lean/OAI/Computability/VertexCover/Analysis/PowerValueRepetition.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.BaseReductionCoherent

namespace OAI

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namespace VertexCover.LabelCover
open UniqueGames.Foundations.Games
noncomputable section

def occurrenceLaw (Φ : LabelCover) : FiniteDistribution (Fin Φ.M) := by
  letI : Nonempty (Fin Φ.M) := ⟨⟨0, Φ.M_pos⟩⟩
  exact FiniteDistribution.uniform (Fin Φ.M)

def questionProjection (Φ : LabelCover) (x : Fin Φ.u) (y : Fin Φ.v) (a : Fin Φ.qU) : Fin Φ.qV :=
  if h : ∃ e, Φ.left e = x ∧ Φ.right e = y then Φ.projection h.choose a
  else ⟨0, Φ.qV_pos⟩

theorem questionProjection_at (Φ : LabelCover) (hc : Φ.ParallelCoherent)
    (e : Fin Φ.M) (a : Fin Φ.qU) :
    Φ.questionProjection (Φ.left e) (Φ.right e) a = Φ.projection e a := by
  unfold questionProjection
  split
  · rename_i h
    exact congrFun (hc h.choose e h.choose_spec.1 h.choose_spec.2) a
  · rename_i h
    exact False.elim (h ⟨e, rfl, rfl⟩)

def asGame (Φ : LabelCover) : Game (Fin Φ.u) (Fin Φ.v) (Fin Φ.qU) (Fin Φ.qV) :=
  { questions := Φ.occurrenceLaw.pushforward (fun e => (Φ.left e, Φ.right e))
    accepts := fun x y a b => decide (Φ.questionProjection x y a = b) }

theorem asGame_success (Φ : LabelCover) (hc : Φ.ParallelCoherent) (A : Φ.Labeling) :
    Φ.asGame.success A = (Φ.satisfiedCount A : ℝ) / Φ.M := by
  classical
  change (Φ.occurrenceLaw.pushforward (fun e => (Φ.left e, Φ.right e))).probability _ = _
  rw [FiniteDistribution.probability_pushforward]
  simp only [FiniteDistribution.probability, Game.wins, asGame, questionProjection_at Φ hc,
    decide_eq_true_eq]
  simp only [occurrenceLaw, FiniteDistribution.uniform, Fintype.card_fin]
  change (∑ e, if Φ.Satisfies A e then 1 / (Φ.M : ℝ) else 0) = _
  calc
    _ = (∑ e, (if Φ.Satisfies A e then 1 else 0 : ℝ)) / Φ.M := by
      rw [Finset.sum_div]
      apply Finset.sum_congr rfl
      intro e he
      split_ifs <;> simp
    _ = _ := by rw [Finset.sum_boole]; rfl

def asGameValue (Φ : LabelCover) : ℝ := by
  letI : Nonempty (Fin Φ.qU) := ⟨⟨0, Φ.qU_pos⟩⟩
  letI : Nonempty (Fin Φ.qV) := ⟨⟨0, Φ.qV_pos⟩⟩
  exact Φ.asGame.value

theorem asGameValue_eq (Φ : LabelCover) (hc : Φ.ParallelCoherent) : Φ.asGameValue = Φ.value := by
  classical
  let : Nonempty (Fin Φ.qU) := ⟨⟨0, Φ.qU_pos⟩⟩
  let : Nonempty (Fin Φ.qV) := ⟨⟨0, Φ.qV_pos⟩⟩
  change Φ.asGame.value = Φ.value
  apply le_antisymm
  · apply (Φ.asGame.value_le_iff _).mpr
    intro A
    rw [asGame_success Φ hc A]
    exact div_le_div_of_nonneg_right (Nat.cast_le.mpr (Φ.satisfiedCount_le_max A)) (by positivity)
  · apply value_le_of_counts
    intro A
    have hs := Φ.asGame.success_le_value A
    rw [asGame_success Φ hc A] at hs
    exact (div_le_iff₀ (Nat.cast_pos.mpr Φ.M_pos)).mp hs

