| import Mathlib |
| import OAI.Geometry.SmoothYau.DifferentialEq.DirectionalCompEquiv |
| import OAI.Geometry.SmoothYau.Geometry.LocalizeRepresentative |
| import OAI.Geometry.SmoothYau.Geometry.ZeroInterior |
| import OAI.Geometry.SmoothYau.Smoothness.BundleRankOne |
| import OAI.Geometry.SmoothYau.Smoothness.ChartDerivativeNeZeroTransfer |
| import OAI.Geometry.SmoothYau.Smoothness.CutoffExactificationSmoothNeighborhood |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold SchwartzMap |
| open Set Filter Manifold Bundle MeasureTheory NNReal |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Topology NNReal |
| open scoped Manifold |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ, E) ∞ M] |
|
|
| lemma laplaceBeltrami_const (g : SmoothMetric E M) (c : ℝ) (x : M) : |
| laplaceBeltrami g (fun _ => c) x = 0 := by |
| simp [laplaceBeltrami, Function.comp_def] |
|
|
| lemma laplaceBeltrami_add {u v : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 u) |
| (hv : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 v) (g : SmoothMetric E M) (x : M) : |
| laplaceBeltrami g (fun y => u y + v y) x = |
| laplaceBeltrami g u x + laplaceBeltrami g v x := by |
| have h := congrArg Complex.re (complexLaplaceBeltrami_add |
| (Complex.ofRealCLM.contMDiff.comp hu) (Complex.ofRealCLM.contMDiff.comp hv) g x) |
| simpa only [complexLaplaceBeltrami, Complex.add_re, Complex.mul_re, Complex.I_re, |
| Complex.ofReal_re, Complex.ofReal_im, zero_mul, mul_zero, sub_zero, add_zero, |
| Function.comp_def, Complex.ofRealCLM_apply, Pi.sub_apply] using h |
|
|
| lemma laplaceBeltrami_sub {u v : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 u) |
| (hv : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 v) (g : SmoothMetric E M) (x : M) : |
| laplaceBeltrami g (fun y => u y - v y) x = |
| laplaceBeltrami g u x - laplaceBeltrami g v x := by |
| have h := congrArg Complex.re (complexLaplaceBeltrami_sub |
| (Complex.ofRealCLM.contMDiff.comp hu) (Complex.ofRealCLM.contMDiff.comp hv) g x) |
| simpa only [complexLaplaceBeltrami, Complex.sub_re, Complex.add_re, Complex.mul_re, Complex.I_re, |
| Complex.ofReal_re, Complex.ofReal_im, zero_mul, mul_zero, sub_zero, add_zero, |
| Function.comp_def, Complex.ofRealCLM_apply, Pi.sub_apply] using h |
|
|
| lemma weightedLaplacian_shift {b h : M → ℝ} |
| (hb : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 b) |
| (hh : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) 2 h) (g : SmoothMetric E M) (x : M) : |
| weightedLaplacian g b (fun y => 1 + h y) x = |
| weightedLaplacian g (fun y => b y - 1) h x + laplaceBeltrami g h x := by |
| rw [weightedLaplacian_product_formula (u := fun y => 1 + h y) hb (contMDiff_const.add hh), |
| weightedLaplacian_product_formula (b := fun y => b y - 1) (hb.sub contMDiff_const) hh] |
| have he1 : (fun y => b y * (1 + h y)) = fun y => b y + b y * h y := by |
| funext y; ring |
| have he2 : (fun y => (b y - 1) * h y) = fun y => b y * h y - h y := by |
| funext y; ring |
| rw [he1, he2, laplaceBeltrami_add (v := fun y => b y * h y) hb (hb.mul hh), |
| laplaceBeltrami_sub (u := fun y => b y * h y) (hb.mul hh) hh, laplaceBeltrami_sub (v := fun _ => 1) hb contMDiff_const, |
| laplaceBeltrami_add (u := fun _ => 1) contMDiff_const hh, laplaceBeltrami_const] |
| ring |
|
|
| end |
|
|
| section |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold SchwartzMap |
| open Set Filter Manifold Bundle MeasureTheory NNReal |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Topology NNReal |
| open Set Filter Module |
| open scoped Topology |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter |
| open scoped Topology ContDiff |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter |
| open scoped Topology |
| open Set Filter Function MeasureTheory FourierTransform TemperedDistribution |
| open scoped Topology SchwartzMap ENNReal Real Laplacian BoundedContinuousFunction |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold |
| open Set Filter Manifold Bundle Matrix |
| open scoped Topology ContDiff |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold InnerProductSpace |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold InnerProductSpace |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| lemma laplaceBeltrami_small_jets_on_compact (g : SmoothMetric E M) (p : M) |
| {K : Set E} (hK : IsCompact K) (hKt : K ⊆ (chartAt E p).target) (h : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (u : M → ℝ), |
| ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u → |
| ∀ (x : E), x ∈ K → ∀ ε : ℝ, 0 ≤ ε → |
| (∀ j ≤ h+2, ‖iteratedFDeriv ℝ j (u ∘ (chartAt E p).symm) x‖ ≤ ε) → |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j ((laplaceBeltrami g u) ∘ (chartAt E p).symm) x‖ ≤ C*ε := by |
| obtain ⟨C,hC,hJ⟩ := weightedLaplacian_small_jets_on_compact g p hK hKt h |
| refine ⟨C,hC,?_⟩ |
| intro u hu x hx ε hε huJ |
| have he : weightedLaplacian g (fun _ => 1) (fun z => u z+1) = laplaceBeltrami g u := by |
| funext z |
| have hw : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (fun z => u z+1) := hu.add contMDiff_const |
| rw [weightedLaplacian_expansion (b := fun _ => 1) (u := fun z => u z+1) contMDiff_const |
| (hw.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2))] |
| rw [laplaceBeltrami_add (u := u) (v := fun _ => 1) |
| (hu.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)) contMDiff_const,laplaceBeltrami_const] |
| simp [coordinateGradientPair,Function.comp_def] |
| have hh := hJ (fun _ => 1) (fun z => u z+1) contMDiff_const (hu.add contMDiff_const) |
| x hx 1 ε zero_le_one hε (by |
| intro j hj |
| cases j with |
| | zero => simp |
| | succ j => simp [Function.comp_def,iteratedFDeriv_succ_const]) (by |
| intro j hj |
| simpa only [add_sub_cancel_right,Function.comp_def] using huJ j hj) |
| rw [he] at hh |
| simpa only [mul_one] using hh |
|
|
| end |
|
|
| section |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold SchwartzMap |
| open Set Filter Manifold Bundle MeasureTheory NNReal |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Topology NNReal |
| open Set Filter Module |
| open scoped Topology |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter |
| open scoped Topology ContDiff |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold Matrix |
| open Set Filter |
| open scoped Topology |
| open Set Filter Function MeasureTheory FourierTransform TemperedDistribution |
| open scoped Topology SchwartzMap ENNReal Real Laplacian BoundedContinuousFunction |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold |
| open Set Filter Manifold Bundle Matrix |
| open scoped Topology ContDiff |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold InnerProductSpace |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold InnerProductSpace |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| lemma laplaceBeltrami_residual_small_jets_on_compact (g : SmoothMetric E M) (p : M) |
| {K : Set E} (hK : IsCompact K) (hKt : K ⊆ (chartAt E p).target) (h : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (u : M → ℝ), |
| ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u → |
| ∀ (x : E), x ∈ K → ∀ ε Λ : ℝ, 0 ≤ ε → 0 ≤ Λ → |
| (∀ j ≤ h+2, ‖iteratedFDeriv ℝ j (u ∘ (chartAt E p).symm) x‖ ≤ ε) → |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j |
| ((fun z => laplaceBeltrami g u z+Λ*u z) ∘ (chartAt E p).symm) x‖ ≤ (C+Λ)*ε := by |
| obtain ⟨C,hC,hJ⟩ := laplaceBeltrami_small_jets_on_compact g p hK hKt h |
| refine ⟨C,hC,?_⟩ |
| intro u hu x hx ε Λ hε hΛ huJ j hj |
| have hc : ContDiffAt ℝ ∞ (u ∘ (chartAt E p).symm) x := contDiffAt_inChart hu p (hKt hx) |
| have hl : ContDiffAt ℝ ∞ ((laplaceBeltrami g u) ∘ (chartAt E p).symm) x := |
| contDiffAt_inChart (contMDiff_laplaceBeltrami hu g) p (hKt hx) |
| have hle : (j:ℕ∞ω) ≤ (∞ : ℕ∞ω) := le_of_lt (WithTop.coe_lt_coe.mpr (ENat.natCast_lt_top j)) |
| have he : ((fun z => laplaceBeltrami g u z+Λ*u z) ∘ (chartAt E p).symm) = |
| (fun y => ((laplaceBeltrami g u) ∘ (chartAt E p).symm) y + Λ • (u ∘ (chartAt E p).symm) y) := rfl |
| rw [he, fun_iteratedFDeriv_add_apply (hl.of_le hle) ((hc.const_smul Λ).of_le hle), |
| iteratedFDeriv_const_smul_apply' (hc.of_le hle)] |
| calc |
| _ ≤ ‖iteratedFDeriv ℝ j ((laplaceBeltrami g u) ∘ (chartAt E p).symm) x‖ + |
| ‖Λ • iteratedFDeriv ℝ j (u ∘ (chartAt E p).symm) x‖ := norm_add_le _ _ |
| _ ≤ C*ε + Λ*ε := by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg hΛ] |
| exact add_le_add (hJ u hu x hx ε hε huJ j hj) |
| (mul_le_mul_of_nonneg_left (huJ j (by omega)) hΛ) |
| _ = (C+Λ)*ε := by ring |
| |
| end |
| |
| section |
| open Set Filter Metric Manifold |
| open scoped Topology ContDiff |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] {M : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
| |
| theorem metric_unique_continuation_general [PreconnectedSpace M] (g : SmoothMetric E M) {u : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) (lam : ℝ) |
| (hsol : ∀ x, -laplaceBeltrami g u x = lam*u x) |
| (hz : ∃ x, u =ᶠ[𝓝 x] 0) : u = 0 := by |
| let F := EuclideanSpace ℝ (Fin (Module.finrank ℝ E)) |
| let L : E ≃L[ℝ] F := toEuclidean |
| let Z := {x : M | u =ᶠ[𝓝 x] 0} |
| have hZ : IsOpen Z := isOpen_setOfPred_eventually_nhds |
| have hcl : closure Z ⊆ Z := by |
| intro p hp |
| obtain ⟨a,V,w,r,hr,ha,hT,hw,hpos,hP⟩ := exists_metric_ucp_model_equiv L g hu lam hsol p |
| let c := chartAt E p |
| have hps : p ∈ c.source := mem_chart_source E p |
| have hb : ball (L (c p)) r ∈ 𝓝 (L (c p)) := ball_mem_nhds _ hr |
| have hN : c.source ∩ (fun x => L (c x)) ⁻¹' ball (L (c p)) r ∈ 𝓝 p := |
| inter_mem (c.open_source.mem_nhds hps) |
| ((L.continuous.continuousAt.comp (c.continuousAt hps)).preimage_mem_nhds hb) |
| obtain ⟨x,hxN,hxZ⟩ := mem_closure_iff_nhds.mp hp _ hN |
| have hct : c x ∈ c.target := c.map_source hxN.1 |
| have hctend : Tendsto (fun y => c.symm (L.symm y)) (𝓝 (L (c x))) (𝓝 x) := by |
| have htL : Tendsto L.symm (𝓝 (L (c x))) (𝓝 (c x)) := by |
| simpa only [L.symm_apply_apply] using L.symm.continuous.continuousAt.tendsto (x := L (c x)) |
| have ht := ((c.continuousAt_symm hct).tendsto).comp htL |
| simpa only [L.symm_apply_apply,Function.comp_def,c.left_inv hxN.1] using ht |
| have hzw : L (c x) ∈ zeroInterior w := by |
| filter_upwards [isOpen_ball.mem_nhds hxN.2, hxZ.comp_tendsto hctend] with y hy hyz |
| change w y = 0 |
| rw [hw y hy] |
| exact hyz |
| let e : Module.Basis (CoordIndex E) ℝ F := (Module.finBasis ℝ E).map L.toLinearEquiv |
| have hwzero := weak_unique_continuation e a ha V w |
| (ball (L (c p)) r) isOpen_ball (convex_ball (L (c p)) r).isPreconnected hpos hP |
| ⟨L (c x),hxN.2,hzw⟩ |
| have hpzero : u =ᶠ[𝓝 p] 0 := by |
| filter_upwards [hN] with x hx |
| have hh := hwzero (L (c x)) hx.2 |
| rw [hw (L (c x)) hx.2,L.symm_apply_apply,c.left_inv hx.1] at hh |
| exact hh |
| exact hpzero |
| have hsub : (univ : Set M) ⊆ Z := isPreconnected_univ.subset_of_closure_inter_subset hZ |
| (by obtain ⟨x,hx⟩ := hz; exact ⟨x,mem_univ _,hx⟩) (fun _ hx => hcl hx.1) |
| funext x |
| exact (hsub (mem_univ x)).eq_of_nhds |
|
|
| lemma eigenfunction_nonzero_on_open_general [PreconnectedSpace M] (g : SmoothMetric E M) {u : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) (lam : ℝ) |
| (hsol : ∀ x, -laplaceBeltrami g u x = lam*u x) (hun : u ≠ 0) |
| (O : Set M) (hO : IsOpen O) (hne : O.Nonempty) : ∃ x ∈ O, u x ≠ 0 := by |
| by_contra hn |
| push Not at hn |
| obtain ⟨p,hp⟩ := hne |
| apply hun |
| apply metric_unique_continuation_general g hu lam hsol |
| refine ⟨p,?_⟩ |
| filter_upwards [hO.mem_nhds hp] with x hx |
| exact hn x hx |
|
|
|
|
| end |
|
|
| open Set Filter Function Manifold |
| open scoped Topology ContDiff |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
| lemma transverse_zero_kernel (g : SmoothMetric E M) (u v : M → ℝ) (x : M) |
| (hp : bundleTransverse g u x (metricGradient g v x) = 0) |
| (w : TangentSpace 𝓘(ℝ,E) x) (hw : mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) u x w = 0) : |
| mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) v x w = 0 := by |
| have hv : metricGradient g v x = |
| (g.inner x (metricGradient g u x) (metricGradient g u x))⁻¹ • |
| (g.inner x (metricGradient g u x) (metricGradient g v x) • metricGradient g u x) := by |
| simpa only [bundleTransverse,sub_apply,ContinuousLinearMap.id_apply, |
| smul_apply,bundleRankOne,ContinuousLinearMap.smulRight_apply, |
| sub_eq_zero] using hp |
| have hzero : g.inner x (metricGradient g u x) w = 0 := |
| (metricGradient_pairing g u x w).trans hw |
| rw [←metricGradient_pairing g v x w,hv] |
| erw [map_smul, smul_apply, map_smul, smul_apply, hzero] |
| simp only [smul_zero] |
| rfl |
| lemma transverse_zero_chart_kernel (g : SmoothMetric E M) {u v : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) |
| (hv : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ v) (p : M) {y : E} |
| (hy : y ∈ (chartAt E p).target) |
| (hp : bundleTransverse g u ((chartAt E p).symm y) |
| (metricGradient g v ((chartAt E p).symm y)) = 0) : |
| ∀ w, fderiv ℝ (u ∘ (chartAt E p).symm) y w = 0 → |
| fderiv ℝ (v ∘ (chartAt E p).symm) y w = 0 := by |
| intro w hw |
| have hd := (contMDiffAt_symm_of_mem_maximalAtlas |
| (IsManifold.chart_mem_maximalAtlas (I := 𝓘(ℝ,E)) (n := ∞) p) hy).mdifferentiableAt (by simp) |
| have he (f : M → ℝ) (hf : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ f) : |
| fderiv ℝ (f ∘ (chartAt E p).symm) y w = |
| mfderiv 𝓘(ℝ,E) 𝓘(ℝ,ℝ) f ((chartAt E p).symm y) |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,E) (chartAt E p).symm y w) := by |
| have hh := mfderiv_comp y (hf.mdifferentiable (by simp)).mdifferentiableAt hd |
| rw [mfderiv_eq_fderiv] at hh |
| exact congrArg (fun L => L w) hh |
| rw [he u hu] at hw |
| rw [he v hv] |
| exact transverse_zero_kernel g u v _ hp _ hw |
| lemma proportional_eigen_global [PreconnectedSpace M] (g : SmoothMetric E M) |
| {u v : M → ℝ} (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) |
| (hv : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ v) (ell c : ℝ) |
| (hue : ∀ x, -laplaceBeltrami g u x = ell*u x) |
| (hve : ∀ x, -laplaceBeltrami g v x = ell*v x) |
| (he : ∃ x, v =ᶠ[𝓝 x] (fun y => c*u y)) : v = fun y => c*u y := by |
| have hcu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (fun y => c*u y) := contMDiff_const.mul hu |
| let w := fun y => v y - c*u y |
| have hw : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ w := hv.sub (contMDiff_const.mul hu) |
| have hscalar (x : M) : laplaceBeltrami g (fun y => c*u y) x = c * laplaceBeltrami g u x := by |
| have hh := laplaceBeltrami_comp_at g x |
| ((contDiff_const.mul contDiff_id : ContDiff ℝ 2 (fun s : ℝ => c*s)).contDiffAt) (hu.of_le (show (2 : WithTop ℕ∞) ≤ ∞ from WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))) |
| have hd : deriv (fun s : ℝ => c*s) = fun _ => c := by |
| funext s; simp |
| simpa only [Function.comp_def,hd,deriv_const,mul_zero,zero_mul,add_zero] using hh |
| have hsub (x : M) : laplaceBeltrami g w x = |
| laplaceBeltrami g v x - laplaceBeltrami g (fun y => c*u y) x := by |
| have hh := congrArg Complex.re (complexLaplaceBeltrami_sub |
| (Complex.ofRealCLM.contMDiff.comp (hv.of_le (show (2 : WithTop ℕ∞) ≤ ∞ from WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top)))) |
| (Complex.ofRealCLM.contMDiff.comp (hcu.of_le (show (2 : WithTop ℕ∞) ≤ ∞ from WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top)))) g x) |
| simpa only [complexLaplaceBeltrami,Complex.sub_re,Complex.add_re,Complex.mul_re, |
| Complex.I_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,mul_zero,sub_zero,add_zero, |
| Function.comp_def,Complex.ofRealCLM_apply,Pi.sub_apply,w] using hh |
| have hwz : w = 0 := metric_unique_continuation_general g hw ell (fun x => by |
| rw [hsub,hscalar] |
| dsimp only [w] |
| rw [show laplaceBeltrami g v x = -ell*v x by linarith [hve x], |
| show laplaceBeltrami g u x = -ell*u x by linarith [hue x]] |
| ring) (by |
| obtain ⟨x,hx⟩ := he |
| refine ⟨x,?_⟩ |
| filter_upwards [hx] with y hy |
| simp only [w,hy,sub_self,Pi.zero_apply]) |
| funext x |
| have hh := congrFun hwz x |
| change v x - c*u x = 0 at hh |
| exact sub_eq_zero.mp hh |
|
|
|
|
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
