open nat

namespace test
inductive {u} lift (A : Type u) : Type (u+1)
| up : A → lift

namespace lift
definition down {A : Type} (a : lift A) : A :=
lift.rec (λ a, a) a

theorem down_up {A : Type} (a : A) : down (up a) = a :=
rfl


theorem up_down {A : Type} (a' : lift A) : up (down a') = a' :=
lift.induction_on a' (λ a, rfl)

end lift

set_option pp.universes true
check nat
check lift nat
open lift
definition one1 : lift nat := up 1
end test
