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f205175 | 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 | import Mathlib
namespace OAI
section
noncomputable section
open scoped Classical BigOperators
namespace CriticalPercolation.HarmonicAnalysis
variable {I X : Type*}
lemma finite_incidence_sum (J : Finset I) (L : I → Finset X) (B : Finset X)
(hsub : ∀ j∈J, L j⊆B) (f : X → ℝ) :
(∑ j∈J, ∑ x∈L j, f x) =
∑ x∈B, ((J.filter (fun j => x∈L j)).card:ℝ)*f x := by
have he (j : I) (hj : j∈J) : (∑ x∈L j, f x)=∑ x∈B, if x∈L j then f x else 0 := by
rw [←Finset.sum_filter]
congr 1
ext x
simp only [Finset.mem_filter]
exact ⟨fun hx => ⟨hsub j hj hx,hx⟩,fun hx => hx.2⟩
calc
_ = ∑ j∈J, ∑ x∈B, if x∈L j then f x else 0 := Finset.sum_congr rfl he
_ = ∑ x∈B, ∑ j∈J, if x∈L j then f x else 0 := Finset.sum_comm
_ = _ := by
apply Finset.sum_congr rfl
intro x hx
rw [←Finset.sum_filter]
simp
lemma sum_le_sum_of_finite_cover (J : Finset I) (L : I → Finset X)
(F : Finset X) (hcover : ∀ x∈F, ∃ j∈J, x∈L j)
(f : X → ℝ) (hf : ∀ x, 0≤f x) :
(∑ x∈F, f x)≤∑ j∈J, ∑ x∈L j, f x := by
let B := F∪J.biUnion L
have hJB : ∀ j∈J, L j⊆B := by
intro j hj x hx
exact Finset.mem_union_right _ (Finset.mem_biUnion.mpr ⟨j,hj,hx⟩)
rw [finite_incidence_sum J L B hJB f]
calc
_ ≤ ∑ x∈F, ((J.filter (fun j => x∈L j)).card:ℝ)*f x := by
apply Finset.sum_le_sum
intro x hx
obtain ⟨j,hj,hxj⟩ := hcover x hx
have hp : 1≤(J.filter (fun j => x∈L j)).card :=
Finset.card_pos.mpr ⟨j,Finset.mem_filter.mpr ⟨hj,hxj⟩⟩
have hp' : (1:ℝ)≤(J.filter (fun j => x∈L j)).card := by exact_mod_cast hp
simpa only [one_mul] using mul_le_mul_of_nonneg_right hp' (hf x)
_ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_union_left) (by
intro x hx hxn
exact mul_nonneg (Nat.cast_nonneg _) (hf x))
lemma sum_le_sum_of_finite_overlap (J : Finset I) (L : I → Finset X)
(B : Finset X) (hsub : ∀ j∈J, L j⊆B) (K : ℕ)
(hover : ∀ x∈B, (J.filter (fun j => x∈L j)).card≤K)
(f : X → ℝ) (hf : ∀ x, 0≤f x) :
(∑ j∈J, ∑ x∈L j, f x)≤(K:ℝ)*(∑ x∈B, f x) := by
rw [finite_incidence_sum J L B hsub f,Finset.mul_sum]
apply Finset.sum_le_sum
intro x hx
exact mul_le_mul_of_nonneg_right (Nat.cast_le.mpr (hover x hx)) (hf x)
end CriticalPercolation.HarmonicAnalysis
end
end
end OAI
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