l3lab/ntp-mathlib huggingface dataset:
instance existsAddOfLE [LE α] [Add α] [ExistsAddOfLE α] : ExistsAddOfLE (WithTop α) :=
  ⟨fun {a} {b} =>
    match a, b with
    | ⊤, ⊤ => by 

Proof state:
α : Type u
β : Type v
inst✝² : LE α
inst✝¹ : Add α
inst✝ : ExistsAddOfLE α
a b : WithTop α
⊢ ⊤ ≤ ⊤ → ∃ c, ⊤ = ⊤ + c

Next tactic:
simp
instance existsAddOfLE [LE α] [Add α] [ExistsAddOfLE α] : ExistsAddOfLE (WithTop α) :=
  ⟨fun {a} {b} =>
    match a, b with
    | ⊤, ⊤ => by simp
    | (a : α), ⊤ => fun _ => ⟨⊤, rfl⟩
    | (a : α), (b : α) => fun h => by
      

Proof state:
α : Type u
β : Type v
inst✝² : LE α
inst✝¹ : Add α
inst✝ : ExistsAddOfLE α
a✝ b✝ : WithTop α
a b : α
h : ↑a ≤ ↑b
⊢ ∃ c, ↑b = ↑a + c

Next tactic:
obtain ⟨c, rfl⟩ := exists_add_of_le (WithTop.coe_le_coe.1 h)
instance existsAddOfLE [LE α] [Add α] [ExistsAddOfLE α] : ExistsAddOfLE (WithTop α) :=
  ⟨fun {a} {b} =>
    match a, b with
    | ⊤, ⊤ => by simp
    | (a : α), ⊤ => fun _ => ⟨⊤, rfl⟩
    | (a : α), (b : α) => fun h => by
      obtain ⟨c, rfl⟩ := exists_add_of_le (WithTop.coe_le_coe.1 h)
      

Proof state:
case intro
α : Type u
β : Type v
inst✝² : LE α
inst✝¹ : Add α
inst✝ : ExistsAddOfLE α
a✝ b : WithTop α
a c : α
h : ↑a ≤ ↑(a + c)
⊢ ∃ c_1, ↑(a + c) = ↑a + c_1

Next tactic:
exact ⟨c, rfl⟩
/-- A version of `WithTop.map` for `OneHom`s. -/
@[to_additive (attr := simps (config := .asFn))
  "A version of `WithTop.map` for `ZeroHom`s"]
protected def _root_.OneHom.withTopMap {M N : Type*} [One M] [One N] (f : OneHom M N) :
    OneHom (WithTop M) (WithTop N) where
  toFun := WithTop.map f
  map_one' := by 

Proof state:
α : Type u
β : Type v
M : Type u_1
N : Type u_2
inst✝¹ : One M
inst✝ : One N
f : OneHom M N
⊢ map (⇑f) 1 = 1

Next tactic:
rw [WithTop.map_one, map_one, coe_one]
@[simp]
theorem add_bot (a : WithBot α) : a + ⊥ = ⊥ := by 

Proof state:
α : Type u
β : Type v
inst✝ : Add α
a✝ b c d : WithBot α
x y : α
a : WithBot α
⊢ a + ⊥ = ⊥

Next tactic:
cases a
@[simp]
theorem add_bot (a : WithBot α) : a + ⊥ = ⊥ := by cases a <;> 

Proof state:
case none
α : Type u
β : Type v
inst✝ : Add α
a b c d : WithBot α
x y : α
⊢ none + ⊥ = ⊥

Next tactic:
rfl
@[simp]
theorem add_bot (a : WithBot α) : a + ⊥ = ⊥ := by cases a <;> 

Proof state:
case some
α : Type u
β : Type v
inst✝ : Add α
a b c d : WithBot α
x y val✝ : α
⊢ Option.some val✝ + ⊥ = ⊥

