-- Mathlib/Algebra/Order/Ring/WithTop.lean
@[simp]
lemma untopD_zero_mul (a b : WithTop α) : (a * b).untopD 0 = a.untopD 0 * b.untopD 0

Proof state:
case neg.top
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithTop α
hb : ¬b = 0
ha : ¬⊤ = 0
⊢ untopD 0 (⊤ * b) = untopD 0 ⊤ * untopD 0 b

Tactic:
rw [top_mul hb, untopD_top, zero_mul]

Result:
no goals

Proof state:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
hb : b = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Tactic:
rw [hb, mul_zero, ← coe_zero, untopD_coe, mul_zero]

Result:
no goals

Proof state:
case neg.coe.coe
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a✝¹ : α
ha : ¬↑a✝¹ = 0
a✝ : α
hb : ¬↑a✝ = 0
⊢ untopD 0 (↑a✝¹ * ↑a✝) = untopD 0 ↑a✝¹ * untopD 0 ↑a✝

Tactic:
rw [← coe_mul, untopD_coe, untopD_coe, untopD_coe]

Result:
no goals

Proof state:
case neg.coe
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithTop α
hb : ¬b = 0
a✝ : α
ha : ¬↑a✝ = 0
⊢ untopD 0 (↑a✝ * b) = untopD 0 ↑a✝ * untopD 0 b

Tactic:
induction b

Result:
case neg.coe.top
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a✝ : α
ha : ¬↑a✝ = 0
hb : ¬⊤ = 0
⊢ untopD 0 (↑a✝ * ⊤) = untopD 0 ↑a✝ * untopD 0 ⊤

case neg.coe.coe
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a✝¹ : α
ha : ¬↑a✝¹ = 0
a✝ : α
hb : ¬↑a✝ = 0
⊢ untopD 0 (↑a✝¹ * ↑a✝) = untopD 0 ↑a✝¹ * untopD 0 ↑a✝

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Tactic:
by_cases ha : a = 0

Result:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : a = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Proof state:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
hb : ¬b = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Tactic:
induction a

Result:
case neg.top
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithTop α
hb : ¬b = 0
ha : ¬⊤ = 0
⊢ untopD 0 (⊤ * b) = untopD 0 ⊤ * untopD 0 b

case neg.coe
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithTop α
hb : ¬b = 0
a✝ : α
ha : ¬↑a✝ = 0
⊢ untopD 0 (↑a✝ * b) = untopD 0 ↑a✝ * untopD 0 b

Proof state:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : a = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Tactic:
rw [ha, zero_mul, ← coe_zero, untopD_coe, zero_mul]

Result:
no goals

Proof state:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Tactic:
by_cases hb : b = 0

Result:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
hb : b = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
hb : ¬b = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b

Proof state:
case neg.coe.top
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a✝ : α
ha : ¬↑a✝ = 0
hb : ¬⊤ = 0
⊢ untopD 0 (↑a✝ * ⊤) = untopD 0 ↑a✝ * untopD 0 ⊤

Tactic:
rw [mul_top ha, untopD_top, mul_zero]

Result:
no goals