| import OAI.Analysis.AncientFlat.Geometry.ParabolicWindows |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| namespace ParabolicLocalization |
| open ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| def windowInv (p0 : V × ℝ) (q : ℝ) (p : V × ℝ) : V × ℝ := |
| (q⁻¹ • (p.1-p0.1),(p.2-p0.2)/q^4) |
| omit [FiniteDimensional ℝ V] in |
| lemma window_windowInv (p0 : V × ℝ) {q : ℝ} (hq : q≠0) (p : V × ℝ) : |
| window p0 q (windowInv p0 q p)=p := by |
| ext |
| · simp [window,windowInv,smul_smul,hq] |
| · dsimp [window,windowInv]; field_simp; ring |
| omit [FiniteDimensional ℝ V] in |
| lemma window_zero (p0 : V × ℝ) (q : ℝ) : window p0 q (0,0)=p0 := by |
| simp [window] |
| lemma pow_quarter_four {r : ℝ} (hr : 0≤r) : (r^((1:ℝ)/4))^4=r := by |
| rw [←Real.rpow_natCast,←Real.rpow_mul hr] |
| norm_num |
| omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in |
| lemma cylinder_pdist_zero {R : ℝ} (hR : 0≤R) {p : V × ℝ} (hp : p∈cylinder R) : |
| pdist (0,0) p≤2*R := by |
| have ht : |p.2|≤R^4 := by rw [abs_of_nonpos hp.2.2]; linarith [hp.2.1] |
| have ht' := Real.rpow_le_rpow (abs_nonneg _) ht (by norm_num : (0:ℝ)≤1/4) |
| rw [pow_four_quarter hR] at ht' |
| simp only [pdist,zero_sub,norm_neg,abs_neg] |
| linarith [hp.1] |
| omit [FiniteDimensional ℝ V] in |
| lemma small_window_maps {r R q : ℝ} (hr : 0≤r) (hq : 0<q) (hgap : 4*q≤R-r) |
| (p0 : V × ℝ) (hp0 : p0∈cylinder r) : |
| MapsTo (window p0 q) (cylinder 4) (cylinder R) := by |
| have ht : r^4+(4*q)^4≤R^4 := (pow_add_pow_le hr (by positivity) (by norm_num : (4:ℕ)≠0)).trans |
| (pow_le_pow_left₀ (by positivity) (by linarith : r+4*q≤R) 4) |
| intro p hp |
| refine ⟨?_,?_,?_⟩ |
| · change ‖p0.1+q • p.1‖<R |
| have hn : ‖q • p.1‖=q*‖p.1‖ := by rw [norm_smul,Real.norm_of_nonneg hq.le] |
| apply (norm_add_le _ _).trans_lt |
| rw [hn] |
| nlinarith [hp0.1,hp.1] |
| · change -(R^4)<p0.2+q^4*p.2 |
| have hp' := hp.2.1 |
| have hp0' := hp0.2.1 |
| have hq4 : 0<q^4 := pow_pos hq _ |
| norm_num at hp' |
| nlinarith |
| · change p0.2+q^4*p.2≤0 |
| have hm := mul_nonpos_of_nonneg_of_nonpos (pow_nonneg hq.le 4) hp.2.2 |
| linarith [hp0.2.2] |
| omit [FiniteDimensional ℝ V] in |
| lemma windowInv_mem_one (p0 : V × ℝ) {q : ℝ} (hq : 0<q) (p : V × ℝ) |
| (hx : ‖p.1-p0.1‖<q) (ht : |p.2-p0.2|<q^4) (ht0 : p.2≤p0.2) : |
| windowInv p0 q p∈cylinder 1 := by |
| refine ⟨?_,?_,?_⟩ |
| · change ‖q⁻¹ • (p.1-p0.1)‖<1 |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hq.le)] |
| have := mul_lt_mul_of_pos_left hx (inv_pos.mpr hq) |
| simpa only [inv_mul_cancel₀ hq.ne'] using this |
| · change -(1:ℝ)^4<(p.2-p0.2)/q^4 |
| rw [lt_div_iff₀ (pow_pos hq 4)] |
| rw [abs_of_nonpos (sub_nonpos.mpr ht0)] at ht |
| norm_num; linarith |
| · change (p.2-p0.2)/q^4≤0 |
| exact div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ht0) (pow_nonneg hq.le 4) |
| omit [FiniteDimensional ℝ V] in |
| lemma close_pair_window {r q : ℝ} (hq : 0<q) (p z : V × ℝ) |
| (hp : p∈cylinder r) (hz : z∈cylinder r) (hd : pdist p z<q) : |
| let p0 : V × ℝ := (p.1,max p.2 z.2) |
| p0∈cylinder r ∧ windowInv p0 q p∈cylinder 1 ∧ windowInv p0 q z∈cylinder 1 := by |
| dsimp only |
| have ht : |p.2-z.2|<q^4 := by |
| have hh : |p.2-z.2|^((1:ℝ)/4)<q := by |
| dsimp [pdist] at hd; linarith [norm_nonneg (p.1-z.1)] |
| have hh' : (|p.2-z.2|^((1:ℝ)/4))^4<q^4 := by gcongr |
| rwa [pow_quarter_four (abs_nonneg _)] at hh' |
| have hx : ‖p.1-z.1‖<q := by |
| dsimp [pdist] at hd; linarith [Real.rpow_nonneg (abs_nonneg (p.2-z.2)) ((1:ℝ)/4)] |
| have ht1 : |p.2-max p.2 z.2|<q^4 := by |
| rcases le_total p.2 z.2 with h | h |
| · simpa only [max_eq_right h] using ht |
| · simp only [max_eq_left h,sub_self,abs_zero]; positivity |
| have ht2 : |z.2-max p.2 z.2|<q^4 := by |
| rcases le_total p.2 z.2 with h | h |
| · simp only [max_eq_right h,sub_self,abs_zero]; positivity |
| · simpa only [max_eq_left h,abs_sub_comm] using ht |
| refine ⟨⟨hp.1,hp.2.1.trans_le (le_max_left _ _),max_le hp.2.2 hz.2.2⟩,?_,?_⟩ |
| · exact windowInv_mem_one _ hq p (by simpa using hq) ht1 (le_max_left _ _) |
| · exact windowInv_mem_one _ hq z (by simpa only [norm_sub_rev] using hx) ht2 (le_max_right _ _) |
| end ParabolicLocalization |
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| end |
|
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| end OAI |
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