l3lab/ntp-mathlib huggingface dataset:
noncomputable instance WithTop.WithBot.completeLattice {α : Type*}
    [ConditionallyCompleteLattice α] : CompleteLattice (WithTop (WithBot α)) :=
  { instInfSet, instSupSet, boundedOrder, lattice with
    le_sSup := fun S a haS => (WithTop.isLUB_sSup' ⟨a, haS⟩).1 haS
    sSup_le := fun S a ha => by
      rcases S.eq_empty_or_nonempty with h | h
      · show ite _ _ _ ≤ a
        split_ifs with h₁ h₂
        · rw [h] at h₁
          cases h₁
        · convert bot_le (a := a)
          -- porting note: previous proof relied on convert unfolding
          -- the definition of ⊥
          apply congr_arg
          simp only [h, preimage_empty, WithBot.csSup_empty]
        · exfalso
          apply h₂
          use ⊥
          rw [h]
          rintro b ⟨⟩
      · refine' (WithTop.isLUB_sSup' h).2 ha
    sInf_le := fun S a haS =>
      show ite _ _ _ ≤ a by
        split_ifs with h₁
        · cases' a with a
          exact le_rfl
          cases h₁ haS
        · cases a
          · exact le_top
          · apply WithTop.some_le_some.2
            refine' csInf_le _ haS
            

Proof state:
case neg.some
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
src✝³ : InfSet (WithTop (WithBot α)) := instInfSet
src✝² : SupSet (WithTop (WithBot α)) := instSupSet
src✝¹ : BoundedOrder (WithTop (WithBot α)) := boundedOrder
src✝ : Lattice (WithTop (WithBot α)) := lattice
S : Set (WithTop (WithBot α))
h₁ : ¬S ⊆ {⊤}
val✝ : WithBot α
haS : Option.some val✝ ∈ S
⊢ BddBelow ((fun a => ↑a) ⁻¹' S)

Next tactic:
use ⊥
noncomputable instance WithTop.WithBot.completeLattice {α : Type*}
    [ConditionallyCompleteLattice α] : CompleteLattice (WithTop (WithBot α)) :=
  { instInfSet, instSupSet, boundedOrder, lattice with
    le_sSup := fun S a haS => (WithTop.isLUB_sSup' ⟨a, haS⟩).1 haS
    sSup_le := fun S a ha => by
      rcases S.eq_empty_or_nonempty with h | h
      · show ite _ _ _ ≤ a
        split_ifs with h₁ h₂
        · rw [h] at h₁
          cases h₁
        · convert bot_le (a := a)
          -- porting note: previous proof relied on convert unfolding
          -- the definition of ⊥
          apply congr_arg
          simp only [h, preimage_empty, WithBot.csSup_empty]
        · exfalso
          apply h₂
          use ⊥
          rw [h]
          rintro b ⟨⟩
      · refine' (WithTop.isLUB_sSup' h).2 ha
    sInf_le := fun S a haS =>
      show ite _ _ _ ≤ a by
        split_ifs with h₁
        · cases' a with a
          exact le_rfl
          cases h₁ haS
        · cases a
          · exact le_top
          · apply WithTop.some_le_some.2
            refine' csInf_le _ haS
            use ⊥
            

Proof state:
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
src✝³ : InfSet (WithTop (WithBot α)) := instInfSet
src✝² : SupSet (WithTop (WithBot α)) := instSupSet
src✝¹ : BoundedOrder (WithTop (WithBot α)) := boundedOrder
src✝ : Lattice (WithTop (WithBot α)) := lattice
S : Set (WithTop (WithBot α))
h₁ : ¬S ⊆ {⊤}
val✝ : WithBot α
haS : Option.some val✝ ∈ S
⊢ ⊥ ∈ lowerBounds ((fun a => ↑a) ⁻¹' S)

Next tactic:
intro b _
noncomputable instance WithTop.WithBot.completeLattice {α : Type*}
    [ConditionallyCompleteLattice α] : CompleteLattice (WithTop (WithBot α)) :=
  { instInfSet, instSupSet, boundedOrder, lattice with
    le_sSup := fun S a haS => (WithTop.isLUB_sSup' ⟨a, haS⟩).1 haS
    sSup_le := fun S a ha => by
      rcases S.eq_empty_or_nonempty with h | h
      · show ite _ _ _ ≤ a
        split_ifs with h₁ h₂
        · rw [h] at h₁
          cases h₁
        · convert bot_le (a := a)
          -- porting note: previous proof relied on convert unfolding
          -- the definition of ⊥
          apply congr_arg
          simp only [h, preimage_empty, WithBot.csSup_empty]
        · exfalso
          apply h₂
          use ⊥
          rw [h]
          rintro b ⟨⟩
      · refine' (WithTop.isLUB_sSup' h).2 ha
    sInf_le := fun S a haS =>
      show ite _ _ _ ≤ a by
        split_ifs with h₁
        · cases' a with a
          exact le_rfl
          cases h₁ haS
        · cases a
          · exact le_top
          · apply WithTop.some_le_some.2
            refine' csInf_le _ haS
            use ⊥
            intro b _
            

Proof state:
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
src✝³ : InfSet (WithTop (WithBot α)) := instInfSet
src✝² : SupSet (WithTop (WithBot α)) := instSupSet
src✝¹ : BoundedOrder (WithTop (WithBot α)) := boundedOrder
src✝ : Lattice (WithTop (WithBot α)) := lattice
S : Set (WithTop (WithBot α))
h₁ : ¬S ⊆ {⊤}
val✝ : WithBot α
haS : Option.some val✝ ∈ S
b : WithBot α
a✝ : b ∈ (fun a => ↑a) ⁻¹' S
⊢ ⊥ ≤ b

Next tactic:
exact bot_le
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
⊢ ⨆ x, ↑(f x) = ⊤ ↔ ¬BddAbove (range f)

Next tactic:
rw [iSup_eq_top, not_bddAbove_iff]
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
⊢ (∀ b < ⊤, ∃ i, b < ↑(f i)) ↔ ∀ (x : α), ∃ y ∈ range f, x < y

Next tactic:
refine' ⟨fun hf r => _, fun hf a ha => _⟩
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  refine' ⟨fun hf r => _, fun hf a ha => _⟩
  · 

Proof state:
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ b < ⊤, ∃ i, b < ↑(f i)
r : α
⊢ ∃ y ∈ range f, r < y

Next tactic:
rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  refine' ⟨fun hf r => _, fun hf a ha => _⟩
  · rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩
    

Proof state:
case refine'_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ b < ⊤, ∃ i, b < ↑(f i)
r : α
i : ι
hi : ↑r < ↑(f i)
⊢ ∃ y ∈ range f, r < y

Next tactic:
exact ⟨f i, ⟨i, rfl⟩, WithTop.coe_lt_coe.mp hi⟩
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  refine' ⟨fun hf r => _, fun hf a ha => _⟩
  · rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩
    exact ⟨f i, ⟨i, rfl⟩, WithTop.coe_lt_coe.mp hi⟩
  · 

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ (x : α), ∃ y ∈ range f, x < y
a : WithTop α
ha : a < ⊤
⊢ ∃ i, a < ↑(f i)

Next tactic:
rcases hf (a.untop ha.ne) with ⟨-, ⟨i, rfl⟩, hi⟩
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  refine' ⟨fun hf r => _, fun hf a ha => _⟩
  · rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩
    exact ⟨f i, ⟨i, rfl⟩, WithTop.coe_lt_coe.mp hi⟩
  · rcases hf (a.untop ha.ne) with ⟨-, ⟨i, rfl⟩, hi⟩
    

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ (x : α), ∃ y ∈ range f, x < y
a : WithTop α
ha : a < ⊤
i : ι
hi : untop a (_ : a ≠ ⊤) < f i
⊢ ∃ i, a < ↑(f i)

Next tactic:
exact ⟨i, by simpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi⟩
lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by
  rw [iSup_eq_top, not_bddAbove_iff]
  refine' ⟨fun hf r => _, fun hf a ha => _⟩
  · rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩
    exact ⟨f i, ⟨i, rfl⟩, WithTop.coe_lt_coe.mp hi⟩
  · rcases hf (a.untop ha.ne) with ⟨-, ⟨i, rfl⟩, hi⟩
    exact ⟨i, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ (x : α), ∃ y ∈ range f, x < y
a : WithTop α
ha : a < ⊤
i : ι
hi : untop a (_ : a ≠ ⊤) < f i
⊢ a < ↑(f i)

