29 lines
944 B
Text
29 lines
944 B
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Finite ordinals.
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-/
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open nat
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inductive fin : nat → Type :=
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| fz : Π n, fin (succ n)
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| fs : Π {n}, fin n → fin (succ n)
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namespace fin
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definition to_nat : Π {n}, fin n → nat
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| @to_nat (succ n) (fz n) := zero
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| @to_nat (succ n) (fs f) := succ (@to_nat n f)
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definition lift : Π {n : nat}, fin n → Π (m : nat), fin (add m n)
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| @lift (succ n) (fz n) m := fz (add m n)
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| @lift (succ n) (@fs n f) m := fs (@lift n f m)
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theorem to_nat_lift : ∀ {n : nat} (f : fin n) (m : nat), to_nat f = to_nat (lift f m)
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| to_nat_lift (fz n) m := rfl
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| to_nat_lift (@fs n f) m := calc
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to_nat (fs f) = (to_nat f) + 1 : rfl
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... = (to_nat (lift f m)) + 1 : to_nat_lift f
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... = to_nat (lift (fs f) m) : rfl
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end fin
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