src/HOL/Divides.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 53199 7a9fe70c8b0a child 54221 56587960e444 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
 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@27651  56 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"  haftmann@27651  57 proof (cases "b = 0")  haftmann@27651  58  case True then show ?thesis by simp  haftmann@27651  59 next  haftmann@27651  60  case False  haftmann@27651  61  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"  haftmann@27651  62  by (simp add: mod_div_equality)  haftmann@27651  63  also from False div_mult_self1 [of b a c] have  haftmann@27651  64  "\ = (c + a div b) * b + (a + c * b) mod b"  nipkow@29667  65  by (simp add: algebra_simps)  haftmann@27651  66  finally have "a = a div b * b + (a + c * b) mod b"  webertj@49962  67  by (simp add: add_commute [of a] add_assoc distrib_right)  haftmann@27651  68  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"  haftmann@27651  69  by (simp add: mod_div_equality)  haftmann@27651  70  then show ?thesis by simp  haftmann@27651  71 qed  haftmann@27651  72 haftmann@27651  73 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"  haftmann@30934  74  by (simp add: mult_commute [of b])  haftmann@27651  75 haftmann@27651  76 lemma div_mult_self1_is_id [simp]: "b \ 0 \ b * a div b = a"  haftmann@27651  77  using div_mult_self2 [of b 0 a] by simp  haftmann@27651  78 haftmann@27651  79 lemma div_mult_self2_is_id [simp]: "b \ 0 \ a * b div b = a"  haftmann@27651  80  using div_mult_self1 [of b 0 a] by simp  haftmann@27651  81 haftmann@27651  82 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"  haftmann@27651  83  using mod_mult_self2 [of 0 b a] by simp  haftmann@27651  84 haftmann@27651  85 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"  haftmann@27651  86  using mod_mult_self1 [of 0 a b] by simp  haftmann@26062  87 haftmann@27651  88 lemma div_by_1 [simp]: "a div 1 = a"  haftmann@27651  89  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp  haftmann@27651  90 haftmann@27651  91 lemma mod_by_1 [simp]: "a mod 1 = 0"  haftmann@27651  92 proof -  haftmann@27651  93  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp  haftmann@27651  94  then have "a + a mod 1 = a + 0" by simp  haftmann@27651  95  then show ?thesis by (rule add_left_imp_eq)  haftmann@27651  96 qed  haftmann@27651  97 haftmann@27651  98 lemma mod_self [simp]: "a mod a = 0"  haftmann@27651  99  using mod_mult_self2_is_0 [of 1] by simp  haftmann@27651  100 haftmann@27651  101 lemma div_self [simp]: "a \ 0 \ a div a = 1"  haftmann@27651  102  using div_mult_self2_is_id [of _ 1] by simp  haftmann@27651  103 haftmann@27676  104 lemma div_add_self1 [simp]:  haftmann@27651  105  assumes "b \ 0"  haftmann@27651  106  shows "(b + a) div b = a div b + 1"  haftmann@27651  107  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)  haftmann@26062  108 haftmann@27676  109 lemma div_add_self2 [simp]:  haftmann@27651  110  assumes "b \ 0"  haftmann@27651  111  shows "(a + b) div b = a div b + 1"  haftmann@27651  112  using assms div_add_self1 [of b a] by (simp add: add_commute)  haftmann@27651  113 haftmann@27676  114 lemma mod_add_self1 [simp]:  haftmann@27651  115  "(b + a) mod b = a mod b"  haftmann@27651  116  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)  haftmann@27651  117 haftmann@27676  118 lemma mod_add_self2 [simp]:  haftmann@27651  119  "(a + b) mod b = a mod b"  haftmann@27651  120  using mod_mult_self1 [of a 1 b] by simp  haftmann@27651  121 haftmann@27651  122 lemma mod_div_decomp:  haftmann@27651  123  fixes a b  haftmann@27651  124  obtains q r where "q = a div b" and "r = a mod b"  haftmann@27651  125  and "a = q * b + r"  haftmann@27651  126 proof -  haftmann@27651  127  from mod_div_equality have "a = a div b * b + a mod b" by simp  haftmann@27651  128  moreover have "a div b = a div b" ..  haftmann@27651  129  moreover have "a mod b = a mod b" ..  haftmann@27651  130  note that ultimately show thesis by blast  haftmann@27651  131 qed  haftmann@27651  132 bulwahn@45231  133 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \ b mod a = 0"  haftmann@25942  134 proof  haftmann@25942  135  assume "b mod a = 0"  haftmann@25942  136  with mod_div_equality [of b a] have "b div a * a = b" by simp  haftmann@25942  137  then have "b = a * (b div a)" unfolding mult_commute ..  haftmann@25942  138  then have "\c. b = a * c" ..  haftmann@25942  139  then show "a dvd b" unfolding dvd_def .  haftmann@25942  140 next  haftmann@25942  141  assume "a dvd b"  haftmann@25942  142  then have "\c. b = a * c" unfolding dvd_def .  haftmann@25942  143  then obtain c where "b = a * c" ..  haftmann@25942  144  then have "b mod a = a * c mod a" by simp  haftmann@25942  145  then have "b mod a = c * a mod a" by (simp add: mult_commute)  haftmann@27651  146  then show "b mod a = 0" by simp  haftmann@25942  147 qed  haftmann@25942  148 huffman@29403  149 lemma mod_div_trivial [simp]: "a mod b div b = 0"  huffman@29403  150 proof (cases "b = 0")  huffman@29403  151  assume "b = 0"  huffman@29403  152  thus ?thesis by simp  huffman@29403  153 next  huffman@29403  154  assume "b \ 0"  huffman@29403  155  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"  huffman@29403  156  by (rule div_mult_self1 [symmetric])  huffman@29403  157  also have "\ = a div b"  huffman@29403  158  by (simp only: mod_div_equality')  huffman@29403  159  also have "\ = a div b + 0"  huffman@29403  160  by simp  huffman@29403  161  finally show ?thesis  huffman@29403  162  by (rule add_left_imp_eq)  huffman@29403  163 qed  huffman@29403  164 huffman@29403  165 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"  huffman@29403  166 proof -  huffman@29403  167  have "a mod b mod b = (a mod b + a div b * b) mod b"  huffman@29403  168  by (simp only: mod_mult_self1)  huffman@29403  169  also have "\ = a mod b"  huffman@29403  170  by (simp only: mod_div_equality')  huffman@29403  171  finally show ?thesis .  huffman@29403  172 qed  huffman@29403  173 nipkow@29925  174 lemma dvd_imp_mod_0: "a dvd b \ b mod a = 0"  nipkow@29948  175 by (rule dvd_eq_mod_eq_0[THEN iffD1])  nipkow@29925  176 nipkow@29925  177 lemma dvd_div_mult_self: "a dvd b \ (b div a) * a = b"  nipkow@29925  178 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)  nipkow@29925  179 haftmann@33274  180 lemma dvd_mult_div_cancel: "a dvd b \ a * (b div a) = b"  haftmann@33274  181 by (drule dvd_div_mult_self) (simp add: mult_commute)  haftmann@33274  182 nipkow@30052  183 lemma dvd_div_mult: "a dvd b \ (b div a) * c = b * c div a"  nipkow@30052  184 apply (cases "a = 0")  nipkow@30052  185  apply simp  nipkow@30052  186 apply (auto simp: dvd_def mult_assoc)  nipkow@30052  187 done  nipkow@30052  188 nipkow@29925  189 lemma div_dvd_div[simp]:  nipkow@29925  190  "a dvd b \ a dvd c \ (b div a dvd c div a) = (b dvd c)"  nipkow@29925  191 apply (cases "a = 0")  nipkow@29925  192  apply simp  nipkow@29925  193 apply (unfold dvd_def)  nipkow@29925  194 apply auto  nipkow@29925  195  apply(blast intro:mult_assoc[symmetric])  nipkow@44890  196 apply(fastforce simp add: mult_assoc)  nipkow@29925  197 done  nipkow@29925  198 huffman@30078  199 lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m"  huffman@30078  200  apply (subgoal_tac "k dvd (m div n) *n + m mod n")  huffman@30078  201  apply (simp add: mod_div_equality)  huffman@30078  202  apply (simp only: dvd_add dvd_mult)  huffman@30078  203  done  huffman@30078  204 huffman@29403  205 text {* Addition respects modular equivalence. *}  huffman@29403  206 huffman@29403  207 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"  huffman@29403  208 proof -  huffman@29403  209  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"  huffman@29403  210  by (simp only: mod_div_equality)  huffman@29403  211  also have "\ = (a mod c + b + a div c * c) mod c"  huffman@29403  212  by (simp only: add_ac)  huffman@29403  213  also have "\ = (a mod c + b) mod c"  huffman@29403  214  by (rule mod_mult_self1)  huffman@29403  215  finally show ?thesis .  huffman@29403  216 qed  huffman@29403  217 huffman@29403  218 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"  huffman@29403  219 proof -  huffman@29403  220  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"  huffman@29403  221  by (simp only: mod_div_equality)  huffman@29403  222  also have "\ = (a + b mod c + b div c * c) mod c"  huffman@29403  223  by (simp only: add_ac)  huffman@29403  224  also have "\ = (a + b mod c) mod c"  huffman@29403  225  by (rule mod_mult_self1)  huffman@29403  226  finally show ?thesis .  huffman@29403  227 qed  huffman@29403  228 huffman@29403  229 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"  huffman@29403  230 by (rule trans [OF mod_add_left_eq mod_add_right_eq])  huffman@29403  231 huffman@29403  232 lemma mod_add_cong:  huffman@29403  233  assumes "a mod c = a' mod c"  huffman@29403  234  assumes "b mod c = b' mod c"  huffman@29403  235  shows "(a + b) mod c = (a' + b') mod c"  huffman@29403  236 proof -  huffman@29403  237  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"  huffman@29403  238  unfolding assms ..  huffman@29403  239  thus ?thesis  huffman@29403  240  by (simp only: mod_add_eq [symmetric])  huffman@29403  241 qed  huffman@29403  242 haftmann@30923  243 lemma div_add [simp]: "z dvd x \ z dvd y  nipkow@30837  244  \ (x + y) div z = x div z + y div z"  haftmann@30923  245 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)  nipkow@30837  246 huffman@29403  247 text {* Multiplication respects modular equivalence. *}  huffman@29403  248 huffman@29403  249 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"  huffman@29403  250 proof -  huffman@29403  251  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"  huffman@29403  252  by (simp only: mod_div_equality)  huffman@29403  253  also have "\ = (a mod c * b + a div c * b * c) mod c"  nipkow@29667  254  by (simp only: algebra_simps)  huffman@29403  255  also have "\ = (a mod c * b) mod c"  huffman@29403  256  by (rule mod_mult_self1)  huffman@29403  257  finally show ?thesis .  huffman@29403  258 qed  huffman@29403  259 huffman@29403  260 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"  huffman@29403  261 proof -  huffman@29403  262  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"  huffman@29403  263  by (simp only: mod_div_equality)  huffman@29403  264  also have "\ = (a * (b mod c) + a * (b div c) * c) mod c"  nipkow@29667  265  by (simp only: algebra_simps)  huffman@29403  266  also have "\ = (a * (b mod c)) mod c"  huffman@29403  267  by (rule mod_mult_self1)  huffman@29403  268  finally show ?thesis .  huffman@29403  269 qed  huffman@29403  270 huffman@29403  271 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"  huffman@29403  272 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])  huffman@29403  273 huffman@29403  274 lemma mod_mult_cong:  huffman@29403  275  assumes "a mod c = a' mod c"  huffman@29403  276  assumes "b mod c = b' mod c"  huffman@29403  277  shows "(a * b) mod c = (a' * b') mod c"  huffman@29403  278 proof -  huffman@29403  279  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"  huffman@29403  280  unfolding assms ..  huffman@29403  281  thus ?thesis  huffman@29403  282  by (simp only: mod_mult_eq [symmetric])  huffman@29403  283 qed  huffman@29403  284 huffman@47164  285 text {* Exponentiation respects modular equivalence. *}  huffman@47164  286 huffman@47164  287 lemma power_mod: "(a mod b)^n mod b = a^n mod b"  huffman@47164  288 apply (induct n, simp_all)  huffman@47164  289 apply (rule mod_mult_right_eq [THEN trans])  huffman@47164  290 apply (simp (no_asm_simp))  huffman@47164  291 apply (rule mod_mult_eq [symmetric])  huffman@47164  292 done  huffman@47164  293 huffman@29404  294 lemma mod_mod_cancel:  huffman@29404  295  assumes "c dvd b"  huffman@29404  296  shows "a mod b mod c = a mod c"  huffman@29404  297 proof -  huffman@29404  298  from c dvd b obtain k where "b = c * k"  huffman@29404  299  by (rule dvdE)  huffman@29404  300  have "a mod b mod c = a mod (c * k) mod c"  huffman@29404  301  by (simp only: b = c * k)  huffman@29404  302  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  303  by (simp only: mod_mult_self1)  huffman@29404  304  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  huffman@29404  305  by (simp only: add_ac mult_ac)  huffman@29404  306  also have "\ = a mod c"  huffman@29404  307  by (simp only: mod_div_equality)  huffman@29404  308  finally show ?thesis .  