| import OAI.Geometry.PinnedDistances.UnboundedContradiction |
|
|
| namespace OAI |
|
|
| universe u1 u2 u3 |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open Filter MeasureTheory |
| namespace WeakPinned.Bounded |
|
|
| / |
| there is no required lower bound on a nonzero archimedean-place weight. -/ |
| lemma weighted_select {J : Type u1} [Fintype J] (w T V : J → ℝ) (C ε : ℝ) |
| (hw : ∀ j, 0 < w j) (hW : ∑ j, w j = 1) |
| (hT : ∀ j, 0 ≤ T j) (hV : ∀ j, 0 ≤ V j) |
| (hC : 0 ≤ C) (hε : 0 < ε) |
| (hTC : ∑ j, w j * T j ≤ C) (hVE : ∑ j, w j * V j ≤ ε) : |
| ∃ j, T j ≤ 2*C+1 ∧ V j ≤ 2*ε := by |
| have hne : (Finset.univ : Finset J).Nonempty := by |
| by_contra h |
| rw [Finset.not_nonempty_iff_eq_empty.mp h, Finset.sum_empty] at hW |
| norm_num at hW |
| have hs : ∑ j, w j*(T j+(C+1)/ε*V j) ≤ ∑ j, w j*(2*C+1) := by |
| simp only [mul_add, Finset.sum_add_distrib, ← Finset.sum_mul] |
| rw [hW, one_mul] |
| have hv := mul_le_mul_of_nonneg_left hVE (show 0≤(C+1)/ε by positivity) |
| have he : (∑ j, w j*((C+1)/ε*V j)) = (C+1)/ε*(∑ j,w j*V j) := by |
| simp only [Finset.mul_sum]; apply Finset.sum_congr rfl; intros; ring |
| rw [he] |
| have he' : (C+1)/ε*ε=C+1 := by field_simp |
| rw [he'] at hv |
| linarith |
| obtain ⟨j, _, hj⟩ := Finset.exists_le_of_sum_le hne hs |
| have hj' : T j+(C+1)/ε*V j ≤ 2*C+1 := (mul_le_mul_iff_right₀ (hw j)).mp hj |
| refine ⟨j, (le_add_of_nonneg_right (mul_nonneg (by positivity) (hV j))).trans hj', ?_⟩ |
| have hv' : (C+1)/ε*V j ≤ 2*C+1 := by linarith [hT j] |
| have he : (C+1)/ε*(2*ε) = 2*(C+1) := by field_simp |
| by_contra h |
| have hh := mul_lt_mul_of_pos_left (lt_of_not_ge h) (show 0 < (C+1)/ε by positivity) |
| rw [he] at hh |
| linarith |
|
|
| end WeakPinned.Bounded |
|
|
| namespace WeakPinned.Bounded |
| open Hierarchy Variance Global NumberField |
| variable {P : Type u2} [Fintype P] [DecidableEq P] |
| variable {K : Type u3} [Field K] [NumberField K] |
| variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂) |
|
|
| omit Z₁ Z₂ h₁ h₂ in |
| lemma massEval_nonneg (q : Finset P → ℝ) (hq : ∀ C, 0≤q C) (d d' : Depth P) : |
| 0 ≤ massEval q d d' := by |
| exact add_nonneg (Finset.sum_nonneg (fun C _ => mul_nonneg (d.mass_nonneg C) (hq C))) |
| (Finset.sum_nonneg (fun C _ => mul_nonneg (d'.mass_nonneg C) (hq C))) |
|
|
| omit Z₁ Z₂ h₁ h₂ in |
| def scaleEval (d d' : Depth P) : ℝ := |
| massEval (pairCell Finset.univ) d d' + potentialEval (prob Finset.univ) d d' |
|
|
| omit Z₁ Z₂ h₁ h₂ in |
| lemma scaleEval_nonneg (d d' : Depth P) : 0 ≤ scaleEval d d' := by |
| apply add_nonneg (massEval_nonneg _ (fun C => pairCell_nonneg _ _) _ _) |
| apply Finset.sum_nonneg |
| intro x _ |
