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720fa25 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import OAI.Geometry.Arnold.MidpointDerivative
namespace OAI
noncomputable section
open scoped BigOperators ContDiff
open scoped Topology NNReal
open Set Filter
namespace ArnoldCounterexample.Construction
open ArnoldCounterexample.Quadric ArnoldCounterexample.Field ArnoldCounterexample.Factors
open ArnoldCounterexample.ThreePoints
open scoped ContDiff Topology Matrix
open Set Filter
def kernelVector : CV := ![0,-1/4,Complex.I/4,1/4,Complex.I/4]
lemma kernelVector_ne_zero : kernelVector ≠ 0 := by
intro h
have := congrFun h 1
norm_num [kernelVector] at this
lemma kernelVector_horizontal : Horizontal zeroVector kernelVector := by
constructor <;> apply Complex.ext <;>
norm_num [Horizontal, zeroVector, kernelVector, Fin.sum_univ_succ,
Complex.mul_re, Complex.mul_im, map_ofNat, vec_five_two, vec_five_three, vec_five_four]
lemma zero_left : leftSphere zeroVector = pole := congrArg Prod.fst zeroVector_factors
lemma zero_right : rightSphere zeroVector = -east := congrArg Prod.snd zeroVector_factors
lemma zero_casimir : casimir zeroVector = 1 := casimir_unit zeroLift (by rfl)
lemma zero_pairing : pairing zeroVector = 0 := by
simp only [pairing, zero_left, zero_right]
norm_num [pole,east,Fin.sum_univ_succ]
lemma zero_leftOperator (i : Fin 3) : leftOperator i zeroVector =
(![![0,Complex.I/2,1/2,-Complex.I/2,1/2],
![0,-1/2,Complex.I/2,1/2,Complex.I/2],
![0,-Complex.I/2,-1/2,-Complex.I/2,1/2]] : Fin 3 → CV) i := by
fin_cases i <;> ext j <;> fin_cases j <;>
norm_num [leftOperator, rotor_apply, Pi.single_apply, zeroVector,
vec_five_two, vec_five_three, vec_five_four, Matrix.cons_val_two, Fin.ext_iff] <;> ring
lemma zero_rightOperator (i : Fin 3) : rightOperator i zeroVector =
(![![0,Complex.I/2,1/2,Complex.I/2,-1/2],
![0,-1/2,-Complex.I/2,1/2,-Complex.I/2],
![0,-Complex.I/2,1/2,Complex.I/2,1/2]] : Fin 3 → CV) i := by
fin_cases i <;> ext j <;> fin_cases j <;>
norm_num [rightOperator, rotor_apply, Pi.single_apply, zeroVector,
vec_five_two, vec_five_three, vec_five_four, Matrix.cons_val_two, Fin.ext_iff] <;> ring
lemma left_kernel (i : Fin 3) : symplectic (leftOperator i zeroVector) kernelVector = east i := by
rw [zero_leftOperator, symplectic_apply]
fin_cases i <;> norm_num [omega, kernelVector, Fin.sum_univ_succ, Complex.mul_im, east,
vec_five_two, vec_five_three, vec_five_four, Matrix.cons_val_two, map_ofNat]
lemma right_kernel (i : Fin 3) : symplectic (rightOperator i zeroVector) kernelVector = 0 := by
rw [zero_rightOperator, symplectic_apply]
fin_cases i <;> norm_num [omega, kernelVector, Fin.sum_univ_succ, Complex.mul_im,
vec_five_two, vec_five_three, vec_five_four, Matrix.cons_val_two, map_ofNat]
lemma zero_operator_cancel : rightOperator 0 zeroVector + leftOperator 2 zeroVector = 0 := by
rw [zero_leftOperator, zero_rightOperator]
ext i
fin_cases i <;> norm_num [vec_five_two, vec_five_three, vec_five_four, Matrix.cons_val_two] <;> ring
def kernelCurve (t : ℝ) : CV := zeroVector + t • kernelVector
lemma kernelCurve_zero : kernelCurve 0 = zeroVector := by simp [kernelCurve]
lemma kernelCurve_derivative : HasDerivAt kernelCurve kernelVector 0 := by
convert! (hasDerivAt_const (0:ℝ) zeroVector).add ((hasDerivAt_id (0:ℝ)).smul_const kernelVector) using 1
simp
lemma left_curve (i : Fin 3) : HasDerivAt (fun t ↦ leftSphere (kernelCurve t) i) (east i) 0 := by
convert! (left_hasFDerivAt i (kernelCurve 0)).comp_hasDerivAt 0 kernelCurve_derivative using 1
simp only [kernelCurve_zero, left_kernel]
