Exact source context

import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
import Mathlib.Tactic

/-!
# Finite power series: exact derivatives and coefficient budgets

This file packages a small analytic tool used throughout statistics and
numerical analysis.  A finitely supported real power series has an exact
`iteratedDeriv` formula at every order.  On the interval `[-1, 1]`, the
absolute value of that derivative is bounded by the corresponding weighted
absolute coefficient mass.

The formulation uses `Nat.descFactorial`, so one theorem handles every order,
including exponents below the requested order: their descending factorial is
zero.  The final lemma records the scalar fourth-order chain-rule estimate
with the exact one-dimensional Faà di Bruno coefficients.
-/

namespace AutoSamplingTheory.TechnicalLemmas.Analysis.FinitePowerSeries

open scoped BigOperators

/-- A real polynomial written as a sum over an arbitrary finite set of
exponents.  Repeated exponents are impossible because the support is a
`Finset`; coefficients outside the support play no role. -/
def finitePowerSeries (support : Finset ℕ) (a : ℕ → ℝ) (u : ℝ) : ℝ :=
  ∑ j ∈ support, a j * u ^ j

/-- The closed form for the `order`-th derivative of a finite power series. -/
def finitePowerSeriesDerivative
    (order : ℕ) (support : Finset ℕ) (a : ℕ → ℝ) (u : ℝ) : ℝ :=
  ∑ j ∈ support,
    (j.descFactorial order : ℝ) * a j * u ^ (j - order)

/-- Absolute coefficient mass controlling the `order`-th derivative on
`[-1, 1]`. -/
def finitePowerSeriesDerivativeMass
    (order : ℕ) (support : Finset ℕ) (a : ℕ → ℝ) : ℝ :=
  ∑ j ∈ support, (j.descFactorial order : ℝ) * |a j|

/-- A finite real power series is smooth to every finite order. -/
theorem finitePowerSeries_contDiff
    (support : Finset ℕ) (a : ℕ → ℝ) :
    ContDiff ℝ ⊤ (finitePowerSeries support a) := by
  unfold finitePowerSeries
  fun_prop

/-- Mathlib's iterated derivative of a finite power series equals the usual
descending-factorial formula. -/
theorem iteratedDeriv_finitePowerSeries
    (order : ℕ) (support : Finset ℕ) (a : ℕ → ℝ) (u : ℝ) :
    iteratedDeriv order (finitePowerSeries support a) u =
      finitePowerSeriesDerivative order support a u := by
  unfold finitePowerSeries finitePowerSeriesDerivative
  rw [iteratedDeriv_fun_sum]
  · apply Finset.sum_congr rfl
    intro j hj
    simp [iteratedDeriv_pow]
    ring
  · intro j hj
    fun_prop

/-- On the full correlation interval, every iterated derivative is controlled
by its weighted absolute coefficient mass.  No sign condition on the
coefficients is required. -/
theorem abs_iteratedDeriv_finitePowerSeries_le_mass
    (order : ℕ) (support : Finset ℕ) (a : ℕ → ℝ) {u : ℝ}
    (hu : |u| ≤ 1) :
    |iteratedDeriv order (finitePowerSeries support a) u| ≤
      finitePowerSeriesDerivativeMass order support a := by
  rw [iteratedDeriv_finitePowerSeries]
  unfold finitePowerSeriesDerivative finitePowerSeriesDerivativeMass
  calc
    |∑ j ∈ support,
        (j.descFactorial order : ℝ) * a j * u ^ (j - order)| ≤
        ∑ j ∈ support,
          |(j.descFactorial order : ℝ) * a j * u ^ (j - order)| :=
      Finset.abs_sum_le_sum_abs _ _
    _ = ∑ j ∈ support,
        (j.descFactorial order : ℝ) * |a j| * |u| ^ (j - order) := by
      apply Finset.sum_congr rfl
      intro j hj
      rw [abs_mul, abs_mul, abs_pow,
        abs_of_nonneg (show 0 ≤ (j.descFactorial order : ℝ) by positivity)]
    _ ≤ ∑ j ∈ support, (j.descFactorial order : ℝ) * |a j| := by
      apply Finset.sum_le_sum
      intro j hj
      exact mul_le_of_le_one_right
        (mul_nonneg (by positivity) (abs_nonneg (a j)))
        (pow_le_one₀ (abs_nonneg u) hu)

/-- Termwise control of the scalar fourth-order chain-rule expression.

For smooth scalar functions `k` and `u`, the left side is the absolute value
of the formal expression for `(k ∘ u)⁽⁴⁾`.  The numerical coefficients
`1, 6, 3, 4, 1` are the exact one-dimensional Faà di Bruno coefficients. -/
theorem fourthOrderCompositionExpression_abs_le
    {k1 k2 k3 k4 u1 u2 u3 u4 A1 A2 A3 A4 : ℝ}
    (hk1 : |k1| ≤ A1) (hk2 : |k2| ≤ A2)
    (hk3 : |k3| ≤ A3) (hk4 : |k4| ≤ A4) :
    |k4 * u1 ^ (4 : ℕ) +
        6 * k3 * u1 ^ (2 : ℕ) * u2 +
        3 * k2 * u2 ^ (2 : ℕ) +
        4 * k2 * u1 * u3 + k1 * u4| ≤
      A4 * |u1| ^ (4 : ℕ) +
        6 * A3 * |u1| ^ (2 : ℕ) * |u2| +
        3 * A2 * |u2| ^ (2 : ℕ) +
        4 * A2 * |u1| * |u3| + A1 * |u4| := by
  let t1 := k4 * u1 ^ (4 : ℕ)
  let t2 := 6 * k3 * u1 ^ (2 : ℕ) * u2
  let t3 := 3 * k2 * u2 ^ (2 : ℕ)
  let t4 := 4 * k2 * u1 * u3
  let t5 := k1 * u4
  have htri : |t1 + t2 + t3 + t4 + t5| ≤
      |t1| + |t2| + |t3| + |t4| + |t5| := by
    calc
      |t1 + t2 + t3 + t4 + t5| ≤ |t1 + t2 + t3 + t4| + |t5| :=
        abs_add_le _ _
      _ ≤ (|t1 + t2 + t3| + |t4|) + |t5| := by
        gcongr
        exact abs_add_le _ _
      _ ≤ ((|t1 + t2| + |t3|) + |t4|) + |t5| := by
        gcongr
        exact abs_add_le _ _
      _ ≤ (((|t1| + |t2|) + |t3|) + |t4|) + |t5| := by
        gcongr
        exact abs_add_le _ _
  have ht1 : |t1| ≤ A4 * |u1| ^ (4 : ℕ) := by
    dsimp [t1]
    rw [abs_mul, abs_pow]
    exact mul_le_mul_of_nonneg_right hk4 (by positivity)
  have ht2 : |t2| ≤ 6 * A3 * |u1| ^ (2 : ℕ) * |u2| := by
    dsimp [t2]
    rw [abs_mul, abs_mul, abs_mul, abs_pow]
    norm_num
    gcongr
  have ht3 : |t3| ≤ 3 * A2 * |u2| ^ (2 : ℕ) := by
    dsimp [t3]
    rw [abs_mul, abs_mul, abs_pow]
    norm_num
    gcongr
  have ht4 : |t4| ≤ 4 * A2 * |u1| * |u3| := by
    dsimp [t4]
    rw [abs_mul, abs_mul, abs_mul]
    norm_num
    gcongr
  have ht5 : |t5| ≤ A1 * |u4| := by
    dsimp [t5]
    rw [abs_mul]
    exact mul_le_mul_of_nonneg_right hk1 (abs_nonneg _)
  dsimp [t1, t2, t3, t4, t5] at htri
  linarith

end AutoSamplingTheory.TechnicalLemmas.Analysis.FinitePowerSeries