import Mathlib import OAI.Geometry.BallPacking.Compactness.GradientBounds namespace OAI noncomputable section namespace HigherDimensionalBallPacking.Rigidity.HolderCompletion section open scoped ContDiff Topology BoundedContinuousFunction open Set Filter variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] local instance ljcInst1 : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup local instance ljcInst2 : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace local instance ljcInst3 : NormedAddCommGroup (COne ℂ E) := inferInstance local instance ljcInst4 : NormedSpace ℝ (COne ℂ E) := inferInstance local instance ljcInst5 : NormedAddCommGroup (HMap ℂ E) := inferInstance local instance ljcInst6 : NormedSpace ℝ (HMap ℂ E) := inferInstance local instance ljcInst7 : NormedAddCommGroup (HMap ℂ (E →L[ℝ] E)) := inferInstance local instance ljcInst8 : NormedSpace ℝ (HMap ℂ (E →L[ℝ] E)) := inferInstance lemma cauchy_coefficient_action {A : ℕ → HMap ℂ (E →L[ℝ] E)} {v : ℕ → HMap ℂ E} (hA : CauchySeq A) (hv : CauchySeq v) : CauchySeq (fun i => holderCoefficientAction (A i) (v i)) := by obtain ⟨a,ha⟩ := cauchySeq_tendsto_of_complete hA obtain ⟨w,hw⟩ := cauchySeq_tendsto_of_complete hv have hAc : Continuous (fun x : HMap ℂ (E →L[ℝ] E) => holderCoefficientAction x) := (bilinCLM (X := ℂ) (α := (1:ℝ)/3) (ContinuousLinearMap.apply (E := E) ℝ E).flip).continuous have hc : Continuous (fun p : HMap ℂ (E →L[ℝ] E) × HMap ℂ E => holderCoefficientAction p.1 p.2) := (hAc.comp continuous_fst).clm_apply continuous_snd exact ((hc.tendsto (a,w)).comp (ha.prodMk_nhds hw)).cauchySeq lemma cutoff_forcing_holder_cauchy (C D : HMap ℂ (E →L[ℝ] E)) {A : ℕ → HMap ℂ (E →L[ℝ] E)} {v : ℕ → HMap ℂ E} (hA : CauchySeq A) (hv : CauchySeq v) : CauchySeq (fun i => holderCoefficientAction C (v i)- holderCoefficientAction (A i) (holderCoefficientAction D (v i))) := by have hC := cauchy_coefficient_action (E := E) (cauchySeq_const (x := C)) hv have hD := cauchy_coefficient_action (E := E) (cauchySeq_const (x := D)) hv obtain ⟨c,hc⟩ := cauchySeq_tendsto_of_complete hC obtain ⟨d,hd⟩ := cauchySeq_tendsto_of_complete (cauchy_coefficient_action hA hD) exact (hc.sub hd).cauchySeq theorem cutoff_jet_cauchy (b : ContDiffBump (0:ℂ)) {L : ℝ} (hL : 0≤L) : ∃ δ : ℝ,0<δ ∧ ∀ (u : ℕ → ℂ → E) (hu : ∀ i,ContDiff ℝ ∞ (u i)) (A : ℕ → ℂ → E →L[ℝ] E), (∀ i,Continuous (A i)) →(∀ i,u i 0=0) → (∀ i z,‖A i z‖≤δ) → (∀ i x y,‖A i x-A i y‖≤L*dist x y) → (∃ M H : ℝ,0≤M ∧ 0≤H ∧ (∀ i z,‖u i z‖≤M) ∧ ∀ i x y,‖u i x-u i y‖≤H*dist x y) → (∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ z,‖u i z-u j z‖≤ε) → (∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ z,‖A i z-A j z‖≤ε) → (∀ i z,b z≠0 →fderiv ℝ (u i) z Complex.I-Complex.I • fderiv ℝ (u i) z 1= A i z (fderiv ℝ (u i) z 1)) → CauchySeq (fun i => cutoffCurveJet b (hu i)) := by obtain ⟨ε,hε,hcauchy⟩ := small_coefficient_cauchy (E := E) (isCompact_closedBall (0:ℂ) b.rOut) obtain ⟨δ,hδ,hsmall⟩ := holder_small_of_small_value (X := ℂ) (E := E →L[ℝ] E) hL hε refine ⟨δ,hδ,?