Module: Mathlib.Data.Finsupp.Option

Goal state before:
pp: case add
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ (b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)) *
      (b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)) =
    b none ((f₁ + f₂) none) * ((f₁.some.prod fun a ↦ b (Option.some a)) * f₂.some.prod fun a ↦ b (Option.some a))

pp: case add.h_zero
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.some.support ∪ f₂.some.support, b (Option.some a) 0 = 1

pp: case add.h_add
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.some.support ∪ f₂.some.support,
    ∀ (b₁ b₂ : M), b (Option.some a) (b₁ + b₂) = b (Option.some a) b₁ * b (Option.some a) b₂

pp: case add.h_zero
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.support ∪ f₂.support, b a 0 = 1

pp: case add.h_add
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.support ∪ f₂.support, ∀ (b₁ b₂ : M), b a (b₁ + b₂) = b a b₁ * b a b₂

Tactic:
· simp only [h_add, Pi.add_apply, Finsupp.coe_add]
        rw [mul_mul_mul_comm]

Goal state after:
case add.h_zero
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.some.support ∪ f₂.some.support, b (Option.some a) 0 = 1

case add.h_add
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.some.support ∪ f₂.some.support,
    ∀ (b₁ b₂ : M), b (Option.some a) (b₁ + b₂) = b (Option.some a) b₁ * b (Option.some a) b₂

case add.h_zero
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.support ∪ f₂.support, b a 0 = 1

case add.h_add
α : Type u_1
M : Type u_2
N : Type u_3
inst✝¹ : AddZeroClass M
inst✝ : CommMonoid N
b : Option α → M → N
h_zero : ∀ (o : Option α), b o 0 = 1
h_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂
f₁ f₂ : Option α →₀ M
h₁ : f₁.prod b = b none (f₁ none) * f₁.some.prod fun a ↦ b (Option.some a)
h₂ : f₂.prod b = b none (f₂ none) * f₂.some.prod fun a ↦ b (Option.some a)
⊢ ∀ a ∈ f₁.support ∪ f₂.support, ∀ (b₁ b₂ : M), b a (b₁ + b₂) = b a b₁ * b a b₂