| import OAI.Geometry.RuledCovering.SingularInverse |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma ceiling_nonneg {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (b : ℝ) : |
| 0 ≤ ceilingPolynomial p b := by |
| unfold ceilingPolynomial |
| have : 0 ≤ 2-p := sub_nonneg.mpr hp2 |
| positivity |
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| lemma ceiling_P_identity (p x y : ℝ) : |
| ceilingPolynomial p x + ceilingPolynomial p y - |
| x^2*(p/4+(2-p)*y^2)-y^2*(p/4+(2-p)*x^2) = (2-p)*(x^2-y^2)^2 := by |
| unfold ceilingPolynomial |
| ring |
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| lemma singular_region_estimates (p : ℝ) : |
| ∃ K : ℝ, 1 ≤ K ∧ ∀ (z : Plane) (B : ℝ), |
| |z 0| ≤ 2 → (1/2:ℝ) ≤ z 1 → z 1 ≤ 2 → |B| ≤ 2 → |
| |singularField p z 0| ≤ K ∧ |
| |singularField p z 1-z 0| ≤ K*(z 1-1/2) ∧ |
| |singularField p z 0-(p/4+(2-p)*B^2)| ≤ K*(|z 0-B|+(z 1-1/2)) ∧ |
| |ceilingPolynomial p (z 0)-ceilingPolynomial p B| ≤ K*|z 0-B| := by |
| let S : Set Plane := (EuclideanSpace.proj 0) ⁻¹' Icc (-2:ℝ) 2 ∩ |
| (EuclideanSpace.proj 1) ⁻¹' Icc (1/2:ℝ) 2 |
| have hSc : IsClosed S := (isClosed_Icc.preimage (by fun_prop)).inter |
| (isClosed_Icc.preimage (by fun_prop)) |
| have hS : IsCompact S := (isCompact_closedBall (0:Plane) 4).of_isClosed_subset hSc (by |
| intro z hz |
| change z 0 ∈ Icc (-2:ℝ) 2 ∧ z 1 ∈ Icc (1/2:ℝ) 2 at hz |
| rw [Metric.mem_closedBall,dist_zero_right] |
| have ha : |z 0| ≤ 2 := abs_le.mpr hz.1 |
| have hb : |z 1| ≤ 2 := (abs_of_nonneg (by linarith [hz.2.1])).le.trans hz.2.2 |
| exact (plane_norm_le_sum z).trans (by linarith)) |
| have hconv : Convex ℝ S := ((convex_Icc (-2:ℝ) 2).linear_preimage (EuclideanSpace.proj 0).toLinearMap).inter |
| ((convex_Icc (1/2:ℝ) 2).linear_preimage (EuclideanSpace.proj 1).toLinearMap) |
| have hsmooth : ∀ z ∈ S, ContDiffAt ℝ ∞ (singularField p) z := by |
| intro z hz |
| apply singularField_contDiffAt |
| have hz1 : (1/2:ℝ) ≤ z 1 := hz.2.1 |
| linarith |
| obtain ⟨C₀,hC₀⟩ := hS.exists_bound_of_continuousOn |
| (fun z hz => (hsmooth z hz).continuousAt.continuousWithinAt) |
| obtain ⟨C₁,hC₁⟩ := hS.exists_bound_of_continuousOn |
| (fun z hz => ((hsmooth z hz).continuousAt_fderiv (by norm_num)).continuousWithinAt) |
| have hdelta : ContDiff ℝ ∞ (ceilingPolynomial p) := by unfold ceilingPolynomial; fun_prop |
| obtain ⟨C₂,hC₂⟩ := (isCompact_Icc (a := (-2:ℝ)) (b := 2)).exists_bound_of_continuousOn |
| (f := fderiv ℝ (ceilingPolynomial p)) |
| (fun _ _ => (hdelta.contDiffAt.continuousAt_fderiv (by norm_num)).continuousWithinAt) |
| let K := max 1 (max C₀ (max C₁ C₂)) |
| have hK : 1 ≤ K := le_max_left _ _ |
