])
  have negative := satisfied #v[⟨0, false⟩, ⟨0, false⟩, ⟨0, false⟩]
    (by simp [rejectBinary])
  cases h : assignment 0 <;>
    simp [BinaryFormula.Clause.eval, BinaryFormula.Literal.eval, h] at positive negative

def rejectFormula : Target.Formula := BinaryOccurrenceRename.renamed rejectBinary

theorem reject_unsatisfiable : ¬rejectFormula.Satisfiable := by
  simpa only [rejectFormula, BinaryOccurrenceRename.renamed_satisfiable_iff] using
    rejectBinary_unsatisfiable

def language (input : List Bool) : Prop :=
  ∃ formula, BinaryEncoding.decodeFormula input = some formula ∧ formula.Satisfiable

def totalParsed (input : List Bool) : BinaryFormula.Formula :=
  (BinaryEncoding.decodeFormula input).getD rejectBinary

def totalRename (input : List Bool) : Target.Formula :=
  BinaryOccurrenceRename.renamed (totalParsed input)

theorem totalParsed_satisfiable_iff (input : List Bool) :
    (totalParsed input).Satisfiable ↔ language input := by
  cases parsed : BinaryEncoding.decodeFormula input with
  | none => simp [totalParsed, parsed, rejectBinary_unsatisfiable, language]
  | some formula => simp [totalParsed, parsed, language]

theorem totalRename_satisfiable_iff (input : List Bool) :
    (totalRename input).Satisfiable ↔ language input := by
  exact (BinaryOccurrenceRename.renamed_satisfiable_iff (totalParsed input)).trans
    (totalParsed_satisfiable_iff input)

theorem totalRename_encoded (formula : BinaryFormula.Formula) :
    totalRename (BinaryEncoding.formulaBits formula) = BinaryOccurrenceRename.renamed formula := by
  simp [totalRename, totalParsed]

theorem language_encoded (formula : BinaryFormula.Formula) :
    language (BinaryEncoding.formulaBits formula) ↔ formula.Satisfiable := by
  simp [language]

theorem reject_bits_length : (Complexity.formulaBits rejectFormula).length = 25 := by decide

theorem totalRename_output_length (input : List Bool) :
    (Complexity.formulaBits (totalRename input)).length ≤
      9 * input.length * input.length + 10 * input.length + 25 := by
  cases parsed : BinaryEncoding.decodeFormula input with
  | none =>
      have output : totalRename input = rejectFormula := by simp [totalRename, totalParsed, parsed, rejectFormula]
      rw [output, reject_bits_length]
      omega
  | some formula =>
      have correct := BinaryParsing.decodeFormula_sound input formula parsed
      have bound := BinaryOccurrenceRename.renamed_encoding_bound_bits formula
      simp only [totalRename, totalParsed, parsed, Option.getD_some]
      rw [correct]
      omega

end DFVSGames.BinaryLanguage


namespace DFVSGames.Explicit.MachineOutputContract

open DFVSGames.Foundations
open Target
open scoped BigOperators

structure SimpleBipartite {q : Nat} (game : Instance q) where
  side : Fin game.vertices → Bool
  sourceSide : ∀ i : Fin game.constraints.length,
    side game.constraints[i].source = false
  targetSide : ∀ i : Fin game.constraints.length,
    side game.constraints[i].target = true
  endpoints_injective : Function.Injective (fun i : Fin game.constraints.length =>
    (game.constraints[i].source, game.constraints[i].target))

def permutationEquiv {q : Nat} (table : PermutationTable q) : Equiv.Perm (Fin q) where
  toFun label := table.images[label]
  invFun label := table.inverseImages[label]
  left_inv := table.leftInverse
  right_inv := table.rightInverse

namespace SimpleBipartite

variable {q : Nat} {game : Instance q} (presentation : SimpleBipartite game)

include presentation in
theorem no_loops (i : Fin game.constraints.length) :
    game.constraints[i].source ≠ game.constraints[i].target := by
  intro same
  have sides := congrArg presentation.side same
  rw [presentation.sourceSide, presentation.targetSide] at sides
  cases sides

include presentation in
theorem no_reverse (i j : Fin game.constraints.length) :
    game.constraints[i].source ≠ game.constraints[j].target := by
  intro same
  have sides := congrArg presentation.side same
  rw [presentation.sourceSide, presentation.targetSide] at sides
  cases sides

include presentation in

theorem undirected_unique (i j : Fin game.constraints.length)
    (same :
      (game.constraints[i].source = game.constraints[j].source ∧
        game.constraints[i].target = game.constraints[j].target) ∨
      (game.constraints[i].source = game.constraints[j].target ∧
        game.constraints[i].target = game.constraints[j].source)) : i = j := by
  rcases same with same | reversed
  · exact presentation.endpoints_injective (Prod.ext same.1 same.2)
  · exact False.elim (presentation.no_reverse i j reversed.1)

