src/HOL/ex/Binary.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 42814 5af15f1e2ef6 child 46236 ae79f2978a67 permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/ex/Binary.thy
```
```     2     Author:     Makarius
```
```     3 *)
```
```     4
```
```     5 header {* Simple and efficient binary numerals *}
```
```     6
```
```     7 theory Binary
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection {* Binary representation of natural numbers *}
```
```    12
```
```    13 definition
```
```    14   bit :: "nat \<Rightarrow> bool \<Rightarrow> nat" where
```
```    15   "bit n b = (if b then 2 * n + 1 else 2 * n)"
```
```    16
```
```    17 lemma bit_simps:
```
```    18     "bit n False = 2 * n"
```
```    19     "bit n True = 2 * n + 1"
```
```    20   unfolding bit_def by simp_all
```
```    21
```
```    22 ML {*
```
```    23   fun dest_bit (Const (@{const_name False}, _)) = 0
```
```    24     | dest_bit (Const (@{const_name True}, _)) = 1
```
```    25     | dest_bit t = raise TERM ("dest_bit", [t]);
```
```    26
```
```    27   fun dest_binary (Const (@{const_name Groups.zero}, Type (@{type_name nat}, _))) = 0
```
```    28     | dest_binary (Const (@{const_name Groups.one}, Type (@{type_name nat}, _))) = 1
```
```    29     | dest_binary (Const (@{const_name bit}, _) \$ bs \$ b) = 2 * dest_binary bs + dest_bit b
```
```    30     | dest_binary t = raise TERM ("dest_binary", [t]);
```
```    31
```
```    32   fun mk_bit 0 = @{term False}
```
```    33     | mk_bit 1 = @{term True}
```
```    34     | mk_bit _ = raise TERM ("mk_bit", []);
```
```    35
```
```    36   fun mk_binary 0 = @{term "0::nat"}
```
```    37     | mk_binary 1 = @{term "1::nat"}
```
```    38     | mk_binary n =
```
```    39         if n < 0 then raise TERM ("mk_binary", [])
```
```    40         else
```
```    41           let val (q, r) = Integer.div_mod n 2
```
```    42           in @{term bit} \$ mk_binary q \$ mk_bit r end;
```
```    43 *}
```
```    44
```
```    45
```
```    46 subsection {* Direct operations -- plain normalization *}
```
```    47
```
```    48 lemma binary_norm:
```
```    49     "bit 0 False = 0"
```
```    50     "bit 0 True = 1"
```
```    51   unfolding bit_def by simp_all
```
```    52
```
```    53 lemma binary_add:
```
```    54     "n + 0 = n"
```
```    55     "0 + n = n"
```
```    56     "1 + 1 = bit 1 False"
```
```    57     "bit n False + 1 = bit n True"
```
```    58     "bit n True + 1 = bit (n + 1) False"
```
```    59     "1 + bit n False = bit n True"
```
```    60     "1 + bit n True = bit (n + 1) False"
```
```    61     "bit m False + bit n False = bit (m + n) False"
```
```    62     "bit m False + bit n True = bit (m + n) True"
```
```    63     "bit m True + bit n False = bit (m + n) True"
```
```    64     "bit m True + bit n True = bit ((m + n) + 1) False"
```
```    65   by (simp_all add: bit_simps)
```
```    66
```
```    67 lemma binary_mult:
```
```    68     "n * 0 = 0"
```
```    69     "0 * n = 0"
```
```    70     "n * 1 = n"
```
```    71     "1 * n = n"
```
```    72     "bit m True * n = bit (m * n) False + n"
```
```    73     "bit m False * n = bit (m * n) False"
```
```    74     "n * bit m True = bit (m * n) False + n"
```
```    75     "n * bit m False = bit (m * n) False"
```
```    76   by (simp_all add: bit_simps)
```
```    77
```
```    78 lemmas binary_simps = binary_norm binary_add binary_mult
```
```    79
```
```    80
```
```    81 subsection {* Indirect operations -- ML will produce witnesses *}
```
```    82
```
```    83 lemma binary_less_eq:
```
```    84   fixes n :: nat
```
```    85   shows "n \<equiv> m + k \<Longrightarrow> (m \<le> n) \<equiv> True"
```
```    86     and "m \<equiv> n + k + 1 \<Longrightarrow> (m \<le> n) \<equiv> False"
```
```    87   by simp_all
```
```    88
```
```    89 lemma binary_less:
```
```    90   fixes n :: nat
```
```    91   shows "m \<equiv> n + k \<Longrightarrow> (m < n) \<equiv> False"
```
```    92     and "n \<equiv> m + k + 1 \<Longrightarrow> (m < n) \<equiv> True"
```
```    93   by simp_all
```
```    94
```
```    95 lemma binary_diff:
```
```    96   fixes n :: nat
```
```    97   shows "m \<equiv> n + k \<Longrightarrow> m - n \<equiv> k"
```
```    98     and "n \<equiv> m + k \<Longrightarrow> m - n \<equiv> 0"
```
