| import OAI.Analysis.BernoulliCone.SobolevCalculus |
|
|
| namespace OAI |
|
|
| universe uE uι |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology ContDiff |
|
|
| noncomputable section |
| open Set Filter Finset |
| open scoped Topology |
| namespace Bernoulli.DeSilvaJerison |
|
|
| / |
| def A (t : ℝ) : ℝ := (Real.log (1+t) - Real.log (1-t))/2 |
|
|
| / |
| def f (t : ℝ) : ℝ := |
| (8-25*t^2+15*t^4)/(8*(1-t^2)^2) - (15/8)*t*A t |
|
|
| / |
| def fp (t : ℝ) : ℝ := |
| -t/(1-t^2)^3 - (5/4)*t/(1-t^2)^2 - (15/8)*t/(1-t^2) - (15/8)*A t |
|
|
| lemma A_neg (t : ℝ) : A (-t) = -A t := by |
| simp only [A, sub_eq_add_neg, neg_neg] |
| ring |
|
|
| @[simp] lemma A_zero : A 0 = 0 := by simp [A] |
| @[simp] lemma f_zero : f 0 = 1 := by norm_num [f] |
| @[simp] lemma fp_zero : fp 0 = 0 := by simp [fp] |
|
|
| lemma f_neg (t : ℝ) : f (-t) = f t := by |
| simp only [f, neg_sq, A_neg] |
| ring |
|
|
| lemma A_eq_half_log {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| A t = 1/2 * Real.log ((1+t)/(1-t)) := by |
| rw [Real.log_div (by linarith [ht.1] : 1+t≠0) (by linarith [ht.2] : 1-t≠0)] |
| unfold A |
| ring |
|
|
| lemma hasDerivAt_A {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| HasDerivAt A (1/(1-t^2)) t := by |
| have hp : 1+t≠0 := by linarith [ht.1] |
| have hm : 1-t≠0 := by linarith [ht.2] |
| have hd : 1-t^2≠0 := by nlinarith [ht.1,ht.2] |
| apply (((((hasDerivAt_const t (1:ℝ)).add (hasDerivAt_id t)).log hp).sub |
| (((hasDerivAt_const t (1:ℝ)).sub (hasDerivAt_id t)).log hm)).div_const 2).congr_deriv |
| norm_num |
| field_simp [hd,hp,hm] |
| ring |
|
|
| lemma hasDerivAt_f {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : HasDerivAt f (fp t) t := by |
| have hd : 1-t^2≠0 := by nlinarith [ht.1,ht.2] |
| have hden : 8*(1-t^2)^2≠0 := mul_ne_zero (by norm_num) (pow_ne_zero _ hd) |
| have hnum := ((hasDerivAt_const t (8:ℝ)).sub |
| (((hasDerivAt_id t).pow 2).const_mul 25)).add |
| (((hasDerivAt_id t).pow 4).const_mul 15) |
| have hden' := (((hasDerivAt_const t (1:ℝ)).sub ((hasDerivAt_id t).pow 2)).pow 2).const_mul 8 |
| apply ((hnum.div hden' hden).sub |
| (((hasDerivAt_id t).const_mul (15/8)).mul (hasDerivAt_A ht))).congr_deriv |
| norm_num [fp] |
| field_simp [hd] |
| ring |
|
|
| lemma hasDerivAt_fp {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| HasDerivAt fp (-6/(1-t^2)^4) t := by |
| have hd : 1-t^2≠0 := by nlinarith [ht.1,ht.2] |
| have hD := (hasDerivAt_const t (1:ℝ)).sub ((hasDerivAt_id t).pow 2) |
| apply (((((hasDerivAt_id t).neg.div (hD.pow 3) (pow_ne_zero _ hd)).sub |
| (((hasDerivAt_id t).const_mul (5/4)).div (hD.pow 2) (pow_ne_zero _ hd))).sub |
| (((hasDerivAt_id t).const_mul (15/8)).div hD hd)).sub |
| ((hasDerivAt_A ht).const_mul (15/8))).congr_deriv |
| norm_num |
| field_simp [hd] |
| ring |
|
|
| lemma f_relation {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| t * fp t - f t = -1/(1-t^2)^3 := by |
| have hd : 1-t^2≠0 := by nlinarith [ht.1,ht.2] |
| unfold fp f |
| field_simp |
| ring |
|
|
| / |
| theorem profile_equation {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| (1-t^2) * deriv (deriv f) t - 6*t*deriv f t + 6*f t = 0 := by |
| have he : deriv f =ᶠ[𝓝 t] fp := by |
| filter_upwards [isOpen_Ioo.mem_nhds ht] with x hx |
| exact (hasDerivAt_f hx).deriv |
| rw [he.deriv_eq, (hasDerivAt_fp ht).deriv, (hasDerivAt_f ht).deriv] |
| have hr := f_relation ht |
| have hd : 1-t^2≠0 := by nlinarith [ht.1,ht.2] |
| field_simp at hr ⊢ |
| nlinarith [hr] |
|
|
| lemma A_nonneg {t : ℝ} (ht : 0 ≤ t) (ht1 : t < 1) : 0 ≤ A t := by |
| rw [A_eq_half_log ⟨by linarith,ht1⟩] |
| exact mul_nonneg (by norm_num) (Real.log_nonneg ((one_le_div (by linarith)).mpr (by linarith))) |
|
|
| lemma fp_neg {t : ℝ} (ht : 0 < t) (ht1 : t < 1) : fp t < 0 := by |
| have hD : 0<1-t^2 := by nlinarith |
| have hA := A_nonneg ht.le ht1 |
| have h1 : 0 < t/(1-t^2)^3 := div_pos ht (pow_pos hD _) |
| have h2 : 0 ≤ (5/4:ℝ)*t/(1-t^2)^2 := by positivity |
| have h3 : 0 ≤ (15/8:ℝ)*t/(1-t^2) := by positivity |
| unfold fp |
| rw [neg_div] |
| nlinarith |
|
|
| lemma f_strictAnti : StrictAntiOn f (Ico (0 : ℝ) 1) := by |
| apply strictAntiOn_of_deriv_neg (convex_Ico 0 1) |
| · intro t ht |
| exact (hasDerivAt_f ⟨by linarith [ht.1],ht.2⟩).continuousAt.continuousWithinAt |
| · intro t ht |
| rw [interior_Ico] at ht |
| rw [(hasDerivAt_f ⟨by linarith [ht.1],ht.2⟩).deriv] |
| exact fp_neg ht.1 ht.2 |
|
|
| / |
| lemma f_lower_pos : 0 < f (517330/1000000) := by |
| have hA : A (517330/1000000) ≤ 5726875/10000000 := by |
| rw [A_eq_half_log (by norm_num)] |
| refine (Real.log_div_le_sum_range_add (by norm_num) (by norm_num) 12).trans ?_ |
| norm_num [Finset.sum_range_succ] |
| unfold f |
| norm_num at hA ⊢ |
| linarith |
|
|
| lemma f_upper_neg : f (517331/1000000) < 0 := by |
| have hA : 572688/1000000 ≤ A (517331/1000000) := by |
| rw [A_eq_half_log (by norm_num)] |
| refine le_trans ?_ (Real.sum_range_le_log_div (by norm_num) (by norm_num) 12) |
| norm_num [Finset.sum_range_succ] |
| unfold f |
| norm_num at hA ⊢ |
| linarith |
|
|
| lemma exists_profile_root : ∃ t ∈ Icc (517330/1000000 : ℝ) (517331/1000000), f t = 0 := by |
| have hc : ContinuousOn f (Icc (517330/1000000 : ℝ) (517331/1000000)) := by |
| intro t ht |
| exact (hasDerivAt_f ⟨by linarith [ht.1],by linarith [ht.2]⟩).continuousAt.continuousWithinAt |
| obtain ⟨t,ht,hft⟩ := intermediate_value_Icc' (by norm_num : (517330/1000000 : ℝ) ≤ 517331/1000000) |
| hc ⟨f_upper_neg.le, f_lower_pos.le⟩ |
| exact ⟨t,ht,hft⟩ |
|
|
| / |
| def t7 : ℝ := exists_profile_root.choose |
|
|
| lemma t7_bounds : t7 ∈ Icc (517330/1000000 : ℝ) (517331/1000000) := |
| exists_profile_root.choose_spec.1 |
|
|
| @[simp] lemma f_t7 : f t7 = 0 := exists_profile_root.choose_spec.2 |
|
|
| lemma t7_pos : 0 < t7 := by linarith [t7_bounds.1] |
| lemma t7_lt_one : t7 < 1 := by linarith [t7_bounds.2] |
|
|
| lemma f_pos_of_abs_lt {t : ℝ} (ht : |t| < t7) : 0 < f t := by |
| have h : 0 < f |t| := by |
| rw [← f_t7] |
| exact f_strictAnti ⟨abs_nonneg t,lt_trans ht t7_lt_one⟩ |
| ⟨t7_pos.le,t7_lt_one⟩ ht |
| by_cases h0 : 0 ≤ t |
| · simpa [abs_of_nonneg h0] using h |
| · simpa [abs_of_neg (lt_of_not_ge h0), f_neg] using h |
|
|
| lemma root_unique {t : ℝ} (ht : t ∈ Ico (0 : ℝ) 1) (hf : f t = 0) : t = t7 := by |
| exact f_strictAnti.injOn ht ⟨t7_pos.le,t7_lt_one⟩ (hf.trans f_t7.symm) |
|
|
| lemma fp_t7 : fp t7 = -1 / (t7 * (1-t7^2)^3) := by |
| have h := f_relation ⟨by linarith [t7_pos],t7_lt_one⟩ |
| rw [f_t7,sub_zero] at h |
| calc |
| fp t7 = (-1 / (1-t7^2)^3) / t7 := |
| (eq_div_iff (ne_of_gt t7_pos)).mpr (by rw [mul_comm]; exact h) |
| _ = _ := by rw [div_div, mul_comm] |
|
|
| / |
| def c7 : ℝ := (1-t7^2) * (fp t7)^2 |
|
|
| lemma c7_pos : 0 < c7 := by |
| have hD : 0 < 1-t7^2 := by nlinarith [t7_pos,t7_lt_one] |
| exact mul_pos hD (sq_pos_of_neg (fp_neg t7_pos t7_lt_one)) |
|
|
| lemma c7_eq : c7 = 1 / (t7^2 * (1-t7^2)^5) := by |
| have ht : t7≠0 := ne_of_gt t7_pos |
| have hd : 1-t7^2≠0 := ne_of_gt (show 0<1-t7^2 by nlinarith [t7_pos,t7_lt_one]) |
| rw [c7,fp_t7] |
| field_simp |
|
|
| lemma contDiffAt_f {n : WithTop ℕ∞} {t : ℝ} (ht : t ∈ Ioo (-1 : ℝ) 1) : |
| ContDiffAt ℝ n f t := by |
| have hd : 8*(1-t^2)^2≠0 := by |
| apply mul_ne_zero (by norm_num) |
| exact pow_ne_zero _ (by nlinarith [ht.1,ht.2]) |
| have hp : 1+t≠0 := by linarith [ht.1] |
| have hm : 1-t≠0 := by linarith [ht.2] |
| unfold f A |
