lean2/tests/lean/run/imp_bang.lean

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import algebra.category.basic
open eq eq.ops category functor natural_transformation
variables {obC obD : Type} {C : category obC} {D : category obD} {F G H : C ⇒ D}
protected definition compose2 (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
natural_transformation.mk
(λ a, η a ∘ θ a)
(λ a b f, calc
H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
... = (η b ∘ G f) ∘ θ a : {naturality η f}
... = η b ∘ (G f ∘ θ a) : symm !assoc
... = η b ∘ (θ b ∘ F f) : {naturality θ f}
... = (η b ∘ θ b) ∘ F f : assoc)