_⟩ intro u hu A hAc h0 hAb hAl hUb hUc hAcc hcr obtain ⟨M,H,hM,hH,hUb,hUl⟩ := hUb obtain ⟨B,hB,hD,hDL,hC,hCL⟩ := cutoff_transport_bounds (E := E) b let V : ℕ → HMap ℂ E := fun i => ofBoundedLipschitz (u i) (hu i).continuous M H hM hH (hUb i) (hUl i) let T : ℕ → HMap ℂ (E →L[ℝ] E) := fun i => ofBoundedLipschitz (A i) (hAc i) δ L hδ.le hL (hAb i) (hAl i) let C : HMap ℂ (E →L[ℝ] E) := ofBoundedLipschitz (cutoffCommute b) (cutoffCommute_smooth b).continuous B B hB.le hB.le hC hCL let D : HMap ℂ (E →L[ℝ] E) := ofBoundedLipschitz (cutoffTransport b 1) (cutoffTransport_smooth b 1).continuous B B hB.le hB.le hD hDL let f : ℕ → HMap ℂ E := fun i => holderCoefficientAction C (V i)- holderCoefficientAction (T i) (holderCoefficientAction D (V i)) have hVc : CauchySeq V := holder_cauchy_of_uniform_values hH hUl hUc have hTc : CauchySeq T := holder_cauchy_of_uniform_values hL hAl hAcc have hfc : CauchySeq f := cutoff_forcing_holder_cauchy C D hTc hVc apply hcauchy (fun i => cutoffCurveJet b (hu i)) T f (fun i => cutoffCurveJet_zero b (hu i) (h0 i)) (fun i => cutoffCurveJet_supported b (hu i)) (fun i => (hsmall (T i) (hAb i) (hAl i)).le) ?_ hTc hfc intro i apply cutoffCurveJet_CR b (hu i) (hcr i) (T i) (fun _ _ => rfl) (f i) intro z rfl end section open scoped ContDiff Topology BoundedContinuousFunction open Set Filter variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] local instance lsoInst1 : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup local instance lsoInst2 : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace local instance lsoInst3 : NormedAddCommGroup (COne ℂ E) := inferInstance local instance lsoInst4 : NormedSpace ℝ (COne ℂ E) := inferInstance local instance lsoInst5 : NormedAddCommGroup (HMap ℂ E) := inferInstance local instance lsoInst6 : NormedSpace ℝ (HMap ℂ E) := inferInstance local instance lsoInst7 : NormedAddCommGroup (HMap ℂ (E →L[ℝ] E)) := inferInstance local instance lsoInst8 : NormedSpace ℝ (HMap ℂ (E →L[ℝ] E)) := inferInstance lemma common_small_value_scale {L K d : ℝ} (hL : 0≤L) (hK : 0≤K) (hd : 0 nlinarith omit [CompleteSpace E] in lemma cutoffCurveJet_deriv_zero (b : ContDiffBump (0:ℂ)) {u : ℂ → E} (hu : ContDiff ℝ ∞ u) (h0 : u 0=0) : cDeriv (cutoffCurveJet b hu) 0=fderiv ℝ u 0 := by ext h rw [cutoffCurveJet_deriv,h0,smul_zero,zero_add, b.one_of_mem_closedBall (Metric.mem_closedBall_self b.rIn_pos.le),one_smul] theorem bounded_cutoff_gradient_small (b : ContDiffBump (0:ℂ)) {L H η : ℝ} (hL : 0≤L) (hH : 0≤H) (hη : 0<η) : ∃ δ : ℝ,0<δ ∧ ∀ (F : E → E →L[ℝ] E),Continuous F → F 0=0 → (∀ x y,‖F x-F y‖≤L*‖x-y‖) → ∀ u : ℂ → E,ContDiff ℝ ∞ u → u 0=0 → (∀ z,‖u z‖≤δ) → (∀ x y,‖u x-u y‖≤H*dist x y) → (∀ z,b z≠0 → fderiv ℝ u z Complex.I-Complex.I • fderiv ℝ u z 1= F (u z) (fderiv ℝ u z 1)) → ‖fderiv ℝ u 0‖<η := by obtain ⟨B,hB,hD,hDL,hC,hCL⟩ := cutoff_transport_bounds (E := E) b let M : ℝ := B*(H+1)+L*(B*(H+1))+L*H*B have hM : 0≤M := by dsimp [M]; positivity obtain ⟨d,hd,hsmall⟩ := small_value_elliptic_estimate (E := E) (isCompact_closedBall (0:ℂ) b.rOut) (mul_nonneg hL hH) hM hη obtain ⟨δ,hδ,hδ1,hδA,hδf⟩ := common_small_value_scale hL (by positivity : 0≤(1+L)*B) hd refine ⟨δ,hδ,?