162 lines
5.2 KiB
Text
162 lines
5.2 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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-- Author: Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic.core.inhabited data.bool general_notation
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open bool
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-- pos_num and num are two auxiliary datatypes used when parsing numerals such as 13, 0, 26.
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-- The parser will generate the terms (pos (bit1 (bit1 (bit0 one)))), zero, and (pos (bit0 (bit1 (bit1 one)))).
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-- This representation can be coerced in whatever we want (e.g., naturals, integers, reals, etc).
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inductive pos_num : Type :=
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one : pos_num,
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bit1 : pos_num → pos_num,
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bit0 : pos_num → pos_num
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theorem pos_num.is_inhabited [instance] : inhabited pos_num :=
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inhabited.mk pos_num.one
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namespace pos_num
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protected theorem induction_on {P : pos_num → Prop} (a : pos_num)
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(H₁ : P one) (H₂ : ∀ (n : pos_num), P n → P (bit1 n)) (H₃ : ∀ (n : pos_num), P n → P (bit0 n)) : P a :=
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rec H₁ H₂ H₃ a
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protected definition rec_on {P : pos_num → Type} (a : pos_num)
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(H₁ : P one) (H₂ : ∀ (n : pos_num), P n → P (bit1 n)) (H₃ : ∀ (n : pos_num), P n → P (bit0 n)) : P a :=
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rec H₁ H₂ H₃ a
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definition succ (a : pos_num) : pos_num :=
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rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
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definition is_one (a : pos_num) : bool :=
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rec_on a tt (λn r, ff) (λn r, ff)
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definition pred (a : pos_num) : pos_num :=
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rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
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definition size (a : pos_num) : pos_num :=
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rec_on a one (λn r, succ r) (λn r, succ r)
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theorem succ_not_is_one {a : pos_num} : is_one (succ a) = ff :=
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induction_on a rfl (take n iH, rfl) (take n iH, rfl)
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theorem pred_succ {a : pos_num} : pred (succ a) = a :=
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rec_on a
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rfl
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(take (n : pos_num) (iH : pred (succ n) = n),
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calc
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pred (succ (bit1 n)) = cond ff one (bit1 (pred (succ n))) : {succ_not_is_one}
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... = bit1 (pred (succ n)) : rfl
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... = bit1 n : {iH})
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(take (n : pos_num) (iH : pred (succ n) = n), rfl)
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definition add (a b : pos_num) : pos_num :=
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rec_on a
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succ
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(λn f b, rec_on b
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(succ (bit1 n))
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(λm r, succ (bit1 (f m)))
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(λm r, bit1 (f m)))
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(λn f b, rec_on b
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(bit1 n)
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(λm r, bit1 (f m))
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(λm r, bit0 (f m)))
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b
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infixl `+` := add
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theorem add_one_one : one + one = bit0 one :=
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rfl
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theorem add_one_bit0 {a : pos_num} : one + (bit0 a) = bit1 a :=
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rfl
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theorem add_one_bit1 {a : pos_num} : one + (bit1 a) = succ (bit1 a) :=
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rfl
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theorem add_bit0_one {a : pos_num} : (bit0 a) + one = bit1 a :=
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rfl
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theorem add_bit1_one {a : pos_num} : (bit1 a) + one = succ (bit1 a) :=
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rfl
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theorem add_bit0_bit0 {a b : pos_num} : (bit0 a) + (bit0 b) = bit0 (a + b) :=
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rfl
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theorem add_bit0_bit1 {a b : pos_num} : (bit0 a) + (bit1 b) = bit1 (a + b) :=
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rfl
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theorem add_bit1_bit0 {a b : pos_num} : (bit1 a) + (bit0 b) = bit1 (a + b) :=
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rfl
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theorem add_bit1_bit1 {a b : pos_num} : (bit1 a) + (bit1 b) = succ (bit1 (a + b)) :=
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rfl
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definition mul (a b : pos_num) : pos_num :=
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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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infixl `*` := mul
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theorem mul_one_left (a : pos_num) : one * a = a :=
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rfl
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theorem mul_one_right (a : pos_num) : a * one = a :=
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induction_on a
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rfl
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(take (n : pos_num) (iH : n * one = n),
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calc bit1 n * one = bit0 (n * one) + one : rfl
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... = bit0 n + one : {iH}
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... = bit1 n : add_bit0_one)
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(take (n : pos_num) (iH : n * one = n),
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calc bit0 n * one = bit0 (n * one) : rfl
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... = bit0 n : {iH})
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end pos_num
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inductive num : Type :=
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zero : num,
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pos : pos_num → num
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theorem num.is_inhabited [instance] : inhabited num :=
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inhabited.mk num.zero
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namespace num
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open pos_num
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protected theorem induction_on {P : num → Prop} (a : num)
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(H₁ : P zero) (H₂ : ∀ (p : pos_num), P (pos p)) : P a :=
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rec H₁ H₂ a
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protected definition rec_on {P : num → Type} (a : num)
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(H₁ : P zero) (H₂ : ∀ (p : pos_num), P (pos p)) : P a :=
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rec H₁ H₂ a
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definition succ (a : num) : num :=
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rec_on a (pos one) (λp, pos (succ p))
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definition pred (a : num) : num :=
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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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rec_on a (pos one) (λp, pos (size p))
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theorem pred_succ (a : num) : pred (succ a) = a :=
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rec_on a
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rfl
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(λp, calc
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pred (succ (pos p)) = pred (pos (pos_num.succ p)) : rfl
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... = cond ff zero (pos (pos_num.pred (pos_num.succ p))) : {succ_not_is_one}
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... = pos (pos_num.pred (pos_num.succ p)) : cond_ff _ _
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... = pos p : {pos_num.pred_succ})
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definition add (a b : num) : num :=
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rec_on a b (λp_a, rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
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definition mul (a b : num) : num :=
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rec_on a zero (λp_a, rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
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infixl `+` := add
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infixl `*` := mul
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end num
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