lean/OAI/Geometry/PinnedDistances/Main.lean

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