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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | import OAI.Dynamics.EntropyCounterexample.ClockCharts
import OAI.Dynamics.EntropyCounterexample.C1Coordinates
namespace OAI
noncomputable section
open scoped ContDiff
namespace Problem340
def clockLocalLift (t₀ : ℝ) (x : ClockAmbient) : ℝ :=
t₀ + (100 / (2 * Real.pi)) *
Complex.arg (complexClockLinearIsometry.symm x /
Complex.exp ((2 * Real.pi / 100 * t₀ : ℝ) * Complex.I))
theorem clockHomeomorph_clockLocalLift (t₀ : ℝ) (x : StandardClock) :
clockHomeomorph ((clockLocalLift t₀ x.1 : ℝ) : AddCircle (100 : ℝ)) = x := by
let c : Circle := circleClockHomeomorph.symm x
let c₀ : Circle := AddCircle.toCircle (t₀ : AddCircle (100 : ℝ))
have hc : (c : ℂ) = complexClockLinearIsometry.symm x.1 := rfl
have hc₀ : (c₀ : ℂ) = Complex.exp ((2 * Real.pi / 100 * t₀ : ℝ) * Complex.I) := rfl
have hlift : clockLocalLift t₀ x.1 =
t₀ + (100 / (2 * Real.pi)) * Complex.arg ((c / c₀ : Circle) : ℂ) := by
simp only [clockLocalLift, Circle.coe_div, hc, hc₀]
have hangle : (2 * Real.pi / 100) *
(t₀ + (100 / (2 * Real.pi)) * Complex.arg ((c / c₀ : Circle) : ℂ)) =
(2 * Real.pi / 100) * t₀ + Complex.arg ((c / c₀ : Circle) : ℂ) := by
field_simp
rw [clockHomeomorph_apply, hlift, AddCircle.toCircle_apply_mk, hangle, Circle.exp_add,
Circle.exp_arg]
change circleClockHomeomorph (c₀ * (c / c₀)) = x
simp [c, div_eq_mul_inv, mul_comm]
theorem contDiffOn_arg_slitPlane {n : ℕ∞ω} :
ContDiffOn ℝ n Complex.arg Complex.slitPlane := by
intro z hz
have hlog : ContDiffAt ℝ n Complex.log z :=
(Complex.contDiffAt_log hz).restrict_scalars ℝ
have h := Complex.imCLM.contDiff.contDiffAt.comp z hlog
have heq : (Complex.imCLM ∘ Complex.log) = Complex.arg := by
funext w
exact Complex.log_im w
rw [heq] at h
exact h.contDiffWithinAt
theorem clockLocalLift_extension (t₀ : ℝ) {n : ℕ∞ω} :
∃ U : Set ClockAmbient, IsOpen U ∧ clockParam t₀ ∈ U ∧
ContDiffOn ℝ n (clockLocalLift t₀) U := by
let D : ClockAmbient → ℂ := fun x => complexClockLinearIsometry.symm x /
Complex.exp ((2 * Real.pi / 100 * t₀ : ℝ) * Complex.I)
have hD : ContDiff ℝ n D :=
complexClockLinearIsometry.symm.toContinuousLinearEquiv.contDiff.div_const _
have hbase : D (clockParam t₀) = 1 := by
simp [D, clockParam, Complex.exp_ne_zero]
refine ⟨D ⁻¹' Complex.slitPlane, Complex.isOpen_slitPlane.preimage hD.continuous,
?_, ?_⟩
· change D (clockParam t₀) ∈ Complex.slitPlane
rw [hbase]
exact Complex.one_mem_slitPlane
· exact contDiffOn_const.add (contDiffOn_const.mul
(contDiffOn_arg_slitPlane.comp hD.contDiffOn fun _ hx => hx))
theorem hasLocalC1Extension_of_clock_lift
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
(f : StandardClock → F)
(hf : ContDiff ℝ 1 (fun t : ℝ => f (clockHomeomorph (t : AddCircle (100 : ℝ))))) :
HasLocalC1Extension (fun x : StandardClock => x.1) f := by
intro x
obtain ⟨t, rfl⟩ := clockHomeomorph.surjective x
induction t using QuotientAddGroup.induction_on with
| H t =>
obtain ⟨U, hU, htU, hlocal⟩ := clockLocalLift_extension t (n := 1)
refine ⟨U, hU, ?_,
fun y => f (clockHomeomorph ((clockLocalLift t y : ℝ) : AddCircle (100 : ℝ))),
hf.comp_contDiffOn hlocal, ?_⟩
· simpa only [clockHomeomorph_coe] using htU
· intro y _
change f y = f (clockHomeomorph ((clockLocalLift t y.1 : ℝ) : AddCircle (100 : ℝ)))
rw [clockHomeomorph_clockLocalLift]
theorem hasLocalC1Extension_of_clock_lift_with_parameters
