src/HOL/Parity.thy
 author haftmann Sat May 12 22:20:46 2018 +0200 (21 months ago ago) changeset 68157 057d5b4ce47e parent 68028 1f9f973eed2a child 68390 1c84a8c513af permissions -rw-r--r--
removed some non-essential rules
```     1 (*  Title:      HOL/Parity.thy
```
```     2     Author:     Jeremy Avigad
```
```     3     Author:     Jacques D. Fleuriot
```
```     4 *)
```
```     5
```
```     6 section \<open>Parity in rings and semirings\<close>
```
```     7
```
```     8 theory Parity
```
```     9   imports Euclidean_Division
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Ring structures with parity and \<open>even\<close>/\<open>odd\<close> predicates\<close>
```
```    13
```
```    14 class semiring_parity = semidom + semiring_char_0 + unique_euclidean_semiring +
```
```    15   assumes of_nat_div: "of_nat (m div n) = of_nat m div of_nat n"
```
```    16     and division_segment_of_nat [simp]: "division_segment (of_nat n) = 1"
```
```    17     and division_segment_euclidean_size [simp]: "division_segment a * of_nat (euclidean_size a) = a"
```
```    18 begin
```
```    19
```
```    20 lemma division_segment_eq_iff:
```
```    21   "a = b" if "division_segment a = division_segment b"
```
```    22     and "euclidean_size a = euclidean_size b"
```
```    23   using that division_segment_euclidean_size [of a] by simp
```
```    24
```
```    25 lemma euclidean_size_of_nat [simp]:
```
```    26   "euclidean_size (of_nat n) = n"
```
```    27 proof -
```
```    28   have "division_segment (of_nat n) * of_nat (euclidean_size (of_nat n)) = of_nat n"
```
```    29     by (fact division_segment_euclidean_size)
```
```    30   then show ?thesis by simp
```
```    31 qed
```
```    32
```
```    33 lemma of_nat_euclidean_size:
```
```    34   "of_nat (euclidean_size a) = a div division_segment a"
```
```    35 proof -
```
```    36   have "of_nat (euclidean_size a) = division_segment a * of_nat (euclidean_size a) div division_segment a"
```
```    37     by (subst nonzero_mult_div_cancel_left) simp_all
```
```    38   also have "\<dots> = a div division_segment a"
```
```    39     by simp
```
```    40   finally show ?thesis .
```
```    41 qed
```
```    42
```
```    43 lemma division_segment_1 [simp]:
```
```    44   "division_segment 1 = 1"
```
```    45   using division_segment_of_nat [of 1] by simp
```
```    46
```
```    47 lemma division_segment_numeral [simp]:
```
```    48   "division_segment (numeral k) = 1"
```
```    49   using division_segment_of_nat [of "numeral k"] by simp
```
```    50
```
```    51 lemma euclidean_size_1 [simp]:
```
```    52   "euclidean_size 1 = 1"
```
```    53   using euclidean_size_of_nat [of 1] by simp
```
```    54
```
```    55 lemma euclidean_size_numeral [simp]:
```
```    56   "euclidean_size (numeral k) = numeral k"
```
```    57   using euclidean_size_of_nat [of "numeral k"] by simp
```
```    58
```
```    59 lemma of_nat_dvd_iff:
```
```    60   "of_nat m dvd of_nat n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
```
```    61 proof (cases "m = 0")
```
```    62   case True
```
```    63   then show ?thesis
```
```    64     by simp
```
```    65 next
```
```    66   case False
```
```    67   show ?thesis
```
```    68   proof
```
```    69     assume ?Q
```
```    70     then show ?P
```
```    71       by (auto elim: dvd_class.dvdE)
```
```    72   next
```
```    73     assume ?P
```
```    74     with False have "of_nat n = of_nat n div of_nat m * of_nat m"
```
```    75       by simp
```
```    76     then have "of_nat n = of_nat (n div m * m)"
```
```    77       by (simp add: of_nat_div)
```
```    78     then have "n = n div m * m"
```
```    79       by (simp only: of_nat_eq_iff)
```
```    80     then have "n = m * (n div m)"
```
```    81       by (simp add: ac_simps)
```
```    82     then show ?Q ..
```
```    83   qed
```
```    84 qed
```
```    85
```
```    86 lemma of_nat_mod:
```
```    87   "of_nat (m mod n) = of_nat m mod of_nat n"
```
```    88 proof -
```
```    89   have "of_nat m div of_nat n * of_nat n + of_nat m mod of_nat n = of_nat m"
```
```    90     by (simp add: div_mult_mod_eq)
```
```    91   also have "of_nat m = of_nat (m div n * n + m mod n)"
```
```    92     by simp
```
```    93   finally show ?thesis
```
```    94     by (simp only: of_nat_div of_nat_mult of_nat_add) simp
```
```    95 qed
```
```    96
```
```    97 lemma one_div_two_eq_zero [simp]:
```
```    98   "1 div 2 = 0"
```
```    99 proof -
```
```   100   from of_nat_div [symmetric] have "of_nat 1 div of_nat 2 = of_nat 0"
```
```   101     by (simp only:) simp
```
```   102   then show ?thesis
```
```   103     by simp
```
```   104 qed
```
```   105
```
```   106 lemma one_mod_two_eq_one [simp]:
```
```   107   "1 mod 2 = 1"
```
```   108 proof -
```
```   109   from of_nat_mod [symmetric] have "of_nat 1 mod of_nat 2 = of_nat 1"
```
```   110     by (simp only:) simp
```
```   111   then show ?thesis
```
```   112     by simp
```
```   113 qed
```
```   114
```
```   115 abbreviation even :: "'a \<Rightarrow> bool"
```
```   116   where "even a \<equiv> 2 dvd a"
```
```   117
```
```   118 abbreviation odd :: "'a \<Rightarrow> bool"
```
```   119   where "odd a \<equiv> \<not> 2 dvd a"
```
```   120
```
```   121 lemma even_iff_mod_2_eq_zero:
```
```   122   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   123   by (fact dvd_eq_mod_eq_0)
```
```   124
```
```   125 lemma odd_iff_mod_2_eq_one:
```
```   126   "odd a \<longleftrightarrow> a mod 2 = 1"
```
```   127 proof
```
```   128   assume "a mod 2 = 1"
```
```   129   then show "odd a"
```
```   130     by auto
```
```   131 next
```
```   132   assume "odd a"
```
```   133   have eucl: "euclidean_size (a mod 2) = 1"
```
```   134   proof (rule order_antisym)
```
```   135     show "euclidean_size (a mod 2) \<le> 1"
```
```   136       using mod_size_less [of 2 a] by simp
```
```   137     show "1 \<le> euclidean_size (a mod 2)"
```
```   138       using \<open>odd a\<close> by (simp add: Suc_le_eq dvd_eq_mod_eq_0)
```
```   139   qed
```
```   140   from \<open>odd a\<close> have "\<not> of_nat 2 dvd division_segment a * of_nat (euclidean_size a)"
```
```   141     by simp
```
```   142   then have "\<not> of_nat 2 dvd of_nat (euclidean_size a)"
```
```   143     by (auto simp only: dvd_mult_unit_iff' is_unit_division_segment)
```
```   144   then have "\<not> 2 dvd euclidean_size a"
```
```   145     using of_nat_dvd_iff [of 2] by simp
```
```   146   then have "euclidean_size a mod 2 = 1"
```
```   147     by (simp add: semidom_modulo_class.dvd_eq_mod_eq_0)
```
```   148   then have "of_nat (euclidean_size a mod 2) = of_nat 1"
```
```   149     by simp
```
```   150   then have "of_nat (euclidean_size a) mod 2 = 1"
```
```   151     by (simp add: of_nat_mod)
```
```   152   from \<open>odd a\<close> eucl
```
```   153   show "a mod 2 = 1"
```
```   154     by (auto intro: division_segment_eq_iff simp add: division_segment_mod)
```
```   155 qed
```
```   156
```
```   157 lemma parity_cases [case_names even odd]:
```
```   158   assumes "even a \<Longrightarrow> a mod 2 = 0 \<Longrightarrow> P"
```
```   159   assumes "odd a \<Longrightarrow> a mod 2 = 1 \<Longrightarrow> P"
```
```   160   shows P
```
```   161   using assms by (cases "even a") (simp_all add: odd_iff_mod_2_eq_one)
```
```   162
```
```   163 lemma not_mod_2_eq_1_eq_0 [simp]:
```
```   164   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
```
```   165   by (cases a rule: parity_cases) simp_all
```
```   166
```
```   167 lemma not_mod_2_eq_0_eq_1 [simp]:
```
```   168   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
```
```   169   by (cases a rule: parity_cases) simp_all
```
```   170
```
```   171 lemma evenE [elim?]:
```
```   172   assumes "even a"
```
```   173   obtains b where "a = 2 * b"
```
```   174   using assms by (rule dvdE)
```
```   175
```
```   176 lemma oddE [elim?]:
```
```   177   assumes "odd a"
```
```   178   obtains b where "a = 2 * b + 1"
```
```   179 proof -
```
```   180   have "a = 2 * (a div 2) + a mod 2"
```
```   181     by (simp add: mult_div_mod_eq)
```
```   182   with assms have "a = 2 * (a div 2) + 1"
```
```   183     by (simp add: odd_iff_mod_2_eq_one)
```
```   184   then show ?thesis ..
```
```   185 qed
```
```   186
```
```   187 lemma mod_2_eq_odd:
```
```   188   "a mod 2 = of_bool (odd a)"
```
```   189   by (auto elim: oddE)
```
```   190
```
```   191 lemma of_bool_odd_eq_mod_2:
```
```   192   "of_bool (odd a) = a mod 2"
```
```   193   by (simp add: mod_2_eq_odd)
```
```   194
```
```   195 lemma one_mod_2_pow_eq [simp]:
```
```   196   "1 mod (2 ^ n) = of_bool (n > 0)"
```
```   197 proof -
```
```   198   have "1 mod (2 ^ n) = of_nat (1 mod (2 ^ n))"
```
```   199     using of_nat_mod [of 1 "2 ^ n"] by simp
```
```   200   also have "\<dots> = of_bool (n > 0)"
```
```   201     by simp
```
```   202   finally show ?thesis .
```
```   203 qed
```
```   204
```
```   205 lemma one_div_2_pow_eq [simp]:
```
```   206   "1 div (2 ^ n) = of_bool (n = 0)"
```
```   207   using div_mult_mod_eq [of 1 "2 ^ n"] by auto
```
```   208
```
```   209 lemma even_of_nat [simp]:
```
```   210   "even (of_nat a) \<longleftrightarrow> even a"
```
```   211 proof -
```
```   212   have "even (of_nat a) \<longleftrightarrow> of_nat 2 dvd of_nat a"
```
```   213     by simp
```
```   214   also have "\<dots> \<longleftrightarrow> even a"
```
```   215     by (simp only: of_nat_dvd_iff)
```
```   216   finally show ?thesis .
```
```   217 qed
```
```   218
```
```   219 lemma even_zero [simp]:
```
```   220   "even 0"
```
```   221   by (fact dvd_0_right)
```
```   222
```
```   223 lemma odd_one [simp]:
```
```   224   "odd 1"
```
```   225 proof -
```
```   226   have "\<not> (2 :: nat) dvd 1"
```
```   227     by simp
```
```   228   then have "\<not> of_nat 2 dvd of_nat 1"
```
```   229     unfolding of_nat_dvd_iff by simp
```
```   230   then show ?thesis
```
```   231     by simp
```
```   232 qed
```
```   233
```
```   234 lemma odd_even_add:
```
```   235   "even (a + b)" if "odd a" and "odd b"
```
```   236 proof -
```
```   237   from that obtain c d where "a = 2 * c + 1" and "b = 2 * d + 1"
```
```   238     by (blast elim: oddE)
```
```   239   then have "a + b = 2 * c + 2 * d + (1 + 1)"
```
```   240     by (simp only: ac_simps)
```
```   241   also have "\<dots> = 2 * (c + d + 1)"
```
```   242     by (simp add: algebra_simps)
```
```   243   finally show ?thesis ..
