| import OAI.Probability.GaussianPropeller.Taylor |
|
|
| 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 |
|
|
| namespace GaussianPropeller.ProbabilityBounds |
| open OneCell Mills Taylor |
| noncomputable def p (x : ℝ) : ℝ := gaussianConst*tail x |
|
|
| lemma gaussianConst_tight : (398942/1000000:ℝ) ≤ gaussianConst ∧ gaussianConst ≤ 398943/1000000 := by |
| refine ⟨?_, gaussianConst_bounds.2⟩ |
| have hs := Real.sq_sqrt (show 0 ≤ 2*Real.pi by positivity) |
| have hspos : 0 < Real.sqrt (2*Real.pi) := by positivity |
| rw [gaussianConst, inv_eq_one_div, le_div_iff₀ hspos] |
| have hh := Real.pi_lt_d6 |
| nlinarith |
|
|
| lemma p_pos (x : ℝ) : 0 < p x := |
| mul_pos (lt_of_lt_of_le (by norm_num) gaussianConst_tight.1) (tail_pos x) |
|
|
| lemma hasDerivAt_p (x : ℝ) : HasDerivAt p (-gaussianConst*density x) x := by |
| convert (hasDerivAt_tail x).const_mul gaussianConst using 1 <;> first | rfl | ring |
|
|
| lemma p_antitone : Antitone p := by |
| apply antitone_of_hasDerivAt_nonpos (fun x => hasDerivAt_p x) |
| intro x |
| exact mul_nonpos_of_nonpos_of_nonneg (by linarith [gaussianConst_tight.1]) (Real.exp_pos _).le |
|
|
| lemma p_le_half {x : ℝ} (hx : 0 ≤ x) : p x ≤ 1/2 := by |
| have hh := p_antitone hx |
| simpa only [p,tail_zero] using hh |
|
|
| lemma p_taylor_lower {x : ℝ} (hx : 0 ≤ x) (n : ℕ) (hJ : 0 ≤ J (2*n) x) : |
| 1/2-(398943/1000000:ℝ)*J (2*n) x ≤ p x := by |
| have hh := (integral_exp_bounds n hx).2 |
| change (∫ t in (0:ℝ)..x, density t) ≤ J (2*n) x at hh |
| have hmul := mul_le_mul_of_nonneg_left hh |
| (lt_of_lt_of_le (by norm_num) gaussianConst_tight.1).le |
| have hmul' := mul_le_mul_of_nonneg_right gaussianConst_tight.2 hJ |
| unfold p |
| rw [tail_sub hx, mul_sub, tail_zero] |
| linarith only [hmul,hmul'] |
|
|
| lemma p_taylor_upper {x : ℝ} (hx : 0 ≤ x) (n : ℕ) (hJ : 0 ≤ J (2*n+1) x) : |
| p x ≤ 1/2-(398942/1000000:ℝ)*J (2*n+1) x := by |
| have hh := (integral_exp_bounds n hx).1 |
| change J (2*n+1) x ≤ ∫ t in (0:ℝ)..x, density t at hh |
| have hmul := mul_le_mul_of_nonneg_left hh |
| (lt_of_lt_of_le (by norm_num) gaussianConst_tight.1).le |
| have hmul' := mul_le_mul_of_nonneg_right gaussianConst_tight.1 hJ |
| unfold p |
| rw [tail_sub hx, mul_sub, tail_zero] |
| linarith only [hmul,hmul'] |
|
|
| lemma lambda_taylor_upper {x : ℝ} (hx : 0 ≤ x) (n : ℕ) {L : ℝ} (hL : 0 ≤ L) |
| (hJ : 0 ≤ J (2*n) x) |
| (hc : (398943/1000000:ℝ)*E (2*n) (x^2/2) ≤ |
| L*(1/2-(398943/1000000:ℝ)*J (2*n) x)) : lambda x ≤ L := by |
| have he := exp_le_E_even n (show 0 ≤ x^2/2 by positivity) |
| rw [← neg_div] at he |
| change density x ≤ E (2*n) (x^2/2) at he |
| have hd : 0 ≤ density x := (Real.exp_pos _).le |
| have hmul := mul_le_mul he gaussianConst_tight.2 |
| (le_trans (by norm_num) gaussianConst_tight.1) (hd.trans he) |
| have ht := mul_le_mul_of_nonneg_left (p_taylor_lower hx n hJ) hL |
| have hpos : 0 < gaussianConst := lt_of_lt_of_le (by norm_num) gaussianConst_tight.1 |
| rw [lambda, div_le_iff₀ (tail_pos x)] |
| apply (mul_le_mul_iff_right₀ hpos).mp |
| dsimp [p] at ht |
| nlinarith only [hmul,hc,ht] |
|
|
| lemma G_upper_of_lambda {x L U : ℝ} (hx : 0 ≤ x) (h : lambda x ≤ L) |
| (hLU : L*(L-x) ≤ U) : G x ≤ U := by |