{E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
(f : StandardClock × E → F)
(hf : ContDiff ℝ 1 (fun p : ℝ × E =>
f (clockHomeomorph (p.1 : AddCircle (100 : ℝ)), p.2))) :
HasLocalC1Extension (fun x : StandardClock × E => (x.1.1, x.2)) f := by
rintro ⟨x, a⟩
obtain ⟨t, rfl⟩ := clockHomeomorph.surjective x
induction t using QuotientAddGroup.induction_on with
| H t =>
obtain ⟨U, hU, htU, hlocal⟩ := clockLocalLift_extension t (n := 1)
refine ⟨U ×ˢ Set.univ, hU.prod isOpen_univ,
⟨?_, Set.mem_univ a⟩,
fun p : ClockAmbient × E =>
f (clockHomeomorph ((clockLocalLift t p.1 : ℝ) : AddCircle (100 : ℝ)), p.2),
?_, ?_⟩
· simpa only [clockHomeomorph_coe] using htU
· have hfst : ContDiffOn ℝ 1 (fun p : ClockAmbient × E => clockLocalLift t p.1)
(U ×ˢ Set.univ) :=
hlocal.comp contDiffOn_fst (fun _ hx => hx.1)
have hpair : ContDiffOn ℝ 1
(fun p : ClockAmbient × E => (clockLocalLift t p.1, p.2))
(U ×ˢ Set.univ) := hfst.prodMk contDiffOn_snd
convert hf.comp_contDiffOn hpair using 1
rfl
· rintro ⟨y, b⟩ _
change f (y, b) =
f (clockHomeomorph ((clockLocalLift t y.1 : ℝ) : AddCircle (100 : ℝ)), b)
rw [clockHomeomorph_clockLocalLift]
def clockFunction {F : Type*} (b : ℝ → F) (hb : Function.Periodic b 100) :
StandardClock → F := hb.lift ∘ clockHomeomorph.symm
@[simp]
theorem clockFunction_param {F : Type*} (b : ℝ → F)
(hb : Function.Periodic b 100) (t : ℝ) :
clockFunction b hb (clockHomeomorph (t : AddCircle (100 : ℝ))) = b t := by
change hb.lift (clockHomeomorph.symm (clockHomeomorph (t : AddCircle (100 : ℝ)))) = b t
rw [clockHomeomorph.symm_apply_apply]
rfl
theorem hasLocalC1Extension_clockFunction
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
(b : ℝ → F) (hb : Function.Periodic b 100) (hcont : ContDiff ℝ 1 b) :
HasLocalC1Extension (fun x : StandardClock => x.1) (clockFunction b hb) := by
apply hasLocalC1Extension_of_clock_lift
simpa only [clockFunction_param] using hcont
theorem hasLocalC1Extension_of_clock_lift_on
{E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{V : Set E} (hV : IsOpen V) (f : StandardClock × E → F)
(hf : ContDiffOn ℝ 1 (fun p : ℝ × E =>
f (clockHomeomorph (p.1 : AddCircle (100 : ℝ)), p.2)) (Set.univ ×ˢ V)) :
HasLocalC1Extension (fun x : StandardClock × V => (x.1.1, x.2.1))
(fun x => f (x.1, x.2.1)) := by
rintro ⟨x, a⟩
obtain ⟨t, rfl⟩ := clockHomeomorph.surjective x
induction t using QuotientAddGroup.induction_on with
| H t =>
obtain ⟨U, hU, htU, hlocal⟩ := clockLocalLift_extension t (n := 1)
refine ⟨U ×ˢ V, hU.prod hV, ⟨?_, a.2⟩,
fun p : ClockAmbient × E =>
f (clockHomeomorph ((clockLocalLift t p.1 : ℝ) : AddCircle (100 : ℝ)), p.2),
?_, ?_⟩
· simpa only [clockHomeomorph_coe] using htU
· have hfst : ContDiffOn ℝ 1 (fun p : ClockAmbient × E => clockLocalLift t p.1)
(U ×ˢ V) := hlocal.comp contDiffOn_fst (fun _ hx => hx.1)
have hpair : ContDiffOn ℝ 1
(fun p : ClockAmbient × E => (clockLocalLift t p.1, p.2)) (U ×ˢ V) :=
hfst.prodMk contDiffOn_snd
have hmaps : Set.MapsTo (fun p : ClockAmbient × E => (clockLocalLift t p.1, p.2))
(U ×ˢ V) (Set.univ ×ˢ V) := fun _ hp => ⟨Set.mem_univ _, hp.2⟩
convert hf.comp hpair hmaps using 1
rfl
· rintro ⟨y, b⟩ _
change f (y, b.1) =
f (clockHomeomorph ((clockLocalLift t y.1 : ℝ) : AddCircle (100 : ℝ)), b.1)
rw [clockHomeomorph_clockLocalLift]
end Problem340
end
end OAI
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