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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.MatroidProphet.SeedLaw
import OAI.Probability.MatroidProphet.Algorithm.Rule
import Mathlib.MeasureTheory.Integral.Prod
namespace OAI
namespace MatroidProphet
open MeasureTheory Finset
abbrev MainSeedBlocks (n : ℕ) := Seed n × (Seed n × (Seed n × (Seed n × Seed 2)))
noncomputable def encodeMainSeed {n : ℕ} (b : MainSeedBlocks n) : Seed (mainSeedBits n) :=
fun i => if hi : i.val < 4*n then
let e : Fin n := ⟨i.val / 4, by omega⟩
if i.val % 4 = 0 then b.1 e else
if i.val % 4 = 1 then b.2.1 e else
if i.val % 4 = 2 then b.2.2.1 e else b.2.2.2.1 e
else b.2.2.2.2 ⟨i.val-4*n, by have := i.isLt; dsimp [mainSeedBits] at this; omega⟩
lemma encodeMainSeed_mask {n : ℕ} (b : MainSeedBlocks n) :
seedMask (encodeMainSeed b) 0 = seedSet b.1 ∧
seedMask (encodeMainSeed b) 1 = seedSet b.2.1 ∧
seedMask (encodeMainSeed b) 2 = seedSet b.2.2.1 ∧
seedMask (encodeMainSeed b) 3 = seedSet b.2.2.2.1 := by
have h0 : ∀ e : Fin n, 4*e.val+0 < 4*n := fun e => by omega
have h1 : ∀ e : Fin n, 4*e.val+1 < 4*n := fun e => by omega
have h2 : ∀ e : Fin n, 4*e.val+2 < 4*n := fun e => by omega
have h3 : ∀ e : Fin n, 4*e.val+3 < 4*n := fun e => by omega
repeat' constructor
all_goals
ext e
simp_all [seedMask, seedSet, encodeMainSeed, Nat.add_mod, Nat.mul_add_div]
lemma encodeMainSeed_branch {n : ℕ} (b : MainSeedBlocks n) :
mainBranch (encodeMainSeed b) = b.2.2.2.2 0 := by
simp [mainBranch, encodeMainSeed]
lemma encodeMainSeed_odd {n : ℕ} (b : MainSeedBlocks n) :
(mainMasks (encodeMainSeed b)).odd = b.2.2.2.2 1 := by
have h : ¬ 4*n+1 < 4*n := by omega
simp [mainMasks, encodeMainSeed, h]
noncomputable def maskCoinLaw (n : ℕ) (p : ℝ) (h0 : 0 ≤ p) (h1 : p ≤ 1) : Measure (Seed n) :=
bernoulliSeedLaw (fun _ => p) (fun _ => h0) (fun _ => h1)
instance maskCoinLaw_probability (n : ℕ) (p : ℝ) (h0 : 0 ≤ p) (h1 : p ≤ 1) :
IsProbabilityMeasure (maskCoinLaw n p h0 h1) := by
unfold maskCoinLaw
infer_instance
noncomputable def mainBlockLaw (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) : Measure (MainSeedBlocks n) :=
(maskCoinLaw n (q 0) (h0 0) (h1 0)).prod
((maskCoinLaw n (q 1) (h0 1) (h1 1)).prod
((maskCoinLaw n (q 2) (h0 2) (h1 2)).prod
((maskCoinLaw n (q 3) (h0 3) (h1 3)).prod
(maskCoinLaw 2 (q 4) (h0 4) (h1 4)))))
instance mainBlockLaw_probability (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) :
IsProbabilityMeasure (mainBlockLaw n q h0 h1) := by
unfold mainBlockLaw
infer_instance
noncomputable def mainSeedLaw (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) : Measure (Seed (mainSeedBits n)) :=
(mainBlockLaw n q h0 h1).map encodeMainSeed
instance mainSeedLaw_probability (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) :
IsProbabilityMeasure (mainSeedLaw n q h0 h1) :=
inferInstanceAs (IsProbabilityMeasure ((mainBlockLaw n q h0 h1).map encodeMainSeed))
lemma integral_mainSeedLaw (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) (f : Seed (mainSeedBits n) → ℝ) :
(∫ r, f r ∂mainSeedLaw n q h0 h1) =
∫ H, ∫ D, ∫ C, ∫ T, ∫ b, f (encodeMainSeed (H, D, C, T, b))
∂maskCoinLaw 2 (q 4) (h0 4) (h1 4)
∂maskCoinLaw n (q 3) (h0 3) (h1 3)
∂maskCoinLaw n (q 2) (h0 2) (h1 2)
∂maskCoinLaw n (q 1) (h0 1) (h1 1)
∂maskCoinLaw n (q 0) (h0 0) (h1 0) := by
rw [mainSeedLaw, integral_map_of_stronglyMeasurable (measurable_of_countable _)
(measurable_of_countable f).stronglyMeasurable]
simp only [mainBlockLaw]
rw [integral_prod _ Integrable.of_finite]
apply integral_congr_ae
exact ae_of_all _ fun H => by
dsimp only
rw [integral_prod _ Integrable.of_finite]
apply integral_congr_ae
exact ae_of_all _ fun D => by
dsimp only
rw [integral_prod _ Integrable.of_finite]
apply integral_congr_ae
exact ae_of_all _ fun C => by
dsimp only
rw [integral_prod _ Integrable.of_finite]
lemma mainSeedLaw_expectation (n : ℕ) (q : Fin 5 → ℝ)
(h0 : ∀ j, 0 ≤ q j) (h1 : ∀ j, q j ≤ 1) (f : Seed (mainSeedBits n) → ℝ) :
(∫ r, f r ∂mainSeedLaw n q h0 h1) =
bitsExpectation (fun _ : Fin n => q 0) Finset.univ (fun H =>
bitsExpectation (fun _ : Fin n => q 1) Finset.univ (fun D =>
bitsExpectation (fun _ : Fin n => q 2) Finset.univ (fun C =>
bitsExpectation (fun _ : Fin n => q 3) Finset.univ (fun T =>
bitsExpectation (fun _ : Fin 2 => q 4) Finset.univ (fun b =>
f (encodeMainSeed (setSeed H, setSeed D, setSeed C, setSeed T, setSeed b))))))) := by
rw [integral_mainSeedLaw]
simp only [maskCoinLaw, integral_bernoulliSeedLaw]
end MatroidProphet
end OAI
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