```
```   244 qed
```
```   245
```
```   246 lemma even_add [simp]:
```
```   247   "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
```
```   248   by (auto simp add: dvd_add_right_iff dvd_add_left_iff odd_even_add)
```
```   249
```
```   250 lemma odd_add [simp]:
```
```   251   "odd (a + b) \<longleftrightarrow> \<not> (odd a \<longleftrightarrow> odd b)"
```
```   252   by simp
```
```   253
```
```   254 lemma even_plus_one_iff [simp]:
```
```   255   "even (a + 1) \<longleftrightarrow> odd a"
```
```   256   by (auto simp add: dvd_add_right_iff intro: odd_even_add)
```
```   257
```
```   258 lemma even_mult_iff [simp]:
```
```   259   "even (a * b) \<longleftrightarrow> even a \<or> even b" (is "?P \<longleftrightarrow> ?Q")
```
```   260 proof
```
```   261   assume ?Q
```
```   262   then show ?P
```
```   263     by auto
```
```   264 next
```
```   265   assume ?P
```
```   266   show ?Q
```
```   267   proof (rule ccontr)
```
```   268     assume "\<not> (even a \<or> even b)"
```
```   269     then have "odd a" and "odd b"
```
```   270       by auto
```
```   271     then obtain r s where "a = 2 * r + 1" and "b = 2 * s + 1"
```
```   272       by (blast elim: oddE)
```
```   273     then have "a * b = (2 * r + 1) * (2 * s + 1)"
```
```   274       by simp
```
```   275     also have "\<dots> = 2 * (2 * r * s + r + s) + 1"
```
```   276       by (simp add: algebra_simps)
```
```   277     finally have "odd (a * b)"
```
```   278       by simp
```
```   279     with \<open>?P\<close> show False
```
```   280       by auto
```
```   281   qed
```
```   282 qed
```
```   283
```
```   284 lemma even_numeral [simp]: "even (numeral (Num.Bit0 n))"
```
```   285 proof -
```
```   286   have "even (2 * numeral n)"
```
```   287     unfolding even_mult_iff by simp
```
```   288   then have "even (numeral n + numeral n)"
```
```   289     unfolding mult_2 .
```
```   290   then show ?thesis
```
```   291     unfolding numeral.simps .
```
```   292 qed
```
```   293
```
```   294 lemma odd_numeral [simp]: "odd (numeral (Num.Bit1 n))"
```
```   295 proof
```
```   296   assume "even (numeral (num.Bit1 n))"
```
```   297   then have "even (numeral n + numeral n + 1)"
```
```   298     unfolding numeral.simps .
```
```   299   then have "even (2 * numeral n + 1)"
```
```   300     unfolding mult_2 .
```
```   301   then have "2 dvd numeral n * 2 + 1"
```
```   302     by (simp add: ac_simps)
```
```   303   then have "2 dvd 1"
```
```   304     using dvd_add_times_triv_left_iff [of 2 "numeral n" 1] by simp
```
```   305   then show False by simp
```
```   306 qed
```
```   307
```
```   308 lemma even_power [simp]: "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
```
```   309   by (induct n) auto
```
```   310
```
```   311 lemma even_succ_div_two [simp]:
```
```   312   "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
```
```   313   by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
```
```   314
```
```   315 lemma odd_succ_div_two [simp]:
```
```   316   "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
```
```   317   by (auto elim!: oddE simp add: add.assoc)
```
```   318
```
```   319 lemma even_two_times_div_two:
```
```   320   "even a \<Longrightarrow> 2 * (a div 2) = a"
```
```   321   by (fact dvd_mult_div_cancel)
```
```   322
```
```   323 lemma odd_two_times_div_two_succ [simp]:
```
```   324   "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
```
```   325   using mult_div_mod_eq [of 2 a]
```
```   326   by (simp add: even_iff_mod_2_eq_zero)
```
```   327
```
```   328 lemma coprime_left_2_iff_odd [simp]:
```
```   329   "coprime 2 a \<longleftrightarrow> odd a"
```
```   330 proof
```
```   331   assume "odd a"
```
```   332   show "coprime 2 a"
```
```   333   proof (rule coprimeI)
```
```   334     fix b
```
```   335     assume "b dvd 2" "b dvd a"
```
```   336     then have "b dvd a mod 2"
```
```   337       by (auto intro: dvd_mod)
```
```   338     with \<open>odd a\<close> show "is_unit b"
```
```   339       by (simp add: mod_2_eq_odd)
```
```   340   qed
```
```   341 next
```
```   342   assume "coprime 2 a"
```
```   343   show "odd a"
```
```   344   proof (rule notI)
```
```   345     assume "even a"
```
```   346     then obtain b where "a = 2 * b" ..