| have hl : 0 ≤ lambda x-x := (mul_nonneg_iff_of_pos_left (lambda_pos x)).mp (G_nonneg hx) |
| have hh := mul_nonneg (sub_nonneg.mpr h) |
| (show 0 ≤ L+lambda x-x by linarith [lambda_pos x]) |
| unfold G |
| nlinarith only [hh,hLU] |
|
|
| lemma certificate_005 : p (2.1) < (0.415+0.15*0.05)*0.05 ∧ G (2.1) ≤ 0.91 := by |
| have hj12 : 0 ≤ J 12 (2.1) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (2.1) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 2.1) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (2.1) < (0.415+0.15*0.05)*0.05 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 2.1) (L := 2.463) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_014 : p (1.6) < (0.415+0.15*0.14)*0.14 ∧ G (1.6) ≤ 0.87 := by |
| have hj12 : 0 ≤ J 12 (1.6) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (1.6) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 1.6) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (1.6) < (0.415+0.15*0.14)*0.14 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 1.6) (L := 2.025) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_024 : p (1.3) < (0.415+0.15*0.24)*0.24 ∧ G (1.3) ≤ 0.836 := by |
| have hj12 : 0 ≤ J 12 (1.3) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (1.3) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 1.3) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (1.3) < (0.415+0.15*0.24)*0.24 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 1.3) (L := 1.771) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_033 : p (1.04) < (0.415+0.15*0.33)*0.33 ∧ G (1.04) ≤ 0.809 := by |
| have hj12 : 0 ≤ J 12 (1.04) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (1.04) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 1.04) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (1.04) < (0.415+0.15*0.33)*0.33 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 1.04) (L := 1.558) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_041 : p (0.87) < (0.415+0.15*0.41)*0.41 ∧ G (0.87) ≤ 0.788 := by |
| have hj12 : 0 ≤ J 12 (0.87) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (0.87) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 0.87) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (0.87) < (0.415+0.15*0.41)*0.41 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 0.87) (L := 1.423) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_048 : p (0.74) < (0.415+0.15*0.48)*0.48 ∧ G (0.74) ≤ 0.771 := by |
| have hj12 : 0 ≤ J 12 (0.74) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (0.74) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 0.74) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (0.74) < (0.415+0.15*0.48)*0.48 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 0.74) (L := 1.322) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_054 : p (0.63) < (0.415+0.15*0.54)*0.54 ∧ G (0.63) ≤ 0.756 := by |
| have hj12 : 0 ≤ J 12 (0.63) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (0.63) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 0.63) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (0.63) < (0.415+0.15*0.54)*0.54 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 0.63) (L := 1.238) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_059 : p (0.55) < (0.415+0.15*0.59)*0.59 ∧ G (0.55) ≤ 0.743 := by |
