openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Estimates /RetainedCardLeft.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.TestedCellFamily | |
| namespace OAI | |
| /-! Quantitative estimates for the distance-set construction. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| namespace CurrentFalconer.TreeND | |
| open Set Metric MeasureTheory PlanarFalconer | |
| open scoped ENNReal RealInnerProductSpace | |
| variable {d : ℕ} | |
| lemma retained_card_left (S T : Finset (CellPath d)) {p : ℕ} | |
| (hT : T ⊆ dyadicPaths p) | |
| (v w x y : EuclideanSpace ℝ (Fin d)) {τ : ℝ} | |
| (hw : ‖w‖=1) (hτ : 0 ≤ τ) : | |
| ((retainedPairs S T v w x y τ).card : ℝ) ≤ | |
| ((S.filter fun P => ‖graphPerpND v (treeCenter P-x)‖ ≤ τ).card : ℝ) * | |
| (4*(2:ℝ)^p*τ+5)^d := by | |
| classical | |
| let E := retainedPairs S T v w x y τ | |
| let F := S.filter fun P => ‖graphPerpND v (treeCenter P-x)‖ ≤ τ | |
| have hEF : ∀ z ∈ E, z.1 ∈ F := by | |
| intro z hz | |
| obtain ⟨hzST,h₁,h₂,h₃⟩ := Finset.mem_filter.mp hz | |
| exact Finset.mem_filter.mpr ⟨(Finset.mem_product.mp hzST).1,h₁⟩ | |
| have hfiber : ∀ P ∈ F, (((E.filter fun z => z.1=P).card) : ℝ) ≤ | |
| (4*(2:ℝ)^p*τ+5)^d := by | |
| intro P _hP | |
| let E₀ := E.filter fun z => z.1=P | |
| let T₀ := E₀.image Prod.snd | |
| have hinj : Set.InjOn Prod.snd (↑E₀ : Set (CellPath d × CellPath d)) := by | |
| intro z hz z' hz' he | |
| apply Prod.ext | |
| · exact (Finset.mem_filter.mp hz).2.trans (Finset.mem_filter.mp hz').2.symm | |
| · exact he | |
| have hdepth : T₀ ⊆ dyadicPaths p := by | |
| intro Q hQ | |
| obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hQ | |
| exact hT (Finset.mem_product.mp (Finset.mem_filter.mp | |
| (Finset.mem_filter.mp hz).1).1).2 | |
| have hball : ∀ Q ∈ T₀, dist (treeCenter Q) | |
| (y-(inner ℝ (treeCenter P-x) v) • w) ≤ 2*τ := by | |
| intro Q hQ | |
| obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hQ | |
| obtain ⟨hzE,hzP⟩ := Finset.mem_filter.mp hz | |
| obtain ⟨_,_,hperp,hscalar⟩ := Finset.mem_filter.mp hzE | |
| rw [hzP] at hscalar | |
| exact graph_scalar_perp_ballND w y (treeCenter z.2) hw hperp hscalar | |
| calc | |
| _ = (T₀.card : ℝ) := by rw [Finset.card_image_of_injOn hinj] | |
| _ ≤ (2*(2:ℝ)^p*(2*τ)+5)^d := | |
| treeCenter_ball_card T₀ hdepth _ (by positivity) hball | |
| _ = _ := by ring | |
| have he : (E.card : ℝ) = ∑ P ∈ F, (((E.filter fun z => z.1=P).card) : ℝ) := by | |
| exact_mod_cast Finset.card_eq_sum_card_fiberwise hEF | |
| calc | |
| _ = ∑ P ∈ F, (((E.filter fun z => z.1=P).card) : ℝ) := he | |
| _ ≤ ∑ _P ∈ F, (4*(2:ℝ)^p*τ+5)^d := Finset.sum_le_sum hfiber | |
| _ = _ := by simp [F] | |
| lemma retained_card_swap (S T : Finset (CellPath d)) | |
| (v w x y : EuclideanSpace ℝ (Fin d)) (τ : ℝ) : | |
| (retainedPairs S T v w x y τ).card = (retainedPairs T S w v y x τ).card := by | |
| classical | |
| have he : retainedPairs S T v w x y τ = | |
| (retainedPairs T S w v y x τ).image Prod.swap := by | |
| ext z | |
| constructor | |
| · intro hz | |
| rcases z with ⟨P,Q⟩ | |
| obtain ⟨hST,hv,hw,hs⟩ := Finset.mem_filter.mp hz | |
| obtain ⟨hP,hQ⟩ := Finset.mem_product.mp hST | |
| refine Finset.mem_image.mpr ⟨(Q,P), ?_, rfl⟩ | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_product.mpr ⟨hQ,hP⟩,hw,hv,?_⟩ | |
| exact (congrArg (fun t : ℝ => |t|) (_root_.add_comm _ _)).trans_le hs | |
| · intro hz | |
| obtain ⟨⟨Q,P⟩,hmem,rfl⟩ := Finset.mem_image.mp hz | |
| obtain ⟨hTS,hw,hv,hs⟩ := Finset.mem_filter.mp hmem | |
| obtain ⟨hQ,hP⟩ := Finset.mem_product.mp hTS | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_product.mpr ⟨hP,hQ⟩,hv,hw,?_⟩ | |
| exact (congrArg (fun t : ℝ => |t|) (_root_.add_comm _ _)).trans_le hs | |
| rw [he,Finset.card_image_of_injective] | |
| exact Prod.swap_injective | |
| theorem retained_degree : RetainedDegreeTarget := by | |
| intro d ρ₁ ρ₂ a g₁ g₂ p X Y S T v w x y δ τ W₁ W₂ K₁ K₂ | |
| hag₁ hag₂ hg₁p hg₂p hv hw hδ hτ hK₁ hK₂ hW₁ hW₂ hS hT | |
| have hpos : 0 ≤ (4*(2:ℝ)^p*τ+5)^d := by positivity | |
| have h₁ := (retained_card_left S T hT.1 v w x y hw hτ).trans | |
| (mul_le_mul_of_nonneg_right | |
| (graph_candidates_card ρ₁ hag₁ hg₁p X S v x hv hδ hK₁ hW₁ hS) hpos) | |
| have h₂ := (retained_card_left T S hS.1 w v y x hv hτ).trans | |
| (mul_le_mul_of_nonneg_right | |
| (graph_candidates_card ρ₂ hag₂ hg₂p Y T w y hw hδ hK₂ hW₂ hT) hpos) | |
| rw [← retained_card_swap S T v w x y τ] at h₂ | |
| rw [min_mul_of_nonneg _ _ hpos] | |
| exact le_min h₁ h₂ | |
| end CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |