| import Mathlib | |
| import OAI.Geometry.SmoothYau.Spectrum.ProfileFrequencyEnergy | |
| namespace OAI | |
| noncomputable section | |
| namespace YauCounterexamples | |
| section | |
| open Set Filter | |
| open scoped Topology ContDiff RealInnerProductSpace | |
| theorem compact_profile_slope_bound (g : SmoothMetric NormalWaveSpace NormalWaveSpace) | |
| {K : Set NormalWaveSpace} (hK : IsCompact K) : | |
| ∃ H > 0, ∀ φ : NormalWaveSpace → ℝ, ContDiff ℝ ∞ φ → | |
| ∀ x ∈ K, ‖fderiv ℝ φ x‖ ≤ H*profileFrequencyScale g φ x := by | |
| obtain ⟨S,hS,hbound⟩ := metricFrameSet_inverse_bound g hK | |
| refine ⟨S,hS,?_⟩ | |
| intro φ hφ x hx | |
| obtain ⟨F,hF⟩ := metricFrameSet_fiber_nonempty g hx | |
| let q : NormalWaveParameter := (x,F) | |
| have hq : q ∈ metricFrameSet g K := hF | |
| let a := gradient (normalWaveProfile g φ q) 0 | |
| have ha : ‖a‖ ≤ profileFrequencyScale g φ x := by | |
| have he := profileFrequencyEnergy_frame g hφ hq | |
| have hs := profileFrequencyScale_sq g φ x | |
| have hp := profileFrequencyScale_ge_one g φ x | |
| change profileFrequencyEnergy g φ x = 1+‖a‖^2 at he | |
| nlinarith [norm_nonneg a] | |
| have heq : fderiv ℝ φ x = (InnerProductSpace.toDual ℝ PhaseSpace a).comp q.2.inverse := by | |
| ext v | |
| change fderiv ℝ φ q.1 v = inner ℝ a (q.2.inverse v) | |
| rw [normalWaveProfile_gradient_pairing g hφ,frame_inverse_apply g hq] | |
| rw [heq] | |
| calc | |
| _ ≤ ‖InnerProductSpace.toDual ℝ PhaseSpace a‖*‖q.2.inverse‖ := | |
| ContinuousLinearMap.opNorm_comp_le _ _ | |
| _ = ‖a‖*‖q.2.inverse‖ := by rw [LinearIsometryEquiv.norm_map] | |
| _ ≤ profileFrequencyScale g φ x*S := | |
| mul_le_mul ha (hbound q hq) (norm_nonneg _) (by linarith [profileFrequencyScale_ge_one g φ x]) | |
| _ = _ := by ring | |
| lemma normalized_real_growth_bounds {L ℓ ε H : ℝ} (hL : 0 < L) (hℓ : L ≤ ℓ) | |
| (hε : 0 ≤ ε) (hε1 : ε ≤ 1) (hH : 0 ≤ H) | |
| (a : NormalWaveSpace →L[ℝ] ℝ) (ha : ‖a‖ ≤ H*L) (i : Fin 3) : | |
| Real.exp (-H) ≤ Real.exp ((ε/ℓ)*a (EuclideanSpace.basisFun (Fin 3) ℝ i)) ∧ | |
| Real.exp ((ε/ℓ)*a (EuclideanSpace.basisFun (Fin 3) ℝ i)) ≤ Real.exp H := by | |
| have hℓ0 : 0 < ℓ := hL.trans_le hℓ | |
| have hi : |a (EuclideanSpace.basisFun (Fin 3) ℝ i)| ≤ H*L := by | |
| have hi' := a.le_opNorm (EuclideanSpace.basisFun (Fin 3) ℝ i) | |
| simpa using hi'.trans (mul_le_mul_of_nonneg_right ha (norm_nonneg _)) | |
| have hεℓ : 0 ≤ ε/ℓ := div_nonneg hε hℓ0.le | |
| have hh : |(ε/ℓ)*a (EuclideanSpace.basisFun (Fin 3) ℝ i)| ≤ H := by | |
| rw [abs_mul,abs_of_nonneg hεℓ] | |
| calc | |
| _ ≤ ε/ℓ*(H*L) := mul_le_mul_of_nonneg_left hi hεℓ | |
| _ = H*ε*(L/ℓ) := by ring | |
| _ ≤ H*ε := mul_le_of_le_one_right (mul_nonneg hH hε) ((div_le_one hℓ0).mpr hℓ) | |
| _ ≤ H := mul_le_of_le_one_right hH hε1 | |
| exact ⟨Real.exp_le_exp.mpr (abs_le.mp hh).1,Real.exp_le_exp.mpr (abs_le.mp hh).2⟩ | |
| lemma endpoint_norm_lower {u v : ℂ} {a ε : ℝ} (ha : 0 < a) | |
| (hε : ε ≤ 1/2) (θ : ℝ) | |
| (h : ‖u-(a : ℂ)*Complex.exp ((θ : ℂ)*Complex.I)*v‖ ≤ | |
| ε*‖(a : ℂ)*Complex.exp ((θ : ℂ)*Complex.I)*v‖) : | |
| (a/2)*‖v‖ ≤ ‖u‖ := by | |
| have he : ‖(a : ℂ)*Complex.exp ((θ : ℂ)*Complex.I)*v‖ = a*‖v‖ := by | |
| simp [Complex.norm_exp,abs_of_pos ha] | |
| have ht := norm_sub_norm_le ((a : ℂ)*Complex.exp ((θ : ℂ)*Complex.I)*v) u | |
| rw [norm_sub_rev,he] at ht | |
| rw [he] at h | |