def toBipartiteGame : BipartiteGame
    {v : Fin game.vertices // presentation.side v = false}
    {v : Fin game.vertices // presentation.side v = true}
    (Fin game.constraints.length) (Fin q) where
  left i := ⟨game.constraints[i].source, presentation.sourceSide i⟩
  right i := ⟨game.constraints[i].target, presentation.targetSide i⟩
  permutation i := permutationEquiv game.constraints[i].permutation
  simple := by
    intro i j same
    apply presentation.endpoints_injective
    apply Prod.ext
    · exact congrArg Subtype.val (congrArg Prod.fst same)
    · exact congrArg Subtype.val (congrArg Prod.snd same)

theorem toBipartiteGame_satisfied_iff (labeling : Fin game.vertices → Fin q)
    (i : Fin game.constraints.length) :
    presentation.toBipartiteGame.Satisfied
      (fun v => labeling v.val) (fun v => labeling v.val) i ↔
      game.constraints[i].satisfied labeling = true := by
  simp [BipartiteGame.Satisfied, toBipartiteGame, permutationEquiv,
    Constraint.satisfied, eq_comm]

def mergeLabelings
    (left : {v : Fin game.vertices // presentation.side v = false} → Fin q)
    (right : {v : Fin game.vertices // presentation.side v = true} → Fin q) :
    Fin game.vertices → Fin q := fun v =>
  if h : presentation.side v = false then left ⟨v, h⟩
  else right ⟨v, by cases hs : presentation.side v <;> simp_all⟩

@[simp] theorem mergeLabelings_left
    (left : {v : Fin game.vertices // presentation.side v = false} → Fin q)
    (right : {v : Fin game.vertices // presentation.side v = true} → Fin q)
    (v : {v : Fin game.vertices // presentation.side v = false}) :
    presentation.mergeLabelings left right v.val = left v := by
  simp [mergeLabelings, v.property]

@[simp] theorem mergeLabelings_right
    (left : {v : Fin game.vertices // presentation.side v = false} → Fin q)
    (right : {v : Fin game.vertices // presentation.side v = true} → Fin q)
    (v : {v : Fin game.vertices // presentation.side v = true}) :
    presentation.mergeLabelings left right v.val = right v := by
  simp [mergeLabelings, v.property]

theorem toBipartiteGame_count (labeling : Fin game.vertices → Fin q) :
    presentation.toBipartiteGame.satisfiedCount
      (fun v => labeling v.val) (fun v => labeling v.val) =
      countSatisfied labeling game.constraints := by
  have count := Integration.GapSemantics.countSatisfied_ofFn labeling
    (fun i : Fin game.constraints.length => game.constraints[i])
  simp only [Fin.getElem_fin, List.ofFn_getElem] at count
  rw [count]
  unfold BipartiteGame.satisfiedCount
  apply Finset.sum_congr rfl
  intro e _
  simp [toBipartiteGame, permutationEquiv, Constraint.satisfied]
  rfl

theorem toBipartiteGame_count_merge
    (left : {v : Fin game.vertices // presentation.side v = false} → Fin q)
    (right : {v : Fin game.vertices // presentation.side v = true} → Fin q) :
    presentation.toBipartiteGame.satisfiedCount left right =
      countSatisfied (presentation.mergeLabelings left right) game.constraints := by
  simpa only [mergeLabelings_left, mergeLabelings_right] using
    presentation.toBipartiteGame_count (presentation.mergeLabelings left right)

theorem toBipartiteGame_maxSatisfied :
    presentation.toBipartiteGame.maxSatisfied =
      Integration.InstanceValue.maxSatisfied game := by
  classical
  apply Nat.le_antisymm
  · unfold BipartiteGame.maxSatisfied
    apply Finset.sup_le
    intro labeling _
    rw [presentation.toBipartiteGame_count_merge labeling.1 labeling.2]
    exact Integration.InstanceValue.countSatisfied_le_maxSatisfied _ _
  · unfold Integration.InstanceValue.maxSatisfied
    apply Finset.sup_le
    intro labeling _
    rw [← presentation.toBipartiteGame_count labeling]
    exact BipartiteGame.satisfiedCount_le_maxSatisfied _ _ _

theorem toBipartiteGame_value :
    presentation.toBipartiteGame.value = Integration.InstanceValue.value game := by
  unfold BipartiteGame.value Integration.InstanceValue.value
  rw [presentation.toBipartiteGame_maxSatisfied]
  simp

