File size: 3,320 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | 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
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