src/HOL/Divides.thy
 author haftmann Thu Aug 06 23:56:48 2015 +0200 (2015-08-06) changeset 60867 86e7560e07d0 parent 60758 d8d85a8172b5 child 60868 dd18c33c001e permissions -rw-r--r--
slight cleanup of lemmas
 paulson@3366  1 (* Title: HOL/Divides.thy  paulson@3366  2  Author: Lawrence C Paulson, Cambridge University Computer Laboratory  paulson@6865  3  Copyright 1999 University of Cambridge  huffman@18154  4 *)  paulson@3366  5 wenzelm@60758  6 section \The division operators div and mod\  paulson@3366  7 nipkow@15131  8 theory Divides  haftmann@58778  9 imports Parity  nipkow@15131  10 begin  paulson@3366  11 wenzelm@60758  12 subsection \Abstract division in commutative semirings.\  haftmann@25942  13 haftmann@60352  14 class div = dvd + divide +  haftmann@60352  15  fixes mod :: "'a \ 'a \ 'a" (infixl "mod" 70)  haftmann@25942  16 haftmann@59833  17 class semiring_div = semidom + div +  haftmann@25942  18  assumes mod_div_equality: "a div b * b + a mod b = a"  haftmann@27651  19  and div_by_0 [simp]: "a div 0 = 0"  haftmann@27651  20  and div_0 [simp]: "0 div a = 0"  haftmann@27651  21  and div_mult_self1 [simp]: "b \ 0 \ (a + c * b) div b = c + a div b"  haftmann@30930  22  and div_mult_mult1 [simp]: "c \ 0 \ (c * a) div (c * b) = a div b"  haftmann@25942  23 begin  haftmann@25942  24 haftmann@60517  25 subclass algebraic_semidom  haftmann@60353  26 proof  haftmann@60353  27  fix b a  haftmann@60353  28  assume "b \ 0"  haftmann@60353  29  then show "a * b div b = a"  haftmann@60353  30  using div_mult_self1 [of b 0 a] by (simp add: ac_simps)  haftmann@60353  31 qed simp  haftmann@58953  32 haftmann@60867  33 lemma div_by_1:  haftmann@60867  34  "a div 1 = a"  haftmann@60867  35  by (fact divide_1)  haftmann@60867  36 haftmann@60867  37 lemma div_mult_self1_is_id:  haftmann@60867  38  "b \ 0 \ b * a div b = a"  haftmann@60867  39  by (fact nonzero_mult_divide_cancel_left)  haftmann@60867  40 haftmann@60867  41 lemma div_mult_self2_is_id:  haftmann@60867  42  "b \ 0 \ a * b div b = a"  haftmann@60867  43  by (fact nonzero_mult_divide_cancel_right)  haftmann@59009  44 wenzelm@60758  45 text \@{const divide} and @{const mod}\  haftmann@26100  46 haftmann@26062  47 lemma mod_div_equality2: "b * (a div b) + a mod b = a"  haftmann@57512  48  unfolding mult.commute [of b]  haftmann@26062  49  by (rule mod_div_equality)  haftmann@26062  50 huffman@29403  51 lemma mod_div_equality': "a mod b + a div b * b = a"  huffman@29403  52  using mod_div_equality [of a b]  haftmann@57514  53  by (simp only: ac_simps)  huffman@29403  54 haftmann@26062  55 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"  haftmann@30934  56  by (simp add: mod_div_equality)  haftmann@26062  57 haftmann@26062  58 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"  haftmann@30934  59  by (simp add: mod_div_equality2)  haftmann@26062  60 haftmann@27651  61 lemma mod_by_0 [simp]: "a mod 0 = a"  haftmann@30934  62  using mod_div_equality [of a zero] by simp  haftmann@27651  63 haftmann@27651  64 lemma mod_0 [simp]: "0 mod a = 0"  haftmann@30934  65  using mod_div_equality [of zero a] div_0 by simp  haftmann@27651  66 haftmann@27651  67 lemma div_mult_self2 [simp]:  haftmann@27651  68  assumes "b \ 0"  haftmann@27651  69  shows "(a + b * c) div b = c + a div b"  haftmann@57512  70  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)  haftmann@26100  71 haftmann@54221  72 lemma div_mult_self3 [simp]:  haftmann@54221  73  assumes "b \ 0"  haftmann@54221  74  shows "(c * b + a) div b = c + a div b"  haftmann@54221  75  using assms by (simp add: add.commute)  haftmann@54221  76 haftmann@54221  77 lemma div_mult_self4 [simp]:  haftmann@54221  78  assumes "b \ 0"  haftmann@54221  79  shows "(b * c + a) div b = c + a div b"  haftmann@54221  80  using assms by (simp add: add.commute)  haftmann@54221  81 haftmann@27651  82 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"  haftmann@27651  83 proof (cases "b = 0")  haftmann@27651  84  case True then show ?thesis by simp  haftmann@27651  85 next  haftmann@27651  86  case False  haftmann@27651  87  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"  haftmann@27651  88  by (simp add: mod_div_equality)  haftmann@27651  89  also from False div_mult_self1 [of b a c] have  haftmann@27651  90  "\ = (c + a div b) * b + (a + c * b) mod b"  nipkow@29667  91  by (simp add: algebra_simps)  haftmann@27651  92  finally have "a = a div b * b + (a + c * b) mod b"  haftmann@57512  93  by (simp add: add.commute [of a] add.assoc distrib_right)  haftmann@27651  94  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"  haftmann@27651  95  by (simp add: mod_div_equality)  haftmann@27651  96  then show ?thesis by simp  haftmann@27651  97 qed  haftmann@27651  98 lp15@60562  99 lemma mod_mult_self2 [simp]:  haftmann@54221  100  "(a + b * c) mod b = a mod b"  haftmann@57512  101  by (simp add: mult.commute [of b])  haftmann@27651  102 haftmann@54221  103 lemma mod_mult_self3 [simp]:  haftmann@54221  104  "(c * b + a) mod b = a mod b"  haftmann@54221  105  by (simp add: add.commute)  haftmann@54221  106 haftmann@54221  107 lemma mod_mult_self4 [simp]:  haftmann@54221  108  "(b * c + a) mod b = a mod b"  haftmann@54221  109  by (simp add: add.commute)  haftmann@54221  110 haftmann@60867  111 lemma mod_mult_self1_is_0 [simp]:  haftmann@60867  112  "b * a mod b = 0"  haftmann@27651  113  using mod_mult_self2 [of 0 b a] by simp  haftmann@27651  114 haftmann@60867  115 lemma mod_mult_self2_is_0 [simp]:  haftmann@60867  116  "a * b mod b = 0"  haftmann@27651  117  using mod_mult_self1 [of 0 a b] by simp  haftmann@26062  118 haftmann@60867  119 lemma mod_by_1 [simp]:  haftmann@60867  120  "a mod 1 = 0"  haftmann@27651  121 proof -  haftmann@27651  122  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp  haftmann@27651  123  then have "a + a mod 1 = a + 0" by simp  haftmann@27651  124  then show ?thesis by (rule add_left_imp_eq)  haftmann@27651  125 qed  haftmann@27651  126 haftmann@60867  127 lemma mod_self [simp]:  haftmann@60867  128  "a mod a = 0"  haftmann@27651  129  using mod_mult_self2_is_0 [of 1] by simp  haftmann@27651  130 haftmann@27676  131 lemma div_add_self1 [simp]:  haftmann@27651  132  assumes "b \ 0"  haftmann@27651  133  shows "(b + a) div b = a div b + 1"  haftmann@57512  134  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)  haftmann@26062  135 haftmann@27676  136 lemma div_add_self2 [simp]:  haftmann@27651  137  assumes "b \ 0"  haftmann@27651  138  shows "(a + b) div b = a div b + 1"  haftmann@57512  139  using assms div_add_self1 [of b a] by (simp add: add.commute)  haftmann@27651  140 haftmann@27676  141 lemma mod_add_self1 [simp]:  haftmann@27651  142  "(b + a) mod b = a mod b"  haftmann@57512  143  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)  haftmann@27651  144 haftmann@27676  145 lemma mod_add_self2 [simp]:  haftmann@27651  146  "(a + b) mod b = a mod b"  haftmann@27651  147  using mod_mult_self1 [of a 1 b] by simp  haftmann@27651  148 haftmann@27651  149 lemma mod_div_decomp:  haftmann@27651  150  fixes a b  haftmann@27651  151  obtains q r where "q = a div b" and "r = a mod b"  haftmann@27651  152  and "a = q * b + r"  haftmann@27651  153 proof -  haftmann@27651  154  from mod_div_equality have "a = a div b * b + a mod b" by simp  haftmann@27651  155  moreover have "a div b = a div b" ..  haftmann@27651  156  moreover have "a mod b = a mod b" ..  haftmann@27651  157  note that ultimately show thesis by blast  haftmann@27651  158 qed  haftmann@27651  159 haftmann@58834  160 lemma dvd_imp_mod_0 [simp]:  haftmann@58834  161  assumes "a dvd b"  haftmann@58834  162  shows "b mod a = 0"  haftmann@58834  163 proof -  haftmann@58834  164  from assms obtain c where "b = a * c" ..  haftmann@58834  165  then have "b mod a = a * c mod a" by simp  haftmann@58834  166  then show "b mod a = 0" by simp  haftmann@58834  167 qed  haftmann@58911  168 haftmann@58911  169 lemma mod_eq_0_iff_dvd:  haftmann@58911  170  "a mod b = 0 \ b dvd a"  haftmann@58911  171 proof  haftmann@58911  172  assume "b dvd a"  haftmann@58911  173  then show "a mod b = 0" by simp  haftmann@58911  174 next  haftmann@58911  175  assume "a mod b = 0"  haftmann@58911  176  with mod_div_equality [of a b] have "a div b * b = a" by simp  haftmann@58911  177  then have "a = b * (a div b)" by (simp add: ac_simps)  haftmann@58911  178  then show "b dvd a" ..  haftmann@58911  179 qed  haftmann@58911  180 haftmann@60867  181 lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:  haftmann@58834  182  "a dvd b \ b mod a = 0"  haftmann@58911  183  by (simp add: mod_eq_0_iff_dvd)  haftmann@58911  184 haftmann@58911  185 lemma mod_div_trivial [simp]:  haftmann@58911  186  "a mod b div b = 0"  huffman@29403  187 proof (cases "b = 0")  huffman@29403  188  assume "b = 0"  huffman@29403  189  thus ?thesis by simp  huffman@29403  190 next  huffman@29403  191  assume "b \ 0"  huffman@29403  192  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"  huffman@29403  193  by (rule div_mult_self1 [symmetric])  huffman@29403  194  also have "\ = a div b"  huffman@29403  195  by (simp only: mod_div_equality')  huffman@29403  196  also have "\ = a div b + 0"  huffman@29403  197  by simp  huffman@29403  198  finally show ?thesis  huffman@29403  199  by (rule add_left_imp_eq)  huffman@29403  200 qed  huffman@29403  201 haftmann@58911  202 lemma mod_mod_trivial [simp]:  haftmann@58911  203  "a mod b mod b = a mod b"  huffman@29403  204 proof -  huffman@29403  205  have "a mod b mod b = (a mod b + a div b * b) mod b"  huffman@29403  206  by (simp only: mod_mult_self1)  huffman@29403  207  also have "\ = a mod b"  huffman@29403  208  by (simp only: mod_div_equality')  huffman@29403  209  finally show ?thesis .  huffman@29403  210 qed  huffman@29403  211 haftmann@58834  212 lemma dvd_mod_imp_dvd:  haftmann@58834  213  assumes "k dvd m mod n" and "k dvd n"  haftmann@58834  214  shows "k dvd m"  haftmann@58834  215 proof -  haftmann@58834  216  from assms have "k dvd (m div n) * n + m mod n"  haftmann@58834  217  by (simp only: dvd_add dvd_mult)  haftmann@58834  218  then show ?thesis by (simp add: mod_div_equality)  haftmann@58834  219 qed  huffman@30078  220 wenzelm@60758  221 text \Addition respects modular equivalence.\  huffman@29403  222 haftmann@60867  223 lemma mod_add_left_eq: -- \FIXME reorient\  haftmann@60867  224  "(a + b) mod c = (a mod c + b) mod c"  huffman@29403  225 proof -  huffman@29403  226  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"  huffman@29403  227  by (simp only: mod_div_equality)  huffman@29403  228  also have "\ = (a mod c + b + a div c * c) mod c"  haftmann@57514  229  by (simp only: ac_simps)  huffman@29403  230  also have "\ = (a mod c + b) mod c"  huffman@29403  231  by (rule mod_mult_self1)  huffman@29403  232  finally show ?thesis .  huffman@29403  233 qed  huffman@29403  234 haftmann@60867  235 lemma mod_add_right_eq: -- \FIXME reorient\  haftmann@60867  236  "(a + b) mod c = (a + b mod c) mod c"  huffman@29403  237 proof -  huffman@29403  238  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"  huffman@29403  239  by (simp only: mod_div_equality)  huffman@29403  240  also have "\ = (a + b mod c + b div c * c) mod c"  haftmann@57514  241  by (simp only: ac_simps)  huffman@29403  242  also have "\ = (a + b mod c) mod c"  huffman@29403  243  by (rule mod_mult_self1)  huffman@29403  244  finally show ?thesis .  huffman@29403  245 qed  huffman@29403  246 haftmann@60867  247 lemma mod_add_eq: -- \FIXME reorient\  haftmann@60867  248  "(a + b) mod c = (a mod c + b mod c) mod c"  huffman@29403  249 by (rule trans [OF mod_add_left_eq mod_add_right_eq])  huffman@29403  250 huffman@29403  251 lemma mod_add_cong:  huffman@29403  252  assumes "a mod c = a' mod c"  huffman@29403  253  assumes "b mod c = b' mod c"  huffman@29403  254  shows "(a + b) mod c = (a' + b') mod c"  huffman@29403  255 proof -  huffman@29403  256  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"  huffman@29403  257  unfolding assms ..  huffman@29403  258  thus ?thesis  huffman@29403  259  by (simp only: mod_add_eq [symmetric])  huffman@29403  260 qed  huffman@29403  261 wenzelm@60758  262 text \Multiplication respects modular equivalence.\  huffman@29403  263 haftmann@60867  264 lemma mod_mult_left_eq: -- \FIXME reorient\  haftmann@60867  265  "(a * b) mod c = ((a mod c) * b) mod c"  huffman@29403  266 proof -  huffman@29403  267  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"  huffman@29403  268  by (simp only: mod_div_equality)  huffman@29403  269  also have "\ = (a mod c * b + a div c * b * c) mod c"  nipkow@29667  270  by (simp only: algebra_simps)  huffman@29403  271  also have "\ = (a mod c * b) mod c"  huffman@29403  272  by (rule mod_mult_self1)  huffman@29403  273  finally show ?thesis .  huffman@29403  274 qed  huffman@29403  275 haftmann@60867  276 lemma mod_mult_right_eq: -- \FIXME reorient\  haftmann@60867  277  "(a * b) mod c = (a * (b mod c)) mod c"  huffman@29403  278 proof -  huffman@29403  279  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"  huffman@29403  280  by (simp only: mod_div_equality)  huffman@29403  281  also have "\ = (a * (b mod c) + a * (b div c) * c) mod c"  nipkow@29667  282  by (simp only: algebra_simps)  huffman@29403  283  also have "\ = (a * (b mod c)) mod c"  huffman@29403  284  by (rule mod_mult_self1)  huffman@29403  285  finally show ?thesis .  huffman@29403  286 qed  huffman@29403  287 haftmann@60867  288 lemma mod_mult_eq: -- \FIXME reorient\  haftmann@60867  289  "(a * b) mod c = ((a mod c) * (b mod c)) mod c"  huffman@29403  290 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])  huffman@29403  291 huffman@29403  292 lemma mod_mult_cong:  huffman@29403  293  assumes "a mod c = a' mod c"  huffman@29403  294  assumes "b mod c = b' mod c"  huffman@29403  295  shows "(a * b) mod c = (a' * b') mod c"  huffman@29403  296 proof -  huffman@29403  297  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"  huffman@29403  298  unfolding assms ..  huffman@29403  299  thus ?thesis  huffman@29403  300  by (simp only: mod_mult_eq [symmetric])  huffman@29403  301 qed  huffman@29403  302 wenzelm@60758  303 text \Exponentiation respects modular equivalence.