lean2/tests/lean/run/finbug2.lean
2015-05-23 20:52:23 +10:00

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/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
Finite ordinals.
-/
open nat
inductive fin : nat → Type :=
| fz : Π n, fin (succ n)
| fs : Π {n}, fin n → fin (succ n)
namespace fin
definition to_nat : Π {n}, fin n → nat
| @to_nat (succ n) (fz n) := zero
| @to_nat (succ n) (fs f) := succ (@to_nat n f)
definition lift : Π {n : nat}, fin n → Π (m : nat), fin (add m n)
| @lift (succ n) (fz n) m := fz (add m n)
| @lift (succ n) (@fs n f) m := fs (@lift n f m)
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
end fin