| have hK₀ : C₀ ≤ K := (le_max_left _ _).trans (le_max_right _ _) |
| have hK₁ : C₁ ≤ K := (le_max_left _ _).trans |
| ((le_max_right _ _).trans (le_max_right _ _)) |
| have hK₂ : C₂ ≤ K := (le_max_right _ _).trans |
| ((le_max_right _ _).trans (le_max_right _ _)) |
| have hKn : 0 ≤ K := by linarith |
| have hlip : ∀ z ∈ S, ∀ w ∈ S, ‖singularField p z-singularField p w‖ ≤ K*‖z-w‖ := by |
| intro z hz w hw |
| exact Convex.norm_image_sub_le_of_norm_fderiv_le |
| (fun q hq => (hsmooth q hq).differentiableAt (by norm_num)) |
| (fun q hq => (hC₁ q hq).trans hK₁) hconv hw hz |
| have hdlip : ∀ b ∈ Icc (-2:ℝ) 2, ∀ B ∈ Icc (-2:ℝ) 2, |
| |ceilingPolynomial p b-ceilingPolynomial p B| ≤ K*|b-B| := by |
| intro b hb B hB |
| simpa only [Real.norm_eq_abs] using Convex.norm_image_sub_le_of_norm_fderiv_le |
| (f := ceilingPolynomial p) (fun _ _ => hdelta.differentiable (by norm_num) _) |
| (fun q hq => (hC₂ q hq).trans hK₂) (convex_Icc _ _) hB hb |
| refine ⟨K,hK,?_⟩ |
| intro z B hz0 hz1 hz2 hB |
| have hz : z ∈ S := ⟨abs_le.mp hz0,⟨hz1,hz2⟩⟩ |
| have hw : vec2 (z 0) (1/2) ∈ S := ⟨abs_le.mp hz0,by norm_num [vec2]⟩ |
| have hwB : vec2 B (1/2) ∈ S := ⟨abs_le.mp hB,by norm_num [vec2]⟩ |
| refine ⟨(plane_coordinate_le _ 0).trans ((hC₀ z hz).trans hK₀),?_,?_,hdlip _ (abs_le.mp hz0) _ (abs_le.mp hB)⟩ |
| · have h := (plane_coordinate_le (singularField p z-singularField p (vec2 (z 0) (1/2))) 1).trans (hlip z hz _ hw) |
| rw [singularField_level] at h |
| change |singularField p z 1-z 0| ≤ K*‖z-vec2 (z 0) (1/2)‖ at h |
| apply h.trans |
| have he := plane_norm_le_sum (z-vec2 (z 0) (1/2)) |
| simpa only [PiLp.sub_apply,vec2_apply_zero,vec2_apply_one,sub_self,abs_zero,zero_add, |
| abs_of_nonneg (sub_nonneg.mpr hz1)] using mul_le_mul_of_nonneg_left he hKn |
| · have h := (plane_coordinate_le (singularField p z-singularField p (vec2 B (1/2))) 0).trans (hlip z hz _ hwB) |
| rw [singularField_level] at h |
| change |singularField p z 0-(p/4+(2-p)*B^2)| ≤ K*‖z-vec2 B (1/2)‖ at h |
| apply h.trans |
| have he := plane_norm_le_sum (z-vec2 B (1/2)) |
| simpa only [PiLp.sub_apply,vec2_apply_zero,vec2_apply_one, |
| abs_of_nonneg (sub_nonneg.mpr hz1)] using mul_le_mul_of_nonneg_left he hKn |
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| lemma abs_unit_mul_le {x A B : ℝ} (hx : |x| ≤ 1) (hA : |A| ≤ B) : |
| |x*A| ≤ B := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_right hx (abs_nonneg A)).trans (by simpa using hA) |
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| lemma singular_overlap_algebra {p K c x y bm bp rm rp Pm Pp Qm Qp : ℝ} |
| (hp2 : p ≤ 2) (hK : 1 ≤ K) (hc : 0 ≤ c) (hc1 : c ≤ 1) |
