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