awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
⊢ UniqueFactorizationMonoid R ↔ ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x

Tactic:
⟨fun _ _ hI hI₂ ↦ hI₂.exists_mem_prime_of_ne_bot hI,
    fun H ↦ of_exists_prime_mem_of_isPrime H⟩

Premises used: _private.0.UniqueFactorizationMonoid.of_exists_prime_mem_of_isPrime, Iff.intro, Ideal.IsPrime.exists_mem_prime_of_ne_bot
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
n : ℕ
e_t : (factors n).card = (ArithmeticFunction.cardFactors : (a : ℕ) → ℕ) n
⊢ (if Squarefree n then HPow.hPow (α := ℤ) (-1) (factors n).card else 0) = (ArithmeticFunction.moebius : (a : ℕ) → ℤ) n

Tactic:
congr
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
n : ℕ
⊢ (factors n).card = (ArithmeticFunction.cardFactors : (a : ℕ) → ℕ) n

Tactic:
simp [Nat.factors_eq, ArithmeticFunction.cardFactors_apply]

Premises used: ArithmeticFunction.cardFactors_apply, Nat.factors_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
ha : Squarefree a
⊢ moebius a = HPow.hPow (α := ℤ) (-1) (factors a).card

Tactic:
if_pos ha

Premises used: if_pos, UniqueFactorizationMonoid.moebius
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
ha : ¬Squarefree a
⊢ moebius a = 0

Tactic:
if_neg ha

Premises used: UniqueFactorizationMonoid.moebius, if_neg
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
⊢ moebius.{u_1} (α := α) 0 = 0

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
⊢ moebius.{u_1} (α := α) 1 = 1

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : Associated a b
rw : (if Squarefree a then HPow.hPow (α := ℤ) (-1) (factors a).card else 0) = moebius b
rw₁ :
  (if Squarefree a then HPow.hPow (α := ℤ) (-1) (factors a).card else 0) =
    if Squarefree b then HPow.hPow (α := ℤ) (-1) (factors b).card else 0
rw₂ :
  (if Squarefree b then HPow.hPow (α := ℤ) (-1) (factors a).card else 0) =
    if Squarefree b then HPow.hPow (α := ℤ) (-1) (factors b).card else 0
rw₃ rw₄ :
  (if Squarefree b then HPow.hPow (α := ℤ) (-1) (factors b).card else 0) =
    if Squarefree b then HPow.hPow (α := ℤ) (-1) (factors b).card else 0
⊢ moebius a = moebius b

Tactic:
rw [moebius, moebius, h.squarefree_iff, h.card_factors_eq]

Premises used: Associated.squarefree_iff, Associated.card_factors_eq, UniqueFactorizationMonoid.moebius
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
ha : IsUnit a
rw : moebius.{u_1} (α := α) 1 = 1
rw₁ rw₂ : Eq (α := ℤ) 1 1
⊢ moebius a = 1

Tactic:
rw [(associated_one_iff_isUnit.mpr ha).moebius_eq, moebius_one]

Premises used: Iff.mpr, UniqueFactorizationMonoid.moebius_one, associated_one_iff_isUnit, Associated.moebius_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
rw : HPow.hPow (α := ℤ) (-1) (factors a).card = -1
rw₁ : HPow.hPow (α := ℤ) (β := ℕ) (-1) 1 = -1
rw₂ rw₃ : Eq (α := ℤ) (-1) (-1)
⊢ moebius a = -1

Tactic:
rw [ha.squarefree.moebius_eq, card_factors_of_irreducible ha, pow_one]

Premises used: Irreducible.squarefree, pow_one, UniqueFactorizationMonoid.card_factors_of_irreducible, Squarefree.moebius_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
inl : Subsingleton α → moebius (a * b) = moebius a * moebius b
inr : Nontrivial α → moebius (a * b) = moebius a * moebius b
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
rcases subsingleton_or_nontrivial α

Premises used: subsingleton_or_nontrivial
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Subsingleton α
inl : moebius (1 * b) = moebius.{u_1} (α := α) 1 * moebius b
inl₁ : moebius (1 * b) = 1 * moebius b
inl₂ : moebius b = 1 * moebius b
inl₃ inl₄ : moebius b = moebius b
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
rw [Subsingleton.elim a 1, moebius_one, one_mul, one_mul]

Premises used: UniqueFactorizationMonoid.moebius_one, Subsingleton.elim, OfNat.ofNat, one_mul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
pos : ∀ (ha : Squarefree a), moebius (a * b) = moebius a * moebius b
neg : ∀ (ha : ¬Squarefree a), moebius (a * b) = moebius a * moebius b
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
by_cases ha : Squarefree a

Premises used: Squarefree
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
ha : ¬Squarefree a
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
simp [ha, mt Squarefree.of_mul_left ha]

Premises used: Squarefree.of_mul_left, mt
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
ha : Squarefree a
pos : ∀ (hb : Squarefree b), moebius (a * b) = moebius a * moebius b
neg : ∀ (hb : ¬Squarefree b), moebius (a * b) = moebius a * moebius b
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
by_cases hb : Squarefree b

Premises used: Squarefree
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
ha : Squarefree a
hb : ¬Squarefree b
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
simp [hb, mt Squarefree.of_mul_right hb]

Premises used: mt, Squarefree.of_mul_right
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
ha : Squarefree a
hb : Squarefree b
⊢ Squarefree (a * b)

Tactic:
have hab : Squarefree (a * b) := squarefree_mul_iff.mpr ⟨h, ha, hb⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Moebius
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : IsRelPrime a b
h_1 : Nontrivial α
ha : Squarefree a
hb : Squarefree b
hab : Squarefree (a * b)
pos : HPow.hPow (α := ℤ) (-1) (factors (a * b)).card = moebius a * moebius b
pos₁ : HPow.hPow (α := ℤ) (-1) (factors a + factors b).card = moebius a * moebius b
pos₂ : HPow.hPow (α := ℤ) (-1) ((factors a).card + (factors b).card) = moebius a * moebius b
pos₃ : HPow.hPow (α := ℤ) (-1) (factors a).card * HPow.hPow (α := ℤ) (-1) (factors b).card = moebius a * moebius b
pos₄ :
  HPow.hPow (α := ℤ) (-1) (factors a).card * HPow.hPow (α := ℤ) (-1) (factors b).card =
    HPow.hPow (α := ℤ) (-1) (factors a).card * moebius b
pos₅ pos₆ :
  HPow.hPow (α := ℤ) (-1) (factors a).card * HPow.hPow (α := ℤ) (-1) (factors b).card =
    HPow.hPow (α := ℤ) (-1) (factors a).card * HPow.hPow (α := ℤ) (-1) (factors b).card
⊢ moebius (a * b) = moebius a * moebius b

Tactic:
rw [hab.moebius_eq, Multiset.card_eq_card_of_rel (factors_mul ha.ne_zero hb.ne_zero),
    Multiset.card_add, pow_add, ha.moebius_eq, hb.moebius_eq]

Premises used: Multiset.card_add, pow_add, Squarefree.ne_zero, UniqueFactorizationMonoid.factors_mul, Multiset.card_eq_card_of_rel, Squarefree.moebius_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
⊢ Prime p

Tactic:
have hp := is_prime _ (Finset.mem_insert_self _ _)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
⊢ IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p')

Tactic:
(isRelPrime_iff_no_prime_factors <| pow_ne_zero _ hp.ne_zero).mpr ?_

Premises used: Iff.mpr, pow_ne_zero, Prime.ne_zero, UniqueFactorizationMonoid.isRelPrime_iff_no_prime_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
refine : ∀ ⦃d : α⦄, d ∣ p ^ i p → d ∣ ∏ p' ∈ s, p' ^ i p' → ¬Prime d
⊢ IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p')

Tactic:
refine (isRelPrime_iff_no_prime_factors <| pow_ne_zero _ hp.ne_zero).mpr ?_

Premises used: Iff.mpr, pow_ne_zero, Prime.ne_zero, UniqueFactorizationMonoid.isRelPrime_iff_no_prime_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdp : d ∣ p ^ i p
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
apply : p ∈ s
⊢ False

Tactic:
apply hps
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdp : d ∣ p ^ i p
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
⊢ d ∣ p

Tactic:
replace hdp := hd.dvd_of_dvd_pow hdp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
obtain :
  ∀ (q : α) (q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) q) (hdq : d ∣ q),
    p ∈ s
⊢ p ∈ s

Tactic:
obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod

Premises used: Prime.exists_mem_multiset_dvd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) q
hdq : d ∣ q
obtain :
  ∀ (q : α) (q_mem : q ∈ s.val)
    (q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q))
    (hdq : d ∣ q ^ i q), p ∈ s
⊢ p ∈ s

Tactic:
obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'

Premises used: Multiset.mem_map, Iff.mp, rfl
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q)
hdq : d ∣ q ^ i q
⊢ d ∣ q

Tactic:
replace hdq := hd.dvd_of_dvd_pow hdq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q)
hdq : d ∣ q
⊢ p ∣ q

Tactic:
have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q)
hdq : d ∣ q
this : p ∣ q
⊢ q ∈ s.val

Tactic:
q_mem
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q)
hdq : d ∣ q
this : p ∣ q
a : p ∈ s ↔ q ∈ s.val
⊢ p ∈ s

Tactic:
convert! q_mem using 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
d : α
hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : Membership.mem (γ := Multiset α) (Multiset.map (fun (p' : α) ↦ p' ^ i p') s.val) (q ^ i q)
hdq : d ∣ q
this : p ∣ q
a : p ∈ s ↔ q ∈ s
a₁ a₂ : q ∈ s ↔ q ∈ s
⊢ p ∈ s ↔ q ∈ s.val

Tactic:
rw [Finset.mem_val,
    is_coprime _ (Finset.mem_insert_self p s) _ (Finset.mem_insert_of_mem q_mem) this]

