| import Mathlib | |
| import OAI.Geometry.BallPacking.Continuation.CompatibleHomotopy | |
| namespace OAI | |
| noncomputable section | |
| namespace HigherDimensionalBallPacking.Rigidity.HolderCompletion | |
| open scoped ContDiff Topology | |
| open Filter | |
| open scoped BoundedContinuousFunction | |
| open Set Function | |
| section | |
| variable {T X Y : Type*} [NormedAddCommGroup T] [NormedSpace ℝ T] [CompleteSpace T] | |
| [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] | |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y] | |
| lemma initial_implicit_branch (F : T × X → Y) (hF : ContDiff ℝ ∞ F) | |
| (L : X ≃L[ℝ] Y) (h0 : ∀ x, F (0,x)=L x) : | |
| ∃ U : T → X, U 0=0 ∧ ContinuousAt U 0 ∧ ∀ᶠ t in 𝓝 (0:T), F (t,U t)=0 := by | |
| let D := fderiv ℝ F (0,0) | |
| have hd : HasStrictFDerivAt F D (0,0) := hF.hasStrictFDerivAt (by simp) | |
| have hdi : HasFDerivAt (fun x : X => F (0,x)) (D.comp (ContinuousLinearMap.inr ℝ T X)) 0 := | |
| hd.hasFDerivAt.comp 0 (ContinuousLinearMap.inr ℝ T X).hasFDerivAt | |
| have hL : HasFDerivAt (fun x : X => F (0,x)) L.toContinuousLinearMap 0 := by | |
| have he : (fun x : X => F (0,x)) = L := funext h0 | |
| rw [he] | |
| exact L.hasFDerivAt | |
| have hDL : D.comp (ContinuousLinearMap.inr ℝ T X) = L.toContinuousLinearMap := hdi.unique hL | |
| have hi : (D.comp (ContinuousLinearMap.inr ℝ T X)).IsInvertible := by | |
| rw [hDL] | |
| exact ⟨L,rfl⟩ | |
| let U := hd.implicitFunctionOfProdDomain hi | |
| have hU0 : U 0=0 := | |
| (hd.eventually_apply_eq_iff_implicitFunctionOfProdDomain hi).self_of_nhds.mp rfl | |
| have hU : ContinuousAt U 0 := by | |
| have ht := hd.tendsto_implicitFunctionOfProdDomain hi | |
| change Tendsto U (𝓝 0) (𝓝 (U 0)) | |
| rw [hU0] | |
| exact ht | |
| refine ⟨U,hU0,hU,?_⟩ | |
| have hz : F (0,0)=0 := (h0 0).trans (map_zero L) | |
| simpa only [hz] using hd.eventually_apply_implicitFunctionOfProdDomain hi | |
| end | |
| section | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] | |
| local instance gmChartInst1 : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance gmChartInst2 : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace | |
| local instance gmChartInst3 : NormedAddCommGroup (COne ℂ E) := inferInstance | |
| local instance gmChartInst4 : NormedSpace ℝ (COne ℂ E) := inferInstance | |
| local instance gmChartInst5 : NormedAddCommGroup (HMap ℂ E) := inferInstance | |
| local instance gmChartInst6 : NormedSpace ℝ (HMap ℂ E) := inferInstance | |
| local instance gmChartInst7 : NormedAddCommGroup (HMap ℂ (E →L[ℝ] E)) := inferInstance | |
| local instance gmChartInst8 : NormedSpace ℝ (HMap ℂ (E →L[ℝ] E)) := inferInstance | |
| variable (b b₁ : ContDiffBump (0:ℂ)) (p q : E) | |
| local instance gmChartInst9 : NormedAddCommGroup (markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst10 : NormedSpace ℝ (markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst11 : AddCommGroup (markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst9 b).toAddCommGroup | |
| local instance gmChartInst12 : Module ℝ (markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst10 b).toModule | |
| local instance gmChartInst13 : NormedAddCommGroup (supportedHolder (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst14 : NormedSpace ℝ (supportedHolder (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst15 : AddCommGroup (supportedHolder (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst13 b).toAddCommGroup | |
| local instance gmChartInst16 : Module ℝ (supportedHolder (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst14 b).toModule | |
| local instance gmChartInst17 : NormedAddCommGroup (ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst18 : NormedSpace ℝ (ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gmChartInst19 : AddCommGroup (ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst17 b).toAddCommGroup | |
