mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : α✝ × β✝
⊢ bitraverse pure pure x✝ = pure x✝

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : α✝ × α'✝
⊢ bitraverse (Functor.Comp.mk ∘ Functor.map f✝ ∘ g✝) (Functor.Comp.mk ∘ Functor.map f'✝ ∘ g'✝) x✝ =
    Functor.Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝ × α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

pp: case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : α✝ × α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : α✝ × β✝
⊢ bitraverse pure pure x✝ = pure x✝

case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : α✝ × α'✝
⊢ bitraverse (Functor.Comp.mk ∘ Functor.map f✝ ∘ g✝) (Functor.Comp.mk ∘ Functor.map f'✝ ∘ g'✝) x✝ =
    Functor.Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝ × α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : α✝ × α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
fst✝ : α✝
snd✝ : α'✝
⊢ Functor.Comp.mk ((fun a ↦ (fun x2 ↦ Prod.mk <$> f✝ a <*> x2) ∘ f'✝) <$> g✝ fst✝ <*> g'✝ snd✝) =
    Functor.Comp.mk ((fun a ↦ Prod.bitraverse f✝ f'✝ ∘ Prod.mk a) <$> g✝ fst✝ <*> g'✝ snd✝)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
fst✝ : α✝
snd✝ : α'✝
⊢ pure (f✝ fst✝, f'✝ snd✝) = pure (bimap f✝ f'✝ (fst✝, snd✝))

Tactic:
<;>

Goal state after:
case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
fst✝ : α✝
snd✝ : α'✝
⊢ Functor.Comp.mk ((fun a ↦ (fun x2 ↦ Prod.mk <$> f✝ a <*> x2) ∘ f'✝) <$> g✝ fst✝ <*> g'✝ snd✝) =
    Functor.Comp.mk ((fun a ↦ Prod.bitraverse f✝ f'✝ ∘ Prod.mk a) <$> g✝ fst✝ <*> g'✝ snd✝)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
fst✝ : α✝
snd✝ : α'✝
⊢ pure (f✝ fst✝, f'✝ snd✝) = pure (bimap f✝ f'✝ (fst✝, snd✝))
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α β : Type u_1} (x : α ⊕ β), bitraverse pure pure x = pure x

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] {α α' β β' γ γ' : Type u_1} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β')
    (x : α ⊕ α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α α' β β' : Type u_1} (f : α → β) (f' : α' → β') (x : α ⊕ α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

pp: case binaturality
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] (η : ApplicativeTransformation F G) {α α' β β' : Type u_1} (f : α → F β) (f' : α' → F β')
    (x : α ⊕ α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α β : Type u_1} (x : α ⊕ β), bitraverse pure pure x = pure x

case comp_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] {α α' β β' γ γ' : Type u_1} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β')
    (x : α ⊕ α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α α' β β' : Type u_1} (f : α → β) (f' : α' → β') (x : α ⊕ α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

case binaturality
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] (η : ApplicativeTransformation F G) {α α' β β' : Type u_1} (f : α → F β) (f' : α' → F β')
    (x : α ⊕ α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : α✝ ⊕ β✝
⊢ bitraverse pure pure x✝ = pure x✝

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : α✝ ⊕ α'✝
⊢ bitraverse (Comp.mk ∘ map f✝ ∘ g✝) (Comp.mk ∘ map f'✝ ∘ g'✝) x✝ = Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝ ⊕ α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

pp: case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : α✝ ⊕ α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : α✝ ⊕ β✝
⊢ bitraverse pure pure x✝ = pure x✝

case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : α✝ ⊕ α'✝
⊢ bitraverse (Comp.mk ∘ map f✝ ∘ g✝) (Comp.mk ∘ map f'✝ ∘ g'✝) x✝ = Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝ ⊕ α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : α✝ ⊕ α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case bitraverse_eq_bimap_id.inl
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝
⊢ pure (Sum.inl (f✝ x✝)) = pure (bimap f✝ f'✝ (Sum.inl x✝))

pp: case bitraverse_eq_bimap_id.inr
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α'✝
⊢ pure (Sum.inr (f'✝ x✝)) = pure (bimap f✝ f'✝ (Sum.inr x✝))

Tactic:
<;>

Goal state after:
case bitraverse_eq_bimap_id.inl
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α✝
⊢ pure (Sum.inl (f✝ x✝)) = pure (bimap f✝ f'✝ (Sum.inl x✝))

case bitraverse_eq_bimap_id.inr
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : α'✝
⊢ pure (Sum.inr (f'✝ x✝)) = pure (bimap f✝ f'✝ (Sum.inr x✝))
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α β : Type u_1} (x : Const α β), bitraverse pure pure x = pure x

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] {α α' β β' γ γ' : Type u_1} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β')
    (x : Const α α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α α' β β' : Type u_1} (f : α → β) (f' : α' → β') (x : Const α α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

pp: case binaturality
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] (η : ApplicativeTransformation F G) {α α' β β' : Type u_1} (f : α → F β) (f' : α' → F β')
    (x : Const α α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α β : Type u_1} (x : Const α β), bitraverse pure pure x = pure x

case comp_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] {α α' β β' γ γ' : Type u_1} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β')
    (x : Const α α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {α α' β β' : Type u_1} (f : α → β) (f' : α' → β') (x : Const α α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

case binaturality
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
⊢ ∀ {F G : Type u_1 → Type u_1} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F]
    [LawfulApplicative G] (η : ApplicativeTransformation F G) {α α' β β' : Type u_1} (f : α → F β) (f' : α' → F β')
    (x : Const α α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : Const α✝ β✝
⊢ bitraverse pure pure x✝ = pure x✝