end SimpleBipartite

structure BinaryGapReduction (ε δ : ℝ) where
  alphabet : Nat
  alphabetAtLeastTwo : 2 ≤ alphabet
  dimension : Nat
  dimensionPositive : 1 ≤ dimension
  coordinates : Fin alphabet ≃ Integration.BinaryLinear.Vector dimension
  construct : List Bool → Instance alphabet
  simpleBipartite : ∀ input, SimpleBipartite (construct input)
  translations : ∀ input,
    Integration.TranslationTarget.IsTranslationInstance coordinates (construct input)
  computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool)
    Complexity.gameBits construct
  finiteAlphabet : Complexity.MachineFiniteAlphabet.FiniteAlphabet computation.tm
  completeness : ∀ input, BinaryLanguage.language input →
    ∃ labeling, 1 - ε ≤
      (countSatisfied labeling (construct input).constraints : ℝ) /
        (construct input).constraints.length
  soundness : ∀ input, ¬BinaryLanguage.language input →
    ∀ labeling, (countSatisfied labeling (construct input).constraints : ℝ) /
      (construct input).constraints.length ≤ δ

namespace BinaryGapReduction

variable {ε δ : ℝ} (reduction : BinaryGapReduction ε δ)

theorem nonempty (input : List Bool) : (reduction.construct input).constraints ≠ [] :=
  (reduction.construct input).nonempty

theorem occurrence_count_positive (input : List Bool) :
    0 < (reduction.construct input).constraints.length :=
  (reduction.construct input).constraintCount_positive

theorem completeness_value (input : List Bool) (yes : BinaryLanguage.language input) :
    1 - ε ≤ Integration.InstanceValue.value (reduction.construct input) := by
  have positive : 0 < reduction.alphabet :=
    lt_of_lt_of_le (by decide : 0 < 2) reduction.alphabetAtLeastTwo
  exact (Integration.InstanceValue.exists_rate_ge_iff _ positive _).mp
    (reduction.completeness input yes)

theorem soundness_value (input : List Bool) (no : ¬BinaryLanguage.language input) :
    Integration.InstanceValue.value (reduction.construct input) ≤ δ := by
  have positive : 0 < reduction.alphabet :=
    lt_of_lt_of_le (by decide : 0 < 2) reduction.alphabetAtLeastTwo
  exact (Integration.InstanceValue.forall_rate_le_iff _ positive _).mp
    (reduction.soundness input no)

theorem alphabet_eq_two_pow : reduction.alphabet = 2 ^ reduction.dimension := by
  have cardinal := Fintype.card_congr reduction.coordinates
  simpa [Integration.BinaryLinear.Vector] using cardinal

theorem no_loops (input : List Bool)
    (i : Fin (reduction.construct input).constraints.length) :
    (reduction.construct input).constraints[i].source ≠
      (reduction.construct input).constraints[i].target :=
  (reduction.simpleBipartite input).no_loops i

theorem undirected_unique (input : List Bool)
    (i j : Fin (reduction.construct input).constraints.length)
    (same :
      ((reduction.construct input).constraints[i].source =
          (reduction.construct input).constraints[j].source ∧
        (reduction.construct input).constraints[i].target =
          (reduction.construct input).constraints[j].target) ∨
      ((reduction.construct input).constraints[i].source =
          (reduction.construct input).constraints[j].target ∧
        (reduction.construct input).constraints[i].target =
          (reduction.construct input).constraints[j].source)) : i = j :=
  (reduction.simpleBipartite input).undirected_unique i j same

theorem full_table_length (input : List Bool)
    (i : Fin (reduction.construct input).constraints.length) :
    (Complexity.tableWords
      (reduction.construct input).constraints[i].permutation).length = reduction.alphabet :=
  Complexity.tableWords_length _

theorem translation_equation (input : List Bool)
    (i : Fin (reduction.construct input).constraints.length) :
    ∃ shift : Integration.BinaryLinear.Vector reduction.dimension,
      ∀ label : Fin reduction.alphabet,
        reduction.coordinates
            ((reduction.construct input).constraints[i].permutation.images[label]) =
          reduction.coordinates label + shift :=
  reduction.translations input _ (List.getElem_mem i.isLt)

theorem decode_output (input : List Bool) :
    Complexity.decodeGameBits reduction.alphabet
      (Complexity.gameBits (reduction.construct input)) =
        some (reduction.construct input) :=
  Complexity.decodeGameBits_encoded _

theorem output_word_count (input : List Bool) :
    (Complexity.gameWords (reduction.construct input)).length =
      3 + (reduction.construct input).constraints.length * (reduction.alphabet + 2) :=
  Complexity.gameWords_length _

theorem malformed_soundness (input : List Bool)
    (malformed : BinaryEncoding.decodeFormula input = none) :
    ∀ labeling, (countSatisfied labeling (reduction.construct input).constraints : ℝ) /
      (reduction.construct input).constraints.length ≤ δ := by
  apply reduction.soundness input
  rintro ⟨formula, decoded, _⟩
  rw [malformed] at decoded
  cases decoded

end BinaryGapReduction
end DFVSGames.Explicit.MachineOutputContract

end OAI