openAI-math-dataset / lean /OAI /Analysis /HardSphere /FormDomain /CoordDerivativeParticleLocality.lean
| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.MollifiedCloseCountMultiplier | |
| 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 | |
| open scoped Topology | |
| attribute [local irreducible] torusDistance | |
| lemma coordDerivative_zero_of_particle_locality {n : ℕ} (L : Length) (f : Configuration n L→ℂ) | |
| (hf : Differentiable ℝ (fun x : LiftedConfiguration n ↦ f (project n L x))) | |
| (v : Torus L) (r : ℝ) | |
| (hloc : ∀ (X Y : Configuration n L) (k : Fin n), (∀ i, i≠k → X i=Y i) → r≤torusDistance L (X k) v → | |
| r≤torusDistance L (Y k) v → f X=f Y) | |
| (p : Direction n) (X : Configuration n L) (hfar : r<torusDistance L (X p.1) v) : | |
| coordDerivative n L f p X=0 := by | |
| let w : LiftedConfiguration n := Pi.single p.1 (Pi.single p.2 1) | |
| let Y : ℝ → Configuration n L := fun t ↦ project n L (representative n L X+t • w) | |
| have hY0 : Y 0=X := by simp only [Y,zero_smul,add_zero,project_representative] | |
| have hY : Continuous Y := by | |
| apply (continuous_project n L).comp | |
| exact continuous_const.add (continuous_id.smul continuous_const) | |
| have hdcont : Continuous (fun t : ℝ ↦ torusDistance L (Y t p.1) v) := | |
| (torusDistance_continuous_pair L).comp (((continuous_apply p.1).comp hY).prodMk continuous_const) | |
| have hevent : ∀ᶠ t in 𝓝 (0:ℝ), r<torusDistance L (Y t p.1) v := | |
| hdcont.continuousAt.eventually_const_lt (by simpa only [hY0] using hfar) | |
| have hrest (t : ℝ) (i : Fin n) (hi : i≠p.1) : X i=Y t i := by | |
| have he : Y t i=project n L (representative n L X) i := by | |
| ext j | |
| simp only [Y,project,Pi.add_apply,Pi.smul_apply,w,Pi.single_eq_of_ne hi,Pi.zero_apply, | |
| smul_zero,add_zero] | |
| rw [he,project_representative] | |
| have heq : (fun t : ℝ ↦ f (Y t))=ᶠ[𝓝 (0:ℝ)] (fun _ ↦ f X) := by | |
| filter_upwards [hevent] with t ht | |
| exact (hloc X (Y t) p.1 (hrest t) hfar.le ht.le).symm | |
| have hz : HasDerivAt (fun t : ℝ ↦ f (Y t)) 0 0 := | |
| (hasDerivAt_const 0 (f X)).congr_of_eventuallyEq heq | |
| have hline := (hasFDerivAt_const (𝕜:=ℝ) (representative n L X) (0:ℝ)).add | |
| ((hasFDerivAt_id (𝕜:=ℝ) (0:ℝ)).smul_const w) | |
| have hf0 : HasFDerivAt (fun x : LiftedConfiguration n ↦ f (project n L x)) | |
| (fderiv ℝ (fun x : LiftedConfiguration n ↦ f (project n L x)) (representative n L X)) | |
| (representative n L X+(0:ℝ) • w) := by | |
| simpa only [zero_smul,add_zero] using (hf (representative n L X)).hasFDerivAt | |
| have hd := (hf0.comp 0 hline).hasDerivAt | |
| have he := hd.unique hz | |
| simpa only [coordDerivative,w,ContinuousLinearMap.comp_apply,add_apply,zero_apply, | |
| ContinuousLinearMap.smulRight_apply,ContinuousLinearMap.id_apply,one_smul,zero_add] using he | |
| variable {n : ℕ} [Nontrivial (Fin n)] | |
| lemma mollifiedCloseCountMultiplier_partial_zero (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1) | |
| (v : Torus L) (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (hφ : φ.rOut≤1) | |
| (p : Direction n) (X : Configuration n L) (hfar : 8<torusDistance L (X p.1) v) : | |
| coordDerivative n L (mollifiedCloseCountMultiplier L s v φ).toFun p X=0 := by | |
| apply coordDerivative_zero_of_particle_locality L _ | |
| ((mollifiedCloseCountMultiplier L s v φ).smooth.differentiable (by simp)) v 8 _ p X hfar | |
| intro X Y k hr hx hy | |
| simp only [mollifiedCloseCountMultiplier_apply, | |
| mollifiedCloseCount_away_eq L hs hs1 v φ hφ X Y k hr hx hy] | |
| lemma mollifiedCloseCutoff_partial_zero (L : Length) {s : ℝ} (hs : 0<s) (hs1 : 2*s≤1) | |
| (v : Torus L) (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (hφ : φ.rOut≤1) | |
| (h : ℝ) (p : Direction n) (X : Configuration n L) (hfar : 8<torusDistance L (X p.1) v) : | |
| coordDerivative n L (mollifiedCloseCutoff L s v φ h).toFun p X=0 := by | |
| rw [mollifiedCloseCutoff_partial,mollifiedCloseCountMultiplier_partial_zero L hs hs1 v φ hφ p X hfar,mul_zero] | |
| end HardSphere | |
| end | |
| end OAI | |