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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | import OAI.Probability.FreeIsing.Estimates.EndpointSums
namespace OAI
noncomputable section
namespace Problem367
theorem endpoint_two_term_le {A B u v : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) :
A * u + B * v ≤ max A B * (u + v) := by
calc
A * u + B * v ≤ max A B * u + max A B * v :=
add_le_add (mul_le_mul_of_nonneg_right (le_max_left _ _) hu)
(mul_le_mul_of_nonneg_right (le_max_right _ _) hv)
_ = _ := by ring
theorem cutoff_square_le {R K x y : ℝ} (hR0 : 0 ≤ R) (hK : 0 ≤ K)
(hx : 0 ≤ x) (hy : 0 ≤ y) (hR : R ≤ K * (x + y)) :
R ^ 2 ≤ 2 * K ^ 2 * (x ^ 2 + y ^ 2) := by
calc
R ^ 2 ≤ (K * (x + y)) ^ 2 :=
(sq_le_sq₀ hR0 (mul_nonneg hK (add_nonneg hx hy))).mpr hR
_ = K ^ 2 * (x + y) ^ 2 := by ring
_ ≤ K ^ 2 * (2 * (x ^ 2 + y ^ 2)) := by
apply mul_le_mul_of_nonneg_left _ (sq_nonneg K)
nlinarith [sq_nonneg (x - y)]
_ = _ := by ring
theorem add_mul_sqrt_add_le {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) :
(x + y) * Real.sqrt (x + y) ≤ 4 * (x * Real.sqrt x + y * Real.sqrt y) := by
have haux : ∀ u v : ℝ, 0 ≤ u → 0 ≤ v → u ≤ v →
(u + v) * Real.sqrt (u + v) ≤ 4 * (u * Real.sqrt u + v * Real.sqrt v) := by
intro u v hu hv huv
have hsum : u + v ≤ 2 * v := by linarith
have hsqrt : Real.sqrt (u + v) ≤ 2 * Real.sqrt v := by
apply Real.sqrt_le_iff.mpr
constructor
· positivity
· rw [mul_pow, Real.sq_sqrt hv]
nlinarith
calc
(u + v) * Real.sqrt (u + v) ≤ (2 * v) * (2 * Real.sqrt v) :=
mul_le_mul hsum hsqrt (Real.sqrt_nonneg _) (by positivity)
_ ≤ 4 * (u * Real.sqrt u + v * Real.sqrt v) := by
nlinarith [mul_nonneg hu (Real.sqrt_nonneg u)]
rcases le_total x y with h | h
· exact haux x y hx hy h
· simpa [add_comm] using haux y x hy hx h
theorem cutoff_mul_sqrt_le {R K x y : ℝ} (_hR0 : 0 ≤ R) (hK : 0 ≤ K)
(hx : 0 ≤ x) (hy : 0 ≤ y) (hR : R ≤ K * (x + y)) :
R * Real.sqrt R ≤
4 * K * Real.sqrt K * (x * Real.sqrt x + y * Real.sqrt y) := by
calc
R * Real.sqrt R ≤ (K * (x + y)) * Real.sqrt (K * (x + y)) :=
mul_le_mul hR (Real.sqrt_le_sqrt hR) (Real.sqrt_nonneg _) (by positivity)
_ = (K * Real.sqrt K) * ((x + y) * Real.sqrt (x + y)) := by
rw [Real.sqrt_mul hK]
ring
_ ≤ (K * Real.sqrt K) * (4 * (x * Real.sqrt x + y * Real.sqrt y)) :=
mul_le_mul_of_nonneg_left (add_mul_sqrt_add_le hx hy) (by positivity)
_ = _ := by ring
theorem quadratic_endpoint_scale {F A B E S V R K x y : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hS : 0 ≤ S)
(hR0 : 0 ≤ R) (hK : 0 ≤ K) (hx : 0 ≤ x) (hy : 0 ≤ y)
(hR : R ≤ K * (x + y))
(hF : F ≤ A * S * R ^ 2 + B * S * x ^ 2 + E * V ^ 2) :
