| import OAI.Probability.GaussianPropeller.SmallCell |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology RealInnerProductSpace |
| open MeasureTheory Set Filter |
| open scoped Topology BigOperators |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology ENNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology ENNReal |
| open Metric |
| open MeasureTheory ProbabilityTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal RealInnerProductSpace |
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| namespace GaussianPropeller.Reduction |
| open FourCertificates ProbabilityBounds |
| open scoped RealInnerProductSpace |
| variable {d:ℕ} |
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| lemma triangle_gram (u v w : Space d) (hs:u+v+w=0) : |
| let D:=‖u‖^2*‖v‖^2-⟪u,v⟫^2 |
| D=‖u‖^2*‖w‖^2-⟪u,w⟫^2 ∧ D=‖v‖^2*‖w‖^2-⟪v,w⟫^2 ∧ |
| 4*D=(‖u‖^2+‖v‖^2+‖w‖^2)^2-2*(‖u‖^4+‖v‖^4+‖w‖^4) := by |
| have hw:w=-(u+v) := eq_neg_of_add_eq_zero_left (by simpa only [add_comm] using hs) |
| subst w |
| simp only [norm_neg,norm_add_sq_real,inner_neg_right,inner_add_right,real_inner_self_eq_norm_sq, |
| real_inner_comm u v] |
| constructor |
| · ring |
| constructor |
| · ring |
| · have h4:‖u+v‖^4=(‖u‖^2+‖v‖^2+2*⟪u,v⟫)^2 := by rw [show (4:ℕ)=2*2 by rfl,pow_mul,norm_add_sq_real]; ring |
| rw [h4] |
| ring |
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| lemma residual_fourth_le (x y:Space d) (hx:x≠0) : |
| ‖residualVector x y‖^4≤‖y‖^4 := by |
| have h:=residual_norm_sq (z:=x) (y:=y) hx |
| have hle:‖residualVector x y‖^2≤‖y‖^2 := by nlinarith only [h,mul_nonneg (sq_nonneg (residualCoefficient x y)) (sq_nonneg ‖x‖)] |
| have hh: (‖residualVector x y‖^2)^2≤(‖y‖^2)^2 := (sq_le_sq₀ (sq_nonneg _) (sq_nonneg _)).mpr hle |
| simpa only [←pow_mul] using hh |
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| lemma four_incompatibility (z:Fin 4→Space d) (P:Fin 4→ℝ) |
| (hsum:∑ j,z j=0) (hnorm:∑ j,‖z j‖^2=1) |
| (hpos:∀ j,z j≠0) (horder:Antitone (fun j=>‖z j‖)) |
| (hneg:Pairwise (fun j l=>⟪z j,z l⟫<0)) (hcap:∀ j,‖z j‖≤2/3) |
| (hprob:∑ j,P j=1) (hPh:∀ j,P j≤1/2) |
| (hcap3:∀ j,‖z j‖≤(8/3:ℝ)*P j*(1-P j)) |
| (hlin:∀ j,0.415*‖z j‖+0.15*‖z j‖^2≤P j) |
| (hloss:∀ j,∃ x≥(0:ℝ),p x=P j ∧ |
| (9/(8*Real.pi))*‖z j‖^2/(1+Real.sqrt (1-(4/3:ℝ)*‖z j‖^2))≤loss x) |
| (hpair:∀ i j l, i≠j → i≠l → |
| ‖residualVector (z i) (z j)‖^2*‖residualVector (z i) (z l)‖^2- |
| ⟪residualVector (z i) (z j),residualVector (z i) (z l)⟫^2≤ |
| (3/Real.pi)^2*‖z i‖^2*(1+residualCoefficient (z i) (z j))^2* |
| (1+residualCoefficient (z i) (z l))^2) : False := by |
| have hn:‖z 0‖^2+‖z 1‖^2+‖z 2‖^2+‖z 3‖^2=1 := by |
| simpa [Fin.sum_univ_succ,add_assoc] using hnorm |
| have hp:P 0+P 1+P 2+P 3=1 := by simpa [Fin.sum_univ_succ,add_assoc] using hprob |
| have hq:∀ j,‖z j‖^2≤(2/3:ℝ)^2 := fun j=>(sq_le_sq₀ (norm_nonneg _) (by norm_num)).mpr (hcap j) |
| have hquartic:∀ j, (147/500:ℝ)≤‖z j‖ → quartic ‖z j‖≤P j := by |
| intro j hj |
| by_cases ht:‖z j‖≤79/200 |
| · obtain ⟨x,hx,hpx,hl⟩:=hloss j |
| rw [←hpx] |
| exact (quartic_of_loss ⟨hj,ht⟩ hx hl).le |
| · exact quartic_from_cap ⟨(le_of_not_ge ht),(hcap j)⟩ (hPh j) (hcap3 j) |
| by_cases ht:‖z 3‖≤39/250 |
| · have ht2:‖z 3‖^2≤(39/250:ℝ)^2 := (sq_le_sq₀ (norm_nonneg _) (by norm_num)).mpr ht |
| have h0:(147/500:ℝ)≤‖z 0‖ := by nlinarith only [hn,hq 1,hq 2,ht2,norm_nonneg (z 0)] |
| have h1:(147/500:ℝ)≤‖z 1‖ := by nlinarith only [hn,hq 0,hq 2,ht2,norm_nonneg (z 1)] |
