l3lab/ntp-mathlib huggingface dataset:
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
⊢ p ∣ a ∨ p ∣ b

Next tactic:
have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
⊢ p ∣ a ∨ p ∣ b

Next tactic:
cases' eif x hx0 with fx hfx
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        

Proof state:
case intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
⊢ p ∣ a ∨ p ∣ b

Next tactic:
cases' eif a ha0 with fa hfa
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        

Proof state:
case intro.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
⊢ p ∣ a ∨ p ∣ b

Next tactic:
cases' eif b hb0 with fb hfb
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ p ∣ a ∨ p ∣ b

Next tactic:
have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _
            _ ~ᵤ fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)
            _ = _ := by rw [Multiset.prod_add]
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ Multiset.Rel Associated (p ::ₘ fx) (fa + fb)

Next tactic:
apply uif
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · 

Proof state:
case a
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ ∀ x ∈ p ::ₘ fx, Irreducible x

Next tactic:
exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · 

Proof state:
case a
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ ∀ x ∈ fa + fb, Irreducible x

Next tactic:
exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          

Proof state:
case a
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ Multiset.prod (p ::ₘ fx) ~ᵤ Multiset.prod (fa + fb)

Next tactic:
calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _
            _ ~ᵤ fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)
            _ = _ := by rw [Multiset.prod_add]
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ Multiset.prod (p ::ₘ fx) ~ᵤ a * b

Next tactic:
rw [hx, Multiset.prod_cons]
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              rw [hx, Multiset.prod_cons]; 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ p * Multiset.prod fx ~ᵤ p * x

Next tactic:
exact hfx.2.mul_left _
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _
            _ ~ᵤ fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)
            _ = _ := by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
⊢ Multiset.prod fa * Multiset.prod fb = Multiset.prod (fa + fb)

Next tactic:
rw [Multiset.prod_add]
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α]
    (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
    (uif :
      ∀ f g : Multiset α,
        (∀ x ∈ f, Irreducible x) →
          (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    (p : α) : Irreducible p ↔ Prime p :=
  letI := Classical.decEq α
  ⟨ fun hpi =>
    ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
      if hab0 : a * b = 0 then
        (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
          simp [hb0]
      else by
        have hx0 : x ≠ 0 := fun hx0 => by simp_all
        have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
        have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
        cases' eif x hx0 with fx hfx
        cases' eif a ha0 with fa hfa
        cases' eif b hb0 with fb hfb
        have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
          apply uif
          · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
          · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
          calc
            Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
              rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _
            _ ~ᵤ fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)
            _ = _ := by rw [Multiset.prod_add]

        

Proof state:
case intro.intro.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ Multiset.prod f ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) →
      (∀ x ∈ g, Irreducible x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
x✝ : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
ha0 : a ≠ 0
hb0 : b ≠ 0
fx : Multiset α
hfx : (∀ b ∈ fx, Irreducible b) ∧ Multiset.prod fx ~ᵤ x
fa : Multiset α
hfa : (∀ b ∈ fa, Irreducible b) ∧ Multiset.prod fa ~ᵤ a
fb : Multiset α
hfb : (∀ b ∈ fb, Irreducible b) ∧ Multiset.prod fb ~ᵤ b
h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb)
⊢ p ∣ a ∨ p ∣ b

Next tactic:
exact
          let ⟨q, hqf, hq⟩ := Multiset.exists_mem_of_rel_of_mem h (Multiset.mem_cons_self p _)
          (Multiset.mem_add.1 hqf).elim
            (fun hqa =>
              Or.inl <| hq.dvd_iff_dvd_left.2 <| hfa.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqa))
            fun hqb =>
            Or.inr <| hq.dvd_iff_dvd_left.2 <| hfb.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqb)