-- Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q

Proof state:
case intro.refine_1.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ p
⊢ Irreducible x

Tactic:
exact hp x h

Result:
no goals

Proof state:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ False

Tactic:
refine not_irreducible_zero (hq _ ?_)

Result:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ p.prod ≤ q.prod → p ≤ q

Tactic:
rintro ⟨c, eqc⟩

Result:
case intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p ≤ q

Proof state:
case intro.refine_1
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
⊢ Irreducible x

Tactic:
obtain h | h := Multiset.mem_add.1 hx

Result:
case intro.refine_1.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ p
⊢ Irreducible x

case intro.refine_1.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ factors c
⊢ Irreducible x

Proof state:
case intro.refine_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ q.prod = (p + factors c).prod

Tactic:
rw [eqc, Multiset.prod_add]

Result:
case intro.refine_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p.prod * c = p.prod * (factors c).prod

Proof state:
case intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p ≤ q

Tactic:
refine Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx ↦ ?_) ?_⟩

Result:
case intro.refine_1
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
⊢ Irreducible x

case intro.refine_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ q.prod = (p + factors c).prod

Proof state:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ c = (factors c).prod

Tactic:
refine associated_iff_eq.mp (factors_prod fun hc => ?_).symm

Result:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ False

Proof state:
case intro.refine_1.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ factors c
⊢ Irreducible x

Tactic:
exact irreducible_of_factor _ h

Result:
no goals

Proof state:
case intro.refine_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ p.prod * c = p.prod * (factors c).prod

Tactic:
congr

Result:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ c = (factors c).prod

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ p.prod ≤ q.prod ↔ p ≤ q

Tactic:
refine ⟨?_, prod_le_prod⟩

Result:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ p.prod ≤ q.prod → p ≤ q

Proof state:
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q

Tactic:
rw [← prod_eq_zero_iff, eqc, hc, mul_zero]

Result:
no goals