awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
K : Type u_2
inst_3 : Field K
inst_4 : Algebra R K
inst_5 : IsFractionRing R K
ϖ : R
hϖ : Irreducible ϖ
y : R
hy : Membership.mem (γ := Submonoid R) (nonZeroDivisors R) y
n : ℕ
u : Rˣ
hx :
  HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
      ((algebraMap R K : (a : R) → K) y : K) ≠
    0
⊢ ∃ (n_1 : ℤ) (u_1 : Rˣ),
    HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
        ((algebraMap R K : (a : R) → K) y : K) =
      u_1 • HPow.hPow (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) n_1
have obtain : ∀ (m : ℕ) (v : Rˣ) (hy : Membership.mem (γ := Submonoid R) (nonZeroDivisors R) ((↑v : R) * ϖ ^ m))
  (hx :
    HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
        ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) ≠
      0),
  ∃ (n_1 : ℤ) (u_1 : Rˣ),
    HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
        ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) =
      u_1 • HPow.hPow (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) n_1
Tactic: obtain ⟨m, v, rfl⟩ := eq_unit_mul_pow_irreducible (by simpa using hy) hϖ
Cleared hypotheses: y, hy, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
K : Type u_2
inst_3 : Field K
inst_4 : Algebra R K
inst_5 : IsFractionRing R K
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
u : Rˣ
m : ℕ
v : Rˣ
hy : Membership.mem (γ := Submonoid R) (nonZeroDivisors R) ((↑v : R) * ϖ ^ m)
hx :
  HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
      ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) ≠
    0
⊢ ∃ (n_1 : ℤ) (u_1 : Rˣ),
    HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
        ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) =
      u_1 • HPow.hPow (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) n_1
have hϖ' : Ne.{u_2 + 1} (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) (0 : K)
Tactic: have hϖ' : algebraMap R K ϖ ≠ 0 := by simpa using hϖ.ne_zero
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
K : Type u_2
inst_3 : Field K
inst_4 : Algebra R K
inst_5 : IsFractionRing R K
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
u : Rˣ
m : ℕ
v : Rˣ
hy : Membership.mem (γ := Submonoid R) (nonZeroDivisors R) ((↑v : R) * ϖ ^ m)
hx :
  HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
      ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) ≠
    0
hϖ' : Ne.{u_2 + 1} (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) (0 : K)
⊢ ∃ (n_1 : ℤ) (u_1 : Rˣ),
    HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
        ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) =
      u_1 • HPow.hPow (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) n_1
have refine : HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) ((↑u : R) * ϖ ^ n) : K)
    ((algebraMap R K : (a : R) → K) ((↑v : R) * ϖ ^ m) : K) =
  (u / v) • HPow.hPow (α := K) ((algebraMap R K : (a : R) → K) ϖ : K) (HSub.hSub (α := ℤ) (↑n : ℤ) (↑m : ℤ))
Tactic: refine ⟨n - m, u / v, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
⊢ generator s ≠ 0
have rw : ¬s = ⊥
Tactic: rw [Ne, ← eq_bot_iff_generator_eq_zero]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
⊢ ∃ (n : ℕ), s = span {ϖ ^ n}
have gen_ne_zero : generator s ≠ 0
Tactic: have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne, ← eq_bot_iff_generator_eq_zero]
    assumption
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
⊢ ∃ (n : ℕ), s = span {ϖ ^ n}
have rcases : ∀ (n : ℕ) (u : Rˣ) (hnu : generator s * (↑u : R) = ϖ ^ n), ∃ (n : ℕ), s = span {ϖ ^ n}
Tactic: rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * (↑u : R) = ϖ ^ n
⊢ ∃ (n : ℕ), s = span {ϖ ^ n}
have h : s = span {ϖ ^ n}
Tactic: use n
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * (↑u : R) = ϖ ^ n
⊢ s = span {ϖ ^ n}
have this : span {generator s} = s
Tactic: have : span _ = _ := Ideal.span_singleton_generator s
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * (↑u : R) = ϖ ^ n
this : span {generator s} = s
⊢ s = span {ϖ ^ n}
have h : Associated (generator s) (generator s * (↑u : R))
Tactic: rw [← this, ← hnu, span_singleton_eq_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * (↑u : R) = ϖ ^ n
this : span {generator s} = s
w : Rˣ
⊢ Associated (generator s) (generator s * (↑u : R))
have h : generator s * (↑u : R) = generator s * (↑u : R)
