src/HOL/Divides.thy
 author haftmann Mon Jun 01 18:59:22 2015 +0200 (2015-06-01) changeset 60353 838025c6e278 parent 60352 d46de31a50c4 child 60429 d3d1e185cd63 permissions -rw-r--r--
implicit partial divison operation in integral domains
 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@58889  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 haftmann@25942  12 subsection {* Syntactic division operations *}  haftmann@25942  13 haftmann@60352  14 class div = dvd + divide +  haftmann@60352  15  fixes mod :: "'a \ 'a \ 'a" (infixl "mod" 70)  haftmann@60352  16 begin  haftmann@60352  17 haftmann@60352  18 abbreviation div :: "'a \ 'a \ 'a" (infixl "div" 70)  haftmann@60352  19 where  haftmann@60352  20  "op div \ divide"  haftmann@60352  21 haftmann@60352  22 end  haftmann@27540  23 haftmann@27651  24 subsection {* Abstract division in commutative semirings. *}  haftmann@25942  25 haftmann@59833  26 class semiring_div = semidom + div +  haftmann@25942  27  assumes mod_div_equality: "a div b * b + a mod b = a"  haftmann@27651  28  and div_by_0 [simp]: "a div 0 = 0"  haftmann@27651  29  and div_0 [simp]: "0 div a = 0"  haftmann@27651  30  and div_mult_self1 [simp]: "b \ 0 \ (a + c * b) div b = c + a div b"  haftmann@30930  31  and div_mult_mult1 [simp]: "c \ 0 \ (c * a) div (c * b) = a div b"  haftmann@25942  32 begin  haftmann@25942  33 haftmann@60353  34 subclass semidom_divide  haftmann@60353  35 proof  haftmann@60353  36  fix b a  haftmann@60353  37  assume "b \ 0"  haftmann@60353  38  then show "a * b div b = a"  haftmann@60353  39  using div_mult_self1 [of b 0 a] by (simp add: ac_simps)  haftmann@60353  40 qed simp  haftmann@58953  41 haftmann@59009  42 lemma power_not_zero: -- \FIXME cf. @{text field_power_not_zero}\  haftmann@59009  43  "a \ 0 \ a ^ n \ 0"  haftmann@59009  44  by (induct n) (simp_all add: no_zero_divisors)  haftmann@59009  45 haftmann@59009  46 lemma semiring_div_power_eq_0_iff: -- \FIXME cf. @{text power_eq_0_iff}, @{text power_eq_0_nat_iff}\  haftmann@59009  47  "n \ 0 \ a ^ n = 0 \ a = 0"  haftmann@59009  48  using power_not_zero [of a n] by (auto simp add: zero_power)  haftmann@59009  49 haftmann@26100  50 text {* @{const div} and @{const mod} *}  haftmann@26100  51 haftmann@26062  52 lemma mod_div_equality2: "b * (a div b) + a mod b = a"  haftmann@57512  53  unfolding mult.commute [of b]  haftmann@26062  54  by (rule mod_div_equality)  haftmann@26062  55 huffman@29403  56 lemma mod_div_equality': "a mod b + a div b * b = a"  huffman@29403  57  using mod_div_equality [of a b]  haftmann@57514  58  by (simp only: ac_simps)  huffman@29403  59 haftmann@26062  60 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"  haftmann@30934  61  by (simp add: mod_div_equality)  haftmann@26062  62 haftmann@26062  63 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"  haftmann@30934  64  by (simp add: mod_div_equality2)  haftmann@26062  65 haftmann@27651  66 lemma mod_by_0 [simp]: "a mod 0 = a"  haftmann@30934  67  using mod_div_equality [of a zero] by simp  haftmann@27651  68 haftmann@27651  69 lemma mod_0 [simp]: "0 mod a = 0"  haftmann@30934  70  using mod_div_equality [of zero a] div_0 by simp  haftmann@27651  71 haftmann@27651  72 lemma div_mult_self2 [simp]:  haftmann@27651  73  assumes "b \ 0"  haftmann@27651  74  shows "(a + b * c) div b = c + a div b"  haftmann@57512  75  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)  haftmann@26100  76 haftmann@54221  77 lemma div_mult_self3 [simp]:  haftmann@54221  78  assumes "b \ 0"  haftmann@54221  79  shows "(c * b + a) div b = c + a div b"  haftmann@54221  80  using assms by (simp add: add.commute)  haftmann@54221  81 haftmann@54221  82 lemma div_mult_self4 [simp]:  haftmann@54221  83  assumes "b \ 0"  haftmann@54221  84  shows "(b * c + a) div b = c + a div b"  haftmann@54221  85  using assms by (simp add: add.commute)  haftmann@54221  86 haftmann@27651  87 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"  haftmann@27651  88 proof (cases "b = 0")  haftmann@27651  89  case True then show ?thesis by simp  haftmann@27651  90 next  haftmann@27651  91  case False  haftmann@27651  92  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"  haftmann@27651  93  by (simp add: mod_div_equality)  haftmann@27651  94  also from False div_mult_self1 [of b a c] have  haftmann@27651  95  "\ = (c + a div b) * b + (a + c * b) mod b"  nipkow@29667  96  by (simp add: algebra_simps)  haftmann@27651  97  finally have "a = a div b * b + (a + c * b) mod b"  haftmann@57512  98  by (simp add: add.commute [of a] add.assoc distrib_right)  haftmann@27651  99  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"  haftmann@27651  100  by (simp add: mod_div_equality)  haftmann@27651  101  then show ?thesis by simp  haftmann@27651  102 qed  haftmann@27651  103 haftmann@54221  104 lemma mod_mult_self2 [simp]:  haftmann@54221  105  "(a + b * c) mod b = a mod b"  haftmann@57512  106  by (simp add: mult.commute [of b])  haftmann@27651  107 haftmann@54221  108 lemma mod_mult_self3 [simp]:  haftmann@54221  109  "(c * b + a) mod b = a mod b"  haftmann@54221  110  by (simp add: add.commute)  haftmann@54221  111 haftmann@54221  112 lemma mod_mult_self4 [simp]:  haftmann@54221  113  "(b * c + a) mod b = a mod b"  haftmann@54221  114  by (simp add: add.commute)  haftmann@54221  115 haftmann@60353  116 lemma div_mult_self1_is_id:  haftmann@60353  117  "b \ 0 \ b * a div b = a"  haftmann@60353  118  by (fact nonzero_mult_divide_cancel_left)  haftmann@60353  119 haftmann@60353  120 lemma div_mult_self2_is_id:  haftmann@60353  121  "b \ 0 \ a * b div b = a"  haftmann@60353  122  by (fact nonzero_mult_divide_cancel_right)  haftmann@27651  123 haftmann@27651  124 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"  haftmann@27651  125  using mod_mult_self2 [of 0 b a] by simp  haftmann@27651  126 haftmann@27651  127 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"  haftmann@27651  128  using mod_mult_self1 [of 0 a b] by simp  haftmann@26062  129 haftmann@27651  130 lemma div_by_1 [simp]: "a div 1 = a"  haftmann@27651  131  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp  haftmann@27651  132 haftmann@27651  133 lemma mod_by_1 [simp]: "a mod 1 = 0"  haftmann@27651  134 proof -  haftmann@27651  135  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp  haftmann@27651  136  then have "a + a mod 1 = a + 0" by simp  haftmann@27651  137  then show ?thesis by (rule add_left_imp_eq)  haftmann@27651  138 qed  haftmann@27651  139 haftmann@27651  140 lemma mod_self [simp]: "a mod a = 0"  haftmann@27651  141  using mod_mult_self2_is_0 [of 1] by simp  haftmann@27651  142 haftmann@27651  143 lemma div_self [simp]: "a \ 0 \ a div a = 1"  haftmann@27651  144  using div_mult_self2_is_id [of _ 1] by simp  haftmann@27651  145 haftmann@27676  146 lemma div_add_self1 [simp]:  haftmann@27651  147  assumes "b \ 0"  haftmann@27651  148  shows "(b + a) div b = a div b + 1"  haftmann@57512  149  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)  haftmann@26062  150 haftmann@27676  151 lemma div_add_self2 [simp]:  haftmann@27651  152  assumes "b \ 0"  haftmann@27651  153  shows "(a + b) div b = a div b + 1"  haftmann@57512  154  using assms div_add_self1 [of b a] by (simp add: add.commute)  haftmann@27651  155 haftmann@27676  156 lemma mod_add_self1 [simp]:  haftmann@27651  157  "(b + a) mod b = a mod b"  haftmann@57512  158  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)  haftmann@27651  159 haftmann@27676  160 lemma mod_add_self2 [simp]:  haftmann@27651  161  "(a + b) mod b = a mod b"  haftmann@27651  162  using mod_mult_self1 [of a 1 b] by simp  haftmann@27651  163 haftmann@27651  164 lemma mod_div_decomp:  haftmann@27651  165  fixes a b  haftmann@27651  166  obtains q r where "q = a div b" and "r = a mod b"  haftmann@27651  167  and "a = q * b + r"  haftmann@27651  168 proof -  haftmann@27651  169  from mod_div_equality have "a = a div b * b + a mod b" by simp  haftmann@27651  170  moreover have "a div b = a div b" ..  haftmann@27651  171  moreover have "a mod b = a mod b" ..  haftmann@27651  172  note that ultimately show thesis by blast  haftmann@27651  173 qed  haftmann@27651  174 haftmann@58834  175 lemma dvd_imp_mod_0 [simp]:  haftmann@58834  176  assumes "a dvd b"  haftmann@58834  177  shows "b mod a = 0"  haftmann@58834  178 proof -  haftmann@58834  179  from assms obtain c where "b = a * c" ..  haftmann@58834  180  then have "b mod a = a * c mod a" by simp  haftmann@58834  181  then show "b mod a = 0" by simp  haftmann@58834  182 qed  haftmann@58911  183 haftmann@58911  184 lemma mod_eq_0_iff_dvd:  haftmann@58911  185  "a mod b = 0 \ b dvd a"  haftmann@58911  186 proof  haftmann@58911  187  assume "b dvd a"  haftmann@58911  188  then show "a mod b = 0" by simp  haftmann@58911  189 next  haftmann@58911  190  assume "a mod b = 0"  haftmann@58911  191  with mod_div_equality [of a b] have "a div b * b = a" by simp  haftmann@58911  192  then have "a = b * (a div b)" by (simp add: ac_simps)  haftmann@58911  193  then show "b dvd a" ..  haftmann@58911  194 qed  haftmann@58911  195 haftmann@58834  196 lemma dvd_eq_mod_eq_0 [code]:  haftmann@58834  197  "a dvd b \ b mod a = 0"  haftmann@58911  198  by (simp add: mod_eq_0_iff_dvd)  haftmann@58911  199 haftmann@58911  200 lemma mod_div_trivial [simp]:  haftmann@58911  201  "a mod b div b = 0"  huffman@29403  202 proof (cases "b = 0")  huffman@29403  203  assume "b = 0"  huffman@29403  204  thus ?thesis by simp  huffman@29403  205 next  huffman@29403  206  assume "b \ 0"  huffman@29403  207  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"  huffman@29403  208  by (rule div_mult_self1 [symmetric])  huffman@29403  209  also have "\ = a div b"  huffman@29403  210  by (simp only: mod_div_equality')  huffman@29403  211  also have "\ = a div b + 0"  huffman@29403  212  by simp  huffman@29403  213  finally show ?thesis  huffman@29403  214  by (rule add_left_imp_eq)  huffman@29403  215 qed  huffman@29403  216 haftmann@58911  217 lemma mod_mod_trivial [simp]:  haftmann@58911  218  "a mod b mod b = a mod b"  huffman@29403  219 proof -  huffman@29403  220  have "a mod b mod b = (a mod b + a div b * b) mod b"  huffman@29403  221  by (simp only: mod_mult_self1)  huffman@29403  222  also have "\ = a mod b"  huffman@29403  223  by (simp only: mod_div_equality')  huffman@29403  224  finally show ?thesis .  huffman@29403  225 qed  huffman@29403  226 haftmann@58834  227 lemma dvd_div_mult_self [simp]:  haftmann@58834  228  "a dvd b \ (b div a) * a = b"  haftmann@58834  229  using mod_div_equality [of b a, symmetric] by simp  haftmann@58834  230 haftmann@58834  231 lemma dvd_mult_div_cancel [simp]:  haftmann@58834  232  "a dvd b \ a * (b div a) = b"  haftmann@58834  233  using dvd_div_mult_self by (simp add: ac_simps)  haftmann@58834  234 haftmann@58834  235 lemma dvd_div_mult:  haftmann@58834  236  "a dvd b \ (b div a) * c = (b * c) div a"  haftmann@58834  237  by (cases "a = 0") (auto elim!: dvdE simp add: mult.assoc)  haftmann@58834  238 haftmann@58834  239 lemma div_dvd_div [simp]:  haftmann@58834  240  assumes "a dvd b" and "a dvd c"  haftmann@58834  241  shows "b div a dvd c div a \ b dvd c"  haftmann@58834  242 using assms apply (cases "a = 0")  haftmann@58834  243 apply auto  nipkow@29925  244 apply (unfold dvd_def)  nipkow@29925  245 apply auto  haftmann@57512  246  apply(blast intro:mult.assoc[symmetric])  haftmann@57512  247 apply(fastforce simp add: mult.assoc)  nipkow@29925  248 done  nipkow@29925  249 haftmann@58834  250 lemma dvd_mod_imp_dvd:  haftmann@58834  251  assumes "k dvd m mod n" and "k dvd n"  haftmann@58834  252  shows "k dvd m"  haftmann@58834  253 proof -  haftmann@58834  254  from assms have "k dvd (m div n) * n + m mod n"  haftmann@58834  255  by (simp only: dvd_add dvd_mult)  haftmann@58834  256  then show ?thesis by (simp add: mod_div_equality)  haftmann@58834  257 qed  huffman@30078  258 huffman@29403  259 text {* Addition respects modular equivalence. *}  huffman@29403  260 huffman@29403  261 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"  huffman@29403  262 proof -  huffman@29403  263  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"  huffman@29403  264  by (simp only: mod_div_equality)  huffman@29403  265  also have "\ = (a mod c + b + a div c * c) mod c"  haftmann@57514  266  by (simp only: ac_simps)  huffman@29403  267  also have "\ = (a mod c + b) mod c"  huffman@29403  268  by (rule mod_mult_self1)  huffman@29403  269  finally show ?thesis .  huffman@29403  270 qed  huffman@29403  271 huffman@29403  272 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"  huffman@29403  273 proof -  huffman@29403  274  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"  huffman@29403  275  by (simp only: mod_div_equality)  huffman@29403  276  also have "\ = (a + b mod c + b div c * c) mod c"  haftmann@57514  277  by (simp only: ac_simps)  huffman@29403  278  also have "\ = (a + b mod c) mod c"  huffman@29403  279  by (rule mod_mult_self1)  huffman@29403  280  finally show ?thesis .  huffman@29403  281 qed  huffman@29403  282 huffman@29403  283 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"  huffman@29403  284 by (rule trans [OF mod_add_left_eq mod_add_right_eq])  huffman@29403  285 huffman@29403  286 lemma mod_add_cong:  huffman@29403  287  assumes "a mod c = a' mod c"  huffman@29403  288  assumes "b mod c = b' mod c"  huffman@29403  289  shows "(a + b) mod c = (a' + b') mod c"  huffman@29403  290 proof -  huffman@29403  291  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"  huffman@29403  292  unfolding assms ..  huffman@29403  293  thus ?thesis  huffman@29403  294  by (simp only: mod_add_eq [symmetric])  huffman@29403  295 qed  huffman@29403  296 haftmann@30923  297 lemma div_add [simp]: "z dvd x \ z dvd y  nipkow@30837  298  \ (x + y) div z = x div z + y div z"  haftmann@30923  299 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)  nipkow@30837  300 huffman@29403  301 text {* Multiplication respects modular equivalence. *}  huffman@29403  302 huffman@29403  303 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"  huffman@29403  304 proof -  huffman@29403  305  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"  huffman@29403  306  by (simp only: mod_div_equality)  huffman@29403  307  also have "\ = (a mod c * b + a div c * b * c) mod c"  nipkow@29667  308  by (simp only: algebra_simps)  huffman@29403  309  also have "\ = (a mod c * b) mod c"  huffman@29403  310  by (rule mod_mult_self1)  huffman@29403  311  finally show ?thesis .  