| fun_prop |
|
|
| / |
| def p (t : ℝ) : ℝ := f (min |t| t7) |
|
|
| lemma f_abs (t : ℝ) : f |t| = f t := by |
| by_cases ht : 0 ≤ t |
| · rw [abs_of_nonneg ht] |
| · rw [abs_of_neg (lt_of_not_ge ht), f_neg] |
|
|
| lemma p_eq_f {t : ℝ} (ht : |t| ≤ t7) : p t = f t := by |
| rw [p, min_eq_left ht, f_abs] |
|
|
| lemma p_eq_zero {t : ℝ} (ht : t7 ≤ |t|) : p t = 0 := by |
| rw [p, min_eq_right ht, f_t7] |
|
|
| lemma p_neg (t : ℝ) : p (-t) = p t := by simp only [p, abs_neg] |
| @[simp] lemma p_zero : p 0 = 1 := by rw [p_eq_f (by simpa using t7_pos.le), f_zero] |
|
|
| lemma p_pos_iff (t : ℝ) : 0 < p t ↔ |t| < t7 := by |
| constructor |
| · intro ht |
| by_contra hn |
| rw [p_eq_zero (le_of_not_gt hn)] at ht |
| exact lt_irrefl 0 ht |
| · intro ht |
| rw [p_eq_f ht.le] |
| exact f_pos_of_abs_lt ht |
|
|
| lemma p_nonneg (t : ℝ) : 0 ≤ p t := by |
| rcases lt_or_ge |t| t7 with ht | ht |
| · exact ((p_pos_iff t).mpr ht).le |
| · rw [p_eq_zero ht] |
|
|
| lemma p_le_one (t : ℝ) : p t ≤ 1 := by |
| rw [p, ← f_zero] |
| apply f_strictAnti.antitoneOn ⟨le_rfl,by norm_num⟩ |
| ⟨le_min (abs_nonneg _) t7_pos.le, (min_le_right _ _).trans_lt t7_lt_one⟩ |
| exact le_min (abs_nonneg _) t7_pos.le |
|
|
| lemma exists_lipschitz_p : ∃ K : NNReal, LipschitzWith K p := by |
| have hc : ContDiffOn ℝ 1 f (Icc 0 t7) := by |
| intro t ht |
| exact (contDiffAt_f ⟨by linarith [ht.1],ht.2.trans_lt t7_lt_one⟩).contDiffWithinAt |
| obtain ⟨K,hK⟩ := hc.exists_lipschitzOnWith (by norm_num) (convex_Icc 0 t7) isCompact_Icc |
| have hclip : LipschitzWith 1 (fun t : ℝ => min |t| t7) := by |
| simpa only [Real.norm_eq_abs] using ((lipschitzWith_one_norm : LipschitzWith 1 (fun t : ℝ => ‖t‖)).min_const t7) |
| refine ⟨K, LipschitzWith.of_dist_le_mul (fun x y => ?_)⟩ |
| have hm (t : ℝ) : min |t| t7 ∈ Icc 0 t7 := |
| ⟨le_min (abs_nonneg _) t7_pos.le, min_le_right _ _⟩ |
| exact (hK.dist_le_mul _ (hm x) _ (hm y)).trans (by |
| have hh := hclip.dist_le_mul x y |
| norm_num at hh |
| exact mul_le_mul_of_nonneg_left hh K.coe_nonneg) |
|
|
| / |
| def κ : ℝ := (Real.sqrt c7)⁻¹ |
|
|
| lemma κ_pos : 0 < κ := inv_pos.mpr (Real.sqrt_pos.mpr c7_pos) |
| lemma κ_sq : κ^2 * c7 = 1 := by |
| rw [κ, inv_pow, Real.sq_sqrt c7_pos.le] |
| exact inv_mul_cancel₀ (ne_of_gt c7_pos) |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| /-! The concrete seven-dimensional cone belonging to the De Silva–Jerison profile. |
| These are actual function/Sobolev constructions, not an assumption of minimality. -/ |
| namespace Bernoulli.DeSilvaJerison |
|
|
| / |
| def Z (x : Space 7) : ℝ := ‖x‖ * p (x 6 / ‖x‖) |
|
|
| def U (x : Space 7) : ℝ := κ * Z x |
|
|
| @[simp] lemma Z_zero : Z 0 = 0 := by simp [Z] |
| @[simp] lemma U_zero : U 0 = 0 := by simp [U] |
| lemma Z_nonneg (x : Space 7) : 0 ≤ Z x := mul_nonneg (norm_nonneg x) (p_nonneg _) |
| lemma U_nonneg (x : Space 7) : 0 ≤ U x := mul_nonneg κ_pos.le (Z_nonneg x) |
| lemma Z_le_norm (x : Space 7) : Z x ≤ ‖x‖ := by |
| calc Z x ≤ ‖x‖ * 1 := mul_le_mul_of_nonneg_left (p_le_one _) (norm_nonneg x) |
| _ = _ := mul_one _ |
| lemma U_even (x : Space 7) : U (-x) = U x := by |
| simp only [U,Z,norm_neg,PiLp.neg_apply,neg_div,p_neg] |
|
|
| lemma coordinate_le_norm (x : Space 7) : |x 6| ≤ ‖x‖ := by |
| simpa only [Real.norm_eq_abs] using PiLp.norm_apply_le x (6 : Fin 7) |
|
|
| lemma angular_estimate {x y : Space 7} (hx : x≠0) (hy : y≠0) : |
| ‖y‖ * |x 6/‖x‖ - y 6/‖y‖| ≤ 2*‖x-y‖ := by |
| have hr : 0<‖x‖ := norm_pos_iff.mpr hx |
| have hs : 0<‖y‖ := norm_pos_iff.mpr hy |
| have hq : |x 6/‖x‖| ≤ 1 := by |
| rw [abs_div,abs_of_pos hr] |
| exact (div_le_one hr).mpr (coordinate_le_norm x) |
| have hxy : |x 6-y 6| ≤ ‖x-y‖ := by |
| simpa only [PiLp.sub_apply] using coordinate_le_norm (x-y) |
| calc |
| ‖y‖ * |x 6/‖x‖ - y 6/‖y‖| = |
| |(‖y‖-‖x‖)*(x 6/‖x‖) + (x 6-y 6)| := by |
| rw [← abs_of_pos hs, ← abs_mul] |
| congr 1 |
| rw [abs_of_pos hs] |
| field_simp |
| ring |
| _ ≤ |‖y‖-‖x‖| * |x 6/‖x‖| + |x 6-y 6| := by |
| simpa only [abs_mul] using abs_add_le ((‖y‖-‖x‖)*(x 6/‖x‖)) (x 6-y 6) |
| _ ≤ |‖y‖-‖x‖| * 1 + ‖x-y‖ := by gcongr |
| _ ≤ ‖x-y‖ + ‖x-y‖ := by |
| rw [mul_one] |
| gcongr |
| simpa only [norm_sub_rev] using abs_norm_sub_norm_le y x |
| _ = _ := by ring |
|
|
| lemma exists_lipschitz_Z : ∃ K : NNReal, LipschitzWith K Z := by |
| obtain ⟨K,hK⟩ := exists_lipschitz_p |
| refine ⟨1+2*K, LipschitzWith.of_dist_le_mul (fun x y => ?_)⟩ |
| simp only [dist_eq_norm, Real.norm_eq_abs, NNReal.coe_add, NNReal.coe_mul, |
| NNReal.coe_one, NNReal.coe_ofNat] |
| by_cases hx : x=0 |
| · subst x |
| rw [Z_zero,zero_sub,abs_neg,abs_of_nonneg (Z_nonneg y),zero_sub,norm_neg] |
| exact (Z_le_norm y).trans (by nlinarith [K.coe_nonneg,norm_nonneg y]) |
| by_cases hy : y=0 |
| · subst y |
| rw [Z_zero,sub_zero,abs_of_nonneg (Z_nonneg x),sub_zero] |
| exact (Z_le_norm x).trans (by nlinarith [K.coe_nonneg,norm_nonneg x]) |
| have hpd : |p (x 6/‖x‖)-p (y 6/‖y‖)| ≤ K * |x 6/‖x‖- y 6/‖y‖| := by |
| simpa only [Real.dist_eq] using hK.dist_le_mul (x 6/‖x‖) (y 6/‖y‖) |
| have hpb : |p (x 6/‖x‖)| ≤ 1 := by |
| rw [abs_of_nonneg (p_nonneg _)] |
| exact p_le_one _ |
| calc |
| |Z x-Z y| = |(‖x‖-‖y‖)*p (x 6/‖x‖) + ‖y‖*(p (x 6/‖x‖)-p (y 6/‖y‖))| := by |
| unfold Z |
| congr 1 |
| ring |
| _ ≤ |‖x‖-‖y‖| * |p (x 6/‖x‖)| + ‖y‖* |p (x 6/‖x‖)-p (y 6/‖y‖)| := by |
| simpa only [abs_mul, abs_of_nonneg (norm_nonneg y)] using |
| abs_add_le ((‖x‖-‖y‖)*p (x 6/‖x‖)) (‖y‖*(p (x 6/‖x‖)-p (y 6/‖y‖))) |
| _ ≤ |‖x‖-‖y‖| * 1 + ‖y‖*(K* |x 6/‖x‖-y 6/‖y‖|) := by gcongr |
| _ = |‖x‖-‖y‖| + K*(‖y‖* |x 6/‖x‖-y 6/‖y‖|) := by ring |
| _ ≤ ‖x-y‖ + K*(2*‖x-y‖) := by |
| exact add_le_add (abs_norm_sub_norm_le x y) |
| (mul_le_mul_of_nonneg_left (angular_estimate hx hy) K.coe_nonneg) |
| _ = _ := by ring |
|
|
| lemma exists_lipschitz_U : ∃ K : NNReal, LipschitzWith K U := by |
| obtain ⟨K,hK⟩ := exists_lipschitz_Z |
| exact ⟨‖κ‖₊*K, (lipschitzWith_smul κ).comp hK⟩ |
|
|
| lemma U_continuous : Continuous U := exists_lipschitz_U.choose_spec.continuous |
|
|
| theorem U_h1loc : H1Loc univ U (gradient U) := by |
| obtain ⟨K,hK⟩ := exists_lipschitz_U |
| intro c r _ _ |
| exact LipschitzSobolev.h1_ball hK c r |
|
|
| lemma U_smul (r : ℝ) (hr : 0<r) (x : Space 7) : U (r • x) = r * U x := by |
| simp only [U,Z,norm_smul,Real.norm_eq_abs,abs_of_pos hr,PiLp.smul_apply,smul_eq_mul, |
| mul_div_mul_left _ _ (ne_of_gt hr)] |
| ring |
|
|
| theorem U_homogeneous : OneHomogeneous U := by |
| intro r hr |
| exact Filter.Eventually.of_forall (U_smul r hr) |
|
|
| / |
| lemma U_of_coordinate_zero {x : Space 7} (hx : x 6=0) : U x = κ*‖x‖ := by |
| simp [U,Z,hx] |
|
|
| theorem U_nonzero : Nonzero U := by |
| intro h |
| have he := congrFun (MeasureTheory.Measure.eq_of_ae_eq h U_continuous continuous_const) |
| (EuclideanSpace.single 0 (1:ℝ) : Space 7) |
| have hv : U (EuclideanSpace.single 0 (1:ℝ)) = κ := by |
| rw [U_of_coordinate_zero (by simp)] |
| simp |
| rw [hv] at he |
| exact (ne_of_gt κ_pos) he |
|
|
| theorem U_not_flat : ¬ Flat U := by |
| rintro ⟨e,he,hae⟩ |
| have hc : Continuous (fun x : Space 7 => max (inner ℝ x e) 0) := by fun_prop |
| have heq := MeasureTheory.Measure.eq_of_ae_eq hae U_continuous hc |
| have hp := congrFun heq e |
| have hn := congrFun heq (-e) |
| rw [U_even] at hn |
| rw [real_inner_self_eq_norm_sq,he] at hp |
| rw [inner_neg_left,real_inner_self_eq_norm_sq,he] at hn |
| norm_num at hp hn |
