Theorem: bernsteinPolynomial.variance
File: Mathlib/RingTheory/Polynomial/Bernstein.lean

Proof steps:
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
state_after: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
tactic: have p : ((((Finset.range (n + 1)).sum fun ν => (ν * (ν - 1)) • bernsteinPolynomial R n ν) +
      (1 - (2 * n) • Polynomial.X) * (Finset.range (n + 1)).sum fun ν =>
        ν • bernsteinPolynomial R n ν) + n ^ 2 • X ^ 2 *
          (Finset.range (n + 1)).sum fun ν => bernsteinPolynomial R n ν) = _ :=
    rfl
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
state_after: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
tactic: conv at p =>
    lhs
    rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
    simp only [← natCast_mul]
    simp only [← mul_assoc]
    simp only [← add_mul]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
state_after: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
tactic: conv at p =>
    rhs
    rw [sum, sum_smul, sum_mul_smul, ← natCast_mul]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
⊢ ∑ ν ∈ Finset.range (n + 1), (n • X - ↑ν) ^ 2 * bernsteinPolynomial R n ν = n • X * (1 - X)
state_after: case calc_1
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
k : ℕ
m : k ∈ Finset.range (n + 1)
⊢ (n • X - ↑k) ^ 2 * bernsteinPolynomial R n k =
    (↑(k * (k - 1)) + (1 - ↑(2 * n) * X) * ↑k + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n k

case calc_2
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ x ∈ Finset.range (n + 1),
      (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =
    ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1
⊢ ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1 = n • X * (1 - X)
tactic: calc
    _ = _ := Finset.sum_congr rfl fun k m => ?_
    _ = _ := p
    _ = _ := ?_