File size: 5,645 Bytes
2fca478 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | 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<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)
/-- Both the local scale and any nonnegative sum of variance cells are
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
|