| linarith |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| /-! Euclidean calculus for the actual De Silva–Jerison axisymmetric construction. |
| No free-boundary or variational conclusion is built into these definitions. -/ |
| noncomputable section |
| open Set Filter Finset |
| open scoped Topology ContDiff Laplacian |
| namespace Bernoulli.Axisymmetric |
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| lemma hasDerivAt_radius (x v : E) (t : ℝ) (h : x+t • v ≠ 0) : |
| HasDerivAt (fun a:ℝ=>‖x+a • v‖) |
| (inner ℝ (x+t • v) v / ‖x+t • v‖) t := by |
| have hc : HasDerivAt (fun a:ℝ=>x+a • v) v t := by |
| simpa using ((hasDerivAt_id t).smul_const v).const_add x |
| have hn : ‖x+t • v‖ ≠ 0 := norm_ne_zero_iff.mpr h |
| have hh := hc.norm_sq.sqrt (pow_ne_zero 2 hn) |
| simp only [Real.sqrt_sq (norm_nonneg _)] at hh |
| convert hh using 1; field_simp |
|
|
| lemma hasDerivAt_axis (x v e : E) (t : ℝ) : |
| HasDerivAt (fun a:ℝ=>inner ℝ (x+a • v) e) (inner ℝ v e) t := by |
| have hc : HasDerivAt (fun a:ℝ=>x+a • v) v t := by |
| simpa using ((hasDerivAt_id t).smul_const v).const_add x |
| simpa using hc.inner ℝ (hasDerivAt_const t e) |
|
|
| / |
| def lift (e : E) (α : ℝ) (φ : ℝ → ℝ) (x : E) : ℝ := |
| ‖x‖ ^ α * φ (inner ℝ x e / ‖x‖) |
|
|
| lemma hasDerivAt_angular (x v e : E) (t : ℝ) (h : x+t • v≠0) : |
| HasDerivAt (fun a:ℝ=>inner ℝ (x+a • v) e / ‖x+a • v‖) |
| ((inner ℝ v e * ‖x+t • v‖^2 - inner ℝ (x+t • v) e * |
| inner ℝ (x+t • v) v) / ‖x+t • v‖^3) t := by |
| have hn := norm_ne_zero_iff.mpr h |
| apply ((hasDerivAt_axis x v e t).div (hasDerivAt_radius x v t h) hn).congr_deriv |
| field_simp |
|
|
| lemma hasDerivAt_lift_line {φ : ℝ → ℝ} {φ' : ℝ} (e x v : E) (α t : ℝ) |
| (hx : x+t • v≠0) |
| (hφ : HasDerivAt φ φ' (inner ℝ (x+t • v) e / ‖x+t • v‖)) : |
| HasDerivAt (fun a:ℝ=>lift e α φ (x+a • v)) |
| (‖x+t • v‖^α * |
| (α * inner ℝ (x+t • v) v / ‖x+t • v‖^2 * φ (inner ℝ (x+t • v) e / ‖x+t • v‖) + |
| φ' * ((inner ℝ v e * ‖x+t • v‖^2 - inner ℝ (x+t • v) e * |
| inner ℝ (x+t • v) v) / ‖x+t • v‖^3))) t := by |
| have hn := norm_ne_zero_iff.mpr hx |
| have hr := (hasDerivAt_radius x v t hx).rpow_const (p:=α) (Or.inl hn) |
| have hp := hφ.comp t (hasDerivAt_angular x v e t hx) |
| apply (hr.mul hp).congr_deriv |
| simp only [Function.comp_def] |
| rw [Real.rpow_sub_one hn] |
| field_simp |
|
|
| / |
| or a second derivative in the Euclidean Laplacian computation. -/ |
| def lineFirst (e x v : E) (α : ℝ) (φ φ₁ : ℝ → ℝ) (t : ℝ) : ℝ := |
| ‖x+t • v‖^α * |
| (α * inner ℝ (x+t • v) v / ‖x+t • v‖^2 * φ (inner ℝ (x+t • v) e / ‖x+t • v‖) + |
| φ₁ (inner ℝ (x+t • v) e / ‖x+t • v‖) * |
| ((inner ℝ v e * ‖x+t • v‖^2 - inner ℝ (x+t • v) e * |
| inner ℝ (x+t • v) v) / ‖x+t • v‖^3)) |
|
|
| lemma hasDerivAt_lineFirst {φ φ₁ : ℝ → ℝ} {φ₂ : ℝ} (e x v : E) (α t : ℝ) |
| (hx : x+t • v≠0) |
| (hφ : HasDerivAt φ (φ₁ (inner ℝ (x+t • v) e / ‖x+t • v‖)) |
| (inner ℝ (x+t • v) e / ‖x+t • v‖)) |
| (hφ₁ : HasDerivAt φ₁ φ₂ (inner ℝ (x+t • v) e / ‖x+t • v‖)) : |
| HasDerivAt (lineFirst e x v α φ φ₁) |
| (‖x+t • v‖^α / ‖x+t • v‖^2 * |
| (α * φ (inner ℝ (x+t • v) e / ‖x+t • v‖) * ‖v‖^2 + |
| α * (α-2) * φ (inner ℝ (x+t • v) e / ‖x+t • v‖) * |
| (inner ℝ (x+t • v) v / ‖x+t • v‖)^2 + |
| 2*(α-1) * φ₁ (inner ℝ (x+t • v) e / ‖x+t • v‖) * |
| (inner ℝ (x+t • v) v / ‖x+t • v‖) * |
| (inner ℝ v e - inner ℝ (x+t • v) e / ‖x+t • v‖ * |
| (inner ℝ (x+t • v) v / ‖x+t • v‖)) - |
| inner ℝ (x+t • v) e / ‖x+t • v‖ * |
| φ₁ (inner ℝ (x+t • v) e / ‖x+t • v‖) * |
| (‖v‖^2-(inner ℝ (x+t • v) v / ‖x+t • v‖)^2) + |
| φ₂ * (inner ℝ v e - inner ℝ (x+t • v) e / ‖x+t • v‖ * |
| (inner ℝ (x+t • v) v / ‖x+t • v‖))^2)) t := by |
| have hn := norm_ne_zero_iff.mpr hx |
| have hr := hasDerivAt_radius x v t hx |
| have ha := hasDerivAt_axis x v e t |
| have hq : HasDerivAt (fun a:ℝ=>inner ℝ (x+a • v) v) (‖v‖^2) t := by |
| simpa only [real_inner_self_eq_norm_sq] using hasDerivAt_axis x v v t |
| have hs := hasDerivAt_angular x v e t hx |
| have hp := hφ.comp t hs |
| have hp₁ := hφ₁.comp t hs |
| have hQ := ((hq.const_mul α).div (hr.pow 2) (pow_ne_zero _ hn)).mul hp |
| have hS := (((hr.pow 2).const_mul (inner ℝ v e)).sub (ha.mul hq)).div |
| (hr.pow 3) (pow_ne_zero _ hn) |
| apply (((hr.rpow_const (p:=α) (Or.inl hn)).mul |
| (hQ.add (hp₁.mul hS)))).congr_deriv |
| simp only [Function.comp_def, Pi.pow_apply, Pi.div_apply, Pi.mul_apply, Pi.sub_apply, Pi.add_apply] |
| rw [Real.rpow_sub_one hn] |
| field_simp |
| ring |
|
|
| / |
| def secondCoeff (α F F₁ F₂ s p b n : ℝ) : ℝ := |
| α*F*n + α*(α-2)*F*p^2 + 2*(α-1)*F₁*p*(b-s*p) - |
| s*F₁*(n-p^2) + F₂*(b-s*p)^2 |
|
|
| lemma secondCoeff_sum {ι : Type uι} [Fintype ι] |
| (α F F₁ F₂ s : ℝ) (p b : ι → ℝ) |
| (hp : ∑ i, (p i)^2 = 1) (hb : ∑ i, (b i)^2 = 1) |
| (hpb : ∑ i, p i*b i = s) : |
| ∑ i, secondCoeff α F F₁ F₂ s (p i) (b i) 1 = |
| α*(α+(Fintype.card ι : ℝ)-2)*F - ((Fintype.card ι : ℝ)-1)*s*F₁ + (1-s^2)*F₂ := by |
| have he (i:ι) : secondCoeff α F F₁ F₂ s (p i) (b i) 1 = |
| (α*F-s*F₁) + (α*(α-2)*F-(2*α-3)*s*F₁+s^2*F₂)*(p i)^2 + |
| (2*(α-1)*F₁-2*s*F₂)*(p i*b i) + F₂*(b i)^2 := by |
| unfold secondCoeff |
| ring |
| simp_rw [he, Finset.sum_add_distrib, ←Finset.mul_sum] |
| rw [hp,hb,hpb] |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] |
| ring |
|
|
| lemma contDiffAt_lift {φ : ℝ → ℝ} {e x : E} {α : ℝ} {n : WithTop ℕ∞} |
| (hx : x ≠ 0) (hf : ContDiffAt ℝ n φ (inner ℝ x e / ‖x‖)) : |
| ContDiffAt ℝ n (lift e α φ) x := by |
| have hn : ContDiffAt ℝ n (fun y:E=>‖y‖) x := contDiffAt_id.norm ℝ hx |
| have ha : ContDiffAt ℝ n (fun y:E=>inner ℝ y e / ‖y‖) x := |
| ((contDiffAt_id.inner ℝ contDiffAt_const).div hn (norm_ne_zero_iff.mpr hx)) |
| exact (hn.rpow_const_of_ne (norm_ne_zero_iff.mpr hx)).mul (hf.comp x ha) |
|
|
| lemma lift_line_second {φ φ₁ : ℝ → ℝ} {φ₂ : ℝ} (e x v : E) (α : ℝ) |
| (hx : x≠0) |
| (hφ : ∀ᶠ s in 𝓝 (inner ℝ x e / ‖x‖), HasDerivAt φ (φ₁ s) s) |
| (hφ₁ : HasDerivAt φ₁ φ₂ (inner ℝ x e / ‖x‖)) : |
| deriv (deriv (fun t:ℝ=>lift e α φ (x+t • v))) 0 = |
| ‖x‖^α / ‖x‖^2 * secondCoeff α (φ (inner ℝ x e / ‖x‖)) |
| (φ₁ (inner ℝ x e / ‖x‖)) φ₂ (inner ℝ x e / ‖x‖) |
| (inner ℝ x v / ‖x‖) (inner ℝ v e) (‖v‖^2) := by |
| have hx0 : x+(0:ℝ) • v≠0 := by simpa |
| have hc : ContinuousAt (fun t:ℝ=>x+t • v) 0 := by fun_prop |
| have hnear : ∀ᶠ t:ℝ in 𝓝 0, x+t • v≠0 := |
| hc.eventually_ne hx0 |
| have hs := (hasDerivAt_angular x v e 0 hx0).continuousAt |
| have hp : ∀ᶠ t in 𝓝 (0:ℝ), HasDerivAt φ (φ₁ (inner ℝ (x+t • v) e / ‖x+t • v‖)) |
| (inner ℝ (x+t • v) e / ‖x+t • v‖) := by |
| exact hs.tendsto.eventually (p:=fun s=>HasDerivAt φ (φ₁ s) s) (by simpa using hφ) |
| have he : deriv (fun t:ℝ=>lift e α φ (x+t • v)) =ᶠ[𝓝 0] lineFirst e x v α φ φ₁ := by |
| filter_upwards [hnear,hp] with t ht hpt |
| exact (hasDerivAt_lift_line e x v α t ht hpt).deriv |
| rw [he.deriv_eq] |
| have hh := hasDerivAt_lineFirst e x v α 0 hx0 (by simpa using hφ.self_of_nhds) |
| (by simpa using hφ₁) |
| simpa only [zero_smul, add_zero, secondCoeff] using hh.deriv |
|
|
| lemma line_second_derivative {f : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (v : E) : |
| deriv (deriv (fun t:ℝ=>f (x+t • v))) 0=(fderiv ℝ (fderiv ℝ f) x v) v := by |
| let c := fun t:ℝ=>x+t • v |
| let q := fun t:ℝ=>(fderiv ℝ f (c t)) v |
| have hc (t:ℝ) : HasDerivAt c v t := by |
| simpa only [one_smul,id_eq,c] using ((hasDerivAt_id t).smul_const v).const_add x |
| have hc0 : c 0=x := by simp [c] |
