| import OAI.Probability.FreeIsing.Energy.MomentCombinatorics | |
| namespace OAI | |
| namespace Problem367 | |
| theorem pow_mul_exp_le_of_log_le | |
| {m a c : ℝ} (hm : 0 < m) (hlog : Real.log m ≤ a) | |
| {q p : ℕ} (hqp : q ≤ p) : | |
| m ^ p * Real.exp (-a * ((p : ℝ) - c)) ≤ | |
| m ^ q * Real.exp (-a * ((q : ℝ) - c)) := by | |
| have hp : m ^ p = Real.exp ((p : ℝ) * Real.log m) := by | |
| rw [Real.exp_nat_mul, Real.exp_log hm] | |
| have hq : m ^ q = Real.exp ((q : ℝ) * Real.log m) := by | |
| rw [Real.exp_nat_mul, Real.exp_log hm] | |
| rw [hp, hq, ← Real.exp_add, ← Real.exp_add, Real.exp_le_exp] | |
| have hqp' : (q : ℝ) ≤ p := by exact_mod_cast hqp | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hqp') (sub_nonneg.mpr hlog)] | |
| theorem critical_moment_error_le_order_three | |
| {m K : ℝ} (hm : 0 < m) (hK : 0 ≤ K) (hlog : Real.log m ≤ 2 * K) | |
| {p : ℕ} (hp : 3 ≤ p) : | |
| Real.exp (-K * (p : ℝ) ^ 2 / 2) + | |
| m ^ p * Real.exp (-2 * K * ((p : ℝ) - 1)) ≤ | |
| Real.exp (-9 * K / 2) + m ^ 3 * Real.exp (-4 * K) := by | |
| have hp' : (3 : ℝ) ≤ p := by exact_mod_cast hp | |
| have hfirst : Real.exp (-K * (p : ℝ) ^ 2 / 2) ≤ Real.exp (-9 * K / 2) := by | |
| apply Real.exp_le_exp.mpr | |
| have hsquare : (9 : ℝ) ≤ (p : ℝ)^2 := by nlinarith | |
| nlinarith [mul_nonneg hK (sub_nonneg.mpr hsquare)] | |
| have hsecond := pow_mul_exp_le_of_log_le (c := (1 : ℝ)) hm hlog hp | |
| norm_num at hsecond | |
| have he1 : -2 * K * ((p : ℝ) - 1) = -(2 * K * ((p : ℝ) - 1)) := by ring | |
| have he2 : -4 * K = -(2 * K * 2) := by ring | |
| rw [he1, he2] | |
| exact add_le_add hfirst hsecond | |
| theorem exists_critical_moment_constant | |
| {m kappa : ℝ} (hm : 0 < m) (hkappa : 0 < kappa) (K0 : ℝ) : | |
| ∃ K : ℝ, K0 ≤ K ∧ 0 ≤ K ∧ Real.log m ≤ 2 * K ∧ | |
| Real.exp (-9 * K / 2) + m ^ 3 * Real.exp (-4 * K) ≤ kappa := by | |
| let K := |K0| + |Real.log m| + |Real.log (kappa / 2)| + | |
| |Real.log (kappa / (2 * m ^ 3))| + 1 | |
| have hK : 0 ≤ K := by | |
| dsimp [K] | |
| positivity | |
| have hK0 : K0 ≤ K := by | |
| dsimp [K] | |
| linarith [le_abs_self K0, abs_nonneg (Real.log m), | |
| abs_nonneg (Real.log (kappa / 2)), abs_nonneg (Real.log (kappa / (2 * m ^ 3)))] | |
| have hlog : Real.log m ≤ 2 * K := by | |
| dsimp [K] | |
| linarith [le_abs_self (Real.log m), abs_nonneg K0, | |
| abs_nonneg (Real.log m), abs_nonneg (Real.log (kappa / 2)), | |
| abs_nonneg (Real.log (kappa / (2 * m ^ 3)))] | |
| have hfirst_exp : -9 * K / 2 ≤ Real.log (kappa / 2) := by | |
| have hh : -Real.log (kappa / 2) ≤ K := by | |
| dsimp [K] | |
| linarith [neg_le_abs (Real.log (kappa / 2)), abs_nonneg K0, | |
| abs_nonneg (Real.log m), abs_nonneg (Real.log (kappa / (2 * m ^ 3)))] | |
| linarith | |
| have hsecond_exp : -4 * K ≤ Real.log (kappa / (2 * m ^ 3)) := by | |
| have hh : -Real.log (kappa / (2 * m ^ 3)) ≤ K := by | |
| dsimp [K] | |
| linarith [neg_le_abs (Real.log (kappa / (2 * m ^ 3))), abs_nonneg K0, | |
| abs_nonneg (Real.log m), abs_nonneg (Real.log (kappa / 2))] | |
| linarith | |
| have hfirst : Real.exp (-9 * K / 2) ≤ kappa / 2 := by | |
| calc | |
| Real.exp (-9 * K / 2) ≤ Real.exp (Real.log (kappa / 2)) := | |
| Real.exp_le_exp.mpr hfirst_exp | |
| _ = kappa / 2 := Real.exp_log (by positivity) | |
| have hsecond : m ^ 3 * Real.exp (-4 * K) ≤ kappa / 2 := by | |
| calc | |
| m ^ 3 * Real.exp (-4 * K) ≤ | |
| m ^ 3 * Real.exp (Real.log (kappa / (2 * m ^ 3))) := | |
| mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hsecond_exp) (by positivity) | |
| _ = kappa / 2 := by rw [Real.exp_log (by positivity)]; field_simp | |
| exact ⟨K, hK0, hK, hlog, by linarith⟩ | |
| theorem critical_moment_absorb | |
| {b theta A B e : ℝ} (hb : 0 ≤ b) (htheta : 0 ≤ theta) (htheta1 : theta < 1) | |
| (hcritical : b * theta ^ 2 = 1) {p : ℕ} (hp : 3 ≤ p) | |
| (hB : 0 ≤ B) (he : e ≤ 1 - theta) | |
| (hrec : (1 - b * theta ^ p) * A ≤ B * e) : A ≤ B := by | |
| have hgap := (critical_moment_absorption hb htheta htheta1 hcritical hp).2 | |
| have hpos : 0 < 1 - b * theta ^ p := by linarith | |
| apply (mul_le_mul_iff_right₀ hpos).mp | |
| calc | |
| (1 - b * theta ^ p) * A ≤ B * e := hrec | |
| _ ≤ B * (1 - theta) := mul_le_mul_of_nonneg_left he hB | |
| _ ≤ (1 - b * theta ^ p) * B := by | |
| simpa only [mul_comm B] using mul_le_mul_of_nonneg_left hgap hB | |
| end Problem367 | |
| end OAI | |