Next tactic:
rfl
/-- A version of `WithBot.map` for `OneHom`s. -/
@[to_additive (attr := simps (config := .asFn))
  "A version of `WithBot.map` for `ZeroHom`s"]
protected def _root_.OneHom.withBotMap {M N : Type*} [One M] [One N] (f : OneHom M N) :
    OneHom (WithBot M) (WithBot N) where
  toFun := WithBot.map f
  map_one' := by 

Proof state:
α : Type u
β : Type v
inst✝² : Add α
a b c d : WithBot α
x y : α
M : Type u_1
N : Type u_2
inst✝¹ : One M
inst✝ : One N
f : OneHom M N
⊢ map (⇑f) 1 = 1

Next tactic:
rw [WithBot.map_one, map_one, coe_one]
theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
    ‖F.eval x - F.eval y‖ ≤ ‖x - y‖ :=
  let ⟨z, hz⟩ := F.evalSubFactor x y
  calc
    ‖F.eval x - F.eval y‖ = ‖z‖ * ‖x - y‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
x y z : ℤ_[p]
hz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)
⊢ ‖Polynomial.eval x F - Polynomial.eval y F‖ = ‖z‖ * ‖x - y‖

Next tactic:
simp [hz]
theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
    ‖F.eval x - F.eval y‖ ≤ ‖x - y‖ :=
  let ⟨z, hz⟩ := F.evalSubFactor x y
  calc
    ‖F.eval x - F.eval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]
    _ ≤ 1 * ‖x - y‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
x y z : ℤ_[p]
hz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)
⊢ ‖z‖ * ‖x - y‖ ≤ 1 * ‖x - y‖

Next tactic:
gcongr
theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
    ‖F.eval x - F.eval y‖ ≤ ‖x - y‖ :=
  let ⟨z, hz⟩ := F.evalSubFactor x y
  calc
    ‖F.eval x - F.eval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]
    _ ≤ 1 * ‖x - y‖ := by gcongr; 

Proof state:
case h
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
x y z : ℤ_[p]
hz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)
⊢ ‖z‖ ≤ 1

Next tactic:
apply PadicInt.norm_le_one
theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
    ‖F.eval x - F.eval y‖ ≤ ‖x - y‖ :=
  let ⟨z, hz⟩ := F.evalSubFactor x y
  calc
    ‖F.eval x - F.eval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]
    _ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one
    _ = ‖x - y‖ := by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
x y z : ℤ_[p]
hz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)
⊢ 1 * ‖x - y‖ = ‖x - y‖

Next tactic:
simp
private theorem ncs_tendsto_const :
    Tendsto (fun i => ‖F.derivative.eval (ncs i)‖) atTop (𝓝 ‖F.derivative.eval a‖) := by
  

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
ncs : CauSeq ℤ_[p] norm
F : Polynomial ℤ_[p]
a : ℤ_[p]
ncs_der_val :
  ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖
⊢ Tendsto (fun i => ‖Polynomial.eval (↑ncs i) (Polynomial.derivative F)‖) atTop
    (𝓝 ‖Polynomial.eval a (Polynomial.derivative F)‖)

Next tactic:
convert @tendsto_const_nhds ℝ ℕ _ _ _
private theorem ncs_tendsto_const :
    Tendsto (fun i => ‖F.derivative.eval (ncs i)‖) atTop (𝓝 ‖F.derivative.eval a‖) := by
  convert @tendsto_const_nhds ℝ ℕ _ _ _; 

Proof state:
case h.e'_3.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
ncs : CauSeq ℤ_[p] norm
F : Polynomial ℤ_[p]
a : ℤ_[p]
ncs_der_val :
  ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖
x✝ : ℕ
⊢ ‖Polynomial.eval (↑ncs x✝) (Polynomial.derivative F)‖ = ‖Polynomial.eval a (Polynomial.derivative F)‖