```
```   347     with \<open>coprime 2 a\<close> have "coprime 2 (2 * b)"
```
```   348       by simp
```
```   349     moreover have "\<not> coprime 2 (2 * b)"
```
```   350       by (rule not_coprimeI [of 2]) simp_all
```
```   351     ultimately show False
```
```   352       by blast
```
```   353   qed
```
```   354 qed
```
```   355
```
```   356 lemma coprime_right_2_iff_odd [simp]:
```
```   357   "coprime a 2 \<longleftrightarrow> odd a"
```
```   358   using coprime_left_2_iff_odd [of a] by (simp add: ac_simps)
```
```   359
```
```   360 lemma div_mult2_eq':
```
```   361   "a div (of_nat m * of_nat n) = a div of_nat m div of_nat n"
```
```   362 proof (cases a "of_nat m * of_nat n" rule: divmod_cases)
```
```   363   case (divides q)
```
```   364   then show ?thesis
```
```   365     using nonzero_mult_div_cancel_right [of "of_nat m" "q * of_nat n"]
```
```   366     by (simp add: ac_simps)
```
```   367 next
```
```   368   case (remainder q r)
```
```   369   then have "division_segment r = 1"
```
```   370     using division_segment_of_nat [of "m * n"] by simp
```
```   371   with division_segment_euclidean_size [of r]
```
```   372   have "of_nat (euclidean_size r) = r"
```
```   373     by simp
```
```   374   have "a mod (of_nat m * of_nat n) div (of_nat m * of_nat n) = 0"
```
```   375     by simp
```
```   376   with remainder(6) have "r div (of_nat m * of_nat n) = 0"
```
```   377     by simp
```
```   378   with \<open>of_nat (euclidean_size r) = r\<close>
```
```   379   have "of_nat (euclidean_size r) div (of_nat m * of_nat n) = 0"
```
```   380     by simp
```
```   381   then have "of_nat (euclidean_size r div (m * n)) = 0"
```
```   382     by (simp add: of_nat_div)
```
```   383   then have "of_nat (euclidean_size r div m div n) = 0"
```
```   384     by (simp add: div_mult2_eq)
```
```   385   with \<open>of_nat (euclidean_size r) = r\<close> have "r div of_nat m div of_nat n = 0"
```
```   386     by (simp add: of_nat_div)
```
```   387   with remainder(1)
```
```   388   have "q = (r div of_nat m + q * of_nat n * of_nat m div of_nat m) div of_nat n"
```
```   389     by simp
```
```   390   with remainder(5) remainder(7) show ?thesis
```
```   391     using div_plus_div_distrib_dvd_right [of "of_nat m" "q * (of_nat m * of_nat n)" r]
```
```   392     by (simp add: ac_simps)
```
```   393 next
```
```   394   case by0
```
```   395   then show ?thesis
```
```   396     by auto
```
```   397 qed
```
```   398
```
```   399 lemma mod_mult2_eq':
```
```   400   "a mod (of_nat m * of_nat n) = of_nat m * (a div of_nat m mod of_nat n) + a mod of_nat m"
```
```   401 proof -
```
```   402   have "a div (of_nat m * of_nat n) * (of_nat m * of_nat n) + a mod (of_nat m * of_nat n) = a div of_nat m div of_nat n * of_nat n * of_nat m + (a div of_nat m mod of_nat n * of_nat m + a mod of_nat m)"
```
```   403     by (simp add: combine_common_factor div_mult_mod_eq)
```
```   404   moreover have "a div of_nat m div of_nat n * of_nat n * of_nat m = of_nat n * of_nat m * (a div of_nat m div of_nat n)"
```
```   405     by (simp add: ac_simps)
```
```   406   ultimately show ?thesis
```
```   407     by (simp add: div_mult2_eq' mult_commute)
```
```   408 qed
```
```   409
```
```   410 lemma div_mult2_numeral_eq:
```
```   411   "a div numeral k div numeral l = a div numeral (k * l)" (is "?A = ?B")
```
```   412 proof -
```
```   413   have "?A = a div of_nat (numeral k) div of_nat (numeral l)"
```
```   414     by simp
```
```   415   also have "\<dots> = a div (of_nat (numeral k) * of_nat (numeral l))"
```
```   416     by (fact div_mult2_eq' [symmetric])
```
```   417   also have "\<dots> = ?B"
```
```   418     by simp
```
```   419   finally show ?thesis .
```
```   420 qed
```
```   421
```
```   422 end
```
```   423
```
```   424 class ring_parity = ring + semiring_parity
```
```   425 begin
```
```   426
```
```   427 subclass comm_ring_1 ..
```
```   428
```
```   429 lemma even_minus:
```
```   430   "even (- a) \<longleftrightarrow> even a"
```
```   431   by (fact dvd_minus_iff)
```
```   432
```
```   433 lemma even_diff [simp]:
```
```   434   "even (a - b) \<longleftrightarrow> even (a + b)"
```
```   435   using even_add [of a "- b"] by simp
```
```   436
```
```   437 lemma minus_1_mod_2_eq [simp]:
```
```   438   "- 1 mod 2 = 1"
```
```   439   by (simp add: mod_2_eq_odd)
```
```   440
```
```   441 lemma minus_1_div_2_eq [simp]:
```
```   442   "- 1 div 2 = - 1"
```
```   443 proof -
```
```   444   from div_mult_mod_eq [of "- 1" 2]
```
```   445   have "- 1 div 2 * 2 = - 1 * 2"
```
```   446     using local.add_implies_diff by fastforce
```
```   447   then show ?thesis
```
```   448     using mult_right_cancel [of 2 "- 1 div 2" "- 1"] by simp
```
```   449 qed
```
```   450
```
```   451 end
```
```   452
```
```   453
```
```   454 subsection \<open>Instance for @{typ nat}\<close>
```
```   455
```
```   456 instance nat :: semiring_parity
```
```   457   by standard (simp_all add: dvd_eq_mod_eq_0)
```
```   458
```
```   459 lemma even_Suc_Suc_iff [simp]:
```
```   460   "even (Suc (Suc n)) \<longleftrightarrow> even n"
```
```   461   using dvd_add_triv_right_iff [of 2 n] by simp
```
```   462
```
```   463 lemma even_Suc [simp]: "even (Suc n) \<longleftrightarrow> odd n"
```
```   464   using even_plus_one_iff [of n] by simp
```
```   465
```
```   466 lemma even_diff_nat [simp]:
```
```   467   "even (m - n) \<longleftrightarrow> m < n \<or> even (m + n)" for m n :: nat