Premises used: Finset.mem_insert_self, Finset.mem_insert_of_mem, Finset.mem_val
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
s : Finset α
i : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
letI : P (∏ p ∈ s, p ^ i p)
⊢ P (∏ p ∈ s, p ^ i p)

Tactic:
letI := Classical.decEq α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
is_prime : ∀ (p : α), Membership.mem (γ := Finset α) ∅ p → Prime p
is_coprime :
  ∀ (p : α), Membership.mem (γ := Finset α) ∅ p → ∀ (q : α), Membership.mem (γ := Finset α) ∅ q → p ∣ q → p = q
⊢ P (∏ p ∈ ∅, p ^ i p)

Tactic:
simpa using h1 isUnit_one

Premises used: isUnit_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf' : p ∉ f'
ih : ∀ (is_prime : ∀ p ∈ f', Prime p) (is_coprime : ∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q), P (∏ p ∈ f', p ^ i p)
is_prime : ∀ p_1 ∈ Insert.insert p f', Prime p_1
is_coprime : ∀ p_1 ∈ Insert.insert p f', ∀ q ∈ Insert.insert p f', p_1 ∣ q → p_1 = q
insert : P (p ^ i p * ∏ x ∈ f', x ^ i x)
⊢ P (∏ p ∈ Insert.insert p f', p ^ i p)

Tactic:
rw [Finset.prod_insert hpf']

Premises used: Finset.prod_insert
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf' : p ∉ f'
ih : ∀ (is_prime : ∀ p ∈ f', Prime p) (is_coprime : ∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q), P (∏ p ∈ f', p ^ i p)
is_prime : ∀ p_1 ∈ insert p f', Prime p_1
is_coprime : ∀ p_1 ∈ insert p f', ∀ q ∈ insert p f', p_1 ∣ q → p_1 = q
⊢ P (p ^ i p * ∏ x ∈ f', x ^ i x)

Tactic:
hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)
        (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))
        (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>
          is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq'))

Premises used: Finset.mem_insert_self, UniqueFactorizationMonoid.prime_pow_coprime_prod_of_coprime_insert, Finset.mem_insert_of_mem
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf' : p ∉ f'
ih : ∀ (is_prime : ∀ p ∈ f', Prime p) (is_coprime : ∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q), P (∏ p ∈ f', p ^ i p)
is_prime : ∀ p_1 ∈ insert p f', Prime p_1
is_coprime : ∀ p_1 ∈ insert p f', ∀ q ∈ insert p f', p_1 ∣ q → p_1 = q
⊢ P (p ^ i p * ∏ x ∈ f', x ^ i x)

Tactic:
exact
      hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)
        (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))
        (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>
          is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq'))

Premises used: Finset.mem_insert_self, UniqueFactorizationMonoid.prime_pow_coprime_prod_of_coprime_insert, Finset.mem_insert_of_mem
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
P : α → Prop
s : Finset α
i : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
empty :
  ∀ (is_prime : ∀ (p : α), Membership.mem (γ := Finset α) ∅ p → Prime p)
    (is_coprime :
      ∀ (p : α), Membership.mem (γ := Finset α) ∅ p → ∀ (q : α), Membership.mem (γ := Finset α) ∅ q → p ∣ q → p = q),
    P (∏ p ∈ ∅, p ^ i p)
insert :
  ∀ (p : α) (f' : Finset α) (hpf' : p ∉ f')
    (ih : ∀ (is_prime : ∀ p ∈ f', Prime p) (is_coprime : ∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q), P (∏ p ∈ f', p ^ i p))
    (is_prime : ∀ p_1 ∈ Insert.insert p f', Prime p_1)
    (is_coprime : ∀ p_1 ∈ Insert.insert p f', ∀ q ∈ Insert.insert p f', p_1 ∣ q → p_1 = q),
    P (∏ p ∈ Insert.insert p f', p ^ i p)
⊢ P (∏ p ∈ s, p ^ i p)

Tactic:
induction s using Finset.induction_on with
  | empty => simpa using h1 isUnit_one
  | insert p f' hpf' ih =>
    rw [Finset.prod_insert hpf']
    exact
      hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)
        (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))
        (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>
          is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq'))

Premises used: Insert.insert, isUnit_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
letI : P a
⊢ P a

Tactic:
letI := Classical.decEq α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
x : α
u : αˣ
hx : P x
⊢ P (x * (↑u : α))

Tactic:
hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)

Premises used: isUnit_of_dvd_unit, Units.isUnit
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
x : α
u : αˣ
hx : P x
⊢ P (x * (↑u : α))

Tactic:
exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)

Premises used: isUnit_of_dvd_unit, Units.isUnit
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
⊢ ∀ {x y : α}, Associated x y → P x → P y

Tactic:
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)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
pos : ∀ (ha0 : a = 0), P a
neg : ∀ (ha0 : ¬a = 0), P a
⊢ P a

Tactic:
by_cases ha0 : a = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
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}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : a = 0
pos pos₁ pos₂ pos₃ : P 0
⊢ P a

Tactic:
rwa [ha0]