Next tactic:
rw [ncs_der_val]
private theorem tendsto_zero_of_norm_tendsto_zero : Tendsto (fun i => F.eval (ncs i)) atTop (𝓝 0) :=
  tendsto_iff_norm_sub_tendsto_zero.2 (by 

Proof state:
p : ℕ
inst✝ : Fact (Nat.Prime p)
ncs : CauSeq ℤ_[p] norm
F : Polynomial ℤ_[p]
hnorm : Tendsto (fun i => ‖Polynomial.eval (↑ncs i) F‖) atTop (𝓝 0)
⊢ Tendsto (fun e => ‖Polynomial.eval (↑ncs e) F - 0‖) atTop (𝓝 0)

Next tactic:
simpa using hnorm
private theorem deriv_norm_ne_zero : ‖F.derivative.eval a‖ ≠ 0 := fun h =>
  deriv_sq_norm_ne_zero hnorm (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
h : ‖Polynomial.eval a (Polynomial.derivative F)‖ = 0
⊢ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 = 0

Next tactic:
simp [*, sq]
private theorem T_def : T = ‖F.eval a‖ / ‖F.derivative.eval a‖ ^ 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
⊢ T_gen p F a = ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2

Next tactic:
simp [T_gen, ← PadicInt.norm_def]
private theorem T_lt_one : T < 1 := 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
⊢ T_gen p F a < 1

Next tactic:
have h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm
private theorem T_lt_one : T < 1 := by
  have h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm
  

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
h : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 < 1
⊢ T_gen p F a < 1

Next tactic:
rw [T_def]
private theorem T_lt_one : T < 1 := by
  have h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm
  rw [T_def]; 

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
h : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 < 1
⊢ ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 < 1

Next tactic:
exact h
private theorem ih_0 : ih 0 a :=
  ⟨rfl, 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
⊢ ‖Polynomial.eval a F‖ ≤ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ 0

Next tactic:
simp [T_def, mul_div_cancel' _ (ne_of_gt (deriv_sq_norm_pos hnorm))]
private theorem calc_norm_le_one {n : ℕ} {z : ℤ_[p]} (hz : ih n z) :
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1 :=
  calc
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ =
        ‖(↑(F.eval z) : ℚ_[p])‖ / ‖(↑(F.derivative.eval z) : ℚ_[p])‖ :=
      norm_div _ _
    _ = ‖F.eval z‖ / ‖F.derivative.eval a‖ := 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
n : ℕ
z : ℤ_[p]
hz : ih_gen n z
⊢ ‖↑(Polynomial.eval z F)‖ / ‖↑(Polynomial.eval z (Polynomial.derivative F))‖ =
    ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖

Next tactic:
simp [hz.1]
private theorem calc_norm_le_one {n : ℕ} {z : ℤ_[p]} (hz : ih n z) :
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1 :=
  calc
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ =
        ‖(↑(F.eval z) : ℚ_[p])‖ / ‖(↑(F.derivative.eval z) : ℚ_[p])‖ :=
      norm_div _ _
    _ = ‖F.eval z‖ / ‖F.derivative.eval a‖ := by simp [hz.1]
    _ ≤ ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n / ‖F.derivative.eval a‖ := 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
n : ℕ
z : ℤ_[p]
hz : ih_gen n z
⊢ ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ≤
    ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n /
      ‖Polynomial.eval a (Polynomial.derivative F)‖

Next tactic:
gcongr
private theorem calc_norm_le_one {n : ℕ} {z : ℤ_[p]} (hz : ih n z) :
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1 :=
  calc
    ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ =
        ‖(↑(F.eval z) : ℚ_[p])‖ / ‖(↑(F.derivative.eval z) : ℚ_[p])‖ :=
      norm_div _ _
    _ = ‖F.eval z‖ / ‖F.derivative.eval a‖ := by simp [hz.1]
    _ ≤ ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n / ‖F.derivative.eval a‖ := by
      gcongr
      

