| import OAI.Analysis.AncientFlat.Geometry.InnerExtensionCore |
| import OAI.Analysis.AncientFlat.Weighted.WeightedScaling |
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| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff ENNReal |
| namespace FlatWeighted |
| open ParabolicInterior |
| variable {V : Type} |
| lemma sigma_sq [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (p : V × ℝ) : (sigma p)^2=‖p.1‖^2+Real.sqrt (-p.2) := by |
| exact Real.sq_sqrt (by positivity) |
| lemma norm_le_sigma [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (p : V × ℝ) : ‖p.1‖≤ sigma p := by |
| have hs := sigma_sq p |
| have hp : 0≤ sigma p := Real.sqrt_nonneg _ |
| nlinarith [norm_nonneg p.1,Real.sqrt_nonneg (-p.2)] |
| lemma pdist_symm [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (p q : V × ℝ) : pdist p q=pdist q p := by |
| simp only [pdist,norm_sub_rev,abs_sub_comm] |
| lemma sigma_le_add_pdist [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (p q : V × ℝ) (hp : p.2<0) (hq : q.2<0) : |
| sigma p≤ sigma q+pdist p q := by |
| let d := |p.2-q.2|^((1:ℝ)/4) |
| have hd : 0≤d := Real.rpow_nonneg (abs_nonneg _) _ |
| have hd4 : d^4=|p.2-q.2| := by |
| dsimp [d] |
| rw [←Real.rpow_natCast,←Real.rpow_mul (abs_nonneg _)] |
| norm_num |
| have hq2 := Real.sq_sqrt (show 0≤ -q.2 by linarith) |
| have ht : Real.sqrt (-p.2)≤Real.sqrt (-q.2)+d^2 := by |
| apply (sq_le_sq₀ (Real.sqrt_nonneg _) (by positivity)).mp |
| rw [Real.sq_sqrt (neg_pos.mpr hp).le] |
| have hdiff : -p.2≤ -q.2+|p.2-q.2| := by linarith [neg_le_abs (p.2-q.2)] |
| nlinarith [mul_nonneg (Real.sqrt_nonneg (-q.2)) (sq_nonneg d)] |
| have hpq : ‖p.1‖≤‖q.1‖+‖p.1-q.1‖ := by |
| calc |
| _ = ‖q.1+(p.1-q.1)‖ := by congr 1; abel |
| _ ≤ _ := norm_add_le _ _ |
| have hpq2 : ‖p.1‖^2≤(‖q.1‖+‖p.1-q.1‖)^2 := pow_le_pow_left₀ (norm_nonneg _) hpq 2 |
| have hqs := sigma_sq q |
| have hnp := norm_le_sigma q |
| have hh : 0≤ sigma q-‖q.1‖ := sub_nonneg.mpr hnp |
| unfold sigma |
| apply Real.sqrt_le_iff.mpr |
| refine ⟨by dsimp [pdist]; positivity,?_⟩ |
| change ‖p.1‖^2+Real.sqrt (-p.2)≤(sigma q+(‖p.1-q.1‖+d))^2 |
| have hpos : 0 ≤ sigma q := Real.sqrt_nonneg _ |
| have hmul : 0 ≤ (sigma q+‖p.1-q.1‖)*d := mul_nonneg (add_nonneg hpos (norm_nonneg _)) hd |
| nlinarith [mul_nonneg hh (norm_nonneg (p.1-q.1))] |
| def timeProject (T : ℝ) (p : V × ℝ) : V × ℝ := (p.1,max p.2 (-T)) |
| def extensionWeight [NormedAddCommGroup V] (ψ : ℝ → ℝ) (T : ℝ) (p : V × ℝ) : ℝ := |
| if -T≤p.2 then 1 else ψ ((-p.2-T)/(‖p.1‖^2+Real.sqrt T)^2) |
| lemma innerExtension_eq [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (ψ : ℝ → ℝ) (T : ℝ) (f : V × ℝ → ℂ) (p : V × ℝ) : |
| innerExtension ψ T f p=extensionWeight ψ T p • f (timeProject T p) := by |
| by_cases h : -T≤p.2 |
| · simp [innerExtension,extensionWeight,timeProject,h] |
| · simp [innerExtension,extensionWeight,timeProject,h,max_eq_right (not_le.mp h).le] |