```
```   468 proof (cases "n \<le> m")
```
```   469   case True
```
```   470   then have "m - n + n * 2 = m + n" by (simp add: mult_2_right)
```
```   471   moreover have "even (m - n) \<longleftrightarrow> even (m - n + n * 2)" by simp
```
```   472   ultimately have "even (m - n) \<longleftrightarrow> even (m + n)" by (simp only:)
```
```   473   then show ?thesis by auto
```
```   474 next
```
```   475   case False
```
```   476   then show ?thesis by simp
```
```   477 qed
```
```   478
```
```   479 lemma odd_pos:
```
```   480   "odd n \<Longrightarrow> 0 < n" for n :: nat
```
```   481   by (auto elim: oddE)
```
```   482
```
```   483 lemma Suc_double_not_eq_double:
```
```   484   "Suc (2 * m) \<noteq> 2 * n"
```
```   485 proof
```
```   486   assume "Suc (2 * m) = 2 * n"
```
```   487   moreover have "odd (Suc (2 * m))" and "even (2 * n)"
```
```   488     by simp_all
```
```   489   ultimately show False by simp
```
```   490 qed
```
```   491
```
```   492 lemma double_not_eq_Suc_double:
```
```   493   "2 * m \<noteq> Suc (2 * n)"
```
```   494   using Suc_double_not_eq_double [of n m] by simp
```
```   495
```
```   496 lemma odd_Suc_minus_one [simp]: "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
```
```   497   by (auto elim: oddE)
```
```   498
```
```   499 lemma even_Suc_div_two [simp]:
```
```   500   "even n \<Longrightarrow> Suc n div 2 = n div 2"
```
```   501   using even_succ_div_two [of n] by simp
```
```   502
```
```   503 lemma odd_Suc_div_two [simp]:
```
```   504   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
```
```   505   using odd_succ_div_two [of n] by simp
```
```   506
```
```   507 lemma odd_two_times_div_two_nat [simp]:
```
```   508   assumes "odd n"
```
```   509   shows "2 * (n div 2) = n - (1 :: nat)"
```
```   510 proof -
```
```   511   from assms have "2 * (n div 2) + 1 = n"
```
```   512     by (rule odd_two_times_div_two_succ)
```
```   513   then have "Suc (2 * (n div 2)) - 1 = n - 1"
```
```   514     by simp
```
```   515   then show ?thesis
```
```   516     by simp
```
```   517 qed
```
```   518
```
```   519 lemma parity_induct [case_names zero even odd]:
```
```   520   assumes zero: "P 0"
```
```   521   assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
```
```   522   assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
```
```   523   shows "P n"
```
```   524 proof (induct n rule: less_induct)
```
```   525   case (less n)
```
```   526   show "P n"
```
```   527   proof (cases "n = 0")
```
```   528     case True with zero show ?thesis by simp
```
```   529   next
```
```   530     case False
```
```   531     with less have hyp: "P (n div 2)" by simp
```
```   532     show ?thesis
```
```   533     proof (cases "even n")
```
```   534       case True
```
```   535       with hyp even [of "n div 2"] show ?thesis
```
```   536         by simp
```
```   537     next
```
```   538       case False
```
```   539       with hyp odd [of "n div 2"] show ?thesis
```
```   540         by simp
```
```   541     qed
```
```   542   qed
```
```   543 qed
```
```   544
```
```   545 lemma not_mod2_eq_Suc_0_eq_0 [simp]:
```
```   546   "n mod 2 \<noteq> Suc 0 \<longleftrightarrow> n mod 2 = 0"
```
```   547   using not_mod_2_eq_1_eq_0 [of n] by simp
```
```   548
```
```   549
```
```   550 subsection \<open>Parity and powers\<close>
```
```   551
```
```   552 context ring_1
```
```   553 begin
```
```   554
```
```   555 lemma power_minus_even [simp]: "even n \<Longrightarrow> (- a) ^ n = a ^ n"
```
```   556   by (auto elim: evenE)
```
```   557
```
```   558 lemma power_minus_odd [simp]: "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
```
```   559   by (auto elim: oddE)
```
```   560
```
```   561 lemma uminus_power_if:
```
```   562   "(- a) ^ n = (if even n then a ^ n else - (a ^ n))"
```
```   563   by auto
```
```   564
```
```   565 lemma neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
```
```   566   by simp
```
```   567
```
```   568 lemma neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
```
```   569   by simp
```
```   570
```
```   571 lemma neg_one_power_add_eq_neg_one_power_diff: "k \<le> n \<Longrightarrow> (- 1) ^ (n + k) = (- 1) ^ (n - k)"
```
```   572   by (cases "even (n + k)") auto
```
```   573
```
```   574 lemma minus_one_power_iff: "(- 1) ^ n = (if even n then 1 else - 1)"
```
```   575   by (induct n) auto
```
```   576
```
```   577 end
```
```   578
```
```   579 context linordered_idom
```
```   580 begin
```
```   581
```
```   582 lemma zero_le_even_power: "even n \<Longrightarrow> 0 \<le> a ^ n"
```
```   583   by (auto elim: evenE)
```
```   584
```
```   585 lemma zero_le_odd_power: "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
```
```   586   by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
```
```   587
```
```   588 lemma zero_le_power_eq: "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
```
```   589   by (auto simp add: zero_le_even_power zero_le_odd_power)
```
```   590
```
```   591 lemma zero_less_power_eq: "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
```
```   592 proof -
```
```   593   have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
```
```   594     unfolding power_eq_0_iff [of a n, symmetric] by blast
```
```   595   show ?thesis
```
```   596     unfolding less_le zero_le_power_eq by auto
```
```   597 qed
```
```   598
```
```   599 lemma power_less_zero_eq [simp]: "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