Proof state:
case h
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
n : ℕ
z : ℤ_[p]
hz : ih_gen n z
⊢ ‖Polynomial.eval z F‖ ≤ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n

Next tactic:
apply hz.2
private theorem calc_deriv_dist {z z' z1 : ℤ_[p]} (hz' : z' = z - z1)
    (hz1 : ‖z1‖ = ‖F.eval z‖ / ‖F.derivative.eval a‖) {n} (hz : ih n z) :
    ‖F.derivative.eval z' - F.derivative.eval z‖ < ‖F.derivative.eval a‖ :=
  calc
    ‖F.derivative.eval z' - F.derivative.eval z‖ ≤ ‖z' - z‖ := padic_polynomial_dist _ _ _
    _ = ‖z1‖ := 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' z1 : ℤ_[p]
hz' : z' = z - z1
hz1 : ‖z1‖ = ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
n : ℕ
hz : ih_gen n z
⊢ ‖z' - z‖ = ‖z1‖

Next tactic:
simp only [sub_eq_add_neg, add_assoc, hz', add_add_neg_cancel'_right, norm_neg]
private theorem calc_deriv_dist {z z' z1 : ℤ_[p]} (hz' : z' = z - z1)
    (hz1 : ‖z1‖ = ‖F.eval z‖ / ‖F.derivative.eval a‖) {n} (hz : ih n z) :
    ‖F.derivative.eval z' - F.derivative.eval z‖ < ‖F.derivative.eval a‖ :=
  calc
    ‖F.derivative.eval z' - F.derivative.eval z‖ ≤ ‖z' - z‖ := padic_polynomial_dist _ _ _
    _ = ‖z1‖ := by simp only [sub_eq_add_neg, add_assoc, hz', add_add_neg_cancel'_right, norm_neg]
    _ = ‖F.eval z‖ / ‖F.derivative.eval a‖ := hz1
    _ ≤ ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n / ‖F.derivative.eval a‖ := 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' z1 : ℤ_[p]
hz' : z' = z - z1
hz1 : ‖z1‖ = ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
n : ℕ
hz : ih_gen n z
⊢ ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖ ≤
    ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n /
      ‖Polynomial.eval a (Polynomial.derivative F)‖

Next tactic:
gcongr
private theorem calc_deriv_dist {z z' z1 : ℤ_[p]} (hz' : z' = z - z1)
    (hz1 : ‖z1‖ = ‖F.eval z‖ / ‖F.derivative.eval a‖) {n} (hz : ih n z) :
    ‖F.derivative.eval z' - F.derivative.eval z‖ < ‖F.derivative.eval a‖ :=
  calc
    ‖F.derivative.eval z' - F.derivative.eval z‖ ≤ ‖z' - z‖ := padic_polynomial_dist _ _ _
    _ = ‖z1‖ := by simp only [sub_eq_add_neg, add_assoc, hz', add_add_neg_cancel'_right, norm_neg]
    _ = ‖F.eval z‖ / ‖F.derivative.eval a‖ := hz1
    _ ≤ ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n / ‖F.derivative.eval a‖ := by
      gcongr
      

Proof state:
case h
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' z1 : ℤ_[p]
hz' : z' = z - z1
hz1 : ‖z1‖ = ‖Polynomial.eval z F‖ / ‖Polynomial.eval a (Polynomial.derivative F)‖
n : ℕ
hz : ih_gen n z
⊢ ‖Polynomial.eval z F‖ ≤ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n

Next tactic:
apply hz.2
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
⊢ { q // Polynomial.eval z' F = q * z1 ^ 2 }

Next tactic:
have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
⊢ ¬‖Polynomial.eval z (Polynomial.derivative F)‖ = 0

Next tactic:
rw [hz.1]
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; 

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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
⊢ ¬‖Polynomial.eval a (Polynomial.derivative F)‖ = 0

Next tactic:
apply deriv_norm_ne_zero
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; 

