openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /FiniteCoverSums.lean
| 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 | |