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : Const α✝ α'✝
⊢ bitraverse (Comp.mk ∘ map f✝ ∘ g✝) (Comp.mk ∘ map f'✝ ∘ g'✝) x✝ = Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : Const α✝ α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

pp: case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : Const α✝ α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : Const α✝ β✝
⊢ bitraverse pure pure x✝ = pure x✝

case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : Const α✝ α'✝
⊢ bitraverse (Comp.mk ∘ map f✝ ∘ g✝) (Comp.mk ∘ map f'✝ ∘ g'✝) x✝ = Comp.mk (bitraverse f✝ f'✝ <$> bitraverse g✝ g'✝ x✝)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : Const α✝ α'✝
⊢ bitraverse (pure ∘ f✝) (pure ∘ f'✝) x✝ = pure (bimap f✝ f'✝ x✝)

case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : Const α✝ α'✝
⊢ (fun {α} ↦ η✝.app α) (bitraverse f✝ f'✝ x✝) = bitraverse ((fun {α} ↦ η✝.app α) ∘ f✝) ((fun {α} ↦ η✝.app α) ∘ f'✝) x✝
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : Const α✝ β✝
⊢ pure x✝ = pure x✝

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : Const α✝ α'✝
⊢ (Comp.mk ∘ map f✝ ∘ g✝) x✝ = Comp.mk (f✝ <$> g✝ x✝)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : Const α✝ α'✝
⊢ (pure ∘ f✝) x✝ = pure (bimap f✝ f'✝ x✝)

pp: case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : Const α✝ α'✝
⊢ η✝.app (Const β✝ β'✝) (f✝ x✝) = (η✝.app β✝ ∘ f✝) x✝

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ β✝ : Type u_1
x✝ : Const α✝ β✝
⊢ pure x✝ = pure x✝

case comp_bitraverse
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ α'✝ β✝ β'✝ γ✝ γ'✝ : Type u_1
f✝ : β✝ → F✝ γ✝
f'✝ : β'✝ → F✝ γ'✝
g✝ : α✝ → G✝ β✝
g'✝ : α'✝ → G✝ β'✝
x✝ : Const α✝ α'✝
⊢ (Comp.mk ∘ map f✝ ∘ g✝) x✝ = Comp.mk (f✝ <$> g✝ x✝)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝¹ : Bitraversable t
F : Type u → Type u
inst✝ : Applicative F
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → β✝
f'✝ : α'✝ → β'✝
x✝ : Const α✝ α'✝
⊢ (pure ∘ f✝) x✝ = pure (bimap f✝ f'✝ x✝)

case binaturality
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F : Type u → Type u
inst✝⁴ : Applicative F
F✝ G✝ : Type u_1 → Type u_1
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η✝ : ApplicativeTransformation F✝ G✝
α✝ α'✝ β✝ β'✝ : Type u_1
f✝ : α✝ → F✝ β✝
f'✝ : α'✝ → F✝ β'✝
x✝ : Const α✝ α'✝
⊢ η✝.app (Const β✝ β'✝) (f✝ x✝) = (η✝.app β✝ ∘ f✝) x✝
Module: Mathlib.Control.Bitraversable.Instances

Goal state before:
pp: case id_bitraverse
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {α β : Type u} (x : _root_.flip t α β), bitraverse pure pure x = pure x

pp: case comp_bitraverse
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {F G : Type u → Type u} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
    {α α' β β' γ γ' : Type u} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β') (x : _root_.flip t α α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

pp: case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {α α' β β' : Type u} (f : α → β) (f' : α' → β') (x : _root_.flip t α α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

pp: case binaturality
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {F G : Type u → Type u} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
    (η : ApplicativeTransformation F G) {α α' β β' : Type u} (f : α → F β) (f' : α' → F β') (x : _root_.flip t α α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x

Tactic:
<;>

Goal state after:
case id_bitraverse
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {α β : Type u} (x : _root_.flip t α β), bitraverse pure pure x = pure x

case comp_bitraverse
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {F G : Type u → Type u} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
    {α α' β β' γ γ' : Type u} (f : β → F γ) (f' : β' → F γ') (g : α → G β) (g' : α' → G β') (x : _root_.flip t α α'),
    bitraverse (Comp.mk ∘ map f ∘ g) (Comp.mk ∘ map f' ∘ g') x = Comp.mk (bitraverse f f' <$> bitraverse g g' x)

case bitraverse_eq_bimap_id
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {α α' β β' : Type u} (f : α → β) (f' : α' → β') (x : _root_.flip t α α'),
    bitraverse (pure ∘ f) (pure ∘ f') x = pure (bimap f f' x)

case binaturality
t : Type u → Type u → Type u
inst✝² : Bitraversable t
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : LawfulBitraversable t
⊢ ∀ {F G : Type u → Type u} [inst : Applicative F] [inst_1 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
    (η : ApplicativeTransformation F G) {α α' β β' : Type u} (f : α → F β) (f' : α' → F β') (x : _root_.flip t α α'),
    (fun {α} ↦ η.app α) (bitraverse f f' x) = bitraverse ((fun {α} ↦ η.app α) ∘ f) ((fun {α} ↦ η.app α) ∘ f') x