--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Binary.thy Sat Jan 20 14:27:46 2007 +0100
@@ -0,0 +1,305 @@
+(* 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
+
+
+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 {*
+ fun dest_bit (Const ("False", _)) = 0
+ | dest_bit (Const ("True", _)) = 1
+ | dest_bit t = raise TERM ("dest_bit", [t]);
+
+ fun dest_binary (Const ("HOL.zero", Type ("nat", _))) = 0
+ | dest_binary (Const ("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]);
+
+ val bit_const = Const ("Binary.bit", HOLogic.natT --> HOLogic.boolT --> HOLogic.natT);
+
+ fun mk_bit 0 = HOLogic.false_const
+ | mk_bit 1 = HOLogic.true_const
+ | mk_bit _ = raise TERM ("mk_bit", []);
+
+ fun mk_binary 0 = Const ("HOL.zero", HOLogic.natT)
+ | mk_binary 1 = Const ("HOL.one", HOLogic.natT)
+ | mk_binary n =
+ if n < 0 then raise TERM ("mk_binary", [])
+ else
+ let val (q, r) = IntInf.divMod (n, 2)
+ in bit_const $ mk_binary q $ mk_bit (IntInf.toInt r) end;
+*}
+
+ML {*
+local
+ val binary_ss = HOL_basic_ss addsimps @{thms binary_simps};
+ val [binary_less_eq1, binary_less_eq2] = @{thms binary_less_eq};
+ val [binary_less1, binary_less2] = @{thms binary_less}
+ val [binary_diff1, binary_diff2] = @{thms binary_diff}
+ val [binary_div, binary_mod] = @{thms binary_divmod}
+
+ infix ==;
+ val op == = Logic.mk_equals;
+
+ fun nat_op c t u = Const (c, HOLogic.natT --> HOLogic.natT --> HOLogic.natT) $ t $ u;
+ val plus = nat_op "HOL.plus";
+ val mult = nat_op "HOL.times";
+
+ fun prove ctxt prop = (* FIXME avoid re-certification *)
+ Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
+
+
+ exception FAIL;
+ fun the_arg t = (t, dest_binary t handle TERM _ => raise FAIL);
+
+ val read =
+ let val thy = the_context () in Thm.cterm_of thy o Sign.read_term thy end;
+ fun mk_proc name pat proc = Simplifier.mk_simproc' name [read pat]
+ (fn ss => fn ct =>
+ (case Thm.term_of ct of
+ _ $ t $ u =>
+ (SOME (proc (Simplifier.the_context ss) (the_arg t) (the_arg u)) handle FAIL => NONE)
+ | _ => NONE));
+
+
+ val less_eq_simproc = mk_proc "binary_nat_less_eq" "?m <= (?n::nat)"
+ (fn ctxt => fn (t, m) => fn (u, n) =>
+ let val k = n - m in
+ if k >= 0 then binary_less_eq1 OF [prove ctxt (u == plus t (mk_binary k))]
+ else binary_less_eq2 OF
+ [prove ctxt (t == plus (plus u (mk_binary (~ k - 1))) (mk_binary 1))]
+ end);
+
+ val less_simproc = mk_proc "binary_nat_less" "?m < (?n::nat)"
+ (fn ctxt => fn (t, m) => fn (u, n) =>
+ let val k = m - n in
+ if k >= 0 then binary_less1 OF [prove ctxt (t == plus u (mk_binary k))]
+ else binary_less2 OF [prove ctxt (u == plus (plus t (mk_binary (~ k - 1))) (mk_binary 1))]
+ end);
+
+ val diff_simproc = mk_proc "binary_nat_diff" "?m - (?n::nat)"
+ (fn ctxt => fn (t, m) => fn (u, n) =>
+ let val k = m - n in
+ if k >= 0 then binary_diff1 OF [prove ctxt (t == plus u (mk_binary k))]
+ else binary_diff2 OF [prove ctxt (u == plus t (mk_binary (~ k)))]
+ end);
+
+ fun divmod_proc rule ctxt (t, m) (u, n) =
+ if n = 0 then raise FAIL
+ else
+ let val (k, l) = IntInf.divMod (m, n)
+ in rule OF [prove ctxt (t == plus (mult u (mk_binary k)) (mk_binary l))] end;
+
+ val div_simproc = mk_proc "binary_nat_div" "?m div (?n::nat)" (divmod_proc binary_div);
+ val mod_simproc = mk_proc "binary_nat_mod" "?m mod (?n::nat)" (divmod_proc binary_mod);
+
+in
+ val binary_nat_simprocs =
+ [less_eq_simproc, less_simproc, diff_simproc, div_simproc, mod_simproc];
+end
+*}
+
+
+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 [t as 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 *}
+
+method_setup binary_simp = {*
+ Method.no_args (Method.SIMPLE_METHOD'
+ (full_simp_tac (HOL_basic_ss addsimps @{thms binary_simps} addsimprocs binary_nat_simprocs)))
+*} "binary simplification"
+
+
+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::int) div 998877665544332211 =
+ 1112359550673033707875"
+ by simp -- {* existing numerals: slower by factor 30 *}
+
+lemma "$1111111111222222222233333333334444444444 mod $998877665544332211 =
+ $42245174317582819"
+ by binary_simp
+
+lemma "(1111111111222222222233333333334444444444::int) mod 998877665544332211 =
+ 42245174317582819"
+ by simp -- {* existing numerals: slower by factor 30 *}
+
+end