l3lab/ntp-mathlib huggingface dataset:
theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
  haveI := Classical.propDecidable
  by_contra nk
  refine' hf fun ε ε0 => _
  simp? [not_forall] at nk says
    simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk
  cases' f.cauchy₃ (half_pos ε0) with i hi
  

Proof state:
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε

Next tactic:
rcases nk _ (half_pos ε0) i with ⟨j, ij, hj⟩
theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
  haveI := Classical.propDecidable
  by_contra nk
  refine' hf fun ε ε0 => _
  simp? [not_forall] at nk says
    simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk
  cases' f.cauchy₃ (half_pos ε0) with i hi
  rcases nk _ (half_pos ε0) i with ⟨j, ij, hj⟩
  

Proof state:
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : i ≤ j
hj : abv (↑f j) < ε / 2
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε

Next tactic:
refine' ⟨j, fun k jk => _⟩
theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
  haveI := Classical.propDecidable
  by_contra nk
  refine' hf fun ε ε0 => _
  simp? [not_forall] at nk says
    simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk
  cases' f.cauchy₃ (half_pos ε0) with i hi
  rcases nk _ (half_pos ε0) i with ⟨j, ij, hj⟩
  refine' ⟨j, fun k jk => _⟩
  

Proof state:
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : i ≤ j
hj : abv (↑f j) < ε / 2
k : ℕ
jk : k ≥ j
⊢ abv (↑f k) < ε

Next tactic:
have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj)
theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
  haveI := Classical.propDecidable
  by_contra nk
  refine' hf fun ε ε0 => _
  simp? [not_forall] at nk says
    simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk
  cases' f.cauchy₃ (half_pos ε0) with i hi
  rcases nk _ (half_pos ε0) i with ⟨j, ij, hj⟩
  refine' ⟨j, fun k jk => _⟩
  have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj)
  

Proof state:
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this✝ : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : i ≤ j
hj : abv (↑f j) < ε / 2
k : ℕ
jk : k ≥ j
this : abv (↑f k - ↑f j + ↑f j) < ε / 2 + ε / 2
⊢ abv (↑f k) < ε

Next tactic:
rwa [sub_add_cancel, add_halves] at this
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  | ε, ε0 =>
    let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
    ⟨i, fun j ij => by
      

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
⊢ abv (f j - f i) < ε

Next tactic:
cases' hi _ le_rfl with h₁ h₂
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  | ε, ε0 =>
    let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
    ⟨i, fun j ij => by
      cases' hi _ le_rfl with h₁ h₂; 

Proof state:
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁ : abv (f i - ↑g i) < ε / 2 / 2
h₂ : ∀ k ≥ i, abv (↑g k - ↑g i) < ε / 2
⊢ abv (f j - f i) < ε

Next tactic:
rw [abv_sub abv] at h₁
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  | ε, ε0 =>
    let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
    ⟨i, fun j ij => by
      cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
      

Proof state:
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁ : abv (↑g i - f i) < ε / 2 / 2
h₂ : ∀ k ≥ i, abv (↑g k - ↑g i) < ε / 2
⊢ abv (f j - f i) < ε

Next tactic:
have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁)
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  | ε, ε0 =>
    let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
    ⟨i, fun j ij => by
      cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
      have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁)
      

Proof state:
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁ : abv (↑g i - f i) < ε / 2 / 2
h₂ : ∀ k ≥ i, abv (↑g k - ↑g i) < ε / 2
this : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2
⊢ abv (f j - f i) < ε

Next tactic:
have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h₂ _ ij))
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  | ε, ε0 =>
    let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
    ⟨i, fun j ij => by
      cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
      have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁)
      have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h₂ _ ij))
      

Proof state:
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁ : abv (↑g i - f i) < ε / 2 / 2
h₂ : ∀ k ≥ i, abv (↑g k - ↑g i) < ε / 2
this✝ : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2
this : abv (f j - ↑g j + (↑g i - f i) + (↑g j - ↑g i)) < ε / 2 / 2 + ε / 2 / 2 + ε / 2
⊢ abv (f j - f i) < ε

Next tactic:
rwa [add_halves, add_halves, add_right_comm, sub_add_sub_cancel, sub_add_sub_cancel] at this
theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
⊢ ¬LimZero f

Next tactic:
intro h
theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
  intro h
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
⊢ False

Next tactic:
have : LimZero (f - 0) := by simp [h]
theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
  intro h
  have : LimZero (f - 0) := by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
⊢ LimZero (f - 0)

Next tactic:
simp [h]
theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
  intro h
  have : LimZero (f - 0) := by simp [h]
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
this : LimZero (f - 0)
⊢ False