Next tactic:
simpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi
lemma iInf_coe_eq_top : ⨅ x, (f x : WithTop α) = ⊤ ↔ IsEmpty ι := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
⊢ ⨅ x, ↑(f x) = ⊤ ↔ IsEmpty ι

Next tactic:
simp [isEmpty_iff]
lemma iInf_coe_lt_top : ⨅ i, (f i : WithTop α) < ⊤ ↔ Nonempty ι := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
⊢ ⨅ i, ↑(f i) < ⊤ ↔ Nonempty ι

Next tactic:
rw [lt_top_iff_ne_top, Ne.def, iInf_coe_eq_top, not_isEmpty_iff]
/-- The metric space uniform structure on ℚ (which presupposes the existence
of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ). -/
theorem Rat.uniformSpace_eq :
    (AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by
  

Proof state:
⊢ AbsoluteValue.uniformSpace AbsoluteValue.abs = PseudoMetricSpace.toUniformSpace

Next tactic:
ext s
/-- The metric space uniform structure on ℚ (which presupposes the existence
of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ). -/
theorem Rat.uniformSpace_eq :
    (AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by
  ext s
  

Proof state:
case a.a
s : Set (ℚ × ℚ)
⊢ s ∈ uniformity ℚ ↔ s ∈ uniformity ℚ

Next tactic:
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
/-- The metric space uniform structure on ℚ (which presupposes the existence
of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ). -/
theorem Rat.uniformSpace_eq :
    (AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by
  ext s
  rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
  

Proof state:
case a.a
s : Set (ℚ × ℚ)
⊢ (∃ i, 0 < i ∧ {p | AbsoluteValue.abs (p.2 - p.1) < i} ⊆ s) ↔ ∃ i, 0 < i ∧ {p | dist p.1 p.2 < ↑i} ⊆ s

Next tactic:
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
    abs_sub_comm]
/-- Cauchy reals packaged as a completion of ℚ using the absolute value route. -/
def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace :=
  @AbstractCompletion.mk
    (space := ℝ)
    (coe := ((↑) : ℚ → ℝ))
    (uniformStruct := by 

Proof state:
⊢ UniformSpace ℝ

Next tactic:
infer_instance
/-- Cauchy reals packaged as a completion of ℚ using the absolute value route. -/
def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace :=
  @AbstractCompletion.mk
    (space := ℝ)
    (coe := ((↑) : ℚ → ℝ))
    (uniformStruct := by infer_instance)
    (complete := by 

Proof state:
⊢ CompleteSpace ℝ

Next tactic:
infer_instance
/-- Cauchy reals packaged as a completion of ℚ using the absolute value route. -/
def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace :=
  @AbstractCompletion.mk
    (space := ℝ)
    (coe := ((↑) : ℚ → ℝ))
    (uniformStruct := by infer_instance)
    (complete := by infer_instance)
    (separation := by 

Proof state:
⊢ SeparatedSpace ℝ

Next tactic:
infer_instance
/-- Cauchy reals packaged as a completion of ℚ using the absolute value route. -/
def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace :=
  @AbstractCompletion.mk
    (space := ℝ)
    (coe := ((↑) : ℚ → ℝ))
    (uniformStruct := by infer_instance)
    (complete := by infer_instance)
    (separation := by infer_instance)
    (uniformInducing := by
      

Proof state:
⊢ UniformInducing Rat.cast

Next tactic:
rw [Rat.uniformSpace_eq]
/-- Cauchy reals packaged as a completion of ℚ using the absolute value route. -/
def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace :=
  @AbstractCompletion.mk
    (space := ℝ)
    (coe := ((↑) : ℚ → ℝ))
    (uniformStruct := by infer_instance)
    (complete := by infer_instance)
    (separation := by infer_instance)
    (uniformInducing := by
      rw [Rat.uniformSpace_eq]
      

Proof state:
⊢ UniformInducing Rat.cast

Next tactic:
exact Rat.uniformEmbedding_coe_real.toUniformInducing
theorem X_dvd_iff {f : R[X]} : X ∣ f ↔ f.coeff 0 = 0 :=
  ⟨fun ⟨g, hfg⟩ => by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
f : R[X]
x✝ : X ∣ f
g : R[X]
hfg : f = X * g
⊢ coeff f 0 = 0

