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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 | import OAI.Dynamics.EntropyCounterexample.ClockPeriodizationCompatibility
namespace OAI
noncomputable section
open Set Function
open scoped ContDiff
namespace Problem340.ClockPeriodization
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem contDiffOn_tsum_of_locallyFinite {ι : Type*} {f : ι → E → F}
{n : ℕ∞ω} {s : Set E} (hf : ∀ i, ContDiffOn ℝ n (f i) s)
(hl : LocallyFinite (fun i => support (f i))) :
ContDiffOn ℝ n (fun x => ∑' i, f i x) s := by
have heq : (fun x => ∑' i, f i x) = (fun x => ∑ᶠ i, f i x) := by
funext x
exact tsum_eq_finsum (hl.point_finite x)
rw [heq]
exact contMDiffOn_iff_contDiffOn.mp (contMDiffOn_finsum hl
(fun i => contMDiffOn_iff_contDiffOn.mpr (hf i)))
theorem contDiffOn_periodize {P a b : ℝ} {n : ℕ∞ω} {U : Set E}
(hP : 0 < P) {f : ℝ × E → F}
(hf : ContDiffOn ℝ n f (univ ×ˢ U))
(hsupport : ∀ x, f x ≠ 0 → x.1 ∈ Icc a b) :
ContDiffOn ℝ n (periodize P f) (univ ×ˢ U) := by
apply contDiffOn_tsum_of_locallyFinite
· intro k
exact hf.comp ((contDiff_fst.sub contDiff_const).prodMk contDiff_snd).contDiffOn
(fun x hx => ⟨mem_univ _, hx.2⟩)
· exact locallyFinite_translates hP hsupport
end Problem340.ClockPeriodization
namespace Problem340.ClockQuotient
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem contDiffOn_periodize {P a b : ℝ} {n : ℕ∞ω} {U : Set E}
(hP : 0 < P) {f : ℝ × E → F}
(hf : ContDiffOn ℝ n f (univ ×ˢ U))
(hsupport : ∀ x, f x ≠ 0 → x.1 ∈ Icc a b) :
ContDiffOn ℝ n (periodize P f) (univ ×ˢ U) := by
rw [periodize_eq_clockPeriodization hP hsupport]
exact ClockPeriodization.contDiffOn_periodize hP hf hsupport
end Problem340.ClockQuotient
end
end OAI
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