| import OAI.NumberTheory.PiExponent.Analysis.LiteralAnalyticSummand |
| import OAI.NumberTheory.PiExponent.Analysis.TranslationCountLimit |
|
|
| namespace OAI |
|
|
| open scoped BigOperators Topology |
| open Filter |
|
|
| namespace PiExponent.LiteralAnalytic |
| open DeterminantContradiction |
|
|
| theorem actual_minor_expansion {nu : ℝ} (d : FixedData nu) {H : ℝ} |
| (selection : Row d H → Column d H) : |
| (actualMinor d H selection).det = |
| ∑ f : ChoiceFamily d H, (∏ r, choiceScalar d r (f r)) * (choiceMatrix d selection f).det := by |
| classical |
| have hp : ∀ r c : Row d H, |
| (MatrixTranslation.periodMonomial r.1.val logarithmicPeriod |
| ((selection c).1 0) (fun i => (selection c).1 i.succ)).support ⊆ |
| MatrixTranslation.transverseIndices (fixedWeights d) H := by |
| intro r c |
| apply MatrixTranslation.periodMonomial_support_subset _ _ _ _ _ _ (fixedWeights_pos d) |
| have hc := InterpolationMatrix.column_weight_le d.w0_pos (fixedWeights_pos d) (selection c) |
| have hz : 0 ≤ (d.w0 : ℝ) * ((selection c).1 0 : ℝ) := |
| mul_nonneg d.w0_pos.le (Nat.cast_nonneg _) |
| linarith |
| have he := MatrixTranslation.det_matrix_translation |
| (MatrixArithmetic.rationalCenters (finiteNumerators d) (finiteDenominators d)) |
| (MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0) |
| (fun r : Row d H => r.1.val) (fun r => r.2.1 0) logarithmicPeriod |
| (fun r i => r.2.1 i.succ) (fun c => (selection c).1 0) |
| (fun c i => (selection c).1 i.succ) (MatrixTranslation.transverseIndices (fixedWeights d) H) hp |
| convert! he using 1 |
|
|
| theorem card_choiceFamily_le {nu : ℝ} (d : FixedData nu) (H : ℝ) : |
| Fintype.card (ChoiceFamily d H) ≤ translationTermCount d.m d.v0 H ^ actualRowCount d H := by |
| classical |
| rw [Fintype.card_pi] |
| calc |
| _ ≤ ∏ _r : Row d H, translationTermCount d.m d.v0 H := by |
| apply Finset.prod_le_prod |
| intro r _ |
| exact MatrixTranslation.row_term_count (fixedWeights d) H d.v0 |
| (fun i => by change 1 ≤ MatrixArithmetic.logWeights (finiteDenominators d) i |
| rw [logWeights_eq]; exact d.x_one_le _) |
| (beta d r) (r.2.1 0) (rowOrder_le d r) |
| _ = _ := by simp only [Finset.prod_const, Finset.card_univ, actualRowCount] |
|
|
| noncomputable def analyticRemainder {nu : ℝ} (d : FixedData nu) (H : ℝ) : ℝ := |
| Collision.collisionRemainder (actualRowCount d H) H + |
| Real.log (translationTermCount d.m d.v0 H : ℝ) / H |
|
|
| theorem actual_minor_analytic_bound {nu : ℝ} (d : FixedData nu) (hnu : 0 ≤ nu) |
| {H : ℝ} (hH : 0 < H) (selection : Row d H → Column d H) |
| (hne : (actualMinor d H selection).det ≠ 0) : |
| Real.log ‖(actualMinor d H selection).det‖ / ((actualRowCount d H : ℝ) * H) ≤ |
| d.analyticError + analyticRemainder d H + |
| max (-collisionRate d H) |
| (-nu * ((d.base.A : ℝ) * (1 - d.base.eta) - actualMean d H)) := by |
| have hQ : (0 : ℝ) < translationTermCount d.m d.v0 H := by |
| unfold translationTermCount |
| positivity |
| have hc : (Fintype.card (ChoiceFamily d H) : ℝ) ≤ |
| (translationTermCount d.m d.v0 H : ℝ) ^ actualRowCount d H := by |
| exact_mod_cast card_choiceFamily_le d H |
| have hb := DeterminantAnalyticBound.log_norm_sum_le |
| (fun f : ChoiceFamily d H => (∏ r, choiceScalar d r (f r)) * (choiceMatrix d selection f).det) |
| (actualMinor d H selection).det (actualRowCount_pos d hH) hH hQ hc |
| (actual_minor_expansion d selection) hne (norm_actual_summand_le d hnu hH selection) |
| convert! hb using 1 |
| unfold analyticRemainder |
| ring |
|
|
| theorem tendsto_analyticRemainder {nu : ℝ} (d : FixedData nu) : |
| Tendsto (analyticRemainder d) atTop (𝓝 0) := by |
| have hv : 0 < d.v0 := by exact_mod_cast d.v0_pos |
| have ht : 0 < d.base.theta := by exact_mod_cast d.base.theta_pos |
| have hcount := MatrixCounting.tendsto_rowCount_normalized d.K d.v0 d.base.theta |
| (finiteDenominators d) hv ht (finiteDenominators_two_le d) |
| have hlead : 0 < (d.K : ℝ) * (d.base.theta : ℝ) ^ d.m / |
| (((d.m + 1).factorial : ℝ) * (d.v0 : ℝ) * |
| ∏ i, MatrixArithmetic.logWeights (finiteDenominators d) i) := by |
| apply div_pos |
| · apply mul_pos |
| · exact_mod_cast lt_of_lt_of_le Nat.zero_lt_one d.K_pos |
| · exact pow_pos d.base.theta_pos _ |
| · apply mul_pos |
| · exact mul_pos (by positivity) d.v0_pos |
| · exact Finset.prod_pos (fun i _ => fixedWeights_pos d i) |
| have hr := tendsto_collisionRemainder_of_normalized_pow |
| (M := actualRowCount d) hcount hlead |
| have hq := tendsto_log_translationTermCount_div d.m d.v0_pos |
| convert! hr.add hq using 1 |
| norm_num |
|
|
| theorem analyticAggregate : AnalyticAggregateStatement := by |
| intro nu hnu d |
| refine ⟨analyticRemainder d, tendsto_analyticRemainder d, ?_⟩ |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with H hH |
| intro selection hne |
| exact actual_minor_analytic_bound d (by linarith) hH selection hne |
|
|
| end PiExponent.LiteralAnalytic |
|
|
| end OAI |
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|
