Module: Mathlib.Algebra.Polynomial.SumIteratedDerivative

Goal state before:
pp: case inr.refine_2.e_a.e_a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Nontrivial A
inst✝ : NoZeroDivisors A
p : R[X]
q : ℕ
hq : 0 < q
inj_amap : Function.Injective ⇑(algebraMap R A)
p0 : p ≠ 0
c : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - k
hc : ∀ (k : ℕ), q ≤ k → ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ (q - 1) * p'
this✝ : range (p.natDegree + 1) = range q ∪ Ico q (p.natDegree + 1)
this : range q = range (q - 1 + 1)
⊢ ∑ x ∈ range (q - 1), (aeval r) ((⇑derivative)^[x] p) = 0

pp: case inr.refine_2.e_a.e_a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Nontrivial A
inst✝ : NoZeroDivisors A
p : R[X]
q : ℕ
hq : 0 < q
inj_amap : Function.Injective ⇑(algebraMap R A)
p0 : p ≠ 0
c : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - k
hc : ∀ (k : ℕ), q ≤ k → ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ (q - 1) * p'
this✝ : range (p.natDegree + 1) = range q ∪ Ico q (p.natDegree + 1)
this : range q = range (q - 1 + 1)
⊢ (aeval r) ((⇑derivative)^[q - 1] p) = (q - 1)! • eval r p'

pp: case inr.refine_2.e_a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Nontrivial A
inst✝ : NoZeroDivisors A
p : R[X]
q : ℕ
hq : 0 < q
inj_amap : Function.Injective ⇑(algebraMap R A)
p0 : p ≠ 0
c : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - k
hc : ∀ (k : ℕ), q ≤ k → ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ (q - 1) * p'
this✝ : range (p.natDegree + 1) = range q ∪ Ico q (p.natDegree + 1)
this : range q = range (q - 1 + 1)
⊢ ∑ x ∈ Ico q (p.natDegree + 1), (aeval r) ((⇑derivative)^[x] p) = q ! • ∑ x ∈ Ico q (p.natDegree + 1), (aeval r) (c x)

Tactic:
· apply sum_eq_zero
      exact fun x hx => aeval_iterate_derivative_of_lt p _ r hp (mem_range.mp hx)

Goal state after:
case inr.refine_2.e_a.e_a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Nontrivial A
inst✝ : NoZeroDivisors A
p : R[X]
q : ℕ
hq : 0 < q
inj_amap : Function.Injective ⇑(algebraMap R A)
p0 : p ≠ 0
c : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - k
hc : ∀ (k : ℕ), q ≤ k → ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ (q - 1) * p'
this✝ : range (p.natDegree + 1) = range q ∪ Ico q (p.natDegree + 1)
this : range q = range (q - 1 + 1)
⊢ (aeval r) ((⇑derivative)^[q - 1] p) = (q - 1)! • eval r p'

case inr.refine_2.e_a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Nontrivial A
inst✝ : NoZeroDivisors A
p : R[X]
q : ℕ
hq : 0 < q
inj_amap : Function.Injective ⇑(algebraMap R A)
p0 : p ≠ 0
c : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - k
hc : ∀ (k : ℕ), q ≤ k → ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
r : A
p' : A[X]
hp : map (algebraMap R A) p = (X - C r) ^ (q - 1) * p'
this✝ : range (p.natDegree + 1) = range q ∪ Ico q (p.natDegree + 1)
this : range q = range (q - 1 + 1)
⊢ ∑ x ∈ Ico q (p.natDegree + 1), (aeval r) ((⇑derivative)^[x] p) = q ! • ∑ x ∈ Ico q (p.natDegree + 1), (aeval r) (c x)