huffman@29403  312 qed  huffman@29403  313 huffman@29403  314 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"  huffman@29403  315 proof -  huffman@29403  316  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"  huffman@29403  317  by (simp only: mod_div_equality)  huffman@29403  318  also have "\ = (a * (b mod c) + a * (b div c) * c) mod c"  nipkow@29667  319  by (simp only: algebra_simps)  huffman@29403  320  also have "\ = (a * (b mod c)) mod c"  huffman@29403  321  by (rule mod_mult_self1)  huffman@29403  322  finally show ?thesis .  huffman@29403  323 qed  huffman@29403  324 huffman@29403  325 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"  huffman@29403  326 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])  huffman@29403  327 huffman@29403  328 lemma mod_mult_cong:  huffman@29403  329  assumes "a mod c = a' mod c"  huffman@29403  330  assumes "b mod c = b' mod c"  huffman@29403  331  shows "(a * b) mod c = (a' * b') mod c"  huffman@29403  332 proof -  huffman@29403  333  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"  huffman@29403  334  unfolding assms ..  huffman@29403  335  thus ?thesis  huffman@29403  336  by (simp only: mod_mult_eq [symmetric])  huffman@29403  337 qed  huffman@29403  338 huffman@47164  339 text {* Exponentiation respects modular equivalence. *}  huffman@47164  340 huffman@47164  341 lemma power_mod: "(a mod b)^n mod b = a^n mod b"  huffman@47164  342 apply (induct n, simp_all)  huffman@47164  343 apply (rule mod_mult_right_eq [THEN trans])  huffman@47164  344 apply (simp (no_asm_simp))  huffman@47164  345 apply (rule mod_mult_eq [symmetric])  huffman@47164  346 done  huffman@47164  347 huffman@29404  348 lemma mod_mod_cancel:  huffman@29404  349  assumes "c dvd b"  huffman@29404  350  shows "a mod b mod c = a mod c"  huffman@29404  351 proof -  huffman@29404  352  from c dvd b obtain k where "b = c * k"  huffman@29404  353  by (rule dvdE)  huffman@29404  354  have "a mod b mod c = a mod (c * k) mod c"  huffman@29404  355  by (simp only: b = c * k)  huffman@29404  356  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  357  by (simp only: mod_mult_self1)  huffman@29404  358  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  haftmann@58786  359  by (simp only: ac_simps)  huffman@29404  360  also have "\ = a mod c"  huffman@29404  361  by (simp only: mod_div_equality)  huffman@29404  362  finally show ?thesis .  huffman@29404  363 qed  huffman@29404  364 haftmann@30930  365 lemma div_mult_div_if_dvd:  haftmann@30930  366  "y dvd x \ z dvd w \ (x div y) * (w div z) = (x * w) div (y * z)"  haftmann@30930  367  apply (cases "y = 0", simp)  haftmann@30930  368  apply (cases "z = 0", simp)  haftmann@30930  369  apply (auto elim!: dvdE simp add: algebra_simps)  haftmann@57512  370  apply (subst mult.assoc [symmetric])  nipkow@30476  371  apply (simp add: no_zero_divisors)  haftmann@30930  372  done  haftmann@30930  373 haftmann@35367  374 lemma div_mult_swap:  haftmann@35367  375  assumes "c dvd b"  haftmann@35367  376  shows "a * (b div c) = (a * b) div c"  haftmann@35367  377 proof -  haftmann@35367  378  from assms have "b div c * (a div 1) = b * a div (c * 1)"  haftmann@35367  379  by (simp only: div_mult_div_if_dvd one_dvd)  haftmann@57512  380  then show ?thesis by (simp add: mult.commute)  haftmann@35367  381 qed  haftmann@35367  382   haftmann@30930  383 lemma div_mult_mult2 [simp]:  haftmann@30930  384  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@57512  385  by (drule div_mult_mult1) (simp add: mult.commute)  haftmann@30930  386 haftmann@30930  387 lemma div_mult_mult1_if [simp]:  haftmann@30930  388  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  389  by simp_all  nipkow@30476  390 haftmann@30930  391 lemma mod_mult_mult1:  haftmann@30930  392  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  393 proof (cases "c = 0")  haftmann@30930  394  case True then show ?thesis by simp  haftmann@30930  395 next  haftmann@30930  396  case False  haftmann@30930  397  from mod_div_equality  haftmann@30930  398  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  399  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  400  = c * a + c * (a mod b)" by (simp add: algebra_simps)  haftmann@30930  401  with mod_div_equality show ?thesis by simp  haftmann@30930  402 qed  haftmann@30930  403   haftmann@30930  404 lemma mod_mult_mult2:  haftmann@30930  405  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@57512  406  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)  haftmann@30930  407 huffman@47159  408 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  huffman@47159  409  by (fact mod_mult_mult2 [symmetric])  huffman@47159  410 huffman@47159  411 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  huffman@47159  412  by (fact mod_mult_mult1 [symmetric])  huffman@47159  413 haftmann@59009  414 lemma dvd_times_left_cancel_iff [simp]: -- \FIXME generalize\  haftmann@59009  415  assumes "c \ 0"  haftmann@59009  416  shows "c * a dvd c * b \ a dvd b"  haftmann@59009  417 proof -  haftmann@59009  418  have "(c * b) mod (c * a) = 0 \ b mod a = 0" (is "?P \ ?Q")  haftmann@59009  419  using assms by (simp add: mod_mult_mult1)  haftmann@59009  420  then show ?thesis by (simp add: mod_eq_0_iff_dvd)  haftmann@59009  421 qed  haftmann@59009  422 haftmann@59009  423 lemma dvd_times_right_cancel_iff [simp]: -- \FIXME generalize\  haftmann@59009  424  assumes "c \ 0"  haftmann@59009  425  shows "a * c dvd b * c \ a dvd b"  haftmann@59009  426  using assms dvd_times_left_cancel_iff [of c a b] by (simp add: ac_simps)  haftmann@59009  427 huffman@31662  428 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  429  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  430 huffman@31662  431 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  432 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  433 haftmann@31009  434 lemma div_power:  huffman@31661  435  "y dvd x \ (x div y) ^ n = x ^ n div y ^ n"  nipkow@30476  436 apply (induct n)  nipkow@30476  437  apply simp  nipkow@30476  438 apply(simp add: div_mult_div_if_dvd dvd_power_same)  nipkow@30476  439 done  nipkow@30476  440 haftmann@35367  441 lemma dvd_div_eq_mult:  haftmann@35367  442  assumes "a \ 0" and "a dvd b"  haftmann@35367  443  shows "b div a = c \ b = c * a"  haftmann@35367  444 proof  haftmann@35367  445  assume "b = c * a"  haftmann@35367  446  then show "b div a = c" by (simp add: assms)  haftmann@35367  447 next  haftmann@35367  448  assume "b div a = c"  haftmann@35367  449  then have "b div a * a = c * a" by simp  haftmann@60353  450  moreover from a dvd b have "b div a * a = b" by simp  haftmann@35367  451  ultimately show "b = c * a" by simp  haftmann@35367  452 qed  haftmann@35367  453   haftmann@35367  454 lemma dvd_div_div_eq_mult:  haftmann@35367  455  assumes "a \ 0" "c \ 0" and "a dvd b" "c dvd d"  haftmann@35367  456  shows "b div a = d div c \ b * c = a * d"  haftmann@60353  457  using assms by (auto simp add: mult.commute [of _ a] dvd_div_eq_mult div_mult_swap intro: sym)  haftmann@35367  458 huffman@31661  459 end  huffman@31661  460 haftmann@59833  461 class ring_div = comm_ring_1 + semiring_div  huffman@29405  462 begin  huffman@29405  463 haftmann@60353  464 subclass idom_divide ..  haftmann@36634  465 huffman@29405  466 text {* Negation respects modular equivalence. *}  huffman@29405  467 huffman@29405  468 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  469 proof -  huffman@29405  470  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  471  by (simp only: mod_div_equality)  huffman@29405  472  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  haftmann@57514  473  by (simp add: ac_simps)  huffman@29405  474  also have "\ = (- (a mod b)) mod b"  huffman@29405  475  by (rule mod_mult_self1)  huffman@29405  476  finally show ?thesis .  huffman@29405  477 qed  huffman@29405  478 huffman@29405  479 lemma mod_minus_cong:  huffman@29405  480  assumes "a mod b = a' mod b"  huffman@29405  481  shows "(- a) mod b = (- a') mod b"  huffman@29405  482 proof -  huffman@29405  483  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  484  unfolding assms ..  huffman@29405  485  thus ?thesis  huffman@29405  486  by (simp only: mod_minus_eq [symmetric])  huffman@29405  487 qed  huffman@29405  488 huffman@29405  489 text {* Subtraction respects modular equivalence. *}  huffman@29405  490 haftmann@54230  491 lemma mod_diff_left_eq:  haftmann@54230  492  "(a - b) mod c = (a mod c - b) mod c"  haftmann@54230  493  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp  haftmann@54230  494 haftmann@54230  495 lemma mod_diff_right_eq:  haftmann@54230  496  "(a - b) mod c = (a - b mod c) mod c"  haftmann@54230  497  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  498 haftmann@54230  499 lemma mod_diff_eq:  haftmann@54230  500  "(a - b) mod c = (a mod c - b mod c) mod c"  haftmann@54230  501  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  502 huffman@29405  503 lemma mod_diff_cong:  huffman@29405  504  assumes "a mod c = a' mod c"  huffman@29405  505  assumes "b mod c = b' mod c"  huffman@29405  506  shows "(a - b) mod c = (a' - b') mod c"  haftmann@54230  507  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp  huffman@29405  508 nipkow@30180  509 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  510 apply (case_tac "y = 0") apply simp  nipkow@30180  511 apply (auto simp add: dvd_def)  nipkow@30180  512 apply (subgoal_tac "-(y * k) = y * - k")  thomas@57492  513  apply (simp only:)  nipkow@30180  514  apply (erule div_mult_self1_is_id)  nipkow@30180  515 apply simp  nipkow@30180  516 done  nipkow@30180  517 nipkow@30180  518 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  519 apply (case_tac "y = 0") apply simp  nipkow@30180  520 apply (auto simp add: dvd_def)  nipkow@30180  521 apply (subgoal_tac "y * k = -y * -k")  thomas@57492  522  apply (erule ssubst, rule div_mult_self1_is_id)  nipkow@30180  523  apply simp  nipkow@30180  524 apply simp  nipkow@30180  525 done  nipkow@30180  526 nipkow@59473  527 lemma div_diff[simp]:  nipkow@59380  528  "\ z dvd x; z dvd y\ \ (x - y) div z = x div z - y div z"  nipkow@59380  529 using div_add[where y = "- z" for z]  nipkow@59380  530 by (simp add: dvd_neg_div)  nipkow@59380  531 huffman@47159  532 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  huffman@47159  533  using div_mult_mult1 [of "- 1" a b]  huffman@47159  534  unfolding neg_equal_0_iff_equal by simp  huffman@47159  535 huffman@47159  536 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  huffman@47159  537  using mod_mult_mult1 [of "- 1" a b] by simp  huffman@47159  538 huffman@47159  539 lemma div_minus_right: "a div (-b) = (-a) div b"  huffman@47159  540  using div_minus_minus [of "-a" b] by simp  huffman@47159  541 huffman@47159  542 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  huffman@47159  543  using mod_minus_minus [of "-a" b] by simp  huffman@47159  544 huffman@47160  545 lemma div_minus1_right [simp]: "a div (-1) = -a"  huffman@47160  546  using div_minus_right [of a 1] by simp  huffman@47160  547 huffman@47160  548 lemma mod_minus1_right [simp]: "a mod (-1) = 0"  huffman@47160  549  using mod_minus_right [of a 1] by simp  huffman@47160  550 haftmann@54221  551 lemma minus_mod_self2 [simp]:  haftmann@54221  552  "(a - b) mod b = a mod b"  haftmann@54221  553  by (simp add: mod_diff_right_eq)  haftmann@54221  554 haftmann@54221  555 lemma minus_mod_self1 [simp]:  haftmann@54221  556  "(b - a) mod b = - a mod b"  haftmann@54230  557  using mod_add_self2 [of "- a" b] by simp  haftmann@54221  558 huffman@29405  559 end  huffman@29405  560 haftmann@58778  561 haftmann@58778  562 subsubsection {* Parity and division *}  haftmann@58778  563 haftmann@59833  564 class semiring_div_parity = semiring_div + comm_semiring_1_diff_distrib + numeral +  haftmann@54226  565  assumes parity: "a mod 2 = 0 \ a mod 2 = 1"  haftmann@58786  566  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"  haftmann@58710  567  assumes zero_not_eq_two: "0 \ 2"  haftmann@54226  568 begin  haftmann@54226  569 haftmann@54226  570 lemma parity_cases [case_names even odd]:  haftmann@54226  571  assumes "a mod 2 = 0 \ P"  haftmann@54226  572  assumes "a mod 2 = 1 \ P"  haftmann@54226  573  shows P  haftmann@54226  574  using assms parity by blast  haftmann@54226  575 haftmann@58786  576 lemma one_div_two_eq_zero [simp]:  haftmann@58778  577  "1 div 2 = 0"  haftmann@58778  578 proof (cases "2 = 0")  haftmann@58778  579  case True then show ?thesis by simp  haftmann@58778  580 next  haftmann@58778  581  case False  haftmann@58778  582  from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .  haftmann@58778  583  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp  haftmann@58953  584  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)  haftmann@58953  585  then have "1 div 2 = 0 \ 2 = 0" by simp  haftmann@58778  586  with False show ?thesis by auto  haftmann@58778  587 qed  haftmann@58778  588 haftmann@58786  589 lemma not_mod_2_eq_0_eq_1 [simp]:  haftmann@58786  590  "a mod 2 \ 0 \ a mod 2 = 1"  haftmann@58786  591  by (cases a rule: parity_cases) simp_all  haftmann@58786  592 haftmann@58786  593 lemma not_mod_2_eq_1_eq_0 [simp]:  haftmann@58786  594  "a mod 2 \ 1 \ a mod 2 = 0"  haftmann@58786  595  by (cases a rule: parity_cases) simp_all  haftmann@58786  596 haftmann@58778  597 subclass semiring_parity  haftmann@58778  598 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)  haftmann@58778  599  show "1 mod 2 = 1"  haftmann@58778  600  by (fact one_mod_two_eq_one)  haftmann@58778  601 next  haftmann@58778  602  fix a b  haftmann@58778  603  assume "a mod 2 = 1"  haftmann@58778  604  moreover assume "b mod 2 = 1"  haftmann@58778  605  ultimately show "(a + b) mod 2 = 0"  haftmann@58778  606  using mod_add_eq [of a b 2] by simp  haftmann@58778  607 next  haftmann@58778  608  fix a b  haftmann@58778  609  assume "(a * b) mod 2 = 0"  haftmann@58778  610  then have "(a mod 2) * (b mod 2) = 0"  haftmann@58778  611  by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])  haftmann@58778  612  then show "a mod 2 = 0 \ b mod 2 = 0"  haftmann@58778  613  by (rule divisors_zero)  haftmann@58778  614 next  haftmann@58778  615  fix a  haftmann@58778  616  assume "a mod 2 = 1"  haftmann@58778  617  then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp  haftmann@58778  618  then show "\b. a = b + 1" ..  