```
```   600   unfolding not_le [symmetric] zero_le_power_eq by auto
```
```   601
```
```   602 lemma power_le_zero_eq: "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
```
```   603   unfolding not_less [symmetric] zero_less_power_eq by auto
```
```   604
```
```   605 lemma power_even_abs: "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
```
```   606   using power_abs [of a n] by (simp add: zero_le_even_power)
```
```   607
```
```   608 lemma power_mono_even:
```
```   609   assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
```
```   610   shows "a ^ n \<le> b ^ n"
```
```   611 proof -
```
```   612   have "0 \<le> \<bar>a\<bar>" by auto
```
```   613   with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close> have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n"
```
```   614     by (rule power_mono)
```
```   615   with \<open>even n\<close> show ?thesis
```
```   616     by (simp add: power_even_abs)
```
```   617 qed
```
```   618
```
```   619 lemma power_mono_odd:
```
```   620   assumes "odd n" and "a \<le> b"
```
```   621   shows "a ^ n \<le> b ^ n"
```
```   622 proof (cases "b < 0")
```
```   623   case True
```
```   624   with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
```
```   625   then have "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
```
```   626   with \<open>odd n\<close> show ?thesis by simp
```
```   627 next
```
```   628   case False
```
```   629   then have "0 \<le> b" by auto
```
```   630   show ?thesis
```
```   631   proof (cases "a < 0")
```
```   632     case True
```
```   633     then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
```
```   634     then have "a ^ n \<le> 0" unfolding power_le_zero_eq using \<open>odd n\<close> by auto
```
```   635     moreover from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
```
```   636     ultimately show ?thesis by auto
```
```   637   next
```
```   638     case False
```
```   639     then have "0 \<le> a" by auto
```
```   640     with \<open>a \<le> b\<close> show ?thesis
```
```   641       using power_mono by auto
```
```   642   qed
```
```   643 qed
```
```   644
```
```   645 text \<open>Simplify, when the exponent is a numeral\<close>
```
```   646
```
```   647 lemma zero_le_power_eq_numeral [simp]:
```
```   648   "0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
```
```   649   by (fact zero_le_power_eq)
```
```   650
```
```   651 lemma zero_less_power_eq_numeral [simp]:
```
```   652   "0 < a ^ numeral w \<longleftrightarrow>
```
```   653     numeral w = (0 :: nat) \<or>
```
```   654     even (numeral w :: nat) \<and> a \<noteq> 0 \<or>
```
```   655     odd (numeral w :: nat) \<and> 0 < a"
```
```   656   by (fact zero_less_power_eq)
```
```   657
```
```   658 lemma power_le_zero_eq_numeral [simp]:
```
```   659   "a ^ numeral w \<le> 0 \<longleftrightarrow>
```
```   660     (0 :: nat) < numeral w \<and>
```
```   661     (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
```
```   662   by (fact power_le_zero_eq)
```
```   663
```
```   664 lemma power_less_zero_eq_numeral [simp]:
```
```   665   "a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
```
```   666   by (fact power_less_zero_eq)
```
```   667
```
```   668 lemma power_even_abs_numeral [simp]:
```
```   669   "even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
```
```   670   by (fact power_even_abs)
```
```   671
```
```   672 end
```
```   673
```
```   674
```
```   675 subsection \<open>Instance for @{typ int}\<close>
```
```   676
```
```   677 instance int :: ring_parity
```
```   678   by standard (simp_all add: dvd_eq_mod_eq_0 divide_int_def division_segment_int_def)
```
```   679
```
```   680 lemma even_diff_iff:
```
```   681   "even (k - l) \<longleftrightarrow> even (k + l)" for k l :: int
```
```   682   by (fact even_diff)
```
```   683
```
```   684 lemma even_abs_add_iff:
```
```   685   "even (\<bar>k\<bar> + l) \<longleftrightarrow> even (k + l)" for k l :: int
```
```   686   by simp
```
```   687
```
```   688 lemma even_add_abs_iff:
```
```   689   "even (k + \<bar>l\<bar>) \<longleftrightarrow> even (k + l)" for k l :: int
```
```   690   by simp
```
```   691
```
```   692 lemma even_nat_iff: "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
```
```   693   by (simp add: even_of_nat [of "nat k", where ?'a = int, symmetric])
```
```   694
```
```   695
```
```   696 subsection \<open>Abstract bit operations\<close>
```
```   697
```
```   698 context semiring_parity
```
```   699 begin
```
```   700
```
```   701 text \<open>The primary purpose of the following operations is
```
```   702   to avoid ad-hoc simplification of concrete expressions @{term "2 ^ n"}\<close>
```
```   703
```
```   704 definition push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   705   where push_bit_eq_mult: "push_bit n a = a * 2 ^ n"
```
```   706
```
```   707 definition take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   708   where take_bit_eq_mod: "take_bit n a = a mod 2 ^ n"
```
```   709
```
```   710 definition drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   711   where drop_bit_eq_div: "drop_bit n a = a div 2 ^ n"
```
```   712
```
```   713 lemma bit_ident:
```
```   714   "push_bit n (drop_bit n a) + take_bit n a = a"
```
```   715   using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div)
```
```   716
```
```   717 lemma push_bit_push_bit [simp]:
```