huffman@29404  309 qed  huffman@29404  310 haftmann@30930  311 lemma div_mult_div_if_dvd:  haftmann@30930  312  "y dvd x \ z dvd w \ (x div y) * (w div z) = (x * w) div (y * z)"  haftmann@30930  313  apply (cases "y = 0", simp)  haftmann@30930  314  apply (cases "z = 0", simp)  haftmann@30930  315  apply (auto elim!: dvdE simp add: algebra_simps)  nipkow@30476  316  apply (subst mult_assoc [symmetric])  nipkow@30476  317  apply (simp add: no_zero_divisors)  haftmann@30930  318  done  haftmann@30930  319 haftmann@35367  320 lemma div_mult_swap:  haftmann@35367  321  assumes "c dvd b"  haftmann@35367  322  shows "a * (b div c) = (a * b) div c"  haftmann@35367  323 proof -  haftmann@35367  324  from assms have "b div c * (a div 1) = b * a div (c * 1)"  haftmann@35367  325  by (simp only: div_mult_div_if_dvd one_dvd)  haftmann@35367  326  then show ?thesis by (simp add: mult_commute)  haftmann@35367  327 qed  haftmann@35367  328   haftmann@30930  329 lemma div_mult_mult2 [simp]:  haftmann@30930  330  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@30930  331  by (drule div_mult_mult1) (simp add: mult_commute)  haftmann@30930  332 haftmann@30930  333 lemma div_mult_mult1_if [simp]:  haftmann@30930  334  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  335  by simp_all  nipkow@30476  336 haftmann@30930  337 lemma mod_mult_mult1:  haftmann@30930  338  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  339 proof (cases "c = 0")  haftmann@30930  340  case True then show ?thesis by simp  haftmann@30930  341 next  haftmann@30930  342  case False  haftmann@30930  343  from mod_div_equality  haftmann@30930  344  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  345  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  346  = c * a + c * (a mod b)" by (simp add: algebra_simps)  haftmann@30930  347  with mod_div_equality show ?thesis by simp  haftmann@30930  348 qed  haftmann@30930  349   haftmann@30930  350 lemma mod_mult_mult2:  haftmann@30930  351  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@30930  352  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)  haftmann@30930  353 huffman@47159  354 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  huffman@47159  355  by (fact mod_mult_mult2 [symmetric])  huffman@47159  356 huffman@47159  357 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  huffman@47159  358  by (fact mod_mult_mult1 [symmetric])  huffman@47159  359 huffman@31662  360 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  361  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  362 huffman@31662  363 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  364 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  365 haftmann@31009  366 lemma div_power:  huffman@31661  367  "y dvd x \ (x div y) ^ n = x ^ n div y ^ n"  nipkow@30476  368 apply (induct n)  nipkow@30476  369  apply simp  nipkow@30476  370 apply(simp add: div_mult_div_if_dvd dvd_power_same)  nipkow@30476  371 done  nipkow@30476  372 haftmann@35367  373 lemma dvd_div_eq_mult:  haftmann@35367  374  assumes "a \ 0" and "a dvd b"  haftmann@35367  375  shows "b div a = c \ b = c * a"  haftmann@35367  376 proof  haftmann@35367  377  assume "b = c * a"  haftmann@35367  378  then show "b div a = c" by (simp add: assms)  haftmann@35367  379 next  haftmann@35367  380  assume "b div a = c"  haftmann@35367  381  then have "b div a * a = c * a" by simp  haftmann@35367  382  moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)  haftmann@35367  383  ultimately show "b = c * a" by simp  haftmann@35367  384 qed  haftmann@35367  385   haftmann@35367  386 lemma dvd_div_div_eq_mult:  haftmann@35367  387  assumes "a \ 0" "c \ 0" and "a dvd b" "c dvd d"  haftmann@35367  388  shows "b div a = d div c \ b * c = a * d"  haftmann@35367  389  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  390 huffman@31661  391 end  huffman@31661  392 haftmann@35673  393 class ring_div = semiring_div + comm_ring_1  huffman@29405  394 begin  huffman@29405  395 haftmann@36634  396 subclass ring_1_no_zero_divisors ..  haftmann@36634  397 huffman@29405  398 text {* Negation respects modular equivalence. *}  huffman@29405  399 huffman@29405  400 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  401 proof -  huffman@29405  402  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  403  by (simp only: mod_div_equality)  huffman@29405  404  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  huffman@29405  405  by (simp only: minus_add_distrib minus_mult_left add_ac)  huffman@29405  406  also have "\ = (- (a mod b)) mod b"  huffman@29405  407  by (rule mod_mult_self1)  huffman@29405  408  finally show ?thesis .  huffman@29405  409 qed  huffman@29405  410 huffman@29405  411 lemma mod_minus_cong:  huffman@29405  412  assumes "a mod b = a' mod b"  huffman@29405  413  shows "(- a) mod b = (- a') mod b"  huffman@29405  414 proof -  huffman@29405  415  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  416  unfolding assms ..  huffman@29405  417  thus ?thesis  huffman@29405  418  by (simp only: mod_minus_eq [symmetric])  huffman@29405  419 qed  huffman@29405  420 huffman@29405  421 text {* Subtraction respects modular equivalence. *}  huffman@29405  422 huffman@29405  423 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"  huffman@29405  424  unfolding diff_minus  huffman@29405  425  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  426 huffman@29405  427 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"  huffman@29405  428  unfolding diff_minus  huffman@29405  429  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  430 huffman@29405  431 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"  huffman@29405  432  unfolding diff_minus  huffman@29405  433  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  434 huffman@29405  435 lemma mod_diff_cong:  huffman@29405  436  assumes "a mod c = a' mod c"  huffman@29405  437  assumes "b mod c = b' mod c"  huffman@29405  438  shows "(a - b) mod c = (a' - b') mod c"  huffman@29405  439  unfolding diff_minus using assms  huffman@29405  440  by (intro mod_add_cong mod_minus_cong)  huffman@29405  441 nipkow@30180  442 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  443 apply (case_tac "y = 0") apply simp  nipkow@30180  444 apply (auto simp add: dvd_def)  nipkow@30180  445 apply (subgoal_tac "-(y * k) = y * - k")  nipkow@30180  446  apply (erule ssubst)  nipkow@30180  447  apply (erule div_mult_self1_is_id)  nipkow@30180  448 apply simp  nipkow@30180  449 done  nipkow@30180  450 nipkow@30180  451 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  452 apply (case_tac "y = 0") apply simp  nipkow@30180  453 apply (auto simp add: dvd_def)  nipkow@30180  454 apply (subgoal_tac "y * k = -y * -k")  nipkow@30180  455  apply (erule ssubst)  nipkow@30180  456  apply (rule div_mult_self1_is_id)  nipkow@30180  457  apply simp  nipkow@30180  458 apply simp  nipkow@30180  459 done  nipkow@30180  460 huffman@47159  461 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  huffman@47159  462  using div_mult_mult1 [of "- 1" a b]  huffman@47159  463  unfolding neg_equal_0_iff_equal by simp  huffman@47159  464 huffman@47159  465 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  huffman@47159  466  using mod_mult_mult1 [of "- 1" a b] by simp  huffman@47159  467 huffman@47159  468 lemma div_minus_right: "a div (-b) = (-a) div b"  huffman@47159  469  using div_minus_minus [of "-a" b] by simp  huffman@47159  470 huffman@47159  471 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  huffman@47159  472  using mod_minus_minus [of "-a" b] by simp  huffman@47159  473 huffman@47160  474 lemma div_minus1_right [simp]: "a div (-1) = -a"  huffman@47160  475  using div_minus_right [of a 1] by simp  huffman@47160  476 huffman@47160  477 lemma mod_minus1_right [simp]: "a mod (-1) = 0"  huffman@47160  478  using mod_minus_right [of a 1] by simp  huffman@47160  479 huffman@29405  480 end  huffman@29405  481 haftmann@25942  482 haftmann@53067  483 subsection {* Generic numeral division with a pragmatic type class *}  haftmann@53067  484 haftmann@53067  485 text {*  haftmann@53067  486  The following type class contains everything necessary to formulate  haftmann@53067  487  a division algorithm in ring structures with numerals, restricted  haftmann@53067  488  to its positive segments. This is its primary motiviation, and it  haftmann@53067  489  could surely be formulated using a more fine-grained, more algebraic  haftmann@53067  490  and less technical class hierarchy.  haftmann@53067  491 *}  haftmann@53067  492 haftmann@53067  493 haftmann@53067  494 class semiring_numeral_div = linordered_semidom + minus + semiring_div +  haftmann@53067  495  assumes diff_invert_add1: "a + b = c \ a = c - b"  haftmann@53067  496  and le_add_diff_inverse2: "b \ a \ a - b + b = a"  haftmann@53067  497  assumes mult_div_cancel: "b * (a div b) = a - a mod b"  haftmann@53067  498  and div_less: "0 \ a \ a < b \ a div b = 0"  haftmann@53067  499  and mod_less: " 0 \ a \ a < b \ a mod b = a"  haftmann@53067  500  and div_positive: "0 < b \ b \ a \ a div b > 0"  haftmann@53067  501  and mod_less_eq_dividend: "0 \ a \ a mod b \ a"  haftmann@53067  502  and pos_mod_bound: "0 < b \ a mod b < b"  haftmann@53067  503  and pos_mod_sign: "0 < b \ 0 \ a mod b"  haftmann@53067  504  and mod_mult2_eq: "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@53067  505  and div_mult2_eq: "0 \ c \ a div (b * c) = a div b div c"  haftmann@53067  506  assumes discrete: "a < b \ a + 1 \ b"  haftmann@53067  507 begin  haftmann@53067  508 haftmann@53067  509 lemma diff_zero [simp]:  haftmann@53067  510  "a - 0 = a"  haftmann@53067  511  by (rule diff_invert_add1 [symmetric]) simp  haftmann@53067  512 haftmann@53067  513 lemma parity:  haftmann@53067  514  "a mod 2 = 0 \ a mod 2 = 1"  haftmann@53067  515 proof (rule ccontr)  haftmann@53067  516  assume "\ (a mod 2 = 0 \ a mod 2 = 1)"  haftmann@53067  517  then have "a mod 2 \ 0" and "a mod 2 \ 1" by simp_all  haftmann@53067  518  have "0 < 2" by simp  haftmann@53067  519  with pos_mod_bound pos_mod_sign have "0 \ a mod 2" "a mod 2 < 2" by simp_all  haftmann@53067  520  with a mod 2 \ 0 have "0 < a mod 2" by simp  haftmann@53067  521  with discrete have "1 \ a mod 2" by simp  haftmann@53067  522  with a mod 2 \ 1 have "1 < a mod 2" by simp  haftmann@53067  523  with discrete have "2 \ a mod 2" by simp  haftmann@53067  524  with a mod 2 < 2 show False by simp  haftmann@53067  525 qed  haftmann@53067  526 haftmann@53067  527 lemma divmod_digit_1:  haftmann@53067  528  assumes "0 \ a" "0 < b" and "b \ a mod (2 * b)"  haftmann@53067  529  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")  haftmann@53067  530  and "a mod (2 * b) - b = a mod b" (is "?Q")  haftmann@53067  531 proof -  haftmann@53067  532  from assms mod_less_eq_dividend [of a "2 * b"] have "b \ a"  haftmann@53067  533  by (auto intro: trans)  haftmann@53067  534  with 0 < b have "0 < a div b" by (auto intro: div_positive)  haftmann@53067  535  then have [simp]: "1 \ a div b" by (simp add: discrete)  haftmann@53067  536  with 0 < b have mod_less: "a mod b < b" by (simp add: pos_mod_bound)  haftmann@53067  537  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  538  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  539  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  540  from assms w_exhaust have "w = 1"  haftmann@53067  541  by (auto simp add: mod_w) (insert mod_less, auto)  haftmann@53067  542  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp  haftmann@53067  543  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  544  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  545  with w = 1 have div: "2 * (a div (2 * b)) = a div b - 1" by simp  haftmann@53067  546  then show ?P and ?Q  haftmann@53067  547  by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)  haftmann@53067  548 qed  haftmann@53067  549 haftmann@53067  550 lemma divmod_digit_0:  haftmann@53067  551  assumes "0 < b" and "a mod (2 * b) < b"  haftmann@53067  552  shows "2 * (a div (2 * b)) = a div b" (is "?P")  haftmann@53067  553  and "a mod (2 * b) = a mod b" (is "?Q")  haftmann@53067  554 proof -  haftmann@53067  555  def w \ "a div b mod 2" with parity have w_exhaust: "w = 0 \ w = 1" by auto  haftmann@53067  556  have mod_w: "a mod (2 * b) = a mod b + b * w"  haftmann@53067  557  by (simp add: w_def mod_mult2_eq ac_simps)  haftmann@53067  558  moreover have "b \ a mod b + b"  haftmann@53067  559  proof -  haftmann@53067  560  from 0 < b pos_mod_sign have "0 \ a mod b" by blast  haftmann@53067  561  then have "0 + b \ a mod b + b" by (rule add_right_mono)  haftmann@53067  562  then show ?thesis by simp  haftmann@53067  563  qed  haftmann@53067  564  moreover note assms w_exhaust  haftmann@53067  565  ultimately have "w = 0" by auto  haftmann@53067  566  with mod_w have mod: "a mod (2 * b) = a mod b" by simp  haftmann@53067  567  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  568  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  569  with w = 0 have div: "2 * (a div (2 * b)) = a div b" by simp  haftmann@53067  570  then show ?P and ?Q  haftmann@53067  571  by (simp_all add: div mod)  haftmann@53067  572 qed  haftmann@53067  573 haftmann@53067  574 definition divmod :: "num \ num \ 'a \ 'a"  haftmann@53067  575 where  haftmann@53067  576  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"  haftmann@53067  577 haftmann@53067  578 lemma fst_divmod [simp]:  haftmann@53067  579  "fst (divmod m n) = numeral m div numeral n"  haftmann@53067  580  by (simp add: divmod_def)  haftmann@53067  581 haftmann@53067  582 lemma snd_divmod [simp]:  haftmann@53067  583  "snd (divmod m n) = numeral m mod numeral n"  haftmann@53067  584  by (simp add: divmod_def)  haftmann@53067  585 haftmann@53067  586 definition divmod_step :: "num \ 'a \ 'a \ 'a \ 'a"  haftmann@53067  587 where  haftmann@53067  588  "divmod_step l qr = (let (q, r) = qr  haftmann@53067  589  in if r \ numeral l then (2 * q + 1, r - numeral l)  haftmann@53067  590  else (2 * q, r))"  haftmann@53067  591 haftmann@53067  592 text {*  haftmann@53067  593  This is a formulation of one step (referring to one digit position)  haftmann@53067  594  in school-method division: compare the dividend at the current  haftmann@53070  595  digit position with the remainder from previous division steps  haftmann@53067  596  and evaluate accordingly.  haftmann@53067  597 *}  haftmann@53067  598 haftmann@53067  599 lemma divmod_step_eq [code]:  haftmann@53067  600  "divmod_step l (q, r) = (if numeral l \ r  haftmann@53067  601  then (2 * q + 1, r - numeral l) else (2 * q, r))"  haftmann@53067  602  by (simp add: divmod_step_def)  haftmann@53067  603 haftmann@53067  604 lemma divmod_step_simps [simp]:  haftmann@53067  605  "r < numeral l \ divmod_step l (q, r) = (2 * q, r)"  haftmann@53067  606  "numeral l \ r \ divmod_step l (q, r) = (2 * q + 1, r - numeral l)"  haftmann@53067  607  by (auto simp add: divmod_step_eq not_le)  haftmann@53067  608 haftmann@53067  609 text {*  haftmann@53067  610  This is a formulation of school-method division.  haftmann@53067  611  If the divisor is smaller than the dividend, terminate.  haftmann@53067  612  If not, shift the dividend to the right until termination  haftmann@53067  613  occurs and then reiterate single division steps in the  haftmann@53067  614  opposite direction.  