\  huffman@47164  304 haftmann@60867  305 lemma power_mod: "(a mod b) ^ n mod b = a ^ n mod b"  huffman@47164  306 apply (induct n, simp_all)  huffman@47164  307 apply (rule mod_mult_right_eq [THEN trans])  huffman@47164  308 apply (simp (no_asm_simp))  huffman@47164  309 apply (rule mod_mult_eq [symmetric])  huffman@47164  310 done  huffman@47164  311 huffman@29404  312 lemma mod_mod_cancel:  huffman@29404  313  assumes "c dvd b"  huffman@29404  314  shows "a mod b mod c = a mod c"  huffman@29404  315 proof -  wenzelm@60758  316  from \c dvd b\ obtain k where "b = c * k"  huffman@29404  317  by (rule dvdE)  huffman@29404  318  have "a mod b mod c = a mod (c * k) mod c"  wenzelm@60758  319  by (simp only: \b = c * k\)  huffman@29404  320  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  321  by (simp only: mod_mult_self1)  huffman@29404  322  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  haftmann@58786  323  by (simp only: ac_simps)  huffman@29404  324  also have "\ = a mod c"  huffman@29404  325  by (simp only: mod_div_equality)  huffman@29404  326  finally show ?thesis .  huffman@29404  327 qed  huffman@29404  328 haftmann@30930  329 lemma div_mult_mult2 [simp]:  haftmann@30930  330  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@57512  331  by (drule div_mult_mult1) (simp add: mult.commute)  haftmann@30930  332 haftmann@30930  333 lemma div_mult_mult1_if [simp]:  haftmann@30930  334  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  335  by simp_all  nipkow@30476  336 haftmann@30930  337 lemma mod_mult_mult1:  haftmann@30930  338  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  339 proof (cases "c = 0")  haftmann@30930  340  case True then show ?thesis by simp  haftmann@30930  341 next  haftmann@30930  342  case False  haftmann@30930  343  from mod_div_equality  haftmann@30930  344  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  345  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  346  = c * a + c * (a mod b)" by (simp add: algebra_simps)  lp15@60562  347  with mod_div_equality show ?thesis by simp  haftmann@30930  348 qed  lp15@60562  349 haftmann@30930  350 lemma mod_mult_mult2:  haftmann@30930  351  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@57512  352  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)  haftmann@30930  353 huffman@47159  354 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  huffman@47159  355  by (fact mod_mult_mult2 [symmetric])  huffman@47159  356 huffman@47159  357 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  huffman@47159  358  by (fact mod_mult_mult1 [symmetric])  huffman@47159  359 huffman@31662  360 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  361  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  362 huffman@31662  363 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  364 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  365 huffman@31661  366 end  huffman@31661  367 haftmann@59833  368 class ring_div = comm_ring_1 + semiring_div  huffman@29405  369 begin  huffman@29405  370 haftmann@60353  371 subclass idom_divide ..  haftmann@36634  372 wenzelm@60758  373 text \Negation respects modular equivalence.\  huffman@29405  374 huffman@29405  375 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  376 proof -  huffman@29405  377  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  378  by (simp only: mod_div_equality)  huffman@29405  379  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  haftmann@57514  380  by (simp add: ac_simps)  huffman@29405  381  also have "\ = (- (a mod b)) mod b"  huffman@29405  382  by (rule mod_mult_self1)  huffman@29405  383  finally show ?thesis .  huffman@29405  384 qed  huffman@29405  385 huffman@29405  386 lemma mod_minus_cong:  huffman@29405  387  assumes "a mod b = a' mod b"  huffman@29405  388  shows "(- a) mod b = (- a') mod b"  huffman@29405  389 proof -  huffman@29405  390  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  391  unfolding assms ..  huffman@29405  392  thus ?thesis  huffman@29405  393  by (simp only: mod_minus_eq [symmetric])  huffman@29405  394 qed  huffman@29405  395 wenzelm@60758  396 text \Subtraction respects modular equivalence.\  huffman@29405  397 haftmann@54230  398 lemma mod_diff_left_eq:  haftmann@54230  399  "(a - b) mod c = (a mod c - b) mod c"  haftmann@54230  400  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp  haftmann@54230  401 haftmann@54230  402 lemma mod_diff_right_eq:  haftmann@54230  403  "(a - b) mod c = (a - b mod c) mod c"  haftmann@54230  404  using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp  haftmann@54230  405 haftmann@54230  406 lemma mod_diff_eq:  haftmann@54230  407  "(a - b) mod c = (a mod c - b mod c) mod c"  haftmann@54230  408  using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp  huffman@29405  409 huffman@29405  410 lemma mod_diff_cong:  huffman@29405  411  assumes "a mod c = a' mod c"  huffman@29405  412  assumes "b mod c = b' mod c"  huffman@29405  413  shows "(a - b) mod c = (a' - b') mod c"  haftmann@54230  414  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp  huffman@29405  415 nipkow@30180  416 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  417 apply (case_tac "y = 0") apply simp  nipkow@30180  418 apply (auto simp add: dvd_def)  nipkow@30180  419 apply (subgoal_tac "-(y * k) = y * - k")  thomas@57492  420  apply (simp only:)  nipkow@30180  421  apply (erule div_mult_self1_is_id)  nipkow@30180  422 apply simp  nipkow@30180  423 done  nipkow@30180  424 nipkow@30180  425 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  426 apply (case_tac "y = 0") apply simp  nipkow@30180  427 apply (auto simp add: dvd_def)  nipkow@30180  428 apply (subgoal_tac "y * k = -y * -k")  thomas@57492  429  apply (erule ssubst, rule div_mult_self1_is_id)  nipkow@30180  430  apply simp  nipkow@30180  431 apply simp  nipkow@30180  432 done  nipkow@30180  433 haftmann@60867  434 lemma div_diff [simp]:  haftmann@60867  435  "z dvd x \ z dvd y \ (x - y) div z = x div z - y div z"  haftmann@60867  436  using div_add [of _ _ "- y"] by (simp add: dvd_neg_div)  nipkow@59380  437 huffman@47159  438 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  huffman@47159  439  using div_mult_mult1 [of "- 1" a b]  huffman@47159  440  unfolding neg_equal_0_iff_equal by simp  huffman@47159  441 huffman@47159  442 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  huffman@47159  443  using mod_mult_mult1 [of "- 1" a b] by simp  huffman@47159  444 huffman@47159  445 lemma div_minus_right: "a div (-b) = (-a) div b"  huffman@47159  446  using div_minus_minus [of "-a" b] by simp  huffman@47159  447 huffman@47159  448 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  huffman@47159  449  using mod_minus_minus [of "-a" b] by simp  huffman@47159  450 huffman@47160  451 lemma div_minus1_right [simp]: "a div (-1) = -a"  huffman@47160  452  using div_minus_right [of a 1] by simp  huffman@47160  453 huffman@47160  454 lemma mod_minus1_right [simp]: "a mod (-1) = 0"  huffman@47160  455  using mod_minus_right [of a 1] by simp  huffman@47160  456 lp15@60562  457 lemma minus_mod_self2 [simp]:  haftmann@54221  458  "(a - b) mod b = a mod b"  haftmann@54221  459  by (simp add: mod_diff_right_eq)  haftmann@54221  460 lp15@60562  461 lemma minus_mod_self1 [simp]:  haftmann@54221  462  "(b - a) mod b = - a mod b"  haftmann@54230  463  using mod_add_self2 [of "- a" b] by simp  haftmann@54221  464 huffman@29405  465 end  huffman@29405  466 haftmann@58778  467 wenzelm@60758  468 subsubsection \Parity and division\  haftmann@58778  469 lp15@60562  470 class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +  haftmann@54226  471  assumes parity: "a mod 2 = 0 \ a mod 2 = 1"  haftmann@58786  472  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"  haftmann@58710  473  assumes zero_not_eq_two: "0 \ 2"  haftmann@54226  474 begin  haftmann@54226  475 haftmann@54226  476 lemma parity_cases [case_names even odd]:  haftmann@54226  477  assumes "a mod 2 = 0 \ P"  haftmann@54226  478  assumes "a mod 2 = 1 \ P"  haftmann@54226  479  shows P  haftmann@54226  480  using assms parity by blast  haftmann@54226  481 haftmann@58786  482 lemma one_div_two_eq_zero [simp]:  haftmann@58778  483  "1 div 2 = 0"  haftmann@58778  484 proof (cases "2 = 0")  haftmann@58778  485  case True then show ?thesis by simp  haftmann@58778  486 next  haftmann@58778  487  case False  haftmann@58778  488  from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .  haftmann@58778  489  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp  haftmann@58953  490  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)  haftmann@58953  491  then have "1 div 2 = 0 \ 2 = 0" by simp  haftmann@58778  492  with False show ?thesis by auto  haftmann@58778  493 qed  haftmann@58778  494 haftmann@58786  495 lemma not_mod_2_eq_0_eq_1 [simp]:  haftmann@58786  496  "a mod 2 \ 0 \ a mod 2 = 1"  haftmann@58786  497  by (cases a rule: parity_cases) simp_all  haftmann@58786  498 haftmann@58786  499 lemma not_mod_2_eq_1_eq_0 [simp]:  haftmann@58786  500  "a mod 2 \ 1 \ a mod 2 = 0"  haftmann@58786  501  by (cases a rule: parity_cases) simp_all  haftmann@58786  502 haftmann@58778  503 subclass semiring_parity  haftmann@58778  504 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)  haftmann@58778  505  show "1 mod 2 = 1"  haftmann@58778  506  by (fact one_mod_two_eq_one)  haftmann@58778  507 next  haftmann@58778  508  fix a b  haftmann@58778  509  assume "a mod 2 = 1"  haftmann@58778  510  moreover assume "b mod 2 = 1"  haftmann@58778  511  ultimately show "(a + b) mod 2 = 0"  haftmann@58778  512  using mod_add_eq [of a b 2] by simp  haftmann@58778  513 next  haftmann@58778  514  fix a b  haftmann@58778  515  assume "(a * b) mod 2 = 0"  haftmann@58778  516  then have "(a mod 2) * (b mod 2) = 0"  haftmann@58778  517  by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])  haftmann@58778  518  then show "a mod 2 = 0 \ b mod 2 = 0"  haftmann@58778  519  by (rule divisors_zero)  haftmann@58778  520 next  haftmann@58778  521  fix a  haftmann@58778  522  assume "a mod 2 = 1"  haftmann@58778  523  then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp  haftmann@58778  524  then show "\b. a = b + 1" ..  haftmann@58778  525 qed  haftmann@58778  526 haftmann@58778  527 lemma even_iff_mod_2_eq_zero:  haftmann@58778  528  "even a \ a mod 2 = 0"  haftmann@58778  529  by (fact dvd_eq_mod_eq_0)  haftmann@58778  530 haftmann@58778  531 lemma even_succ_div_two [simp]:  haftmann@58778  532  "even a \ (a + 1) div 2 = a div 2"  haftmann@58778  533  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)  haftmann@58778  534 haftmann@58778  535 lemma odd_succ_div_two [simp]:  haftmann@58778  536  "odd a \ (a + 1) div 2 = a div 2 + 1"  haftmann@58778  537  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)  haftmann@58778  538 haftmann@58778  539 lemma even_two_times_div_two:  haftmann@58778  540  "even a \ 2 * (a div 2) = a"  haftmann@58778  541  by (fact dvd_mult_div_cancel)  haftmann@58778  542 haftmann@58834  543 lemma odd_two_times_div_two_succ [simp]:  haftmann@58778  544  "odd a \ 2 * (a div 2) + 1 = a"  haftmann@58778  545  using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)  haftmann@58778  546 haftmann@54226  547 end  haftmann@54226  548 haftmann@25942  549 wenzelm@60758  550 subsection \Generic numeral division with a pragmatic type class\  wenzelm@60758  551 wenzelm@60758  552 text \  haftmann@53067  553  The following type class contains everything necessary to formulate  haftmann@53067  554  a division algorithm in ring structures with numerals, restricted  haftmann@53067  555  to its positive segments. This is its primary motiviation, and it  haftmann@53067  556  could surely be formulated using a more fine-grained, more algebraic  haftmann@53067  557  and less technical class hierarchy.  wenzelm@60758  558 \  haftmann@53067  559 lp15@60562  560 class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +  haftmann@59816  561  assumes div_less: "0 \ a \ a < b \ a div b = 0"  haftmann@53067  562  and mod_less: " 0 \ a \ a < b \ a mod b = a"  haftmann@53067  563  and div_positive: "0 < b \ b \ a \ a div b > 0"  haftmann@53067  564  and mod_less_eq_dividend: "0 \ a \ a mod b \ a"  haftmann@53067  565  and pos_mod_bound: "0 < b \ a mod b < b"  haftmann@53067  566  and pos_mod_sign: "0 < b \ 0 \ a mod b"  haftmann@53067  567  and mod_mult2_eq: "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@53067  568  and div_mult2_eq: "0 \ c \ a div (b * c) = a div b div c"  haftmann@53067  569  assumes discrete: "a < b \ a + 1 \ b"  haftmann@53067  570 begin  haftmann@53067  571 haftmann@59816  572 lemma mult_div_cancel:  haftmann@59816  573  "b * (a div b) = a - a mod b"  haftmann@59816  574 proof -  haftmann@59816  575  have "b * (a div b) + a mod b = a"  haftmann@59816  576  using mod_div_equality [of a b] by (simp add: ac_simps)  haftmann@59816  577  then have "b * (a div b) + a mod b - a mod b = a - a mod b"  haftmann@59816  578  by simp  haftmann@59816  579  then show ?thesis  haftmann@59816  580  by simp  haftmann@59816  581 qed  haftmann@53067  582 haftmann@54226  583 subclass semiring_div_parity  haftmann@54226  584 proof  haftmann@54226  585  fix a  haftmann@54226  586  show "a mod 2 = 0 \ a mod 2 = 1"  haftmann@54226  587  proof (rule ccontr)  haftmann@54226  588  assume "\ (a mod 2 = 0 \ a mod 2 = 1)"  haftmann@54226  589  then have "a mod 2 \ 0" and "a mod 2 \ 1" by simp_all  haftmann@54226  590  have "0 < 2" by simp  haftmann@54226  591  with pos_mod_bound pos_mod_sign have "0 \ a mod 2" "a mod 2 < 2" by simp_all  wenzelm@60758  592  with \a mod 2 \ 0\ have "0 < a mod 2" by simp  haftmann@54226  593  with discrete have "1 \ a mod 2" by simp  wenzelm@60758  594  with \a mod 2 \ 1\ have "1 < a mod 2" by simp  haftmann@54226  595  with discrete have "2 \ a mod 2" by simp  wenzelm@60758  596  with \a mod 2 < 2\ show False by simp  haftmann@54226  597  qed  haftmann@58646  598 next  haftmann@58646  599  show "1 mod 2 = 1"  haftmann@58646  600  by (rule mod_less) simp_all  haftmann@58710  601 next  haftmann@58710  602  show "0 \ 2"  haftmann@58710  603  by simp  haftmann@53067  604 qed  haftmann@53067  605 haftmann@53067  606 lemma divmod_digit_1:  haftmann@53067  607  assumes "0 \ a" "0 < b" and "b \ a mod (2 * b)"  haftmann@53067  608  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")  haftmann@53067  609  and "a mod (2 * b) - b = a mod b" (is "?Q")  haftmann@53067  610 proof -  