86 lines
2 KiB
Text
86 lines
2 KiB
Text
/-
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Authors: Leonardo de Moura
|
|
-/
|
|
prelude
|
|
import init.bool
|
|
|
|
namespace pos_num
|
|
protected definition mul (a b : pos_num) : pos_num :=
|
|
pos_num.rec_on a
|
|
b
|
|
(λ n r, bit0 r + b)
|
|
(λ n r, bit0 r)
|
|
|
|
definition lt (a b : pos_num) : bool :=
|
|
pos_num.rec_on a
|
|
(λ b, pos_num.cases_on b
|
|
ff
|
|
(λ m, tt)
|
|
(λ m, tt))
|
|
(λ n f b, pos_num.cases_on b
|
|
ff
|
|
(λ m, f m)
|
|
(λ m, f m))
|
|
(λ n f b, pos_num.cases_on b
|
|
ff
|
|
(λ m, f (succ m))
|
|
(λ m, f m))
|
|
b
|
|
|
|
definition le (a b : pos_num) : bool :=
|
|
pos_num.lt a (succ b)
|
|
end pos_num
|
|
|
|
attribute [instance]
|
|
definition pos_num_has_mul : has_mul pos_num :=
|
|
⟨pos_num.mul⟩
|
|
|
|
namespace num
|
|
open pos_num
|
|
|
|
definition pred (a : num) : num :=
|
|
num.rec_on a zero (λ p, bool.cond (is_one p) zero (pos (pred p)))
|
|
|
|
definition size (a : num) : num :=
|
|
num.rec_on a (pos pos_num.one) (λ p, pos (size p))
|
|
|
|
protected definition mul (a b : num) : num :=
|
|
num.rec_on a zero (λ pa, num.rec_on b zero (λ pb, pos (pos_num.mul pa pb)))
|
|
end num
|
|
|
|
attribute [instance]
|
|
definition num_has_mul : has_mul num :=
|
|
⟨num.mul⟩
|
|
|
|
namespace num
|
|
protected definition le (a b : num) : bool :=
|
|
num.rec_on a tt (λ pa, num.rec_on b ff (λ pb, pos_num.le pa pb))
|
|
|
|
private definition psub (a b : pos_num) : num :=
|
|
pos_num.rec_on a
|
|
(λ b, zero)
|
|
(λ n f b,
|
|
bool.cond (pos_num.le (bit1 n) b)
|
|
zero
|
|
(pos_num.cases_on b
|
|
(pos (bit0 n))
|
|
(λ m, 2 * f m)
|
|
(λ m, 2 * f m + 1)))
|
|
(λ n f b,
|
|
bool.cond (pos_num.le (bit0 n) b)
|
|
zero
|
|
(pos_num.cases_on b
|
|
(pos (pos_num.pred (bit0 n)))
|
|
(λ m, pred (2 * f m))
|
|
(λ m, 2 * f m)))
|
|
b
|
|
|
|
protected definition sub (a b : num) : num :=
|
|
num.rec_on a zero (λ pa, num.rec_on b a (λ pb, psub pa pb))
|
|
end num
|
|
|
|
attribute [instance]
|
|
definition num_has_sub : has_sub num :=
|
|
⟨num.sub⟩
|