openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Harmonic /HarmonicFunctionSpace.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace CriticalPercolation.HarmonicAnalysis | |
| variable {X : Type*} | |
| def vanishingSubspace (D : Submodule ℝ (X → ℝ)) (F : Set X) : Submodule ℝ D where | |
| carrier := {u | ∀ x∈F, (u:X → ℝ) x=0} | |
| zero_mem' := by intro x hx; rfl | |
| add_mem' := by intro u v hu hv x hx; change u.val x+v.val x=0; rw [hu x hx,hv x hx,add_zero] | |
| smul_mem' := by intro a u hu x hx; change a*u.val x=0; rw [hu x hx,mul_zero] | |
| lemma vanishingSubspace_antitone (D : Submodule ℝ (X → ℝ)) : | |
| Antitone (vanishingSubspace D) := by | |
| intro s t h u hu x hx | |
| exact hu x (h hx) | |
| theorem exists_detecting_ball (F : ℕ → Set X) (hmono : Monotone F) | |
| (hex : ∀ x, ∃ n, x∈F n) (D : Submodule ℝ (X → ℝ)) | |
| [FiniteDimensional ℝ D] : | |
| ∃ n, ∀ u : D, (∀ x∈F n, (u:X → ℝ) x=0) → u=0 := by | |
| let K : ℕ →o (Submodule ℝ D)ᵒᵈ := | |
| ⟨fun n => vanishingSubspace D (F n), | |
| fun m n h => vanishingSubspace_antitone D (hmono h)⟩ | |
| obtain ⟨n,hn⟩ := IsArtinian.monotone_stabilizes K | |
| refine ⟨n,?_⟩ | |
| intro u hu | |
| apply Subtype.ext | |
| funext x | |
| obtain ⟨m,hm⟩ := hex x | |
| have he : vanishingSubspace D (F n)=vanishingSubspace D (F (max n m)) := | |
| hn (max n m) (le_max_left _ _) | |
| have hum : u∈vanishingSubspace D (F (max n m)) := he ▸ hu | |
| exact hum x (hmono (le_max_right _ _) hm) | |
| def squareMass (F : Finset X) (u : X → ℝ) : ℝ := ∑ x∈F, (u x)^2 | |
| lemma squareMass_nonneg (F : Finset X) (u : X → ℝ) : 0≤ squareMass F u := | |
| Finset.sum_nonneg (fun vertex _ => sq_nonneg (u vertex)) | |
| lemma squareMass_eq_zero_iff (F : Finset X) (u : X → ℝ) : | |
| squareMass F u=0 ↔ ∀ x∈F, u x=0 := by | |
| rw [squareMass,Finset.sum_eq_zero_iff_of_nonneg (fun x _ => sq_nonneg (u x))] | |
| simp only [sq_eq_zero_iff] | |
| lemma squareMass_mono {F H : Finset X} (h : F⊆H) (u : X → ℝ) : | |
| squareMass F u≤ squareMass H u := | |
| Finset.sum_le_sum_of_subset_of_nonneg h (by intros; positivity) | |
| lemma squareMass_pos (D : Submodule ℝ (X → ℝ)) (F : Finset X) | |
| (hF : ∀ u : D, (∀ x∈F, (u:X → ℝ) x=0) → u=0) (u : D) (hu : u≠0) : | |
| 0<squareMass F u := by | |
| apply lt_of_le_of_ne (squareMass_nonneg _ _) | |
| intro hz | |
| exact hu (hF u ((squareMass_eq_zero_iff _ _).mp hz.symm)) | |
| end CriticalPercolation.HarmonicAnalysis | |
| end | |
| end | |
| end OAI | |