| import OAI.Probability.GaussianPropeller.CapCertificates |
|
|
| namespace OAI |
|
|
| 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 |
|
|
| namespace GaussianPropeller.FourAlgebra |
| open scoped RealInnerProductSpace |
|
|
| lemma quartic_positive {t L c₀ c₁ c₂ c₄:ℝ} (ht:0≤t) (htL:t≤L) (hL:0<L) |
| (hc₄:0≤c₄) (hconc:c₂+3*c₄*L^2≤0) (hc₀:0<c₀) |
| (hend:0<c₀+c₁*L+c₂*L^2+c₄*L^4) : 0<c₀+c₁*t+c₂*t^2+c₄*t^4 := by |
| have ht2 : t^2≤L^2 := (sq_le_sq₀ ht hL.le).mpr htL |
| have htL' : L*t≤L^2 := by nlinarith only [mul_le_mul_of_nonneg_left htL hL.le] |
| have hb : 0≤-c₂-c₄*(t^2+L*t+L^2) := by |
| have hh := mul_le_mul_of_nonneg_left (show t^2+L*t+L^2≤3*L^2 by linarith only [ht2,htL']) hc₄ |
| linarith only [hh,hconc] |
| have hp:=mul_nonneg (mul_nonneg ht (sub_nonneg.mpr htL)) hb |
| have hid : L*(c₀+c₁*t+c₂*t^2+c₄*t^4) = |
| (L-t)*c₀+t*(c₀+c₁*L+c₂*L^2+c₄*L^4)+L*(t*(L-t)*(-c₂-c₄*(t^2+L*t+L^2))) := by ring |
| have hlin : 0<(L-t)*c₀+t*(c₀+c₁*L+c₂*L^2+c₄*L^4) := by |
| by_cases htz:t=0 |
| · subst t; simpa using mul_pos hL hc₀ |
| · exact add_pos_of_nonneg_of_pos (mul_nonneg (sub_nonneg.mpr htL) hc₀.le) |
| (mul_pos (lt_of_le_of_ne ht (Ne.symm htz)) hend) |
| have hm:=mul_nonneg hL.le hp |
| nlinarith only [hid,hlin,hm,hL] |
|
|
| lemma small_numeric_one {t:ℝ} (ht:t∈Set.Icc (0:ℝ) (39/250:ℝ)) : |
| 4*(150/157:ℝ)^2*t^2*(13/10:ℝ)^4 < |
| (1-2*t^2)^2-(10/3:ℝ)*((10003/40000:ℝ)-(83/200:ℝ)*t+(81/250:ℝ)*t^2) := by |
| have hh:=quartic_positive (c₀:=(1997/12000:ℝ)) (c₁:=(83/60:ℝ)) |
| (c₂:=-(9556648/616225:ℝ)) (c₄:=4) ht.1 ht.2 |
| (by norm_num) (by norm_num) (by norm_num) (by norm_num) (by norm_num) |
| nlinarith only [hh] |
|
|
| lemma small_numeric_two {t:ℝ} (ht:t∈Set.Icc (0:ℝ) (39/250:ℝ)) : |
| 4*(150/157:ℝ)^2*t^2*(4/3:ℝ)^4 < |
| (1-(14/9:ℝ)*t^2)^2-(10/3:ℝ)*((10003/40000:ℝ)-(83/200:ℝ)*t+(81/250:ℝ)*t^2) := by |
| have hh:=quartic_positive (c₀:=(1997/12000:ℝ)) (c₁:=(83/60:ℝ)) |
| (c₂:=-(87244007/5546025:ℝ)) (c₄:=(196/81:ℝ)) ht.1 ht.2 |
| (by norm_num) (by norm_num) (by norm_num) (by norm_num) (by norm_num) |
| nlinarith only [hh] |
|
|
| lemma pair_product {a b B:ℝ} (ha:0≤a) (hb:0≤b) (hab:a+b≤B) : |
| (1+a)^2*(1+b)^2≤(1+B/2)^4 := by |
| have hB:0≤B := (add_nonneg ha hb).trans hab |
| have hh : (1+a)*(1+b)≤(1+B/2)^2 := by |
| have hs:(a+b)^2≤B^2 := (sq_le_sq₀ (add_nonneg ha hb) hB).mpr hab |
| nlinarith only [hs,hab,sq_nonneg (a-b)] |
| have hq:((1+a)*(1+b))^2≤((1+B/2)^2)^2 := |
| (sq_le_sq₀ (by positivity) (sq_nonneg _)).mpr hh |
| nlinarith only [hq] |
|
|
| lemma small_core {t a b c H V D:ℝ} (ht:t∈Set.Icc (0:ℝ) (39/250:ℝ)) |
| (ha:0≤a) (hb:0≤b) (hc:0≤c) (hs:a+b+c=1) (hac:a≤c) (hbc:b≤c) |
| (hH:H=1-t^2-(a^2+b^2+c^2)*t^2) |
| (hV:(3/5:ℝ)*V≤(10003/40000:ℝ)-(83/200:ℝ)*t+(81/250:ℝ)*t^2) |
| (hD:4*D=H^2-2*V) |
| (hpair:D≤(3/Real.pi)^2*t^2*(1+a)^2*(1+b)^2) : False := by |
| have hk : (3/Real.pi)^2≤(150/157:ℝ)^2 := by |
| apply (sq_le_sq₀ (by positivity) (by norm_num)).mpr |
| rw [div_le_iff₀ Real.pi_pos] |
