| import OAI.Dynamics.ConditionalShuffle.HybridCycles |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| open Filter Topology |
| namespace Thorp.Conditional.OverlayEnergy |
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| def prepError (g : ℕ) : ℝ := (2^g : ℝ) * Real.sqrt |
| ((Fintype.card (Position (g+3)) : ℝ) * |
| ((1/2 : ℝ)^(8*(g+3)) + 32*(9/10 : ℝ)^(2^g))) |
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| lemma prepError_nonneg (g : ℕ) : 0 ≤ prepError g := by unfold prepError; positivity |
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| lemma error_scaled_sq (g : ℕ) : ((2^g : ℝ) * prepError g)^2 = |
| (8 / 2^24 : ℝ) * (1/8 : ℝ)^g + 256 * (2^g : ℝ)^5 * (9/10 : ℝ)^(2^g) := by |
| unfold prepError |
| rw [mul_pow, mul_pow, Real.sq_sqrt (by positivity)] |
| have hN : (Fintype.card (Position (g+3)) : ℝ) = 8 * (2^g : ℝ) := by |
| rw [card_position]; push_cast; rw [pow_add]; norm_num; ring |
| rw [hN] |
| have hp : (1/2 : ℝ)^(8*(g+3)) = (1/2 : ℝ)^(8*g) * (1/2 : ℝ)^24 := by |
| rw [show 8*(g+3) = 8*g+24 by omega, pow_add] |
| have he : (2^g : ℝ)^5 * (1/2 : ℝ)^(8*g) = (1/8 : ℝ)^g := by |
| rw [← pow_mul, Nat.mul_comm g 5, pow_mul, pow_mul, ← mul_pow] |
| norm_num |
| rw [hp] |
| calc |
| _ = 8 * ((2^g : ℝ)^5 * (1/2 : ℝ)^(8*g)) * (1/2 : ℝ)^24 + |
| 256 * (2^g : ℝ)^5 * (9/10 : ℝ)^(2^g) := by ring |
| _ = _ := by rw [he]; norm_num; ring |
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| lemma error_scaled_tendsto : Tendsto (fun g : ℕ => (2^g : ℝ) * prepError g) atTop (nhds 0) := by |
| have h₁ := (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1/8) |
| (by norm_num : (1/8 : ℝ) < 1)).const_mul (8/2^24 : ℝ) |
| have h₂ := ((tendsto_pow_const_mul_const_pow_of_lt_one 5 |
| (by norm_num : (0 : ℝ) ≤ 9/10) (by norm_num : (9/10 : ℝ) < 1)).comp |
| (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))).const_mul (256 : ℝ) |
| have hh := (h₁.add h₂).sqrt |
| have he (g : ℕ) : Real.sqrt ((8/2^24 : ℝ)*(1/8 : ℝ)^g + |
| 256*(2^g : ℝ)^5*(9/10 : ℝ)^(2^g)) = (2^g : ℝ)*prepError g := by |
| rw [← error_scaled_sq, Real.sqrt_sq (mul_nonneg (by positivity) (prepError_nonneg g))] |
| simpa only [Function.comp_def, Nat.cast_pow, Nat.cast_ofNat, mul_zero, |
| add_zero, Real.sqrt_zero, ← mul_assoc, he] using hh |
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| lemma prepError_tendsto : Tendsto prepError atTop (nhds 0) := by |
| apply squeeze_zero prepError_nonneg (fun g => ?_) error_scaled_tendsto |
| exact le_mul_of_one_le_left (prepError_nonneg g) (one_le_pow₀ (by norm_num : (1 : ℝ) ≤ 2)) |
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| lemma pow_add_bound (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (n : ℕ) : |
| (a+b)^n ≤ (2 : ℝ)^n * (a^n+b^n) := by |
| rcases le_total a b with hab | hba |
| · calc |
| _ ≤ (2*b)^n := pow_le_pow_left₀ (add_nonneg ha hb) (by linarith) n |
| _ = (2 : ℝ)^n*b^n := mul_pow _ _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left (le_add_of_nonneg_left (pow_nonneg ha _)) (by positivity) |
| · calc |
| _ ≤ (2*a)^n := pow_le_pow_left₀ (add_nonneg ha hb) (by linarith) n |
