(* Title: HOL/ex/Binary.thy
ID: $Id$
Author: Makarius
*)
header {* Simple and efficient binary numerals *}
theory Binary
imports Main
begin
subsection {* Binary representation of natural numbers *}
definition
bit :: "nat \<Rightarrow> bool \<Rightarrow> nat" where
"bit n b = (if b then 2 * n + 1 else 2 * n)"
lemma bit_simps:
"bit n False = 2 * n"
"bit n True = 2 * n + 1"
unfolding bit_def by simp_all
ML {*
fun dest_bit (Const ("False", _)) = 0
| dest_bit (Const ("True", _)) = 1
| dest_bit t = raise TERM ("dest_bit", [t]);
fun dest_binary (Const (@{const_name HOL.zero}, Type ("nat", _))) = 0
| dest_binary (Const (@{const_name HOL.one}, Type ("nat", _))) = 1
| dest_binary (Const ("Binary.bit", _) $ bs $ b) =
2 * dest_binary bs + IntInf.fromInt (dest_bit b)
| dest_binary t = raise TERM ("dest_binary", [t]);
fun mk_bit 0 = @{term False}
| mk_bit 1 = @{term True}
| mk_bit _ = raise TERM ("mk_bit", []);
fun mk_binary 0 = @{term "0::nat"}
| mk_binary 1 = @{term "1::nat"}
| mk_binary n =
if n < 0 then raise TERM ("mk_binary", [])
else
let val (q, r) = IntInf.divMod (n, 2)
in @{term bit} $ mk_binary q $ mk_bit (IntInf.toInt r) end;
*}
subsection {* Direct operations -- plain normalization *}
lemma binary_norm:
"bit 0 False = 0"
"bit 0 True = 1"
unfolding bit_def by simp_all
lemma binary_add:
"n + 0 = n"
"0 + n = n"
"1 + 1 = bit 1 False"
"bit n False + 1 = bit n True"
"bit n True + 1 = bit (n + 1) False"
"1 + bit n False = bit n True"
"1 + bit n True = bit (n + 1) False"
"bit m False + bit n False = bit (m + n) False"
"bit m False + bit n True = bit (m + n) True"
"bit m True + bit n False = bit (m + n) True"
"bit m True + bit n True = bit ((m + n) + 1) False"
by (simp_all add: bit_simps)
lemma binary_mult:
"n * 0 = 0"
"0 * n = 0"
"n * 1 = n"
"1 * n = n"
"bit m True * n = bit (m * n) False + n"
"bit m False * n = bit (m * n) False"
"n * bit m True = bit (m * n) False + n"
"n * bit m False = bit (m * n) False"
by (simp_all add: bit_simps)
lemmas binary_simps = binary_norm binary_add binary_mult
subsection {* Indirect operations -- ML will produce witnesses *}
lemma binary_less_eq:
fixes n :: nat
shows "n \<equiv> m + k \<Longrightarrow> (m \<le> n) \<equiv> True"
and "m \<equiv> n + k + 1 \<Longrightarrow> (m \<le> n) \<equiv> False"
by simp_all
lemma binary_less:
fixes n :: nat
shows "m \<equiv> n + k \<Longrightarrow> (m < n) \<equiv> False"
and "n \<equiv> m + k + 1 \<Longrightarrow> (m < n) \<equiv> True"
by simp_all
lemma binary_diff:
fixes n :: nat
shows "m \<equiv> n + k \<Longrightarrow> m - n \<equiv> k"
and "n \<equiv> m + k \<Longrightarrow> m - n \<equiv> 0"
by simp_all
lemma binary_divmod:
fixes n :: nat
assumes "m \<equiv> n * k + l" and "0 < n" and "l < n"
shows "m div n \<equiv> k"
and "m mod n \<equiv> l"
proof -
from `m \<equiv> n * k + l` have "m = l + k * n" by simp
with `0 < n` and `l < n` show "m div n \<equiv> k" and "m mod n \<equiv> l" by simp_all
qed
ML {*
local
infix ==;
val op == = Logic.mk_equals;
fun plus m n = @{term "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"} $ m $ n;
fun mult m n = @{term "times :: nat \<Rightarrow> nat \<Rightarrow> nat"} $ m $ n;
val binary_ss = HOL_basic_ss addsimps @{thms binary_simps};
fun prove ctxt prop =
Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
fun binary_proc proc ss ct =
(case Thm.term_of ct of
_ $ t $ u =>
(case try (pairself (`dest_binary)) (t, u) of
SOME args => proc (Simplifier.the_context ss) args
| NONE => NONE)
| _ => NONE);
in
val less_eq_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
let val k = n - m in
if k >= 0 then
SOME (@{thm binary_less_eq(1)} OF [prove ctxt (u == plus t (mk_binary k))])
else
SOME (@{thm binary_less_eq(2)} OF
[prove ctxt (t == plus (plus u (mk_binary (~ k - 1))) (mk_binary 1))])
end);
val less_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
let val k = m - n in
if k >= 0 then
SOME (@{thm binary_less(1)} OF [prove ctxt (t == plus u (mk_binary k))])
else
SOME (@{thm binary_less(2)} OF
[prove ctxt (u == plus (plus t (mk_binary (~ k - 1))) (mk_binary 1))])
