l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
instance : LawfulTraversable FreeSemigroup.{u} :=
  { instLawfulMonad with
    id_traverse := fun x ↦
      FreeSemigroup.recOnMul x (fun x ↦ rfl) fun x y ih1 ih2 ↦ by
        rw [traverse_mul, ih1, ih2, mul_map_seq]
    comp_traverse := fun f g x ↦
      recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, (· ∘ ·)])
        fun x y ih1 ih2 ↦ by (rw [traverse_mul, ih1, ih2,
          traverse_mul, Functor.Comp.map_mk]; simp only [Function.comp, functor_norm, traverse_mul])
    naturality := fun η α β f x ↦
      recOnPure x (fun x ↦ by 

Proof state:
α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
src✝ : LawfulMonad FreeSemigroup := instLawfulMonad
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeSemigroup α
x : α
⊢ (fun {α} => ApplicativeTransformation.app η α) (traverse f (pure x)) =
    traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) (pure x)

Next tactic:
simp only [traverse_pure, functor_norm, Function.comp]
@[to_additive]
instance : LawfulTraversable FreeSemigroup.{u} :=
  { instLawfulMonad with
    id_traverse := fun x ↦
      FreeSemigroup.recOnMul x (fun x ↦ rfl) fun x y ih1 ih2 ↦ by
        rw [traverse_mul, ih1, ih2, mul_map_seq]
    comp_traverse := fun f g x ↦
      recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, (· ∘ ·)])
        fun x y ih1 ih2 ↦ by (rw [traverse_mul, ih1, ih2,
          traverse_mul, Functor.Comp.map_mk]; simp only [Function.comp, functor_norm, traverse_mul])
    naturality := fun η α β f x ↦
      recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, Function.comp])
          (fun x y ih1 ih2 ↦ by 

Proof state:
α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
src✝ : LawfulMonad FreeSemigroup := instLawfulMonad
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeSemigroup α
x : α
y : FreeSemigroup α
ih1 :
  (fun {α} => ApplicativeTransformation.app η α) (traverse f (pure x)) =
    traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) (pure x)
ih2 :
  (fun {α} => ApplicativeTransformation.app η α) (traverse f y) =
    traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) y
⊢ (fun {α} => ApplicativeTransformation.app η α) (traverse f (pure x * y)) =
    traverse ((fun {α} => ApplicativeTransformation.app η α) ∘ f) (pure x * y)

Next tactic:
simp only [traverse_mul, functor_norm, ih1, ih2]
@[to_additive]
theorem toFreeSemigroup_comp_map (f : α → β) :
    toFreeSemigroup.comp (map f) = (FreeSemigroup.map f).comp toFreeSemigroup := by 

Proof state:
α : Type u
β : Type v
f : α → β
⊢ MulHom.comp toFreeSemigroup (map f) = MulHom.comp (FreeSemigroup.map f) toFreeSemigroup

Next tactic:
ext1
@[to_additive]
theorem toFreeSemigroup_comp_map (f : α → β) :
    toFreeSemigroup.comp (map f) = (FreeSemigroup.map f).comp toFreeSemigroup := by ext1; 

Proof state:
case h
α : Type u
β : Type v
f : α → β
⊢ ⇑(MulHom.comp toFreeSemigroup (map f)) ∘ of = ⇑(MulHom.comp (FreeSemigroup.map f) toFreeSemigroup) ∘ of

Next tactic:
rfl
@[to_additive (attr := simp)]
theorem length_toFreeSemigroup (x : FreeMagma α) : (toFreeSemigroup x).length = x.length :=
  FreeMagma.recOnMul x (fun x ↦ rfl) fun x y hx hy ↦ by
    

Proof state:
α : Type u
β : Type v
x✝ x y : FreeMagma α
hx : FreeSemigroup.length (toFreeSemigroup x) = length x
hy : FreeSemigroup.length (toFreeSemigroup y) = length y
⊢ FreeSemigroup.length (toFreeSemigroup (x * y)) = length (x * y)

Next tactic:
rw [map_mul, FreeSemigroup.length_mul, hx, hy]
@[to_additive (attr := simp)]
theorem length_toFreeSemigroup (x : FreeMagma α) : (toFreeSemigroup x).length = x.length :=
  FreeMagma.recOnMul x (fun x ↦ rfl) fun x y hx hy ↦ by
    rw [map_mul, FreeSemigroup.length_mul, hx, hy]; 

Proof state:
α : Type u
β : Type v
x✝ x y : FreeMagma α
hx : FreeSemigroup.length (toFreeSemigroup x) = length x
hy : FreeSemigroup.length (toFreeSemigroup y) = length y
⊢ length x + length y = length (x * y)

Next tactic:
rfl
/-- Isomorphism between `Magma.AssocQuotient (FreeMagma α)` and `FreeSemigroup α`. -/
@[to_additive "Isomorphism between `AddMagma.AssocQuotient (FreeAddMagma α)` and
`FreeAddSemigroup α`."]
def FreeMagmaAssocQuotientEquiv (α : Type u) :
    Magma.AssocQuotient (FreeMagma α) ≃* FreeSemigroup α :=
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).toMulEquiv
      (FreeSemigroup.lift (Magma.AssocQuotient.of ∘ FreeMagma.of))
      (by 

Proof state:
α : Type u
⊢ MulHom.comp (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of))
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup) =
    MulHom.id (Magma.AssocQuotient (FreeMagma α))

Next tactic:
ext
/-- Isomorphism between `Magma.AssocQuotient (FreeMagma α)` and `FreeSemigroup α`. -/
@[to_additive "Isomorphism between `AddMagma.AssocQuotient (FreeAddMagma α)` and
`FreeAddSemigroup α`."]
def FreeMagmaAssocQuotientEquiv (α : Type u) :
    Magma.AssocQuotient (FreeMagma α) ≃* FreeSemigroup α :=
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).toMulEquiv
      (FreeSemigroup.lift (Magma.AssocQuotient.of ∘ FreeMagma.of))
      (by ext; 

Proof state:
case h.h.h
α : Type u
x✝ : α
⊢ (⇑(MulHom.comp
            (MulHom.comp (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of))
              (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup))
            Magma.AssocQuotient.of) ∘
        FreeMagma.of)
      x✝ =
    (⇑(MulHom.comp (MulHom.id (Magma.AssocQuotient (FreeMagma α))) Magma.AssocQuotient.of) ∘ FreeMagma.of) x✝

Next tactic:
rfl
/-- Isomorphism between `Magma.AssocQuotient (FreeMagma α)` and `FreeSemigroup α`. -/
@[to_additive "Isomorphism between `AddMagma.AssocQuotient (FreeAddMagma α)` and
`FreeAddSemigroup α`."]
def FreeMagmaAssocQuotientEquiv (α : Type u) :
    Magma.AssocQuotient (FreeMagma α) ≃* FreeSemigroup α :=
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).toMulEquiv
      (FreeSemigroup.lift (Magma.AssocQuotient.of ∘ FreeMagma.of))
      (by ext; rfl)
      (by 

Proof state:
α : Type u
⊢ MulHom.comp (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup)
      (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of)) =
    MulHom.id (FreeSemigroup α)

Next tactic:
ext1
/-- Isomorphism between `Magma.AssocQuotient (FreeMagma α)` and `FreeSemigroup α`. -/
@[to_additive "Isomorphism between `AddMagma.AssocQuotient (FreeAddMagma α)` and
`FreeAddSemigroup α`."]
def FreeMagmaAssocQuotientEquiv (α : Type u) :
    Magma.AssocQuotient (FreeMagma α) ≃* FreeSemigroup α :=
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).toMulEquiv
      (FreeSemigroup.lift (Magma.AssocQuotient.of ∘ FreeMagma.of))
      (by ext; rfl)
      (by ext1; 

Proof state:
case h
α : Type u
⊢ ⇑(MulHom.comp (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup)
          (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of))) ∘
      FreeSemigroup.of =
    ⇑(MulHom.id (FreeSemigroup α)) ∘ FreeSemigroup.of

Next tactic:
rfl
theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
    A.det = (Matrix.charpoly A).roots.prod := by
  

Proof state:
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
R : Type u_2
inst✝ : Field R
A : Matrix n n R
hAps : Splits (RingHom.id R) (charpoly A)
⊢ det A = Multiset.prod (roots (charpoly A))

Next tactic:
rw [det_eq_sign_charpoly_coeff, ← charpoly_natDegree_eq_dim A,
    Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic hAps, ← mul_assoc,
    ← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
    A.trace = (Matrix.charpoly A).roots.sum := by
  

