src/HOL/Hyperreal/EvenOdd.ML
author wenzelm
Tue Aug 27 11:03:05 2002 +0200 (2002-08-27)
changeset 13524 604d0f3622d6
parent 13517 42efec18f5b2
permissions -rw-r--r--
*** empty log message ***
     1 (*  Title       : Even.ML
     2     Author      : Jacques D. Fleuriot  
     3     Copyright   : 1999  University of Edinburgh
     4     Description : Even numbers defined
     5 *)
     6 
     7 Goal "even(Suc(Suc n)) = (even(n))";
     8 by (Auto_tac);
     9 qed "even_Suc_Suc";
    10 Addsimps [even_Suc_Suc];
    11 
    12 Goal "(even(n)) = (~odd(n))";
    13 by (induct_tac "n" 1);
    14 by (Auto_tac);
    15 qed "even_not_odd";
    16 
    17 Goal "(odd(n)) = (~even(n))";
    18 by (simp_tac (simpset() addsimps [even_not_odd]) 1);
    19 qed "odd_not_even";
    20 
    21 Goal "even(2*n)";
    22 by (induct_tac "n" 1);
    23 by (Auto_tac);
    24 qed "even_mult_two";
    25 Addsimps [even_mult_two];
    26 
    27 Goal "even(n*2)";
    28 by (induct_tac "n" 1);
    29 by (Auto_tac);
    30 qed "even_mult_two'";
    31 Addsimps [even_mult_two'];
    32 
    33 Goal "even(n + n)";
    34 by (induct_tac "n" 1);
    35 by (Auto_tac);
    36 qed "even_sum_self";
    37 Addsimps [even_sum_self];
    38 
    39 Goal "~odd(2*n)";
    40 by (induct_tac "n" 1);
    41 by Auto_tac;
    42 qed "not_odd_self_mult2";
    43 Addsimps [not_odd_self_mult2];
    44 
    45 Goal "~odd(n + n)";
    46 by (induct_tac "n" 1);
    47 by Auto_tac;
    48 qed "not_odd_sum_self";
    49 Addsimps [not_odd_sum_self];
    50 
    51 Goal "~even(Suc(n + n))";
    52 by (induct_tac "n" 1);
    53 by Auto_tac;
    54 qed "not_even_Suc_sum_self";
    55 Addsimps [not_even_Suc_sum_self];
    56 
    57 Goal "odd(Suc(2*n))";
    58 by (induct_tac "n" 1);
    59 by (Auto_tac);
    60 qed "odd_mult_two_add_one";
    61 Addsimps [odd_mult_two_add_one];
    62 
    63 Goal "odd(Suc(n + n))";
    64 by (induct_tac "n" 1);
    65 by (Auto_tac);
    66 qed "odd_sum_Suc_self";
    67 Addsimps [odd_sum_Suc_self];
    68 
    69 Goal "even(Suc n) = odd(n)";
    70 by (induct_tac "n" 1);
    71 by (Auto_tac);
    72 qed "even_Suc_odd_iff";
    73 
    74 Goal "odd(Suc n) = even(n)";
    75 by (induct_tac "n" 1);
    76 by (Auto_tac);
    77 qed "odd_Suc_even_iff";
    78 
    79 Goal "even n | odd n";
    80 by (simp_tac (simpset() addsimps [even_not_odd]) 1);
    81 qed "even_odd_disj";
    82 
    83 (* could be proved automatically before: spoiled by numeral arith! *)
    84 Goal "EX m. (n = 2*m | n = Suc(2*m))";
    85 by (induct_tac "n" 1 THEN Auto_tac);
    86 by (res_inst_tac [("x","Suc m")] exI 1 THEN Auto_tac);
    87 qed "nat_mult_two_Suc_disj";
    88 
    89 Goal "even(n) = (EX m. n = 2*m)";
    90 by (cut_inst_tac [("n","n")] nat_mult_two_Suc_disj 1);
    91 by (Auto_tac);
    92 qed "even_mult_two_ex";
    93 
    94 Goal "odd(n) = (EX m. n = Suc (2*m))";
    95 by (cut_inst_tac [("n","n")] nat_mult_two_Suc_disj 1);
    96 by (Auto_tac);
    97 qed "odd_Suc_mult_two_ex";
    98 
    99 Goal "even(n) ==> even(m*n)";
   100 by (auto_tac (claset(),
   101               simpset() addsimps [add_mult_distrib2, even_mult_two_ex]));
   102 qed "even_mult_even";
   103 Addsimps [even_mult_even];
   104 
   105 Goal "(m + m) div 2 = (m::nat)";
   106 by (arith_tac 1);
   107 qed "div_add_self_two_is_m";
   108 Addsimps [div_add_self_two_is_m];
   109 
   110 Goal "Suc(Suc(m*2)) div 2 = Suc m";
   111 by (arith_tac 1);
   112 qed "div_mult_self_Suc_Suc";
   113 Addsimps [div_mult_self_Suc_Suc];
   114 
   115 Goal "Suc(Suc(2*m)) div 2 = Suc m";
   116 by (arith_tac 1);
   117 qed "div_mult_self_Suc_Suc2";
   118 Addsimps [div_mult_self_Suc_Suc2];
   119 
   120 Goal "Suc(Suc(m + m)) div 2 = Suc m";
   121 by (arith_tac 1);
   122 qed "div_add_self_Suc_Suc";
   123 Addsimps [div_add_self_Suc_Suc];
   124 
   125 Goal "~ even n ==> (Suc n) div 2 = Suc((n - 1) div 2)";
   126 by (auto_tac (claset(),simpset() addsimps [odd_not_even RS sym,
   127     odd_Suc_mult_two_ex]));
   128 qed "not_even_imp_Suc_Suc_diff_one_eq";
   129 Addsimps [not_even_imp_Suc_Suc_diff_one_eq];
   130 
   131 Goal "even(m + n) = (even m = even n)";
   132 by (induct_tac "m" 1);
   133 by Auto_tac;
   134 qed "even_add";
   135 AddIffs [even_add];
   136 
   137 Goal "even(m * n) = (even m | even n)";
   138 by (induct_tac "m" 1);
   139 by Auto_tac;
   140 qed "even_mult";
   141 
   142 Goal "even (m ^ n) = (even m & n ~= 0)";
   143 by (induct_tac "n" 1);
   144 by (auto_tac (claset(),simpset() addsimps [even_mult]));
   145 qed "even_pow";
   146 AddIffs [even_pow];
   147 
   148 Goal "odd(m + n) = (odd m ~= odd n)";
   149 by (induct_tac "m" 1);
   150 by Auto_tac;
   151 qed "odd_add";
   152 AddIffs [odd_add];
   153 
   154 Goal "odd(m * n) = (odd m & odd n)";
   155 by (induct_tac "m" 1);
   156 by Auto_tac;
   157 qed "odd_mult";
   158 AddIffs [odd_mult];
   159 
   160 Goal "odd (m ^ n) = (odd m | n = 0)";
   161 by (induct_tac "n" 1);
   162 by Auto_tac;
   163 qed "odd_pow";
   164 
   165 Goal "0 < n --> ~even (n + n - 1)";
   166 by (induct_tac "n" 1);
   167 by Auto_tac;
   168 qed_spec_mp "not_even_2n_minus_1";
   169 Addsimps [not_even_2n_minus_1];
   170 
   171 Goal "Suc (2 * m) mod 2 = 1";
   172 by (induct_tac "m" 1);
   173 by (auto_tac (claset(),simpset() addsimps [mod_Suc]));
   174 qed "mod_two_Suc_2m";
   175 Addsimps [mod_two_Suc_2m];
   176 
   177 Goal "(Suc (Suc (2 * m)) div 2) = Suc m";
   178 by (arith_tac 1);
   179 qed "div_two_Suc_Suc_2m";
   180 Addsimps [div_two_Suc_Suc_2m];
   181 
   182 Goal "even n ==> 2 * ((n + 1) div 2) = n";
   183 by (auto_tac (claset(),simpset() addsimps [mult_div_cancel,
   184     even_mult_two_ex]));
   185 qed "lemma_even_div";
   186 Addsimps [lemma_even_div];
   187 
   188 Goal "~even n ==> 2 * ((n + 1) div 2) = Suc n";
   189 by (auto_tac (claset(),simpset() addsimps [even_not_odd,
   190     odd_Suc_mult_two_ex]));
   191 qed "lemma_not_even_div";
   192 Addsimps [lemma_not_even_div];
   193 
   194 Goal "Suc n div 2 <= Suc (Suc n) div 2";
   195 by (arith_tac 1);
   196 qed "Suc_n_le_Suc_Suc_n_div_2";
   197 Addsimps [Suc_n_le_Suc_Suc_n_div_2];
   198 
   199 Goal "(0::nat) < n --> 0 < (n + 1) div 2";
   200 by (arith_tac 1);
   201 qed_spec_mp "Suc_n_div_2_gt_zero";
   202 Addsimps [Suc_n_div_2_gt_zero];
   203 
   204 Goal "0 < n & even n --> 1 < n";
   205 by (induct_tac "n" 1);
   206 by Auto_tac;
   207 qed_spec_mp "even_gt_zero_gt_one_aux";
   208 bind_thm ("even_gt_zero_gt_one",conjI RS even_gt_zero_gt_one_aux);
   209 
   210 (* more general *)
   211 Goal "n div 2 <= (Suc n) div 2";
   212 by (arith_tac 1);
   213 qed "le_Suc_n_div_2";
   214 Addsimps [le_Suc_n_div_2];
   215 
   216 Goal "(1::nat) < n --> 0 < n div 2";
   217 by (arith_tac 1);
   218 qed_spec_mp "div_2_gt_zero";
   219 Addsimps [div_2_gt_zero];
   220 
   221 Goal "even n ==> (n + 1) div 2 = n div 2";
   222 by (rtac (CLAIM "2 * x = 2 * y ==> x = (y::nat)") 1);
   223 by (stac lemma_even_div 1);
   224 by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex]));
   225 qed "lemma_even_div2";
   226 Addsimps [lemma_even_div2];
   227 
   228 Goal "~even n ==> (n + 1) div 2 = Suc (n div 2)";
   229 by (rtac (CLAIM "2 * x = 2 * y ==> x = (y::nat)") 1);
   230 by (stac lemma_not_even_div 1);
   231 by (auto_tac (claset(),simpset() addsimps [even_not_odd,
   232     odd_Suc_mult_two_ex])); 
   233 by (cut_inst_tac [("m","Suc(2*m)"),("n","2")] mod_div_equality 1); 
   234 by Auto_tac; 
   235 qed "lemma_not_even_div2";
   236 Addsimps [lemma_not_even_div2];
   237 
   238 Goal "(Suc n) div 2 = 0 ==> even n";
   239 by (rtac ccontr 1);
   240 by (dtac lemma_not_even_div2 1 THEN Auto_tac);
   241 qed "Suc_n_div_two_zero_even";
   242 
   243 Goal "0 < n ==> even n = (~ even(n - 1))";
   244 by (case_tac "n" 1);
   245 by Auto_tac;
   246 qed "even_num_iff";
   247 
   248 Goal "0 < n ==> odd n = (~ odd(n - 1))";
   249 by (case_tac "n" 1);
   250 by Auto_tac;
   251 qed "odd_num_iff";
   252 
   253 (* Some mod and div stuff *)
   254 
   255 Goal "n ~= (0::nat) ==> (m = m div n * n + m mod n) & m mod n < n";
   256 by (auto_tac (claset() addIs [mod_less_divisor],simpset()));
   257 qed "mod_div_eq_less";
   258 
   259 Goal "(k*n + m) mod n = m mod (n::nat)";
   260 by (simp_tac (simpset() addsimps mult_ac @ add_ac) 1);
   261 qed "mod_mult_self3";
   262 Addsimps [mod_mult_self3];
   263 
   264 Goal "Suc (k*n + m) mod n = Suc m mod n";
   265 by (rtac (CLAIM "Suc (m + n) = (m + Suc n)" RS ssubst) 1);
   266 by (rtac mod_mult_self3 1);
   267 qed "mod_mult_self4";
   268 Addsimps [mod_mult_self4];
   269 
   270 Goal "Suc m mod n = Suc (m mod n) mod n";
   271 by (cut_inst_tac [("d'","Suc (m mod n) mod n")] (CLAIM "EX d. d' = d") 1);
   272 by (etac exE 1);
   273 by (Asm_simp_tac 1);
   274 by (res_inst_tac [("t","m"),("n1","n")] (mod_div_equality RS subst) 1);
   275 by (auto_tac (claset(),simpset() delsimprocs [cancel_div_mod_proc]));
   276 qed "mod_Suc_eq_Suc_mod";
   277 
   278 Goal "even n = (even (n mod 4))";
   279 by (cut_inst_tac [("d'","(even (n mod 4))")] (CLAIM "EX d. d' = d") 1);
   280 by (etac exE 1);
   281 by (Asm_simp_tac 1);
   282 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   283 by (auto_tac (claset(),simpset() addsimps [even_mult,even_num_iff] delsimprocs [cancel_div_mod_proc]));
   284 qed "even_even_mod_4_iff";
   285 
   286 Goal "odd n = (odd (n mod 4))";
   287 by (auto_tac (claset(),simpset() addsimps [odd_not_even,
   288     even_even_mod_4_iff RS sym]));
   289 qed "odd_odd_mod_4_iff";
   290 
   291 Goal "odd n ==> ((-1) ^ n) = (-1::real)";
   292 by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
   293 qed "odd_realpow_minus_one";
   294 Addsimps [odd_realpow_minus_one];
   295 
   296 Goal "even(4*n)";
   297 by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex]));
   298 qed "even_4n";
   299 Addsimps [even_4n];
   300 
   301 Goal "n mod 4 = 0 ==> even(n div 2)";
   302 by Auto_tac;
   303 qed "lemma_even_div_2";
   304 
   305 Goal "n mod 4 = 0 ==> even(n)";
   306 by Auto_tac;
   307 qed "lemma_mod_4_even_n";
   308 
   309 Goal "n mod 4 = 1 ==> odd(n)";
   310 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   311 by (auto_tac (claset(),simpset() addsimps [odd_num_iff] delsimprocs [cancel_div_mod_proc]));
   312 qed "lemma_mod_4_odd_n";
   313 
   314 Goal "n mod 4 = 2 ==> even(n)";
   315 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   316 by (auto_tac (claset(),simpset() addsimps [even_num_iff] delsimprocs [cancel_div_mod_proc]));
   317 qed "lemma_mod_4_even_n2";
   318 
   319 Goal "n mod 4 = 3 ==> odd(n)";
   320 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   321 by (auto_tac (claset(),simpset() addsimps [odd_num_iff] delsimprocs [cancel_div_mod_proc]));
   322 qed "lemma_mod_4_odd_n2";
   323 
   324 Goal "even n ==> ((-1) ^ n) = (1::real)";
   325 by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex]));
   326 qed "even_realpow_minus_one";
   327 Addsimps [even_realpow_minus_one];
   328 
   329 Goal "((4 * n) + 2) div 2 = (2::nat) * n + 1";
   330 by (arith_tac 1);
   331 qed "lemma_4n_add_2_div_2_eq";
   332 Addsimps [lemma_4n_add_2_div_2_eq];
   333 
   334 Goal "(Suc(Suc(4 * n))) div 2 = (2::nat) * n + 1";
   335 by (arith_tac 1);
   336 qed "lemma_Suc_Suc_4n_div_2_eq";
   337 Addsimps [lemma_Suc_Suc_4n_div_2_eq];
   338 
   339 Goal "(Suc(Suc(n * 4))) div 2 = (2::nat) * n + 1";
   340 by (arith_tac 1);
   341 qed "lemma_Suc_Suc_4n_div_2_eq2";
   342 Addsimps [lemma_Suc_Suc_4n_div_2_eq2];
   343 
   344 Goal "n mod 4 = 3 ==> odd((n - 1) div 2)";
   345 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   346 by (asm_full_simp_tac (simpset() addsimps [odd_num_iff] delsimprocs [cancel_div_mod_proc]) 1);
   347 val lemma_odd_mod_4_div_2 = result();
   348 
   349 Goal "(4 * n) div 2 = (2::nat) * n";
   350 by (arith_tac 1);
   351 qed "lemma_4n_div_2_eq";
   352 Addsimps [lemma_4n_div_2_eq];
   353 
   354 Goal "(n  * 4) div 2 = (2::nat) * n";
   355 by (arith_tac 1);
   356 qed "lemma_4n_div_2_eq2";
   357 Addsimps [lemma_4n_div_2_eq2];
   358 
   359 Goal "n mod 4 = 1 ==> even ((n - 1) div 2)";
   360 by (res_inst_tac [("t","n"),("n1","4")] (mod_div_equality RS subst) 1);
   361 by (dtac ssubst 1 THEN assume_tac 2);
   362 by (rtac ((CLAIM "(n::nat) + 1 - 1 = n") RS ssubst) 1);
   363 by Auto_tac;
   364 val lemma_even_mod_4_div_2 = result();
   365 
   366