haftmann@53067  611  from assms mod_less_eq_dividend [of a "2 * b"] have "b \ a"  haftmann@53067  612  by (auto intro: trans)  wenzelm@60758  613  with \0 < b\ have "0 < a div b" by (auto intro: div_positive)  haftmann@53067  614  then have [simp]: "1 \ a div b" by (simp add: discrete)  wenzelm@60758  615  with \0 < b\ have mod_less: "a mod b < b" by (simp add: pos_mod_bound)  haftmann@53067  616  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  617  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  618  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  619  from assms w_exhaust have "w = 1"  haftmann@53067  620  by (auto simp add: mod_w) (insert mod_less, auto)  haftmann@53067  621  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp  haftmann@53067  622  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  623  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  wenzelm@60758  624  with \w = 1\ have div: "2 * (a div (2 * b)) = a div b - 1" by simp  haftmann@53067  625  then show ?P and ?Q  haftmann@60867  626  by (simp_all add: div mod add_implies_diff [symmetric])  haftmann@53067  627 qed  haftmann@53067  628 haftmann@53067  629 lemma divmod_digit_0:  haftmann@53067  630  assumes "0 < b" and "a mod (2 * b) < b"  haftmann@53067  631  shows "2 * (a div (2 * b)) = a div b" (is "?P")  haftmann@53067  632  and "a mod (2 * b) = a mod b" (is "?Q")  haftmann@53067  633 proof -  haftmann@53067  634  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  635  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  636  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  637  moreover have "b \ a mod b + b"  haftmann@53067  638  proof -  wenzelm@60758  639  from \0 < b\ pos_mod_sign have "0 \ a mod b" by blast  haftmann@53067  640  then have "0 + b \ a mod b + b" by (rule add_right_mono)  haftmann@53067  641  then show ?thesis by simp  haftmann@53067  642  qed  haftmann@53067  643  moreover note assms w_exhaust  haftmann@53067  644  ultimately have "w = 0" by auto  haftmann@53067  645  with mod_w have mod: "a mod (2 * b) = a mod b" by simp  haftmann@53067  646  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  647  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  wenzelm@60758  648  with \w = 0\ have div: "2 * (a div (2 * b)) = a div b" by simp  haftmann@53067  649  then show ?P and ?Q  haftmann@53067  650  by (simp_all add: div mod)  haftmann@53067  651 qed  haftmann@53067  652 haftmann@53067  653 definition divmod :: "num \ num \ 'a \ 'a"  haftmann@53067  654 where  haftmann@53067  655  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"  haftmann@53067  656 haftmann@60867  657 lemma fst_divmod:  haftmann@53067  658  "fst (divmod m n) = numeral m div numeral n"  haftmann@53067  659  by (simp add: divmod_def)  haftmann@53067  660 haftmann@60867  661 lemma snd_divmod:  haftmann@53067  662  "snd (divmod m n) = numeral m mod numeral n"  haftmann@53067  663  by (simp add: divmod_def)  haftmann@53067  664 haftmann@53067  665 definition divmod_step :: "num \ 'a \ 'a \ 'a \ 'a"  haftmann@53067  666 where  haftmann@53067  667  "divmod_step l qr = (let (q, r) = qr  haftmann@53067  668  in if r \ numeral l then (2 * q + 1, r - numeral l)  haftmann@53067  669  else (2 * q, r))"  haftmann@53067  670 wenzelm@60758  671 text \  haftmann@53067  672  This is a formulation of one step (referring to one digit position)  haftmann@53067  673  in school-method division: compare the dividend at the current  haftmann@53070  674  digit position with the remainder from previous division steps  haftmann@53067  675  and evaluate accordingly.  wenzelm@60758  676 \  haftmann@53067  677 haftmann@60867  678 lemma divmod_step_eq [code, simp]:  haftmann@53067  679  "divmod_step l (q, r) = (if numeral l \ r  haftmann@53067  680  then (2 * q + 1, r - numeral l) else (2 * q, r))"  haftmann@53067  681  by (simp add: divmod_step_def)  haftmann@53067  682 wenzelm@60758  683 text \  haftmann@53067  684  This is a formulation of school-method division.  haftmann@53067  685  If the divisor is smaller than the dividend, terminate.  haftmann@53067  686  If not, shift the dividend to the right until termination  haftmann@53067  687  occurs and then reiterate single division steps in the  haftmann@53067  688  opposite direction.  wenzelm@60758  689 \  haftmann@53067  690 haftmann@60867  691 lemma divmod_divmod_step:  haftmann@53067  692  "divmod m n = (if m < n then (0, numeral m)  haftmann@53067  693  else divmod_step n (divmod m (Num.Bit0 n)))"  haftmann@53067  694 proof (cases "m < n")  haftmann@53067  695  case True then have "numeral m < numeral n" by simp  haftmann@53067  696  then show ?thesis  haftmann@60867  697  by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)  haftmann@53067  698 next  haftmann@53067  699  case False  haftmann@53067  700  have "divmod m n =  haftmann@53067  701  divmod_step n (numeral m div (2 * numeral n),  haftmann@53067  702  numeral m mod (2 * numeral n))"  haftmann@53067  703  proof (cases "numeral n \ numeral m mod (2 * numeral n)")  haftmann@53067  704  case True  haftmann@60867  705  with divmod_step_eq  haftmann@53067  706  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  707  (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"  haftmann@60867  708  by simp  haftmann@53067  709  moreover from True divmod_digit_1 [of "numeral m" "numeral n"]  haftmann@53067  710  have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"  haftmann@53067  711  and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"  haftmann@53067  712  by simp_all  haftmann@53067  713  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  714  next  haftmann@53067  715  case False then have *: "numeral m mod (2 * numeral n) < numeral n"  haftmann@53067  716  by (simp add: not_le)  haftmann@60867  717  with divmod_step_eq  haftmann@53067  718  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  719  (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"  haftmann@60867  720  by auto  haftmann@53067  721  moreover from * divmod_digit_0 [of "numeral n" "numeral m"]  haftmann@53067  722  have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"  haftmann@53067  723  and "numeral m mod (2 * numeral n) = numeral m mod numeral n"  haftmann@53067  724  by (simp_all only: zero_less_numeral)  haftmann@53067  725  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  726  qed  haftmann@53067  727  then have "divmod m n =  haftmann@53067  728  divmod_step n (numeral m div numeral (Num.Bit0 n),  haftmann@53067  729  numeral m mod numeral (Num.Bit0 n))"  lp15@60562  730  by (simp only: numeral.simps distrib mult_1)  haftmann@53067  731  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@53067  732  by (simp add: divmod_def)  haftmann@53067  733  with False show ?thesis by simp  haftmann@53067  734 qed  haftmann@53067  735 haftmann@60867  736 text \The division rewrite proper â€“ first, trivial results involving @{text 1}\  haftmann@60867  737 haftmann@60867  738 lemma divmod_trivial [simp, code]:  haftmann@60867  739  "divmod Num.One Num.One = (numeral Num.One, 0)"  haftmann@60867  740  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"  haftmann@60867  741  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"  haftmann@60867  742  "divmod num.One (num.Bit0 n) = (0, Numeral1)"  haftmann@60867  743  "divmod num.One (num.Bit1 n) = (0, Numeral1)"  haftmann@60867  744  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)  haftmann@60867  745 haftmann@60867  746 text \Division by an even number is a right-shift\  haftmann@58953  747 haftmann@58953  748 lemma divmod_cancel [simp, code]:  haftmann@53069  749  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r))" (is ?P)  haftmann@53069  750  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r + 1))" (is ?Q)  haftmann@53069  751 proof -  haftmann@53069  752  have *: "\q. numeral (Num.Bit0 q) = 2 * numeral q"  haftmann@53069  753  "\q. numeral (Num.Bit1 q) = 2 * numeral q + 1"  haftmann@53069  754  by (simp_all only: numeral_mult numeral.simps distrib) simp_all  haftmann@53069  755  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)  haftmann@53069  756  then show ?P and ?Q  haftmann@60867  757  by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1  haftmann@60867  758  div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]  haftmann@60867  759  add.commute del: numeral_times_numeral)  haftmann@58953  760 qed  haftmann@58953  761 haftmann@60867  762 text \The really hard work\  haftmann@60867  763 haftmann@60867  764 lemma divmod_steps [simp, code]:  haftmann@60867  765  "divmod (num.Bit0 m) (num.Bit1 n) =  haftmann@60867  766  (if m \ n then (0, numeral (num.Bit0 m))  haftmann@60867  767  else divmod_step (num.Bit1 n)  haftmann@60867  768  (divmod (num.Bit0 m)  haftmann@60867  769  (num.Bit0 (num.Bit1 n))))"  haftmann@60867  770  "divmod (num.Bit1 m) (num.Bit1 n) =  haftmann@60867  771  (if m < n then (0, numeral (num.Bit1 m))  haftmann@60867  772  else divmod_step (num.Bit1 n)  haftmann@60867  773  (divmod (num.Bit1 m)  haftmann@60867  774  (num.Bit0 (num.Bit1 n))))"  haftmann@60867  775  by (simp_all add: divmod_divmod_step)  haftmann@60867  776 wenzelm@60758  777 text \Special case: divisibility\  haftmann@58953  778 haftmann@58953  779 definition divides_aux :: "'a \ 'a \ bool"  haftmann@58953  780 where  haftmann@58953  781  "divides_aux qr \ snd qr = 0"  haftmann@58953  782 haftmann@58953  783 lemma divides_aux_eq [simp]:  haftmann@58953  784  "divides_aux (q, r) \ r = 0"  haftmann@58953  785  by (simp add: divides_aux_def)  haftmann@58953  786 haftmann@58953  787 lemma dvd_numeral_simp [simp]:  haftmann@58953  788  "numeral m dvd numeral n \ divides_aux (divmod n m)"  haftmann@58953  789  by (simp add: divmod_def mod_eq_0_iff_dvd)  haftmann@53069  790 haftmann@60867  791 text \Generic computation of quotient and remainder\  haftmann@60867  792 haftmann@60867  793 lemma numeral_div_numeral [simp]:  haftmann@60867  794  "numeral k div numeral l = fst (divmod k l)"  haftmann@60867  795  by (simp add: fst_divmod)  haftmann@60867  796 haftmann@60867  797 lemma numeral_mod_numeral [simp]:  haftmann@60867  798  "numeral k mod numeral l = snd (divmod k l)"  haftmann@60867  799  by (simp add: snd_divmod)  haftmann@60867  800 haftmann@60867  801 lemma one_div_numeral [simp]:  haftmann@60867  802  "1 div numeral n = fst (divmod num.One n)"  haftmann@60867  803  by (simp add: fst_divmod)  haftmann@60867  804 haftmann@60867  805 lemma one_mod_numeral [simp]:  haftmann@60867  806  "1 mod numeral n = snd (divmod num.One n)"  haftmann@60867  807  by (simp add: snd_divmod)  haftmann@60867  808   haftmann@53067  809 end  haftmann@53067  810 lp15@60562  811 wenzelm@60758  812 subsection \Division on @{typ nat}\  wenzelm@60758  813 wenzelm@60758  814 text \  haftmann@60429  815  We define @{const divide} and @{const mod} on @{typ nat} by means  haftmann@26100  816  of a characteristic relation with two input arguments  haftmann@26100  817  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  818  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  wenzelm@60758  819 \  haftmann@26100  820 haftmann@33340  821 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  822  "divmod_nat_rel m n qr \  haftmann@30923  823  m = fst qr * n + snd qr \  haftmann@30923  824  (if n = 0 then fst qr = 0 else if n > 0 then 0 \ snd qr \ snd qr < n else n < snd qr \ snd qr \ 0)"  haftmann@26100  825 wenzelm@60758  826 text \@{const divmod_nat_rel} is total:\  haftmann@26100  827 haftmann@33340  828 lemma divmod_nat_rel_ex:  haftmann@33340  829  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  830 proof (cases "n = 0")  haftmann@30923  831  case True with that show thesis  haftmann@33340  832  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  833 next  haftmann@26100  834  case False  haftmann@26100  835  have "\q r. m = q * n + r \ r < n"  haftmann@26100  836  proof (induct m)  wenzelm@60758  837  case 0 with \n \ 0\  haftmann@26100  838  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  839  then show ?case by blast  haftmann@26100  840  next  haftmann@26100  841  case (Suc m) then obtain q' r'  haftmann@26100  842  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  843  then show ?case proof (cases "Suc r' < n")  haftmann@26100  844  case True  haftmann@26100  845  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  846  with True show ?thesis by blast  haftmann@26100  847  next  haftmann@26100  848  case False then have "n \ Suc r'" by auto  haftmann@26100  849  moreover from n have "Suc r' \ n" by auto  haftmann@26100  850  ultimately have "n = Suc r'" by auto  haftmann@26100  851  with m have "Suc m = Suc q' * n + 0" by simp  wenzelm@60758  852  with \n \ 0\ show ?thesis by blast  haftmann@26100  853  qed  haftmann@26100  854  qed  haftmann@26100  855  with that show thesis  wenzelm@60758  856  using \n \ 0\ by (auto simp add: divmod_nat_rel_def)  haftmann@26100  857 qed  haftmann@26100  858 wenzelm@60758  859 text \@{const divmod_nat_rel} is injective:\  haftmann@26100  860 haftmann@33340  861 lemma divmod_nat_rel_unique:  haftmann@33340  862  assumes "divmod_nat_rel m n qr"  haftmann@33340  863  and "divmod_nat_rel m n qr'"  haftmann@30923  864  shows "qr = qr'"  haftmann@26100  865 proof (cases "n = 0")  haftmann@26100  866  case True with assms show ?thesis  haftmann@30923  867  by (cases qr, cases qr')  haftmann@33340  868  (simp add: divmod_nat_rel_def)  haftmann@26100  869 next  haftmann@26100  870  case False  haftmann@26100  871  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  872  apply (rule leI)  haftmann@26100  873  apply (subst less_iff_Suc_add)  haftmann@26100  874  apply (auto simp add: add_mult_distrib)  haftmann@26100  875  done  wenzelm@60758  876  from \n \ 0\ assms have *: "fst qr = fst qr'"  haftmann@33340  877  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  wenzelm@53374  878  with assms have "snd qr = snd qr'"  haftmann@33340  879  by (simp add: divmod_nat_rel_def)  wenzelm@53374  880  with * show ?thesis by (cases qr, cases qr') simp  haftmann@26100  881 qed  haftmann@26100  882 wenzelm@60758  883 text \  haftmann@26100  884  We instantiate divisibility on the natural numbers by  haftmann@33340  885  means of @{const divmod_nat_rel}:  wenzelm@60758  886 \  haftmann@25942  887 haftmann@33340  888 