Proof state:
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
R : Type u_2
inst✝ : Field R
A : Matrix n n R
hAps : Splits (RingHom.id R) (charpoly A)
⊢ trace A = Multiset.sum (roots (charpoly A))

Next tactic:
cases' isEmpty_or_nonempty n with h
theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
    A.trace = (Matrix.charpoly A).roots.sum := by
  cases' isEmpty_or_nonempty n with h
  · 

Proof state:
case inl
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
R : Type u_2
inst✝ : Field R
A : Matrix n n R
hAps : Splits (RingHom.id R) (charpoly A)
h : IsEmpty n
⊢ trace A = Multiset.sum (roots (charpoly A))

Next tactic:
rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,
      det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,
      Multiset.empty_eq_zero, Multiset.sum_zero]
theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
    A.trace = (Matrix.charpoly A).roots.sum := by
  cases' isEmpty_or_nonempty n with h
  · rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,
      det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,
      Multiset.empty_eq_zero, Multiset.sum_zero]
  · 

Proof state:
case inr
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
R : Type u_2
inst✝ : Field R
A : Matrix n n R
hAps : Splits (RingHom.id R) (charpoly A)
h✝ : Nonempty n
⊢ trace A = Multiset.sum (roots (charpoly A))

Next tactic:
rw [trace_eq_neg_charpoly_coeff, neg_eq_iff_eq_neg,
      ← Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split A.charpoly_monic hAps, nextCoeff,
      charpoly_natDegree_eq_dim, if_neg (Fintype.card_ne_zero : Fintype.card n ≠ 0)]
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ IsUnit a ↔ a = 1

Next tactic:
refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ a = 1 → IsUnit a

Next tactic:
rintro rfl
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      

Proof state:
b : Cardinal.{u}
n m : ℕ
⊢ IsUnit 1

Next tactic:
exact isUnit_one
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
h : IsUnit a
⊢ a = 1

Next tactic:
rcases eq_or_ne a 0 with (rfl | ha)
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · 

Proof state:
case inl
b : Cardinal.{u}
n m : ℕ
h : IsUnit 0
⊢ 0 = 1

Next tactic:
exact (not_isUnit_zero h).elim
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  

Proof state:
case inr
a b : Cardinal.{u}
n m : ℕ
h : IsUnit a
ha : a ≠ 0
⊢ a = 1

Next tactic:
rw [isUnit_iff_forall_dvd] at h
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  

Proof state:
case inr
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
⊢ a = 1

Next tactic:
cases' h 1 with t ht
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  

Proof state:
case inr.intro
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : 1 = a * t
⊢ a = 1

Next tactic:
rw [eq_comm, mul_eq_one_iff'] at ht
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · 

Proof state:
case inr.intro
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a = 1 ∧ t = 1
⊢ a = 1

Next tactic:
exact ht.1
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · exact ht.1
  · 

Proof state:
case inr.intro.ha
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a * t = 1
⊢ 1 ≤ a

Next tactic:
exact one_le_iff_ne_zero.mpr ha
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · exact ht.1
  · exact one_le_iff_ne_zero.mpr ha
  · 

Proof state:
case inr.intro.hb
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a * t = 1
⊢ 1 ≤ t

Next tactic:
apply one_le_iff_ne_zero.mpr
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · exact ht.1
  · exact one_le_iff_ne_zero.mpr ha
  · apply one_le_iff_ne_zero.mpr
    

Proof state:
case inr.intro.hb
a b : Cardinal.{u}
n m : ℕ
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a * t = 1
⊢ t ≠ 0

Next tactic:
intro h
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · exact ht.1
  · exact one_le_iff_ne_zero.mpr ha
  · apply one_le_iff_ne_zero.mpr
    intro h
    

Proof state:
case inr.intro.hb
a b : Cardinal.{u}
n m : ℕ
h✝ : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a * t = 1
h : t = 0
⊢ False

Next tactic:
rw [h, mul_zero] at ht
@[simp]
theorem isUnit_iff : IsUnit a ↔ a = 1 := by
  refine'
    ⟨fun h => _, by
      rintro rfl
      exact isUnit_one⟩
  rcases eq_or_ne a 0 with (rfl | ha)
  · exact (not_isUnit_zero h).elim
  rw [isUnit_iff_forall_dvd] at h
  cases' h 1 with t ht
  rw [eq_comm, mul_eq_one_iff'] at ht
  · exact ht.1
  · exact one_le_iff_ne_zero.mpr ha
  · apply one_le_iff_ne_zero.mpr
    intro h
    rw [h, mul_zero] at ht
    

