import Mathlib import OAI.Geometry.BallPacking.Necessity.NonlinearCR namespace OAI noncomputable section open scoped ContDiff Topology open Set Function Filter open scoped ContDiff Topology Manifold open Set Function Filter MeasureTheory open Set Function MeasureTheory open Set Function open SymplecticBallPacking.Hamiltonian (Plane planarCurl) open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm radiusSq planarArea planarArea_apply) open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm) open SymplecticBallPacking.Hamiltonian (Plane angularOneForm) open SymplecticBallPacking.Hamiltonian open SymplecticBallPacking.Hamiltonian (Plane) open Set Filter Function open Set Filter MeasureTheory open scoped Topology open Set Filter Finset open scoped ContDiff Topology Classical open Set Filter open scoped BoundedContinuousFunction ContDiff Topology open Set Function Filter Topology open scoped NNReal open scoped ContDiff Topology BoundedContinuousFunction namespace HigherDimensionalBallPacking.Rigidity open scoped BoundedContinuousFunction ContDiff Topology variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] local instance crFDerivC1NormedAddCommGroup (α : ℝ) : NormedAddCommGroup (C1HolderSpace E α) := inferInstance local instance crFDerivC1NormedSpace (α : ℝ) : NormedSpace ℝ (C1HolderSpace E α) := inferInstance local instance crFDerivHolderNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance local instance crFDerivHolderNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance theorem c1HolderCR_fderiv (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (J : E → E →L[ℝ] E) (J₀ : E →L[ℝ] E) (hJ : ContDiff ℝ ∞ J) (hc : HasCompactSupport (fun x => J x-J₀)) (c : E) (A : ℂ →L[ℝ] E) (u h : C1HolderSpace E α) (z : ℂ) : holderValue α (fderiv ℝ (c1HolderCR α hα₀ hα₁ J J₀ hJ hc c A) u h) z = holderValue α (c1HolderDeriv α h) z Complex.I - (fderiv ℝ J (c1HolderAffineCurve α c A u z) (holderValue α (c1HolderValue α h) z)) (fderiv ℝ (c1HolderAffineCurve α c A u) z 1) - J (c1HolderAffineCurve α c A u z) (holderValue α (c1HolderDeriv α h) z 1) := by let ev : C1HolderSpace E α →L[ℝ] E := (BoundedContinuousFunction.evalCLM ℝ z).comp ((holderValue α).comp (c1HolderValue α)) let dev : C1HolderSpace E α →L[ℝ] (ℂ →L[ℝ] E) := (BoundedContinuousFunction.evalCLM ℝ z).comp ((holderValue α).comp (c1HolderDeriv α)) have hf : HasFDerivAt (fun v : C1HolderSpace E α => c1HolderAffineCurve α c A v z) ev u := ev.hasFDerivAt.const_add (c+A z) have hs : HasFDerivAt (fun v : C1HolderSpace E α => A+holderValue α (c1HolderDeriv α v) z) dev u := dev.hasFDerivAt.const_add A have hdJ := ((hJ.differentiable (by simp)) (c1HolderAffineCurve α c A u z)).hasFDerivAt.comp u hf have hdI := hs.clm_apply (hasFDerivAt_const Complex.I u) have hd1 := hs.clm_apply (hasFDerivAt_const (1:ℂ) u) have hd := hdI.sub (hdJ.clm_apply hd1) let hev : HolderSpace ℂ E α →L[ℝ] E := (BoundedContinuousFunction.evalCLM ℝ z).comp (holderValue α) have hout := hev.hasFDerivAt.comp u (((c1HolderCR_contDiff α hα₀ hα₁ J J₀ hJ hc c A).differentiable (by simp)) u).hasFDerivAt change HasFDerivAt (fun v : C1HolderSpace E α => holderValue α (c1HolderCR α hα₀ hα₁ J J₀ hJ hc c A v) z) _ u at hout