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8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.Modulus.Cost
namespace OAI
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open Set Topology
open Set Topology Filter Metric unitInterval
open scoped InnerProductSpace
open Set unitInterval
open Set Metric Topology
open Set Topology unitInterval
namespace CannonCells
open CannonDuality Set Filter
variable {X ι J : Type*} [Fintype ι] [Fintype J]
lemma modulus_le_energy (C : ι → Set X) (F : Set (Set X)) (w : Weights ι)
(hw : Admissible (familyVectors C F : Set (Weights ι)) w) :
modulus C F ≤ ‖w‖^2 := by
apply csInf_le
· refine ⟨0,?_⟩
rintro a ⟨v,_,rfl⟩
exact sq_nonneg _
· exact ⟨w,hw,rfl⟩
theorem modulus_le_card_mul_sum (C : ι → Set X)
(hC : ∀ x, ∃ i, x ∈ C i) (F : Set (Set X)) (H : J → Set (Set X))
(hH : ∀ j, (H j).Nonempty) (hHn : ∀ j A, A ∈ H j → A.Nonempty)
(hsub : ∀ A ∈ F, ∃ j, ∃ B ∈ H j, B ⊆ A) :
modulus C F ≤ Fintype.card J * ∑ j, modulus C (H j) := by
classical
choose w hw hwe using fun j => (modulus_minimum C hC (hH j) (hHn j)).2
have hwp (j : J) : Nonnegative (w j) := (hw j).1
let W : Weights ι := ∑ j, w j
have hW : Admissible (familyVectors C F : Set (Weights ι)) W := by
rw [admissible_iff]
have hpoint (i : ι) : W i = ∑ j, w j i := by simp [W]
have hnon : Nonnegative W := by
intro i
rw [hpoint]
exact Finset.sum_nonneg (fun j _ => hwp j i)
refine ⟨hnon,?_⟩
intro A hA
obtain ⟨j,B,hB,hBA⟩ := hsub A hA
have hwj := (admissible_iff C (H j) (w j)).mp (hw j)
calc
1 ≤ ∑ i ∈ indices C B, w j i := hwj.2 B hB
_ ≤ ∑ i ∈ indices C A, w j i := Finset.sum_le_sum_of_subset_of_nonneg
(indices_mono C hBA) (fun i _ _ => hwp j i)
_ ≤ ∑ i ∈ indices C A, W i := by
apply Finset.sum_le_sum
intro i _
rw [hpoint]
exact Finset.single_le_sum (fun k _ => hwp k i) (Finset.mem_univ j)
calc
modulus C F ≤ ‖W‖^2 := modulus_le_energy C F W hW
_ ≤ (∑ j, ‖w j‖)^2 := by
apply pow_le_pow_left₀ (norm_nonneg W)
exact norm_sum_le _ _
_ ≤ (Fintype.card J : ℝ)*∑ j, ‖w j‖^2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun j => ‖w j‖))
_ = _ := by simp only [hwe]
theorem exists_modulus_localization [Nonempty J] (C : ι → Set X)
(hC : ∀ x, ∃ i, x ∈ C i) (F : Set (Set X)) (H : J → Set (Set X))
(hH : ∀ j, (H j).Nonempty) (hHn : ∀ j A, A ∈ H j → A.Nonempty)
(hsub : ∀ A ∈ F, ∃ j, ∃ B ∈ H j, B ⊆ A) :
∃ j, modulus C F ≤ (Fintype.card J : ℝ)^2 * modulus C (H j) := by
obtain ⟨j,_,hj⟩ := Finset.univ.exists_max_image (fun j => modulus C (H j)) Finset.univ_nonempty
refine ⟨j,(modulus_le_card_mul_sum C hC F H hH hHn hsub).trans ?_⟩
calc
(Fintype.card J : ℝ)*∑ k, modulus C (H k) ≤
(Fintype.card J : ℝ)*∑ _k : J, modulus C (H j) := by
apply mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun k hk => hj k hk))
exact Nat.cast_nonneg _
_ = _ := by simp [pow_two,mul_assoc]
theorem fixed_localization_subsequence (M : ℕ → ℝ) (m : J → ℕ → ℝ) (C : ℝ)
(h : ∀ n, ∃ j, M n ≤ C*m j n) :
∃ j, ∃ N : ℕ → ℕ, StrictMono N ∧ ∀ n, M (N n) ≤ C*m j (N n) := by
have hf : ∃ᶠ n in atTop, ∃ j, M n ≤ C*m j n :=
(Eventually.of_forall h).frequently
obtain ⟨j,hj⟩ := frequently_exists.mp hf
obtain ⟨N,hN,hNj⟩ := extraction_of_frequently_atTop hj
exact ⟨j,N,hN,hNj⟩
end CannonCells
end
end
end
end
end
end
end
end
end
end OAI
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