| have hj12 : 0 ≤ J 12 (0.55) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (0.55) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 0.55) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (0.55) < (0.415+0.15*0.59)*0.59 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 0.55) (L := 1.178) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma certificate_064 : p (0.47) < (0.415+0.15*0.64)*0.64 ∧ G (0.47) ≤ 0.73 := by |
| have hj12 : 0 ≤ J 12 (0.47) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| have hj13 : 0 ≤ J 13 (0.47) := by norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| constructor |
| · have ht := p_taylor_upper (by norm_num : (0:ℝ) ≤ 0.47) 6 hj13 |
| have hc : 1/2-(398942/1000000:ℝ)*J 13 (0.47) < (0.415+0.15*0.64)*0.64 := by |
| norm_num [J, Finset.sum_range_succ, Nat.factorial] |
| exact lt_of_le_of_lt ht hc |
| · apply G_upper_of_lambda (by norm_num : (0:ℝ) ≤ 0.47) (L := 1.120) |
| · apply lambda_taylor_upper (by norm_num) 6 (by norm_num) hj12 |
| norm_num [E,J, Finset.sum_range_succ, Nat.factorial] |
| · norm_num |
|
|
| lemma exists_p {P : ℝ} (hP : 0 < P) (hPh : P ≤ 1/2) : ∃ x ≥ (0:ℝ), p x=P := by |
| have ht : Tendsto p atTop (𝓝 0) := by |
| convert! (tendsto_integral_Ioi_zero (f := density) (μ := volume) tendsto_id).const_mul gaussianConst using 1 |
| simp only [mul_zero] |
| have hh := isPreconnected_Ici.intermediate_value_Ioc (a := (0:ℝ)) self_mem_Ici |
| (le_principal_iff.mpr (Ici_mem_atTop (0:ℝ))) |
| (show ContinuousOn p (Ici 0) from fun x _ => (hasDerivAt_p x).continuousAt.continuousWithinAt) ht |
| have hm : P ∈ Ioc 0 (p 0) := by exact ⟨hP, by simpa [p,tail_zero] using hPh⟩ |
| exact hh hm |
|
|
| noncomputable def loss (x : ℝ) : ℝ := p x^2*G x |
|
|
| lemma loss_eq (x : ℝ) : loss x=gaussianConst^2*density x*(density x-x*tail x) := by |
| unfold loss p G lambda |
| field_simp [(tail_pos x).ne'] |
| |
| lemma hasDerivAt_loss (x : ℝ) : HasDerivAt loss |
| (gaussianConst^2*density x*(x^2*tail x-x*density x-tail x)) x := by |
| have he : loss = fun x => gaussianConst^2*density x*(density x-x*tail x) := funext loss_eq |
| rw [he] |
| convert ((hasDerivAt_density x).const_mul (gaussianConst^2)).mul |
| ((hasDerivAt_density x).sub ((hasDerivAt_id x).mul (hasDerivAt_tail x))) using 1 <;> |
| first | rfl | (simp only [Pi.sub_apply, Pi.mul_apply, id_eq]; ring) |
| |
| lemma loss_antitone : AntitoneOn loss (Ici 0) := by |
| apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ici 0) |
| (fun x _ => (hasDerivAt_loss x).continuousAt.continuousWithinAt) |
| (fun x _ => (hasDerivAt_loss x).hasDerivWithinAt) |
| intro x hx |
| have hx' : 0 ≤ x := interior_subset hx |
| have hl : x ≤ lambda x := by |
| have hh := (mul_nonneg_iff_of_pos_left (lambda_pos x)).mp (G_nonneg hx') |
| linarith only [hh] |
| rw [lambda, le_div_iff₀ (tail_pos x)] at hl |
| apply mul_nonpos_of_nonneg_of_nonpos (mul_nonneg (sq_nonneg _) (Real.exp_pos _).le) |
| have hh := mul_le_mul_of_nonneg_left hl hx' |
| have ht : 0 < tail x := tail_pos x |
| have hm : 0 ≤ x*density x := mul_nonneg hx' (Real.exp_pos _).le |
