awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
a b : ℕ
⊢ Eq (α := ℕ∞)
    (if h : ∃ (n : ℕ), ¬HPow.hPow (α := ℤ) (↑a : ℤ) (n + 1) ∣ (↑b : ℤ) then
      (↑(Nat.find (p := fun (n : ℕ) ↦ ¬HPow.hPow (α := ℤ) (↑a : ℤ) (n + 1) ∣ (↑b : ℤ)) h) : ℕ∞)
    else ⊤)
    (if h : ∃ (n : ℕ), ¬a ^ (n + 1) ∣ b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
have h₂ : (∀ {b c : Prop} {α : Type} {x : Decidable b} [inst : Decidable c] {x_1 : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
    (h₁ : b = c) (h₂ : ∀ (h : c), x_1 sorry = u h) (h₃ : ∀ (h : ¬c), y sorry = v h), dite b x_1 y = dite c u v) →
  (∃ (n : ℕ), ¬a ^ (n + 1) ∣ b) → ∀ (x : ℕ), HPow.hPow (α := ℤ) (↑a : ℤ) (x + 1) ∣ (↑b : ℤ) ↔ a ^ (x + 1) ∣ b
Tactic: congr!
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
a b : ℕ
h₂ :
  (∀ {b c : Prop} {α : Type} {x : Decidable b} [inst : Decidable c] {x_1 : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
      (h₁ : b = c) (h₂ : ∀ (h : c), x_1 ⋯ = u h) (h₃ : ∀ (h : ¬c), y ⋯ = v h), dite b x_1 y = dite c u v) →
    (∃ (n : ℕ), ¬a ^ (n + 1) ∣ b) → ∀ (x : ℕ), HPow.hPow (α := ℤ) (↑a : ℤ) (x + 1) ∣ (↑b : ℤ) ↔ a ^ (x + 1) ∣ b
⊢ Eq (α := ℕ∞)
    (if h : ∃ (n : ℕ), ¬HPow.hPow (α := ℤ) (↑a : ℤ) (n + 1) ∣ (↑b : ℤ) then
      (↑(Nat.find (p := fun (n : ℕ) ↦ ¬HPow.hPow (α := ℤ) (↑a : ℤ) (n + 1) ∣ (↑b : ℤ)) h) : ℕ∞)
    else ⊤)
    (if h : ∃ (n : ℕ), ¬a ^ (n + 1) ∣ b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
have h₁ : ∀ (x : ℕ), HPow.hPow (α := ℤ) (↑a : ℤ) (x + 1) ∣ (↑b : ℤ) ↔ a ^ (x + 1) ∣ b
Tactic: congr!
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
a b : ℕ
h_congr_thm :
  ∀ {b c : Prop} {α : Type} {x : Decidable b} [inst : Decidable c] {x_1 : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
    (h₁ : b = c) (h₂ : ∀ (h : c), x_1 ⋯ = u h) (h₃ : ∀ (h : ¬c), y ⋯ = v h), dite b x_1 y = dite c u v
h : ∃ (n : ℕ), ¬a ^ (n + 1) ∣ b
x : ℕ
⊢ HPow.hPow (α := ℤ) (↑a : ℤ) (x + 1) ∣ (↑b : ℤ) ↔ a ^ (x + 1) ∣ b
have h₂ : a ^ (x + 1) ∣ b ↔ a ^ (x + 1) ∣ b
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
a b x : ℕ
⊢ HPow.hPow (α := ℤ) (↑a : ℤ) (x + 1) ∣ (↑b : ℤ) ↔ a ^ (x + 1) ∣ b
have h₁ : a ^ (x + 1) ∣ b ↔ a ^ (x + 1) ∣ b
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
h : ¬FiniteMultiplicity a b
n : ℕ
⊢ a ^ zero ∣ b
have rw : 1 ∣ b
Tactic: rw [_root_.pow_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
⊢ (∀ (n : ℕ), a ^ n ∣ b) → ¬FiniteMultiplicity a b
have simp : (∀ (n : ℕ), a ^ n ∣ b) → ∀ (x : ℕ), a ^ (x + 1) ∣ b
Tactic: simp [FiniteMultiplicity]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
⊢ a ^ k ∣ b
have zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
Tactic: cases k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
⊢ a ^ k ∣ b
have succ : ∀ (n : ℕ) (hk : (↑(n + 1) : ℕ∞) ≤ emultiplicity a b), a ^ (n + 1) ∣ b
Tactic: cases k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
hk :
  LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞)
    (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
⊢ a ^ (n + 1) ∣ b
have succ : ∀ (h : FiniteMultiplicity a b)
  (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)), a ^ (n + 1) ∣ b
Tactic: split at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
hk :
  LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞)
    (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
succ :
  ∀ (h : FiniteMultiplicity a b)
    (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)), a ^ (n + 1) ∣ b
⊢ a ^ (n + 1) ∣ b
have succ₁ : ¬FiniteMultiplicity a b → ∀ (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) ⊤), a ^ (n + 1) ∣ b
Tactic: split at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (hypothesis)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : FiniteMultiplicity a b
hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)
⊢ a ^ (n + 1) ∣ b
have hk₁ : n + 1 ≤ Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h
Tactic: norm_cast at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
⊢ a ^ k ∣ b
have zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
Tactic: classical
  cases k
  · simp
  unfold emultiplicity at hk
  split at hk
  · norm_cast at hk
    simpa using (Nat.find_min _ (lt_of_succ_le hk))
  · apply FiniteMultiplicity.not_iff_forall.mp ‹_›
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
⊢ a ^ k ∣ b
have succ : ∀ (n : ℕ) (hk : (↑(n + 1) : ℕ∞) ≤ emultiplicity a b), a ^ (n + 1) ∣ b
Tactic: classical
  cases k
  · simp
  unfold emultiplicity at hk
  split at hk
  · norm_cast at hk
    simpa using (Nat.find_min _ (lt_of_succ_le hk))
  · apply FiniteMultiplicity.not_iff_forall.mp ‹_›
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
succ succ₁ : ∀ (n : ℕ) (hk : (↑(n + 1) : ℕ∞) ≤ emultiplicity a b), a ^ (n + 1) ∣ b
⊢ a ^ k ∣ b
have succ₂ : ∀ (n : ℕ)
  (hk :
    LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞)
      (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)),
  a ^ (n + 1) ∣ b
Tactic: classical
  cases k
  · simp
  unfold emultiplicity at hk
  split at hk
  · norm_cast at hk
    simpa using (Nat.find_min _ (lt_of_succ_le hk))
  · apply FiniteMultiplicity.not_iff_forall.mp ‹_›
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
succ succ₁ : ∀ (n : ℕ) (hk : (↑(n + 1) : ℕ∞) ≤ emultiplicity a b), a ^ (n + 1) ∣ b
succ₂ :
  ∀ (n : ℕ)
    (hk :
      LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞)
        (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)),
    a ^ (n + 1) ∣ b
⊢ a ^ k ∣ b
have succ₃ : ∀ (n : ℕ) (h : FiniteMultiplicity a b)
  (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)), a ^ (n + 1) ∣ b
Tactic: classical
  cases k
  · simp
  unfold emultiplicity at hk
  split at hk
  · norm_cast at hk
    simpa using (Nat.find_min _ (lt_of_succ_le hk))
  · apply FiniteMultiplicity.not_iff_forall.mp ‹_›
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : (↑k : ℕ∞) ≤ emultiplicity a b
zero : ∀ (hk : (↑0 : ℕ∞) ≤ emultiplicity a b), HPow.hPow (β := ℕ) a 0 ∣ b
succ succ₁ : ∀ (n : ℕ) (hk : (↑(n + 1) : ℕ∞) ≤ emultiplicity a b), a ^ (n + 1) ∣ b
succ₂ :
  ∀ (n : ℕ)
    (hk :
      LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞)
        (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)),
    a ^ (n + 1) ∣ b
succ₃ :
  ∀ (n : ℕ) (h : FiniteMultiplicity a b)
    (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)), a ^ (n + 1) ∣ b
⊢ a ^ k ∣ b
have succ₄ : ∀ (n : ℕ), ¬FiniteMultiplicity a b → ∀ (hk : LE.le (α := ℕ∞) (↑(n + 1) : ℕ∞) ⊤), a ^ (n + 1) ∣ b
Tactic: classical
  cases k
  · simp
  unfold emultiplicity at hk
  split at hk
  · norm_cast at hk
    simpa using (Nat.find_min _ (lt_of_succ_le hk))
  · apply FiniteMultiplicity.not_iff_forall.mp ‹_›
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
m : ℕ
hm :
  LT.lt (α := ℕ∞) (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
    (↑m : ℕ∞)
nh : a ^ m ∣ b
⊢ False
have isTrue : ∀ (h : FiniteMultiplicity a b)
  (hm : LT.lt (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) (↑m : ℕ∞)), False
Tactic: split at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
m : ℕ
hm :
  LT.lt (α := ℕ∞) (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
    (↑m : ℕ∞)
nh : a ^ m ∣ b
isTrue :
  ∀ (h : FiniteMultiplicity a b)
    (hm : LT.lt (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) (↑m : ℕ∞)), False
⊢ False
have isFalse : ¬FiniteMultiplicity a b → ∀ (hm : LT.lt (α := ℕ∞) ⊤ (↑m : ℕ∞)), False
Tactic: split at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (hypothesis)
