Theorem: Multiset.comp_traverse
File: Mathlib/Data/Multiset/Functor.lean

Proof steps:
state_before: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
⊢ traverse (Comp.mk ∘ Functor.map h ∘ g) x = Comp.mk (traverse h <$> traverse g x)
state_after: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
⊢ ∀ (a : List α), traverse (Comp.mk ∘ Functor.map h ∘ g) ⟦a⟧ = Comp.mk (traverse h <$> traverse g ⟦a⟧)
tactic: refine Quotient.inductionOn x ?_
state_before: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
⊢ ∀ (a : List α), traverse (Comp.mk ∘ Functor.map h ∘ g) ⟦a⟧ = Comp.mk (traverse h <$> traverse g ⟦a⟧)
state_after: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
a✝ : List α
⊢ traverse (Comp.mk ∘ Functor.map h ∘ g) ⟦a✝⟧ = Comp.mk (traverse h <$> traverse g ⟦a✝⟧)
tactic: intro
state_before: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
a✝ : List α
⊢ traverse (Comp.mk ∘ Functor.map h ∘ g) ⟦a✝⟧ = Comp.mk (traverse h <$> traverse g ⟦a✝⟧)
state_after: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
a✝ : List α
⊢ Comp.mk (((fun x => ofList <$> x) ∘ Traversable.traverse h) <$> Traversable.traverse g a✝) =
    Comp.mk ((Quotient.lift (Functor.map ofList ∘ Traversable.traverse h) ⋯ ∘ ofList) <$> Traversable.traverse g a✝)
tactic: simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,
    functor_norm]
state_before: F : Type u → Type u
inst✝⁵ : Applicative F
inst✝⁴ : CommApplicative F
α' β' : Type u
f : α' → F β'
G H : Type u_1 → Type u_1
inst✝³ : Applicative G
inst✝² : Applicative H
inst✝¹ : CommApplicative G
inst✝ : CommApplicative H
α β γ : Type u_1
g : α → G β
h : β → H γ
x : Multiset α
a✝ : List α
⊢ Comp.mk (((fun x => ofList <$> x) ∘ Traversable.traverse h) <$> Traversable.traverse g a✝) =
    Comp.mk ((Quotient.lift (Functor.map ofList ∘ Traversable.traverse h) ⋯ ∘ ofList) <$> Traversable.traverse g a✝)
state_after: no goals
tactic: simp only [Function.comp, lift_coe]