| import OAI.Geometry.RuledCovering.Main |
| import OAI.Geometry.CylinderCovering.Singular.Cutoff |
| import OAI.Geometry.CylinderCovering.Singular.BasicDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter Topology Metric |
| open scoped ContDiff |
| namespace CylinderCovering.FlexibleSingular |
| open RuledApproximation |
|
|
| lemma labelDomain_isCompact (p c d q : ℝ) : IsCompact (labelDomain p c d q) := by |
| have hb : Continuous (fun z : Plane => z 0) := by fun_prop |
| have hr : Continuous (fun z : Plane => z 1) := by fun_prop |
| have hcap : Continuous (fun z : Plane => (1/2:ℝ)+c^2*ceilingPolynomial p (z 0)+q) := by |
| unfold ceilingPolynomial |
| fun_prop |
| have hclosed : IsClosed (labelDomain p c d q) := |
| (isClosed_le continuous_const hr).inter ((isClosed_le hr continuous_const).inter |
| ((isClosed_le hb.abs hr).inter (isClosed_le hr hcap))) |
| apply (isCompact_closedBall (0 : Plane) 4).of_isClosed_subset hclosed |
| intro z hz |
| have hz0 : |z 0| ≤ 2 := hz.2.2.1.trans hz.2.1 |
| have hz1 : |z 1| ≤ 2 := (abs_of_nonneg ((abs_nonneg _).trans hz.2.2.1)).le.trans hz.2.1 |
| rw [Metric.mem_closedBall,dist_zero_right] |
| exact (plane_norm_le_sum z).trans (by linarith) |
|
|
| lemma labelDomain_smoothNear (p c q : ℝ) {d : ℝ} (hd : 0 < d) : |
| SmoothNear ∞ (labelDomain p c d q) (singularField p) := by |
| refine ⟨{z | 0 < z 1},isOpen_lt continuous_const (by fun_prop),?_,singularField_contDiffOn p⟩ |
| intro z hz |
| exact hd.trans_le hz.1 |
|
|
| lemma overlap_algebra {p K c q 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*q) |
| (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+q ∨ |
| rp ≤ 1/2+c^2*ceilingPolynomial p bp+q := 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*q < 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] |
|
|
| lemma overlap_small {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) : |
| ∃ K : ℝ, 1 ≤ K ∧ ∀ (c q x y : ℝ) (zm zp : Plane), |
| 0 ≤ c → 0 ≤ q → c ≤ 1 → c*K ≤ 1/2 → 16*K^2*c^3 ≤ 2*q → |
| |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)+q ∨ |
| zp 1 ≤ 1/2+c^2*ceilingPolynomial p (zp 0)+q := by |
| obtain ⟨K,hK,he⟩ := singular_region_estimates p |
| refine ⟨K,hK,?_⟩ |
| intro c q x y zm zp hc hq 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)+q |
| · exact Or.inl hm |
| by_cases hp' : zp 1 ≤ 1/2+c^2*ceilingPolynomial p (zp 0)+q |
| · 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)+q := 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)+q := 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 overlap_algebra hp2 hK hc hc1 hcK hsmall hx hy hrm hrp |
| hPm hPp hQm hQp hPmy hPpx hdm hdp hezm hezp hesum |
|
|
| theorem geometric_cover {p : ℝ} {ρ q : ℝ → ℝ} (hp : 1 < p) (hp2 : p < 2) |
| (hadm : SingularCutoff.Admissible p ρ q) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, |
| tetrahedron ⊆ lowerFamily p c (ρ c) (q c) ∪ upperFamily p c (ρ c) (q c) ∪ |
| lowerTip (ρ c) ∪ upperTip (ρ c) := by |
| obtain ⟨C,_,hinv⟩ := singular_inverse_small hp2 |
