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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.Probability.FreeIsing.Trees.PathCutoff
namespace OAI
noncomputable section
open Filter
namespace Problem367
theorem tsum_le_prefix_add_majorant {f g : ℕ → ℝ} (R : ℕ)
(hf0 : ∀ n, 0 ≤ f n) (hg0 : ∀ n, 0 ≤ g n) (hg : Summable g)
(htail : ∀ n, R ≤ n → f n ≤ g n) :
(∑' n, f n) ≤ (∑ n ∈ Finset.range R, f n) + ∑' n, g n := by
have hf : Summable f := hg.of_norm_bounded_eventually_nat <|
eventually_atTop.mpr ⟨R, fun n hn => by
rw [Real.norm_eq_abs, abs_of_nonneg (hf0 n)]
exact htail n hn⟩
have hft : Summable (fun n => f (n + R)) := (summable_nat_add_iff R).mpr hf
have hgt : Summable (fun n => g (n + R)) := (summable_nat_add_iff R).mpr hg
calc
(∑' n, f n) = (∑ n ∈ Finset.range R, f n) + ∑' n, f (n + R) :=
(hf.sum_add_tsum_nat_add R).symm
_ ≤ (∑ n ∈ Finset.range R, f n) + ∑' n, g (n + R) := by
exact add_le_add le_rfl (hft.tsum_le_tsum
(fun n => htail (n + R) (Nat.le_add_left R n)) hgt)
_ ≤ (∑ n ∈ Finset.range R, f n) + ∑' n, g n := by
apply add_le_add le_rfl
exact tsum_comp_le_tsum_of_inj (i := fun n : ℕ => n + R) hg hg0
(fun m n h => Nat.add_right_cancel h)
theorem sum_range_nat_pow_le (R k : ℕ) :
(∑ n ∈ Finset.range R, (n : ℝ) ^ k) ≤ (R : ℝ) ^ (k + 1) := by
calc
_ ≤ ∑ _n ∈ Finset.range R, (R : ℝ) ^ k := by
apply Finset.sum_le_sum
intro n hn
have h : (n : ℝ) ≤ R := by exact_mod_cast (Finset.mem_range.mp hn).le
exact pow_le_pow_left₀ (Nat.cast_nonneg n) h k
_ = (R : ℝ) ^ (k + 1) := by simp [pow_succ, mul_comm]
theorem sum_range_sqrt_le (R : ℕ) :
(∑ n ∈ Finset.range R, Real.sqrt n) ≤ (R : ℝ) * Real.sqrt R := by
calc
_ ≤ ∑ _n ∈ Finset.range R, Real.sqrt R := by
apply Finset.sum_le_sum
intro n hn
apply Real.sqrt_le_sqrt
exact_mod_cast (Finset.mem_range.mp hn).le
_ = (R : ℝ) * Real.sqrt R := by simp
theorem tsum_integer_path_endpoint_le {f : ℕ → ℝ} (R k m : ℕ)
{A B E c a q : ℝ} (hf0 : ∀ n, 0 ≤ f n)
(hA : 0 ≤ A) (hB : 0 ≤ B) (hE : 0 ≤ E)
(hc : 0 < c) (hc1 : c ≤ 1) (ha : 0 < a) (hq : 1 < q)
(hshort : ∀ n, n < R → f n ≤ A * (n : ℝ) ^ k)
(htail : ∀ n, R ≤ n → f n ≤
B * ((n : ℝ) ^ k * Real.exp (-c * n)) +
E * ((n : ℝ) ^ m * Real.exp (-a * q ^ n))) :
(∑' n, f n) ≤ A * (R : ℝ) ^ (k + 1) +
B * ((k.factorial : ℝ) * (2 / c) ^ (k + 1)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n)) := by
have hpoly := summable_path_attenuation k hc
have hexception := summable_superexponential_path m ha hq
have hg := (hpoly.mul_left B).add (hexception.mul_left E)
have hprefix : (∑ n ∈ Finset.range R, f n) ≤ A * (R : ℝ) ^ (k + 1) := by
calc
_ ≤ ∑ n ∈ Finset.range R, A * (n : ℝ) ^ k := by
apply Finset.sum_le_sum
intro n hn
exact hshort n (Finset.mem_range.mp hn)
