| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.FlowInitialFderiv |
| import OAI.Geometry.SmoothYau.Smoothness.NormIteratedFDerivScalarDirection |
|
|
| namespace OAI |
|
|
| noncomputable 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 |
| namespace YauCounterexamples |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| {ι : Type*} [Fintype ι] |
|
|
| lemma smooth_extension_at {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {O : Set E} (hO : IsOpen O) {x : E} (hx : x ∈ O) |
| {f : E → F} (hf : ContDiffOn ℝ ∞ f O) : |
| ∃ F : E → F, ContDiff ℝ ∞ F ∧ f =ᶠ[𝓝 x] F := by |
| obtain ⟨F,hF,he⟩ := smooth_extension_near_compact isCompact_singleton hO |
| (singleton_subset_iff.mpr hx) hf |
| exact ⟨F,hF,he.filter_mono (nhds_le_nhdsSet (mem_singleton x))⟩ |
|
|
| lemma positive_smooth_extension_at {O : Set E} (hO : IsOpen O) {x : E} (hx : x ∈ O) |
| {f : E → ℝ} (hf : ContDiffOn ℝ ∞ f O) (hfx : 0 < f x) : |
| ∃ F : E → ℝ, ContDiff ℝ ∞ F ∧ (∀ y, 0 < F y) ∧ f =ᶠ[𝓝 x] F := by |
| have hp : {y | 0 < f y} ∈ 𝓝 x := |
| (hf.continuousOn.continuousAt (hO.mem_nhds hx)).eventually (isOpen_Ioi.mem_nhds hfx) |
| obtain ⟨V,hV,hVo,hxV⟩ := mem_nhds_iff.mp hp |
| obtain ⟨F,hF,he⟩ := smooth_extension_at (hO.inter hVo) ⟨hx,hxV⟩ |
| ((hf.mono inter_subset_left).log (fun y hy => (hV hy.2).ne')) |
| refine ⟨fun y => Real.exp (F y), hF.exp,fun _ => Real.exp_pos _,?_⟩ |
| filter_upwards [he, hp] with y hy hpy |
| rw [← hy,Real.exp_log hpy] |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma coordinateDivergence_eventuallyEq (e : ι → E) {ρ σ : E → ℝ} |
| {X Y : ι → E → ℝ} {x : E} (hρ : ρ =ᶠ[𝓝 x] σ) |
| (hX : ∀ i, X i =ᶠ[𝓝 x] Y i) : |
| coordinateDivergence e ρ X =ᶠ[𝓝 x] coordinateDivergence e σ Y := by |
| have hd (i : ι) := (hρ.mul (hX i)).fderiv (𝕜:=ℝ) |
| filter_upwards [hρ,eventually_all.mpr hd] with y hy hdy |
| simp only [coordinateDivergence,hy] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact congrArg (fun L : E →L[ℝ] ℝ => L (e i)) (hdy i) |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma scalarCorrectionDenominator_eventuallyEq (e : ι → E) (n : ℝ) |
| {U V : E → ℝ} {X Y : ι → E → ℝ} {x : E} |
| (hU : U =ᶠ[𝓝 x] V) (hX : ∀ i, X i =ᶠ[𝓝 x] Y i) : |
| scalarCorrectionDenominator e n U X =ᶠ[𝓝 x] scalarCorrectionDenominator e n V Y := by |
| filter_upwards [hU,hU.fderiv (𝕜:=ℝ),eventually_all.mpr hX] with y hy hdy hxy |
| simp only [scalarCorrectionDenominator,hy,hdy,hxy] |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma scalarCorrectionRatio_eventuallyEq (e : ι → E) (n Λ : ℝ) |
| {ρ σ U V : E → ℝ} {X Y : ι → E → ℝ} {x : E} |
