import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Topology.MetricSpace.Cauchy

namespace OAI

open Filter
open scoped Topology

namespace Invariant233.ProductDensity

theorem cauchySeq_of_prefix_inner
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    (f : ℕ → E)
    (hinner : ∀ n m, n ≤ m → inner ℝ (f n) (f m) = ‖f n‖ ^ 2)
    (hbound : BddAbove (Set.range fun n => ‖f n‖ ^ 2)) : CauchySeq f := by
  have hdist : ∀ n m, n ≤ m → dist (f n) (f m) ^ 2 = ‖f m‖ ^ 2 - ‖f n‖ ^ 2 := by
    intro n m hnm
    rw [dist_eq_norm, norm_sub_sq_real, hinner n m hnm]
    ring
  have hmono : Monotone (fun n => ‖f n‖ ^ 2) := by
    intro n m hnm
    have := sq_nonneg (dist (f n) (f m))
    rw [hdist n m hnm] at this
    linarith
  have hc : CauchySeq (fun n => ‖f n‖ ^ 2) :=
    (tendsto_atTop_ciSup hmono hbound).cauchySeq
  apply Metric.cauchySeq_iff.2
  intro ε hε
  obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.1 hc (ε ^ 2) (sq_pos_of_pos hε)
  refine ⟨N, fun n hn m hm => ?_⟩
  have hreal := hN n hn m hm
  rw [Real.dist_eq] at hreal
  rcases le_total n m with hnm | hmn
  · have hs := hdist n m hnm
    have hl := (abs_lt.mp hreal).1
    nlinarith [dist_nonneg (x := f n) (y := f m)]
  · have hs := hdist m n hmn
    have hl := (abs_lt.mp hreal).2
    rw [dist_comm (f m) (f n)] at hs
    nlinarith [dist_nonneg (x := f n) (y := f m)]

theorem exists_limit_of_prefix_inner
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
    (f : ℕ → E)
    (hinner : ∀ n m, n ≤ m → inner ℝ (f n) (f m) = ‖f n‖ ^ 2)
    (hbound : BddAbove (Set.range fun n => ‖f n‖ ^ 2)) :
    ∃ g : E, Tendsto f atTop (𝓝 g) :=
  cauchySeq_tendsto_of_complete (cauchySeq_of_prefix_inner f hinner hbound)

end Invariant233.ProductDensity

end OAI