end);
val diff_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
let val k = m - n in
if k >= 0 then
SOME (@{thm binary_diff(1)} OF [prove ctxt (t == plus u (mk_binary k))])
else
SOME (@{thm binary_diff(2)} OF [prove ctxt (u == plus t (mk_binary (~ k)))])
end);
fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
if n = 0 then NONE
else
let val (k, l) = IntInf.divMod (m, n)
in SOME (rule OF [prove ctxt (t == plus (mult u (mk_binary k)) (mk_binary l))]) end);
end;
*}
simproc_setup binary_nat_less_eq ("m <= (n::nat)") = {* K less_eq_proc *}
simproc_setup binary_nat_less ("m < (n::nat)") = {* K less_proc *}
simproc_setup binary_nat_diff ("m - (n::nat)") = {* K diff_proc *}
simproc_setup binary_nat_div ("m div (n::nat)") = {* K (divmod_proc @{thm binary_divmod(1)}) *}
simproc_setup binary_nat_mod ("m mod (n::nat)") = {* K (divmod_proc @{thm binary_divmod(2)}) *}
method_setup binary_simp = {*
Method.no_args (Method.SIMPLE_METHOD'
(full_simp_tac
(HOL_basic_ss
addsimps @{thms binary_simps}
addsimprocs
[@{simproc binary_nat_less_eq},
@{simproc binary_nat_less},
@{simproc binary_nat_diff},
@{simproc binary_nat_div},
@{simproc binary_nat_mod}])))
*} "binary simplification"
subsection {* Concrete syntax *}
syntax
"_Binary" :: "num_const \<Rightarrow> 'a" ("$_")
parse_translation {*
let
val syntax_consts = map_aterms (fn Const (c, T) => Const (Syntax.constN ^ c, T) | a => a);
fun binary_tr [Const (num, _)] =
let
val {leading_zeros = z, value = n, ...} = Syntax.read_xnum num;
val _ = z = 0 andalso n >= 0 orelse error ("Bad binary number: " ^ num);
in syntax_consts (mk_binary n) end
| binary_tr ts = raise TERM ("binary_tr", ts);
in [("_Binary", binary_tr)] end
*}
subsection {* Examples *}
lemma "$6 = 6"
by (simp add: bit_simps)
lemma "bit (bit (bit 0 False) False) True = 1"
by (simp add: bit_simps)
lemma "bit (bit (bit 0 False) False) True = bit 0 True"
by (simp add: bit_simps)
lemma "$5 + $3 = $8"
by binary_simp
lemma "$5 * $3 = $15"
by binary_simp
lemma "$5 - $3 = $2"
by binary_simp
lemma "$3 - $5 = 0"
by binary_simp
lemma "$123456789 - $123 = $123456666"
by binary_simp
lemma "$1111111111222222222233333333334444444444 - $998877665544332211 =
$1111111111222222222232334455668900112233"
by binary_simp
lemma "(1111111111222222222233333333334444444444::nat) - 998877665544332211 =
1111111111222222222232334455668900112233"
by simp
lemma "(1111111111222222222233333333334444444444::int) - 998877665544332211 =
1111111111222222222232334455668900112233"
by simp
lemma "$1111111111222222222233333333334444444444 * $998877665544332211 =
$1109864072938022197293802219729380221972383090160869185684"
by binary_simp
lemma "$1111111111222222222233333333334444444444 * $998877665544332211 -
$5555555555666666666677777777778888888888 =
$1109864072938022191738246664062713555294605312381980296796"
by binary_simp
lemma "$42 < $4 = False"
by binary_simp
lemma "$4 < $42 = True"
by binary_simp
lemma "$42 <= $4 = False"
by binary_simp
lemma "$4 <= $42 = True"
by binary_simp
lemma "$1111111111222222222233333333334444444444 < $998877665544332211 = False"
by binary_simp
lemma "$998877665544332211 < $1111111111222222222233333333334444444444 = True"
by binary_simp
lemma "$1111111111222222222233333333334444444444 <= $998877665544332211 = False"
by binary_simp
lemma "$998877665544332211 <= $1111111111222222222233333333334444444444 = True"
by binary_simp
lemma "$1234 div $23 = $53"
by binary_simp
lemma "$1234 mod $23 = $15"
by binary_simp
lemma "$1111111111222222222233333333334444444444 div $998877665544332211 =
$1112359550673033707875"
by binary_simp
lemma "$1111111111222222222233333333334444444444 mod $998877665544332211 =
$42245174317582819"
by binary_simp
lemma "(1111111111222222222233333333334444444444::int) div 998877665544332211 =
1112359550673033707875"
by simp -- {* legacy numerals: 30 times slower *}
lemma "(1111111111222222222233333333334444444444::int) mod 998877665544332211 =
42245174317582819"
by simp -- {* legacy numerals: 30 times slower *}
end