| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.ThreeCoupledCleanChartJets |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
|
|
| lemma norm_power_variable_weight_control |
| {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [InnerProductSpace ℝ F] |
| {f : E → F} (hf : ContDiff ℝ 1 f) {φ : E → ℝ} (hφ : Continuous φ) |
| {K : Set E} (hK : IsCompact K) : |
| ∃ C>0, ∀ k : ℕ, 2≤k → |
| ContDiff ℝ 1 (fun x => ‖f x‖^k) ∧ |
| ∀ x∈K, ‖fderiv ℝ (fun y => ‖f y‖^k) x‖ ≤ |
| C*productWaveFrequency k*(Real.exp (productWaveFrequency k*φ x)+‖f x‖^k) := by |
| obtain ⟨A,hA⟩ := hK.exists_bound_of_continuousOn (hf.continuous_fderiv one_ne_zero).continuousOn |
| obtain ⟨R,hR⟩ := hK.exists_bound_of_continuousOn hφ.continuousOn |
| let D := max R 0 |
| let B := max A 1 |
| let m := Real.exp (-2*D) |
| have hB : 0<B := zero_lt_one.trans_le (le_max_right _ _) |
| have hm : 0 < m := Real.exp_pos _ |
| refine ⟨B/m,div_pos hB hm,?_⟩ |
| intro k hk |
| have hk1 : 1≤k := by omega |
| have hkR : (1:ℝ)<k := by exact_mod_cast (show 1<k by omega) |
| have hν := productWaveFrequency_bounds hk1 |
| have hν0 : 0≤productWaveFrequency k := (Nat.cast_nonneg _).trans hν.1 |
| have hfn : ContDiff ℝ 1 (fun x => ‖f x‖^k) := by |
| simpa only [Real.rpow_natCast] using hf.norm_rpow hkR |
| refine ⟨hfn,?_⟩ |
| intro x hx |
| have hφlow : -D≤φ x := by |
| have h := hR x hx |
| rw [Real.norm_eq_abs] at h |
| exact (neg_le_neg (le_max_left R 0)).trans ((neg_le_neg h).trans (neg_abs_le _)) |
| have hem : m^k≤Real.exp (productWaveFrequency k*φ x) := by |
| dsimp [m] |
| rw [←Real.exp_nat_mul] |
| apply Real.exp_le_exp.mpr |
| have h1 := mul_le_mul_of_nonpos_right hν.2 (neg_nonpos.mpr (le_max_right R 0)) |
| have h2 := mul_le_mul_of_nonneg_left hφlow hν0 |
| dsimp [D] at * |
| nlinarith |
| let W := Real.exp (productWaveFrequency k*φ x)+‖f x‖^k |
| have hpred : ‖f x‖^(k-1)≤W/m := by |
| apply (le_div_iff₀ hm).mpr |
| have hh := scalar_pred_power_weight_bound (norm_nonneg (f x)) hm (le_refl m) hk1 |
| have ht := hh.trans (add_le_add hem le_rfl) |
| simpa only [W,mul_comm] using ht |
| have hnorm : ‖fderiv ℝ (fun y => ‖f y‖^k) x‖ ≤ (k:ℝ)*‖f x‖^(k-1)*B := by |
| have hh := norm_fderiv_norm_rpow_le (x:=x) (hf.differentiable one_ne_zero) hkR |
| have hcast : (k:ℝ)-1=((k-1:ℕ):ℝ) := by rw [Nat.cast_sub hk1]; norm_num |
| rw [hcast,Real.rpow_natCast] at hh |
| simp only [Real.rpow_natCast] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left ((hA x hx).trans (le_max_left _ _)) (by positivity)) |
| calc |
| _ ≤ (k:ℝ)*(W/m)*B := hnorm.trans (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hpred (Nat.cast_nonneg _)) hB.le) |
| _ ≤ productWaveFrequency k*(W/m)*B := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hν.1 (by dsimp [W]; positivity)) hB.le |
| _ = _ := by dsimp [W]; ring |
|
|
| lemma two_power_variable_weight_control |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {a b : E → ℂ} (ha : ContDiff ℝ ∞ a) (hb : ContDiff ℝ ∞ b) |
| {φ : E → ℝ} (hφ : ContDiff ℝ ∞ φ) {K : Set E} (hK : IsCompact K) : |
| ∃ T>0, ∃ H>0, ∀ k : ℕ, 2≤k → |
| Differentiable ℝ (fun x => Real.exp (productWaveFrequency k*φ x)+‖a x‖^k+‖b x‖^k) ∧ |
| (∀ x, 0<Real.exp (productWaveFrequency k*φ x)+‖a x‖^k+‖b x‖^k) ∧ |
| (∀ x∈K, ∀ j≤2, ‖iteratedFDeriv ℝ j (fun y => (a y^k).re+(b y^k).re) x‖ ≤ |
| T*(productWaveFrequency k)^j*(Real.exp (productWaveFrequency k*φ x)+‖a x‖^k+‖b x‖^k)) ∧ |
