-- Mathlib/RingTheory/Polynomial/Bernstein.lean
theorem sum_mul_smul (n : ℕ) :
    (∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν) =
      (n * (n - 1)) • X ^ 2

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
w :
  ∀ (k : ℕ),
    (k * (k - 1)) • bernsteinPolynomial R n k =
      ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν =
    ∑ i ∈ Finset.range (n + 1),
      (MvPolynomial.aeval e) ((pderiv true) ((pderiv true) (x ^ i * y ^ (n - i) * ↑(n.choose i)))) * X ^ 2

Tactic:
refine Finset.sum_congr rfl fun k _ => (w k).trans ?_

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
w :
  ∀ (k : ℕ),
    (k * (k - 1)) • bernsteinPolynomial R n k =
      ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2
k : ℕ
x✝ : k ∈ Finset.range (n + 1)
⊢ ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2 =
    (MvPolynomial.aeval e) ((pderiv true) ((pderiv true) (x ^ k * y ^ (n - k) * ↑(n.choose k)))) * X ^ 2

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
⊢ Pi.single true 1 false = 0

Tactic:
rfl

Result:
no goals

Proof state:
case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
k : ℕ
⊢ ((k + 1 + 1) * (k + 1 + 1 - 1)) • (↑(n.choose (k + 1 + 1)) * X ^ (k + 1 + 1) * (1 - X) ^ (n - (k + 1 + 1))) =
    ↑(n.choose (k + 1 + 1)) *
        ((1 - X) ^ (n - (k + 1 + 1)) * (↑(k + 1 + 1) * (↑(k + 1 + 1 - 1) * X ^ (k + 1 + 1 - 1 - 1)))) *
      X ^ 2

Tactic:
simp only [← natCast_mul, Nat.succ_eq_add_one, Nat.add_succ_sub_one, add_zero, pow_succ]

Result:
case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
k : ℕ
⊢ ↑((k + 1 + 1) * (k + 1)) * (↑(n.choose (k + 1 + 1)) * (X ^ k * X * X) * (1 - X) ^ (n - (k + 1 + 1))) =
    ↑(n.choose (k + 1 + 1)) * ((1 - X) ^ (n - (k + 1 + 1)) * (↑(k + 1 + 1) * (↑(k + 1) * X ^ k))) * (X ^ 0 * X * X)

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
w :
  ∀ (k : ℕ),
    (k * (k - 1)) • bernsteinPolynomial R n k =
      ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν =
    (MvPolynomial.aeval e) ((pderiv true) ((pderiv true) ((x + y) ^ n))) * X ^ 2

Tactic:
rw [add_pow, map_sum (pderiv true), map_sum (pderiv true), map_sum (MvPolynomial.aeval e),
  Finset.sum_mul]

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
w :
  ∀ (k : ℕ),
    (k * (k - 1)) • bernsteinPolynomial R n k =
      ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν =
    ∑ i ∈ Finset.range (n + 1),
      (MvPolynomial.aeval e) ((pderiv true) ((pderiv true) (x ^ i * y ^ (n - i) * ↑(n.choose i)))) * X ^ 2

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • X ^ 2

Tactic:
have pderiv_true_x : pderiv true x = 1 := by rw [pderiv_X]; rfl

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • X ^ 2

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ (pderiv true) x = 1

Tactic:
rw [pderiv_X]

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
⊢ Pi.single true 1 true = 1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
⊢ ∀ (k : ℕ),
    (k * (k - 1)) • bernsteinPolynomial R n k =
      ↑(n.choose k) * ((1 - X) ^ (n - k) * (↑k * (↑(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2

Tactic:
rintro (_ | _ | k)

Result:
case zero
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
⊢ (0 * (0 - 1)) • bernsteinPolynomial R n 0 =
    ↑(n.choose 0) * ((1 - X) ^ (n - 0) * (↑0 * (↑(0 - 1) * X ^ (0 - 1 - 1)))) * X ^ 2

case succ.zero
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
⊢ ((0 + 1) * (0 + 1 - 1)) • bernsteinPolynomial R n (0 + 1) =
    ↑(n.choose (0 + 1)) * ((1 - X) ^ (n - (0 + 1)) * (↑(0 + 1) * (↑(0 + 1 - 1) * X ^ (0 + 1 - 1 - 1)))) * X ^ 2

case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i => bif i then X else 1 - X
k : ℕ
⊢ ((k + 1 + 1) * (k + 1 + 1 - 1)) • bernsteinPolynomial R n (k + 1 + 1) =
    ↑(n.choose (k + 1 + 1)) *
        ((1 - X) ^ (n - (k + 1 + 1)) * (↑(k + 1 + 1) * (↑(k + 1 + 1 - 1) * X ^ (k + 1 + 1 - 1 - 1)))) *
      X ^ 2