| have hnear : ∀ᶠ t in 𝓝 (0:ℝ), ContDiffAt ℝ 2 f (c t) := |
| (hc 0).continuousAt.tendsto.eventually (hc0 ▸ hf.eventually (by norm_num)) |
| have hq : deriv (fun t:ℝ=>f (x+t • v)) =ᶠ[𝓝 0] q := by |
| filter_upwards [hnear] with t ht |
| exact ((ht.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt t (hc t)).deriv |
| have hff := (hf.fderiv_right (m:=1) (by norm_num)).differentiableAt (by norm_num) |
| have hff' : HasFDerivAt (fderiv ℝ f) (fderiv ℝ (fderiv ℝ f) x) (c 0) := hc0.symm ▸ hff.hasFDerivAt |
| have hh := (hff'.comp_hasDerivAt 0 (hc 0)).clm_apply (hasDerivAt_const 0 v) |
| have hqq : HasDerivAt q ((fderiv ℝ (fderiv ℝ f) x v) v) 0 := by |
| simpa only [Function.comp_def,ContinuousLinearMap.map_zero,add_zero,q] using hh |
| rw [hq.deriv_eq,hqq.deriv] |
|
|
| lemma laplacian_lift [FiniteDimensional ℝ E] {φ φ₁ : ℝ → ℝ} {φ₂ : ℝ} |
| (e x : E) (α : ℝ) (he : ‖e‖=1) (hx : x≠0) |
| (hc : ContDiffAt ℝ 2 φ (inner ℝ x e / ‖x‖)) |
| (hφ : ∀ᶠ s in 𝓝 (inner ℝ x e / ‖x‖), HasDerivAt φ (φ₁ s) s) |
| (hφ₁ : HasDerivAt φ₁ φ₂ (inner ℝ x e / ‖x‖)) : |
| Δ (lift e α φ) x = ‖x‖^α / ‖x‖^2 * |
| (α*(α+(Module.finrank ℝ E : ℝ)-2)*φ (inner ℝ x e / ‖x‖) - |
| ((Module.finrank ℝ E : ℝ)-1)*(inner ℝ x e / ‖x‖)*φ₁ (inner ℝ x e / ‖x‖) + |
| (1-(inner ℝ x e / ‖x‖)^2)*φ₂) := by |
| let b := stdOrthonormalBasis ℝ E |
| have hn := norm_ne_zero_iff.mpr hx |
| have hp : ∑ i, (inner ℝ x (b i) / ‖x‖)^2 = 1 := by |
| simp_rw [div_pow, ←Finset.sum_div] |
| rw [b.sum_sq_inner_left] |
| exact div_self (pow_ne_zero _ hn) |
| have hb : ∑ i, (inner ℝ (b i) e)^2 = 1 := by |
| rw [b.sum_sq_inner_right,he] |
| norm_num |
| have hpb : ∑ i, (inner ℝ x (b i) / ‖x‖) * inner ℝ (b i) e = inner ℝ x e / ‖x‖ := by |
| simp_rw [div_mul_eq_mul_div, ←Finset.sum_div] |
| rw [b.sum_inner_mul_inner] |
| have hs := secondCoeff_sum α (φ (inner ℝ x e / ‖x‖)) (φ₁ (inner ℝ x e / ‖x‖)) φ₂ |
| (inner ℝ x e / ‖x‖) (fun i=>inner ℝ x (b i) / ‖x‖) (fun i=>inner ℝ (b i) e) hp hb hpb |
| rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis] |
| simp_rw [iteratedFDeriv_two_apply] |
| simp only [Fin.isValue,Matrix.cons_val_zero,Matrix.cons_val_one] |
| have hcc := contDiffAt_lift (α:=α) hx hc |
| simp_rw [←line_second_derivative hcc, lift_line_second e x _ α hx hφ hφ₁] |
| have hbunit (i:Fin (Module.finrank ℝ E)) : ‖b i‖^2 = (1:ℝ) := by simp |
| change (∑ i, ‖x‖^α / ‖x‖^2 * secondCoeff α (φ (inner ℝ x e / ‖x‖)) |
| (φ₁ (inner ℝ x e / ‖x‖)) φ₂ (inner ℝ x e / ‖x‖) |
| (inner ℝ x (b i) / ‖x‖) (inner ℝ (b i) e) (‖b i‖^2)) = _ |
| simp_rw [hbunit, ←Finset.mul_sum] |
| rw [hs] |
| simp |
|
|
| / |
| def radial (x : E) : E := ‖x‖⁻¹ • x |
|
|
| def polarGradient (e x : E) (a b : ℝ) : E := |
| a • radial x + b • (e - (inner ℝ x e / ‖x‖) • radial x) |
|
|
| lemma radial_norm {x : E} (hx : x≠0) : ‖radial x‖=1 := by |
| rw [radial,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)] |
|
|
| lemma radial_inner (x e : E) : inner ℝ (radial x) e = inner ℝ x e / ‖x‖ := by |
| simp only [radial,real_inner_smul_left,div_eq_mul_inv,mul_comm] |
|
|
| lemma norm_sq_polarGradient {e x : E} (he : ‖e‖=1) (hx : x≠0) (a b : ℝ) : |
| ‖polarGradient e x a b‖^2 = a^2 + (1-(inner ℝ x e / ‖x‖)^2)*b^2 := by |
| have hn : inner ℝ (radial x) (radial x)=1 := by |
| rw [real_inner_self_eq_norm_sq,radial_norm hx]; norm_num |
| have het : inner ℝ e e=1 := by rw [real_inner_self_eq_norm_sq,he]; norm_num |
| have hrev : inner ℝ e (radial x) = inner ℝ x e / ‖x‖ := |
| (real_inner_comm e (radial x)).symm.trans (radial_inner x e) |
| rw [←real_inner_self_eq_norm_sq] |
| simp only [polarGradient,inner_add_left,inner_add_right,inner_sub_left,inner_sub_right, |
| real_inner_smul_left,real_inner_smul_right,hn,het,hrev,radial_inner] |
| ring |
|
|
| lemma hasGradientAt_lift [CompleteSpace E] {φ : ℝ → ℝ} {φ₁ : ℝ} |
| (e x : E) (α : ℝ) (hx : x≠0) |
| (hφ : HasDerivAt φ φ₁ (inner ℝ x e / ‖x‖)) : |
| HasGradientAt (lift e α φ) |
| ((‖x‖^α / ‖x‖) • polarGradient e x (α*φ (inner ℝ x e / ‖x‖)) φ₁) x := by |
| have hn := norm_ne_zero_iff.mpr hx |
| have hdN : DifferentiableAt ℝ (fun y:E=>‖y‖) x := differentiableAt_id.norm ℝ hx |
| have ha : ContDiffAt ℝ 1 (fun y:E=>inner ℝ y e / ‖y‖) x := |
| (contDiffAt_id.inner ℝ contDiffAt_const).div (contDiffAt_id.norm ℝ hx) hn |
| have hdA := ha.differentiableAt (by norm_num : (1 : WithTop ℕ∞) ≠ 0) |
| have hd : DifferentiableAt ℝ (lift e α φ) x := by |
| exact (hdN.rpow_const (Or.inl hn)).fun_mul (hφ.differentiableAt.comp x hdA) |
| apply hasGradientAt_iff_hasFDerivAt.mpr |
| apply hd.hasFDerivAt.congr_fderiv |
| ext v |
| have hl : HasDerivAt (fun t:ℝ=>x+t • v) v 0 := by |
| simpa using ((hasDerivAt_id (0:ℝ)).smul_const v).const_add x |
| have hd0 : HasFDerivAt (lift e α φ) (fderiv ℝ (lift e α φ) x) (x+(0:ℝ) • v) := by |
| simpa using hd.hasFDerivAt |
| have hcomp := hd0.comp_hasDerivAt (0:ℝ) hl |
| have hline := hasDerivAt_lift_line e x v α 0 (by simpa using hx) (by simpa using hφ) |
| simp only [zero_smul,add_zero] at hline |
| rw [hcomp.unique hline] |
| simp only [InnerProductSpace.toDual_apply_apply,real_inner_smul_left,polarGradient, |
| inner_add_left,inner_sub_left,real_inner_smul_left,radial,real_inner_smul_left, |
| real_inner_comm e v] |
| field_simp |
|
|
| lemma gradient_lift_norm_sq [CompleteSpace E] {φ : ℝ → ℝ} {φ₁ : ℝ} |
| (e x : E) (α : ℝ) (he : ‖e‖=1) (hx : x≠0) |
| (hφ : HasDerivAt φ φ₁ (inner ℝ x e / ‖x‖)) : |
| ‖gradient (lift e α φ) x‖^2 = (‖x‖^α / ‖x‖)^2 * |
| ((α*φ (inner ℝ x e / ‖x‖))^2+(1-(inner ℝ x e / ‖x‖)^2)*φ₁^2) := by |
| rw [(hasGradientAt_lift e x α hx hφ).gradient,norm_smul,mul_pow] |
| rw [norm_sq_polarGradient he hx] |
| simp only [Real.norm_eq_abs,sq_abs] |
|
|
| end Bernoulli.Axisymmetric |
|
|
| namespace Bernoulli.DeSilvaJerison |
| open Axisymmetric InnerProductSpace |
| open scoped Laplacian ContDiff |
|
|
| def axis : Space 7 := EuclideanSpace.single 6 1 |
|
|
| def angle (x : Space 7) : ℝ := x 6 / ‖x‖ |
|
|
| @[simp] lemma axis_norm : ‖axis‖=1 := by simp [axis] |
| @[simp] lemma inner_axis (x : Space 7) : inner ℝ x axis=x 6 := by |
| simp [axis, EuclideanSpace.inner_single_right] |
|
|
| lemma angle_contDiffAt {x : Space 7} {n : WithTop ℕ∞} (hx : x≠0) : |
| ContDiffAt ℝ n angle x := by |
| have hn : ContDiffAt ℝ n (fun y:Space 7=>‖y‖) x := contDiffAt_id.norm ℝ hx |
| have hi : ContDiffAt ℝ n (fun y:Space 7=>inner ℝ y axis) x := |
| contDiffAt_id.inner ℝ contDiffAt_const |
| unfold angle |
| simpa only [inner_axis] using hi.fun_div hn (norm_ne_zero_iff.mpr hx) |
|
|
| lemma U_pos_iff (x : Space 7) : 0<U x ↔ x≠0 ∧ |angle x|<t7 := by |
| rw [U, mul_pos_iff_of_pos_left κ_pos,Z] |
| constructor |
| · intro h |
| have hx : x≠0 := by rintro rfl; simp at h |
| exact ⟨hx, (p_pos_iff _).mp ((mul_pos_iff_of_pos_left (norm_pos_iff.mpr hx)).mp h)⟩ |
| · rintro ⟨hx,ht⟩ |
| exact mul_pos (norm_pos_iff.mpr hx) ((p_pos_iff _).mpr ht) |
|
|
| / |
| def Zsigned : Space 7 → ℝ := lift axis 1 f |
|
|
| def Usigned (x : Space 7) : ℝ := κ*Zsigned x |
|
|
| lemma Zsigned_eq (x : Space 7) : Zsigned x=‖x‖*f (angle x) := by |
| simp [Zsigned,lift,angle] |
|
|
| lemma Z_eq_Zsigned {x : Space 7} (ht : |angle x|≤t7) : Z x=Zsigned x := by |
| rw [Zsigned_eq,Z,p_eq_f (show |x 6 / ‖x‖|≤t7 from ht)] |
| rfl |
|
|
| lemma U_eq_Usigned {x : Space 7} (ht : |angle x|≤t7) : U x=Usigned x := by |
| unfold U Usigned |
| rw [Z_eq_Zsigned ht] |
|
|
| lemma signed_contDiffAt {x : Space 7} {n : WithTop ℕ∞} (hx : x≠0) |
| (ha : |angle x|<1) : ContDiffAt ℝ n Zsigned x := by |