| local instance gmChartInst20 : Module ℝ (ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) := (gmChartInst18 b).toModule | |
| variable {P : ℝ × E → E →L[ℝ] E} (hP : BoundedCThree P) | |
| def globalNonlinearity (v : ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) : HMap ℂ E := | |
| bilinCLM (X := ℂ) (α := (1:ℝ)/3) (ContinuousLinearMap.apply (E := E) ℝ E).flip | |
| (globalCoefficient b₁ p q hP (v.1,v.2.val)) (markedDX p q 0 v.2.val) | |
| def globalChart (v : ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) : | |
| supportedHolder (E := E) (Metric.closedBall (0:ℂ) b.rOut) := | |
| (markedCR (E := E) (K := Metric.closedBall (0:ℂ) b.rOut)) v.2- | |
| supportedSourceCutoff b (globalNonlinearity b b₁ p q hP v) | |
| lemma globalChart_contDiff (hPc : HasCompactSupport P) : ContDiff ℝ ∞ (globalChart b b₁ p q hP) := by | |
| have hv : ContDiff ℝ ∞ (fun v : ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut) => (v.1,v.2.val)) := | |
| contDiff_fst.prodMk ((markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)).subtypeL.contDiff.comp contDiff_snd) | |
| have hc := (globalCoefficient_contDiff b₁ p q hP hPc).comp hv | |
| have hd : ContDiff ℝ ∞ (fun v : ℝ × markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut) => markedDX p q 0 v.2.val) := | |
| (markedDX_contDiff p q 0).comp ((markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)).subtypeL.contDiff.comp contDiff_snd) | |
| have hN : ContDiff ℝ ∞ (globalNonlinearity b b₁ p q hP) := | |
| (((bilinCLM (X := ℂ) (α := (1:ℝ)/3) (ContinuousLinearMap.apply (E := E) ℝ E).flip).contDiff.comp hc).clm_apply hd) | |
| exact ((markedCR (E := E) (K := Metric.closedBall (0:ℂ) b.rOut)).contDiff.comp contDiff_snd).sub | |
| ((supportedSourceCutoff (E := E) b).contDiff.comp hN) | |
| lemma globalChart_value (t : ℝ) (w : markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) (z : ℂ) : | |
| valueCLM _ (globalChart b b₁ p q hP (t,w)).val z=valueCLM _ (standardJetCR w.val) z- | |
| b z • P (t,cValue (globalMarkedJet b₁ p q w.val) z) (fderiv ℝ (markedCurve p q w.val) z 1) := by | |
| change valueCLM _ (standardJetCR w.val) z-b z • valueCLM _ (globalNonlinearity b b₁ p q hP (t,w)) z=_ | |
| simp only [globalNonlinearity,bilinCLM_value,globalCoefficient_value,markedDX_value,zero_add] | |
| rfl | |
| lemma globalChart_at_zero (hP0 : ∀ x,P (0,x)=0) | |
| (w : markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) : | |
| globalChart b b₁ p q hP (0,w)=markedCR (E := E) (K := Metric.closedBall (0:ℂ) b.rOut) w := by | |
| apply Subtype.ext | |
| apply value_ext | |
| intro z | |
| have hV := globalChart_value b b₁ p q hP 0 w z | |
| have hz := hP0 (cValue (globalMarkedJet b₁ p q w.val) z) | |
| have hzero : P (0,cValue (globalMarkedJet b₁ p q w.val) z) | |
| (fderiv ℝ (markedCurve p q w.val) z 1)=0 := by | |
| rw [hz,zero_apply] | |
| rw [hzero,smul_zero,sub_zero] at hV | |
| exact hV | |
| lemma globalChart_actual_value | |
| (hb₁ : Metric.closedBall (0:ℂ) b.rOut⊆Metric.closedBall (0:ℂ) b₁.rIn) | |
| {B : ℝ} (hPB : ∀ t x,B<‖x‖ → P (t,x)=0) (t : ℝ) | |
| (w : markedModel (E := E) (Metric.closedBall (0:ℂ) b.rOut)) | |
| (he : ∀ z,b.rIn≤‖z‖ → B<‖markedCurve p q w.val z‖) (z : ℂ) : | |
| valueCLM _ (globalChart b b₁ p q hP (t,w)).val z= | |
| fderiv ℝ (markedCurve p q w.val) z Complex.I- | |
| (Complex.I • fderiv ℝ (markedCurve p q w.val) z 1+ | |
| P (t,markedCurve p q w.val z) (fderiv ℝ (markedCurve p q w.val) z 1)) := by | |
| have hz0 : z∉Metric.closedBall (0:ℂ) b.rIn → P (t,markedCurve p q w.val z)=0 := by | |
| intro hz | |
| apply hPB | |
| apply he | |