| (hcK : c*K ≤ 1/2) (hcsmall : 16*K^2*c^3 ≤ 2*c^(5/2:ℝ)) |
| (hx : |x| ≤ 1) (hy : |y| ≤ 1) (hrm : 1/2 ≤ rm) (hrp : 1/2 ≤ rp) |
| (hPm : |Pm| ≤ K) (hPp : |Pp| ≤ K) |
| (hQm : |Qm-bm| ≤ K*(rm-1/2)) (hQp : |Qp-bp| ≤ K*(rp-1/2)) |
| (hPmy : |Pm-(p/4+(2-p)*y^2)| ≤ K*(|bm-y|+(rm-1/2))) |
| (hPpx : |Pp-(p/4+(2-p)*x^2)| ≤ K*(|bp-x|+(rp-1/2))) |
| (hdm : |ceilingPolynomial p bm-ceilingPolynomial p y| ≤ K*|bm-y|) |
| (hdp : |ceilingPolynomial p bp-ceilingPolynomial p x| ≤ K*|bp-x|) |
| (hbm : bm=y-c*x*Pm) (hbp : bp=x+c*y*Pp) |
| (hsum : rm+rp=1+c*(y*Qp-x*Qm)) : |
| rm ≤ 1/2+c^2*ceilingPolynomial p bm+c^(5/2:ℝ) ∨ |
| rp ≤ 1/2+c^2*ceilingPolynomial p bp+c^(5/2:ℝ) := by |
| have hKn : 0 ≤ K := by linarith |
| have hxs : |x^2| ≤ 1 := by rw [abs_of_nonneg (sq_nonneg x)]; nlinarith only [(abs_le.mp hx).1,(abs_le.mp hx).2] |
| have hys : |y^2| ≤ 1 := by rw [abs_of_nonneg (sq_nonneg y)]; nlinarith only [(abs_le.mp hy).1,(abs_le.mp hy).2] |
| have hbm' : |bm-y| ≤ c*K := by |
| rw [hbm,sub_sub_cancel_left,abs_neg,abs_mul,abs_mul,abs_of_nonneg hc] |
| simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (by simpa only [abs_mul] using abs_unit_mul_le hx hPm) hc |
| have hbp' : |bp-x| ≤ c*K := by |
| rw [hbp,add_sub_cancel_left,abs_mul,abs_mul,abs_of_nonneg hc] |
| simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (by simpa only [abs_mul] using abs_unit_mul_le hy hPp) hc |
| let D := rm+rp-1 |
| have hD : 0 ≤ D := by dsimp [D]; linarith |
| have hEd : |y*(Qp-bp)-x*(Qm-bm)| ≤ K*D := by |
| calc |
| _ ≤ |y*(Qp-bp)|+|x*(Qm-bm)| := abs_sub _ _ |
| _ ≤ K*(rp-1/2)+K*(rm-1/2) := add_le_add (abs_unit_mul_le hy hQp) (abs_unit_mul_le hx hQm) |
| _ = _ := by dsimp [D]; ring |
| have hPe : |y^2*Pp+x^2*Pm| ≤ 2*K := by |
| calc |
| _ ≤ |y^2*Pp|+|x^2*Pm| := abs_add_le _ _ |
| _ ≤ K+K := add_le_add (abs_unit_mul_le hys hPp) (abs_unit_mul_le hxs hPm) |
| _ = _ := by ring |
| have hDeq : D=c^2*(y^2*Pp+x^2*Pm)+c*(y*(Qp-bp)-x*(Qm-bm)) := by |
| dsimp [D] |
| rw [hsum,hbm,hbp] |
| ring |
| have hDb : D ≤ 4*K*c^2 := by |
| have hu : D ≤ c^2*(2*K)+c*(K*D) := by |
| calc |
| D = c^2*(y^2*Pp+x^2*Pm)+c*(y*(Qp-bp)-x*(Qm-bm)) := hDeq |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left (le_trans (le_abs_self _) hPe) (sq_nonneg c)) |
| (mul_le_mul_of_nonneg_left (le_trans (le_abs_self _) hEd) hc) |
| have hmul := mul_le_mul_of_nonneg_right hcK hD |
| nlinarith only [hmul,hu] |
| have hrmb : rm-1/2 ≤ 4*K*c^2 := by dsimp [D] at hDb; linarith |
| have hrpb : rp-1/2 ≤ 4*K*c^2 := by dsimp [D] at hDb; linarith |
| have hc2 : c^2 ≤ c := by nlinarith only [hc,hc1] |
| have hPmy' : |Pm-(p/4+(2-p)*y^2)| ≤ 5*K^2*c := by |