| have hp := mul_le_mul_of_nonneg_right hε (mul_nonneg ha.le (norm_nonneg v)) | |
| nlinarith | |
| end | |
| open Set Filter | |
| open scoped Topology ContDiff RealInnerProductSpace | |
| lemma compact_profile_unit_lipschitz {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] (φ : E → ℝ) (hφ : ContDiff ℝ ∞ φ) | |
| {K : Set E} (hK : IsCompact K) : | |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ q ∈ K, ∀ x : E, ‖x-q‖ ≤ 1 → | |
| ‖fderiv ℝ φ x‖ ≤ C ∧ |φ x-φ q| ≤ C*‖x-q‖ := by | |
| let L := (fun p : E×E => p.1+p.2) '' (K ×ˢ Metric.closedBall 0 1) | |
| have hL : IsCompact L := (hK.prod (isCompact_closedBall 0 1)).image | |
| (continuous_fst.add continuous_snd) | |
| obtain ⟨C,hbound⟩ := hL.exists_bound_of_continuousOn | |
| (hφ.fderiv_right (m := ∞) (by simp)).continuous.continuousOn | |
| let H := max 1 C | |
| have hb (q) (hq : q ∈ K) (x) (hx : ‖x-q‖ ≤ 1) : ‖fderiv ℝ φ x‖ ≤ H := by | |
| apply (hbound x ?_).trans (le_max_right _ _) | |
| refine ⟨(q,x-q),⟨hq,?_⟩,add_sub_cancel q x⟩ | |
| simpa only [Metric.mem_closedBall,dist_zero_right] using hx | |
| refine ⟨H,le_max_left _ _,?_⟩ | |
| intro q hq x hx | |
| refine ⟨hb q hq x hx,?_⟩ | |
| have hh := (convex_closedBall q (1 : ℝ)).norm_image_sub_le_of_norm_fderiv_le | |
| (fun y _ => hφ.differentiable (by simp) y) | |
| (fun y hy => hb q hq y (by simpa only [Metric.mem_closedBall,dist_eq_norm] using hy)) | |
| (Metric.mem_closedBall_self zero_le_one) | |
| (show x ∈ Metric.closedBall q 1 by simpa only [Metric.mem_closedBall,dist_eq_norm] using hx) | |
| exact hh | |
| lemma packet_value_lower_of_endpoint (φ : NormalWaveSpace → ℝ) | |
| {q x : NormalWaveSpace} {n H M r : ℝ} (hn : 1 ≤ n) (hH : 0 ≤ H) | |
| (hr : 0 ≤ r) (hMr : M*r ≤ 1/2) (hstep : n*‖x-q‖ ≤ 1) | |
| (hd : ‖fderiv ℝ φ q‖ ≤ H) (hval : |φ x-φ q| ≤ H*‖x-q‖) | |
| (u : NormalWaveSpace → ℂ) (β : NormalWaveSpace →L[ℝ] ℝ) | |
| (huq : u q = Complex.exp ((n : ℂ)*(φ q : ℂ))) | |
| (hend : ‖u x-(Real.exp (n*fderiv ℝ φ q (x-q)) : ℂ)* | |
| Complex.exp (((n*β (x-q) : ℝ) : ℂ)*Complex.I)*u q‖ ≤ | |
| (M*r)*‖(Real.exp (n*fderiv ℝ φ q (x-q)) : ℂ)* | |
| Complex.exp (((n*β (x-q) : ℝ) : ℂ)*Complex.I)*u q‖) : | |
| (Real.exp (-2*H)/2)*Real.exp (n*φ x) ≤ ‖u x‖ := by | |
| have hn0 : 0 ≤ n := zero_le_one.trans hn | |
| have hd' : |n*fderiv ℝ φ q (x-q)| ≤ H := by | |
| rw [abs_mul,abs_of_nonneg hn0] | |
| calc | |
| _ ≤ n*(‖fderiv ℝ φ q‖*‖x-q‖) := mul_le_mul_of_nonneg_left ((fderiv ℝ φ q).le_opNorm _) hn0 | |
| _ ≤ n*(H*‖x-q‖) := by gcongr | |
| _ = H*(n*‖x-q‖) := by ring | |
| _ ≤ H := mul_le_of_le_one_right hH hstep | |
| have hv' : |n*(φ x-φ q)| ≤ H := by | |
| rw [abs_mul,abs_of_nonneg hn0] | |
| calc | |
| _ ≤ n*(H*‖x-q‖) := mul_le_mul_of_nonneg_left hval hn0 | |
| _ = H*(n*‖x-q‖) := by ring | |
| _ ≤ H := mul_le_of_le_one_right hH hstep | |
| rw [← abs_of_nonneg hr] at hMr hend | |
| have hl := endpoint_norm_lower (Real.exp_pos (n*fderiv ℝ φ q (x-q))) hMr | |
| (n*β (x-q)) hend | |
| have hnorm : ‖u q‖ = Real.exp (n*φ q) := by | |
| rw [huq,Complex.norm_exp] | |
| simp | |
| rw [hnorm] at hl | |
| refine le_trans ?_ hl | |
| have he : -2*H+n*φ x ≤ n*fderiv ℝ φ q (x-q)+n*φ q := by | |
| have h1 := (abs_le.mp hd').1 | |
| have h2 := (abs_le.mp hv').2 | |
| nlinarith | |
| have hh := Real.exp_le_exp.mpr he | |
| rw [Real.exp_add,Real.exp_add] at hh | |
| nlinarith | |
| end YauCounterexamples | |
| end | |
| end OAI | |