| import OAI.Dynamics.EntropyCounterexample.ClockRotation |
| import OAI.Dynamics.EntropyCounterexample.ClockModel |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340.Clock |
|
|
| lemma clockHomeomorph_val_real (t : ℝ) : |
| (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))).val = parametrization t := by |
| ext i |
| fin_cases i |
| · change (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))).val 0 = |
| Real.cos (2 * Real.pi * t / 100) |
| simp [ClockModel.clockHomeomorph, Homeomorph.trans_apply, |
| AddCircle.homeomorphCircle_apply, |
| AddCircle.toCircle_apply_mk, Circle.coe_exp, Complex.exp_re] |
| congr 1 |
| ring |
| · change (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))).val 1 = |
| Real.sin (2 * Real.pi * t / 100) |
| simp [ClockModel.clockHomeomorph, Homeomorph.trans_apply, |
| AddCircle.homeomorphCircle_apply, |
| AddCircle.toCircle_apply_mk, Circle.coe_exp, Complex.exp_im] |
| congr 1 |
| ring |
|
|
| lemma ambientBarrier_clockHomeomorph (t : ℝ) : |
| ambientBarrier (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))).val = barrier t := by |
| rw [clockHomeomorph_val_real] |
| exact ambientBarrier_on_circle t (parametrization t) rfl |
|
|
| lemma update_clockHomeomorph_real (t e : ℝ) : |
| update (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))) e = |
| ClockModel.clockHomeomorph ((t + increment t e : ℝ) : AddCircle (100 : ℝ)) := by |
| apply Subtype.ext |
| change ambientUpdate _ e = _ |
| rw [clockHomeomorph_val_real, clockHomeomorph_val_real, ambientUpdate_parametrization] |
|
|
| lemma update_clockHomeomorph (t : AddCircle (100 : ℝ)) (e : ℝ) : |
| update (ClockModel.clockHomeomorph t) e = |
| ClockModel.clockHomeomorph |
| (t + (ambientIncrement (ClockModel.clockHomeomorph t).val e : ℝ)) := by |
| obtain ⟨t, rfl⟩ := QuotientAddGroup.mk_surjective t |
| have hi : ambientIncrement (ClockModel.clockHomeomorph (t : AddCircle (100 : ℝ))).val e = |
| increment t e := by |
| rw [clockHomeomorph_val_real] |
| exact ambientIncrement_parametrization t e |
| rw [hi] |
| exact update_clockHomeomorph_real t e |
|
|
| end Problem340.Clock |
|
|
| end |
|
|
| end OAI |
|
|
