feat(analysis/calculus/specific_functions): smoothness of exp(-1/x) (#…

…2087)

* feat(analysis/calculus/specific_functions): smoothness of exp(-1/x)

* use namespace; shorter names

* fix field_simp

Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>

src/algebra/field.lean

@@ -170,10 +170,18 @@ lemma add_div' (a b c : α) (hc : c ≠ 0) :
   b + a / c = (b * c + a) / c :=
 by simpa using div_add_div b a one_ne_zero hc
 
+lemma sub_div' (a b c : α) (hc : c ≠ 0) :
+  b - a / c = (b * c - a) / c :=
+by simpa using div_sub_div b a one_ne_zero hc
+
 lemma div_add' (a b c : α) (hc : c ≠ 0) :
   a / c + b = (a + b * c) / c :=
 by rwa [add_comm, add_div', add_comm]
 
+lemma div_sub' (a b c : α) (hc : c ≠ 0) :
+  a / c - b = (a - c * b) / c :=
+by simpa using div_sub_div a b hc one_ne_zero
+
 end
 
 section
@@ -275,6 +283,6 @@ by simp [neg_div]
 
 attribute [field_simps] div_add_div_same inv_eq_one_div div_mul_eq_mul_div div_add' add_div'
 div_div_eq_div_mul mul_div_assoc' div_eq_div_iff div_eq_iff eq_div_iff mul_ne_zero'
-div_div_eq_mul_div neg_div' two_ne_zero
+div_div_eq_mul_div neg_div' two_ne_zero div_sub_div div_sub' sub_div'
 
