| import OAI.Geometry.RuledCovering.Nilpotent |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma singular_parameter_bounds {p gamma : ℝ} (hp : 1 < p) |
| (hp' : p < (Real.sqrt 51-5)/2) (hg : 1 < gamma) (hg' : gamma < 1/(2-p)) : |
| p < 2 ∧ 0 < 1+gamma*(p-2) ∧ 0 < 2*gamma-2 ∧ singularCoefficient p < 0 := by |
| have hs : (Real.sqrt 51)^2 = (51:ℝ) := Real.sq_sqrt (by norm_num) |
| have hsn := Real.sqrt_nonneg (51:ℝ) |
| have hs9 : Real.sqrt 51 < 9 := by nlinarith |
| have hp2 : p < 2 := by linarith |
| have hgam : gamma*(2-p) < 1 := (lt_div_iff₀ (sub_pos.mpr hp2)).mp hg' |
| have hroot : 2*p+5 < Real.sqrt 51 := by linarith |
| have hnum : 2*p^2+10*p-13 < 0 := by nlinarith |
| refine ⟨hp2, by nlinarith, by linarith, ?_⟩ |
| exact div_neg_of_neg_of_pos (mul_neg_of_pos_of_neg (by norm_num) hnum) (by linarith) |
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| lemma singularField_contDiffAt (p : ℝ) {z : Plane} (hz : 0 < z 1) : |
| ContDiffAt ℝ ∞ (singularField p) z := by |
| apply contDiffAt_euclidean.mpr |
| intro i |
| fin_cases i |
| · change ContDiffAt ℝ ∞ (fun b : Plane => ((1/2:ℝ)/2)*(b 1/(1/2))^(p-1)* |
| (p+(2-p)*(b 0/b 1)^2)) z |
| have h1 : ContDiffAt ℝ ∞ (fun b : Plane => b 1/(1/2:ℝ)) z := by fun_prop |
| have hpow := h1.rpow_const_of_ne (p := p-1) (by positivity) |
| exact (contDiffAt_const.mul hpow).mul |
| (contDiffAt_const.add (contDiffAt_const.mul (((by fun_prop : |
| ContDiffAt ℝ ∞ (fun b : Plane => b 0) z).div (by fun_prop) hz.ne').pow 2))) |
| · change ContDiffAt ℝ ∞ (fun b : Plane => (b 1/(1/2:ℝ))^(p-2)*b 0) z |
| have h1 : ContDiffAt ℝ ∞ (fun b : Plane => b 1/(1/2:ℝ)) z := by fun_prop |
| exact (h1.rpow_const_of_ne (p := p-2) (by positivity)).mul (by fun_prop) |
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| lemma singularField_contDiffOn (p : ℝ) : |
| ContDiffOn ℝ ∞ (singularField p) {z : Plane | 0 < z 1} := |
| fun _ hz => (singularField_contDiffAt p hz).contDiffWithinAt |
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| lemma singularField_smul (p : ℝ) {r : ℝ} (hr : 0 < r) {z : Plane} (hz : 0 < z 1) : |
| singularField p (r • z) = r^(p-1) • singularField p z := by |
| have he : (r*z 1)/(1/2:ℝ) = r*(z 1/(1/2)) := by ring |
| have hpow : r^(p-1) = r^(p-2)*r := by |
| rw [show p-1 = p-2+1 by ring,Real.rpow_add_one hr.ne'] |
| ext i |
| fin_cases i |
| · change ((1/2:ℝ)/2)*((r*z 1)/(1/2))^(p-1)*(p+(2-p)*(r*z 0/(r*z 1))^2) = |
| r^(p-1)*(((1/2:ℝ)/2)*(z 1/(1/2))^(p-1)*(p+(2-p)*(z 0/z 1)^2)) |
| rw [he,Real.mul_rpow hr.le (by positivity),mul_div_mul_left _ _ hr.ne'] |
| ring |
| · change ((r*z 1)/(1/2:ℝ))^(p-2)*(r*z 0) = r^(p-1)*((z 1/(1/2))^(p-2)*z 0) |
| rw [he,Real.mul_rpow hr.le (by positivity),hpow] |
| ring |
