60 lines
1.7 KiB
Text
60 lines
1.7 KiB
Text
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/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Finite ordinal types.
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-/
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import types.fin
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open eq nat fin algebra
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inductive is_succ [class] : ℕ → Type :=
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| mk : Πn, is_succ (succ n)
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attribute is_succ.mk [instance]
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definition is_succ_add_right [instance] (n m : ℕ) [H : is_succ m] : is_succ (n+m) :=
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by induction H with m; constructor
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definition is_succ_add_left (n m : ℕ) [H : is_succ n] : is_succ (n+m) :=
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by induction H with n; cases m with m: constructor
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definition is_succ_bit0 [instance] (n : ℕ) [H : is_succ n] : is_succ (bit0 n) :=
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by induction H with n; constructor
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namespace fin
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definition my_succ {n : ℕ} (x : fin n) : fin n :=
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begin
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cases n with n,
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{ exfalso, apply not_lt_zero _ (is_lt x)},
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{ exact
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if H : val x = n
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then fin.mk 0 !zero_lt_succ
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else fin.mk (nat.succ (val x)) (succ_lt_succ (lt_of_le_of_ne (le_of_lt_succ (is_lt x)) H))}
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end
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protected definition add {n : ℕ} (x y : fin n) : fin n :=
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iterate my_succ (val y) x
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definition has_zero_fin [instance] (n : ℕ) [H : is_succ n] : has_zero (fin n) :=
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by induction H with n; exact has_zero.mk (fin.zero n)
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definition has_one_fin [instance] (n : ℕ) [H : is_succ n] : has_one (fin n) :=
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by induction H with n; exact has_one.mk (my_succ (fin.zero n))
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definition has_add_fin [instance] (n : ℕ) : has_add (fin n) :=
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has_add.mk fin.add
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-- definition my_succ_eq_zero {n : ℕ} (x : fin (nat.succ n)) (p : val x = n) : my_succ x = 0 :=
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-- if H : val x = n
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-- then fin.mk 0 !zero_lt_succ
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-- else fin.mk (nat.succ (val x)) (succ_lt_succ (lt_of_le_of_ne (le_of_lt_succ (is_lt x)) H))
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end fin
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