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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 | import OAI.Dynamics.EntropyCounterexample.C1Coordinates
namespace OAI
noncomputable section
namespace Problem340
namespace HasLocalC1Extension
theorem contDiffOn_comp {X D E F : Type*}
[NormedAddCommGroup D] [NormedSpace ℝ D]
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{e : X → E} {f : X → F} (hf : HasLocalC1Extension e f)
(r : D → X) {V : Set D} (hV : IsOpen V)
(hr : ContDiffOn ℝ 1 (e ∘ r) V) :
ContDiffOn ℝ 1 (f ∘ r) V := by
intro y hy
obtain ⟨U, hU, hryU, F, hF, hcomm⟩ := hf (r y)
have hrAt : ContDiffAt ℝ 1 (e ∘ r) y := hr.contDiffAt (hV.mem_nhds hy)
have hFAt : ContDiffAt ℝ 1 F (e (r y)) := hF.contDiffAt (hU.mem_nhds hryU)
have hcomp : ContDiffAt ℝ 1 (F ∘ e ∘ r) y := hFAt.comp y hrAt
apply ContDiffAt.contDiffWithinAt
apply hcomp.congr_of_eventuallyEq
filter_upwards [hrAt.continuousAt.preimage_mem_nhds (hU.mem_nhds hryU)] with z hz
exact hcomm (r z) hz
theorem exists_fderiv_bound_on_compact {X D E F : Type*}
[NormedAddCommGroup D] [NormedSpace ℝ D]
[NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{e : X → E} {f : X → F} (hf : HasLocalC1Extension e f)
(r : D → X) {V K : Set D} (hV : IsOpen V)
(hr : ContDiffOn ℝ 1 (e ∘ r) V)
(hK : IsCompact K) (hKV : K ⊆ V) :
∃ B : ℝ, 1 ≤ B ∧ ∀ y ∈ K, ‖fderiv ℝ (f ∘ r) y‖ ≤ B := by
have hcont := (hf.contDiffOn_comp r hV hr).continuousOn_fderiv_of_isOpen hV le_rfl
obtain ⟨B, hB⟩ := hK.bddAbove_image (hcont.norm.mono hKV)
refine ⟨max 1 B, le_max_left _ _, ?_⟩
intro y hy
exact (hB (Set.mem_image_of_mem _ hy)).trans (le_max_right _ _)
end HasLocalC1Extension
end Problem340
end
end OAI
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