| import OAI.Probability.GaussianPropeller.QuarticCertificates |
|
|
| 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.FourCertificates |
| open OneCell ProbabilityBounds Mills Taylor |
|
|
| lemma small_0 {r x:ℝ} (hr:r∈Icc (39/250:ℝ) (157/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (157/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (9139/6250:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (9139/6250:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (157/1000:ℝ) ≤ p (9139/6250:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (9139/6250:ℝ)≤(449/6250:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (9139/6250:ℝ)≤(953337/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (9139/6250:ℝ)≤(847391/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(39/250:ℝ)) (q:=(9139/6250:ℝ)) (v:=(983643/1000000:ℝ)) |
| (u:=(449/6250:ℝ)^2*(847391/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_1 {r x:ℝ} (hr:r∈Icc (157/1000:ℝ) (79/500:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (79/500:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (72937/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (72937/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (79/500:ℝ) ≤ p (72937/50000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (72937/50000:ℝ)≤(72321/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (72937/50000:ℝ)≤(1903709/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (72937/50000:ℝ)≤(211773/250000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(157/1000:ℝ)) (q:=(72937/50000:ℝ)) (v:=(983431/1000000:ℝ)) |
| (u:=(72321/1000000:ℝ)^2*(211773/250000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_2 {r x:ℝ} (hr:r∈Icc (79/500:ℝ) (159/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (159/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (5821/4000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (5821/4000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (159/1000:ℝ) ≤ p (5821/4000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (5821/4000:ℝ)≤(72803/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (5821/4000:ℝ)≤(950377/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (5821/4000:ℝ)≤(423397/500000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(79/500:ℝ)) (q:=(5821/4000:ℝ)) (v:=(983217/1000000:ℝ)) |
| (u:=(72803/1000000:ℝ)^2*(423397/500000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_3 {r x:ℝ} (hr:r∈Icc (159/1000:ℝ) (4/25:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (4/25:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (72589/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (72589/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (4/25:ℝ) ≤ p (72589/50000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (72589/50000:ℝ)≤(18321/250000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (72589/50000:ℝ)≤(1897817/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (72589/50000:ℝ)≤(846497/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(159/1000:ℝ)) (q:=(72589/50000:ℝ)) (v:=(491501/500000:ℝ)) |
| (u:=(18321/250000:ℝ)^2*(846497/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_4 {r x:ℝ} (hr:r∈Icc (4/25:ℝ) (161/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (161/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (4526/3125:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (4526/3125:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (161/1000:ℝ) ≤ p (4526/3125:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (4526/3125:ℝ)≤(36883/500000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (4526/3125:ℝ)≤(1894889/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (4526/3125:ℝ)≤(846199/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(4/25:ℝ)) (q:=(4526/3125:ℝ)) (v:=(491393/500000:ℝ)) |
| (u:=(36883/500000:ℝ)^2*(846199/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_5 {r x:ℝ} (hr:r∈Icc (161/1000:ℝ) (81/500:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (81/500:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (144487/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (144487/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (81/500:ℝ) ≤ p (144487/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (144487/100000:ℝ)≤(297/4000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (144487/100000:ℝ)≤(189197/100000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (144487/100000:ℝ)≤(8459/10000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(161/1000:ℝ)) (q:=(144487/100000:ℝ)) (v:=(122821/125000:ℝ)) |
| (u:=(297/4000:ℝ)^2*(8459/10000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_6 {r x:ℝ} (hr:r∈Icc (81/500:ℝ) (163/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (163/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (28829/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (28829/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (163/1000:ℝ) ≤ p (28829/20000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (28829/20000:ℝ)≤(74731/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (28829/20000:ℝ)≤(944539/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (28829/20000:ℝ)≤(169121/200000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(81/500:ℝ)) (q:=(28829/20000:ℝ)) (v:=(982349/1000000:ℝ)) |