| exact (lt_of_not_ge (by simpa only [Metric.mem_closedBall,dist_zero_right] using hz)).le | |
| have hN : b z • P (t,cValue (globalMarkedJet b₁ p q w.val) z) | |
| (fderiv ℝ (markedCurve p q w.val) z 1)= | |
| P (t,markedCurve p q w.val z) (fderiv ℝ (markedCurve p q w.val) z 1) := by | |
| by_cases hz : z∈Metric.closedBall (0:ℂ) b.rOut | |
| · apply cutoff_action_eq b _ hz0 | |
| rw [globalMarkedJet_eq_marked b₁ p q w.val (hb₁ hz)] | |
| · have hb0 : b z=0 := image_eq_zero_of_notMem_tsupport (by rwa [b.tsupport_eq]) | |
| have hz' : z∉Metric.closedBall (0:ℂ) b.rIn := | |
| fun h => hz (Metric.closedBall_subset_closedBall b.rIn_lt_rOut.le h) | |
| rw [hb0,hz0 hz',zero_apply,zero_smul] | |
| have hS := markedCurve_standardResidual p q w.val z | |
| calc | |
| _ = valueCLM _ (standardJetCR w.val) z-b z • | |
| P (t,cValue (globalMarkedJet b₁ p q w.val) z) (fderiv ℝ (markedCurve p q w.val) z 1) := | |
| globalChart_value b b₁ p q hP t w z | |
| _ = (fderiv ℝ (markedCurve p q w.val) z Complex.I- | |
| Complex.I • fderiv ℝ (markedCurve p q w.val) z 1)- | |
| P (t,markedCurve p q w.val z) (fderiv ℝ (markedCurve p q w.val) z 1) := | |
| congrArg₂ (· - ·) hS.symm hN | |
| _ = _ := sub_sub _ _ _ | |
| end | |
| section | |
| variable {n : ℕ} | |
| local instance gle1 : NormedAddCommGroup (End n) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance gle2 : NormedSpace ℝ (End n) := ContinuousLinearMap.toNormedSpace | |
| local instance gle3 : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance | |
| local instance gle4 : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance | |
| local instance gle5 : NormedAddCommGroup (HMap ℂ (Phase n)) := inferInstance | |
| local instance gle6 : NormedSpace ℝ (HMap ℂ (Phase n)) := inferInstance | |
| variable {J : Phase n → End n} (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) | |
| include hJs hJ hJc in | |
| theorem fullCoefficientBound : BoundedCThree (fullCoefficient J) := | |
| boundedCThree_of_compactSupport (fullCoefficient_smooth hJs hJ) (fullCoefficient_compact hJc) | |
| variable (b b₁ : ContDiffBump (0:ℂ)) (p q : Phase n) | |
| local instance gle7 : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gle8 : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gle9 : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gle10 : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gle11 : AddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (gle7 (n := n) b).toAddCommGroup | |
| local instance gle12 : Module ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (gle8 (n := n) b).toModule | |
| local instance gle13 : AddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (gle9 (n := n) b).toAddCommGroup | |
| local instance gle14 : Module ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (gle10 (n := n) b).toModule | |
| def fullLineEquation := globalChart b b₁ p q (fullCoefficientBound hJs hJ hJc) | |
| lemma fullLineEquation_contDiff : ContDiff ℝ ∞ (fullLineEquation hJs hJ hJc b b₁ p q) := | |
| globalChart_contDiff b b₁ p q (fullCoefficientBound hJs hJ hJc) (fullCoefficient_compact hJc) | |
| lemma fullLineEquation_initial | |
| (w : markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) : | |
| fullLineEquation hJs hJ hJc b b₁ p q (0,w)= | |
| markedCR (E := Phase n) (K := Metric.closedBall (0:ℂ) b.rOut) w := | |
| globalChart_at_zero b b₁ p q (fullCoefficientBound hJs hJ hJc) (fullCoefficient_zero hJ) w | |
| variable (hb₁ : Metric.closedBall (0:ℂ) b.rOut⊆Metric.closedBall (0:ℂ) b₁.rIn) | |
| {B : ℝ} (hstd : ∀ x,B<‖x‖ → J x=standardJ n) | |
| {t : ℝ} (ht : t∈Icc (0:ℝ) 1) | |
| (w : markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) | |
| (he : ∀ z,b.rIn≤‖z‖ → B<‖markedCurve p q w.val z‖) | |
| include hb₁ hstd ht he in | |
| lemma fullLineEquation_value (z : ℂ) : | |