haftmann@53067  615 *}  haftmann@53067  616 haftmann@53067  617 lemma divmod_divmod_step [code]:  haftmann@53067  618  "divmod m n = (if m < n then (0, numeral m)  haftmann@53067  619  else divmod_step n (divmod m (Num.Bit0 n)))"  haftmann@53067  620 proof (cases "m < n")  haftmann@53067  621  case True then have "numeral m < numeral n" by simp  haftmann@53067  622  then show ?thesis  haftmann@53067  623  by (simp add: prod_eq_iff div_less mod_less)  haftmann@53067  624 next  haftmann@53067  625  case False  haftmann@53067  626  have "divmod m n =  haftmann@53067  627  divmod_step n (numeral m div (2 * numeral n),  haftmann@53067  628  numeral m mod (2 * numeral n))"  haftmann@53067  629  proof (cases "numeral n \ numeral m mod (2 * numeral n)")  haftmann@53067  630  case True  haftmann@53067  631  with divmod_step_simps  haftmann@53067  632  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  633  (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"  haftmann@53067  634  by blast  haftmann@53067  635  moreover from True divmod_digit_1 [of "numeral m" "numeral n"]  haftmann@53067  636  have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"  haftmann@53067  637  and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"  haftmann@53067  638  by simp_all  haftmann@53067  639  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  640  next  haftmann@53067  641  case False then have *: "numeral m mod (2 * numeral n) < numeral n"  haftmann@53067  642  by (simp add: not_le)  haftmann@53067  643  with divmod_step_simps  haftmann@53067  644  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  645  (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"  haftmann@53067  646  by blast  haftmann@53067  647  moreover from * divmod_digit_0 [of "numeral n" "numeral m"]  haftmann@53067  648  have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"  haftmann@53067  649  and "numeral m mod (2 * numeral n) = numeral m mod numeral n"  haftmann@53067  650  by (simp_all only: zero_less_numeral)  haftmann@53067  651  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  652  qed  haftmann@53067  653  then have "divmod m n =  haftmann@53067  654  divmod_step n (numeral m div numeral (Num.Bit0 n),  haftmann@53067  655  numeral m mod numeral (Num.Bit0 n))"  haftmann@53067  656  by (simp only: numeral.simps distrib mult_1)  haftmann@53067  657  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@53067  658  by (simp add: divmod_def)  haftmann@53067  659  with False show ?thesis by simp  haftmann@53067  660 qed  haftmann@53067  661 haftmann@53069  662 lemma divmod_cancel [code]:  haftmann@53069  663  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r))" (is ?P)  haftmann@53069  664  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \ (q, 2 * r + 1))" (is ?Q)  haftmann@53069  665 proof -  haftmann@53069  666  have *: "\q. numeral (Num.Bit0 q) = 2 * numeral q"  haftmann@53069  667  "\q. numeral (Num.Bit1 q) = 2 * numeral q + 1"  haftmann@53069  668  by (simp_all only: numeral_mult numeral.simps distrib) simp_all  haftmann@53069  669  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)  haftmann@53069  670  then show ?P and ?Q  haftmann@53069  671  by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1  haftmann@53069  672  div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)  haftmann@53069  673  qed  haftmann@53069  674 haftmann@53067  675 end  haftmann@53067  676 haftmann@53067  677 hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero  haftmann@53067  678  -- {* restore simple accesses for more general variants of theorems *}  haftmann@53067  679 haftmann@53067  680   haftmann@26100  681 subsection {* Division on @{typ nat} *}  haftmann@26100  682 haftmann@26100  683 text {*  haftmann@26100  684  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  685  of a characteristic relation with two input arguments  haftmann@26100  686  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  687  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  688 *}  haftmann@26100  689 haftmann@33340  690 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  691  "divmod_nat_rel m n qr \  haftmann@30923  692  m = fst qr * n + snd qr \  haftmann@30923  693  (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  694 haftmann@33340  695 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  696 haftmann@33340  697 lemma divmod_nat_rel_ex:  haftmann@33340  698  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  699 proof (cases "n = 0")  haftmann@30923  700  case True with that show thesis  haftmann@33340  701  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  702 next  haftmann@26100  703  case False  haftmann@26100  704  have "\q r. m = q * n + r \ r < n"  haftmann@26100  705  proof (induct m)  haftmann@26100  706  case 0 with n \ 0  haftmann@26100  707  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  708  then show ?case by blast  haftmann@26100  709  next  haftmann@26100  710  case (Suc m) then obtain q' r'  haftmann@26100  711  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  712  then show ?case proof (cases "Suc r' < n")  haftmann@26100  713  case True  haftmann@26100  714  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  715  with True show ?thesis by blast  haftmann@26100  716  next  haftmann@26100  717  case False then have "n \ Suc r'" by auto  haftmann@26100  718  moreover from n have "Suc r' \ n" by auto  haftmann@26100  719  ultimately have "n = Suc r'" by auto  haftmann@26100  720  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  721  with n \ 0 show ?thesis by blast  haftmann@26100  722  qed  haftmann@26100  723  qed  haftmann@26100  724  with that show thesis  haftmann@33340  725  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  726 qed  haftmann@26100  727 haftmann@33340  728 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  729 haftmann@33340  730 lemma divmod_nat_rel_unique:  haftmann@33340  731  assumes "divmod_nat_rel m n qr"  haftmann@33340  732  and "divmod_nat_rel m n qr'"  haftmann@30923  733  shows "qr = qr'"  haftmann@26100  734 proof (cases "n = 0")  haftmann@26100  735  case True with assms show ?thesis  haftmann@30923  736  by (cases qr, cases qr')  haftmann@33340  737  (simp add: divmod_nat_rel_def)  haftmann@26100  738 next  haftmann@26100  739  case False  haftmann@26100  740  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  741  apply (rule leI)  haftmann@26100  742  apply (subst less_iff_Suc_add)  haftmann@26100  743  apply (auto simp add: add_mult_distrib)  haftmann@26100  744  done  wenzelm@53374  745  from n \ 0 assms have *: "fst qr = fst qr'"  haftmann@33340  746  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  wenzelm@53374  747  with assms have "snd qr = snd qr'"  haftmann@33340  748  by (simp add: divmod_nat_rel_def)  wenzelm@53374  749  with * show ?thesis by (cases qr, cases qr') simp  haftmann@26100  750 qed  haftmann@26100  751 haftmann@26100  752 text {*  haftmann@26100  753  We instantiate divisibility on the natural numbers by  haftmann@33340  754  means of @{const divmod_nat_rel}:  haftmann@26100  755 *}  haftmann@25942  756 haftmann@33340  757 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  758  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  759 haftmann@33340  760 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  761  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  762 proof -  haftmann@33340  763  from divmod_nat_rel_ex  haftmann@33340  764  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  765  then show ?thesis  haftmann@33340  766  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  767 qed  haftmann@30923  768 huffman@47135  769 lemma divmod_nat_unique:  haftmann@33340  770  assumes "divmod_nat_rel m n qr"  haftmann@33340  771  shows "divmod_nat m n = qr"  haftmann@33340  772  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  773 huffman@46551  774 instantiation nat :: semiring_div  huffman@46551  775 begin  huffman@46551  776 haftmann@26100  777 definition div_nat where  haftmann@33340  778  "m div n = fst (divmod_nat m n)"  haftmann@26100  779 huffman@46551  780 lemma fst_divmod_nat [simp]:  huffman@46551  781  "fst (divmod_nat m n) = m div n"  huffman@46551  782  by (simp add: div_nat_def)  huffman@46551  783 haftmann@26100  784 definition mod_nat where  haftmann@33340  785  "m mod n = snd (divmod_nat m n)"  haftmann@25571  786 huffman@46551  787 lemma snd_divmod_nat [simp]:  huffman@46551  788  "snd (divmod_nat m n) = m mod n"  huffman@46551  789  by (simp add: mod_nat_def)  huffman@46551  790 haftmann@33340  791 lemma divmod_nat_div_mod:  haftmann@33340  792  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  793  by (simp add: prod_eq_iff)  haftmann@26100  794 huffman@47135  795 lemma div_nat_unique:  haftmann@33340  796  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  797  shows "m div n = q"  huffman@47135  798  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  799 huffman@47135  800 lemma mod_nat_unique:  haftmann@33340  801  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  802  shows "m mod n = r"  huffman@47135  803  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  804 haftmann@33340  805 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  806  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  807 huffman@47136  808 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  809  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  810 huffman@47136  811 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  812  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  813 huffman@47137  814 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  815  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  816 haftmann@33340  817 lemma divmod_nat_step:  haftmann@26100  818  assumes "0 < n" and "n \ m"  haftmann@33340  819  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  820 proof (rule divmod_nat_unique)  huffman@47134  821  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  822  by (rule divmod_nat_rel)  huffman@47134  823  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  824  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  825 qed  haftmann@25942  826 wenzelm@26300  827 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  828 haftmann@26100  829 lemma div_less [simp]:  haftmann@26100  830  fixes m n :: nat  haftmann@26100  831  assumes "m < n"  haftmann@26100  832  shows "m div n = 0"  huffman@46551  833  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  834 haftmann@26100  835 lemma le_div_geq:  haftmann@26100  836  fixes m n :: nat  haftmann@26100  837  assumes "0 < n" and "n \ m"  haftmann@26100  838  shows "m div n = Suc ((m - n) div n)"  huffman@46551  839  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  840 haftmann@26100  841 lemma mod_less [simp]:  haftmann@26100  842  fixes m n :: nat  haftmann@26100  843  assumes "m < n"  haftmann@26100  844  shows "m mod n = m"  huffman@46551  845  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  846 haftmann@26100  847 lemma le_mod_geq:  haftmann@26100  848  fixes m n :: nat  haftmann@26100  849  assumes "n \ m"  haftmann@26100  850  shows "m mod n = (m - n) mod n"  huffman@46551  851  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  852 huffman@47136  853 instance proof  huffman@47136  854  fix m n :: nat  huffman@47136  855  show "m div n * n + m mod n = m"  huffman@47136  856  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  857 next  huffman@47136  858  fix m n q :: nat  huffman@47136  859  assume "n \ 0"  huffman@47136  860  then show "(q + m * n) div n = m + q div n"  huffman@47136  861  by (induct m) (simp_all add: le_div_geq)  huffman@47136  862 next  huffman@47136  863  fix m n q :: nat  huffman@47136  864  assume "m \ 0"  huffman@47136  865  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  866  unfolding divmod_nat_rel_def  huffman@47136  867  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  868  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  869  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  870  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  871 next  huffman@47136  872  fix n :: nat show "n div 0 = 0"  haftmann@33340  873  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  874 next  huffman@47136  875  fix n :: nat show "0 div n = 0"  huffman@47136  876  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  877 qed  haftmann@26100  878 haftmann@25942  879 end  paulson@14267  880 haftmann@33361  881 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  882  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  huffman@46551  883  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)  haftmann@33361  884 haftmann@26100  885 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  886 wenzelm@51299  887 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"  wenzelm@51299  888 haftmann@30934  889 ML {*  wenzelm@43594  890 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  891 (  haftmann@30934  892  val div_name = @{const_name div};  haftmann@30934  893  val mod_name = @{const_name mod};  haftmann@30934  894  val mk_binop = HOLogic.mk_binop;  huffman@48561  895  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  huffman@48561  896  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  huffman@48561  897  fun mk_sum [] = HOLogic.zero  huffman@48561  898  | mk_sum [t] = t  huffman@48561  899  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  huffman@48561  900  fun dest_sum tm =  huffman@48561  901  if HOLogic.is_zero tm then []  huffman@48561  902  else  huffman@48561  903  (case try HOLogic.dest_Suc tm of  huffman@48561  904  SOME t => HOLogic.Suc_zero :: dest_sum t  huffman@48561  905  | NONE =>  huffman@48561  906  (case try dest_plus tm of  huffman@48561  907  SOME (t, u) => dest_sum t @ dest_sum u  huffman@48561  908  | NONE => [tm]));  haftmann@25942  909 haftmann@30934  910  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  911 haftmann@30934  912  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@35050  913  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))  wenzelm@41550  914 )  haftmann@25942  915 *}  haftmann@25942  916 wenzelm@43594  917 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  918 haftmann@26100  919 haftmann@26100  920 subsubsection {* Quotient *}  haftmann@26100  921 haftmann@26100  922 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  923 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  924 haftmann@26100  925 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  926 by (simp add: div_geq)  haftmann@26100  927 haftmann@26100  928 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  929 by simp  haftmann@26100  930 haftmann@26100  931 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  932 by simp  haftmann@26100  933 haftmann@53066  934 lemma div_positive:  haftmann@53066  935  fixes m n :: nat  haftmann@53066  936  assumes "n > 0"  haftmann@53066  937  assumes "m \ n"  haftmann@53066  938  shows "m div n > 0"  haftmann@53066  939 proof -  haftmann@53066  940  from m \ n obtain q where "m = n + q"  haftmann@53066  941  by (auto simp add: le_iff_add)  haftmann@53066  942  with n > 0 show ?thesis by simp  haftmann@53066  943 qed  haftmann@53066  944 haftmann@25942  945 haftmann@25942  946 subsubsection {* Remainder *}  haftmann@25942  947 haftmann@26100  948 lemma mod_less_divisor [simp]:  haftmann@26100  949  fixes m n :: nat  haftmann@26100  950  assumes "n > 0"  haftmann@26100  951  shows "m mod n < (n::nat)"  haftmann@33340  952  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  953 haftmann@51173  954 lemma mod_Suc_le_divisor [simp]:  haftmann@51173  955  "m mod Suc n \ n"  haftmann@51173  956  using mod_less_divisor [of "Suc n" m] by arith  haftmann@51173  957 haftmann@26100  958 lemma mod_less_eq_dividend [simp]:  haftmann@26100  959  fixes m n :: nat  haftmann@26100  960  shows "m mod n \ m"  haftmann@26100  961 proof (rule add_leD2)  haftmann@26100  962  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  963  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  964 qed  haftmann@26100  965 haftmann@26100  966 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  967 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  968 haftmann@26100  969 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  970 by (simp add: le_mod_geq)  haftmann@26100  971 paulson@14267  972 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  973 by (induct m) (simp_all add: mod_geq)  paulson@14267  974 paulson@14267  975 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  976 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  977  using mod_div_equality2 [of n m] by arith  paulson@14267  978 nipkow@15439  979 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  980  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  981  apply simp  wenzelm@22718  982  done  paulson@14267  983 haftmann@26100  984 subsubsection {* Quotient and Remainder *}  paulson@14267  985 haftmann@33340  986 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  987  "divmod_nat_rel b c (q, r)  haftmann@33340  988  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  989 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  990 haftmann@30923  991 lemma div_mult1_eq:  haftmann@30923  992  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  993 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  994 haftmann@33340  995 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  996  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  997  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  998 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  999 paulson@14267  1000 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  1001 lemma div_add1_eq:  nipkow@25134  1002  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  1003 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  1004 paulson@14267  1005 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@22718  1006  apply (cut_tac m = q and n = c in mod_less_divisor)  wenzelm@22718  1007  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  wenzelm@22718  1008  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)  wenzelm@22718  1009  apply (simp add: add_mult_distrib2)  wenzelm@22718  1010  done  paulson@10559  1011 haftmann@33340  1012 lemma divmod_nat_rel_mult2_eq:  bulwahn@46552  1013  "divmod_nat_rel a b (q, r)  haftmann@33340  1014  \ divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@33340  1015 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)  paulson@14267  1016 paulson@14267  1017 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"  huffman@47135  1018 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  1019 paulson@14267  1020 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"  huffman@47135  1021 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  1022 paulson@14267  1023 huffman@46551  1024 subsubsection {* Further Facts about Quotient and Remainder *}  paulson@14267  1025 paulson@14267  1026 lemma div_1 [simp]: "m div Suc 0 = m"  nipkow@29667  1027 by (induct m) (simp_all add: div_geq)  paulson@14267  1028 paulson@14267  1029 (* Monotonicity of div in first argument *)  haftmann@30923  1030 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  1031  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  1032 apply (case_tac "k=0", simp)  paulson@15251  1033 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  1034 apply (case_tac "n= k *)  paulson@14267  1038 apply (case_tac "m=k *)  nipkow@15439  1042 apply (simp add: div_geq diff_le_mono)  paulson@14267  1043 done  paulson@14267  1044 paulson@14267  1045 (* Antimonotonicity of div in second argument *)  paulson@14267  1046 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  1047 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  1056  prefer 2  paulson@14267  1057  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  1058 apply (rule le_trans, simp)  nipkow@15439  1059 apply (simp)  paulson@14267  1060 done  paulson@14267  1061 paulson@14267  1062 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  1063 apply (case_tac "n=0", simp)  paulson@14267  1064 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  1065 apply (rule div_le_mono2)  paulson@14267  1066 apply (simp_all (no_asm_simp))  paulson@14267  1067 done  paulson@14267  1068 wenzelm@22718  1069 (* Similar for "less than" *)  huffman@47138  1070 lemma div_less_dividend [simp]:  huffman@47138  1071  "\(1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  1072 apply (induct m rule: nat_less_induct)  paulson@14267  1073 apply (rename_tac "m")  paulson@14267  1074 apply (case_tac "m Suc(na) *)  paulson@16796  1093 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  1094 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  1095 done  paulson@14267  1096 paulson@14267  1097 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  1098 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  1099 wenzelm@22718  1100 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  1101 paulson@14267  1102 (*Loses information, namely we also have r \q::nat. m = r + q*d"  haftmann@27651  1104  apply (cut_tac a = m in mod_div_equality)  wenzelm@22718  1105  apply (simp only: add_ac)  wenzelm@22718  1106  apply (blast intro: sym)  wenzelm@22718  1107  done  paulson@14267  1108 nipkow@13152  1109 lemma split_div:  nipkow@13189  1110  "P(n div k :: nat) =  nipkow@13189  1111  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  1112  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1113 proof  nipkow@13189  1114  assume P: ?P  nipkow@13189  1115  show ?Q  nipkow@13189  1116  proof (cases)  nipkow@13189  1117  assume "k = 0"  haftmann@27651  1118  with P show ?Q by simp  nipkow@13189  1119  next  nipkow@13189  1120  assume not0: "k \ 0"  nipkow@13189  1121  thus ?Q  nipkow@13189  1122  proof (simp, intro allI impI)  nipkow@13189  1123  fix i j  nipkow@13189  1124  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  1125  show "P i"  nipkow@13189  1126  proof (cases)  wenzelm@22718  1127  assume "i = 0"  wenzelm@22718  1128  with n j P show "P i" by simp  nipkow@13189  1129  next  wenzelm@22718  1130  assume "i \ 0"  wenzelm@22718  1131  with not0 n j P show "P i" by(simp add:add_ac)  nipkow@13189  1132  qed  nipkow@13189  1133  qed  nipkow@13189  1134  qed  nipkow@13189  1135 next  nipkow@13189  1136  assume Q: ?Q  nipkow@13189  1137  show ?P  nipkow@13189  1138  proof (cases)  nipkow@13189  1139  assume "k = 0"  haftmann@27651  1140  with Q show ?P by simp  nipkow@13189  1141  next  nipkow@13189  1142  assume not0: "k \ 0"  nipkow@13189  1143  with Q have R: ?R by simp  nipkow@13189  1144  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1145  show ?P by simp  nipkow@13189  1146  qed  nipkow@13189  1147 qed  nipkow@13189  1148 berghofe@13882  1149 lemma split_div_lemma:  haftmann@26100  1150  assumes "0 < n"  haftmann@26100  1151  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  1152 proof  haftmann@26100  1153  assume ?rhs  haftmann@26100  1154  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  1155  then have A: "n * q \ m" by simp  haftmann@26100  1156  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  1157  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  1158  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  1159  with nq have "m < n + n * q" by simp  haftmann@26100  1160  then have B: "m < n * Suc q" by simp  haftmann@26100  1161  from A B show ?lhs ..  haftmann@26100  1162 next  haftmann@26100  1163  assume P: ?lhs  haftmann@33340  1164  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@33340  1165  unfolding divmod_nat_rel_def by (auto simp add: mult_ac)  haftmann@33340  1166  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  1167  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  1168  then show ?rhs by simp  haftmann@26100  1169 qed  berghofe@13882  1170 berghofe@13882  1171 theorem split_div':  berghofe@13882  1172  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  1173  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  1174  apply (case_tac "0 < n")  berghofe@13882  1175  apply (simp only: add: split_div_lemma)  haftmann@27651  1176  apply simp_all  berghofe@13882  1177  done  berghofe@13882  1178 nipkow@13189  1179 lemma split_mod:  nipkow@13189  1180  "P(n mod k :: nat) =  nipkow@13189  1181  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  1182  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1183 proof  nipkow@13189  1184  assume P: ?P  nipkow@13189  1185  show ?Q  nipkow@13189  1186  proof (cases)  nipkow@13189  1187  assume "k = 0"  haftmann@27651  1188  with P show ?Q by simp  nipkow@13189  1189  next  nipkow@13189  1190  assume not0: "k \ 0"  nipkow@13189  1191  thus ?Q  nipkow@13189  1192  proof (simp, intro allI impI)  nipkow@13189  1193  fix i j  nipkow@13189  1194  assume "n = k*i + j" "j < k"  nipkow@13189  1195  thus "P j" using not0 P by(simp add:add_ac mult_ac)  nipkow@13189  1196  qed  nipkow@13189  1197  qed  nipkow@13189  1198 next  nipkow@13189  1199  assume Q: ?Q  nipkow@13189  1200  show ?P  nipkow@13189  1201  proof (cases)  nipkow@13189  1202  assume "k = 0"  haftmann@27651  1203  with Q show ?P by simp  nipkow@13189  1204  next  nipkow@13189  1205  assume not0: "k \ 0"  nipkow@13189  1206  with Q have R: ?R by simp  nipkow@13189  1207  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1208  show ?P by simp  nipkow@13189  1209  qed  nipkow@13189  1210 qed  nipkow@13189  1211 berghofe@13882  1212 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  1213  using mod_div_equality [of m n] by arith  huffman@47138  1214 huffman@47138  1215 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  1216  using mod_div_equality [of m n] by arith  huffman@47138  1217 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  1218 noschinl@52398  1219 lemma div_eq_dividend_iff: "a \ 0 \ (a :: nat) div b = a \ b = 1"  noschinl@52398  1220  apply rule  noschinl@52398  1221  apply (cases "b = 0")  noschinl@52398  1222  apply simp_all  noschinl@52398  1223  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)  noschinl@52398  1224  done  noschinl@52398  1225 haftmann@22800  1226 huffman@46551  1227 subsubsection {* An induction'' law for modulus arithmetic. *}  paulson@14640  1228 paulson@14640  1229 lemma mod_induct_0:  paulson@14640  1230  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1231  and base: "P i" and i: "i(P 0)"  paulson@14640  1235  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1237  proof  paulson@14640  1238  fix k  paulson@14640  1239  show "?A k"  paulson@14640  1240  proof (induct k)  paulson@14640  1241  show "?A 0" by simp -- "by contradiction"  paulson@14640  1242  next  paulson@14640  1243  fix n  paulson@14640  1244  assume ih: "?A n"  paulson@14640  1245  show "?A (Suc n)"  paulson@14640  1246  proof (clarsimp)  wenzelm@22718  1247  assume y: "P (p - Suc n)"  wenzelm@22718  1248  have n: "Suc n < p"  wenzelm@22718  1249  proof (rule ccontr)  wenzelm@22718  1250  assume "\(Suc n < p)"  wenzelm@22718  1251  hence "p - Suc n = 0"  wenzelm@22718  1252  by simp  wenzelm@22718  1253  with y contra show "False"  wenzelm@22718  1254  by simp  wenzelm@22718  1255  qed  wenzelm@22718  1256  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1257  from p have "p - Suc n < p" by arith  wenzelm@22718  1258  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1259  by blast  wenzelm@22718  1260  show "False"  wenzelm@22718  1261  proof (cases "n=0")  wenzelm@22718  1262  case True  wenzelm@22718  1263  with z n2 contra show ?thesis by simp  wenzelm@22718  1264  next  wenzelm@22718  1265  case False  wenzelm@22718  1266  with p have "p-n < p" by arith  wenzelm@22718  1267  with z n2 False ih show ?thesis by simp  wenzelm@22718  1268  qed  paulson@14640  1269  qed  paulson@14640  1270  qed  paulson@14640  1271  qed  paulson@14640  1272  moreover  paulson@14640  1273  from i obtain k where "0 i+k=p"  paulson@14640  1274  by (blast dest: less_imp_add_positive)  paulson@14640  1275  hence "0 i=p-k" by auto  paulson@14640  1276  moreover  paulson@14640  1277  note base  paulson@14640  1278  ultimately  paulson@14640  1279  show "False" by blast  paulson@14640  