| 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 | |