2015-01-14 02:45:58 +00:00
|
|
|
/-
|
|
|
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
|
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
|
|
|
|
|
|
Module: data.fin
|
|
|
|
|
Author: Leonardo de Moura
|
|
|
|
|
|
|
|
|
|
Finite ordinals.
|
|
|
|
|
-/
|
|
|
|
|
open nat
|
|
|
|
|
|
|
|
|
|
inductive fin : nat → Type :=
|
2015-02-26 01:00:10 +00:00
|
|
|
| fz : Π n, fin (succ n)
|
|
|
|
|
| fs : Π {n}, fin n → fin (succ n)
|
2015-01-14 02:45:58 +00:00
|
|
|
|
|
|
|
|
namespace fin
|
2015-02-26 00:20:44 +00:00
|
|
|
definition to_nat : Π {n}, fin n → nat
|
|
|
|
|
| to_nat (fz n) := zero
|
|
|
|
|
| to_nat (@fs n f) := succ (@to_nat n f)
|
2015-01-14 02:45:58 +00:00
|
|
|
|
2015-02-26 00:20:44 +00:00
|
|
|
definition lift : Π {n : nat}, fin n → Π (m : nat), fin (add m n)
|
|
|
|
|
| lift (fz n) m := fz (add m n)
|
|
|
|
|
| lift (@fs n f) m := fs (@lift n f m)
|
2015-01-14 02:45:58 +00:00
|
|
|
|
2015-02-26 00:20:44 +00:00
|
|
|
theorem to_nat_lift : ∀ {n : nat} (f : fin n) (m : nat), to_nat f = to_nat (lift f m)
|
|
|
|
|
| to_nat_lift (fz n) m := rfl
|
|
|
|
|
| to_nat_lift (@fs n f) m := calc
|
|
|
|
|
to_nat (fs f) = (to_nat f) + 1 : rfl
|
|
|
|
|
... = (to_nat (lift f m)) + 1 : to_nat_lift f
|
|
|
|
|
... = to_nat (lift (fs f) m) : rfl
|
2015-01-14 02:45:58 +00:00
|
|
|
end fin
|