1280 qed  paulson@14640  1281 paulson@14640  1282 lemma mod_induct:  paulson@14640  1283  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1284  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1291  proof (induct j)  paulson@14640  1292  from step base i show "?A 0"  wenzelm@22718  1293  by (auto elim: mod_induct_0)  paulson@14640  1294  next  paulson@14640  1295  fix k  paulson@14640  1296  assume ih: "?A k"  paulson@14640  1297  show "?A (Suc k)"  paulson@14640  1298  proof  wenzelm@22718  1299  assume suc: "Suc k < p"  wenzelm@22718  1300  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1325 proof -  boehmes@35815  1326  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1327  moreover have "m mod 2 < 2" by simp  haftmann@33296  1328  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1329  then show ?thesis by auto  haftmann@33296  1330 qed  haftmann@33296  1331 haftmann@33296  1332 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1333  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1334  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1335 haftmann@33296  1336 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1337 by (simp add: Suc3_eq_add_3)  haftmann@33296  1338 haftmann@33296  1339 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1340 by (simp add: Suc3_eq_add_3)  haftmann@33296  1341 haftmann@33296  1342 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1343 by (simp add: Suc3_eq_add_3)  haftmann@33296  1344 haftmann@33296  1345 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1346 by (simp add: Suc3_eq_add_3)  haftmann@33296  1347 huffman@47108  1348 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1349 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1350 haftmann@33361  1351 haftmann@33361  1352 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1353 apply (induct "m")  haftmann@33361  1354 apply (simp_all add: mod_Suc)  haftmann@33361  1355 done  haftmann@33361  1356 huffman@47108  1357 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1358 huffman@47138  1359 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1360 by (simp add: div_le_mono)  haftmann@33361  1361 haftmann@33361  1362 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1363 by (cases n) simp_all  haftmann@33361  1364 boehmes@35815  1365 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1366 proof -  boehmes@35815  1367  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1368  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1369 qed  haftmann@33361  1370 haftmann@33361  1371  (* Potential use of algebra : Equality modulo n*)  haftmann@33361  1372 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"  haftmann@33361  1373 by (simp add: mult_ac add_ac)  haftmann@33361  1374 haftmann@33361  1375 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1376 proof -  haftmann@33361  1377  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1378  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1379  finally show ?thesis .  haftmann@33361  1380 qed  haftmann@33361  1381 haftmann@33361  1382 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1383 apply (subst mod_Suc [of m])  haftmann@33361  1384 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1385 done  haftmann@33361  1386 huffman@47108  1387 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1388  fixes n :: nat  huffman@47108  1389  shows "n mod 2 \ 0 \ n mod 2 = 1"  huffman@47108  1390  by simp  huffman@47108  1391 haftmann@53067  1392 instance nat :: semiring_numeral_div  haftmann@53067  1393  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)  haftmann@53067  1394 haftmann@33361  1395 haftmann@33361  1396 subsection {* Division on @{typ int} *}  haftmann@33361  1397 haftmann@33361  1398 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1399  --{*definition of quotient and remainder*}  huffman@47139  1400  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1401  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1402 haftmann@53067  1403 text {*  haftmann@53067  1404  The following algorithmic devlopment actually echos what has already  haftmann@53067  1405  been developed in class @{class semiring_numeral_div}. In the long  haftmann@53067  1406  run it seems better to derive division on @{typ int} just from  haftmann@53067  1407  division on @{typ nat} and instantiate @{class semiring_numeral_div}  haftmann@53067  1408  accordingly.  haftmann@53067  1409 *}  haftmann@53067  1410 haftmann@33361  1411 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1412  --{*for the division algorithm*}  huffman@47108  1413  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1414  else (2 * q, r))"  haftmann@33361  1415 haftmann@33361  1416 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1417 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1418  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1419  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1420 by auto  haftmann@33361  1421 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1422  (auto simp add: mult_2)  haftmann@33361  1423 haftmann@33361  1424 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1425 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1426  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1427  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1428 by auto  haftmann@33361  1429 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1430  (auto simp add: mult_2)  haftmann@33361  1431 haftmann@33361  1432 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1433 haftmann@33361  1434 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1435  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1436  including the special case @{text "a=0, b<0"} because  haftmann@33361  1437  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1438  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1439  else if a = 0 then (0, 0)  huffman@46560  1440  else apsnd uminus (negDivAlg (-a) (-b))  haftmann@33361  1441  else  haftmann@33361  1442  if 0 < b then negDivAlg a b  huffman@46560  1443  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1444 haftmann@33361  1445 instantiation int :: Divides.div  haftmann@33361  1446 begin  haftmann@33361  1447 huffman@46551  1448 definition div_int where  haftmann@33361  1449  "a div b = fst (divmod_int a b)"  haftmann@33361  1450 huffman@46551  1451 lemma fst_divmod_int [simp]:  huffman@46551  1452  "fst (divmod_int a b) = a div b"  huffman@46551  1453  by (simp add: div_int_def)  huffman@46551  1454 huffman@46551  1455 definition mod_int where  huffman@46560  1456  "a mod b = snd (divmod_int a b)"  haftmann@33361  1457 huffman@46551  1458 lemma snd_divmod_int [simp]:  huffman@46551  1459  "snd (divmod_int a b) = a mod b"  huffman@46551  1460  by (simp add: mod_int_def)  huffman@46551  1461 haftmann@33361  1462 instance ..  haftmann@33361  1463 paulson@3366  1464 end  haftmann@33361  1465 haftmann@33361  1466 lemma divmod_int_mod_div:  haftmann@33361  1467  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1468  by (simp add: prod_eq_iff)  haftmann@33361  1469 haftmann@33361  1470 text{*  haftmann@33361  1471 Here is the division algorithm in ML:  haftmann@33361  1472 haftmann@33361  1473 \begin{verbatim}  haftmann@33361  1474  fun posDivAlg (a,b) =  haftmann@33361  1475  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1478  end  haftmann@33361  1479 haftmann@33361  1480  fun negDivAlg (a,b) =  haftmann@33361  1481  if 0\a+b then (~1,a+b)  haftmann@33361  1482  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1483  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1484  end;  haftmann@33361  1485 haftmann@33361  1486  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1487 haftmann@33361  1488  fun divmod (a,b) = if 0\a then  haftmann@33361  1489  if b>0 then posDivAlg (a,b)  haftmann@33361  1490  else if a=0 then (0,0)  haftmann@33361  1491  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1492  else  haftmann@33361  1493  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1503  ==> q' \ (q::int)"  haftmann@33361  1504 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1505  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1506 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1507 apply (erule_tac [2] order_le_less_trans)  webertj@49962  1508  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1509 apply (subgoal_tac "b * q' < b * (1 + q) ")  webertj@49962  1510  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1511 apply (simp add: mult_less_cancel_left)  haftmann@33361  1512 done  haftmann@33361  1513 haftmann@33361  1514 lemma unique_quotient_lemma_neg:  haftmann@33361  1515  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1516  ==> q \ (q'::int)"  haftmann@33361  1517 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1518  auto)  haftmann@33361  1519 haftmann@33361  1520 lemma unique_quotient:  bulwahn@46552  1521  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1522  ==> q = q'"  haftmann@33361  1523 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1524 apply (blast intro: order_antisym  haftmann@33361  1525  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1526  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1527 done  haftmann@33361  1528 haftmann@33361  1529 haftmann@33361  1530 lemma unique_remainder:  bulwahn@46552  1531  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1532  ==> r = r'"  haftmann@33361  1533 apply (subgoal_tac "q = q'")  haftmann@33361  1534  apply (simp add: divmod_int_rel_def)  haftmann@33361  1535 apply (blast intro: unique_quotient)  haftmann@33361  1536 done  haftmann@33361  1537 haftmann@33361  1538 huffman@46551  1539 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}  haftmann@33361  1540 haftmann@33361  1541 text{*And positive divisors*}  haftmann@33361  1542 haftmann@33361  1543 lemma adjust_eq [simp]:  huffman@47108  1544  "adjust b (q, r) =  huffman@47108  1545  (let diff = r - b in  huffman@47108  1546  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1547  else (2*q, r))"  huffman@47108  1548  by (simp add: Let_def adjust_def)  haftmann@33361  1549 haftmann@33361  1550 declare posDivAlg.simps [simp del]  haftmann@33361  1551 haftmann@33361  1552 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1553 lemma posDivAlg_eqn:  haftmann@33361  1554  "0 < b ==>  haftmann@33361  1555  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1561  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1562  using assms  wenzelm@41550  1563  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1564  apply auto  wenzelm@41550  1565  apply (simp add: divmod_int_rel_def)  webertj@49962  1566  apply (subst posDivAlg_eqn, simp add: distrib_left)  wenzelm@41550  1567  apply (case_tac "a < b")  wenzelm@41550  1568  apply simp_all  wenzelm@41550  1569  apply (erule splitE)  webertj@49962  1570  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1571  done  haftmann@33361  1572 haftmann@33361  1573 huffman@46551  1574 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}  haftmann@33361  1575 haftmann@33361  1576 text{*And positive divisors*}  haftmann@33361  1577 haftmann@33361  1578 declare negDivAlg.simps [simp del]  haftmann@33361  1579 haftmann@33361  1580 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1581 lemma negDivAlg_eqn:  haftmann@33361  1582  "0 < b ==>  haftmann@33361  1583  negDivAlg a b =  haftmann@33361  1584  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1585 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1586 haftmann@33361  1587 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1588  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1589 lemma negDivAlg_correct:  haftmann@33361  1590  assumes "a < 0" and "b > 0"  haftmann@33361  1591  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1592  using assms  wenzelm@41550  1593  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1594  apply (auto simp add: linorder_not_le)  wenzelm@41550  1595  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1596  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1597  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1598  apply simp_all  wenzelm@41550  1599  apply (erule splitE)  webertj@49962  1600  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1601  done  haftmann@33361  1602 haftmann@33361  1603 huffman@46551  1604 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}  haftmann@33361  1605 haftmann@33361  1606 (*the case a=0*)  huffman@47139  1607 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1608 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1609 haftmann@33361  1610 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1611 by (subst posDivAlg.simps, auto)  haftmann@33361  1612 huffman@47139  1613 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1614 by (subst posDivAlg.simps, auto)  huffman@47139  1615 haftmann@33361  1616 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  haftmann@33361  1617 by (subst negDivAlg.simps, auto)  haftmann@33361  1618 huffman@46560  1619 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1620 by (auto simp add: divmod_int_rel_def)  huffman@47139  1621 huffman@47139  1622 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1623 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1624 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1625  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1626 huffman@47141  1627 lemma divmod_int_unique:  huffman@47141  1628  assumes "divmod_int_rel a b qr"  huffman@47141  1629  shows "divmod_int a b = qr"  huffman@47141  1630  using assms divmod_int_correct [of a b]  huffman@47141  1631  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1632  by (metis pair_collapse)  huffman@47141  1633 huffman@47141  1634 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1635  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1636 huffman@47141  1637 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1638  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1639 huffman@47141  1640 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1641  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1642 huffman@47141  1643 instance int :: ring_div  huffman@47141  1644 proof  huffman@47141  1645  fix a b :: int  huffman@47141  1646  show "a div b * b + a mod b = a"  huffman@47141  1647  using divmod_int_rel_div_mod [of a b]  huffman@47141  1648  unfolding divmod_int_rel_def by (simp add: mult_commute)  huffman@47141  1649 next  huffman@47141  1650  fix a b c :: int  huffman@47141  1651  assume "b \ 0"  huffman@47141  1652  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1653  using divmod_int_rel_div_mod [of a b]  huffman@47141  1654  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1655  thus "(a + c * b) div b = c + a div b"  huffman@47141  1656  by (rule div_int_unique)  huffman@47141  1657 next  huffman@47141  1658  fix a b c :: int  huffman@47141  1659  assume "c \ 0"  huffman@47141  1660  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1661  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1662  unfolding divmod_int_rel_def  huffman@47141  1663  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1664  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1665  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1666  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1667  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1668  thus "(c * a) div (c * b) = a div b"  huffman@47141  1669  by (rule div_int_unique)  huffman@47141  1670 next  huffman@47141  1671  fix a :: int show "a div 0 = 0"  huffman@47141  1672  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1673 next  huffman@47141  1674  fix a :: int show "0 div a = 0"  huffman@47141  1675  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1676 qed  huffman@47141  1677 haftmann@33361  1678 text{*Basic laws about division and remainder*}  haftmann@33361  1679 haftmann@33361  1680 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1681  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1682 haftmann@33361  1683 text {* Tool setup *}  haftmann@33361  1684 huffman@47108  1685 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)  huffman@47108  1686 lemmas add_0s = add_0_left add_0_right  huffman@47108  1687 haftmann@33361  1688 ML {*  wenzelm@43594  1689 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1690 (  haftmann@33361  1691  val div_name = @{const_name div};  haftmann@33361  1692  val mod_name = @{const_name mod};  haftmann@33361  1693  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1694  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1695  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1696 huffman@47165  1697  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1698 haftmann@33361  1699  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@33361  1700  (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))  wenzelm@41550  1701 )  haftmann@33361  1702 *}  haftmann@33361  1703 wenzelm@43594  1704 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1705 huffman@47141  1706 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1707  using divmod_int_correct [of a b]  huffman@47141  1708  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1709 wenzelm@45607  1710 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1711  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1712 huffman@47141  1713 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1714  using divmod_int_correct [of a b]  huffman@47141  1715  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1716 wenzelm@45607  1717 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1718  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1719 haftmann@33361  1720 huffman@46551  1721 subsubsection {* General Properties of div and mod *}  haftmann@33361  1722 haftmann@33361  1723 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1724 apply (rule div_int_unique)  haftmann@33361  1725 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1726 done  haftmann@33361  1727 haftmann@33361  1728 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1729 apply (rule div_int_unique)  haftmann@33361  1730 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1731 done  haftmann@33361  1732 haftmann@33361  1733 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1734 apply (rule div_int_unique)  haftmann@33361  1735 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1736 done  haftmann@33361  1737 haftmann@33361  1738 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1739 haftmann@33361  1740 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1741 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1742 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1743 done  haftmann@33361  1744 haftmann@33361  1745 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  1746 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1747 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1748 done  haftmann@33361  1749 haftmann@33361  1750 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  1751 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  1752 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1753 done  haftmann@33361  1754 haftmann@33361  1755 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  1756 haftmann@33361  1757 huffman@46551  1758 subsubsection {* Laws for div and mod with Unary Minus *}  haftmann@33361  1759 haftmann@33361  1760 lemma zminus1_lemma:  huffman@47139  1761  "divmod_int_rel a b (q, r) ==> b \ 0  haftmann@33361  1762  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1763  if r=0 then 0 else b-r)"  haftmann@33361  1764 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1765 haftmann@33361  1766 haftmann@33361  1767 lemma zdiv_zminus1_eq_if:  haftmann@33361  1768  "b \ (0::int)  haftmann@33361  1769  ==> (-a) div b =  haftmann@33361  1770  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  1771 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  1772 haftmann@33361  1773 lemma zmod_zminus1_eq_if:  haftmann@33361  1774  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1775 apply (case_tac "b = 0", simp)  huffman@47140  1776 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  1777 done  haftmann@33361  1778 haftmann@33361  1779 lemma zmod_zminus1_not_zero:  haftmann@33361  1780  fixes k l :: int  haftmann@33361  1781  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  1782  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  1783 haftmann@33361  1784 lemma zdiv_zminus2_eq_if:  haftmann@33361  1785  "b \ (0::int)  haftmann@33361  1786  ==> a div (-b) =  haftmann@33361  1787  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  1788 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  1789 haftmann@33361  1790 lemma zmod_zminus2_eq_if:  haftmann@33361  1791  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  1792 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  1793 haftmann@33361  1794 lemma zmod_zminus2_not_zero:  haftmann@33361  1795  fixes k l :: int  haftmann@33361  1796  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  1797  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  1798 haftmann@33361  1799 huffman@46551  1800 subsubsection {* Computation of Division and Remainder *}  haftmann@33361  1801 haftmann@33361  1802 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  1803 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1804 haftmann@33361  1805 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  1806 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1807 haftmann@33361  1808 text{*a positive, b positive *}  haftmann@33361  1809 haftmann@33361  1810 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  1811 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1812 haftmann@33361  1813 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  1814 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1815 haftmann@33361  1816 text{*a negative, b positive *}  haftmann@33361  1817 haftmann@33361  1818 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  1819 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1820 haftmann@33361  1821 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  1822 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1823 haftmann@33361  1824 text{*a positive, b negative *}  haftmann@33361  1825 haftmann@33361  1826 lemma div_pos_neg:  huffman@46560  1827  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1828 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1829 haftmann@33361  1830 lemma mod_pos_neg:  huffman@46560  1831  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1832 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1833 haftmann@33361  1834 text{*a negative, b negative *}  haftmann@33361  1835 haftmann@33361  1836 lemma div_neg_neg:  huffman@46560  1837  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1838 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1839 haftmann@33361  1840 lemma mod_neg_neg:  huffman@46560  1841  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1842 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1843 haftmann@33361  1844 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  1845 huffman@45530  1846 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  1847  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  1848 huffman@45530  1849 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  1850  by (rule div_int_unique [of a b q r],  bulwahn@46552  1851  simp add: divmod_int_rel_def)  huffman@45530  1852 huffman@45530  1853 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  1854  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1855  simp add: divmod_int_rel_def)  huffman@45530  1856 huffman@45530  1857 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  1858  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1859  simp add: divmod_int_rel_def)  huffman@45530  1860 haftmann@53069  1861 text {*  haftmann@53069  1862  numeral simprocs -- high chance that these can be replaced  haftmann@53069  1863  by divmod algorithm from @{class semiring_numeral_div}  haftmann@53069  1864 *}  haftmann@53069  1865 haftmann@33361  1866 ML {*  haftmann@33361  1867 local  huffman@45530  1868  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  1869  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  1870  val times = @{term "times :: int \ int \ int"}  huffman@45530  1871  val zero = @{term "0 :: int"}  huffman@45530  1872  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  1873  val le = @{term "op \ :: int \ int \ bool"}  huffman@45530  1874  val simps = @{thms arith_simps} @ @{thms rel_simps} @  huffman@47108  1875  map (fn th => th RS sym) [@{thm numeral_1_eq_1}]  huffman@45530  1876  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  wenzelm@51717  1877  (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));  wenzelm@51717  1878  fun binary_proc proc ctxt ct =  haftmann@33361  1879  (case Thm.term_of ct of  haftmann@33361  1880  _ $t$ u =>  haftmann@33361  1881  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  wenzelm@51717  1882  SOME args => proc ctxt args  haftmann@33361  1883  | NONE => NONE)  haftmann@33361  1884  | _ => NONE);  haftmann@33361  1885 in  huffman@45530  1886  fun divmod_proc posrule negrule =  huffman@45530  1887  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  huffman@45530  1888  if b = 0 then NONE else let  huffman@45530  1889  val (q, r) = pairself mk_number (Integer.div_mod a b)  huffman@45530  1890  val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)  huffman@45530  1891  val (goal2, goal3, rule) = if b > 0  huffman@45530  1892  then (le $zero$ r, less $r$ u, posrule RS eq_reflection)  huffman@45530  1893  else (le $r$ zero, less $u$ r, negrule RS eq_reflection)  huffman@45530  1894  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  1895 end  haftmann@33361  1896 *}  haftmann@33361  1897 huffman@47108  1898 simproc_setup binary_int_div  huffman@47108  1899  ("numeral m div numeral n :: int" |  huffman@47108  1900  "numeral m div neg_numeral n :: int" |  huffman@47108  1901  "neg_numeral m div numeral n :: int" |  huffman@47108  1902  "neg_numeral m div neg_numeral n :: int") =  huffman@45530  1903  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  1904 huffman@47108  1905 simproc_setup binary_int_mod  huffman@47108  1906  ("numeral m mod numeral n :: int" |  huffman@47108  1907  "numeral m mod neg_numeral n :: int" |  huffman@47108  1908  "neg_numeral m mod numeral n :: int" |  huffman@47108  1909  "neg_numeral m mod neg_numeral n :: int") =  huffman@45530  1910  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  1911 huffman@47108  1912 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  1913  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  1914 huffman@47108  1915 lemmas negDivAlg_eqn_numeral [simp] =  huffman@47108  1916  negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w  haftmann@33361  1917 haftmann@33361  1918 haftmann@33361  1919 text{*Special-case simplification *}  haftmann@33361  1920 haftmann@33361  1921 (** The last remaining special cases for constant arithmetic:  haftmann@33361  1922  1 div z and 1 mod z **)  haftmann@33361  1923 huffman@47108  1924 lemmas div_pos_pos_1_numeral [simp] =  huffman@47108  1925  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1926 huffman@47108  1927 lemmas div_pos_neg_1_numeral [simp] =  huffman@47108  1928  div_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1929  OF neg_numeral_less_zero] for w  huffman@47108  1930 huffman@47108  1931 lemmas mod_pos_pos_1_numeral [simp] =  huffman@47108  1932  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1933 huffman@47108  1934 lemmas mod_pos_neg_1_numeral [simp] =  huffman@47108  1935  mod_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1936  OF neg_numeral_less_zero] for w  huffman@47108  1937 huffman@47108  1938 lemmas posDivAlg_eqn_1_numeral [simp] =  huffman@47108  1939  posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  huffman@47108  