def tupleEquiv (a n : ℕ) : (Fin n → Fin a) ≃ Fin (a ^ n) :=
  Fintype.equivFinOfCardEq (by simp)

def parallelPower (Φ : LabelCover) (n : ℕ) : LabelCover where
  u := Φ.u ^ n
  v := Φ.v ^ n
  qU := Φ.qU ^ n
  qV := Φ.qV ^ n
  M := Φ.M ^ n
  qU_pos := pow_pos Φ.qU_pos _
  qV_pos := pow_pos Φ.qV_pos _
  M_pos := pow_pos Φ.M_pos _
  left e := tupleEquiv Φ.u n (fun i => Φ.left ((tupleEquiv Φ.M n).symm e i))
  right e := tupleEquiv Φ.v n (fun i => Φ.right ((tupleEquiv Φ.M n).symm e i))
  projection e a := tupleEquiv Φ.qV n (fun i => Φ.projection ((tupleEquiv Φ.M n).symm e i)
    ((tupleEquiv Φ.qU n).symm a i))

def repeatLabeling (Φ : LabelCover) (n : ℕ) (A : Φ.Labeling) : (Φ.parallelPower n).Labeling :=
  (fun x => tupleEquiv Φ.qU n (fun i => A.1 ((tupleEquiv Φ.u n).symm x i)),
   fun y => tupleEquiv Φ.qV n (fun i => A.2 ((tupleEquiv Φ.v n).symm y i)))

theorem repeat_perfect (Φ : LabelCover) (n : ℕ) (A : Φ.Labeling) (hA : ∀ e, Φ.Satisfies A e) :
    ∀ e, (Φ.parallelPower n).Satisfies (Φ.repeatLabeling n A) e := by
  intro e
  change tupleEquiv Φ.qV n _ = tupleEquiv Φ.qV n _
  apply congrArg (tupleEquiv Φ.qV n)
  funext i
  simpa only [Satisfies, repeatLabeling, parallelPower, Equiv.symm_apply_apply] using hA ((tupleEquiv Φ.M n).symm e i)

def repeatedStrategy (Φ : LabelCover) (n : ℕ) (A : (Φ.parallelPower n).Labeling) :
    Strategy (Fin n → Fin Φ.u) (Fin n → Fin Φ.v) (Fin n → Fin Φ.qU) (Fin n → Fin Φ.qV) :=
  (fun x => (tupleEquiv Φ.qU n).symm (A.1 (tupleEquiv Φ.u n x)),
   fun y => (tupleEquiv Φ.qV n).symm (A.2 (tupleEquiv Φ.v n y)))

theorem repetition_success_probability (Φ : LabelCover) (hc : Φ.ParallelCoherent) (n : ℕ)
    (B : Strategy (Fin n → Fin Φ.u) (Fin n → Fin Φ.v)
      (Fin n → Fin Φ.qU) (Fin n → Fin Φ.qV)) :
    (Φ.asGame.repetition n).success B = (Φ.occurrenceLaw.iid n).probability
      (fun es => decide (∀ i, Φ.projection (es i)
        (B.1 (fun j => Φ.left (es j)) i) = B.2 (fun j => Φ.right (es j)) i)) := by
  classical
  unfold Game.success
  change (((Φ.occurrenceLaw.pushforward (fun e => (Φ.left e, Φ.right e))).iid n).transport
    (Game.tupleQuestionEquiv n)).probability _ = _
  rw [FiniteDistribution.probability_transport, FiniteDistribution.iid_pushforward,
    FiniteDistribution.probability_pushforward]
  congr 1
  funext es
  simp only [Game.wins, Game.repetition, Game.tupleQuestionEquiv, asGame,
    Equiv.coe_fn_mk, questionProjection_at Φ hc, decide_eq_true_eq]

