| import Mathlib |
| import OAI.Geometry.BallPacking.SurfaceArea.BoundedPlanarResidue |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped ContDiff Manifold Topology BigOperators |
| open Set Function Filter Manifold |
| open DiagonalQuadrics DiagonalQuadrics.Explicit Hamiltonian FiniteMoment |
| open MeasureTheory |
|
|
| theorem radialEndProbability_contDiffAt {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| (D m : TrapezoidWeight A B → ℕ) {p : Radial.Plane} |
| (hp : p∈Radial.lowerTrapezoid A B) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) (i : TrapezoidWeight A B) : |
| ContDiffAt ℝ ∞ (fun z => radialEndProbability Radial.latticeIndex D m p ε z i) y := |
| ((radialProbability_smooth hA hAB D m hp i).contMDiffAt.comp y |
| ((infinityDiffeomorph (parameters 0) ε).contMDiffOn.contMDiffAt |
| ((infinityDiffeomorph (parameters 0) ε).open_source.mem_nhds hy))).contDiffAt |
|
|
| theorem radialEndRegularPrimitive_contDiffAt {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| (D m : TrapezoidWeight A B → ℕ) |
| (c : ℝ) {p : Radial.Plane} (hp : p∈Radial.lowerTrapezoid A B) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| ContDiffAt ℝ ∞ (radialEndRegularPrimitive Radial.latticeIndex D m c p ε) y := |
| ContDiffAt.sum (fun i _ => (radialEndProbability_contDiffAt hA hAB D m hp ε hy i).smul |
| (weightedRegularEndPrimitive_contDiffAt (D i) (m i) ε c hR)) |
|
|
| theorem radialEndResidue_contDiffAt {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| (D m : TrapezoidWeight A B → ℕ) |
| (c : ℝ) {p : Radial.Plane} (hp : p∈Radial.lowerTrapezoid A B) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) : |
| ContDiffAt ℝ ∞ (radialEndResidue Radial.latticeIndex D m c p ε) y := |
| ContDiffAt.sum (fun i _ => (radialEndProbability_contDiffAt hA hAB D m hp ε hy i).mul contDiffAt_const) |
|
|
| local instance cubicRadialEndSmoothSigmaCompact : SigmaCompactSpace BaseCurve := |
| curveSigmaCompact (parameters 0) |
|
|
| theorem tendsto_radial_cubic_cutoff_mass {A B D S : ℕ} (hA : 0<A) (hS0 : 0<S) |
| (hAB : A≤B) (hSB : S≤B) (hB : 3*B<D) (hS : 3*S<D) |
| {p : Radial.Plane} (hp : p∈Radial.lowerTrapezoid A B) |
| {g : ℝ → ℝ} {r R : ℝ} (hg : ContDiff ℝ ∞ g) (hgc : HasCompactSupport g) |
| (hr : 0<r) (hRp : 0<R) (hR1 : R<1) |
| (hR : {s : ℂ | Complex.normSq s≤R}⊆endRegularRegion) |
| (h1 : ∀ t∈Ioo (-r) r,g t=1) (h0 : tsupport g⊆Iic R) (c : ℝ) : |
| Tendsto (fun n => surfaceCutoffMass (shrinking_curveEndCutoff_compact (parameters 0) hr h1 n) |
| (radialForm (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p)) atTop |
| (𝓝 (c*(3*((D:ℝ)-3*(p.1+p.2))-2*markedHeight S (planeVector p.1 p.2))*Real.pi)) := by |
| obtain ⟨Mlog,hMlog,hlog⟩ := exists_regularCoefficient_log_bound |
| (cubicDegree D (A := A) (B := B)) (cubicMarkedOrder S) hR |
| obtain ⟨M,hM,hbound⟩ := exists_regularPrimitive_bound |