| lemma timeProject_mem [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| {T : ℝ} (hT : 0<T) (p : V × ℝ) (hp : p.2<0) : |
| -T≤(timeProject T p).2 ∧ (timeProject T p).2<0 := |
| ⟨le_max_right _ _,max_lt hp (by linarith)⟩ |
| lemma timeProject_pdist [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (T : ℝ) (p q : V × ℝ) : pdist (timeProject T p) (timeProject T q)≤pdist p q := by |
| dsimp [pdist,timeProject] |
| apply add_le_add_right |
| exact Real.rpow_le_rpow (abs_nonneg _) (abs_max_sub_max_le_abs _ _ _) (by norm_num) |
| lemma sigma_project_le [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (T : ℝ) (p : V × ℝ) : sigma (timeProject T p)≤ sigma p := by |
| apply Real.sqrt_le_sqrt |
| exact add_le_add_right (Real.sqrt_le_sqrt (neg_le_neg (le_max_left _ _))) _ |
| lemma extensionWeight_range [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| {ψ : ℝ → ℝ} (hψ : ExtensionCutoff ψ) {T : ℝ} (hT : 0<T) (p : V × ℝ) : |
| 0≤extensionWeight ψ T p ∧ extensionWeight ψ T p≤1 := by |
| by_cases h : -T≤p.2 |
| · simp [extensionWeight,h] |
| · rw [extensionWeight,ite_eq_right h] |
| apply hψ.2.1 |
| apply div_nonneg |
| · linarith [not_le.mp h] |
| · exact (sq_pos_of_pos (add_pos_of_nonneg_of_pos (sq_nonneg _) (Real.sqrt_pos.mpr hT))).le |
| lemma sigma_le_project_of_active [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| {ψ : ℝ → ℝ} (hψ : ExtensionCutoff ψ) {T : ℝ} (hT : 0<T) |
| (p : V × ℝ) (hp : p.2<0) (hactive : extensionWeight ψ T p≠0) : |
| sigma p≤2*sigma (timeProject T p) := by |
| by_cases h : -T≤p.2 |
| · have he : timeProject T p=p := by ext <;> simp [timeProject,max_eq_left h] |
| rw [he] |
| have hs : 0 ≤ sigma p := (sigma_pos p hp).le |
| linarith |
| have hb : 0<‖p.1‖^2+Real.sqrt T := add_pos_of_nonneg_of_pos (sq_nonneg _) (Real.sqrt_pos.2 hT) |
| have harg : (-p.2-T)/(‖p.1‖^2+Real.sqrt T)^2<1 := by |
| by_contra hn |
| exact hactive (by rw [extensionWeight,ite_eq_right h,hψ.2.2.2 _ (not_lt.mp hn)]) |
| have hbnd : -p.2-T<(‖p.1‖^2+Real.sqrt T)^2 := by |
| simpa only [one_mul] using (div_lt_iff₀ (sq_pos_of_pos hb)).mp harg |
| have hTsq := Real.sq_sqrt hT.le |
| have hrt : Real.sqrt (-p.2)≤2*(‖p.1‖^2+Real.sqrt T) := by |
| apply Real.sqrt_le_iff.mpr |
| refine ⟨by positivity,?_⟩ |
| nlinarith [sq_nonneg (‖p.1‖^2),mul_nonneg (sq_nonneg ‖p.1‖) (Real.sqrt_nonneg T)] |
| change Real.sqrt (‖p.1‖^2+Real.sqrt (-p.2))≤2*Real.sqrt (‖p.1‖^2+Real.sqrt (-max p.2 (-T))) |
| rw [max_eq_right (not_le.mp h).le,neg_neg] |
| apply Real.sqrt_le_iff.mpr |
| refine ⟨by positivity,?_⟩ |
| rw [mul_pow,Real.sq_sqrt hb.le] |
| nlinarith [sq_nonneg ‖p.1‖,Real.sqrt_nonneg T] |
| end FlatWeighted |
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| end |
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| end OAI |
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