```   718   "push_bit m (push_bit n a) = push_bit (m + n) a"
```
```   719   by (simp add: push_bit_eq_mult power_add ac_simps)
```
```   720
```
```   721 lemma take_bit_take_bit [simp]:
```
```   722   "take_bit m (take_bit n a) = take_bit (min m n) a"
```
```   723 proof (cases "m \<le> n")
```
```   724   case True
```
```   725   then show ?thesis
```
```   726     by (simp add: take_bit_eq_mod not_le min_def mod_mod_cancel le_imp_power_dvd)
```
```   727 next
```
```   728   case False
```
```   729   then have "n < m" and "min m n = n"
```
```   730     by simp_all
```
```   731   then have "2 ^ m = of_nat (2 ^ n) * of_nat (2 ^ (m - n))"
```
```   732     by (simp add: power_add [symmetric])
```
```   733   then have "a mod 2 ^ n mod 2 ^ m = a mod 2 ^ n mod (of_nat (2 ^ n) * of_nat (2 ^ (m - n)))"
```
```   734     by simp
```
```   735   also have "\<dots> = of_nat (2 ^ n) * (a mod 2 ^ n div of_nat (2 ^ n) mod of_nat (2 ^ (m - n))) + a mod 2 ^ n mod of_nat (2 ^ n)"
```
```   736     by (simp only: mod_mult2_eq')
```
```   737   finally show ?thesis
```
```   738     using \<open>min m n = n\<close> by (simp add: take_bit_eq_mod)
```
```   739 qed
```
```   740
```
```   741 lemma drop_bit_drop_bit [simp]:
```
```   742   "drop_bit m (drop_bit n a) = drop_bit (m + n) a"
```
```   743 proof -
```
```   744   have "a div (2 ^ m * 2 ^ n) = a div (of_nat (2 ^ n) * of_nat (2 ^ m))"
```
```   745     by (simp add: ac_simps)
```
```   746   also have "\<dots> = a div of_nat (2 ^ n) div of_nat (2 ^ m)"
```
```   747     by (simp only: div_mult2_eq')
```
```   748   finally show ?thesis
```
```   749     by (simp add: drop_bit_eq_div power_add)
```
```   750 qed
```
```   751
```
```   752 lemma push_bit_take_bit:
```
```   753   "push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)"
```
```   754   by (simp add: push_bit_eq_mult take_bit_eq_mod power_add mult_mod_right ac_simps)
```
```   755
```
```   756 lemma take_bit_push_bit:
```
```   757   "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)"
```
```   758 proof (cases "m \<le> n")
```
```   759   case True
```
```   760   then show ?thesis
```
```   761     by (simp_all add: push_bit_eq_mult take_bit_eq_mod mod_eq_0_iff_dvd dvd_power_le)
```
```   762 next
```
```   763   case False
```
```   764   then show ?thesis
```
```   765     using push_bit_take_bit [of n "m - n" a]
```
```   766     by simp
```
```   767 qed
```
```   768
```
```   769 lemma take_bit_drop_bit:
```
```   770   "take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)"
```
```   771   using mod_mult2_eq' [of a "2 ^ n" "2 ^ m"]
```
```   772   by (simp add: drop_bit_eq_div take_bit_eq_mod power_add ac_simps)
```
```   773
```
```   774 lemma drop_bit_take_bit:
```
```   775   "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
```
```   776 proof (cases "m \<le> n")
```
```   777   case True
```
```   778   then show ?thesis
```
```   779     using take_bit_drop_bit [of "n - m" m a] by simp
```
```   780 next
```
```   781   case False
```
```   782   then have "a mod 2 ^ n div 2 ^ m = a mod 2 ^ n div 2 ^ (n + (m - n))"
```
```   783     by simp
```
```   784   also have "\<dots> = a mod 2 ^ n div (2 ^ n * 2 ^ (m - n))"
```
```   785     by (simp add: power_add)
```
```   786   also have "\<dots> = a mod 2 ^ n div (of_nat (2 ^ n) * of_nat (2 ^ (m - n)))"
```
```   787     by simp
```
```   788   also have "\<dots> = a mod 2 ^ n div of_nat (2 ^ n) div of_nat (2 ^ (m - n))"
```
```   789     by (simp only: div_mult2_eq')
```
```   790   finally show ?thesis
```
```   791     using False by (simp add: take_bit_eq_mod drop_bit_eq_div)
```
```   792 qed
```
```   793
```
```   794 lemma push_bit_0_id [simp]:
```
```   795   "push_bit 0 = id"
```
```   796   by (simp add: fun_eq_iff push_bit_eq_mult)
```
```   797
```
```   798 lemma push_bit_of_0 [simp]:
```
```   799   "push_bit n 0 = 0"
```
```   800   by (simp add: push_bit_eq_mult)
```
```   801
```
```   802 lemma push_bit_of_1:
```
```   803   "push_bit n 1 = 2 ^ n"
```
```   804   by (simp add: push_bit_eq_mult)
```
```   805
```
```   806 lemma push_bit_Suc [simp]:
```
```   807   "push_bit (Suc n) a = push_bit n (a * 2)"
```
```   808   by (simp add: push_bit_eq_mult ac_simps)
```
```   809
```
```   810 lemma push_bit_double:
```
```   811   "push_bit n (a * 2) = push_bit n a * 2"
```
```   812   by (simp add: push_bit_eq_mult ac_simps)
```
```   813
```
```   814 lemma push_bit_eq_0_iff [simp]:
```
```   815   "push_bit n a = 0 \<longleftrightarrow> a = 0"
```
```   816   by (simp add: push_bit_eq_mult)
```
```   817
```
```   818 lemma push_bit_add:
```
```   819   "push_bit n (a + b) = push_bit n a + push_bit n b"
```
```   820   by (simp add: push_bit_eq_mult algebra_simps)
```
```   821
```
```   822 lemma push_bit_numeral [simp]:
```
```   823   "push_bit (numeral l) (numeral k) = push_bit (pred_numeral l) (numeral (Num.Bit0 k))"
```
```   824   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric]) (simp add: ac_simps)
```
```   825
```
```   826 lemma push_bit_of_nat:
```
```   827   "push_bit n (of_nat m) = of_nat (push_bit n m)"
```
```   828   by (simp add: push_bit_eq_mult Parity.push_bit_eq_mult)
```
```   829
```
```   830 lemma take_bit_0 [simp]:
```
```   831   "take_bit 0 a = 0"
```
```   832   by (simp add: take_bit_eq_mod)
```
```   833
```
```   834 lemma take_bit_Suc [simp]:
```
```   835   "take_bit (Suc n) a = take_bit n (a div 2) * 2 + of_bool (odd a)"
```
```   836 proof -
