src/HOL/Divides.thy
 author haftmann Sun Aug 18 15:29:50 2013 +0200 (2013-08-18) changeset 53068 41fc65da66f1 parent 53067 ee0b7c2315d2 child 53069 d165213e3924 permissions -rw-r--r--
relaxed preconditions
 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  find_theorems "_ + ?b < _ + ?b"  haftmann@53067  567  with mod_w have mod: "a mod (2 * b) = a mod b" by simp  haftmann@53067  568  have "2 * (a div (2 * b)) = a div b - w"  haftmann@53067  569  by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)  haftmann@53067  570  with w = 0 have div: "2 * (a div (2 * b)) = a div b" by simp  haftmann@53067  571  then show ?P and ?Q  haftmann@53067  572  by (simp_all add: div mod)  haftmann@53067  573 qed  haftmann@53067  574 haftmann@53067  575 definition divmod :: "num \ num \ 'a \ 'a"  haftmann@53067  576 where  haftmann@53067  577  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"  haftmann@53067  578 haftmann@53067  579 lemma fst_divmod [simp]:  haftmann@53067  580  "fst (divmod m n) = numeral m div numeral n"  haftmann@53067  581  by (simp add: divmod_def)  haftmann@53067  582 haftmann@53067  583 lemma snd_divmod [simp]:  haftmann@53067  584  "snd (divmod m n) = numeral m mod numeral n"  haftmann@53067  585  by (simp add: divmod_def)  haftmann@53067  586 haftmann@53067  587 definition divmod_step :: "num \ 'a \ 'a \ 'a \ 'a"  haftmann@53067  588 where  haftmann@53067  589  "divmod_step l qr = (let (q, r) = qr  haftmann@53067  590  in if r \ numeral l then (2 * q + 1, r - numeral l)  haftmann@53067  591  else (2 * q, r))"  haftmann@53067  592 haftmann@53067  593 text {*  haftmann@53067  594  This is a formulation of one step (referring to one digit position)  haftmann@53067  595  in school-method division: compare the dividend at the current  haftmann@53067  596  digit position with the remained from previous division steps  haftmann@53067  597  and evaluate accordingly.  haftmann@53067  598 *}  haftmann@53067  599 haftmann@53067  600 lemma divmod_step_eq [code]:  haftmann@53067  601  "divmod_step l (q, r) = (if numeral l \ r  haftmann@53067  602  then (2 * q + 1, r - numeral l) else (2 * q, r))"  haftmann@53067  603  by (simp add: divmod_step_def)  haftmann@53067  604 haftmann@53067  605 lemma divmod_step_simps [simp]:  haftmann@53067  606  "r < numeral l \ divmod_step l (q, r) = (2 * q, r)"  haftmann@53067  607  "numeral l \ r \ divmod_step l (q, r) = (2 * q + 1, r - numeral l)"  haftmann@53067  608  by (auto simp add: divmod_step_eq not_le)  haftmann@53067  609 haftmann@53067  610 text {*  haftmann@53067  611  This is a formulation of school-method division.  haftmann@53067  612  If the divisor is smaller than the dividend, terminate.  haftmann@53067  613  If not, shift the dividend to the right until termination  haftmann@53067  614  occurs and then reiterate single division steps in the  haftmann@53067  615  opposite direction.  haftmann@53067  616 *}  haftmann@53067  617 haftmann@53067  618 lemma divmod_divmod_step [code]:  haftmann@53067  619  "divmod m n = (if m < n then (0, numeral m)  haftmann@53067  620  else divmod_step n (divmod m (Num.Bit0 n)))"  haftmann@53067  621 proof (cases "m < n")  haftmann@53067  622  case True then have "numeral m < numeral n" by simp  haftmann@53067  623  then show ?thesis  haftmann@53067  624  by (simp add: prod_eq_iff div_less mod_less)  haftmann@53067  625 next  haftmann@53067  626  case False  haftmann@53067  627  have "divmod m n =  haftmann@53067  628  divmod_step n (numeral m div (2 * numeral n),  haftmann@53067  629  numeral m mod (2 * numeral n))"  haftmann@53067  630  proof (cases "numeral n \ numeral m mod (2 * numeral n)")  haftmann@53067  631  case True  haftmann@53067  632  with divmod_step_simps  haftmann@53067  633  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  634  (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"  haftmann@53067  635  by blast  haftmann@53067  636  moreover from True divmod_digit_1 [of "numeral m" "numeral n"]  haftmann@53067  637  have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"  haftmann@53067  638  and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"  haftmann@53067  639  by simp_all  haftmann@53067  640  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  641  next  haftmann@53067  642  case False then have *: "numeral m mod (2 * numeral n) < numeral n"  haftmann@53067  643  by (simp add: not_le)  haftmann@53067  644  with divmod_step_simps  haftmann@53067  645  have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =  haftmann@53067  646  (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"  haftmann@53067  647  by blast  haftmann@53067  648  moreover from * divmod_digit_0 [of "numeral n" "numeral m"]  haftmann@53067  649  have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"  haftmann@53067  650  and "numeral m mod (2 * numeral n) = numeral m mod numeral n"  haftmann@53067  651  by (simp_all only: zero_less_numeral)  haftmann@53067  652  ultimately show ?thesis by (simp only: divmod_def)  haftmann@53067  653  qed  haftmann@53067  654  then have "divmod m n =  haftmann@53067  655  divmod_step n (numeral m div numeral (Num.Bit0 n),  haftmann@53067  656  numeral m mod numeral (Num.Bit0 n))"  haftmann@53067  657  by (simp only: numeral.simps distrib mult_1)  haftmann@53067  658  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"  haftmann@53067  659  by (simp add: divmod_def)  haftmann@53067  660  with False show ?thesis by simp  haftmann@53067  661 qed  haftmann@53067  662 haftmann@53067  663 end  haftmann@53067  664 haftmann@53067  665 hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero  haftmann@53067  666  -- {* restore simple accesses for more general variants of theorems *}  haftmann@53067  667 haftmann@53067  668   haftmann@26100  669 subsection {* Division on @{typ nat} *}  haftmann@26100  670 haftmann@26100  671 text {*  haftmann@26100  672  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  673  of a characteristic relation with two input arguments  haftmann@26100  674  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  675  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  676 *}  haftmann@26100  677 haftmann@33340  678 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  679  "divmod_nat_rel m n qr \  haftmann@30923  680  m = fst qr * n + snd qr \  haftmann@30923  681  (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  682 haftmann@33340  683 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  684 haftmann@33340  685 lemma divmod_nat_rel_ex:  haftmann@33340  686  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  687 proof (cases "n = 0")  haftmann@30923  688  case True with that show thesis  haftmann@33340  689  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  690 next  haftmann@26100  691  case False  haftmann@26100  692  have "\q r. m = q * n + r \ r < n"  haftmann@26100  693  proof (induct m)  haftmann@26100  694  case 0 with n \ 0  haftmann@26100  695  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  696  then show ?case by blast  haftmann@26100  697  next  haftmann@26100  698  case (Suc m) then obtain q' r'  haftmann@26100  699  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  700  then show ?case proof (cases "Suc r' < n")  haftmann@26100  701  case True  haftmann@26100  702  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  703  with True show ?thesis by blast  haftmann@26100  704  next  haftmann@26100  705  case False then have "n \ Suc r'" by auto  haftmann@26100  706  moreover from n have "Suc r' \ n" by auto  haftmann@26100  707  ultimately have "n = Suc r'" by auto  haftmann@26100  708  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  709  with n \ 0 show ?thesis by blast  haftmann@26100  710  qed  haftmann@26100  711  qed  haftmann@26100  712  with that show thesis  haftmann@33340  713  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  714 qed  haftmann@26100  715 haftmann@33340  716 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  717 haftmann@33340  718 lemma divmod_nat_rel_unique:  haftmann@33340  719  assumes "divmod_nat_rel m n qr"  haftmann@33340  720  and "divmod_nat_rel m n qr'"  haftmann@30923  721  shows "qr = qr'"  haftmann@26100  722 proof (cases "n = 0")  haftmann@26100  723  case True with assms show ?thesis  haftmann@30923  724  by (cases qr, cases qr')  haftmann@33340  725  (simp add: divmod_nat_rel_def)  haftmann@26100  726 next  haftmann@26100  727  case False  haftmann@26100  728  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  729  apply (rule leI)  haftmann@26100  730  apply (subst less_iff_Suc_add)  haftmann@26100  731  apply (auto simp add: add_mult_distrib)  haftmann@26100  732  done  haftmann@30923  733  from n \ 0 assms have "fst qr = fst qr'"  haftmann@33340  734  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  haftmann@30923  735  moreover from this assms have "snd qr = snd qr'"  haftmann@33340  736  by (simp add: divmod_nat_rel_def)  haftmann@30923  737  ultimately show ?thesis by (cases qr, cases qr') simp  haftmann@26100  738 qed  haftmann@26100  739 haftmann@26100  740 text {*  haftmann@26100  741  We instantiate divisibility on the natural numbers by  haftmann@33340  742  means of @{const divmod_nat_rel}:  haftmann@26100  743 *}  haftmann@25942  744 haftmann@33340  745 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  746  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  747 haftmann@33340  748 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  749  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  750 proof -  haftmann@33340  751  from divmod_nat_rel_ex  haftmann@33340  752  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  753  then show ?thesis  haftmann@33340  754  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  755 qed  haftmann@30923  756 huffman@47135  757 lemma divmod_nat_unique:  haftmann@33340  758  assumes "divmod_nat_rel m n qr"  haftmann@33340  759  shows "divmod_nat m n = qr"  haftmann@33340  760  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  761 huffman@46551  762 instantiation nat :: semiring_div  huffman@46551  763 begin  huffman@46551  764 haftmann@26100  765 definition div_nat where  haftmann@33340  766  "m div n = fst (divmod_nat m n)"  haftmann@26100  767 huffman@46551  768 lemma fst_divmod_nat [simp]:  huffman@46551  769  "fst (divmod_nat m n) = m div n"  huffman@46551  770  by (simp add: div_nat_def)  huffman@46551  771 haftmann@26100  772 definition mod_nat where  haftmann@33340  773  "m mod n = snd (divmod_nat m n)"  haftmann@25571  774 huffman@46551  775 lemma snd_divmod_nat [simp]:  huffman@46551  776  "snd (divmod_nat m n) = m mod n"  huffman@46551  777  by (simp add: mod_nat_def)  huffman@46551  778 haftmann@33340  779 lemma divmod_nat_div_mod:  haftmann@33340  780  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  781  by (simp add: prod_eq_iff)  haftmann@26100  782 huffman@47135  783 lemma div_nat_unique:  haftmann@33340  784  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  785  shows "m div n = q"  huffman@47135  786  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  787 huffman@47135  788 lemma mod_nat_unique:  haftmann@33340  789  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  790  shows "m mod n = r"  huffman@47135  791  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  792 haftmann@33340  793 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  794  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  795 huffman@47136  796 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  797  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  798 huffman@47136  799 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  800  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  801 huffman@47137  802 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  803  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  804 haftmann@33340  805 lemma divmod_nat_step:  haftmann@26100  806  assumes "0 < n" and "n \ m"  haftmann@33340  807  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  808 proof (rule divmod_nat_unique)  huffman@47134  809  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  810  by (rule divmod_nat_rel)  huffman@47134  811  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  812  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  813 qed  haftmann@25942  814 wenzelm@26300  815 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  816 haftmann@26100  817 lemma div_less [simp]:  haftmann@26100  818  fixes m n :: nat  haftmann@26100  819  assumes "m < n"  haftmann@26100  820  shows "m div n = 0"  huffman@46551  821  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  822 haftmann@26100  823 lemma le_div_geq:  haftmann@26100  824  fixes m n :: nat  haftmann@26100  825  assumes "0 < n" and "n \ m"  haftmann@26100  826  shows "m div n = Suc ((m - n) div n)"  huffman@46551  827  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  828 haftmann@26100  829 lemma mod_less [simp]:  haftmann@26100  830  fixes m n :: nat  haftmann@26100  831  assumes "m < n"  haftmann@26100  832  shows "m mod n = m"  huffman@46551  833  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  834 haftmann@26100  835 lemma le_mod_geq:  haftmann@26100  836  fixes m n :: nat  haftmann@26100  837  assumes "n \ m"  haftmann@26100  838  shows "m mod n = (m - n) mod n"  huffman@46551  839  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  840 huffman@47136  841 instance proof  huffman@47136  842  fix m n :: nat  huffman@47136  843  show "m div n * n + m mod n = m"  huffman@47136  844  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  845 next  huffman@47136  846  fix m n q :: nat  huffman@47136  847  assume "n \ 0"  huffman@47136  848  then show "(q + m * n) div n = m + q div n"  huffman@47136  849  by (induct m) (simp_all add: le_div_geq)  huffman@47136  850 next  huffman@47136  851  fix m n q :: nat  huffman@47136  852  assume "m \ 0"  huffman@47136  853  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  854  unfolding divmod_nat_rel_def  huffman@47136  855  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  856  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  857  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  858  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  859 next  huffman@47136  860  fix n :: nat show "n div 0 = 0"  haftmann@33340  861  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  862 next  huffman@47136  863  fix n :: nat show "0 div n = 0"  huffman@47136  864  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  865 qed  haftmann@26100  866 haftmann@25942  867 end  paulson@14267  868 haftmann@33361  869 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  870  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  huffman@46551  871  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)  haftmann@33361  872 haftmann@26100  873 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  874 wenzelm@51299  875 ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"  wenzelm@51299  876 haftmann@30934  877 ML {*  wenzelm@43594  878 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  879 (  haftmann@30934  880  val div_name = @{const_name div};  haftmann@30934  881  val mod_name = @{const_name mod};  haftmann@30934  882  val mk_binop = HOLogic.mk_binop;  huffman@48561  883  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  huffman@48561  884  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  huffman@48561  885  fun mk_sum [] = HOLogic.zero  huffman@48561  886  | mk_sum [t] = t  huffman@48561  887  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  huffman@48561  888  fun dest_sum tm =  huffman@48561  889  if HOLogic.is_zero tm then []  huffman@48561  890  else  huffman@48561  891  (case try HOLogic.dest_Suc tm of  huffman@48561  892  SOME t => HOLogic.Suc_zero :: dest_sum t  huffman@48561  893  | NONE =>  huffman@48561  894  (case try dest_plus tm of  huffman@48561  895  SOME (t, u) => dest_sum t @ dest_sum u  huffman@48561  896  | NONE => [tm]));  haftmann@25942  897 haftmann@30934  898  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  899 haftmann@30934  900  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@35050  901  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))  wenzelm@41550  902 )  haftmann@25942  903 *}  haftmann@25942  904 wenzelm@43594  905 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  906 haftmann@26100  907 haftmann@26100  908 subsubsection {* Quotient *}  haftmann@26100  909 haftmann@26100  910 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  911 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  912 haftmann@26100  913 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  914 by (simp add: div_geq)  haftmann@26100  915 haftmann@26100  916 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  917 by simp  haftmann@26100  918 haftmann@26100  919 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  920 by simp  haftmann@26100  921 haftmann@53066  922 lemma div_positive:  haftmann@53066  923  fixes m n :: nat  haftmann@53066  924  assumes "n > 0"  haftmann@53066  925  assumes "m \ n"  haftmann@53066  926  shows "m div n > 0"  haftmann@53066  927 proof -  haftmann@53066  928  from m \ n obtain q where "m = n + q"  haftmann@53066  929  by (auto simp add: le_iff_add)  haftmann@53066  930  with n > 0 show ?thesis by simp  haftmann@53066  931 qed  haftmann@53066  932 haftmann@25942  933 haftmann@25942  934 subsubsection {* Remainder *}  haftmann@25942  935 haftmann@26100  936 lemma mod_less_divisor [simp]:  haftmann@26100  937  fixes m n :: nat  haftmann@26100  938  assumes "n > 0"  haftmann@26100  939  shows "m mod n < (n::nat)"  haftmann@33340  940  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  941 haftmann@51173  942 lemma mod_Suc_le_divisor [simp]:  haftmann@51173  943  "m mod Suc n \ n"  haftmann@51173  944  using mod_less_divisor [of "Suc n" m] by arith  haftmann@51173  945 haftmann@26100  946 lemma mod_less_eq_dividend [simp]:  haftmann@26100  947  fixes m n :: nat  haftmann@26100  948  shows "m mod n \ m"  haftmann@26100  949 proof (rule add_leD2)  haftmann@26100  950  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  951  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  952 qed  haftmann@26100  953 haftmann@26100  954 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  955 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  956 haftmann@26100  957 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  958 by (simp add: le_mod_geq)  haftmann@26100  959 paulson@14267  960 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  961 by (induct m) (simp_all add: mod_geq)  paulson@14267  962 paulson@14267  963 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  964 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  965  using mod_div_equality2 [of n m] by arith  paulson@14267  966 nipkow@15439  967 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  968  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  969  apply simp  wenzelm@22718  970  done  paulson@14267  971 haftmann@26100  972 subsubsection {* Quotient and Remainder *}  paulson@14267  973 haftmann@33340  974 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  975  "divmod_nat_rel b c (q, r)  haftmann@33340  976  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  977 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  978 haftmann@30923  979 lemma div_mult1_eq:  haftmann@30923  980  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  981 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  982 haftmann@33340  983 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  984  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  985  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  986 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  987 paulson@14267  988 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  989 lemma div_add1_eq:  nipkow@25134  990  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  991 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  992 paulson@14267  993 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@22718  994  apply (cut_tac m = q and n = c in mod_less_divisor)  wenzelm@22718  995  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  wenzelm@22718  996  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)  wenzelm@22718  997  apply (simp add: add_mult_distrib2)  wenzelm@22718  998  done  paulson@10559  999 haftmann@33340  1000 lemma divmod_nat_rel_mult2_eq:  bulwahn@46552  1001  "divmod_nat_rel a b (q, r)  haftmann@33340  1002  \ divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@33340  1003 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)  paulson@14267  1004 paulson@14267  1005 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"  huffman@47135  1006 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  1007 paulson@14267  1008 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"  huffman@47135  1009 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  1010 paulson@14267  1011 huffman@46551  1012 subsubsection {* Further Facts about Quotient and Remainder *}  paulson@14267  1013 paulson@14267  1014 lemma div_1 [simp]: "m div Suc 0 = m"  nipkow@29667  1015 by (induct m) (simp_all add: div_geq)  paulson@14267  1016 paulson@14267  1017 (* Monotonicity of div in first argument *)  haftmann@30923  1018 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  1019  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  1020 apply (case_tac "k=0", simp)  paulson@15251  1021 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  1022 apply (case_tac "n= k *)  paulson@14267  1026 apply (case_tac "m=k *)  nipkow@15439  1030 apply (simp add: div_geq diff_le_mono)  paulson@14267  1031 done  paulson@14267  1032 paulson@14267  1033 (* Antimonotonicity of div in second argument *)  paulson@14267  1034 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  1035 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  1044  prefer 2  paulson@14267  1045  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  1046 apply (rule le_trans, simp)  nipkow@15439  1047 apply (simp)  paulson@14267  1048 done  paulson@14267  1049 paulson@14267  1050 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  1051 apply (case_tac "n=0", simp)  paulson@14267  1052 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  1053 apply (rule div_le_mono2)  paulson@14267  1054 apply (simp_all (no_asm_simp))  paulson@14267  1055 done  paulson@14267  1056 wenzelm@22718  1057 (* Similar for "less than" *)  huffman@47138  1058 lemma div_less_dividend [simp]:  huffman@47138  1059  "\(1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  1060 apply (induct m rule: nat_less_induct)  paulson@14267  1061 apply (rename_tac "m")  paulson@14267  1062 apply (case_tac "m Suc(na) *)  paulson@16796  1081 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  1082 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  1083 done  paulson@14267  1084 paulson@14267  1085 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  1086 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  1087 wenzelm@22718  1088 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  1089 paulson@14267  1090 (*Loses information, namely we also have r \q::nat. m = r + q*d"  haftmann@27651  1092  apply (cut_tac a = m in mod_div_equality)  wenzelm@22718  1093  apply (simp only: add_ac)  wenzelm@22718  1094  apply (blast intro: sym)  wenzelm@22718  1095  done  paulson@14267  1096 nipkow@13152  1097 lemma split_div:  nipkow@13189  1098  "P(n div k :: nat) =  nipkow@13189  1099  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  1100  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1101 proof  nipkow@13189  1102  assume P: ?P  nipkow@13189  1103  show ?Q  nipkow@13189  1104  proof (cases)  nipkow@13189  1105  assume "k = 0"  haftmann@27651  1106  with P show ?Q by simp  nipkow@13189  1107  next  nipkow@13189  1108  assume not0: "k \ 0"  nipkow@13189  1109  thus ?Q  nipkow@13189  1110  proof (simp, intro allI impI)  nipkow@13189  1111  fix i j  nipkow@13189  1112  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  1113  show "P i"  nipkow@13189  1114  proof (cases)  wenzelm@22718  1115  assume "i = 0"  wenzelm@22718  1116  with n j P show "P i" by simp  nipkow@13189  1117  next  wenzelm@22718  1118  assume "i \ 0"  wenzelm@22718  1119  with not0 n j P show "P i" by(simp add:add_ac)  nipkow@13189  1120  qed  nipkow@13189  1121  qed  nipkow@13189  1122  qed  nipkow@13189  1123 next  nipkow@13189  1124  assume Q: ?Q  nipkow@13189  1125  show ?P  nipkow@13189  1126  proof (cases)  nipkow@13189  1127  assume "k = 0"  haftmann@27651  1128  with Q show ?P by simp  nipkow@13189  1129  next  nipkow@13189  1130  assume not0: "k \ 0"  nipkow@13189  1131  with Q have R: ?R by simp  nipkow@13189  1132  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1133  show ?P by simp  nipkow@13189  1134  qed  nipkow@13189  1135 qed  nipkow@13189  1136 berghofe@13882  1137 lemma split_div_lemma:  haftmann@26100  1138  assumes "0 < n"  haftmann@26100  1139  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  1140 proof  haftmann@26100  1141  assume ?rhs  haftmann@26100  1142  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  1143  then have A: "n * q \ m" by simp  haftmann@26100  1144  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  1145  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  1146  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  1147  with nq have "m < n + n * q" by simp  haftmann@26100  1148  then have B: "m < n * Suc q" by simp  haftmann@26100  1149  from A B show ?lhs ..  haftmann@26100  1150 next  haftmann@26100  1151  assume P: ?lhs  haftmann@33340  1152  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@33340  1153  unfolding divmod_nat_rel_def by (auto simp add: mult_ac)  haftmann@33340  1154  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  1155  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  1156  then show ?rhs by simp  haftmann@26100  1157 qed  berghofe@13882  1158 berghofe@13882  1159 theorem split_div':  berghofe@13882  1160  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  1161  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  1162  apply (case_tac "0 < n")  berghofe@13882  1163  apply (simp only: add: split_div_lemma)  haftmann@27651  1164  apply simp_all  berghofe@13882  1165  done  berghofe@13882  1166 nipkow@13189  1167 lemma split_mod:  nipkow@13189  1168  "P(n mod k :: nat) =  nipkow@13189  1169  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  1170  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  1171 proof  nipkow@13189  1172  assume P: ?P  nipkow@13189  1173  show ?Q  nipkow@13189  1174  proof (cases)  nipkow@13189  1175  assume "k = 0"  haftmann@27651  1176  with P show ?Q by simp  nipkow@13189  1177  next  nipkow@13189  1178  assume not0: "k \ 0"  nipkow@13189  1179  thus ?Q  nipkow@13189  1180  proof (simp, intro allI impI)  nipkow@13189  1181  fix i j  nipkow@13189  1182  assume "n = k*i + j" "j < k"  nipkow@13189  1183  thus "P j" using not0 P by(simp add:add_ac mult_ac)  nipkow@13189  1184  qed  nipkow@13189  1185  qed  nipkow@13189  1186 next  nipkow@13189  1187  assume Q: ?Q  nipkow@13189  1188  show ?P  nipkow@13189  1189  proof (cases)  nipkow@13189  1190  assume "k = 0"  haftmann@27651  1191  with Q show ?P by simp  nipkow@13189  1192  next  nipkow@13189  1193  assume not0: "k \ 0"  nipkow@13189  1194  with Q have R: ?R by simp  nipkow@13189  1195  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  1196  show ?P by simp  nipkow@13189  1197  qed  nipkow@13189  1198 qed  nipkow@13189  1199 berghofe@13882  1200 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  1201  using mod_div_equality [of m n] by arith  huffman@47138  1202 huffman@47138  1203 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  1204  using mod_div_equality [of m n] by arith  huffman@47138  1205 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  1206 noschinl@52398  1207 lemma div_eq_dividend_iff: "a \ 0 \ (a :: nat) div b = a \ b = 1"  noschinl@52398  1208  apply rule  noschinl@52398  1209  apply (cases "b = 0")  noschinl@52398  1210  apply simp_all  noschinl@52398  1211  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)  noschinl@52398  1212  done  noschinl@52398  1213 haftmann@22800  1214 huffman@46551  1215 subsubsection {* An induction'' law for modulus arithmetic. *}  paulson@14640  1216 paulson@14640  1217 lemma mod_induct_0:  paulson@14640  1218  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1219  and base: "P i" and i: "i(P 0)"  paulson@14640  1223  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1225  proof  paulson@14640  1226  fix k  paulson@14640  1227  show "?A k"  paulson@14640  1228  proof (induct k)  paulson@14640  1229  show "?A 0" by simp -- "by contradiction"  paulson@14640  1230  next  paulson@14640  1231  fix n  paulson@14640  1232  assume ih: "?A n"  paulson@14640  1233  show "?A (Suc n)"  paulson@14640  1234  proof (clarsimp)  wenzelm@22718  1235  assume y: "P (p - Suc n)"  wenzelm@22718  1236  have n: "Suc n < p"  wenzelm@22718  1237  proof (rule ccontr)  wenzelm@22718  1238  assume "\(Suc n < p)"  wenzelm@22718  1239  hence "p - Suc n = 0"  wenzelm@22718  1240  by simp  wenzelm@22718  1241  with y contra show "False"  wenzelm@22718  1242  by simp  wenzelm@22718  1243  qed  wenzelm@22718  1244  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1245  from p have "p - Suc n < p" by arith  wenzelm@22718  1246  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1247  by blast  wenzelm@22718  1248  show "False"  wenzelm@22718  1249  proof (cases "n=0")  wenzelm@22718  1250  case True  wenzelm@22718  1251  with z n2 contra show ?thesis by simp  wenzelm@22718  1252  next  wenzelm@22718  1253  case False  wenzelm@22718  1254  with p have "p-n < p" by arith  wenzelm@22718  1255  with z n2 False ih show ?thesis by simp  wenzelm@22718  1256  qed  paulson@14640  1257  qed  paulson@14640  1258  qed  paulson@14640  1259  qed  paulson@14640  1260  moreover  paulson@14640  1261  from i obtain k where "0 i+k=p"  paulson@14640  1262  by (blast dest: less_imp_add_positive)  paulson@14640  1263  hence "0 i=p-k" by auto  paulson@14640  1264  moreover  paulson@14640  1265  note base  paulson@14640  1266  ultimately  paulson@14640  1267  show "False" by blast  paulson@14640  1268 qed  paulson@14640  1269 paulson@14640  1270 lemma mod_induct:  paulson@14640  1271  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1272  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1279  proof (induct j)  paulson@14640  1280  from step base i show "?A 0"  wenzelm@22718  1281  by (auto elim: mod_induct_0)  paulson@14640  1282  next  paulson@14640  1283  fix k  paulson@14640  1284  assume ih: "?A k"  paulson@14640  1285  show "?A (Suc k)"  paulson@14640  1286  proof  wenzelm@22718  1287  assume suc: "Suc k < p"  wenzelm@22718  1288  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1313 proof -  boehmes@35815  1314  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1315  moreover have "m mod 2 < 2" by simp  haftmann@33296  1316  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1317  then show ?thesis by auto  haftmann@33296  1318 qed  haftmann@33296  1319 haftmann@33296  1320 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1321  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1322  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1323 haftmann@33296  1324 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1325 by (simp add: Suc3_eq_add_3)  haftmann@33296  1326 haftmann@33296  1327 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1328 by (simp add: Suc3_eq_add_3)  haftmann@33296  1329 haftmann@33296  1330 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1331 by (simp add: Suc3_eq_add_3)  haftmann@33296  1332 haftmann@33296  1333 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1334 by (simp add: Suc3_eq_add_3)  haftmann@33296  1335 huffman@47108  1336 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1337 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1338 haftmann@33361  1339 haftmann@33361  1340 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1341 apply (induct "m")  haftmann@33361  1342 apply (simp_all add: mod_Suc)  haftmann@33361  1343 done  haftmann@33361  1344 huffman@47108  1345 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1346 huffman@47138  1347 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1348 by (simp add: div_le_mono)  haftmann@33361  1349 haftmann@33361  1350 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1351 by (cases n) simp_all  haftmann@33361  1352 boehmes@35815  1353 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1354 proof -  boehmes@35815  1355  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1356  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1357 qed  haftmann@33361  1358 haftmann@33361  1359  (* Potential use of algebra : Equality modulo n*)  haftmann@33361  1360 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"  haftmann@33361  1361 by (simp add: mult_ac add_ac)  haftmann@33361  1362 haftmann@33361  1363 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1364 proof -  haftmann@33361  1365  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1366  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1367  finally show ?thesis .  haftmann@33361  1368 qed  haftmann@33361  1369 haftmann@33361  1370 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1371 apply (subst mod_Suc [of m])  haftmann@33361  1372 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1373 done  haftmann@33361  1374 huffman@47108  1375 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1376  fixes n :: nat  huffman@47108  1377  shows "n mod 2 \ 0 \ n mod 2 = 1"  huffman@47108  1378  by simp  huffman@47108  1379 haftmann@53067  1380 instance nat :: semiring_numeral_div  haftmann@53067  1381  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)  haftmann@53067  1382 haftmann@33361  1383 haftmann@33361  1384 subsection {* Division on @{typ int} *}  haftmann@33361  1385 haftmann@33361  1386 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1387  --{*definition of quotient and remainder*}  huffman@47139  1388  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1389  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1390 haftmann@53067  1391 text {*  haftmann@53067  1392  The following algorithmic devlopment actually echos what has already  haftmann@53067  1393  been developed in class @{class semiring_numeral_div}. In the long  haftmann@53067  1394  run it seems better to derive division on @{typ int} just from  haftmann@53067  1395  division on @{typ nat} and instantiate @{class semiring_numeral_div}  haftmann@53067  1396  accordingly.  haftmann@53067  1397 *}  haftmann@53067  1398 haftmann@33361  1399 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1400  --{*for the division algorithm*}  huffman@47108  1401  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1402  else (2 * q, r))"  haftmann@33361  1403 haftmann@33361  1404 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1405 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1406  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1407  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1408 by auto  haftmann@33361  1409 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1410  (auto simp add: mult_2)  haftmann@33361  1411 haftmann@33361  1412 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1413 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1414  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1415  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1416 by auto  haftmann@33361  1417 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1418  (auto simp add: mult_2)  haftmann@33361  1419 haftmann@33361  1420 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1421 haftmann@33361  1422 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1423  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1424  including the special case @{text "a=0, b<0"} because  haftmann@33361  1425  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1426  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1427  else if a = 0 then (0, 0)  huffman@46560  1428  else apsnd uminus (negDivAlg (-a) (-b))  haftmann@33361  1429  else  haftmann@33361  1430  if 0 < b then negDivAlg a b  huffman@46560  1431  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1432 haftmann@33361  1433 instantiation int :: Divides.div  haftmann@33361  1434 begin  haftmann@33361  1435 huffman@46551  1436 definition div_int where  haftmann@33361  1437  "a div b = fst (divmod_int a b)"  haftmann@33361  1438 huffman@46551  1439 lemma fst_divmod_int [simp]:  huffman@46551  1440  "fst (divmod_int a b) = a div b"  huffman@46551  1441  by (simp add: div_int_def)  huffman@46551  1442 huffman@46551  1443 definition mod_int where  huffman@46560  1444  "a mod b = snd (divmod_int a b)"  haftmann@33361  1445 huffman@46551  1446 lemma snd_divmod_int [simp]:  huffman@46551  1447  "snd (divmod_int a b) = a mod b"  huffman@46551  1448  by (simp add: mod_int_def)  huffman@46551  1449 haftmann@33361  1450 instance ..  haftmann@33361  1451 paulson@3366  1452 end  haftmann@33361  1453 haftmann@33361  1454 lemma divmod_int_mod_div:  haftmann@33361  1455  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1456  by (simp add: prod_eq_iff)  haftmann@33361  1457 haftmann@33361  1458 text{*  haftmann@33361  1459 Here is the division algorithm in ML:  haftmann@33361  1460 haftmann@33361  1461 \begin{verbatim}  haftmann@33361  1462  fun posDivAlg (a,b) =  haftmann@33361  1463  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1466  end  haftmann@33361  1467 haftmann@33361  1468  fun negDivAlg (a,b) =  haftmann@33361  1469  if 0\a+b then (~1,a+b)  haftmann@33361  1470  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1471  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1472  end;  haftmann@33361  1473 haftmann@33361  1474  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1475 haftmann@33361  1476  fun divmod (a,b) = if 0\a then  haftmann@33361  1477  if b>0 then posDivAlg (a,b)  haftmann@33361  1478  else if a=0 then (0,0)  haftmann@33361  1479  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1480  else  haftmann@33361  1481  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1491  ==> q' \ (q::int)"  haftmann@33361  1492 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1493  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1494 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1495 apply (erule_tac [2] order_le_less_trans)  webertj@49962  1496  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1497 apply (subgoal_tac "b * q' < b * (1 + q) ")  webertj@49962  1498  prefer 2 apply (simp add: right_diff_distrib distrib_left)  haftmann@33361  1499 apply (simp add: mult_less_cancel_left)  haftmann@33361  1500 done  haftmann@33361  1501 haftmann@33361  1502 lemma unique_quotient_lemma_neg:  haftmann@33361  1503  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1504  ==> q \ (q'::int)"  haftmann@33361  1505 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1506  auto)  haftmann@33361  1507 haftmann@33361  1508 lemma unique_quotient:  bulwahn@46552  1509  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1510  ==> q = q'"  haftmann@33361  1511 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1512 apply (blast intro: order_antisym  haftmann@33361  1513  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1514  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1515 done  haftmann@33361  1516 haftmann@33361  1517 haftmann@33361  1518 lemma unique_remainder:  bulwahn@46552  1519  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1520  ==> r = r'"  haftmann@33361  1521 apply (subgoal_tac "q = q'")  haftmann@33361  1522  apply (simp add: divmod_int_rel_def)  haftmann@33361  1523 apply (blast intro: unique_quotient)  haftmann@33361  1524 done  haftmann@33361  1525 haftmann@33361  1526 huffman@46551  1527 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}  haftmann@33361  1528 haftmann@33361  1529 text{*And positive divisors*}  haftmann@33361  1530 haftmann@33361  1531 lemma adjust_eq [simp]:  huffman@47108  1532  "adjust b (q, r) =  huffman@47108  1533  (let diff = r - b in  huffman@47108  1534  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1535  else (2*q, r))"  huffman@47108  1536  by (simp add: Let_def adjust_def)  haftmann@33361  1537 haftmann@33361  1538 declare posDivAlg.simps [simp del]  haftmann@33361  1539 haftmann@33361  1540 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1541 lemma posDivAlg_eqn:  haftmann@33361  1542  "0 < b ==>  haftmann@33361  1543  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1549  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1550  using assms  wenzelm@41550  1551  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1552  apply auto  wenzelm@41550  1553  apply (simp add: divmod_int_rel_def)  webertj@49962  1554  apply (subst posDivAlg_eqn, simp add: distrib_left)  wenzelm@41550  1555  apply (case_tac "a < b")  wenzelm@41550  1556  apply simp_all  wenzelm@41550  1557  apply (erule splitE)  webertj@49962  1558  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1559  done  haftmann@33361  1560 haftmann@33361  1561 huffman@46551  1562 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}  haftmann@33361  1563 haftmann@33361  1564 text{*And positive divisors*}  haftmann@33361  1565 haftmann@33361  1566 declare negDivAlg.simps [simp del]  haftmann@33361  1567 haftmann@33361  1568 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1569 lemma negDivAlg_eqn:  haftmann@33361  1570  "0 < b ==>  haftmann@33361  1571  negDivAlg a b =  haftmann@33361  1572  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1573 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1574 haftmann@33361  1575 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1576  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1577 lemma negDivAlg_correct:  haftmann@33361  1578  assumes "a < 0" and "b > 0"  haftmann@33361  1579  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1580  using assms  wenzelm@41550  1581  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1582  apply (auto simp add: linorder_not_le)  wenzelm@41550  1583  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1584  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1585  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1586  apply simp_all  wenzelm@41550  1587  apply (erule splitE)  webertj@49962  1588  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  wenzelm@41550  1589  done  haftmann@33361  1590 haftmann@33361  1591 huffman@46551  1592 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}  haftmann@33361  1593 haftmann@33361  1594 (*the case a=0*)  huffman@47139  1595 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1596 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1597 haftmann@33361  1598 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1599 by (subst posDivAlg.simps, auto)  haftmann@33361  1600 huffman@47139  1601 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1602 by (subst posDivAlg.simps, auto)  huffman@47139  1603 haftmann@33361  1604 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  haftmann@33361  1605 by (subst negDivAlg.simps, auto)  haftmann@33361  1606 huffman@46560  1607 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1608 by (auto simp add: divmod_int_rel_def)  huffman@47139  1609 huffman@47139  1610 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1611 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1612 