src/HOL/Divides.thy
 author huffman Fri Mar 30 11:16:35 2012 +0200 (2012-03-30) changeset 47217 501b9bbd0d6e parent 47167 099397de21e3 child 47255 30a1692557b0 permissions -rw-r--r--
removed redundant nat-specific copies of theorems
 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  haftmann@33318  9 imports Nat_Numeral Nat_Transfer  haftmann@33340  10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"  nipkow@15131  11 begin  paulson@3366  12 haftmann@25942  13 subsection {* Syntactic division operations *}  haftmann@25942  14 haftmann@27651  15 class div = dvd +  haftmann@27540  16  fixes div :: "'a \ 'a \ 'a" (infixl "div" 70)  haftmann@27651  17  and mod :: "'a \ 'a \ 'a" (infixl "mod" 70)  haftmann@27540  18 haftmann@27540  19 haftmann@27651  20 subsection {* Abstract division in commutative semirings. *}  haftmann@25942  21 haftmann@30930  22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +  haftmann@25942  23  assumes mod_div_equality: "a div b * b + a mod b = a"  haftmann@27651  24  and div_by_0 [simp]: "a div 0 = 0"  haftmann@27651  25  and div_0 [simp]: "0 div a = 0"  haftmann@27651  26  and div_mult_self1 [simp]: "b \ 0 \ (a + c * b) div b = c + a div b"  haftmann@30930  27  and div_mult_mult1 [simp]: "c \ 0 \ (c * a) div (c * b) = a div b"  haftmann@25942  28 begin  haftmann@25942  29 haftmann@26100  30 text {* @{const div} and @{const mod} *}  haftmann@26100  31 haftmann@26062  32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"  haftmann@26062  33  unfolding mult_commute [of b]  haftmann@26062  34  by (rule mod_div_equality)  haftmann@26062  35 huffman@29403  36 lemma mod_div_equality': "a mod b + a div b * b = a"  huffman@29403  37  using mod_div_equality [of a b]  huffman@29403  38  by (simp only: add_ac)  huffman@29403  39 haftmann@26062  40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"  haftmann@30934  41  by (simp add: mod_div_equality)  haftmann@26062  42 haftmann@26062  43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"  haftmann@30934  44  by (simp add: mod_div_equality2)  haftmann@26062  45 haftmann@27651  46 lemma mod_by_0 [simp]: "a mod 0 = a"  haftmann@30934  47  using mod_div_equality [of a zero] by simp  haftmann@27651  48 haftmann@27651  49 lemma mod_0 [simp]: "0 mod a = 0"  haftmann@30934  50  using mod_div_equality [of zero a] div_0 by simp  haftmann@27651  51 haftmann@27651  52 lemma div_mult_self2 [simp]:  haftmann@27651  53  assumes "b \ 0"  haftmann@27651  54  shows "(a + b * c) div b = c + a div b"  haftmann@27651  55  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)  haftmann@26100  56 haftmann@27651  57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"  haftmann@27651  58 proof (cases "b = 0")  haftmann@27651  59  case True then show ?thesis by simp  haftmann@27651  60 next  haftmann@27651  61  case False  haftmann@27651  62  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"  haftmann@27651  63  by (simp add: mod_div_equality)  haftmann@27651  64  also from False div_mult_self1 [of b a c] have  haftmann@27651  65  "\ = (c + a div b) * b + (a + c * b) mod b"  nipkow@29667  66  by (simp add: algebra_simps)  haftmann@27651  67  finally have "a = a div b * b + (a + c * b) mod b"  haftmann@27651  68  by (simp add: add_commute [of a] add_assoc left_distrib)  haftmann@27651  69  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"  haftmann@27651  70  by (simp add: mod_div_equality)  haftmann@27651  71  then show ?thesis by simp  haftmann@27651  72 qed  haftmann@27651  73 haftmann@27651  74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"  haftmann@30934  75  by (simp add: mult_commute [of b])  haftmann@27651  76 haftmann@27651  77 lemma div_mult_self1_is_id [simp]: "b \ 0 \ b * a div b = a"  haftmann@27651  78  using div_mult_self2 [of b 0 a] by simp  haftmann@27651  79 haftmann@27651  80 