awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n : ℕ
⊢ norm (E := ℤ_[p])
      (((Polynomial.aeval (newton_seq n) : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) ≤
    HPow.hPow (β := ℕ)
        (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]))
        2 *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n

Tactic:
apply newton_seq_norm_le

Premises used: _private.0.newton_seq_norm_le
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n : ℕ
⊢ ‖newton_seq (n + 0) - newton_seq n‖ ≤
    norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]) *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n

Tactic:
simp [T_pow_nonneg, mul_nonneg]

Premises used: mul_nonneg, _private.0.T_pow_nonneg
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
h : LE.le (α := ℕ) 1 2
a₁ : n ≤ n + k
⊢ HPow.hPow (α := ℕ) 2 n ≤ HPow.hPow (α := ℕ) 2 (n + k)

Tactic:
apply pow_right_mono₀

Premises used: pow_right_mono₀
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
⊢ LE.le (α := ℕ) 1 2

Tactic:
simp
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
⊢ n ≤ n + k

Tactic:
apply Nat.le_add_right

Premises used: Nat.le_add_right
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
this : HPow.hPow (α := ℕ) 2 n ≤ HPow.hPow (α := ℕ) 2 (n + k)
rw rw₁ : ‖newton_seq (n + (k + 1)) - newton_seq n‖ = ‖newton_seq (n + (k + 1)) - newton_seq n‖
⊢ ‖newton_seq (n + (k + 1)) - newton_seq n‖ = ‖newton_seq (n + k + 1) - newton_seq n‖

Tactic:
rw [add_assoc]

Premises used: add_assoc
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
this : HPow.hPow (α := ℕ) 2 n ≤ HPow.hPow (α := ℕ) 2 (n + k)
rw :
  ‖newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)‖ =
    ‖newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)‖
b : ℤ_[p]
rw₁ :
  ‖newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)‖ =
    ‖newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)‖
b₁ : ℤ_[p]
⊢ ‖newton_seq (n + k + 1) - newton_seq n‖ =
    ‖newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)‖

Tactic:
rw [← sub_add_sub_cancel]

Premises used: sub_add_sub_cancel
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
hnk : n ≤ k
⊢ ∃ (m : ℕ), k = n + m

Tactic:
have hex : ∃ m, k = n + m := Nat.exists_eq_add_of_le hnk
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
hnk : n ≤ k
hex : ∃ (m : ℕ), k = n + m
let :
  ∀ (w : ℕ) (hex' : k = n + w),
    ‖newton_seq k - newton_seq n‖ ≤
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n
⊢ ‖newton_seq k - newton_seq n‖ ≤
    norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]) *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n

Tactic:
let ⟨_, hex'⟩ := hex
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
hnk : n ≤ k
hex : ∃ (m : ℕ), k = n + m
w : ℕ
hex' : k = n + w
rw :
  ‖newton_seq (n + w) - newton_seq n‖ ≤
    norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]) *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n
⊢ ‖newton_seq k - newton_seq n‖ ≤
    norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]) *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n

Tactic:
rw [hex']
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
n k : ℕ
hnk : n ≤ k
hex : ∃ (m : ℕ), k = n + m
w : ℕ
hex' : k = n + w
⊢ ‖newton_seq (n + w) - newton_seq n‖ ≤
    norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]) *
      T_gen p F a ^ HPow.hPow (α := ℕ) 2 n

Tactic:
apply newton_seq_dist_aux

Premises used: _private.0.newton_seq_dist_aux
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
rw :
  Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))
⊢ Tendsto (β := ℝ)
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop (𝓝 0)

Tactic:
rw [← mul_zero ‖F.derivative.aeval a‖]

Premises used: Polynomial.derivative, Norm.norm, MulZeroClass.mul_zero, Polynomial.aeval, DFunLike.coe
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ LT.lt (α := ℕ) 1 2

Tactic:
simp
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))

Tactic:
exact
    tendsto_const_nhds.mul
      (Tendsto.comp (tendsto_pow_atTop_nhds_zero_of_lt_one (norm_nonneg _) (T_lt_one hnorm))
        (tendsto_pow_atTop_atTop_of_one_lt (by simp)))

Premises used: _private.0.T_lt_one, tendsto_const_nhds, Filter.Tendsto.mul, tendsto_pow_atTop_atTop_of_one_lt, tendsto_pow_atTop_nhds_zero_of_lt_one, Filter.Tendsto.comp, norm_nonneg
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
ε : ℝ
⊢ ε > 0 →
    ∃ (N : ℕ),
      ∀ {n : ℕ},
        n ≥ N →
          norm (E := ℤ_[p])
                (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                    ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
                  ℤ_[p]) *
              T_gen p F a ^ HPow.hPow (α := ℕ) 2 n <
            ε

Tactic:
fun hε ↦
  eventually_atTop.1 <| (bound' hnorm).eventually <| gt_mem_nhds hε

Premises used: gt_mem_nhds, _private.0.bound', Filter.Tendsto.eventually, Iff.mp, Filter.eventually_atTop
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
rw :
  Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
          norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))
⊢ Tendsto (β := ℝ)
    (fun (n : ℕ) ↦
      HPow.hPow (β := ℕ)
          (norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]))
          2 *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop (𝓝 0)

Tactic:
rw [← mul_zero ‖F.derivative.aeval a‖, sq]

Premises used: Polynomial.derivative, Norm.norm, sq, MulZeroClass.mul_zero, Polynomial.aeval, DFunLike.coe
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
simp :
  Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        (norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
          T_gen p F a ^ HPow.hPow (α := ℕ) 2 n))
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))
⊢ Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
          norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 n)
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))

Tactic:
simp only [mul_assoc]

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
hf :
  Tendsto
    (fun (x : ℕ) ↦
      norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
    atTop
    (𝓝
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])))
hg :
  Tendsto (β := ℝ)
    (fun (x : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 x)
    atTop (𝓝 0)
⊢ Tendsto
    (fun (n : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        (norm (E := ℤ_[p])
            (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
              ℤ_[p]) *
          T_gen p F a ^ HPow.hPow (α := ℕ) 2 n))
    atTop
    (𝓝
      (norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        0))

Tactic:
apply Tendsto.mul

Premises used: Filter.Tendsto.mul
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ Tendsto
    (fun (x : ℕ) ↦
      norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
    atTop
    (𝓝
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])))

Tactic:
apply tendsto_const_nhds

Premises used: tendsto_const_nhds
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm hg :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ Tendsto (β := ℝ)
    (fun (x : ℕ) ↦
      norm (E := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p]) *
        T_gen p F a ^ HPow.hPow (α := ℕ) 2 x)
    atTop (𝓝 0)

Tactic:
apply bound'

Premises used: _private.0.bound'
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2

Tactic:
assumption
-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
hnorm :
  norm (E := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) <
    HPow.hPow (β := ℕ)
      (norm (E := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p]))
      2
⊢ IsCauSeq norm newton_seq

Tactic:
fun _ε hε ↦
  (bound hnorm hε).imp fun _N hN _j hj ↦ (newton_seq_dist hnorm hj).trans_lt <| hN le_rfl

Premises used: _private.0.newton_seq_dist, Exists.imp, _private.0.bound, LE.le.trans_lt, IsCauSeq