l3lab/ntp-mathlib huggingface dataset:
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
⊢ ∀ {x y : α}, x ~ᵤ y → P x → P y

Next tactic:
rintro x y ⟨u, rfl⟩ hx
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    

Proof state:
case intro
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
x : α
u : αˣ
hx : P x
⊢ P (x * ↑u)

Next tactic:
exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
⊢ P a

Next tactic:
by_cases ha0 : a = 0
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · 

Proof state:
case pos
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : a = 0
⊢ P a

Next tactic:
rwa [ha0]
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
⊢ P a

Next tactic:
haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this : Nontrivial α
⊢ P a

Next tactic:
letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ P a

Next tactic:
refine' P_of_associated (normalizedFactors_prod ha0) _
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ P (Multiset.prod (normalizedFactors a))

Next tactic:
rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ P (∏ m in Multiset.toFinset (normalizedFactors a), id m ^ Multiset.count m (normalizedFactors a))

Next tactic:
refine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
  refine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp <;> 

Proof state:
case neg.refine'_1
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ ∀ p ∈ Multiset.toFinset (normalizedFactors a), Prime p

Next tactic:
simp only [Multiset.mem_toFinset]
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
  refine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp <;> 

Proof state:
case neg.refine'_2
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ ∀ p ∈ Multiset.toFinset (normalizedFactors a), ∀ q ∈ Multiset.toFinset (normalizedFactors a), p ∣ q → p = q

Next tactic:
simp only [Multiset.mem_toFinset]
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
  refine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp <;> simp only [Multiset.mem_toFinset]
  · 

Proof state:
case neg.refine'_1
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, Prime p

Next tactic:
apply prime_of_normalized_factor
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
  by_cases ha0 : a = 0
  · rwa [ha0]
  haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
  letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
  refine' P_of_associated (normalizedFactors_prod ha0) _
  rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count]
  refine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp <;> simp only [Multiset.mem_toFinset]
  · apply prime_of_normalized_factor
  · 

Proof state:
case neg.refine'_2
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, ∀ q ∈ normalizedFactors a, p ∣ q → p = q

Next tactic:
apply normalizedFactors_eq_of_dvd
/-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f`
is multiplicative on coprime elements, then `f` is multiplicative on all products of primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem multiplicative_prime_power {f : α → β} (s : Finset α) (i j : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y)
    (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i)
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y) :
    f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p) := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
f : α → β
s : Finset α
i j : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y
⊢ f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p)

Next tactic:
letI := Classical.decEq α
/-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f`
is multiplicative on coprime elements, then `f` is multiplicative on all products of primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem multiplicative_prime_power {f : α → β} (s : Finset α) (i j : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y)
    (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i)
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y) :
    f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p) := by
  letI := Classical.decEq α
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
f : α → β
s : Finset α
i j : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y
this : DecidableEq α := Classical.decEq α
⊢ f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p)

Next tactic:
induction' s using Finset.induction_on with p s hps ih
/-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f`
is multiplicative on coprime elements, then `f` is multiplicative on all products of primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem multiplicative_prime_power {f : α → β} (s : Finset α) (i j : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y)
    (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i)
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y) :
    f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p) := by
  letI := Classical.decEq α
  induction' s using Finset.induction_on with p s hps ih
  · 

Proof state:
case empty
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
f : α → β
i j : α → ℕ
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y
this : DecidableEq α := Classical.decEq α
is_prime : ∀ p ∈ ∅, Prime p
is_coprime : ∀ p ∈ ∅, ∀ q ∈ ∅, p ∣ q → p = q
⊢ f (∏ p in ∅, p ^ (i p + j p)) = f (∏ p in ∅, p ^ i p) * f (∏ p in ∅, p ^ j p)

Next tactic:
simpa using h1 isUnit_one
/-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f`
is multiplicative on coprime elements, then `f` is multiplicative on all products of primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem multiplicative_prime_power {f : α → β} (s : Finset α) (i j : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y)
    (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i)
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y) :
    f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p) := by
  letI := Classical.decEq α
  induction' s using Finset.induction_on with p s hps ih
  · simpa using h1 isUnit_one
  

Proof state:
case insert
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
f : α → β
i j : α → ℕ
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y
this : DecidableEq α := Classical.decEq α
p : α
s : Finset α
hps : p ∉ s
ih :
  (∀ p ∈ s, Prime p) →
    (∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q) → f (∏ p in s, p ^ (i p + j p)) = f (∏ p in s, p ^ i p) * f (∏ p in s, p ^ j p)
is_prime : ∀ p_1 ∈ insert p s, Prime p_1
is_coprime : ∀ p_1 ∈ insert p s, ∀ q ∈ insert p s, p_1 ∣ q → p_1 = q
⊢ f (∏ p in insert p s, p ^ (i p + j p)) = f (∏ p in insert p s, p ^ i p) * f (∏ p in insert p s, p ^ j p)

Next tactic:
have hpr_p := is_prime _ (Finset.mem_insert_self _ _)