| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace PeriodicCutoff |
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| lemma transition_derivative (x : ℝ) : |
| HasDerivAt Real.smoothTransition (deriv Real.smoothTransition x) x := |
| ((Real.smoothTransition.contDiff (n := 1)).differentiable (by norm_num) x).hasDerivAt |
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| lemma transition_deriv_eq_zero {x : ℝ} (hx : x∉Icc (0:ℝ) 1) : |
| deriv Real.smoothTransition x=0 := by |
| have he : (Real.smoothTransition =ᶠ[𝓝 x] fun _ ↦ (Real.smoothTransition x)) := by |
| rcases not_and_or.mp hx with h | h |
| · have hx : x<0 := lt_of_not_ge h |
| filter_upwards [isOpen_Iio.mem_nhds hx] with y hy |
| rw [Real.smoothTransition.zero_of_nonpos hy.le,Real.smoothTransition.zero_of_nonpos hx.le] |
| · have hx : 1<x := lt_of_not_ge h |
| filter_upwards [isOpen_Ioi.mem_nhds hx] with y hy |
| rw [Real.smoothTransition.one_of_one_le hy.le,Real.smoothTransition.one_of_one_le hx.le] |
| exact ((hasDerivAt_const x (Real.smoothTransition x)).congr_of_eventuallyEq he).deriv |
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| lemma exists_transition_bound : ∃ M : ℝ, 0<M ∧ ∀ x : ℝ, |deriv Real.smoothTransition x|≤M := by |
| obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn |
| ((Real.smoothTransition.contDiff (n := 2)).continuous_deriv (by norm_num)).continuousOn |
| (s := Icc (0:ℝ) 1) |
| refine ⟨max C 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),fun x ↦ ?_⟩ |
| by_cases hx : x∈Icc (0:ℝ) 1 |
| · exact (hC x hx).trans (le_max_left _ _) |
| · rw [transition_deriv_eq_zero hx,abs_zero] |
| exact (zero_le_one : (0:ℝ)≤1).trans (le_max_right _ _) |
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| end PeriodicCutoff |
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| end |
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| end OAI |
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