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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | 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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