| (cubicDegree D (A := A) (B := B)) (cubicMarkedOrder S) c hR |
| obtain ⟨K,hK⟩ := (hg.continuous_deriv (by simp)).bounded_above_of_compact_support hgc.deriv |
| have hK0 : 0≤K := (norm_nonneg (deriv g 0)).trans (hK 0) |
| have hKg (t : ℝ) : |deriv g t|≤K := by simpa only [Real.norm_eq_abs] using hK t |
| have hpoint (ε : EndIndex) (n : ℕ) (y : Plane) |
| (hy : y∈radialAnnulus (r/((n:ℝ)+1)) (R/((n:ℝ)+1))) : |
| y∈(infinityDiffeomorph (parameters 0) ε).source ∧ |
| Complex.equivRealProdCLM.symm y∈endRegularRegion ∧ |
| y∈radialAnnulus 0 R ∧ 0<radiusSq y ∧ radiusSq y<1 := by |
| have hn : 1≤(n:ℝ)+1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith |
| have hys : y∈radialAnnulus 0 R := ⟨radiusSq_nonneg _,hy.2.trans (div_le_self hRp.le hn)⟩ |
| have hxR : Complex.equivRealProdCLM.symm y∈endRegularRegion := by |
| apply hR |
| change Complex.normSq (Complex.equivRealProdCLM.symm y)≤R |
| rw [radiusSq_complex] |
| exact hys.2 |
| have hRI : {s : ℂ | Complex.normSq s≤R/((n:ℝ)+1)}⊆infinityRegion (parameters 0) := |
| fun s hs => (hR (hs.trans (div_le_self hRp.le hn))).1 |
| exact ⟨radialAnnulus_in_end (parameters 0) (div_pos hr (by positivity)) hRI ε hy, |
| hxR,hys,(div_pos hr (by positivity)).trans_le hy.1,hys.2.trans_lt hR1⟩ |
| have ht := tendsto_curveCutoff_bounded_residue (parameters 0) |
| (radialPrimitive (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p) |
| (radialPrimitive_smooth hA hAB _ _ c hp) |
| (radialForm (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p) |
| (radialForm_smooth hA hAB _ _ c hp) (radialForm_skew _ _ _ c p) rfl |
| hg hgc hr hRp.le (fun s hs => (hR hs).1) h1 h0 hK0 hM hKg |
| (fun ε => c*limitingEndOrder D S (planeVector p.1 p.2) ε) |
| (logarithmicError (|c| * (2*Mlog+Fintype.card (TrapezoidWeight A B))) R) |
| (fun n => logarithmicError_nonneg (by positivity) hRp hR1 n) |
| (tendsto_logarithmicError hRp _) |
| (radialEndRegularPrimitive (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p) |
| (radialEndResidue (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p) |
| (fun ε n y hy => (radialEndRegularPrimitive_contDiffAt hA hAB _ _ c hp ε |
| (hpoint ε n y hy).1 (hpoint ε n y hy).2.1).continuousAt.continuousWithinAt) |
| (fun ε n y hy => (radialEndResidue_contDiffAt hA hAB _ _ c hp ε |
| (hpoint ε n y hy).1).continuousAt.continuousWithinAt) |
| (fun ε n y hy => radialEndRegularPrimitive_norm_le (Radial.lattice_zero A B) _ _ c p ε y |
| (fun k => hbound k ε y (hpoint ε n y hy).2.2.1)) |
| (fun ε n y hy => by |
| have hh := radialEndResidue_error hA hS0 hAB hSB hB hS hp ε |
| (hpoint ε n y hy).1 (hpoint ε n y hy).2.1 |
| (hpoint ε n y hy).2.2.2.1 (hpoint ε n y hy).2.2.2.2 hMlog |
| (fun k => hlog k ε y (hpoint ε n y hy).2.2.1) c |
| apply hh.trans |
| rw [←mul_div_assoc] |
| exact logarithmicError_bound (by positivity) hRp hR1 n (hpoint ε n y hy).2.2.2.1 hy.2) |
| (fun ε n y hy => radialPrimitive_infinity |