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  889  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  890 haftmann@33340  891 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  892  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  893 proof -  haftmann@33340  894  from divmod_nat_rel_ex  haftmann@33340  895  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  896  then show ?thesis  haftmann@33340  897  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  898 qed  haftmann@30923  899 huffman@47135  900 lemma divmod_nat_unique:  lp15@60562  901  assumes "divmod_nat_rel m n qr"  haftmann@33340  902  shows "divmod_nat m n = qr"  haftmann@33340  903  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  904 haftmann@60429  905 instantiation nat :: semiring_div  haftmann@60352  906 begin  haftmann@60352  907 haftmann@60352  908 definition divide_nat where  haftmann@60429  909  div_nat_def: "m div n = fst (divmod_nat m n)"  haftmann@60352  910 haftmann@60352  911 definition mod_nat where  haftmann@60352  912  "m mod n = snd (divmod_nat m n)"  huffman@46551  913 huffman@46551  914 lemma fst_divmod_nat [simp]:  huffman@46551  915  "fst (divmod_nat m n) = m div n"  huffman@46551  916  by (simp add: div_nat_def)  huffman@46551  917 huffman@46551  918 lemma snd_divmod_nat [simp]:  huffman@46551  919  "snd (divmod_nat m n) = m mod n"  huffman@46551  920  by (simp add: mod_nat_def)  huffman@46551  921 haftmann@33340  922 lemma divmod_nat_div_mod:  haftmann@33340  923  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  924  by (simp add: prod_eq_iff)  haftmann@26100  925 huffman@47135  926 lemma div_nat_unique:  lp15@60562  927  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  928  shows "m div n = q"  huffman@47135  929  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  930 huffman@47135  931 lemma mod_nat_unique:  lp15@60562  932  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  933  shows "m mod n = r"  huffman@47135  934  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  935 haftmann@33340  936 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  937  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  938 huffman@47136  939 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  940  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  941 huffman@47136  942 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  943  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  944 huffman@47137  945 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  946  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  947 haftmann@33340  948 lemma divmod_nat_step:  haftmann@26100  949  assumes "0 < n" and "n \ m"  haftmann@33340  950  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  951 proof (rule divmod_nat_unique)  huffman@47134  952  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  953  by (rule divmod_nat_rel)  huffman@47134  954  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  955  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  956 qed  haftmann@25942  957 wenzelm@60758  958 text \The ''recursion'' equations for @{const divide} and @{const mod}\  haftmann@26100  959 haftmann@26100  960 lemma div_less [simp]:  haftmann@26100  961  fixes m n :: nat  haftmann@26100  962  assumes "m < n"  haftmann@26100  963  shows "m div n = 0"  huffman@46551  964  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  965 haftmann@26100  966 lemma le_div_geq:  haftmann@26100  967  fixes m n :: nat  haftmann@26100  968  assumes "0 < n" and "n \ m"  haftmann@26100  969  shows "m div n = Suc ((m - n) div n)"  huffman@46551  970  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  971 haftmann@26100  972 lemma mod_less [simp]:  haftmann@26100  973  fixes m n :: nat  haftmann@26100  974  assumes "m < n"  haftmann@26100  975  shows "m mod n = m"  huffman@46551  976  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  977 haftmann@26100  978 lemma le_mod_geq:  haftmann@26100  979  fixes m n :: nat  haftmann@26100  980  assumes "n \ m"  haftmann@26100  981  shows "m mod n = (m - n) mod n"  huffman@46551  982  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  983 huffman@47136  984 instance proof  huffman@47136  985  fix m n :: nat  huffman@47136  986  show "m div n * n + m mod n = m"  huffman@47136  987  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  988 next  huffman@47136  989  fix m n q :: nat  huffman@47136  990  assume "n \ 0"  huffman@47136  991  then show "(q + m * n) div n = m + q div n"  huffman@47136  992  by (induct m) (simp_all add: le_div_geq)  huffman@47136  993 next  huffman@47136  994  fix m n q :: nat  huffman@47136  995  assume "m \ 0"  huffman@47136  996  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  997  unfolding divmod_nat_rel_def  huffman@47136  998  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  999  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  1000  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  1001  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  1002 next  huffman@47136  1003  fix n :: nat show "n div 0 = 0"  haftmann@33340  1004  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  1005 next  huffman@47136  1006  fix n :: nat show "0 div n = 0"  huffman@47136  1007  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  1008 qed  haftmann@26100  1009 haftmann@25942  1010 end  paulson@14267  1011 haftmann@60685  1012 instantiation nat :: normalization_semidom  haftmann@60685  1013 begin  haftmann@60685  1014 haftmann@60685  1015 definition normalize_nat  haftmann@60685  1016  where [simp]: "normalize = (id :: nat \ nat)"  haftmann@60685  1017 haftmann@60685  1018 definition unit_factor_nat  haftmann@60685  1019  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"  haftmann@60685  1020 haftmann@60685  1021 lemma unit_factor_simps [simp]:  haftmann@60685  1022  "unit_factor 0 = (0::nat)"  haftmann@60685  1023  "unit_factor (Suc n) = 1"  haftmann@60685  1024  by (simp_all add: unit_factor_nat_def)  haftmann@60685  1025 haftmann@60685  1026 instance  haftmann@60685  1027  by standard (simp_all add: unit_factor_nat_def)  haftmann@60685  1028   haftmann@60685  1029 end  haftmann@60685  1030 haftmann@33361  1031 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  1032  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  blanchet@55414  1033  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)  haftmann@33361  1034 wenzelm@60758  1035 text \Simproc for cancelling @{const divide} and @{const mod}\  haftmann@25942  1036 wenzelm@51299  1037 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"  wenzelm@51299  1038 wenzelm@60758  1039 ML \  wenzelm@43594  1040 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  1041 (  haftmann@60352  1042  val div_name = @{const_name divide};  haftmann@30934  1043  val mod_name = @{const_name mod};  haftmann@30934  1044  val mk_binop = HOLogic.mk_binop;  huffman@48561  1045  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  huffman@48561  1046  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  huffman@48561  1047  fun mk_sum [] = HOLogic.zero  huffman@48561  1048  | mk_sum [t] = t  huffman@48561  1049  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  huffman@48561  1050  fun dest_sum tm =  huffman@48561  1051  if HOLogic.is_zero tm then []  huffman@48561  1052  else  huffman@48561  1053  (case try HOLogic.dest_Suc tm of  huffman@48561  1054  SOME t => HOLogic.Suc_zero :: dest_sum t  huffman@48561  1055  | NONE =>  huffman@48561  1056  (case try dest_plus tm of  huffman@48561  1057  SOME (t, u) => dest_sum t @ dest_sum u  huffman@48561  1058  | NONE => [tm]));  haftmann@25942  1059 haftmann@30934  1060  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  1061 haftmann@30934  1062  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@57514  1063  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))  wenzelm@41550  1064 )  wenzelm@60758  1065 \  wenzelm@60758  1066 wenzelm@60758  1067 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = \K Cancel_Div_Mod_Nat.proc\  wenzelm@60758  1068 wenzelm@60758  1069 wenzelm@60758  1070 subsubsection \Quotient\  haftmann@26100  1071 haftmann@26100  1072 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  1073 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  1074 haftmann@26100  1075 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  1076 by (simp add: div_geq)  haftmann@26100  1077 haftmann@26100  1078 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  1079 by simp  haftmann@26100  1080 haftmann@26100  1081 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  1082 by simp  haftmann@26100  1083 haftmann@53066  1084 lemma div_positive:  haftmann@53066  1085  fixes m n :: nat  haftmann@53066  1086  assumes "n > 0"  haftmann@53066  1087  assumes "m \ n"  haftmann@53066  1088  shows "m div n > 0"  haftmann@53066  1089 proof -  wenzelm@60758  1090  from \m \ n\ obtain q where "m = n + q"  haftmann@53066  1091  by (auto simp add: le_iff_add)  wenzelm@60758  1092  with \n > 0\ show ?thesis by simp  haftmann@53066  1093 qed  haftmann@53066  1094 hoelzl@59000  1095 lemma div_eq_0_iff: "(a div b::nat) = 0 \ a < b \ b = 0"  hoelzl@59000  1096  by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)  haftmann@25942  1097 wenzelm@60758  1098 subsubsection \Remainder\  haftmann@25942  1099 haftmann@26100  1100 lemma mod_less_divisor [simp]:  haftmann@26100  1101  fixes m n :: nat  haftmann@26100  1102  assumes "n > 0"  haftmann@26100  1103  shows "m mod n < (n::nat)"  haftmann@33340  1104  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  1105 haftmann@51173  1106 lemma mod_Suc_le_divisor [simp]:  haftmann@51173  1107  "m mod Suc n \ n"  haftmann@51173  1108  using mod_less_divisor [of "Suc n" m] by arith  haftmann@51173  1109 haftmann@26100  1110 lemma mod_less_eq_dividend [simp]:  haftmann@26100  1111  fixes m n :: nat  haftmann@26100  1112  shows "m mod n \ m"  haftmann@26100  1113 proof (rule add_leD2)  haftmann@26100  1114  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  1115  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  1116 qed  haftmann@26100  1117 haftmann@26100  1118 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  1119 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  1120 haftmann@26100  1121 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  1122 by (simp add: le_mod_geq)  haftmann@26100  1123 paulson@14267  1124 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  1125 by (induct m) (simp_all add: mod_geq)  paulson@14267  1126 paulson@14267  1127 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  1128 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  1129  using mod_div_equality2 [of n m] by arith  paulson@14267  1130 nipkow@15439  1131 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  1132  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  1133  apply simp  wenzelm@22718  1134  done  paulson@14267  1135 wenzelm@60758  1136 subsubsection \Quotient and Remainder\  paulson@14267  1137 haftmann@33340  1138 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  1139  "divmod_nat_rel b c (q, r)  haftmann@33340  1140  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  1141 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1142 haftmann@30923  1143 lemma div_mult1_eq:  haftmann@30923  1144  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  1145 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1146 haftmann@33340  1147 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  1148  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  1149  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  1150 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1151 paulson@14267  1152 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  1153 lemma div_add1_eq:  nipkow@25134  1154  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  1155 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1156 haftmann@33340  1157 lemma divmod_nat_rel_mult2_eq:  haftmann@60352  1158  assumes "divmod_nat_rel a b (q, r)"  haftmann@60352  1159  shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@60352  1160 proof -  lp15@60562  1161  { assume "r < b" and "0 < c"  haftmann@60352  1162  then have "b * (q mod c) + r < b * c"  haftmann@60352  1163  apply (cut_tac m = q and n = c in mod_less_divisor)  haftmann@60352  1164  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  haftmann@60352  1165  apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)  haftmann@60352  1166  apply (simp add: add_mult_distrib2)  haftmann@60352  1167  done  haftmann@60352  1168  then have "r + b * (q mod c) < b * c"  haftmann@60352  1169  by (simp add: ac_simps)  haftmann@60352  1170  } with assms show ?thesis  haftmann@60352  1171  by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])  haftmann@60352  1172 qed  lp15@60562  1173 blanchet@55085  1174 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"  huffman@47135  1175 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  1176 blanchet@55085  1177 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"  haftmann@57512  1178 by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  1179 haftmann@58786  1180 instance nat :: semiring_numeral_div  haftmann@58786  1181  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)  haftmann@58786  1182 paulson@14267  1183 wenzelm@60758  1184 subsubsection \Further Facts about Quotient and Remainder\  paulson@14267  1185 haftmann@58786  1186 lemma div_1 [simp]:  haftmann@58786  1187  "m div Suc 0 = m"  haftmann@58786  1188  using div_by_1 [of m] by simp  paulson@14267  1189 paulson@14267  1190 (* Monotonicity of div in first argument *)  haftmann@30923  1191 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  1192  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  1193 apply (case_tac "k=0", simp)  paulson@15251  1194 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  1195 apply (case_tac "n= k *)  paulson@14267  1199 apply (case_tac "m=k *)  nipkow@15439  1203 apply (simp add: div_geq diff_le_mono)  paulson@14267  1204 done  paulson@14267  1205 paulson@14267  1206 (* Antimonotonicity of div in second argument *)  paulson@14267  1207 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  1208 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  1217  prefer 2  paulson@14267  1218  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  1219 apply (rule le_trans, simp)  nipkow@15439  1220 apply (simp)  paulson@14267  1221 done  paulson@14267  1222 paulson@14267  1223 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  1224 apply (case_tac "n=0", simp)  paulson@14267  1225 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  1226 apply (rule div_le_mono2)  paulson@14267  1227 apply (simp_all (no_asm_simp))  paulson@14267  1228 done  paulson@14267  1229 wenzelm@22718  1230 (* Similar for "less than" *)  huffman@47138  1231 lemma div_less_dividend [simp]:  huffman@47138  1232  "\(1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  1233 apply (induct m rule: nat_less_induct)  paulson@14267  1234 apply (rename_tac "m")  paulson@14267  1235 apply (case_tac "mA fact for the mutilated chess board\  paulson@14267  1247 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"  paulson@14267  1248 apply (case_tac "n=0", simp)  paulson@15251  1249 apply (induct "m" rule: nat_less_induct)  paulson@14267  1250 apply (case_tac "Suc (na) Suc(na) *)  paulson@16796  1254 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  1255 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  1256 done  paulson@14267  1257 paulson@14267  1258 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  1259 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  1260 wenzelm@22718  1261 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  1262 paulson@14267  1263 (*Loses information, namely we also have rq. m = r + q * d"  haftmann@57514  1268 proof -  haftmann@57514  1269  from mod_div_equality obtain q where "q * d + m mod d = m" by blast  haftmann@57514  1270  with assms have "m = r + q * d" by simp  haftmann@57514  1271  then show ?thesis ..  haftmann@57514  1272 qed  paulson@14267  1273 nipkow@13152  1274 lemma split_div:  nipkow@13189  1275  "P(n div k :: nat) =  nipkow@13189  1276  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  1277  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1278 proof  nipkow@13189  1279  assume P: ?P  nipkow@13189  1280  show ?Q  nipkow@13189  1281  proof (cases)  nipkow@13189  1282  assume "k = 0"  haftmann@27651  1283  with P show ?Q by simp  nipkow@13189  1284  next  nipkow@13189  1285  assume not0: "k \ 0"  nipkow@13189  1286  thus ?Q  nipkow@13189  1287  proof (simp, intro allI impI)  nipkow@13189  1288  fix i j  nipkow@13189  1289  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  1290  show "P i"  nipkow@13189  1291  proof (cases)  wenzelm@22718  1292  assume "i = 0"  wenzelm@22718  1293  with n j P show "P i" by simp  nipkow@13189  1294  next  wenzelm@22718  1295  assume "i \ 0"  haftmann@57514  1296  with not0 n j P show "P i" by(simp add:ac_simps)  nipkow@13189  1297  qed  nipkow@13189  1298  qed  nipkow@13189  1299  qed  nipkow@13189  1300 next  nipkow@13189  1301  assume Q: ?Q  nipkow@13189  1302  show ?P  nipkow@13189  1303  proof (cases)  nipkow@13189  1304  assume "k = 0"  haftmann@27651  1305  with Q show ?P by simp  nipkow@13189  1306  next  nipkow@13189  1307  assume not0: "k \ 0"  nipkow@13189  1308  with Q have R: ?R by simp  nipkow@13189  1309  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1310  show ?P by simp  nipkow@13189  1311  qed  nipkow@13189  1312 qed  nipkow@13189  1313 berghofe@13882  1314 lemma split_div_lemma:  haftmann@26100  1315  assumes "0 < n"  haftmann@26100  1316  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  1317 proof  haftmann@26100  1318  assume ?rhs  haftmann@26100  1319  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  1320  then have A: "n * q \ m" by simp  haftmann@26100  1321  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  1322  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  1323  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  1324  with nq have "m < n + n * q" by simp  haftmann@26100  1325  then have B: "m < n * Suc q" by simp  haftmann@26100  1326  from A B show ?lhs ..  haftmann@26100  1327 next  haftmann@26100  1328  assume P: ?lhs  haftmann@33340  1329  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@57514  1330  unfolding divmod_nat_rel_def by (auto simp add: ac_simps)  haftmann@33340  1331  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  1332  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  1333  then show ?rhs by simp  haftmann@26100  1334 qed  berghofe@13882  1335 berghofe@13882  1336 theorem split_div':  berghofe@13882  1337  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  1338  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  1339  apply (case_tac "0 < n")  berghofe@13882  1340  apply (simp only: add: split_div_lemma)  haftmann@27651  1341  apply simp_all  berghofe@13882  1342  done  berghofe@13882  1343 nipkow@13189  1344 lemma split_mod:  nipkow@13189  1345  "P(n mod k :: nat) =  nipkow@13189  1346  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  1347  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1348 proof  nipkow@13189  1349  assume P: ?P  nipkow@13189  1350  show ?Q  nipkow@13189  1351  proof (cases)  nipkow@13189  1352  assume "k = 0"  haftmann@27651  1353  with P show ?Q by simp  nipkow@13189  1354  next  nipkow@13189  1355  assume not0: "k \ 0"  nipkow@13189  1356  thus ?Q  nipkow@13189  1357  proof (simp, intro allI impI)  nipkow@13189  1358  fix i j  nipkow@13189  1359  assume "n = k*i + j" "j < k"  haftmann@58786  1360  thus "P j" using not0 P by (simp add: ac_simps)  nipkow@13189  1361  qed  nipkow@13189  1362  qed  nipkow@13189  1363 next  nipkow@13189  1364  assume Q: ?Q  nipkow@13189  1365  show ?P  nipkow@13189  1366  proof (cases)  nipkow@13189  1367  assume "k = 0"  haftmann@27651  1368  with Q show ?P by simp  nipkow@13189  1369  next  nipkow@13189  1370  assume not0: "k \ 0"  nipkow@13189  1371  with Q have R: ?R by simp  nipkow@13189  1372  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1373  show ?P by simp  nipkow@13189  1374  qed  nipkow@13189  1375 qed  nipkow@13189  1376 berghofe@13882  1377 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  1378  using mod_div_equality [of m n] by arith  huffman@47138  1379 huffman@47138  1380 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  1381  using mod_div_equality [of m n] by arith  huffman@47138  1382 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  1383 noschinl@52398  1384 lemma div_eq_dividend_iff: "a \ 0 \ (a :: nat) div b = a \ b = 1"  noschinl@52398  1385  apply rule  noschinl@52398  1386  apply (cases "b = 0")  noschinl@52398  1387  apply simp_all  noschinl@52398  1388  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)  noschinl@52398  1389  done  noschinl@52398  1390 haftmann@22800  1391 wenzelm@60758  1392 subsubsection \An induction'' law for modulus arithmetic.\  paulson@14640  1393 paulson@14640  1394 lemma mod_induct_0:  paulson@14640  1395  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1396  and base: "P i" and i: "i(P 0)"  paulson@14640  1400  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1402  proof  paulson@14640  1403  fix k  paulson@14640  1404  show "?A k"  paulson@14640  1405  proof (induct k)  paulson@14640  1406  show "?A 0" by simp -- "by contradiction"  paulson@14640  1407  next  paulson@14640  1408  fix n  paulson@14640  1409  assume ih: "?A n"  paulson@14640  1410  show "?A (Suc n)"  paulson@14640  1411  proof (clarsimp)  wenzelm@22718  1412  assume y: "P (p - Suc n)"  wenzelm@22718  1413  have n: "Suc n < p"  wenzelm@22718  1414  proof (rule ccontr)  wenzelm@22718  1415  assume "\(Suc n < p)"  wenzelm@22718  1416  hence "p - Suc n = 0"  wenzelm@22718  1417  by simp  wenzelm@22718  1418  with y contra show "False"  wenzelm@22718  1419  by simp  wenzelm@22718  1420  qed  wenzelm@22718  1421  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1422  from p have "p - Suc n < p" by arith  wenzelm@22718  1423  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1424  by blast  wenzelm@22718  1425  show "False"  wenzelm@22718  1426  proof (cases "n=0")  wenzelm@22718  1427  case True  wenzelm@22718  1428  with z n2 contra show ?thesis by simp  wenzelm@22718  1429  next  wenzelm@22718  1430  case False  wenzelm@22718  1431  with p have "p-n < p" by arith  wenzelm@22718  1432  with z n2 False ih show ?thesis by simp  wenzelm@22718  1433  qed  paulson@14640  1434  qed  paulson@14640  1435  qed  paulson@14640  1436  qed  paulson@14640  1437  moreover  paulson@14640  1438  from i obtain k where "0 i+k=p"  paulson@14640  1439  by (blast dest: less_imp_add_positive)  paulson@14640  1440  hence "0 i=p-k" by auto  paulson@14640  1441  moreover  paulson@14640  1442  note base  paulson@14640  1443  ultimately  paulson@14640  1444  show "False" by blast  paulson@14640  1445 qed  paulson@14640  1446 paulson@14640  1447 lemma mod_induct:  paulson@14640  1448  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1449  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1456  proof (induct j)  paulson@14640  1457  from step base i show "?A 0"  wenzelm@22718  1458  by (auto elim: mod_induct_0)  paulson@14640  1459  next  paulson@14640  1460  fix k  paulson@14640  1461  assume ih: "?A k"  paulson@14640  1462  show "?A (Suc k)"  paulson@14640  1463  proof  wenzelm@22718  1464  assume suc: "Suc k < p"  wenzelm@22718  1465  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1490 proof -  boehmes@35815  1491  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1492  moreover have "m mod 2 < 2" by simp  haftmann@33296  1493  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1494  then show ?thesis by auto  haftmann@33296  1495 qed  haftmann@33296  1496 wenzelm@60758  1497 text\These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1498  at least three Sucs, since two and fewer are rewritten back to Suc again!  wenzelm@60758  1499  We already have some rules to simplify operands smaller than 3.\  haftmann@33296  1500 haftmann@33296  1501 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1502 by (simp add: Suc3_eq_add_3)  haftmann@33296  1503 haftmann@33296  1504 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1505 by (simp add: Suc3_eq_add_3)  haftmann@33296  1506 haftmann@33296  1507 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1508 by (simp add: Suc3_eq_add_3)  haftmann@33296  1509 haftmann@33296  1510 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1511 by (simp add: Suc3_eq_add_3)  haftmann@33296  1512 huffman@47108  1513 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1514 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1515 lp15@60562  1516 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1517 apply (induct "m")  haftmann@33361  1518 apply (simp_all add: mod_Suc)  haftmann@33361  1519 done  haftmann@33361  1520 huffman@47108  1521 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1522 huffman@47138  1523 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1524 by (simp add: div_le_mono)  haftmann@33361  1525 haftmann@33361  1526 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1527 by (cases n) simp_all  haftmann@33361  1528 boehmes@35815  1529 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1530 proof -  boehmes@35815  1531  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  lp15@60562  1532  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1533 qed  haftmann@33361  1534 haftmann@33361  1535 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1536 proof -  haftmann@33361  1537  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  lp15@60562  1538  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1539  finally show ?thesis .  haftmann@33361  1540 qed  haftmann@33361  1541 haftmann@33361  1542 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  lp15@60562  1543 apply (subst mod_Suc [of m])  lp15@60562  1544 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1545 done  haftmann@33361  1546 huffman@47108  1547 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1548  fixes n :: nat  huffman@47108  1549  shows "n mod 2 \ 0 \ n mod 2 = 1"  haftmann@58786  1550  by (fact not_mod_2_eq_0_eq_1)  lp15@60562  1551 haftmann@58778  1552 lemma even_Suc_div_two [simp]:  haftmann@58778  1553  "even n \ Suc n div 2 = n div 2"  haftmann@58778  1554  using even_succ_div_two [of n] by simp  lp15@60562  1555 haftmann@58778  1556 lemma odd_Suc_div_two [simp]:  haftmann@58778  1557  "odd n \ Suc n div 2 = Suc (n div 2)"  haftmann@58778  1558  using odd_succ_div_two [of n] by simp  haftmann@58778  1559 haftmann@58834  1560 lemma odd_two_times_div_two_nat [simp]:  haftmann@60352  1561  assumes "odd n"  haftmann@60352  1562  shows "2 * (n div 2) = n - (1 :: nat)"  haftmann@60352  1563 proof -  haftmann@60352  1564  from assms have "2 * (n div 2) + 1 = n"  haftmann@60352  1565  by (rule odd_two_times_div_two_succ)  haftmann@60352  1566  then have "Suc (2 * (n div 2)) - 1 = n - 1"  haftmann@60352  1567  by simp  haftmann@60352  1568  then show ?thesis  haftmann@60352  1569  by simp  haftmann@60352  1570 qed  haftmann@58778  1571 haftmann@58778  1572 lemma parity_induct [case_names zero even odd]:  haftmann@58778  1573  assumes zero: "P 0"  haftmann@58778  1574  assumes even: "\n. P n \ P (2 * n)"  haftmann@58778  1575  assumes odd: "\n. P n \ P (Suc (2 * n))"  haftmann@58778  1576  shows "P n"  haftmann@58778  1577 proof (induct n rule: less_induct)  haftmann@58778  1578  case (less n)  haftmann@58778  1579  show "P n"  haftmann@58778  1580  proof (cases "n = 0")  haftmann@58778  1581  case True with zero show ?thesis by simp  haftmann@58778  1582  next  haftmann@58778  1583  case False  haftmann@58778  1584  with less have hyp: "P (n div 2)" by simp  haftmann@58778  1585  show ?thesis  haftmann@58778  1586  proof (cases "even n")  haftmann@58778  1587  case True  haftmann@58778  1588  with hyp even [of "n div 2"] show ?thesis  haftmann@58834  1589  by simp  haftmann@58778  1590  next  haftmann@58778  1591  case False  lp15@60562  1592  with hyp odd [of "n div 2"] show ?thesis  haftmann@58834  1593  by simp  haftmann@58778  1594  qed  haftmann@58778  1595  qed  haftmann@58778  1596 qed  haftmann@58778  1597 haftmann@33361  1598 wenzelm@60758  1599 subsection \Division on @{typ int}\  haftmann@33361  1600 haftmann@33361  1601 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  wenzelm@60758  1602  --\definition of quotient and remainder\  huffman@47139  1603  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1604  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1605 wenzelm@60758  1606 text \  haftmann@53067  1607  The following algorithmic devlopment actually echos what has already  haftmann@53067  1608  been developed in class @{class semiring_numeral_div}. In the long  haftmann@53067  1609  run it seems better to derive division on @{typ int} just from  haftmann@53067  1610  division on @{typ nat} and instantiate @{class semiring_numeral_div}  haftmann@53067  1611  accordingly.  