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92416a2 | 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 | import OAI.Dynamics.EntropyCounterexample.Definitions
namespace OAI
noncomputable section
open scoped ContDiff
namespace Problem340
def complexClockLinearIsometry : ℂ ≃ₗᵢ[ℝ] ClockAmbient :=
Complex.orthonormalBasisOneI.repr
@[simp]
theorem complexClockLinearIsometry_apply (z : ℂ) :
complexClockLinearIsometry z = ![z.re, z.im] := rfl
def circleClockHomeomorph : Circle ≃ₜ StandardClock :=
complexClockLinearIsometry.toHomeomorph.subtype fun z => by
change z ∈ Metric.sphere (0 : ℂ) 1 ↔
complexClockLinearIsometry z ∈ Metric.sphere (0 : ClockAmbient) 1
simp only [Metric.mem_sphere, dist_zero_right,
complexClockLinearIsometry.norm_map]
@[simp]
theorem circleClockHomeomorph_coe (z : Circle) :
(circleClockHomeomorph z).1 = complexClockLinearIsometry (z : ℂ) := rfl
def clockHomeomorph : AddCircle (100 : ℝ) ≃ₜ StandardClock :=
(AddCircle.homeomorphCircle (by norm_num : (100 : ℝ) ≠ 0)).trans
circleClockHomeomorph
@[simp]
theorem clockHomeomorph_apply (t : AddCircle (100 : ℝ)) :
clockHomeomorph t = circleClockHomeomorph (AddCircle.toCircle t) := by
simp only [clockHomeomorph, Homeomorph.trans_apply, AddCircle.homeomorphCircle_apply]
def clockParam (t : ℝ) : ClockAmbient :=
complexClockLinearIsometry (Complex.exp ((2 * Real.pi / 100 * t : ℝ) * Complex.I))
@[simp]
theorem clockHomeomorph_coe (t : ℝ) :
(clockHomeomorph (t : AddCircle (100 : ℝ))).1 = clockParam t := by
simp only [clockHomeomorph_apply, circleClockHomeomorph_coe,
AddCircle.toCircle_apply_mk, Circle.coe_exp, clockParam]
theorem contDiff_clockParam {n : ℕ∞ω} : ContDiff ℝ n clockParam := by
unfold clockParam
apply complexClockLinearIsometry.toContinuousLinearEquiv.contDiff.comp
exact ((Complex.ofRealCLM.contDiff.comp (contDiff_const.mul contDiff_id)).mul
contDiff_const).cexp
theorem clockParam_periodic : Function.Periodic clockParam 100 := by
intro t
rw [← clockHomeomorph_coe, ← clockHomeomorph_coe]
congr 2
exact AddCircle.coe_add_period 100 t
def ambientClockRotate (s : ℝ) (x : ClockAmbient) : ClockAmbient :=
complexClockLinearIsometry
(Complex.exp ((2 * Real.pi / 100 * s : ℝ) * Complex.I) *
complexClockLinearIsometry.symm x)
theorem contDiff_ambientClockRotate {n : ℕ∞ω} :
ContDiff ℝ n (fun p : ℝ × ClockAmbient => ambientClockRotate p.1 p.2) := by
unfold ambientClockRotate
apply complexClockLinearIsometry.toContinuousLinearEquiv.contDiff.comp
apply ContDiff.mul
· exact ((Complex.ofRealCLM.contDiff.comp (contDiff_const.mul contDiff_fst)).mul
contDiff_const).cexp
· exact complexClockLinearIsometry.symm.toContinuousLinearEquiv.contDiff.comp contDiff_snd
def clockRotate (s : ℝ) (x : StandardClock) : StandardClock :=
⟨ambientClockRotate s x.1, by
apply mem_sphere_zero_iff_norm.mpr
simp only [ambientClockRotate, complexClockLinearIsometry.norm_map, norm_mul,
complexClockLinearIsometry.symm.norm_map]
have hphase : ‖Complex.exp ((2 * Real.pi / 100 * s : ℝ) * Complex.I)‖ = 1 := by
simp [Complex.norm_exp]
rw [hphase, one_mul]
exact mem_sphere_zero_iff_norm.mp x.2⟩
theorem clockHomeomorph_add (t : AddCircle (100 : ℝ)) (s : ℝ) :
clockHomeomorph (t + (s : AddCircle (100 : ℝ))) =
clockRotate s (clockHomeomorph t) := by
apply Subtype.ext
simp only [clockHomeomorph_apply, circleClockHomeomorph_coe,
AddCircle.toCircle_add, Circle.coe_mul, AddCircle.toCircle_apply_mk,
Circle.coe_exp, clockRotate, ambientClockRotate,
LinearIsometryEquiv.symm_apply_apply]
rw [mul_comm]
end Problem340
end
end OAI
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