| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
| lemma linear_prod_surjective_of_minor {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (L R : E →L[ℝ] ℝ) (v w : E) (h : L v*R w-L w*R v ≠ 0) : |
| Function.Surjective (L.prod R) := by |
| rintro ⟨a,b⟩ |
| let D := L v*R w-L w*R v |
| refine ⟨((a*R w-b*L w)/D) • v+((b*L v-a*R v)/D) • w, ?_⟩ |
| apply Prod.ext |
| · change L (_ • v+_ • w)=a |
| simp only [map_add,map_smul,smul_eq_mul] |
| calc |
| _ = a*D/D := by dsimp only [D]; ring |
| _ = a := mul_div_cancel_right₀ a h |
| · change R (_ • v+_ • w)=b |
| simp only [map_add,map_smul,smul_eq_mul] |
| calc |
| _ = b*D/D := by dsimp only [D]; ring |
| _ = b := mul_div_cancel_right₀ b h |
| lemma analytic_rank_dense {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (f g : E → ℝ) (hf : ContDiff ℝ ω f) (hg : ContDiff ℝ ω g) |
| {x₀ : E} (hr : Function.Surjective (fderiv ℝ (fun x => (f x,g x)) x₀)) |
| {O : Set E} (hO : IsOpen O) (hne : O.Nonempty) : |
| ∃ y ∈ O, Function.Surjective (fderiv ℝ (fun x => (f x,g x)) y) := by |
| have hd (x : E) : fderiv ℝ (fun x => (f x,g x)) x = (fderiv ℝ f x).prod (fderiv ℝ g x) := |
| (hf.differentiable (by simp) x).fderiv_prodMk (hg.differentiable (by simp) x) |
| obtain ⟨v,hv⟩ := hr (1,0) |
| obtain ⟨w,hw⟩ := hr (0,1) |
| rw [hd] at hv hw |
| have hv1 := congrArg Prod.fst hv |
| have hv2 := congrArg Prod.snd hv |
| have hw1 := congrArg Prod.fst hw |
| have hw2 := congrArg Prod.snd hw |
| simp only [ContinuousLinearMap.prod_apply] at hv1 hv2 hw1 hw2 |
| let D : E → ℝ := fun x => fderiv ℝ f x v*fderiv ℝ g x w-fderiv ℝ f x w*fderiv ℝ g x v |
| have hD : ContDiff ℝ ω D := by |
| have H (q : E → ℝ) (hq : ContDiff ℝ ω q) (z : E) : |
| ContDiff ℝ ω (fun x => fderiv ℝ q x z) := |
| (hq.contDiff_fderiv_apply (by simp)).comp (contDiff_id.prodMk contDiff_const) |
| exact ((H f hf v).mul (H g hg w)).sub ((H f hf w).mul (H g hg v)) |
| by_contra! hn |
| obtain ⟨z,hz⟩ := hne |
| have he : D =ᶠ[𝓝 z] (fun _ => (0:ℝ)) := by |
| filter_upwards [hO.mem_nhds hz] with x hx |
| by_contra hh |
| apply hn x hx |
| rw [hd] |
| exact linear_prod_surjective_of_minor _ _ v w hh |
| have hglo := hD.analyticOnNhd.eq_of_eventuallyEq contDiff_const.analyticOnNhd he |
| have := congrFun hglo x₀ |
| simp only [D,hv1,hv2,hw1,hw2] at this |
| norm_num at this |
| end YauCounterexamples |
| end |
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| end OAI |
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