Goal:
α : Type u_1
inst : Monoid α
a b : α
m : ℕ
nh : a ^ m ∣ b
h : FiniteMultiplicity a b
hm : LT.lt (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) (↑m : ℕ∞)
⊢ False
have hm₁ : ∃ m_1 < m, ¬a ^ (m_1 + 1) ∣ b
Tactic: simp only [cast_lt, find_lt_iff] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
m : ℕ
nh : a ^ m ∣ b
h : FiniteMultiplicity a b
hm : ∃ m_1 < m, ¬a ^ (m_1 + 1) ∣ b
⊢ False
have isTrue : ∀ (n : ℕ) (hn1 : n < m) (hn2 : ¬a ^ (n + 1) ∣ b), False
Tactic: obtain ⟨n, hn1, hn2⟩ := hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hf : FiniteMultiplicity a b
m : ℕ
hm : multiplicity a b < m
⊢ ¬a ^ m ∣ b
have hm₁ : emultiplicity a b < (↑m : ℕ∞)
Tactic: apply not_pow_dvd_of_emultiplicity_lt
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hf : FiniteMultiplicity a b
m : ℕ
hm : multiplicity a b < m
⊢ emultiplicity a b < (↑m : ℕ∞)
have hm₁ : LT.lt (α := ℕ∞) (↑(multiplicity a b) : ℕ∞) (↑m : ℕ∞)
Tactic: rw [hf.emultiplicity_eq_multiplicity]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hdiv : a ∣ b
⊢ 0 < multiplicity a b
have refine : ∀ (h : multiplicity a b = 0), False
Tactic: refine Nat.pos_iff_ne_zero.2 fun h => ?_
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hdiv : a ∣ b
h : multiplicity a b = 0
⊢ FiniteMultiplicity a b
have by_contra! : ∀ (nh : ∀ (n : ℕ), a ^ (n + 1) ∣ b), False
Tactic: by_contra! nh
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (hypothesis)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
⊢ emultiplicity a b = (↑k : ℕ∞)
have this : FiniteMultiplicity a b
Tactic: have : FiniteMultiplicity a b := ⟨k, hsucc⟩
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
this : FiniteMultiplicity a b
⊢ emultiplicity a b = (↑k : ℕ∞)
have simp : ∀ n < k, a ^ (n + 1) ∣ b
Tactic: simp only [emultiplicity, this, ↓reduceDIte, Nat.cast_inj, find_eq_iff, hsucc, not_false_eq_true,
    Decidable.not_not, true_and]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
⊢ emultiplicity a b = (↑k : ℕ∞)
have have : ∀ (this : FiniteMultiplicity a b), emultiplicity a b = (↑k : ℕ∞)
Tactic: classical
  have : FiniteMultiplicity a b := ⟨k, hsucc⟩
  simp only [emultiplicity, this, ↓reduceDIte, Nat.cast_inj, find_eq_iff, hsucc, not_false_eq_true,
    Decidable.not_not, true_and]
  exact fun n hn ↦ (pow_dvd_pow _ hn).trans hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hk : a ^ k ∣ b
hsucc : ¬a ^ (k + 1) ∣ b
have have₁ have₂ : ∀ (this : FiniteMultiplicity a b), emultiplicity a b = (↑k : ℕ∞)
⊢ emultiplicity a b = (↑k : ℕ∞)
have have₃ : ∀ (this : FiniteMultiplicity a b), ∀ n < k, a ^ (n + 1) ∣ b
Tactic: classical
  have : FiniteMultiplicity a b := ⟨k, hsucc⟩
  simp only [emultiplicity, this, ↓reduceDIte, Nat.cast_inj, find_eq_iff, hsucc, not_false_eq_true,
    Decidable.not_not, true_and]
  exact fun n hn ↦ (pow_dvd_pow _ hn).trans hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
⊢ emultiplicity a b < (↑k : ℕ∞) ↔ ¬a ^ k ∣ b
have rw : emultiplicity a b < (↑k : ℕ∞) ↔ ¬(↑k : ℕ∞) ≤ emultiplicity a b
Tactic: rw [pow_dvd_iff_le_emultiplicity, not_le]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
rw : emultiplicity a b < (↑k : ℕ∞) ↔ ¬(↑k : ℕ∞) ≤ emultiplicity a b
⊢ emultiplicity a b < (↑k : ℕ∞) ↔ ¬a ^ k ∣ b
have rw₁ : emultiplicity a b < (↑k : ℕ∞) ↔ emultiplicity a b < (↑k : ℕ∞)
Tactic: rw [pow_dvd_iff_le_emultiplicity, not_le]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hf : FiniteMultiplicity a b
⊢ multiplicity a b < k ↔ ¬a ^ k ∣ b
have rw : multiplicity a b < k ↔ ¬k ≤ multiplicity a b
Tactic: rw [hf.pow_dvd_iff_le_multiplicity, not_le]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
k : ℕ
hf : FiniteMultiplicity a b
rw : multiplicity a b < k ↔ ¬k ≤ multiplicity a b