| valueCLM _ (fullLineEquation hJs hJ hJc b b₁ p q (t,w)).val z= | |
| fderiv ℝ (markedCurve p q w.val) z Complex.I- | |
| lineHomotopy J (fullTime t) (markedCurve p q w.val z) | |
| (fderiv ℝ (markedCurve p q w.val) z 1) := by | |
| have hx := globalChart_actual_value b b₁ p q (fullCoefficientBound hJs hJ hJc) | |
| hb₁ (fullCoefficient_outside hstd) t w he z | |
| have hc := congrArg (fun A : End n => A (fderiv ℝ (markedCurve p q w.val) z 1)) | |
| (fullCoefficient_exact (J := J) ht (markedCurve p q w.val z)) | |
| change Complex.I • fderiv ℝ (markedCurve p q w.val) z 1+ | |
| fullCoefficient J (t,markedCurve p q w.val z) (fderiv ℝ (markedCurve p q w.val) z 1)=_ at hc | |
| exact hx.trans (congrArg (fun v : Phase n => fderiv ℝ (markedCurve p q w.val) z Complex.I-v) hc) | |
| include hb₁ hstd ht he in | |
| lemma fullLineEquation_zero_iff | |
| (ha : markedSlope p q w.val≠0) : | |
| fullLineEquation hJs hJ hJc b b₁ p q (t,w)=0 ↔ | |
| AffineLineCurve (lineHomotopy J (fullTime t)) p q (markedCurve p q w.val) := by | |
| have hc := lineHomotopy_compatible hJ (fullTime_range t) | |
| have hs := lineHomotopy_slice_contDiff hJs hJ (fullTime_range t) | |
| have hstd' : ∀ x,B<‖x‖ → lineHomotopy J (fullTime t) x=standardJ n := by | |
| intro x hx | |
| change interpolateJ (J x) (fullTime t)=standardJ n | |
| rw [hstd x hx] | |
| exact interpolateJ_standard_initial | |
| ⟨lt_of_lt_of_le (by norm_num) (fullTime_range t).1,(fullTime_range t).2⟩ | |
| constructor | |
| · intro hz | |
| have hcr (z : ℂ) : fderiv ℝ (markedCurve p q w.val) z Complex.I= | |
| lineHomotopy J (fullTime t) (markedCurve p q w.val z) | |
| (fderiv ℝ (markedCurve p q w.val) z 1) := by | |
| have hv := fullLineEquation_value hJs hJ hJc b b₁ p q hb₁ hstd ht w he z | |
| have hzero : valueCLM _ (fullLineEquation hJs hJ hJc b b₁ p q (t,w)).val z=0 := by | |
| rw [hz]; rfl | |
| exact sub_eq_zero.mp (hv.symm.trans hzero) | |
| refine ⟨completed_markedCurve_contDiff hs hc hstd' p q w.val ha hcr,?_, | |
| markedCurve_zero p q w.property.1,markedCurve_one p q w.val, | |
| markedSlope p q w.val,ha,markedCurve_asymptote p q w.val⟩ | |
| intro z | |
| exact ⟨(markedCurve_hasFDerivAt p q w.val z).differentiableAt, | |
| realCR_axis_implies_all _ (hc _).1 _ (hcr z)⟩ | |
| · intro hu | |
| apply Subtype.ext | |
| apply value_ext | |
| intro z | |
| have hv := fullLineEquation_value hJs hJ hJc b b₁ p q hb₁ hstd ht w he z | |
| have hcr := (hu.2.1 z).2 1 | |
| simp only [mul_one] at hcr | |
| exact hv.trans (sub_eq_zero.mpr hcr) | |
| end | |
| section | |
| section Translation | |
| variable {X Y : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] | |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] | |
| lemma spatial_derivative_translation {f : X → Y} (hf : ContDiff ℝ ∞ f) (w : X) : | |
| fderiv ℝ (fun h => f (w+h)) 0=fderiv ℝ f w := by | |
| have hdw : HasFDerivAt f (fderiv ℝ f w) (w+0) := by | |
| simpa only [add_zero] using (hf.differentiable (by simp) w).hasFDerivAt | |
| have hh := hdw.comp 0 ((hasFDerivAt_id (0:X)).const_add w) | |
| simpa only [Function.comp_def,ContinuousLinearMap.comp_id] using hh.fderiv | |
| end Translation | |
| variable {n : ℕ} | |
| local instance glf1 : NormedAddCommGroup (End n) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance glf2 : NormedSpace ℝ (End n) := ContinuousLinearMap.toNormedSpace | |
| local instance glf3 : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance | |
| local instance glf4 : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance | |
| variable {J : Phase n → End n} (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) | |
| (b b₁ : ContDiffBump (0:ℂ)) (p q : Phase n) | |
| local instance glf5 : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance glf6 : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance glf7 : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance glf8 : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance glf9 : AddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (glf5 (n := n) b).toAddCommGroup | |
| local instance glf10 : Module ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (glf6 (n := n) b).toModule | |
| local instance glf11 : AddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (glf7 (n := n) b).toAddCommGroup | |
| local instance glf12 : Module ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (glf8 (n := n) b).toModule | |
| local instance glf13 : TopologicalSpace (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := | |
| (glf5 (n := n) b).toPseudoMetricSpace.toUniformSpace.toTopologicalSpace | |
| local instance glf14 : TopologicalSpace (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := | |
| (glf7 (n := n) b).toPseudoMetricSpace.toUniformSpace.toTopologicalSpace | |
| variable (hb₁ : Metric.closedBall (0:ℂ) b.rOut⊆Metric.closedBall (0:ℂ) b₁.rIn) | |
| {B : ℝ} (hstd : ∀ x,B<‖x‖ →J x=standardJ n) | |
| {t : ℝ} (ht : t∈Icc (0:ℝ) 1) | |
| (w : markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) | |
| (he : MarkedEscapeNeighborhood b p q w.val B) | |
| include hstd in | |
| lemma fullTime_slice_outside : ∀ x,B<‖x‖ →lineHomotopy J (fullTime t) x=standardJ n := by | |
| intro x hx | |
| change interpolateJ (J x) (fullTime t)=standardJ n | |
| rw [hstd x hx] | |
| exact interpolateJ_standard (fullTime_range t) | |
| include hb₁ hstd ht he in | |
| lemma fullLineEquation_local_chart_eq : | |
| (fun h => fullLineEquation hJs hJ hJc b b₁ p q (t,w+h)) =ᶠ[𝓝 0] | |
| actualMarkedChart (lineHomotopy_slice_contDiff hJs hJ (fullTime_range t)) | |
| (fullTime_slice_outside hstd (t := t)) b p q w := by | |
| filter_upwards [he] with h hh | |
| apply Subtype.ext | |
| apply value_ext | |
| intro z | |
| have hv := fullLineEquation_value hJs hJ hJc b b₁ p q hb₁ hstd ht (w+h) | |
| (fun z hz => (hh z hz).1) z | |
| have ha := actualMarkedChart_value (lineHomotopy_slice_contDiff hJs hJ (fullTime_range t)) | |
| (fullTime_slice_outside hstd (t := t)) b p q w h hh z | |
| exact hv.trans ha.symm | |
| include hb₁ hstd ht he in | |
| lemma fullLineEquation_spatial_fredholm : IsIndexZeroFredholm | |
| (fderiv ℝ (fun v => fullLineEquation hJs hJ hJc b b₁ p q (t,v)) w) := by | |
| let f := fun v => fullLineEquation hJs hJ hJc b b₁ p q (t,v) | |
| have hf : ContDiff ℝ ∞ f := (fullLineEquation_contDiff hJs hJ hJc b b₁ p q).comp | |
| (contDiff_const.prodMk contDiff_id) | |
| have hs := lineHomotopy_slice_contDiff hJs hJ (fullTime_range t) | |
| have ho := fullTime_slice_outside hstd (t := t) | |
| have hEq := fullLineEquation_local_chart_eq hJs hJ hJc b b₁ p q hb₁ hstd ht w he | |
| have hder := (spatial_derivative_translation hf w).symm.trans hEq.fderiv_eq | |
| have hfred := actualMarkedChart_derivative_fredholm hs | |
| (lineHomotopy_compatible hJ (fullTime_range t)) ho b p q w he | |
| change IsIndexZeroFredholm (fderiv ℝ f w) | |
| rw [hder] | |
| exact hfred | |
| end | |
| variable {n : ℕ} | |
| local instance gid1 : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance | |
| local instance gid2 : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance | |
| variable (b : ContDiffBump (0:ℂ)) | |
| local instance gid3 : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gid4 : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gid5 : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance gid6 : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| def fullLineDomain (p q : Phase n) (M δ : ℝ) : | |
| Set (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := | |
| {w | w.val∈markedBoundedDomain p q M δ} | |