| apply hPmy.trans |
| calc |
| _ ≤ K*(c*K+4*K*c^2) := mul_le_mul_of_nonneg_left (add_le_add hbm' hrmb) hKn |
| _ ≤ K*(c*K+4*K*c) := mul_le_mul_of_nonneg_left (add_le_add le_rfl (mul_le_mul_of_nonneg_left hc2 (by positivity : 0 ≤ 4*K))) hKn |
| _ = _ := by ring |
| have hPpx' : |Pp-(p/4+(2-p)*x^2)| ≤ 5*K^2*c := by |
| apply hPpx.trans |
| calc |
| _ ≤ K*(c*K+4*K*c^2) := mul_le_mul_of_nonneg_left (add_le_add hbp' hrpb) hKn |
| _ ≤ K*(c*K+4*K*c) := mul_le_mul_of_nonneg_left (add_le_add le_rfl (mul_le_mul_of_nonneg_left hc2 (by positivity : 0 ≤ 4*K))) hKn |
| _ = _ := by ring |
| let A := x^2*(p/4+(2-p)*y^2)+y^2*(p/4+(2-p)*x^2) |
| have hAe : |y^2*Pp+x^2*Pm-A| ≤ 10*K^2*c := by |
| calc |
| _ = |y^2*(Pp-(p/4+(2-p)*x^2))+x^2*(Pm-(p/4+(2-p)*y^2))| := by dsimp [A]; congr 1; ring |
| _ ≤ |y^2*(Pp-(p/4+(2-p)*x^2))|+|x^2*(Pm-(p/4+(2-p)*y^2))| := abs_add_le _ _ |
| _ ≤ 5*K^2*c+5*K^2*c := add_le_add (abs_unit_mul_le hys hPpx') (abs_unit_mul_le hxs hPmy') |
| _ = _ := by ring |
| have hDelta : D ≤ c^2*A+14*K^2*c^3 := by |
| have hh : D-c^2*A ≤ c^2*(10*K^2*c)+c*(K*(4*K*c^2)) := by |
| calc |
| _ = c^2*(y^2*Pp+x^2*Pm-A)+c*(y*(Qp-bp)-x*(Qm-bm)) := by rw [hDeq]; ring |
| _ ≤ c^2*(10*K^2*c)+c*(K*D) := add_le_add |
| (mul_le_mul_of_nonneg_left ((le_abs_self _).trans hAe) (sq_nonneg c)) |
| (mul_le_mul_of_nonneg_left ((le_abs_self _).trans hEd) hc) |
| _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hDb hKn) hc) |
| nlinarith only [hh] |
| have hdm' : ceilingPolynomial p y-K^2*c ≤ ceilingPolynomial p bm := by |
| have hh := hdm.trans (mul_le_mul_of_nonneg_left hbm' hKn) |
| have hh' := (abs_le.mp hh).1 |
| nlinarith only [hh'] |
| have hdp' : ceilingPolynomial p x-K^2*c ≤ ceilingPolynomial p bp := by |
| have hh := hdp.trans (mul_le_mul_of_nonneg_left hbp' hKn) |
| have hh' := (abs_le.mp hh).1 |
| nlinarith only [hh'] |
| have hAid : A ≤ ceilingPolynomial p y+ceilingPolynomial p x := by |
| have hh := ceiling_P_identity p x y |
| have hz : 0 ≤ (2-p)*(x^2-y^2)^2 := mul_nonneg (by linarith) (sq_nonneg _) |
| dsimp [A] |
| linarith |
| by_contra h |
| push Not at h |
| have hfail : c^2*(ceilingPolynomial p bm+ceilingPolynomial p bp)+2*c^(5/2:ℝ) < D := by |
| dsimp [D] |
| linarith [h.1,h.2] |
| have hdlow : A-2*K^2*c ≤ ceilingPolynomial p bm+ceilingPolynomial p bp := by linarith |
| have hh := mul_le_mul_of_nonneg_left hdlow (sq_nonneg c) |
| nlinarith only [hh,hfail,hDelta,hcsmall] |
|
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| lemma singular_overlap_small {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) : |
| ∃ K : ℝ, 1 ≤ K ∧ ∀ (c x y : ℝ) (zm zp : Plane), |
