lean2/library/data/num.lean
2014-10-05 13:47:51 -07:00

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
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import logic.inhabited data.bool general_notation
open bool
-- pos_num and num are two auxiliary datatypes used when parsing numerals such as 13, 0, 26.
-- The parser will generate the terms (pos (bit1 (bit1 (bit0 one)))), zero, and (pos (bit0 (bit1 (bit1 one)))).
-- This representation can be coerced in whatever we want (e.g., naturals, integers, reals, etc).
inductive pos_num : Type :=
one : pos_num,
bit1 : pos_num → pos_num,
bit0 : pos_num → pos_num
definition pos_num.is_inhabited [instance] : inhabited pos_num :=
inhabited.mk pos_num.one
namespace pos_num
protected theorem induction_on {P : pos_num → Prop} (a : pos_num)
(H₁ : P one) (H₂ : ∀ (n : pos_num), P n → P (bit1 n)) (H₃ : ∀ (n : pos_num), P n → P (bit0 n)) : P a :=
rec H₁ H₂ H₃ a
protected definition rec_on {P : pos_num → Type} (a : pos_num)
(H₁ : P one) (H₂ : ∀ (n : pos_num), P n → P (bit1 n)) (H₃ : ∀ (n : pos_num), P n → P (bit0 n)) : P a :=
rec H₁ H₂ H₃ a
definition succ (a : pos_num) : pos_num :=
rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
definition is_one (a : pos_num) : bool :=
rec_on a tt (λn r, ff) (λn r, ff)
definition pred (a : pos_num) : pos_num :=
rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
definition size (a : pos_num) : pos_num :=
rec_on a one (λn r, succ r) (λn r, succ r)
theorem succ_not_is_one (a : pos_num) : is_one (succ a) = ff :=
induction_on a rfl (take n iH, rfl) (take n iH, rfl)
theorem pred.succ (a : pos_num) : pred (succ a) = a :=
rec_on a
rfl
(take (n : pos_num) (iH : pred (succ n) = n),
calc
pred (succ (bit1 n)) = cond ff one (bit1 (pred (succ n))) : {!succ_not_is_one}
... = bit1 (pred (succ n)) : rfl
... = bit1 n : {iH})
(take (n : pos_num) (iH : pred (succ n) = n), rfl)
definition add (a b : pos_num) : pos_num :=
rec_on a
succ
(λn f b, rec_on b
(succ (bit1 n))
(λm r, succ (bit1 (f m)))
(λm r, bit1 (f m)))
(λn f b, rec_on b
(bit1 n)
(λm r, bit1 (f m))
(λm r, bit0 (f m)))
b
infixl `+` := add
section
variables (a b : pos_num)
theorem add.one_one : one + one = bit0 one :=
rfl
theorem add.one_bit0 : one + (bit0 a) = bit1 a :=
rfl
theorem add.one_bit1 : one + (bit1 a) = succ (bit1 a) :=
rfl
theorem add.bit0_one : (bit0 a) + one = bit1 a :=
rfl
theorem add.bit1_one : (bit1 a) + one = succ (bit1 a) :=
rfl
theorem add.bit0_bit0 : (bit0 a) + (bit0 b) = bit0 (a + b) :=
rfl
theorem add.bit0_bit1 : (bit0 a) + (bit1 b) = bit1 (a + b) :=
rfl
theorem add.bit1_bit0 : (bit1 a) + (bit0 b) = bit1 (a + b) :=
rfl
theorem add.bit1_bit1 : (bit1 a) + (bit1 b) = succ (bit1 (a + b)) :=
rfl
end
definition mul (a b : pos_num) : pos_num :=
rec_on a
b
(λn r, bit0 r + b)
(λn r, bit0 r)
infixl `*` := mul
theorem mul.one_left (a : pos_num) : one * a = a :=
rfl
theorem mul.one_right (a : pos_num) : a * one = a :=
induction_on a
rfl
(take (n : pos_num) (iH : n * one = n),
calc bit1 n * one = bit0 (n * one) + one : rfl
... = bit0 n + one : {iH}
... = bit1 n : !add.bit0_one)
(take (n : pos_num) (iH : n * one = n),
calc bit0 n * one = bit0 (n * one) : rfl
... = bit0 n : {iH})
end pos_num
inductive num : Type :=
zero : num,
pos : pos_num → num
definition num.is_inhabited [instance] : inhabited num :=
inhabited.mk num.zero
namespace num
open pos_num
protected theorem induction_on {P : num → Prop} (a : num)
(H₁ : P zero) (H₂ : ∀ (p : pos_num), P (pos p)) : P a :=
rec H₁ H₂ a
protected definition rec_on {P : num → Type} (a : num)
(H₁ : P zero) (H₂ : ∀ (p : pos_num), P (pos p)) : P a :=
rec H₁ H₂ a
definition succ (a : num) : num :=
rec_on a (pos one) (λp, pos (succ p))
definition pred (a : num) : num :=
rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
definition size (a : num) : num :=
rec_on a (pos one) (λp, pos (size p))
theorem pred.succ (a : num) : pred (succ a) = a :=
rec_on a
rfl
(λp, calc
pred (succ (pos p)) = pred (pos (pos_num.succ p)) : rfl
... = cond ff zero (pos (pos_num.pred (pos_num.succ p))) : {!succ_not_is_one}
... = pos (pos_num.pred (pos_num.succ p)) : !cond.ff
... = pos p : {!pos_num.pred.succ})
definition add (a b : num) : num :=
rec_on a b (λp_a, rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
definition mul (a b : num) : num :=
rec_on a zero (λp_a, rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
infixl `+` := add
infixl `*` := mul
end num