| lemma fullLineDomain_open (p q : Phase n) (M δ : ℝ) : IsOpen (fullLineDomain b p q M δ) := | |
| (markedBoundedDomain_open p q M δ).preimage continuous_subtype_val | |
| lemma marked_pair_closedEmbedding : Topology.IsClosedEmbedding | |
| (fun v : ℝ × markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut) => (v.1,v.2.val)) := by | |
| exact Topology.IsClosedEmbedding.id.prodMap | |
| ((markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)).isClosedEmbedding_subtypeL (markedModel_closed _)) | |
| variable {J : Phase n → End n} (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) | |
| (b₁ : ContDiffBump (0:ℂ)) (p q : Phase n) | |
| (hb₁ : Metric.closedBall (0:ℂ) b.rOut⊆Metric.closedBall (0:ℂ) b₁.rIn) | |
| {B M δ : ℝ} (hstd : ∀ x,B<‖x‖ →J x=standardJ n) (hδ : 0<δ) | |
| (he : ∀ w∈markedBoundedDomain p q M δ,MarkedEscapeNeighborhood b p q w B) | |
| (hbound : ∀ v∈exactMarkedZeroSet J p q,v.2∈markedBoundedDomain p q M δ) | |
| def isolatedFullLineZeros : Set (ℝ × markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := | |
| {v | v.1∈Icc (0:ℝ) 1 ∧ v.2∈fullLineDomain b p q M δ ∧ | |
| fullLineEquation hJs hJ hJc b b₁ p q v=0} | |
| include hb₁ hstd hδ he hbound in | |
| lemma isolatedFullLineZeros_eq_preimage : | |
| isolatedFullLineZeros b hJs hJ hJc b₁ p q (M := M) (δ := δ)= | |
| (fun v : ℝ × markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut) => (v.1,v.2.val)) ⁻¹' | |
| fullMarkedZeroSet J p q := by | |
| ext v | |
| constructor | |
| · intro hv | |
| have hesc := marked_escape_neighborhood_base b p q (he v.2.val hv.2.1) | |
| have hline := (fullLineEquation_zero_iff hJs hJ hJc b b₁ p q hb₁ hstd hv.1 v.2 hesc | |
| (markedBoundedDomain_slope hδ hv.2.1)).mp hv.2.2 | |
| exact ⟨hv.1,fullTime_range v.1,v.2.property.1,hline⟩ | |
| · intro hv | |
| have hw : v.2∈fullLineDomain b p q M δ := hbound (fullTime v.1,v.2.val) hv.2 | |
| have hesc := marked_escape_neighborhood_base b p q (he v.2.val hw) | |
| exact ⟨hv.1,hw,(fullLineEquation_zero_iff hJs hJ hJc b b₁ p q hb₁ hstd hv.1 v.2 hesc | |
| (markedBoundedDomain_slope hδ hw)).mpr hv.2.2.2⟩ | |
| include hb₁ hstd hδ he hbound in | |
| lemma isolatedFullLineZeros_compact | |
| (hc : IsCompact (fullMarkedZeroSet J p q)) : | |
| IsCompact (isolatedFullLineZeros b hJs hJ hJc b₁ p q (M := M) (δ := δ)) := by | |
| rw [isolatedFullLineZeros_eq_preimage b hJs hJ hJc b₁ p q hb₁ hstd hδ he hbound] | |
| exact (marked_pair_closedEmbedding b).isCompact_preimage hc | |
| include hb₁ hstd he in | |
| lemma fullLineEquation_fredholm_on_domain {t : ℝ} (ht : t∈Icc (0:ℝ) 1) | |
| (w : markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) | |
| (hw : w∈fullLineDomain b p q M δ) : IsIndexZeroFredholm | |
| (fderiv ℝ (fun v => fullLineEquation hJs hJ hJc b b₁ p q (t,v)) w) := | |
| fullLineEquation_spatial_fredholm hJs hJ hJc b b₁ p q hb₁ hstd ht w (he w.val hw) | |
| end HigherDimensionalBallPacking.Rigidity.HolderCompletion | |
| namespace HigherDimensionalBallPacking.Rigidity | |
| open scoped ContDiff Topology BoundedContinuousFunction | |
| open Set Filter | |
| open HolderCompletion | |
| section | |
| structure ProperIndexZeroPath {X Y : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] | |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] (F : ℝ × X → Y) (U : Set X) : Prop where | |
| domain_open : IsOpen U | |
| origin_mem : (0:X)∈U | |
| smooth : ContDiff ℝ ∞ F | |
| initial : ∃ L : X ≃L[ℝ] Y,∀ x,F (0,x)=L x | |
| zeroSet_compact : IsCompact {v : ℝ × X | v.1∈Icc (0:ℝ) 1 ∧ v.2∈U ∧ F v=0} | |
| fredholm : ∀ t∈Icc (0:ℝ) 1,∀ x∈U,IsIndexZeroFredholm (fderiv ℝ (fun y => F (t,y)) x) | |
| variable {n : ℕ} | |
| local instance lcdEnd1 : NormedAddCommGroup (End n) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance lcdEnd2 : NormedSpace ℝ (End n) := ContinuousLinearMap.toNormedSpace | |