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| lemma singularField_fderiv_smul (p : ℝ) {r : ℝ} (hr : 0 < r) {z : Plane} (hz : 0 < z 1) : |
| fderiv ℝ (singularField p) (r • z) = r^(p-2) • fderiv ℝ (singularField p) z := by |
| have he : (fun w : Plane => singularField p (r • w)) =ᶠ[𝓝 z] |
| (fun w : Plane => r^(p-1) • singularField p w) := by |
| filter_upwards [(isOpen_lt continuous_const (by fun_prop : Continuous (fun w : Plane => w 1))).mem_nhds hz] |
| with w hw |
| exact singularField_smul p hr hw |
| have hd := (singularField_contDiffAt p hz).differentiableAt (by norm_num) |
| have he' := he.fderiv_eq (𝕜 := ℝ) |
| rw [fderiv_comp_smul r, fderiv_fun_const_smul hd] at he' |
| have hp : r^(p-1) = r*r^(p-2) := by |
| rw [show p-1 = p-2+1 by ring,Real.rpow_add_one hr.ne'] |
| ring |
| rw [hp,mul_smul] at he' |
| exact (smul_right_injective (Plane →L[ℝ] Plane) hr.ne') he' |
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| lemma singularField_level (p b : ℝ) : |
| singularField p (vec2 b (1/2)) = vec2 (p/4+(2-p)*b^2) b := by |
| ext i |
| fin_cases i <;> simp [singularField,vec2]; ring |
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| lemma singularField_wall (p : ℝ) {b r : ℝ} (hr : 0 < r) : |
| singularField p (vec2 b r) 1 + singularField p (vec2 b r) 0 = |
| (r+b)/2*(r/(1/2:ℝ))^(p-2)*((2-p)*b/r+p) ∧ |
| singularField p (vec2 b r) 1 - singularField p (vec2 b r) 0 = |
| (r-b)/2*(r/(1/2:ℝ))^(p-2)*((2-p)*b/r-p) := by |
| have hp : (r/(1/2:ℝ))^(p-1) = (r/(1/2:ℝ))^(p-2)*(r/(1/2)) := by |
| rw [show p-1 = p-2+1 by ring,Real.rpow_add_one (by positivity : r/(1/2:ℝ)≠0)] |
| simp only [singularField,vec2_apply_one,vec2_apply_zero,hp] |
| constructor <;> field_simp <;> ring |
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| def matrixCLM (a b c d : ℝ) : Plane →L[ℝ] Plane := |
| (a • EuclideanSpace.proj 0 + b • EuclideanSpace.proj 1 : Plane →L[ℝ] ℝ).smulRight (vec2 1 0) + |
| (c • EuclideanSpace.proj 0 + d • EuclideanSpace.proj 1 : Plane →L[ℝ] ℝ).smulRight (vec2 0 1) |
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| @[simp] lemma matrixCLM_apply (a b c d : ℝ) (z : Plane) : |
| matrixCLM a b c d z = vec2 (a*z 0+b*z 1) (c*z 0+d*z 1) := by |
| ext i |
| fin_cases i <;> simp [matrixCLM,vec2] |
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| lemma hasFDerivAt_vec2 {f g : Plane → ℝ} {f' g' : Plane →L[ℝ] ℝ} {z : Plane} |
| (hf : HasFDerivAt f f' z) (hg : HasFDerivAt g g' z) : |
| HasFDerivAt (fun w => vec2 (f w) (g w)) |
| (f'.smulRight (vec2 1 0) + g'.smulRight (vec2 0 1)) z := by |
| have he : (fun w => vec2 (f w) (g w)) = |
| (fun w => f w • vec2 1 0 + g w • vec2 0 1) := by |
| funext w |
| ext i |
| fin_cases i <;> simp [vec2] |
| rw [he] |
| exact (hf.smul_const (vec2 1 0)).add (hg.smul_const (vec2 0 1)) |
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| lemma singularField_fderiv (p : ℝ) {z : Plane} (hz : 0 < z 1) : |
