/- 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 open bool algebra namespace pos_num protected definition mul (a b : pos_num) : pos_num := pos_num.rec_on a b (λn r, bit0 r + b) (λn r, bit0 r) definition lt (a b : pos_num) : bool := pos_num.rec_on a (λ b, pos_num.cases_on b ff (λm, tt) (λm, tt)) (λn f b, pos_num.cases_on b ff (λm, f m) (λm, f m)) (λn f b, pos_num.cases_on b ff (λm, f (succ m)) (λm, f m)) b definition le (a b : pos_num) : bool := pos_num.lt a (succ b) end pos_num definition pos_num_has_mul [instance] [reducible] : has_mul pos_num := has_mul.mk pos_num.mul namespace num open pos_num definition pred (a : num) : num := num.rec_on a zero (λp, cond (is_one p) zero (pos (pred p))) definition size (a : num) : num := num.rec_on a (pos one) (λp, pos (size p)) protected definition mul (a b : num) : num := num.rec_on a zero (λpa, num.rec_on b zero (λpb, pos (pos_num.mul pa pb))) end num definition num_has_mul [instance] [reducible] : has_mul num := has_mul.mk num.mul namespace num protected definition le (a b : num) : bool := num.rec_on a tt (λpa, num.rec_on b ff (λpb, pos_num.le pa pb)) private definition psub (a b : pos_num) : num := pos_num.rec_on a (λb, zero) (λn f b, cond (pos_num.le (bit1 n) b) zero (pos_num.cases_on b (pos (bit0 n)) (λm, 2 * f m) (λm, 2 * f m + 1))) (λn f b, cond (pos_num.le (bit0 n) b) zero (pos_num.cases_on b (pos (pos_num.pred (bit0 n))) (λm, pred (2 * f m)) (λm, 2 * f m))) b protected definition sub (a b : num) : num := num.rec_on a zero (λpa, num.rec_on b a (λpb, psub pa pb)) end num definition num_has_sub [instance] [reducible] : has_sub num := has_sub.mk num.sub