| 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 | |