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1613  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1614 huffman@47141  1615 lemma divmod_int_unique:  huffman@47141  1616  assumes "divmod_int_rel a b qr"  huffman@47141  1617  shows "divmod_int a b = qr"  huffman@47141  1618  using assms divmod_int_correct [of a b]  huffman@47141  1619  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1620  by (metis pair_collapse)  huffman@47141  1621 huffman@47141  1622 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1623  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1624 huffman@47141  1625 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1626  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1627 huffman@47141  1628 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1629  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1630 huffman@47141  1631 instance int :: ring_div  huffman@47141  1632 proof  huffman@47141  1633  fix a b :: int  huffman@47141  1634  show "a div b * b + a mod b = a"  huffman@47141  1635  using divmod_int_rel_div_mod [of a b]  huffman@47141  1636  unfolding divmod_int_rel_def by (simp add: mult_commute)  huffman@47141  1637 next  huffman@47141  1638  fix a b c :: int  huffman@47141  1639  assume "b \ 0"  huffman@47141  1640  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1641  using divmod_int_rel_div_mod [of a b]  huffman@47141  1642  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1643  thus "(a + c * b) div b = c + a div b"  huffman@47141  1644  by (rule div_int_unique)  huffman@47141  1645 next  huffman@47141  1646  fix a b c :: int  huffman@47141  1647  assume "c \ 0"  huffman@47141  1648  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1649  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1650  unfolding divmod_int_rel_def  huffman@47141  1651  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1652  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1653  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1654  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1655  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1656  thus "(c * a) div (c * b) = a div b"  huffman@47141  1657  by (rule div_int_unique)  huffman@47141  1658 next  huffman@47141  1659  fix a :: int show "a div 0 = 0"  huffman@47141  1660  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1661 next  huffman@47141  1662  fix a :: int show "0 div a = 0"  huffman@47141  1663  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1664 qed  huffman@47141  1665 haftmann@33361  1666 text{*Basic laws about division and remainder*}  haftmann@33361  1667 haftmann@33361  1668 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1669  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1670 haftmann@33361  1671 text {* Tool setup *}  haftmann@33361  1672 huffman@47108  1673 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)  huffman@47108  1674 lemmas add_0s = add_0_left add_0_right  huffman@47108  1675 haftmann@33361  1676 ML {*  wenzelm@43594  1677 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1678 (  haftmann@33361  1679  val div_name = @{const_name div};  haftmann@33361  1680  val mod_name = @{const_name mod};  haftmann@33361  1681  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1682  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1683  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1684 huffman@47165  1685  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1686 haftmann@33361  1687  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@33361  1688  (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))  wenzelm@41550  1689 )  haftmann@33361  1690 *}  haftmann@33361  1691 wenzelm@43594  1692 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1693 huffman@47141  1694 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1695  using divmod_int_correct [of a b]  huffman@47141  1696  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1697 wenzelm@45607  1698 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1699  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1700 huffman@47141  1701 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1702  using divmod_int_correct [of a b]  huffman@47141  1703  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1704 wenzelm@45607  1705 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1706  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1707 haftmann@33361  1708 huffman@46551  1709 subsubsection {* General Properties of div and mod *}  haftmann@33361  1710 haftmann@33361  1711 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1712 apply (rule div_int_unique)  haftmann@33361  1713 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1714 done  haftmann@33361  1715 haftmann@33361  1716 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1717 apply (rule div_int_unique)  haftmann@33361  1718 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1719 done  haftmann@33361  1720 haftmann@33361  1721 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1722 apply (rule div_int_unique)  haftmann@33361  1723 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1724 done  haftmann@33361  1725 haftmann@33361  1726 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1727 haftmann@33361  1728 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1729 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1730 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1731 done  haftmann@33361  1732 haftmann@33361  1733 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  1734 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1735 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1736 done  haftmann@33361  1737 haftmann@33361  1738 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  1739 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  1740 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1741 done  haftmann@33361  1742 haftmann@33361  1743 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  1744 haftmann@33361  1745 huffman@46551  1746 subsubsection {* Laws for div and mod with Unary Minus *}  haftmann@33361  1747 haftmann@33361  1748 lemma zminus1_lemma:  huffman@47139  1749  "divmod_int_rel a b (q, r) ==> b \ 0  haftmann@33361  1750  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1751  if r=0 then 0 else b-r)"  haftmann@33361  1752 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1753 haftmann@33361  1754 haftmann@33361  1755 lemma zdiv_zminus1_eq_if:  haftmann@33361  1756  "b \ (0::int)  haftmann@33361  1757  ==> (-a) div b =  haftmann@33361  1758  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  1759 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  1760 haftmann@33361  1761 lemma zmod_zminus1_eq_if:  haftmann@33361  1762  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1763 apply (case_tac "b = 0", simp)  huffman@47140  1764 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  1765 done  haftmann@33361  1766 haftmann@33361  1767 lemma zmod_zminus1_not_zero:  haftmann@33361  1768  fixes k l :: int  haftmann@33361  1769  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  1770  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  1771 haftmann@33361  1772 lemma zdiv_zminus2_eq_if:  haftmann@33361  1773  "b \ (0::int)  haftmann@33361  1774  ==> a div (-b) =  haftmann@33361  1775  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  1776 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  1777 haftmann@33361  1778 lemma zmod_zminus2_eq_if:  haftmann@33361  1779  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  1780 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  1781 haftmann@33361  1782 lemma zmod_zminus2_not_zero:  haftmann@33361  1783  fixes k l :: int  haftmann@33361  1784  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  1785  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  1786 haftmann@33361  1787 huffman@46551  1788 subsubsection {* Computation of Division and Remainder *}  haftmann@33361  1789 haftmann@33361  1790 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  1791 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1792 haftmann@33361  1793 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  1794 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1795 haftmann@33361  1796 text{*a positive, b positive *}  haftmann@33361  1797 haftmann@33361  1798 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  1799 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1800 haftmann@33361  1801 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  1802 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1803 haftmann@33361  1804 text{*a negative, b positive *}  haftmann@33361  1805 haftmann@33361  1806 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  1807 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1808 haftmann@33361  1809 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  1810 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1811 haftmann@33361  1812 text{*a positive, b negative *}  haftmann@33361  1813 haftmann@33361  1814 lemma div_pos_neg:  huffman@46560  1815  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1816 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1817 haftmann@33361  1818 lemma mod_pos_neg:  huffman@46560  1819  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1820 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1821 haftmann@33361  1822 text{*a negative, b negative *}  haftmann@33361  1823 haftmann@33361  1824 lemma div_neg_neg:  huffman@46560  1825  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1826 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1827 haftmann@33361  1828 lemma mod_neg_neg:  huffman@46560  1829  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1830 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1831 haftmann@33361  1832 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  1833 huffman@45530  1834 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  1835  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  1836 huffman@45530  1837 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  1838  by (rule div_int_unique [of a b q r],  bulwahn@46552  1839  simp add: divmod_int_rel_def)  huffman@45530  1840 huffman@45530  1841 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  1842  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1843  simp add: divmod_int_rel_def)  huffman@45530  1844 huffman@45530  1845 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  1846  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1847  simp add: divmod_int_rel_def)  huffman@45530  1848 haftmann@33361  1849 (* simprocs adapted from HOL/ex/Binary.thy *)  haftmann@33361  1850 ML {*  haftmann@33361  1851 local  huffman@45530  1852  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  1853  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  1854  val times = @{term "times :: int \ int \ int"}  huffman@45530  1855  val zero = @{term "0 :: int"}  huffman@45530  1856  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  1857  val le = @{term "op \ :: int \ int \ bool"}  huffman@45530  1858  val simps = @{thms arith_simps} @ @{thms rel_simps} @  huffman@47108  1859  map (fn th => th RS sym) [@{thm numeral_1_eq_1}]  huffman@45530  1860  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  wenzelm@51717  1861  (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));  wenzelm@51717  1862  fun binary_proc proc ctxt ct =  haftmann@33361  1863  (case Thm.term_of ct of  haftmann@33361  1864  _ $t$ u =>  haftmann@33361  1865  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  wenzelm@51717  1866  SOME args => proc ctxt args  haftmann@33361  1867  | NONE => NONE)  haftmann@33361  1868  | _ => NONE);  haftmann@33361  1869 in  huffman@45530  1870  fun divmod_proc posrule negrule =  huffman@45530  1871  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  huffman@45530  1872  if b = 0 then NONE else let  huffman@45530  1873  val (q, r) = pairself mk_number (Integer.div_mod a b)  huffman@45530  1874  val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)  huffman@45530  1875  val (goal2, goal3, rule) = if b > 0  huffman@45530  1876  then (le $zero$ r, less $r$ u, posrule RS eq_reflection)  huffman@45530  1877  else (le $r$ zero, less $u$ r, negrule RS eq_reflection)  huffman@45530  1878  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  1879 end  haftmann@33361  1880 *}  haftmann@33361  1881 huffman@47108  1882 simproc_setup binary_int_div  huffman@47108  1883  ("numeral m div numeral n :: int" |  huffman@47108  1884  "numeral m div neg_numeral n :: int" |  huffman@47108  1885  "neg_numeral m div numeral n :: int" |  huffman@47108  1886  "neg_numeral m div neg_numeral n :: int") =  huffman@45530  1887  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  1888 huffman@47108  1889 simproc_setup binary_int_mod  huffman@47108  1890  ("numeral m mod numeral n :: int" |  huffman@47108  1891  "numeral m mod neg_numeral n :: int" |  huffman@47108  1892  "neg_numeral m mod numeral n :: int" |  huffman@47108  1893  "neg_numeral m mod neg_numeral n :: int") =  huffman@45530  1894  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  1895 huffman@47108  1896 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  1897  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  1898 huffman@47108  1899 lemmas negDivAlg_eqn_numeral [simp] =  huffman@47108  1900  negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w  haftmann@33361  1901 haftmann@33361  1902 haftmann@33361  1903 text{*Special-case simplification *}  haftmann@33361  1904 haftmann@33361  1905 (** The last remaining special cases for constant arithmetic:  haftmann@33361  1906  1 div z and 1 mod z **)  haftmann@33361  1907 huffman@47108  1908 lemmas div_pos_pos_1_numeral [simp] =  huffman@47108  1909  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1910 huffman@47108  1911 lemmas div_pos_neg_1_numeral [simp] =  huffman@47108  1912  div_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1913  OF neg_numeral_less_zero] for w  huffman@47108  1914 huffman@47108  1915 lemmas mod_pos_pos_1_numeral [simp] =  huffman@47108  1916  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1917 huffman@47108  1918 lemmas mod_pos_neg_1_numeral [simp] =  huffman@47108  1919  mod_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1920  OF neg_numeral_less_zero] for w  huffman@47108  1921 huffman@47108  1922 lemmas posDivAlg_eqn_1_numeral [simp] =  huffman@47108  1923  posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  huffman@47108  1924 huffman@47108  1925 lemmas negDivAlg_eqn_1_numeral [simp] =  huffman@47108  1926  negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  haftmann@33361  1927 haftmann@33361  1928 huffman@46551  1929 subsubsection {* Monotonicity in the First Argument (Dividend) *}  haftmann@33361  1930 haftmann@33361  1931 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  1932 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1933 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1934 apply (rule unique_quotient_lemma)  haftmann@33361  1935 apply (erule subst)  haftmann@33361  1936 apply (erule subst, simp_all)  haftmann@33361  1937 done  haftmann@33361  1938 haftmann@33361  1939 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  1940 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1941 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1942 apply (rule unique_quotient_lemma_neg)  haftmann@33361  1943 apply (erule subst)  haftmann@33361  1944 apply (erule subst, simp_all)  haftmann@33361  1945 done  haftmann@33361  1946 haftmann@33361  1947 huffman@46551  1948 subsubsection {* Monotonicity in the Second Argument (Divisor) *}  haftmann@33361  1949 haftmann@33361  1950 lemma q_pos_lemma:  haftmann@33361  1951  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  1952 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  1953  apply (simp add: zero_less_mult_iff)  webertj@49962  1954 apply (simp add: distrib_left)  haftmann@33361  1955 done  haftmann@33361  1956 haftmann@33361  1957 lemma zdiv_mono2_lemma:  haftmann@33361  1958  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  1959  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  1960  ==> q \ (q'::int)"  haftmann@33361  1961 apply (frule q_pos_lemma, assumption+)  haftmann@33361  1962 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  1963  apply (simp add: mult_less_cancel_left)  haftmann@33361  1964 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  1965  prefer 2 apply simp  webertj@49962  1966 apply (simp (no_asm_simp) add: distrib_left)  huffman@44766  1967 apply (subst add_commute, rule add_less_le_mono, arith)  haftmann@33361  1968 apply (rule mult_right_mono, auto)  haftmann@33361  1969 done  haftmann@33361  1970 haftmann@33361  1971 lemma zdiv_mono2:  haftmann@33361  1972  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  1973 apply (subgoal_tac "b \ 0")  haftmann@33361  1974  prefer 2 apply arith  haftmann@33361  1975 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1976 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  1977 apply (rule zdiv_mono2_lemma)  haftmann@33361  1978 apply (erule subst)  haftmann@33361  1979 apply (erule subst, simp_all)  haftmann@33361  1980 done  haftmann@33361  1981 haftmann@33361  1982 lemma q_neg_lemma:  haftmann@33361  1983  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  1984 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  1985  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  1986 done  haftmann@33361  1987 haftmann@33361  1988 lemma zdiv_mono2_neg_lemma:  haftmann@33361  1989  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  1990  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  1991  ==> q' \ (q::int)"  haftmann@33361  1992 apply (frule q_neg_lemma, assumption+)  haftmann@33361  1993 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  1994  apply (simp add: mult_less_cancel_left)  webertj@49962  1995 apply (simp add: distrib_left)  haftmann@33361  1996 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  1997  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  1998 done  haftmann@33361  1999 haftmann@33361  2000 lemma zdiv_mono2_neg:  haftmann@33361  2001  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  2002 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  2003 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  2004 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  2005 apply (erule subst)  haftmann@33361  2006 apply (erule subst, simp_all)  haftmann@33361  2007 done  haftmann@33361  2008 haftmann@33361  2009 huffman@46551  2010 subsubsection {* More Algebraic Laws for div and mod *}  haftmann@33361  2011 haftmann@33361  2012 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  2013 haftmann@33361  2014 lemma zmult1_lemma:  bulwahn@46552  2015  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  2016  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  webertj@49962  2017 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)  haftmann@33361  2018 haftmann@33361  2019 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  2020 apply (case_tac "c = 0", simp)  huffman@47140  2021 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  2022 done  haftmann@33361  2023 haftmann@33361  2024 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  2025 haftmann@33361  2026 lemma zadd1_lemma:  bulwahn@46552  2027  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  2028  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  webertj@49962  2029 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)  haftmann@33361  2030 haftmann@33361  2031 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  2032 lemma zdiv_zadd1_eq:  haftmann@33361  2033  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  2034 apply (case_tac "c = 0", simp)  huffman@47140  2035 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  2036 done  haftmann@33361  2037 haftmann@33361  2038 lemma posDivAlg_div_mod:  haftmann@33361  2039  assumes "k \ 0"  haftmann@33361  2040  and "l \ 0"  haftmann@33361  2041  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  2042 proof (cases "l = 0")  haftmann@33361  2043  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  2044 next  haftmann@33361  2045  case False with assms posDivAlg_correct  haftmann@33361  2046  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  2047  by simp  huffman@47140  2048  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2049  show ?thesis by simp  haftmann@33361  2050 qed  haftmann@33361  2051 haftmann@33361  2052 lemma negDivAlg_div_mod:  haftmann@33361  2053  assumes "k < 0"  haftmann@33361  2054  and "l > 0"  haftmann@33361  2055  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  2056 proof -  haftmann@33361  2057  from assms have "l \ 0" by simp  haftmann@33361  2058  from assms negDivAlg_correct  haftmann@33361  2059  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  2060  by simp  huffman@47140  2061  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  2062  show ?thesis by simp  haftmann@33361  2063 qed  haftmann@33361  2064 haftmann@33361  2065 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  2066 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  2067 haftmann@33361  2068 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  2069 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  2070 huffman@47108  2071 lemma zmod_zdiv_equality':  huffman@47108  2072  "(m\int) mod n = m - (m div n) * n"  huffman@47141  2073  using mod_div_equality [of m n] by arith  huffman@47108  2074 haftmann@33361  2075 huffman@46551  2076 subsubsection {* Proving @{term "a div (b*c) = (a div b) div c"} *}  haftmann@33361  2077 haftmann@33361  2078 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  2079  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  2080  to cause particular problems.*)  haftmann@33361  2081 haftmann@33361  2082 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  2083 haftmann@33361  2084 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  haftmann@33361  2085 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  2086  apply (simp add: algebra_simps)  haftmann@33361  2087 apply (rule order_le_less_trans)  haftmann@33361  2088  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  2089  apply (rule mult_left_mono_neg)  huffman@35216  2090  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  2091  apply (simp)  haftmann@33361  2092 apply (simp)  haftmann@33361  2093 done  haftmann@33361  2094 haftmann@33361  2095 lemma zmult2_lemma_aux2:  haftmann@33361  2096  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  2097 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  2098  apply arith  haftmann@33361  2099 apply (simp add: mult_le_0_iff)  haftmann@33361  2100 done  haftmann@33361  2101 haftmann@33361  2102 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  2103 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  2104 apply arith  haftmann@33361  2105 apply (simp add: zero_le_mult_iff)  haftmann@33361  2106 done  haftmann@33361  2107 haftmann@33361  2108 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  2109 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  2110  apply (simp add: right_diff_distrib)  haftmann@33361  2111 apply (rule order_less_le_trans)  haftmann@33361  2112  apply (erule mult_strict_right_mono)  haftmann@33361  2113  apply (rule_tac [2] mult_left_mono)  haftmann@33361  2114  apply simp  huffman@35216  2115  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  2116 apply simp  haftmann@33361  2117 done  haftmann@33361  2118 bulwahn@46552  2119 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  2120  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@33361  2121 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff  webertj@49962  2122  zero_less_mult_iff distrib_left [symmetric]  huffman@47139  2123  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  2124 haftmann@53068  2125 lemma zdiv_zmult2_eq:  haftmann@53068  2126  fixes a b c :: int  haftmann@53068  2127  shows "0 \ c \ a div (b * c) = (a div b) div c"  haftmann@33361  2128 apply (case_tac "b = 0", simp)  haftmann@53068  2129 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  2130 done  haftmann@33361  2131 haftmann@33361  2132 lemma zmod_zmult2_eq:  haftmann@53068  2133  fixes a b c :: int  haftmann@53068  2134  shows "0 \ c \ a mod (b * c) = b * (a div b mod c) + a mod b"  haftmann@33361  2135 apply (case_tac "b = 0", simp)  haftmann@53068  2136 apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  2137 done  haftmann@33361  2138 huffman@47108  2139 lemma div_pos_geq:  huffman@47108  2140  fixes k l :: int  huffman@47108  2141  assumes "0 < l" and "l \ k"  huffman@47108  2142  shows "k div l = (k - l) div l + 1"  huffman@47108  2143 proof -  huffman@47108  2144  have "k = (k - l) + l" by simp  huffman@47108  2145  then obtain j where k: "k = j + l" ..  huffman@47108  2146  with assms show ?thesis by simp  huffman@47108  2147 qed  huffman@47108  2148 huffman@47108  2149 lemma mod_pos_geq:  huffman@47108  2150  fixes k l :: int  huffman@47108  2151  assumes "0 < l" and "l \ k"  huffman@47108  2152  shows "k mod l = (k - l) mod l"  huffman@47108  2153 proof -  huffman@47108  2154  have "k = (k - l) + l" by simp  huffman@47108  2155  then obtain j where k: "k = j + l" ..  huffman@47108  2156  with assms show ?thesis by simp  huffman@47108  2157 qed  huffman@47108  2158 haftmann@33361  2159 huffman@46551  2160 subsubsection {* Splitting Rules for div and mod *}  haftmann@33361  2161 haftmann@33361  2162 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  2163 haftmann@33361  2164 lemma split_pos_lemma:  haftmann@33361  2165  "0  haftmann@33361  2166  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2167 apply (rule iffI, clarify)  haftmann@33361  2168  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2169  apply (subst mod_add_eq)  haftmann@33361  2170  apply (subst zdiv_zadd1_eq)  haftmann@33361  2171  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  2172 txt{*converse direction*}  haftmann@33361  2173 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2174 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2175 done  haftmann@33361  2176 haftmann@33361  2177 lemma split_neg_lemma:  haftmann@33361  2178  "k<0 ==>  haftmann@33361  2179  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2180 apply (rule iffI, clarify)  haftmann@33361  2181  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2182  apply (subst mod_add_eq)  haftmann@33361  2183  apply (subst zdiv_zadd1_eq)  haftmann@33361  2184  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  2185 txt{*converse direction*}  haftmann@33361  2186 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2187 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2188 done  haftmann@33361  2189 haftmann@33361  2190 lemma split_zdiv:  haftmann@33361  2191  "P(n div k :: int) =  haftmann@33361  2192  ((k = 0 --> P 0) &  haftmann@33361  2193  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2194  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2195 apply (case_tac "k=0", simp)  haftmann@33361  2196 apply (simp only: linorder_neq_iff)  haftmann@33361  2197 apply (erule disjE)  haftmann@33361  2198  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2199  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2200 done  haftmann@33361  2201 haftmann@33361  2202 lemma split_zmod:  haftmann@33361  2203  "P(n mod k :: int) =  haftmann@33361  2204  ((k = 0 --> P n) &  haftmann@33361  2205  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2206  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2207 apply (case_tac "k=0", simp)  haftmann@33361  2208 apply (simp only: linorder_neq_iff)  haftmann@33361  2209 apply (erule disjE)  haftmann@33361  2210  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2211  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2212 done  haftmann@33361  2213 webertj@33730  2214 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  2215  when these are applied to some constant that is of the form  huffman@47108  2216  @{term "numeral k"}: *}  huffman@47108  2217 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  2218 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  2219 haftmann@33361  2220 huffman@47166  2221 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}  huffman@47166  2222 huffman@47166  2223 lemma pos_divmod_int_rel_mult_2:  huffman@47166  2224  assumes "0 \ b"  huffman@47166  2225  assumes "divmod_int_rel a b (q, r)"  huffman@47166  2226  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  2227  using assms unfolding divmod_int_rel_def by auto  huffman@47166  2228 huffman@47166  2229 lemma neg_divmod_int_rel_mult_2:  huffman@47166  2230  assumes "b \ 0"  huffman@47166  2231  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  2232  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  2233  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  2234 haftmann@33361  2235 text{*computing div by shifting *}  haftmann@33361  2236 haftmann@33361  2237 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2238  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2239  by (rule div_int_unique)  haftmann@33361  2240 boehmes@35815  2241 lemma neg_zdiv_mult_2:  boehmes@35815  2242  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2243  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2244  by (rule div_int_unique)  haftmann@33361  2245 huffman@47108  2246 (* FIXME: add rules for negative numerals *)  huffman@47108  2247 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2248  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2249  numeral v div (numeral w :: int)"  huffman@47108  2250  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2251  by (rule div_mult_mult1, simp)  huffman@47108  2252 huffman@47108  2253 lemma zdiv_numeral_Bit1 [simp]:  huffman@47108  2254  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  huffman@47108  2255  (numeral v div (numeral w :: int))"  huffman@47108  2256  unfolding numeral.simps  huffman@47108  2257  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2258  by (rule pos_zdiv_mult_2, simp)  haftmann@33361  2259 haftmann@33361  2260 lemma pos_zmod_mult_2:  haftmann@33361  2261  fixes a b :: int  haftmann@33361  2262  assumes "0 \ a"  haftmann@33361  2263  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  huffman@47166  2264  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2265  by (rule mod_int_unique)  haftmann@33361  2266 haftmann@33361  2267 lemma neg_zmod_mult_2:  haftmann@33361  2268  fixes a b :: int  haftmann@33361  2269  assumes "a \ 0"  haftmann@33361  2270  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  huffman@47166  2271  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2272  by (rule mod_int_unique)  haftmann@33361  2273 huffman@47108  2274 (* FIXME: add rules for negative numerals *)  huffman@47108  2275 lemma zmod_numeral_Bit0 [simp]:  huffman@47108  2276  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  huffman@47108  2277  (2::int) * (numeral v mod numeral w)"  huffman@47108  2278  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  huffman@47108  2279  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)  huffman@47108  2280 huffman@47108  2281 lemma zmod_numeral_Bit1 [simp]:  huffman@47108  2282  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =  huffman@47108  2283  2 * (numeral v mod numeral w) + (1::int)"  huffman@47108  2284  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]  huffman@47108  2285  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2286  by (rule pos_zmod_mult_2, simp)  haftmann@33361  2287 nipkow@39489  2288 lemma zdiv_eq_0_iff:  nipkow@39489  2289  "(i::int) div k = 0 \ k=0 \ 0\i \ i i\0 \ k ?R" by (rule split_zdiv[THEN iffD2]) simp  nipkow@39489  2293  with ?L show ?R by blast  nipkow@39489  2294 next  nipkow@39489  2295  assume ?R thus ?L  nipkow@39489  2296  by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)  nipkow@39489  2297 qed  nipkow@39489  2298 nipkow@39489  2299 huffman@46551  2300 subsubsection {* Quotients of Signs *}  haftmann@33361  2301 haftmann@33361  2302 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  haftmann@33361  2303 apply (subgoal_tac "a div b \ -1", force)  haftmann@33361  2304 apply (rule order_trans)  haftmann@33361  2305 apply (rule_tac a' = "-1" in zdiv_mono1)  haftmann@33361  2306 apply (auto simp add: div_eq_minus1)  haftmann@33361  2307 done  haftmann@33361  2308 haftmann@33361  2309 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  haftmann@33361  2310 by (drule zdiv_mono1_neg, auto)  haftmann@33361  2311 haftmann@33361  2312 