| (hρ : ρ =ᶠ[𝓝 x] σ) (hU : U =ᶠ[𝓝 x] V) (hX : ∀ i, X i =ᶠ[𝓝 x] Y i) : |
| scalarCorrectionRatio e ρ n Λ U X =ᶠ[𝓝 x] scalarCorrectionRatio e σ n Λ V Y := by |
| exact ((coordinateDivergence_eventuallyEq e hρ hX).add |
| (EventuallyEq.rfl.mul hU)).div (scalarCorrectionDenominator_eventuallyEq e n hU hX) |
|
|
| lemma contDiffOn_coordinateDivergence (e : ι → E) {O : Set E} (hO : IsOpen O) |
| {ρ : E → ℝ} {X : ι → E → ℝ} (hρ : ContDiffOn ℝ ∞ ρ O) |
| (hX : ∀ i, ContDiffOn ℝ ∞ (X i) O) (hρp : ∀ y ∈ O, 0 < ρ y) : |
| ContDiffOn ℝ ∞ (coordinateDivergence e ρ X) O := by |
| intro x hx |
| obtain ⟨σ,hσ,hσp,hσe⟩ := positive_smooth_extension_at hO hx hρ (hρp x hx) |
| choose Y hY hYe using fun i => smooth_extension_at hO hx (hX i) |
| exact ((contDiff_coordinateDivergence e σ Y hσ hY (fun y => (hσp y).ne')).contDiffAt.congr_of_eventuallyEq |
| (coordinateDivergence_eventuallyEq e hσe hYe)).contDiffWithinAt |
|
|
| lemma contDiffOn_scalarCorrectionDenominator (e : ι → E) (n : ℝ) |
| {O : Set E} (hO : IsOpen O) {U : E → ℝ} {X : ι → E → ℝ} |
| (hU : ContDiffOn ℝ ∞ U O) (hX : ∀ i, ContDiffOn ℝ ∞ (X i) O) : |
| ContDiffOn ℝ ∞ (scalarCorrectionDenominator e n U X) O := by |
| intro x hx |
| obtain ⟨V,hV,hVe⟩ := smooth_extension_at hO hx hU |
| choose Y hY hYe using fun i => smooth_extension_at hO hx (hX i) |
| exact ((contDiff_scalarCorrectionDenominator e n V Y hV hY).contDiffAt.congr_of_eventuallyEq |
| (scalarCorrectionDenominator_eventuallyEq e n hVe hYe)).contDiffWithinAt |
|
|
| lemma contDiffOn_scalarCorrectionRatio (e : ι → E) (n Λ : ℝ) |
| {O : Set E} (hO : IsOpen O) {ρ U : E → ℝ} {X : ι → E → ℝ} |
| (hρ : ContDiffOn ℝ ∞ ρ O) (hU : ContDiffOn ℝ ∞ U O) |
| (hX : ∀ i, ContDiffOn ℝ ∞ (X i) O) (hρp : ∀ y ∈ O, 0 < ρ y) |
| (hden : ∀ y ∈ O, scalarCorrectionDenominator e n U X y ≠ 0) : |
| ContDiffOn ℝ ∞ (scalarCorrectionRatio e ρ n Λ U X) O := by |
| exact ((contDiffOn_coordinateDivergence e hO hρ hX hρp).add |
| (contDiffOn_const.mul hU)).div (contDiffOn_scalarCorrectionDenominator e n hO hU hX) hden |
|
|
| theorem frequency_weighted_quotient_uniform_on |
| {O : Set E} (hO : IsOpen O) {x : E} (hxO : x ∈ O) |
| {f d : E → ℝ} (hf : ContDiffOn ℝ ∞ f O) (hd : ContDiffOn ℝ ∞ d O) |
| (h B a D : ℕ) {n W F C : ℝ} (hn : 1 ≤ n) (hW : 0 < W) (hF : 0 ≤ F) |
| (hx : W^2/(2*n^(2*B)) ≤ d x) |
| (hfj : ∀ j ≤ h, ‖iteratedFDeriv ℝ j f x‖ ≤ F*n^(a+j)/(n^D)*W^2) |
| (hdj : ∀ j : ℕ, 1 ≤ j → j ≤ h → ‖iteratedFDeriv ℝ j d x‖ ≤ C*n^(j+10)*W^2) : |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j (fun y => f y/d y) x‖ ≤ |
| (2*F*(h.factorial:ℝ)^2*(1+max 1 (2*C))^h)* |