theorem power_satisfies_iff (Φ : LabelCover) (n : ℕ) (A : (Φ.parallelPower n).Labeling)
    (es : Fin n → Fin Φ.M) :
    (Φ.parallelPower n).Satisfies A (tupleEquiv Φ.M n es) ↔
      ∀ i, Φ.projection (es i)
        ((Φ.repeatedStrategy n A).1 (fun j => Φ.left (es j)) i) =
        (Φ.repeatedStrategy n A).2 (fun j => Φ.right (es j)) i := by
  simp only [Satisfies, parallelPower, repeatedStrategy, Equiv.symm_apply_apply]
  rw [← Equiv.eq_symm_apply]
  exact funext_iff

theorem repetition_success (Φ : LabelCover) (hc : Φ.ParallelCoherent) (n : ℕ)
    (A : (Φ.parallelPower n).Labeling) :
    (Φ.asGame.repetition n).success (Φ.repeatedStrategy n A) =
      ((Φ.parallelPower n).satisfiedCount A : ℝ) / (Φ.M ^ n : ℕ) := by
  classical
  let : Nonempty (Fin Φ.M) := ⟨⟨0, Φ.M_pos⟩⟩
  rw [repetition_success_probability Φ hc]
  have hlaw : Φ.occurrenceLaw.iid n = FiniteDistribution.uniform (Fin n → Fin Φ.M) :=
    FiniteDistribution.iid_uniform n
  rw [hlaw]
  simp only [← power_satisfies_iff Φ n A, FiniteDistribution.probability,
    FiniteDistribution.uniform, Fintype.card_fun, Fintype.card_fin, decide_eq_true_eq]
  calc
    _ = ∑ e : Fin (Φ.M ^ n), if (Φ.parallelPower n).Satisfies A e then
        1 / ((Φ.M ^ n : ℕ) : ℝ) else 0 :=
      Fintype.sum_equiv (tupleEquiv Φ.M n) _ _ (fun _ => rfl)
    _ = (∑ e : Fin (Φ.M ^ n), (if (Φ.parallelPower n).Satisfies A e then 1 else 0 : ℝ)) /
        (Φ.M ^ n : ℕ) := by
      rw [Finset.sum_div]
      apply Finset.sum_congr rfl
      intro e he
      split_ifs <;> simp
    _ = _ := by
      have hsum : (∑ e : Fin (Φ.parallelPower n).M,
          (@ite ℝ ((Φ.parallelPower n).Satisfies A e) (Classical.propDecidable _) 1 0)) =
          ((Φ.parallelPower n).satisfiedCount A : ℝ) := by
        rw [satisfiedCount, Finset.card_filter, Nat.cast_sum]
        apply Finset.sum_congr rfl
        intro e he
        by_cases h : (Φ.parallelPower n).Satisfies A e <;> simp only [h, ite_true, ite_false, Nat.cast_zero, Nat.cast_one]
      exact congrArg (fun x : ℝ => x / ((Φ.M ^ n : ℕ) : ℝ)) hsum

theorem power_value_le_repetition (Φ : LabelCover) (hc : Φ.ParallelCoherent) (n : ℕ) :
    (Φ.parallelPower n).value ≤
      (letI : Nonempty (Fin Φ.qU) := ⟨⟨0, Φ.qU_pos⟩⟩;
       letI : Nonempty (Fin Φ.qV) := ⟨⟨0, Φ.qV_pos⟩⟩;
       (Φ.asGame.repetition n).value) := by
  classical
  let : Nonempty (Fin Φ.qU) := ⟨⟨0, Φ.qU_pos⟩⟩
  let : Nonempty (Fin Φ.qV) := ⟨⟨0, Φ.qV_pos⟩⟩
  apply value_le_of_counts
  intro A
  have hs := (Φ.asGame.repetition n).success_le_value (Φ.repeatedStrategy n A)
  rw [repetition_success Φ hc] at hs
  exact (div_le_iff₀ (Nat.cast_pos.mpr (pow_pos Φ.M_pos n))).mp hs

end
end VertexCover.LabelCover


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end OAI