wenzelm@60758  1612 \  haftmann@53067  1613 haftmann@33361  1614 definition adjust :: "int \ int \ int \ int \ int" where  wenzelm@60758  1615  --\for the division algorithm\  huffman@47108  1616  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1617  else (2 * q, r))"  haftmann@33361  1618 wenzelm@60758  1619 text\algorithm for the case @{text "a\0, b>0"}\  haftmann@33361  1620 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1621  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1622  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1623 by auto  haftmann@33361  1624 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1625  (auto simp add: mult_2)  haftmann@33361  1626 wenzelm@60758  1627 text\algorithm for the case @{text "a<0, b>0"}\  haftmann@33361  1628 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1629  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1630  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1631 by auto  haftmann@33361  1632 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1633  (auto simp add: mult_2)  haftmann@33361  1634 wenzelm@60758  1635 text\algorithm for the general case @{term "b\0"}\  haftmann@33361  1636 haftmann@33361  1637 definition divmod_int :: "int \ int \ int \ int" where  wenzelm@60758  1638  --\The full division algorithm considers all possible signs for a, b  lp15@60562  1639  including the special case @{text "a=0, b<0"} because  wenzelm@60758  1640  @{term negDivAlg} requires @{term "a<0"}.\  haftmann@33361  1641  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1642  else if a = 0 then (0, 0)  huffman@46560  1643  else apsnd uminus (negDivAlg (-a) (-b))  lp15@60562  1644  else  haftmann@33361  1645  if 0 < b then negDivAlg a b  huffman@46560  1646  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1647 haftmann@60429  1648 instantiation int :: ring_div  haftmann@33361  1649 begin  haftmann@33361  1650 haftmann@60352  1651 definition divide_int where  haftmann@60429  1652  div_int_def: "a div b = fst (divmod_int a b)"  haftmann@60352  1653 haftmann@60352  1654 definition mod_int where  haftmann@60352  1655  "a mod b = snd (divmod_int a b)"  haftmann@60352  1656 huffman@46551  1657 lemma fst_divmod_int [simp]:  huffman@46551  1658  "fst (divmod_int a b) = a div b"  huffman@46551  1659  by (simp add: div_int_def)  huffman@46551  1660 huffman@46551  1661 lemma snd_divmod_int [simp]:  huffman@46551  1662  "snd (divmod_int a b) = a mod b"  huffman@46551  1663  by (simp add: mod_int_def)  huffman@46551  1664 haftmann@33361  1665 lemma divmod_int_mod_div:  haftmann@33361  1666  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1667  by (simp add: prod_eq_iff)  haftmann@33361  1668 wenzelm@60758  1669 text\  haftmann@33361  1670 Here is the division algorithm in ML:  haftmann@33361  1671 haftmann@33361  1672 \begin{verbatim}  haftmann@33361  1673  fun posDivAlg (a,b) =  haftmann@33361  1674  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1677  end  haftmann@33361  1678 haftmann@33361  1679  fun negDivAlg (a,b) =  haftmann@33361  1680  if 0\a+b then (~1,a+b)  haftmann@33361  1681  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1682  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1683  end;  haftmann@33361  1684 haftmann@33361  1685  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1686 lp15@60562  1687  fun divmod (a,b) = if 0\a then  lp15@60562  1688  if b>0 then posDivAlg (a,b)  haftmann@33361  1689  else if a=0 then (0,0)  haftmann@33361  1690  else negateSnd (negDivAlg (~a,~b))  lp15@60562  1691  else  haftmann@33361  1692  if 0  wenzelm@60758  1696 wenzelm@60758  1697 wenzelm@60758  1698 subsubsection \Uniqueness and Monotonicity of Quotients and Remainders\  haftmann@33361  1699 haftmann@33361  1700 lemma unique_quotient_lemma:  lp15@60562  1701  "[| b*q' + r' \ b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1702  ==> q' \ (q::int)"  haftmann@33361  1703 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1704  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1705 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1706 apply (erule_tac [2] order_le_less_trans)  webertj@49962  1707  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1708 apply (subgoal_tac "b * q' < b * (1 + q) ")  webertj@49962  1709  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1710 apply (simp add: mult_less_cancel_left)  haftmann@33361  1711 done  haftmann@33361  1712 haftmann@33361  1713 lemma unique_quotient_lemma_neg:  lp15@60562  1714  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1715  ==> q \ (q'::int)"  lp15@60562  1716 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1717  auto)  haftmann@33361  1718 haftmann@33361  1719 lemma unique_quotient:  lp15@60562  1720  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1721  ==> q = q'"  haftmann@33361  1722 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1723 apply (blast intro: order_antisym  lp15@60562  1724  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1725  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1726 done  haftmann@33361  1727 haftmann@33361  1728 haftmann@33361  1729 lemma unique_remainder:  lp15@60562  1730  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1731  ==> r = r'"  haftmann@33361  1732 apply (subgoal_tac "q = q'")  haftmann@33361  1733  apply (simp add: divmod_int_rel_def)  haftmann@33361  1734 apply (blast intro: unique_quotient)  haftmann@33361  1735 done  haftmann@33361  1736 haftmann@33361  1737 wenzelm@60758  1738 subsubsection \Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends\  wenzelm@60758  1739 wenzelm@60758  1740 text\And positive divisors\  haftmann@33361  1741 haftmann@33361  1742 lemma adjust_eq [simp]:  lp15@60562  1743  "adjust b (q, r) =  lp15@60562  1744  (let diff = r - b in  lp15@60562  1745  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1746  else (2*q, r))"  huffman@47108  1747  by (simp add: Let_def adjust_def)  haftmann@33361  1748 haftmann@33361  1749 declare posDivAlg.simps [simp del]  haftmann@33361  1750 wenzelm@60758  1751 text\use with a simproc to avoid repeatedly proving the premise\  haftmann@33361  1752 lemma posDivAlg_eqn:  lp15@60562  1753  "0 < b ==>  haftmann@33361  1754  posDivAlg a b = (if aCorrectness of @{term posDivAlg}: it computes quotients correctly\  haftmann@33361  1758 theorem posDivAlg_correct:  haftmann@33361  1759  assumes "0 \ a" and "0 < b"  haftmann@33361  1760  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1761  using assms  wenzelm@41550  1762  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1763  apply auto  wenzelm@41550  1764  apply (simp add: divmod_int_rel_def)  webertj@49962  1765  apply (subst posDivAlg_eqn, simp add: distrib_left)  wenzelm@41550  1766  apply (case_tac "a < b")  wenzelm@41550  1767  apply simp_all  wenzelm@41550  1768  apply (erule splitE)  haftmann@57514  1769  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)  wenzelm@41550  1770  done  haftmann@33361  1771 haftmann@33361  1772 wenzelm@60758  1773 subsubsection \Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends\  wenzelm@60758  1774 wenzelm@60758  1775 text\And positive divisors\  haftmann@33361  1776 haftmann@33361  1777 declare negDivAlg.simps [simp del]  haftmann@33361  1778 wenzelm@60758  1779 text\use with a simproc to avoid repeatedly proving the premise\  haftmann@33361  1780 lemma negDivAlg_eqn:  lp15@60562  1781  "0 < b ==>  lp15@60562  1782  negDivAlg a b =  haftmann@33361  1783  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1784 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1785 haftmann@33361  1786 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1787  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1788 lemma negDivAlg_correct:  haftmann@33361  1789  assumes "a < 0" and "b > 0"  haftmann@33361  1790  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1791  using assms  wenzelm@41550  1792  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1793  apply (auto simp add: linorder_not_le)  wenzelm@41550  1794  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1795  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1796  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1797  apply simp_all  wenzelm@41550  1798  apply (erule splitE)  haftmann@57514  1799  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)  wenzelm@41550  1800  done  haftmann@33361  1801 haftmann@33361  1802 wenzelm@60758  1803 subsubsection \Existence Shown by Proving the Division Algorithm to be Correct\  haftmann@33361  1804 haftmann@33361  1805 (*the case a=0*)  huffman@47139  1806 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1807 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1808 haftmann@33361  1809 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1810 by (subst posDivAlg.simps, auto)  haftmann@33361  1811 huffman@47139  1812 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1813 by (subst posDivAlg.simps, auto)  huffman@47139  1814 haftmann@58410  1815 lemma negDivAlg_minus1 [simp]: "negDivAlg (- 1) b = (- 1, b - 1)"  haftmann@33361  1816 by (subst negDivAlg.simps, auto)  haftmann@33361  1817 huffman@46560  1818 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1819 by (auto simp add: divmod_int_rel_def)  huffman@47139  1820 huffman@47139  1821 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1822 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1823 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1824  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1825 huffman@47141  1826 lemma divmod_int_unique:  lp15@60562  1827  assumes "divmod_int_rel a b qr"  huffman@47141  1828  shows "divmod_int a b = qr"  huffman@47141  1829  using assms divmod_int_correct [of a b]  huffman@47141  1830  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1831  by (metis pair_collapse)  huffman@47141  1832 huffman@47141  1833 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1834  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1835 huffman@47141  1836 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1837  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1838 huffman@47141  1839 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1840  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1841 haftmann@60429  1842 instance  huffman@47141  1843 proof  huffman@47141  1844  fix a b :: int  huffman@47141  1845  show "a div b * b + a mod b = a"  huffman@47141  1846  using divmod_int_rel_div_mod [of a b]  haftmann@57512  1847  unfolding divmod_int_rel_def by (simp add: mult.commute)  huffman@47141  1848 next  huffman@47141  1849  fix a b c :: int  huffman@47141  1850  assume "b \ 0"  huffman@47141  1851  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1852  using divmod_int_rel_div_mod [of a b]  huffman@47141  1853  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1854  thus "(a + c * b) div b = c + a div b"  huffman@47141  1855  by (rule div_int_unique)  huffman@47141  1856 next  huffman@47141  1857  fix a b c :: int  huffman@47141  1858  assume "c \ 0"  huffman@47141  1859  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1860  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1861  unfolding divmod_int_rel_def  huffman@47141  1862  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1863  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1864  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1865  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1866  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1867  thus "(c * a) div (c * b) = a div b"  huffman@47141  1868  by (rule div_int_unique)  huffman@47141  1869 next  huffman@47141  1870  fix a :: int show "a div 0 = 0"  huffman@47141  1871  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1872 next  huffman@47141  1873  fix a :: int show "0 div a = 0"  huffman@47141  1874  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1875 qed  huffman@47141  1876 haftmann@60429  1877 end  haftmann@60429  1878 haftmann@60517  1879 lemma is_unit_int:  haftmann@60517  1880  "is_unit (k::int) \ k = 1 \ k = - 1"  haftmann@60517  1881  by auto  haftmann@60517  1882 haftmann@60685  1883 instantiation int :: normalization_semidom  haftmann@60685  1884 begin  haftmann@60685  1885 haftmann@60685  1886 definition normalize_int  haftmann@60685  1887  where [simp]: "normalize = (abs :: int \ int)"  haftmann@60685  1888 haftmann@60685  1889 definition unit_factor_int  haftmann@60685  1890  where [simp]: "unit_factor = (sgn :: int \ int)"  haftmann@60685  1891 haftmann@60685  1892 instance  haftmann@60685  1893 proof  haftmann@60685  1894  fix k :: int  haftmann@60685  1895  assume "k \ 0"  haftmann@60685  1896  then have "\sgn k\ = 1"  haftmann@60685  1897  by (cases "0::int" k rule: linorder_cases) simp_all  haftmann@60685  1898  then show "is_unit (unit_factor k)"  haftmann@60685  1899  by simp  haftmann@60685  1900 qed (simp_all add: sgn_times mult_sgn_abs)  haftmann@60685  1901   haftmann@60685  1902 end  haftmann@60685  1903   wenzelm@60758  1904 text\Basic laws about division and remainder\  haftmann@33361  1905 haftmann@33361  1906 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1907  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1908 wenzelm@60758  1909 text \Tool setup\  wenzelm@60758  1910 wenzelm@60758  1911 ML \  wenzelm@43594  1912 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1913 (  haftmann@60352  1914  val div_name = @{const_name Rings.divide};  haftmann@33361  1915  val mod_name = @{const_name mod};  haftmann@33361  1916  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1917  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1918  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1919 huffman@47165  1920  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1921 