_⟩ intro F hF hF0 hFL u hu h0 hub hul hcr let A : ℂ → E →L[ℝ] E := fun z => F (u z) have hAc : Continuous A := hF.comp hu.continuous have hAb : ∀ z,‖A z‖≤L*δ := by intro z have hh := hFL (u z) 0 rw [hF0,sub_zero,sub_zero] at hh exact hh.trans (mul_le_mul_of_nonneg_left (hub z) hL) have hAl : ∀ x y,‖A x-A y‖≤(L*H)*dist x y := by intro x y exact (hFL (u x) (u y)).trans ((mul_le_mul_of_nonneg_left (hul x y) hL).trans_eq (by ring)) let ac : HMap ℂ (E →L[ℝ] E) := ofBoundedLipschitz A hAc (L*δ) (L*H) (mul_nonneg hL hδ.le) (mul_nonneg hL hH) hAb hAl have hbds := cutoff_forcing_bounds hδ.le hδ1 hH hL hB.le hub hul hAb hAl hC hCL hD hDL let g : ℂ → E := fun z => cutoffCommute b z (u z)-A z (cutoffTransport b 1 z (u z)) have hgc : Continuous g := ((cutoffCommute_smooth b).continuous.clm_apply hu.continuous).sub (hAc.clm_apply ((cutoffTransport_smooth b 1).continuous.clm_apply hu.continuous)) let f : HMap ℂ E := ofBoundedLipschitz g hgc (((1+L)*B)*δ) M (by positivity) hM hbds.1 hbds.2 have hAe : ∀ z,valueCLM _ ac z=A z := fun _ => rfl have hfe : ∀ z,valueCLM _ f z=g z := fun _ => rfl have heq : standardJetCR (cutoffCurveJet b hu)= holderCoefficientAction ac (jetDirectional 1 (cutoffCurveJet b hu))+f := by apply cutoffCurveJet_CR b hu hcr ac (fun z _ => hAe z) f intro z rw [hfe,hAe] exact rfl have hh := hsmall (cutoffCurveJet b hu) (cutoffCurveJet_zero b hu h0) (cutoffCurveJet_supported b hu) ac f (fun z => (hAb z).trans hδA) hAl (fun z => (hbds.1 z).trans hδf) hbds.2 heq have hder : ‖fderiv ℝ u 0‖≤‖cutoffCurveJet b hu‖ := by rw [←cutoffCurveJet_deriv_zero b hu h0] exact c_deriv_bound _ _ exact hder.trans_lt hh end section open scoped ContDiff Topology BoundedContinuousFunction open Set Filter variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] local instance legInst1 : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup local instance legInst2 : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace omit [CompleteSpace E] in lemma smooth_cutoff_local_bounds (b : ContDiffBump (0:ℂ)) {u : ℂ → E} (hu : ContDiff ℝ ∞ u) {R δ H : ℝ} (hR : b.rOut≤R) (hδ : 0≤δ) (hδ1 : δ≤1) (hH : 0≤H) (hb : ∀ z,‖z‖≤R → ‖u z‖≤δ) (hd : ∀ z,‖z‖≤R → ‖fderiv ℝ u z‖≤H) : ∃ K : ℝ,0≤K ∧ K=(boundedCThree_of_compactSupport b.contDiff b.hasCompactSupport).bound ∧ (∀ z,‖b z • u z‖≤δ) ∧ (∀ z,‖fderiv ℝ (fun z => b z • u z) z‖≤H+K) := by let hs := boundedCThree_of_compactSupport b.contDiff b.hasCompactSupport let K := hs.bound have hK : 0≤K := hs.bound_nonneg have hbn (z : ℂ) : ‖b z‖≤1 := by rw [Real.norm_eq_abs,abs_of_nonneg b.nonneg] exact b.le_one have hder (z : ℂ) : fderiv ℝ (fun z => b z • u z) z= b z • fderiv ℝ u z+(fderiv ℝ b z).smulRight (u z) := fderiv_fun_smul (b.contDiff.differentiable (by simp : (∞ : WithTop ℕ∞)≠0) z) (hu.differentiable (by simp : (∞ : WithTop ℕ∞)≠0) z) have hzouts (z : ℂ) (hz : ¬‖z‖≤R) : b z=0 ∧ fderiv ℝ b z=0 := by have hn : z∉tsupport b := by rw [b.tsupport_eq,Metric.mem_closedBall,dist_zero_right] exact fun hh => hz (hh.trans hR) exact ⟨image_eq_zero_of_notMem_tsupport hn,fderiv_of_notMem_tsupport ℝ hn⟩ refine ⟨K,hK,rfl,?_,?_⟩ · intro z by_cases hz : ‖z‖≤R · rw [norm_smul] exact (mul_le_mul (hbn z) (hb z hz) (norm_nonneg _) zero_le_one).trans_eq (one_mul _) · rw [(hzouts z hz).1,zero_smul,norm_zero]; exact hδ · intro z by_cases hz : ‖z‖≤R · rw [hder] have h1 : ‖b z • fderiv ℝ u z‖≤H := by rw [norm_smul] exact (mul_le_mul (hbn z) (hd z hz) (norm_nonneg _) zero_le_one).trans_eq (one_mul _) have h2 : ‖(fderiv ℝ b z).smulRight (u z)‖≤K := by rw [ContinuousLinearMap.norm_smulRight_apply] have h := mul_le_mul (hs.norm_fderiv_le z) (hb z hz) (norm_nonneg _) hK exact h.trans (by nlinarith) exact (norm_add_le _ _).trans (add_le_add h1 h2) · rw [hder,(hzouts z hz).1,(hzouts z hz).2] simp only [zero_smul,ContinuousLinearMap.zero_smulRight,add_zero,norm_zero] positivity theorem local_nonlinear_gradient_small {L H η : ℝ} (hL : 0≤L) (hH : 0≤H) (hη : 0<η) : ∃ δ : ℝ,0<δ ∧ ∀ (F : E → E →L[ℝ] E),Continuous F → F 0=0 → (∀ x y,‖F x-F y‖≤L*‖x-y‖) → ∀ u : ℂ → E,ContDiff ℝ ∞ u → u 0=0 → (∀ z,‖z‖≤2 → ‖u z‖≤δ) → (∀ z,‖z‖≤2 → ‖fderiv ℝ u z‖≤H) → (∀ z,‖z‖<2 → fderiv ℝ u z Complex.I-Complex.I • fderiv ℝ u z 1= F (u z) (fderiv ℝ u z 1)) → ‖fderiv ℝ u 0‖<η := by let b : ContDiffBump (0:ℂ) := ⟨1/4,1/2,by norm_num,by norm_num⟩ let c : ContDiffBump (0:ℂ) := ⟨1,3/2,by norm_num,by norm_num⟩ let hc := boundedCThree_of_compactSupport c.contDiff c.hasCompactSupport have hH' : 0≤H+hc.bound := add_nonneg hH hc.bound_nonneg obtain ⟨d,hd,hsmall⟩ := bounded_cutoff_gradient_small (E := E) b hL hH' hη let δ := min d 1 have hδ : 0<δ := lt_min hd zero_lt_one refine ⟨δ,hδ,?_⟩ intro F hF hF0 hFL u hu h0 hb hdu hcr let w : ℂ → E := fun z => c z • u z have hw : ContDiff ℝ ∞ w := c.contDiff.smul hu have hw0 : w 0=0 := by simp only [w,h0,smul_zero] obtain ⟨K,hK,hKe,hwb,hwd⟩ := smooth_cutoff_local_bounds c hu (by change (3:ℝ)/2≤2; norm_num) hδ.le (min_le_right _ _) hH hb hdu have hwl : ∀ x y,‖w x-w y‖≤(H+hc.bound)*dist x y := by intro x y rw [dist_eq_norm] apply norm_sub_le_of_deriv_bound (hw.differentiable (by simp)) intro z simpa only [hKe] using hwd z have hwne (z : ℂ) (hz : ‖z‖<1) : w =ᶠ[𝓝 z] u := by have he := c.eventuallyEq_one_of_mem_ball (show z∈Metric.ball (0:ℂ) c.rIn by simpa only [Metric.mem_ball,dist_zero_right] using hz) filter_upwards [he] with y hy change c y • u y=u y rw [hy]; exact one_smul ℝ (u y) have hcrw : ∀ z,b z≠0 → fderiv ℝ w z Complex.I-Complex.I • fderiv ℝ w z 1= F (w z) (fderiv ℝ w z 1) := by intro z hz have