| have h2:(147/500:ℝ)≤‖z 2‖ := by nlinarith only [hn,hq 0,hq 1,ht2,norm_nonneg (z 2)] |
| let a:=residualCoefficient (z 3) (z 0) |
| let b:=residualCoefficient (z 3) (z 1) |
| let c:=residualCoefficient (z 3) (z 2) |
| let u:=residualVector (z 3) (z 0) |
| let v:=residualVector (z 3) (z 1) |
| let w:=residualVector (z 3) (z 2) |
| have ha:0≤a := (residualCoefficient_pos (hneg (by decide : (3:Fin 4)≠0))).le |
| have hb:0≤b := (residualCoefficient_pos (hneg (by decide : (3:Fin 4)≠1))).le |
| have hc:0≤c := (residualCoefficient_pos (hneg (by decide : (3:Fin 4)≠2))).le |
| have habc:a+b+c=1 := by |
| have hh:=residual_coefficient_sum z hsum 3 (hpos 3) |
| have he:(Finset.univ.erase (3:Fin 4))={0,1,2} := by decide |
| rw [he] at hh |
| simpa [a,b,c,Finset.sum_insert,add_assoc] using hh |
| have hs:u+v+w=0 := by |
| have hh:=residual_sum z hsum 3 (hpos 3) |
| have he:(Finset.univ.erase (3:Fin 4))={0,1,2} := by decide |
| rw [he] at hh |
| simpa [u,v,w,Finset.sum_insert,add_assoc] using hh |
| obtain ⟨hD1,hD2,hD3⟩:=triangle_gram u v w hs |
| have hu:‖u‖^2=‖z 0‖^2-a^2*‖z 3‖^2 := residual_norm_sq (hpos 3) |
| have hv:‖v‖^2=‖z 1‖^2-b^2*‖z 3‖^2 := residual_norm_sq (hpos 3) |
| have hw:‖w‖^2=‖z 2‖^2-c^2*‖z 3‖^2 := residual_norm_sq (hpos 3) |
| have hH:‖u‖^2+‖v‖^2+‖w‖^2=1-‖z 3‖^2-(a^2+b^2+c^2)*‖z 3‖^2 := by |
| nlinarith only [hu,hv,hw,hn] |
| have he0:=hquartic 0 h0 |
| have he1:=hquartic 1 h1 |
| have he2:=hquartic 2 h2 |
| unfold quartic at he0 he1 he2 |
| have hV:(3/5:ℝ)*(‖u‖^4+‖v‖^4+‖w‖^4)≤(10003/40000:ℝ)-(83/200:ℝ)*‖z 3‖+(81/250:ℝ)*‖z 3‖^2 := by |
| nlinarith only [he0,he1,he2,hlin 3,hn,hp, |
| residual_fourth_le (z 3) (z 0) (hpos 3),residual_fourth_le (z 3) (z 1) (hpos 3),residual_fourth_le (z 3) (z 2) (hpos 3)] |
| apply FourAlgebra.small_exclusion ⟨norm_nonneg _,ht⟩ ha hb hc habc hH hV hD3 |
| · exact hpair 3 0 1 (by decide) (by decide) |
| · rw [hD1]; exact hpair 3 0 2 (by decide) (by decide) |
| · rw [hD2]; exact hpair 3 1 2 (by decide) (by decide) |
| have htlo:(39/250:ℝ)≤‖z 3‖ := (lt_of_not_ge ht).le |
| have hmin:∀ j, ‖z 3‖≤‖z j‖ := fun j=>horder (show j≤3 by omega) |
| have haff:∀ j,affine ‖z j‖≤P j := fun j=> affine_from_cap ⟨htlo.trans (hmin j),hcap j⟩ (hPh j) (hcap3 j) |
| have ha0:=haff 0; have ha1:=haff 1; have ha2:=haff 2 |
| unfold affine at ha0 ha1 ha2 |
| by_cases ht18:‖z 3‖≤9/50 |
| · obtain ⟨x,hx,hpx,hl⟩:=hloss 3 |
| have hsmall:=small_of_loss ⟨htlo,ht18⟩ hx hl |
| rw [hpx] at hsmall |
| unfold small at hsmall |
| have hprod:=mul_nonneg (sub_nonneg.mpr htlo) (sub_nonneg.mpr ht18) |
| nlinarith only [ha0,ha1,ha2,hsmall,hn,hp,hprod,htlo,ht18] |
| by_cases ht40:‖z 3‖≤2/5 |
| · have hlo:(9/50:ℝ)≤‖z 3‖ := (lt_of_not_ge ht18).le |
| have hprod:=mul_nonneg (sub_nonneg.mpr hlo) (sub_nonneg.mpr ht40) |
| nlinarith only [ha0,ha1,ha2,hlin 3,hn,hp,hprod,hlo,ht40] |
| have hlarge:∀ j,large ‖z j‖≤P j := by |
| intro j |
| have hj:(2/5:ℝ)≤‖z j‖ := (le_of_not_ge ht40).trans (hmin j) |
| by_cases hj525:‖z j‖≤21/40 |
| · obtain ⟨x,hx,hpx,hl⟩:=hloss j |
| rw [←hpx] |
| exact (large_of_loss ⟨hj,hj525⟩ hx hl).le |
| · exact large_from_cap ⟨le_of_not_ge hj525,hcap j⟩ (hPh j) (hcap3 j) |
| have hl0:=hlarge 0; have hl1:=hlarge 1; have hl2:=hlarge 2; have hl3:=hlarge 3 |
| unfold large at hl0 hl1 hl2 hl3 |
| nlinarith only [hl0,hl1,hl2,hl3,hn,hp] |
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| end GaussianPropeller.Reduction |
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| end OAI |
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