| import OAI.Analysis.BernoulliCone.PositiveTrace | |
| namespace OAI | |
| universe uE | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace Bernoulli.Foliation | |
| variable {E : Type uE} [TopologicalSpace E] | |
| /-- Compact zero-fiber control. No convergence of the selected root is needed. -/ | |
| lemma eventually_near_zero_fiber {K:Set E} (hK:IsCompact K) {f g:E→ℝ} {C:ℝ} | |
| (hf:∀x∈K,ContinuousAt f x) (hg:∀x∈K,ContinuousAt g x) | |
| (hzero:∀x∈K,f x=0 → g x<C) : | |
| ∀ᶠ t:ℝ in 𝓝 0,∀x∈K,f x=t → g x<C := by | |
| apply hK.eventually_forall_of_forall_eventually | |
| intro x hx | |
| by_cases hz:f x=0 | |
| · have hh:∀ᶠ p:ℝ×E in 𝓝 (0,x),g p.2<C:= | |
| ((hg x hx).comp continuous_snd.continuousAt).eventually (gt_mem_nhds (hzero x hx hz)) | |
| exact hh.mono fun p hp _=>hp | |
| · have hn:f x-(0:ℝ)≠0:=by simpa using hz | |
| have hh:∀ᶠ p:ℝ×E in 𝓝 (0,x),f p.2-p.1≠0:= | |
| (((hf x hx).comp continuous_snd.continuousAt).sub continuous_fst.continuousAt).eventually_ne hn | |
| exact hh.mono fun p hp he=>False.elim (hp (sub_eq_zero.mpr he)) | |
| end Bernoulli.Foliation | |
| end | |
| end | |
| end OAI | |