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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 99 | import Mathlib
namespace OAI
namespace Problem367
theorem critical_estimator_cutoff {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) :
∃ n : ℕ, Real.sqrt t / 5 ≤
t * ((n : ℝ) + 1) / (1 + t * ((n : ℝ) + 1) ^ 2) := by
let n : ℕ := ⌊1 / Real.sqrt t⌋₊
have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
have hs1 : Real.sqrt t ≤ 1 := Real.sqrt_le_one.mpr ht1
have hsq : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht.le
have hinv : 1 ≤ 1 / Real.sqrt t := (le_div_iff₀ hs).mpr (by simpa using hs1)
have hnlow : 1 / Real.sqrt t ≤ (n : ℝ) + 1 :=
(Nat.lt_floor_add_one (1 / Real.sqrt t)).le
have hnhigh : (n : ℝ) + 1 ≤ 2 / Real.sqrt t := by
have hfloor := Nat.floor_le (by positivity : 0 ≤ 1 / Real.sqrt t)
change (n : ℝ) ≤ 1 / Real.sqrt t at hfloor
calc
(n : ℝ) + 1 ≤ 1 / Real.sqrt t + 1 := by linarith
_ ≤ 1 / Real.sqrt t + 1 / Real.sqrt t := by linarith
_ = 2 / Real.sqrt t := by ring
have hmul_low : 1 ≤ Real.sqrt t * ((n : ℝ) + 1) := by
have h := (div_le_iff₀ hs).mp hnlow
nlinarith
have hmul_high : Real.sqrt t * ((n : ℝ) + 1) ≤ 2 := by
have h := (le_div_iff₀ hs).mp hnhigh
nlinarith
have hnum : Real.sqrt t ≤ t * ((n : ℝ) + 1) := by
have h := mul_le_mul_of_nonneg_left hmul_low hs.le
nlinarith [hsq]
have hprod_sq : (Real.sqrt t * ((n : ℝ) + 1)) ^ 2 =
t * ((n : ℝ) + 1) ^ 2 := by rw [mul_pow, hsq]
have hden : 1 + t * ((n : ℝ) + 1) ^ 2 ≤ 5 := by
have h := mul_self_le_mul_self (by positivity :
0 ≤ Real.sqrt t * ((n : ℝ) + 1)) hmul_high
nlinarith [hprod_sq]
have hdenpos : 0 < 1 + t * ((n : ℝ) + 1) ^ 2 := by positivity
refine ⟨n, (le_div_iff₀ hdenpos).mpr ?_⟩
calc
Real.sqrt t / 5 * (1 + t * ((n : ℝ) + 1) ^ 2) ≤
Real.sqrt t / 5 * 5 :=
mul_le_mul_of_nonneg_left hden (by positivity)
_ ≤ t * ((n : ℝ) + 1) := by nlinarith
theorem critical_information_lower_of_estimators {t M : ℝ} (ht : 0 < t)
(h : ∀ n : ℕ,
t * ((n : ℝ) + 1) / (1 + t * ((n : ℝ) + 1) ^ 2) ≤ M) :
min (Real.sqrt t) 1 / 5 ≤ M := by
by_cases ht1 : t ≤ 1
· obtain ⟨n, hn⟩ := critical_estimator_cutoff ht ht1
rw [min_eq_left (Real.sqrt_le_one.mpr ht1)]
exact hn.trans (h n)
· have ht1' : 1 ≤ t := le_of_lt (lt_of_not_ge ht1)
rw [min_eq_right (Real.one_le_sqrt.mpr ht1')]
have h0 : t / (1 + t) ≤ M := by simpa using h 0
have hhalf : (1 / 2 : ℝ) ≤ t / (1 + t) := by
apply (le_div_iff₀ (by positivity : 0 < 1 + t)).mpr
linarith
linarith
theorem sqrt_bound_of_quadratic_information_loss {M c K t : ℝ}
(hc : 0 < c) (hK : 0 ≤ K)
(h : M ≤ M - c * M ^ 2 + K * t) :
M ≤ Real.sqrt (K / c) * Real.sqrt t := by
have hquad : M ^ 2 ≤ K * t / c := by
apply (le_div_iff₀ hc).mpr
nlinarith
calc
M ≤ Real.sqrt (K * t / c) := Real.le_sqrt_of_sq_le hquad
_ = Real.sqrt (K / c) * Real.sqrt t := by
rw [show K * t / c = (K / c) * t by ring,
Real.sqrt_mul (div_nonneg hK hc.le)]
theorem critical_information_upper_of_fusion {b theta M t : ℝ}
(hθ0 : 0 < theta) (hθ1 : theta < 1)
(hcritical : b * theta ^ 2 = 1) (ht : 0 ≤ t) (ht1 : t ≤ 1)
(h : M ≤ b * theta ^ 2 * M -
2 * (1 - theta ^ 2) * theta ^ 4 * M ^ 2 + (t + t ^ 2)) :
M ≤ Real.sqrt (1 / ((1 - theta ^ 2) * theta ^ 4)) * Real.sqrt t := by
have hθsq : theta ^ 2 < 1 :=
(sq_lt_one_iff_abs_lt_one theta).mpr (abs_lt.mpr ⟨by linarith, hθ1⟩)
have hc : 0 < (1 - theta ^ 2) * theta ^ 4 :=
mul_pos (sub_pos.mpr hθsq) (pow_pos hθ0 4)
have htsq : t ^ 2 ≤ t := by nlinarith
rw [hcritical, one_mul] at h
have hrec : M ≤ M - (2 * ((1 - theta ^ 2) * theta ^ 4)) * M ^ 2 + 2 * t := by
nlinarith
have hbound := sqrt_bound_of_quadratic_information_loss
(mul_pos (by norm_num : (0 : ℝ) < 2) hc) (by norm_num : (0 : ℝ) ≤ 2) hrec
have hratio : (2 : ℝ) / (2 * ((1 - theta ^ 2) * theta ^ 4)) =
1 / ((1 - theta ^ 2) * theta ^ 4) := by field_simp
simpa only [hratio] using hbound
end Problem367
end OAI
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