lemma div_mult_self2_is_id [simp]: "b \ 0 \ a * b div b = a"  haftmann@27651  81  using div_mult_self1 [of b 0 a] by simp  haftmann@27651  82 haftmann@27651  83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"  haftmann@27651  84  using mod_mult_self2 [of 0 b a] by simp  haftmann@27651  85 haftmann@27651  86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"  haftmann@27651  87  using mod_mult_self1 [of 0 a b] by simp  haftmann@26062  88 haftmann@27651  89 lemma div_by_1 [simp]: "a div 1 = a"  haftmann@27651  90  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp  haftmann@27651  91 haftmann@27651  92 lemma mod_by_1 [simp]: "a mod 1 = 0"  haftmann@27651  93 proof -  haftmann@27651  94  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp  haftmann@27651  95  then have "a + a mod 1 = a + 0" by simp  haftmann@27651  96  then show ?thesis by (rule add_left_imp_eq)  haftmann@27651  97 qed  haftmann@27651  98 haftmann@27651  99 lemma mod_self [simp]: "a mod a = 0"  haftmann@27651  100  using mod_mult_self2_is_0 [of 1] by simp  haftmann@27651  101 haftmann@27651  102 lemma div_self [simp]: "a \ 0 \ a div a = 1"  haftmann@27651  103  using div_mult_self2_is_id [of _ 1] by simp  haftmann@27651  104 haftmann@27676  105 lemma div_add_self1 [simp]:  haftmann@27651  106  assumes "b \ 0"  haftmann@27651  107  shows "(b + a) div b = a div b + 1"  haftmann@27651  108  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)  haftmann@26062  109 haftmann@27676  110 lemma div_add_self2 [simp]:  haftmann@27651  111  assumes "b \ 0"  haftmann@27651  112  shows "(a + b) div b = a div b + 1"  haftmann@27651  113  using assms div_add_self1 [of b a] by (simp add: add_commute)  haftmann@27651  114 haftmann@27676  115 lemma mod_add_self1 [simp]:  haftmann@27651  116  "(b + a) mod b = a mod b"  haftmann@27651  117  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)  haftmann@27651  118 haftmann@27676  119 lemma mod_add_self2 [simp]:  haftmann@27651  120  "(a + b) mod b = a mod b"  haftmann@27651  121  using mod_mult_self1 [of a 1 b] by simp  haftmann@27651  122 haftmann@27651  123 lemma mod_div_decomp:  haftmann@27651  124  fixes a b  haftmann@27651  125  obtains q r where "q = a div b" and "r = a mod b"  haftmann@27651  126  and "a = q * b + r"  haftmann@27651  127 proof -  haftmann@27651  128  from mod_div_equality have "a = a div b * b + a mod b" by simp  haftmann@27651  129  moreover have "a div b = a div b" ..  haftmann@27651  130  moreover have "a mod b = a mod b" ..  haftmann@27651  131  note that ultimately show thesis by blast  haftmann@27651  132 qed  haftmann@27651  133 bulwahn@45231  134 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \ b mod a = 0"  haftmann@25942  135 proof  haftmann@25942  136  assume "b mod a = 0"  haftmann@25942  137  with mod_div_equality [of b a] have "b div a * a = b" by simp  haftmann@25942  138  then have "b = a * (b div a)" unfolding mult_commute ..  haftmann@25942  139  then have "\c. b = a * c" ..  haftmann@25942  140  then show "a dvd b" unfolding dvd_def .  haftmann@25942  141 next  haftmann@25942  142  assume "a dvd b"  haftmann@25942  143  then have "\c. b = a * c" unfolding dvd_def .  haftmann@25942  144  then obtain c where "b = a * c" ..  haftmann@25942  145  then have "b mod a = a * c mod a" by simp  haftmann@25942  146  then have "b mod a = c * a mod a" by (simp add: mult_commute)  haftmann@27651  147  then show "b mod a = 0" by simp  haftmann@25942  148 qed  haftmann@25942  149 huffman@29403  150 lemma mod_div_trivial [simp]: "a mod b div b = 0"  huffman@29403  151 proof (cases "b = 0")  huffman@29403  152  assume "b = 0"  huffman@29403  153  thus ?thesis by simp  huffman@29403  154 next  huffman@29403  155  assume "b \ 0"  huffman@29403  156  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"  huffman@29403  157  by (rule