hz' : ‖z‖<(1:ℝ)/2 := by have hs : z∈Function.support b := hz rw [b.support_eq] at hs simpa only [Metric.mem_ball,dist_zero_right] using hs have he := hwne z (by linarith) rw [he.fderiv_eq,he.eq_of_nhds] exact hcr z (by linarith) have hh := hsmall F hF hF0 hFL w hw hw0 (fun z => (hwb z).trans (min_le_left _ _)) hwl hcrw rwa [(hwne 0 (by simp)).fderiv_eq] at hh end section open scoped ContDiff Topology BoundedContinuousFunction open Set Filter variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] local instance djcInst1 : NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup local instance djcInst2 : NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace local instance djcInst3 : NormedAddCommGroup (COne ℂ E) := inferInstance local instance djcInst4 : NormedSpace ℝ (COne ℂ E) := inferInstance lemma uniform_cauchy_comp_lipschitz {X Y Z : Type*} [NormedAddCommGroup Y] [NormedAddCommGroup Z] {F : ℕ → Y → Z} {u : ℕ → X → Y} {L M : ℝ} (hL : 0≤L) (hF : ∀ i x y,‖F i x-F i y‖≤L*‖x-y‖) (hb : ∀ i x,‖u i x‖≤M) (hu : ∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ x,‖u i x-u j x‖≤ε) (hFc : ∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ y,‖y‖≤M →‖F i y-F j y‖≤ε) : ∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ x,‖F i (u i x)-F j (u j x)‖≤ε := by intro ε hε obtain ⟨N,hN⟩ := hu (ε/(2*(L+1))) (div_pos hε (by linarith)) obtain ⟨M,hM⟩ := hFc (ε/2) (half_pos hε) refine ⟨max N M,fun i hi j hj x => ?_⟩ have h1 := hN i ((le_max_left _ _).trans hi) j ((le_max_left _ _).trans hj) x have h2 := hM i ((le_max_right _ _).trans hi) j ((le_max_right _ _).trans hj) (u j x) (hb j x) have hp : 0 ?_⟩ by_cases hz : ‖z‖≤R · rw [←smul_sub,norm_smul,Real.norm_eq_abs,abs_of_nonneg c.nonneg] exact (mul_le_mul c.le_one (hN i hi j hj z hz) (norm_nonneg _) zero_le_one).trans_eq (one_mul _) · have hz' : z∉tsupport c := by rw [c.tsupport_eq,Metric.mem_closedBall,dist_zero_right] exact fun hh => hz (hh.trans hc) simp only [image_eq_zero_of_notMem_tsupport hz',zero_smul,sub_self,norm_zero] exact hε.le def smallDiskBump : ContDiffBump (0:ℂ) := ⟨1/4,1/2,by norm_num,by norm_num⟩ def largerDiskBump : ContDiffBump (0:ℂ) := ⟨1,3/2,by norm_num,by norm_num⟩ omit [CompleteSpace E] in lemma double_disk_cutoff {u : ℂ → E} (hu : ContDiff ℝ ∞ u) : cutoffCurveJet smallDiskBump (largerDiskBump.contDiff.smul hu)=cutoffCurveJet smallDiskBump hu := by apply cValue_ext intro z simp only [cutoffCurveJet_value] by_cases hz : smallDiskBump z=0 · rw [hz,zero_smul,zero_smul] · have hr : ‖z‖<(1:ℝ)/2 := by have hh : z∈Function.support smallDiskBump := hz rw [smallDiskBump.support_eq,Metric.mem_ball,dist_zero_right] at hh exact hh have hc : largerDiskBump z=1 := largerDiskBump.one_of_mem_closedBall (by rw [Metric.mem_closedBall,dist_zero_right] change ‖z‖≤(1:ℝ) linarith) change smallDiskBump z • (largerDiskBump z • u z)=_ rw [hc,one_smul] theorem disk_nonlinear_jet_cauchy {L H : ℝ} (hL : 0≤L) (hH : 0≤H) : ∃ δ : ℝ,0<δ ∧ ∀ (u : ℕ → ℂ → E) (hu : ∀ i,ContDiff ℝ ∞ (u i)) (F : ℕ → E → E →L[ℝ] E), (∀ i,Continuous (F i)) →(∀ i,F i 0=0) → (∀ i x y,‖F i x-F i y‖≤L*‖x-y‖) →(∀ i,u i 0=0) → (∀ i z,‖z‖≤2 →‖u i z‖≤δ) → (∀ i z,‖z‖≤2 →‖fderiv ℝ (u i) z‖≤H) → (∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ z,‖z‖≤2 →‖u i z-u j z‖≤ε) → (∀ ε : ℝ,0<ε →∃ N : ℕ,∀ i≥N,∀ j≥N,∀ v,‖v‖≤δ →‖F i v-F j v‖≤ε) → (∀ i z,‖z‖<2 →fderiv ℝ (u i) z Complex.I-Complex.I • fderiv ℝ (u i) z 1= F i (u i z) (fderiv ℝ (u i) z 1)) → CauchySeq (fun i => cutoffCurveJet smallDiskBump (hu i)) := by let c := largerDiskBump let B := (boundedCThree_of_compactSupport c.contDiff c.hasCompactSupport).bound have hB : 0≤B := (boundedCThree_of_compactSupport c.contDiff c.hasCompactSupport).bound_nonneg obtain ⟨d,hd,hcauchy⟩ := cutoff_jet_cauchy (E := E) smallDiskBump (mul_nonneg hL (add_nonneg hH hB)) let δ := min 1 (d/(L+1)) have hδ : 0<δ := lt_min zero_lt_one (div_pos hd (by linarith)) have hδd : L*δ≤d := by have hh := (le_div_iff₀ (by linarith : 0 c z • u i z have hw : ∀ i,ContDiff ℝ ∞ (w i) := fun i => c.contDiff.smul (hu i) have hw0 : ∀ i,w i 0=0 := by intro i; simp only [w,h0 i,smul_zero] have hwb : ∀ i z,‖w i z‖≤δ := by intro i exact (smooth_cutoff_local_bounds c (hu i) (by change (3:ℝ)/2≤2; norm_num) hδ.le (min_le_left _ _) hH (hUb i) (hUd i)).choose_spec.2.2.1 have hwd : ∀ i z,‖fderiv ℝ (w i) z‖≤H+B := by intro i obtain ⟨K,hK,hKe,_,hh⟩ := smooth_cutoff_local_bounds c (hu i) (by change (3:ℝ)/2≤2; norm_num) hδ.le (min_le_left _ _) hH (hUb i) (hUd i) simpa only [hKe] using hh have hwl : ∀ i x y,‖w i x-w i y‖≤(H+B)*dist x y := by intro i x y rw [dist_eq_norm] exact norm_sub_le_of_deriv_bound ((hw i).differentiable (by simp)) (hwd i) x y have hwc := cutoff_uniform_cauchy (E := E) c (by change (3:ℝ)/2≤2; norm_num) hUc let A : ℕ → ℂ → E →L[ℝ] E := fun i z => F i (w i z) have hAb : ∀ i z,‖A i z‖≤d := by intro i z have hh := hFL i (w i z) 0 rw [hF0 i,sub_zero,sub_zero] at hh exact hh.trans ((mul_le_mul_of_nonneg_left (hwb i z) hL).trans hδd) have hAl : ∀ i x y,‖A i x-A i y‖≤(L*(H+B))*dist x y := by intro i x y exact (hFL i _ _).trans ((mul_le_mul_of_nonneg_left (hwl i x y) hL).trans_eq (by ring)) have hAc := uniform_cauchy_comp_lipschitz hL hFL hwb hwc hFuc have heq (i : ℕ) (z : ℂ) (hz : ‖z‖<1) : w i =ᶠ[𝓝 z] u i := by have he := c.eventuallyEq_one_of_mem_ball (show z∈Metric.ball (0:ℂ) c.rIn by rw [Metric.mem_ball,dist_zero_right] change ‖z‖<(1:ℝ) exact hz) filter_upwards [he] with y hy change c y • u i y=u i y rw [hy]; exact one_smul ℝ (u i y) have hres : ∀ i z,smallDiskBump z≠0 → fderiv ℝ (w i) z Complex.I-Complex.I • fderiv ℝ (w i) z 1=A i z (fderiv ℝ (w i) z 1) := by intro i z hz have hr : ‖z‖<(1:ℝ)/2 := by have hh : z∈Function.support smallDiskBump := hz rw [smallDiskBump.support_eq,Metric.mem_ball,dist_zero_right] at hh exact hh have he := heq i z (by linarith) change _=F i (w i z) _ rw [he.fderiv_eq,he.eq_of_nhds] exact