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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 | 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
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