| apply contDiffAt_lift hx |
| simpa only [inner_axis,angle] using contDiffAt_f (abs_lt.mp ha) |
|
|
| lemma signed_laplacian {x : Space 7} (hx : x≠0) (ha : |angle x|<1) : Δ Zsigned x=0 := by |
| have ht : angle x ∈ Ioo (-1:ℝ) 1 := abs_lt.mp ha |
| have hp : ∀ᶠ s in 𝓝 (inner ℝ x axis/‖x‖), HasDerivAt f (fp s) s := by |
| simp only [inner_axis] |
| filter_upwards [isOpen_Ioo.mem_nhds ht] with s hs |
| exact hasDerivAt_f hs |
| have ht' : inner ℝ x axis / ‖x‖ ∈ Ioo (-1:ℝ) 1 := by simpa only [inner_axis,angle] using ht |
| have hl := laplacian_lift axis x 1 axis_norm hx (contDiffAt_f ht') hp (hasDerivAt_fp ht') |
| have hd : 1-(angle x)^2≠0 := by nlinarith [sq_nonneg (angle x),ht.1,ht.2] |
| have hr := f_relation ht |
| have he : 6*f (angle x)-6*angle x*fp (angle x)+(1-(angle x)^2)*(-6/(1-(angle x)^2)^4)=0 := by |
| field_simp at hr ⊢ |
| nlinarith [hr] |
| change Δ (lift axis 1 f) x=0 |
| simp only [inner_axis,Space,finrank_euclideanSpace_fin, Nat.cast_ofNat] at hl |
| norm_num only at hl |
| change Δ (lift axis 1 f) x = ‖x‖^ (1:ℝ) / ‖x‖^2 * |
| (6*f (angle x)-6*angle x*fp (angle x)+(1-(angle x)^2)*(-6/(1-(angle x)^2)^4)) at hl |
| simpa only [he,mul_zero] using hl |
|
|
| lemma signed_harmonicAt {x : Space 7} (hx : x≠0) (ha : |angle x|<1) : |
| HarmonicAt Zsigned x := by |
| refine ⟨signed_contDiffAt hx ha,?_⟩ |
| have hN : ∀ᶠ y in 𝓝 x, y≠0 := isOpen_compl_singleton.mem_nhds hx |
| have hA : ∀ᶠ y in 𝓝 x, |angle y|<1 := |
| (angle_contDiffAt (n:=1) hx).continuousAt.abs.eventually (gt_mem_nhds ha) |
| filter_upwards [hN,hA] with y hy hya |
| exact signed_laplacian hy hya |
|
|
| lemma U_eventuallyEq_Usigned {x : Space 7} (hu : 0<U x) : U =ᶠ[𝓝 x] Usigned := by |
| obtain ⟨hx,ht⟩ := (U_pos_iff x).mp hu |
| have hA : ∀ᶠ y in 𝓝 x, |angle y|<t7 := |
| (angle_contDiffAt (n:=1) hx).continuousAt.abs.eventually (gt_mem_nhds ht) |
| exact hA.mono (fun y hy=>U_eq_Usigned hy.le) |
|
|
| / |
| theorem U_harmonic : HarmonicOnNhd U {x|0<U x} := by |
| intro x hu |
| obtain ⟨hx,ht⟩ := (U_pos_iff x).mp hu |
| apply (harmonicAt_congr_nhds (U_eventuallyEq_Usigned hu)).mpr |
| exact (signed_harmonicAt hx (ht.trans t7_lt_one)).const_smul (c:=κ) |
|
|
| lemma fp_neg_arg (t : ℝ) : fp (-t)= -fp t := by |
| simp only [fp,neg_sq,A_neg] |
| ring |
|
|
| lemma f_zero_at_boundary {t : ℝ} (ht : |t|=t7) : f t=0 := by |
| rw [←f_abs,ht,f_t7] |
|
|
| lemma fp_sq_at_boundary {t : ℝ} (ht : |t|=t7) : (fp t)^2=(fp t7)^2 := by |
| have h : (fp |t|)^2=(fp t)^2 := by |
| rcases le_total 0 t with h|h |
| · rw [abs_of_nonneg h] |
| · rw [abs_of_nonpos h,fp_neg_arg,neg_sq] |
| rw [ht] at h |
| exact h.symm |
|
|
| lemma signed_gradient_norm_sq {x : Space 7} (hx : x≠0) (ha : |angle x|<1) : |
| ‖gradient Zsigned x‖^2=(f (angle x))^2+(1-(angle x)^2)*(fp (angle x))^2 := by |
| have ha' : |inner ℝ x axis/‖x‖|<1 := by simpa only [inner_axis,angle] using ha |
| have hh := gradient_lift_norm_sq axis x 1 axis_norm hx (hasDerivAt_f (abs_lt.mp ha')) |
| simpa only [Zsigned,inner_axis,angle,Real.rpow_one,div_self (norm_ne_zero_iff.mpr hx),one_pow,one_mul] using hh |
|
|
| / |
| lemma Zsigned_boundary_slope {x : Space 7} (hx : x≠0) (ha : |angle x|=t7) : |
| ‖gradient Zsigned x‖^2=c7 := by |
| rw [signed_gradient_norm_sq hx (ha.trans_lt t7_lt_one),f_zero_at_boundary ha, |
| fp_sq_at_boundary ha] |
| have hs : (angle x)^2=t7^2 := by rw [←sq_abs,ha] |
| simp only [hs,zero_pow (by norm_num : 2≠0),zero_add,c7] |
|
|
| lemma gradient_const_mul {F : Space 7 → ℝ} {x : Space 7} (c : ℝ) |
| (hF : DifferentiableAt ℝ F x) : |
| gradient (fun y=>c*F y) x=c • gradient F x := by |
| rw [gradient, fderiv_const_mul hF] |
| simp only [map_smul,gradient] |
|
|
| / |
| nonvertex free-boundary point. This does not claim its energy comparison. -/ |
| theorem Usigned_boundary_slope {x : Space 7} (hx : x≠0) (ha : |angle x|=t7) : |
| ‖gradient Usigned x‖=1 := by |
| have hd : DifferentiableAt ℝ Zsigned x := |
| (signed_contDiffAt (n:=1) hx (ha.trans_lt t7_lt_one)).differentiableAt (by norm_num) |
| have he : gradient Usigned x=κ • gradient Zsigned x := gradient_const_mul κ hd |
| have hs : ‖gradient Usigned x‖^2=1 := by |
| rw [he,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs,Zsigned_boundary_slope hx ha,κ_sq] |
| nlinarith [norm_nonneg (gradient Usigned x)] |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| namespace Bernoulli.DeSilvaJerison |
| open scoped BigOperators |
|
|
| def B (n:ℕ) (u:ℝ) : ℝ := ∑ i ∈ Finset.range n, u^i / (2*i+1) |
|
|
| lemma B_nonneg (n:ℕ) {u:ℝ} (hu:0≤u) : 0≤B n u := by |
| apply Finset.sum_nonneg |
| intro i hi |
| positivity |
|
|
| lemma odd_sum_eq (n:ℕ) (t:ℝ) : |
| (∑ i ∈ Finset.range n, t^(2*i+1)/(2*i+1))=t*B n (t^2) := by |
| rw [B,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [pow_add,pow_mul] |
| ring |
|
|
| lemma A_series_bounds (n:ℕ) {t:ℝ} (ht:0≤t) (ht1:t<1) : |
| t*B n (t^2) ≤ A t ∧ A t ≤ t*(B n (t^2)+(t^2)^n/(1-t^2)) := by |
| have hI : t∈Ioo (-1:ℝ) 1 := ⟨by linarith,ht1⟩ |
| have hL := Real.sum_range_le_log_div ht ht1 n |
| have hU := Real.log_div_le_sum_range_add ht ht1 n |
| rw [odd_sum_eq,A_eq_half_log hI] at * |
| constructor |
| · exact hL |
| · convert hU using 1 |
| rw [pow_add,pow_mul] |
| ring |
|
|
| lemma f_series_lower (n:ℕ) {t:ℝ} (ht:0≤t) (ht1:t<1) : |
| (8-25*t^2+15*(t^2)^2)/(8*(1-t^2)^2) |
| -15/8*t^2*(B n (t^2)+(t^2)^n/(1-t^2)) ≤ f t := by |
| have ha := (A_series_bounds n ht ht1).2 |
| have hp := mul_le_mul_of_nonneg_left ha (show (0:ℝ)≤15/8*t from by positivity) |
| unfold f |
| convert! sub_le_sub_left hp ((8-25*t^2+15*t^4)/(8*(1-t^2)^2)) using 1 |
| ring |
|
|
| lemma fp_series_upper (n:ℕ) {t:ℝ} (ht:0≤t) (ht1:t<1) : |
| fp t ≤ -t*(1/(1-t^2)^3+5/4/(1-t^2)^2+15/8/(1-t^2)+15/8*B n (t^2)) := by |
| have ha := (A_series_bounds n ht ht1).1 |
| unfold fp |
| have hp := mul_le_mul_of_nonneg_left ha (show (0:ℝ)≤15/8 from by norm_num) |
| convert! sub_le_sub_left hp (-t/(1-t^2)^3-5/4*t/(1-t^2)^2-15/8*t/(1-t^2)) using 1 |
| ring |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Bernoulli.DeSilvaJerison.Algebra |
| def g (u:ℝ) : ℝ := (1+u*((2869387/1000000:ℝ)+u*((589981/125000:ℝ)+u*((6566689/1000000:ℝ)+u*((4206127/500000:ℝ)+u*((10257227/1000000:ℝ)+u*((6050939/500000:ℝ)+u*((2789267/200000:ℝ)+u*(7895333/500000:ℝ))))))))) |
|
|
| def g1 (u:ℝ) : ℝ := ((2869387/1000000:ℝ)+u*((589981/62500:ℝ)+u*((19700067/1000000:ℝ)+u*((4206127/125000:ℝ)+u*((10257227/200000:ℝ)+u*((18152817/250000:ℝ)+u*((19524869/200000:ℝ)+u*(7895333/62500:ℝ)))))))) |
|
|
| def g2 (u:ℝ) : ℝ := ((589981/62500:ℝ)+u*((19700067/500000:ℝ)+u*((12618381/125000:ℝ)+u*((10257227/50000:ℝ)+u*((18152817/50000:ℝ)+u*((58574607/100000:ℝ)+u*(55267331/62500:ℝ))))))) |
|
|
| def FlowNum (u:ℝ) : ℝ := (8+u*(-40+u*(40+u*(-8+u*((-8/7:ℝ)+u*((-8/21:ℝ)+u*((-40/231:ℝ)+u*((-460/33:ℝ)+u*(150/11:ℝ))))))))) |
|
|
| def JlowNum (u:ℝ) : ℝ := (48+u*(-80+u*(48+u*((-48/7:ℝ)+u*((-16/21:ℝ)+u*((-16/77:ℝ)+u*((-95/77:ℝ)+u*((80/33:ℝ)+u*(-15/11:ℝ))))))))) |
|
|
| def H (u:ℝ) : ℝ := |
| (1-(-643/200:ℝ))^2 * (FlowNum u)^2*(1-u)^2*(g u)^2 |
| +u*(1-u)*(JlowNum u*g u+2*FlowNum u*(1-u)*g1 u)^2 |
| -64*(177344/10000:ℝ)*(1-u)^6*(g u)^2 |
|
|
| def bern (x:ℝ) :ℝ := |
| (1273/625:ℝ)*x^0*(1-x)^34 + |
| (18301700752813/39062500000:ℝ)*x^1*(1-x)^33 + |
| (215052290412518053937/15625000000000000:ℝ)*x^2*(1-x)^32 + |
| (4011660524023528296977/19531250000000000:ℝ)*x^3*(1-x)^31 + |
| (2737813200177456418780827/1367187500000000000:ℝ)*x^4*(1-x)^30 + |