| local instance lcd1 : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance | |
| local instance lcd2 : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance | |
| lemma markedCurve_zero_correction (p q : Phase n) : | |
| markedCurve p q (0:COne ℂ (Phase n))=(fun z : ℂ => p+z • (q-p)) := by | |
| funext z | |
| have hz (z : ℂ) : cValue (0:COne ℂ (Phase n)) z=0 := rfl | |
| simp only [markedCurve,markedSlope,hz,sub_zero,add_zero] | |
| lemma exact_marked_initial {J : Phase n → End n} (hJ : ∀ x,Compatible (J x)) | |
| {p q : Phase n} (hpq : p≠q) : (0,0)∈exactMarkedZeroSet J p q := by | |
| refine ⟨by constructor <;> norm_num,?_,?_⟩ | |
| · change cValue (0:COne ℂ (Phase n)) 0=0 | |
| rfl | |
| · rw [markedCurve_zero_correction] | |
| have he : lineHomotopy J 0=(fun _ => standardJ n) := funext (lineHomotopy_zero hJ) | |
| rw [he] | |
| exact standard_affine_line p q hpq | |
| variable (b : ContDiffBump (0:ℂ)) | |
| local instance lcd3 : NormedAddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance lcd4 : NormedSpace ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance lcd5 : NormedAddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance lcd6 : NormedSpace ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance lcd7 : AddCommGroup (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (lcd3 (n := n) b).toAddCommGroup | |
| local instance lcd8 : Module ℝ (markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (lcd4 (n := n) b).toModule | |
| local instance lcd9 : AddCommGroup (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (lcd5 (n := n) b).toAddCommGroup | |
| local instance lcd10 : Module ℝ (supportedHolder (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := (lcd6 (n := n) b).toModule | |
| local instance lcd11 : NormedAddCommGroup (ℝ × markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| local instance lcd12 : NormedSpace ℝ (ℝ × markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut)) := inferInstance | |
| variable {J : Phase n → End n} (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) (b₁ : ContDiffBump (0:ℂ)) (p q : Phase n) | |
| (hb₁ : Metric.closedBall (0:ℂ) b.rOut⊆Metric.closedBall (0:ℂ) b₁.rIn) | |
| {B M δ : ℝ} (hstd : ∀ x,B<‖x‖ →J x=standardJ n) (hδ : 0<δ) | |
| (he : ∀ w∈markedBoundedDomain p q M δ,MarkedEscapeNeighborhood b p q w B) | |
| include hb₁ hstd hδ he in | |
| lemma fullLineEndpointMap : ∀ w∈fullLineDomain b p q M δ, | |
| fullLineEquation hJs hJ hJc b b₁ p q (1,w)=0 → | |
| AffineLineCurve J p q (markedCurve p q w.val) := by | |
| intro w hw hz | |
| have hesc := marked_escape_neighborhood_base b p q (he w.val hw) | |
| have hl := (fullLineEquation_zero_iff hJs hJ hJc b b₁ p q hb₁ hstd (t := 1) | |
| (by constructor <;> norm_num) w hesc (markedBoundedDomain_slope hδ hw)).mp hz | |
| have ht : lineHomotopy J (fullTime 1)=J := by | |
| funext x | |
| rw [fullTime_one,lineHomotopy_one hJ] | |
| rwa [ht] at hl | |
| include hb₁ hstd hδ he in | |
| lemma fullLineProperPath | |
| (hbound : ∀ v∈exactMarkedZeroSet J p q,v.2∈markedBoundedDomain p q M δ) | |
| (hc : IsCompact (fullMarkedZeroSet J p q)) (hpq : p≠q) : | |
| ProperIndexZeroPath (fullLineEquation hJs hJ hJc b b₁ p q) (fullLineDomain b p q M δ) := by | |
| have h0 : (0:markedModel (E := Phase n) (Metric.closedBall (0:ℂ) b.rOut))∈fullLineDomain b p q M δ := | |
| hbound (0,0) (exact_marked_initial hJ hpq) | |
| refine ⟨fullLineDomain_open b p q M δ,h0, | |
| fullLineEquation_contDiff hJs hJ hJc b b₁ p q,?_,?_,?_⟩ | |
| · exact ⟨markedCREquiv (E := Phase n) (isCompact_closedBall (0:ℂ) b.rOut), | |
| fullLineEquation_initial hJs hJ hJc b b₁ p q⟩ | |
