| import OAI.Geometry.Relativity.CKS.InducedSphereCalculus |
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| namespace OAI |
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| noncomputable section |
| namespace CKSSphericalChart |
| noncomputable section |
| open Set Filter Finset CKSCalculus CKSRealizedRound |
| open CKSInducedSphere (E Ix) |
| open scoped Topology ContDiff |
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| lemma D_line {F : Point → ℝ} {x : Point} (hF : DifferentiableAt ℝ F x) (v : Point) : |
| D v F x = deriv (fun s : ℝ => F (x + s • v)) 0 := by |
| have hd : HasDerivAt (fun s : ℝ => x + s • v) v 0 := by |
| convert! ((hasDerivAt_id (0:ℝ)).smul_const v).const_add x using 1 |
| simp |
| have hf : HasFDerivAt F (fderiv ℝ F x) (x + (0:ℝ) • v) := by simpa using hF.hasFDerivAt |
| have hh := hf.comp_hasDerivAt 0 hd |
| simpa only [zero_smul,add_zero,D,Function.comp_def] using hh.deriv.symm |
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| lemma D_line_congr {F G : Point → ℝ} {x : Point} |
| (hF : DifferentiableAt ℝ F x) (hG : DifferentiableAt ℝ G x) (v : Point) |
| (he : ∀ s : ℝ, F (x+s•v) = G (x+s•v)) : D v F x = D v G x := by |
| rw [D_line hF,D_line hG] |
| congr 1 |
| funext s |
| exact he s |
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| lemma angular_line_radial (x : Point) (k : Ix) (hk : k ≠ 0) (s : ℝ) : |
| (x+s•basis k) 0 = x 0 := by |
| simp [basis,Ne.symm hk] |
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| lemma angular_first_congr_leaf {F G : Point → ℝ} {x : Point} |
| (hF : DifferentiableAt ℝ F x) (hG : DifferentiableAt ℝ G x) |
| (he : ∀ y : Point, y 0 = x 0 → F y = G y) (k : Ix) (hk : k ≠ 0) : |
| D (basis k) F x = D (basis k) G x := by |
| apply D_line_congr hF hG |
| intro s |
| exact he _ (angular_line_radial x k hk s) |
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| lemma angular_second_congr_leaf {F G : Point → ℝ} {x : Point} |
| (hF : ∀ y : Point, y 0 = x 0 → ContDiffAt ℝ 2 F y) |
| (hG : ∀ y : Point, y 0 = x 0 → ContDiffAt ℝ 2 G y) |
| (he : ∀ y : Point, y 0 = x 0 → F y = G y) |
| (i j : Ix) (hi : i ≠ 0) (hj : j ≠ 0) : |
| D (basis j) (D (basis i) F) x = D (basis j) (D (basis i) G) x := by |
| have hFd := (CKSCalculus.contDiffAt_D (hF x rfl) (m := 1) (by norm_num) (basis i)).differentiableAt (by norm_num) |
| have hGd := (CKSCalculus.contDiffAt_D (hG x rfl) (m := 1) (by norm_num) (basis i)).differentiableAt (by norm_num) |
| apply D_line_congr hFd hGd |
| intro s |
| have hy := angular_line_radial x j hj s |
| apply angular_first_congr_leaf ((hF _ hy).differentiableAt (by norm_num)) |
| ((hG _ hy).differentiableAt (by norm_num)) _ i hi |
| intro z hz |
| exact he z (hz.trans hy) |
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| end |
| end CKSSphericalChart |
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| end |
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| end OAI |
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