haftmann@58778  619 qed  haftmann@58778  620 haftmann@58778  621 lemma even_iff_mod_2_eq_zero:  haftmann@58778  622  "even a \ a mod 2 = 0"  haftmann@58778  623  by (fact dvd_eq_mod_eq_0)  haftmann@58778  624 haftmann@58778  625 lemma even_succ_div_two [simp]:  haftmann@58778  626  "even a \ (a + 1) div 2 = a div 2"  haftmann@58778  627  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)  haftmann@58778  628 haftmann@58778  629 lemma odd_succ_div_two [simp]:  haftmann@58778  630  "odd a \ (a + 1) div 2 = a div 2 + 1"  haftmann@58778  631  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)  haftmann@58778  632 haftmann@58778  633 lemma even_two_times_div_two:  haftmann@58778  634  "even a \ 2 * (a div 2) = a"  haftmann@58778  635  by (fact dvd_mult_div_cancel)  haftmann@58778  636 haftmann@58834  637 lemma odd_two_times_div_two_succ [simp]:  haftmann@58778  638  "odd a \ 2 * (a div 2) + 1 = a"  haftmann@58778  639  using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)  haftmann@58778  640 haftmann@54226  641 end  haftmann@54226  642 haftmann@25942  643 haftmann@53067  644 subsection {* Generic numeral division with a pragmatic type class *}  haftmann@53067  645 haftmann@53067  646 text {*  haftmann@53067  647  The following type class contains everything necessary to formulate  haftmann@53067  648  a division algorithm in ring structures with numerals, restricted  haftmann@53067  649  to its positive segments. This is its primary motiviation, and it  haftmann@53067  650  could surely be formulated using a more fine-grained, more algebraic  haftmann@53067  651  and less technical class hierarchy.  haftmann@53067  652 *}  haftmann@53067  653 haftmann@59833  654 class semiring_numeral_div = semiring_div + comm_semiring_1_diff_distrib + linordered_semidom +  haftmann@59816  655  assumes le_add_diff_inverse2: "b \ a \ a - b + b = a"  haftmann@59816  656  assumes div_less: "0 \ a \ a < b \ a div b = 0"  haftmann@53067  657  and mod_less: " 0 \ a \ a < b \ a mod b = a"  haftmann@53067  658  and div_positive: "0 < b \ b \ a \ a div b > 0"  haftmann@53067  659  and mod_less_eq_dividend: "0 \ a \ a mod b \ a"  haftmann@53067  660  and pos_mod_bound: "0 < b \ a mod b < b"  haftmann@53067  661  and pos_mod_sign: "0 < b \ 0 \ a mod b"  haftmann@53067  662  and mod_mult2_eq: "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@53067  663  and div_mult2_eq: "0 \ c \ a div (b * c) = a div b div c"  haftmann@53067  664  assumes discrete: "a < b \ a + 1 \ b"  haftmann@53067  665 begin  haftmann@53067  666 haftmann@59816  667 lemma mult_div_cancel:  haftmann@59816  668  "b * (a div b) = a - a mod b"  haftmann@59816  669 proof -  haftmann@59816  670  have "b * (a div b) + a mod b = a"  haftmann@59816  671  using mod_div_equality [of a b] by (simp add: ac_simps)  haftmann@59816  672  then have "b * (a div b) + a mod b - a mod b = a - a mod b"  haftmann@59816  673  by simp  haftmann@59816  674  then show ?thesis  haftmann@59816  675  by simp  haftmann@59816  676 qed  haftmann@53067  677 haftmann@54226  678 subclass semiring_div_parity  haftmann@54226  679 proof  haftmann@54226  680  fix a  haftmann@54226  681  show "a mod 2 = 0 \ a mod 2 = 1"  haftmann@54226  682  proof (rule ccontr)  haftmann@54226  683  assume "\ (a mod 2 = 0 \ a mod 2 = 1)"  haftmann@54226  684  then have "a mod 2 \ 0" and "a mod 2 \ 1" by simp_all  haftmann@54226  685  have "0 < 2" by simp  haftmann@54226  686  with pos_mod_bound pos_mod_sign have "0 \ a mod 2" "a mod 2 < 2" by simp_all  haftmann@54226  687  with a mod 2 \ 0 have "0 < a mod 2" by simp  haftmann@54226  688  with discrete have "1 \ a mod 2" by simp  haftmann@54226  689  with a mod 2 \ 1 have "1 < a mod 2" by simp  haftmann@54226  690  with discrete have "2 \ a mod 2" by simp  haftmann@54226  691  with a mod 2 < 2 show False by simp  haftmann@54226  692  qed  haftmann@58646  693 next  haftmann@58646  694  show "1 mod 2 = 1"  haftmann@58646  695  by (rule mod_less) simp_all  haftmann@58710  696 next  haftmann@58710  697  show "0 \ 2"  haftmann@58710  698  by simp  haftmann@53067  699 qed  haftmann@53067  700 haftmann@53067  701 lemma divmod_digit_1:  haftmann@53067  702  assumes "0 \ a" "0 < b" and "b \ a mod (2 * b)"  haftmann@53067  703  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")  haftmann@53067  704  and "a mod (2 * b) - b = a mod b" (is "?Q")  haftmann@53067  705 proof -  haftmann@53067  706  from assms mod_less_eq_dividend [of a "2 * b"] have "b \ a"  haftmann@53067  707  by (auto intro: trans)  haftmann@53067  708  with 0 < b have "0 < a div b" by (auto intro: div_positive)  haftmann@53067  709  then have [simp]: "1 \ a div b" by (simp add: discrete)  haftmann@53067  710  with 0 < b have mod_less: "a mod b < b" by (simp add: pos_mod_bound)  haftmann@53067  711  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  712  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  713  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  714  from assms w_exhaust have "w = 1"  haftmann@53067  715  by (auto simp add: mod_w) (insert mod_less, auto)  haftmann@53067  716  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp  haftmann@53067  717  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  718  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  719  with w = 1 have div: "2 * (a div (2 * b)) = a div b - 1" by simp  haftmann@53067  720  then show ?P and ?Q  haftmann@59816  721  by (simp_all add: div mod add_implies_diff [symmetric] le_add_diff_inverse2)  haftmann@53067  722 qed  haftmann@53067  723 haftmann@53067  724 lemma divmod_digit_0:  haftmann@53067  725  assumes "0 < b" and "a mod (2 * b) < b"  haftmann@53067  726  shows "2 * (a div (2 * b)) = a div b" (is "?P")  haftmann@53067  727  and "a mod (2 * b) = a mod b" (is "?Q")  haftmann@53067  728 proof -  haftmann@53067  729  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  730  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  731  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  732  moreover have "b \ a mod b + b"  haftmann@53067  733  proof -  haftmann@53067  734  from 0 < b pos_mod_sign have "0 \ a mod b" by blast  haftmann@53067  735  then have "0 + b \ a mod b + b" by (rule add_right_mono)  haftmann@53067  736  then show ?thesis by simp  haftmann@53067  737  qed  haftmann@53067  738  moreover note assms w_exhaust  haftmann@53067  739  ultimately have "w = 0" by auto  haftmann@53067  740  with mod_w have mod: "a mod (2 * b) = a mod b" by simp  haftmann@53067  741  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  742  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  743  with w = 0 have div: "2 * (a div (2 * b)) = a div b" by simp  haftmann@53067  744  then show ?P and ?Q  haftmann@53067  745  by (simp_all add: div mod)  haftmann@53067  746 qed  haftmann@53067  747 haftmann@53067  748 definition divmod :: "num \ num \ 'a \ 'a"  haftmann@53067  749 where  haftmann@53067  750  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"  haftmann@53067  751 haftmann@53067  752 lemma fst_divmod [simp]:  haftmann@53067  753  "fst (divmod m n) = numeral m div numeral n"  haftmann@53067  754  by (simp add: divmod_def)  haftmann@53067  755 haftmann@53067  756 lemma snd_divmod [simp]:  haftmann@53067  757  "snd (divmod m n) = numeral m mod numeral n"  haftmann@53067  758  by (simp add: divmod_def)  haftmann@53067  759 haftmann@53067  760 definition divmod_step :: "num \ 'a \ 'a \ 'a \ 'a"  haftmann@53067  761 where  haftmann@53067  762  "divmod_step l qr = (let (q, r) = qr  haftmann@53067  763  in if r \ numeral l then (2 * q + 1, r - numeral l)  haftmann@53067  764  else (2 * q, r))"  haftmann@53067  765 haftmann@53067  766 text {*  haftmann@53067  767  This is a formulation of one step (referring to one digit position)  haftmann@53067  768  in school-method division: compare the dividend at the current  haftmann@53070  769  digit position with the remainder from previous division steps  haftmann@53067  770  and evaluate accordingly.  haftmann@53067  771 *}  haftmann@53067  772 haftmann@53067  773 lemma divmod_step_eq [code]:  haftmann@53067  774  "divmod_step l (q, r) = (if numeral l \ r  haftmann@53067  775  then (2 * q + 1, r - numeral l) else (2 * q, r))"  haftmann@53067  776  by (simp add: divmod_step_def)  haftmann@53067  777 haftmann@53067  778 lemma divmod_step_simps [simp]:  haftmann@53067  779  "r < numeral l \ divmod_step l (q, r) = (2 * q, r)"  haftmann@53067  780  "numeral l \ r \ divmod_step l (q, r) = (2 * q + 1, r - numeral l)"  haftmann@53067  781  by (auto simp add: divmod_step_eq not_le)  haftmann@53067  782 haftmann@53067  783 text {*  haftmann@53067  784  This is a formulation of school-method division.  haftmann@53067  785  If the divisor is smaller than the dividend, terminate.  haftmann@53067  786  If not, shift the dividend to the right until termination  haftmann@53067  787  occurs and then reiterate single division steps in the  haftmann@53067  788  opposite direction.  haftmann@53067  789 *}  haftmann@53067  790 haftmann@53067  791 lemma divmod_divmod_step [code]:  haftmann@53067  792  "divmod m n = (if m < n then (0, numeral m)  haftmann@53067  793  else divmod_step n (divmod m (Num.Bit0 n)))"  haftmann@53067  794 proof (cases "m < n")  haftmann@53067  795  case True then have "numeral m < numeral n" by simp  haftmann@53067  796  then show ?thesis  haftmann@53067  797  by (simp add: prod_eq_iff div_less mod_less)  haftmann@53067  798 next  haftmann@53067  799  case False  haftmann@53067  800  have "divmod m n =  haftmann@53067  801  divmod_step n (numeral m div (2 * numeral n),  haftmann@53067  802  numeral m mod (2 * numeral n))"  haftmann@53067  803  proof (cases "numeral n \ numeral m mod (2 * numeral n)")  haftmann@53067  804  case True  haftmann@53067  805  with divmod_step_simps  haftmann@53067  806  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  807  (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"  haftmann@53067  808  by blast  haftmann@53067  809  moreover from True divmod_digit_1 [of "numeral m" "numeral n"]  haftmann@53067  810  have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"  haftmann@53067  811  and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"  haftmann@53067  812  by simp_all  haftmann@53067  813  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  814  next  haftmann@53067  815  case False then have *: "numeral m mod (2 * numeral n) < numeral n"  haftmann@53067  816  by (simp add: not_le)  haftmann@53067  817  with divmod_step_simps  haftmann@53067  818  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  819  (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"  haftmann@53067  820  by blast  haftmann@53067  821  moreover from * divmod_digit_0 [of "numeral n" "numeral m"]  haftmann@53067  822  have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"  haftmann@53067  823  and "numeral m mod (2 * numeral n) = numeral m mod numeral n"  haftmann@53067  824  by (simp_all only: zero_less_numeral)  haftmann@53067  825  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  826  qed  haftmann@53067  827  then have "divmod m n =  haftmann@53067  828  divmod_step n (numeral m div numeral (Num.Bit0 n),  haftmann@53067  829  numeral m mod numeral (Num.Bit0 n))"  haftmann@53067  830  by (simp only: numeral.simps distrib mult_1)  haftmann@53067  831  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@53067  832  by (simp add: divmod_def)  haftmann@53067  833  with False show ?thesis by simp  haftmann@53067  834 qed  haftmann@53067  835 haftmann@58953  836 lemma divmod_eq [simp]:  haftmann@58953  837  "m < n \ divmod m n = (0, numeral m)"  haftmann@58953  838  "n \ m \ divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@58953  839  by (auto simp add: divmod_divmod_step [of m n])  haftmann@58953  840 haftmann@58953  841 lemma divmod_cancel [simp, code]:  haftmann@53069  842  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r))" (is ?P)  haftmann@53069  843  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r + 1))" (is ?Q)  haftmann@53069  844 proof -  haftmann@53069  845  have *: "\q. numeral (Num.Bit0 q) = 2 * numeral q"  haftmann@53069  846  "\q. numeral (Num.Bit1 q) = 2 * numeral q + 1"  haftmann@53069  847  by (simp_all only: numeral_mult numeral.simps distrib) simp_all  haftmann@53069  848  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)  haftmann@53069  849  then show ?P and ?Q  haftmann@53069  850  by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1  haftmann@53069  851  div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)  haftmann@58953  852 qed  haftmann@58953  853 haftmann@58953  854 text {* Special case: divisibility *}  haftmann@58953  855 haftmann@58953  856 definition divides_aux :: "'a \ 'a \ bool"  haftmann@58953  857 where  haftmann@58953  858  "divides_aux qr \ snd qr = 0"  haftmann@58953  859 haftmann@58953  860 lemma divides_aux_eq [simp]:  haftmann@58953  861  "divides_aux (q, r) \ r = 0"  haftmann@58953  862  by (simp add: divides_aux_def)  haftmann@58953  863 haftmann@58953  864 lemma dvd_numeral_simp [simp]:  haftmann@58953  865  "numeral m dvd numeral n \ divides_aux (divmod n m)"  haftmann@58953  866  by (simp add: divmod_def mod_eq_0_iff_dvd)  haftmann@53069  867 haftmann@53067  868 end  haftmann@53067  869 haftmann@59816  870 hide_fact (open) le_add_diff_inverse2  haftmann@53067  871  -- {* restore simple accesses for more general variants of theorems *}  haftmann@53067  872 haftmann@53067  873   haftmann@26100  874 subsection {* Division on @{typ nat} *}  haftmann@26100  875 haftmann@26100  876 text {*  haftmann@26100  877  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  878  of a characteristic relation with two input arguments  haftmann@26100  879  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  880  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  881 *}  haftmann@26100  882 haftmann@33340  883 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  884  "divmod_nat_rel m n qr \  haftmann@30923  885  m = fst qr * n + snd qr \  haftmann@30923  886  (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  887 haftmann@33340  888 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  889 haftmann@33340  890 lemma divmod_nat_rel_ex:  haftmann@33340  891  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  892 proof (cases "n = 0")  haftmann@30923  893  case True with that show thesis  haftmann@33340  894  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  895 next  haftmann@26100  896  case False  haftmann@26100  897  have "\q r. m = q * n + r \ r < n"  haftmann@26100  898  proof (induct m)  haftmann@26100  899  case 0 with n \ 0  haftmann@26100  900  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  901  then show ?case by blast  haftmann@26100  902  next  haftmann@26100  903  case (Suc m) then obtain q' r'  haftmann@26100  904  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  905  then show ?case proof (cases "Suc r' < n")  haftmann@26100  906  case True  haftmann@26100  907  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  908  with True show ?thesis by blast  haftmann@26100  909  next  haftmann@26100  910  case False then have "n \ Suc r'" by auto  haftmann@26100  911  moreover from n have "Suc r' \ n" by auto  haftmann@26100  912  ultimately have "n = Suc r'" by auto  haftmann@26100  913  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  914  with n \ 0 show ?thesis by blast  haftmann@26100  915  qed  haftmann@26100  916  qed  haftmann@26100  917  with that show thesis  haftmann@33340  918  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  919 qed  haftmann@26100  920 haftmann@33340  921 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  922 haftmann@33340  923 lemma divmod_nat_rel_unique:  haftmann@33340  924  assumes "divmod_nat_rel m n qr"  haftmann@33340  925  and "divmod_nat_rel m n qr'"  haftmann@30923  926  shows "qr = qr'"  haftmann@26100  927 proof (cases "n = 0")  haftmann@26100  928  case True with assms show ?thesis  haftmann@30923  929  by (cases qr, cases qr')  haftmann@33340  930  (simp add: divmod_nat_rel_def)  haftmann@26100  931 next  haftmann@26100  932  case False  haftmann@26100  933  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  934  apply (rule leI)  haftmann@26100  935  apply (subst less_iff_Suc_add)  haftmann@26100  936  apply (auto simp add: add_mult_distrib)  haftmann@26100  937  done  wenzelm@53374  938  from n \ 0 assms have *: "fst qr = fst qr'"  haftmann@33340  939  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  wenzelm@53374  940  with assms have "snd qr = snd qr'"  haftmann@33340  941  by (simp add: divmod_nat_rel_def)  wenzelm@53374  942  with * show ?thesis by (cases qr, cases qr') simp  haftmann@26100  943 qed  haftmann@26100  944 haftmann@26100  945 text {*  haftmann@26100  946  We instantiate divisibility on the natural numbers by  haftmann@33340  947  means of @{const divmod_nat_rel}:  haftmann@26100  948 *}  haftmann@25942  949 haftmann@33340  950 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  951  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  952 haftmann@33340  953 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  954  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  955 proof -  haftmann@33340  956  from divmod_nat_rel_ex  haftmann@33340  957  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  958  then show ?thesis  haftmann@33340  959  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  960 qed  haftmann@30923  961 huffman@47135  962 lemma divmod_nat_unique:  haftmann@33340  963  assumes "divmod_nat_rel m n qr"  haftmann@33340  964  shows "divmod_nat m n = qr"  haftmann@33340  965  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  966 haftmann@60352  967 instantiation nat :: "Divides.div"  haftmann@60352  968 begin  haftmann@60352  969 haftmann@60352  970 definition divide_nat where  haftmann@60352  971  div_nat_def: "divide m n = fst (divmod_nat m n)"  haftmann@60352  972 haftmann@60352  973 definition mod_nat where  haftmann@60352  974  "m mod n = snd (divmod_nat m n)"  haftmann@60352  975   haftmann@60352  976 instance ..  haftmann@60352  977 haftmann@60352  978 end  haftmann@60352  979 huffman@46551  980 instantiation nat :: semiring_div  huffman@46551  981 begin  huffman@46551  982 huffman@46551  983 lemma fst_divmod_nat [simp]:  huffman@46551  984  "fst (divmod_nat m n) = m div n"  huffman@46551  985  by (simp add: div_nat_def)  huffman@46551  986 huffman@46551  987 lemma snd_divmod_nat [simp]:  huffman@46551  988  "snd (divmod_nat m n) = m mod n"  huffman@46551  989  by (simp add: mod_nat_def)  huffman@46551  990 haftmann@33340  991 lemma divmod_nat_div_mod:  haftmann@33340  992  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  993  by (simp add: prod_eq_iff)  haftmann@26100  994 huffman@47135  995 lemma div_nat_unique:  haftmann@33340  996  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  997  shows "m div n = q"  huffman@47135  998  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  999 huffman@47135  1000 lemma mod_nat_unique:  haftmann@33340  1001  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  1002  shows "m mod n = r"  huffman@47135  1003  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  1004 haftmann@33340  1005 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  1006  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  1007 huffman@47136  1008 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  1009  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  1010 huffman@47136  1011 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  1012  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  1013 huffman@47137  1014 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  1015  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  1016 haftmann@33340  1017 lemma divmod_nat_step:  haftmann@26100  1018  assumes "0 < n" and "n \ m"  haftmann@33340  1019  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  1020 proof (rule divmod_nat_unique)  huffman@47134  1021  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  1022  by (rule divmod_nat_rel)  huffman@47134  1023  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  1024  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  1025 qed  haftmann@25942  1026 wenzelm@26300  1027 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  1028 haftmann@26100  1029 lemma div_less [simp]:  haftmann@26100  1030  fixes m n :: nat  haftmann@26100  1031  assumes "m < n"  haftmann@26100  1032  shows "m div n = 0"  huffman@46551  1033  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  1034 haftmann@26100  1035 lemma le_div_geq:  haftmann@26100  1036  fixes m n :: nat  haftmann@26100  1037  assumes "0 < n" and "n \ m"  haftmann@26100  1038  shows "m div n = Suc ((m - n) div n)"  huffman@46551  1039  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  1040 haftmann@26100  1041 lemma mod_less [simp]:  haftmann@26100  1042  fixes m n :: nat  haftmann@26100  1043  assumes "m < n"  haftmann@26100  1044  shows "m mod n = m"  huffman@46551  1045  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  1046 haftmann@26100  1047 lemma le_mod_geq:  haftmann@26100  1048  fixes m n :: nat  haftmann@26100  1049  assumes "n \ m"  haftmann@26100  1050  shows "m mod n = (m - n) mod n"  huffman@46551  1051  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  1052 huffman@47136  1053 instance proof  huffman@47136  1054  fix m n :: nat  huffman@47136  1055  show "m div n * n + m mod n = m"  huffman@47136  1056  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  1057 next  huffman@47136  1058  fix m n q :: nat  huffman@47136  1059  assume "n \ 0"  huffman@47136  1060  then show "(q + m * n) div n = m + q div n"  huffman@47136  1061  by (induct m) (simp_all add: le_div_geq)  huffman@47136  1062 next  huffman@47136  1063  fix m n q :: nat  huffman@47136  1064  assume "m \ 0"  huffman@47136  1065  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  1066  unfolding divmod_nat_rel_def  huffman@47136  1067  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  1068  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  1069  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  1070  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  1071 next  huffman@47136  1072  fix n :: nat show "n div 0 = 0"  haftmann@33340  1073  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  1074 next  huffman@47136  1075  fix n :: nat show "0 div n = 0"  huffman@47136  1076  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  1077 qed  haftmann@26100  1078 haftmann@25942  1079 end  paulson@14267  1080 haftmann@33361  1081 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  1082  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  blanchet@55414  1083  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)  haftmann@33361  1084 haftmann@26100  1085 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  1086 wenzelm@51299  1087 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"  wenzelm@51299  1088 haftmann@30934  1089 ML {*  wenzelm@43594  1090 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  1091 (  haftmann@60352  1092  val div_name = @{const_name divide};  haftmann@30934  1093  val mod_name = @{const_name mod};  haftmann@30934  1094  val mk_binop = HOLogic.mk_binop;  huffman@48561  1095  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  huffman@48561  1096  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  huffman@48561  1097  fun mk_sum [] = HOLogic.zero  huffman@48561  1098  | mk_sum [t] = t  huffman@48561  1099  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  huffman@48561  1100  fun dest_sum tm =  huffman@48561  1101  if HOLogic.is_zero tm then []  huffman@48561  1102  else  huffman@48561  1103  (case try HOLogic.dest_Suc tm of  huffman@48561  1104  SOME t => HOLogic.Suc_zero :: dest_sum t  huffman@48561  1105  | NONE =>  huffman@48561  1106  (case try dest_plus tm of  huffman@48561  1107  SOME (t, u) => dest_sum t @ dest_sum u  huffman@48561  1108  | NONE => [tm]));  haftmann@25942  1109 haftmann@30934  1110  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  1111 haftmann@30934  1112  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@57514  1113  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))  wenzelm@41550  1114 )  haftmann@25942  1115 *}  haftmann@25942  1116 wenzelm@43594  1117 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  1118 haftmann@26100  1119 haftmann@26100  1120 subsubsection {* Quotient *}  haftmann@26100  1121 haftmann@26100  1122 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  1123 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  1124 haftmann@26100  1125 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  1126 by (simp add: div_geq)  haftmann@26100  1127 haftmann@26100  1128 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  1129 by simp  haftmann@26100  1130 haftmann@26100  1131 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  1132 by simp  haftmann@26100  1133 haftmann@53066  1134 lemma div_positive:  haftmann@53066  1135  fixes m n :: nat  haftmann@53066  1136  assumes "n > 0"  haftmann@53066  1137  assumes "m \ n"  haftmann@53066  1138  shows "m div n > 0"  haftmann@53066  1139 proof -  haftmann@53066  1140  from m \ n obtain q where "m = n + q"  haftmann@53066  1141  by (auto simp add: le_iff_add)  haftmann@53066  1142  with n > 0 show ?thesis by simp  haftmann@53066  1143 qed  haftmann@53066  1144 hoelzl@59000  1145 lemma div_eq_0_iff: "(a div b::nat) = 0 \ a < b \ b = 0"  hoelzl@59000  1146  by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)  haftmann@25942  1147 haftmann@25942  1148 subsubsection {* Remainder *}  haftmann@25942  1149 haftmann@26100  1150 lemma mod_less_divisor [simp]:  haftmann@26100  1151  fixes m n :: nat  haftmann@26100  1152  assumes "n > 0"  haftmann@26100  1153  shows "m mod n < (n::nat)"  haftmann@33340  1154  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  1155 haftmann@51173  1156 lemma mod_Suc_le_divisor [simp]:  haftmann@51173  1157  "m mod Suc n \ n"  haftmann@51173  1158  using mod_less_divisor [of "Suc n" m] by arith  haftmann@51173  1159 haftmann@26100  1160 lemma mod_less_eq_dividend [simp]:  haftmann@26100  1161  fixes m n :: nat  haftmann@26100  1162  shows "m mod n \ m"  haftmann@26100  1163 proof (rule add_leD2)  haftmann@26100  1164  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  1165  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  1166 qed  haftmann@26100  1167 haftmann@26100  1168 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  1169 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  1170 haftmann@26100  1171 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  1172 by (simp add: le_mod_geq)  haftmann@26100  1173 paulson@14267  1174 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  1175 by (induct m) (simp_all add: mod_geq)  paulson@14267  1176 paulson@14267  1177 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  1178 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  1179  using mod_div_equality2 [of n m] by arith  paulson@14267  1180 nipkow@15439  1181 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  1182  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  1183  apply simp  wenzelm@22718  1184  done  paulson@14267  1185 haftmann@26100  1186 subsubsection {* Quotient and Remainder *}  paulson@14267  1187 haftmann@33340  1188 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  1189  "divmod_nat_rel b c (q, r)  haftmann@33340  1190  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  1191 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1192 haftmann@30923  1193 lemma div_mult1_eq:  haftmann@30923  1194  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  1195 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1196 haftmann@33340  1197 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  1198  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  1199  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  1200 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1201 paulson@14267  1202 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  1203 lemma div_add1_eq:  nipkow@25134  1204  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  1205 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1206 haftmann@33340  1207 lemma divmod_nat_rel_mult2_eq:  haftmann@60352  1208  assumes "divmod_nat_rel a b (q, r)"  haftmann@60352  1209  shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@60352  1210 proof -  haftmann@60352  1211  { assume "r < b" and "0 < c"  haftmann@60352  1212  then have "b * (q mod c) + r < b * c"  haftmann@60352  1213  apply (cut_tac m = q and n = c in mod_less_divisor)  haftmann@60352  1214  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  haftmann@60352  1215  apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)  haftmann@60352  1216  apply (simp add: add_mult_distrib2)  haftmann@60352  1217  done  haftmann@60352  1218  then have "r + b * (q mod c) < b * c"  haftmann@60352  1219  by (simp add: ac_simps)  haftmann@60352  1220  } with assms show ?thesis  haftmann@60352  1221  by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])  haftmann@60352  1222 qed  haftmann@60352  1223   blanchet@55085  1224 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"  huffman@47135  1225 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  1226 blanchet@55085  1227 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"  haftmann@57512  1228 by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  1229 haftmann@58786  1230 instance nat :: semiring_numeral_div  haftmann@58786  1231  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)  haftmann@58786  1232 paulson@14267  1233 huffman@46551  1234 subsubsection {* Further Facts about Quotient and Remainder *}  paulson@14267  1235 haftmann@58786  1236 lemma div_1 [simp]:  haftmann@58786  1237  "m div Suc 0 = m"  haftmann@58786  1238  using div_by_1 [of m] by simp  paulson@14267  1239 paulson@14267  1240 (* Monotonicity of div in first argument *)  haftmann@30923  1241 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  1242  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  1243 apply (case_tac "k=0", simp)  paulson@15251  1244 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  1245 apply (case_tac "n= k *)  paulson@14267  1249 apply (case_tac "m=k *)  nipkow@15439  1253 apply (simp add: div_geq diff_le_mono)  paulson@14267  1254 done  paulson@14267  1255 paulson@14267  1256 (* Antimonotonicity of div in second argument *)  paulson@14267  1257 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  1258 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  1267  prefer 2  paulson@14267  1268  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  1269 apply (rule le_trans, simp)  nipkow@15439  1270 apply (simp)  paulson@14267  1271 done  paulson@14267  1272 paulson@14267  1273 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  1274 apply (case_tac "n=0", simp)  paulson@14267  1275 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  1276 apply (rule div_le_mono2)  paulson@14267  1277 apply (simp_all (no_asm_simp))  paulson@14267  1278 done  paulson@14267  1279 wenzelm@22718  1280 (* Similar for "less than" *)  huffman@47138  1281 lemma div_less_dividend [simp]:  huffman@47138  1282  "\(1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  1283 apply (induct m rule: nat_less_induct)  paulson@14267  1284 apply (rename_tac "m")  paulson@14267  1285 apply (case_tac "m Suc(na) *)  paulson@16796  1304 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  1305 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  1306 done  paulson@14267  1307 paulson@14267  1308 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  1309 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  1310 wenzelm@22718  1311 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  1312 paulson@14267  1313 (*Loses information, namely we also have rq. m = r + q * d"  haftmann@57514  1318 proof -  haftmann@57514  1319  from mod_div_equality obtain q where "q * d + m mod d = m" by blast  haftmann@57514  1320  with assms have "m = r + q * d" by simp  haftmann@57514  1321  then show ?thesis ..  haftmann@57514  1322 qed  paulson@14267  1323 nipkow@13152  1324 lemma split_div:  nipkow@13189  1325  "P(n div k :: nat) =  nipkow@13189  1326  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  1327  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1328 proof  nipkow@13189  1329  assume P: ?P  nipkow@13189  1330  show ?Q  nipkow@13189  1331  proof (cases)  nipkow@13189  1332  assume "k = 0"  haftmann@27651  1333  with P show ?Q by simp  nipkow@13189  1334  next  nipkow@13189  1335  assume not0: "k \ 0"  nipkow@13189  1336  thus ?Q  nipkow@13189  1337  proof (simp, intro allI impI)  nipkow@13189  1338  fix i j  nipkow@13189  1339  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  1340  show "P i"  nipkow@13189  1341  proof (cases)  wenzelm@22718  1342  assume "i = 0"  wenzelm@22718  1343  with n j P show "P i" by simp  nipkow@13189  1344  next  wenzelm@22718  1345  assume "i \ 0"  haftmann@57514  1346  with not0 n j P show "P i" by(simp add:ac_simps)  nipkow@13189  1347  qed  nipkow@13189  1348  qed  nipkow@13189  1349  qed  nipkow@13189  1350 next  nipkow@13189  1351  assume Q: ?Q  nipkow@13189  1352  show ?P  nipkow@13189  1353  proof (cases)  nipkow@13189  1354  assume "k = 0"  haftmann@27651  1355  with Q show ?P by simp  nipkow@13189  1356  next  nipkow@13189  1357  assume not0: "k \ 0"  nipkow@13189  1358  with Q have R: ?R by simp  nipkow@13189  1359  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1360  show ?P by simp  nipkow@13189  1361  qed  nipkow@13189  1362 qed  nipkow@13189  1363 berghofe@13882  1364 lemma split_div_lemma:  haftmann@26100  1365  assumes "0 < n"  haftmann@26100  1366  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  1367 proof  haftmann@26100  1368  assume ?rhs  haftmann@26100  1369  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  1370  then have A: "n * q \ m" by simp  haftmann@26100  1371  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  1372  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  1373  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  1374  with nq have "m < n + n * q" by simp  haftmann@26100  1375  then have B: "m < n * Suc q" by simp  haftmann@26100  1376  from A B show ?lhs ..  haftmann@26100  1377 next  haftmann@26100  1378  assume P: ?lhs  haftmann@33340  1379  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@57514  1380  unfolding divmod_nat_rel_def by (auto simp add: ac_simps)  haftmann@33340  1381  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  1382  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  1383  then show ?rhs by simp  haftmann@26100  1384 qed  berghofe@13882  1385 berghofe@13882  1386 theorem split_div':  berghofe@13882  1387  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  1388  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  1389  apply (case_tac "0 < n")  berghofe@13882  1390  apply (simp only: add: split_div_lemma)  haftmann@27651  1391  apply simp_all  berghofe@13882  1392  done  berghofe@13882  1393 nipkow@13189  1394 lemma split_mod:  nipkow@13189  1395  "P(n mod k :: nat) =  nipkow@13189  1396  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  1397  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1398 proof  nipkow@13189  1399  assume P: ?P  nipkow@13189  1400  show ?Q  nipkow@13189  1401  proof (cases)  nipkow@13189  1402  assume "k = 0"  haftmann@27651  1403  with P show ?Q by simp  nipkow@13189  1404  next  nipkow@13189  1405  assume not0: "k \ 0"  nipkow@13189  1406  thus ?Q  nipkow@13189  1407  proof (simp, intro allI impI)  nipkow@13189  1408  fix i j  nipkow@13189  1409  assume "n = k*i + j" "j < k"  haftmann@58786  1410  thus "P j" using not0 P by (simp add: ac_simps)  nipkow@13189  1411  qed  nipkow@13189  1412  qed  nipkow@13189  1413 next  nipkow@13189  1414  assume Q: ?Q  nipkow@13189  1415  show ?P  nipkow@13189  1416  proof (cases)  nipkow@13189  1417  assume "k = 0"  haftmann@27651  1418  with Q show ?P by simp  nipkow@13189  1419  next  nipkow@13189  1420  assume not0: "k \ 0"  nipkow@13189  1421  with Q have R: ?R by simp  nipkow@13189  1422  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1423  show ?P by simp  nipkow@13189  1424  qed  nipkow@13189  1425 qed  nipkow@13189  1426 berghofe@13882  1427 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  1428  using mod_div_equality [of m n] by arith  huffman@47138  1429 huffman@47138  1430 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  1431  using mod_div_equality [of m n] by arith  huffman@47138  1432 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  1433 noschinl@52398  1434 lemma div_eq_dividend_iff: "a \ 0 \ (a :: nat) div b = a \ b = 1"  noschinl@52398  1435  apply rule  noschinl@52398  1436  apply (cases "b = 0")  noschinl@52398  1437  apply simp_all  noschinl@52398  1438  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)  noschinl@52398  1439  done  noschinl@52398  1440 haftmann@22800  1441 huffman@46551  1442 subsubsection {* An induction'' law for modulus arithmetic. *}  paulson@14640  1443 paulson@14640  1444 lemma mod_induct_0:  paulson@14640  1445  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1446  and base: "P i" and i: "i(P 0)"  paulson@14640  1450  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1452  proof  paulson@14640  1453  fix k  paulson@14640  1454  show "?A k"  paulson@14640  1455  proof (induct k)  paulson@14640  1456  show "?A 0" by simp -- "by contradiction"  paulson@14640  1457  next  paulson@14640  1458  fix n  paulson@14640  1459  assume ih: "?A n"  paulson@14640  1460  show "?A (Suc n)"  paulson@14640  1461  proof (clarsimp)  wenzelm@22718  1462  assume y: "P (p - Suc n)"  wenzelm@22718  1463  have n: "Suc n < p"  wenzelm@22718  1464  proof (rule ccontr)  wenzelm@22718  1465  assume "\(Suc n < p)"  wenzelm@22718  1466  hence "p - Suc n = 0"  wenzelm@22718  1467  by simp  wenzelm@22718  1468  with y contra show "False"  wenzelm@22718  1469  by simp  wenzelm@22718  1470  qed  wenzelm@22718  1471  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1472  from p have "p - Suc n < p" by arith  wenzelm@22718  1473  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1474  by blast  wenzelm@22718  1475  show "False"  wenzelm@22718  1476  proof (cases "n=0")  wenzelm@22718  1477  case True  wenzelm@22718  1478  with z n2 contra show ?thesis by simp  wenzelm@22718  1479  next  wenzelm@22718  1480  case False  wenzelm@22718  1481  with p have "p-n < p" by arith  wenzelm@22718  1482  with z n2 False ih show ?thesis by simp  wenzelm@22718  1483  qed  paulson@14640  1484  qed  paulson@14640  1485  qed  paulson@14640  1486  qed  paulson@14640  1487  moreover  paulson@14640  1488  from i obtain k where "0 i+k=p"  paulson@14640  1489  by (blast dest: less_imp_add_positive)  paulson@14640  1490  hence "0 i=p-k" by auto  paulson@14640  1491  moreover  paulson@14640  1492  note base  paulson@14640  1493  ultimately  paulson@14640  1494  show "False" by blast  paulson@14640  1495 qed  paulson@14640  1496 paulson@14640  1497 lemma mod_induct:  paulson@14640  1498  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1499  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1506  proof (induct j)  paulson@14640  1507  from step base i show "?A 0"  wenzelm@22718  1508  by (auto elim: mod_induct_0)  paulson@14640  1509  next  paulson@14640  1510  fix k  paulson@14640  1511  assume ih: "?A k"  paulson@14640  1512  show "?A (Suc k)"  paulson@14640  1513  proof  wenzelm@22718  1514  assume suc: "Suc k < p"  wenzelm@22718  1515  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1540 proof -  boehmes@35815  1541  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1542  moreover have "m mod 2 < 2" by simp  haftmann@33296  1543  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1544  then show ?thesis by auto  haftmann@33296  1545 qed  haftmann@33296  1546 haftmann@33296  1547 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1548  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1549  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1550 haftmann@33296  1551 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1552 by (simp add: Suc3_eq_add_3)  haftmann@33296  1553 haftmann@33296  1554 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1555 by (simp add: Suc3_eq_add_3)  haftmann@33296  1556 haftmann@33296  1557 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1558 by (simp add: Suc3_eq_add_3)  haftmann@33296  1559 haftmann@33296  1560 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1561 by (simp add: Suc3_eq_add_3)  haftmann@33296  1562 huffman@47108  1563 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1564 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1565 haftmann@33361  1566 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1567 apply (induct "m")  haftmann@33361  1568 apply (simp_all add: mod_Suc)  haftmann@33361  1569 done  haftmann@33361  1570 huffman@47108  1571 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1572 huffman@47138  1573 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1574 by (simp add: div_le_mono)  haftmann@33361  1575 haftmann@33361  1576 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1577 by (cases n) simp_all  haftmann@33361  1578 boehmes@35815  1579 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1580 proof -  boehmes@35815  1581  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1582  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1583 qed  haftmann@33361  1584 haftmann@33361  1585 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1586 proof -  haftmann@33361  1587  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1588  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1589  finally show ?thesis .  haftmann@33361  1590 qed  haftmann@33361  1591 haftmann@33361  1592 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1593 apply (subst mod_Suc [of m])  haftmann@33361  1594 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1595 done  haftmann@33361  1596 huffman@47108  1597 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1598  fixes n :: nat  huffman@47108  1599  shows "n mod 2 \ 0 \ n mod 2 = 1"  haftmann@58786  1600  by (fact not_mod_2_eq_0_eq_1)  haftmann@58786  1601   haftmann@58778  1602 lemma even_Suc_div_two [simp]:  haftmann@58778  1603  "even n \ Suc n div 2 = n div 2"  haftmann@58778  1604  using even_succ_div_two [of n] by simp  haftmann@58778  1605   haftmann@58778  1606 lemma odd_Suc_div_two [simp]:  haftmann@58778  1607  "odd n \ Suc n div 2 = Suc (n div 2)"  haftmann@58778  1608  using odd_succ_div_two [of n] by simp  haftmann@58778  1609 haftmann@58834  1610 lemma odd_two_times_div_two_nat [simp]:  haftmann@60352  1611  assumes "odd n"  haftmann@60352  1612  shows "2 * (n div 2) = n - (1 :: nat)"  haftmann@60352  1613 proof -  haftmann@60352  1614  from assms have "2 * (n div 2) + 1 = n"  haftmann@60352  1615  by (rule odd_two_times_div_two_succ)  haftmann@60352  1616  then have "Suc (2 * (n div 2)) - 1 = n - 1"  haftmann@60352  1617  by simp  haftmann@60352  1618  then show ?thesis  haftmann@60352  1619  by simp  haftmann@60352  1620 qed  haftmann@58778  1621 haftmann@58834  1622 lemma odd_Suc_minus_one [simp]:  haftmann@58834  1623  "odd n \ Suc (n - Suc 0) = n"  haftmann@58834  1624  by (auto elim: oddE)  haftmann@58834  1625 haftmann@58778  1626 lemma parity_induct [case_names zero even odd]:  haftmann@58778  1627  assumes zero: "P 0"  haftmann@58778  1628  assumes even: "\n. P n \ P (2 * n)"  haftmann@58778  1629  assumes odd: "\n. P n \ P (Suc (2 * n))"  haftmann@58778  1630  shows "P n"  haftmann@58778  1631 proof (induct n rule: less_induct)  haftmann@58778  1632  case (less n)  haftmann@58778  1633  show "P n"  haftmann@58778  1634  proof (cases "n = 0")  haftmann@58778  1635  case True with zero show ?thesis by simp  haftmann@58778  1636  next  haftmann@58778  1637  case False  haftmann@58778  1638  with less have hyp: "P (n div 2)" by simp  haftmann@58778  1639  show ?thesis  haftmann@58778  1640  proof (cases "even n")  haftmann@58778  1641  case True  haftmann@58778  1642  with hyp even [of "n div 2"] show ?thesis  haftmann@58834  1643  by simp  haftmann@58778  1644  next  haftmann@58778  1645  case False  haftmann@58778  1646  with hyp odd [of "n div 2"] show ?thesis  haftmann@58834  1647  by simp  haftmann@58778  1648  qed  haftmann@58778  1649  qed  haftmann@58778  1650 qed  haftmann@58778  1651 haftmann@33361  1652 haftmann@33361  1653 subsection {* Division on @{typ int} *}  haftmann@33361  1654 haftmann@33361  1655 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1656  --{*definition of quotient and remainder*}  huffman@47139  1657  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1658  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1659 haftmann@53067  1660 text {*  haftmann@53067  1661  The following algorithmic devlopment actually echos what has already  haftmann@53067  1662  been developed in class @{class semiring_numeral_div}. In the long  haftmann@53067  1663  run it seems better to derive division on @{typ int} just from  haftmann@53067  1664  division on @{typ nat} and instantiate @{class semiring_numeral_div}  haftmann@53067  1665  accordingly.  haftmann@53067  1666 *}  haftmann@53067  1667 haftmann@33361  1668 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1669  --{*for the division algorithm*}  huffman@47108  1670  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1671  else (2 * q, r))"  haftmann@33361  1672 haftmann@33361  1673 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1674 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1675  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1676  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1677 by auto  haftmann@33361  1678 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1679  (auto simp add: mult_2)  haftmann@33361  1680 haftmann@33361  1681 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1682 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1683  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1684  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1685 by auto  haftmann@33361  1686 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1687  (auto simp add: mult_2)  haftmann@33361  1688 haftmann@33361  1689 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1690 haftmann@33361  1691 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1692  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1693  including the special case @{text "a=0, b<0"} because  haftmann@33361  1694  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1695  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1696  else if a = 0 then (0, 0)  huffman@46560  1697  else apsnd uminus (negDivAlg (-a) (-b))  haftmann@33361  1698  else  haftmann@33361  1699  if 0 < b then negDivAlg a b  huffman@46560  1700  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1701 haftmann@33361  1702 instantiation int :: Divides.div  haftmann@33361  1703 begin  haftmann@33361  1704 haftmann@60352  1705 definition divide_int where  haftmann@60352  1706  div_int_def: "divide a b = fst (divmod_int a b)"  haftmann@60352  1707 haftmann@60352  1708 definition mod_int where  haftmann@60352  1709  "a mod b = snd (divmod_int a b)"  haftmann@60352  1710 haftmann@60352  1711 instance ..  haftmann@60352  1712 haftmann@60352  1713 end  haftmann@33361  1714 huffman@46551  1715 lemma fst_divmod_int [simp]:  huffman@46551  1716  "fst (divmod_int a b) = a div b"  huffman@46551  1717  by (simp add: div_int_def)  huffman@46551  1718 huffman@46551  1719 lemma snd_divmod_int [simp]:  huffman@46551  1720  "snd (divmod_int a b) = a mod b"  huffman@46551  1721  by (simp add: mod_int_def)  huffman@46551  1722 haftmann@33361  1723 lemma divmod_int_mod_div:  haftmann@33361  1724  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1725  by (simp add: prod_eq_iff)  haftmann@33361  1726 haftmann@33361  1727 text{*  haftmann@33361  1728 Here is the division algorithm in ML:  haftmann@33361  1729 haftmann@33361  1730 \begin{verbatim}  haftmann@33361  1731  fun posDivAlg (a,b) =  haftmann@33361  1732  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1735  end  haftmann@33361  1736 haftmann@33361  1737  fun negDivAlg (a,b) =  haftmann@33361  1738  if 0\a+b then (~1,a+b)  haftmann@33361  1739  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1740  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1741  end;  haftmann@33361  1742 haftmann@33361  1743  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1744 haftmann@33361  1745  fun divmod (a,b) = if 0\a then  haftmann@33361  1746  if b>0 then posDivAlg (a,b)  haftmann@33361  1747  else if a=0 then (0,0)  haftmann@33361  1748  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1749  else  haftmann@33361  1750  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1760  ==> q' \ (q::int)"  haftmann@33361  1761 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1762  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1763 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1764 apply (erule_tac [2] order_le_less_trans)  webertj@49962  1765  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1766 apply (subgoal_tac "b * q' < b * (1 + q) ")  webertj@49962  1767  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1768 apply (simp add: mult_less_cancel_left)  haftmann@33361  1769 done  haftmann@33361  1770 haftmann@33361  1771 lemma unique_quotient_lemma_neg:  haftmann@33361  1772  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1773  ==> q \ (q'::int)"  haftmann@33361  1774 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1775  auto)  haftmann@33361  1776 haftmann@33361  1777 lemma unique_quotient:  bulwahn@46552  1778  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1779  ==> q = q'"  haftmann@33361  1780 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1781 apply (blast intro: order_antisym  haftmann@33361  1782  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1783  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1784 done  haftmann@33361  1785 haftmann@33361  1786 haftmann@33361  1787 lemma unique_remainder:  bulwahn@46552  1788  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1789  ==> r = r'"  haftmann@33361  1790 apply (subgoal_tac "q = q'")  haftmann@33361  1791  apply (simp add: divmod_int_rel_def)  haftmann@33361  1792 apply (blast intro: unique_quotient)  haftmann@33361  1793 done  haftmann@33361  1794 haftmann@33361  1795 huffman@46551  1796 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}  haftmann@33361  1797 haftmann@33361  1798 text{*And positive divisors*}  haftmann@33361  1799 haftmann@33361  1800 lemma adjust_eq [simp]:  huffman@47108  1801  "adjust b (q, r) =  huffman@47108  1802  (let diff = r - b in  huffman@47108  1803  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1804  else (2*q, r))"  huffman@47108  1805  by (simp add: Let_def adjust_def)  haftmann@33361  1806 haftmann@33361  1807 declare posDivAlg.simps [simp del]  haftmann@33361  1808 haftmann@33361  1809 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1810 lemma posDivAlg_eqn:  haftmann@33361  1811  "0 < b ==>  haftmann@33361  1812  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1818  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1819  using assms  wenzelm@41550  1820  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1821  apply auto  wenzelm@41550  1822  apply (simp add: divmod_int_rel_def)  webertj@49962  1823  apply (subst posDivAlg_eqn, simp add: distrib_left)  wenzelm@41550  1824  apply (case_tac "a < b")  wenzelm@41550  1825  apply simp_all  wenzelm@41550  1826  apply (erule splitE)  haftmann@57514  1827  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)  wenzelm@41550  1828  done  haftmann@33361  1829 haftmann@33361  1830 huffman@46551  1831 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}  haftmann@33361  1832 haftmann@33361  1833 text{*And positive divisors*}  haftmann@33361  1834 haftmann@33361  1835 declare negDivAlg.simps [simp del]  haftmann@33361  1836 haftmann@33361  1837 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1838 lemma negDivAlg_eqn:  haftmann@33361  1839  "0 < b ==>  haftmann@33361  1840  negDivAlg a b =  haftmann@33361  1841  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1842 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1843 haftmann@33361  1844 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1845  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1846 lemma negDivAlg_correct:  haftmann@33361  1847  assumes "a < 0" and "b > 0"  haftmann@33361  1848  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1849  using assms  wenzelm@41550  1850  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1851  apply (auto simp add: linorder_not_le)  wenzelm@41550  1852  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1853  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1854  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1855  apply simp_all  wenzelm@41550  1856  apply (erule splitE)  haftmann@57514  1857  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)  wenzelm@41550  1858  done  haftmann@33361  1859 haftmann@33361  1860 huffman@46551  1861 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}  haftmann@33361  1862 haftmann@33361  1863 (*the case a=0*)  huffman@47139  1864 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1865 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1866 haftmann@33361  1867 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1868 by (subst posDivAlg.simps, auto)  haftmann@33361  1869 huffman@47139  1870 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1871 by (subst posDivAlg.simps, auto)  huffman@47139  1872 haftmann@58410  1873 lemma negDivAlg_minus1 [simp]: "negDivAlg (- 1) b = (- 1, b - 1)"  haftmann@33361  1874 by (subst negDivAlg.simps, auto)  haftmann@33361  1875 huffman@46560  1876 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1877 by (auto simp add: divmod_int_rel_def)  huffman@47139  1878 huffman@47139  1879 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1880 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1881 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1882  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1883 huffman@47141  1884 lemma divmod_int_unique:  huffman@47141  1885  assumes "divmod_int_rel a b qr"  huffman@47141  1886  shows "divmod_int a b = qr"  huffman@47141  1887  using assms divmod_int_correct [of a b]  huffman@47141  1888  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1889  by (metis pair_collapse)  huffman@47141  1890 huffman@47141  1891 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1892  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1893 huffman@47141  1894 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1895  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1896 huffman@47141  1897 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1898  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1899 huffman@47141  1900 instance int :: ring_div  huffman@47141  1901 proof  huffman@47141  1902  fix a b :: int  huffman@47141  1903  show "a div b * b + a mod b = a"  huffman@47141  1904  using divmod_int_rel_div_mod [of a b]  haftmann@57512  1905  unfolding divmod_int_rel_def by (simp add: mult.commute)  huffman@47141  1906 next  huffman@47141  1907  fix a b c :: int  huffman@47141  1908  assume "b \ 0"  huffman@47141  1909  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1910  using divmod_int_rel_div_mod [of a b]  huffman@47141  1911  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1912  thus "(a + c * b) div b = c + a div b"  huffman@47141  1913  by (rule div_int_unique)  huffman@47141  1914 next  huffman@47141  1915  fix a b c :: int  huffman@47141  1916  assume "c \ 0"  huffman@47141  1917  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1918  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1919  unfolding divmod_int_rel_def  huffman@47141  1920  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1921  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1922  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1923  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1924  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1925  thus "(c * a) div (c * b) = a div b"  huffman@47141  1926  by (rule div_int_unique)  huffman@47141  1927 next  huffman@47141  1928  fix a :: int show "a div 0 = 0"  huffman@47141  1929  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1930 next  huffman@47141  1931  fix a :: int show "0 div a = 0"  huffman@47141  1932  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1933 qed  huffman@47141  1934 haftmann@33361  1935 text{*Basic laws about division and remainder*}  haftmann@33361  1936 haftmann@33361  1937 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1938  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1939 haftmann@33361  1940 text {* Tool setup *}  haftmann@33361  1941 haftmann@33361  1942 ML {*  wenzelm@43594  1943 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1944 (  haftmann@60352  1945  val div_name = @{const_name Rings.divide};  haftmann@33361  1946  val mod_name = @{const_name mod};  haftmann@33361  1947  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1948  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1949  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1950 huffman@47165  1951  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1952 haftmann@33361  1953  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@59556  1954  (@{thm diff_conv_add_uminus} :: @{thms add_0_left add_0_right} @ @{thms ac_simps}))  wenzelm@41550  1955 )  haftmann@33361  1956 *}  haftmann@33361  1957 wenzelm@43594  1958 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1959 huffman@47141  1960 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1961  using divmod_int_correct [of a b]  huffman@47141  1962  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1963 wenzelm@45607  1964 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1965  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1966 huffman@47141  1967 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1968  using divmod_int_correct [of a b]  huffman@47141  1969  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1970 wenzelm@45607  1971 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1972  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1973 haftmann@33361  1974 huffman@46551  1975 subsubsection {* General Properties of div and mod *}  haftmann@33361  1976 haftmann@33361  1977 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1978 apply (rule div_int_unique)  haftmann@33361  1979 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1980 done  haftmann@33361  1981 haftmann@33361  1982 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1983 apply (rule div_int_unique)  haftmann@33361  1984 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1985 done  haftmann@33361  1986 haftmann@33361  1987 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1988 apply (rule div_int_unique)  haftmann@33361  1989 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1990 done  haftmann@33361  1991 haftmann@33361  1992 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1993 haftmann@33361  1994 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1995 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1996 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1997 done  haftmann@33361  1998 haftmann@33361  1999 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  2000 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  2001 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  2002 done  haftmann@33361  2003 haftmann@33361  2004 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  2005 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  2006 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  2007 done  haftmann@33361  2008 haftmann@33361  2009 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  2010 haftmann@33361  2011 huffman@46551  2012 subsubsection {* Laws for div and mod with Unary Minus *}  haftmann@33361  2013 haftmann@33361  2014 lemma zminus1_lemma:  huffman@47139  2015  "divmod_int_rel a b (q, r) ==> b \ 0  haftmann@33361  2016  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  2017  if r=0 then 0 else b-r)"  haftmann@33361  2018 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  2019 haftmann@33361  2020 haftmann@33361  2021 lemma zdiv_zminus1_eq_if:  haftmann@33361  2022  "b \ (0::int)  haftmann@33361  2023  ==> (-a) div b =  haftmann@33361  2024  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  2025 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  2026 haftmann@33361  2027 lemma zmod_zminus1_eq_if:  haftmann@33361  2028  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  2029 apply (case_tac "b = 0", simp)  huffman@47140  2030 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  2031 done  haftmann@33361  2032 haftmann@33361  2033 lemma zmod_zminus1_not_zero:  haftmann@33361  2034  fixes k l :: int  haftmann@33361  2035  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  2036  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  2037 haftmann@33361  2038 lemma zdiv_zminus2_eq_if:  haftmann@33361  2039  "b \ (0::int)  haftmann@33361  2040  ==> a div (-b) =  haftmann@33361  2041  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  2042 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  2043 haftmann@33361  2044 lemma zmod_zminus2_eq_if:  haftmann@33361  2045  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  2046 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  2047 haftmann@33361  2048 lemma zmod_zminus2_not_zero:  haftmann@33361  2049  fixes k l :: int  haftmann@33361  2050  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  2051  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  2052 haftmann@33361  2053 huffman@46551  2054 subsubsection {* Computation of Division and Remainder *}  haftmann@33361  2055 haftmann@33361  2056 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  2057 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2058 haftmann@33361  2059 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  2060 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2061 haftmann@33361  2062 text{*a positive, b positive *}  haftmann@33361  2063 haftmann@33361  2064 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  2065 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2066 haftmann@33361  2067 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  2068 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2069 haftmann@33361  2070 text{*a negative, b positive *}  haftmann@33361  2071 haftmann@33361  2072 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  2073 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2074 haftmann@33361  2075 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  2076 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2077 haftmann@33361  2078 text{*a positive, b negative *}  haftmann@33361  2079 haftmann@33361  2080 lemma div_pos_neg:  huffman@46560  2081  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  2082 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2083 haftmann@33361  2084 lemma mod_pos_neg:  huffman@46560  2085  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  2086 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2087 haftmann@33361  2088 text{*a negative, b negative *}  haftmann@33361  2089 haftmann@33361  2090 lemma div_neg_neg:  huffman@46560  2091  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  2092 by (simp add: div_int_def divmod_int_def)  haftmann@33361  2093 haftmann@33361  2094 lemma mod_neg_neg:  huffman@46560  2095  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  2096 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  2097 haftmann@33361  2098 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  2099 huffman@45530  2100 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  2101  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  2102 huffman@45530  2103 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  2104  by (rule div_int_unique [of a b q r],  bulwahn@46552  2105  simp add: divmod_int_rel_def)  huffman@45530  2106 huffman@45530  2107 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  2108  by (rule mod_int_unique [of a b q r],  bulwahn@46552  2109  simp add: divmod_int_rel_def)  huffman@45530  2110 huffman@45530  2111 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  2112  by (rule mod_int_unique [of a b q r],  bulwahn@46552  2113  simp add: divmod_int_rel_def)  huffman@45530  2114 haftmann@53069  2115 text {*  haftmann@53069  2116  numeral simprocs -- high chance that these can be replaced  haftmann@53069  2117  by divmod algorithm from @{class semiring_numeral_div}  haftmann@53069  2118 *}  haftmann@53069  2119 haftmann@33361  2120 ML {*  haftmann@33361  2121 local  huffman@45530  2122  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  2123  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  2124  val times = @{term "times :: int \ int \ int"}  huffman@45530  2125  val zero = @{term "0 :: int"}  huffman@45530  2126  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  2127  val le = @{term "op \ :: int \ int \ bool"}  haftmann@54489  2128  val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]  wenzelm@58847  2129  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  wenzelm@58847  2130  (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));  wenzelm@51717  2131  fun binary_proc proc ctxt ct =  haftmann@33361  2132  (case Thm.term_of ct of  haftmann@33361  2133  _ $t$ u =>  wenzelm@59058  2134  (case try (apply2 ((snd o HOLogic.dest_number))) (t, u) of  wenzelm@51717  2135  SOME args => proc ctxt args  haftmann@33361  2136  | NONE => NONE)  haftmann@33361  2137  | _ => NONE);  haftmann@33361  2138 in  huffman@45530  2139  fun divmod_proc posrule negrule =  huffman@45530  2140  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  wenzelm@59058  2141  if b = 0 then NONE  wenzelm@59058  2142  else  wenzelm@59058  2143  let  wenzelm@59058  2144  val (q, r) = apply2 mk_number (Integer.div_mod a b)  wenzelm@59058  2145  val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)  wenzelm@59058  2146  val (goal2, goal3, rule) =  wenzelm@59058  2147  if b > 0  wenzelm@59058  2148  then (le $zero$ r, less $r$ u, posrule RS eq_reflection)  wenzelm@59058  2149  else (le $r$ zero, less $u$ r, negrule RS eq_reflection)  wenzelm@59058  2150  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  2151 end  haftmann@33361  2152 *}  haftmann@33361  2153 huffman@47108  2154 simproc_setup binary_int_div  huffman@47108  2155  ("numeral m div numeral n :: int" |  haftmann@54489  2156  "numeral m div - numeral n :: int" |  haftmann@54489  2157  "- numeral m div numeral n :: int" |  haftmann@54489  2158  "- numeral m div - numeral n :: int") =  huffman@45530  2159  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  2160 huffman@47108  2161 simproc_setup binary_int_mod  huffman@47108  2162  ("numeral m mod numeral n :: int" |  haftmann@54489  2163  "numeral m mod - numeral n :: int" |  haftmann@54489  2164  "- numeral m mod numeral n :: int" |  haftmann@54489  2165  "- numeral m mod - numeral n :: int") =  huffman@45530  2166  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  2167 huffman@47108  2168 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  2169  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  2170 huffman@47108  2171 lemmas negDivAlg_eqn_numeral [simp] =  haftmann@54489  2172  negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w  haftmann@33361  2173 haftmann@33361  2174 haftmann@55172  2175 text {* Special-case simplification: @{text "\1 div z"} and @{text "\1 mod z"} *}  haftmann@55172  2176 haftmann@55172  2177 lemma [simp]:  haftmann@55172  2178  shows div_one_bit0: "1 div numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  2179  and mod_one_bit0: "1 mod numeral (Num.Bit0 v) = (1 :: int)"  wenzelm@55439  2180  and div_one_bit1: "1 div numeral (Num.Bit1 v) = (0 :: int)"  wenzelm@55439  2181  and mod_one_bit1: "1 mod numeral (Num.Bit1 v) = (1 :: int)"  wenzelm@55439  2182  and div_one_neg_numeral: "1 div - numeral v = (- 1 :: int)"  wenzelm@55439  2183  and mod_one_neg_numeral: "1 mod - numeral v = (1 :: int) - numeral v"  haftmann@55172  2184  by (simp_all del: arith_special  haftmann@55172  2185  add: div_pos_pos mod_pos_pos div_pos_neg mod_pos_neg posDivAlg_eqn)  wenzelm@55439  2186 haftmann@55172  2187 lemma [simp]:  haftmann@55172  2188  shows div_neg_one_numeral: "- 1 div numeral v = (- 1 :: int)"  haftmann@55172  2189  and mod_neg_one_numeral: "- 1 mod numeral v = numeral v - (1 :: int)"  haftmann@55172  2190  and div_neg_one_neg_bit0: "- 1 div - numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  2191  and mod_neg_one_neb_bit0: "- 1 mod - numeral (Num.Bit0 v) = (- 1 :: int)"  haftmann@55172  2192  and div_neg_one_neg_bit1: "- 1 div - numeral (Num.Bit1 v) = (0 :: int)"  haftmann@55172  2193  and mod_neg_one_neb_bit1: "- 1 mod - numeral (Num.Bit1 v) = (- 1 :: int)"  haftmann@55172  2194  by (simp_all add: div_eq_minus1 zmod_minus1)  haftmann@33361  2195 haftmann@33361  2196 huffman@46551  2197 subsubsection {* Monotonicity in the First Argument (Dividend) *}  haftmann@33361  2198 haftmann@33361  2199 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  2200 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2201 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  2202 apply (rule unique_quotient_lemma)  haftmann@33361  2203 apply (erule subst)  haftmann@33361  2204 apply (erule subst, simp_all)  haftmann@33361  2205 done  haftmann@33361  2206 haftmann@33361  2207 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  2208 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2209 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  2210 apply (rule unique_quotient_lemma_neg)  haftmann@33361  2211 apply (erule subst)  haftmann@33361  2212 apply (erule subst, simp_all)  haftmann@33361  2213 done  haftmann@33361  2214 haftmann@33361  2215 huffman@46551  2216 subsubsection {* Monotonicity in the Second Argument (Divisor) *}  haftmann@33361  2217 haftmann@33361  2218 lemma q_pos_lemma:  haftmann@33361  2219  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  2220 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  2221  apply (simp add: zero_less_mult_iff)  webertj@49962  2222 apply (simp add: distrib_left)  haftmann@33361  2223 done  haftmann@33361  2224 haftmann@33361  2225 lemma zdiv_mono2_lemma:  haftmann@33361  2226  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  2227  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  2228  ==> q \ (q'::int)"  haftmann@33361  2229 apply (frule q_pos_lemma, assumption+)  haftmann@33361  2230 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  2231  apply (simp add: mult_less_cancel_left)  haftmann@33361  2232 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  2233  prefer 2 apply simp  webertj@49962  2234 apply (simp (no_asm_simp) add: distrib_left)  haftmann@57512  2235 apply (subst add.commute, rule add_less_le_mono, arith)  haftmann@33361  2236 apply (rule mult_right_mono, auto)  haftmann@33361  2237 done  haftmann@33361  2238 haftmann@33361  2239 lemma zdiv_mono2:  haftmann@33361  2240  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  2241 apply (subgoal_tac "b \ 0")  haftmann@33361  2242  prefer 2 apply arith  haftmann@33361  2243 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2244 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2245 apply (rule zdiv_mono2_lemma)  haftmann@33361  2246 apply (erule subst)  haftmann@33361  2247 apply (erule subst, simp_all)  haftmann@33361  2248 done  haftmann@33361  2249 haftmann@33361  2250 lemma q_neg_lemma:  haftmann@33361  2251  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  2252 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  2253  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  2254 done  haftmann@33361  2255 haftmann@33361  2256 lemma zdiv_mono2_neg_lemma:  haftmann@33361  2257  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  2258  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  2259  ==> q' \ (q::int)"  haftmann@33361  2260 apply (frule q_neg_lemma, assumption+)  haftmann@33361  2261 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  2262  apply (simp add: mult_less_cancel_left)  webertj@49962  2263 apply (simp add: distrib_left)  haftmann@33361  2264 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  2265  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  2266 done  haftmann@33361  2267 haftmann@33361  2268 lemma zdiv_mono2_neg:  haftmann@33361  2269  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  2270 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2271 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2272 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  2273 apply (erule subst)  haftmann@33361  2274 apply (erule subst, simp_all)  haftmann@33361  2275 done  haftmann@33361  2276 haftmann@33361  2277 huffman@46551  2278 subsubsection {* More Algebraic Laws for div and mod *}  haftmann@33361  2279 haftmann@33361  2280 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  2281 haftmann@33361  2282 lemma zmult1_lemma:  bulwahn@46552  2283  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  2284  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@57514  2285 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)  haftmann@33361  2286 haftmann@33361  2287 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  2288 apply (case_tac "c = 0", simp)  huffman@47140  2289 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  2290 done  haftmann@33361  2291 haftmann@33361  2292 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  2293 haftmann@33361  2294 lemma zadd1_lemma:  bulwahn@46552  2295  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  2296  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  webertj@49962  2297 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)  haftmann@33361  2298 haftmann@33361  2299 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  2300 lemma zdiv_zadd1_eq:  haftmann@33361  2301  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  2302 apply (case_tac "c = 0", simp)  huffman@47140  2303 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  2304 done  haftmann@33361  2305 haftmann@33361  2306 lemma posDivAlg_div_mod:  haftmann@33361  2307  assumes "k \ 0"  haftmann@33361  2308  and "l \ 0"  haftmann@33361  2309  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  2310 proof (cases "l = 0")  haftmann@33361  2311  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  2312 next  haftmann@33361  2313  case False with assms posDivAlg_correct  haftmann@33361  2314  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  2315  by simp  huffman@47140  2316  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2317  show ?thesis by simp  haftmann@33361  2318 qed  haftmann@33361  2319 haftmann@33361  2320 lemma negDivAlg_div_mod:  haftmann@33361  2321  assumes "k < 0"  haftmann@33361  2322  and "l > 0"  haftmann@33361  2323  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  2324 proof -  haftmann@33361  2325  from assms have "l \ 0" by simp  haftmann@33361  2326  from assms negDivAlg_correct  haftmann@33361  2327  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  2328  by simp  huffman@47140  2329  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2330  show ?thesis by simp  haftmann@33361  2331 qed  haftmann@33361  2332 haftmann@33361  2333 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  2334 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  2335 haftmann@33361  2336 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  2337 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  2338 huffman@47108  2339 lemma zmod_zdiv_equality':  huffman@47108  2340  "(m\int) mod n = m - (m div n) * n"  huffman@47141  2341  using mod_div_equality [of m n] by arith  huffman@47108  2342 haftmann@33361  2343 blanchet@55085  2344 subsubsection {* Proving @{term "a div (b * c) = (a div b) div c"} *}  haftmann@33361  2345 haftmann@33361  2346 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  2347  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  2348  to cause particular problems.*)  haftmann@33361  2349 haftmann@33361  2350 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  2351 blanchet@55085  2352 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b * c < b * (q mod c) + r"  haftmann@33361  2353 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  2354  apply (simp add: algebra_simps)  haftmann@33361  2355 apply (rule order_le_less_trans)  haftmann@33361  2356  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  2357  apply (rule mult_left_mono_neg)  huffman@35216  2358  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  2359  apply (simp)  haftmann@33361  2360 apply (simp)  haftmann@33361  2361 done  haftmann@33361  2362 haftmann@33361  2363 lemma zmult2_lemma_aux2:  haftmann@33361  2364  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  2365 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  2366  apply arith  haftmann@33361  2367 apply (simp add: mult_le_0_iff)  haftmann@33361  2368 done  haftmann@33361  2369 haftmann@33361  2370 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  2371 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  2372 apply arith  haftmann@33361  2373 apply (simp add: zero_le_mult_iff)  haftmann@33361  2374 done  haftmann@33361  2375 haftmann@33361  2376 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  2377 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  2378  apply (simp add: right_diff_distrib)  haftmann@33361  2379 apply (rule order_less_le_trans)  haftmann@33361  2380  apply (erule mult_strict_right_mono)  haftmann@33361  2381  apply (rule_tac [2] mult_left_mono)  haftmann@33361  2382  apply simp  huffman@35216  2383  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  2384 apply simp  haftmann@33361  2385 done  haftmann@33361  2386 bulwahn@46552  2387 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  2388  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@57514  2389 by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff  webertj@49962  2390  zero_less_mult_iff distrib_left [symmetric]  huffman@47139  2391  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  2392 haftmann@53068  2393 lemma zdiv_zmult2_eq:  haftmann@53068  2394  fixes a b c :: int  haftmann@53068  2395  shows "0 \ c \ a div (b * c) = (a div b) div c"  haftmann@33361  2396 apply (case_tac "b = 0", simp)  haftmann@53068  2397 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  2398 done  haftmann@33361  2399 haftmann@33361  2400 lemma zmod_zmult2_eq:  haftmann@53068  2401  fixes a b c :: int  haftmann@53068  2402  shows "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@33361  2403 apply (case_tac "b = 0", simp)  haftmann@53068  2404 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  2405 done  haftmann@33361  2406 huffman@47108  2407 lemma div_pos_geq:  huffman@47108  2408  fixes k l :: int  huffman@47108  2409  assumes "0 < l" and "l \ k"  huffman@47108  2410  shows "k div l = (k - l) div l + 1"  huffman@47108  2411 proof -  huffman@47108  2412  have "k = (k - l) + l" by simp  huffman@47108  2413  then obtain j where k: "k = j + l" ..  huffman@47108  2414  with assms show ?thesis by simp  huffman@47108  2415 qed  huffman@47108  2416 huffman@47108  2417 lemma mod_pos_geq:  huffman@47108  2418  fixes k l :: int  huffman@47108  2419  assumes "0 < l" and "l \ k"  huffman@47108  2420  shows "k mod l = (k - l) mod l"  huffman@47108  2421 proof -  huffman@47108  2422  have "k = (k - l) + l" by simp  huffman@47108  2423  then obtain j where k: "k = j + l" ..  huffman@47108  2424  with assms show ?thesis by simp  huffman@47108  2425 qed  huffman@47108  2426 haftmann@33361  2427 huffman@46551  2428 subsubsection {* Splitting Rules for div and mod *}  haftmann@33361  2429 haftmann@33361  2430 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  2431 haftmann@33361  2432 lemma split_pos_lemma:  haftmann@33361  2433  "0  haftmann@33361  2434  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2435 apply (rule iffI, clarify)  wenzelm@59807  2436  apply (erule_tac P="P x y" for x y in rev_mp)  haftmann@33361  2437  apply (subst mod_add_eq)  haftmann@33361  2438  apply (subst zdiv_zadd1_eq)  haftmann@33361  2439  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  2440 txt{*converse direction*}  haftmann@33361  2441 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2442 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2443 done  haftmann@33361  2444 haftmann@33361  2445 lemma split_neg_lemma:  haftmann@33361  2446  "k<0 ==>  haftmann@33361  2447  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2448 apply (rule iffI, clarify)  wenzelm@59807  2449  apply (erule_tac P="P x y" for x y in rev_mp)  haftmann@33361  2450  apply (subst mod_add_eq)  haftmann@33361  2451  apply (subst zdiv_zadd1_eq)  haftmann@33361  2452  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  2453 txt{*converse direction*}  haftmann@33361  2454 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2455 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2456 done  haftmann@33361  2457 haftmann@33361  2458 lemma split_zdiv:  haftmann@33361  2459  "P(n div k :: int) =  haftmann@33361  2460  ((k = 0 --> P 0) &  haftmann@33361  2461  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2462  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2463 apply (case_tac "k=0", simp)  haftmann@33361  2464 apply (simp only: linorder_neq_iff)  haftmann@33361  2465 apply (erule disjE)  haftmann@33361  2466  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2467  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2468 done  haftmann@33361  2469 haftmann@33361  2470 lemma split_zmod:  haftmann@33361  2471  "P(n mod k :: int) =  haftmann@33361  2472  ((k = 0 --> P n) &  haftmann@33361  2473  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2474  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2475 apply (case_tac "k=0", simp)  haftmann@33361  2476 apply (simp only: linorder_neq_iff)  haftmann@33361  2477 apply (erule disjE)  haftmann@33361  2478  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2479  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2480 done  haftmann@33361  2481 webertj@33730  2482 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  2483  when these are applied to some constant that is of the form  huffman@47108  2484  @{term "numeral k"}: *}  huffman@47108  2485 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  2486 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  2487 haftmann@33361  2488 huffman@47166  2489 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}  huffman@47166  2490 huffman@47166  2491 lemma pos_divmod_int_rel_mult_2:  huffman@47166  2492  assumes "0 \ b"  huffman@47166  2493  assumes "divmod_int_rel a b (q, r)"  huffman@47166  2494  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  2495  using assms unfolding divmod_int_rel_def by auto  huffman@47166  2496 haftmann@54489  2497 declaration {* K (Lin_Arith.add_simps @{thms uminus_numeral_One}) *}  haftmann@54489  2498 huffman@47166  2499 lemma neg_divmod_int_rel_mult_2:  huffman@47166  2500  assumes "b \ 0"  huffman@47166  2501  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  2502  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  2503  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  2504 haftmann@33361  2505 text{*computing div by shifting *}  haftmann@33361  2506 haftmann@33361  2507 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2508  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2509  by (rule div_int_unique)  haftmann@33361  2510 boehmes@35815  2511 lemma neg_zdiv_mult_2:  boehmes@35815  2512  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2513  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2514  by (rule div_int_unique)  haftmann@33361  2515 huffman@47108  2516 (* FIXME: add rules for negative numerals *)  huffman@47108  2517 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2518  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2519  numeral v div (numeral w :: int)"  huffman@47108  2520  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2`