Next tactic:
rw [hfg, coeff_X_mul_zero]
theorem X_dvd_iff {f : R[X]} : X ∣ f ↔ f.coeff 0 = 0 :=
  ⟨fun ⟨g, hfg⟩ => by rw [hfg, coeff_X_mul_zero], fun hf =>
    ⟨f.divX, by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
f : R[X]
hf : coeff f 0 = 0
⊢ f = X * divX f

Next tactic:
rw [← add_zero (X * f.divX), ← C_0, ← hf, X_mul_divX_add]
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
x✝ : X ^ n ∣ f
d : ℕ
hd : d < n
g : R[X]
hgf : f = X ^ n * g
⊢ coeff f d = 0

Next tactic:
simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff]
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hd : ∀ d < n, coeff f d = 0
⊢ X ^ n ∣ f

Next tactic:
induction' n with n hn
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · 

Proof state:
case zero
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
f : R[X]
hd : ∀ d < Nat.zero, coeff f d = 0
⊢ X ^ Nat.zero ∣ f

Next tactic:
simp [pow_zero, one_dvd]
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · 

Proof state:
case succ
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
⊢ X ^ Nat.succ n ∣ f

Next tactic:
obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
      

Proof state:
case succ.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
g : R[X]
hgf : f = X ^ n * g
⊢ X ^ Nat.succ n ∣ f

Next tactic:
have := coeff_X_pow_mul g n 0
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
      have := coeff_X_pow_mul g n 0
      

Proof state:
case succ.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
g : R[X]
hgf : f = X ^ n * g
this : coeff (X ^ n * g) (0 + n) = coeff g 0
⊢ X ^ Nat.succ n ∣ f

Next tactic:
rw [zero_add, ← hgf, hd n (Nat.lt_succ_self n)] at this
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
      have := coeff_X_pow_mul g n 0
      rw [zero_add, ← hgf, hd n (Nat.lt_succ_self n)] at this
      

Proof state:
case succ.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
g : R[X]
hgf : f = X ^ n * g
this : 0 = coeff g 0
⊢ X ^ Nat.succ n ∣ f

Next tactic:
obtain ⟨k, hgk⟩ := Polynomial.X_dvd_iff.mpr this.symm
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
      have := coeff_X_pow_mul g n 0
      rw [zero_add, ← hgf, hd n (Nat.lt_succ_self n)] at this
      obtain ⟨k, hgk⟩ := Polynomial.X_dvd_iff.mpr this.symm
      

Proof state:
case succ.intro.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
g : R[X]
hgf : f = X ^ n * g
this : 0 = coeff g 0
k : R[X]
hgk : g = X * k
⊢ X ^ Nat.succ n ∣ f

Next tactic:
use k
theorem X_pow_dvd_iff {f : R[X]} {n : ℕ} : X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0 :=
  ⟨fun ⟨g, hgf⟩ d hd => by
    simp only [hgf, coeff_X_pow_mul', ite_eq_right_iff, not_le_of_lt hd, IsEmpty.forall_iff],
    fun hd => by
    induction' n with n hn
    · simp [pow_zero, one_dvd]
    · obtain ⟨g, hgf⟩ := hn fun d : ℕ => fun H : d < n => hd _ (Nat.lt_succ_of_lt H)
      have := coeff_X_pow_mul g n 0
      rw [zero_add, ← hgf, hd n (Nat.lt_succ_self n)] at this
      obtain ⟨k, hgk⟩ := Polynomial.X_dvd_iff.mpr this.symm
      use k
      

Proof state:
case h
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
f : R[X]
n : ℕ
hn : (∀ d < n, coeff f d = 0) → X ^ n ∣ f
hd : ∀ d < Nat.succ n, coeff f d = 0
g : R[X]
hgf : f = X ^ n * g
this : 0 = coeff g 0
k : R[X]
hgk : g = X * k
⊢ f = X ^ Nat.succ n * k

Next tactic:
rwa [pow_succ', mul_assoc, ← hgk]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
⊢ False

Next tactic:
have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p = 0
⊢ False

Next tactic:
simp [hp0] at hp
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
⊢ False

Next tactic:
have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r = 0
⊢ False

Next tactic:
subst hr0
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
hp0 : p ≠ 0
hr : q = p ^ (natDegree q + 1) * 0
⊢ False

Next tactic:
simp [hq] at hr
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
⊢ False

Next tactic:
have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
⊢ leadingCoeff p ^ (natDegree q + 1) = 1

Next tactic:
simp [show _ = _ from hmp]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
⊢ False

Next tactic:
have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm