File size: 2,520 Bytes
9eeb9c7 | 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 | import Mathlib
namespace OAI
open Filter
open scoped Topology
namespace Invariant233.ProductDensity
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
theorem norm_sub_sq_of_nested_inner (u : ℕ → E)
(hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2)
{m n : ℕ} (hmn : m ≤ n) :
‖u m - u n‖ ^ 2 = ‖u n‖ ^ 2 - ‖u m‖ ^ 2 := by
rw [norm_sub_sq_real, hinner m n hmn]
ring
theorem monotone_norm_sq_of_nested_inner (u : ℕ → E)
(hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) :
Monotone (fun n => ‖u n‖ ^ 2) := by
intro m n hmn
have h := norm_sub_sq_of_nested_inner u hinner hmn
nlinarith [sq_nonneg ‖u m - u n‖]
theorem norm_sub_sq_eq_abs_of_nested_inner (u : ℕ → E)
(hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2)
(m n : ℕ) :
‖u m - u n‖ ^ 2 = |‖u m‖ ^ 2 - ‖u n‖ ^ 2| := by
rcases le_total m n with hmn | hnm
· rw [norm_sub_sq_of_nested_inner u hinner hmn,
abs_of_nonpos (sub_nonpos.mpr (monotone_norm_sq_of_nested_inner u hinner hmn))]
ring
· rw [norm_sub_rev, norm_sub_sq_of_nested_inner u hinner hnm,
abs_of_nonneg (sub_nonneg.mpr (monotone_norm_sq_of_nested_inner u hinner hnm))]
theorem cauchySeq_of_nested_inner (u : ℕ → E)
(hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2)
(hbound : BddAbove (Set.range (fun n => ‖u n‖ ^ 2))) :
CauchySeq u := by
have hlim := tendsto_atTop_ciSup (monotone_norm_sq_of_nested_inner u hinner) hbound
have hc : CauchySeq (fun n => ‖u n‖ ^ 2) := hlim.cauchySeq
apply Metric.cauchySeq_iff.mpr
intro ε hε
obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.mp hc (ε ^ 2) (sq_pos_of_pos hε)
refine ⟨N, ?_⟩
intro m hm n hn
have hs := hN m hm n hn
rw [Real.dist_eq, ← norm_sub_sq_eq_abs_of_nested_inner u hinner m n] at hs
rw [dist_eq_norm]
nlinarith [norm_nonneg (u m - u n)]
theorem exists_limit_of_nested_inner [CompleteSpace E] (u : ℕ → E)
(hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2)
(C : ℝ) (hbound : ∀ n, ‖u n‖ ^ 2 ≤ C) :
∃ v : E, Tendsto u atTop (𝓝 v) ∧ ‖v‖ ^ 2 ≤ C := by
obtain ⟨v, hv⟩ := cauchySeq_tendsto_of_complete
(cauchySeq_of_nested_inner u hinner ⟨C, by rintro _ ⟨n, rfl⟩; exact hbound n⟩)
refine ⟨v, hv, ?_⟩
exact le_of_tendsto (hv.norm.pow 2) (Eventually.of_forall hbound)
end Invariant233.ProductDensity
end OAI
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