l3lab/ntp-mathlib huggingface dataset:
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x y : Associates α
hx : ¬x = 0
⊢ OneHom.toFun
      {
        toFun := fun a =>
          if a = 0 then 0
          else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
        map_one' :=
          (_ :
            (if 1 = 0 then 0
              else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
              1) }
      (x * y) =
    OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        x *
      OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        y

Next tactic:
by_cases hy : y = 0
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · 

Proof state:
case pos
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x y : Associates α
hx : ¬x = 0
hy : y = 0
⊢ OneHom.toFun
      {
        toFun := fun a =>
          if a = 0 then 0
          else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
        map_one' :=
          (_ :
            (if 1 = 0 then 0
              else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
              1) }
      (x * y) =
    OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        x *
      OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        y

Next tactic:
simp [hy]
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x y : Associates α
hx : ¬x = 0
hy : ¬y = 0
⊢ OneHom.toFun
      {
        toFun := fun a =>
          if a = 0 then 0
          else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
        map_one' :=
          (_ :
            (if 1 = 0 then 0
              else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
              1) }
      (x * y) =
    OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        x *
      OneHom.toFun
        {
          toFun := fun a =>
            if a = 0 then 0
            else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),
          map_one' :=
            (_ :
              (if 1 = 0 then 0
                else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =
                1) }
        y

Next tactic:
simp [hx, hy]
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
⊢ Associates.mk
      (if x = 0 then 0
      else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =
    x

Next tactic:
by_cases hx : x = 0
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · 

Proof state:
case pos
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : x = 0
⊢ Associates.mk
      (if x = 0 then 0
      else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =
    x

Next tactic:
simp [hx]
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : ¬x = 0
⊢ Associates.mk
      (if x = 0 then 0
      else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =
    x

Next tactic:
have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        ext x
        rw [Function.comp_apply, mkMonoidHom_apply,
          Classical.choose_spec mk_surjective.hasRightInverse x]
        rfl
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : ¬x = 0
⊢ ⇑Associates.mkMonoidHom ∘ Classical.choose (_ : Function.HasRightInverse Associates.mk) = id

Next tactic:
ext x
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        ext x
        

Proof state:
case h
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x✝ : Associates α
hx : ¬x✝ = 0
x : Associates α
⊢ (⇑Associates.mkMonoidHom ∘ Classical.choose (_ : Function.HasRightInverse Associates.mk)) x = id x

Next tactic:
rw [Function.comp_apply, mkMonoidHom_apply,
          Classical.choose_spec mk_surjective.hasRightInverse x]
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        ext x
        rw [Function.comp_apply, mkMonoidHom_apply,
          Classical.choose_spec mk_surjective.hasRightInverse x]
        

Proof state:
case h
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x✝ : Associates α
hx : ¬x✝ = 0
x : Associates α
⊢ x = id x

Next tactic:
rfl
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        ext x
        rw [Function.comp_apply, mkMonoidHom_apply,
          Classical.choose_spec mk_surjective.hasRightInverse x]
        rfl
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : ¬x = 0
h : ⇑Associates.mkMonoidHom ∘ Classical.choose (_ : Function.HasRightInverse Associates.mk) = id
⊢ Associates.mk
      (if x = 0 then 0
      else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =
    x

Next tactic:
rw [if_neg hx, ← mkMonoidHom_apply, MonoidHom.map_multiset_prod, map_map, h, map_id, ←
        associated_iff_eq]
/-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/
protected noncomputable def normalizationMonoid : NormalizationMonoid α :=
  normalizationMonoidOfMonoidHomRightInverse
    { toFun := fun a : Associates α =>
        if a = 0 then 0
        else
          ((normalizedFactors a).map
              (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod
      map_one' := by simp
      map_mul' := fun x y => by
        by_cases hx : x = 0
        · simp [hx]
        by_cases hy : y = 0
        · simp [hy]
        simp [hx, hy] }
    (by
      intro x
      dsimp
      by_cases hx : x = 0
      · simp [hx]
      have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse =
          (id : Associates α → Associates α) := by
        ext x
        rw [Function.comp_apply, mkMonoidHom_apply,
          Classical.choose_spec mk_surjective.hasRightInverse x]
        rfl
      rw [if_neg hx, ← mkMonoidHom_apply, MonoidHom.map_multiset_prod, map_map, h, map_id, ←
        associated_iff_eq]
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : Nontrivial α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : ¬x = 0
h : ⇑Associates.mkMonoidHom ∘ Classical.choose (_ : Function.HasRightInverse Associates.mk) = id
⊢ prod (normalizedFactors x) ~ᵤ x