| n^(a+2*B+(2*B+11)*h)/(n^D) := by |
| have hn0 : 0 < n := zero_lt_one.trans_le hn |
| obtain ⟨f',hf',hfe⟩ := smooth_extension_at hO hxO hf |
| obtain ⟨d',hd',hdp,hde⟩ := positive_smooth_extension_at hO hxO hd |
| ((show 0 < W^2/(2*n^(2*B)) by positivity).trans_le hx) |
| have hq := frequency_weighted_quotient_uniform hf' hd' (fun y => (hdp y).ne') x h B a D |
| hn hW hF (hde.self_of_nhds ▸ hx) |
| (fun j hj => (hfe.iteratedFDeriv ℝ j).self_of_nhds ▸ hfj j hj) |
| (fun j hj hjh => (hde.iteratedFDeriv ℝ j).self_of_nhds ▸ hdj j hj hjh) |
| intro j hj |
| change ‖iteratedFDeriv ℝ j (f/d) x‖ ≤ _ |
| rw [(hfe.div hde).iteratedFDeriv ℝ j |>.self_of_nhds] |
| exact hq j hj |
|
|
| theorem frequency_weighted_product_bound_on |
| {O : Set E} (hO : IsOpen O) {x : E} (hxO : x ∈ O) |
| {f g : E → ℝ} (hf : ContDiffOn ℝ ∞ f O) (hg : ContDiffOn ℝ ∞ g O) |
| (h a b D : ℕ) {n W F G : ℝ} |
| (hn : 1 ≤ n) (hW : 0 < W) (hF : 0 ≤ F) (hG : 0 ≤ G) |
| (hfj : ∀ j ≤ h, ‖iteratedFDeriv ℝ j f x‖ ≤ F*n^(a+j)/(n^D)*W) |
| (hgj : ∀ j ≤ h, ‖iteratedFDeriv ℝ j g x‖ ≤ G*n^(b+j)*W) : |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j (fun y => f y*g y) x‖ ≤ |
| (2^h*F*G)*n^(a+b+j)/(n^D)*W^2 := by |
| obtain ⟨f',hf',hfe⟩ := smooth_extension_at hO hxO hf |
| obtain ⟨g',hg',hge⟩ := smooth_extension_at hO hxO hg |
| have hq := frequency_weighted_product_bound hf' hg' x h a b D hn hW hF hG |
| (fun j hj => (hfe.iteratedFDeriv ℝ j).self_of_nhds ▸ hfj j hj) |
| (fun j hj => (hge.iteratedFDeriv ℝ j).self_of_nhds ▸ hgj j hj) |
| intro j hj |
| change ‖iteratedFDeriv ℝ j (f*g) x‖ ≤ _ |
| rw [(hfe.mul hge).iteratedFDeriv ℝ j |>.self_of_nhds] |
| exact hq j hj |
|
|
| theorem scalar_denominator_jet_bound_on (e : ι → E) (U : E → ℝ) (X : ι → E → ℝ) |
| {O : Set E} (hO : IsOpen O) {x : E} (hxO : x ∈ O) |
| (hU : ContDiffOn ℝ ∞ U O) (hX : ∀ i, ContDiffOn ℝ ∞ (X i) O) |
| (h : ℕ) {n W C : ℝ} (hn : 1 ≤ n) (hW : 0 < W) (hC : 0 ≤ C) |
| (hUj : ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j U x‖ ≤ C*n^(j+4)*W) |
| (hXj : ∀ i, ∀ j ≤ h, ‖iteratedFDeriv ℝ j (X i) x‖ ≤ C*n^(j+5)*W) : |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j (scalarCorrectionDenominator e n U X) x‖ ≤ |
| (2^h*C*C*(1+∑ i, ‖e i‖))*n^(j+10)*W^2 := by |
| obtain ⟨V,hV,hVe⟩ := smooth_extension_at hO hxO hU |
| choose Y hY hYe using fun i => smooth_extension_at hO hxO (hX i) |
| have hq := scalar_denominator_jet_bound e V Y hV hY x h hn hW hC |
| (fun j hj => (hVe.iteratedFDeriv ℝ j).self_of_nhds ▸ hUj j hj) |
| (fun i j hj => ((hYe i).iteratedFDeriv ℝ j).self_of_nhds ▸ hXj i j hj) |