```
```   837   have "1 + 2 * (a div 2) mod (2 * 2 ^ n) = (a div 2 * 2 + a mod 2) mod (2 * 2 ^ n)"
```
```   838     if "odd a"
```
```   839     using that mod_mult2_eq' [of "1 + 2 * (a div 2)" 2 "2 ^ n"]
```
```   840     by (simp add: ac_simps odd_iff_mod_2_eq_one mult_mod_right)
```
```   841   also have "\<dots> = a mod (2 * 2 ^ n)"
```
```   842     by (simp only: div_mult_mod_eq)
```
```   843   finally show ?thesis
```
```   844     by (simp add: take_bit_eq_mod algebra_simps mult_mod_right)
```
```   845 qed
```
```   846
```
```   847 lemma take_bit_of_0 [simp]:
```
```   848   "take_bit n 0 = 0"
```
```   849   by (simp add: take_bit_eq_mod)
```
```   850
```
```   851 lemma take_bit_of_1 [simp]:
```
```   852   "take_bit n 1 = of_bool (n > 0)"
```
```   853   by (simp add: take_bit_eq_mod)
```
```   854
```
```   855 lemma take_bit_add:
```
```   856   "take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)"
```
```   857   by (simp add: take_bit_eq_mod mod_simps)
```
```   858
```
```   859 lemma take_bit_eq_0_iff:
```
```   860   "take_bit n a = 0 \<longleftrightarrow> 2 ^ n dvd a"
```
```   861   by (simp add: take_bit_eq_mod mod_eq_0_iff_dvd)
```
```   862
```
```   863 lemma take_bit_of_1_eq_0_iff [simp]:
```
```   864   "take_bit n 1 = 0 \<longleftrightarrow> n = 0"
```
```   865   by (simp add: take_bit_eq_mod)
```
```   866
```
```   867 lemma even_take_bit_eq [simp]:
```
```   868   "even (take_bit n a) \<longleftrightarrow> n = 0 \<or> even a"
```
```   869   by (cases n) (simp_all add: take_bit_eq_mod dvd_mod_iff)
```
```   870
```
```   871 lemma take_bit_numeral_bit0 [simp]:
```
```   872   "take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2"
```
```   873   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric] take_bit_Suc
```
```   874     ac_simps even_mult_iff nonzero_mult_div_cancel_right [OF numeral_neq_zero]) simp
```
```   875
```
```   876 lemma take_bit_numeral_bit1 [simp]:
```
```   877   "take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1"
```
```   878   by (simp only: numeral_eq_Suc power_Suc numeral_Bit1 [of k] mult_2 [symmetric] take_bit_Suc
```
```   879     ac_simps even_add even_mult_iff div_mult_self1 [OF numeral_neq_zero]) (simp add: ac_simps)
```
```   880
```
```   881 lemma take_bit_of_nat:
```
```   882   "take_bit n (of_nat m) = of_nat (take_bit n m)"
```
```   883   by (simp add: take_bit_eq_mod Parity.take_bit_eq_mod of_nat_mod [of m "2 ^ n"])
```
```   884
```
```   885 lemma drop_bit_0 [simp]:
```
```   886   "drop_bit 0 = id"
```
```   887   by (simp add: fun_eq_iff drop_bit_eq_div)
```
```   888
```
```   889 lemma drop_bit_of_0 [simp]:
```
```   890   "drop_bit n 0 = 0"
```
```   891   by (simp add: drop_bit_eq_div)
```
```   892
```
```   893 lemma drop_bit_of_1 [simp]:
```
```   894   "drop_bit n 1 = of_bool (n = 0)"
```
```   895   by (simp add: drop_bit_eq_div)
```
```   896
```
```   897 lemma drop_bit_Suc [simp]:
```
```   898   "drop_bit (Suc n) a = drop_bit n (a div 2)"
```
```   899 proof (cases "even a")
```
```   900   case True
```
```   901   then obtain b where "a = 2 * b" ..
```
```   902   moreover have "drop_bit (Suc n) (2 * b) = drop_bit n b"
```
```   903     by (simp add: drop_bit_eq_div)
```
```   904   ultimately show ?thesis
```
```   905     by simp
```
```   906 next
```
```   907   case False
```
```   908   then obtain b where "a = 2 * b + 1" ..
```
```   909   moreover have "drop_bit (Suc n) (2 * b + 1) = drop_bit n b"
```
```   910     using div_mult2_eq' [of "1 + b * 2" 2 "2 ^ n"]
```
```   911     by (auto simp add: drop_bit_eq_div ac_simps)
```
```   912   ultimately show ?thesis
```
```   913     by simp
```
```   914 qed
```
```   915
```
```   916 lemma drop_bit_half:
```
```   917   "drop_bit n (a div 2) = drop_bit n a div 2"
```
```   918   by (induction n arbitrary: a) simp_all
```
```   919
```
```   920 lemma drop_bit_of_bool [simp]:
```
```   921   "drop_bit n (of_bool d) = of_bool (n = 0 \<and> d)"
```
```   922   by (cases n) simp_all
```
```   923
```
```   924 lemma drop_bit_numeral_bit0 [simp]:
```
```   925   "drop_bit (numeral l) (numeral (Num.Bit0 k)) = drop_bit (pred_numeral l) (numeral k)"
```
```   926   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric] drop_bit_Suc
```
```   927     nonzero_mult_div_cancel_left [OF numeral_neq_zero])
```
```   928
```
```   929 lemma drop_bit_numeral_bit1 [simp]:
```
```   930   "drop_bit (numeral l) (numeral (Num.Bit1 k)) = drop_bit (pred_numeral l) (numeral k)"
```
```   931   by (simp only: numeral_eq_Suc power_Suc numeral_Bit1 [of k] mult_2 [symmetric] drop_bit_Suc
```
```   932     div_mult_self4 [OF numeral_neq_zero]) simp
```
```   933
```
```   934 lemma drop_bit_of_nat:
```
```   935   "drop_bit n (of_nat m) = of_nat (drop_bit n m)"
```
```   936 	by (simp add: drop_bit_eq_div Parity.drop_bit_eq_div of_nat_div [of m "2 ^ n"])
```
```   937
```
```   938 end
```
```   939
```
```   940 lemma push_bit_of_Suc_0 [simp]:
```
```   941   "push_bit n (Suc 0) = 2 ^ n"
```
```   942   using push_bit_of_1 [where ?'a = nat] by simp
```
```   943
```
```   944 lemma take_bit_of_Suc_0 [simp]:
```
```   945   "take_bit n (Suc 0) = of_bool (0 < n)"
```
```   946   using take_bit_of_1 [where ?'a = nat] by simp
```
```   947
```
```   948 lemma drop_bit_of_Suc_0 [simp]:
```
```   949   "drop_bit n (Suc 0) = of_bool (n = 0)"
```
```   950   using drop_bit_of_1 [where ?'a = nat] by simp
```
```   951
```
```   952 end
```