import OAI.Geometry.PinnedDistances.UnboundedContradiction namespace OAI universe u1 u2 u3 section noncomputable section open scoped BigOperators Topology open Filter MeasureTheory namespace WeakPinned.Bounded /-- Simultaneous selection from a normalized weighted family. In particular, 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