| import OAI.Geometry.Relativity.CKS.BendingDerivative | |
| namespace OAI | |
| noncomputable section | |
| namespace CKSBending | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| lemma bendingV_deriv {R r : ℝ} (hR : 0 < R) (hr : 0 < r) : | |
| deriv (bendingV R) r = | |
| (phi (r/R) + r * (deriv phi (r/R)/R)) * xi (r/R^2) + | |
| r * phi (r/R) * (deriv xi (r/R^2)/R^2) := by | |
| have hp := (phi_smooth.contDiffAt (isOpen_Ioi.mem_nhds (div_pos hr hR))).differentiableAt (by simp) | |
| have hpd := hp.hasDerivAt.comp r ((hasDerivAt_id r).div_const R) | |
| have hxd := (xi_smooth.differentiable (by simp) (r/R^2)).hasDerivAt.comp r | |
| ((hasDerivAt_id r).div_const (R^2)) | |
| convert! (((hasDerivAt_id r).mul hpd).mul hxd).deriv using 1 | |
| simp | |
| ring | |
| lemma bendingV_reduced_deriv {R r : ℝ} (hR : 0 < R) (hr : 0 < r) : | |
| r * deriv (bendingV R) r - bendingV R r = | |
| r * ((r/R) * deriv phi (r/R)) * xi (r/R^2) + | |
| r * phi (r/R) * ((r/R^2) * deriv xi (r/R^2)) := by | |
| rw [bendingV_deriv hR hr] | |
| unfold bendingV | |
| ring | |
| lemma base_tail {R r : ℝ} (hR : 12 ≤ R) (hr : R^2 ≤ r) : | |
| r * phi (r/R) = R^3/r^2 := by | |
| have hRp : 0 < R := by linarith | |
| have hrp : 0 < r := lt_of_lt_of_le (sq_pos_of_pos hRp) hr | |
| have hx : 12 ≤ r/R := (le_div_iff₀ hRp).mpr (by nlinarith) | |
| rw [phi_tail hx] | |
| field_simp | |
| lemma base_tail_le_one {R r : ℝ} (hR : 12 ≤ R) (hr : R^2 ≤ r) : | |
| r * phi (r/R) ≤ 1 := by | |
| have hRp : 0 < R := by linarith | |
| have hrp : 0 < r := lt_of_lt_of_le (sq_pos_of_pos hRp) hr | |
| rw [base_tail hR hr] | |
| apply (div_le_iff₀ (sq_pos_of_pos hrp)).mpr | |
| have hR1 : 1 ≤ R := by linarith | |
| have hrr : R^2 * R^2 ≤ r*r := mul_self_le_mul_self (sq_nonneg R) hr | |
| have hp := mul_le_mul_of_nonneg_left hR1 (pow_nonneg hRp.le 3) | |
| nlinarith | |
| lemma v_le_sqrt (z : ℝ) (hz : 0 ≤ z) : z ≤ Real.sqrt (1+z^2) := by | |
| apply (Real.le_sqrt hz (by positivity)).mpr | |
| nlinarith | |
| lemma one_le_sqrt (z : ℝ) : 1 ≤ Real.sqrt (1+z^2) := by | |
| apply (Real.le_sqrt (by norm_num) (by positivity)).mpr | |
| nlinarith [sq_nonneg z] | |
| theorem bendingV_first_scaled : ∃ C : ℝ, 0 ≤ C ∧ ∀ R : ℝ, 12 ≤ R → | |
| ∀ r : ℝ, 0 < r → |r * deriv (bendingV R) r - bendingV R r| ≤ | |
| C * Real.sqrt (1 + (bendingV R r)^2) := by | |
| obtain ⟨C,hC,hphi⟩ := phi_scaled_deriv_bound | |
| obtain ⟨K,hK,hxi⟩ := xi_scaled_deriv_bound | |
| have hC0 : 0 ≤ C := by linarith | |
| refine ⟨C+K,add_nonneg hC0 hK,fun R hR r hr => ?_⟩ | |
| have hRp : 0 < R := by linarith | |
| have hv0 := bendingV_nonneg hRp hr | |
| have hvs := v_le_sqrt _ hv0 | |
| have hs1 := one_le_sqrt (bendingV R r) | |
| have hmain : |r * ((r/R) * deriv phi (r/R)) * xi (r/R^2)| ≤ C * bendingV R r := by | |
| rw [abs_mul,abs_mul,abs_of_pos hr,abs_of_nonneg (xi_nonneg _)] | |
| have hh := mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (hphi (r/R) (div_pos hr hRp)) hr.le) (xi_nonneg (r/R^2)) | |
| convert! hh using 1 | |
| simp only [bendingV] | |
| ring | |
| have hcut : |r * phi (r/R) * ((r/R^2) * deriv xi (r/R^2))| ≤ K * Real.sqrt (1+(bendingV R r)^2) := by | |
| by_cases hc : r ≤ R^2 | |
| · have hy : r/R^2 ≤ 1 := (div_le_iff₀ (sq_pos_of_pos hRp)).mpr (by simpa using hc) | |
| rw [xi_deriv_initial hy,mul_zero,mul_zero,abs_zero] | |
| exact mul_nonneg hK (Real.sqrt_nonneg _) | |
| · rw [abs_mul,abs_mul,abs_of_pos hr,abs_of_pos (phi_pos (div_pos hr hRp))] | |
| calc | |
| _ ≤ r * phi (r/R) * K := mul_le_mul_of_nonneg_left (hxi _) (mul_nonneg hr.le (phi_pos (div_pos hr hRp)).le) | |
| _ ≤ K := by nlinarith [base_tail_le_one hR (le_of_not_ge hc)] | |
| _ ≤ K * Real.sqrt (1+(bendingV R r)^2) := by nlinarith | |
| rw [bendingV_reduced_deriv hRp hr] | |
| calc | |
| _ ≤ |r * ((r/R) * deriv phi (r/R)) * xi (r/R^2)| + | |
| |r * phi (r/R) * ((r/R^2) * deriv xi (r/R^2))| := abs_add_le _ _ | |
| _ ≤ C * bendingV R r + K * Real.sqrt (1+(bendingV R r)^2) := add_le_add hmain hcut | |
| _ ≤ (C+K) * Real.sqrt (1+(bendingV R r)^2) := by nlinarith | |
| end | |
| end CKSBending | |
| end | |
| end OAI | |