1940 huffman@47108  1941 lemmas negDivAlg_eqn_1_numeral [simp] =  huffman@47108  1942  negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  haftmann@33361  1943 haftmann@33361  1944 huffman@46551  1945 subsubsection {* Monotonicity in the First Argument (Dividend) *}  haftmann@33361  1946 haftmann@33361  1947 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  1948 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1949 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1950 apply (rule unique_quotient_lemma)  haftmann@33361  1951 apply (erule subst)  haftmann@33361  1952 apply (erule subst, simp_all)  haftmann@33361  1953 done  haftmann@33361  1954 haftmann@33361  1955 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  1956 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1957 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1958 apply (rule unique_quotient_lemma_neg)  haftmann@33361  1959 apply (erule subst)  haftmann@33361  1960 apply (erule subst, simp_all)  haftmann@33361  1961 done  haftmann@33361  1962 haftmann@33361  1963 huffman@46551  1964 subsubsection {* Monotonicity in the Second Argument (Divisor) *}  haftmann@33361  1965 haftmann@33361  1966 lemma q_pos_lemma:  haftmann@33361  1967  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  1968 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  1969  apply (simp add: zero_less_mult_iff)  webertj@49962  1970 apply (simp add: distrib_left)  haftmann@33361  1971 done  haftmann@33361  1972 haftmann@33361  1973 lemma zdiv_mono2_lemma:  haftmann@33361  1974  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  1975  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  1976  ==> q \ (q'::int)"  haftmann@33361  1977 apply (frule q_pos_lemma, assumption+)  haftmann@33361  1978 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  1979  apply (simp add: mult_less_cancel_left)  haftmann@33361  1980 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  1981  prefer 2 apply simp  webertj@49962  1982 apply (simp (no_asm_simp) add: distrib_left)  huffman@44766  1983 apply (subst add_commute, rule add_less_le_mono, arith)  haftmann@33361  1984 apply (rule mult_right_mono, auto)  haftmann@33361  1985 done  haftmann@33361  1986 haftmann@33361  1987 lemma zdiv_mono2:  haftmann@33361  1988  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  1989 apply (subgoal_tac "b \ 0")  haftmann@33361  1990  prefer 2 apply arith  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 zdiv_mono2_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 q_neg_lemma:  haftmann@33361  1999  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  2000 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  2001  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  2002 done  haftmann@33361  2003 haftmann@33361  2004 lemma zdiv_mono2_neg_lemma:  haftmann@33361  2005  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  2006  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  2007  ==> q' \ (q::int)"  haftmann@33361  2008 apply (frule q_neg_lemma, assumption+)  haftmann@33361  2009 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  2010  apply (simp add: mult_less_cancel_left)  webertj@49962  2011 apply (simp add: distrib_left)  haftmann@33361  2012 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  2013  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  2014 done  haftmann@33361  2015 haftmann@33361  2016 lemma zdiv_mono2_neg:  haftmann@33361  2017  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  2018 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2019 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2020 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  2021 apply (erule subst)  haftmann@33361  2022 apply (erule subst, simp_all)  haftmann@33361  2023 done  haftmann@33361  2024 haftmann@33361  2025 huffman@46551  2026 subsubsection {* More Algebraic Laws for div and mod *}  haftmann@33361  2027 haftmann@33361  2028 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  2029 haftmann@33361  2030 lemma zmult1_lemma:  bulwahn@46552  2031  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  2032  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  webertj@49962  2033 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)  haftmann@33361  2034 haftmann@33361  2035 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  2036 apply (case_tac "c = 0", simp)  huffman@47140  2037 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  2038 done  haftmann@33361  2039 haftmann@33361  2040 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  2041 haftmann@33361  2042 lemma zadd1_lemma:  bulwahn@46552  2043  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  2044  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  webertj@49962  2045 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)  haftmann@33361  2046 haftmann@33361  2047 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  2048 lemma zdiv_zadd1_eq:  haftmann@33361  2049  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  2050 apply (case_tac "c = 0", simp)  huffman@47140  2051 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  2052 done  haftmann@33361  2053 haftmann@33361  2054 lemma posDivAlg_div_mod:  haftmann@33361  2055  assumes "k \ 0"  haftmann@33361  2056  and "l \ 0"  haftmann@33361  2057  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  2058 proof (cases "l = 0")  haftmann@33361  2059  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  2060 next  haftmann@33361  2061  case False with assms posDivAlg_correct  haftmann@33361  2062  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  2063  by simp  huffman@47140  2064  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2065  show ?thesis by simp  haftmann@33361  2066 qed  haftmann@33361  2067 haftmann@33361  2068 lemma negDivAlg_div_mod:  haftmann@33361  2069  assumes "k < 0"  haftmann@33361  2070  and "l > 0"  haftmann@33361  2071  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  2072 proof -  haftmann@33361  2073  from assms have "l \ 0" by simp  haftmann@33361  2074  from assms negDivAlg_correct  haftmann@33361  2075  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  2076  by simp  huffman@47140  2077  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2078  show ?thesis by simp  haftmann@33361  2079 qed  haftmann@33361  2080 haftmann@33361  2081 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  2082 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  2083 haftmann@33361  2084 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  2085 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  2086 huffman@47108  2087 lemma zmod_zdiv_equality':  huffman@47108  2088  "(m\int) mod n = m - (m div n) * n"  huffman@47141  2089  using mod_div_equality [of m n] by arith  huffman@47108  2090 haftmann@33361  2091 huffman@46551  2092 subsubsection {* Proving @{term "a div (b*c) = (a div b) div c"} *}  haftmann@33361  2093 haftmann@33361  2094 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  2095  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  2096  to cause particular problems.*)  haftmann@33361  2097 haftmann@33361  2098 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  2099 haftmann@33361  2100 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  haftmann@33361  2101 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  2102  apply (simp add: algebra_simps)  haftmann@33361  2103 apply (rule order_le_less_trans)  haftmann@33361  2104  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  2105  apply (rule mult_left_mono_neg)  huffman@35216  2106  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  2107  apply (simp)  haftmann@33361  2108 apply (simp)  haftmann@33361  2109 done  haftmann@33361  2110 haftmann@33361  2111 lemma zmult2_lemma_aux2:  haftmann@33361  2112  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  2113 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  2114  apply arith  haftmann@33361  2115 apply (simp add: mult_le_0_iff)  haftmann@33361  2116 done  haftmann@33361  2117 haftmann@33361  2118 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  2119 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  2120 apply arith  haftmann@33361  2121 apply (simp add: zero_le_mult_iff)  haftmann@33361  2122 done  haftmann@33361  2123 haftmann@33361  2124 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  2125 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  2126  apply (simp add: right_diff_distrib)  haftmann@33361  2127 apply (rule order_less_le_trans)  haftmann@33361  2128  apply (erule mult_strict_right_mono)  haftmann@33361  2129  apply (rule_tac [2] mult_left_mono)  haftmann@33361  2130  apply simp  huffman@35216  2131  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  2132 apply simp  haftmann@33361  2133 done  haftmann@33361  2134 bulwahn@46552  2135 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  2136  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@33361  2137 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff  webertj@49962  2138  zero_less_mult_iff distrib_left [symmetric]  huffman@47139  2139  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  2140 haftmann@53068  2141 lemma zdiv_zmult2_eq:  haftmann@53068  2142  fixes a b c :: int  haftmann@53068  2143  shows "0 \ c \ a div (b * c) = (a div b) div c"  haftmann@33361  2144 apply (case_tac "b = 0", simp)  haftmann@53068  2145 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  2146 done  haftmann@33361  2147 haftmann@33361  2148 lemma zmod_zmult2_eq:  haftmann@53068  2149  fixes a b c :: int  haftmann@53068  2150  shows "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@33361  2151 apply (case_tac "b = 0", simp)  haftmann@53068  2152 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  2153 done  haftmann@33361  2154 huffman@47108  2155 lemma div_pos_geq:  huffman@47108  2156  fixes k l :: int  huffman@47108  2157  assumes "0 < l" and "l \ k"  huffman@47108  2158  shows "k div l = (k - l) div l + 1"  huffman@47108  2159 proof -  huffman@47108  2160  have "k = (k - l) + l" by simp  huffman@47108  2161  then obtain j where k: "k = j + l" ..  huffman@47108  2162  with assms show ?thesis by simp  huffman@47108  2163 qed  huffman@47108  2164 huffman@47108  2165 lemma mod_pos_geq:  huffman@47108  2166  fixes k l :: int  huffman@47108  2167  assumes "0 < l" and "l \ k"  huffman@47108  2168  shows "k mod l = (k - l) mod l"  huffman@47108  2169 proof -  huffman@47108  2170  have "k = (k - l) + l" by simp  huffman@47108  2171  then obtain j where k: "k = j + l" ..  huffman@47108  2172  with assms show ?thesis by simp  huffman@47108  2173 qed  huffman@47108  2174 haftmann@33361  2175 huffman@46551  2176 subsubsection {* Splitting Rules for div and mod *}  haftmann@33361  2177 haftmann@33361  2178 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  2179 haftmann@33361  2180 lemma split_pos_lemma:  haftmann@33361  2181  "0  haftmann@33361  2182  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2183 apply (rule iffI, clarify)  haftmann@33361  2184  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2185  apply (subst mod_add_eq)  haftmann@33361  2186  apply (subst zdiv_zadd1_eq)  haftmann@33361  2187  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  2188 txt{*converse direction*}  haftmann@33361  2189 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2190 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2191 done  haftmann@33361  2192 haftmann@33361  2193 lemma split_neg_lemma:  haftmann@33361  2194  "k<0 ==>  haftmann@33361  2195  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2196 apply (rule iffI, clarify)  haftmann@33361  2197  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2198  apply (subst mod_add_eq)  haftmann@33361  2199  apply (subst zdiv_zadd1_eq)  haftmann@33361  2200  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  2201 txt{*converse direction*}  haftmann@33361  2202 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2203 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2204 done  haftmann@33361  2205 haftmann@33361  2206 lemma split_zdiv:  haftmann@33361  2207  "P(n div k :: int) =  haftmann@33361  2208  ((k = 0 --> P 0) &  haftmann@33361  2209  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2210  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2211 apply (case_tac "k=0", simp)  haftmann@33361  2212 apply (simp only: linorder_neq_iff)  haftmann@33361  2213 apply (erule disjE)  haftmann@33361  2214  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2215  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2216 done  haftmann@33361  2217 haftmann@33361  2218 lemma split_zmod:  haftmann@33361  2219  "P(n mod k :: int) =  haftmann@33361  2220  ((k = 0 --> P n) &  haftmann@33361  2221  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2222  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2223 apply (case_tac "k=0", simp)  haftmann@33361  2224 apply (simp only: linorder_neq_iff)  haftmann@33361  2225 apply (erule disjE)  haftmann@33361  2226  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2227  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2228 done  haftmann@33361  2229 webertj@33730  2230 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  2231  when these are applied to some constant that is of the form  huffman@47108  2232  @{term "numeral k"}: *}  huffman@47108  2233 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  2234 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  2235 haftmann@33361  2236 huffman@47166  2237 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}  huffman@47166  2238 huffman@47166  2239 lemma pos_divmod_int_rel_mult_2:  huffman@47166  2240  assumes "0 \ b"  huffman@47166  2241  assumes "divmod_int_rel a b (q, r)"  huffman@47166  2242  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  2243  using assms unfolding divmod_int_rel_def by auto  huffman@47166  2244 huffman@47166  2245 lemma neg_divmod_int_rel_mult_2:  