| fderiv ℝ (singularField p) z = |
| matrixCLM ((2-p)*(z 1/(1/2:ℝ))^(p-2)*(z 0/z 1)) |
| ((z 1/(1/2:ℝ))^(p-2)/2*(p*(p-1)+(2-p)*(p-3)*(z 0/z 1)^2)) |
| ((z 1/(1/2:ℝ))^(p-2)) ((p-2)*(z 1/(1/2:ℝ))^(p-2)*(z 0/z 1)) := by |
| let b : Plane →L[ℝ] ℝ := EuclideanSpace.proj 0 |
| let r : Plane →L[ℝ] ℝ := EuclideanSpace.proj 1 |
| have hb := b.hasFDerivAt (x := z) |
| have hr := r.hasFDerivAt (x := z) |
| have hri := (hasDerivAt_inv hz.ne').comp_hasFDerivAt z hr |
| have hquot := hb.mul hri |
| have hscale := hr.const_mul 2 |
| have hpow1 := hscale.rpow_const (p := p-1) (Or.inl (by change 2*z 1 ≠ 0; positivity)) |
| have hpow2 := hscale.rpow_const (p := p-2) (Or.inl (by change 2*z 1 ≠ 0; positivity)) |
| have hP := (hpow1.const_mul ((1/2:ℝ)/2)).mul |
| (((hquot.pow 2).const_mul (2-p)).const_add p) |
| have hQ := hpow2.mul hb |
| have hf := (hasFDerivAt_vec2 hP hQ).fderiv |
| have he : (fun w : Plane => vec2 (((1/2:ℝ)/2)*(2*r w)^(p-1)* |
| (p+(2-p)*(b w*(r w)⁻¹)^2)) ((2*r w)^(p-2)*b w)) = singularField p := by |
| funext w |
| ext i |
| fin_cases i <;> simp [singularField,vec2,b,r,div_eq_mul_inv,mul_comm] |
| change fderiv ℝ (fun w : Plane => vec2 (((1/2:ℝ)/2)*(2*r w)^(p-1)* |
| (p+(2-p)*(b w*(r w)⁻¹)^2)) ((2*r w)^(p-2)*b w)) z = _ at hf |
| rw [he] at hf |
| rw [hf] |
| have h2 : 0 < 2*z 1 := by positivity |
| have hp1 : (2*z 1)^(p-1-1) = (2*z 1)^(p-2) := by congr 1; ring |
| have hp2 : (2*z 1)^(p-2-1) = (2*z 1)^(p-2)/(2*z 1) := by |
| rw [Real.rpow_sub_one h2.ne'] |
| have hp3 : (2*z 1)^(p-1) = (2*z 1)^(p-2)*(2*z 1) := by |
| rw [show p-1 = p-2+1 by ring,Real.rpow_add_one h2.ne'] |
| ext w i |
| fin_cases i <;> |
| simp [matrixCLM,b,r,ContinuousLinearMap.smulRight_apply,vec2,hp1,hp2,hp3] <;> |
| (rw [show z 1*2=2*z 1 by ring]; field_simp [hz.ne'] <;> ring) |
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| lemma matrixCLM_trace (a b c d : ℝ) : |
| LinearMap.trace ℝ Plane (matrixCLM a b c d).toLinearMap = a+d := by |
| rw [LinearMap.trace_eq_matrix_trace ℝ (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis, |
| Matrix.trace_fin_two] |
| simp [LinearMap.toMatrix_apply, EuclideanSpace.basisFun_apply, |
| EuclideanSpace.basisFun_repr,matrixCLM_apply] |
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| lemma matrixCLM_det (a b c d : ℝ) : |
| LinearMap.det (matrixCLM a b c d).toLinearMap = a*d-b*c := by |
| rw [← LinearMap.det_toMatrix (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis, |
| Matrix.det_fin_two] |
| simp [LinearMap.toMatrix_apply, EuclideanSpace.basisFun_apply, |
| EuclideanSpace.basisFun_repr,matrixCLM_apply] |
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| lemma singularField_trace_det (p : ℝ) {z : Plane} (hz : 0 < z 1) : |
