| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| lemma normalized_nearest_separated {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {x y : E} (hx : x≠0) (hy : y≠0) (hxy : ‖x‖≤‖x-y‖) (hyx : ‖y‖≤‖x-y‖) : |
| 1≤‖‖x‖⁻¹ • x-‖y‖⁻¹ • y‖ := by |
| have aux {x y : E} (hx : x≠0) (hy : y≠0) (h : ‖y‖≤‖x‖) (hd : ‖x‖≤‖x-y‖) : |
| 1≤‖‖x‖⁻¹ • x-‖y‖⁻¹ • y‖ := by |
| have hx0 : 0<‖x‖ := norm_pos_iff.mpr hx |
| have hy0 : 0<‖y‖ := norm_pos_iff.mpr hy |
| have he : x-y=‖y‖ • (‖x‖⁻¹ • x-‖y‖⁻¹ • y)+(‖x‖-‖y‖) • (‖x‖⁻¹ • x) := by |
| rw [smul_sub,sub_smul,smul_smul,smul_smul,smul_smul, |
| mul_inv_cancel₀ hy0.ne',mul_inv_cancel₀ hx0.ne',one_smul,one_smul] |
| abel |
| have ht := norm_add_le (‖y‖ • (‖x‖⁻¹ • x-‖y‖⁻¹ • y)) ((‖x‖-‖y‖) • (‖x‖⁻¹ • x)) |
| rw [←he,norm_smul,norm_smul,Real.norm_of_nonneg hy0.le, |
| Real.norm_of_nonneg (sub_nonneg.mpr h)] at ht |
| have hn : ‖‖x‖⁻¹ • x‖=1 := by |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hx0.le),inv_mul_cancel₀ hx0.ne'] |
| rw [hn,mul_one] at ht |
| nlinarith only [ht,hd,hy0] |
| rcases le_total ‖y‖ ‖x‖ with h | h |
| · exact aux hx hy h hxy |
| · have hd : ‖y‖≤‖y-x‖ := by simpa only [norm_sub_rev] using hyx |
| simpa only [norm_sub_rev] using aux hy hx h hd |
| |
| lemma nearest_incoming_card {ι E : Type*} [DecidableEq ι] [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [FiniteDimensional ℝ E] (I : Finset ι) (v : ι→E) |
| (h0 : ∀ i∈I, v i≠0) |
| (hne : ∀ i∈I, ∀ j∈I, i≠j → ‖v i‖≤‖v i-v j‖) : |
| I.card≤5^Module.finrank ℝ E := by |
| classical |
| let d : ι→E := fun i ↦ ‖v i‖⁻¹ • v i |
| have hsep : ∀ i∈I, ∀ j∈I, i≠j → 1≤‖d i-d j‖ := by |
| intro i hi j hj hij |
| exact normalized_nearest_separated (h0 i hi) (h0 j hj) (hne i hi j hj hij) |
| (by simpa only [norm_sub_rev] using hne j hj i hi hij.symm) |
| have hinj : Set.InjOn d I := by |
| intro i hi j hj hd |
| by_contra hn |
| have hh := hsep i hi j hj hn |
| rw [hd,sub_self,norm_zero] at hh |
| norm_num at hh |
| rw [←Finset.card_image_of_injOn hinj] |
| apply Besicovitch.card_le_of_separated |
| · intro x hx |
| obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hx |
| dsimp [d] |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr (h0 i hi))] |
| norm_num |
| · intro x hx y hy hxy |
| obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hx |
| obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hy |
| exact hsep i hi j hj (fun hij ↦ hxy (congrArg d hij)) |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