huffman@47166  2246  assumes "b \ 0"  huffman@47166  2247  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  2248  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  2249  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  2250 haftmann@33361  2251 text{*computing div by shifting *}  haftmann@33361  2252 haftmann@33361  2253 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2254  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2255  by (rule div_int_unique)  haftmann@33361  2256 boehmes@35815  2257 lemma neg_zdiv_mult_2:  boehmes@35815  2258  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2259  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2260  by (rule div_int_unique)  haftmann@33361  2261 huffman@47108  2262 (* FIXME: add rules for negative numerals *)  huffman@47108  2263 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2264  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2265  numeral v div (numeral w :: int)"  huffman@47108  2266  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2267  by (rule div_mult_mult1, simp)  huffman@47108  2268 huffman@47108  2269 lemma zdiv_numeral_Bit1 [simp]:  huffman@47108  2270  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  huffman@47108  2271  (numeral v div (numeral w :: int))"  huffman@47108  2272  unfolding numeral.simps  huffman@47108  2273  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2274  by (rule pos_zdiv_mult_2, simp)  haftmann@33361  2275 haftmann@33361  2276 lemma pos_zmod_mult_2:  haftmann@33361  2277  fixes a b :: int  haftmann@33361  2278  assumes "0 \ a"  haftmann@33361  2279  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  huffman@47166  2280  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2281  by (rule mod_int_unique)  haftmann@33361  2282 haftmann@33361  2283 lemma neg_zmod_mult_2:  haftmann@33361  2284  fixes a b :: int  haftmann@33361  2285  assumes "a \ 0"  haftmann@33361  2286  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  huffman@47166  2287  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2288  by (rule mod_int_unique)  haftmann@33361  2289 huffman@47108  2290 (* FIXME: add rules for negative numerals *)  huffman@47108  2291 lemma zmod_numeral_Bit0 [simp]:  huffman@47108  2292  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  huffman@47108  2293  (2::int) * (numeral v mod numeral w)"  huffman@47108  2294  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  huffman@47108  2295  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)  huffman@47108  2296 huffman@47108  2297 lemma zmod_numeral_Bit1 [simp]:  huffman@47108  2298  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =  huffman@47108  2299  2 * (numeral v mod numeral w) + (1::int)"  huffman@47108  2300  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]  huffman@47108  2301  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2302  by (rule pos_zmod_mult_2, simp)  haftmann@33361  2303 nipkow@39489  2304 lemma zdiv_eq_0_iff:  nipkow@39489  2305  "(i::int) div k = 0 \ k=0 \ 0\i \ i i\0 \ k ?R" by (rule split_zdiv[THEN iffD2]) simp  nipkow@39489  2309  with ?L show ?R by blast  nipkow@39489  2310 next  nipkow@39489  2311  assume ?R thus ?L  nipkow@39489  2312  by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)  nipkow@39489  2313 qed  nipkow@39489  2314 nipkow@39489  2315 huffman@46551  2316 subsubsection {* Quotients of Signs *}  haftmann@33361  2317 haftmann@33361  2318 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  haftmann@33361  2319 apply (subgoal_tac "a div b \ -1", force)  haftmann@33361  2320 apply (rule order_trans)  haftmann@33361  2321 apply (rule_tac a' = "-1" in zdiv_mono1)  haftmann@33361  2322 apply (auto simp add: div_eq_minus1)  haftmann@33361  2323 done  haftmann@33361  2324 haftmann@33361  2325 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  haftmann@33361  2326 by (drule zdiv_mono1_neg, auto)  haftmann@33361  2327 haftmann@33361  2328 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  haftmann@33361  2329 by (drule zdiv_mono1, auto)  haftmann@33361  2330 nipkow@33804  2331 text{* Now for some equivalences of the form @{text"a div b >=< 0 \ \"}  nipkow@33804  2332 conditional upon the sign of @{text a} or @{text b}. There are many more.  nipkow@33804  2333 They should all be simp rules unless that causes too much search. *}  nipkow@33804  2334 haftmann@33361  2335 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  haftmann@33361  2336 apply auto  haftmann@33361  2337 apply (drule_tac [2] zdiv_mono1)  haftmann@33361  2338 apply (auto simp add: linorder_neq_iff)  haftmann@33361  2339 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  haftmann@33361  2340 apply (blast intro: div_neg_pos_less0)  haftmann@33361  2341 done  haftmann@33361  2342 haftmann@33361  2343 lemma neg_imp_zdiv_nonneg_iff:  nipkow@33804  2344  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  huffman@47159  2345 apply (subst div_minus_minus [symmetric])  haftmann@33361  2346 apply (subst pos_imp_zdiv_nonneg_iff, auto)  haftmann@33361  2347 done  haftmann@33361  2348 haftmann@33361  2349 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  haftmann@33361  2350 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  haftmann@33361  2351 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  haftmann@33361  2352 nipkow@39489  2353 lemma pos_imp_zdiv_pos_iff:  nipkow@39489  2354  "0 0 < (i::int) div k \ k \ i"  nipkow@39489  2355 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]  nipkow@39489  2356 by arith  nipkow@39489  2357 haftmann@33361  2358 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  haftmann@33361  2359 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  haftmann@33361  2360 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  haftmann@33361  2361 nipkow@33804  2362 lemma nonneg1_imp_zdiv_pos_iff:  nipkow@33804  2363  "(0::int) <= a \ (a div b > 0) = (a >= b & b>0)"  nipkow@33804  2364 apply rule  nipkow@33804  2365  apply rule  nipkow@33804  2366  using div_pos_pos_trivial[of a b]apply arith  nipkow@33804  2367  apply(cases "b=0")apply simp  nipkow@33804  2368  using div_nonneg_neg_le0[of a b]apply arith  nipkow@33804  2369 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp  nipkow@33804  2370 done  nipkow@33804  2371 nipkow@39489  2372 lemma zmod_le_nonneg_dividend: "(m::int) \ 0 ==> m mod k \ m"  nipkow@39489  2373 apply (rule split_zmod[THEN iffD2])  nipkow@44890  2374 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)  nipkow@39489  2375 done  nipkow@39489  2376 nipkow@39489  2377 haftmann@33361  2378 subsubsection {* The Divides Relation *}  haftmann@33361  2379 huffman@47268  2380 lemma dvd_neg_numeral_left [simp]:  huffman@47268  2381  fixes y :: "'a::comm_ring_1"  huffman@47268  2382  shows "(neg_numeral k) dvd y \ (numeral k) dvd y"  huffman@47268  2383  unfolding neg_numeral_def minus_dvd_iff ..  huffman@47268  2384 huffman@47268  2385 lemma dvd_neg_numeral_right [simp]:  huffman@47268  2386  fixes x :: "'a::comm_ring_1"  huffman@47268  2387  shows "x dvd (neg_numeral k) \ x dvd (numeral k)"  huffman@47268  2388  unfolding neg_numeral_def dvd_minus_iff ..  haftmann@33361  2389 huffman@47108  2390 lemmas dvd_eq_mod_eq_0_numeral [simp] =  huffman@47108  2391  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y  huffman@47108  2392 huffman@47108  2393 huffman@47108  2394 subsubsection {* Further properties *}  huffman@47108  2395 haftmann@33361  2396 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  haftmann@33361  2397  using zmod_zdiv_equality[where a="m" and b="n"]  huffman@47142  2398  by (simp add: algebra_simps) (* FIXME: generalize *)  haftmann@33361  2399 haftmann@33361  2400 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  haftmann@33361  2401 apply (subst split_div, auto)  haftmann@33361  2402 apply (subst split_zdiv, auto)  haftmann@33361  2403 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  2404 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2405 done  haftmann@33361  2406 haftmann@33361  2407 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  haftmann@33361  2408 apply (subst split_mod, auto)  haftmann@33361  2409 apply (subst split_zmod, auto)  haftmann@33361  2410 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  haftmann@33361  2411  in unique_remainder)  haftmann@33361  2412 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2413 done  haftmann@33361  2414 haftmann@33361  2415 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  haftmann@33361  2416 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  haftmann@33361  2417 haftmann@33361  2418 text{*Suggested by Matthias Daum*}  haftmann@33361  2419 lemma int_power_div_base:  haftmann@33361  2420  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  haftmann@33361  2421 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  haftmann@33361  2422  apply (erule ssubst)  haftmann@33361  2423  apply (simp only: power_add)  haftmann@33361  2424  apply simp_all  haftmann@33361  2425 done  haftmann@33361  2426 haftmann@33361  2427 text {* by Brian Huffman *}  haftmann@33361  2428 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@33361  2429 by (rule mod_minus_eq [symmetric])  haftmann@33361  2430 haftmann@33361  2431 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@33361  2432 by (rule mod_diff_left_eq [symmetric])  haftmann@33361  2433 haftmann@33361  2434 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@33361  2435 by (rule mod_diff_right_eq [symmetric])  haftmann@33361  2436 haftmann@33361  2437 lemmas zmod_simps =  haftmann@33361  2438  mod_add_left_eq [symmetric]  haftmann@33361  2439  mod_add_right_eq [symmetric]  huffman@47142  2440  mod_mult_right_eq[symmetric]  haftmann@33361  2441  mod_mult_left_eq [symmetric]  huffman@47164  2442  power_mod  haftmann@33361  2443  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@33361  2444 haftmann@33361  2445 text {* Distributive laws for function @{text nat}. *}  haftmann@33361  2446 haftmann@33361  2447 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  haftmann@33361  2448 apply (rule linorder_cases [of y 0])  haftmann@33361  2449 apply (simp add: div_nonneg_neg_le0)  haftmann@33361  2450 apply simp  haftmann@33361  2451 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  haftmann@33361  2452 done  haftmann@33361  2453 haftmann@33361  2454 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  haftmann@33361  2455 lemma nat_mod_distrib:  haftmann@33361  2456  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  haftmann@33361  2457 apply (case_tac "y = 0", simp)  haftmann@33361  2458 apply (simp add: nat_eq_iff zmod_int)  haftmann@33361  2459 done  haftmann@33361  2460 haftmann@33361  2461 text {* transfer setup *}  haftmann@33361  2462 haftmann@33361  2463 lemma transfer_nat_int_functions:  haftmann@33361  2464  "(x::int) >= 0 \ y >= 0 \ (nat x) div (nat y) = nat (x div y)"  haftmann@33361  2465  "(x::int) >= 0 \ y >= 0 \ (nat x) mod (nat y) = nat (x mod y)"  haftmann@33361  2466  by (auto simp add: nat_div_distrib nat_mod_distrib)  haftmann@33361  2467 haftmann@33361  2468 lemma transfer_nat_int_function_closures:  haftmann@33361  2469  "(x::int) >= 0 \ y >= 0 \ x div y >= 0"  haftmann@33361  2470  "(x::int) >= 0 \ y >= 0 \ x mod y >= 0"  haftmann@33361  2471  apply (cases "y = 0")  haftmann@33361  2472  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  haftmann@33361  2473  apply (cases "y = 0")  haftmann@33361  2474  apply auto  haftmann@33361  2475 done  haftmann@33361  2476 haftmann@35644  2477 declare transfer_morphism_nat_int [transfer add return:  haftmann@33361  2478  transfer_nat_int_functions  haftmann@33361  2479  transfer_nat_int_function_closures  haftmann@33361  2480 ]  haftmann@33361  2481 haftmann@33361  2482 lemma transfer_int_nat_functions:  haftmann@33361  2483  "(int x) div (int y) = int (x div y)"  haftmann@33361  2484  "(int x) mod (int y) = int (x mod y)"  haftmann@33361  2485  by (auto simp add: zdiv_int zmod_int)  haftmann@33361  2486 haftmann@33361  2487 lemma transfer_int_nat_function_closures:  haftmann@33361  2488  "is_nat x \ is_nat y \ is_nat (x div y)"  haftmann@33361  2489  "is_nat x \ is_nat y \ is_nat (x mod y)"  haftmann@33361  2490  by (simp_all only: is_nat_def transfer_nat_int_function_closures)  haftmann@33361  2491 haftmann@35644  2492 declare transfer_morphism_int_nat [transfer add return:  haftmann@33361  2493  transfer_int_nat_functions  haftmann@33361  2494  transfer_int_nat_function_closures  haftmann@33361  2495 ]  haftmann@33361  2496 haftmann@33361  2497 text{*Suggested by Matthias Daum*}  haftmann@33361  2498 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  haftmann@33361  2499 apply (subgoal_tac "nat x div nat k < nat x")  nipkow@34225  2500  apply (simp add: nat_div_distrib [symmetric])  haftmann@33361  2501 apply (rule Divides.div_less_dividend, simp_all)  haftmann@33361  2502 done  haftmann@33361  2503 haftmann@33361  2504 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  haftmann@33361  2505 proof  haftmann@33361  2506  assume H: "x mod n = y mod n"  haftmann@33361  2507  hence "x mod n - y mod n = 0" by simp  haftmann@33361  2508  hence "(x mod n - y mod n) mod n = 0" by simp  haftmann@33361  2509  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])  haftmann@33361  2510  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)  haftmann@33361  2511 next  haftmann@33361  2512  assume H: "n dvd x - y"  haftmann@33361  2513  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  haftmann@33361  2514  hence "x = n*k + y" by simp  haftmann@33361  2515  hence "x mod n = (n*k + y) mod n" by simp  haftmann@33361  2516  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)  haftmann@33361  2517 qed  haftmann@33361  2518 haftmann@33361  2519 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  haftmann@33361  2520  shows "\q. x = y + n * q"  haftmann@33361  2521 proof-  haftmann@33361  2522  from xy have th: "int x - int y = int (x - y)" by simp  haftmann@33361  2523  from xyn have "int x mod int n = int y mod `