⊢ multiplicity a b < k ↔ ¬a ^ k ∣ b
have rw₁ : multiplicity a b < k ↔ multiplicity a b < k
Tactic: rw [hf.pow_dvd_iff_le_multiplicity, not_le]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
⊢ emultiplicity a b = (↑n : ℕ∞) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
have mp : emultiplicity a b = (↑n : ℕ∞) → a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
mp : emultiplicity a b = (↑n : ℕ∞) → a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
⊢ emultiplicity a b = (↑n : ℕ∞) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
have mpr : a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b → emultiplicity a b = (↑n : ℕ∞)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
⊢ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
have mp : a ^ n ∣ b
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
mp : a ^ n ∣ b
⊢ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b
have mp₁ : ¬a ^ (n + 1) ∣ b
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
⊢ a ^ n ∣ b
have mp : (↑n : ℕ∞) ≤ emultiplicity a b
Tactic: apply pow_dvd_of_le_emultiplicity
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
⊢ ¬a ^ (n + 1) ∣ b
have mp : emultiplicity a b < (↑(n + 1) : ℕ∞)
Tactic: apply not_pow_dvd_of_emultiplicity_lt
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
⊢ emultiplicity a b < (↑(n + 1) : ℕ∞)
have mp : LT.lt (α := ℕ∞) (↑n : ℕ∞) (↑(n + 1) : ℕ∞)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
h : emultiplicity a b = (↑n : ℕ∞)
⊢ LT.lt (α := ℕ∞) (↑n : ℕ∞) (↑(n + 1) : ℕ∞)
have mp : n < n + 1
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
n : ℕ
⊢ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b → emultiplicity a b = (↑n : ℕ∞)
have mpr : a ^ n ∣ b → ¬a ^ (n + 1) ∣ b → emultiplicity a b = (↑n : ℕ∞)
Tactic: rw [and_imp]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
⊢ emultiplicity a b = 0 ↔ ¬a ∣ b
have pos : ∀ (hf : FiniteMultiplicity a b), emultiplicity a b = 0 ↔ ¬a ∣ b
Tactic: by_cases hf : FiniteMultiplicity a b
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
pos : ∀ (hf : FiniteMultiplicity a b), emultiplicity a b = 0 ↔ ¬a ∣ b
⊢ emultiplicity a b = 0 ↔ ¬a ∣ b
have neg : ∀ (hf : ¬FiniteMultiplicity a b), emultiplicity a b = 0 ↔ ¬a ∣ b
Tactic: by_cases hf : FiniteMultiplicity a b
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hf : FiniteMultiplicity a b
⊢ emultiplicity a b = 0 ↔ ¬a ∣ b
have pos : HPow.hPow (β := ℕ) a 0 ∣ b ∧ ¬a ^ HAdd.hAdd (α := ℕ) 0 1 ∣ b ↔ ¬a ∣ b
Tactic: rw [← ENat.coe_zero, emultiplicity_eq_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hfin : FiniteMultiplicity a b
⊢ ∃ (c : α), b = a ^ multiplicity a b * c ∧ ¬a ∣ c
have obtain : ∀ (c : α) (hc : b = a ^ multiplicity a b * c), ∃ (c : α), b = a ^ multiplicity a b * c ∧ ¬a ∣ c
Tactic: obtain ⟨c, hc⟩ := pow_multiplicity_dvd a b
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hfin : FiniteMultiplicity a b
c : α
hc : b = a ^ multiplicity a b * c
⊢ ∃ (c : α), b = a ^ multiplicity a b * c ∧ ¬a ∣ c
have refine : ¬a ∣ c
Tactic: refine ⟨c, hc, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (hypothesis)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hfin : FiniteMultiplicity a b
c : α
hc : b = a ^ multiplicity a b * c
k : α
hk : c = a * k
⊢ False
have hc₁ : b = a ^ (multiplicity a b + 1) * k
Tactic: rw [hk, ← mul_assoc, ← _root_.pow_succ] at hc
Cleared hypotheses: hc, hk
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (hypothesis)
Goal:
α : Type u_1
inst : Monoid α
a b : α
hfin : FiniteMultiplicity a b
c k : α
hc : b = a ^ (multiplicity a b + 1) * k
hk : c = a * k
⊢ False
have h₁ : a ^ (multiplicity a b + 1) ∣ b
Tactic: have h₁ : a ^ (multiplicity a b + 1) ∣ b := ⟨k, hc⟩
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