| nlinarith only [hh,ht,hm] |
| |
| lemma G_test_bound {a r x w U : ℝ} (ha : 0 ≤ a) (har : a ≤ r) |
| (hx : 0 ≤ x) (hw : 0 ≤ w) (hp : p x=(0.415+0.15*r)*r) |
| (hc : p w < (0.415+0.15*a)*a ∧ G w ≤ U) : G x ≤ U := by |
| have hpw : p w < p x := by |
| rw [hp] |
| have hm := mul_nonneg (sub_nonneg.mpr har) (show 0 ≤ r+a by linarith) |
| nlinarith only [hc.1,har,hm] |
| have hxw : x ≤ w := by |
| by_contra! hgt |
| exact (not_lt_of_ge (p_antitone hgt.le)) hpw |
| exact (G_monotone hx hw hxw).trans hc.2 |
| |
| lemma row_margin {a b r x U S : ℝ} (ha : 0 ≤ a) (har : a ≤ r) (hrb : r ≤ b) |
| (hb : b ≤ 2/3) (hU : 0 ≤ U) (hG : G x ≤ U) (hS : 0 ≤ S) |
| (hSsq : 1-a^2 ≤ S^2) |
| (hc : U*(0.415+0.15*b)^2 < (9/(8*3.1416:ℝ))/(1+S)) : |
| (0.415+0.15*r)^2*G x < (9/(8*Real.pi))/(1+Real.sqrt (1-r^2)) := by |
| have hr : 0 ≤ r := ha.trans har |
| have hr2 : r^2 ≤ (1:ℝ) := by nlinarith only [hr,hrb,hb] |
| have hs := Real.sq_sqrt (show 0 ≤ 1-r^2 by linarith only [hr2]) |
| have hsp := Real.sqrt_nonneg (1-r^2) |
| have hsq : Real.sqrt (1-r^2) ≤ S := by |
| have hm := mul_nonneg (sub_nonneg.mpr har) (show 0 ≤ r+a by positivity) |
| nlinarith only [hs,hsp,hSsq,hS,hm] |
| have hbcoef : (0.415+0.15*r)^2 ≤ (0.415+0.15*b)^2 := by |
| nlinarith only [hr,hrb, sq_nonneg (b-r)] |
| have hh₁ := mul_le_mul_of_nonneg_left hG (sq_nonneg (0.415+0.15*r)) |
| have hh₂ := mul_le_mul_of_nonneg_right hbcoef hU |
| have hB : (9/(8*3.1416:ℝ)) ≤ 9/(8*Real.pi) := |
| div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith [Real.pi_lt_d4]) |
| have hd₁ := div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 9/(8*3.1416)) |
| (show 0 < 1+Real.sqrt (1-r^2) by positivity) (show 1+Real.sqrt (1-r^2) ≤ 1+S by linarith only [hsq]) |
| have hd₂ := div_le_div_of_nonneg_right hB (show 0 ≤ 1+Real.sqrt (1-r^2) by positivity) |
| have hc' : (0.415+0.15*b)^2*U < (9/(8*3.1416:ℝ))/(1+S) := by simpa only [mul_comm] using hc |
| exact lt_of_le_of_lt (hh₁.trans hh₂) (lt_of_lt_of_le hc' (hd₁.trans hd₂)) |
| |
| lemma test_probability {r x : ℝ} (hr : 0 < r) (hrcap : r ≤ 2/3) (hx : 0 ≤ x) |
| (hp : p x=(0.415+0.15*r)*r) : |
| (0.415+0.15*r)^2*G x < (9/(8*Real.pi))/(1+Real.sqrt (1-r^2)) := by |
| by_cases h0 : r ≤ 0.05 |
| · apply row_margin (a := 0) (b := 0.05) (U := 1) (S := 1) |
| · norm_num |
| · exact hr.le |
| · exact h0 |
| · norm_num |
| · norm_num |
| · exact G_le_one x |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h1 : r ≤ 0.14 |
| · apply row_margin (a := 0.05) (b := 0.14) (U := 0.91) (S := 0.99875) |
| · norm_num |
| · exact (lt_of_not_ge h0).le |
| · exact h1 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h0).le hx (by norm_num) hp certificate_005 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h2 : r ≤ 0.24 |
| · apply row_margin (a := 0.14) (b := 0.24) (U := 0.87) (S := 0.99016) |
| · norm_num |
| · exact (lt_of_not_ge h1).le |
| · exact h2 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h1).le hx (by norm_num) hp certificate_014 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h3 : r ≤ 0.33 |
| · apply row_margin (a := 0.24) (b := 0.33) (U := 0.836) (S := 0.97078) |
| · norm_num |
| · exact (lt_of_not_ge h2).le |