Proof state:
case hnorm
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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
⊢ ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2

Next tactic:
assumption
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
  

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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
hdzne : Polynomial.eval z (Polynomial.derivative F) ≠ 0
⊢ { q // Polynomial.eval z' F = q * z1 ^ 2 }

Next tactic:
have hdzne' : (↑(F.derivative.eval z) : ℚ_[p]) ≠ 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
  have hdzne' : (↑(F.derivative.eval z) : ℚ_[p]) ≠ 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)
  

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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
hdzne : Polynomial.eval z (Polynomial.derivative F) ≠ 0
hdzne' : ↑(Polynomial.eval z (Polynomial.derivative F)) ≠ 0
⊢ { q // Polynomial.eval z' F = q * z1 ^ 2 }

Next tactic:
obtain ⟨q, hq⟩ := F.binomExpansion z (-z1)
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
  have hdzne' : (↑(F.derivative.eval z) : ℚ_[p]) ≠ 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)
  obtain ⟨q, hq⟩ := F.binomExpansion z (-z1)
  

Proof state:
case mk
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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
hdzne : Polynomial.eval z (Polynomial.derivative F) ≠ 0
hdzne' : ↑(Polynomial.eval z (Polynomial.derivative F)) ≠ 0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (Polynomial.derivative F) * -z1 + q * (-z1) ^ 2
⊢ { q // Polynomial.eval z' F = q * z1 ^ 2 }

Next tactic:
have : ‖(↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)) : ℚ_[p])‖ ≤ 1 := by
    rw [padicNormE.mul]
    exact mul_le_one (PadicInt.norm_le_one _) (norm_nonneg _) h1
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
  have hdzne' : (↑(F.derivative.eval z) : ℚ_[p]) ≠ 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)
  obtain ⟨q, hq⟩ := F.binomExpansion z (-z1)
  have : ‖(↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)) : ℚ_[p])‖ ≤ 1 := 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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
hdzne : Polynomial.eval z (Polynomial.derivative F) ≠ 0
hdzne' : ↑(Polynomial.eval z (Polynomial.derivative F)) ≠ 0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (Polynomial.derivative F) * -z1 + q * (-z1) ^ 2
⊢ ‖↑(Polynomial.eval z (Polynomial.derivative F)) *
        (↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)))‖ ≤
    1

Next tactic:
rw [padicNormE.mul]
private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)
    (h1 : ‖(↑(F.eval z) : ℚ_[p]) / ↑(F.derivative.eval z)‖ ≤ 1) (hzeq : z1 = ⟨_, h1⟩) :
    { q : ℤ_[p] // F.eval z' = q * z1 ^ 2 } := by
  have hdzne : F.derivative.eval z ≠ 0 :=
    mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)
  have hdzne' : (↑(F.derivative.eval z) : ℚ_[p]) ≠ 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)
  obtain ⟨q, hq⟩ := F.binomExpansion z (-z1)
  have : ‖(↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)) : ℚ_[p])‖ ≤ 1 := by
    rw [padicNormE.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' z1 : ℤ_[p]
hz' : z' = z - z1
n : ℕ
hz : ih_gen n z
h1 : ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤ 1
hzeq : z1 = { val := ↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F)), property := h1 }
hdzne : Polynomial.eval z (Polynomial.derivative F) ≠ 0
hdzne' : ↑(Polynomial.eval z (Polynomial.derivative F)) ≠ 0
q : ℤ_[p]
hq :
  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (Polynomial.derivative F) * -z1 + q * (-z1) ^ 2
⊢ ‖↑(Polynomial.eval z (Polynomial.derivative F))‖ *
      ‖↑(Polynomial.eval z F) / ↑(Polynomial.eval z (Polynomial.derivative F))‖ ≤
    1

Next tactic:
exact mul_le_one (PadicInt.norm_le_one _) (norm_nonneg _) h1