openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Pressure /PressureCoordinateMinimum.lean
| import OAI.Probability.InvariantIsing.Arrays.PerturbationWeights | |
| namespace OAI | |
| /-! Exact conversion of a pressure-coordinate minimum to its selected-field CGF. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| namespace InvariantIsing | |
| lemma pressureCoordinate_minimum_rescale {M L : ℝ → ℝ} {N a w s c : ℝ} | |
| (hN : 0 ≤ N) (ha : 0 < a) (hw : w ∈ Set.Icc (1 : ℝ) 2) | |
| (hlo : 1 ≤ w - s) (hhi : w + s ≤ 2) | |
| (hmean : ∀ v ∈ Set.Icc (1 : ℝ) 2, L (a * v) = N * (M v - M 0)) | |
| (hmin : ∀ v ∈ Set.Icc (1 : ℝ) 2, | |
| -M w + c * (w - 3 / 2) ^ 2 ≤ -M v + c * (v - 3 / 2) ^ 2) : | |
| ∀ t ∈ Set.Icc (a * w - a * s) (a * w + a * s), | |
| -L (a * w) + (N * c / a ^ 2) * (a * w - a * (3 / 2)) ^ 2 ≤ | |
| -L t + (N * c / a ^ 2) * (t - a * (3 / 2)) ^ 2 := by | |
| intro t ht | |
| have htv : t / a ∈ Set.Icc (1 : ℝ) 2 := by | |
| constructor | |
| · apply (le_div_iff₀ ha).mpr | |
| nlinarith [ht.1] | |
| · apply (div_le_iff₀ ha).mpr | |
| nlinarith [ht.2] | |
| have hm := hmean (t / a) htv | |
| rw [mul_div_cancel₀ _ ha.ne'] at hm | |
| rw [hm, hmean w hw] | |
| have hc := mul_le_mul_of_nonneg_left (hmin (t / a) htv) hN | |
| have he1 : (N * c / a ^ 2) * (a * w - a * (3 / 2)) ^ 2 = | |
| N * (c * (w - 3 / 2) ^ 2) := by field_simp | |
| have he2 : (N * c / a ^ 2) * (t - a * (3 / 2)) ^ 2 = | |
| N * (c * (t / a - 3 / 2) ^ 2) := by field_simp | |
| rw [he1, he2] | |
| linarith | |
| /-- Comparing a Gaussian-coordinate minimum with its centered choice | |
| puts it in the interior, using the Gaussian variance cap of the frozen field. -/ | |
| lemma gaussianCoordinate_minimum_penalty {N : ℕ} (hN : 0 < N) (j : Fin N) | |
| (M : ℝ → ℝ) (u : ℝ) (hu : |u| ≤ 2) | |
| (hcost : 0 ≤ M (3 / 2) - M 0 ∧ M u - M 0 ≤ (perturbationAmplitude N j * u) ^ 2 / (2 * N)) | |
| (hmin : -M u + perturbationWeight j * (u - 3 / 2) ^ 2 ≤ | |
| -M (3 / 2) + perturbationWeight j * (3 / 2 - 3 / 2) ^ 2) : | |
| (u - 3 / 2) ^ 2 ≤ 2 * perturbationScale N ^ 2 * perturbationWeight j := by | |
| have hn : (0 : ℝ) < N := by exact_mod_cast hN | |
| have hw := perturbationWeight_pos j | |
| have ha := perturbationAmplitude_sq N j | |
| have hu2 : u ^ 2 ≤ 4 := by | |
| have hb := abs_le.mp hu | |
| nlinarith [sq_nonneg (u + 2), sq_nonneg (u - 2)] | |
| have hanu : (perturbationAmplitude N j * u) ^ 2 / (2 * N) ≤ | |
| 2 * perturbationScale N ^ 2 * perturbationWeight j ^ 2 := by | |
| apply (div_le_iff₀ (show 0 < 2 * (N : ℝ) by positivity)).mpr | |
| rw [mul_pow, ha] | |
| nlinarith [mul_le_mul_of_nonneg_left hu2 | |
| (show 0 ≤ (N : ℝ) * perturbationScale N ^ 2 * perturbationWeight j ^ 2 by positivity)] | |
| have hm : perturbationWeight j * (u - 3 / 2) ^ 2 ≤ | |
| perturbationWeight j * (2 * perturbationScale N ^ 2 * perturbationWeight j) := by | |
| nlinarith [hcost.1, hcost.2] | |
| exact (mul_le_mul_iff_right₀ hw).mp hm | |
| lemma gaussianCoordinate_minimum_interior {N : ℕ} (j : Fin N) {u s : ℝ} | |
| (hpen : (u - 3 / 2) ^ 2 ≤ 2 * perturbationScale N ^ 2 * perturbationWeight j) | |
| (he : perturbationScale N ≤ 1 / 8) (hs : s ≤ 1 / 4) : | |
| 1 ≤ u - s ∧ u + s ≤ 2 := by | |
| have he0 := perturbationScale_nonneg N | |
| have hw := perturbationWeight_le_one j | |
| have hbnd : (u - 3 / 2) ^ 2 ≤ 1 / 32 := by | |
| have hm := mul_le_mul_of_nonneg_left hw (show 0 ≤ 2 * perturbationScale N ^ 2 by positivity) | |
| nlinarith [sq_nonneg (perturbationScale N - 1 / 8)] | |
| constructor <;> nlinarith [sq_nonneg (u - 3 / 2 - 1 / 4), sq_nonneg (u - 3 / 2 + 1 / 4)] | |
| lemma diagonalCoordinate_minimum_penalty {M : ℝ → ℝ} {w e : ℝ} | |
| (hcost : 0 ≤ M (3 / 2) - M 0 ∧ M w - M 0 ≤ 2 * e) | |
| (hmin : -M w + (w - 3 / 2) ^ 2 ≤ -M (3 / 2) + (3 / 2 - 3 / 2) ^ 2) : | |
| (w - 3 / 2) ^ 2 ≤ 2 * e := by | |
| nlinarith [hcost.1, hcost.2] | |
| lemma diagonalCoordinate_minimum_interior {w e s : ℝ} | |
| (hpen : (w - 3 / 2) ^ 2 ≤ 2 * e) (he : e ≤ 1 / 32) (hs : s ≤ 1 / 4) : | |
| 1 ≤ w - s ∧ w + s ≤ 2 := by | |
| constructor <;> nlinarith [sq_nonneg (w - 3 / 2 - 1 / 4), sq_nonneg (w - 3 / 2 + 1 / 4)] | |
| end InvariantIsing | |
| end | |
| end OAI | |