| (u:=(74731/1000000:ℝ)^2*(169121/200000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_7 {r x:ℝ} (hr:r∈Icc (163/1000:ℝ) (41/250:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (41/250:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (143803/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (143803/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (41/250:ℝ) ≤ p (143803/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (143803/100000:ℝ)≤(15043/200000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (143803/100000:ℝ)≤(1886187/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (143803/100000:ℝ)≤(211327/250000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(163/1000:ℝ)) (q:=(143803/100000:ℝ)) (v:=(61383/62500:ℝ)) |
| (u:=(15043/200000:ℝ)^2*(211327/250000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_8 {r x:ℝ} (hr:r∈Icc (41/250:ℝ) (33/200:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (33/200:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (143463/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (143463/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (33/200:ℝ) ≤ p (143463/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (143463/100000:ℝ)≤(75699/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (143463/100000:ℝ)≤(941657/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (143463/100000:ℝ)≤(845013/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(41/250:ℝ)) (q:=(143463/100000:ℝ)) (v:=(490953/500000:ℝ)) |
| (u:=(75699/1000000:ℝ)^2*(845013/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_9 {r x:ℝ} (hr:r∈Icc (33/200:ℝ) (83/500:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (83/500:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (229/160:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (229/160:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (83/500:ℝ) ≤ p (229/160:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (229/160:ℝ)≤(38091/500000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (229/160:ℝ)≤(1880459/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (229/160:ℝ)≤(10559/12500:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(33/200:ℝ)) (q:=(229/160:ℝ)) (v:=(981683/1000000:ℝ)) |
| (u:=(38091/500000:ℝ)^2*(10559/12500:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_10 {r x:ℝ} (hr:r∈Icc (83/500:ℝ) (167/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (167/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (35697/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (35697/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (167/1000:ℝ) ≤ p (35697/25000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (35697/25000:ℝ)≤(38333/500000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (35697/25000:ℝ)≤(1877613/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (35697/25000:ℝ)≤(33777/40000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(83/500:ℝ)) (q:=(35697/25000:ℝ)) (v:=(490729/500000:ℝ)) |
| (u:=(38333/500000:ℝ)^2*(33777/40000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_11 {r x:ℝ} (hr:r∈Icc (167/1000:ℝ) (21/125:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (21/125:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (35613/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (35613/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (21/125:ℝ) ≤ p (35613/25000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (35613/25000:ℝ)≤(1543/20000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (35613/25000:ℝ)≤(234347/125000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (35613/25000:ℝ)≤(84413/100000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(167/1000:ℝ)) (q:=(35613/25000:ℝ)) (v:=(61327/62500:ℝ)) |
| (u:=(1543/20000:ℝ)^2*(84413/100000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_12 {r x:ℝ} (hr:r∈Icc (21/125:ℝ) (169/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (169/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (71059/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (71059/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (169/1000:ℝ) ≤ p (71059/50000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (71059/50000:ℝ)≤(15527/200000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (71059/50000:ℝ)≤(935979/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (71059/50000:ℝ)≤(421919/500000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(21/125:ℝ)) (q:=(71059/50000:ℝ)) (v:=(245251/250000:ℝ)) |
| (u:=(15527/200000:ℝ)^2*(421919/500000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_13 {r x:ℝ} (hr:r∈Icc (169/1000:ℝ) (17/100:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (17/100:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (28357/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (28357/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (17/100:ℝ) ≤ p (28357/20000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (28357/20000:ℝ)≤(1953/25000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (28357/20000:ℝ)≤(1869149/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (28357/20000:ℝ)≤(421773/500000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(169/1000:ℝ)) (q:=(28357/20000:ℝ)) (v:=(39231/40000:ℝ)) |