div_mult_self1 [symmetric])  huffman@29403  158  also have "\ = a div b"  huffman@29403  159  by (simp only: mod_div_equality')  huffman@29403  160  also have "\ = a div b + 0"  huffman@29403  161  by simp  huffman@29403  162  finally show ?thesis  huffman@29403  163  by (rule add_left_imp_eq)  huffman@29403  164 qed  huffman@29403  165 huffman@29403  166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"  huffman@29403  167 proof -  huffman@29403  168  have "a mod b mod b = (a mod b + a div b * b) mod b"  huffman@29403  169  by (simp only: mod_mult_self1)  huffman@29403  170  also have "\ = a mod b"  huffman@29403  171  by (simp only: mod_div_equality')  huffman@29403  172  finally show ?thesis .  huffman@29403  173 qed  huffman@29403  174 nipkow@29925  175 lemma dvd_imp_mod_0: "a dvd b \ b mod a = 0"  nipkow@29948  176 by (rule dvd_eq_mod_eq_0[THEN iffD1])  nipkow@29925  177 nipkow@29925  178 lemma dvd_div_mult_self: "a dvd b \ (b div a) * a = b"  nipkow@29925  179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)  nipkow@29925  180 haftmann@33274  181 lemma dvd_mult_div_cancel: "a dvd b \ a * (b div a) = b"  haftmann@33274  182 by (drule dvd_div_mult_self) (simp add: mult_commute)  haftmann@33274  183 nipkow@30052  184 lemma dvd_div_mult: "a dvd b \ (b div a) * c = b * c div a"  nipkow@30052  185 apply (cases "a = 0")  nipkow@30052  186  apply simp  nipkow@30052  187 apply (auto simp: dvd_def mult_assoc)  nipkow@30052  188 done  nipkow@30052  189 nipkow@29925  190 lemma div_dvd_div[simp]:  nipkow@29925  191  "a dvd b \ a dvd c \ (b div a dvd c div a) = (b dvd c)"  nipkow@29925  192 apply (cases "a = 0")  nipkow@29925  193  apply simp  nipkow@29925  194 apply (unfold dvd_def)  nipkow@29925  195 apply auto  nipkow@29925  196  apply(blast intro:mult_assoc[symmetric])  nipkow@44890  197 apply(fastforce simp add: mult_assoc)  nipkow@29925  198 done  nipkow@29925  199 huffman@30078  200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m"  huffman@30078  201  apply (subgoal_tac "k dvd (m div n) *n + m mod n")  huffman@30078  202  apply (simp add: mod_div_equality)  huffman@30078  203  apply (simp only: dvd_add dvd_mult)  huffman@30078  204  done  huffman@30078  205 huffman@29403  206 text {* Addition respects modular equivalence. *}  huffman@29403  207 huffman@29403  208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"  huffman@29403  209 proof -  huffman@29403  210  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"  huffman@29403  211  by (simp only: mod_div_equality)  huffman@29403  212  also have "\ = (a mod c + b + a div c * c) mod c"  huffman@29403  213  by (simp only: add_ac)  huffman@29403  214  also have "\ = (a mod c + b) mod c"  huffman@29403  215  by (rule mod_mult_self1)  huffman@29403  216  finally show ?thesis .  huffman@29403  217 qed  huffman@29403  218 huffman@29403  219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"  huffman@29403  220 proof -  huffman@29403  221  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"  huffman@29403  222  by (simp only: mod_div_equality)  huffman@29403  223  also have "\ = (a + b mod c + b div c * c) mod c"  huffman@29403  224  by (simp only: add_ac)  huffman@29403  225  also have "\ = (a + b mod c) mod c"  huffman@29403  226  by (rule mod_mult_self1)  huffman@29403  227  finally show ?thesis .  huffman@29403  228 qed  huffman@29403  229 huffman@29403  230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"  huffman@29403  231 by (rule trans [OF mod_add_left_eq mod_add_right_eq])  huffman@29403  232 huffman@29403  233 lemma mod_add_cong:  huffman@29403  234  assumes "a mod c = a' mod c"  huffman@29403  235  assumes "b mod c = b' mod c"  huffman@29403  236  shows "(a + b) mod c = (a' + b') mod c"  huffman@29403  237 proof -  huffman@29403  238  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"  huffman@29403  239  unfolding assms ..  