| import OAI.Analysis.AncientFlat.Geometry.ExtensionGeometry |
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| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff |
| namespace FlatWeighted |
| open ParabolicInterior ParabolicLocalization |
| variable {V : Type} [NormedAddCommGroup V] |
| lemma pdist_triangle [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| (p q r : V × ℝ) : pdist p r≤pdist p q+pdist q r := by |
| have hx : ‖p.1-r.1‖≤‖p.1-q.1‖+‖q.1-r.1‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _ |
| have ht : |p.2-r.2|≤|p.2-q.2|+|q.2-r.2| := abs_sub_le _ _ _ |
| have hh := (Real.rpow_le_rpow (abs_nonneg _) ht (by norm_num : (0:ℝ)≤1/4)).trans |
| (Real.rpow_add_le_add_rpow (abs_nonneg _) (abs_nonneg _) (by norm_num) (by norm_num)) |
| dsimp [pdist] |
| linarith |
| variable [InnerProductSpace ℝ V] |
| lemma chart_pdist [FiniteDimensional ℝ V] (p q r : V × ℝ) : |
| pdist (chart p q) (chart p r)=sigma p*pdist q r := |
| pdist_window p (Real.sqrt_nonneg _) q r |
| lemma localDomain_pdist [FiniteDimensional ℝ V] {a : ℝ} (ha : 0<a) |
| (p q : V × ℝ) (hq : q∈localDomain a p) : |
| pdist (0:V × ℝ) q<2*a := by |
| have habs : |q.2|<a^4 := abs_lt.mpr ⟨hq.2.1,hq.2.2.1⟩ |
| have ht : |q.2|^((1:ℝ)/4)<a := by |
| have hh := Real.rpow_lt_rpow (abs_nonneg _) habs (by norm_num : (0:ℝ)<1/4) |
| rw [←Real.rpow_natCast,←Real.rpow_mul ha.le] at hh |
| norm_num at hh |
| exact hh |
| simpa only [pdist,Prod.fst_zero,Prod.snd_zero,zero_sub,norm_neg,abs_neg, ←two_mul] using |
| (add_lt_add hq.1 ht) |
| def normBox (a : ℝ) (p : V × ℝ) : Set (V × ℝ) := chart p '' localDomain a p |
| lemma normBox_time [FiniteDimensional ℝ V] {a : ℝ} {p q : V × ℝ} |
| (hq : q∈normBox a p) : q.2<0 := by |
| obtain ⟨r,hr,rfl⟩ := hq |
| exact hr.2.2.2 |
| variable [FiniteDimensional ℝ V] |
| lemma normBox_dist {a : ℝ} (ha : 0<a) {p q : V × ℝ} (hs : sigma p=1) (hq : q∈normBox a p) : |
| pdist p q<2*a := by |
| obtain ⟨r,hr,rfl⟩ := hq |
| have h := chart_pdist p 0 r |
| rw [chart_zero,hs,one_mul] at h |
| rw [h] |
| exact localDomain_pdist ha p r hr |
| lemma normBox_diam {a : ℝ} (ha : 0<a) {p q r : V × ℝ} (hs : sigma p=1) |
| (hq : q∈normBox a p) (hr : r∈normBox a p) : pdist q r<4*a := by |
| have hq' := normBox_dist ha hs hq |
| have hr' := normBox_dist ha hs hr |
| have hh := pdist_triangle q p r |
| rw [pdist_symm q p] at hh |
| linarith |
| lemma normBox_sigma {a : ℝ} (ha : 0<a) (ha1 : a≤1/16) {p q : V × ℝ} |
| (hp : p.2<0) (hs : sigma p=1) (hq : q∈normBox a p) : |
| (7:ℝ)/8≤ sigma q ∧ sigma q≤9/8 := by |
| have hq0 := normBox_time hq |
| have hd := normBox_dist ha hs hq |
| have h1 := sigma_le_add_pdist p q hp hq0 |
| have h2 := sigma_le_add_pdist q p hq0 hp |
| rw [pdist_symm q p] at h2 |
| rw [hs] at h1 h2 |
| constructor <;> linarith |
| lemma normBox_bounds {a : ℝ} (ha : 0<a) (ha1 : a≤1/16) {p q : V × ℝ} |
| (hp : p.2<0) (hs : sigma p=1) (hq : q∈normBox a p) : |
| ‖q.1‖≤2 ∧ -2≤q.2 ∧ q.2<0 := by |
| have hq0 := normBox_time hq |
| have hsig := normBox_sigma ha ha1 hp hs hq |
| have hsq := sigma_sq q |
| have ht := Real.sq_sqrt (neg_pos.mpr hq0).le |
| have hroot : Real.sqrt (-q.2)≤(81:ℝ)/64 := by |
| nlinarith [sq_nonneg ‖q.1‖,Real.sqrt_nonneg (-q.2)] |
| refine ⟨(norm_le_sigma q).trans (by linarith),?_,hq0⟩ |
| nlinarith [Real.sqrt_nonneg (-q.2)] |
| lemma normBox_project_sigma {ψ : ℝ → ℝ} (hψ : ExtensionCutoff ψ) {T a : ℝ} |
| (hT : 0<T) (ha : 0<a) (ha1 : a≤1/16) {p : V × ℝ} (hp : p.2<0) (hs : sigma p=1) |
| (hactive : ∃ q∈normBox a p, extensionWeight ψ T q≠0) : |
| ∀ q∈normBox a p, (1:ℝ)/8≤ sigma (timeProject T q) := by |
| obtain ⟨r,hr,hr0⟩ := hactive |
| have hsr := (normBox_sigma ha ha1 hp hs hr).1 |
| have hsr' := sigma_le_project_of_active hψ hT r (normBox_time hr) hr0 |
| intro q hq |
| have hd := normBox_diam ha hs hr hq |
| have hpdt := timeProject_pdist T r q |
| have hsig := sigma_le_add_pdist (timeProject T r) (timeProject T q) |
| (timeProject_mem hT r (normBox_time hr)).2 (timeProject_mem hT q (normBox_time hq)).2 |
| linarith |
| end FlatWeighted |
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| end |
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| end OAI |
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