| (u:=(1953/25000:ℝ)^2*(421773/500000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_14 {r x:ℝ} (hr:r∈Icc (17/100:ℝ) (171/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (171/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (70727/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (70727/50000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (171/1000:ℝ) ≤ p (70727/50000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (70727/50000:ℝ)≤(19651/250000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (70727/50000:ℝ)≤(933179/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (70727/50000:ℝ)≤(168651/200000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(17/100:ℝ)) (q:=(70727/50000:ℝ)) (v:=(196109/200000:ℝ)) |
| (u:=(19651/250000:ℝ)^2*(168651/200000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_15 {r x:ℝ} (hr:r∈Icc (171/1000:ℝ) (43/250:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (43/250:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (141123/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (141123/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (43/250:ℝ) ≤ p (141123/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (141123/100000:ℝ)≤(79091/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (141123/100000:ℝ)≤(1863567/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (141123/100000:ℝ)≤(842961/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(171/1000:ℝ)) (q:=(141123/100000:ℝ)) (v:=(980313/1000000:ℝ)) |
| (u:=(79091/1000000:ℝ)^2*(842961/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_16 {r x:ℝ} (hr:r∈Icc (43/250:ℝ) (173/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (173/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (28159/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (28159/20000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (173/1000:ℝ) ≤ p (28159/20000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (28159/20000:ℝ)≤(3183/40000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (28159/20000:ℝ)≤(1860803/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (28159/20000:ℝ)≤(842671/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(43/250:ℝ)) (q:=(28159/20000:ℝ)) (v:=(980079/1000000:ℝ)) |
| (u:=(3183/40000:ℝ)^2*(842671/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_17 {r x:ℝ} (hr:r∈Icc (173/1000:ℝ) (87/500:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (87/500:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (140467/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (140467/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (87/500:ℝ) ≤ p (140467/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (140467/100000:ℝ)≤(40031/500000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (140467/100000:ℝ)≤(46451/25000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (140467/100000:ℝ)≤(42119/50000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(173/1000:ℝ)) (q:=(140467/100000:ℝ)) (v:=(195969/200000:ℝ)) |
| (u:=(40031/500000:ℝ)^2*(42119/50000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_18 {r x:ℝ} (hr:r∈Icc (87/500:ℝ) (7/40:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (7/40:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (140141/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (140141/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (7/40:ℝ) ≤ p (140141/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (140141/100000:ℝ)≤(20137/250000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (140141/100000:ℝ)≤(371059/200000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (140141/100000:ℝ)≤(842091/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(87/500:ℝ)) (q:=(140141/100000:ℝ)) (v:=(979609/1000000:ℝ)) |
| (u:=(20137/250000:ℝ)^2*(842091/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_19 {r x:ℝ} (hr:r∈Icc (7/40:ℝ) (22/125:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (22/125:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (17477/12500:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (17477/12500:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (22/125:ℝ) ≤ p (17477/12500:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (17477/12500:ℝ)≤(16207/200000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (17477/12500:ℝ)≤(1852559/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (17477/12500:ℝ)≤(841801/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(7/40:ℝ)) (q:=(17477/12500:ℝ)) (v:=(979371/1000000:ℝ)) |
| (u:=(16207/200000:ℝ)^2*(841801/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_20 {r x:ℝ} (hr:r∈Icc (22/125:ℝ) (177/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (177/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (34873/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (34873/25000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (177/1000:ℝ) ≤ p (34873/25000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (34873/25000:ℝ)≤(40761/500000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (34873/25000:ℝ)≤(231229/125000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (34873/25000:ℝ)≤(841511/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(22/125:ℝ)) (q:=(34873/25000:ℝ)) (v:=(244783/250000:ℝ)) |