huffman@29403  240  thus ?thesis  huffman@29403  241  by (simp only: mod_add_eq [symmetric])  huffman@29403  242 qed  huffman@29403  243 haftmann@30923  244 lemma div_add [simp]: "z dvd x \ z dvd y  nipkow@30837  245  \ (x + y) div z = x div z + y div z"  haftmann@30923  246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)  nipkow@30837  247 huffman@29403  248 text {* Multiplication respects modular equivalence. *}  huffman@29403  249 huffman@29403  250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"  huffman@29403  251 proof -  huffman@29403  252  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"  huffman@29403  253  by (simp only: mod_div_equality)  huffman@29403  254  also have "\ = (a mod c * b + a div c * b * c) mod c"  nipkow@29667  255  by (simp only: algebra_simps)  huffman@29403  256  also have "\ = (a mod c * b) mod c"  huffman@29403  257  by (rule mod_mult_self1)  huffman@29403  258  finally show ?thesis .  huffman@29403  259 qed  huffman@29403  260 huffman@29403  261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"  huffman@29403  262 proof -  huffman@29403  263  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"  huffman@29403  264  by (simp only: mod_div_equality)  huffman@29403  265  also have "\ = (a * (b mod c) + a * (b div c) * c) mod c"  nipkow@29667  266  by (simp only: algebra_simps)  huffman@29403  267  also have "\ = (a * (b mod c)) mod c"  huffman@29403  268  by (rule mod_mult_self1)  huffman@29403  269  finally show ?thesis .  huffman@29403  270 qed  huffman@29403  271 huffman@29403  272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"  huffman@29403  273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])  huffman@29403  274 huffman@29403  275 lemma mod_mult_cong:  huffman@29403  276  assumes "a mod c = a' mod c"  huffman@29403  277  assumes "b mod c = b' mod c"  huffman@29403  278  shows "(a * b) mod c = (a' * b') mod c"  huffman@29403  279 proof -  huffman@29403  280  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"  huffman@29403  281  unfolding assms ..  huffman@29403  282  thus ?thesis  huffman@29403  283  by (simp only: mod_mult_eq [symmetric])  huffman@29403  284 qed  huffman@29403  285 huffman@47164  286 text {* Exponentiation respects modular equivalence. *}  huffman@47164  287 huffman@47164  288 lemma power_mod: "(a mod b)^n mod b = a^n mod b"  huffman@47164  289 apply (induct n, simp_all)  huffman@47164  290 apply (rule mod_mult_right_eq [THEN trans])  huffman@47164  291 apply (simp (no_asm_simp))  huffman@47164  292 apply (rule mod_mult_eq [symmetric])  huffman@47164  293 done  huffman@47164  294 huffman@29404  295 lemma mod_mod_cancel:  huffman@29404  296  assumes "c dvd b"  huffman@29404  297  shows "a mod b mod c = a mod c"  huffman@29404  298 proof -  huffman@29404  299  from c dvd b obtain k where "b = c * k"  huffman@29404  300  by (rule dvdE)  huffman@29404  301  have "a mod b mod c = a mod (c * k) mod c"  huffman@29404  302  by (simp only: b = c * k)  huffman@29404  303  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  304  by (simp only: mod_mult_self1)  huffman@29404  305  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  huffman@29404  306  by (simp only: add_ac mult_ac)  huffman@29404  307  also have "\ = a mod c"  huffman@29404  308  by (simp only: mod_div_equality)  huffman@29404  309  finally show ?thesis .  