| LinearMap.trace ℝ Plane (fderiv ℝ (singularField p) z).toLinearMap = 0 ∧ |
| LinearMap.det (fderiv ℝ (singularField p) z).toLinearMap = |
| ((z 1/(1/2:ℝ))^(p-2))^2*(p-1)/2*((2-p)*(z 0/z 1)^2-p) := by |
| rw [singularField_fderiv p hz,matrixCLM_trace,matrixCLM_det] |
| constructor <;> ring |
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| lemma singularField_hyperbolic {p : ℝ} (hp : 1 < p) (hp2 : p < 2) |
| {z : Plane} (hz : 0 < z 1) (hb : |z 0| ≤ z 1) : |
| LinearMap.trace ℝ Plane (fderiv ℝ (singularField p) z).toLinearMap = 0 ∧ |
| LinearMap.det (fderiv ℝ (singularField p) z).toLinearMap < 0 := by |
| obtain ⟨ht,hd⟩ := singularField_trace_det p hz |
| refine ⟨ht,?_⟩ |
| rw [hd] |
| have hq : |z 0/z 1| ≤ 1 := by |
| rw [abs_div,abs_of_pos hz,div_le_one hz] |
| exact hb |
| have hq2 : (z 0/z 1)^2 ≤ 1 := by |
| have := mul_self_le_mul_self (abs_nonneg (z 0/z 1)) hq |
| nlinarith [sq_abs (z 0/z 1)] |
| apply mul_neg_of_pos_of_neg |
| · exact div_pos (mul_pos (sq_pos_of_pos (Real.rpow_pos_of_pos (by positivity) _)) (by linarith)) (by norm_num) |
| · have := mul_le_mul_of_nonneg_left hq2 (sub_nonneg.mpr hp2.le) |
| nlinarith |
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| lemma singularField_scaled_hyperbolic {p c : ℝ} (hp : 1 < p) (hp2 : p < 2) |
| (hc : c ≠ 0) {z : Plane} (hz : 0 < z 1) (hb : |z 0| ≤ z 1) |
| (hd : |c| *‖fderiv ℝ (singularField p) z‖ < 1) : |
| HyperbolicAt 1 (fun w => c • singularField p w) z := by |
| have hdiff := (singularField_contDiffAt p hz).differentiableAt (by norm_num) |
| rw [HyperbolicAt,fderiv_fun_const_smul hdiff] |
| have htd := singularField_hyperbolic hp hp2 hz hb |
| have heig := plane_hyperbolic_eigenvalues _ htd.1 htd.2 |
| let lam := Real.sqrt (-LinearMap.det (fderiv ℝ (singularField p) z).toLinearMap) |
| have hlam : 0 < lam := Real.sqrt_pos.mpr (by linarith [htd.2]) |
| have hnorm : lam ≤ ‖fderiv ℝ (singularField p) z‖ := eigenvalue_le_norm _ hlam.le heig.1 |
| have hnonneg : 0 < |c| := abs_pos.mpr hc |
| refine ⟨|c| *lam,by simpa using mul_pos hnonneg hlam,?_,?_,?_⟩ |
| · simpa using (mul_le_mul_of_nonneg_left hnorm hnonneg.le).trans_lt hd |
| · by_cases hcp : 0 ≤ c |
| · simpa [abs_of_nonneg hcp] using eigenvalue_smul _ heig.1 c |
| · have hcn : c < 0 := lt_of_not_ge hcp |
| simpa [abs_of_neg hcn] using eigenvalue_smul _ heig.2 c |
| · by_cases hcp : 0 ≤ c |
| · simpa [abs_of_nonneg hcp] using eigenvalue_smul _ heig.2 c |
| · have hcn : c < 0 := lt_of_not_ge hcp |
| simpa [abs_of_neg hcn] using eigenvalue_smul _ heig.1 c |
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| end RuledApproximation |
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| end |
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| end OAI |
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