Theorem: ApplicativeTransformation.coe_inj
File: Mathlib/Control/Traversable/Basic.lean

Proof steps:
state_before: F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η η' : ApplicativeTransformation F G
h : (fun {α} => η.app α) = fun {α} => η'.app α
⊢ η = η'
state_after: case mk
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η' : ApplicativeTransformation F G
app✝ : (α : Type u) → F α → G α
preserves_pure'✝ : ∀ {α : Type u} (x : α), app✝ α (pure x) = pure x
preserves_seq'✝ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α), app✝ β (Seq.seq x fun x => y) = Seq.seq (app✝ (α → β) x) fun x => app✝ α y
h :
  (fun {α} => { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }.app α) =
    fun {α} => η'.app α
⊢ { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ } = η'
tactic: cases η
state_before: case mk
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η' : ApplicativeTransformation F G
app✝ : (α : Type u) → F α → G α
preserves_pure'✝ : ∀ {α : Type u} (x : α), app✝ α (pure x) = pure x
preserves_seq'✝ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α), app✝ β (Seq.seq x fun x => y) = Seq.seq (app✝ (α → β) x) fun x => app✝ α y
h :
  (fun {α} => { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }.app α) =
    fun {α} => η'.app α
⊢ { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ } = η'
state_after: case mk.mk
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
app✝¹ : (α : Type u) → F α → G α
preserves_pure'✝¹ : ∀ {α : Type u} (x : α), app✝¹ α (pure x) = pure x
preserves_seq'✝¹ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α),
    app✝¹ β (Seq.seq x fun x => y) = Seq.seq (app✝¹ (α → β) x) fun x => app✝¹ α y
app✝ : (α : Type u) → F α → G α
preserves_pure'✝ : ∀ {α : Type u} (x : α), app✝ α (pure x) = pure x
preserves_seq'✝ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α), app✝ β (Seq.seq x fun x => y) = Seq.seq (app✝ (α → β) x) fun x => app✝ α y
h :
  (fun {α} => { app := app✝¹, preserves_pure' := preserves_pure'✝¹, preserves_seq' := preserves_seq'✝¹ }.app α) =
    fun {α} => { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }.app α
⊢ { app := app✝¹, preserves_pure' := preserves_pure'✝¹, preserves_seq' := preserves_seq'✝¹ } =
    { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }
tactic: cases η'
state_before: case mk.mk
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
app✝¹ : (α : Type u) → F α → G α
preserves_pure'✝¹ : ∀ {α : Type u} (x : α), app✝¹ α (pure x) = pure x
preserves_seq'✝¹ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α),
    app✝¹ β (Seq.seq x fun x => y) = Seq.seq (app✝¹ (α → β) x) fun x => app✝¹ α y
app✝ : (α : Type u) → F α → G α
preserves_pure'✝ : ∀ {α : Type u} (x : α), app✝ α (pure x) = pure x
preserves_seq'✝ :
  ∀ {α β : Type u} (x : F (α → β)) (y : F α), app✝ β (Seq.seq x fun x => y) = Seq.seq (app✝ (α → β) x) fun x => app✝ α y
h :
  (fun {α} => { app := app✝¹, preserves_pure' := preserves_pure'✝¹, preserves_seq' := preserves_seq'✝¹ }.app α) =
    fun {α} => { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }.app α
⊢ { app := app✝¹, preserves_pure' := preserves_pure'✝¹, preserves_seq' := preserves_seq'✝¹ } =
    { app := app✝, preserves_pure' := preserves_pure'✝, preserves_seq' := preserves_seq'✝ }
state_after: no goals
tactic: congr