_ = A * (∑ n ∈ Finset.range R, (n : ℝ) ^ k) := by rw [Finset.mul_sum]
_ ≤ A * (R : ℝ) ^ (k + 1) :=
mul_le_mul_of_nonneg_left (sum_range_nat_pow_le R k) hA
calc
_ ≤ (∑ n ∈ Finset.range R, f n) +
∑' n : ℕ, (B * ((n : ℝ) ^ k * Real.exp (-c * n)) +
E * ((n : ℝ) ^ m * Real.exp (-a * q ^ n))) :=
tsum_le_prefix_add_majorant R hf0 (fun n => by positivity) hg htail
_ = (∑ n ∈ Finset.range R, f n) +
(B * (∑' n : ℕ, (n : ℝ) ^ k * Real.exp (-c * n)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n))) := by
rw [(hpoly.mul_left B).tsum_add (hexception.mul_left E), tsum_mul_left, tsum_mul_left]
_ ≤ A * (R : ℝ) ^ (k + 1) +
(B * ((k.factorial : ℝ) * (2 / c) ^ (k + 1)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n))) := by
exact add_le_add hprefix (add_le_add
(mul_le_mul_of_nonneg_left (tsum_path_attenuation_small_le k hc hc1) hB) le_rfl)
_ = _ := by ring
theorem tsum_sqrt_path_endpoint_le {f : ℕ → ℝ} (R m : ℕ)
{A B E c a q : ℝ} (hf0 : ∀ n, 0 ≤ f n)
(hA : 0 ≤ A) (hB : 0 ≤ B) (hE : 0 ≤ E)
(hc : 0 < c) (hc1 : c ≤ 1) (ha : 0 < a) (hq : 1 < q)
(hshort : ∀ n, n < R → f n ≤ A * Real.sqrt n)
(htail : ∀ n, R ≤ n → f n ≤
B * (Real.sqrt n * Real.exp (-c * n)) +
E * ((n : ℝ) ^ m * Real.exp (-a * q ^ n))) :
(∑' n, f n) ≤ A * ((R : ℝ) * Real.sqrt R) +
B * (3 / (c * Real.sqrt c)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n)) := by
have hpoly := summable_sqrt_path_attenuation hc
have hexception := summable_superexponential_path m ha hq
have hg := (hpoly.mul_left B).add (hexception.mul_left E)
have hprefix : (∑ n ∈ Finset.range R, f n) ≤ A * ((R : ℝ) * Real.sqrt R) := by
calc
_ ≤ ∑ n ∈ Finset.range R, A * Real.sqrt n := by
apply Finset.sum_le_sum
intro n hn
exact hshort n (Finset.mem_range.mp hn)
_ = A * (∑ n ∈ Finset.range R, Real.sqrt n) := by rw [Finset.mul_sum]
_ ≤ A * ((R : ℝ) * Real.sqrt R) :=
mul_le_mul_of_nonneg_left (sum_range_sqrt_le R) hA
calc
_ ≤ (∑ n ∈ Finset.range R, f n) +
∑' n : ℕ, (B * (Real.sqrt n * Real.exp (-c * n)) +
E * ((n : ℝ) ^ m * Real.exp (-a * q ^ n))) :=
tsum_le_prefix_add_majorant R hf0 (fun n => by positivity) hg htail
_ = (∑ n ∈ Finset.range R, f n) +
(B * (∑' n : ℕ, Real.sqrt n * Real.exp (-c * n)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n))) := by
rw [(hpoly.mul_left B).tsum_add (hexception.mul_left E), tsum_mul_left, tsum_mul_left]
_ ≤ A * ((R : ℝ) * Real.sqrt R) +
(B * (3 / (c * Real.sqrt c)) +
E * (∑' n : ℕ, (n : ℝ) ^ m * Real.exp (-a * q ^ n))) := by
exact add_le_add hprefix (add_le_add
(mul_le_mul_of_nonneg_left (tsum_sqrt_path_attenuation_le hc hc1) hB) le_rfl)
_ = _ := by ring
end Problem367
end
end OAI
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