| import Mathlib.Basic.Real.Basic |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SmoothLocal.Weighted |
|
|
| theorem normalized_T_lower_bound {H Is epsilon t B MI Mt MB : ℝ} |
| (heps : 0 ≤ epsilon) (hMt : 0 ≤ Mt) (hIs : |Is| ≤ MI) |
| (ht : |t| ≤ Mt) (hB : |B| ≤ MB) |
| (hbudget : 2 * (MI + epsilon + epsilon * Mt * MB) ≤ H) : |
| H / 4 ≤ (H - Is - epsilon + epsilon * t * B) / 2 := by |
| have hprod : |epsilon * t * B| ≤ epsilon * Mt * MB := by |
| rw [abs_mul, abs_mul, abs_of_nonneg heps] |
| exact mul_le_mul (mul_le_mul_of_nonneg_left ht heps) hB |
| (abs_nonneg B) (mul_nonneg heps hMt) |
| have hlower := (abs_le.mp hprod).1 |
| have hupper := (abs_le.mp hIs).2 |
| linarith |
|
|
| theorem quadratic_cross_absorption {T S J u v : ℝ} (hT : 0 < T) : |
| T / 2 * u ^ 2 + (S - J ^ 2 / (2 * T)) * v ^ 2 ≤ |
| T * u ^ 2 + S * v ^ 2 + J * u * v := by |
| have hs : 0 ≤ (T * u + J * v) ^ 2 / (2 * T) := |
| div_nonneg (sq_nonneg _) (by positivity) |
| have he : T * u ^ 2 + S * v ^ 2 + J * u * v - |
| (T / 2 * u ^ 2 + (S - J ^ 2 / (2 * T)) * v ^ 2) = |
| (T * u + J * v) ^ 2 / (2 * T) := by |
| field_simp |
| ring |
| linarith |
|
|
| theorem cross_loss_bound {T J W H A C epsilon : ℝ} |
| (hW : 0 < W) (hH : 0 < H) (hC : 0 ≤ C) (heps : 0 ≤ epsilon) |
| (hT : W * H / 4 ≤ T) |
| (hJ : |J| ≤ W * C * epsilon * (1 + H * |A|)) : |
| J ^ 2 / (T * W) ≤ |
| 8 * C ^ 2 * epsilon ^ 2 * (H⁻¹ + H * A ^ 2) := by |
| have hTpos : 0 < T := lt_of_lt_of_le (by positivity) hT |
| have hmajor : 0 ≤ W * C * epsilon * (1 + H * |A|) := by positivity |
| have hj2 : J ^ 2 ≤ (W * C * epsilon * (1 + H * |A|)) ^ 2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg J) hmajor).mpr hJ |
| have hshape : (1 + H * |A|) ^ 2 ≤ 2 * (1 + H ^ 2 * A ^ 2) := by |
| nlinarith [sq_nonneg (H * |A| - 1), sq_abs A] |
| have hnum : J ^ 2 ≤ 2 * W ^ 2 * C ^ 2 * epsilon ^ 2 * (1 + H ^ 2 * A ^ 2) := by |
| have hm := mul_le_mul_of_nonneg_left hshape (sq_nonneg (W * C * epsilon)) |
| nlinarith [hj2] |
| have hden : W ^ 2 * H / 4 ≤ T * W := by |
| have hm := mul_le_mul_of_nonneg_right hT hW.le |
| nlinarith [hm] |
| have hR : 0 ≤ 8 * C ^ 2 * epsilon ^ 2 * (H⁻¹ + H * A ^ 2) := by positivity |
| apply (div_le_iff₀ (mul_pos hTpos hW)).mpr |
| have hm := mul_le_mul_of_nonneg_left hden hR |
| have he : (8 * C ^ 2 * epsilon ^ 2 * (H⁻¹ + H * A ^ 2)) * |
| (W ^ 2 * H / 4) = |
| 2 * W ^ 2 * C ^ 2 * epsilon ^ 2 * (1 + H ^ 2 * A ^ 2) := by |
| field_simp |
| ring |
| rw [he] at hm |
| exact hnum.trans hm |
|
|
| theorem positive_strip_weight_bound {d dmax lambda c1 K G M : ℝ} |
| (hd : 0 < d) (hdd : d ≤ dmax) (hlambda : 0 ≤ lambda) |
| (hc : 0 ≤ c1) (hK : 0 ≤ K) (hKd : K ≤ c1 * d) |
| (hG : 0 ≤ G) (hGM : G ≤ M) : |
| (lambda + 8 / d) * (G * K) ≤ M * c1 * (lambda * dmax + 8) := by |
| have hH : 0 ≤ lambda + 8 / d := by positivity |
| have hGK : G * K ≤ M * (c1 * d) := |
| (mul_le_mul_of_nonneg_right hGM hK).trans |
| (mul_le_mul_of_nonneg_left hKd (hG.trans hGM)) |
| have hm := mul_le_mul_of_nonneg_left hGK hH |
| have he : (lambda + 8 / d) * (M * (c1 * d)) = |
| M * c1 * (lambda * d + 8) := by |
| field_simp |
| rw [he] at hm |
| exact hm.trans (mul_le_mul_of_nonneg_left |
| (add_le_add (mul_le_mul_of_nonneg_left hdd hlambda) (le_refl 8)) |
| (mul_nonneg (hG.trans hGM) hc)) |
|
|
| theorem exists_positive_strip_scale {M lambda dmax margin : ℝ} |
| (hM : 0 ≤ M) (hlambda : 0 ≤ lambda) (hdmax : 0 ≤ dmax) |
| (hmargin : 0 < margin) : |
| ∃ c1 : ℝ, 0 < c1 ∧ M * c1 * (lambda * dmax + 8) ≤ margin := by |
| let D := M * (lambda * dmax + 8) + 1 |
| have hD : 0 < D := by dsimp [D]; positivity |
| refine ⟨margin / D, div_pos hmargin hD, ?_⟩ |
| have he : M * (margin / D) * (lambda * dmax + 8) = |
| margin / D * (D - 1) := by dsimp [D]; ring |
| rw [he] |
| have hle : D - 1 ≤ D := by linarith |
| have hm := mul_le_mul_of_nonneg_left hle (div_pos hmargin hD).le |
| simpa only [div_mul_cancel₀ margin hD.ne'] using hm |
|
|
| end SmoothLocal.Weighted |
|
|
| end |
|
|
| end OAI |
|
|
