| import OAI.Dynamics.EntropyCounterexample.Definitions |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff |
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| namespace Problem340 |
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| def complexClockLinearIsometry : ℂ ≃ₗᵢ[ℝ] ClockAmbient := |
| Complex.orthonormalBasisOneI.repr |
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| @[simp] |
| theorem complexClockLinearIsometry_apply (z : ℂ) : |
| complexClockLinearIsometry z = ![z.re, z.im] := rfl |
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| 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] |
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| @[simp] |
| theorem circleClockHomeomorph_coe (z : Circle) : |
| (circleClockHomeomorph z).1 = complexClockLinearIsometry (z : ℂ) := rfl |
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| def clockHomeomorph : AddCircle (100 : ℝ) ≃ₜ StandardClock := |
| (AddCircle.homeomorphCircle (by norm_num : (100 : ℝ) ≠ 0)).trans |
| circleClockHomeomorph |
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| @[simp] |
| theorem clockHomeomorph_apply (t : AddCircle (100 : ℝ)) : |
| clockHomeomorph t = circleClockHomeomorph (AddCircle.toCircle t) := by |
| simp only [clockHomeomorph, Homeomorph.trans_apply, AddCircle.homeomorphCircle_apply] |
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| def clockParam (t : ℝ) : ClockAmbient := |
| complexClockLinearIsometry (Complex.exp ((2 * Real.pi / 100 * t : ℝ) * Complex.I)) |
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| @[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] |
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| 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 |
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| theorem clockParam_periodic : Function.Periodic clockParam 100 := by |
| intro t |
| rw [← clockHomeomorph_coe, ← clockHomeomorph_coe] |
| congr 2 |
| exact AddCircle.coe_add_period 100 t |
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| def ambientClockRotate (s : ℝ) (x : ClockAmbient) : ClockAmbient := |
| complexClockLinearIsometry |
| (Complex.exp ((2 * Real.pi / 100 * s : ℝ) * Complex.I) * |
| complexClockLinearIsometry.symm x) |
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| 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 |
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| 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⟩ |
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| 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] |
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| end Problem340 |
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| end |
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| end OAI |
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