 end field_simp

src/analysis/calculus/specific_functions.lean

@@ -0,0 +1,185 @@
+/-
+Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sébastien Gouëzel
+-/
+
+import analysis.calculus.extend_deriv analysis.calculus.iterated_deriv analysis.complex.exponential
+
+/-!
+# Smoothness of specific functions
+
+The real function `exp_neg_inv_glue` given by `x ↦ exp (-1/x)` for `x > 0` and `0`
+for `x ≤ 0` is a basic building block to construct smooth partitions of unity. We prove that it
+is `C^∞` in `exp_neg_inv_glue.smooth`.
+-/
+
+noncomputable theory
+open_locale classical topological_space
+
+open polynomial real filter set
+
+/-- `exp_neg_inv_glue` is the real function given by `x ↦ exp (-1/x)` for `x > 0` and `0`
+for `x ≤ 0`. is a basic building block to construct smooth partitions of unity. Its main property
+is that it vanishes for `x ≤ 0`, it is positive for `x > 0`, and the junction between the two
+behaviors is flat enough to retain smoothness. The fact that this function is `C^∞` is proved in
+`exp_neg_inv_glue.smooth`. -/
+def exp_neg_inv_glue (x : ℝ) : ℝ := if x ≤ 0 then 0 else exp (-x⁻¹)
+
+namespace exp_neg_inv_glue
+
+/-- Our goal is to prove that `exp_neg_inv_glue` is `C^∞`. For this, we compute its successive
+derivatives for `x > 0`. The `n`-th derivative is of the form `P_aux n (x) exp(-1/x) / x^(2 n)`,
+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)
+
+/-- 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 :=
+if x ≤ 0 then 0 else (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n)
+
+/-- The `0`-th auxiliary function `f_aux 0` coincides with `exp_neg_inv_glue`, by definition. -/
+lemma f_aux_zero_eq : f_aux 0 = exp_neg_inv_glue :=
+begin
+   ext x,
+   by_cases h : x ≤ 0,
+   { simp [exp_neg_inv_glue, f_aux, h] },
+   { simp [h, exp_neg_inv_glue, f_aux, ne_of_gt (not_le.1 h), P_aux] }
+end
+
+/-- For positive values, the derivative of the `n`-th auxiliary function `f_aux n`
+(given in this statement in unfolded form) is the `n+1`-th auxiliary function, since
+the polynomial `P_aux (n+1)` was chosen precisely to ensure this. -/
+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, by omega,
+  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],
+  cases n; simp [nat.succ_eq_add_one, A]; ring_exp
+end
+
+/-- For positive values, the derivative of the `n`-th auxiliary function `f_aux n`
+is the `n+1`-th auxiliary function. -/
+lemma f_aux_deriv_pos (n : ℕ) (x : ℝ) (hx : 0 < x) :
+  has_deriv_at (f_aux n) ((P_aux (n+1)).eval x * exp (-x⁻¹) / x^(2 * (n + 1))) x :=
+begin
+  apply (f_aux_deriv n x (ne_of_gt hx)).congr_of_mem_nhds,
+  have : Ioi (0 : ℝ) ∈ 𝓝 x := lt_mem_nhds hx,
+  filter_upwards [this],
+  assume y hy,
+  have : ¬(y ≤ 0), by simpa using hy,
+  simp [f_aux, this]
+end
+
+/-- To get differentiability at `0` of the auxiliary functions, we need to know that their limit
+is `0`, to be able to apply general differentiability extension theorems. This limit is checked in
+this lemma. -/
+lemma f_aux_limit (n : ℕ) :
+  tendsto (λx, (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n)) (nhds_within 0 (Ioi 0)) (𝓝 0) :=
+begin
+  have A : tendsto (λx, (P_aux n).eval x) (nhds_within 0 (Ioi 0)) (𝓝 ((P_aux n).eval 0)) :=
+  (P_aux n).continuous_within_at,
+  have B : tendsto (λx, exp (-x⁻¹) / x^(2 * n)) (nhds_within 0 (Ioi 0)) (𝓝 0),
+  { convert (tendsto_pow_mul_exp_neg_at_top_nhds_0 (2 * n)).comp tendsto_inv_zero_at_top,
+    ext x,
+    field_simp },
+  convert A.mul B;
+  simp [mul_div_assoc]
+end
+
+/-- Deduce from the limiting behavior at `0` of its derivative and general differentiability
+extension theorems that the auxiliary function `f_aux n` is differentiable at `0`,
+with derivative `0`. -/
+lemma f_aux_deriv_zero (n : ℕ) : has_deriv_at (f_aux n) 0 0 :=
+begin
+  -- we check separately differentiability on the left and on the right
+  have A : has_deriv_within_at (f_aux n) (0 : ℝ) (Iic 0) 0,
+  { apply (has_deriv_at_const (0 : ℝ) (0 : ℝ)).has_deriv_within_at.congr,
+    { assume y hy,
+      simp at hy,
+      simp [f_aux, hy] },
+    { simp [f_aux, le_refl] } },
+  have B : has_deriv_within_at (f_aux n) (0 : ℝ) (Ici 0) 0,
+  { have diff : differentiable_on ℝ (f_aux n) (Ioi 0) :=
+      λx hx, (f_aux_deriv_pos n x hx).differentiable_at.differentiable_within_at,
+    -- next line is the nontrivial bit of this proof, appealing to differentiability
+    -- extension results.
+    apply has_deriv_at_interval_left_endpoint_of_tendsto_deriv diff _ self_mem_nhds_within,
+    { refine (f_aux_limit (n+1)).congr' _,
+      apply mem_sets_of_superset self_mem_nhds_within (λx hx, _),
+      simp [(f_aux_deriv_pos n x hx).deriv] },
+    { have : f_aux n 0 = 0, by simp [f_aux, le_refl],
+      simp only [continuous_within_at, this],
+      refine (f_aux_limit n).congr' _,
+      apply mem_sets_of_superset self_mem_nhds_within (λx hx, _),
+      have : ¬(x ≤ 0), by simpa using hx,
+      simp [f_aux, this] } },
+  simpa using A.union B,
+end
+
+/-- At every point, the auxiliary function `f_aux n` has a derivative which is
+equal to `f_aux (n+1)`. -/
+lemma f_aux_has_deriv_at (n : ℕ) (x : ℝ) : has_deriv_at (f_aux n) (f_aux (n+1) x) x :=
+begin
+  -- check separately the result for `x < 0`, where it is trivial, for `x > 0`, where it is done
+  -- in `f_aux_deriv_pos`, and for `x = 0`, done in
+  -- `f_aux_deriv_zero`.
+  rcases lt_trichotomy x 0 with hx|hx|hx,
+  { have : f_aux (n+1) x = 0, by simp [f_aux, le_of_lt hx],
+    rw this,
+    apply (has_deriv_at_const x (0 : ℝ)).congr_of_mem_nhds,
+    have : Iio (0 : ℝ) ∈ 𝓝 x := gt_mem_nhds hx,
+    filter_upwards [this],
+    assume y hy,
+    simp [f_aux, le_of_lt hy] },
+  { have : f_aux (n + 1) 0 = 0, by simp [f_aux, le_refl],
+    rw [hx, this],
+    exact f_aux_deriv_zero n },
+  { have : f_aux (n+1) x = (P_aux (n+1)).eval x * exp (-x⁻¹) / x^(2 * (n+1)),
+      by simp [f_aux, not_le_of_gt hx],
+    rw this,
+    exact f_aux_deriv_pos n x hx },
+end
+
+/-- The successive derivatives of the auxiliary function `f_aux 0` are the
+functions `f_aux n`, by induction. -/
+lemma f_aux_iterated_deriv (n : ℕ) : iterated_deriv n (f_aux 0) = f_aux n :=
+begin
+  induction n with n IH,
+  { simp },
+  { simp [iterated_deriv_succ, IH],
+    ext x,
+    exact (f_aux_has_deriv_at n x).deriv }
+end
+
+/-- The function `exp_neg_inv_glue` is smooth. -/
+theorem smooth : times_cont_diff ℝ ⊤ (exp_neg_inv_glue) :=
+begin
+  rw ← f_aux_zero_eq,
+  apply times_cont_diff_of_differentiable_iterated_deriv (λ m hm, _),
+  rw f_aux_iterated_deriv m,
+  exact λ x, (f_aux_has_deriv_at m x).differentiable_at
+end
+
+/-- The function `exp_neg_inv_glue` vanishes on `(-∞, 0]`. -/
+lemma zero_of_nonpos {x : ℝ} (hx : x ≤ 0) : exp_neg_inv_glue x = 0 :=
+by simp [exp_neg_inv_glue, hx]
+
+/-- The function `exp_neg_inv_glue` is positive on `(0, +∞)`. -/
+lemma pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < exp_neg_inv_glue x :=
+by simp [exp_neg_inv_glue, not_le.2 hx, exp_pos]
+
+/-- The function exp_neg_inv_glue` is nonnegative. -/
+lemma nonneg (x : ℝ) : 0 ≤ exp_neg_inv_glue x :=
+begin
+  cases le_or_gt x 0,
+  { exact ge_of_eq (zero_of_nonpos h) },
+  { exact le_of_lt (pos_of_pos h) }
+end
+
+end exp_neg_inv_glue

test/ring.lean

@@ -32,7 +32,7 @@ begin
 end
 
 example (a b c d x y : ℚ) (hx : x ≠ 0) (hy : y ≠ 0) :
-  a + b / x + c / x^2 + d / x^3 = a + x⁻¹ * (y * b / y + (d / x + c) / x) :=
+  a + b / x - c / x^2 + d / x^3 = a + x⁻¹ * (y * b / y + (d / x - c) / x) :=
 begin
   field_simp [hx, hy],
   ring