| have hp:=Real.pi_gt_d2 |
| linarith only [hp] |
| have hab : a+b≤2/3 := by linarith only [hac,hbc,hs] |
| have ht2 : t^2≤(39/250:ℝ)^2 := (sq_le_sq₀ ht.1 (by norm_num)).mpr ht.2 |
| have hcoef : a^2+b^2+c^2≤1 := by |
| nlinarith only [hs,mul_nonneg ha hb,mul_nonneg ha hc,mul_nonneg hb hc] |
| have hpair' : D≤(150/157:ℝ)^2*t^2*(1+a)^2*(1+b)^2 := hpair.trans |
| (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hk (sq_nonneg t)) (sq_nonneg (1+a))) (sq_nonneg (1+b))) |
| by_cases hsmall : a+b≤3/5 |
| · have hprod := pair_product ha hb hsmall |
| have hd : D≤(150/157:ℝ)^2*t^2*(13/10:ℝ)^4 := by |
| have hh:=mul_le_mul_of_nonneg_left hprod (mul_nonneg (sq_nonneg (150/157:ℝ)) (sq_nonneg t)) |
| norm_num at hh |
| nlinarith only [hpair',hh] |
| have hl : 1-2*t^2≤H := by |
| have hh:=mul_le_mul_of_nonneg_right hcoef (sq_nonneg t) |
| nlinarith only [hh,hH] |
| have hl0 : 0≤1-2*t^2 := by nlinarith only [ht2] |
| have hsq : (1-2*t^2)^2≤H^2 := (sq_le_sq₀ hl0 (hl0.trans hl)).mpr hl |
| have hn:=small_numeric_one ht |
| nlinarith only [hV,hD,hd,hsq,hn] |
| · have hablo : 3/5≤a+b := (lt_of_not_ge hsmall).le |
| have hcoef' : a^2+b^2+c^2≤5/9 := by |
| have hh:=mul_nonneg (sub_nonneg.mpr hab) (show 0≤2*(a+b)-2/3 by linarith only [hablo]) |
| nlinarith only [hs,hh,mul_nonneg ha hb] |
| have hprod := pair_product ha hb hab |
| have hd : D≤(150/157:ℝ)^2*t^2*(4/3:ℝ)^4 := by |
| have hh:=mul_le_mul_of_nonneg_left hprod (mul_nonneg (sq_nonneg (150/157:ℝ)) (sq_nonneg t)) |
| norm_num at hh |
| nlinarith only [hpair',hh] |
| have hl : 1-(14/9:ℝ)*t^2≤H := by |
| have hh:=mul_le_mul_of_nonneg_right hcoef' (sq_nonneg t) |
| nlinarith only [hh,hH] |
| have hl0 : 0≤1-(14/9:ℝ)*t^2 := by nlinarith only [ht2] |
| have hsq : (1-(14/9:ℝ)*t^2)^2≤H^2 := (sq_le_sq₀ hl0 (hl0.trans hl)).mpr hl |
| have hn:=small_numeric_two ht |
| nlinarith only [hV,hD,hd,hsq,hn] |
| |
| lemma small_exclusion {t a b c H V D:ℝ} (ht:t∈Set.Icc (0:ℝ) (39/250:ℝ)) |
| (ha:0≤a) (hb:0≤b) (hc:0≤c) (hs:a+b+c=1) |
| (hH:H=1-t^2-(a^2+b^2+c^2)*t^2) |
| (hV:(3/5:ℝ)*V≤(10003/40000:ℝ)-(83/200:ℝ)*t+(81/250:ℝ)*t^2) |
| (hD:4*D=H^2-2*V) |
| (hab:D≤(3/Real.pi)^2*t^2*(1+a)^2*(1+b)^2) |
| (hac:D≤(3/Real.pi)^2*t^2*(1+a)^2*(1+c)^2) |
| (hbc:D≤(3/Real.pi)^2*t^2*(1+b)^2*(1+c)^2) : False := by |
| by_cases hac':a≤c |
| · by_cases hbc':b≤c |
| · exact small_core ht ha hb hc hs hac' hbc' hH hV hD hab |
| · apply small_core (a:=a) (b:=c) (c:=b) ht ha hc hb (by linarith only [hs]) |
| (hac'.trans (le_of_not_ge hbc')) (le_of_not_ge hbc') |
| (by nlinarith only [hH]) hV hD hac |
| · by_cases hba:b≤a |
| · exact small_core (a:=b) (b:=c) (c:=a) ht hb hc ha (by linarith only [hs]) hba |
| (le_of_not_ge hac') (by nlinarith only [hH]) hV hD hbc |
| · exact small_core (a:=a) (b:=c) (c:=b) ht ha hc hb (by linarith only [hs]) |
| (le_of_not_ge hba) ((le_of_not_ge hac').trans (le_of_not_ge hba)) |
| (by nlinarith only [hH]) hV hD hac |
|
|
| end GaussianPropeller.FourAlgebra |
|
|
| end OAI |
|
|
