File size: 3,421 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.Combinatorics.EditApproximation.Geometry.PublishedInputs
namespace OAI
/-! The scalar exponential estimate used to discharge the finite Bernstein
input. Its rational majorant is proved by a derivative comparison, avoiding
any assumption of a concentration theorem. -/
namespace EditApproximation
noncomputable def bernsteinEnvelope (x : ℝ) : ℝ :=
Real.exp (-x) * ((x ^ 2 + 4 * x + 6) / (6 - 2 * x))
private theorem bernsteinEnvelope_deriv (x : ℝ) (hx : x < 3) :
HasDerivAt bernsteinEnvelope (Real.exp (-x) * (2 * x ^ 3 / (6 - 2 * x) ^ 2)) x := by
have hd : (6 : ℝ) - 2 * x ≠ 0 := by linarith
have hn := (((hasDerivAt_pow 2 x).add ((hasDerivAt_id x).const_mul 4)).add_const 6)
have hb := (hasDerivAt_const x (6 : ℝ)).sub ((hasDerivAt_id x).const_mul 2)
have h := ((hasDerivAt_id x).neg.exp).mul (hn.div hb hd)
dsimp only [Pi.mul_apply, Pi.div_apply, Pi.add_apply, Pi.sub_apply, Pi.neg_apply, id_eq] at h
unfold bernsteinEnvelope
convert h using 1
field_simp [hd]
ring
private theorem bernsteinEnvelope_zero : bernsteinEnvelope 0 = 1 := by
norm_num [bernsteinEnvelope]
theorem bernsteinEnvelope_one_le (x : ℝ) (hx : x < 3) : 1 ≤ bernsteinEnvelope x := by
by_cases hpos : 0 ≤ x
· have hc : ContinuousOn bernsteinEnvelope (Set.Icc 0 x) := by
intro y hy
exact (bernsteinEnvelope_deriv y (hy.2.trans_lt hx)).continuousAt.continuousWithinAt
have hm := monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 x) hc
(fun y hy => (bernsteinEnvelope_deriv y ((interior_subset hy).2.trans_lt hx)).hasDerivWithinAt)
(fun y hy => mul_nonneg (Real.exp_nonneg _) (div_nonneg
(mul_nonneg (by norm_num) (pow_nonneg (interior_subset hy).1 3)) (sq_nonneg _)))
simpa only [bernsteinEnvelope_zero] using hm ⟨le_rfl, hpos⟩ ⟨hpos, le_rfl⟩ hpos
· have hneg : x ≤ 0 := le_of_not_ge hpos
have hc : ContinuousOn bernsteinEnvelope (Set.Icc x 0) := by
intro y hy
exact (bernsteinEnvelope_deriv y (by linarith [hy.2])).continuousAt.continuousWithinAt
have hm := antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc x 0) hc
(fun y hy => (bernsteinEnvelope_deriv y (by linarith [(interior_subset hy).2])).hasDerivWithinAt)
(fun y hy => mul_nonpos_of_nonneg_of_nonpos (Real.exp_nonneg _) (div_nonpos_of_nonpos_of_nonneg
(mul_nonpos_of_nonneg_of_nonpos (by norm_num) (show y ^ 3 ≤ 0 by
have h := mul_nonpos_of_nonneg_of_nonpos (sq_nonneg y) (interior_subset hy).2
nlinarith only [h]))
(sq_nonneg _)))
simpa only [bernsteinEnvelope_zero] using hm ⟨le_rfl, hneg⟩ ⟨hneg, le_rfl⟩ hneg
/-- Exact Bernstein exponential majorant, valid for all `x < 3`. -/
theorem exp_le_bernstein_envelope (x : ℝ) (hx : x < 3) :
Real.exp x ≤ 1 + x + x ^ 2 / (2 * (1 - x / 3)) := by
have h := mul_le_mul_of_nonneg_left (bernsteinEnvelope_one_le x hx) (Real.exp_nonneg x)
have hd : (6 : ℝ) - 2 * x ≠ 0 := by linarith
have he : (2 : ℝ) * (1 - x / 3) ≠ 0 := by linarith
have hc : (x ^ 2 + 4 * x + 6) / (6 - 2 * x) = 1 + x + x ^ 2 / (2 * (1 - x / 3)) := by
have h3 : (3 : ℝ) - x ≠ 0 := by linarith
field_simp [hd, he, h3]
field_simp [show (6 : ℝ) - x * 2 ≠ 0 by linarith]
ring
simpa only [bernsteinEnvelope, mul_one, ← mul_assoc, ← Real.exp_add, add_neg_cancel,
Real.exp_zero, one_mul, hc] using h
end EditApproximation
end OAI
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