Tactic: use u
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
⊢ Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
have rw : ∃ (u : Rˣ), p ^ m * (↑u : R) = q ^ n
Tactic: rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
⊢ ∃ (u : Rˣ), p ^ m * (↑u : R) = q ^ n
have refine : p ^ m * (↑(u * v⁻¹) : R) = q ^ n
Tactic: refine ⟨u * v⁻¹, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
⊢ p ^ m * (↑(u * v⁻¹) : R) = q ^ n
have simp : p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
Tactic: simp only [Units.val_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw₁ : (↑u : R) * p ^ m * (↑v⁻¹ : R) = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
rw₁ : (↑u : R) * p ^ m * (↑v⁻¹ : R) = q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw₂ : (↑v : R) * q ^ n * (↑v⁻¹ : R) = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
rw₁ : (↑u : R) * p ^ m * (↑v⁻¹ : R) = q ^ n
rw₂ : (↑v : R) * q ^ n * (↑v⁻¹ : R) = q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw₃ : (↑v : R) * (↑v⁻¹ : R) * q ^ n = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
rw₁ : (↑u : R) * p ^ m * (↑v⁻¹ : R) = q ^ n
rw₂ : (↑v : R) * q ^ n * (↑v⁻¹ : R) = q ^ n
rw₃ : (↑v : R) * (↑v⁻¹ : R) * q ^ n = q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw₄ : 1 * q ^ n = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
rw : (↑u : R) * (p ^ m * (↑v⁻¹ : R)) = q ^ n
rw₁ : (↑u : R) * p ^ m * (↑v⁻¹ : R) = q ^ n
rw₂ : (↑v : R) * q ^ n * (↑v⁻¹ : R) = q ^ n
rw₃ : (↑v : R) * (↑v⁻¹ : R) * q ^ n = q ^ n
rw₄ : 1 * q ^ n = q ^ n
⊢ p ^ m * ((↑u : R) * (↑v⁻¹ : R)) = q ^ n
have rw₅ : q ^ n = q ^ n
Tactic: rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
⊢ m = n
have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
Tactic: have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine ⟨u * v⁻¹, ?_⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
⊢ (Multiset.replicate m p).card = (Multiset.replicate n q).card
have refine_1 : ∀ x ∈ Multiset.replicate m p, Irreducible x
Tactic: refine Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ key)
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
refine_1 : ∀ x ∈ Multiset.replicate m p, Irreducible x
⊢ (Multiset.replicate m p).card = (Multiset.replicate n q).card
have refine_2 : ∀ x ∈ Multiset.replicate n q, Irreducible x
Tactic: refine Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ key)
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate m p
hp₁ : Irreducible x
⊢ Irreducible x
have h₁ : (↑u : R) * x ^ m = (↑v : R) * q ^ n
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: p, hp, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate m p
hp₁ : Irreducible x
h₁ : (↑u : R) * x ^ m = (↑v : R) * q ^ n
⊢ Irreducible x
have key₁ : Associated (Multiset.replicate m x).prod (Multiset.replicate n q).prod
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: p, hp, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate m p
hp₁ : Irreducible x
h₁ : (↑u : R) * x ^ m = (↑v : R) * q ^ n
key₁ : Associated (Multiset.replicate m x).prod (Multiset.replicate n q).prod
⊢ Irreducible x
have hx₁ : x ∈ Multiset.replicate m x
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: p, hp, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate n q
hq₁ : Irreducible x
⊢ Irreducible x
have h₁ : (↑u : R) * p ^ m = (↑v : R) * x ^ n
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: q, hq, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate n q
hq₁ : Irreducible x
h₁ : (↑u : R) * p ^ m = (↑v : R) * x ^ n
⊢ Irreducible x
have key₁ : Associated (Multiset.replicate m p).prod (Multiset.replicate n x).prod
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: q, hq, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
x : R
hx : x ∈ Multiset.replicate n q
hq₁ : Irreducible x
h₁ : (↑u : R) * p ^ m = (↑v : R) * x ^ n
key₁ : Associated (Multiset.replicate m p).prod (Multiset.replicate n x).prod
⊢ Irreducible x
have hx₁ : x ∈ Multiset.replicate n x
Tactic: obtain rfl := Multiset.eq_of_mem_replicate hx
Cleared hypotheses: q, hq, h, key, hx
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : (↑u : R) * p ^ m = (↑v : R) * q ^ n
key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod
⊢ m = n
have this : (Multiset.replicate m p).card = (Multiset.replicate n q).card
Tactic: have := by