| _ = (2 : ℝ)^n*a^n := mul_pow _ _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left (le_add_of_nonneg_right (pow_nonneg hb _)) (by positivity) |
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| lemma gamma_power : 2*(255/256 : ℝ)^200 < 1 := by norm_num |
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| lemma coefficient_scaled_tendsto : |
| Tendsto (fun g : ℕ => (2^g : ℝ) * coefficient g ^ 200) atTop (nhds 0) := by |
| have ha : Tendsto (fun g : ℕ => (2^g : ℝ) * ((255/256 : ℝ)^(g+3))^200) atTop (nhds 0) := by |
| have hh := (tendsto_pow_atTop_nhds_zero_of_lt_one |
| (by positivity : (0 : ℝ) ≤ 2*(255/256 : ℝ)^200) gamma_power).mul_const ((255/256 : ℝ)^600) |
| convert hh using 1 |
| · funext g |
| rw [← pow_mul, Nat.add_mul, pow_add, Nat.mul_comm g 200, pow_mul, mul_pow] |
| generalize (255/256 : ℝ)^600 = C |
| ring |
| · simp only [zero_mul] |
| have hb : Tendsto (fun g : ℕ => (2^g : ℝ) * prepError g ^ 200) atTop (nhds 0) := by |
| apply squeeze_zero' (Filter.Eventually.of_forall (fun g => by positivity)) ?_ error_scaled_tendsto |
| filter_upwards [(tendsto_order.1 prepError_tendsto).2 1 (by norm_num)] with g hg |
| exact mul_le_mul_of_nonneg_left |
| (by simpa only [pow_one] using pow_le_pow_of_le_one (prepError_nonneg g) (le_of_lt hg) (by norm_num : 1 ≤ 200)) (by positivity) |
| have hh := (ha.add hb).const_mul ((2 : ℝ)^200) |
| have hc : Tendsto (fun g : ℕ => (2 : ℝ)^200 * |
| ((2^g : ℝ)*((255/256 : ℝ)^(g+3))^200 + (2^g : ℝ)*prepError g^200)) atTop (nhds 0) := by |
| simpa only [add_zero, mul_zero] using hh |
| apply squeeze_zero (fun g => by positivity) (fun g => ?_) hc |
| have hh := mul_le_mul_of_nonneg_left (pow_add_bound ((255/256 : ℝ)^(g+3)) |
| (prepError g) (by positivity) (prepError_nonneg g) 200) (by positivity : (0 : ℝ) ≤ 2^g) |
| change (2^g : ℝ) * (((255/256 : ℝ)^(g+3)) + prepError g)^200 ≤ _ |
| exact hh.trans_eq (by ring) |
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| lemma total_energy_tendsto : |
| Tendsto (fun g : ℕ => (2^g : ℝ)^3 * coefficient g^800) atTop (nhds 0) := by |
| have hh := coefficient_scaled_tendsto.pow 4 |
| simp only [zero_pow (by norm_num : 4 ≠ 0)] at hh |
| apply squeeze_zero (fun g => by positivity) (fun g => ?_) hh |
| rw [mul_pow, ← pow_mul (coefficient g), show 200*4=800 by omega] |
| apply mul_le_mul_of_nonneg_right _ (pow_nonneg (coefficient_nonneg g) _) |
| exact pow_le_pow_right₀ (one_le_pow₀ (by norm_num : (1 : ℝ) ≤ 2)) (by norm_num : 3 ≤ 4) |
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| def marginalRate (g : ℕ) : ℝ := Real.sqrt (8 * (2^g : ℝ)^3 * coefficient g^800) |
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| lemma marginalRate_tendsto : Tendsto marginalRate atTop (nhds 0) := by |
| change Tendsto (fun g : ℕ => Real.sqrt (8 * (2^g : ℝ)^3 * coefficient g^800)) atTop (nhds 0) |
| simpa only [mul_assoc, mul_zero, Real.sqrt_zero] using |
| (total_energy_tendsto.const_mul 8).sqrt |
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| end Thorp.Conditional.OverlayEnergy |
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| end |
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| end OAI |
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