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  haftmann@33361  2313 by (drule zdiv_mono1, auto)  haftmann@33361  2314 nipkow@33804  2315 text{* Now for some equivalences of the form @{text"a div b >=< 0 \ \"}  nipkow@33804  2316 conditional upon the sign of @{text a} or @{text b}. There are many more.  nipkow@33804  2317 They should all be simp rules unless that causes too much search. *}  nipkow@33804  2318 haftmann@33361  2319 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  haftmann@33361  2320 apply auto  haftmann@33361  2321 apply (drule_tac [2] zdiv_mono1)  haftmann@33361  2322 apply (auto simp add: linorder_neq_iff)  haftmann@33361  2323 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  haftmann@33361  2324 apply (blast intro: div_neg_pos_less0)  haftmann@33361  2325 done  haftmann@33361  2326 haftmann@33361  2327 lemma neg_imp_zdiv_nonneg_iff:  nipkow@33804  2328  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  huffman@47159  2329 apply (subst div_minus_minus [symmetric])  haftmann@33361  2330 apply (subst pos_imp_zdiv_nonneg_iff, auto)  haftmann@33361  2331 done  haftmann@33361  2332 haftmann@33361  2333 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  haftmann@33361  2334 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  haftmann@33361  2335 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  haftmann@33361  2336 nipkow@39489  2337 lemma pos_imp_zdiv_pos_iff:  nipkow@39489  2338  "0 0 < (i::int) div k \ k \ i"  nipkow@39489  2339 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]  nipkow@39489  2340 by arith  nipkow@39489  2341 haftmann@33361  2342 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  haftmann@33361  2343 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  haftmann@33361  2344 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  haftmann@33361  2345 nipkow@33804  2346 lemma nonneg1_imp_zdiv_pos_iff:  nipkow@33804  2347  "(0::int) <= a \ (a div b > 0) = (a >= b & b>0)"  nipkow@33804  2348 apply rule  nipkow@33804  2349  apply rule  nipkow@33804  2350  using div_pos_pos_trivial[of a b]apply arith  nipkow@33804  2351  apply(cases "b=0")apply simp  nipkow@33804  2352  using div_nonneg_neg_le0[of a b]apply arith  nipkow@33804  2353 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp  nipkow@33804  2354 done  nipkow@33804  2355 nipkow@39489  2356 lemma zmod_le_nonneg_dividend: "(m::int) \ 0 ==> m mod k \ m"  nipkow@39489  2357 apply (rule split_zmod[THEN iffD2])  nipkow@44890  2358 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)  nipkow@39489  2359 done  nipkow@39489  2360 nipkow@39489  2361 haftmann@33361  2362 subsubsection {* The Divides Relation *}  haftmann@33361  2363 huffman@47268  2364 lemma dvd_neg_numeral_left [simp]:  huffman@47268  2365  fixes y :: "'a::comm_ring_1"  huffman@47268  2366  shows "(neg_numeral k) dvd y \ (numeral k) dvd y"  huffman@47268  2367  unfolding neg_numeral_def minus_dvd_iff ..  huffman@47268  2368 huffman@47268  2369 lemma dvd_neg_numeral_right [simp]:  huffman@47268  2370  fixes x :: "'a::comm_ring_1"  huffman@47268  2371  shows "x dvd (neg_numeral k) \ x dvd (numeral k)"  huffman@47268  2372  unfolding neg_numeral_def dvd_minus_iff ..  haftmann@33361  2373 huffman@47108  2374 lemmas dvd_eq_mod_eq_0_numeral [simp] =  huffman@47108  2375  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y  huffman@47108  2376 huffman@47108  2377 huffman@47108  2378 subsubsection {* Further properties *}  huffman@47108  2379 haftmann@33361  2380 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  haftmann@33361  2381  using zmod_zdiv_equality[where a="m" and b="n"]  huffman@47142  2382  by (simp add: algebra_simps) (* FIXME: generalize *)  haftmann@33361  2383 haftmann@33361  2384 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  haftmann@33361  2385 apply (subst split_div, auto)  haftmann@33361  2386 apply (subst split_zdiv, auto)  haftmann@33361  2387 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  2388 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2389 done  haftmann@33361  2390 haftmann@33361  2391 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  haftmann@33361  2392 apply (subst split_mod, auto)  haftmann@33361  2393 apply (subst split_zmod, auto)  haftmann@33361  2394 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  haftmann@33361  2395  in unique_remainder)  haftmann@33361  2396 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2397 done  haftmann@33361  2398 haftmann@33361  2399 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  haftmann@33361  2400 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  haftmann@33361  2401 haftmann@33361  2402 text{*Suggested by Matthias Daum*}  haftmann@33361  2403 lemma int_power_div_base:  haftmann@33361  2404  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  haftmann@33361  2405 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  haftmann@33361  2406  apply (erule ssubst)  haftmann@33361  2407  apply (simp only: power_add)  haftmann@33361  2408  apply simp_all  haftmann@33361  2409 done  haftmann@33361  2410 haftmann@33361  2411 text {* by Brian Huffman *}  haftmann@33361  2412 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@33361  2413 by (rule mod_minus_eq [symmetric])  haftmann@33361  2414 haftmann@33361  2415 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@33361  2416 by (rule mod_diff_left_eq [symmetric])  haftmann@33361  2417 haftmann@33361  2418 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@33361  2419 by (rule mod_diff_right_eq [symmetric])  haftmann@33361  2420 haftmann@33361  2421 lemmas zmod_simps =  haftmann@33361  2422  mod_add_left_eq [symmetric]  haftmann@33361  2423  mod_add_right_eq [symmetric]  huffman@47142  2424  mod_mult_right_eq[symmetric]  haftmann@33361  2425  mod_mult_left_eq [symmetric]  huffman@47164  2426  power_mod  haftmann@33361  2427  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@33361  2428 haftmann@33361  2429 text {* Distributive laws for function @{text nat}. *}  haftmann@33361  2430 haftmann@33361  2431 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  haftmann@33361  2432 apply (rule linorder_cases [of y 0])  haftmann@33361  2433 apply (simp add: div_nonneg_neg_le0)  haftmann@33361  2434 apply simp  haftmann@33361  2435 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  haftmann@33361  2436 done  haftmann@33361  2437 haftmann@33361  2438 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  haftmann@33361  2439 lemma nat_mod_distrib:  haftmann@33361  2440  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  haftmann@33361  2441 apply (case_tac "y = 0", simp)  haftmann@33361  2442 apply (simp add: nat_eq_iff zmod_int)  haftmann@33361  2443 done  haftmann@33361  2444 haftmann@33361  2445 text {* transfer setup *}  haftmann@33361  2446 haftmann@33361  2447 lemma transfer_nat_int_functions:  haftmann@33361  2448  "(x::int) >= 0 \ y >= 0 \ (nat x) div (nat y) = nat (x div y)"  haftmann@33361  2449  "(x::int) >= 0 \ y >= 0 \ (nat x) mod (nat y) = nat (x mod y)"  haftmann@33361  2450  by (auto simp add: nat_div_distrib nat_mod_distrib)  haftmann@33361  2451 haftmann@33361  2452 lemma transfer_nat_int_function_closures:  haftmann@33361  2453  "(x::int) >= 0 \ y >= 0 \ x div y >= 0"  haftmann@33361  2454  "(x::int) >= 0 \ y >= 0 \ x mod y >= 0"  haftmann@33361  2455  apply (cases "y = 0")  haftmann@33361  2456  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  haftmann@33361  2457  apply (cases "y = 0")  haftmann@33361  2458  apply auto  haftmann@33361  2459 done  haftmann@33361  2460 haftmann@35644  2461 declare transfer_morphism_nat_int [transfer add return:  haftmann@33361  2462  transfer_nat_int_functions  haftmann@33361  2463  transfer_nat_int_function_closures  haftmann@33361  2464 ]  haftmann@33361  2465 haftmann@33361  2466 lemma transfer_int_nat_functions:  haftmann@33361  2467  "(int x) div (int y) = int (x div y)"  haftmann@33361  2468  "(int x) mod (int y) = int (x mod y)"  haftmann@33361  2469  by (auto simp add: zdiv_int zmod_int)  haftmann@33361  2470 haftmann@33361  2471 lemma transfer_int_nat_function_closures:  haftmann@33361  2472  "is_nat x \ is_nat y \ is_nat (x div y)"  haftmann@33361  2473  "is_nat x \ is_nat y \ is_nat (x mod y)"  haftmann@33361  2474  by (simp_all only: is_nat_def transfer_nat_int_function_closures)  haftmann@33361  2475 haftmann@35644  2476 declare transfer_morphism_int_nat [transfer add return:  haftmann@33361  2477  transfer_int_nat_functions  haftmann@33361  2478  transfer_int_nat_function_closures  haftmann@33361  2479 ]  haftmann@33361  2480 haftmann@33361  2481 text{*Suggested by Matthias Daum*}  haftmann@33361  2482 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  haftmann@33361  2483 apply (subgoal_tac "nat x div nat k < nat x")  nipkow@34225  2484  apply (simp add: nat_div_distrib [symmetric])  haftmann@33361  2485 apply (rule Divides.div_less_dividend, simp_all)  haftmann@33361  2486 done  haftmann@33361  2487 haftmann@33361  2488 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  haftmann@33361  2489 proof  haftmann@33361  2490  assume H: "x mod n = y mod n"  haftmann@33361  2491  hence "x mod n - y mod n = 0" by simp  haftmann@33361  2492  hence "(x mod n - y mod n) mod n = 0" by simp  haftmann@33361  2493  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])  haftmann@33361  2494  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)  haftmann@33361  2495 next  haftmann@33361  2496  assume H: "n dvd x - y"  haftmann@33361  2497  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  haftmann@33361  2498  hence "x = n*k + y" by simp  haftmann@33361  2499  hence "x mod n = (n*k + y) mod n" by simp  haftmann@33361  2500  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)  haftmann@33361  2501 qed  haftmann@33361  2502 haftmann@33361  2503 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  haftmann@33361  2504  shows "\q. x = y + n * q"  haftmann@33361  2505 proof-  haftmann@33361  2506  from xy have th: "int x - int y = int (x - y)" by simp  haftmann@33361  2507  from xyn have "int x mod int n = int y mod int n"  huffman@46551  2508  by (simp add: zmod_int [symmetric])  haftmann@33361  2509  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  haftmann@33361  2510  hence "n dvd x - y" by (simp add: th zdvd_int)  haftmann@33361  2511  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  haftmann@33361  2512 qed  haftmann@33361  2513 haftmann@33361  2514 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  haftmann@33361  2515  (is "?lhs = ?rhs")  haftmann@33361  2516 proof  haftmann@33361  2517  assume H: "x mod n = y mod n"  haftmann@33361  2518  {assume xy: "x \ y"  haftmann@33361  2519  from H have th: "y mod n = x mod n" by simp  haftmann@33361  2520  from nat_mod_eq_lemma[OF th xy] have ?rhs  haftmann@33361  2521  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  haftmann@33361  2522  moreover  haftmann@33361  2523  {assume xy: "y \ x" ` haftmann@33361