Next tactic:
apply normalizedFactors_prod hx
theorem no_factors_of_no_prime_factors {a b : R} (ha : a ≠ 0)
    (h : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : ∀ {d}, d ∣ a → d ∣ b → IsUnit d := fun {d} =>
  induction_on_prime d
    (by
      

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d
d : R
⊢ 0 ∣ a → 0 ∣ b → IsUnit 0

Next tactic:
simp only [zero_dvd_iff]
theorem no_factors_of_no_prime_factors {a b : R} (ha : a ≠ 0)
    (h : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : ∀ {d}, d ∣ a → d ∣ b → IsUnit d := fun {d} =>
  induction_on_prime d
    (by
      simp only [zero_dvd_iff]
      

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d
d : R
⊢ a = 0 → b = 0 → IsUnit 0

Next tactic:
intros
theorem no_factors_of_no_prime_factors {a b : R} (ha : a ≠ 0)
    (h : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : ∀ {d}, d ∣ a → d ∣ b → IsUnit d := fun {d} =>
  induction_on_prime d
    (by
      simp only [zero_dvd_iff]
      intros
      

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d
d : R
a✝¹ : a = 0
a✝ : b = 0
⊢ IsUnit 0

Next tactic:
contradiction
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
⊢ (∀ {d : R}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b

Next tactic:
refine' induction_on_prime c _ _ _
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · 

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
⊢ (∀ {d : R}, d ∣ a → d ∣ 0 → ¬Prime d) → a ∣ b * 0 → a ∣ b

Next tactic:
intro no_factors
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
no_factors : ∀ {d : R}, d ∣ a → d ∣ 0 → ¬Prime d
⊢ a ∣ b * 0 → a ∣ b

Next tactic:
simp only [dvd_zero, mul_zero, forall_prop_of_true]
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
no_factors : ∀ {d : R}, d ∣ a → d ∣ 0 → ¬Prime d
⊢ a ∣ b

Next tactic:
haveI := Classical.propDecidable
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
no_factors : ∀ {d : R}, d ∣ a → d ∣ 0 → ¬Prime d
this : (a : Prop) → Decidable a
⊢ a ∣ b

Next tactic:
exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · 

Proof state:
case refine'_2
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
⊢ ∀ (x : R), IsUnit x → (∀ {d : R}, d ∣ a → d ∣ x → ¬Prime d) → a ∣ b * x → a ∣ b

Next tactic:
rintro _ ⟨x, rfl⟩ _ a_dvd_bx
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    

Proof state:
case refine'_2.intro
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
x : Rˣ
a✝ : ∀ {d : R}, d ∣ a → d ∣ ↑x → ¬Prime d
a_dvd_bx : a ∣ b * ↑x
⊢ a ∣ b

Next tactic:
apply Units.dvd_mul_right.mp a_dvd_bx
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    apply Units.dvd_mul_right.mp a_dvd_bx
  · 

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
⊢ ∀ (a_1 p : R),
    a_1 ≠ 0 →
      Prime p →
        ((∀ {d : R}, d ∣ a → d ∣ a_1 → ¬Prime d) → a ∣ b * a_1 → a ∣ b) →
          (∀ {d : R}, d ∣ a → d ∣ p * a_1 → ¬Prime d) → a ∣ b * (p * a_1) → a ∣ b

Next tactic:
intro c p _ hp ih no_factors a_dvd_bpc
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    apply Units.dvd_mul_right.mp a_dvd_bx
  · intro c p _ hp ih no_factors a_dvd_bpc
    

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c✝ : R
ha : a ≠ 0
c p : R
a✝ : c ≠ 0
hp : Prime p
ih : (∀ {d : R}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b
no_factors : ∀ {d : R}, d ∣ a → d ∣ p * c → ¬Prime d
a_dvd_bpc : a ∣ b * (p * c)
⊢ a ∣ b

Next tactic:
apply ih fun {q} dvd_a dvd_c hq => no_factors dvd_a (dvd_c.mul_left _) hq