| (36470062299690915838876891/2563476562500000000:ℝ)*x^5*(1-x)^29 + |
| (25224069904037561558116841153/322265625000000000000:ℝ)*x^6*(1-x)^28 + |
| (1140064408982925692415284885429/3289794921875000000000:ℝ)*x^7*(1-x)^27 + |
| (41702430755303183867323989789361/32897949218750000000000:ℝ)*x^8*(1-x)^26 + |
| (3302410109850186979935464577903227/845947265625000000000000:ℝ)*x^9*(1-x)^25 + |
| (607809096133375762153335847523839391/59216308593750000000000000:ℝ)*x^10*(1-x)^24 + |
| (8427099588852868346099742136229588957/361877441406250000000000000:ℝ)*x^11*(1-x)^23 + |
| (83162165720757436741420385026341254509/1809387207031250000000000000:ℝ)*x^12*(1-x)^22 + |
| (6465714023183326120974320513863787790989/81422424316406250000000000000:ℝ)*x^13*(1-x)^21 + |
| (3070489019357153411700150072588091455833/25444507598876953125000000000:ℝ)*x^14*(1-x)^20 + |
| (82369624574464912186052678411201103823739/508890151977539062500000000000:ℝ)*x^15*(1-x)^19 + |
| (3910314359275519861283649778682853171355229/20355606079101562500000000000000:ℝ)*x^16*(1-x)^18 + |
| (979175253390516279186463881299949127320211/4846572875976562500000000000000:ℝ)*x^17*(1-x)^17 + |
| (18263450287363530970672089631693726965414361/96931457519531250000000000000000:ℝ)*x^18*(1-x)^16 + |
| (17617068143172995639202711944743442184849557/113086700439453125000000000000000:ℝ)*x^19*(1-x)^15 + |
| (5805157568137808862325200615665450856083238073/50889015197753906250000000000000000:ℝ)*x^20*(1-x)^14 + |
| (106751395374573342818643134988112428956196041/1445710659027099609375000000000000:ℝ)*x^21*(1-x)^13 + |
| (13399476602886887470912830751358415258789263239/318056344985961914062500000000000000:ℝ)*x^22*(1-x)^12 + |
| (203394843320074906765348971241270582814374973/9638071060180664062500000000000000:ℝ)*x^23*(1-x)^11 + |
| (73412104728736551372464217800669117696986097123/7951408624649047851562500000000000000:ℝ)*x^24*(1-x)^10 + |
| (139309476218905417280721075386752271276084623033/39757043123245239257812500000000000000:ℝ)*x^25*(1-x)^9 + |
| (32477127068915750179291210097408098171730074813/28397887945175170898437500000000000000:ℝ)*x^26*(1-x)^8 + |
| (39433145196824909814460561789894422591933034093/124240759760141372680664062500000000000:ℝ)*x^27*(1-x)^7 + |
| (1852317908162190170035216935656039577364200647/25099143385887145996093750000000000000:ℝ)*x^28*(1-x)^6 + |
| (5836982899230309706402487066150347002468521567/414135865867137908935546875000000000000:ℝ)*x^29*(1-x)^5 + |
| (53440683331803983473324825254354074715316174661/24848151952028274536132812500000000000000:ℝ)*x^30*(1-x)^4 + |
| (1564303979409844850797520400352897238700085837/6212037988007068634033203125000000000000:ℝ)*x^31*(1-x)^3 + |
| (219633445088755317503000659291685432421229847/10353396646678447723388671875000000000000:ℝ)*x^32*(1-x)^2 + |
| (4437106302213137739667714564011045560602003/3882523742504417896270751953125000000000:ℝ)*x^33*(1-x)^1 + |
| (57307964213152837654157884692498315776209/1941261871252208948135375976562500000000:ℝ)*x^34*(1-x)^0 |
|
|
| lemma identity (x:ℝ) : H ((1/5:ℝ)*x)=bern x := by |
| unfold H FlowNum JlowNum g g1 bern |
| ring |
|
|
| lemma bern_nonneg {x:ℝ} (hx:0≤x) (hx1:x≤1) : 0≤bern x := by |
| have hy : 0≤1-x := sub_nonneg.mpr hx1 |
| unfold bern |
| positivity |
|
|
| lemma H_nonneg {u:ℝ} (hu:0≤u) (hu1:u≤1/5) : 0≤H u := by |
| have he : u=(1/5:ℝ)*(u/(1/5)) := by ring |
| rw [he,identity] |
| apply bern_nonneg |
| · positivity |
| · rw [div_le_one (by norm_num : (0:ℝ)<1/5)] |
| exact hu1 |
|
|
| lemma hasDerivAt_g (u:ℝ) : HasDerivAt g (g1 u) u := by |
| have hh := ((hasDerivAt_const u 1).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (2869387/1000000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (589981/125000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (6566689/1000000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (4206127/500000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (10257227/1000000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (6050939/500000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (2789267/200000:ℝ)).add ((hasDerivAt_id u).mul (hasDerivAt_const u (7895333/500000:ℝ)))))))))))))))))) |
| convert! hh using 1 |
| simp only [g1, Pi.add_apply, Pi.mul_apply, id_eq] |
| ring |
|
|
| lemma hasDerivAt_g1 (u:ℝ) : HasDerivAt g1 (g2 u) u := by |
| have hh := ((hasDerivAt_const u (2869387/1000000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (589981/62500:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (19700067/1000000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (4206127/125000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (10257227/200000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (18152817/250000:ℝ)).add ((hasDerivAt_id u).mul ((hasDerivAt_const u (19524869/200000:ℝ)).add ((hasDerivAt_id u).mul (hasDerivAt_const u (7895333/62500:ℝ)))))))))))))))) |
| convert! hh using 1 |
| simp only [g2, Pi.add_apply, Pi.mul_apply, id_eq] |
| ring |
|
|
| lemma g_ge_one {u:ℝ} (hu:0≤u) : 1≤g u := by |
| unfold g |
| have : 0≤u*((2869387/1000000:ℝ)+u*((589981/125000:ℝ)+u*((6566689/1000000:ℝ)+u*((4206127/500000:ℝ)+u*((10257227/1000000:ℝ)+u*((6050939/500000:ℝ)+u*((2789267/200000:ℝ)+u*(7895333/500000:ℝ)))))))) := by positivity |
| linarith |
|
|
| lemma g1_nonneg {u:ℝ} (hu:0≤u) : 0≤g1 u := by unfold g1; positivity |
|
|
| lemma g_residual_nonpos {u:ℝ} (hu:0≤u) : |
| (1-u)*(2*g1 u+4*u*g2 u)-12*u*g1 u+(-643/200:ℝ)*(-643/200+5)*g u ≤0 := by |
| have he : -((1-u)*(2*g1 u+4*u*g2 u)-12*u*g1 u+(-643/200:ℝ)*(-643/200+5)*g u) = |
| ((1/1000000:ℝ)+u*((335237/40000000000:ℝ)+u*((18531/5000000000:ℝ)+u*((26639/40000000000:ℝ)+u*((378977/20000000000:ℝ)+u*((2875077/40000000000:ℝ)+u*((1938389/20000000000:ℝ)+u*((1189117/8000000000:ℝ)+u*(107925657105483/20000000000:ℝ))))))))) := by unfold g g1 g2; ring |
| have hz : 0≤ -((1-u)*(2*g1 u+4*u*g2 u)-12*u*g1 u+(-643/200:ℝ)*(-643/200+5)*g u) := by |
| rw [he] |
| positivity |
| linarith |
|
|
| lemma lower_logder {u:ℝ} (hu:0≤u) (hub:u≤(27/100:ℝ)) : 0≤2*(1-u)*g1 u-5*g u := by |
| let x:=u/(27/100:ℝ) |
| have hx:0≤x := by dsimp [x]; positivity |
| have hy:0≤1-x := by dsimp [x]; linarith |
| have he : 2*(1-u)*g1 u-5*g u = |
| (369387/500000:ℝ)*x^0*(1-x)^8 + (558448641/100000000:ℝ)*x^1*(1-x)^7 + (90906551829/5000000000:ℝ)*x^2*(1-x)^6 + (33087845213199/1000000000000:ℝ)*x^3*(1-x)^5 + (910658646456597/25000000000000:ℝ)*x^4*(1-x)^4 + (243550667389510017/10000000000000000:ℝ)*x^5*(1-x)^3 + (1156300362730287837/125000000000000000:ℝ)*x^6*(1-x)^2 + (160712254758181821273/100000000000000000000:ℝ)*x^7*(1-x)^1 + (141611539097570077017/5000000000000000000000:ℝ)*x^8*(1-x)^0 := by |
| dsimp [x] |
| unfold g g1 |
| ring |
| rw [he] |
| positivity |
|
|
| lemma upper_logder {u:ℝ} (hu:0≤u) (hub:u≤(27/100:ℝ)) : 0≤2*g u-(13/25)*g1 u := by |
| let x:=u/(27/100:ℝ) |
| have hx:0≤x := by dsimp [x]; positivity |
| have hy:0≤1-x := by dsimp [x]; linarith |
| have he : 2*g u-(13/25)*g1 u = |
| (12697969/25000000:ℝ)*x^0*(1-x)^8 + (5359357177/1250000000:ℝ)*x^1*(1-x)^7 + (3933009652441/250000000000:ℝ)*x^2*(1-x)^6 + (408907285769443/12500000000000:ℝ)*x^3*(1-x)^5 + (21018853182775349/500000000000000:ℝ)*x^4*(1-x)^4 + (4252809530492624731/125000000000000000:ℝ)*x^5*(1-x)^3 + (83958125315148478907/5000000000000000000:ℝ)*x^6*(1-x)^2 + (5678700560736300182329/1250000000000000000000:ℝ)*x^7*(1-x)^1 + (1242771118592263674331/2500000000000000000000:ℝ)*x^8*(1-x)^0 := by |
| dsimp [x] |