Next tactic:
exact hf this
theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 :=
  have : LimZero (f - 0) := hf
  have : LimZero (g * f) := mul_limZero_right _ <| by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this : LimZero (f - 0)
⊢ LimZero f

Next tactic:
simpa
theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 :=
  have : LimZero (f - 0) := hf
  have : LimZero (g * f) := mul_limZero_right _ <| by simpa
  show LimZero (g * f - 0) by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this✝ : LimZero (f - 0)
this : LimZero (g * f)
⊢ LimZero (g * f - 0)

Next tactic:
simpa
theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 :=
  have : LimZero (f - 0) := hf
  have : LimZero (f * g) := mul_limZero_left _ <| by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this : LimZero (f - 0)
⊢ LimZero f

Next tactic:
simpa
theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 :=
  have : LimZero (f - 0) := hf
  have : LimZero (f * g) := mul_limZero_left _ <| by simpa
  show LimZero (f * g - 0) by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this✝ : LimZero (f - 0)
this : LimZero (f * g)
⊢ LimZero (f * g - 0)

Next tactic:
simpa
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
⊢ False

Next tactic:
have hlz : LimZero (f * g) := by simpa
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
⊢ LimZero (f * g)

Next tactic:
simpa
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
⊢ False

Next tactic:
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
⊢ ¬LimZero f

Next tactic:
simpa using show ¬LimZero (f - 0) from hf
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
⊢ False

Next tactic:
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by 

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
⊢ ¬LimZero g

Next tactic:
simpa using show ¬LimZero (g - 0) from hg
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  

Proof state:
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
⊢ False

Next tactic:
rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  

Proof state:
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
⊢ False

Next tactic:
rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
⊢ False

Next tactic:
have : 0 < a1 * a2 := mul_pos ha1 ha2
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
⊢ False

Next tactic:
cases' hlz _ this with N hN
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  cases' hlz _ this with N hN
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
N : ℕ
hN : ∀ j ≥ N, abv (↑(f * g) j) < a1 * a2
⊢ False

Next tactic:
let i := max N (max N1 N2)
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  cases' hlz _ this with N hN
  let i := max N (max N1 N2)
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
N : ℕ
hN : ∀ j ≥ N, abv (↑(f * g) j) < a1 * a2
i : ℕ := max N (max N1 N2)
⊢ False

Next tactic:
have hN' := hN i (le_max_left _ _)
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  cases' hlz _ this with N hN
  let i := max N (max N1 N2)
  have hN' := hN i (le_max_left _ _)
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
N : ℕ
hN : ∀ j ≥ N, abv (↑(f * g) j) < a1 * a2
i : ℕ := max N (max N1 N2)
hN' : abv (↑(f * g) i) < a1 * a2
⊢ False

Next tactic:
have hN1' := hN1 i (le_trans (le_max_left _ _) (le_max_right _ _))
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  cases' hlz _ this with N hN
  let i := max N (max N1 N2)
  have hN' := hN i (le_max_left _ _)
  have hN1' := hN1 i (le_trans (le_max_left _ _) (le_max_right _ _))
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
N : ℕ
hN : ∀ j ≥ N, abv (↑(f * g) j) < a1 * a2
i : ℕ := max N (max N1 N2)
hN' : abv (↑(f * g) i) < a1 * a2
hN1' : a1 ≤ abv (↑f i)
⊢ False

Next tactic:
have hN1' := hN2 i (le_trans (le_max_right _ _) (le_max_right _ _))
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
  fun (this : LimZero (f * g - 0)) => by
  have hlz : LimZero (f * g) := by simpa
  have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
  have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
  rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩
  rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩
  have : 0 < a1 * a2 := mul_pos ha1 ha2
  cases' hlz _ this with N hN
  let i := max N (max N1 N2)
  have hN' := hN i (le_max_left _ _)
  have hN1' := hN1 i (le_trans (le_max_left _ _) (le_max_right _ _))
  have hN1' := hN2 i (le_trans (le_max_right _ _) (le_max_right _ _))
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
a2 : α
ha2 : a2 > 0
N2 : ℕ
hN2 : ∀ j ≥ N2, a2 ≤ abv (↑g j)
this : 0 < a1 * a2
N : ℕ
hN : ∀ j ≥ N, abv (↑(f * g) j) < a1 * a2
i : ℕ := max N (max N1 N2)
hN' : abv (↑(f * g) i) < a1 * a2
hN1'✝ : a1 ≤ abv (↑f i)
hN1' : a2 ≤ abv (↑g i)
⊢ False

Next tactic:
apply not_le_of_lt hN'