src/HOL/Divides.thy
 author blanchet Wed Feb 12 08:35:57 2014 +0100 (2014-02-12) changeset 55414 eab03e9cee8a parent 55172 92735f0d5302 child 55440 721b4561007a permissions -rw-r--r--
renamed '{prod,sum,bool,unit}_case' to 'case_...'
 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 haftmann@27651  6 header {* The division operators div and mod *}  paulson@3366  7 nipkow@15131  8 theory Divides  huffman@47255  9 imports Nat_Transfer  nipkow@15131  10 begin  paulson@3366  11 haftmann@25942  12 subsection {* Syntactic division operations *}  haftmann@25942  13 haftmann@27651  14 class div = dvd +  haftmann@27540  15  fixes div :: "'a \ 'a \ 'a" (infixl "div" 70)  haftmann@27651  16  and mod :: "'a \ 'a \ 'a" (infixl "mod" 70)  haftmann@27540  17 haftmann@27540  18 haftmann@27651  19 subsection {* Abstract division in commutative semirings. *}  haftmann@25942  20 haftmann@30930  21 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +  haftmann@25942  22  assumes mod_div_equality: "a div b * b + a mod b = a"  haftmann@27651  23  and div_by_0 [simp]: "a div 0 = 0"  haftmann@27651  24  and div_0 [simp]: "0 div a = 0"  haftmann@27651  25  and div_mult_self1 [simp]: "b \ 0 \ (a + c * b) div b = c + a div b"  haftmann@30930  26  and div_mult_mult1 [simp]: "c \ 0 \ (c * a) div (c * b) = a div b"  haftmann@25942  27 begin  haftmann@25942  28 haftmann@26100  29 text {* @{const div} and @{const mod} *}  haftmann@26100  30 haftmann@26062  31 lemma mod_div_equality2: "b * (a div b) + a mod b = a"  haftmann@26062  32  unfolding mult_commute [of b]  haftmann@26062  33  by (rule mod_div_equality)  haftmann@26062  34 huffman@29403  35 lemma mod_div_equality': "a mod b + a div b * b = a"  huffman@29403  36  using mod_div_equality [of a b]  huffman@29403  37  by (simp only: add_ac)  huffman@29403  38 haftmann@26062  39 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"  haftmann@30934  40  by (simp add: mod_div_equality)  haftmann@26062  41 haftmann@26062  42 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"  haftmann@30934  43  by (simp add: mod_div_equality2)  haftmann@26062  44 haftmann@27651  45 lemma mod_by_0 [simp]: "a mod 0 = a"  haftmann@30934  46  using mod_div_equality [of a zero] by simp  haftmann@27651  47 haftmann@27651  48 lemma mod_0 [simp]: "0 mod a = 0"  haftmann@30934  49  using mod_div_equality [of zero a] div_0 by simp  haftmann@27651  50 haftmann@27651  51 lemma div_mult_self2 [simp]:  haftmann@27651  52  assumes "b \ 0"  haftmann@27651  53  shows "(a + b * c) div b = c + a div b"  haftmann@27651  54  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)  haftmann@26100  55 haftmann@54221  56 lemma div_mult_self3 [simp]:  haftmann@54221  57  assumes "b \ 0"  haftmann@54221  58  shows "(c * b + a) div b = c + a div b"  haftmann@54221  59  using assms by (simp add: add.commute)  haftmann@54221  60 haftmann@54221  61 lemma div_mult_self4 [simp]:  haftmann@54221  62  assumes "b \ 0"  haftmann@54221  63  shows "(b * c + a) div b = c + a div b"  haftmann@54221  64  using assms by (simp add: add.commute)  haftmann@54221  65 haftmann@27651  66 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"  haftmann@27651  67 proof (cases "b = 0")  haftmann@27651  68  case True then show ?thesis by simp  haftmann@27651  69 next  haftmann@27651  70  case False  haftmann@27651  71  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"  haftmann@27651  72  by (simp add: mod_div_equality)  haftmann@27651  73  also from False div_mult_self1 [of b a c] have  haftmann@27651  74  "\ = (c + a div b) * b + (a + c * b) mod b"  nipkow@29667  75  by (simp add: algebra_simps)  haftmann@27651  76  finally have "a = a div b * b + (a + c * b) mod b"  webertj@49962  77  by (simp add: add_commute [of a] add_assoc distrib_right)  haftmann@27651  78  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"  haftmann@27651  79  by (simp add: mod_div_equality)  haftmann@27651  80  then show ?thesis by simp  haftmann@27651  81 qed  haftmann@27651  82 haftmann@54221  83 lemma mod_mult_self2 [simp]:  haftmann@54221  84  "(a + b * c) mod b = a mod b"  haftmann@30934  85  by (simp add: mult_commute [of b])  haftmann@27651  86 haftmann@54221  87 lemma mod_mult_self3 [simp]:  haftmann@54221  88  "(c * b + a) mod b = a mod b"  haftmann@54221  89  by (simp add: add.commute)  haftmann@54221  90 haftmann@54221  91 lemma mod_mult_self4 [simp]:  haftmann@54221  92  "(b * c + a) mod b = a mod b"  haftmann@54221  93  by (simp add: add.commute)  haftmann@54221  94 haftmann@27651  95 lemma div_mult_self1_is_id [simp]: "b \ 0 \ b * a div b = a"  haftmann@27651  96  using div_mult_self2 [of b 0 a] by simp  haftmann@27651  97 haftmann@27651  98 lemma div_mult_self2_is_id [simp]: "b \ 0 \ a * b div b = a"  haftmann@27651  99  using div_mult_self1 [of b 0 a] by simp  haftmann@27651  100 haftmann@27651  101 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"  haftmann@27651  102  using mod_mult_self2 [of 0 b a] by simp  haftmann@27651  103 haftmann@27651  104 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"  haftmann@27651  105  using mod_mult_self1 [of 0 a b] by simp  haftmann@26062  106 haftmann@27651  107 lemma div_by_1 [simp]: "a div 1 = a"  haftmann@27651  108  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp  haftmann@27651  109 haftmann@27651  110 lemma mod_by_1 [simp]: "a mod 1 = 0"  haftmann@27651  111 proof -  haftmann@27651  112  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp  haftmann@27651  113  then have "a + a mod 1 = a + 0" by simp  haftmann@27651  114  then show ?thesis by (rule add_left_imp_eq)  haftmann@27651  115 qed  haftmann@27651  116 haftmann@27651  117 lemma mod_self [simp]: "a mod a = 0"  haftmann@27651  118  using mod_mult_self2_is_0 [of 1] by simp  haftmann@27651  119 haftmann@27651  120 lemma div_self [simp]: "a \ 0 \ a div a = 1"  haftmann@27651  121  using div_mult_self2_is_id [of _ 1] by simp  haftmann@27651  122 haftmann@27676  123 lemma div_add_self1 [simp]:  haftmann@27651  124  assumes "b \ 0"  haftmann@27651  125  shows "(b + a) div b = a div b + 1"  haftmann@27651  126  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)  haftmann@26062  127 haftmann@27676  128 lemma div_add_self2 [simp]:  haftmann@27651  129  assumes "b \ 0"  haftmann@27651  130  shows "(a + b) div b = a div b + 1"  haftmann@27651  131  using assms div_add_self1 [of b a] by (simp add: add_commute)  haftmann@27651  132 haftmann@27676  133 lemma mod_add_self1 [simp]:  haftmann@27651  134  "(b + a) mod b = a mod b"  haftmann@27651  135  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)  haftmann@27651  136 haftmann@27676  137 lemma mod_add_self2 [simp]:  haftmann@27651  138  "(a + b) mod b = a mod b"  haftmann@27651  139  using mod_mult_self1 [of a 1 b] by simp  haftmann@27651  140 haftmann@27651  141 lemma mod_div_decomp:  haftmann@27651  142  fixes a b  haftmann@27651  143  obtains q r where "q = a div b" and "r = a mod b"  haftmann@27651  144  and "a = q * b + r"  haftmann@27651  145 proof -  haftmann@27651  146  from mod_div_equality have "a = a div b * b + a mod b" by simp  haftmann@27651  147  moreover have "a div b = a div b" ..  haftmann@27651  148  moreover have "a mod b = a mod b" ..  haftmann@27651  149  note that ultimately show thesis by blast  haftmann@27651  150 qed  haftmann@27651  151 bulwahn@45231  152 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \ b mod a = 0"  haftmann@25942  153 proof  haftmann@25942  154  assume "b mod a = 0"  haftmann@25942  155  with mod_div_equality [of b a] have "b div a * a = b" by simp  haftmann@25942  156  then have "b = a * (b div a)" unfolding mult_commute ..  haftmann@25942  157  then have "\c. b = a * c" ..  haftmann@25942  158  then show "a dvd b" unfolding dvd_def .  haftmann@25942  159 next  haftmann@25942  160  assume "a dvd b"  haftmann@25942  161  then have "\c. b = a * c" unfolding dvd_def .  haftmann@25942  162  then obtain c where "b = a * c" ..  haftmann@25942  163  then have "b mod a = a * c mod a" by simp  haftmann@25942  164  then have "b mod a = c * a mod a" by (simp add: mult_commute)  haftmann@27651  165  then show "b mod a = 0" by simp  haftmann@25942  166 qed  haftmann@25942  167 huffman@29403  168 lemma mod_div_trivial [simp]: "a mod b div b = 0"  huffman@29403  169 proof (cases "b = 0")  huffman@29403  170  assume "b = 0"  huffman@29403  171  thus ?thesis by simp  huffman@29403  172 next  huffman@29403  173  assume "b \ 0"  huffman@29403  174  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"  huffman@29403  175  by (rule div_mult_self1 [symmetric])  huffman@29403  176  also have "\ = a div b"  huffman@29403  177  by (simp only: mod_div_equality')  huffman@29403  178  also have "\ = a div b + 0"  huffman@29403  179  by simp  huffman@29403  180  finally show ?thesis  huffman@29403  181  by (rule add_left_imp_eq)  huffman@29403  182 qed  huffman@29403  183 huffman@29403  184 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"  huffman@29403  185 proof -  huffman@29403  186  have "a mod b mod b = (a mod b + a div b * b) mod b"  huffman@29403  187  by (simp only: mod_mult_self1)  huffman@29403  188  also have "\ = a mod b"  huffman@29403  189  by (simp only: mod_div_equality')  huffman@29403  190  finally show ?thesis .  huffman@29403  191 qed  huffman@29403  192 nipkow@29925  193 lemma dvd_imp_mod_0: "a dvd b \ b mod a = 0"  nipkow@29948  194 by (rule dvd_eq_mod_eq_0[THEN iffD1])  nipkow@29925  195 nipkow@29925  196 lemma dvd_div_mult_self: "a dvd b \ (b div a) * a = b"  nipkow@29925  197 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)  nipkow@29925  198 haftmann@33274  199 lemma dvd_mult_div_cancel: "a dvd b \ a * (b div a) = b"  haftmann@33274  200 by (drule dvd_div_mult_self) (simp add: mult_commute)  haftmann@33274  201 nipkow@30052  202 lemma dvd_div_mult: "a dvd b \ (b div a) * c = b * c div a"  nipkow@30052  203 apply (cases "a = 0")  nipkow@30052  204  apply simp  nipkow@30052  205 apply (auto simp: dvd_def mult_assoc)  nipkow@30052  206 done  nipkow@30052  207 nipkow@29925  208 lemma div_dvd_div[simp]:  nipkow@29925  209  "a dvd b \ a dvd c \ (b div a dvd c div a) = (b dvd c)"  nipkow@29925  210 apply (cases "a = 0")  nipkow@29925  211  apply simp  nipkow@29925  212 apply (unfold dvd_def)  nipkow@29925  213 apply auto  nipkow@29925  214  apply(blast intro:mult_assoc[symmetric])  nipkow@44890  215 apply(fastforce simp add: mult_assoc)  nipkow@29925  216 done  nipkow@29925  217 huffman@30078  218 lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m"  huffman@30078  219  apply (subgoal_tac "k dvd (m div n) *n + m mod n")  huffman@30078  220  apply (simp add: mod_div_equality)  huffman@30078  221  apply (simp only: dvd_add dvd_mult)  huffman@30078  222  done  huffman@30078  223 huffman@29403  224 text {* Addition respects modular equivalence. *}  huffman@29403  225 huffman@29403  226 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"  huffman@29403  227 proof -  huffman@29403  228  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"  huffman@29403  229  by (simp only: mod_div_equality)  huffman@29403  230  also have "\ = (a mod c + b + a div c * c) mod c"  huffman@29403  231  by (simp only: add_ac)  huffman@29403  232  also have "\ = (a mod c + b) mod c"  huffman@29403  233  by (rule mod_mult_self1)  huffman@29403  234  finally show ?thesis .  huffman@29403  235 qed  huffman@29403  236 huffman@29403  237 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"  huffman@29403  238 proof -  huffman@29403  239  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"  huffman@29403  240  by (simp only: mod_div_equality)  huffman@29403  241  also have "\ = (a + b mod c + b div c * c) mod c"  huffman@29403  242  by (simp only: add_ac)  huffman@29403  243  also have "\ = (a + b mod c) mod c"  huffman@29403  244  by (rule mod_mult_self1)  huffman@29403  245  finally show ?thesis .  huffman@29403  246 qed  huffman@29403  247 huffman@29403  248 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"  huffman@29403  249 by (rule trans [OF mod_add_left_eq mod_add_right_eq])  huffman@29403  250 huffman@29403  251 lemma mod_add_cong:  huffman@29403  252  assumes "a mod c = a' mod c"  huffman@29403  253  assumes "b mod c = b' mod c"  huffman@29403  254  shows "(a + b) mod c = (a' + b') mod c"  huffman@29403  255 proof -  huffman@29403  256  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"  huffman@29403  257  unfolding assms ..  huffman@29403  258  thus ?thesis  huffman@29403  259  by (simp only: mod_add_eq [symmetric])  huffman@29403  260 qed  huffman@29403  261 haftmann@30923  262 lemma div_add [simp]: "z dvd x \ z dvd y  nipkow@30837  263  \ (x + y) div z = x div z + y div z"  haftmann@30923  264 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)  nipkow@30837  265 huffman@29403  266 text {* Multiplication respects modular equivalence. *}  huffman@29403  267 huffman@29403  268 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"  huffman@29403  269 proof -  huffman@29403  270  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"  huffman@29403  271  by (simp only: mod_div_equality)  huffman@29403  272  also have "\ = (a mod c * b + a div c * b * c) mod c"  nipkow@29667  273  by (simp only: algebra_simps)  huffman@29403  274  also have "\ = (a mod c * b) mod c"  huffman@29403  275  by (rule mod_mult_self1)  huffman@29403  276  finally show ?thesis .  huffman@29403  277 qed  huffman@29403  278 huffman@29403  279 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"  huffman@29403  280 proof -  huffman@29403  281  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"  huffman@29403  282  by (simp only: mod_div_equality)  huffman@29403  283  also have "\ = (a * (b mod c) + a * (b div c) * c) mod c"  nipkow@29667  284  by (simp only: algebra_simps)  huffman@29403  285  also have "\ = (a * (b mod c)) mod c"  huffman@29403  286  by (rule mod_mult_self1)  huffman@29403  287  finally show ?thesis .  huffman@29403  288 qed  huffman@29403  289 huffman@29403  290 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"  huffman@29403  291 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])  huffman@29403  292 huffman@29403  293 lemma mod_mult_cong:  huffman@29403  294  assumes "a mod c = a' mod c"  huffman@29403  295  assumes "b mod c = b' mod c"  huffman@29403  296  shows "(a * b) mod c = (a' * b') mod c"  huffman@29403  297 proof -  huffman@29403  298  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"  huffman@29403  299  unfolding assms ..  huffman@29403  300  thus ?thesis  huffman@29403  301  by (simp only: mod_mult_eq [symmetric])  huffman@29403  302 qed  huffman@29403  303 huffman@47164  304 text {* Exponentiation respects modular equivalence. *}  huffman@47164  305 huffman@47164  306 lemma power_mod: "(a mod b)^n mod b = a^n mod b"  huffman@47164  307 apply (induct n, simp_all)  huffman@47164  308 apply (rule mod_mult_right_eq [THEN trans])  huffman@47164  309 apply (simp (no_asm_simp))  huffman@47164  310 apply (rule mod_mult_eq [symmetric])  huffman@47164  311 done  huffman@47164  312 huffman@29404  313 lemma mod_mod_cancel:  huffman@29404  314  assumes "c dvd b"  huffman@29404  315  shows "a mod b mod c = a mod c"  huffman@29404  316 proof -  huffman@29404  317  from c dvd b obtain k where "b = c * k"  huffman@29404  318  by (rule dvdE)  huffman@29404  319  have "a mod b mod c = a mod (c * k) mod c"  huffman@29404  320  by (simp only: b = c * k)  huffman@29404  321  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  322  by (simp only: mod_mult_self1)  huffman@29404  323  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  huffman@29404  324  by (simp only: add_ac mult_ac)  huffman@29404  325  also have "\ = a mod c"  huffman@29404  326  by (simp only: mod_div_equality)  huffman@29404  327  finally show ?thesis .  huffman@29404  328 qed  huffman@29404  329 haftmann@30930  330 lemma div_mult_div_if_dvd:  haftmann@30930  331  "y dvd x \ z dvd w \ (x div y) * (w div z) = (x * w) div (y * z)"  haftmann@30930  332  apply (cases "y = 0", simp)  haftmann@30930  333  apply (cases "z = 0", simp)  haftmann@30930  334  apply (auto elim!: dvdE simp add: algebra_simps)  nipkow@30476  335  apply (subst mult_assoc [symmetric])  nipkow@30476  336  apply (simp add: no_zero_divisors)  haftmann@30930  337  done  haftmann@30930  338 haftmann@35367  339 lemma div_mult_swap:  haftmann@35367  340  assumes "c dvd b"  haftmann@35367  341  shows "a * (b div c) = (a * b) div c"  haftmann@35367  342 proof -  haftmann@35367  343  from assms have "b div c * (a div 1) = b * a div (c * 1)"  haftmann@35367  344  by (simp only: div_mult_div_if_dvd one_dvd)  haftmann@35367  345  then show ?thesis by (simp add: mult_commute)  haftmann@35367  346 qed  haftmann@35367  347   haftmann@30930  348 lemma div_mult_mult2 [simp]:  haftmann@30930  349  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@30930  350  by (drule div_mult_mult1) (simp add: mult_commute)  haftmann@30930  351 haftmann@30930  352 lemma div_mult_mult1_if [simp]:  haftmann@30930  353  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  354  by simp_all  nipkow@30476  355 haftmann@30930  356 lemma mod_mult_mult1:  haftmann@30930  357  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  358 proof (cases "c = 0")  haftmann@30930  359  case True then show ?thesis by simp  haftmann@30930  360 next  haftmann@30930  361  case False  haftmann@30930  362  from mod_div_equality  haftmann@30930  363  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  364  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  365  = c * a + c * (a mod b)" by (simp add: algebra_simps)  haftmann@30930  366  with mod_div_equality show ?thesis by simp  haftmann@30930  367 qed  haftmann@30930  368   haftmann@30930  369 lemma mod_mult_mult2:  haftmann@30930  370  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@30930  371  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)  haftmann@30930  372 huffman@47159  373 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  huffman@47159  374  by (fact mod_mult_mult2 [symmetric])  huffman@47159  375 huffman@47159  376 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  huffman@47159  377  by (fact mod_mult_mult1 [symmetric])  huffman@47159  378 huffman@31662  379 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  380  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  381 huffman@31662  382 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  383 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  384 haftmann@31009  385 lemma div_power:  huffman@31661  386  "y dvd x \ (x div y) ^ n = x ^ n div y ^ n"  nipkow@30476  387 apply (induct n)  nipkow@30476  388  apply simp  nipkow@30476  389 apply(simp add: div_mult_div_if_dvd dvd_power_same)  nipkow@30476  390 done  nipkow@30476  391 haftmann@35367  392 lemma dvd_div_eq_mult:  haftmann@35367  393  assumes "a \ 0" and "a dvd b"  haftmann@35367  394  shows "b div a = c \ b = c * a"  haftmann@35367  395 proof  haftmann@35367  396  assume "b = c * a"  haftmann@35367  397  then show "b div a = c" by (simp add: assms)  haftmann@35367  398 next  haftmann@35367  399  assume "b div a = c"  haftmann@35367  400  then have "b div a * a = c * a" by simp  haftmann@35367  401  moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)  haftmann@35367  402  ultimately show "b = c * a" by simp  haftmann@35367  403 qed  haftmann@35367  404   haftmann@35367  405 lemma dvd_div_div_eq_mult:  haftmann@35367  406  assumes "a \ 0" "c \ 0" and "a dvd b" "c dvd d"  haftmann@35367  407  shows "b div a = d div c \ b * c = a * d"  haftmann@35367  408  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)  haftmann@35367  409 huffman@31661  410 end  huffman@31661  411 haftmann@35673  412 class ring_div = semiring_div + comm_ring_1  huffman@29405  413 begin  huffman@29405  414 haftmann@36634  415 subclass ring_1_no_zero_divisors ..  haftmann@36634  416 huffman@29405  417 text {* Negation respects modular equivalence. *}  huffman@29405  418 huffman@29405  419 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  420 proof -  huffman@29405  421  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  422  by (simp only: mod_div_equality)  huffman@29405  423  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  huffman@29405  424  by (simp only: minus_add_distrib minus_mult_left add_ac)  huffman@29405  425  also have "\ = (- (a mod b)) mod b"  huffman@29405  426  by (rule mod_mult_self1)  huffman@29405  427  finally show ?thesis .  huffman@29405  428 qed  huffman@29405  429 huffman@29405  430 lemma mod_minus_cong:  huffman@29405  431  assumes "a mod b = a' mod b"  huffman@29405  432  shows "(- a) mod b = (- a') mod b"  huffman@29405  433 proof -  huffman@29405  434  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  435  unfolding assms ..  huffman@29405  436  thus ?thesis  huffman@29405  437  by (simp only: mod_minus_eq [symmetric])  huffman@29405  438 qed  huffman@29405  439 huffman@29405  440 text {* Subtraction respects modular equivalence. *}  huffman@29405  441 haftmann@54230  442 lemma mod_diff_left_eq:  haftmann@54230  443  "(a - b) mod c = (a mod c - b) mod c"  haftmann@54230  444  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp  haftmann@54230  445 haftmann@54230  446 lemma mod_diff_right_eq:  haftmann@54230  447  "(a - b) mod c = (a - b mod c) mod c"  haftmann@54230  448  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  449 haftmann@54230  450 lemma mod_diff_eq:  haftmann@54230  451  "(a - b) mod c = (a mod c - b mod c) mod c"  haftmann@54230  452  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  453 huffman@29405  454 lemma mod_diff_cong:  huffman@29405  455  assumes "a mod c = a' mod c"  huffman@29405  456  assumes "b mod c = b' mod c"  huffman@29405  457  shows "(a - b) mod c = (a' - b') mod c"  haftmann@54230  458  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp  huffman@29405  459 nipkow@30180  460 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  461 apply (case_tac "y = 0") apply simp  nipkow@30180  462 apply (auto simp add: dvd_def)  nipkow@30180  463 apply (subgoal_tac "-(y * k) = y * - k")  nipkow@30180  464  apply (erule ssubst)  nipkow@30180  465  apply (erule div_mult_self1_is_id)  nipkow@30180  466 apply simp  nipkow@30180  467 done  nipkow@30180  468 nipkow@30180  469 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  470 apply (case_tac "y = 0") apply simp  nipkow@30180  471 apply (auto simp add: dvd_def)  nipkow@30180  472 apply (subgoal_tac "y * k = -y * -k")  nipkow@30180  473  apply (erule ssubst)  nipkow@30180  474  apply (rule div_mult_self1_is_id)  nipkow@30180  475  apply simp  nipkow@30180  476 apply simp  nipkow@30180  477 done  nipkow@30180  478 huffman@47159  479 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  huffman@47159  480  using div_mult_mult1 [of "- 1" a b]  huffman@47159  481  unfolding neg_equal_0_iff_equal by simp  huffman@47159  482 huffman@47159  483 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  huffman@47159  484  using mod_mult_mult1 [of "- 1" a b] by simp  huffman@47159  485 huffman@47159  486 lemma div_minus_right: "a div (-b) = (-a) div b"  huffman@47159  487  using div_minus_minus [of "-a" b] by simp  huffman@47159  488 huffman@47159  489 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  huffman@47159  490  using mod_minus_minus [of "-a" b] by simp  huffman@47159  491 huffman@47160  492 lemma div_minus1_right [simp]: "a div (-1) = -a"  huffman@47160  493  using div_minus_right [of a 1] by simp  huffman@47160  494 huffman@47160  495 lemma mod_minus1_right [simp]: "a mod (-1) = 0"  huffman@47160  496  using mod_minus_right [of a 1] by simp  huffman@47160  497 haftmann@54221  498 lemma minus_mod_self2 [simp]:  haftmann@54221  499  "(a - b) mod b = a mod b"  haftmann@54221  500  by (simp add: mod_diff_right_eq)  haftmann@54221  501 haftmann@54221  502 lemma minus_mod_self1 [simp]:  haftmann@54221  503  "(b - a) mod b = - a mod b"  haftmann@54230  504  using mod_add_self2 [of "- a" b] by simp  haftmann@54221  505 huffman@29405  506 end  huffman@29405  507 haftmann@54226  508 class semiring_div_parity = semiring_div + semiring_numeral +  haftmann@54226  509  assumes parity: "a mod 2 = 0 \ a mod 2 = 1"  haftmann@54226  510 begin  haftmann@54226  511 haftmann@54226  512 lemma parity_cases [case_names even odd]:  haftmann@54226  513  assumes "a mod 2 = 0 \ P"  haftmann@54226  514  assumes "a mod 2 = 1 \ P"  haftmann@54226  515  shows P  haftmann@54226  516  using assms parity by blast  haftmann@54226  517 haftmann@54226  518 lemma not_mod_2_eq_0_eq_1 [simp]:  haftmann@54226  519  "a mod 2 \ 0 \ a mod 2 = 1"  haftmann@54226  520  by (cases a rule: parity_cases) simp_all  haftmann@54226  521 haftmann@54226  522 lemma not_mod_2_eq_1_eq_0 [simp]:  haftmann@54226  523  "a mod 2 \ 1 \ a mod 2 = 0"  haftmann@54226  524  by (cases a rule: parity_cases) simp_all  haftmann@54226  525 haftmann@54226  526 end  haftmann@54226  527 haftmann@25942  528 haftmann@53067  529 subsection {* Generic numeral division with a pragmatic type class *}  haftmann@53067  530 haftmann@53067  531 text {*  haftmann@53067  532  The following type class contains everything necessary to formulate  haftmann@53067  533  a division algorithm in ring structures with numerals, restricted  haftmann@53067  534  to its positive segments. This is its primary motiviation, and it  haftmann@53067  535  could surely be formulated using a more fine-grained, more algebraic  haftmann@53067  536  and less technical class hierarchy.  haftmann@53067  537 *}  haftmann@53067  538 haftmann@53067  539 class semiring_numeral_div = linordered_semidom + minus + semiring_div +  haftmann@53067  540  assumes diff_invert_add1: "a + b = c \ a = c - b"  haftmann@53067  541  and le_add_diff_inverse2: "b \ a \ a - b + b = a"  haftmann@53067  542  assumes mult_div_cancel: "b * (a div b) = a - a mod b"  haftmann@53067  543  and div_less: "0 \ a \ a < b \ a div b = 0"  haftmann@53067  544  and mod_less: " 0 \ a \ a < b \ a mod b = a"  haftmann@53067  545  and div_positive: "0 < b \ b \ a \ a div b > 0"  haftmann@53067  546  and mod_less_eq_dividend: "0 \ a \ a mod b \ a"  haftmann@53067  547  and pos_mod_bound: "0 < b \ a mod b < b"  haftmann@53067  548  and pos_mod_sign: "0 < b \ 0 \ a mod b"  haftmann@53067  549  and mod_mult2_eq: "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@53067  550  and div_mult2_eq: "0 \ c \ a div (b * c) = a div b div c"  haftmann@53067  551  assumes discrete: "a < b \ a + 1 \ b"  haftmann@53067  552 begin  haftmann@53067  553 haftmann@53067  554 lemma diff_zero [simp]:  haftmann@53067  555  "a - 0 = a"  haftmann@53067  556  by (rule diff_invert_add1 [symmetric]) simp  haftmann@53067  557 haftmann@54226  558 subclass semiring_div_parity  haftmann@54226  559 proof  haftmann@54226  560  fix a  haftmann@54226  561  show "a mod 2 = 0 \ a mod 2 = 1"  haftmann@54226  562  proof (rule ccontr)  haftmann@54226  563  assume "\ (a mod 2 = 0 \ a mod 2 = 1)"  haftmann@54226  564  then have "a mod 2 \ 0" and "a mod 2 \ 1" by simp_all  haftmann@54226  565  have "0 < 2" by simp  haftmann@54226  566  with pos_mod_bound pos_mod_sign have "0 \ a mod 2" "a mod 2 < 2" by simp_all  haftmann@54226  567  with a mod 2 \ 0 have "0 < a mod 2" by simp  haftmann@54226  568  with discrete have "1 \ a mod 2" by simp  haftmann@54226  569  with a mod 2 \ 1 have "1 < a mod 2" by simp  haftmann@54226  570  with discrete have "2 \ a mod 2" by simp  haftmann@54226  571  with a mod 2 < 2 show False by simp  haftmann@54226  572  qed  haftmann@53067  573 qed  haftmann@53067  574 haftmann@53067  575 lemma divmod_digit_1:  haftmann@53067  576  assumes "0 \ a" "0 < b" and "b \ a mod (2 * b)"  haftmann@53067  577  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")  haftmann@53067  578  and "a mod (2 * b) - b = a mod b" (is "?Q")  haftmann@53067  579 proof -  haftmann@53067  580  from assms mod_less_eq_dividend [of a "2 * b"] have "b \ a"  haftmann@53067  581  by (auto intro: trans)  haftmann@53067  582  with 0 < b have "0 < a div b" by (auto intro: div_positive)  haftmann@53067  583  then have [simp]: "1 \ a div b" by (simp add: discrete)  haftmann@53067  584  with 0 < b have mod_less: "a mod b < b" by (simp add: pos_mod_bound)  haftmann@53067  585  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  586  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  587  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  588  from assms w_exhaust have "w = 1"  haftmann@53067  589  by (auto simp add: mod_w) (insert mod_less, auto)  haftmann@53067  590  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp  haftmann@53067  591  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  592  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  593  with w = 1 have div: "2 * (a div (2 * b)) = a div b - 1" by simp  haftmann@53067  594  then show ?P and ?Q  haftmann@53067  595  by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)  haftmann@53067  596 qed  haftmann@53067  597 haftmann@53067  598 lemma divmod_digit_0:  haftmann@53067  599  assumes "0 < b" and "a mod (2 * b) < b"  haftmann@53067  600  shows "2 * (a div (2 * b)) = a div b" (is "?P")  haftmann@53067  601  and "a mod (2 * b) = a mod b" (is "?Q")  haftmann@53067  602 proof -  haftmann@53067  603  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  604  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  605  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  606  moreover have "b \ a mod b + b"  haftmann@53067  607  proof -  haftmann@53067  608  from 0 < b pos_mod_sign have "0 \ a mod b" by blast  haftmann@53067  609  then have "0 + b \ a mod b + b" by (rule add_right_mono)  haftmann@53067  610  then show ?thesis by simp  haftmann@53067  611  qed  haftmann@53067  612  moreover note assms w_exhaust  haftmann@53067  613  ultimately have "w = 0" by auto  haftmann@53067  614  with mod_w have mod: "a mod (2 * b) = a mod b" by simp  haftmann@53067  615  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  616  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  617  with w = 0 have div: "2 * (a div (2 * b)) = a div b" by simp  haftmann@53067  618  then show ?P and ?Q  haftmann@53067  619  by (simp_all add: div mod)  haftmann@53067  620 qed  haftmann@53067  621 haftmann@53067  622 definition divmod :: "num \ num \ 'a \ 'a"  haftmann@53067  623 where  haftmann@53067  624  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"  haftmann@53067  625 haftmann@53067  626 lemma fst_divmod [simp]:  haftmann@53067  627  "fst (divmod m n) = numeral m div numeral n"  haftmann@53067  628  by (simp add: divmod_def)  haftmann@53067  629 haftmann@53067  630 lemma snd_divmod [simp]:  haftmann@53067  631  "snd (divmod m n) = numeral m mod numeral n"  haftmann@53067  632  by (simp add: divmod_def)  haftmann@53067  633 haftmann@53067  634 definition divmod_step :: "num \ 'a \ 'a \ 'a \ 'a"  haftmann@53067  635 where  haftmann@53067  636  "divmod_step l qr = (let (q, r) = qr  haftmann@53067  637  in if r \ numeral l then (2 * q + 1, r - numeral l)  haftmann@53067  638  else (2 * q, r))"  haftmann@53067  639 haftmann@53067  640 text {*  haftmann@53067  641  This is a formulation of one step (referring to one digit position)  haftmann@53067  642  in school-method division: compare the dividend at the current  haftmann@53070  643  digit position with the remainder from previous division steps  haftmann@53067  644  and evaluate accordingly.  haftmann@53067  645 *}  haftmann@53067  646 haftmann@53067  647 lemma divmod_step_eq [code]:  haftmann@53067  648  "divmod_step l (q, r) = (if numeral l \ r  haftmann@53067  649  then (2 * q + 1, r - numeral l) else (2 * q, r))"  haftmann@53067  650  by (simp add: divmod_step_def)  haftmann@53067  651 haftmann@53067  652 lemma divmod_step_simps [simp]:  haftmann@53067  653  "r < numeral l \ divmod_step l (q, r) = (2 * q, r)"  haftmann@53067  654  "numeral l \ r \ divmod_step l (q, r) = (2 * q + 1, r - numeral l)"  haftmann@53067  655  by (auto simp add: divmod_step_eq not_le)  haftmann@53067  656 haftmann@53067  657 text {*  haftmann@53067  658  This is a formulation of school-method division.  haftmann@53067  659  If the divisor is smaller than the dividend, terminate.  haftmann@53067  660  If not, shift the dividend to the right until termination  haftmann@53067  661  occurs and then reiterate single division steps in the  haftmann@53067  662  opposite direction.  haftmann@53067  663 *}  haftmann@53067  664 haftmann@53067  665 lemma divmod_divmod_step [code]:  haftmann@53067  666  "divmod m n = (if m < n then (0, numeral m)  haftmann@53067  667  else divmod_step n (divmod m (Num.Bit0 n)))"  haftmann@53067  668 proof (cases "m < n")  haftmann@53067  669  case True then have "numeral m < numeral n" by simp  haftmann@53067  670  then show ?thesis  haftmann@53067  671  by (simp add: prod_eq_iff div_less mod_less)  haftmann@53067  672 next  haftmann@53067  673  case False  haftmann@53067  674  have "divmod m n =  haftmann@53067  675  divmod_step n (numeral m div (2 * numeral n),  haftmann@53067  676  numeral m mod (2 * numeral n))"  haftmann@53067  677  proof (cases "numeral n \ numeral m mod (2 * numeral n)")  haftmann@53067  678  case True  haftmann@53067  679  with divmod_step_simps  haftmann@53067  680  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  681  (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"  haftmann@53067  682  by blast  haftmann@53067  683  moreover from True divmod_digit_1 [of "numeral m" "numeral n"]  haftmann@53067  684  have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"  haftmann@53067  685  and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"  haftmann@53067  686  by simp_all  haftmann@53067  687  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  688  next  haftmann@53067  689  case False then have *: "numeral m mod (2 * numeral n) < numeral n"  haftmann@53067  690  by (simp add: not_le)  haftmann@53067  691  with divmod_step_simps  haftmann@53067  692  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  693  (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"  haftmann@53067  694  by blast  haftmann@53067  695  moreover from * divmod_digit_0 [of "numeral n" "numeral m"]  haftmann@53067  696  have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"  haftmann@53067  697  and "numeral m mod (2 * numeral n) = numeral m mod numeral n"  haftmann@53067  698  by (simp_all only: zero_less_numeral)  haftmann@53067  699  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  700  qed  haftmann@53067  701  then have "divmod m n =  haftmann@53067  702  divmod_step n (numeral m div numeral (Num.Bit0 n),  haftmann@53067  703  numeral m mod numeral (Num.Bit0 n))"  haftmann@53067  704  by (simp only: numeral.simps distrib mult_1)  haftmann@53067  705  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@53067  706  by (simp add: divmod_def)  haftmann@53067  707  with False show ?thesis by simp  haftmann@53067  708 qed  haftmann@53067  709 haftmann@53069  710 lemma divmod_cancel [code]:  haftmann@53069  711  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r))" (is ?P)  haftmann@53069  712  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r + 1))" (is ?Q)  haftmann@53069  713 proof -  haftmann@53069  714  have *: "\q. numeral (Num.Bit0 q) = 2 * numeral q"  haftmann@53069  715  "\q. numeral (Num.Bit1 q) = 2 * numeral q + 1"  haftmann@53069  716  by (simp_all only: numeral_mult numeral.simps distrib) simp_all  haftmann@53069  717  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)  haftmann@53069  718  then show ?P and ?Q  haftmann@53069  719  by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1  haftmann@53069  720  div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)  haftmann@53069  721  qed  haftmann@53069  722 haftmann@53067  723 end  haftmann@53067  724 haftmann@53067  725 hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero  haftmann@53067  726  -- {* restore simple accesses for more general variants of theorems *}  haftmann@53067  727 haftmann@53067  728   haftmann@26100  729 subsection {* Division on @{typ nat} *}  haftmann@26100  730 haftmann@26100  731 text {*  haftmann@26100  732  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  733  of a characteristic relation with two input arguments  haftmann@26100  734  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  735  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  736 *}  haftmann@26100  737 haftmann@33340  738 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  739  "divmod_nat_rel m n qr \  haftmann@30923  740  m = fst qr * n + snd qr \  haftmann@30923  741  (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  742 haftmann@33340  743 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  744 haftmann@33340  745 lemma divmod_nat_rel_ex:  haftmann@33340  746  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  747 proof (cases "n = 0")  haftmann@30923  748  case True with that show thesis  haftmann@33340  749  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  750 next  haftmann@26100  751  case False  haftmann@26100  752  have "\q r. m = q * n + r \ r < n"  haftmann@26100  753  proof (induct m)  haftmann@26100  754  case 0 with n \ 0  haftmann@26100  755  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  756  then show ?case by blast  haftmann@26100  757  next  haftmann@26100  758  case (Suc m) then obtain q' r'  haftmann@26100  759  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  760  then show ?case proof (cases "Suc r' < n")  haftmann@26100  761  case True  haftmann@26100  762  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  763  with True show ?thesis by blast  haftmann@26100  764  next  haftmann@26100  765  case False then have "n \ Suc r'" by auto  haftmann@26100  766  moreover from n have "Suc r' \ n" by auto  haftmann@26100  767  ultimately have "n = Suc r'" by auto  haftmann@26100  768  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  769  with n \ 0 show ?thesis by blast  haftmann@26100  770  qed  haftmann@26100  771  qed  haftmann@26100  772  with that show thesis  haftmann@33340  773  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  774 qed  haftmann@26100  775 haftmann@33340  776 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  777 haftmann@33340  778 lemma divmod_nat_rel_unique:  haftmann@33340  779  assumes "divmod_nat_rel m n qr"  haftmann@33340  780  and "divmod_nat_rel m n qr'"  haftmann@30923  781  shows "qr = qr'"  haftmann@26100  782 proof (cases "n = 0")  haftmann@26100  783  case True with assms show ?thesis  haftmann@30923  784  by (cases qr, cases qr')  haftmann@33340  785  (simp add: divmod_nat_rel_def)  haftmann@26100  786 next  haftmann@26100  787  case False  haftmann@26100  788  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  789  apply (rule leI)  haftmann@26100  790  apply (subst less_iff_Suc_add)  haftmann@26100  791  apply (auto simp add: add_mult_distrib)  haftmann@26100  792  done  wenzelm@53374  793  from n \ 0 assms have *: "fst qr = fst qr'"  haftmann@33340  794  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  wenzelm@53374  795  with assms have "snd qr = snd qr'"  haftmann@33340  796  by (simp add: divmod_nat_rel_def)  wenzelm@53374  797  with * show ?thesis by (cases qr, cases qr') simp  haftmann@26100  798 qed  haftmann@26100  799 haftmann@26100  800 text {*  haftmann@26100  801  We instantiate divisibility on the natural numbers by  haftmann@33340  802  means of @{const divmod_nat_rel}:  haftmann@26100  803 *}  haftmann@25942  804 haftmann@33340  805 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  806  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  807 haftmann@33340  808 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  809  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  810 proof -  haftmann@33340  811  from divmod_nat_rel_ex  haftmann@33340  812  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  813  then show ?thesis  haftmann@33340  814  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  815 qed  haftmann@30923  816 huffman@47135  817 lemma divmod_nat_unique:  haftmann@33340  818  assumes "divmod_nat_rel m n qr"  haftmann@33340  819  shows "divmod_nat m n = qr"  haftmann@33340  820  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  821 huffman@46551  822 instantiation nat :: semiring_div  huffman@46551  823 begin  huffman@46551  824 haftmann@26100  825 definition div_nat where  haftmann@33340  826  "m div n = fst (divmod_nat m n)"  haftmann@26100  827 huffman@46551  828 lemma fst_divmod_nat [simp]:  huffman@46551  829  "fst (divmod_nat m n) = m div n"  huffman@46551  830  by (simp add: div_nat_def)  huffman@46551  831 haftmann@26100  832 definition mod_nat where  haftmann@33340  833  "m mod n = snd (divmod_nat m n)"  haftmann@25571  834 huffman@46551  835 lemma snd_divmod_nat [simp]:  huffman@46551  836  "snd (divmod_nat m n) = m mod n"  huffman@46551  837  by (simp add: mod_nat_def)  huffman@46551  838 haftmann@33340  839 lemma divmod_nat_div_mod:  haftmann@33340  840  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  841  by (simp add: prod_eq_iff)  haftmann@26100  842 huffman@47135  843 lemma div_nat_unique:  haftmann@33340  844  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  845  shows "m div n = q"  huffman@47135  846  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  847 huffman@47135  848 lemma mod_nat_unique:  haftmann@33340  849  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  850  shows "m mod n = r"  huffman@47135  851  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  852 haftmann@33340  853 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  854  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  855 huffman@47136  856 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  857  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  858 huffman@47136  859 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  860  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  861 huffman@47137  862 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  863  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  864 haftmann@33340  865 lemma divmod_nat_step:  haftmann@26100  866  assumes "0 < n" and "n \ m"  haftmann@33340  867  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  868 proof (rule divmod_nat_unique)  huffman@47134  869  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  870  by (rule divmod_nat_rel)  huffman@47134  871  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  872  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  873 qed  haftmann@25942  874 wenzelm@26300  875 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  876 haftmann@26100  877 lemma div_less [simp]:  haftmann@26100  878  fixes m n :: nat  haftmann@26100  879  assumes "m < n"  haftmann@26100  880  shows "m div n = 0"  huffman@46551  881  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  882 haftmann@26100  883 lemma le_div_geq:  haftmann@26100  884  fixes m n :: nat  haftmann@26100  885  assumes "0 < n" and "n \ m"  haftmann@26100  886  shows "m div n = Suc ((m - n) div n)"  huffman@46551  887  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  888 haftmann@26100  889 lemma mod_less [simp]:  