| (Radial.latticeIndex (A := A) (B := B)) (fun k => (cubicMarked_lt_degree hB hS k).le) |
| (fun k => cubicMarked_twice_complement hB hS k) c p ε |
| (hpoint ε n y hy).1 (hpoint ε n y hy).2.1) |
| rw [←Finset.mul_sum] at ht |
| rw [sum_limitingEndOrder] at ht |
| exact ht |
|
|
| end PackingSufficiencySupport.CubicModel |
|
|
| namespace PackingSufficiencySupport.FiniteMoment.Radial |
| open scoped BigOperators Topology ContDiff |
| open Set Filter Function |
|
|
| def latticeOrigin (A B : ℕ) : TrapezoidWeight A B := |
| ⟨(⟨0,by omega⟩,⟨0,by omega⟩),by simp⟩ |
|
|
| theorem latticeIndex_injective (A B : ℕ) : Injective (latticeIndex (A:=A) (B:=B)) := by |
| intro i j h |
| apply Subtype.ext |
| apply Prod.ext |
| · exact Fin.ext (congrArg Prod.fst h) |
| · exact Fin.ext (congrArg Prod.snd h) |
|
|
| theorem monomial_origin (k : ℕ × ℕ) : monomial k (0,0)=if k=(0,0) then 1 else 0 := by |
| classical |
| by_cases h0 : k.1=0 <;> by_cases h1 : k.2=0 |
| · have hk : k=(0,0) := Prod.ext h0 h1 |
| simp [hk,monomial] |
| · have hk : k≠(0,0) := fun h => h1 (congrArg Prod.snd h) |
| simp [monomial,h0,h1,hk] |
| · have hk : k≠(0,0) := fun h => h0 (congrArg Prod.fst h) |
| simp [monomial,h0,h1,hk] |
| · have hk : k≠(0,0) := fun h => h0 (congrArg Prod.fst h) |
| simp [monomial,h0,h1,hk] |
|
|
| theorem lattice_monomial_origin {A B : ℕ} (i : TrapezoidWeight A B) : |
| monomial (latticeIndex i) (0,0)=if i=latticeOrigin A B then 1 else 0 := by |
| rw [monomial_origin] |
| have he : latticeIndex i=(0,0) ↔ i=latticeOrigin A B := |
| ⟨fun h => latticeIndex_injective A B h,fun h => by simp [h,latticeOrigin,latticeIndex]⟩ |
| simp only [he] |
|
|
| theorem lattice_polynomial_origin {A B : ℕ} (a : TrapezoidWeight A B → ℝ) : |
| polynomial latticeIndex a (0,0)=a (latticeOrigin A B) := by |
| classical |
| simp [polynomial,lattice_monomial_origin] |
|
|
| theorem lattice_probability_origin {A B : ℕ} {a : TrapezoidWeight A B → ℝ} |
| (ha : 0<a (latticeOrigin A B)) (i : TrapezoidWeight A B) : |
| probability latticeIndex a (0,0) i=if i=latticeOrigin A B then 1 else 0 := by |
| rw [probability,lattice_monomial_origin,lattice_polynomial_origin] |
| split_ifs with hi |
| · subst i |
| simp [ha.ne'] |
| · simp |
|
|
| theorem lattice_selected_origin {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| {a : TrapezoidWeight A B → ℝ} (ha : ∀ i,0<a i) (i : TrapezoidWeight A B) : |
| selected latticeIndex a (0,0) i=if i=latticeOrigin A B then 1 else 0 := by |
| have hi := nonnegativeInverse_of_moment ha (lattice_zero A B) |
| (lattice_unit_fst hA hAB) (lattice_unit_snd hA hAB) (r:=(0,0)) ⟨le_rfl,le_rfl⟩ |
| rw [moment_origin] at hi |
| rw [selected,hi] |
| exact lattice_probability_origin (ha _) i |
|
|
| end PackingSufficiencySupport.FiniteMoment.Radial |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped ContDiff Manifold Topology BigOperators |