huffman@29404  310 qed  huffman@29404  311 haftmann@30930  312 lemma div_mult_div_if_dvd:  haftmann@30930  313  "y dvd x \ z dvd w \ (x div y) * (w div z) = (x * w) div (y * z)"  haftmann@30930  314  apply (cases "y = 0", simp)  haftmann@30930  315  apply (cases "z = 0", simp)  haftmann@30930  316  apply (auto elim!: dvdE simp add: algebra_simps)  nipkow@30476  317  apply (subst mult_assoc [symmetric])  nipkow@30476  318  apply (simp add: no_zero_divisors)  haftmann@30930  319  done  haftmann@30930  320 haftmann@35367  321 lemma div_mult_swap:  haftmann@35367  322  assumes "c dvd b"  haftmann@35367  323  shows "a * (b div c) = (a * b) div c"  haftmann@35367  324 proof -  haftmann@35367  325  from assms have "b div c * (a div 1) = b * a div (c * 1)"  haftmann@35367  326  by (simp only: div_mult_div_if_dvd one_dvd)  haftmann@35367  327  then show ?thesis by (simp add: mult_commute)  haftmann@35367  328 qed  haftmann@35367  329   haftmann@30930  330 lemma div_mult_mult2 [simp]:  haftmann@30930  331  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@30930  332  by (drule div_mult_mult1) (simp add: mult_commute)  haftmann@30930  333 haftmann@30930  334 lemma div_mult_mult1_if [simp]:  haftmann@30930  335  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  336  by simp_all  nipkow@30476  337 haftmann@30930  338 lemma mod_mult_mult1:  haftmann@30930  339  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  340 proof (cases "c = 0")  haftmann@30930  341  case True then show ?thesis by simp  haftmann@30930  342 next  haftmann@30930  343  case False  haftmann@30930  344  from mod_div_equality  haftmann@30930  345  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  346  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  347  = c * a + c * (a mod b)" by (simp add: algebra_simps)  haftmann@30930  348  with mod_div_equality show ?thesis by simp  haftmann@30930  349 qed  haftmann@30930  350   haftmann@30930  351 lemma mod_mult_mult2:  haftmann@30930  352  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@30930  353  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)  haftmann@30930  354 huffman@47159  355 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  huffman@47159  356  by (fact mod_mult_mult2 [symmetric])  huffman@47159  357 huffman@47159  358 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  huffman@47159  359  by (fact mod_mult_mult1 [symmetric])  huffman@47159  360 huffman@31662  361 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  362  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  363 huffman@31662  364 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  365 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  366 haftmann@31009  367 lemma div_power:  huffman@31661  368  "y dvd x \ (x div y) ^ n = x ^ n div y ^ n"  nipkow@30476  369 apply (induct n)  nipkow@30476  370  apply simp  nipkow@30476  371 apply(simp add: div_mult_div_if_dvd dvd_power_same)  nipkow@30476  372 done  nipkow@30476  373 haftmann@35367  374 lemma dvd_div_eq_mult:  haftmann@35367  375  assumes "a \ 0" and "a dvd b"  haftmann@35367  376  shows "b div a = c \ b = c * a"  haftmann@35367  377 proof  haftmann@35367  378  assume "b = c * a"  haftmann@35367  379  then show "b div a = c" by (simp add: assms)  haftmann@35367  380 next  haftmann@35367  381  assume "b div a = c"  haftmann@35367  382  then have "b div a * a = c * a" by simp  haftmann@35367  383  moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)  haftmann@35367  384  ultimately show "b = c * a" by simp  haftmann@35367  385 qed  haftmann@35367  386   haftmann@35367  387 lemma dvd_div_div_eq_mult:  haftmann@35367  388  assumes "a \ 0" "c \ 0" and "a dvd b" "c dvd d"  haftmann@35367  389  shows "b div a = d div c \ b * c = a * d"  haftmann@35367  390  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  391 huffman@31661  392 end  huffman@31661  393 haftmann@35673  394 class ring_div = semiring_div + comm_ring_1  huffman@29405  395 begin  huffman@29405  396 haftmann@36634  397 subclass ring_1_no_zero_divisors ..  