haftmann@26100  890  fixes m n :: nat  haftmann@26100  891  assumes "m < n"  haftmann@26100  892  shows "m mod n = m"  huffman@46551  893  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  894 haftmann@26100  895 lemma le_mod_geq:  haftmann@26100  896  fixes m n :: nat  haftmann@26100  897  assumes "n \ m"  haftmann@26100  898  shows "m mod n = (m - n) mod n"  huffman@46551  899  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  900 huffman@47136  901 instance proof  huffman@47136  902  fix m n :: nat  huffman@47136  903  show "m div n * n + m mod n = m"  huffman@47136  904  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  905 next  huffman@47136  906  fix m n q :: nat  huffman@47136  907  assume "n \ 0"  huffman@47136  908  then show "(q + m * n) div n = m + q div n"  huffman@47136  909  by (induct m) (simp_all add: le_div_geq)  huffman@47136  910 next  huffman@47136  911  fix m n q :: nat  huffman@47136  912  assume "m \ 0"  huffman@47136  913  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  914  unfolding divmod_nat_rel_def  huffman@47136  915  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  916  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  917  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  918  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  919 next  huffman@47136  920  fix n :: nat show "n div 0 = 0"  haftmann@33340  921  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  922 next  huffman@47136  923  fix n :: nat show "0 div n = 0"  huffman@47136  924  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  925 qed  haftmann@26100  926 haftmann@25942  927 end  paulson@14267  928 haftmann@33361  929 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  930  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  blanchet@55414  931  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)  haftmann@33361  932 haftmann@26100  933 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  934 wenzelm@51299  935 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"  wenzelm@51299  936 haftmann@30934  937 ML {*  wenzelm@43594  938 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  939 (  haftmann@30934  940  val div_name = @{const_name div};  haftmann@30934  941  val mod_name = @{const_name mod};  haftmann@30934  942  val mk_binop = HOLogic.mk_binop;  huffman@48561  943  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  huffman@48561  944  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  huffman@48561  945  fun mk_sum [] = HOLogic.zero  huffman@48561  946  | mk_sum [t] = t  huffman@48561  947  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  huffman@48561  948  fun dest_sum tm =  huffman@48561  949  if HOLogic.is_zero tm then []  huffman@48561  950  else  huffman@48561  951  (case try HOLogic.dest_Suc tm of  huffman@48561  952  SOME t => HOLogic.Suc_zero :: dest_sum t  huffman@48561  953  | NONE =>  huffman@48561  954  (case try dest_plus tm of  huffman@48561  955  SOME (t, u) => dest_sum t @ dest_sum u  huffman@48561  956  | NONE => [tm]));  haftmann@25942  957 haftmann@30934  958  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  959 haftmann@30934  960  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@35050  961  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))  wenzelm@41550  962 )  haftmann@25942  963 *}  haftmann@25942  964 wenzelm@43594  965 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  966 haftmann@26100  967 haftmann@26100  968 subsubsection {* Quotient *}  haftmann@26100  969 haftmann@26100  970 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  971 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  972 haftmann@26100  973 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  974 by (simp add: div_geq)  haftmann@26100  975 haftmann@26100  976 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  977 by simp  haftmann@26100  978 haftmann@26100  979 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  980 by simp  haftmann@26100  981 haftmann@53066  982 lemma div_positive:  haftmann@53066  983  fixes m n :: nat  haftmann@53066  984  assumes "n > 0"  haftmann@53066  985  assumes "m \ n"  haftmann@53066  986  shows "m div n > 0"  haftmann@53066  987 proof -  haftmann@53066  988  from m \ n obtain q where "m = n + q"  haftmann@53066  989  by (auto simp add: le_iff_add)  haftmann@53066  990  with n > 0 show ?thesis by simp  haftmann@53066  991 qed  haftmann@53066  992 haftmann@25942  993 haftmann@25942  994 subsubsection {* Remainder *}  haftmann@25942  995 haftmann@26100  996 lemma mod_less_divisor [simp]:  haftmann@26100  997  fixes m n :: nat  haftmann@26100  998  assumes "n > 0"  haftmann@26100  999  shows "m mod n < (n::nat)"  haftmann@33340  1000  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  1001 haftmann@51173  1002 lemma mod_Suc_le_divisor [simp]:  haftmann@51173  1003  "m mod Suc n \ n"  haftmann@51173  1004  using mod_less_divisor [of "Suc n" m] by arith  haftmann@51173  1005 haftmann@26100  1006 lemma mod_less_eq_dividend [simp]:  haftmann@26100  1007  fixes m n :: nat  haftmann@26100  1008  shows "m mod n \ m"  haftmann@26100  1009 proof (rule add_leD2)  haftmann@26100  1010  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  1011  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  1012 qed  haftmann@26100  1013 haftmann@26100  1014 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  1015 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  1016 haftmann@26100  1017 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  1018 by (simp add: le_mod_geq)  haftmann@26100  1019 paulson@14267  1020 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  1021 by (induct m) (simp_all add: mod_geq)  paulson@14267  1022 paulson@14267  1023 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  1024 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  1025  using mod_div_equality2 [of n m] by arith  paulson@14267  1026 nipkow@15439  1027 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  1028  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  1029  apply simp  wenzelm@22718  1030  done  paulson@14267  1031 haftmann@26100  1032 subsubsection {* Quotient and Remainder *}  paulson@14267  1033 haftmann@33340  1034 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  1035  "divmod_nat_rel b c (q, r)  haftmann@33340  1036  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  1037 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1038 haftmann@30923  1039 lemma div_mult1_eq:  haftmann@30923  1040  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  1041 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1042 haftmann@33340  1043 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  1044  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  1045  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  1046 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  1047 paulson@14267  1048 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  1049 lemma div_add1_eq:  nipkow@25134  1050  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  1051 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1052 paulson@14267  1053 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@22718  1054  apply (cut_tac m = q and n = c in mod_less_divisor)  wenzelm@22718  1055  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  wenzelm@22718  1056  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)  wenzelm@22718  1057  apply (simp add: add_mult_distrib2)  wenzelm@22718  1058  done  paulson@10559  1059 haftmann@33340  1060 lemma divmod_nat_rel_mult2_eq:  bulwahn@46552  1061  "divmod_nat_rel a b (q, r)  haftmann@33340  1062  \ divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@33340  1063 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)  paulson@14267  1064 blanchet@55085  1065 lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"  huffman@47135  1066 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  1067 blanchet@55085  1068 lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"  huffman@47135  1069 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  1070 paulson@14267  1071 huffman@46551  1072 subsubsection {* Further Facts about Quotient and Remainder *}  paulson@14267  1073 paulson@14267  1074 lemma div_1 [simp]: "m div Suc 0 = m"  nipkow@29667  1075 by (induct m) (simp_all add: div_geq)  paulson@14267  1076 paulson@14267  1077 (* Monotonicity of div in first argument *)  haftmann@30923  1078 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  1079  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  1080 apply (case_tac "k=0", simp)  paulson@15251  1081 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  1082 apply (case_tac "n= k *)  paulson@14267  1086 apply (case_tac "m=k *)  nipkow@15439  1090 apply (simp add: div_geq diff_le_mono)  paulson@14267  1091 done  paulson@14267  1092 paulson@14267  1093 (* Antimonotonicity of div in second argument *)  paulson@14267  1094 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  1095 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  1104  prefer 2  paulson@14267  1105  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  1106 apply (rule le_trans, simp)  nipkow@15439  1107 apply (simp)  paulson@14267  1108 done  paulson@14267  1109 paulson@14267  1110 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  1111 apply (case_tac "n=0", simp)  paulson@14267  1112 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  1113 apply (rule div_le_mono2)  paulson@14267  1114 apply (simp_all (no_asm_simp))  paulson@14267  1115 done  paulson@14267  1116 wenzelm@22718  1117 (* Similar for "less than" *)  huffman@47138  1118 lemma div_less_dividend [simp]:  huffman@47138  1119  "\(1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  1120 apply (induct m rule: nat_less_induct)  paulson@14267  1121 apply (rename_tac "m")  paulson@14267  1122 apply (case_tac "m Suc(na) *)  paulson@16796  1141 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  1142 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  1143 done  paulson@14267  1144 paulson@14267  1145 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  1146 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  1147 wenzelm@22718  1148 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  1149 paulson@14267  1150 (*Loses information, namely we also have r \q::nat. m = r + q*d"  haftmann@27651  1152  apply (cut_tac a = m in mod_div_equality)  wenzelm@22718  1153  apply (simp only: add_ac)  wenzelm@22718  1154  apply (blast intro: sym)  wenzelm@22718  1155  done  paulson@14267  1156 nipkow@13152  1157 lemma split_div:  nipkow@13189  1158  "P(n div k :: nat) =  nipkow@13189  1159  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  1160  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1161 proof  nipkow@13189  1162  assume P: ?P  nipkow@13189  1163  show ?Q  nipkow@13189  1164  proof (cases)  nipkow@13189  1165  assume "k = 0"  haftmann@27651  1166  with P show ?Q by simp  nipkow@13189  1167  next  nipkow@13189  1168  assume not0: "k \ 0"  nipkow@13189  1169  thus ?Q  nipkow@13189  1170  proof (simp, intro allI impI)  nipkow@13189  1171  fix i j  nipkow@13189  1172  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  1173  show "P i"  nipkow@13189  1174  proof (cases)  wenzelm@22718  1175  assume "i = 0"  wenzelm@22718  1176  with n j P show "P i" by simp  nipkow@13189  1177  next  wenzelm@22718  1178  assume "i \ 0"  wenzelm@22718  1179  with not0 n j P show "P i" by(simp add:add_ac)  nipkow@13189  1180  qed  nipkow@13189  1181  qed  nipkow@13189  1182  qed  nipkow@13189  1183 next  nipkow@13189  1184  assume Q: ?Q  nipkow@13189  1185  show ?P  nipkow@13189  1186  proof (cases)  nipkow@13189  1187  assume "k = 0"  haftmann@27651  1188  with Q show ?P by simp  nipkow@13189  1189  next  nipkow@13189  1190  assume not0: "k \ 0"  nipkow@13189  1191  with Q have R: ?R by simp  nipkow@13189  1192  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1193  show ?P by simp  nipkow@13189  1194  qed  nipkow@13189  1195 qed  nipkow@13189  1196 berghofe@13882  1197 lemma split_div_lemma:  haftmann@26100  1198  assumes "0 < n"  haftmann@26100  1199  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  1200 proof  haftmann@26100  1201  assume ?rhs  haftmann@26100  1202  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  1203  then have A: "n * q \ m" by simp  haftmann@26100  1204  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  1205  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  1206  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  1207  with nq have "m < n + n * q" by simp  haftmann@26100  1208  then have B: "m < n * Suc q" by simp  haftmann@26100  1209  from A B show ?lhs ..  haftmann@26100  1210 next  haftmann@26100  1211  assume P: ?lhs  haftmann@33340  1212  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@33340  1213  unfolding divmod_nat_rel_def by (auto simp add: mult_ac)  haftmann@33340  1214  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  1215  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  1216  then show ?rhs by simp  haftmann@26100  1217 qed  berghofe@13882  1218 berghofe@13882  1219 theorem split_div':  berghofe@13882  1220  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  1221  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  1222  apply (case_tac "0 < n")  berghofe@13882  1223  apply (simp only: add: split_div_lemma)  haftmann@27651  1224  apply simp_all  berghofe@13882  1225  done  berghofe@13882  1226 nipkow@13189  1227 lemma split_mod:  nipkow@13189  1228  "P(n mod k :: nat) =  nipkow@13189  1229  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  1230  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1231 proof  nipkow@13189  1232  assume P: ?P  nipkow@13189  1233  show ?Q  nipkow@13189  1234  proof (cases)  nipkow@13189  1235  assume "k = 0"  haftmann@27651  1236  with P show ?Q by simp  nipkow@13189  1237  next  nipkow@13189  1238  assume not0: "k \ 0"  nipkow@13189  1239  thus ?Q  nipkow@13189  1240  proof (simp, intro allI impI)  nipkow@13189  1241  fix i j  nipkow@13189  1242  assume "n = k*i + j" "j < k"  nipkow@13189  1243  thus "P j" using not0 P by(simp add:add_ac mult_ac)  nipkow@13189  1244  qed  nipkow@13189  1245  qed  nipkow@13189  1246 next  nipkow@13189  1247  assume Q: ?Q  nipkow@13189  1248  show ?P  nipkow@13189  1249  proof (cases)  nipkow@13189  1250  assume "k = 0"  haftmann@27651  1251  with Q show ?P by simp  nipkow@13189  1252  next  nipkow@13189  1253  assume not0: "k \ 0"  nipkow@13189  1254  with Q have R: ?R by simp  nipkow@13189  1255  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1256  show ?P by simp  nipkow@13189  1257  qed  nipkow@13189  1258 qed  nipkow@13189  1259 berghofe@13882  1260 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  1261  using mod_div_equality [of m n] by arith  huffman@47138  1262 huffman@47138  1263 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  1264  using mod_div_equality [of m n] by arith  huffman@47138  1265 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  1266 noschinl@52398  1267 lemma div_eq_dividend_iff: "a \ 0 \ (a :: nat) div b = a \ b = 1"  noschinl@52398  1268  apply rule  noschinl@52398  1269  apply (cases "b = 0")  noschinl@52398  1270  apply simp_all  noschinl@52398  1271  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)  noschinl@52398  1272  done  noschinl@52398  1273 haftmann@22800  1274 huffman@46551  1275 subsubsection {* An induction'' law for modulus arithmetic. *}  paulson@14640  1276 paulson@14640  1277 lemma mod_induct_0:  paulson@14640  1278  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1279  and base: "P i" and i: "i(P 0)"  paulson@14640  1283  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1285  proof  paulson@14640  1286  fix k  paulson@14640  1287  show "?A k"  paulson@14640  1288  proof (induct k)  paulson@14640  1289  show "?A 0" by simp -- "by contradiction"  paulson@14640  1290  next  paulson@14640  1291  fix n  paulson@14640  1292  assume ih: "?A n"  paulson@14640  1293  show "?A (Suc n)"  paulson@14640  1294  proof (clarsimp)  wenzelm@22718  1295  assume y: "P (p - Suc n)"  wenzelm@22718  1296  have n: "Suc n < p"  wenzelm@22718  1297  proof (rule ccontr)  wenzelm@22718  1298  assume "\(Suc n < p)"  wenzelm@22718  1299  hence "p - Suc n = 0"  wenzelm@22718  1300  by simp  wenzelm@22718  1301  with y contra show "False"  wenzelm@22718  1302  by simp  wenzelm@22718  1303  qed  wenzelm@22718  1304  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1305  from p have "p - Suc n < p" by arith  wenzelm@22718  1306  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1307  by blast  wenzelm@22718  1308  show "False"  wenzelm@22718  1309  proof (cases "n=0")  wenzelm@22718  1310  case True  wenzelm@22718  1311  with z n2 contra show ?thesis by simp  wenzelm@22718  1312  next  wenzelm@22718  1313  case False  wenzelm@22718  1314  with p have "p-n < p" by arith  wenzelm@22718  1315  with z n2 False ih show ?thesis by simp  wenzelm@22718  1316  qed  paulson@14640  1317  qed  paulson@14640  1318  qed  paulson@14640  1319  qed  paulson@14640  1320  moreover  paulson@14640  1321  from i obtain k where "0 i+k=p"  paulson@14640  1322  by (blast dest: less_imp_add_positive)  paulson@14640  1323  hence "0 i=p-k" by auto  paulson@14640  1324  moreover  paulson@14640  1325  note base  paulson@14640  1326  ultimately  paulson@14640  1327  show "False" by blast  paulson@14640  1328 qed  paulson@14640  1329 paulson@14640  1330 lemma mod_induct:  paulson@14640  1331  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1332  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1339  proof (induct j)  paulson@14640  1340  from step base i show "?A 0"  wenzelm@22718  1341  by (auto elim: mod_induct_0)  paulson@14640  1342  next  paulson@14640  1343  fix k  paulson@14640  1344  assume ih: "?A k"  paulson@14640  1345  show "?A (Suc k)"  paulson@14640  1346  proof  wenzelm@22718  1347  assume suc: "Suc k < p"  wenzelm@22718  1348  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1373 proof -  boehmes@35815  1374  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1375  moreover have "m mod 2 < 2" by simp  haftmann@33296  1376  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1377  then show ?thesis by auto  haftmann@33296  1378 qed  haftmann@33296  1379 haftmann@33296  1380 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1381  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1382  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1383 haftmann@33296  1384 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1385 by (simp add: Suc3_eq_add_3)  haftmann@33296  1386 haftmann@33296  1387 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1388 by (simp add: Suc3_eq_add_3)  haftmann@33296  1389 haftmann@33296  1390 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1391 by (simp add: Suc3_eq_add_3)  haftmann@33296  1392 haftmann@33296  1393 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1394 by (simp add: Suc3_eq_add_3)  haftmann@33296  1395 huffman@47108  1396 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1397 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1398 haftmann@33361  1399 haftmann@33361  1400 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1401 apply (induct "m")  haftmann@33361  1402 apply (simp_all add: mod_Suc)  haftmann@33361  1403 done  haftmann@33361  1404 huffman@47108  1405 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1406 huffman@47138  1407 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1408 by (simp add: div_le_mono)  haftmann@33361  1409 haftmann@33361  1410 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1411 by (cases n) simp_all  haftmann@33361  1412 boehmes@35815  1413 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1414 proof -  boehmes@35815  1415  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1416  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1417 qed  haftmann@33361  1418 haftmann@33361  1419  (* Potential use of algebra : Equality modulo n*)  haftmann@33361  1420 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"  haftmann@33361  1421 by (simp add: mult_ac add_ac)  haftmann@33361  1422 haftmann@33361  1423 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1424 proof -  haftmann@33361  1425  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1426  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1427  finally show ?thesis .  haftmann@33361  1428 qed  haftmann@33361  1429 haftmann@33361  1430 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1431 apply (subst mod_Suc [of m])  haftmann@33361  1432 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1433 done  haftmann@33361  1434 huffman@47108  1435 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1436  fixes n :: nat  huffman@47108  1437  shows "n mod 2 \ 0 \ n mod 2 = 1"  huffman@47108  1438  by simp  huffman@47108  1439 haftmann@53067  1440 instance nat :: semiring_numeral_div  haftmann@53067  1441  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)  haftmann@53067  1442 haftmann@33361  1443 haftmann@33361  1444 subsection {* Division on @{typ int} *}  haftmann@33361  1445 haftmann@33361  1446 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1447  --{*definition of quotient and remainder*}  huffman@47139  1448  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1449  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1450 haftmann@53067  1451 text {*  haftmann@53067  1452  The following algorithmic devlopment actually echos what has already  haftmann@53067  1453  been developed in class @{class semiring_numeral_div}. In the long  haftmann@53067  1454  run it seems better to derive division on @{typ int} just from  haftmann@53067  1455  division on @{typ nat} and instantiate @{class semiring_numeral_div}  haftmann@53067  1456  accordingly.  haftmann@53067  1457 *}  haftmann@53067  1458 haftmann@33361  1459 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1460  --{*for the division algorithm*}  huffman@47108  1461  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1462  else (2 * q, r))"  haftmann@33361  1463 haftmann@33361  1464 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1465 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1466  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1467  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1468 by auto  haftmann@33361  1469 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1470  (auto simp add: mult_2)  haftmann@33361  1471 haftmann@33361  1472 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1473 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1474  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1475  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1476 by auto  haftmann@33361  1477 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1478  (auto simp add: mult_2)  haftmann@33361  1479 haftmann@33361  1480 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1481 haftmann@33361  1482 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1483  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1484  including the special case @{text "a=0, b<0"} because  haftmann@33361  1485  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1486  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1487  else if a = 0 then (0, 0)  huffman@46560  1488  else apsnd uminus (negDivAlg (-a) (-b))  haftmann@33361  1489  else  haftmann@33361  1490  if 0 < b then negDivAlg a b  huffman@46560  1491  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1492 haftmann@33361  1493 instantiation int :: Divides.div  haftmann@33361  1494 begin  haftmann@33361  1495 huffman@46551  1496 definition div_int where  haftmann@33361  1497  "a div b = fst (divmod_int a b)"  haftmann@33361  1498 huffman@46551  1499 lemma fst_divmod_int [simp]:  huffman@46551  1500  "fst (divmod_int a b) = a div b"  huffman@46551  1501  by (simp add: div_int_def)  huffman@46551  1502 huffman@46551  1503 definition mod_int where  huffman@46560  1504  "a mod b = snd (divmod_int a b)"  haftmann@33361  1505 huffman@46551  1506 lemma snd_divmod_int [simp]:  huffman@46551  1507  "snd (divmod_int a b) = a mod b"  huffman@46551  1508  by (simp add: mod_int_def)  huffman@46551  1509 haftmann@33361  1510 instance ..  haftmann@33361  1511 paulson@3366  1512 end  haftmann@33361  1513 haftmann@33361  1514 lemma divmod_int_mod_div:  haftmann@33361  1515  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1516  by (simp add: prod_eq_iff)  haftmann@33361  1517 haftmann@33361  1518 text{*  haftmann@33361  1519 Here is the division algorithm in ML:  haftmann@33361  1520 haftmann@33361  1521 \begin{verbatim}  haftmann@33361  1522  fun posDivAlg (a,b) =  haftmann@33361  1523  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1526  end  haftmann@33361  1527 haftmann@33361  1528  fun negDivAlg (a,b) =  haftmann@33361  1529  if 0\a+b then (~1,a+b)  haftmann@33361  1530  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1531  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1532  end;  haftmann@33361  1533 haftmann@33361  1534  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1535 haftmann@33361  1536  fun divmod (a,b) = if 0\a then  haftmann@33361  1537  if b>0 then posDivAlg (a,b)  haftmann@33361  1538  else if a=0 then (0,0)  haftmann@33361  1539  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1540  else  haftmann@33361  1541  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1551  ==> q' \ (q::int)"  haftmann@33361  1552 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1553  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1554 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1555 apply (erule_tac [2] order_le_less_trans)  webertj@49962  1556  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1557 apply (subgoal_tac "b * q' < b * (1 + q) ")  webertj@49962  1558  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1559 apply (simp add: mult_less_cancel_left)  haftmann@33361  1560 done  haftmann@33361  1561 haftmann@33361  1562 lemma unique_quotient_lemma_neg:  haftmann@33361  1563  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1564  ==> q \ (q'::int)"  haftmann@33361  1565 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1566  auto)  haftmann@33361  1567 haftmann@33361  1568 lemma unique_quotient:  bulwahn@46552  1569  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1570  ==> q = q'"  haftmann@33361  1571 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1572 apply (blast intro: order_antisym  haftmann@33361  1573  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1574  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1575 done  haftmann@33361  1576 haftmann@33361  1577 haftmann@33361  1578 lemma unique_remainder:  bulwahn@46552  1579  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1580  ==> r = r'"  haftmann@33361  1581 apply (subgoal_tac "q = q'")  haftmann@33361  1582  apply (simp add: divmod_int_rel_def)  haftmann@33361  1583 apply (blast intro: unique_quotient)  haftmann@33361  1584 done  haftmann@33361  1585 haftmann@33361  1586 huffman@46551  1587 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}  haftmann@33361  1588 haftmann@33361  1589 text{*And positive divisors*}  haftmann@33361  1590 haftmann@33361  1591 lemma adjust_eq [simp]:  huffman@47108  1592  "adjust b (q, r) =  huffman@47108  1593  (let diff = r - b in  huffman@47108  1594  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1595  else (2*q, r))"  huffman@47108  1596  by (simp add: Let_def adjust_def)  haftmann@33361  1597 haftmann@33361  1598 declare posDivAlg.simps [simp del]  haftmann@33361  1599 haftmann@33361  1600 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1601 lemma posDivAlg_eqn:  haftmann@33361  1602  "0 < b ==>  haftmann@33361  1603  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1609  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1610  using assms  wenzelm@41550  1611  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1612  apply auto  wenzelm@41550  1613  apply (simp add: divmod_int_rel_def)  webertj@49962  1614  apply (subst posDivAlg_eqn, simp add: distrib_left)  wenzelm@41550  1615  apply (case_tac "a < b")  wenzelm@41550  1616  apply simp_all  wenzelm@41550  1617  apply (erule splitE)  webertj@49962  1618  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1619  done  haftmann@33361  1620 haftmann@33361  1621 huffman@46551  1622 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}  haftmann@33361  1623 haftmann@33361  1624 text{*And positive divisors*}  haftmann@33361  1625 haftmann@33361  1626 declare negDivAlg.simps [simp del]  haftmann@33361  1627 haftmann@33361  1628 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1629 