| unfold g g1 |
| ring |
| rw [he] |
| positivity |
|
|
| lemma FlowNum_nonneg {u:ℝ} (hu:0≤u) (hub:u≤(1/5:ℝ)) : 0≤FlowNum u := by |
| let x:=u/(1/5:ℝ) |
| have hx:0≤x := by dsimp [x]; positivity |
| have hy:0≤1-x := by dsimp [x]; linarith |
| have he : FlowNum u = |
| 8*x^0*(1-x)^8 + 56*x^1*(1-x)^7 + (848/5:ℝ)*x^2*(1-x)^6 + (36192/125:ℝ)*x^3*(1-x)^5 + (1328592/4375:ℝ)*x^4*(1-x)^4 + (13082512/65625:ℝ)*x^5*(1-x)^3 + (57279808/721875:ℝ)*x^6*(1-x)^2 + (62341556/3609375:ℝ)*x^7*(1-x)^1 + (5536402/3609375:ℝ)*x^8*(1-x)^0 := by |
| dsimp [x] |
| unfold FlowNum |
| ring |
| rw [he] |
| positivity |
|
|
| end Bernoulli.DeSilvaJerison.Algebra |
|
|
| namespace Bernoulli.DeSilvaJerison |
|
|
| def gSub (t:ℝ) : ℝ := Algebra.g (t^2) |
| def gSub1 (t:ℝ) : ℝ := 2*t*Algebra.g1 (t^2) |
| def gSub2 (t:ℝ) : ℝ := 2*Algebra.g1 (t^2)+4*t^2*Algebra.g2 (t^2) |
| def qSub (t:ℝ) : ℝ := gSub1 t/gSub t |
|
|
| lemma gSub_pos (t:ℝ) : 0<gSub t := by |
| have h:=Algebra.g_ge_one (sq_nonneg t) |
| unfold gSub |
| linarith |
|
|
| lemma hasDerivAt_gSub (t:ℝ) : HasDerivAt gSub (gSub1 t) t := by |
| have hh:=(Algebra.hasDerivAt_g (t^2)).comp t (hasDerivAt_pow 2 t) |
| convert! hh using 1 |
| dsimp only [gSub1,Function.comp_apply,id_eq] |
| norm_num only |
| ring |
|
|
| lemma hasDerivAt_gSub1 (t:ℝ) : HasDerivAt gSub1 (gSub2 t) t := by |
| have hh:=((hasDerivAt_id t).const_mul 2).mul |
| ((Algebra.hasDerivAt_g1 (t^2)).comp t (hasDerivAt_pow 2 t)) |
| convert! hh using 1 |
| dsimp only [gSub2,Function.comp_apply,id_eq] |
| norm_num only |
| ring |
|
|
| lemma gSub_residual (t:ℝ) : |
| (1-t^2)*gSub2 t-6*t*gSub1 t+(-643/200:ℝ)*(-643/200+5)*gSub t ≤0 := by |
| convert! Algebra.g_residual_nonpos (sq_nonneg t) using 1 |
| unfold gSub gSub1 gSub2 |
| ring |
|
|
| lemma qSub_nonneg {t:ℝ} (ht:0≤t) : 0≤qSub t := by |
| unfold qSub gSub1 |
| exact div_nonneg (by have:=Algebra.g1_nonneg (sq_nonneg t); positivity) (gSub_pos t).le |
|
|
| lemma qSub_bounds {t:ℝ} (ht:0≤t) (hT:t≤t7) : |
| 5*t≤(1-t^2)*qSub t ∧ qSub t≤4 := by |
| have htl:t≤517331/1000000 := hT.trans t7_bounds.2 |
| have hu:t^2≤27/100 := by nlinarith |
| have hd:=Algebra.lower_logder (sq_nonneg t) hu |
| have hm:=Algebra.upper_logder (sq_nonneg t) hu |
| have hp:=Algebra.g1_nonneg (sq_nonneg t) |
| have hg:=gSub_pos t |
| unfold qSub gSub1 gSub at * |
| constructor |
| · rw [←mul_div_assoc] |
| apply (le_div_iff₀ hg).mpr |
| have hh:=mul_nonneg ht hd |
| nlinarith |
| · apply (div_le_iff₀ hg).mpr |
| have hh:=mul_le_mul_of_nonneg_right (show 2*t≤26/25 by linarith) hp |
| linarith |
|
|
| lemma c7_upper : c7≤177344/10000 := by |
| have hl:t7^2≥(517330/1000000:ℝ)^2 := by nlinarith [t7_bounds.1,t7_pos] |
| have hu:1-t7^2≥1-(517331/1000000:ℝ)^2 := by nlinarith [t7_bounds.2,t7_pos] |
| have hp:=pow_le_pow_left₀ (by norm_num : (0:ℝ)≤1-(517331/1000000:ℝ)^2) hu 5 |
| have hm:=mul_le_mul hl hp (by norm_num : (0:ℝ)≤(1-(517331/1000000:ℝ)^2)^5) (sq_nonneg t7) |
| have ht : 0<1-t7^2 := by nlinarith [t7_pos,t7_lt_one] |
| have hT:=t7_pos |
| rw [c7_eq] |
| apply (div_le_iff₀ (by positivity : 0<t7^2*(1-t7^2)^5)).mpr |
| norm_num at hm |
| linarith |
|
|
| lemma fp_strictAnti : StrictAntiOn fp (Ico (0:ℝ) 1) := by |
| apply strictAntiOn_of_deriv_neg (convex_Ico 0 1) |
| · intro t ht |
| exact (hasDerivAt_fp ⟨by linarith [ht.1],ht.2⟩).continuousAt.continuousWithinAt |
| · intro t ht |
| rw [interior_Ico] at ht |
| rw [(hasDerivAt_fp ⟨by linarith [ht.1],ht.2⟩).deriv] |
| have hp : 0<1-t^2 := by nlinarith [ht.1,ht.2] |
| exact div_neg_of_neg_of_pos (by norm_num) (pow_pos hp 4) |
|
|
| lemma sub_endpoint_bounds : f (11/25)≤1/3 ∧ fp (11/25)≤-7/2 := by |
| have ha : 4722/10000 ≤ A (11/25) := by |
| rw [A_eq_half_log (by norm_num)] |
| refine le_trans ?_ (Real.sum_range_le_log_div (by norm_num) (by norm_num) 6) |
| norm_num [Finset.sum_range_succ] |
| constructor <;> dsimp only [f,fp] <;> norm_num at ha ⊢ <;> linarith |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| namespace Bernoulli.DeSilvaJerison |
|
|
| / |
| def GSub (t:ℝ) : ℝ := (1-(-643/200:ℝ))^2*(f t)^2+ |
| (1-t^2)*(fp t-f t*qSub t)^2 |
|
|
| def FLo (t:ℝ) : ℝ := Algebra.FlowNum (t^2)/(8*(1-t^2)^2) |
| def JLo (t:ℝ) : ℝ := Algebra.JlowNum (t^2)/(8*(1-t^2)^3) |
|
|
| lemma FLo_le {t:ℝ} (ht:0≤t) (ht1:t<1) : FLo t≤f t := by |
| have hd:1-t^2≠0 := by nlinarith |
| convert! f_series_lower 6 ht ht1 using 1 |
| unfold FLo Algebra.FlowNum B |
| norm_num [Finset.sum_range_succ] |
| field_simp [hd] |
| ring |
|
|
| lemma fp_le_JLo {t:ℝ} (ht:0≤t) (ht1:t<1) : fp t≤-t*JLo t := by |
| have hd:1-t^2≠0 := by nlinarith |
| convert! fp_series_upper 6 ht ht1 using 1 |
| unfold JLo Algebra.JlowNum B |
| norm_num [Finset.sum_range_succ] |
| field_simp [hd] |
| ring_nf |
| simp |
|
|
| lemma JLo_nonneg {t:ℝ} (ht:0≤t) (ht1:t<1) : 0≤JLo t := by |
| have hd:0<1-t^2 := by nlinarith |
| have he : JLo t=1/(1-t^2)^3+5/4/(1-t^2)^2+15/8/(1-t^2)+15/8*B 6 (t^2) := by |
| unfold JLo Algebra.JlowNum B |
| norm_num [Finset.sum_range_succ] |
| field_simp |
| ring |
| rw [he] |
| have hh:=B_nonneg 6 (sq_nonneg t) |
| positivity |
|
|
| lemma GSub_low {t:ℝ} (ht:0≤t) (htb:t≤11/25) : c7≤GSub t := by |
| have ht1:t<1 := by linarith |
| have hd:0<1-t^2 := by nlinarith |
| have hu:t^2≤1/5 := by nlinarith |
| have hg:=gSub_pos t |
| have hF:0≤FLo t := div_nonneg (Algebra.FlowNum_nonneg (sq_nonneg t) hu) (by positivity) |
| have hL:=FLo_le ht ht1 |
| have hp:=fp_le_JLo ht ht1 |
| have hJ:=JLo_nonneg ht ht1 |
| have hq:=qSub_nonneg ht |
| have hB: t*JLo t+FLo t*qSub t ≤ -(fp t-f t*qSub t) := by |
| have hh:=mul_le_mul_of_nonneg_right hL hq |
| linarith |
| have hB0:0≤t*JLo t+FLo t*qSub t := by positivity |
| have hs1 := sq_le_sq₀ hF (hF.trans hL) |>.mpr hL |
| have hs2 := sq_le_sq₀ hB0 (hB0.trans hB) |>.mpr hB |
| have hH:=Algebra.H_nonneg (sq_nonneg t) hu |
| have he : Algebra.H (t^2) = |
| 64*(1-t^2)^6*(gSub t)^2 * |
| ((1-(-643/200:ℝ))^2*(FLo t)^2+(1-t^2)*(t*JLo t+FLo t*qSub t)^2-177344/10000) := by |
| unfold Algebra.H FLo JLo qSub gSub gSub1 |
| have hgn : Algebra.g (t^2) ≠ 0 := ne_of_gt hg |
| field_simp [ne_of_gt hd,hgn] |
| ring |
| rw [he] at hH |
| have hn := (mul_nonneg_iff_of_pos_left (by positivity : 0<64*(1-t^2)^6*(gSub t)^2)).mp hH |
| have hu1:=mul_le_mul_of_nonneg_left hs1 (show (0:ℝ)≤(1-(-643/200:ℝ))^2 by positivity) |
| have hu2:=mul_le_mul_of_nonneg_left hs2 hd.le |
| dsimp [GSub] |
| nlinarith [c7_upper] |
|
|
| lemma hasDerivAt_qSub (t:ℝ) : |
| HasDerivAt qSub (gSub2 t/gSub t-(qSub t)^2) t := by |
| have hh := (hasDerivAt_gSub1 t).div (hasDerivAt_gSub t) (ne_of_gt (gSub_pos t)) |
| convert! hh using 1 |
| unfold qSub |
| field_simp |
|
|
| lemma qSub_deriv_bound {t:ℝ} (_ht:0≤t) (_ht1:t<1) : |
| (1-t^2)*(gSub2 t/gSub t-(qSub t)^2) ≤ |
| 6*t*qSub t-(-643/200:ℝ)*(-643/200+5)-(1-t^2)*(qSub t)^2 := by |
| have hg:=gSub_pos t |
| have hh:=gSub_residual t |
| have he : (1-t^2)*(gSub2 t/gSub t-(qSub t)^2)- |
| (6*t*qSub t-(-643/200:ℝ)*(-643/200+5)-(1-t^2)*(qSub t)^2) = |
| ((1-t^2)*gSub2 t-6*t*gSub1 t+(-643/200:ℝ)*(-643/200+5)*gSub t)/gSub t := by |
| unfold qSub |
| field_simp |
| ring |
| have hz:=div_nonpos_of_nonpos_of_nonneg hh hg.le |
| linarith |
|
|
| lemma hasDerivAt_GSub {t:ℝ} (ht:t∈Ioo (-1:ℝ) 1) : |
| HasDerivAt GSub |
| (2*((1-(-643/200:ℝ))^2*f t*fp t-t*(fp t-f t*qSub t)^2+ |
| (1-t^2)*(fp t-f t*qSub t)*(-6/(1-t^2)^4-fp t*qSub t-f t*(gSub2 t/gSub t-(qSub t)^2)))) t := by |
| have hh:=(((hasDerivAt_f ht).pow 2).const_mul ((1-(-643/200:ℝ))^2)).add |
