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|
Set Printing Universes.
Module unidecls.
Universes a b.
End unidecls.
Universe a.
Constraint a < unidecls.a.
Print Universes.
(** These are different universes *)
Check Type@{a}.
Check Type@{unidecls.a}.
Check Type@{unidecls.b}.
Fail Check Type@{unidecls.c}.
Fail Check Type@{i}.
Universe foo.
Module Foo.
(** Already declared globaly: but universe names are scoped at the module level *)
Universe foo.
Universe bar.
Check Type@{Foo.foo}.
Definition bar := 0.
End Foo.
(** Already declared in the module *)
Universe bar.
(** Accessible outside the module: universe declarations are global *)
Check Type@{bar}.
Check Type@{Foo.bar}.
Check Type@{Foo.foo}.
(** The same *)
Check Type@{foo}.
Check Type@{Top.foo}.
Universe secfoo.
Section Foo'.
Fail Universe secfoo.
Universe secfoo2.
Check Type@{Foo'.secfoo2}.
Constraint secfoo2 < a.
End Foo'.
Check Type@{secfoo2}.
Fail Check Type@{Foo'.secfoo2}.
Fail Check eq_refl : Type@{secfoo2} = Type@{a}.
(** Below, u and v are global, fixed universes *)
Module Type Arg.
Universe u.
Parameter T: Type@{u}.
End Arg.
Module Fn(A : Arg).
Universes v.
Check Type@{A.u}.
Constraint A.u < v.
Definition foo : Type@{v} := nat.
Definition bar : Type@{A.u} := nat.
Fail Definition foo(A : Type@{v}) : Type@{A.u} := A.
End Fn.
Module ArgImpl : Arg.
Definition T := nat.
End ArgImpl.
Module ArgImpl2 : Arg.
Definition T := bool.
End ArgImpl2.
(** Two applications of the functor result in the exact same universes *)
Module FnApp := Fn(ArgImpl).
Check Type@{FnApp.v}.
Check FnApp.foo.
Check FnApp.bar.
Check (eq_refl : Type@{ArgImpl.u} = Type@{ArgImpl2.u}).
Module FnApp2 := Fn(ArgImpl).
Check Type@{FnApp2.v}.
Check FnApp2.foo.
Check FnApp2.bar.
Import ArgImpl2.
(** Now u refers to ArgImpl.u and ArgImpl2.u *)
Check FnApp2.bar.
(** It can be shadowed *)
Universe u.
(** This refers to the qualified name *)
Check FnApp2.bar.
Constraint u = ArgImpl.u.
Print Universes.
Set Universe Polymorphism.
Section PS.
Universe poly.
Definition id (A : Type@{poly}) (a : A) : A := a.
End PS.
(** The universe is polymorphic and discharged, does not persist *)
Fail Check Type@{poly}.
Print Universes.
Check id nat.
Check id@{Set}.
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