-- Mathlib/Dynamics/Newton.lean
theorem aeval_pow_two_pow_dvd_aeval_iterate_newtonMap
    (h : IsNilpotent (aeval x P)) (h' : IsUnit (aeval x <| derivative P)) (n : ℕ) :
    (aeval x P) ^ (2 ^ n) ∣ aeval (P.newtonMap^[n] x) P

Proof state:
case succ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  eval (P.newtonMap^[n] x + -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)
      (map (algebraMap R S) P) =
    eval (P.newtonMap^[n] x) (map (algebraMap R S) P) +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P

Tactic:
rw [eval_map_algebraMap, eval_map_algebraMap] at hd

Result:
case succ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P

Proof state:
case succ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P

Tactic:
have ⟨d, hd⟩ := binomExpansion (P.map (algebraMap R S)) (P.newtonMap^[n] x)
  (-Ring.inverse (aeval (P.newtonMap^[n] x) <| derivative P) * aeval (P.newtonMap^[n] x) P)

Result:
case succ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  eval (P.newtonMap^[n] x + -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)
      (map (algebraMap R S) P) =
    eval (P.newtonMap^[n] x) (map (algebraMap R S) P) +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P

Proof state:
case succ.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣
    (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2

Tactic:
rw [neg_mul, even_two.neg_pow, mul_pow, pow_succ, pow_mul]

Result:
case succ.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ ((aeval x) P ^ 2 ^ n) ^ 2 ∣
    Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) ^ 2 * (aeval (P.newtonMap^[n] x)) P ^ 2

Proof state:
case zero
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
⊢ (aeval x) P ^ 2 ^ 0 ∣ (aeval (P.newtonMap^[0] x)) P

Tactic:
simp

Result:
no goals

Proof state:
case succ.refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval x) P ^ 2 ^ (n + 1) ∣
    (aeval (P.newtonMap^[n] x)) P +
      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)

Tactic:
convert dvd_zero _

Result:
case h.e'_4
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ (aeval (P.newtonMap^[n] x)) P +
      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) =
    0