lemma negDivAlg_eqn:  haftmann@33361  1630  "0 < b ==>  haftmann@33361  1631  negDivAlg a b =  haftmann@33361  1632  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1633 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1634 haftmann@33361  1635 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1636  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1637 lemma negDivAlg_correct:  haftmann@33361  1638  assumes "a < 0" and "b > 0"  haftmann@33361  1639  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1640  using assms  wenzelm@41550  1641  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1642  apply (auto simp add: linorder_not_le)  wenzelm@41550  1643  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1644  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1645  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1646  apply simp_all  wenzelm@41550  1647  apply (erule splitE)  webertj@49962  1648  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1649  done  haftmann@33361  1650 haftmann@33361  1651 huffman@46551  1652 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}  haftmann@33361  1653 haftmann@33361  1654 (*the case a=0*)  huffman@47139  1655 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1656 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1657 haftmann@33361  1658 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1659 by (subst posDivAlg.simps, auto)  haftmann@33361  1660 huffman@47139  1661 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1662 by (subst posDivAlg.simps, auto)  huffman@47139  1663 haftmann@33361  1664 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  haftmann@33361  1665 by (subst negDivAlg.simps, auto)  haftmann@33361  1666 huffman@46560  1667 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1668 by (auto simp add: divmod_int_rel_def)  huffman@47139  1669 huffman@47139  1670 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1671 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1672 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1673  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1674 huffman@47141  1675 lemma divmod_int_unique:  huffman@47141  1676  assumes "divmod_int_rel a b qr"  huffman@47141  1677  shows "divmod_int a b = qr"  huffman@47141  1678  using assms divmod_int_correct [of a b]  huffman@47141  1679  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1680  by (metis pair_collapse)  huffman@47141  1681 huffman@47141  1682 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1683  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1684 huffman@47141  1685 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1686  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1687 huffman@47141  1688 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1689  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1690 huffman@47141  1691 instance int :: ring_div  huffman@47141  1692 proof  huffman@47141  1693  fix a b :: int  huffman@47141  1694  show "a div b * b + a mod b = a"  huffman@47141  1695  using divmod_int_rel_div_mod [of a b]  huffman@47141  1696  unfolding divmod_int_rel_def by (simp add: mult_commute)  huffman@47141  1697 next  huffman@47141  1698  fix a b c :: int  huffman@47141  1699  assume "b \ 0"  huffman@47141  1700  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1701  using divmod_int_rel_div_mod [of a b]  huffman@47141  1702  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1703  thus "(a + c * b) div b = c + a div b"  huffman@47141  1704  by (rule div_int_unique)  huffman@47141  1705 next  huffman@47141  1706  fix a b c :: int  huffman@47141  1707  assume "c \ 0"  huffman@47141  1708  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1709  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1710  unfolding divmod_int_rel_def  huffman@47141  1711  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1712  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1713  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1714  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1715  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1716  thus "(c * a) div (c * b) = a div b"  huffman@47141  1717  by (rule div_int_unique)  huffman@47141  1718 next  huffman@47141  1719  fix a :: int show "a div 0 = 0"  huffman@47141  1720  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1721 next  huffman@47141  1722  fix a :: int show "0 div a = 0"  huffman@47141  1723  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1724 qed  huffman@47141  1725 haftmann@33361  1726 text{*Basic laws about division and remainder*}  haftmann@33361  1727 haftmann@33361  1728 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1729  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1730 haftmann@33361  1731 text {* Tool setup *}  haftmann@33361  1732 huffman@47108  1733 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)  huffman@47108  1734 lemmas add_0s = add_0_left add_0_right  huffman@47108  1735 haftmann@33361  1736 ML {*  wenzelm@43594  1737 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1738 (  haftmann@33361  1739  val div_name = @{const_name div};  haftmann@33361  1740  val mod_name = @{const_name mod};  haftmann@33361  1741  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1742  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1743  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1744 huffman@47165  1745  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1746 haftmann@33361  1747  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@54230  1748  (@{thm diff_conv_add_uminus} :: @{thms add_0s} @ @{thms add_ac}))  wenzelm@41550  1749 )  haftmann@33361  1750 *}  haftmann@33361  1751 wenzelm@43594  1752 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1753 huffman@47141  1754 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1755  using divmod_int_correct [of a b]  huffman@47141  1756  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1757 wenzelm@45607  1758 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1759  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1760 huffman@47141  1761 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1762  using divmod_int_correct [of a b]  huffman@47141  1763  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1764 wenzelm@45607  1765 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1766  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1767 haftmann@33361  1768 huffman@46551  1769 subsubsection {* General Properties of div and mod *}  haftmann@33361  1770 haftmann@33361  1771 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1772 apply (rule div_int_unique)  haftmann@33361  1773 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1774 done  haftmann@33361  1775 haftmann@33361  1776 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1777 apply (rule div_int_unique)  haftmann@33361  1778 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1779 done  haftmann@33361  1780 haftmann@33361  1781 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1782 apply (rule div_int_unique)  haftmann@33361  1783 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1784 done  haftmann@33361  1785 haftmann@33361  1786 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1787 haftmann@33361  1788 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1789 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1790 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1791 done  haftmann@33361  1792 haftmann@33361  1793 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  1794 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1795 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1796 done  haftmann@33361  1797 haftmann@33361  1798 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  1799 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  1800 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1801 done  haftmann@33361  1802 haftmann@33361  1803 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  1804 haftmann@33361  1805 huffman@46551  1806 subsubsection {* Laws for div and mod with Unary Minus *}  haftmann@33361  1807 haftmann@33361  1808 lemma zminus1_lemma:  huffman@47139  1809  "divmod_int_rel a b (q, r) ==> b \ 0  haftmann@33361  1810  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1811  if r=0 then 0 else b-r)"  haftmann@33361  1812 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1813 haftmann@33361  1814 haftmann@33361  1815 lemma zdiv_zminus1_eq_if:  haftmann@33361  1816  "b \ (0::int)  haftmann@33361  1817  ==> (-a) div b =  haftmann@33361  1818  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  1819 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  1820 haftmann@33361  1821 lemma zmod_zminus1_eq_if:  haftmann@33361  1822  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1823 apply (case_tac "b = 0", simp)  huffman@47140  1824 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  1825 done  haftmann@33361  1826 haftmann@33361  1827 lemma zmod_zminus1_not_zero:  haftmann@33361  1828  fixes k l :: int  haftmann@33361  1829  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  1830  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  1831 haftmann@33361  1832 lemma zdiv_zminus2_eq_if:  haftmann@33361  1833  "b \ (0::int)  haftmann@33361  1834  ==> a div (-b) =  haftmann@33361  1835  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  1836 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  1837 haftmann@33361  1838 lemma zmod_zminus2_eq_if:  haftmann@33361  1839  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  1840 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  1841 haftmann@33361  1842 lemma zmod_zminus2_not_zero:  haftmann@33361  1843  fixes k l :: int  haftmann@33361  1844  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  1845  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  1846 haftmann@33361  1847 huffman@46551  1848 subsubsection {* Computation of Division and Remainder *}  haftmann@33361  1849 haftmann@33361  1850 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  1851 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1852 haftmann@33361  1853 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  1854 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1855 haftmann@33361  1856 text{*a positive, b positive *}  haftmann@33361  1857 haftmann@33361  1858 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  1859 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1860 haftmann@33361  1861 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  1862 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1863 haftmann@33361  1864 text{*a negative, b positive *}  haftmann@33361  1865 haftmann@33361  1866 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  1867 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1868 haftmann@33361  1869 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  1870 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1871 haftmann@33361  1872 text{*a positive, b negative *}  haftmann@33361  1873 haftmann@33361  1874 lemma div_pos_neg:  huffman@46560  1875  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1876 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1877 haftmann@33361  1878 lemma mod_pos_neg:  huffman@46560  1879  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1880 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1881 haftmann@33361  1882 text{*a negative, b negative *}  haftmann@33361  1883 haftmann@33361  1884 lemma div_neg_neg:  huffman@46560  1885  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1886 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1887 haftmann@33361  1888 lemma mod_neg_neg:  huffman@46560  1889  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1890 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1891 haftmann@33361  1892 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  1893 huffman@45530  1894 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  1895  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  1896 huffman@45530  1897 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  1898  by (rule div_int_unique [of a b q r],  bulwahn@46552  1899  simp add: divmod_int_rel_def)  huffman@45530  1900 huffman@45530  1901 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  1902  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1903  simp add: divmod_int_rel_def)  huffman@45530  1904 huffman@45530  1905 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  1906  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1907  simp add: divmod_int_rel_def)  huffman@45530  1908 haftmann@53069  1909 text {*  haftmann@53069  1910  numeral simprocs -- high chance that these can be replaced  haftmann@53069  1911  by divmod algorithm from @{class semiring_numeral_div}  haftmann@53069  1912 *}  haftmann@53069  1913 haftmann@33361  1914 ML {*  haftmann@33361  1915 local  huffman@45530  1916  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  1917  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  1918  val times = @{term "times :: int \ int \ int"}  huffman@45530  1919  val zero = @{term "0 :: int"}  huffman@45530  1920  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  1921  val le = @{term "op \ :: int \ int \ bool"}  haftmann@54489  1922  val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]  haftmann@54489  1923  fun prove ctxt goal = (writeln "prove"; Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  haftmann@54489  1924  (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps)))));  wenzelm@51717  1925  fun binary_proc proc ctxt ct =  haftmann@33361  1926  (case Thm.term_of ct of  haftmann@33361  1927  _ $t$ u =>  haftmann@33361  1928  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  wenzelm@51717  1929  SOME args => proc ctxt args  haftmann@33361  1930  | NONE => NONE)  haftmann@33361  1931  | _ => NONE);  haftmann@33361  1932 in  huffman@45530  1933  fun divmod_proc posrule negrule =  huffman@45530  1934  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  huffman@45530  1935  if b = 0 then NONE else let  huffman@45530  1936  val (q, r) = pairself mk_number (Integer.div_mod a b)  huffman@45530  1937  val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)  huffman@45530  1938  val (goal2, goal3, rule) = if b > 0  huffman@45530  1939  then (le $zero$ r, less $r$ u, posrule RS eq_reflection)  huffman@45530  1940  else (le $r$ zero, less $u$ r, negrule RS eq_reflection)  huffman@45530  1941  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  1942 end  haftmann@33361  1943 *}  haftmann@33361  1944 huffman@47108  1945 simproc_setup binary_int_div  huffman@47108  1946  ("numeral m div numeral n :: int" |  haftmann@54489  1947  "numeral m div - numeral n :: int" |  haftmann@54489  1948  "- numeral m div numeral n :: int" |  haftmann@54489  1949  "- numeral m div - numeral n :: int") =  huffman@45530  1950  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  1951 huffman@47108  1952 simproc_setup binary_int_mod  huffman@47108  1953  ("numeral m mod numeral n :: int" |  haftmann@54489  1954  "numeral m mod - numeral n :: int" |  haftmann@54489  1955  "- numeral m mod numeral n :: int" |  haftmann@54489  1956  "- numeral m mod - numeral n :: int") =  huffman@45530  1957  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  1958 huffman@47108  1959 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  1960  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  1961 huffman@47108  1962 lemmas negDivAlg_eqn_numeral [simp] =  haftmann@54489  1963  negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w  haftmann@33361  1964 haftmann@33361  1965 haftmann@55172  1966 text {* Special-case simplification: @{text "\1 div z"} and @{text "\1 mod z"} *}  haftmann@55172  1967 haftmann@55172  1968 lemma [simp]:  haftmann@55172  1969  shows div_one_bit0: "1 div numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  1970  and mod_one_bit0: "1 mod numeral (Num.Bit0 v) = (1 :: int)"  haftmann@55172  1971  and div_one_bit1: "1 div numeral (Num.Bit1 v) = (0 :: int)"  haftmann@55172  1972  and mod_one_bit1: "1 mod numeral (Num.Bit1 v) = (1 :: int)"  haftmann@55172  1973  and div_one_neg_numeral: "1 div - numeral v = (- 1 :: int)"  haftmann@55172  1974  and mod_one_neg_numeral: "1 mod - numeral v = (1 :: int) - numeral v"  haftmann@55172  1975  by (simp_all del: arith_special  haftmann@55172  1976  add: div_pos_pos mod_pos_pos div_pos_neg mod_pos_neg posDivAlg_eqn)  haftmann@55172  1977   haftmann@55172  1978 lemma [simp]:  haftmann@55172  1979  shows div_neg_one_numeral: "- 1 div numeral v = (- 1 :: int)"  haftmann@55172  1980  and mod_neg_one_numeral: "- 1 mod numeral v = numeral v - (1 :: int)"  haftmann@55172  1981  and div_neg_one_neg_bit0: "- 1 div - numeral (Num.Bit0 v) = (0 :: int)"  haftmann@55172  1982  and mod_neg_one_neb_bit0: "- 1 mod - numeral (Num.Bit0 v) = (- 1 :: int)"  haftmann@55172  1983  and div_neg_one_neg_bit1: "- 1 div - numeral (Num.Bit1 v) = (0 :: int)"  haftmann@55172  1984  and mod_neg_one_neb_bit1: "- 1 mod - numeral (Num.Bit1 v) = (- 1 :: int)"  haftmann@55172  1985  by (simp_all add: div_eq_minus1 zmod_minus1)  haftmann@33361  1986 haftmann@33361  1987 huffman@46551  1988 subsubsection {* Monotonicity in the First Argument (Dividend) *}  haftmann@33361  1989 haftmann@33361  1990 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  1991 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1992 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1993 apply (rule unique_quotient_lemma)  haftmann@33361  1994 apply (erule subst)  haftmann@33361  1995 apply (erule subst, simp_all)  haftmann@33361  1996 done  haftmann@33361  1997 haftmann@33361  1998 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  1999 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2000 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  2001 apply (rule unique_quotient_lemma_neg)  haftmann@33361  2002 apply (erule subst)  haftmann@33361  2003 apply (erule subst, simp_all)  haftmann@33361  2004 done  haftmann@33361  2005 haftmann@33361  2006 huffman@46551  2007 subsubsection {* Monotonicity in the Second Argument (Divisor) *}  haftmann@33361  2008 haftmann@33361  2009 lemma q_pos_lemma:  haftmann@33361  2010  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  2011 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  2012  apply (simp add: zero_less_mult_iff)  webertj@49962  2013 apply (simp add: distrib_left)  haftmann@33361  2014 done  haftmann@33361  2015 haftmann@33361  2016 lemma zdiv_mono2_lemma:  haftmann@33361  2017  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  2018  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  2019  ==> q \ (q'::int)"  haftmann@33361  2020 apply (frule q_pos_lemma, assumption+)  haftmann@33361  2021 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  2022  apply (simp add: mult_less_cancel_left)  haftmann@33361  2023 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  2024  prefer 2 apply simp  webertj@49962  2025 apply (simp (no_asm_simp) add: distrib_left)  huffman@44766  2026 apply (subst add_commute, rule add_less_le_mono, arith)  haftmann@33361  2027 apply (rule mult_right_mono, auto)  haftmann@33361  2028 done  haftmann@33361  2029 haftmann@33361  2030 lemma zdiv_mono2:  haftmann@33361  2031  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  2032 apply (subgoal_tac "b \ 0")  haftmann@33361  2033  prefer 2 apply arith  haftmann@33361  2034 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2035 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2036 apply (rule zdiv_mono2_lemma)  haftmann@33361  2037 apply (erule subst)  haftmann@33361  2038 apply (erule subst, simp_all)  haftmann@33361  2039 done  haftmann@33361  2040 haftmann@33361  2041 lemma q_neg_lemma:  haftmann@33361  2042  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  2043 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  2044  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  2045 done  haftmann@33361  2046 haftmann@33361  2047 lemma zdiv_mono2_neg_lemma:  haftmann@33361  2048  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  2049  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  2050  ==> q' \ (q::int)"  haftmann@33361  2051 apply (frule q_neg_lemma, assumption+)  haftmann@33361  2052 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  2053  apply (simp add: mult_less_cancel_left)  webertj@49962  2054 apply (simp add: distrib_left)  haftmann@33361  2055 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  2056  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  2057 done  haftmann@33361  2058 haftmann@33361  2059 lemma zdiv_mono2_neg:  haftmann@33361  2060  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  2061 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2062 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2063 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  2064 apply (erule subst)  haftmann@33361  2065 apply (erule subst, simp_all)  haftmann@33361  2066 done  haftmann@33361  2067 haftmann@33361  2068 huffman@46551  2069 subsubsection {* More Algebraic Laws for div and mod *}  haftmann@33361  2070 haftmann@33361  2071 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  2072 haftmann@33361  2073 lemma zmult1_lemma:  bulwahn@46552  2074  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  2075  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  webertj@49962  2076 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)  haftmann@33361  2077 haftmann@33361  2078 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  2079 apply (case_tac "c = 0", simp)  huffman@47140  2080 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  2081 done  haftmann@33361  2082 haftmann@33361  2083 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  2084 haftmann@33361  2085 lemma zadd1_lemma:  bulwahn@46552  2086  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  2087  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  webertj@49962  2088 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)  haftmann@33361  2089 haftmann@33361  2090 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  2091 lemma zdiv_zadd1_eq:  haftmann@33361  2092  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  2093 apply (case_tac "c = 0", simp)  huffman@47140  2094 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  2095 done  haftmann@33361  2096 haftmann@33361  2097 lemma posDivAlg_div_mod:  haftmann@33361  2098  assumes "k \ 0"  haftmann@33361  2099  and "l \ 0"  haftmann@33361  2100  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  2101 proof (cases "l = 0")  haftmann@33361  2102  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  2103 next  haftmann@33361  2104  case False with assms posDivAlg_correct  haftmann@33361  2105  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  2106  by simp  huffman@47140  2107  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2108  show ?thesis by simp  haftmann@33361  2109 qed  haftmann@33361  2110 haftmann@33361  2111 lemma negDivAlg_div_mod:  haftmann@33361  2112  assumes "k < 0"  haftmann@33361  2113  and "l > 0"  haftmann@33361  2114  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  2115 proof -  haftmann@33361  2116  from assms have "l \ 0" by simp  haftmann@33361  2117  from assms negDivAlg_correct  haftmann@33361  2118  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  2119  by simp  huffman@47140  2120  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2121  show ?thesis by simp  haftmann@33361  2122 qed  haftmann@33361  2123 haftmann@33361  2124 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  2125 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  2126 haftmann@33361  2127 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  2128 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  2129 huffman@47108  2130 lemma zmod_zdiv_equality':  huffman@47108  2131  "(m\int) mod n = m - (m div n) * n"  huffman@47141  2132  using mod_div_equality [of m n] by arith  huffman@47108  2133 haftmann@33361  2134 blanchet@55085  2135 subsubsection {* Proving @{term "a div (b * c) = (a div b) div c"} *}  haftmann@33361  2136 haftmann@33361  2137 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  2138  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  2139  to cause particular problems.*)  haftmann@33361  2140 haftmann@33361  2141 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  2142 blanchet@55085  2143 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b * c < b * (q mod c) + r"  haftmann@33361  2144 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  2145  apply (simp add: algebra_simps)  haftmann@33361  2146 apply (rule order_le_less_trans)  haftmann@33361  2147  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  2148  apply (rule mult_left_mono_neg)  huffman@35216  2149  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  2150  apply (simp)  haftmann@33361  2151 apply (simp)  haftmann@33361  2152 done  haftmann@33361  2153 haftmann@33361  2154 lemma zmult2_lemma_aux2:  haftmann@33361  2155  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  2156 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  2157  apply arith  haftmann@33361  2158 apply (simp add: mult_le_0_iff)  haftmann@33361  2159 done  haftmann@33361  2160 haftmann@33361  2161 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  2162 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  2163 apply arith  haftmann@33361  2164 apply (simp add: zero_le_mult_iff)  haftmann@33361  2165 done  haftmann@33361  2166 haftmann@33361  2167 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  2168 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  2169  apply (simp add: right_diff_distrib)  haftmann@33361  2170 apply (rule order_less_le_trans)  haftmann@33361  2171  apply (erule mult_strict_right_mono)  haftmann@33361  2172  apply (rule_tac [2] mult_left_mono)  haftmann@33361  2173  apply simp  huffman@35216  2174  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  2175 apply simp  haftmann@33361  2176 done  haftmann@33361  2177 bulwahn@46552  2178 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  2179  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@33361  2180 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff  webertj@49962  2181  zero_less_mult_iff distrib_left [symmetric]  huffman@47139  2182  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  2183 haftmann@53068  2184 lemma zdiv_zmult2_eq:  haftmann@53068  2185  fixes a b c :: int  haftmann@53068  2186  shows "0 \ c \ a div (b * c) = (a div b) div c"  haftmann@33361  2187 apply (case_tac "b = 0", simp)  haftmann@53068  2188 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  2189 done  haftmann@33361  2190 haftmann@33361  2191 lemma zmod_zmult2_eq:  haftmann@53068  2192  fixes a b c :: int  haftmann@53068  2193  shows "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@33361  2194 apply (case_tac "b = 0", simp)  haftmann@53068  2195 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  2196 done  haftmann@33361  2197 huffman@47108  2198 lemma div_pos_geq:  huffman@47108  2199  fixes k l :: int  huffman@47108  2200  assumes "0 < l" and "l \ k"  huffman@47108  2201  shows "k div l = (k - l) div l + 1"  huffman@47108  2202 proof -  huffman@47108  2203  have "k = (k - l) + l" by simp  huffman@47108  2204  then obtain j where k: "k = j + l" ..  huffman@47108  2205  with assms show ?thesis by simp  huffman@47108  2206 qed  huffman@47108  2207 huffman@47108  2208 lemma mod_pos_geq:  huffman@47108  2209  fixes k l :: int  huffman@47108  2210  assumes "0 < l" and "l \ k"  huffman@47108  2211  shows "k mod l = (k - l) mod l"  huffman@47108  2212 proof -  huffman@47108  2213  have "k = (k - l) + l" by simp  huffman@47108  2214  then obtain j where k: "k = j + l" ..  huffman@47108  2215  with assms show ?thesis by simp  huffman@47108  2216 qed  huffman@47108  2217 haftmann@33361  2218 huffman@46551  2219 subsubsection {* Splitting Rules for div and mod *}  haftmann@33361  2220 haftmann@33361  2221 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  2222 haftmann@33361  2223 lemma split_pos_lemma:  haftmann@33361  2224  "0  haftmann@33361  2225  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2226 apply (rule iffI, clarify)  haftmann@33361  2227  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2228  apply (subst mod_add_eq)  haftmann@33361  2229  apply (subst zdiv_zadd1_eq)  haftmann@33361  2230  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  2231 txt{*converse direction*}  haftmann@33361  2232 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2233 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2234 done  haftmann@33361  2235 haftmann@33361  2236 lemma split_neg_lemma:  haftmann@33361  2237  "k<0 ==>  haftmann@33361  2238  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2239 apply (rule iffI, clarify)  haftmann@33361  2240  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2241  apply (subst mod_add_eq)  haftmann@33361  2242  apply (subst zdiv_zadd1_eq)  haftmann@33361  2243  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  2244 txt{*converse direction*}  haftmann@33361  2245 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2246 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2247 done  haftmann@33361  2248 haftmann@33361  2249 lemma split_zdiv:  haftmann@33361  2250  "P(n div k :: int) =  haftmann@33361  2251  ((k = 0 --> P 0) &  haftmann@33361  2252  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2253  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2254 apply (case_tac "k=0", simp)  haftmann@33361  2255 apply (simp only: linorder_neq_iff)  haftmann@33361  2256 apply (erule disjE)  haftmann@33361  2257  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2258  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2259 done  haftmann@33361  2260 haftmann@33361  2261 lemma split_zmod:  haftmann@33361  2262  "P(n mod k :: int) =  haftmann@33361  2263  ((k = 0 --> P n) &  haftmann@33361  2264  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2265  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2266 apply (case_tac "k=0", simp)  haftmann@33361  2267 apply (simp only: linorder_neq_iff)  haftmann@33361  2268 apply (erule disjE)  haftmann@33361  2269  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2270  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2271 done  haftmann@33361  2272 webertj@33730  2273 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  2274  when these are applied to some constant that is of the form  huffman@47108  2275  @{term "numeral k"}: *}  huffman@47108  2276 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  2277 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  2278 haftmann@33361  2279 huffman@47166  2280 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}  huffman@47166  2281 huffman@47166  2282 lemma pos_divmod_int_rel_mult_2:  huffman@47166  2283  assumes "0 \ b"  huffman@47166  2284  assumes "divmod_int_rel a b (q, r)"  huffman@47166  2285  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  2286  using assms unfolding divmod_int_rel_def by auto  huffman@47166  2287 haftmann@54489  2288 declaration {* K (Lin_Arith.add_simps @{thms uminus_numeral_One}) *}  haftmann@54489  2289 huffman@47166  2290 lemma neg_divmod_int_rel_mult_2:  huffman@47166  2291  assumes "b \ 0"  huffman@47166  2292  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  2293  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  2294  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  2295 haftmann@33361  2296 text{*computing div by shifting *}  haftmann@33361  2297 haftmann@33361  2298 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2299  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2300  by (rule div_int_unique)  haftmann@33361  2301 boehmes@35815  2302 lemma neg_zdiv_mult_2:  boehmes@35815  2303  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2304  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2305  by (rule div_int_unique)  haftmann@33361  2306 huffman@47108  2307 (* FIXME: add rules for negative numerals *)  huffman@47108  2308 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2309  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2310  numeral v div (numeral w :: int)"  huffman@47108  2311  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2312  by (rule div_mult_mult1, simp)  huffman@47108  2313 huffman@47108  2314 lemma zdiv_numeral_Bit1 [simp]:  huffman@47108  2315  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  huffman@47108  2316  (numeral v div (numeral w :: int))"  huffman@47108  2317  unfolding numeral.simps  huffman@47108  2318  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2319  by (rule pos_zdiv_mult_2, simp)  haftmann@33361  2320 haftmann@33361  2321 lemma pos_zmod_mult_2:  haftmann@33361  2322  fixes a b :: int  haftmann@33361  2323  assumes "0 \ a"  haftmann@33361  2324  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  huffman@47166  2325  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2326  by (rule mod_int_unique)  haftmann@33361  2327 haftmann@33361  2328 lemma neg_zmod_mult_2:  haftmann@33361  2329  fixes a b :: int  haftmann@33361  2330  assumes "a \ 0"  haftmann@33361  2331  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  huffman@47166  2332  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2333  by (rule mod_int_unique)  haftmann@33361  2334 huffman@47108  2335 (* FIXME: add rules for negative numerals *)  huffman@47108  2336 lemma zmod_numeral_Bit0 [simp]:  huffman@47108  2337  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  huffman@47108  2338  (2::int) * (numeral v mod numeral w)"  huffman@47108  2339  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  huffman@47108  2340  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)  huffman@47108  2341 huffman@47108  2342 lemma zmod_numeral_Bit1 [simp]:  huffman@47108  2343  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =  huffman@47108  2344  2 * (numeral v mod numeral w) + (1::int)"  huffman@47108  2345  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]  huffman@47108  2346  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2347  by (rule pos_zmod_mult_2, simp)  haftmann@33361  2348 nipkow@39489  2349 lemma zdiv_eq_0_iff:  nipkow@39489  2350  "(i::int) div k = 0 \ k=0 \ 0\i \ i i\0 \ k ?R" by (rule split_zdiv[THEN iffD2]) simp  nipkow@39489  2354  with ?L show ?R by blast  nipkow@39489  2355 next  nipkow@39489  2356  assume ?R thus ?L  nipkow@39489  2357  by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)  nipkow@39489  2358 qed  nipkow@39489  2359 nipkow@39489  2360 huffman@46551  2361 subsubsection {* Quotients of Signs *}  haftmann@33361  2362 haftmann@33361  2363 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  haftmann@33361  2364 apply (subgoal_tac "a div b \ -1", force)  haftmann@33361  2365 apply (rule order_trans)  haftmann@33361  2366 apply (rule_tac a' = "-1" in zdiv_mono1)  haftmann@33361  2367 apply (auto simp add: div_eq_minus1)  haftmann@33361  2368 done  haftmann@33361  2369 haftmann@33361  2370 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  haftmann@33361  2371 by (drule zdiv_mono1_neg, auto)  haftmann@33361  2372 haftmann@33361  2373 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  haftmann@33361  2374 by (drule zdiv_mono1, auto)  haftmann@33361  2375 nipkow@33804  2376 text{* Now for some equivalences of the form @{text"a div b >=< 0 \ \"}  nipkow@33804  2377 conditional upon the sign of @{text a} or @{text b}. There are many more.  nipkow@33804  2378 They should all be simp rules unless that causes too much search. *}  nipkow@33804  2379 haftmann@33361  2380 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  haftmann@33361  2381 apply auto  haftmann@33361  2382 apply (drule_tac [2] zdiv_mono1)  haftmann@33361  2383 apply (auto simp add: linorder_neq_iff)  haftmann@33361  2384 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  haftmann@33361  2385 apply (blast intro: div_neg_pos_less0)  haftmann@33361  2386 done  haftmann@33361  2387 haftmann@33361  2388 lemma neg_imp_zdiv_nonneg_iff:  nipkow@33804  2389  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  huffman@47159  2390 apply (subst div_minus_minus [symmetric])  haftmann@33361  2391 apply (subst pos_imp_zdiv_nonneg_iff, auto)  haftmann@33361  2392 done  haftmann@33361  2393 haftmann@33361  2394 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  haftmann@33361  2395 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  haftmann@33361  2396 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  haftmann@33361  2397 nipkow@39489  2398 lemma pos_imp_zdiv_pos_iff:  nipkow@39489  2399  "0 0 < (i::int) div k \ k \ i"  nipkow@39489  2400 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]  nipkow@39489  2401 by arith  nipkow@39489  2402 haftmann@33361  2403 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  haftmann@33361  2404 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  haftmann@33361  2405 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  haftmann@33361  2406 nipkow@33804  2407 lemma nonneg1_imp_zdiv_pos_iff:  nipkow@33804  2408  "(0::int) <= a \ (a div b > 0) = (a >= b & b>0)"  nipkow@33804  2409 apply rule  nipkow@33804  2410  apply rule  nipkow@33804  2411  using div_pos_pos_trivial[of a b]apply arith  nipkow@33804  2412  apply(cases "b=0")apply simp  nipkow@33804  2413  using div_nonneg_neg_le0[of a b]apply arith  nipkow@33804  2414 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp  nipkow@33804  2415 done  nipkow@33804  2416 nipkow@39489  2417 lemma zmod_le_nonneg_dividend: "(m::int) \ 0 ==> m mod k \ m"  nipkow@39489  2418 apply (rule split_zmod[THEN iffD2])  nipkow@44890  2419 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)  nipkow@39489  2420 done  nipkow@39489  2421 nipkow@39489  2422 haftmann@33361  2423 subsubsection {* The Divides Relation *}  haftmann@33361  2424 huffman@47268  2425 lemma dvd_neg_numeral_left [simp]:  huffman@47268  2426  fixes y :: "'a::comm_ring_1"  haftmann@54489  2427  shows "(- numeral k) dvd y \ (numeral k) dvd y"  haftmann@54489  2428  by (fact minus_dvd_iff)  huffman@47268  2429 huffman@47268  2430 lemma dvd_neg_numeral_right [simp]:  huffman@47268  2431  fixes x :: "'a::comm_ring_1"  haftmann@54489  2432  shows "x dvd (- numeral k) \ x dvd (numeral k)"  haftmann@54489  2433  by (fact dvd_minus_iff)  haftmann@33361  2434 huffman@47108  2435 lemmas dvd_eq_mod_eq_0_numeral [simp] =  huffman@47108  2436  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y  huffman@47108  2437 huffman@47108  2438 huffman@47108  2439 subsubsection {* Further properties *}  huffman@47108  2440 haftmann@33361  2441 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  haftmann@33361  2442  using zmod_zdiv_equality[where a="m" and b="n"]  huffman@47142  2443  by (simp add: algebra_simps) (* FIXME: generalize *)  haftmann@33361  2444 haftmann@33361  2445 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  haftmann@33361  2446 apply (subst split_div, auto)  haftmann@33361  2447 apply (subst split_zdiv, auto)  haftmann@33361  2448 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)  haftmann@33361  2449 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2450 done  haftmann@33361  2451 haftmann@33361  2452 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  haftmann@33361  2453 apply (subst split_mod, auto)  haftmann@33361  2454 apply (subst split_zmod, auto)  haftmann@33361  2455 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  haftmann@33361  2456  in unique_remainder)  haftmann@33361  2457 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2458 done  haftmann@33361  2459 haftmann@33361  2460 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  haftmann@33361  2461 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  haftmann@33361  2462 haftmann@33361  2463 text{*Suggested by Matthias Daum*}  haftmann@33361  2464 lemma int_power_div_base:  haftmann@33361  2465  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  haftmann@33361  2466 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  haftmann@33361  2467  apply (erule ssubst)  haftmann@33361  2468  apply (simp only: power_add)  haftmann@33361  2469  apply simp_all  haftmann@33361  2470 done  haftmann@33361  2471 haftmann@33361  2472 text {* by Brian Huffman *}  haftmann@33361  2473 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@33361  2474 by (rule mod_minus_eq [symmetric])  haftmann@33361  2475 haftmann@33361  2476 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@33361  2477 by (rule mod_diff_left_eq [symmetric])  haftmann@33361  2478 haftmann@33361  2479 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@33361  2480 by (rule mod_diff_right_eq [symmetric])  haftmann@33361  2481 haftmann@33361  2482 lemmas zmod_simps =  haftmann@33361  2483  mod_add_left_eq [symmetric]  haftmann@33361  2484  mod_add_right_eq [symmetric]  huffman@47142  2485  mod_mult_right_eq[symmetric]  haftmann@33361  2486  mod_mult_left_eq [symmetric]  huffman@47164  2487  power_mod  haftmann@33361  2488  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@33361  2489 haftmann@33361  2490 text {* Distributive laws for function @{text nat}. *}  haftmann@33361  2491 haftmann@33361  2492 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  haftmann@33361  2493 apply (rule linorder_cases [of y 0])  haftmann@33361  2494 apply (simp add: div_nonneg_neg_le0)  haftmann@33361  2495 apply simp  haftmann@33361  2496 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  haftmann@33361  2497 done  haftmann@33361  2498 haftmann@33361  2499 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  haftmann@33361  2500 lemma nat_mod_distrib:  haftmann@33361  2501  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  haftmann@33361  2502 apply (case_tac "y = 0", simp)  haftmann@33361  2503 apply (simp add: nat_eq_iff zmod_int)  haftmann@33361  2504 done  haftmann@33361  2505 haftmann@33361  2506 text {* transfer setup *}  haftmann@33361  2507 haftmann@33361  2508 lemma transfer_nat_int_functions:  haftmann@33361  2509  "(x::int) >= 0 \ y >= 0 \ (nat x) div (nat y) = nat (x div y)"  haftmann@33361  2510  "(x::int) >= 0 \ y >= 0 \ (nat x) mod (nat y) = nat (x mod y)"  haftmann@33361  2511  by (auto simp add: nat_div_distrib nat_mod_distrib)  haftmann@33361  2512 haftmann@33361  2513 lemma transfer_nat_int_function_closures:  haftmann@33361  2514  "(x::int) >= 0 \ y >= 0 \ x div y >= 0"  haftmann@33361  2515  "(x::int) >= 0 \ y >= 0 \ x mod y >= 0"  haftmann@33361  2516  apply (cases "y = 0")  haftmann@33361  2517  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  haftmann@33361  2518  apply (cases "y = 0")  haftmann@33361  2519  apply auto  haftmann@33361  2520 done  haftmann@33361  2521 haftmann@35644  2522 declare transfer_morphism_nat_int [transfer add return:  haftmann@33361  2523  transfer_nat_int_functions  haftmann@33361  2524  transfer_nat_int_function_closures ` haftmann@33361