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[Merged by Bors] - perf(analysis/calculus/specific_functions): speed up exp_neg_inv_glue.f_aux_deriv #16812

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[Merged by Bors] - perf(analysis/calculus/specific_functions): speed up exp_neg_inv_glue.f_aux_deriv #16812

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WIP
alreadydone Oct 4, 2022
2e7dfec
golf `f_aux_deriv`
alreadydone Oct 5, 2022
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13 changes: 8 additions & 5 deletions src/analysis/calculus/specific_functions.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -64,7 +64,7 @@ derivatives for `x > 0`. The `n`-th derivative is of the form `P_aux n (x) exp(-
where `P_aux n` is computed inductively. -/
noncomputable def P_aux : ℕ → polynomial ℝ
| 0 := 1
| (n+1) := X^2 * (P_aux n).derivative + (1 - C ↑(2 * n) * X) * (P_aux n)
| (n+1) := X^2 * (P_aux n).derivative + (1 - C ↑(2 * n) * X) * (P_aux n)

/-- Formula for the `n`-th derivative of `exp_neg_inv_glue`, as an auxiliary function `f_aux`. -/
def f_aux (n : ℕ) (x : ℝ) : ℝ :=
Expand All @@ -86,13 +86,16 @@ lemma f_aux_deriv (n : ℕ) (x : ℝ) (hx : x ≠ 0) :
has_deriv_at (λx, (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n))
((P_aux (n+1)).eval x * exp (-x⁻¹) / x^(2 * (n + 1))) x :=
begin
have A : ∀ k : ℕ, 2 * (k + 1) - 1 = 2 * k + 1 := λ k, rfl,
simp only [P_aux, eval_add, eval_sub, eval_mul, eval_pow, eval_X, eval_C, eval_one],
convert (((P_aux n).has_deriv_at x).mul
(((has_deriv_at_exp _).comp x (has_deriv_at_inv hx).neg))).div
(has_deriv_at_pow (2 * n) x) (pow_ne_zero _ hx) using 1,
field_simp [hx, P_aux],
-- `ring_exp` can't solve `p ∨ q` goal generated by `mul_eq_mul_right_iff`
cases n; simp [nat.succ_eq_add_one, A, -mul_eq_mul_right_iff]; ring_exp
rw div_eq_div_iff,
{ have := pow_ne_zero 2 hx, field_simp only,
cases n,
{ simp only [mul_zero, nat.cast_zero, mul_one], ring },
{ rw (id rfl : 2 * n.succ - 1 = 2 * n + 1), ring_exp } },
all_goals { apply_rules [pow_ne_zero] },
end

/-- For positive values, the derivative of the `n`-th auxiliary function `f_aux n`
Expand Down