F ≤ max (2 * A * K ^ 2 + B) E *
(S * (x ^ 2 + y ^ 2) + V ^ 2) := by
have hfirst := mul_le_mul_of_nonneg_left (cutoff_square_le hR0 hK hx hy hR)
(mul_nonneg hA hS)
have hsecond : B * S * x ^ 2 ≤ B * S * (x ^ 2 + y ^ 2) := by
apply mul_le_mul_of_nonneg_left _ (mul_nonneg hB hS)
nlinarith [sq_nonneg y]
calc
F ≤ A * S * (2 * K ^ 2 * (x ^ 2 + y ^ 2)) +
B * S * (x ^ 2 + y ^ 2) + E * V ^ 2 :=
hF.trans (add_le_add (add_le_add hfirst hsecond) le_rfl)
_ = (2 * A * K ^ 2 + B) * (S * (x ^ 2 + y ^ 2)) + E * V ^ 2 := by ring
_ ≤ _ := endpoint_two_term_le (by positivity) (sq_nonneg V)
theorem linear_endpoint_scale {F A B E V R K x y : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hV : 0 ≤ V)
(hx : 0 ≤ x) (hy : 0 ≤ y) (hR : R ≤ K * (x + y))
(hF : F ≤ A * V * R + B * V * x + E * V ^ 2) :
F ≤ max (A * K + B) E * (V * (x + y) + V ^ 2) := by
have hfirst := mul_le_mul_of_nonneg_left hR (mul_nonneg hA hV)
have hsecond : B * V * x ≤ B * V * (x + y) := by
apply mul_le_mul_of_nonneg_left _ (mul_nonneg hB hV)
linarith
calc
F ≤ A * V * (K * (x + y)) + B * V * (x + y) + E * V ^ 2 :=
hF.trans (add_le_add (add_le_add hfirst hsecond) le_rfl)
_ = (A * K + B) * (V * (x + y)) + E * V ^ 2 := by ring
_ ≤ _ := endpoint_two_term_le (by positivity) (sq_nonneg V)
theorem sqrt_endpoint_scale {F A B E S V R K x y : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hV : 0 ≤ V)
(hR0 : 0 ≤ R) (hK : 0 ≤ K) (hx : 0 ≤ x) (hy : 0 ≤ y)
(hR : R ≤ K * (x + y))
(hF : F ≤ A * Real.sqrt S * (R * Real.sqrt R) +
B * Real.sqrt S * (x * Real.sqrt x) + E * V) :
F ≤ max (4 * A * K * Real.sqrt K + B) E *
(Real.sqrt S * (x * Real.sqrt x + y * Real.sqrt y) + V) := by
have hfirst := mul_le_mul_of_nonneg_left (cutoff_mul_sqrt_le hR0 hK hx hy hR)
(mul_nonneg hA (Real.sqrt_nonneg S))
have hsecond : B * Real.sqrt S * (x * Real.sqrt x) ≤
B * Real.sqrt S * (x * Real.sqrt x + y * Real.sqrt y) := by
apply mul_le_mul_of_nonneg_left _ (mul_nonneg hB (Real.sqrt_nonneg S))
nlinarith [mul_nonneg hy (Real.sqrt_nonneg y)]
calc
F ≤ A * Real.sqrt S *
(4 * K * Real.sqrt K * (x * Real.sqrt x + y * Real.sqrt y)) +
B * Real.sqrt S * (x * Real.sqrt x + y * Real.sqrt y) + E * V :=
hF.trans (add_le_add (add_le_add hfirst hsecond) le_rfl)
_ = (4 * A * K * Real.sqrt K + B) *
(Real.sqrt S * (x * Real.sqrt x + y * Real.sqrt y)) + E * V := by ring
_ ≤ _ := endpoint_two_term_le (by positivity) hV
theorem tsum_scaled_path_attenuation_le (k : ℕ) {c ω : ℝ}
(hc : 0 < c) (hω : 0 < ω) (hω1 : ω ≤ 1) :
(∑' n : ℕ, (n : ℝ) ^ k * Real.exp (-(c * ω) * n)) ≤
(k.factorial : ℝ) * (2 / min c 1) ^ (k + 1) * (ω⁻¹) ^ (k + 1) := by