| intro j hj |
| rw [(scalarCorrectionDenominator_eventuallyEq e n hVe hYe).iteratedFDeriv ℝ j |>.self_of_nhds] |
| exact hq j hj |
|
|
| theorem coordinate_divergence_jet_bound_on (e : ι → E) (ρ : E → ℝ) (Z : ι → E → ℝ) |
| {O : Set E} (hO : IsOpen O) {x : E} (hxO : x ∈ O) |
| (hρ : ContDiffOn ℝ ∞ ρ O) (hρp : ∀ y ∈ O, 0 < ρ y) |
| (hZ : ∀ i, ContDiffOn ℝ ∞ (Z i) O) |
| (h : ℕ) {C F : ℝ} (hC : 0 ≤ C) (hF : 0 ≤ F) |
| (hρj : ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j ρ x‖ ≤ C) |
| (hρij : ∀ j ≤ h, ‖iteratedFDeriv ℝ j (fun y => (ρ y)⁻¹) x‖ ≤ C) |
| (hZj : ∀ i, ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j (Z i) x‖ ≤ F) : |
| ∀ j ≤ h, ‖iteratedFDeriv ℝ j (coordinateDivergence e ρ Z) x‖ ≤ |
| (2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F := by |
| obtain ⟨σ,hσ,hσp,hσe⟩ := positive_smooth_extension_at hO hxO hρ (hρp x hxO) |
| choose Y hY hYe using fun i => smooth_extension_at hO hxO (hZ i) |
| have hq := coordinate_divergence_jet_bound e σ Y hσ (fun y => (hσp y).ne') hY x h hC hF |
| (fun j hj => (hσe.iteratedFDeriv ℝ j).self_of_nhds ▸ hρj j hj) |
| (fun j hj => (hσe.inv.iteratedFDeriv ℝ j).self_of_nhds ▸ hρij j hj) |
| (fun i j hj => ((hYe i).iteratedFDeriv ℝ j).self_of_nhds ▸ hZj i j hj) |
| intro j hj |
| rw [(coordinateDivergence_eventuallyEq e hσe hYe).iteratedFDeriv ℝ j |>.self_of_nhds] |
| exact hq j hj |
|
|
| theorem scalar_correction_error_from_quotients_on |
| (e : ι → E) (ρ U : E → ℝ) (X : ι → E → ℝ) |
| {O : Set E} (hO : IsOpen O) {x : E} (hxO : x ∈ O) |
| (hρ : ContDiffOn ℝ ∞ ρ O) (hρp : ∀ y ∈ O, 0 < ρ y) |
| (hU : ContDiffOn ℝ ∞ U O) (hX : ∀ i, ContDiffOn ℝ ∞ (X i) O) |
| {n : ℝ} (hn : 1 ≤ n) |
| (hden : ∀ y ∈ O, scalarCorrectionDenominator e n U X y ≠ 0) |
| (h : ℕ) {C F : ℝ} (hC : 0 ≤ C) (hF : 0 ≤ F) |
| (hρj : ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j ρ x‖ ≤ C) |
| (hρij : ∀ j ≤ h, ‖iteratedFDeriv ℝ j (fun y => (ρ y)⁻¹) x‖ ≤ C) |
| (hqU : ∀ j ≤ h, ‖iteratedFDeriv ℝ j |
| (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*U y) x‖ ≤ F) |
| (hqX : ∀ i, ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j |
| (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*X i y) x‖ ≤ F) : |
| ∀ j ≤ h, |
| ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionB e ρ n (n*(n+2)) U X y-1) x‖ + |
| ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionS e ρ n (n*(n+2)) U X y-1) x‖ ≤ |
| (2+2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F := by |
| let f := scalarCorrectionRatio e ρ n (n*(n+2)) U X |
| have hf : ContDiffOn ℝ ∞ f O := contDiffOn_scalarCorrectionRatio e n (n*(n+2)) hO |
| hρ hU hX hρp hden |
| let Z : ι → E → ℝ := fun i y => f y*X i y |
| have hZ (i : ι) : ContDiffOn ℝ ∞ (Z i) O := hf.mul (hX i) |
| have hd := contDiffOn_coordinateDivergence e hO hρ hZ hρp |
| have hq := hf.mul hU |
| have hdiv := coordinate_divergence_jet_bound_on e ρ Z hO hxO hρ hρp hZ h hC hF hρj hρij hqX |
| have hn0 : 0 < n := zero_lt_one.trans_le hn |
| have hΛ : 0 < n*(n+2) := by positivity |
| have hΛ1 : 1 ≤ n*(n+2) := by nlinarith |
| have hnratio : (n*(n+2))⁻¹*n^2 ≤ 1 := by |
| rw [← div_eq_inv_mul,div_le_one hΛ] |
| nlinarith |
| have hΛinv : (n*(n+2))⁻¹ ≤ 1 := inv_le_one_of_one_le₀ hΛ1 |
| have hCb : 0 ≤ 2^(2*h+1)*C*C*(∑ i, ‖e i‖) := by positivity |
| intro j hj |
| have hj' : (j:ℕ∞ω) ≤ (∞ : ℕ∞ω) := le_of_lt (WithTop.coe_lt_coe.mpr (ENat.natCast_lt_top j)) |
| have heb : (fun y => scalarCorrectionB e ρ n (n*(n+2)) U X y-1) = |
| fun y => -(f y*U y) := by funext y; dsimp only [scalarCorrectionB,f]; ring |
| have hes : (fun y => scalarCorrectionS e ρ n (n*(n+2)) U X y-1) = |
| fun y => -((n*(n+2))⁻¹ • (n^2 • (f y*U y)-coordinateDivergence e ρ Z y)) := by |
| funext y; dsimp only [scalarCorrectionS,f,Z,smul_eq_mul]; ring |
| have hb : ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionB e ρ n (n*(n+2)) U X y-1) x‖ ≤ F := by |
| rw [heb] |
| change ‖iteratedFDeriv ℝ j (-(fun y => f y*U y)) x‖ ≤ F |
| rw [iteratedFDeriv_neg_apply,norm_neg] |
| exact hqU j hj |
| have hs : ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionS e ρ n (n*(n+2)) U X y-1) x‖ ≤ |
| F+(2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F := by |
| have hz : ContDiffOn ℝ ∞ (fun y => n^2 • (f y*U y)-coordinateDivergence e ρ Z y) O := |
| (hq.const_smul (n^2)).sub hd |
| rw [hes] |
| change ‖iteratedFDeriv ℝ j (-(fun y => (n*(n+2))⁻¹ • |
| (n^2 • (f y*U y)-coordinateDivergence e ρ Z y))) x‖ ≤ _ |
| rw [iteratedFDeriv_neg_apply,norm_neg,iteratedFDeriv_const_smul_apply' ((hz.contDiffAt (hO.mem_nhds hxO)).of_le hj'), |
| norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hΛ), |
| fun_iteratedFDeriv_sub_apply (((hq.const_smul (n^2)).contDiffAt (hO.mem_nhds hxO)).of_le hj') ((hd.contDiffAt (hO.mem_nhds hxO)).of_le hj')] |
| calc |
| _ ≤ (n*(n+2))⁻¹*(‖iteratedFDeriv ℝ j (fun y => n^2 • (f y*U y)) x‖+ |
| ‖iteratedFDeriv ℝ j (coordinateDivergence e ρ Z) x‖) := |
| mul_le_mul_of_nonneg_left (norm_sub_le _ _) (inv_nonneg.mpr hΛ.le) |
| _ = (n*(n+2))⁻¹*(n^2*‖iteratedFDeriv ℝ j (fun y => f y*U y) x‖+ |
| ‖iteratedFDeriv ℝ j (coordinateDivergence e ρ Z) x‖) := by |
| rw [iteratedFDeriv_const_smul_apply' ((hq.contDiffAt (hO.mem_nhds hxO)).of_le hj'),norm_smul, |
| Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| _ ≤ (n*(n+2))⁻¹*(n^2*F+(2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F) := by |
| exact mul_le_mul_of_nonneg_left |
| (add_le_add (mul_le_mul_of_nonneg_left (hqU j hj) (sq_nonneg _)) (hdiv j hj)) |
| (inv_nonneg.mpr hΛ.le) |
| _ = ((n*(n+2))⁻¹*n^2)*F + (n*(n+2))⁻¹*((2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F) := by ring |
| _ ≤ _ := add_le_add |
| (by simpa only [one_mul] using mul_le_mul_of_nonneg_right hnratio hF) |
| (by simpa only [one_mul] using mul_le_mul_of_nonneg_right hΛinv (mul_nonneg hCb hF)) |
| calc |
| _ ≤ F+(F+(2^(2*h+1)*C*C*(∑ i, ‖e i‖))*F) := add_le_add hb hs |
| _ = _ := by ring |
|
|
| theorem quantitative_scalar_correction_jets_on |
| (e : ι → E) (ρ : E → ℝ) {O : Set E} (hO : IsOpen O) |
| (hρ : ContDiffOn ℝ ∞ ρ O) (hρp : ∀ y ∈ O, 0 < ρ y) |
| (h B : ℕ) {C : ℝ} (hC : 0 < C) : |
| ∃ K > 0, ∀ (n : ℝ), 1 ≤ n → ∀ (D : ℕ) (U : E → ℝ) (X : ι → E → ℝ), |
| ContDiffOn ℝ ∞ U O → (∀ i, ContDiffOn ℝ ∞ (X i) O) → |
| (∀ y ∈ O, scalarCorrectionDenominator e n U X y ≠ 0) → |
| ∀ x ∈ O, ∀ (W : ℝ), 0 < W → |
| W^2/(2*n^(2*B)) ≤ scalarCorrectionDenominator e n U X x → |
| (∀ j ≤ h+2, ‖iteratedFDeriv ℝ j U x‖ ≤ C*n^(j+4)*W) → |
| (∀ i, ∀ j ≤ h+1, ‖iteratedFDeriv ℝ j (X i) x‖ ≤ C*n^(j+5)*W) → |
| (∀ j ≤ h+1, ‖iteratedFDeriv ℝ j |
| (fun y => coordinateDivergence e ρ X y+n*(n+2)*U y) x‖ ≤ C*n^4/(n^D)*W) → |
| (∀ j ≤ h+1, ‖iteratedFDeriv ℝ j ρ x‖ ≤ C) → |
| (∀ j ≤ h, ‖iteratedFDeriv ℝ j (fun y => (ρ y)⁻¹) x‖ ≤ C) → |
| ∀ j ≤ h, |
| ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionB e ρ n (n*(n+2)) U X y-1) x‖ + |
| ‖iteratedFDeriv ℝ j (fun y => scalarCorrectionS e ρ n (n*(n+2)) U X y-1) x‖ ≤ |
| K*n^((2*B+21)*(h+2)+10)/(n^D) := by |
| let Cd : ℝ := 2^(h+1)*C*C*(1+∑ i, ‖e i‖) |
| let A : ℝ := 2^(h+1)*C*C |
| let Kq : ℝ := 2*A*((h+1).factorial:ℝ)^2*(1+max 1 (2*Cd))^(h+1) |
| let Kv : ℝ := 2+2^(2*h+1)*C*C*(∑ i, ‖e i‖) |
| have hA : 0 < A := by dsimp only [A]; positivity |
| have hKq : 0 < Kq := by |
| have hc : 0 < (1+max 1 (2*Cd)) := by have := le_max_left (1:ℝ) (2*Cd); linarith |
| dsimp only [Kq] |
| positivity |
| have hKv : 0 < Kv := by dsimp only [Kv]; positivity |
| refine ⟨Kv*Kq, mul_pos hKv hKq, ?_⟩ |
| intro n hn D U X hU hX hden x hxO W hW hx hUj hXj hRj hρj hρij j hj |
| have hn0 : 0 < n := zero_lt_one.trans_le hn |
| let R : E → ℝ := fun y => coordinateDivergence e ρ X y+n*(n+2)*U y |
| let d := scalarCorrectionDenominator e n U X |
| have hR : ContDiffOn ℝ ∞ R O := |
| (contDiffOn_coordinateDivergence e hO hρ hX hρp).add (contDiffOn_const.mul hU) |
| have hd : ContDiffOn ℝ ∞ d O := contDiffOn_scalarCorrectionDenominator e n hO hU hX |
| have hRj' : ∀ k ≤ h+1, ‖iteratedFDeriv ℝ k R x‖ ≤ C*n^(4+k)/(n^D)*W := by |
| intro k hk |
| apply (hRj k hk).trans |
| gcongr |
| omega |
| have hUj' : ∀ k ≤ h+1, ‖iteratedFDeriv ℝ k U x‖ ≤ C*n^(4+k)*W := by |
| intro k hk |
| simpa only [Nat.add_comm] using hUj k (by omega) |
| have hXj' (i : ι) : ∀ k ≤ h+1, ‖iteratedFDeriv ℝ k (X i) x‖ ≤ C*n^(5+k)*W := by |
| intro k hk |
| simpa only [Nat.add_comm] using hXj i k hk |
| have hdj : ∀ k ≤ h+1, ‖iteratedFDeriv ℝ k d x‖ ≤ Cd*n^(k+10)*W^2 := |
| scalar_denominator_jet_bound_on e U X hO hxO hU hX (h+1) hn hW hC.le |
| (fun k hk => hUj k (by omega)) hXj |
| have hRU := frequency_weighted_product_bound_on hO hxO hR hU (h+1) 4 4 D hn hW hC.le hC.le hRj' hUj' |
| have hRX (i : ι) := frequency_weighted_product_bound_on hO hxO hR (hX i) (h+1) 4 5 D hn hW hC.le hC.le hRj' (hXj' i) |
| have hqU := frequency_weighted_quotient_uniform_on hO hxO (hR.mul hU) hd (h+1) B 8 D hn hW hA.le hx hRU |
| (fun k _ hk => hdj k hk) |
| have hqX (i : ι) := frequency_weighted_quotient_uniform_on hO hxO (hR.mul (hX i)) hd (h+1) B 9 D hn hW hA.le hx (hRX i) |
| (fun k _ hk => hdj k hk) |
| let T : ℕ := 9+2*B+(2*B+11)*(h+1) |
| let F : ℝ := Kq*n^T/(n^D) |
| have hF : 0 ≤ F := by dsimp only [F]; positivity |
| have hqu : ∀ k ≤ h, ‖iteratedFDeriv ℝ k |
| (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*U y) x‖ ≤ F := by |
| intro k hk |
| have he : (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*U y) = |
| fun y => (R y*U y)/d y := by |
| funext y |
| dsimp only [scalarCorrectionRatio,R,d] |
| ring |
| rw [he] |
| apply (hqU k (by omega)).trans |
| change Kq*n^(8+2*B+(2*B+11)*(h+1))/(n^D) ≤ Kq*n^T/(n^D) |
| gcongr |
| dsimp only [T]; omega |
| have hqx : ∀ i, ∀ k ≤ h+1, ‖iteratedFDeriv ℝ k |
| (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*X i y) x‖ ≤ F := by |
| intro i k hk |
| have he : (fun y => scalarCorrectionRatio e ρ n (n*(n+2)) U X y*X i y) = |
| fun y => (R y*X i y)/d y := by |
| funext y |
| dsimp only [scalarCorrectionRatio,R,d] |
| ring |
| rw [he] |
| exact hqX i k hk |
| have hb := scalar_correction_error_from_quotients_on e ρ U X hO hxO hρ hρp hU hX hn hden h hC.le hF hρj hρij hqu hqx j hj |
| apply hb.trans |
| change Kv*(Kq*n^T/(n^D)) ≤ Kv*Kq*n^((2*B+21)*(h+2)+10)/(n^D) |
| rw [show Kv*(Kq*n^T/(n^D))=Kv*Kq*n^T/(n^D) by ring] |
| gcongr |
| dsimp only [T]; nlinarith |
|
|
| end YauCounterexamples |
|
|
| end |
|
|
| end OAI |
|
|