| · exact h3 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h2).le hx (by norm_num) hp certificate_024 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h4 : r ≤ 0.41 |
| · apply row_margin (a := 0.33) (b := 0.41) (U := 0.809) (S := 0.94399) |
| · norm_num |
| · exact (lt_of_not_ge h3).le |
| · exact h4 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h3).le hx (by norm_num) hp certificate_033 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h5 : r ≤ 0.48 |
| · apply row_margin (a := 0.41) (b := 0.48) (U := 0.788) (S := 0.91209) |
| · norm_num |
| · exact (lt_of_not_ge h4).le |
| · exact h5 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h4).le hx (by norm_num) hp certificate_041 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h6 : r ≤ 0.54 |
| · apply row_margin (a := 0.48) (b := 0.54) (U := 0.771) (S := 0.87727) |
| · norm_num |
| · exact (lt_of_not_ge h5).le |
| · exact h6 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h5).le hx (by norm_num) hp certificate_048 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h7 : r ≤ 0.59 |
| · apply row_margin (a := 0.54) (b := 0.59) (U := 0.756) (S := 0.84167) |
| · norm_num |
| · exact (lt_of_not_ge h6).le |
| · exact h7 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h6).le hx (by norm_num) hp certificate_054 |
| · norm_num |
| · norm_num |
| · norm_num |
| by_cases h8 : r ≤ 0.64 |
| · apply row_margin (a := 0.59) (b := 0.64) (U := 0.743) (S := 0.80741) |
| · norm_num |
| · exact (lt_of_not_ge h7).le |
| · exact h8 |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h7).le hx (by norm_num) hp certificate_059 |
| · norm_num |
| · norm_num |
| · norm_num |
| apply row_margin (a := 0.64) (b := (2/3)) (U := 0.73) (S := 0.76838) |
| · norm_num |
| · exact (lt_of_not_ge h8).le |
| · exact hrcap |
| · norm_num |
| · norm_num |
| · exact G_test_bound (by norm_num) (lt_of_not_ge h8).le hx (by norm_num) hp certificate_064 |
| · norm_num |
| · norm_num |
| · norm_num |
| |
| lemma linear_quadratic_of_loss {r x : ℝ} (hr : 0 < r) (hrcap : r ≤ 2/3) (hx : 0 ≤ x) |
| (hc : (9/(8*Real.pi))*r^2/(1+Real.sqrt (1-r^2)) ≤ loss x) : |
| 0.415*r+0.15*r^2 < p x := by |
| let P₀ := (0.415+0.15*r)*r |
| have hP₀ : 0 < P₀ := by dsimp [P₀]; positivity |
| have hP₀h : P₀ ≤ 1/2 := by dsimp [P₀]; nlinarith only [hr,hrcap] |
| obtain ⟨y,hy,hPy⟩ := exists_p hP₀ hP₀h |
| have ht := test_probability hr hrcap hy hPy |
| have ht' := mul_lt_mul_of_pos_right ht (sq_pos_of_pos hr) |
| have htest : loss y < (9/(8*Real.pi))*r^2/(1+Real.sqrt (1-r^2)) := by |
| unfold loss |
| rw [hPy] |
| dsimp [P₀] at * |
| calc |
| _ = (0.415+0.15*r)^2*G y*r^2 := by ring |
| _ < (9/(8*Real.pi))/(1+Real.sqrt (1-r^2))*r^2 := ht' |
| _ = _ := by ring |
| by_contra! hh |
| have hpx : p x ≤ p y := by rw [hPy]; dsimp [P₀]; nlinarith only [hh] |
| have hyx : y ≤ x := by |
| by_contra! hgt |
| have hd : StrictAnti p := strictAnti_of_hasDerivAt_neg (fun z => hasDerivAt_p z) |
| (fun z => mul_neg_of_neg_of_pos (by linarith [gaussianConst_tight.1]) (Real.exp_pos _)) |
| exact (not_lt_of_ge hpx) (hd hgt) |
| have hh' := loss_antitone hy hx hyx |
| exact (not_lt_of_ge (hc.trans hh')) htest |
| |
| end GaussianPropeller.ProbabilityBounds |
| |
| end OAI |
| |
