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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 | import OAI.Dynamics.EntropyCounterexample.Definitions
namespace OAI
noncomputable section
open Topology
namespace Problem340
theorem isEmbedding_standardEmbedding (q : ℕ) :
IsEmbedding (standardEmbedding q) := by
exact IsEmbedding.subtypeVal.prodMap
(IsEmbedding.piMap fun _ => IsEmbedding.subtypeVal)
@[continuity, fun_prop]
theorem continuous_standardEmbedding (q : ℕ) : Continuous (standardEmbedding q) :=
(isEmbedding_standardEmbedding q).continuous
theorem standardEmbedding_injective (q : ℕ) : Function.Injective (standardEmbedding q) :=
(isEmbedding_standardEmbedding q).injective
theorem continuous_iff_standardEmbedding {X : Type*} [TopologicalSpace X]
(q : ℕ) (f : X → CounterexampleManifold q) :
Continuous f ↔ Continuous (standardEmbedding q ∘ f) :=
(isEmbedding_standardEmbedding q).continuous_iff
theorem isC1SelfMap_of_contDiff {q : ℕ}
(f : CounterexampleManifold q → CounterexampleManifold q)
(F : CounterexampleAmbient q → CounterexampleAmbient q)
(hF : ContDiff ℝ 1 F)
(hcomm : ∀ y, standardEmbedding q (f y) = F (standardEmbedding q y)) :
IsC1SelfMap q f := by
refine ⟨?_, ?_⟩
· apply (continuous_iff_standardEmbedding q f).2
have heq : standardEmbedding q ∘ f = F ∘ standardEmbedding q := funext hcomm
rw [heq]
exact hF.continuous.comp (continuous_standardEmbedding q)
· intro x
exact ⟨Set.univ, isOpen_univ, Set.mem_univ _, F, hF.contDiffOn,
fun y _ => hcomm y⟩
theorem continuous_of_local_contDiff_extension {q : ℕ}
(f : CounterexampleManifold q → CounterexampleManifold q)
(hlocal : ∀ x : CounterexampleManifold q,
∃ U : Set (CounterexampleAmbient q),
IsOpen U ∧ standardEmbedding q x ∈ U ∧
∃ F : CounterexampleAmbient q → CounterexampleAmbient q,
ContDiffOn ℝ 1 F U ∧
∀ y : CounterexampleManifold q,
standardEmbedding q y ∈ U →
standardEmbedding q (f y) = F (standardEmbedding q y)) :
Continuous f := by
apply (continuous_iff_standardEmbedding q f).2
rw [continuous_iff_continuousAt]
intro x
obtain ⟨U, hU, hxU, F, hF, hcomm⟩ := hlocal x
have hcont : ContinuousAt (F ∘ standardEmbedding q) x :=
(hF.contDiffAt (hU.mem_nhds hxU)).continuousAt.comp
(continuous_standardEmbedding q).continuousAt
apply hcont.congr_of_eventuallyEq
filter_upwards [(continuous_standardEmbedding q).continuousAt.preimage_mem_nhds
(hU.mem_nhds hxU)] with y hy
exact hcomm y hy
theorem IsC1SelfMap.comp {q : ℕ}
{f g : CounterexampleManifold q → CounterexampleManifold q}
(hg : IsC1SelfMap q g) (hf : IsC1SelfMap q f) :
IsC1SelfMap q (g ∘ f) := by
refine ⟨hg.1.comp hf.1, ?_⟩
intro x
obtain ⟨U, hU, hxU, F, hF, hcommF⟩ := hf.2 x
obtain ⟨V, hV, hfxV, G, hG, hcommG⟩ := hg.2 (f x)
refine ⟨U ∩ F ⁻¹' V, hF.continuousOn.isOpen_inter_preimage hU hV, ?_,
G ∘ F, hG.comp_inter hF, ?_⟩
· exact ⟨hxU, by simpa only [Set.mem_preimage, ← hcommF x hxU] using hfxV⟩
· intro y hy
change standardEmbedding q (g (f y)) = G (F (standardEmbedding q y))
rw [hcommG (f y), hcommF y hy.1]
simpa only [hcommF y hy.1, Set.mem_preimage] using hy.2
theorem isC1SelfMap_id (q : ℕ) :
IsC1SelfMap q (id : CounterexampleManifold q → CounterexampleManifold q) :=
isC1SelfMap_of_contDiff id id contDiff_id (fun _ => rfl)
theorem isC1SelfMap_const {q : ℕ} (c : CounterexampleManifold q) :
IsC1SelfMap q (fun _ : CounterexampleManifold q => c) :=
isC1SelfMap_of_contDiff (fun _ => c) (fun _ => standardEmbedding q c)
contDiff_const (fun _ => rfl)
theorem IsC1SelfMap.iterate {q : ℕ}
{f : CounterexampleManifold q → CounterexampleManifold q}
(hf : IsC1SelfMap q f) (n : ℕ) : IsC1SelfMap q (f^[n]) := by
induction n with
| zero => exact isC1SelfMap_id q
| succ n hn =>
rw [Function.iterate_succ']
exact hf.comp hn
end Problem340
end
end OAI
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