have hm : 0 < min c 1 := lt_min hc zero_lt_one
have hrate : 0 < min c 1 * ω := mul_pos hm hω
have hrate1 : min c 1 * ω ≤ 1 :=
(mul_le_mul_of_nonneg_right (min_le_right c 1) hω.le).trans (by simpa using hω1)
calc
_ ≤ ∑' n : ℕ, (n : ℝ) ^ k * Real.exp (-(min c 1 * ω) * n) := by
apply Summable.tsum_le_tsum _ (summable_path_attenuation k (mul_pos hc hω))
(summable_path_attenuation k hrate)
intro n
apply mul_le_mul_of_nonneg_left _ (by positivity)
apply Real.exp_le_exp.mpr
have h := mul_le_mul_of_nonneg_right (min_le_left c 1) hω.le
have := mul_le_mul_of_nonneg_right h (Nat.cast_nonneg (α := ℝ) n)
linarith
_ ≤ (k.factorial : ℝ) * (2 / (min c 1 * ω)) ^ (k + 1) :=
tsum_path_attenuation_small_le k hrate hrate1
_ = _ := by simp [div_eq_mul_inv, mul_inv_rev, mul_pow, mul_assoc, mul_comm, mul_left_comm]
theorem tsum_scaled_sqrt_path_attenuation_le {c ω : ℝ}
(hc : 0 < c) (hω : 0 < ω) (hω1 : ω ≤ 1) :
(∑' n : ℕ, Real.sqrt n * Real.exp (-(c * ω) * n)) ≤
(3 / (min c 1 * Real.sqrt (min c 1))) * (ω⁻¹ * Real.sqrt (ω⁻¹)) := by
have hm : 0 < min c 1 := lt_min hc zero_lt_one
have hrate : 0 < min c 1 * ω := mul_pos hm hω
have hrate1 : min c 1 * ω ≤ 1 :=
(mul_le_mul_of_nonneg_right (min_le_right c 1) hω.le).trans (by simpa using hω1)
calc
_ ≤ ∑' n : ℕ, Real.sqrt n * Real.exp (-(min c 1 * ω) * n) := by
apply Summable.tsum_le_tsum _ (summable_sqrt_path_attenuation (mul_pos hc hω))
(summable_sqrt_path_attenuation hrate)
intro n
apply mul_le_mul_of_nonneg_left _ (Real.sqrt_nonneg _)
apply Real.exp_le_exp.mpr
have h := mul_le_mul_of_nonneg_right (min_le_left c 1) hω.le
have := mul_le_mul_of_nonneg_right h (Nat.cast_nonneg (α := ℝ) n)
linarith
_ ≤ 3 / ((min c 1 * ω) * Real.sqrt (min c 1 * ω)) :=
tsum_sqrt_path_attenuation_le hrate hrate1
_ = _ := by
rw [Real.sqrt_mul hm.le, Real.sqrt_inv]
field_simp
theorem pathCutoff_square_le {C ω I : ℝ}
(hC : 0 ≤ C) (hω : 0 < ω) (hω1 : ω ≤ 1) (hI : 1 ≤ I) :
(pathCutoff C ω I : ℝ) ^ 2 ≤
2 * (C + 1) ^ 2 * ((ω⁻¹) ^ 2 + (Real.log I) ^ 2) := by
exact cutoff_square_le (Nat.cast_nonneg _) (by positivity) (by positivity)
(Real.log_nonneg hI) (pathCutoff_le hC hω hω1 hI)
theorem pathCutoff_mul_sqrt_le {C ω I : ℝ}
(hC : 0 ≤ C) (hω : 0 < ω) (hω1 : ω ≤ 1) (hI : 1 ≤ I) :
(pathCutoff C ω I : ℝ) * Real.sqrt (pathCutoff C ω I) ≤
4 * (C + 1) * Real.sqrt (C + 1) *
(ω⁻¹ * Real.sqrt (ω⁻¹) + Real.log I * Real.sqrt (Real.log I)) := by
exact cutoff_mul_sqrt_le (Nat.cast_nonneg _) (by positivity) (by positivity)
(Real.log_nonneg hI) (pathCutoff_le hC hω hω1 hI)
end Problem367
end
end OAI
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