```    99   by simp_all
```
```   100
```
```   101 lemma binary_divmod:
```
```   102   fixes n :: nat
```
```   103   assumes "m \<equiv> n * k + l" and "0 < n" and "l < n"
```
```   104   shows "m div n \<equiv> k"
```
```   105     and "m mod n \<equiv> l"
```
```   106 proof -
```
```   107   from `m \<equiv> n * k + l` have "m = l + k * n" by simp
```
```   108   with `0 < n` and `l < n` show "m div n \<equiv> k" and "m mod n \<equiv> l" by simp_all
```
```   109 qed
```
```   110
```
```   111 ML {*
```
```   112 local
```
```   113   infix ==;
```
```   114   val op == = Logic.mk_equals;
```
```   115   fun plus m n = @{term "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"} \$ m \$ n;
```
```   116   fun mult m n = @{term "times :: nat \<Rightarrow> nat \<Rightarrow> nat"} \$ m \$ n;
```
```   117
```
```   118   val binary_ss = HOL_basic_ss addsimps @{thms binary_simps};
```
```   119   fun prove ctxt prop =
```
```   120     Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
```
```   121
```
```   122   fun binary_proc proc ss ct =
```
```   123     (case Thm.term_of ct of
```
```   124       _ \$ t \$ u =>
```
```   125       (case try (pairself (`dest_binary)) (t, u) of
```
```   126         SOME args => proc (Simplifier.the_context ss) args
```
```   127       | NONE => NONE)
```
```   128     | _ => NONE);
```
```   129 in
```
```   130
```
```   131 val less_eq_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
```
```   132   let val k = n - m in
```
```   133     if k >= 0 then
```
```   134       SOME (@{thm binary_less_eq(1)} OF [prove ctxt (u == plus t (mk_binary k))])
```
```   135     else
```
```   136       SOME (@{thm binary_less_eq(2)} OF
```
```   137         [prove ctxt (t == plus (plus u (mk_binary (~ k - 1))) (mk_binary 1))])
```
```   138   end);
```
```   139
```
```   140 val less_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
```
```   141   let val k = m - n in
```
```   142     if k >= 0 then
```
```   143       SOME (@{thm binary_less(1)} OF [prove ctxt (t == plus u (mk_binary k))])
```
```   144     else
```
```   145       SOME (@{thm binary_less(2)} OF
```
```   146         [prove ctxt (u == plus (plus t (mk_binary (~ k - 1))) (mk_binary 1))])
```
```   147   end);
```
```   148
```
```   149 val diff_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
```
```   150   let val k = m - n in
```
```   151     if k >= 0 then
```
```   152       SOME (@{thm binary_diff(1)} OF [prove ctxt (t == plus u (mk_binary k))])
```
```   153     else
```
```   154       SOME (@{thm binary_diff(2)} OF [prove ctxt (u == plus t (mk_binary (~ k)))])
```
```   155   end);
```
```   156
```
```   157 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
```
```   158   if n = 0 then NONE
```
```   159   else
```
```   160     let val (k, l) = Integer.div_mod m n
```
```   161     in SOME (rule OF [prove ctxt (t == plus (mult u (mk_binary k)) (mk_binary l))]) end);
```
```   162
```
```   163 end;
```
```   164 *}
```
```   165
```
```   166 simproc_setup binary_nat_less_eq ("m <= (n::nat)") = {* K less_eq_proc *}
```
```   167 simproc_setup binary_nat_less ("m < (n::nat)") = {* K less_proc *}
```
```   168 simproc_setup binary_nat_diff ("m - (n::nat)") = {* K diff_proc *}
```
```   169 simproc_setup binary_nat_div ("m div (n::nat)") = {* K (divmod_proc @{thm binary_divmod(1)}) *}
```
```   170 simproc_setup binary_nat_mod ("m mod (n::nat)") = {* K (divmod_proc @{thm binary_divmod(2)}) *}
```
```   171
```
```   172 method_setup binary_simp = {*
```
```   173   Scan.succeed (K (SIMPLE_METHOD'
```
```   174     (full_simp_tac
```
```   175       (HOL_basic_ss
```
```   176         addsimps @{thms binary_simps}
```
```   177         addsimprocs
```
```   178          [@{simproc binary_nat_less_eq},
```
```   179           @{simproc binary_nat_less},
```
```   180           @{simproc binary_nat_diff},
```
```   181           @{simproc binary_nat_div},
```
```   182           @{simproc binary_nat_mod}]))))
```
```   183 *}
```
```   184
```
```   185
```
```   186 subsection {* Concrete syntax *}
```
```   187
```
```   188 syntax
```
```   189   "_Binary" :: "num_const \<Rightarrow> 'a"    ("\$_")
```
```   190
```
```   191 parse_translation {*
```
```   192 let
```
```   193   val syntax_consts =
```
```   194     map_aterms (fn Const (c, T) => Const (Lexicon.mark_const c, T) | a => a);
```
```   195
```
```   196   fun binary_tr [Const (num, _)] =
```
```   197         let
```
```   198           val {leading_zeros = z, value = n, ...} = Lexicon.read_xnum num;
```
```   199           val _ = z = 0 andalso n >= 0 orelse error ("Bad binary number: " ^ num);
```
```   200         in syntax_consts (mk_binary n) end
```
```   201     | binary_tr ts = raise TERM ("binary_tr", ts);
```
```   202
```
```   203 in [(@{syntax_const "_Binary"}, binary_tr)] end
```
```   204 *}
```
```   205
```
```   206
```
```   207 subsection {* Examples *}
```
```   208
```
```   209 lemma "\$6 = 6"
```
```   210   by (simp add: bit_simps)
```
```   211
```
```   212 lemma "bit (bit (bit 0 False) False) True = 1"
```
```   213   by (simp add: bit_simps)
```
```   214
```
```   215 lemma "bit (bit (bit 0 False) False) True = bit 0 True"
```
```   216   by (simp add: bit_simps)
```
```   217
```
```   218 lemma "\$5 + \$3 = \$8"
```
```   219   by binary_simp
```
```   220
```
```   221 lemma "\$5 * \$3 = \$15"
```
```   222   by binary_simp
```
```   223
```
```   224 lemma "\$5 - \$3 = \$2"
```
```   225   by binary_simp
```
```   226
```
```   227 lemma "\$3 - \$5 = 0"
```
```   228   by binary_simp
```
```   229
```
```   230 lemma "\$123456789 - \$123 = \$123456666"
```
```   231   by binary_simp
```
```   232
```
```   233 lemma "\$1111111111222222222233333333334444444444 - \$998877665544332211 =
```
```   234   \$1111111111222222222232334455668900112233"
```
```   235   by binary_simp
```
```   236
```
```   237 lemma "(1111111111222222222233333333334444444444::nat) - 998877665544332211 =
```
```   238   1111111111222222222232334455668900112233"
```
```   239   by simp
```
```   240
```
```   241 lemma "(1111111111222222222233333333334444444444::int) - 998877665544332211 =
```
```   242   1111111111222222222232334455668900112233"
```
```   243   by simp
```
```   244
```
```   245 lemma "\$1111111111222222222233333333334444444444 * \$998877665544332211 =
```
```   246     \$1109864072938022197293802219729380221972383090160869185684"
```
```   247   by binary_simp
```
```   248
```
```   249 lemma "\$1111111111222222222233333333334444444444 * \$998877665544332211 -
```
```   250       \$5555555555666666666677777777778888888888 =
```
```   251     \$1109864072938022191738246664062713555294605312381980296796"
```
```   252   by binary_simp
```
```   253
```
```   254 lemma "\$42 < \$4 = False"
```
```   255   by binary_simp
```
```   256
```
```   257 lemma "\$4 < \$42 = True"
```
```   258   by binary_simp
```
```   259
```
```   260 lemma "\$42 <= \$4 = False"
```
```   261   by binary_simp
```
```   262
```
```   263 lemma "\$4 <= \$42 = True"
```
```   264   by binary_simp
```
```   265
```
```   266 lemma "\$1111111111222222222233333333334444444444 < \$998877665544332211 = False"
```
```   267   by binary_simp
```
```   268
```
```   269 lemma "\$998877665544332211 < \$1111111111222222222233333333334444444444 = True"
```
```   270   by binary_simp
```
```   271
```
```   272 lemma "\$1111111111222222222233333333334444444444 <= \$998877665544332211 = False"
```
```   273   by binary_simp
```
```   274
```
```   275 lemma "\$998877665544332211 <= \$1111111111222222222233333333334444444444 = True"
```
```   276   by binary_simp
```
```   277
```
```   278 lemma "\$1234 div \$23 = \$53"
```
```   279   by binary_simp
```
```   280
```
```   281 lemma "\$1234 mod \$23 = \$15"
```
```   282   by binary_simp
```
```   283
```
```   284 lemma "\$1111111111222222222233333333334444444444 div \$998877665544332211 =
```
```   285     \$1112359550673033707875"
```
```   286   by binary_simp
```
```   287
```
```   288 lemma "\$1111111111222222222233333333334444444444 mod \$998877665544332211 =
```
```   289     \$42245174317582819"
```
```   290   by binary_simp
```
```   291
```
```   292 lemma "(1111111111222222222233333333334444444444::int) div 998877665544332211 =
```
```   293     1112359550673033707875"
```
```   294   by simp  -- {* legacy numerals: 30 times slower *}
```
```   295
```
```   296 lemma "(1111111111222222222233333333334444444444::int) mod 998877665544332211 =
```
```   297     42245174317582819"
```
```   298   by simp  -- {* legacy numerals: 30 times slower *}
```
```   299
```
```   300 end
```