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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 | import Mathlib
import OAI.Geometry.Cannon.Modulus.ModulusEnergy
namespace OAI
section
section
section
section
noncomputable section
open scoped InnerProductSpace
open Set
namespace CannonCells
open Set CannonDuality
variable {X ι J : Type*} [Fintype ι] [Fintype J]
theorem modulus_serial_bound [Nonempty J]
(C : ι → Set X) (F : Set (Set X)) (H : J → Set (Set X))
(w : J → Weights ι) (hw : ∀ j, Admissible (familyVectors C (H j) : Set (Weights ι)) (w j))
(horth : ∀ i j k, j ≠ k → w j i*w k i=0)
(hcross : ∀ A ∈ F, ∀ j, ∃ B ∈ H j, B ⊆ A) :
modulus C F ≤ (∑ j, ‖w j‖^2)/(Fintype.card J : ℝ)^2 := by
classical
have hcard : 0 < (Fintype.card J : ℝ) := by exact_mod_cast Fintype.card_pos
let W : Weights ι := (Fintype.card J : ℝ)⁻¹ • ∑ j, w j
have hw' (j : J) := (admissible_iff C (H j) (w j)).mp (hw j)
have hpoint (i : ι) : W i=(Fintype.card J : ℝ)⁻¹ * ∑ j, w j i := by simp [W]
have hW : Admissible (familyVectors C F : Set (Weights ι)) W := by
rw [admissible_iff]
refine ⟨?_,?_⟩
· intro i
rw [hpoint]
exact mul_nonneg (inv_nonneg.mpr hcard.le) (Finset.sum_nonneg (fun j _ => (hw' j).1 i))
· intro A hA
have hsum (j : J) : 1 ≤ ∑ i ∈ indices C A, w j i := by
obtain ⟨B,hB,hBA⟩ := hcross A hA j
exact ((hw' j).2 B hB).trans (Finset.sum_le_sum_of_subset_of_nonneg
(indices_mono C hBA) (fun i _ _ => (hw' j).1 i))
have hh := Finset.sum_le_sum (s:=Finset.univ) (fun j _ => hsum j)
simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hh
calc
1 = (Fintype.card J : ℝ)⁻¹ * Fintype.card J := (inv_mul_cancel₀ (ne_of_gt hcard)).symm
_ ≤ (Fintype.card J : ℝ)⁻¹ * ∑ j, ∑ i ∈ indices C A, w j i :=
mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr hcard.le)
_ = ∑ i ∈ indices C A, W i := by
simp only [hpoint,Finset.mul_sum]
rw [Finset.sum_comm]
have hsquare (i : ι) : (∑ j, w j i)^2=∑ j, (w j i)^2 := by
rw [pow_two,Finset.sum_mul_sum]
apply Finset.sum_congr rfl
intro j hj
rw [Finset.sum_eq_single j]
· rw [pow_two]
· intro k hk hkj
exact horth i j k hkj.symm
· simp
have henergy : ‖W‖^2=(∑ j, ‖w j‖^2)/(Fintype.card J : ℝ)^2 := by
rw [EuclideanSpace.real_norm_sq_eq]
simp only [hpoint,mul_pow,hsquare]
rw [← Finset.mul_sum]
simp only [EuclideanSpace.real_norm_sq_eq]
rw [Finset.sum_comm]
simp only [div_eq_mul_inv,inv_pow,mul_comm]
exact (modulus_le_energy C F W hW).trans henergy.le
theorem modulus_serial_uniform [Nonempty J]
(C : ι → Set X) (F : Set (Set X)) (H : J → Set (Set X))
(w : J → Weights ι) (hw : ∀ j, Admissible (familyVectors C (H j) : Set (Weights ι)) (w j))
(horth : ∀ i j k, j ≠ k → w j i*w k i=0)
(hcross : ∀ A ∈ F, ∀ j, ∃ B ∈ H j, B ⊆ A)
{B : ℝ} (henergy : ∀ j, ‖w j‖^2 ≤ B) :
modulus C F ≤ B/(Fintype.card J : ℝ) := by
have hcard : 0 < (Fintype.card J : ℝ) := by exact_mod_cast Fintype.card_pos
calc
_ ≤ (∑ j, ‖w j‖^2)/(Fintype.card J : ℝ)^2 :=
modulus_serial_bound C F H w hw horth hcross
_ ≤ ((Fintype.card J : ℝ)*B)/(Fintype.card J : ℝ)^2 := by
apply div_le_div_of_nonneg_right _ (sq_nonneg _)
simpa only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] using
Finset.sum_le_sum (s:=Finset.univ) (fun j _ => henergy j)
_ = _ := by field_simp
end CannonCells
end
end
section
noncomputable section
namespace CannonCapacity
open Set Metric Finset
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
end
end
end
end
end OAI
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