| import OAI.Probability.MatroidProphet.Algorithm.SeedLaw |
| import OAI.Probability.MatroidProphet.SafeParity |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
|
|
| open MeasureTheory Finset |
|
|
| noncomputable def sourceCoinProbability (j : Fin 5) : ℝ := |
| if j.val = 0 then 1/2 else if j.val = 1 then 1/4 else if j.val = 4 then 1/2 else thinningRate |
|
|
| lemma sourceCoinProbability_nonneg (j : Fin 5) : 0 ≤ sourceCoinProbability j := by |
| unfold sourceCoinProbability |
| split_ifs <;> norm_num [thinningRate] |
|
|
| lemma sourceCoinProbability_le_one (j : Fin 5) : sourceCoinProbability j ≤ 1 := by |
| unfold sourceCoinProbability |
| split_ifs <;> norm_num [thinningRate] |
|
|
| noncomputable def sourceSeedLaw (n : ℕ) : Measure (Seed (mainSeedBits n)) := |
| mainSeedLaw n sourceCoinProbability sourceCoinProbability_nonneg sourceCoinProbability_le_one |
|
|
| instance sourceSeedLaw_probability (n : ℕ) : IsProbabilityMeasure (sourceSeedLaw n) := by |
| unfold sourceSeedLaw |
| infer_instance |
|
|
| lemma mainMasks_encodeMainSeed {n : ℕ} (b : MainSeedBlocks n) : |
| mainMasks (encodeMainSeed b) = |
| ⟨seedSet b.1, seedSet b.2.1, seedSet b.2.2.1, seedSet b.2.2.2.1, b.2.2.2.2 1⟩ := by |
| obtain ⟨hH, hD, hC, hT⟩ := encodeMainSeed_mask b |
| have ho := encodeMainSeed_odd b |
| change (encodeMainSeed b) _ = _ at ho |
| simp only [mainMasks, hH, hD, hC, hT, ho] |
|
|
| lemma fairTwoBits (f : Bool → Bool → ℝ) : |
| bitsExpectation (fun _ : Fin 2 => (1/2 : ℝ)) Finset.univ |
| (fun S => f (setSeed S 0) (setSeed S 1)) = |
| fairParityExpectation (fun p => fairParityExpectation (fun b => f b p)) := by |
| have hu : (Finset.univ : Finset (Fin 2)) = insert 0 (insert 1 ∅) := by decide |
| rw [hu, bitsExpectation_insert _ _ 0 (by decide)] |
| rw [bitsExpectation_insert _ _ 1 (by decide), bitsExpectation_insert _ _ 1 (by decide)] |
| simp [bitsExpectation_empty, setSeed, fairParityExpectation] |
| ring |
|
|
| lemma integral_sourceSeedLaw {n : ℕ} (f : MainMasks n → Bool → ℝ) : |
| (∫ r, f (mainMasks r) (mainBranch r) ∂sourceSeedLaw n) = |
| bitsExpectation (fun _ : Fin n => (1/2 : ℝ)) Finset.univ (fun H => |
| bitsExpectation (fun _ : Fin n => (1/4 : ℝ)) Finset.univ (fun D => |
| bitsExpectation (fun _ : Fin n => thinningRate) Finset.univ (fun C => |
| bitsExpectation (fun _ : Fin n => thinningRate) Finset.univ (fun T => |
| fairParityExpectation (fun p => fairParityExpectation (fun b => f ⟨H,D,C,T,p⟩ b)))))) := by |
| rw [sourceSeedLaw, mainSeedLaw_expectation] |
| simp only [sourceCoinProbability] |
| norm_num only [Fin.val_zero, Fin.val_one, Fin.reduceFinMk, ite_true, ite_false, |
| OfNat.ofNat_ne_zero] |
| simp only [mainMasks_encodeMainSeed, encodeMainSeed_branch, seedSet_setSeed] |
| apply bitsExpectation_congr |
| intro H _ |
| apply bitsExpectation_congr |
| intro D _ |
| apply bitsExpectation_congr |
| intro C _ |
| apply bitsExpectation_congr |
| intro T _ |
| exact fairTwoBits (fun b p => f ⟨H,D,C,T,p⟩ b) |
|
|
| end MatroidProphet |
|
|
| end OAI |
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|