hcr i z (by linarith) have hh := hcauchy w hw A (fun i => (hFc i).comp (hw i).continuous) hw0 hAb hAl ⟨δ,H+B,hδ.le,add_nonneg hH hB,hwb,hwl⟩ hwc hAc hres have he (i : ℕ) : cutoffCurveJet smallDiskBump (hw i)=cutoffCurveJet smallDiskBump (hu i) := by change cutoffCurveJet smallDiskBump (largerDiskBump.contDiff.smul (hu i))=_ exact double_disk_cutoff (hu i) simpa only [he] using hh end open scoped ContDiff Topology BoundedContinuousFunction open Set Filter variable {n : ℕ} def compatibleFrame (J : End n) : End n := cayleyP J 1 1 @[simp] lemma compatibleFrame_apply (J : End n) (v : Phase n) : compatibleFrame J v = -standardJ n (J v)+v := by simp only [compatibleFrame,cayleyP_apply,one_smul,relativeOperator_apply] lemma compatibleFrame_intertwine {J : End n} (hJ : Compatible J) (v : Phase n) : compatibleFrame J (J v) = standardJ n (compatibleFrame J v) := by simp only [compatibleFrame_apply,hJ.1,map_neg,map_add,standardJ_sq,neg_neg] abel lemma compatibleFrame_invertible {J : End n} (hJ : Compatible J) : (compatibleFrame J).IsInvertible := cayleyP_invertible hJ zero_lt_one zero_le_one lemma compatibleFrame_standard : compatibleFrame (standardJ n) = (2:ℝ) • ContinuousLinearMap.id ℝ (Phase n) := by ext v i simp only [compatibleFrame_apply,standardJ_sq,neg_neg,Pi.add_apply, smul_apply,ContinuousLinearMap.id_apply,Pi.smul_apply] exact (two_smul ℝ (v i)).symm lemma compatibleFrame_smooth {D : Type*} [NormedAddCommGroup D] [NormedSpace ℝ D] {A : D → End n} (hA : ContDiff ℝ ∞ A) : ContDiff ℝ ∞ (fun z => compatibleFrame (A z)) := by have hI : ContDiff ℝ ∞ (fun _ : D => standardJ n) := contDiff_const simpa only [compatibleFrame,cayleyP,one_smul,relativeOperator] using ((hI.clm_comp hA).neg.add (contDiff_const (c := ContinuousLinearMap.id ℝ (Phase n)))) lemma compatibleFrame_inverse_smooth {D : Type*} [NormedAddCommGroup D] [NormedSpace ℝ D] {A : D → End n} (hA : ContDiff ℝ ∞ A) (hc : ∀ z, Compatible (A z)) : ContDiff ℝ ∞ (fun z => (compatibleFrame (A z)).inverse) := by apply contDiff_iff_contDiffAt.mpr intro z exact (compatibleFrame_invertible (hc z)).contDiffAt_map_inverse.comp z (compatibleFrame_smooth hA).contDiffAt lemma compatibleFrame_inverse_left {J : End n} (hJ : Compatible J) (v : Phase n) : (compatibleFrame J).inverse (compatibleFrame J v) = v := by obtain ⟨e,he⟩ := compatibleFrame_invertible hJ rw [←he] simp only [ContinuousLinearMap.inverse_equiv,ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_apply_apply] lemma compatibleFrame_inverse_right {J : End n} (hJ : Compatible J) (v : Phase n) : compatibleFrame J ((compatibleFrame J).inverse v) = v := by obtain ⟨e,he⟩ := compatibleFrame_invertible hJ rw [←he] simp only [ContinuousLinearMap.inverse_equiv,ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply] lemma compatibleFrame_inverse_standard : (compatibleFrame (standardJ n)).inverse = (1/2:ℝ) • ContinuousLinearMap.id ℝ (Phase n) := by apply