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f61e547 | 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 | 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
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