| open Set Function Filter Manifold |
| open DiagonalQuadrics DiagonalQuadrics.Explicit Hamiltonian FiniteMoment |
| section |
|
|
| variable {ι : Type*} [Fintype ι] |
|
|
| theorem radialPrimitive_swap {k : ι → ℕ × ℕ} |
| (h0 : ∃ i,k i=(0,0)) (he0 : ∃ i,k i=(1,0)) (he1 : ∃ i,k i=(0,1)) |
| (D m : ι → ℕ) (c : ℝ) {p : Radial.Plane} |
| (hp : ∀ x : BaseCurve,∃ r : Radial.Plane,(0≤r.1 ∧ 0≤r.2) ∧ |
| Radial.moment k (fun j => coefficientNorm (D j) (m j) x) r=p) : |
| radialPrimitive (fun j => (k j).swap) D m c p.swap=radialPrimitive k D m c p := by |
| funext x |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [radialProbability,Radial.selected_swap |
| (fun j => coefficientNorm_pos (D j) (m j) x) h0 he0 he1 (hp x)] |
| rfl |
|
|
| theorem radialPrimitive_origin {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| (D m : TrapezoidWeight A B → ℕ) (c : ℝ) : |
| radialPrimitive Radial.latticeIndex D m c (0,0)= |
| weightedFSPrimitive (D (Radial.latticeOrigin A B)) (m (Radial.latticeOrigin A B)) c := by |
| classical |
| funext x |
| simp only [radialPrimitive,radialProbability,Radial.lattice_selected_origin hA hAB |
| (fun j => coefficientNorm_pos (D j) (m j) x)] |
| simp |
|
|
| theorem radialForm_face_positive {k : ι → ℕ × ℕ} |
| (h0 : ∃ i,k i=(0,0)) (he0 : ∃ i,k i=(1,0)) (he1 : ∃ i,k i=(0,1)) |
| {D m : ι → ℕ} (hm : ∀ i,m i<D i) {c : ℝ} (hc : 0<c) |
| {p : ℝ} (hp : Surrounds (Radial.faceWeight k) p) (b : BaseCurve) {y : RealModel} |
| (hy : y∈(extChartAt 𝓘(ℝ,RealModel) b).target) : |
| 0<chartTwoForm (radialForm k D m c (0,p)) b y (1,0) (0,1) := by |
| obtain ⟨i0,hi0⟩ := h0 |
| let : Nonempty (Radial.Face k) := ⟨⟨i0,by simp [hi0]⟩⟩ |
| rw [radialForm,radialPrimitive_eq_face ⟨i0,hi0⟩ he0 he1 D m c hp] |
| have hh := selectedForm_positive (D := fun i : Radial.Face k => D i.val) |
| (m := fun i : Radial.Face k => m i.val) (fun i => hm i.val) hc hp b hy |
| exact hh |
|
|
| theorem radialForm_positive {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| {D m : TrapezoidWeight A B → ℕ} (hm : ∀ i,m i<D i) {c : ℝ} (hc : 0<c) |
| {p : Radial.Plane} (hp : p∈Radial.lowerTrapezoid A B) (b : BaseCurve) {y : RealModel} |
| (hy : y∈(extChartAt 𝓘(ℝ,RealModel) b).target) : |
| 0<chartTwoForm (radialForm Radial.latticeIndex D m c p) b y (1,0) (0,1) := by |
| have h0 := Radial.lattice_zero A B |
| have he0 := Radial.lattice_unit_fst hA hAB |
| have he1 := Radial.lattice_unit_snd hA hAB |
| by_cases hp0 : p.1=0 |
| · by_cases hp1 : p.2=0 |
| · have he : p=(0,0) := Prod.ext hp0 hp1 |
| rw [he,radialForm,radialPrimitive_origin hA hAB,weightedFSPrimitive_exterior] |
| exact weightedFSForm_positive (hm _) hc b hy |
| · have hpos : 0<p.2 := lt_of_le_of_ne hp.2.2.1 (Ne.symm hp1) |
| have he : p=(0,p.2) := Prod.ext hp0 rfl |
| rw [he] |
| apply radialForm_face_positive h0 he0 he1 hm hc _ b hy |
| exact Radial.face_surrounds h0 (Radial.lattice_end_snd A B) |
