openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Geometry /CommonProjectionConfiguration.lean
| import OAI.MeasureTheory.Falconer.Geometry.SeparatedProjectionBases | |
| import OAI.MeasureTheory.Falconer.Incidence.FiniteBallBounds | |
| import OAI.MeasureTheory.Falconer.Geometry.ProjectionCells | |
| namespace OAI | |
| /-! The two bases and every graph slice are selected from the original | |
| common projection family, retaining explicit relative mass and regularity. -/ | |
| namespace Falconer | |
| attribute [local instance] Classical.propDecidable | |
| theorem common_projection_configuration {δ t K ρ r : ℝ} | |
| (hδ : 0 ≤ δ) (hK : 0 ≤ K) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hr : 0 < r) | |
| (I : Finset ℝ) (U : Finset (Space 2)) (A : ℝ → Finset (Space 2)) | |
| (hI : I.Nonempty) (hU : U.Nonempty) | |
| (hsub : ∀ i ∈ I, A i ⊆ U) | |
| (hsize : ∀ i ∈ I, ρ*(U.card:ℝ) ≤ ((A i).card:ℝ)) | |
| (hball : ∀ z : ℝ, scalarBallMass I z r ≤ (ρ^6/128)*(I.card:ℝ)) | |
| (hUreg : NormalizedFiniteBallBound δ t K U) | |
| (P : ℕ) (hproj : ∀ i ∈ I, (projectionCells δ i (A i)).card ≤ P) : | |
| ∃ a ∈ I, ∃ b ∈ I, r < |b-a| ∧ b ≠ a ∧ ∃ J ⊆ I, | |
| J.Nonempty ∧ (ρ^6/64)*(I.card:ℝ) ≤ (J.card:ℝ) ∧ | |
| let V := A a ∩ A b | |
| V.Nonempty ∧ V ⊆ U ∧ (U.card:ℝ) ≤ (8/ρ^4)*(V.card:ℝ) ∧ | |
| NormalizedFiniteBallBound δ t (K*(8/ρ^4)) V ∧ | |
| (projectionCells δ a V).card ≤ P ∧ (projectionCells δ b V).card ≤ P ∧ | |
| ∀ j ∈ J, r < |j-a| ∧ r < |j-b| ∧ | |
| (ρ^4/8)*(V.card:ℝ) ≤ ((V ∩ A j).card:ℝ) ∧ | |
| (projectionCells δ j (V ∩ A j)).card ≤ P := by | |
| classical | |
| obtain ⟨a,ha,b,hb,hab,hba,J,hJI,hJsize,htriple⟩ := | |
| separated_projection_bases I U A hI hU hρ hρ1 hr hsub hsize hball | |
| have hJ : J.Nonempty := by | |
| apply Finset.card_pos.mp | |
| have hp : 0 < (ρ^6/64)*(I.card:ℝ) := | |
| mul_pos (by positivity) (Nat.cast_pos.mpr hI.card_pos) | |
| exact_mod_cast hp.trans_le hJsize | |
| let V := A a ∩ A b | |
| have hVU : V ⊆ U := Finset.inter_subset_left.trans (hsub a ha) | |
| obtain ⟨j,hj⟩ := hJ | |
| have hVlower : (ρ^4/8)*(U.card:ℝ) ≤ (V.card:ℝ) := | |
| (htriple j hj).2.2.trans (Nat.cast_le.mpr (Finset.card_le_card Finset.inter_subset_left)) | |
| have hV : V.Nonempty := by | |
| apply Finset.card_pos.mp | |
| have hp : 0 < (ρ^4/8)*(U.card:ℝ) := | |
| mul_pos (by positivity) (Nat.cast_pos.mpr hU.card_pos) | |
| exact_mod_cast hp.trans_le hVlower | |
| have hVUsize : (U.card:ℝ) ≤ (8/ρ^4)*(V.card:ℝ) := by | |
| have hpos : 0 < ρ^4 := by positivity | |
| have hh := mul_le_mul_of_nonneg_left hVlower (show 0 ≤ 8/ρ^4 by positivity) | |
| have he : (8/ρ^4)*((ρ^4/8)*(U.card:ℝ)) = (U.card:ℝ) := by field_simp | |
| rwa [he] at hh | |
| refine ⟨a,ha,b,hb,hab,hba,J,hJI,⟨j,hj⟩,hJsize,hV,hVU,hVUsize, | |
| normalized_finite_ball_bound_subset hδ hK hVU hVUsize hUreg,?_,?_,?_⟩ | |
| · exact (Finset.card_le_card (projectionCells_mono δ a Finset.inter_subset_left)).trans (hproj a ha) | |
| · exact (Finset.card_le_card (projectionCells_mono δ b Finset.inter_subset_right)).trans (hproj b hb) | |
| · intro j hj | |
| obtain ⟨hja,hjb,hcard⟩ := htriple j hj | |
| refine ⟨hja,hjb,?_,?_⟩ | |
| · exact (mul_le_mul_of_nonneg_left (Nat.cast_le.mpr (Finset.card_le_card hVU)) | |
| (show 0 ≤ ρ^4/8 by positivity)).trans hcard | |
| · exact (Finset.card_le_card (projectionCells_mono δ j Finset.inter_subset_right)).trans (hproj j (hJI hj)) | |
| end Falconer | |
| end OAI | |