src/HOL/ex/Bit_Operations.thy
 author haftmann Tue Feb 11 19:03:57 2020 +0100 (7 weeks ago ago) changeset 71654 d45495e897f4 parent 71638 745e518d3d0b child 71749 b612edee9b0c permissions -rw-r--r--
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```     1 (*  Author:  Florian Haftmann, TUM
```
```     2 *)
```
```     3
```
```     4 section \<open>Proof of concept for purely algebraically founded lists of bits\<close>
```
```     5
```
```     6 theory Bit_Operations
```
```     7   imports
```
```     8     "HOL-Library.Boolean_Algebra"
```
```     9     Main
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Bit operations in suitable algebraic structures\<close>
```
```    13
```
```    14 class semiring_bit_operations = semiring_bit_shifts +
```
```    15   fixes "and" :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr \<open>AND\<close> 64)
```
```    16     and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr \<open>OR\<close>  59)
```
```    17     and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr \<open>XOR\<close> 59)
```
```    18   assumes bit_and_iff: \<open>\<And>n. bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close>
```
```    19     and bit_or_iff: \<open>\<And>n. bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close>
```
```    20     and bit_xor_iff: \<open>\<And>n. bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close>
```
```    21 begin
```
```    22
```
```    23 text \<open>
```
```    24   We want the bitwise operations to bind slightly weaker
```
```    25   than \<open>+\<close> and \<open>-\<close>.
```
```    26   For the sake of code generation
```
```    27   the operations \<^const>\<open>and\<close>, \<^const>\<open>or\<close> and \<^const>\<open>xor\<close>
```
```    28   are specified as definitional class operations.
```
```    29 \<close>
```
```    30
```
```    31 lemma stable_imp_drop_eq:
```
```    32   \<open>drop_bit n a = a\<close> if \<open>a div 2 = a\<close>
```
```    33   by (induction n) (simp_all add: that)
```
```    34
```
```    35 sublocale "and": semilattice \<open>(AND)\<close>
```
```    36   by standard (auto simp add: bit_eq_iff bit_and_iff)
```
```    37
```
```    38 sublocale or: semilattice_neutr \<open>(OR)\<close> 0
```
```    39   by standard (auto simp add: bit_eq_iff bit_or_iff)
```
```    40
```
```    41 sublocale xor: comm_monoid \<open>(XOR)\<close> 0
```
```    42   by standard (auto simp add: bit_eq_iff bit_xor_iff)
```
```    43
```
```    44 lemma zero_and_eq [simp]:
```
```    45   "0 AND a = 0"
```
```    46   by (simp add: bit_eq_iff bit_and_iff)
```
```    47
```
```    48 lemma and_zero_eq [simp]:
```
```    49   "a AND 0 = 0"
```
```    50   by (simp add: bit_eq_iff bit_and_iff)
```
```    51
```
```    52 lemma one_and_eq [simp]:
```
```    53   "1 AND a = of_bool (odd a)"
```
```    54   by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff)
```
```    55
```
```    56 lemma and_one_eq [simp]:
```
```    57   "a AND 1 = of_bool (odd a)"
```
```    58   using one_and_eq [of a] by (simp add: ac_simps)
```
```    59
```
```    60 lemma one_or_eq [simp]:
```
```    61   "1 OR a = a + of_bool (even a)"
```
```    62   by (simp add: bit_eq_iff bit_or_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff)
```
```    63
```
```    64 lemma or_one_eq [simp]:
```
```    65   "a OR 1 = a + of_bool (even a)"
```
```    66   using one_or_eq [of a] by (simp add: ac_simps)
```
```    67
```
```    68 lemma one_xor_eq [simp]:
```
```    69   "1 XOR a = a + of_bool (even a) - of_bool (odd a)"
```
```    70   by (simp add: bit_eq_iff bit_xor_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff odd_bit_iff_bit_pred elim: oddE)
```
```    71
```
```    72 lemma xor_one_eq [simp]:
```
```    73   "a XOR 1 = a + of_bool (even a) - of_bool (odd a)"
```
```    74   using one_xor_eq [of a] by (simp add: ac_simps)
```
```    75
```
```    76 lemma take_bit_and [simp]:
```
```    77   \<open>take_bit n (a AND b) = take_bit n a AND take_bit n b\<close>
```
```    78   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_and_iff)
```
```    79
```
```    80 lemma take_bit_or [simp]:
```
```    81   \<open>take_bit n (a OR b) = take_bit n a OR take_bit n b\<close>
```
```    82   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_or_iff)
```
```    83
```
```    84 lemma take_bit_xor [simp]:
```
```    85   \<open>take_bit n (a XOR b) = take_bit n a XOR take_bit n b\<close>
```
```    86   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_xor_iff)
```
```    87
```
```    88 end
```
```    89
```
```    90 class ring_bit_operations = semiring_bit_operations + ring_parity +
```
```    91   fixes not :: \<open>'a \<Rightarrow> 'a\<close>  (\<open>NOT\<close>)
```
```    92   assumes bit_not_iff: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
```
```    93   assumes minus_eq_not_minus_1: \<open>- a = NOT (a - 1)\<close>
```
```    94 begin
```
```    95
```
```    96 text \<open>
```
```    97   For the sake of code generation \<^const>\<open>not\<close> is specified as
```
```    98   definitional class operation.  Note that \<^const>\<open>not\<close> has no
```
```    99   sensible definition for unlimited but only positive bit strings
```
```   100   (type \<^typ>\<open>nat\<close>).
```
```   101 \<close>
```
```   102
```
```   103 lemma bits_minus_1_mod_2_eq [simp]:
```
```   104   \<open>(- 1) mod 2 = 1\<close>
```
```   105   by (simp add: mod_2_eq_odd)
```
```   106
```
```   107 lemma not_eq_complement:
```
```   108   \<open>NOT a = - a - 1\<close>
```
```   109   using minus_eq_not_minus_1 [of \<open>a + 1\<close>] by simp
```
```   110
```
```   111 lemma minus_eq_not_plus_1:
```
```   112   \<open>- a = NOT a + 1\<close>
```
```   113   using not_eq_complement [of a] by simp
```
```   114
```
```   115 lemma bit_minus_iff:
```
```   116   \<open>bit (- a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit (a - 1) n\<close>
```
```   117   by (simp add: minus_eq_not_minus_1 bit_not_iff)
```
```   118
```
```   119 lemma even_not_iff [simp]:
```
```   120   "even (NOT a) \<longleftrightarrow> odd a"
```
```   121   using bit_not_iff [of a 0] by auto
```
```   122
```
```   123 lemma bit_not_exp_iff:
```
```   124   \<open>bit (NOT (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<noteq> m\<close>
```
```   125   by (auto simp add: bit_not_iff bit_exp_iff)
```
```   126
```
```   127 lemma bit_minus_1_iff [simp]:
```
```   128   \<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close>
```
```   129   by (simp add: bit_minus_iff)
```
```   130
```
```   131 lemma bit_minus_exp_iff:
```
```   132   \<open>bit (- (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<ge> m\<close>
```
```   133   oops
```
```   134
```
```   135 lemma bit_minus_2_iff [simp]:
```
```   136   \<open>bit (- 2) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n > 0\<close>
```
```   137   by (simp add: bit_minus_iff bit_1_iff)
```
```   138
```
```   139 lemma not_one [simp]:
```
```   140   "NOT 1 = - 2"
```
```   141   by (simp add: bit_eq_iff bit_not_iff) (simp add: bit_1_iff)
```
```   142
```
```   143 sublocale "and": semilattice_neutr \<open>(AND)\<close> \<open>- 1\<close>
```
```   144   apply standard
```
```   145   apply (simp add: bit_eq_iff bit_and_iff)
```
```   146   apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff)
```
```   147   done
```
```   148
```
```   149 sublocale bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
```
```   150   rewrites \<open>bit.xor = (XOR)\<close>
```
```   151 proof -
```
```   152   interpret bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
```
```   153     apply standard
```
```   154          apply (simp_all add: bit_eq_iff bit_and_iff bit_or_iff bit_not_iff)
```
```   155       apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff)
```
```   156     done
```
```   157   show \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close>
```
```   158     by standard
```
```   159   show \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close>
```
```   160     apply (auto simp add: fun_eq_iff bit.xor_def bit_eq_iff bit_and_iff bit_or_iff bit_not_iff bit_xor_iff)
```
```   161          apply (simp_all add: bit_exp_iff, simp_all add: bit_def)
```
```   162         apply (metis local.bit_exp_iff local.bits_div_by_0)
```
```   163        apply (metis local.bit_exp_iff local.bits_div_by_0)
```
```   164     done
```
```   165 qed
```
```   166
```
```   167 lemma push_bit_minus:
```
```   168   \<open>push_bit n (- a) = - push_bit n a\<close>
```
```   169   by (simp add: push_bit_eq_mult)
```
```   170
```
```   171 lemma take_bit_not_take_bit:
```
```   172   \<open>take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)\<close>
```
```   173   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff)
```
```   174
```
```   175 lemma take_bit_not_iff:
```
```   176   "take_bit n (NOT a) = take_bit n (NOT b) \<longleftrightarrow> take_bit n a = take_bit n b"
```
```   177   apply (simp add: bit_eq_iff bit_not_iff bit_take_bit_iff)
```
```   178   apply (simp add: bit_exp_iff)
```
```   179   apply (use local.exp_eq_0_imp_not_bit in blast)
```
```   180   done
```
```   181
```
```   182 definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
```
```   183   where \<open>set_bit n a = a OR 2 ^ n\<close>
```
```   184
```
```   185 definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
```
```   186   where \<open>unset_bit n a = a AND NOT (2 ^ n)\<close>
```
```   187
```
```   188 definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
```
```   189   where \<open>flip_bit n a = a XOR 2 ^ n\<close>
```
```   190
```
```   191 lemma bit_set_bit_iff:
```
```   192   \<open>bit (set_bit m a) n \<longleftrightarrow> bit a n \<or> (m = n \<and> 2 ^ n \<noteq> 0)\<close>
```
```   193   by (auto simp add: set_bit_def bit_or_iff bit_exp_iff)
```
```   194
```
```   195 lemma even_set_bit_iff:
```
```   196   \<open>even (set_bit m a) \<longleftrightarrow> even a \<and> m \<noteq> 0\<close>
```
```   197   using bit_set_bit_iff [of m a 0] by auto
```
```   198
```
```   199 lemma bit_unset_bit_iff:
```
```   200   \<open>bit (unset_bit m a) n \<longleftrightarrow> bit a n \<and> m \<noteq> n\<close>
```
```   201   by (auto simp add: unset_bit_def bit_and_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit)
```
```   202
```
```   203 lemma even_unset_bit_iff:
```
```   204   \<open>even (unset_bit m a) \<longleftrightarrow> even a \<or> m = 0\<close>
```
```   205   using bit_unset_bit_iff [of m a 0] by auto
```
```   206
```
```   207 lemma bit_flip_bit_iff:
```
```   208   \<open>bit (flip_bit m a) n \<longleftrightarrow> (m = n \<longleftrightarrow> \<not> bit a n) \<and> 2 ^ n \<noteq> 0\<close>
```
```   209   by (auto simp add: flip_bit_def bit_xor_iff bit_exp_iff exp_eq_0_imp_not_bit)
```
```   210
```
```   211 lemma even_flip_bit_iff:
```
```   212   \<open>even (flip_bit m a) \<longleftrightarrow> \<not> (even a \<longleftrightarrow> m = 0)\<close>
```
```   213   using bit_flip_bit_iff [of m a 0] by auto
```
```   214
```
```   215 lemma set_bit_0 [simp]:
```
```   216   \<open>set_bit 0 a = 1 + 2 * (a div 2)\<close>
```
```   217 proof (rule bit_eqI)
```
```   218   fix m
```
```   219   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   220   then show \<open>bit (set_bit 0 a) m = bit (1 + 2 * (a div 2)) m\<close>
```
```   221     by (simp add: bit_set_bit_iff bit_double_iff even_bit_succ_iff)
```
```   222       (cases m, simp_all)
```
```   223 qed
```
```   224
```
```   225 lemma set_bit_Suc [simp]:
```
```   226   \<open>set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\<close>
```
```   227 proof (rule bit_eqI)
```
```   228   fix m
```
```   229   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   230   show \<open>bit (set_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * set_bit n (a div 2)) m\<close>
```
```   231   proof (cases m)
```
```   232     case 0
```
```   233     then show ?thesis
```
```   234       by (simp add: even_set_bit_iff)
```
```   235   next
```
```   236     case (Suc m)
```
```   237     with * have \<open>2 ^ m \<noteq> 0\<close>
```
```   238       using mult_2 by auto
```
```   239     show ?thesis
```
```   240       by (cases a rule: parity_cases)
```
```   241         (simp_all add: bit_set_bit_iff bit_double_iff even_bit_succ_iff *,
```
```   242         simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close>)
```
```   243   qed
```
```   244 qed
```
```   245
```
```   246 lemma unset_bit_0 [simp]:
```
```   247   \<open>unset_bit 0 a = 2 * (a div 2)\<close>
```
```   248 proof (rule bit_eqI)
```
```   249   fix m
```
```   250   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   251   then show \<open>bit (unset_bit 0 a) m = bit (2 * (a div 2)) m\<close>
```
```   252     by (simp add: bit_unset_bit_iff bit_double_iff)
```
```   253       (cases m, simp_all)
```
```   254 qed
```
```   255
```
```   256 lemma unset_bit_Suc [simp]:
```
```   257   \<open>unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\<close>
```
```   258 proof (rule bit_eqI)
```
```   259   fix m
```
```   260   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   261   then show \<open>bit (unset_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * unset_bit n (a div 2)) m\<close>
```
```   262   proof (cases m)
```
```   263     case 0
```
```   264     then show ?thesis
```
```   265       by (simp add: even_unset_bit_iff)
```
```   266   next
```
```   267     case (Suc m)
```
```   268     show ?thesis
```
```   269       by (cases a rule: parity_cases)
```
```   270         (simp_all add: bit_unset_bit_iff bit_double_iff even_bit_succ_iff *,
```
```   271          simp_all add: Suc)
```
```   272   qed
```
```   273 qed
```
```   274
```
```   275 lemma flip_bit_0 [simp]:
```
```   276   \<open>flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\<close>
```
```   277 proof (rule bit_eqI)
```
```   278   fix m
```
```   279   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   280   then show \<open>bit (flip_bit 0 a) m = bit (of_bool (even a) + 2 * (a div 2)) m\<close>
```
```   281     by (simp add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff)
```
```   282       (cases m, simp_all)
```
```   283 qed
```
```   284
```
```   285 lemma flip_bit_Suc [simp]:
```
```   286   \<open>flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\<close>
```
```   287 proof (rule bit_eqI)
```
```   288   fix m
```
```   289   assume *: \<open>2 ^ m \<noteq> 0\<close>
```
```   290   show \<open>bit (flip_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * flip_bit n (a div 2)) m\<close>
```
```   291   proof (cases m)
```
```   292     case 0
```
```   293     then show ?thesis
```
```   294       by (simp add: even_flip_bit_iff)
```
```   295   next
```
```   296     case (Suc m)
```
```   297     with * have \<open>2 ^ m \<noteq> 0\<close>
```
```   298       using mult_2 by auto
```
```   299     show ?thesis
```
```   300       by (cases a rule: parity_cases)
```
```   301         (simp_all add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff,
```
```   302         simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close>)
```
```   303   qed
```
```   304 qed
```
```   305
```
```   306 end
```
```   307
```
```   308
```
```   309 subsubsection \<open>Instance \<^typ>\<open>nat\<close>\<close>
```
```   310
```
```   311 locale zip_nat = single: abel_semigroup f
```
```   312     for f :: "bool \<Rightarrow> bool \<Rightarrow> bool"  (infixl \<open>\<^bold>*\<close> 70) +
```
```   313   assumes end_of_bits: \<open>\<not> False \<^bold>* False\<close>
```
```   314 begin
```
```   315
```
```   316 function F :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>  (infixl \<open>\<^bold>\<times>\<close> 70)
```
```   317   where \<open>m \<^bold>\<times> n = (if m = 0 \<and> n = 0 then 0
```
```   318     else of_bool (odd m \<^bold>* odd n) + 2 * ((m div 2) \<^bold>\<times> (n div 2)))\<close>
```
```   319   by auto
```
```   320
```
```   321 termination
```
```   322   by (relation "measure (case_prod (+))") auto
```
```   323
```
```   324 declare F.simps [simp del]
```
```   325
```
```   326 lemma rec:
```
```   327   "m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2"
```
```   328 proof (cases \<open>m = 0 \<and> n = 0\<close>)
```
```   329   case True
```
```   330   then have \<open>m \<^bold>\<times> n = 0\<close>
```
```   331     using True by (simp add: F.simps [of 0 0])
```
```   332   moreover have \<open>(m div 2) \<^bold>\<times> (n div 2) = m \<^bold>\<times> n\<close>
```
```   333     using True by simp
```
```   334   ultimately show ?thesis
```
```   335     using True by (simp add: end_of_bits)
```
```   336 next
```
```   337   case False
```
```   338   then show ?thesis
```
```   339     by (auto simp add: ac_simps F.simps [of m n])
```
```   340 qed
```
```   341
```
```   342 lemma bit_eq_iff:
```
```   343   \<open>bit (m \<^bold>\<times> n) q \<longleftrightarrow> bit m q \<^bold>* bit n q\<close>
```
```   344 proof (induction q arbitrary: m n)
```
```   345   case 0
```
```   346   then show ?case
```
```   347     by (simp add: rec [of m n])
```
```   348 next
```
```   349   case (Suc n)
```
```   350   then show ?case
```
```   351     by (simp add: rec [of m n])
```
```   352 qed
```
```   353
```
```   354 sublocale abel_semigroup F
```
```   355   by standard (simp_all add: Parity.bit_eq_iff bit_eq_iff ac_simps)
```
```   356
```
```   357 end
```
```   358
```
```   359 instantiation nat :: semiring_bit_operations
```
```   360 begin
```
```   361
```
```   362 global_interpretation and_nat: zip_nat \<open>(\<and>)\<close>
```
```   363   defines and_nat = and_nat.F
```
```   364   by standard auto
```
```   365
```
```   366 global_interpretation and_nat: semilattice \<open>(AND) :: nat \<Rightarrow> nat \<Rightarrow> nat\<close>
```
```   367 proof (rule semilattice.intro, fact and_nat.abel_semigroup_axioms, standard)
```
```   368   show \<open>n AND n = n\<close> for n :: nat
```
```   369     by (simp add: bit_eq_iff and_nat.bit_eq_iff)
```
```   370 qed
```
```   371
```
```   372 global_interpretation or_nat: zip_nat \<open>(\<or>)\<close>
```
```   373   defines or_nat = or_nat.F
```
```   374   by standard auto
```
```   375
```
```   376 global_interpretation or_nat: semilattice \<open>(OR) :: nat \<Rightarrow> nat \<Rightarrow> nat\<close>
```
```   377 proof (rule semilattice.intro, fact or_nat.abel_semigroup_axioms, standard)
```
```   378   show \<open>n OR n = n\<close> for n :: nat
```
```   379     by (simp add: bit_eq_iff or_nat.bit_eq_iff)
```
```   380 qed
```
```   381
```
```   382 global_interpretation xor_nat: zip_nat \<open>(\<noteq>)\<close>
```
```   383   defines xor_nat = xor_nat.F
```
```   384   by standard auto
```
```   385
```
```   386 instance proof
```
```   387   fix m n q :: nat
```
```   388   show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
```
```   389     by (fact and_nat.bit_eq_iff)
```
```   390   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
```
```   391     by (fact or_nat.bit_eq_iff)
```
```   392   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
```
```   393     by (fact xor_nat.bit_eq_iff)
```
```   394 qed
```
```   395
```
```   396 end
```
```   397
```
```   398 lemma Suc_0_and_eq [simp]:
```
```   399   \<open>Suc 0 AND n = of_bool (odd n)\<close>
```
```   400   using one_and_eq [of n] by simp
```
```   401
```
```   402 lemma and_Suc_0_eq [simp]:
```
```   403   \<open>n AND Suc 0 = of_bool (odd n)\<close>
```
```   404   using and_one_eq [of n] by simp
```
```   405
```
```   406 lemma Suc_0_or_eq [simp]:
```
```   407   \<open>Suc 0 OR n = n + of_bool (even n)\<close>
```
```   408   using one_or_eq [of n] by simp
```
```   409
```
```   410 lemma or_Suc_0_eq [simp]:
```
```   411   \<open>n OR Suc 0 = n + of_bool (even n)\<close>
```
```   412   using or_one_eq [of n] by simp
```
```   413
```
```   414 lemma Suc_0_xor_eq [simp]:
```
```   415   \<open>Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\<close>
```
```   416   using one_xor_eq [of n] by simp
```
```   417
```
```   418 lemma xor_Suc_0_eq [simp]:
```
```   419   \<open>n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\<close>
```
```   420   using xor_one_eq [of n] by simp
```
```   421
```
```   422
```
```   423 subsubsection \<open>Instance \<^typ>\<open>int\<close>\<close>
```
```   424
```
```   425 locale zip_int = single: abel_semigroup f
```
```   426   for f :: \<open>bool \<Rightarrow> bool \<Rightarrow> bool\<close>  (infixl \<open>\<^bold>*\<close> 70)
```
```   427 begin
```
```   428
```
```   429 function F :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>  (infixl \<open>\<^bold>\<times>\<close> 70)
```
```   430   where \<open>k \<^bold>\<times> l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1}
```
```   431     then - of_bool (odd k \<^bold>* odd l)
```
```   432     else of_bool (odd k \<^bold>* odd l) + 2 * ((k div 2) \<^bold>\<times> (l div 2)))\<close>
```
```   433   by auto
```
```   434
```
```   435 termination
```
```   436   by (relation "measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))") auto
```
```   437
```
```   438 declare F.simps [simp del]
```
```   439
```
```   440 lemma rec:
```
```   441   \<open>k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + 2 * ((k div 2) \<^bold>\<times> (l div 2))\<close>
```
```   442 proof (cases \<open>k \<in> {0, - 1} \<and> l \<in> {0, - 1}\<close>)
```
```   443   case True
```
```   444   then have \<open>(k div 2) \<^bold>\<times> (l div 2) = k \<^bold>\<times> l\<close>
```
```   445     by auto
```
```   446   moreover have \<open>of_bool (odd k \<^bold>* odd l) = - (k \<^bold>\<times> l)\<close>
```
```   447     using True by (simp add: F.simps [of k l])
```
```   448   ultimately show ?thesis by simp
```
```   449 next
```
```   450   case False
```
```   451   then show ?thesis
```
```   452     by (auto simp add: ac_simps F.simps [of k l])
```
```   453 qed
```
```   454
```
```   455 lemma bit_eq_iff:
```
```   456   \<open>bit (k \<^bold>\<times> l) n \<longleftrightarrow> bit k n \<^bold>* bit l n\<close>
```
```   457 proof (induction n arbitrary: k l)
```
```   458   case 0
```
```   459   then show ?case
```
```   460     by (simp add: rec [of k l])
```
```   461 next
```
```   462   case (Suc n)
```
```   463   then show ?case
```
```   464     by (simp add: rec [of k l])
```
```   465 qed
```
```   466
```
```   467 sublocale abel_semigroup F
```
```   468   by standard (simp_all add: Parity.bit_eq_iff bit_eq_iff ac_simps)
```
```   469
```
```   470 end
```
```   471
```
```   472 instantiation int :: ring_bit_operations
```
```   473 begin
```
```   474
```
```   475 global_interpretation and_int: zip_int "(\<and>)"
```
```   476   defines and_int = and_int.F
```
```   477   by standard
```
```   478
```
```   479 global_interpretation and_int: semilattice "(AND) :: int \<Rightarrow> int \<Rightarrow> int"
```
```   480 proof (rule semilattice.intro, fact and_int.abel_semigroup_axioms, standard)
```
```   481   show "k AND k = k" for k :: int
```
```   482     by (simp add: bit_eq_iff and_int.bit_eq_iff)
```
```   483 qed
```
```   484
```
```   485 global_interpretation or_int: zip_int "(\<or>)"
```
```   486   defines or_int = or_int.F
```
```   487   by standard
```
```   488
```
```   489 global_interpretation or_int: semilattice "(OR) :: int \<Rightarrow> int \<Rightarrow> int"
```
```   490 proof (rule semilattice.intro, fact or_int.abel_semigroup_axioms, standard)
```
```   491   show "k OR k = k" for k :: int
```
```   492     by (simp add: bit_eq_iff or_int.bit_eq_iff)
```
```   493 qed
```
```   494
```
```   495 global_interpretation xor_int: zip_int "(\<noteq>)"
```
```   496   defines xor_int = xor_int.F
```
```   497   by standard
```
```   498
```
```   499 definition not_int :: \<open>int \<Rightarrow> int\<close>
```
```   500   where \<open>not_int k = - k - 1\<close>
```
```   501
```
```   502 lemma not_int_rec:
```
```   503   "NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int
```
```   504   by (auto simp add: not_int_def elim: oddE)
```
```   505
```
```   506 lemma even_not_iff_int:
```
```   507   \<open>even (NOT k) \<longleftrightarrow> odd k\<close> for k :: int
```
```   508   by (simp add: not_int_def)
```
```   509
```
```   510 lemma not_int_div_2:
```
```   511   \<open>NOT k div 2 = NOT (k div 2)\<close> for k :: int
```
```   512   by (simp add: not_int_def)
```
```   513
```
```   514 lemma bit_not_iff_int:
```
```   515   \<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close>
```
```   516     for k :: int
```
```   517   by (induction n arbitrary: k) (simp_all add: not_int_div_2 even_not_iff_int)
```
```   518
```
```   519 instance proof
```
```   520   fix k l :: int and n :: nat
```
```   521   show \<open>- k = NOT (k - 1)\<close>
```
```   522     by (simp add: not_int_def)
```
```   523   show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close>
```
```   524     by (fact and_int.bit_eq_iff)
```
```   525   show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
```
```   526     by (fact or_int.bit_eq_iff)
```
```   527   show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close>
```
```   528     by (fact xor_int.bit_eq_iff)
```
```   529 qed (simp_all add: bit_not_iff_int)
```
```   530
```
```   531 end
```
```   532
```
```   533
```
```   534 subsubsection \<open>Instances for \<^typ>\<open>integer\<close> and \<^typ>\<open>natural\<close>\<close>
```
```   535
```
```   536 unbundle integer.lifting natural.lifting
```
```   537
```
```   538 context
```
```   539   includes lifting_syntax
```
```   540 begin
```
```   541
```
```   542 lemma transfer_rule_bit_integer [transfer_rule]:
```
```   543   \<open>((pcr_integer :: int \<Rightarrow> integer \<Rightarrow> bool) ===> (=)) bit bit\<close>
```
```   544   by (unfold bit_def) transfer_prover
```
```   545
```
```   546 lemma transfer_rule_bit_natural [transfer_rule]:
```
```   547   \<open>((pcr_natural :: nat \<Rightarrow> natural \<Rightarrow> bool) ===> (=)) bit bit\<close>
```
```   548   by (unfold bit_def) transfer_prover
```
```   549
```
```   550 end
```
```   551
```
```   552 instantiation integer :: ring_bit_operations
```
```   553 begin
```
```   554
```
```   555 lift_definition not_integer :: \<open>integer \<Rightarrow> integer\<close>
```
```   556   is not .
```
```   557
```
```   558 lift_definition and_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
```
```   559   is \<open>and\<close> .
```
```   560
```
```   561 lift_definition or_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
```
```   562   is or .
```
```   563
```
```   564 lift_definition xor_integer ::  \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
```
```   565   is xor .
```
```   566
```
```   567 instance proof
```
```   568   fix k l :: \<open>integer\<close> and n :: nat
```
```   569   show \<open>- k = NOT (k - 1)\<close>
```
```   570     by transfer (simp add: minus_eq_not_minus_1)
```
```   571   show \<open>bit (NOT k) n \<longleftrightarrow> (2 :: integer) ^ n \<noteq> 0 \<and> \<not> bit k n\<close>
```
```   572     by transfer (fact bit_not_iff)
```
```   573   show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close>
```
```   574     by transfer (fact and_int.bit_eq_iff)
```
```   575   show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
```
```   576     by transfer (fact or_int.bit_eq_iff)
```
```   577   show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close>
```
```   578     by transfer (fact xor_int.bit_eq_iff)
```
```   579 qed
```
```   580
```
```   581 end
```
```   582
```
```   583 instantiation natural :: semiring_bit_operations
```
```   584 begin
```
```   585
```
```   586 lift_definition and_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
```
```   587   is \<open>and\<close> .
```
```   588
```
```   589 lift_definition or_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
```
```   590   is or .
```
```   591
```
```   592 lift_definition xor_natural ::  \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
```
```   593   is xor .
```
```   594
```
```   595 instance proof
```
```   596   fix m n :: \<open>natural\<close> and q :: nat
```
```   597   show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
```
```   598     by transfer (fact and_nat.bit_eq_iff)
```
```   599   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
```
```   600     by transfer (fact or_nat.bit_eq_iff)
```
```   601   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
```
```   602     by transfer (fact xor_nat.bit_eq_iff)
```
```   603 qed
```
```   604
```
```   605 end
```
```   606
```
```   607 lifting_update integer.lifting
```
```   608 lifting_forget integer.lifting
```
```   609
```
```   610 lifting_update natural.lifting
```
```   611 lifting_forget natural.lifting
```
```   612
```
```   613 end
```