| ⟨hpos,by simpa [hp0] using hp.2.2.2⟩ |
| · have hpos : 0<p.1 := lt_of_le_of_ne hp.1 (Ne.symm hp0) |
| by_cases hp1 : p.2=0 |
| · have h0' : ∃ i : TrapezoidWeight A B,(Radial.latticeIndex i).swap=(0,0) := by |
| obtain ⟨i,hi⟩ := h0; exact ⟨i,by simp [hi]⟩ |
| have he0' : ∃ i : TrapezoidWeight A B,(Radial.latticeIndex i).swap=(1,0) := by |
| obtain ⟨i,hi⟩ := he1; exact ⟨i,by simp [hi]⟩ |
| have he1' : ∃ i : TrapezoidWeight A B,(Radial.latticeIndex i).swap=(0,1) := by |
| obtain ⟨i,hi⟩ := he0; exact ⟨i,by simp [hi]⟩ |
| have hend : ∃ i : TrapezoidWeight A B,(Radial.latticeIndex i).swap=(0,A) := by |
| obtain ⟨i,hi⟩ := Radial.lattice_end_fst hAB; exact ⟨i,by simp [hi]⟩ |
| rw [radialForm,←radialPrimitive_swap h0 he0 he1 D m c |
| (fun x => Radial.trapezoid_radial_surjective hAB |
| (fun j => coefficientNorm_pos (D j) (m j) x) hp)] |
| have he : p.swap=(0,p.1) := Prod.ext hp1 rfl |
| rw [he] |
| exact radialForm_face_positive h0' he0' he1' hm hc |
| (Radial.face_surrounds h0' hend ⟨hpos,hp.2.1⟩) b hy |
| · let : Nonempty (TrapezoidWeight A B) := ⟨Radial.latticeOrigin A B⟩ |
| have hs : Surrounds (Radial.weight Radial.latticeIndex) (planeVector p.1 p.2) := |
| trapezoidWeight_surrounds hAB ⟨hpos,hp.2.1,lt_of_le_of_ne hp.2.2.1 (Ne.symm hp1),hp.2.2.2⟩ |
| rw [radialForm,radialPrimitive_eq_reduced h0 he0 he1 D m c hs] |
| exact reducedForm_positive hm hc hs b hy |
|
|
| end |
|
|
| def regularEndPhase (ε : EndIndex) (y : Plane) : PlanePhase HomogeneousIndex := |
| complexCartesian (endRegular ε (Complex.equivRealProdCLM.symm y)) |
|
|
| def regularEndPrimitive (ε : EndIndex) (c : ℝ) : Plane → Plane →L[ℝ] ℝ := |
| staticPullbackOneForm (hopfPrimitive c) (regularEndPhase ε) |
|
|
| def endResidue (ε : EndIndex) (c : ℝ) : ℝ := if ε=zeroEnd then 0 else c |
|
|
| theorem regularEndPhase_contDiffAt (ε : EndIndex) {y : Plane} |
| (hy : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| ContDiffAt ℝ ∞ (regularEndPhase ε) y := |
| (complexCartesian (ι := HomogeneousIndex)).contDiff.contDiffAt.comp y |
| (((endRegular_contDiffAt ε hy).restrict_scalars ℝ).comp y |
| Complex.equivRealProdCLM.symm.contDiff.contDiffAt) |
|
|
| theorem regularEndPrimitive_contDiffAt (ε : EndIndex) (c : ℝ) {y : Plane} |
| (hy : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| ContDiffAt ℝ ∞ (regularEndPrimitive ε c) y := |
| staticPullbackOneForm_smoothAt |
| (hopfPrimitive_smoothAt c (endRegular_cartesian_ne_zero ε _)) |
| (regularEndPhase_contDiffAt ε hy) |
|
|
| theorem endRegular_scale {ε : EndIndex} (hε : ε≠zeroEnd) {s : ℂ} (hs : s≠0) : |
| endRegular ε s=s•complexAffineLift (endQuotient ε s) := by |
| rw [endQuotient_homogeneous hε hs,smul_smul,mul_inv_cancel₀ hs,one_smul] |
|
|
| theorem endQuotientPrimitive_residue (ε : EndIndex) (c : ℝ) {y : Plane} |
| (hy : Complex.equivRealProdCLM.symm y∈endRegularRegion) |
| (h0 : Complex.equivRealProdCLM.symm y≠0) (v : Plane) : |