lemma right_curve (i : Fin 3) : HasDerivAt (fun t ↦ rightSphere (kernelCurve t) i) 0 0 := by
convert! (right_hasFDerivAt i (kernelCurve 0)).comp_hasDerivAt 0 kernelCurve_derivative using 1
simp only [kernelCurve_zero, right_kernel]
lemma casimir_curve : HasDerivAt (fun t ↦ casimir (kernelCurve t)) 0 0 := by
have h := HasDerivAt.fun_sum (u := Finset.univ) (fun i : Fin 3 ↦ fun _ ↦ (left_curve i).pow 2)
convert! h using 1
norm_num [kernelCurve_zero, zero_left, pole, east, Fin.sum_univ_succ]
lemma pairing_curve : HasDerivAt (fun t ↦ pairing (kernelCurve t)) (-1) 0 := by
have h := HasDerivAt.fun_sum (u := Finset.univ) (fun i : Fin 3 ↦ fun _ ↦ (left_curve i).mul (right_curve i))
convert! h using 1
norm_num [kernelCurve_zero, zero_left, zero_right, pole, east, Fin.sum_univ_succ]
lemma mixed_curve : HasDerivAt (fun t ↦ mixedField (kernelCurve t))
(-leftOperator 0 kernelVector + rightOperator 2 kernelVector + rightOperator 0 zeroVector) 0 := by
have h := HasDerivAt.fun_sum (u := Finset.univ) (fun i : Fin 3 ↦ fun _ ↦
((right_curve i).smul ((leftOperator i).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative)).add
((left_curve i).smul ((rightOperator i).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative)))
convert! h using 1
simp [kernelCurve_zero, zero_left, zero_right, pole, east, Fin.sum_univ_succ]
module
lemma core_curve : HasDerivAt (fun t ↦ core (kernelCurve t)) 0 0 := by
have hb : HasDerivAt (fun t ↦ radial (casimir (kernelCurve t))) 0 0 := by
convert! (radial_smooth.differentiable (by simp) (casimir (kernelCurve 0))).hasDerivAt.comp 0 casimir_curve using 1
simp only [mul_zero]
have hds : ContDiff ℝ ∞ (deriv radial) := (contDiff_infty_iff_deriv.mp radial_smooth).2
have hdb : HasDerivAt (fun t ↦ deriv radial (casimir (kernelCurve t))) 0 0 := by
convert! (hds.differentiable (by simp) (casimir (kernelCurve 0))).hasDerivAt.comp 0 casimir_curve using 1
simp only [mul_zero]
have hc : HasDerivAt (fun t ↦ 2 * deriv radial (casimir (kernelCurve t)) * pairing (kernelCurve t)) 1 0 := by
convert! (hdb.const_mul 2).mul pairing_curve using 1
simp [kernelCurve_zero, zero_casimir, zero_pairing, radial_at_one.2]
have hs : HasDerivAt (fun t ↦ selfField (kernelCurve t))
(fderiv ℝ selfField zeroVector kernelVector) 0 := by
convert! (self_smooth.differentiable (by simp) (kernelCurve 0)).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative using 1
simp only [kernelCurve_zero]
have h := (((leftOperator 0).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative).sub
((rightOperator 2).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative)).add (hb.smul mixed_curve) |>.add (hc.smul hs)
convert! h using 1
simp [kernelCurve_zero, zero_casimir, zero_pairing, radial_at_one.1, selfField,
zero_left, pole, Fin.sum_univ_succ]
calc
_ = rightOperator 0 zeroVector + leftOperator 2 zeroVector := zero_operator_cancel.symm
_ = _ := by module
lemma core_kernel : fderiv ℝ core zeroVector kernelVector = 0 := by
have h := (core_smooth.differentiable (by simp) (kernelCurve 0)).hasFDerivAt.comp_hasDerivAt 0 kernelCurve_derivative
simpa only [kernelCurve_zero] using h.unique core_curve
lemma vectorField_zeroVector : vectorField zeroVector = 0 := by
rw [vectorField, core_zeroVector, smul_zero]
lemma vectorField_kernel : fderiv ℝ vectorField zeroVector kernelVector = 0 := by
have h := vectorField_local_core (by rw [zeroVector_unit.1]; norm_num : normSquared zeroVector < 4)
rw [h.fderiv_eq]
exact core_kernel
end ArnoldCounterexample.Construction
end
end OAI
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