| apply mul_nonneg _ (add_nonneg (d.potential_nonneg x) (d'.potential_nonneg x)) |
| simp only [prob,Finset.mem_univ,ite_true]; positivity |
|
|
| omit [NumberField K] in |
| include h₁ h₂ in |
| lemma integrable_scaleEval (hn : 2≤Fintype.card P) : |
| AverageIntegrable Z₁ Z₂ (scaleEval (P:=P)) := by |
| intro v |
| exact (integrable_massEval Z₁ Z₂ h₁ h₂ hn _ v).add |
| (integrable_potentialEval Z₁ Z₂ h₁ h₂ hn _ v) |
|
|
| lemma average_scaleEval (hn : 2≤Fintype.card P) : |
| average Z₁ Z₂ h₁ h₂ scaleEval = overlapScale Z₁ Z₂ h₁ h₂ + potentialScale Z₁ Z₂ h₁ h₂ := by |
| unfold scaleEval |
| rw [average_add Z₁ Z₂ h₁ h₂ |
| (integrable_massEval Z₁ Z₂ h₁ h₂ hn _) (integrable_potentialEval Z₁ Z₂ h₁ h₂ hn _), |
| average_massEval Z₁ Z₂ h₁ h₂ hn,average_potentialEval Z₁ Z₂ h₁ h₂ hn] |
| rfl |
|
|
| def archMean (f : Depth P → Depth P → ℝ) (v : InfinitePlace K) : ℝ := |
| ∫ ω, f (arch Z₁ v ω) (arch Z₂ v ω) ∂ Grid.fullLaw |
|
|
| omit [Fintype P] [NumberField K] in |
| lemma archMean_nonneg (f : Depth P → Depth P → ℝ) (hf : ∀ d d', 0≤f d d') |
| (v : InfinitePlace K) : 0≤archMean Z₁ Z₂ f v := integral_nonneg (fun _ => hf _ _) |
|
|
| lemma archMean_le_average (f : Depth P → Depth P → ℝ) (hf : ∀ d d', 0≤f d d') : |
| (∑ v, Arithmetic.archWeight v * archMean Z₁ Z₂ f v) ≤ average Z₁ Z₂ h₁ h₂ f := by |
| apply le_add_of_nonneg_right |
| exact mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => hf _ _)) |
|
|
| omit Z₁ Z₂ h₁ h₂ in |
| lemma archWeight_pos (v : InfinitePlace K) : 0<Arithmetic.archWeight v := by |
| unfold Arithmetic.archWeight |
| apply div_pos |
| · exact_mod_cast (InfinitePlace.mult_pos (w := v)) |
| · exact_mod_cast Module.finrank_pos (R:=ℚ) (M:=K) |
|
|
| / |
| controlled at one genuine complex embedding. -/ |
| lemma select_arch (hn : 2≤Fintype.card P) (q : Finset P → ℝ) (hq : ∀ C, 0≤q C) |
| (C ε : ℝ) (hC : 0≤C) (hε : 0<ε) |
| (hscale : overlapScale Z₁ Z₂ h₁ h₂ + potentialScale Z₁ Z₂ h₁ h₂ ≤ C) |
| (hvar : weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) q ≤ ε) : |
| ∃ v : InfinitePlace K, |
| archMean Z₁ Z₂ scaleEval v ≤ 2*C+1 ∧ archMean Z₁ Z₂ (massEval q) v ≤ 2*ε := by |
| apply weighted_select Arithmetic.archWeight _ _ C ε archWeight_pos Arithmetic.sum_archWeight |
| (archMean_nonneg Z₁ Z₂ _ scaleEval_nonneg) |
| (archMean_nonneg Z₁ Z₂ _ (massEval_nonneg q hq)) hC hε |
| · exact (archMean_le_average Z₁ Z₂ h₁ h₂ _ scaleEval_nonneg).trans |
| (by rw [average_scaleEval Z₁ Z₂ h₁ h₂ hn]; exact hscale) |
| · exact (archMean_le_average Z₁ Z₂ h₁ h₂ _ (massEval_nonneg q hq)).trans |
| (by rw [average_massEval Z₁ Z₂ h₁ h₂ hn]; exact hvar) |
|
|
| end WeakPinned.Bounded |
|
|
| end |
| end |
|
|
| end OAI |
|
|
