| import OAI.Computability.UniqueGames.Foundations.FiniteTrialsLemmas |
|
|
| namespace OAI |
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| section |
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|
| /-! |
| Conditioning of actual finite distributions and the selected-coordinate chain |
| rule used in parallel repetition. A normalized conditional distribution is only |
| constructed when the conditioning event has strictly positive probability. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Foundations.Games |
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| open scoped BigOperators |
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| noncomputable section |
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| namespace FiniteDistribution |
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|
| variable {Ω : Type*} [Fintype Ω] |
| |
| def condition (μ : FiniteDistribution Ω) (given : Ω → Bool) |
| (positive : 0 < μ.probability given) : FiniteDistribution Ω where |
| weight x := if given x then μ.weight x / μ.probability given else 0 |
| nonnegative x := by |
| split |
| · exact div_nonneg (μ.nonnegative x) positive.le |
| · exact le_rfl |
| normalized := by |
| calc |
| _ = (∑ x, if given x then μ.weight x else 0) / μ.probability given := by |
| simp only [div_eq_mul_inv, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro x _ |
| by_cases hx : given x = true <;> simp [hx] |
| _ = 1 := div_self (ne_of_gt positive) |
|
|
| theorem probability_condition (μ : FiniteDistribution Ω) (given event : Ω → Bool) |
| (positive : 0 < μ.probability given) : |
| (μ.condition given positive).probability event = |
| μ.probability (fun x => given x && event x) / μ.probability given := by |
| simp only [probability, condition, div_eq_mul_inv, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro x _ |
| by_cases hg : given x = true <;> by_cases he : event x = true <;> simp [hg, he] |
| |
| theorem probability_inter_eq_mul_conditional (μ : FiniteDistribution Ω) |
| (given event : Ω → Bool) (positive : 0 < μ.probability given) : |
| μ.probability (fun x => given x && event x) = |
| μ.probability given * (μ.condition given positive).probability event := by |
| have h := (eq_div_iff (ne_of_gt positive)).mp |
| (μ.probability_condition given event positive) |
| simpa only [mul_comm] using h.symm |
|
|
| theorem probability_and_le_left (μ : FiniteDistribution Ω) (event event' : Ω → Bool) : |
| μ.probability (fun x => event x && event' x) ≤ μ.probability event := by |
| apply μ.probability_mono |
| intro x hx |
| have h : event x = true ∧ event' x = true := by simpa using hx |
| exact h.1 |
|
|
| theorem probability_and_le_right (μ : FiniteDistribution Ω) (event event' : Ω → Bool) : |
| μ.probability (fun x => event x && event' x) ≤ μ.probability event' := by |
| apply μ.probability_mono |
| intro x hx |
| have h : event x = true ∧ event' x = true := by simpa using hx |
| exact h.2 |
|
|
| theorem probability_and_eq_zero_of_probability_eq_zero (μ : FiniteDistribution Ω) |
| (given event : Ω → Bool) (zero : μ.probability given = 0) : |
| μ.probability (fun x => given x && event x) = 0 := by |
| apply le_antisymm |
| · exact (μ.probability_and_le_left given event).trans_eq zero |
| · exact μ.probability_nonnegative _ |
|
|
| @[simp] theorem probability_condition_given (μ : FiniteDistribution Ω) |
| (given : Ω → Bool) (positive : 0 < μ.probability given) : |
| (μ.condition given positive).probability given = 1 := by |
| rw [μ.probability_condition] |
| simpa only [Bool.and_self] using div_self (ne_of_gt positive) |
|
|
| end FiniteDistribution |
|
|
| namespace Game |
|
|
| variable {Q₁ Q₂ A₁ A₂ : Type*} |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] |
| {n : Nat} |
|
|
| theorem selectedWins_insert (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (coordinate : Fin n) : |
| G.selectedWins strategy (insert coordinate selected) = |
| fun questions => |
| G.selectedWins strategy selected questions && G.coordinateWin strategy coordinate questions := by |
| classical |
| funext questions |
| apply Bool.eq_iff_iff.mpr |
| simp [selectedWins, and_comm] |
|
|
| def selectedSuccess (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) : ℝ := |
| (G.repetition n).questions.probability (G.selectedWins strategy selected) |
|
|
| theorem selectedSuccess_nonnegative (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) : 0 ≤ G.selectedSuccess strategy selected := |
| (G.repetition n).questions.probability_nonnegative _ |
|
|
| theorem selectedSuccess_le_one (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) : G.selectedSuccess strategy selected ≤ 1 := |
| (G.repetition n).questions.probability_le_one _ |
|
|
| @[simp] theorem selectedSuccess_empty (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) : |
| G.selectedSuccess strategy ∅ = 1 := by |
| have h : G.selectedWins strategy ∅ = fun _ => true := |
| funext (G.selectedWins_empty strategy) |
| rw [selectedSuccess, h, FiniteDistribution.probability_true] |
|
|
| theorem selectedSuccess_univ (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) : |
| G.selectedSuccess strategy Finset.univ = (G.repetition n).success strategy := by |
| rw [selectedSuccess, G.selectedWins_univ] |
| rfl |
|
|
| theorem selectedSuccess_antitone (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) : |
| Antitone (G.selectedSuccess strategy) := by |
| intro selected selected' h |
| exact G.selectedWins_mono strategy h |
|
|
| theorem selectedSuccess_insert (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (coordinate : Fin n) |
| (positive : 0 < G.selectedSuccess strategy selected) : |
| G.selectedSuccess strategy (insert coordinate selected) = |
| G.selectedSuccess strategy selected * |
| ((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).probability |
| (G.coordinateWin strategy coordinate) := by |
| unfold selectedSuccess |
| rw [G.selectedWins_insert] |
| exact (G.repetition n).questions.probability_inter_eq_mul_conditional _ _ positive |
|
|
| theorem selectedSuccess_insert_eq_zero (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (coordinate : Fin n) |
| (zero : G.selectedSuccess strategy selected = 0) : |
| G.selectedSuccess strategy (insert coordinate selected) = 0 := by |
| apply le_antisymm |
| · exact (G.selectedSuccess_antitone strategy (Finset.subset_insert coordinate selected)).trans_eq zero |
| · exact G.selectedSuccess_nonnegative _ _ |
|
|
| end Game |
|
|
| end |
|
|
| end UniqueGamesTheorem.Foundations.Games |
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| end |
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| end OAI |
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|