| obtain ⟨K,_,hover⟩ := overlap_small (by linarith : 0 ≤ p) hp2.le |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| have hcut : Tendsto (fun c : ℝ => c*C*(ρ c)^(p-2)) (𝓝[>] 0) (𝓝 0) := by |
| simpa only [mul_assoc,mul_left_comm,mul_zero] using (hadm.2.2.1).const_mul C |
| have hKlim : Tendsto (fun c : ℝ => c*K) (𝓝[>] 0) (𝓝 0) := by simpa using h0.mul_const K |
| filter_upwards [self_mem_nhdsWithin,h0.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)), |
| hcut.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/8)), |
| hKlim.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2)),SingularCutoff.overlap_error_eventually hadm K, hadm.1] |
| with c hc hc1 hcut hcK herr hpos |
| have hc : 0 < c := hc |
| intro z hz |
| obtain ⟨t,ht0,ht1,hx,hy,hz2⟩ := hz |
| by_cases htip : t ≤ 2*ρ c |
| · exact Or.inl (Or.inr ⟨t,ht0,htip,hy,hz2⟩) |
| by_cases htip' : 1-t ≤ 2*ρ c |
| · exact Or.inr ⟨1-t,by linarith,htip',hx,by rw [hz2]; congr 1; ring⟩ |
| have hx1 : |z 0| ≤ 1 := hx.trans (by linarith) |
| have hy1 : |z 1| ≤ 1 := hy.trans ht1 |
| have hkx : |c*z 0| ≤ c := by simpa only [abs_mul,abs_of_pos hc,mul_one] using mul_le_mul_of_nonneg_left hx1 hc.le |
| have hky : |-c*z 1| ≤ c := by simpa only [abs_mul,abs_neg,abs_of_pos hc,mul_one] using mul_le_mul_of_nonneg_left hy1 hc.le |
| obtain ⟨zm,hm0,hm2,hmb,hem⟩ := hinv (ρ c) c (z 1) t (c*z 0) |
| hpos.1 hc.le hcut.le (by linarith) ht1 hy hkx |
| obtain ⟨zp,hp0,hp2',hpb,hep⟩ := hinv (ρ c) c (z 0) (1-t) (-c*z 1) |
| hpos.1 hc.le hcut.le (by linarith) (by linarith) hx hky |
| have hem0 := congrArg (fun w : Plane => w 0) hem |
| have hem1 := congrArg (fun w : Plane => w 1) hem |
| have hep0 := congrArg (fun w : Plane => w 0) hep |
| have hep1 := congrArg (fun w : Plane => w 1) hep |
| change z 1 = zm 0+(c*z 0)*singularField p zm 0 at hem0 |
| change t = zm 1+(c*z 0)*singularField p zm 1 at hem1 |
| change z 0 = zp 0+(-c*z 1)*singularField p zp 0 at hep0 |
| change 1-t = zp 1+(-c*z 1)*singularField p zp 1 at hep1 |
| have hceil := hover c (q c) (z 0) (z 1) zm zp hc.le hpos.2.le hc1.le hcK.le herr hx1 hy1 hm2 hp2' hmb hpb |
| (by linarith only [hem0]) (by linarith only [hep0]) (by nlinarith only [hem1,hep1]) |
| rcases hceil with hm | hp' |
| · apply Or.inl ∘ Or.inl ∘ Or.inl |
| refine ⟨zm,⟨hm0,hm2,hmb,hm⟩,z 0,hx1,?_⟩ |
| ext i |
| fin_cases i |
| · rfl |
| · change z 1 = zm 0+z 0*(c*singularField p zm 0) |
| nlinarith only [hem0] |
| · change z 2 = Real.sqrt 2*(zm 1+z 0*(c*singularField p zm 1)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [hem1] |
| · apply Or.inl ∘ Or.inl ∘ Or.inr |
| refine ⟨zp,⟨hp0,hp2',hpb,hp'⟩,z 1,hy1,?_⟩ |
| ext i |
| fin_cases i |
| · change z 0 = zp 0-c*z 1*singularField p zp 0 |
| nlinarith only [hep0] |
| · rfl |
| · change z 2 = Real.sqrt 2*(1-(zp 1-c*z 1*singularField p zp 1)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [hep1] |
|
|
| end CylinderCovering.FlexibleSingular |
|
|
| end |
|
|
| end OAI |
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