| import Mathlib |
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| namespace OAI |
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| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal |
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|
| namespace SharpIntegralFillings.SmoothCutoff |
| lemma deriv_transition_eq_zero {x : ℝ} (hx : x ∉ Icc (0:ℝ) 1) : |
| deriv Real.smoothTransition x = 0 := by |
| have hcases : x < 0 ∨ 1 < x := by simpa only [mem_Icc,not_and_or,not_le] using hx |
| rcases hcases with hx | hx |
| · have heq : Real.smoothTransition =ᶠ[𝓝 x] fun _ => (0:ℝ) := by |
| filter_upwards [eventually_lt_nhds hx] with y hy |
| exact Real.smoothTransition.zero_of_nonpos hy.le |
| rw [heq.deriv_eq] |
| simp |
| · have heq : Real.smoothTransition =ᶠ[𝓝 x] fun _ => (1:ℝ) := by |
| filter_upwards [eventually_gt_nhds hx] with y hy |
| exact Real.smoothTransition.one_of_one_le hy.le |
| rw [heq.deriv_eq] |
| simp |
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| lemma tsupport_deriv_transition : |
| tsupport (deriv Real.smoothTransition) ⊆ Icc (0:ℝ) 1 := by |
| apply closure_minimal _ isClosed_Icc |
| intro x hx |
| by_contra h |
| exact hx (deriv_transition_eq_zero h) |
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| lemma exists_derivative_bound : ∃ C : ℝ≥0, (∀ x, ‖deriv Real.smoothTransition x‖ ≤ C) ∧ |
| (∀ x, ‖deriv (deriv Real.smoothTransition) x‖ ≤ C) := by |
| have hd : ContDiff ℝ (1:ℕ) (deriv Real.smoothTransition) := |
| (@Real.smoothTransition.contDiff 2).deriv' |
| have hc := hd.continuous |
| have hc2 := hd.continuous_deriv (by norm_num) |
| obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn hc.continuousOn |
| obtain ⟨D,hD⟩ := isCompact_Icc.exists_bound_of_continuousOn hc2.continuousOn |
| refine ⟨⟨max 0 (max C D),le_max_left _ _⟩,?_,?_⟩ |
| · intro x |
| by_cases hx : x ∈ Icc (0:ℝ) 1 |
| · exact (hC x hx).trans ((le_max_left C D).trans (le_max_right _ _)) |
| · rw [deriv_transition_eq_zero hx,norm_zero] |
| exact le_max_left _ _ |
| · intro x |
| by_cases hx : x ∈ Icc (0:ℝ) 1 |
| · exact (hD x hx).trans ((le_max_right C D).trans (le_max_right _ _)) |
| · have hz : deriv (deriv Real.smoothTransition) x = 0 := by |
| exact deriv_of_notMem_tsupport (fun h => hx (tsupport_deriv_transition h)) |
| rw [hz,norm_zero] |
| exact le_max_left _ _ |
| |
| lemma exists_lipschitz_transition : ∃ C : ℝ≥0, LipschitzWith C Real.smoothTransition ∧ |
| LipschitzWith C (deriv Real.smoothTransition) ∧ |
| ∀ x, |deriv Real.smoothTransition x| ≤ C := by |
| obtain ⟨C,hC,hD⟩ := exists_derivative_bound |
| refine ⟨C,?_,?_,?_⟩ |
| · apply lipschitzWith_of_nnnorm_deriv_le ((@Real.smoothTransition.contDiff 1).differentiable (by norm_num)) |
| exact fun x => hC x |
| · apply lipschitzWith_of_nnnorm_deriv_le ((show ContDiff ℝ (1:ℕ) (deriv Real.smoothTransition) from (@Real.smoothTransition.contDiff 2).deriv').differentiable (by norm_num)) |
| exact fun x => hD x |
| · simpa only [Real.norm_eq_abs] using hC |
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| end SharpIntegralFillings.SmoothCutoff |
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| end |
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| end OAI |
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