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Inductive listn : nat-> Set :=
niln : (listn O)
| consn : (n:nat)nat->(listn n) -> (listn (S n)).
Inductive empty : (n:nat)(listn n)-> Prop :=
intro_empty: (empty O niln).
Parameter inv_empty : (n,a:nat)(l:(listn n)) ~(empty (S n) (consn n a l)).
Type
[n:nat] [l:(listn n)]
<[n:nat] [l:(listn n)](empty n l) \/ ~(empty n l)>Cases l of
niln => (or_introl ? ~(empty O niln) intro_empty)
| ((consn n O y) as b) => (or_intror (empty (S n) b) ? (inv_empty n O y))
| ((consn n a y) as b) => (or_intror (empty (S n) b) ? (inv_empty n a y))
end.
Type
[n:nat] [l:(listn n)]
<[n:nat] [l:(listn n)](empty n l) \/ ~(empty n l)>Cases l of
niln => (or_introl ? ~(empty O niln) intro_empty)
| (consn n O y) => (or_intror (empty (S n) (consn n O y)) ?
(inv_empty n O y))
| (consn n a y) => (or_intror (empty (S n) (consn n a y)) ?
(inv_empty n a y))
end.
Type
[n:nat] [l:(listn n)]
<[n:nat] [l:(listn n)](empty n l) \/ ~(empty n l)>Cases l of
niln => (or_introl ? ~(empty O niln) intro_empty)
| ((consn O a y) as b) => (or_intror (empty (S O) b) ? (inv_empty O a y))
| ((consn n a y) as b) => (or_intror (empty (S n) b) ? (inv_empty n a y))
end.
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