| import OAI.Probability.MatroidProphet.GroupConditioning |
| import OAI.Probability.MatroidProphet.SafeParity |
| import OAI.Probability.MatroidProphet.Residual.Global |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset |
|
|
| lemma fairParityExpectation_mono {f g : Bool → ℝ} (h : ∀ p, f p ≤ g p) : |
| fairParityExpectation f ≤ fairParityExpectation g := by |
| have h0 := h false |
| have h1 := h true |
| unfold fairParityExpectation |
| linarith |
|
|
| lemma fairParityExpectation_const (c : ℝ) : fairParityExpectation (fun _ => c) = c := by |
| unfold fairParityExpectation |
| ring |
|
|
| lemma fairParityExpectation_add (f g : Bool → ℝ) : |
| fairParityExpectation (fun p => f p + g p) = fairParityExpectation f + fairParityExpectation g := by |
| unfold fairParityExpectation |
| ring |
|
|
| lemma fairParityExpectation_mul (c : ℝ) (f : Bool → ℝ) : |
| fairParityExpectation (fun p => c * f p) = c * fairParityExpectation f := by |
| unfold fairParityExpectation |
| ring |
|
|
| lemma bitsExpectation_fairParity {α : Type*} [DecidableEq α] (q : α → ℝ) |
| (V : Finset α) (f : Finset α → Bool → ℝ) : |
| bitsExpectation q V (fun T => fairParityExpectation (f T)) = |
| fairParityExpectation (fun p => bitsExpectation q V (fun T => f T p)) := by |
| simp only [fairParityExpectation, div_eq_mul_inv] |
| rw [bitsExpectation_mul_right, bitsExpectation_add] |
|
|
| namespace MainAlgorithm |
| variable {n : ℕ} |
|
|
| noncomputable def mainMean (d : MainMasks n) (f : MainMasks n → ℝ) : ℝ := |
| tripleMaskExpectation (fun _ => (1:ℝ)/4) (fun _ => thinningRate) (fun _ => thinningRate) |
| univ (fun D C T => fairParityExpectation (fun p => f {withMasks d D C T with odd := p})) |
|
|
| lemma mainMean_mono (d : MainMasks n) {f g : MainMasks n → ℝ} |
| (h : ∀ d', d'.H = d.H → f d' ≤ g d') : mainMean d f ≤ mainMean d g := by |
| apply tripleMaskExpectation_mono _ _ _ (fun _ => by norm_num) (fun _ => by norm_num) |
| (fun _ => constants_positive.2.2.1.le) (fun _ => by norm_num [thinningRate]) |
| (fun _ => constants_positive.2.2.1.le) (fun _ => by norm_num [thinningRate]) |
| intro D hD C hC T hT |
| exact fairParityExpectation_mono (fun p => h _ rfl) |
|
|
| lemma mainMean_const (d : MainMasks n) (c : ℝ) : mainMean d (fun _ => c) = c := by |
| simp only [mainMean, fairParityExpectation_const, tripleMaskExpectation_const] |
|
|
| lemma mainMean_add (d : MainMasks n) (f g : MainMasks n → ℝ) : |
| mainMean d (fun d' => f d' + g d') = mainMean d f + mainMean d g := by |
| simp only [mainMean, fairParityExpectation_add, tripleMaskExpectation_add] |
|
|
| lemma mainMean_mul (d : MainMasks n) (c : ℝ) (f : MainMasks n → ℝ) : |
| mainMean d (fun d' => c * f d') = c * mainMean d f := by |
| simp only [mainMean, fairParityExpectation_mul, tripleMaskExpectation_mul] |
|
|
| lemma mainMean_guard_count (d : MainMasks n) (K : Finset (Fin n)) : |
| mainMean d (fun d' => ((d'.C ∩ K).card : ℝ)) = thinningRate * K.card := by |
| unfold mainMean tripleMaskExpectation MainAlgorithm.withMasks |
| simp only [fairParityExpectation_const, bitsExpectation_const] |
| have hc : bitsExpectation (fun _ : Fin n => thinningRate) univ |
| (fun C => ((C ∩ K).card : ℝ)) = thinningRate * K.card := by |
| simp_rw [Finset.inter_comm _ K] |
| rw [bitsExpectation_card_inter _ univ K (subset_univ _)] |
| simp [mul_comm] |
| rw [hc] |
| |
| end MainAlgorithm |
| end MatroidProphet |
| |
| end OAI |
| |
