openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Certificates /ContinuousHalfLineBound.lean
| import Mathlib.Topology.ContinuousMap.Bounded.Normed | |
| import Mathlib.Topology.Instances.Real.Lemmas | |
| import Mathlib.Analysis.Normed.Group.Tannery | |
| namespace OAI | |
| /-! Bounded extensions of continuous functions with a bounded tail. -/ | |
| open Filter Set | |
| open scoped BoundedContinuousFunction | |
| namespace DefocusingNLS | |
| variable {E : Type*} [NormedAddCommGroup E] | |
| theorem continuous_halfLine_bound (f : ℝ → E) (hf : Continuous f) | |
| (h : ∃ C : ℝ, ∀ᶠ t in atTop, ‖f t‖ ≤ C) : | |
| ∃ M : ℝ, ∀ t, 0 ≤ t → ‖f t‖ ≤ M := by | |
| obtain ⟨C,hC⟩ := h | |
| obtain ⟨T,hT⟩ := eventually_atTop.mp hC | |
| obtain ⟨D,hD⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ) (max T 0))).exists_bound_of_continuousOn | |
| hf.continuousOn | |
| refine ⟨max C D,?_⟩ | |
| intro t ht | |
| by_cases hTt : T ≤ t | |
| · exact (hT t hTt).trans (le_max_left _ _) | |
| · exact (hD t ⟨ht,(le_of_lt (lt_of_not_ge hTt)).trans (le_max_left _ _)⟩).trans | |
| (le_max_right _ _) | |
| theorem exists_bounded_extension_nonneg (f : ℝ → E) (hf : Continuous f) | |
| (h : ∃ C : ℝ, ∀ᶠ t in atTop, ‖f t‖ ≤ C) : | |
| ∃ F : ℝ →ᵇ E, ∀ t, 0 ≤ t → F t=f t := by | |
| obtain ⟨M,hM⟩ := continuous_halfLine_bound f hf h | |
| let F : ℝ →ᵇ E := BoundedContinuousFunction.ofNormedAddCommGroup | |
| (fun t => f (max t 0)) (hf.comp (continuous_id.max continuous_const)) M | |
| (fun t => hM _ (le_max_right t 0)) | |
| refine ⟨F,?_⟩ | |
| intro t ht | |
| change f (max t 0)=f t | |
| rw [max_eq_left ht] | |
| theorem continuous_tendsto_halfLine_bounded (f : ℝ → E) (hf : Continuous f) | |
| (a : E) (ha : Tendsto f atTop (nhds a)) : | |
| ∃ F : ℝ →ᵇ E, ∀ t, 0 ≤ t → F t=f t := by | |
| apply exists_bounded_extension_nonneg f hf | |
| refine ⟨‖a‖+1,?_⟩ | |
| filter_upwards [ha.norm.eventually (gt_mem_nhds (show ‖a‖<‖a‖+1 by linarith))] with t ht | |
| exact ht.le | |
| end DefocusingNLS | |
| end OAI | |