| 0 ≤ c → c ≤ 1 → c*K ≤ 1/2 → 16*K^2*c^3 ≤ 2*c^(5/2:ℝ) → |
| |x| ≤ 1 → |y| ≤ 1 → zm 1 ≤ 2 → zp 1 ≤ 2 → |
| |zm 0| ≤ zm 1 → |zp 0| ≤ zp 1 → |
| zm 0=y-c*x*singularField p zm 0 → zp 0=x+c*y*singularField p zp 0 → |
| zm 1+zp 1=1+c*(y*singularField p zp 1-x*singularField p zm 1) → |
| zm 1 ≤ 1/2+c^2*ceilingPolynomial p (zm 0)+c^(5/2:ℝ) ∨ |
| zp 1 ≤ 1/2+c^2*ceilingPolynomial p (zp 0)+c^(5/2:ℝ) := by |
| obtain ⟨K,hK,he⟩ := singular_region_estimates p |
| refine ⟨K,hK,?_⟩ |
| intro c x y zm zp hc hc1 hcK hsmall hx hy hzm hzp hbzm hbzp hezm hezp hesum |
| by_cases hm : zm 1 ≤ 1/2+c^2*ceilingPolynomial p (zm 0)+c^(5/2:ℝ) |
| · exact Or.inl hm |
| by_cases hp' : zp 1 ≤ 1/2+c^2*ceilingPolynomial p (zp 0)+c^(5/2:ℝ) |
| · exact Or.inr hp' |
| have hrm : 1/2 ≤ zm 1 := by |
| have hn := ceiling_nonneg hp hp2 (zm 0) |
| have hn' : 0 ≤ c^2*ceilingPolynomial p (zm 0)+c^(5/2:ℝ) := by positivity |
| linarith |
| have hrp : 1/2 ≤ zp 1 := by |
| have hn := ceiling_nonneg hp hp2 (zp 0) |
| have hn' : 0 ≤ c^2*ceilingPolynomial p (zp 0)+c^(5/2:ℝ) := by positivity |
| linarith |
| obtain ⟨hPm,hQm,hPmy,hdm⟩ := he zm y (hbzm.trans hzm) hrm hzm (hy.trans (by norm_num)) |
| obtain ⟨hPp,hQp,hPpx,hdp⟩ := he zp x (hbzp.trans hzp) hrp hzp (hx.trans (by norm_num)) |
| exact singular_overlap_algebra hp2 hK hc hc1 hcK hsmall hx hy hrm hrp |
| hPm hPp hQm hQp hPmy hPpx hdm hdp hezm hezp hesum |
|
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| lemma right_rpow_tendsto_zero {q : ℝ} (hq : 0 < q) : |
| Tendsto (fun c : ℝ => c^q) (𝓝[>] 0) (𝓝 0) := by |
| have h : Tendsto (fun c : ℝ => c^q) (𝓝[>] 0) (𝓝 ((0:ℝ)^q)) := |
| ((Real.continuous_rpow_const hq.le).tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds |
| simpa only [Real.zero_rpow hq.ne'] using h |
|
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| lemma overlap_error_eventually (K : ℝ) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, 16*K^2*c^3 ≤ 2*c^(5/2:ℝ) := by |
| have h : Tendsto (fun c : ℝ => 16*K^2*c^(1/2:ℝ)) (𝓝[>] 0) (𝓝 0) := by |
| simpa using (right_rpow_tendsto_zero (q := (1/2:ℝ)) (by norm_num)).const_mul (16*K^2) |
| filter_upwards [self_mem_nhdsWithin,h.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<2))] with c hc hb |
| have hc : 0 < c := hc |
| have he : c^3=c^(1/2:ℝ)*c^(5/2:ℝ) := by |
| rw [← Real.rpow_add hc] |
| norm_num |
| rw [he] |
| nlinarith only [mul_le_mul_of_nonneg_right hb.le (Real.rpow_nonneg hc.le (5/2:ℝ))] |
|
|
| end RuledApproximation |
|
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| end |
|
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| end OAI |
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