haftmann@36634  398 huffman@29405  399 text {* Negation respects modular equivalence. *}  huffman@29405  400 huffman@29405  401 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  402 proof -  huffman@29405  403  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  404  by (simp only: mod_div_equality)  huffman@29405  405  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  huffman@29405  406  by (simp only: minus_add_distrib minus_mult_left add_ac)  huffman@29405  407  also have "\ = (- (a mod b)) mod b"  huffman@29405  408  by (rule mod_mult_self1)  huffman@29405  409  finally show ?thesis .  huffman@29405  410 qed  huffman@29405  411 huffman@29405  412 lemma mod_minus_cong:  huffman@29405  413  assumes "a mod b = a' mod b"  huffman@29405  414  shows "(- a) mod b = (- a') mod b"  huffman@29405  415 proof -  huffman@29405  416  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  417  unfolding assms ..  huffman@29405  418  thus ?thesis  huffman@29405  419  by (simp only: mod_minus_eq [symmetric])  huffman@29405  420 qed  huffman@29405  421 huffman@29405  422 text {* Subtraction respects modular equivalence. *}  huffman@29405  423 huffman@29405  424 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"  huffman@29405  425  unfolding diff_minus  huffman@29405  426  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  427 huffman@29405  428 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"  huffman@29405  429  unfolding diff_minus  huffman@29405  430  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  431 huffman@29405  432 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"  huffman@29405  433  unfolding diff_minus  huffman@29405  434  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  435 huffman@29405  436 lemma mod_diff_cong:  huffman@29405  437  assumes "a mod c = a' mod c"  huffman@29405  438  assumes "b mod c = b' mod c"  huffman@29405  439  shows "(a - b) mod c = (a' - b') mod c"  huffman@29405  440  unfolding diff_minus using assms  huffman@29405  441  by (intro mod_add_cong mod_minus_cong)  huffman@29405  442 nipkow@30180  443 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  444 apply (case_tac "y = 0") apply simp  nipkow@30180  445 apply (auto simp add: dvd_def)  nipkow@30180  446 apply (subgoal_tac "-(y * k) = y * - k")  nipkow@30180  447  apply (erule ssubst)  nipkow@30180  448  apply (erule div_mult_self1_is_id)  nipkow@30180  449 apply simp  nipkow@30180  450 done  nipkow@30180  451 nipkow@30180  452 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  453 apply (case_tac "y = 0") apply simp  nipkow@30180  454 apply (auto simp add: dvd_def)  nipkow@30180  455 apply (subgoal_tac "y * k = -y * -k")  nipkow@30180  456  apply (erule ssubst)  nipkow@30180  457  apply (rule div_mult_self1_is_id)  nipkow@30180  458  apply simp  nipkow@30180  459 apply simp  nipkow@30180  460 done  nipkow@30180  461 huffman@47159  462 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  huffman@47159  463  using div_mult_mult1 [of "- 1" a b]  huffman@47159  464  unfolding neg_equal_0_iff_equal by simp  huffman@47159  465 huffman@47159  466 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  huffman@47159  467  using mod_mult_mult1 [of "- 1" a b] by simp  huffman@47159  468 huffman@47159  469 lemma div_minus_right: "a div (-b) = (-a) div b"  huffman@47159  470  using div_minus_minus [of "-a" b] by simp  huffman@47159  471 huffman@47159  472 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  huffman@47159  473  using mod_minus_minus [of "-a" b] by simp  huffman@47159  474 huffman@47160  475 lemma div_minus1_right [simp]: "a div (-1) = -a"  huffman@47160  476  using div_minus_right [of a 1] by simp  huffman@47160  477 huffman@47160  478 lemma mod_minus1_right [simp]: "a mod (-1) = 0"  huffman@47160  479  using mod_minus_right [of a 1] by simp  huffman@47160  480 huffman@29405  481 end  huffman@29405  482 haftmann@25942  483 haftmann@26100  484 subsection {* Division on @{typ nat} *}  haftmann@26100  485 haftmann@26100  486 text {*  haftmann@26100  487  