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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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Dynamics.EntropyCounterexample.Definitions
namespace OAI
noncomputable section
namespace Problem340.ClockModel
def circleHomeomorph : Circle ≃ₜ StandardClock :=
Complex.orthonormalBasisOneI.repr.toHomeomorph.subtype (by
intro z
change z ∈ Metric.sphere (0 : ℂ) 1 ↔
Complex.orthonormalBasisOneI.repr z ∈ Metric.sphere (0 : ClockAmbient) 1
simp only [Metric.mem_sphere, dist_zero_right, LinearIsometryEquiv.norm_map])
@[simp] theorem circleHomeomorph_val (z : Circle) :
(circleHomeomorph z).val = Complex.orthonormalBasisOneI.repr (z : ℂ) := rfl
@[simp] theorem circleHomeomorph_coord_zero (z : Circle) :
(circleHomeomorph z).val 0 = (z : ℂ).re := by
simp [circleHomeomorph_val, Complex.orthonormalBasisOneI_repr_apply]
@[simp] theorem circleHomeomorph_coord_one (z : Circle) :
(circleHomeomorph z).val 1 = (z : ℂ).im := by
simp [circleHomeomorph_val, Complex.orthonormalBasisOneI_repr_apply]
def clockHomeomorph : AddCircle (100 : ℝ) ≃ₜ StandardClock :=
(AddCircle.homeomorphCircle (T := (100 : ℝ)) (by norm_num)).trans circleHomeomorph
def rotationAmbient (p : ℝ × ClockAmbient) : ClockAmbient :=
Complex.orthonormalBasisOneI.repr
((Circle.exp (2 * Real.pi / 100 * p.1) : ℂ) *
Complex.orthonormalBasisOneI.repr.symm p.2)
theorem contDiff_rotationAmbient : ContDiff ℝ ⊤ rotationAmbient := by
unfold rotationAmbient
apply Complex.orthonormalBasisOneI.repr.contDiff.comp
apply ContDiff.mul
· simp only [Circle.coe_exp]
exact Complex.contDiff_exp.comp
((Complex.ofRealCLM.contDiff.comp (contDiff_const.mul contDiff_fst)).mul contDiff_const)
· exact Complex.orthonormalBasisOneI.repr.symm.contDiff.comp contDiff_snd
def clockRotation (s : ℝ) (z : StandardClock) : StandardClock :=
circleHomeomorph
(Circle.exp (2 * Real.pi / 100 * s) * circleHomeomorph.symm z)
@[simp] theorem clockRotation_val (s : ℝ) (z : StandardClock) :
(clockRotation s z).val = rotationAmbient (s, z.val) := rfl
theorem clockHomeomorph_add (t : AddCircle (100 : ℝ)) (s : ℝ) :
clockHomeomorph (t + (s : AddCircle (100 : ℝ))) =
clockRotation s (clockHomeomorph t) := by
simp [clockHomeomorph, clockRotation, Homeomorph.trans_apply,
AddCircle.homeomorphCircle_apply, AddCircle.toCircle_add,
AddCircle.toCircle_apply_mk, mul_comm]
theorem contDiff_rotationAmbient_comp {E : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
{n : WithTop ℕ∞} {h : E → ℝ} {x : E → ClockAmbient}
(hh : ContDiff ℝ n h) (hx : ContDiff ℝ n x) :
ContDiff ℝ n (fun y => rotationAmbient (h y, x y)) :=
(contDiff_rotationAmbient.of_le le_top).comp (hh.prodMk hx)
def prependHomeomorph (q : ℕ) (S : Type*) [TopologicalSpace S] :
S × (Fin q → S) ≃ₜ (Fin (q + 1) → S) where
toFun p := Fin.cons p.1 p.2
invFun x := (x 0, fun i => x i.succ)
left_inv p := by cases p; rfl
right_inv x := by
funext i
exact Fin.cases rfl (fun _ => rfl) i
continuous_toFun := by
apply continuous_pi
intro i
refine Fin.cases ?_ (fun j => ?_) i
· exact continuous_fst
· exact (continuous_apply j).comp continuous_snd
continuous_invFun :=
(continuous_apply 0).prodMk (continuous_pi fun i => continuous_apply i.succ)
@[simp] theorem prependHomeomorph_zero (q : ℕ) {S : Type*} [TopologicalSpace S]
(p : S × (Fin q → S)) : prependHomeomorph q S p 0 = p.1 := rfl
@[simp] theorem prependHomeomorph_succ (q : ℕ) {S : Type*} [TopologicalSpace S]
(p : S × (Fin q → S)) (i : Fin q) :
prependHomeomorph q S p i.succ = p.2 i := rfl
def productHomeomorph (q : ℕ) {S : Type*} [TopologicalSpace S]
(eSphere : S ≃ₜ StandardSphere2) :
(AddCircle (100 : ℝ) × (S × (Fin q → S))) ≃ₜ CounterexampleManifold q :=
clockHomeomorph.prodCongr
((prependHomeomorph q S).trans (Homeomorph.piCongrRight fun _ => eSphere))
@[simp] theorem productHomeomorph_clock (q : ℕ) {S : Type*} [TopologicalSpace S]
(eSphere : S ≃ₜ StandardSphere2)
(x : AddCircle (100 : ℝ) × (S × (Fin q → S))) :
(productHomeomorph q eSphere x).1 = clockHomeomorph x.1 := rfl
@[simp] theorem productHomeomorph_main (q : ℕ) {S : Type*} [TopologicalSpace S]
(eSphere : S ≃ₜ StandardSphere2)
(x : AddCircle (100 : ℝ) × (S × (Fin q → S))) :
(productHomeomorph q eSphere x).2 0 = eSphere x.2.1 := rfl
@[simp] theorem productHomeomorph_register (q : ℕ) {S : Type*} [TopologicalSpace S]
(eSphere : S ≃ₜ StandardSphere2)
(x : AddCircle (100 : ℝ) × (S × (Fin q → S))) (i : Fin q) :
(productHomeomorph q eSphere x).2 i.succ = eSphere (x.2.2 i) := rfl
end Problem340.ClockModel
end
end OAI
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