simp only [c1HolderCR_apply,c1HolderAffineCurve_fderiv] at hout have he := congrArg (fun L : C1HolderSpace E α →L[ℝ] E => L h) (hout.unique hd) simp only [hev,ev,dev,ContinuousLinearMap.comp_apply,BoundedContinuousFunction.evalCLM_apply, sub_apply,add_apply,ContinuousLinearMap.flip_apply,zero_apply,map_zero,zero_add, Function.comp_def,sub_add_eq_sub_sub_swap] at he simpa only [c1HolderAffineCurve_fderiv, add_apply] using he end HigherDimensionalBallPacking.Rigidity namespace HigherDimensionalBallPacking.Rigidity open scoped BoundedContinuousFunction ContDiff Topology open Set Filter universe u v variable {K : Type u} [MetricSpace K] variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] local instance cutoffHolderNormedAddCommGroup (G : Type) [NormedAddCommGroup G] [NormedSpace ℝ G] (α : ℝ) : NormedAddCommGroup (HolderSpace K G α) := inferInstance local instance cutoffHolderNormedSpace (G : Type) [NormedAddCommGroup G] [NormedSpace ℝ G] (α : ℝ) : NormedSpace ℝ (HolderSpace K G α) := inferInstance def holderConstCLM (α : ℝ) : E →L[ℝ] HolderSpace K E α := LinearMap.mkContinuous { toFun := holderConst α map_add' := fun c d => by apply holderValue_injective α; ext z; rfl map_smul' := fun c d => by apply holderValue_injective α; ext z; rfl } 1 (fun c => by change max ‖BoundedContinuousFunction.const K c‖ ‖(0 : OffDiagonal K →ᵇ E)‖ ≤ 1*‖c‖ rw [norm_zero,one_mul] exact max_le (BoundedContinuousFunction.norm_const_le c) (norm_nonneg c)) @[simp] theorem holderConstCLM_apply (α : ℝ) (c : E) : holderConstCLM (K := K) α c = holderConst α c := rfl def holderSourceCutoff (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (b : ContDiffBump (0:ℂ)) : HolderSpace ℂ ℝ α := compactSmoothHolder α hα₀ hα₁ b b.contDiff b.hasCompactSupport def holderSourceCutoffLinear (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (b : ContDiffBump (0:ℂ)) : HolderSpace ℂ ℂ α := compactSmoothHolder α hα₀ hα₁ (fun z => b z • z) (b.contDiff.smul contDiff_id) b.hasCompactSupport.smul_right def holderCutoffAffine (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (b : ContDiffBump (0:ℂ)) (p : E × (ℂ →L[ℝ] E)) : HolderSpace ℂ E α := holderCLM α (holderLinearPost α (ContinuousLinearMap.lsmul ℝ ℝ) (holderSourceCutoff α hα₀ hα₁ b)) (holderConst α p.1) + holderLinearPost α p.2 (holderSourceCutoffLinear α hα₀ hα₁ b) @[simp] theorem holderCutoffAffine_apply (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (b : ContDiffBump (0:ℂ)) (p : E × (ℂ →L[ℝ] E)) (z : ℂ) : holderValue α (holderCutoffAffine α hα₀ hα₁ b p) z = b z • (p.1+p.2 z) := by change b z • p.1+p.2 (b z • z) = _ rw [map_smul,smul_add] theorem holderCutoffAffine_contDiff (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (b : ContDiffBump (0:ℂ)) : ContDiff ℝ ∞ (holderCutoffAffine (E := E) α hα₀ hα₁ b) := by have hconst : ContDiff ℝ ∞ (fun p : E × (ℂ →L[ℝ] E) => holderConst (K := ℂ) α p.1) := (holderConstCLM (K := ℂ) (E := E) α).contDiff.comp contDiff_fst have hlin : ContDiff ℝ ∞ (fun p : E × (ℂ →L[ℝ] E) => holderLinearPost α p.2 (holderSourceCutoffLinear α hα₀ hα₁ b)) := by exact ((holderOperatorCLM (K := ℂ) (E := ℂ) (F := E) α).contDiff.comp ((holderConstCLM (K := ℂ) (E := ℂ →L[ℝ] E) α).contDiff.comp