-- Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean
theorem aeval_sumIDeriv (p : R[X]) (q : ℕ) :
    ∃ gp : R[X], gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ p.map (algebraMap R A) →
        aeval r (sumIDeriv p) = q ! • aeval r gp

Proof state:
case refine_2
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (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 * p'
⊢ (aeval r) (sumIDeriv p) = q ! • (aeval r) ((range (p.natDegree + 1)).sum c)

Tactic:
rw [sumIDeriv_apply, map_sum]

Result:
case refine_2
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (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 * p'
⊢ ∑ x ∈ range (p.natDegree + 1), (aeval r) ((⇑derivative)^[x] p) = q ! • (aeval r) ((range (p.natDegree + 1)).sum c)

Proof state:
case h
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k < q
⊢ natDegree 0 ≤ p.natDegree - q ∧
    ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) 0

Tactic:
rw [natDegree_zero]

Result:
case h
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k < q
⊢ 0 ≤ p.natDegree - q ∧
    ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) 0

Proof state:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
h :
  ∀ (k : ℕ),
    (c k).natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) gp

Tactic:
choose c_le hc using h

Result:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) gp

Proof state:
case h
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k < q
⊢ 0 ≤ p.natDegree - q ∧
    ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) 0

Tactic:
use Nat.zero_le _

Result:
case right
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k < q
⊢ ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) 0

Proof state:
case inr
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k ≥ q
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp

Tactic:
obtain ⟨gp, gp_le, h⟩ := aeval_iterate_derivative_of_ge A p q hk

Result:
case inr.intro.intro
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
hk : k ≥ q
gp : R[X]
gp_le : gp.natDegree ≤ p.natDegree - k
h : ∀ (r : A), (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp

Proof state:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) gp

Tactic:
refine ⟨(range (p.natDegree + 1)).sum c, ?_, ?_⟩

Result:
case refine_1
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ((range (p.natDegree + 1)).sum c).natDegree ≤ p.natDegree - q

case refine_2
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ∀ (r : A),
    (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) ((range (p.natDegree + 1)).sum c)

Proof state:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q k : ℕ
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp

Tactic:
cases lt_or_ge k q with
| inl hk =>
  use 0
  rw [natDegree_zero]
  use Nat.zero_le _
  intro r ⟨p', hp⟩
  rw [map_zero, smul_zero, aeval_iterate_derivative_of_lt p q r hp hk]
| inr hk =>
  obtain ⟨gp, gp_le, h⟩ := aeval_iterate_derivative_of_ge A p q hk
  exact ⟨gp, gp_le.trans (tsub_le_tsub_left hk _), fun r _ => h r⟩

Result:
no goals

Proof state:
case refine_1
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
c_le : ∀ (k : ℕ), (c k).natDegree ≤ p.natDegree - q
hc : ∀ (k : ℕ) (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ 0 ≤ p.natDegree - q ∧ ∀ x ∈ range (p.natDegree + 1), (natDegree ∘ c) x ≤ p.natDegree - q

Tactic:
exact ⟨Nat.zero_le _, fun i _ => c_le i⟩

Result:
no goals

Proof state:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
h :
  ∀ (k : ℕ),
    ∃ gp,
      gp.natDegree ≤ p.natDegree - q ∧
        ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) gp
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) gp

Tactic:
choose c h using h

Result:
R : Type u_1
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
p : R[X]
q : ℕ
c : ℕ → R[X]
h :
  ∀ (k : ℕ),
    (c k).natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) ((⇑derivative)^[k] p) = q ! • (aeval r) (c k)
⊢ ∃ gp,
    gp.natDegree ≤ p.natDegree - q ∧
      ∀ (r : A), (X - C r) ^ q ∣ map (algebraMap R A) p → (aeval r) (sumIDeriv p) = q ! • (aeval r) gp