File size: 5,095 Bytes
c270adf | 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 | import Mathlib
import OAI.Combinatorics.IndependentSets.Repetition.ActualGame
import OAI.Combinatorics.IndependentSets.Repetition.RepetitionUpper
namespace OAI
namespace IndependentSetsGames.Integration.SourceParameters
def rate (gap : ℚ) : ℚ := 1 - gap ^ 3 / 100000
theorem rate_bounds {gap : ℚ} (hgap : 0 < gap) (hgap1 : gap ≤ 1) :
0 < rate gap ∧ rate gap < 1 := by
have hcube : gap ^ 3 ≤ 1 := pow_le_one₀ hgap.le hgap1
have hcubePos : 0 < gap ^ 3 := pow_pos hgap _
constructor <;> unfold rate <;> linarith
theorem fourth_power_comparison {gap : ℚ} (hgap : 0 ≤ gap) (hgap1 : gap ≤ 1) :
1 - (gap : ℝ) ^ 3 / 6000 ≤ (rate gap : ℝ) ^ 4 := by
have hg : (0 : ℝ) ≤ gap := by exact_mod_cast hgap
have hg1 : (gap : ℝ) ≤ 1 := by exact_mod_cast hgap1
have hc : (gap : ℝ) ^ 3 ≤ 1 := pow_le_one₀ hg hg1
have hc0 : 0 ≤ (gap : ℝ) ^ 3 := pow_nonneg hg _
have hb := one_add_mul_le_pow (a := -(gap : ℝ) ^ 3 / 100000)
(by linarith : (-2 : ℝ) ≤ -(gap : ℝ) ^ 3 / 100000) 4
have hr : (rate gap : ℝ) = 1 - (gap : ℝ) ^ 3 / 100000 := by
simp only [rate, Rat.cast_sub, Rat.cast_one, Rat.cast_div,
Rat.cast_pow, Rat.cast_ofNat]
rw [hr]
calc
_ ≤ 1 + 4 * (-(gap : ℝ) ^ 3 / 100000) := by linarith
_ ≤ _ := by simpa only [Nat.cast_ofNat, sub_eq_add_neg, neg_div] using hb
theorem four_bit_rate_comparison {gap : ℚ}
(hgap : 0 < gap) (hgap1 : gap ≤ 1) (n : Nat) :
(1 - (gap : ℝ) ^ 3 / 6000) ^ ((n : ℝ) / 4) ≤ (rate gap : ℝ) ^ n := by
have hg : (0 : ℝ) ≤ gap := by exact_mod_cast hgap.le
have hg1 : (gap : ℝ) ≤ 1 := by exact_mod_cast hgap1
have hc : (gap : ℝ) ^ 3 ≤ 1 := pow_le_one₀ hg hg1
have hr : (0 : ℝ) ≤ rate gap := by exact_mod_cast (rate_bounds hgap hgap1).1.le
calc
_ ≤ ((rate gap : ℝ) ^ 4) ^ ((n : ℝ) / 4) :=
Real.rpow_le_rpow (by linarith) (fourth_power_comparison hgap.le hgap1) (by positivity)
_ = (rate gap : ℝ) ^ ((4 : ℝ) * ((n : ℝ) / 4)) :=
(Real.rpow_natCast_mul hr 4 ((n : ℝ) / 4)).symm
_ = (rate gap : ℝ) ^ (n : ℝ) := by congr 1; ring
_ = _ := Real.rpow_natCast _ _
theorem logTwo_sixteen : IndependentSetsGames.Foundations.Repetition.logTwo 16 = 4 := by
unfold IndependentSetsGames.Foundations.Repetition.logTwo
rw [show (16 : ℝ) = 2 ^ 4 by norm_num, Real.log_pow]
have hlog : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num : (1 : ℝ) < 2))
simp [hlog]
theorem game_repetition_rate {Q₁ Q₂ A₁ A₂ : Type*}
[Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
[Nonempty A₁] [Nonempty A₂] [DecidableEq Q₁] [DecidableEq Q₂]
(G : IndependentSetsGames.Foundations.Games.Game Q₁ Q₂ A₁ A₂)
{gap : ℚ} (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hcards : Fintype.card A₁ * Fintype.card A₂ = 16)
(hvalue : G.value ≤ 1 - (gap : ℝ)) (n : Nat) :
(G.repetition n).value ≤ (rate gap : ℝ) ^ n := by
have hg : (0 : ℝ) < gap := by exact_mod_cast hgap
have halphabet : IndependentSetsGames.Foundations.Repetition.logTwo
((Fintype.card A₁ : ℝ) * (Fintype.card A₂ : ℝ)) ≤ 4 := by
rw [← Nat.cast_mul, hcards]
exact logTwo_sixteen.le
have h := IndependentSetsGames.Foundations.Repetition.holenstein_repetition_value
G n hvalue (by linarith) (by norm_num : (1 : ℝ) ≤ 4) halphabet
simp only [sub_sub_cancel] at h
exact h.trans (four_bit_rate_comparison hgap hgap1 n)
def sourceThreshold (D : Nat) (δ : ℚ) : ℚ := 4 * (D : ℚ)⁻¹ * δ ^ 2
theorem sourceThreshold_pos {D : Nat} (hD : 0 < D) {δ : ℚ} (hδ : 0 < δ) :
0 < sourceThreshold D δ := by
unfold sourceThreshold
positivity
theorem exists_repetition_length {gap δ : ℚ} (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hδ : 0 < δ) {D : Nat} (hD : 0 < D) :
∃ u : Nat, 0 < u ∧ (rate gap) ^ u < sourceThreshold D δ :=
IndependentSetsGames.Soundness.RepetitionUpper.exists_power_lt (rate gap) (sourceThreshold D δ)
(rate_bounds hgap hgap1).1.le (rate_bounds hgap hgap1).2 (sourceThreshold_pos hD hδ)
def repetitionLength (gap δ : ℚ) (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hδ : 0 < δ) (D : Nat) (hD : 0 < D) : Nat :=
Nat.find (exists_repetition_length hgap hgap1 hδ hD)
theorem repetitionLength_spec (gap δ : ℚ) (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hδ : 0 < δ) (D : Nat) (hD : 0 < D) :
0 < repetitionLength gap δ hgap hgap1 hδ D hD ∧
rate gap ^ repetitionLength gap δ hgap hgap1 hδ D hD < sourceThreshold D δ :=
Nat.find_spec (exists_repetition_length hgap hgap1 hδ hD)
theorem repetitionLength_real_bound (gap δ : ℚ) (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hδ : 0 < δ) (D : Nat) (hD : 0 < D) :
(rate gap : ℝ) ^ repetitionLength gap δ hgap hgap1 hδ D hD ≤
4 * (D : ℝ)⁻¹ * (δ : ℝ) ^ 2 := by
have h := (repetitionLength_spec gap δ hgap hgap1 hδ D hD).2.le
unfold sourceThreshold at h
have hr := (Rat.cast_le (K := ℝ)).mpr h
push_cast at hr
exact hr
end IndependentSetsGames.Integration.SourceParameters
end OAI
|