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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
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