File size: 2,937 Bytes
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 | import OAI.Dynamics.EntropyCounterexample.ClockLocalExtension
namespace OAI
noncomputable section
namespace Problem340
namespace HasLocalC1Extension
theorem precomp_ambient {X Y D E F : Type*}
[NormedAddCommGroup D] [NormedSpace ℝ D]
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{eX : X → E} {eY : Y → D} {f : X → F}
(hf : HasLocalC1Extension eX f) (k : Y → X) (K : D → E)
(hK : ContDiff ℝ 1 K) (hcomm : ∀ y, eX (k y) = K (eY y)) :
HasLocalC1Extension eY (f ∘ k) := by
intro y
obtain ⟨U, hU, hyU, F, hF, hcommF⟩ := hf (k y)
refine ⟨K ⁻¹' U, hU.preimage hK.continuous, ?_, F ∘ K,
hF.comp hK.contDiffOn (fun _ hx => hx), ?_⟩
· simpa only [Set.mem_preimage, ← hcomm y] using hyU
· intro z hz
change f (k z) = F (K (eY z))
rw [← hcomm z]
apply hcommF
simpa only [Set.mem_preimage, ← hcomm z] using hz
end HasLocalC1Extension
theorem HasLocalC1Extension.clock_coordinate {q : ℕ}
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : StandardClock → F}
(hf : HasLocalC1Extension (fun x : StandardClock => x.1) f) :
HasLocalC1Extension (standardEmbedding q)
(fun x : CounterexampleManifold q => f x.1) :=
hf.precomp_ambient Prod.fst Prod.fst contDiff_fst (fun _ => rfl)
theorem HasLocalC1Extension.sphere_coordinate {q : ℕ}
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : StandardSphere2 → F}
(hf : HasLocalC1Extension (fun x : StandardSphere2 => x.1) f)
(i : Fin (q + 1)) :
HasLocalC1Extension (standardEmbedding q)
(fun x : CounterexampleManifold q => f (x.2 i)) :=
hf.precomp_ambient (fun x : CounterexampleManifold q => x.2 i)
(fun x : CounterexampleAmbient q => x.2 i)
((contDiff_apply ℝ SphereAmbient i).comp contDiff_snd) (fun _ => rfl)
theorem hasLocalC1Extension_clock_coordinate (q : ℕ) :
HasLocalC1Extension (standardEmbedding q)
(fun x : CounterexampleManifold q => x.1.1) :=
.of_contDiffOn isOpen_univ (fun _ => Set.mem_univ _) Prod.fst
contDiff_fst.contDiffOn (fun _ => rfl)
theorem hasLocalC1Extension_clock_update {q : ℕ}
{h : CounterexampleManifold q → ℝ}
(hh : HasLocalC1Extension (standardEmbedding q) h) :
HasLocalC1Extension (standardEmbedding q)
(fun x : CounterexampleManifold q => (clockRotate (h x) x.1).1) :=
(hh.prodMk (hasLocalC1Extension_clock_coordinate q)).comp_contDiff
contDiff_ambientClockRotate
theorem hasLocalC1Extension_manifold_clock_lift {q : ℕ}
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
(f : StandardClock → F)
(hf : ContDiff ℝ 1 (fun t : ℝ => f (clockHomeomorph (t : AddCircle (100 : ℝ))))) :
HasLocalC1Extension (standardEmbedding q)
(fun x : CounterexampleManifold q => f x.1) :=
(hasLocalC1Extension_of_clock_lift f hf).clock_coordinate
end Problem340
end
end OAI
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