| (((hasDerivAt_const t (1:ℝ)).sub ((hasDerivAt_id t).pow 2)).mul |
| (((hasDerivAt_fp ht).sub ((hasDerivAt_f ht).mul (hasDerivAt_qSub t))).pow 2)) |
| convert! hh using 1 |
| norm_num only [id_eq,pow_one,Pi.add_apply,Pi.sub_apply,Pi.mul_apply,Pi.pow_apply] |
| ring |
|
|
| lemma GSub_nonpos_deriv {t:ℝ} (ht:11/25≤t) (ht7:t≤t7) : deriv GSub t ≤0 := by |
| have ht0:0≤t := by linarith |
| have ht1:t<1:=ht7.trans_lt t7_lt_one |
| have hI:t∈Ioo (-1:ℝ) 1:=⟨by linarith,ht1⟩ |
| have hd:0<1-t^2 := by nlinarith |
| have hf0:0≤f t := by |
| have hh:=f_strictAnti.antitoneOn ⟨ht0,ht1⟩ ⟨t7_pos.le,t7_lt_one⟩ ht7 |
| simpa only [f_t7] using hh |
| have hfall : f t ≤ f (11/25) := f_strictAnti.antitoneOn ⟨by norm_num,by norm_num⟩ ⟨ht0,ht1⟩ ht |
| have hfp : fp t ≤ fp (11/25) := fp_strictAnti.antitoneOn ⟨by norm_num,by norm_num⟩ ⟨ht0,ht1⟩ ht |
| have hf:f t≤1/3:=hfall.trans sub_endpoint_bounds.1 |
| have hp:fp t≤-7/2:=hfp.trans sub_endpoint_bounds.2 |
| have hq:=qSub_nonneg ht0 |
| obtain ⟨hqL,hqU⟩:=qSub_bounds ht0 ht7 |
| have hb:fp t-f t*qSub t≤0 := by nlinarith [mul_nonneg hf0 hq] |
| have hres:=qSub_deriv_bound ht0 ht1 |
| have hODE: (1-t^2)*(-6/(1-t^2)^4)=6*t*fp t-6*f t := by |
| have hh:=f_relation hI |
| field_simp at hh ⊢ |
| nlinarith |
| have hbd : (1-t^2)*(-6/(1-t^2)^4-fp t*qSub t-f t*(gSub2 t/gSub t-(qSub t)^2)) ≥ |
| 6*t*fp t-6*f t-(1-t^2)*fp t*qSub t- |
| f t*(6*t*qSub t-(-643/200:ℝ)*(-643/200+5)-(1-t^2)*(qSub t)^2) := by |
| have hh:=mul_le_mul_of_nonneg_left hres hf0 |
| nlinarith only [hh,hODE] |
| have hmul:=mul_le_mul_of_nonpos_left hbd hb |
| rw [(hasDerivAt_GSub hI).deriv] |
| have hfirst : (5*t-(1-t^2)*qSub t)*(fp t-f t*qSub t)^2 ≤0 := |
| mul_nonpos_of_nonpos_of_nonneg (by linarith) (sq_nonneg _) |
| have hlast : (6.02745:ℝ)*fp t+(11.738775:ℝ)*f t*qSub t≤0 := by |
| have hmul:=mul_le_mul hf hqU hq (by norm_num : (0:ℝ)≤1/3) |
| linarith |
| have hlast2:=mul_nonpos_of_nonneg_of_nonpos hf0 hlast |
| nlinarith only [hmul,hfirst,hlast2] |
|
|
| lemma GSub_high {t:ℝ} (ht:11/25≤t) (ht7:t≤t7) : c7≤GSub t := by |
| have hcont:ContinuousOn GSub (Icc (11/25) t7) := by |
| intro x hx |
| exact (hasDerivAt_GSub ⟨by linarith [hx.1],hx.2.trans_lt t7_lt_one⟩).continuousAt.continuousWithinAt |
| have hanti:AntitoneOn GSub (Icc (11/25) t7) := by |
| apply antitoneOn_of_deriv_nonpos (convex_Icc _ _) hcont |
| · intro x hx |
| rw [interior_Icc] at hx |
| exact (hasDerivAt_GSub ⟨by linarith [hx.1],hx.2.trans t7_lt_one⟩).differentiableAt.differentiableWithinAt |
| · intro x hx |
| rw [interior_Icc] at hx |
| exact GSub_nonpos_deriv hx.1.le hx.2.le |
| have he:GSub t7=c7 := by simp [GSub,f_t7,c7] |
| rw [←he] |
| exact hanti ⟨ht,ht7⟩ ⟨by linarith [t7_bounds.1],le_rfl⟩ ht7 |
|
|
| / |
| theorem GSub_ge_c7 {t:ℝ} (ht:0≤t) (ht7:t≤t7) : c7≤GSub t := by |
| by_cases h:t≤11/25 |
| · exact GSub_low ht h |
| · exact GSub_high (le_of_lt (lt_of_not_ge h)) ht7 |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| namespace Bernoulli.DeSilvaJerison |
| open Axisymmetric InnerProductSpace |
| open scoped Laplacian ContDiff |
|
|
| lemma qSub_neg (t:ℝ) : qSub (-t)= -qSub t := by |
| simp only [qSub,gSub,gSub1,neg_sq] |
| ring |
|
|
| lemma GSub_neg (t:ℝ) : GSub (-t)=GSub t := by |
| rw [GSub,GSub,f_neg,fp_neg_arg,qSub_neg,neg_sq] |
| ring |
|
|
| lemma GSub_ge_c7_abs {t:ℝ} (ht:|t|≤t7) : c7≤GSub t := by |
| have hh:=GSub_ge_c7 (abs_nonneg t) ht |
| rcases le_total 0 t with hp|hn |
| · simpa only [abs_of_nonneg hp] using hh |
| · simpa only [abs_of_nonpos hn,GSub_neg] using hh |
|
|
| lemma gSub_contDiff {n:WithTop ℕ∞} : ContDiff ℝ n gSub := by |
| unfold gSub Algebra.g |
| fun_prop |
|
|
| / |
| def YSub : Space 7 → ℝ := lift axis (-643/200) gSub |
|
|
| lemma YSub_pos {x:Space 7} (hx:x≠0) : 0<YSub x := by |
| unfold YSub lift |
| exact mul_pos (Real.rpow_pos_of_pos (norm_pos_iff.mpr hx) _) (gSub_pos _) |
|
|
| lemma YSub_contDiffAt {x:Space 7} {n:WithTop ℕ∞} (hx:x≠0) : |
| ContDiffAt ℝ n YSub x := contDiffAt_lift hx gSub_contDiff.contDiffAt |
|
|
| / |
| theorem YSub_superharmonic {x:Space 7} (hx:x≠0) : Δ YSub x≤0 := by |
| have hp : ∀ᶠ s in 𝓝 (inner ℝ x axis/‖x‖), HasDerivAt gSub (gSub1 s) s := |
| Filter.Eventually.of_forall hasDerivAt_gSub |
| have hh:=laplacian_lift axis x (-643/200) axis_norm hx |
| (gSub_contDiff (n:=2)).contDiffAt hp (hasDerivAt_gSub1 _) |
| rw [YSub,hh] |
| have hR:=gSub_residual (inner ℝ x axis/‖x‖) |
| have he : (-643/200:ℝ)*(-643/200+(Module.finrank ℝ (Space 7):ℝ)-2)* |
| gSub (inner ℝ x axis/‖x‖)-((Module.finrank ℝ (Space 7):ℝ)-1)* |
| (inner ℝ x axis/‖x‖)*gSub1 (inner ℝ x axis/‖x‖)+ |
| (1-(inner ℝ x axis/‖x‖)^2)*gSub2 (inner ℝ x axis/‖x‖)≤0 := by |
| simp only [Space,finrank_euclideanSpace_fin,Nat.cast_ofNat] |
| nlinarith [hR] |
| exact mul_nonpos_of_nonneg_of_nonpos (by positivity) he |
|
|
| lemma Usigned_gradient {x:Space 7} (hx:x≠0) (ha:|angle x|<1) : |
| gradient Usigned x=κ • polarGradient axis x (f (angle x)) (fp (angle x)) := by |
| have hi : inner ℝ x axis/‖x‖∈Ioo (-1:ℝ) 1 := by |
| simpa only [inner_axis,angle,Set.mem_Ioo] using abs_lt.mp ha |
| rw [show gradient Usigned x=κ • gradient Zsigned x from |
| gradient_const_mul κ ((signed_contDiffAt (n:=1) hx ha).differentiableAt (by norm_num))] |
| rw [Zsigned,(hasGradientAt_lift axis x 1 hx (hasDerivAt_f hi)).gradient] |
| simp only [Real.rpow_one,div_self (norm_ne_zero_iff.mpr hx),one_smul,one_mul,inner_axis,angle] |
|
|
| lemma YSub_gradient {x:Space 7} (hx:x≠0) : |
| gradient YSub x=(‖x‖^(-643/200:ℝ)/‖x‖) • |
| polarGradient axis x ((-643/200:ℝ)*gSub (angle x)) (gSub1 (angle x)) := by |
| simpa only [YSub,inner_axis,angle] using |
| (hasGradientAt_lift axis x (-643/200) hx (hasDerivAt_gSub _)).gradient |
|
|
| lemma U_gradient {x:Space 7} (hu:0<U x) : |
| gradient U x=κ • polarGradient axis x (f (angle x)) (fp (angle x)) := by |
| obtain ⟨hx,ha⟩:=(U_pos_iff x).mp hu |
| have he:gradient U x=gradient Usigned x := by |
| rw [gradient,(U_eventuallyEq_Usigned hu).fderiv_eq] |
| rfl |
| rw [he,Usigned_gradient hx (ha.trans t7_lt_one)] |
|
|
| / |
| throughout the entire positive phase of the concrete seven-dimensional cone. -/ |
| theorem lower_groundstate_slope {x:Space 7} (hu:0<U x) : |
| 1≤‖gradient U x-(U x/YSub x) • gradient YSub x‖^2 := by |
| obtain ⟨hx,ha⟩:=(U_pos_iff x).mp hu |
| have hn:=norm_ne_zero_iff.mpr hx |
| have hg:=ne_of_gt (gSub_pos (angle x)) |
| have hr:=ne_of_gt (Real.rpow_pos_of_pos (norm_pos_iff.mpr hx) (-643/200:ℝ)) |
| have hU : U x=κ*‖x‖*f (angle x) := by |
| rw [U_eq_Usigned ha.le,Usigned,Zsigned_eq] |
| ring |
| have he : gradient U x-(U x/YSub x) • gradient YSub x= |
| κ • polarGradient axis x ((1-(-643/200:ℝ))*f (angle x)) |
| (fp (angle x)-f (angle x)*qSub (angle x)) := by |
| rw [U_gradient hu,YSub_gradient hx,hU] |
| have hy : YSub x = ‖x‖^(-643/200:ℝ)*gSub (angle x) := by |
| simp only [YSub,lift,inner_axis,angle] |
| rw [hy] |
| unfold polarGradient qSub |
| simp only [inner_axis] |
| ext i |
| simp only [PiLp.sub_apply,PiLp.add_apply,PiLp.smul_apply,smul_eq_mul] |
| unfold angle at * |
| field_simp [hn,hg,hr] |
| ring |
| rw [he,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs,norm_sq_polarGradient axis_norm hx] |
| have hG:=mul_le_mul_of_nonneg_left (GSub_ge_c7_abs ha.le) (sq_nonneg κ) |
| rw [κ_sq] at hG |
| simpa only [GSub,inner_axis,angle,mul_pow] using hG |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