| · exact isolatedFullLineZeros_compact b hJs hJ hJc b₁ p q hb₁ hstd hδ he hbound hc | |
| · intro t ht w hw | |
| exact fullLineEquation_fredholm_on_domain b hJs hJ hJc b₁ p q hb₁ hstd he ht w hw | |
| end | |
| section | |
| variable {n : ℕ} {J : Phase n → End n} | |
| local instance lcdFinal1 : NormedAddCommGroup (End n) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance lcdFinal2 : NormedSpace ℝ (End n) := ContinuousLinearMap.toNormedSpace | |
| local instance lcdFinal3 : NormedAddCommGroup (COne ℂ (Phase n)) := inferInstance | |
| local instance lcdFinal4 : NormedSpace ℝ (COne ℂ (Phase n)) := inferInstance | |
| variable (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) (p q : Phase n) | |
| def FullLineReady (b b₁ : ContDiffBump (0:ℂ)) (M δ : ℝ) : Prop := | |
| ProperIndexZeroPath (fullLineEquation hJs hJ hJc b b₁ p q) (fullLineDomain b p q M δ) ∧ | |
| (∀ w∈fullLineDomain b p q M δ,fullLineEquation hJs hJ hJc b b₁ p q (1,w)=0 → | |
| AffineLineCurve J p q (markedCurve p q w.val)) | |
| variable {S T : ℝ} (hST : S<T) (hT : T<1) (hout : ∀ x,S<capacity x →J x=standardJ n) | |
| (hpq : p≠q) | |
| include hJs hJ hJc hST hT hout hpq in | |
| theorem full_line_continuation_data : | |
| ∃ (b b₁ : ContDiffBump (0:ℂ)) (M δ : ℝ), | |
| FullLineReady hJs hJ hJc p q b b₁ M δ := by | |
| obtain ⟨B,M,δ,b,hB,hM,hδ,hstd,hbound,he⟩ := | |
| exact_marked_common_neighborhood (J := J) (p := p) (q := q) hJs hJ hJc hST hT hout hpq | |
| obtain ⟨b₁,hb₁⟩ := compact_subset_bump_inner (K := Metric.closedBall (0:ℂ) b.rOut) | |
| (isCompact_closedBall (0:ℂ) b.rOut) | |
| have hbound' : ∀ v∈exactMarkedZeroSet J p q,v.2∈markedBoundedDomain p q M δ := | |
| fun v hv => (hbound v hv).1 | |
| refine ⟨b,b₁,M,δ,?_,?_⟩ | |
| · exact fullLineProperPath b hJs hJ hJc b₁ p q hb₁ hstd hδ he hbound' | |
| (full_marked_zeroSet_compact hJs hJ hJc hST hT hout hpq) hpq | |
| · exact fullLineEndpointMap b hJs hJ hJc b₁ p q hb₁ hstd hδ he | |
| end | |
| variable {n : ℕ} {J : Phase n → End n} | |
| local instance alc1 : NormedAddCommGroup (End n) := ContinuousLinearMap.toNormedAddCommGroup | |
| local instance alc2 : NormedSpace ℝ (End n) := ContinuousLinearMap.toNormedSpace | |
| lemma adapted_structure_capacity_bound | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) | |
| (hinside : tsupport (fun x => J x-standardJ n)⊆openBall n 1) : | |
| ∃ S T : ℝ,S<T ∧ T<1 ∧ ∀ x,S<capacity x →J x=standardJ n := by | |
| let K := tsupport (fun x => J x-standardJ n) ∪ {(0:Phase n)} | |
| have hK : IsCompact K := hJc.union (isCompact_singleton) | |
| have hne : K.Nonempty := ⟨0,Or.inr (by rfl)⟩ | |
| have hKi : K⊆openBall n 1 := by | |
| intro x hx | |
| rcases hx with hx|hx | |
| · exact hinside hx | |
| · have hx0 : x=0 := hx | |
| subst x | |
| simp [openBall,capacity] | |
| obtain ⟨x,hx,hmax⟩ := hK.exists_isMaxOn hne (capacity_smooth n).continuous.continuousOn | |
| have hS : capacity x<1 := hKi hx | |
| refine ⟨capacity x,(capacity x+1)/2,by linarith,by linarith,?_⟩ | |
| intro y hy | |
| have hn : y∉tsupport (fun z => J z-standardJ n) := by | |
| intro hh | |
| exact (not_le_of_gt hy) (hmax (Or.inl hh)) | |
| exact sub_eq_zero.mp (image_eq_zero_of_notMem_tsupport (f := fun z => J z-standardJ n) hn) | |
| lemma adapted_line_continuation_data (hJs : ContDiff ℝ ∞ J) (hJ : ∀ x,Compatible (J x)) | |
| (hJc : HasCompactSupport (fun x => J x-standardJ n)) | |
| (hinside : tsupport (fun x => J x-standardJ n)⊆openBall n 1) (p q : Phase n) (hpq : p≠q) : | |
| ∃ (b b₁ : ContDiffBump (0:ℂ)) (M δ : ℝ), | |
| FullLineReady hJs hJ hJc p q b b₁ M δ := by | |
| obtain ⟨S,T,hST,hT,hout⟩ := adapted_structure_capacity_bound hJc hinside | |
| exact full_line_continuation_data hJs hJ hJc p q hST hT hout hpq | |
| end HigherDimensionalBallPacking.Rigidity | |
| end | |
| end OAI | |