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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 | import OAI.Geometry.PinnedDistances.ArithmeticSequence
namespace OAI
section
noncomputable section
open Filter
open scoped Topology
namespace WeakPinned
/-- Uniform weak pinned planar distances for arbitrary real configurations. -/
theorem mainStatement : MainStatement := by
classical
by_contra hfail
obtain ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hcard,hG,hmodels⟩ := sequence_arithmetic_model hfail
choose L hL ii Z₁ Z₂ t hii h₁ h₂ ht0 ht using hmodels
let K := fun j => L j
let (j : ℕ) : NumberField (K j) := hL j
have hn : Tendsto (fun j => (Fintype.card (C j):ℝ)) atTop atTop := by
simpa only [Fintype.card_coe, Function.comp_def] using tendsto_natCast_atTop_atTop.comp (graph_card_tendsto hcard)
apply arithmetic_sequence_impossible (fun j => coreGraph (P j) s (C j)) (Theta s) γ c
(fun j => Theta s/((j:ℝ)+2)) Z₁ Z₂ h₁ h₂ hθ (hc.trans hcγ) (hγs.trans hs) hc hcγ hn
(graph_error_tendsto (Theta s)) (fun j => by positivity) _ hG t ht
intro j
apply div_le_div_of_nonneg_left hθ.le (by norm_num : (0:ℝ)<2)
have : 0 ≤ (j : ℝ) := Nat.cast_nonneg j
linarith
theorem main (s : ℝ) (hs : 0<s) : Tendsto (fun n : ℕ => F n s) atTop (𝓝 0) :=
mainStatement s hs
end WeakPinned
end
end
end OAI
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