contDiff_snd)).clm_apply contDiff_const let M : HolderSpace ℂ (E →L[ℝ] E) α := holderLinearPost α (ContinuousLinearMap.lsmul ℝ ℝ) (holderSourceCutoff α hα₀ hα₁ b) have hmul := (holderCLM (K := ℂ) (E := E) (F := E) α M).contDiff.comp hconst exact hmul.add hlin end HigherDimensionalBallPacking.Rigidity namespace HigherDimensionalBallPacking.Rigidity open scoped BoundedContinuousFunction ContDiff Topology open Set Filter variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] local instance affineFamilyDomainNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance local instance affineFamilyDomainNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance local instance affineFamilyCodomainNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ F α) := inferInstance local instance affineFamilyCodomainNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ F α) := inferInstance omit [CompleteSpace E] in theorem affineProfile_lower_bound (c d v : E) (A B : ℂ →L[ℝ] E) (m U R t : ℝ) (_hm : 0 < m) (hA : ∀ z, m*‖z‖≤‖A z‖) (hd : ‖d‖≤1) (hB : ‖B‖≤ m/2) (hv : ‖v‖≤U) (ht : t∈Icc (0:ℝ) 1) (z : ℂ) (hz : R+‖c‖+1+U ≤ m/2*‖z‖) : R ≤ ‖c+A z+t • (d+B z)+v‖ := by have hb : ‖t • (d+B z)‖ ≤ 1+(m/2)*‖z‖ := by rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg ht.1] calc t*‖d+B z‖ ≤ ‖d+B z‖ := mul_le_of_le_one_left (norm_nonneg _) ht.2 _ ≤ ‖d‖+‖B z‖ := norm_add_le _ _ _ ≤ 1+‖B‖*‖z‖ := add_le_add hd (B.le_opNorm z) _ ≤ 1+(m/2)*‖z‖ := add_le_add (le_refl 1) (mul_le_mul_of_nonneg_right hB (norm_nonneg z)) have hrev : ‖A z‖ ≤ ‖c+A z+t • (d+B z)+v‖+‖c‖+1+(m/2)*‖z‖+U := by calc ‖A z‖ = ‖(c+A z+t • (d+B z)+v)-(c+t • (d+B z)+v)‖ := by congr 1; abel _ ≤ ‖c+A z+t • (d+B z)+v‖+‖c+t • (d+B z)+v‖ := norm_sub_le _ _ _ ≤ ‖c+A z+t • (d+B z)+v‖+((‖c‖+‖t • (d+B z)‖)+‖v‖) := add_le_add (le_refl _) ((norm_add_le _ _).trans (add_le_add (norm_add_le _ _) (le_refl _))) _ ≤ ‖c+A z+t • (d+B z)+v‖+‖c‖+1+(m/2)*‖z‖+U := by linarith have ha := hA z linarith omit [NormedSpace ℝ E] [CompleteSpace E] [NormedSpace ℝ F] [CompleteSpace F] in theorem compactCoefficient_zero_outside {f : E → F} (hc : HasCompactSupport f) : ∃ R : ℝ, 0 ?_⟩ apply image_eq_zero_of_notMem_tsupport intro h have hh := hs h simp only [Metric.mem_ball,dist_zero_right] at hh exact (not_lt_of_ge hx) hh def holderAffineFamily (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) : HolderSpace ℂ F α := holderProfilePost α hα₀ hα₁ (realAffineProfile p.1.1 p.1.2) (realAffineProfile_lipschitz p.1.1 p.1.2) f hf hc p.2 omit [CompleteSpace E] in @[simp] theorem holderAffineFamily_apply (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) (z : ℂ) : holderValue α (holderAffineFamily α hα₀ hα₁ f hf hc p) z = f (p.1.1+p.1.2 z+holderValue α p.2 z) := rfl def localizedAffineFamily (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (c : E) (A : ℂ →L[ℝ] E) (b : ContDiffBump (0:ℂ)) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) : HolderSpace ℂ F α := holderProfilePost α hα₀ hα₁ (realAffineProfile c A) (realAffineProfile_lipschitz c A) f hf hc (holderCutoffAffine α hα₀ hα₁ b (p.1.1-c,p.1.2-A)+p.2) omit [CompleteSpace E] in theorem localizedAffineFamily_contDiff (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (c : E) (A : ℂ →L[ℝ] E) (b : ContDiffBump (0:ℂ)) : ContDiff ℝ ∞ (localizedAffineFamily α hα₀ hα₁ f hf hc c A b) := by apply (holderProfilePost_contDiff α hα₀ hα₁ (realAffineProfile c A) (realAffineProfile_lipschitz c A) f hf hc).comp exact ((holderCutoffAffine_contDiff α hα₀ hα₁ b).comp (((contDiff_fst.fst).sub contDiff_const).prodMk ((contDiff_fst.snd).sub contDiff_const))).add contDiff_snd omit [CompleteSpace E] in @[simp] theorem localizedAffineFamily_apply (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (c : E) (A : ℂ →L[ℝ] E) (b : ContDiffBump (0:ℂ)) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) (z : ℂ) : holderValue α (localizedAffineFamily α hα₀ hα₁ f hf hc c A b p) z = f (c+A z+b z • ((p.1.1-c)+(p.1.2-A) z)+holderValue α p.2 z) := by change f (c+A z+(holderValue α (holderCutoffAffine α hα₀ hα₁ b (p.1.1-c,p.1.2-A)) z+ holderValue α p.2 z)) = _ rw [holderCutoffAffine_apply] simp only [add_assoc] omit [CompleteSpace E] in theorem localizedAffineFamily_eq (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (c : E) (A : ℂ →L[ℝ] E) (b : ContDiffBump (0:ℂ)) (m U R : ℝ) (hm : 0 < m) (hA : ∀ z, m*‖z‖≤‖A z‖) (hfR : ∀ x, R≤‖x‖ → f x=0) (hT : R+‖c‖+1+U ≤ m/2*b.rIn) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) (hd : ‖p.1.1-c‖≤1) (hB : ‖p.1.2-A‖≤ m/2) (hu : ‖p.2‖≤U) : localizedAffineFamily α hα₀ hα₁ f hf hc c A b p = holderAffineFamily α hα₀ hα₁ f hf hc p := by apply holderValue_injective α ext z rw [localizedAffineFamily_apply,holderAffineFamily_apply] by_cases hz : ‖z‖≤b.rIn · have hb : b z=1 := b.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right] using hz) rw [hb,one_smul] congr 1 simp only [sub_apply] abel · have hz' : R+‖c‖+1+U ≤ m/2*‖z‖ := hT.trans (mul_le_mul_of_nonneg_left (le_of_not_ge hz) (half_pos hm).le) have hv : ‖holderValue α p.2 z‖≤U := ((holderValue α p.2).norm_coe_le_norm z).trans ((holderValue_norm α p.2).trans hu) have ht : b z∈Icc (0:ℝ) 1 := ⟨b.nonneg,b.le_one⟩ rw [hfR _ (affineProfile_lower_bound c (p.1.1-c) (holderValue α p.2 z) A (p.1.2-A) m U R (b z) hm hA hd hB hv ht z hz')] symm apply hfR convert affineProfile_lower_bound c (p.1.1-c) (holderValue α p.2 z) A (p.1.2-A) m U R 1 hm hA hd hB hv (by constructor <;> norm_num) z hz' using 1 congr 1 simp only [one_smul,sub_apply] abel omit [CompleteSpace E] in theorem holderAffineFamily_contDiffAt (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : E → F) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (p : (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α) (m : ℝ) (hm : 0 < m) (hA : ∀ z, m*‖z‖≤‖p.1.2 z‖) : ContDiffAt ℝ ∞ (holderAffineFamily α hα₀ hα₁ f hf hc) p := by obtain ⟨R,hR,hfR⟩ := compactCoefficient_zero_outside hc let U : ℝ := ‖p.2‖+1 let T : ℝ := (R+‖p.1.1‖+1+U)/(m/2) have hTpos : 0 q.1.1) from continuous_fst.fst).continuousAt.tendsto.eventually (Metric.ball_mem_nhds p.1.1 (by norm_num : (0:ℝ)<1)) simpa only [Metric.mem_ball,dist_eq_norm] using h have hdA : ∀ᶠ q : P in 𝓝 p, ‖q.1.2-p.1.2‖< m/2 := by have h := (show Continuous (fun q : P => q.1.2) from continuous_fst.snd).continuousAt.tendsto.eventually (Metric.ball_mem_nhds p.1.2 (half_pos hm)) simpa only [Metric.mem_ball,dist_eq_norm] using h have hdu : ∀ᶠ q : P in 𝓝 p, ‖q.2-p.2‖<1 := by have h := (show Continuous (fun q : P => q.2) from continuous_snd).continuousAt.tendsto.eventually (Metric.ball_mem_nhds p.2 (by norm_num : (0:ℝ)<1)) simpa only [Metric.mem_ball,dist_eq_norm] using h apply (localizedAffineFamily_contDiff α hα₀ hα₁ f hf hc p.1.1 p.1.2 b).contDiffAt.congr_of_eventuallyEq filter_upwards [hdc,hdA,hdu] with q hqc hqA hqu have hqu' : ‖q.2‖≤U := by calc ‖q.2‖ ≤ ‖q.2-p.2‖+‖p.2‖ := norm_le_norm_sub_add q.2 p.2 _ ≤ 1+‖p.2‖ := add_le_add (le_of_lt hqu) (le_refl _) _ = U := add_comm _ _ exact (localizedAffineFamily_eq α hα₀ hα₁ f hf hc p.1.1 p.1.2 b m U R hm hA hfR hT q hqc.le hqA.le hqu').symm end HigherDimensionalBallPacking.Rigidity namespace HigherDimensionalBallPacking.Rigidity open scoped BoundedContinuousFunction ContDiff Topology open Set Filter variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] local instance crFamilyHolderNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance local instance crFamilyHolderNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance local instance crFamilyC1NormedAddCommGroup (α : ℝ) : NormedAddCommGroup (C1HolderSpace E α) := inferInstance local instance crFamilyC1NormedSpace (α : ℝ) : NormedSpace ℝ (C1HolderSpace E α) := inferInstance def c1HolderCRFamily (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (J : E → E →L[ℝ] E) (J₀ : E →L[ℝ] E) (hJ : ContDiff ℝ ∞ J) (hc : HasCompactSupport (fun x => J x-J₀)) (p : (E × (ℂ →L[ℝ] E)) × C1HolderSpace E α) : HolderSpace ℂ E α := c1HolderCR α hα₀ hα₁ J J₀ hJ hc p.1.1 p.1.2 p.2 theorem c1HolderCRFamily_contDiffAt (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (J : E → E →L[ℝ] E) (J₀ : E →L[ℝ] E) (hJ : ContDiff ℝ ∞ J) (hc : HasCompactSupport (fun x => J x-J₀)) (p : (E × (ℂ →L[ℝ] E)) × C1HolderSpace E α) (m : ℝ) (hm : 0 < m) (hA : ∀ z, m*‖z‖≤‖p.1.2 z‖) : ContDiffAt ℝ ∞ (c1HolderCRFamily α hα₀ hα₁ J J₀ hJ hc) p := by let P := (E × (ℂ →L[ℝ] E)) × C1HolderSpace E α let Q : P → (E × (ℂ →L[ℝ] E)) × HolderSpace ℂ E α := fun q => (q.1,c1HolderValue α q.2) have hQ : ContDiff ℝ ∞ Q := contDiff_fst.prodMk ((c1HolderValue (E := E) α).contDiff.comp contDiff_snd) let B : P → HolderSpace ℂ (E →L[ℝ] E) α := fun q => holderConst α J₀ + holderAffineFamily α hα₀ hα₁ (fun x => J x-J₀) (hJ.sub contDiff_const) hc (Q q) have hB : ContDiffAt ℝ ∞ B p := contDiffAt_const.add ((holderAffineFamily_contDiffAt α hα₀ hα₁ (fun x => J x-J₀) (hJ.sub contDiff_const) hc (Q p) m hm hA).comp p hQ.contDiffAt) let D : P → HolderSpace ℂ (ℂ →L[ℝ] E) α := fun q => holderConst α q.1.2+c1HolderDeriv α q.2 have hD : ContDiff ℝ ∞ D := ((holderConstCLM (K := ℂ) (E := ℂ →L[ℝ] E) α).contDiff.comp contDiff_fst.snd).add ((c1HolderDeriv (E := E) α).contDiff.comp contDiff_snd) have hop := (holderOperatorCLM (K := ℂ) (E := E) (F := E) α).contDiff.contDiffAt.comp p hB have hI := (holderJetEval (K := ℂ) (E := E) α Complex.I).contDiff.contDiffAt.comp p hD.contDiffAt have h1 := (holderJetEval (K := ℂ) (E := E) α 1).contDiff.contDiffAt.comp p hD.contDiffAt exact hI.sub (hop.clm_apply h1) end HigherDimensionalBallPacking.Rigidity open scoped ContDiff Topology BoundedContinuousFunction open Set Function Filter namespace HigherDimensionalBallPacking.Rigidity variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] local instance graphHolderNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance local instance graphHolderNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance local instance graphC1NormedAddCommGroup (α : ℝ) : NormedAddCommGroup (C1HolderSpace E α) := inferInstance local instance graphC1NormedSpace (α : ℝ) : NormedSpace ℝ (C1HolderSpace E α) := inferInstance lemma c1HolderGraph_mem_of_hasFDerivAt (α : ℝ) (f : HolderSpace ℂ E α) (D : HolderSpace ℂ (ℂ →L[ℝ] E) α) (hD : ∀ z, HasFDerivAt (holderValue α f) (holderValue α D z) z) : (f,D) ∈ c1HolderGraph α := by intro x y change holderValue α f y-holderValue α f x = linearSegmentIntegral x y (holderValue α D) rw [linearSegmentIntegral_apply] have hd (t : ℝ) : HasDerivAt (fun s : ℝ => holderValue α f (x+s • (y-x))) (holderValue α D (x+t • (y-x)) (y-x)) t := by simpa only [id_eq,one_smul,Function.comp_def] using ((hD (x+t • (y-x))).comp_hasDerivAt t (((hasDerivAt_id t).smul_const (y-x)).const_add x)) symm simpa only [one_smul,zero_smul,add_zero,add_sub_cancel] using intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t) ((segmentIntegrand_continuous (holderValue α D) x y).intervalIntegrable (0:ℝ) 1) def compactSmoothC1Holder (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : ℂ → E) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) : C1HolderSpace E α := ⟨(compactSmoothHolder α hα₀ hα₁ f hf hc, compactSmoothHolder α hα₀ hα₁ (fderiv ℝ f) (hf.fderiv_right (by simp)) (hc.fderiv ℝ)), c1HolderGraph_mem_of_hasFDerivAt α _ _ (fun z => (hf.differentiable (by simp) z).hasFDerivAt)⟩ @[simp] lemma compactSmoothC1Holder_value (α : ℝ) (hα₀ : 0≤α) (hα₁ : α≤1) (f : ℂ → E) (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (z : ℂ) : holderValue α (c1HolderValue α (compactSmoothC1Holder α hα₀ hα₁ f hf hc)) z=f z := rfl def c1HolderMultiply (α : ℝ) (P : C1HolderSpace (E →L[ℝ] F) α) : C1HolderSpace E α →L[ℝ] C1HolderSpace F α := by let V := (holderCLM α (c1HolderValue α P)).comp (c1HolderValue α) let A : HolderSpace ℂ ((ℂ →L[ℝ] E) →L[ℝ] (ℂ →L[ℝ] F)) α := holderMap α (ContinuousLinearMap.compL ℝ ℂ E F) (c1HolderValue α P) let B : HolderSpace ℂ (E →L[ℝ] (ℂ →L[ℝ] F)) α := holderMap α (ContinuousLinearMap.flipₗᵢ ℝ ℂ E F).toContinuousLinearEquiv.toContinuousLinearMap (c1HolderDeriv α P) let D := ((holderCLM α A).comp (c1HolderDeriv α))+((holderCLM α B).comp (c1HolderValue α)) exact (V.prod D).codRestrict (c1HolderGraph α) (by intro u apply c1HolderGraph_mem_of_hasFDerivAt α intro z exact (c1Holder_hasFDerivAt α P z).clm_apply (c1Holder_hasFDerivAt α u z)) @[simp] lemma c1HolderMultiply_value (α : ℝ) (P : C1HolderSpace (E →L[ℝ] F) α) (u : C1HolderSpace E α) (z : ℂ) : holderValue α (c1HolderValue α (c1HolderMultiply α P u)) z = holderValue α (c1HolderValue α P) z (holderValue α (c1HolderValue α u) z) := rfl def c1HolderConstantCLM (α : ℝ) : E →L[ℝ] C1HolderSpace E α := ((holderConstantCLM α).prod 0).codRestrict (c1HolderGraph α) (by intro c apply c1HolderGraph_mem_of_hasFDerivAt α intro z exact hasFDerivAt_const c z) @[simp] lemma c1HolderConstantCLM_value (α : ℝ) (c : E) (z : ℂ) : holderValue α (c1HolderValue α (c1HolderConstantCLM α c)) z = c := rfl end HigherDimensionalBallPacking.Rigidity open scoped ContDiff Topology BoundedContinuousFunction open Set Function Filter namespace HigherDimensionalBallPacking.Rigidity variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] local instance markedProfileHolderNormedAddCommGroup (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance local instance markedProfileHolderNormedSpace (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance local instance markedProfileC1NormedAddCommGroup (α : ℝ) : NormedAddCommGroup (C1HolderSpace E α) := inferInstance local instance markedProfileC1NormedSpace (α : ℝ) : NormedSpace ℝ (C1HolderSpace E α) := inferInstance local instance markedProfileCompactNormedAddCommGroup (R : ℝ) : NormedAddCommGroup (CompactHolderSpace E R) := inferInstance local instance markedProfileCompactNormedSpace (R : ℝ) : NormedSpace ℝ (CompactHolderSpace E R) := inferInstance def markedFramedProfile (R : ℝ) (Ainf : E →L[ℝ] E) (P : ℂ → E →L[ℝ] E) (hP : ContDiff ℝ ∞ P) (hc : HasCompactSupport P) : CompactHolderSpace E R →L[ℝ] (E × (ℂ →L[ℝ] E)) × C1HolderSpace E ((1:ℝ)/3) := by let C := compactCRInverse (E := E) R let C₀ := (c1HolderEval ((1:ℝ)/3) 0).comp C let D := ((c1HolderEval ((1:ℝ)/3) 1).comp C)-C₀ let p := compactSmoothC1Holder ((1:ℝ)/3) (by norm_num) (by norm_num) P hP hc let N := compactSmoothC1Holder ((1:ℝ)/3) (by norm_num) (by norm_num) (slopeCoefficient P) (slopeCoefficient_smooth P hP) (slopeCoefficient_compact P hc) let A := c1HolderConstantCLM ((1:ℝ)/3) Ainf+p let B := (c1HolderMultiply ((1:ℝ)/3) A).comp C- ((c1HolderMultiply ((1:ℝ)/3) p).comp (c1HolderConstantCLM ((1:ℝ)/3))).comp C₀- ((c1HolderMultiply ((1:ℝ)/3) N).comp (c1HolderConstantCLM ((1:ℝ)/3))).comp D let a := -(Ainf.comp C₀) let d := -(((ContinuousLinearMap.compL ℝ ℂ E E) Ainf).comp (((ContinuousLinearMap.lsmul ℝ ℂ : ℂ →L[ℝ] E →L[ℝ] E).flip).comp D)) exact (a.prod d).prod B lemma markedFramedProfile_value (R : ℝ) (Ainf : E →L[ℝ] E) (P : ℂ → E →L[ℝ] E) (hP : ContDiff ℝ ∞ P) (hc : HasCompactSupport P) (g : CompactHolderSpace E R) (z : ℂ) : (markedFramedProfile R Ainf P hP hc g).1.1+ (markedFramedProfile R Ainf P hP hc g).1.2 z+ holderValue ((1:ℝ)/3) (c1HolderValue ((1:ℝ)/3) (markedFramedProfile R Ainf P hP hc g).2) z = (Ainf+P z) (markedCRInverse R g z) := by change -Ainf (compactCRInverseValue R g 0) + -Ainf (z • (compactCRInverseValue R g 1-compactCRInverseValue R g 0))+ ((Ainf+P z) (compactCRInverseValue R g z)-P z (compactCRInverseValue R g 0)- P z (z • (compactCRInverseValue R g 1-compactCRInverseValue R g 0))) = _ simp only [markedCRInverse,map_sub,add_apply] abel lemma markedFramedProfile_marks (R : ℝ) (Ainf : E →L[ℝ] E) (P : ℂ → E →L[ℝ] E) (hP : ContDiff ℝ ∞ P) (hc : HasCompactSupport P) (g : CompactHolderSpace E R) : (∀ z ∈ ({0,1} : Set ℂ), (markedFramedProfile R Ainf P hP hc g).1.1+ (markedFramedProfile R Ainf P hP hc g).1.2 z+ holderValue ((1:ℝ)/3) (c1HolderValue ((1:ℝ)/3) (markedFramedProfile R Ainf P hP hc g).2) z=0) := by intro z hz rw [markedFramedProfile_value] simp only [mem_insert_iff,mem_singleton_iff] at hz rcases hz with rfl|rfl <;> simp end HigherDimensionalBallPacking.Rigidity end end OAI