-- Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors

Proof state:
case neg.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb

Result:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h_sb : b.factors = ↑sb
⊢ p.count a.factors ≤ p.count b.factors

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha

Result:
case neg.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
⊢ p.count a.factors ≤ p.count b.factors

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb

Tactic:
rw [WithTop.coe_le_coe] at h

Result:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : sa ≤ sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ p.count ↑sa ≤ p.count ↑sb

Tactic:
rw [count_some hp, count_some hp]

Result:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h_sb : b.factors = ↑sb
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
rw [h_sa, h_sb] at h ⊢

Result:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ p.count ↑sa ≤ p.count ↑sb

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : sa ≤ sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb

Tactic:
exact Multiset.count_le_of_le _ h

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
by_cases ha : a = 0

Result:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : a = 0
⊢ p.count a.factors ≤ p.count b.factors

case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : ¬a = 0
⊢ p.count a.factors ≤ p.count b.factors

Proof state:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
ha : a = 0
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
simp_all

Result:
no goals