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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Probability.FreeIsing.Estimates.EndpointScale
namespace OAI
noncomputable section
namespace Problem367
theorem inv_sqrt_sq_time {t : ℝ} (ht : 0 ≤ t) :
(Real.sqrt t)⁻¹ ^ 2 = 1 / t := by
rw [inv_pow, Real.sq_sqrt ht, one_div]
theorem mul_sqrt_eq_three_halves {x : ℝ} (hx : 0 ≤ x) :
x * Real.sqrt x = x ^ ((3 : ℝ) / 2) := by
rcases eq_or_lt_of_le hx with rfl | hx
· norm_num
calc
x * Real.sqrt x = x ^ (1 : ℝ) * x ^ ((1 : ℝ) / 2) := by
rw [Real.rpow_one, Real.sqrt_eq_rpow]
_ = x ^ ((1 : ℝ) + 1 / 2) := (Real.rpow_add hx _ _).symm
_ = x ^ ((3 : ℝ) / 2) := by norm_num
theorem inv_sqrt_mul_sqrt_time {t : ℝ} (ht : 0 < t) :
(Real.sqrt t)⁻¹ * Real.sqrt ((Real.sqrt t)⁻¹) = t ^ (-(3 : ℝ) / 4) := by
rw [mul_sqrt_eq_three_halves (by positivity), Real.sqrt_eq_rpow,
← Real.rpow_neg_one, ← Real.rpow_mul ht.le, ← Real.rpow_mul ht.le]
norm_num
theorem inv_sqrt_le_time_three_quarters {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) :
(Real.sqrt t)⁻¹ ≤ t ^ (-(3 : ℝ) / 4) := by
have hs : 0 < Real.sqrt t := Real.sqrt_pos.2 ht
have hs1 : Real.sqrt t ≤ 1 := by
simpa using Real.sqrt_le_sqrt ht1
have hi : 1 ≤ (Real.sqrt t)⁻¹ := (one_le_inv₀ hs).2 hs1
have hsqrt1 : 1 ≤ Real.sqrt ((Real.sqrt t)⁻¹) := by
simpa using Real.sqrt_le_sqrt hi
rw [← inv_sqrt_mul_sqrt_time ht]
simpa using mul_le_mul_of_nonneg_left hsqrt1 (inv_nonneg.2 hs.le)
theorem quadratic_endpoint_time_scale {F A B E S V R K I t : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hS : 0 ≤ S)
(hR0 : 0 ≤ R) (hK : 0 ≤ K) (hI : 1 ≤ I) (ht : 0 < t)
(hR : R ≤ K * ((Real.sqrt t)⁻¹ + Real.log I))
(hF : F ≤ A * S * R ^ 2 + B * S * (1 / t) + E * V ^ 2) :
F ≤ max (2 * A * K ^ 2 + B) E *
(S * (1 / t + (Real.log I) ^ 2) + V ^ 2) := by
have hs := quadratic_endpoint_scale (F := F) (E := E) (V := V) hA hB hS hR0 hK
(show 0 ≤ (Real.sqrt t)⁻¹ by positivity) (Real.log_nonneg hI) hR
rw [inv_sqrt_sq_time ht.le] at hs
exact hs hF
theorem linear_endpoint_time_scale {F A B E V R K I t : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hV : 0 ≤ V) (hK : 0 ≤ K)
(hI : 1 ≤ I) (ht : 0 < t) (ht1 : t ≤ 1)
(hR : R ≤ K * ((Real.sqrt t)⁻¹ + Real.log I))
(hF : F ≤ A * V * R + B * V * (Real.sqrt t)⁻¹ + E * V ^ 2) :
F ≤ max (A * K + B) E *
(V * (t ^ (-(3 : ℝ) / 4) + Real.log I) + V ^ 2) := by
have hscale := linear_endpoint_scale hA hB hV
(show 0 ≤ (Real.sqrt t)⁻¹ by positivity) (Real.log_nonneg hI) hR hF
have hC : 0 ≤ max (A * K + B) E :=
(add_nonneg (mul_nonneg hA hK) hB).trans (le_max_left _ _)
apply hscale.trans
apply mul_le_mul_of_nonneg_left _ hC
apply add_le_add _ le_rfl
apply mul_le_mul_of_nonneg_left _ hV
exact add_le_add (inv_sqrt_le_time_three_quarters ht ht1) le_rfl
theorem sqrt_endpoint_time_scale {F A B E S V R K I t : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hV : 0 ≤ V)
(hR0 : 0 ≤ R) (hK : 0 ≤ K) (hI : 1 ≤ I) (ht : 0 < t)
(hR : R ≤ K * ((Real.sqrt t)⁻¹ + Real.log I))
(hF : F ≤ A * Real.sqrt S * (R * Real.sqrt R) +
B * Real.sqrt S * t ^ (-(3 : ℝ) / 4) + E * V) :
F ≤ max (4 * A * K * Real.sqrt K + B) E *
(Real.sqrt S * (t ^ (-(3 : ℝ) / 4) + (Real.log I) ^ ((3 : ℝ) / 2)) + V) := by
have hs := sqrt_endpoint_scale (F := F) (E := E) (S := S) hA hB hV hR0 hK
(show 0 ≤ (Real.sqrt t)⁻¹ by positivity) (Real.log_nonneg hI) hR
rw [inv_sqrt_mul_sqrt_time ht, mul_sqrt_eq_three_halves (Real.log_nonneg hI)] at hs
exact hs hF
def endpointTimeConstant (A B E K : ℝ) : ℝ :=
max 1 (max (max (2 * A * K ^ 2 + B) E)
(max (max (A * K + B) E) (max (4 * A * K * Real.sqrt K + B) E)))
theorem endpointTimeConstant_ge_one (A B E K : ℝ) :
1 ≤ endpointTimeConstant A B E K := le_max_left _ _
theorem endpoint_time_scales {Q L Z A B E S V R K I t C : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hS : 0 ≤ S) (hV : 0 ≤ V)
(hR0 : 0 ≤ R) (hK : 0 ≤ K) (hI : 1 ≤ I) (ht : 0 < t) (ht1 : t ≤ 1)
(hC : endpointTimeConstant A B E K ≤ C)
(hR : R ≤ K * ((Real.sqrt t)⁻¹ + Real.log I))
(hQ : Q ≤ A * S * R ^ 2 + B * S * (1 / t) + E * V ^ 2)
(hL : L ≤ A * V * R + B * V * (Real.sqrt t)⁻¹ + E * V ^ 2)
(hZ : Z ≤ A * Real.sqrt S * (R * Real.sqrt R) +
B * Real.sqrt S * t ^ (-(3 : ℝ) / 4) + E * V) :
Q ≤ C * (S * (1 / t + (Real.log I) ^ 2) + V ^ 2) ∧
L ≤ C * (V * (t ^ (-(3 : ℝ) / 4) + Real.log I) + V ^ 2) ∧
Z ≤ C * (Real.sqrt S *
(t ^ (-(3 : ℝ) / 4) + (Real.log I) ^ ((3 : ℝ) / 2)) + V) := by
have hQC : max (2 * A * K ^ 2 + B) E ≤ C := by
apply le_trans _ hC
exact (le_max_left _ _).trans (le_max_right _ _)
have hLC : max (A * K + B) E ≤ C := by
apply le_trans _ hC
exact ((le_max_left _ _).trans (le_max_right _ _)).trans (le_max_right _ _)
have hZC : max (4 * A * K * Real.sqrt K + B) E ≤ C := by
apply le_trans _ hC
exact ((le_max_right _ _).trans (le_max_right _ _)).trans (le_max_right _ _)
have hlog : 0 ≤ Real.log I := Real.log_nonneg hI
refine ⟨?_, ?_, ?_⟩
· exact (quadratic_endpoint_time_scale hA hB hS hR0 hK hI ht hR hQ).trans
(mul_le_mul_of_nonneg_right hQC (by positivity))
· exact (linear_endpoint_time_scale hA hB hV hK hI ht ht1 hR hL).trans
(mul_le_mul_of_nonneg_right hLC (by positivity))
· exact (sqrt_endpoint_time_scale hA hB hV hR0 hK hI ht hR hZ).trans
(mul_le_mul_of_nonneg_right hZC (by positivity))
end Problem367
end
end OAI
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