| affineFSPrimitive c (complexCartesian (endQuotient ε (Complex.equivRealProdCLM.symm y))) |
| (fderiv ℝ (fun z : Plane => complexCartesian |
| (endQuotient ε (Complex.equivRealProdCLM.symm z))) y v)= |
| regularEndPrimitive ε c y v-(endResidue ε c/2)*angularOneForm y v := by |
| let Q : Plane → AffineIndex → ℂ := fun z => endQuotient ε (Complex.equivRealProdCLM.symm z) |
| have hQ : DifferentiableAt ℝ Q y := |
| (((endQuotient_contDiffAt ε ⟨h0,hy.1⟩).restrict_scalars ℝ).differentiableAt (by simp)).comp y |
| Complex.equivRealProdCLM.symm.differentiableAt |
| have hp := affineFSPrimitive_hopf_pullback hQ c v |
| change _ =_ at hp |
| have hN : complexCartesian (complexAffineLift (Q y))≠0 := by |
| intro hz |
| have hf : complexAffineLift (Q y)=0 := by |
| apply (complexCartesian (ι := HomogeneousIndex)).injective |
| simpa only [map_zero] using hz |
| exact one_ne_zero (congrFun hf none) |
| by_cases hε : ε=zeroEnd |
| · have he : regularEndPhase ε=ᶠ[𝓝 y] fun z => complexCartesian (complexAffineLift (Q z)) := by |
| have hR := endRegularRegion_open.preimage Complex.equivRealProdCLM.symm.continuous |
| have hZ := (isOpen_ne_fun Complex.equivRealProdCLM.symm.continuous continuous_const).mem_nhds h0 |
| filter_upwards [hR.mem_nhds hy,hZ] with z hz hz0 |
| simp only [regularEndPhase,endRegular,ite_eq_left hε] |
| congr 2 |
| exact (endQuotient_zeroEnd hz0 hz).symm.trans (congrArg (fun ε => endQuotient ε _) hε.symm) |
| change _=hopfPrimitive c (regularEndPhase ε y) (fderiv ℝ (regularEndPhase ε) y v)-_ |
| rw [he.eq_of_nhds,he.fderiv_eq] |
| rw [show endResidue ε c=0 from ite_eq_left hε] |
| simpa only [zero_div,zero_mul,sub_zero] using hp.symm |
| · have he : regularEndPhase ε=ᶠ[𝓝 y] fun z => complexCartesian |
| ((Complex.equivRealProdCLM.symm z)•complexAffineLift (Q z)) := by |
| filter_upwards [(isOpen_ne_fun Complex.equivRealProdCLM.symm.continuous continuous_const).mem_nhds h0] |
| with z hz |
| exact congrArg complexCartesian (endRegular_scale hε hz) |
| have hG : DifferentiableAt ℝ (fun z => complexAffineLift (Q z)) y := |
| (complexAffineLift_smooth.differentiable (by simp) _).comp y hQ |
| have hg := hopfPrimitive_variable_gauge Complex.equivRealProdCLM.symm.differentiableAt |
| hG h0 hN c v |
| change _=hopfPrimitive c (regularEndPhase ε y) (fderiv ℝ (regularEndPhase ε) y v)-_ |
| rw [he.eq_of_nhds,he.fderiv_eq,hg,hp,show endResidue ε c=c from ite_eq_right hε] |
| rw [ContinuousLinearEquiv.fderiv] |
| simp only [ContinuousLinearEquiv.coe_coe] |
| have hns : Complex.normSq (Complex.equivRealProdCLM.symm y)=radiusSq y := by |
| change y.1*y.1+y.2*y.2=y.1^2+y.2^2 |
| ring |
| have hca : complexArea (Complex.equivRealProdCLM.symm y) |
| (Complex.equivRealProdCLM.symm v)=planarArea y v := by |
| rw [complexArea_apply] |
| rfl |
| rw [hns,hca] |
| simp only [angularOneForm,smul_apply,smul_eq_mul] |
| ring |