Proof state:
case inr.intro.hb
a b : Cardinal.{u}
n m : ℕ
h✝ : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : 0 = 1
h : t = 0
⊢ False

Next tactic:
exact zero_ne_one ht
theorem le_of_dvd : ∀ {a b : Cardinal}, b ≠ 0 → a ∣ b → a ≤ b
  | a, x, b0, ⟨b, hab⟩ => by
    

Proof state:
a✝ b✝ : Cardinal.{u}
n m : ℕ
a x : Cardinal.{u_1}
b0 : x ≠ 0
b : Cardinal.{u_1}
hab : x = a * b
⊢ a ≤ x

Next tactic:
simpa only [hab, mul_one] using
      mul_le_mul_left' (one_le_iff_ne_zero.2 fun h : b = 0 => b0 (by rwa [h, mul_zero] at hab)) a
theorem le_of_dvd : ∀ {a b : Cardinal}, b ≠ 0 → a ∣ b → a ≤ b
  | a, x, b0, ⟨b, hab⟩ => by
    simpa only [hab, mul_one] using
      mul_le_mul_left' (one_le_iff_ne_zero.2 fun h : b = 0 => b0 (by 

Proof state:
a✝ b✝ : Cardinal.{u}
n m : ℕ
a x : Cardinal.{u_1}
b0 : x ≠ 0
b : Cardinal.{u_1}
hab : x = a * b
h : b = 0
⊢ x = 0

Next tactic:
rwa [h, mul_zero] at hab
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
⊢ Prime a

Next tactic:
refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · 

Proof state:
case refine'_1
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
⊢ ¬IsUnit a

Next tactic:
rw [isUnit_iff]
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    

Proof state:
case refine'_1
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
⊢ ¬a = 1

Next tactic:
exact (one_lt_aleph0.trans_le ha).ne'
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  

Proof state:
case refine'_2
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
⊢ a ∣ b ∨ a ∣ c

Next tactic:
rcases eq_or_ne (b * c) 0 with hz | hz
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · 

Proof state:
case refine'_2.inl
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c = 0
⊢ a ∣ b ∨ a ∣ c

Next tactic:
rcases mul_eq_zero.mp hz with (rfl | rfl)
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> 

Proof state:
case refine'_2.inl.inl
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
c : Cardinal.{u}
hbc : a ∣ 0 * c
hz : 0 * c = 0
⊢ a ∣ 0 ∨ a ∣ c

Next tactic:
simp
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> 

Proof state:
case refine'_2.inl.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b : Cardinal.{u}
hbc : a ∣ b * 0
hz : b * 0 = 0
⊢ a ∣ b ∨ a ∣ 0

Next tactic:
simp
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  

Proof state:
case refine'_2.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
⊢ a ∣ b ∨ a ∣ c

Next tactic:
wlog h : c ≤ b
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · 

Proof state:
case refine'_2.inr.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
⊢ a ∣ b ∨ a ∣ c

Next tactic:
cases le_total c b <;> [skip; rw [or_comm]]
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · 

Proof state:
case refine'_2.inr.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
⊢ a ∣ b ∨ a ∣ c

Next tactic:
cases le_total c b
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [

Proof state:
case refine'_2.inr.inr.inl
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : c ≤ b
⊢ a ∣ b ∨ a ∣ c

Next tactic:
skip
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; 

Proof state:
case refine'_2.inr.inr.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
⊢ a ∣ b ∨ a ∣ c

Next tactic:
rw [or_comm]
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> 

Proof state:
case refine'_2.inr.inr.inl
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : c ≤ b
⊢ a ∣ b ∨ a ∣ c

Next tactic:
apply_assumption
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> 

Proof state:
case refine'_2.inr.inr.inr
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
⊢ a ∣ c ∨ a ∣ b

Next tactic:
apply_assumption
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    

Proof state:
case refine'_2.inr.inr.inr.hbc
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
hz_symm : 0 ≠ b * c
⊢ a ∣ c * b
case refine'_2.inr.inr.inr.hz
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
hz_symm : 0 ≠ b * c
⊢ c * b ≠ 0

Next tactic:
all_goals rwa [mul_comm]