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
z z' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hdzne :
  Ne (α := ℤ_[p])
    (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
        ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
      ℤ_[p])
    (0 : ℤ_[p])
hdzne' :
  Ne (α := ℚ_[p])
    (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
      ℚ_[p])
    0
obtain :
  (q : ℤ_[p]) →
    (hq :
        Polynomial.eval (z + -z1) (Polynomial.map (algebraMap R ℤ_[p]) F) =
          Polynomial.eval z (Polynomial.map (algebraMap R ℤ_[p]) F) +
              Polynomial.eval z
                  ((Polynomial.derivative : (a : Polynomial ℤ_[p]) → Polynomial ℤ_[p])
                    (Polynomial.map (algebraMap R ℤ_[p]) F)) *
                -z1 +
            q * HPow.hPow (β := ℕ) (-z1) 2) →
      { q : ℤ_[p] //
        ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F =
          q * HPow.hPow (β := ℕ) z1 2 }
⊢ { q : ℤ_[p] //
    ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2 }

Tactic:
obtain ⟨q, hq⟩ := (F.map (algebraMap R ℤ_[p])).binomExpansion z (-z1)

Premises used: PadicInt, Algebra.algebraMap, Polynomial.binomExpansion, Polynomial.map, Neg.neg
-- 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
z z' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hdzne :
  Ne (α := ℤ_[p])
    (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
        ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
      ℤ_[p])
    (0 : ℤ_[p])
hdzne' :
  Ne (α := ℚ_[p])
    (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
      ℚ_[p])
    0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) (Polynomial.map (algebraMap R ℤ_[p]) F) =
    Polynomial.eval z (Polynomial.map (algebraMap R ℤ_[p]) F) +
        Polynomial.eval z
            ((Polynomial.derivative : (a : Polynomial ℤ_[p]) → Polynomial ℤ_[p])
              (Polynomial.map (algebraMap R ℤ_[p]) F)) *
          -z1 +
      q * HPow.hPow (β := ℕ) (-z1) 2
⊢ ‖(↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]) *
        HDiv.hDiv (α := ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
            ℚ_[p])‖ ≤
    1

Tactic:
simpa using mul_le_one₀ (PadicInt.norm_le_one _) (norm_nonneg _) h1

Premises used: mul_le_one₀, PadicInt.norm_le_one, 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
z z' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hdzne :
  Ne (α := ℤ_[p])
    (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
        ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
      ℤ_[p])
    (0 : ℤ_[p])
hdzne' :
  Ne (α := ℚ_[p])
    (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
      ℚ_[p])
    0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) (Polynomial.map (algebraMap R ℤ_[p]) F) =
    Polynomial.eval z (Polynomial.map (algebraMap R ℤ_[p]) F) +
        Polynomial.eval z
            ((Polynomial.derivative : (a : Polynomial ℤ_[p]) → Polynomial ℤ_[p])
              (Polynomial.map (algebraMap R ℤ_[p]) F)) *
          -z1 +
      q * HPow.hPow (β := ℕ) (-z1) 2
⊢ ‖(↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]) *
        HDiv.hDiv (α := ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
            ℚ_[p])‖ ≤
    1

Tactic:
have : ‖(↑(F.derivative.aeval z) * (↑(F.aeval z) / ↑(F.derivative.aeval z)) : ℚ_[p])‖ ≤ 1 := by
    simpa using mul_le_one₀ (PadicInt.norm_le_one _) (norm_nonneg _) h1
-- 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
z z' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hdzne :
  Ne (α := ℤ_[p])
    (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
        ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
      ℤ_[p])
    (0 : ℤ_[p])
hdzne' :
  Ne (α := ℚ_[p])
    (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
      ℚ_[p])
    0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) (Polynomial.map (algebraMap R ℤ_[p]) F) =
    Polynomial.eval z (Polynomial.map (algebraMap R ℤ_[p]) F) +
        Polynomial.eval z
            ((Polynomial.derivative : (a : Polynomial ℤ_[p]) → Polynomial ℤ_[p])
              (Polynomial.map (algebraMap R ℤ_[p]) F)) *
          -z1 +
      q * HPow.hPow (β := ℕ) (-z1) 2
this_1 :
  ‖(↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]) *
        HDiv.hDiv (α := ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
          (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
                ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
            ℚ_[p])‖ ≤
    1
this :
  HMul.hMul (α := ℤ_[p])
      (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
        ℤ_[p])
      (-z1) =
    -((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F
⊢ { q : ℤ_[p] //
    ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2 }

Tactic:
exact ⟨q, by simpa [sub_eq_add_neg, neg_mul_eq_mul_neg, this, hz'] using hq⟩
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
⊢ norm (E := ℤ_[p]) (((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) =
    ‖q‖ * HPow.hPow (β := ℕ) ‖z1‖ 2

Tactic:
simp [heq]
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hbc : ‖q‖ ≤ 1
⊢ ‖q‖ * HPow.hPow (β := ℕ) ‖z1‖ 2 ≤ 1 * HPow.hPow (β := ℕ) ‖z1‖ 2

Tactic:
gcongr
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
⊢ ‖q‖ ≤ 1

Tactic:
apply PadicInt.norm_le_one

Premises used: PadicInt.norm_le_one
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
⊢ 1 * HPow.hPow (β := ℕ) ‖z1‖ 2 =
    HPow.hPow (β := ℕ)
        (norm (E := ℤ_[p]) (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]))
        2 /
      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:
simp [hzeq, hz.1, div_pow]

Premises used: And.left, div_pow
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
hab :
  norm (E := ℤ_[p]) (((Polynomial.aeval z : 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
⊢ HPow.hPow (β := ℕ)
        (norm (E := ℤ_[p]) (((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]))
        2 /
      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 ≤
    HPow.hPow (β := 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)
        2 /
      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:
gcongr
-- 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
z z' z1 : ℤ_[p]
n : ℕ
hz : ih_gen.{u_1} (R := R) (F := F) (a := a) n z
q : ℤ_[p]
heq : ((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F = q * HPow.hPow (β := ℕ) z1 2
h1 :
  ‖HDiv.hDiv (α := ℚ_[p]) (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p])‖ ≤
    1
hzeq :
  z1 =
    Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1)
      (HDiv.hDiv (α := ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F) : ℚ_[p])
        (↑(((Polynomial.aeval z : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F)) :
          ℚ_[p]))
      h1
⊢ norm (E := ℤ_[p]) (((Polynomial.aeval z : 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:
hz.2

Premises used: And.right