| (u:=(40761/500000:ℝ)^2*(841511/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_21 {r x:ℝ} (hr:r∈Icc (177/1000:ℝ) (89/500:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (89/500:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (13917/10000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (13917/10000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (89/500:ℝ) ≤ p (13917/10000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (13917/10000:ℝ)≤(82009/1000000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (13917/10000:ℝ)≤(1847123/1000000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (13917/10000:ℝ)≤(841223/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(177/1000:ℝ)) (q:=(13917/10000:ℝ)) (v:=(244723/250000:ℝ)) |
| (u:=(82009/1000000:ℝ)^2*(841223/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_22 {r x:ℝ} (hr:r∈Icc (89/500:ℝ) (179/1000:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (179/1000:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (138849/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (138849/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (179/1000:ℝ) ≤ p (138849/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (138849/100000:ℝ)≤(1289/15625:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (138849/100000:ℝ)≤(230553/125000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (138849/100000:ℝ)≤(105117/125000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(89/500:ℝ)) (q:=(138849/100000:ℝ)) (v:=(19573/20000:ℝ)) |
| (u:=(1289/15625:ℝ)^2*(105117/125000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_23 {r x:ℝ} (hr:r∈Icc (179/1000:ℝ) (9/50:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_contra hn |
| have hP : p x ≤ small (9/50:ℝ) := (le_of_not_gt hn).trans (small_mono (by change 0≤r; linarith only [hr.1]) (by norm_num) hr.2) |
| have hj : 0≤J 12 (138529/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hj' : 0≤J 13 (138529/100000:ℝ) := by norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hpL : small (9/50:ℝ) ≤ p (138529/100000:ℝ) := by |
| apply le_trans ?_ (p_taylor_lower (by norm_num) 6 hj) |
| norm_num [small,J,Finset.sum_range_succ,Nat.factorial] |
| have hpU : p (138529/100000:ℝ)≤(10373/125000:ℝ) := by |
| apply le_trans (p_taylor_upper (by norm_num) 6 hj') |
| norm_num [J,Finset.sum_range_succ,Nat.factorial] |
| have hla : lambda (138529/100000:ℝ)≤(920867/500000:ℝ) := by |
| apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj |
| norm_num [E,J,Finset.sum_range_succ,Nat.factorial] |
| have hg : G (138529/100000:ℝ)≤(840649/1000000:ℝ) := G_upper_of_lambda (by norm_num) hla (by norm_num) |
| exact loss_row (lo:=(179/1000:ℝ)) (q:=(138529/100000:ℝ)) (v:=(978407/1000000:ℝ)) |
| (u:=(10373/125000:ℝ)^2*(840649/1000000:ℝ)) (by norm_num) hr.1 hx (by norm_num) (by norm_num) |
| (by norm_num) (by norm_num) (loss_bound (by norm_num) hpU hg) (hP.trans hpL) hl |
|
|
| lemma small_of_loss {r x:ℝ} (hr:r∈Icc (39/250:ℝ) (9/50:ℝ)) (hx:0≤x) |
| (hl:(9/(8*Real.pi))*r^2/(1+Real.sqrt (1-(4/3:ℝ)*r^2))≤loss x) : small r < p x := by |
| by_cases h0 : r≤(157/1000:ℝ) |
| · exact small_0 ⟨by linarith only [hr.1],h0⟩ hx hl |
| by_cases h1 : r≤(79/500:ℝ) |
| · exact small_1 ⟨by linarith only [h0],h1⟩ hx hl |
| by_cases h2 : r≤(159/1000:ℝ) |
| · exact small_2 ⟨by linarith only [h1],h2⟩ hx hl |
| by_cases h3 : r≤(4/25:ℝ) |
| · exact small_3 ⟨by linarith only [h2],h3⟩ hx hl |
| by_cases h4 : r≤(161/1000:ℝ) |
| · exact small_4 ⟨by linarith only [h3],h4⟩ hx hl |
| by_cases h5 : r≤(81/500:ℝ) |
| · exact small_5 ⟨by linarith only [h4],h5⟩ hx hl |
| by_cases h6 : r≤(163/1000:ℝ) |
| · exact small_6 ⟨by linarith only [h5],h6⟩ hx hl |
| by_cases h7 : r≤(41/250:ℝ) |
| · exact small_7 ⟨by linarith only [h6],h7⟩ hx hl |
| by_cases h8 : r≤(33/200:ℝ) |
| · exact small_8 ⟨by linarith only [h7],h8⟩ hx hl |
| by_cases h9 : r≤(83/500:ℝ) |
| · exact small_9 ⟨by linarith only [h8],h9⟩ hx hl |
| by_cases h10 : r≤(167/1000:ℝ) |
| · exact small_10 ⟨by linarith only [h9],h10⟩ hx hl |
| by_cases h11 : r≤(21/125:ℝ) |
| · exact small_11 ⟨by linarith only [h10],h11⟩ hx hl |
| by_cases h12 : r≤(169/1000:ℝ) |
| · exact small_12 ⟨by linarith only [h11],h12⟩ hx hl |
| by_cases h13 : r≤(17/100:ℝ) |
| · exact small_13 ⟨by linarith only [h12],h13⟩ hx hl |
| by_cases h14 : r≤(171/1000:ℝ) |
| · exact small_14 ⟨by linarith only [h13],h14⟩ hx hl |
| by_cases h15 : r≤(43/250:ℝ) |
| · exact small_15 ⟨by linarith only [h14],h15⟩ hx hl |
| by_cases h16 : r≤(173/1000:ℝ) |
| · exact small_16 ⟨by linarith only [h15],h16⟩ hx hl |
| by_cases h17 : r≤(87/500:ℝ) |
| · exact small_17 ⟨by linarith only [h16],h17⟩ hx hl |
| by_cases h18 : r≤(7/40:ℝ) |
| · exact small_18 ⟨by linarith only [h17],h18⟩ hx hl |
| by_cases h19 : r≤(22/125:ℝ) |
| · exact small_19 ⟨by linarith only [h18],h19⟩ hx hl |
| by_cases h20 : r≤(177/1000:ℝ) |
| · exact small_20 ⟨by linarith only [h19],h20⟩ hx hl |
| by_cases h21 : r≤(89/500:ℝ) |
| · exact small_21 ⟨by linarith only [h20],h21⟩ hx hl |
| by_cases h22 : r≤(179/1000:ℝ) |
| · exact small_22 ⟨by linarith only [h21],h22⟩ hx hl |
| exact small_23 ⟨by linarith only [h22],hr.2⟩ hx hl |
|
|
| end GaussianPropeller.FourCertificates |
|
|
| end OAI |
|
|