| (∀ x∈K, ‖fderiv ℝ (fun y => Real.exp (productWaveFrequency k*φ y)+‖a y‖^k+‖b y‖^k) x‖ ≤ |
| H*productWaveFrequency k*(Real.exp (productWaveFrequency k*φ x)+‖a x‖^k+‖b x‖^k)) := by |
| obtain ⟨A,hA,hAJ⟩ := real_power_variable_weighted_jets ha hK φ hφ.continuous.continuousOn 2 |
| obtain ⟨B,hB,hBJ⟩ := real_power_variable_weighted_jets hb hK φ hφ.continuous.continuousOn 2 |
| obtain ⟨C,hC,hCj⟩ := norm_power_variable_weight_control (ha.of_le (by simp)) hφ.continuous hK |
| obtain ⟨D,hD,hDj⟩ := norm_power_variable_weight_control (hb.of_le (by simp)) hφ.continuous hK |
| obtain ⟨H,hH,hHe⟩ := compact_exponential_envelope_log_bound hK hφ |
| refine ⟨A+B,add_pos hA hB,H+C+D,by positivity,?_⟩ |
| intro k hk |
| have hk1 : 1≤k := by omega |
| have hν := productWaveFrequency_bounds hk1 |
| have hν0 : 0≤productWaveFrequency k := (Nat.cast_nonneg _).trans hν.1 |
| obtain ⟨hak,hakJ⟩ := hCj k hk |
| obtain ⟨hbk,hbkJ⟩ := hDj k hk |
| have he : ContDiff ℝ ∞ (fun x => Real.exp (productWaveFrequency k*φ x)) := (contDiff_const.mul hφ).exp |
| have hWe : Differentiable ℝ (fun x => Real.exp (productWaveFrequency k*φ x)) := he.differentiable (by simp) |
| refine ⟨(hWe.add (hak.differentiable one_ne_zero)).add (hbk.differentiable one_ne_zero),fun x => by positivity,?_,?_⟩ |
| · intro x hx j hj |
| let W := Real.exp (productWaveFrequency k*φ x)+‖a x‖^k+‖b x‖^k |
| have hwA : Real.exp (productWaveFrequency k*φ x)+‖a x‖^k≤W := le_add_of_nonneg_right (by positivity) |
| have hwB : Real.exp (productWaveFrequency k*φ x)+‖b x‖^k≤W := by dsimp [W]; nlinarith [pow_nonneg (norm_nonneg (a x)) k] |
| have hpow := pow_le_pow_left₀ (Nat.cast_nonneg k) hν.1 j |
| have h1 := (hAJ k hk1 x hx j hj).trans (mul_le_mul |
| (mul_le_mul_of_nonneg_left hpow hA.le) hwA (by positivity) (by positivity)) |
| have h2 := (hBJ k hk1 x hx j hj).trans (mul_le_mul |
| (mul_le_mul_of_nonneg_left hpow hB.le) hwB (by positivity) (by positivity)) |
| exact (real_smooth_add_jet_bound (Complex.reCLM.contDiff.comp (ha.pow k)) |
| (Complex.reCLM.contDiff.comp (hb.pow k)) j x).trans (by |
| change ‖iteratedFDeriv ℝ j (fun y => (a y^k).re) x‖+ |
| ‖iteratedFDeriv ℝ j (fun y => (b y^k).re) x‖ ≤ _ |
| calc |
| _ ≤ A*(productWaveFrequency k)^j*W+B*(productWaveFrequency k)^j*W := add_le_add h1 h2 |
| _ = _ := by dsimp [W]; ring) |
| · intro x hx |
| have hea := hWe x |
| have haa := hak.differentiable one_ne_zero x |
| have hba := hbk.differentiable one_ne_zero x |
| rw [fderiv_fun_add (hea.fun_add haa) hba,fderiv_fun_add hea haa] |
| apply (norm_add_le _ _).trans |
| apply (add_le_add (norm_add_le _ _) le_rfl).trans |
| have h1 := hHe (productWaveFrequency k) hν0 x hx |
| have h2 := hakJ x hx |
| have h3 := hbkJ x hx |
| calc |
| _ ≤ H*productWaveFrequency k*Real.exp (productWaveFrequency k*φ x)+ |
| C*productWaveFrequency k*(Real.exp (productWaveFrequency k*φ x)+‖a x‖^k)+ |
| D*productWaveFrequency k*(Real.exp (productWaveFrequency k*φ x)+‖b x‖^k) := by linarith |
| _ ≤ _ := by nlinarith [mul_nonneg (mul_nonneg hH hν0) (pow_nonneg (norm_nonneg (a x)) k), |
| mul_nonneg (mul_nonneg hH hν0) (pow_nonneg (norm_nonneg (b x)) k), |
| mul_nonneg (mul_nonneg hC.le hν0) (pow_nonneg (norm_nonneg (b x)) k), |
| mul_nonneg (mul_nonneg hD.le hν0) (pow_nonneg (norm_nonneg (a x)) k)] |
| end YauCounterexamples |
| end |
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| end OAI |
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