|
|
| theorem cubic_curveFSPrimitive_infinity (ε : EndIndex) (c : ℝ) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| euclideanPullbackOneForm (fun _ => curveFSPrimitive c) |
| (infinityDiffeomorph (parameters 0) ε) (0,y)= |
| regularEndPrimitive ε c y-(endResidue ε c/2) • angularOneForm y := by |
| have hD : Complex.equivRealProdCLM.symm y∈infinityDomain (parameters 0) := by |
| simpa only [infinityDiffeomorph_source,mem_preimage] using hy |
| have hg := cubic_infinityPhase_germ ε hy |
| change euclideanPullbackOneForm (fun _ => manifoldPullbackOneForm |
| (fun _ => affineFSPrimitive c) phaseInclusion 0) (infinityDiffeomorph (parameters 0) ε) (0,y)=_ |
| rw [euclideanPullbackOneForm_comp (phaseInclusion_smooth.mdifferentiable (by simp) _) |
| ((infinityDiffeomorph (parameters 0) ε).mdifferentiableAt (by simp) hy)] |
| apply ContinuousLinearMap.ext |
| intro v |
| change affineFSPrimitive c ((phaseInclusion ∘ infinityDiffeomorph (parameters 0) ε) y) |
| (manifoldMapDifferential (E := PlanePhase AffineIndex) (F := Plane) |
| (phaseInclusion ∘ infinityDiffeomorph (parameters 0) ε) y v)=_ |
| rw [manifoldMapDifferential,mfderiv_eq_fderiv,hg.eq_of_nhds,hg.fderiv_eq] |
| exact endQuotientPrimitive_residue ε c hR hD.1 v |
|
|
| theorem sum_endResidue (c : ℝ) : (∑ ε : EndIndex,endResidue ε c)=3*c := by |
| have h (ε : EndIndex) : endResidue ε c=c-(if ε=zeroEnd then c else 0) := by |
| by_cases he : ε=zeroEnd <;> simp [endResidue,he] |
| simp_rw [h] |
| rw [Finset.sum_sub_distrib] |
| norm_num [EndIndex,Fintype.card_fun] |
| ring |
|
|
| theorem exists_regularEndPrimitive_extension {R : ℝ} |
| (hR : {s : ℂ | Complex.normSq s≤R}⊆endRegularRegion) (ε : EndIndex) (c : ℝ) : |
| ∃ β : Plane → Plane →L[ℝ] ℝ,ContDiff ℝ ∞ β ∧ HasCompactSupport β ∧ |
| β=ᶠ[𝓝ˢ (radialAnnulus 0 R)] regularEndPrimitive ε c := by |
| let U : Set Plane := Complex.equivRealProdCLM.symm ⁻¹' endRegularRegion |
| have hU : IsOpen U := endRegularRegion_open.preimage Complex.equivRealProdCLM.symm.continuous |
| have hKU : radialAnnulus 0 R⊆U := by |
| intro y hy |
| apply hR |
| change Complex.normSq (Complex.equivRealProdCLM.symm y)≤R |
| rw [radiusSq_complex] |
| exact hy.2 |
| have hf : ContDiffOn ℝ ∞ (regularEndPrimitive ε c) U := |
| fun y hy => (regularEndPrimitive_contDiffAt ε c hy).contDiffWithinAt |
| obtain ⟨χ,_hχ,_hχc,_hχU,_hr,_h1,hf,hc,he⟩ := |
| exists_smooth_compact_extension (radialAnnulus_compact 0 R) hU hKU hf |
| exact ⟨fun y => χ y • regularEndPrimitive ε c y,hf,hc,he⟩ |
|
|
| theorem cubic_regular_extension_germ {R : ℝ} {ε : EndIndex} {c : ℝ} |
| {β : Plane → Plane →L[ℝ] ℝ} |
| (he : β=ᶠ[𝓝ˢ (radialAnnulus 0 R)] regularEndPrimitive ε c) |
| {z : Plane} (hz : radiusSq z≤R) : β=ᶠ[𝓝 z] regularEndPrimitive ε c := |
| he.filter_mono (nhds_le_nhdsSet ⟨radiusSq_nonneg z,hz⟩) |
|
|
| end PackingSufficiencySupport.CubicModel |
| end |
|
|
| end OAI |
|
|