lp15@60562  1922  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@59556  1923  (@{thm diff_conv_add_uminus} :: @{thms add_0_left add_0_right} @ @{thms ac_simps}))  wenzelm@41550  1924 )  wenzelm@60758  1925 \  wenzelm@60758  1926 wenzelm@60758  1927 simproc_setup cancel_div_mod_int ("(k::int) + l") = \K Cancel_Div_Mod_Int.proc\  wenzelm@43594  1928 huffman@47141  1929 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1930  using divmod_int_correct [of a b]  huffman@47141  1931  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1932 wenzelm@45607  1933 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1934  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1935 huffman@47141  1936 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1937  using divmod_int_correct [of a b]  huffman@47141  1938  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1939 wenzelm@45607  1940 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1941  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1942 haftmann@33361  1943 wenzelm@60758  1944 subsubsection \General Properties of div and mod\  haftmann@33361  1945 haftmann@33361  1946 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1947 apply (rule div_int_unique)  haftmann@33361  1948 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1949 done  haftmann@33361  1950 haftmann@33361  1951 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1952 apply (rule div_int_unique)  haftmann@33361  1953 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1954 done  haftmann@33361  1955 haftmann@33361  1956 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1957 apply (rule div_int_unique)  haftmann@33361  1958 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1959 done  haftmann@33361  1960 haftmann@33361  1961 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1962 haftmann@33361  1963 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1964 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1965 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1966 done  haftmann@33361  1967 haftmann@33361  1968 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  1969 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1970 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1971 done  haftmann@33361  1972 haftmann@33361  1973 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  1974 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  1975 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1976 done  haftmann@33361  1977 wenzelm@60758  1978 text\There is no @{text mod_neg_pos_trivial}.\  wenzelm@60758  1979 wenzelm@60758  1980 wenzelm@60758  1981 subsubsection \Laws for div and mod with Unary Minus\  haftmann@33361  1982 haftmann@33361  1983 lemma zminus1_lemma:  huffman@47139  1984  "divmod_int_rel a b (q, r) ==> b \ 0  lp15@60562  1985  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1986  if r=0 then 0 else b-r)"  haftmann@33361  1987 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1988 haftmann@33361  1989 haftmann@33361  1990 lemma zdiv_zminus1_eq_if:  lp15@60562  1991  "b \ (0::int)  lp15@60562  1992  ==> (-a) div b =  haftmann@33361  1993  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  1994 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  1995 haftmann@33361  1996 lemma zmod_zminus1_eq_if:  haftmann@33361  1997  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1998 apply (case_tac "b = 0", simp)  huffman@47140  1999 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  2000 done  haftmann@33361  2001 haftmann@33361  2002 lemma zmod_zminus1_not_zero:  haftmann@33361  2003  fixes k l :: int  haftmann@33361  2004  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  2005  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  2006 haftmann@33361  2007 lemma zdiv_zminus2_eq_if:  lp15@60562  2008  "b \ (0::int)  lp15@60562  2009  ==> a div (-b) =  haftmann@33361  2010  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  2011 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  2012 haftmann@33361  2013 lemma zmod_zminus2_eq_if:  haftmann@33361  2014  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  2015 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  2016 haftmann@33361  2017 lemma zmod_zminus2_not_zero:  haftmann@33361  2018  fixes k l :: int  haftmann@33361  2019  shows "k mod - l \ 0 \ k mod l \ 0"  lp15@60562  2020  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  2021 haftmann@33361  2022 wenzelm@60758  2023 subsubsection \Computation of Division and Remainder\  haftmann@33361  2024 haftmann@33361  2025 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  2026 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2027 haftmann@33361  2028 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  2029 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2030 wenzelm@60758  2031 text\a positive, b positive\  haftmann@33361  2032 haftmann@33361  2033 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  2034 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2035 haftmann@33361  2036 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  2037 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2038 wenzelm@60758  2039 text\a negative, b positive\  haftmann@33361  2040 haftmann@33361  2041 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  2042 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2043 haftmann@33361  2044 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  2045 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2046 wenzelm@60758  2047 text\a positive, b negative\  haftmann@33361  2048 haftmann@33361  2049 lemma div_pos_neg:  huffman@46560  2050  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  2051 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2052 haftmann@33361  2053 lemma mod_pos_neg:  huffman@46560  2054  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  2055 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2056 wenzelm@60758  2057 text\a negative, b negative\  haftmann@33361  2058 haftmann@33361  2059 lemma div_neg_neg:  huffman@46560  2060  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  2061 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2062 haftmann@33361  2063 lemma mod_neg_neg:  huffman@46560  2064  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  2065 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2066 wenzelm@60758  2067 text \Simplify expresions in which div and mod combine numerical constants\  haftmann@33361  2068 huffman@45530  2069 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  2070  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  2071 huffman@45530  2072 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  2073  by (rule div_int_unique [of a b q r],  bulwahn@46552  2074  simp add: divmod_int_rel_def)  huffman@45530  2075 huffman@45530  2076 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  2077  by (rule mod_int_unique [of a b q r],  bulwahn@46552  2078  simp add: divmod_int_rel_def)  huffman@45530  2079 huffman@45530  2080 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  2081  by (rule mod_int_unique [of a b q r],  bulwahn@46552  2082  simp add: divmod_int_rel_def)  huffman@45530  2083 wenzelm@60758  2084 text \  haftmann@53069  2085  numeral simprocs -- high chance that these can be replaced  haftmann@53069  2086  by divmod algorithm from @{class semiring_numeral_div}  wenzelm@60758  2087 \  wenzelm@60758  2088 wenzelm@60758  2089 ML \  haftmann@33361  2090 local  huffman@45530  2091  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  2092  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  2093  val times = @{term "times :: int \ int \ int"}  huffman@45530  2094  val zero = @{term "0 :: int"}  huffman@45530  2095  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  2096  val le = @{term "op \ :: int \ int \ bool"}  haftmann@54489  2097  val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]  wenzelm@58847  2098  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  wenzelm@58847  2099  (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));  wenzelm@51717  2100  fun binary_proc proc ctxt ct =  haftmann@33361  2101  (case Thm.term_of ct of  haftmann@33361  2102  _ $t$ u =>  wenzelm@59058  2103  (case try (apply2 ((snd o HOLogic.dest_number))) (t, u) of  wenzelm@51717  2104  SOME args => proc ctxt args  haftmann@33361  2105  | NONE => NONE)  haftmann@33361  2106  | _ => NONE);  haftmann@33361  2107 in  huffman@45530  2108  fun divmod_proc posrule negrule =  huffman@45530  2109  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  wenzelm@59058  2110  if b = 0 then NONE  wenzelm@59058  2111  else  wenzelm@59058  2112  let  wenzelm@59058  2113  val (q, r) = apply2 mk_number (Integer.div_mod a b)  wenzelm@59058  2114  val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)  wenzelm@59058  2115  val (goal2, goal3, rule) =  wenzelm@59058  2116  if b > 0  wenzelm@59058  2117  then (le $zero$ r, less $r$ u, posrule RS eq_reflection)  wenzelm@59058  2118  else (le $r$ zero, less $u$ r, negrule RS eq_reflection)  wenzelm@59058  2119  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  2120 end  wenzelm@60758  2121 \  haftmann@33361  2122 huffman@47108  2123 simproc_setup binary_int_div  huffman@47108  2124  ("numeral m div numeral n :: int" |  haftmann@54489  2125  "numeral m div - numeral n :: int" |  haftmann@54489  2126  "- numeral m div numeral n :: int" |  haftmann@54489  2127  "- numeral m div - numeral n :: int") =  wenzelm@60758  2128  \K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq})\  haftmann@33361  2129 huffman@47108  2130 simproc_setup binary_int_mod  huffman@47108  2131  ("numeral m mod numeral n :: int" |  haftmann@54489  2132  "numeral m mod - numeral n :: int" |  haftmann@54489  2133  "- numeral m mod numeral n :: int" |  haftmann@54489  2134  "- numeral m mod - numeral n :: int") =  wenzelm@60758  2135  \K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq})\  haftmann@33361  2136 huffman@47108  2137 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  2138  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  2139 huffman@47108  2140 lemmas negDivAlg_eqn_numeral [simp] =  haftmann@54489  2141  negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w  haftmann@33361  2142 haftmann@33361  2143 wenzelm@60758  2144 text \Special-case simplification: @{text "\1 div z"} and @{text "\1 mod z"}\  haftmann@55172  2145 haftmann@55172  2146 lemma [simp]:  haftmann@55172  2147  shows div_one_bit0: "1 div numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  2148  and mod_one_bit0: "1 mod numeral (Num.Bit0 v) = (1 :: int)"  wenzelm@55439  2149  and div_one_bit1: "1 div numeral (Num.Bit1 v) = (0 :: int)"  wenzelm@55439  2150  and mod_one_bit1: "1 mod numeral (Num.Bit1 v) = (1 :: int)"  wenzelm@55439  2151  and div_one_neg_numeral: "1 div - numeral v = (- 1 :: int)"  wenzelm@55439  2152  and mod_one_neg_numeral: "1 mod - numeral v = (1 :: int) - numeral v"  haftmann@55172  2153  by (simp_all del: arith_special  haftmann@55172  2154  add: div_pos_pos mod_pos_pos div_pos_neg mod_pos_neg posDivAlg_eqn)  wenzelm@55439  2155 haftmann@55172  2156 lemma [simp]:  haftmann@55172  2157  shows div_neg_one_numeral: "- 1 div numeral v = (- 1 :: int)"  haftmann@55172  2158  and mod_neg_one_numeral: "- 1 mod numeral v = numeral v - (1 :: int)"  haftmann@55172  2159  and div_neg_one_neg_bit0: "- 1 div - numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  2160  and mod_neg_one_neb_bit0: "- 1 mod - numeral (Num.Bit0 v) = (- 1 :: int)"  haftmann@55172  2161  and div_neg_one_neg_bit1: "- 1 div - numeral (Num.Bit1 v) = (0 :: int)"  haftmann@55172  2162  and mod_neg_one_neb_bit1: "- 1 mod - numeral (Num.Bit1 v) = (- 1 :: int)"  haftmann@55172  2163  by (simp_all add: div_eq_minus1 zmod_minus1)  haftmann@33361  2164 haftmann@33361  2165 wenzelm@60758  2166 subsubsection \Monotonicity in the First Argument (Dividend)\  haftmann@33361  2167 haftmann@33361  2168 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  2169 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2170 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  2171 apply (rule unique_quotient_lemma)  haftmann@33361  2172 apply (erule subst)  haftmann@33361  2173 apply (erule subst, simp_all)  haftmann@33361  2174 done  haftmann@33361  2175 haftmann@33361  2176 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  2177 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2178 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  2179 apply (rule unique_quotient_lemma_neg)  haftmann@33361  2180 apply (erule subst)  haftmann@33361  2181 apply (erule subst, simp_all)  haftmann@33361  2182 done  haftmann@33361  2183 haftmann@33361  2184 wenzelm@60758  2185 subsubsection \Monotonicity in the Second Argument (Divisor)\  haftmann@33361  2186 haftmann@33361  2187 lemma q_pos_lemma:  haftmann@33361  2188  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  2189 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  2190  apply (simp add: zero_less_mult_iff)  webertj@49962  2191 apply (simp add: distrib_left)  haftmann@33361  2192 done  haftmann@33361  2193 haftmann@33361  2194 lemma zdiv_mono2_lemma:  lp15@60562  2195  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  lp15@60562  2196  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  2197  ==> q \ (q'::int)"  lp15@60562  2198 apply (frule q_pos_lemma, assumption+)  haftmann@33361  2199 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  2200  apply (simp add: mult_less_cancel_left)  haftmann@33361  2201 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  2202  prefer 2 apply simp  webertj@49962  2203 apply (simp (no_asm_simp) add: distrib_left)  haftmann@57512  2204 apply (subst add.commute, rule add_less_le_mono, arith)  haftmann@33361  2205 apply (rule mult_right_mono, auto)  haftmann@33361  2206 done  haftmann@33361  2207 haftmann@33361  2208 lemma zdiv_mono2:  haftmann@33361  2209  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  2210 apply (subgoal_tac "b \ 0")  haftmann@33361  2211  prefer 2 apply arith  haftmann@33361  2212 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2213 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2214 apply (rule zdiv_mono2_lemma)  haftmann@33361  2215 apply (erule subst)  haftmann@33361  2216 apply (erule subst, simp_all)  haftmann@33361  2217 done  haftmann@33361  2218 haftmann@33361  2219 lemma q_neg_lemma:  haftmann@33361  2220  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  2221 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  2222  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  2223 done  haftmann@33361  2224 haftmann@33361  2225 lemma zdiv_mono2_neg_lemma:  lp15@60562  2226  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  lp15@60562  2227  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  2228  ==> q' \ (q::int)"  lp15@60562  2229 apply (frule q_neg_lemma, assumption+)  haftmann@33361  2230 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  2231  apply (simp add: mult_less_cancel_left)  webertj@49962  2232 apply (simp add: distrib_left)  haftmann@33361  2233 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  2234  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  2235 done  haftmann@33361  2236 haftmann@33361  2237 lemma zdiv_mono2_neg:  haftmann@33361  2238  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  2239 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2240 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2241 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  2242 apply (erule subst)  haftmann@33361  2243 apply (erule subst, simp_all)  haftmann@33361  2244 done  haftmann@33361  2245 haftmann@33361  2246 wenzelm@60758  2247 subsubsection \More Algebraic Laws for div and mod\  wenzelm@60758  2248 wenzelm@60758  2249 text\proving (a*b) div c = a * (b div c) + a * (b mod c)\  haftmann@33361  2250 haftmann@33361  2251 lemma zmult1_lemma:  lp15@60562  2252  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  2253  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@57514  2254 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)  haftmann@33361  2255 haftmann@33361  2256 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  2257 apply (case_tac "c = 0", simp)  huffman@47140  2258 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  2259 done  haftmann@33361  2260 wenzelm@60758  2261 text\proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\  haftmann@33361  2262 haftmann@33361  2263 lemma zadd1_lemma:  lp15@60562  2264  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  2265  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  webertj@49962  2266 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)  haftmann@33361  2267 haftmann@33361  2268 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  2269 lemma zdiv_zadd1_eq:  haftmann@33361  2270  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  2271 apply (case_tac "c = 0", simp)  huffman@47140  2272 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  2273 done  haftmann@33361  2274 haftmann@33361  2275 lemma posDivAlg_div_mod:  haftmann@33361  2276  assumes "k \ 0"  haftmann@33361  2277  and "l \ 0"  haftmann@33361  2278  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  2279 proof (cases "l = 0")  haftmann@33361  2280  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  2281 next  haftmann@33361  2282  case False with assms posDivAlg_correct  haftmann@33361  2283  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  2284  by simp  huffman@47140  2285  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2286  show ?thesis by simp  haftmann@33361  2287 qed  haftmann@33361  2288 haftmann@33361  2289 lemma negDivAlg_div_mod:  haftmann@33361  2290  assumes "k < 0"  haftmann@33361  2291  and "l > 0"  haftmann@33361  2292  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  2293 proof -  haftmann@33361  2294  from assms have "l \ 0" by simp  haftmann@33361  2295  from assms negDivAlg_correct  haftmann@33361  2296  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  2297  by simp  huffman@47140  2298  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2299  show ?thesis by simp  haftmann@33361  2300 qed  haftmann@33361  2301 haftmann@33361  2302 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  2303 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  2304 haftmann@33361  2305 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  2306 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  2307 huffman@47108  2308 lemma zmod_zdiv_equality':  huffman@47108  2309  "(m\int) mod n = m - (m div n) * n"  huffman@47141  2310  using mod_div_equality [of m n] by arith  huffman@47108  2311 haftmann@33361  2312 wenzelm@60758  2313 subsubsection \Proving @{term "a div (b * c) = (a div b) div c"}\  haftmann@33361  2314 haftmann@33361  2315 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  2316  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  2317  to cause particular problems.*)  haftmann@33361  2318 wenzelm@60758  2319 text\first, four lemmas to bound the remainder for the cases b<0 and b>0\  haftmann@33361  2320 blanchet@55085  2321 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b * c < b * (q mod c) + r"  haftmann@33361  2322 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  2323  apply (simp add: algebra_simps)  haftmann@33361  2324 apply (rule order_le_less_trans)  haftmann@33361  2325  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  2326  apply (rule mult_left_mono_neg)  huffman@35216  2327  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  2328  apply (simp)  haftmann@33361  2329 apply (simp)  haftmann@33361  2330 done  haftmann@33361  2331 haftmann@33361  2332 lemma zmult2_lemma_aux2:  haftmann@33361  2333  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  2334 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  2335  apply arith  haftmann@33361  2336 apply (simp add: mult_le_0_iff)  haftmann@33361  2337 done  haftmann@33361  2338 haftmann@33361  2339 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  2340 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  2341 apply arith  haftmann@33361  2342 apply (simp add: zero_le_mult_iff)  haftmann@33361  2343 done  haftmann@33361  2344 haftmann@33361  2345 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  2346 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  2347  apply (simp add: right_diff_distrib)  haftmann@33361  2348 apply (rule order_less_le_trans)  haftmann@33361  2349  apply (erule mult_strict_right_mono)  haftmann@33361  2350  apply (rule_tac [2] mult_left_mono)  haftmann@33361  2351  apply simp  huffman@35216  2352  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  2353 apply simp  haftmann@33361  2354 done  haftmann@33361  2355 lp15@60562  2356 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  2357  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@57514  2358 by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff  lp15@60562  2359  zero_less_mult_iff distrib_left [symmetric]  huffman@47139  2360  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  2361 haftmann@53068  2362 lemma zdiv_zmult2_eq:  haftmann@53068  2363  fixes a b c :: int  haftmann@53068  2364  shows "0 \ c \ a div (b * c) = (a div b) div c"  haftmann@33361  2365 apply (case_tac "b = 0", simp)  haftmann@53068  2366 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  2367 done  haftmann@33361  2368 haftmann@33361  2369 lemma zmod_zmult2_eq:  haftmann@53068  2370  fixes a b c :: int  haftmann@53068  2371  shows "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@33361  2372 apply (case_tac "b = 0", simp)  haftmann@53068  2373 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  2374 done  haftmann@33361  2375 huffman@47108  2376 lemma div_pos_geq:  huffman@47108  2377  fixes k l :: int  huffman@47108  2378  assumes "0 < l" and "l \ k"  huffman@47108  2379  shows "k div l = (k - l) div l + 1"  huffman@47108  2380 proof -  huffman@47108  2381  have "k = (k - l) + l" by simp  huffman@47108  2382  then obtain j where k: "k = j + l" ..  huffman@47108  2383  with assms show ?thesis by simp  huffman@47108  2384 qed  huffman@47108  2385 huffman@47108  2386 lemma mod_pos_geq:  huffman@47108  2387  fixes k l :: int  huffman@47108  2388  assumes "0 < l" and "l \ k"  huffman@47108  2389  shows "k mod l = (k - l) mod l"  huffman@47108  2390 proof -  huffman@47108  2391  have "k = (k - l) + l" by simp  huffman@47108  2392  then obtain j where k: "k = j + l" ..  huffman@47108  2393  with assms show ?thesis by simp  huffman@47108  2394 qed  huffman@47108  2395 haftmann@33361  2396 wenzelm@60758  2397 subsubsection \Splitting Rules for div and mod\  wenzelm@60758  2398 wenzelm@60758  2399 text\The proofs of the two lemmas below are essentially identical\  haftmann@33361  2400 haftmann@33361  2401 lemma split_pos_lemma:  lp15@60562  2402  "0  haftmann@33361  2403  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2404 apply (rule iffI, clarify)  lp15@60562  2405  apply (erule_tac P="P x y" for x y in rev_mp)  lp15@60562  2406  apply (subst mod_add_eq)  lp15@60562  2407  apply (subst zdiv_zadd1_eq)  lp15@60562  2408  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@60758  2409 txt\converse direction\  lp15@60562  2410 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2411 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2412 done  haftmann@33361  2413 haftmann@33361  2414 lemma split_neg_lemma:  haftmann@33361  2415  "k<0 ==>  haftmann@33361  2416  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2417 apply (rule iffI, clarify)  lp15@60562  2418  apply (erule_tac P="P x y" for x y in rev_mp)  lp15@60562  2419  apply (subst mod_add_eq)  lp15@60562  2420  apply (subst zdiv_zadd1_eq)  lp15@60562  2421  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@60758  2422 txt\converse direction\  lp15@60562  2423 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2424 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2425 done  haftmann@33361  2426 haftmann@33361  2427 lemma split_zdiv:  haftmann@33361  2428  "P(n div k :: int) =  lp15@60562  2429  ((k = 0 --> P 0) &  lp15@60562  2430  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2431  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2432 apply (case_tac "k=0", simp)  haftmann@33361  2433 apply (simp only: linorder_neq_iff)  lp15@60562  2434 apply (erule disjE)  lp15@60562  2435  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2436  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2437 done  haftmann@33361  2438 haftmann@33361  2439 lemma split_zmod:  haftmann@33361  2440  "P(n mod k :: int) =  lp15@60562  2441  ((k = 0 --> P n) &  lp15@60562  2442  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2443  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2444 apply (case_tac "k=0", simp)  haftmann@33361  2445 apply (simp only: linorder_neq_iff)  lp15@60562  2446 apply (erule disjE)  lp15@60562  2447  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2448  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2449 done  haftmann@33361  2450 wenzelm@60758  2451 text \Enable (lin)arith to deal with @{const divide} and @{const mod}  webertj@33730  2452  when these are applied to some constant that is of the form  wenzelm@60758  2453  @{term "numeral k"}:\  huffman@47108  2454 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  2455 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  2456 haftmann@33361  2457 wenzelm@60758  2458 subsubsection \Computing @{text "div"} and @{text "mod"} with shifting\  huffman@47166  2459 huffman@47166  2460 lemma pos_divmod_int_rel_mult_2:  huffman@47166  2461  assumes "0 \ b"  huffman@47166  2462  assumes "divmod_int_rel a b (q, r)"  huffman@47166  2463  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  2464  using assms unfolding divmod_int_rel_def by auto  huffman@47166  2465 wenzelm@60758  2466 declaration \K (Lin_Arith.add_simps @{thms uminus_numeral_One})\  haftmann@54489  2467 huffman@47166  2468 lemma neg_divmod_int_rel_mult_2:  huffman@47166  2469  assumes "b \ 0"  huffman@47166  2470  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  2471  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  2472  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  2473 wenzelm@60758  2474 text\computing div by shifting\  haftmann@33361  2475 haftmann@33361  2476 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2477  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2478  by (rule div_int_unique)  haftmann@33361  2479 lp15@60562  2480 lemma neg_zdiv_mult_2:  boehmes@35815  2481  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2482  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2483  by (rule div_int_unique)  haftmann@33361  2484 huffman@47108  2485 (* FIXME: add rules for negative numerals *)  huffman@47108  2486 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2487  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2488  numeral v div (numeral w :: int)"  huffman@47108  2489  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2490  by (rule div_mult_mult1, simp)  huffman@47108  2491 huffman@47108  2492 lemma zdiv_numeral_Bit1 [simp]:  lp15@60562  2493  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  huffman@47108  2494  (numeral v div (numeral w :: int))"  huffman@47108  2495  unfolding numeral.simps  haftmann@57512  2496  unfolding mult_2 [symmetric] add.commute [of _ 1]  huffman@47108  2497  by (rule pos_zdiv_mult_2, simp)  haftmann@33361  2498 haftmann@33361  2499 lemma pos_zmod_mult_2:  haftmann@33361  2500  fixes a b :: int  haftmann@33361  2501  assumes "0 \ a"  haftmann@33361  2502  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  huffman@47166  2503  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2504  by (rule mod_int_unique)  haftmann@33361  2505 haftmann@33361  2506 lemma neg_zmod_mult_2:  haftmann@33361  2507  fixes a b :: int  haftmann@33361  2508  assumes "a \ 0"  haftmann@33361  2509  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  huffman@47166  2510  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2511  by (rule mod_int_unique)  haftmann@33361  2512 huffman@47108  2513 (* FIXME: add rules for negative numerals *)  huffman@47108  2514 lemma zmod_numeral_Bit0 [simp]:  lp15@60562  2515  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  huffman@47108  2516  (2::int) * (numeral v mod numeral w)"  huffman@47108  2517  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  huffman@47108  2518  unfolding mult_2 [symmetric] by (rule mod_mult_mult1) `