    refine Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ key)
    all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      assumption
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m n : ℕ
h : (↑u : R) * ϖ ^ m = (↑v : R) * ϖ ^ n
⊢ u = v
have h₁ : (↑u : R) * ϖ ^ m = (↑v : R) * ϖ ^ m
Tactic: obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
Cleared hypotheses: n, h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : (↑u : R) * ϖ ^ m = (↑v : R) * ϖ ^ m
⊢ u = v
have h₁ : (↑u : R) * ϖ ^ m - (↑v : R) * ϖ ^ m = 0
Tactic: rw [← sub_eq_zero] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : (↑u : R) * ϖ ^ m - (↑v : R) * ϖ ^ m = 0
⊢ u = v
have h₁ : (↑u : R) - (↑v : R) = 0 ∨ ϖ ^ m = 0
Tactic: rw [← sub_mul, mul_eq_zero] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : (↑u : R) - (↑v : R) = 0 ∨ ϖ ^ m = 0
⊢ u = v
have inl : ∀ (h : (↑u : R) - (↑v : R) = 0), u = v
Tactic: rcases h with h | h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : (↑u : R) - (↑v : R) = 0 ∨ ϖ ^ m = 0
inl : ∀ (h : (↑u : R) - (↑v : R) = 0), u = v
⊢ u = v
have inr : ∀ (h : ϖ ^ m = 0), u = v
Tactic: rcases h with h | h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : (↑u : R) - (↑v : R) = 0
⊢ u = v
have h₁ : (↑u : R) = (↑v : R)
Tactic: rw [sub_eq_zero] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = (↑u : R) * ϖ ^ n
⊢ Eq (α := ℕ∞) (((addVal R : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞) ((↑n : ℕ∞) : ℕ∞)
have rw : Eq (α := ℕ∞)
  (((multiplicity_addValuation (p := Classical.choose (p := Prime) sorry) sorry : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞)
  ((↑n : ℕ∞) : ℕ∞)
Tactic: rw [addVal, multiplicity_addValuation_apply, hr, emultiplicity_eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = (↑u : R) * ϖ ^ n
rw :
  Eq (α := ℕ∞)
    (((multiplicity_addValuation (p := Classical.choose (p := Prime) ⋯) ⋯ : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞)
    ((↑n : ℕ∞) : ℕ∞)
⊢ Eq (α := ℕ∞) (((addVal R : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞) ((↑n : ℕ∞) : ℕ∞)
have rw₁ : emultiplicity (Classical.choose (p := Prime) sorry) r = (↑n : ℕ∞)
Tactic: rw [addVal, multiplicity_addValuation_apply, hr, emultiplicity_eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = (↑u : R) * ϖ ^ n
rw :
  Eq (α := ℕ∞)
    (((multiplicity_addValuation (p := Classical.choose (p := Prime) ⋯) ⋯ : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞)
    ((↑n : ℕ∞) : ℕ∞)
rw₁ : emultiplicity (Classical.choose (p := Prime) ⋯) r = (↑n : ℕ∞)
⊢ Eq (α := ℕ∞) (((addVal R : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞) ((↑n : ℕ∞) : ℕ∞)
have rw₂ : emultiplicity (Classical.choose (p := Prime) sorry) ((↑u : R) * ϖ ^ n) = (↑n : ℕ∞)
Tactic: rw [addVal, multiplicity_addValuation_apply, hr, emultiplicity_eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = (↑u : R) * ϖ ^ n
rw :
  Eq (α := ℕ∞)
    (((multiplicity_addValuation (p := Classical.choose (p := Prime) ⋯) ⋯ : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞)
    ((↑n : ℕ∞) : ℕ∞)
rw₁ : emultiplicity (Classical.choose (p := Prime) ⋯) r = (↑n : ℕ∞)
rw₂ : emultiplicity (Classical.choose (p := Prime) ⋯) ((↑u : R) * ϖ ^ n) = (↑n : ℕ∞)
⊢ Eq (α := ℕ∞) (((addVal R : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞) ((↑n : ℕ∞) : ℕ∞)
have rw₃ : emultiplicity ϖ ((↑u : R) * ϖ ^ n) = (↑n : ℕ∞)
Tactic: rw [addVal, multiplicity_addValuation_apply, hr, emultiplicity_eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]
-- Lean 4 proof step from Mathlib.RingTheory.DiscreteValuationRing.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = (↑u : R) * ϖ ^ n
rw :
  Eq (α := ℕ∞)
    (((multiplicity_addValuation (p := Classical.choose (p := Prime) ⋯) ⋯ : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞)
    ((↑n : ℕ∞) : ℕ∞)
rw₁ : emultiplicity (Classical.choose (p := Prime) ⋯) r = (↑n : ℕ∞)
rw₂ : emultiplicity (Classical.choose (p := Prime) ⋯) ((↑u : R) * ϖ ^ n) = (↑n : ℕ∞)
rw₃ : emultiplicity ϖ ((↑u : R) * ϖ ^ n) = (↑n : ℕ∞)
⊢ Eq (α := ℕ∞) (((addVal R : AddValuation R ℕ∞) : (a : R) → ℕ∞) r : ℕ∞) ((↑n : ℕ∞) : ℕ∞)
have rw₄ : emultiplicity ϖ (ϖ ^ n) = (↑n : ℕ∞)
Tactic: rw [addVal, multiplicity_addValuation_apply, hr, emultiplicity_eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    emultiplicity_eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ)]