⊢ emultiplicity a b ≤ emultiplicity c d ↔ ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
have mp : emultiplicity a b ≤ emultiplicity c d → ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
mp : emultiplicity a b ≤ emultiplicity c d → ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
⊢ emultiplicity a b ≤ emultiplicity c d ↔ ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
have mpr : (∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d) → emultiplicity a b ≤ emultiplicity c d
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
⊢ LE.le (α := ℕ∞) (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
have mpr : ∀ (h : FiniteMultiplicity a b),
  LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
Tactic: split
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
mpr :
  ∀ (h : FiniteMultiplicity a b),
    LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞)
      (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
⊢ LE.le (α := ℕ∞) (if h : FiniteMultiplicity a b then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h) : ℕ∞) else ⊤)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
have mpr₁ : ¬FiniteMultiplicity a b →
  LE.le (α := ℕ∞) ⊤
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
Tactic: split
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
h_1 : FiniteMultiplicity a b
⊢ LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) h_1) : ℕ∞)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
have obtain : ∀ (w : ℕ) (h_1 : ¬a ^ (w + 1) ∣ b),
  LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) sorry) : ℕ∞)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
Tactic: obtain ⟨w, h_1⟩ := h_1
Cleared hypotheses: h_1
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
w : ℕ
h_1 : ¬a ^ (w + 1) ∣ b
⊢ LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) ⋯) : ℕ∞)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
have isTrue : ∀ (h : FiniteMultiplicity c d),
  LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) sorry) : ℕ∞)
    (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞)
Tactic: split
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
w : ℕ
h_1 : ¬a ^ (w + 1) ∣ b
isTrue :
  ∀ (h : FiniteMultiplicity c d),
    LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) ⋯) : ℕ∞)
      (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞)
⊢ LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) ⋯) : ℕ∞)
    (if h : FiniteMultiplicity c d then (↑(Nat.find (p := fun (n : ℕ) ↦ ¬c ^ (n + 1) ∣ d) h) : ℕ∞) else ⊤)
have isFalse : ¬FiniteMultiplicity c d → LE.le (α := ℕ∞) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬a ^ (n + 1) ∣ b) sorry) : ℕ∞) ⊤
Tactic: split
-- Lean 4 proof step from Mathlib.RingTheory.Multiplicity (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Monoid α
inst_1 : Monoid β
a b : α
c d : β
⊢ emultiplicity a b ≤ emultiplicity c d ↔ ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
have mp : emultiplicity a b ≤ emultiplicity c d → ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
Tactic: classical
  constructor
  · exact fun h n hab ↦ pow_dvd_of_le_emultiplicity (le_trans (le_emultiplicity_of_pow_dvd hab) h)
  · intro h
    unfold emultiplicity
    -- aesop? says
    split
    next h_1 =>
      obtain ⟨w, h_1⟩ := h_1
      split
      next h_2 =>
        simp_all only [cast_le, le_find_iff, lt_find_iff, Decidable.not_not, le_refl,
          not_true_eq_false, not_false_eq_true, implies_true]
      next h_2 => simp_all only [not_exists, Decidable.not_not, le_top]
    next h_1 =>
      simp_all only [not_exists, Decidable.not_not, not_true_eq_false, top_le_iff,
        dite_eq_right_iff, ENat.coe_ne_top, imp_false, not_false_eq_true, implies_true]