ContinuousLinearMap.ext intro v obtain ⟨e,he⟩ := compatibleFrame_invertible (compatible_standardJ n) apply (show Function.Injective (compatibleFrame (standardJ n)) by rw [←he]; exact e.injective) rw [compatibleFrame_inverse_right (compatible_standardJ n)] change v = compatibleFrame (standardJ n) ((1/2:ℝ) • v) rw [compatibleFrame_standard] simp only [smul_apply,ContinuousLinearMap.id_apply,smul_smul] norm_num variable {K : Set ℂ} (hK : IsCompact K) {A : ℂ → End n} (hA : ContDiff ℝ ∞ A) (hc : ∀ z, Compatible (A z)) (hstd : ∀ z ∉ K, A z = standardJ n) include hK hstd in lemma compatibleFrame_compactPerturbation : HasCompactSupport (fun z => compatibleFrame (A z) - (2:ℝ) • ContinuousLinearMap.id ℝ (Phase n)) := by apply HasCompactSupport.of_support_subset_isCompact hK intro z hz by_contra hzK apply hz change compatibleFrame (A z) - (2:ℝ) • ContinuousLinearMap.id ℝ (Phase n) = 0 rw [hstd z hzK,compatibleFrame_standard,sub_self] include hK hstd in lemma compatibleFrame_inverse_compactPerturbation : HasCompactSupport (fun z => (compatibleFrame (A z)).inverse - (1/2:ℝ) • ContinuousLinearMap.id ℝ (Phase n)) := by apply HasCompactSupport.of_support_subset_isCompact hK intro z hz by_contra hzK apply hz change (compatibleFrame (A z)).inverse - (1/2:ℝ) • ContinuousLinearMap.id ℝ (Phase n) = 0 rw [hstd z hzK,compatibleFrame_inverse_standard,sub_self] def compatibleFrameJet : COne ℂ (End n) := ofBoundedCThree (boundedCThree_of_compactPerturbation (compatibleFrame_smooth hA) _ (compatibleFrame_compactPerturbation hK hstd)) def compatibleInverseJet : COne ℂ (End n) := ofBoundedCThree (boundedCThree_of_compactPerturbation (compatibleFrame_inverse_smooth hA hc) _ (compatibleFrame_inverse_compactPerturbation hK hstd)) @[simp] lemma compatibleFrameJet_value (z : ℂ) : cValue (compatibleFrameJet hK hA hstd) z = compatibleFrame (A z) := rfl @[simp] lemma compatibleInverseJet_value (z : ℂ) : cValue (compatibleInverseJet hK hA hc hstd) z = (compatibleFrame (A z)).inverse := rfl def compatibleMarkedEquiv : markedModel (E := Phase n) K ≃L[ℝ] markedModel (E := Phase n) K := markedCoefficientEquiv hK.isClosed (compatibleFrameJet hK hA hstd) (compatibleInverseJet hK hA hc hstd) 2 (1/2) (by intro z hz; rw [compatibleFrameJet_value,hstd z hz,compatibleFrame_standard]) (by intro z hz; rw [compatibleInverseJet_value,hstd z hz,compatibleFrame_inverse_standard]) (by intro z v; exact compatibleFrame_inverse_left (hc z) v) (by intro z v; exact compatibleFrame_inverse_right (hc z) v) def compatibleResidualEquiv : supportedHolder (E := Phase n) K ≃L[ℝ] supportedHolder (E := Phase n) K := supportedCoefficientEquiv (jetValueCLM _ (compatibleFrameJet hK hA hstd)) (jetValueCLM _ (compatibleInverseJet hK hA hc hstd)) (by intro z v; exact compatibleFrame_inverse_left (hc z) v) (by intro z v; exact compatibleFrame_inverse_right (hc z) v) end HigherDimensionalBallPacking.Rigidity.HolderCompletion end end OAI