l3lab/ntp-mathlib huggingface dataset:
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · 

Proof state:
case pos
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : b = 0
⊢ a * ↑(normUnit a) * ↑(normUnit (a * ↑(normUnit a))) = a * ↑(normUnit a)

Next tactic:
apply normalize_idem
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
⊢ Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))) *
      ↑(normUnit (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))))) =
    Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b)))

Next tactic:
have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · exact absurd ‹b = 0› h_1
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
⊢ lcm a b ≠ 0

Next tactic:
intro con
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
⊢ False

Next tactic:
have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h✝ : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
h : lcm a b ∣ a * b
⊢ False

Next tactic:
rw [con, zero_dvd_iff, mul_eq_zero] at h
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h✝ : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
h : a = 0 ∨ b = 0
⊢ False

Next tactic:
cases h
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · 

Proof state:
case inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
h✝ : a = 0
⊢ False

Next tactic:
exact absurd ‹a = 0› h
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · 

Proof state:
case inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
h✝ : b = 0
⊢ False

Next tactic:
exact absurd ‹b = 0› h_1
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · exact absurd ‹b = 0› h_1
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))) *
      ↑(normUnit (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))))) =
    Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b)))

Next tactic:
apply mul_left_cancel₀ h0
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · exact absurd ‹b = 0› h_1
      apply mul_left_cancel₀ h0
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ lcm a b *
      (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))) *
        ↑(normUnit (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b)))))) =
    lcm a b * Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b)))

Next tactic:
refine' _root_.trans _ (Classical.choose_spec (exists_gcd a b))
/-- Define `NormalizedGCDMonoid` on a structure just from the `lcm` and its properties. -/
noncomputable def normalizedGCDMonoidOfLCM [NormalizationMonoid α] [DecidableEq α] (lcm : α → α → α)
    (dvd_lcm_left : ∀ a b, a ∣ lcm a b) (dvd_lcm_right : ∀ a b, b ∣ lcm a b)
    (lcm_dvd : ∀ {a b c}, c ∣ a → b ∣ a → lcm c b ∣ a)
    (normalize_lcm : ∀ a b, normalize (lcm a b) = lcm a b) : NormalizedGCDMonoid α :=
  let exists_gcd a b := dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
  { (inferInstance : NormalizationMonoid α) with
    lcm
    gcd := fun a b =>
      if a = 0 then normalize b
      else if b = 0 then normalize a else Classical.choose (exists_gcd a b)
    gcd_mul_lcm := fun a b => by
      dsimp only
      split_ifs with h h_1
      · rw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]
      · rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
      rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
      exact normalize_associated (a * b)
    normalize_lcm
    normalize_gcd := fun a b => by
      dsimp [normalize]
      split_ifs with h h_1
      · apply normalize_idem
      · apply normalize_idem
      have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · exact absurd ‹b = 0› h_1
      apply mul_left_cancel₀ h0
      refine' _root_.trans _ (Classical.choose_spec (exists_gcd a b))
      

Proof state:
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
normalize_lcm : ∀ (a b : α), normalize (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ normalize (a * b) :=
  fun a b => dvd_normalize_iff.mpr (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))
src✝ : NormalizationMonoid α := inferInstance
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ lcm a b *
      (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b))) *
        ↑(normUnit (Classical.choose (_ : lcm a b ∣ a * b * ↑(normUnit (a * b)))))) =
    normalize (a * b)

Next tactic:
conv_lhs =>
        congr
        rw [← normalize_lcm a b]