l3lab/ntp-mathlib huggingface dataset:
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
⊢ Polynomial.eval (soln + h) F = Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2

Next tactic:
rw [hq, eval_soln, zero_add]
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
⊢ Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2 =
    (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h

Next tactic:
rw [sq, right_distrib, mul_assoc]
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
⊢ Polynomial.eval soln (Polynomial.derivative F) = -q * h

Next tactic:
simpa using eq_neg_of_add_eq_zero_left this
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖Polynomial.eval soln (Polynomial.derivative F)‖ = ‖-q * h‖

Next tactic:
rw [this]
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖-q * h‖ ≤ 1 * ‖h‖

Next tactic:
rw [PadicInt.norm_mul]
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            rw [PadicInt.norm_mul]
            

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖-q‖ * ‖h‖ ≤ 1 * ‖h‖

Next tactic:
exact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            rw [PadicInt.norm_mul]
            exact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)
          _ = ‖z - soln‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ 1 * ‖h‖ = ‖z - soln‖

Next tactic:
simp
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            rw [PadicInt.norm_mul]
            exact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)
          _ = ‖z - soln‖ := by simp
          _ < ‖F.derivative.eval soln‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖z - soln‖ < ‖Polynomial.eval soln (Polynomial.derivative F)‖

Next tactic:
rw [soln_deriv_norm]
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            rw [PadicInt.norm_mul]
            exact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)
          _ = ‖z - soln‖ := by simp
          _ < ‖F.derivative.eval soln‖ := by rw [soln_deriv_norm]; 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖

Next tactic:
apply soln_dist
private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
    (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
  have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
    calc
      ‖z - soln‖ = ‖z - a + (a - soln)‖ := by rw [sub_add_sub_cancel]
      _ ≤ max ‖z - a‖ ‖a - soln‖ := (PadicInt.nonarchimedean _ _)
      _ < ‖F.derivative.eval a‖ :=
        max_lt hnlt ((norm_sub_rev soln a ▸ (soln_dist_to_a_lt_deriv hnorm)) hnsol)

  let h := z - soln
  let ⟨q, hq⟩ := F.binomExpansion soln h
  have : (F.derivative.eval soln + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (soln + h) := by simp [hev]
        _ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
        _ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval soln + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval soln = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
      lt_irrefl ‖F.derivative.eval soln‖
        (calc
          ‖F.derivative.eval soln‖ = ‖-q * h‖ := by rw [this]
          _ ≤ 1 * ‖h‖ := by
            rw [PadicInt.norm_mul]
            exact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)
          _ = ‖z - soln‖ := by simp
          _ < ‖F.derivative.eval soln‖ := by rw [soln_deriv_norm]; apply soln_dist
          )
  eq_of_sub_eq_zero (by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
  Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
this : h = 0
⊢ z - soln = 0

Next tactic:
rw [← this]
private theorem a_soln_is_unique (ha : F.eval a = 0) (z' : ℤ_[p]) (hz' : F.eval z' = 0)
    (hnormz' : ‖z' - a‖ < ‖F.derivative.eval a‖) : z' = a :=
  let h := z' - a
  let ⟨q, hq⟩ := F.binomExpansion a h
  have : (F.derivative.eval a + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (a + h) := show 0 = F.eval (a + (z' - a)) by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
⊢ 0 = Polynomial.eval (a + (z' - a)) F

Next tactic:
rw [add_comm]
private theorem a_soln_is_unique (ha : F.eval a = 0) (z' : ℤ_[p]) (hz' : F.eval z' = 0)
    (hnormz' : ‖z' - a‖ < ‖F.derivative.eval a‖) : z' = a :=
  let h := z' - a
  let ⟨q, hq⟩ := F.binomExpansion a h
  have : (F.derivative.eval a + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (a + h) := show 0 = F.eval (a + (z' - a)) by rw [add_comm]; 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
⊢ 0 = Polynomial.eval (z' - a + a) F

Next tactic:
simp [hz']
private theorem a_soln_is_unique (ha : F.eval a = 0) (z' : ℤ_[p]) (hz' : F.eval z' = 0)
    (hnormz' : ‖z' - a‖ < ‖F.derivative.eval a‖) : z' = a :=
  let h := z' - a
  let ⟨q, hq⟩ := F.binomExpansion a h
  have : (F.derivative.eval a + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (a + h) := show 0 = F.eval (a + (z' - a)) by rw [add_comm]; simp [hz']
        _ = F.derivative.eval a * h + q * h ^ 2 := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
⊢ Polynomial.eval (a + h) F = Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2

Next tactic:
rw [hq, ha, zero_add]
private theorem a_soln_is_unique (ha : F.eval a = 0) (z' : ℤ_[p]) (hz' : F.eval z' = 0)
    (hnormz' : ‖z' - a‖ < ‖F.derivative.eval a‖) : z' = a :=
  let h := z' - a
  let ⟨q, hq⟩ := F.binomExpansion a h
  have : (F.derivative.eval a + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (a + h) := show 0 = F.eval (a + (z' - a)) by rw [add_comm]; simp [hz']
        _ = F.derivative.eval a * h + q * h ^ 2 := by rw [hq, ha, zero_add]
        _ = (F.derivative.eval a + q * h) * h := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
⊢ Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2 =
    (Polynomial.eval a (Polynomial.derivative F) + q * h) * h

Next tactic:
rw [sq, right_distrib, mul_assoc]
private theorem a_soln_is_unique (ha : F.eval a = 0) (z' : ℤ_[p]) (hz' : F.eval z' = 0)
    (hnormz' : ‖z' - a‖ < ‖F.derivative.eval a‖) : z' = a :=
  let h := z' - a
  let ⟨q, hq⟩ := F.binomExpansion a h
  have : (F.derivative.eval a + q * h) * h = 0 :=
    Eq.symm
      (calc
        0 = F.eval (a + h) := show 0 = F.eval (a + (z' - a)) by rw [add_comm]; simp [hz']
        _ = F.derivative.eval a * h + q * h ^ 2 := by rw [hq, ha, zero_add]
        _ = (F.derivative.eval a + q * h) * h := by rw [sq, right_distrib, mul_assoc]
        )
  have : h = 0 :=
    by_contra fun hne =>
      have : F.derivative.eval a + q * h = 0 :=
        (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
      have : F.derivative.eval a = -q * h := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
this✝ : (Polynomial.eval a (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this : Polynomial.eval a (Polynomial.derivative F) + q * h = 0
⊢ Polynomial.eval a (Polynomial.derivative F) = -q * h

Next tactic:
simpa using eq_neg_of_add_eq_zero_left this