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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.CoordinateLine
import OAI.Analysis.HardSphere.Geometry.NormSqIntegral
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
namespace HardSphere
open MeasureTheory Filter Set
lemma smooth_insertNth {n : ℕ} (j : Fin (n+1)) (x : Fin 3 → ℝ) :
ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞)
(fun Y : LiftedConfiguration n ↦ Fin.insertNth (α := fun _ ↦ Fin 3 → ℝ) j x Y) := by
apply contDiff_pi.mpr
intro i
induction i using j.succAboveCases with
| x => simpa only [Fin.insertNth_apply_same] using
(contDiff_const : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun _ : LiftedConfiguration n ↦ x))
| p i => simpa only [Fin.insertNth_apply_succAbove] using
(contDiff_apply ℝ (Fin 3 → ℝ) i)
lemma smooth_slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a)
(j : Fin (n+1)) (x : Torus L) :
ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞)
(fun Y : LiftedConfiguration n ↦ f.toFun (j.insertNth x (project n L Y))) := by
have h := f.smooth.comp (smooth_insertNth j (representative 1 L (fun _ ↦ x) 0))
convert h using 1
funext Y
congr 1
ext i q
induction i using j.succAboveCases with
| x =>
simp only [project,Fin.insertNth_apply_same]
exact (congr_fun (congr_fun (project_representative 1 L (fun _ ↦ x)) 0) q).symm
| p i => simp only [project,Fin.insertNth_apply_succAbove]
lemma insertNth_coordinateLine {n : ℕ} {L : Length} (j : Fin (n+1)) (x : Torus L)
(Y : Configuration n L) (p : Direction n) (t : AddCircle L.val) :
j.insertNth x (coordinateLine Y p t)=coordinateLine (j.insertNth x Y) (j.succAbove p.1,p.2) t := by
ext i q
induction i using j.succAboveCases with
| x => simp [coordinateLine]
| p i => simp [coordinateLine,Pi.single_apply,Fin.succAbove_inj]
lemma coordDerivative_slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a)
(j : Fin (n+1)) (x : Torus L) (p : Direction n) :
coordDerivative n L (fun Y ↦ f.toFun (j.insertNth x Y)) p=
fun Y ↦ coordDerivative (n+1) L f.toFun (j.succAbove p.1,p.2) (j.insertNth x Y) := by
funext Y
have h1 := hasDerivAt_coordinateLine (f := fun Y ↦ f.toFun (j.insertNth x Y)) ((smooth_slice f j x).differentiable (by simp)) Y p 0
have h2 := hasDerivAt_coordinateLine (f.smooth.differentiable (by simp))
(j.insertNth x Y) (j.succAbove p.1,p.2) 0
simp only [insertNth_coordinateLine] at h1
have h := h1.unique h2
simpa [coordinateLine] using h
def SmoothTest.slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a)
(j : Fin (n+1)) (x : Torus L) : SmoothTest n L a where
toFun Y := f.toFun (j.insertNth x Y)
smooth := smooth_slice f j x
compact := HasCompactSupport.of_compactSpace _
support_allowed := by
intro Y hY i k hik
have h := f.support_allowed (tsupport_comp_subset_preimage f.toFun
(Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id) hY)
simpa only [Fin.insertNth_apply_succAbove,id_eq] using h (j.succAbove i) (j.succAbove k)
(Fin.strictMono_succAbove j hik)
memLp := (f.continuous.comp (Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id)).memLp_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
partial_memLp := fun p ↦ by
rw [coordDerivative_slice]
exact ((f.continuous_partial (j.succAbove p.1,p.2)).comp
(Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id)).memLp_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
lemma smoothTest_norm_sq {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest n L a) :
‖f.jet.1‖^2=∫ X : Configuration n L, ‖f.toFun X‖^2 := by
rw [IndicatorStrong.norm_sq_integral]
exact integral_congr_ae ((MemLp.coeFn_toLp f.memLp).fun_comp (fun z : ℂ ↦ ‖z‖^2))
lemma smoothTest_partial_norm_sq {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest n L a)
(p : Direction n) :
‖f.jet.2 p‖^2=∫ X : Configuration n L, ‖coordDerivative n L f.toFun p X‖^2 := by
rw [IndicatorStrong.norm_sq_integral]
exact integral_congr_ae ((MemLp.coeFn_toLp (f.partial_memLp p)).fun_comp (fun z : ℂ ↦ ‖z‖^2))
end HardSphere
end
end OAI
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