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  488  of a characteristic relation with two input arguments  haftmann@26100  489  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  490  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  491 *}  haftmann@26100  492 haftmann@33340  493 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  494  "divmod_nat_rel m n qr \  haftmann@30923  495  m = fst qr * n + snd qr \  haftmann@30923  496  (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  497 haftmann@33340  498 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  499 haftmann@33340  500 lemma divmod_nat_rel_ex:  haftmann@33340  501  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  502 proof (cases "n = 0")  haftmann@30923  503  case True with that show thesis  haftmann@33340  504  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  505 next  haftmann@26100  506  case False  haftmann@26100  507  have "\q r. m = q * n + r \ r < n"  haftmann@26100  508  proof (induct m)  haftmann@26100  509  case 0 with n \ 0  haftmann@26100  510  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  511  then show ?case by blast  haftmann@26100  512  next  haftmann@26100  513  case (Suc m) then obtain q' r'  haftmann@26100  514  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  515  then show ?case proof (cases "Suc r' < n")  haftmann@26100  516  case True  haftmann@26100  517  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  518  with True show ?thesis by blast  haftmann@26100  519  next  haftmann@26100  520  case False then have "n \ Suc r'" by auto  haftmann@26100  521  moreover from n have "Suc r' \ n" by auto  haftmann@26100  522  ultimately have "n = Suc r'" by auto  haftmann@26100  523  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  524  with n \ 0 show ?thesis by blast  haftmann@26100  525  qed  haftmann@26100  526  qed  haftmann@26100  527  with that show thesis  haftmann@33340  528  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  529 qed  haftmann@26100  530 haftmann@33340  531 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  532 haftmann@33340  533 lemma divmod_nat_rel_unique:  haftmann@33340  534  assumes "divmod_nat_rel m n qr"  haftmann@33340  535  and "divmod_nat_rel m n qr'"  haftmann@30923  536  shows "qr = qr'"  haftmann@26100  537 proof (cases "n = 0")  haftmann@26100  538  case True with assms show ?thesis  haftmann@30923  539  by (cases qr, cases qr')  haftmann@33340  540  (simp add: divmod_nat_rel_def)  haftmann@26100  541 next  haftmann@26100  542  case False  haftmann@26100  543  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  544  apply (rule leI)  haftmann@26100  545  apply (subst less_iff_Suc_add)  haftmann@26100  546  apply (auto simp add: add_mult_distrib)  haftmann@26100  547  done  haftmann@30923  548  from n \ 0 assms have "fst qr = fst qr'"  haftmann@33340  549  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  haftmann@30923  550  moreover from this assms have "snd qr = snd qr'"  haftmann@33340  551  by (simp add: divmod_nat_rel_def)  haftmann@30923  552  ultimately show ?thesis by (cases qr, cases qr') simp  haftmann@26100  553 qed  haftmann@26100  554 haftmann@26100  555 text {*  haftmann@26100  556  We instantiate divisibility on the natural numbers by  haftmann@33340  557  means of @{const divmod_nat_rel}:  haftmann@26100  558 *}  haftmann@25942  559 haftmann@33340  560 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  561  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  562 haftmann@33340  563 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  564  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  565 proof -  haftmann@33340  566  from divmod_nat_rel_ex  haftmann@33340  567  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  568  then show ?thesis  haftmann@33340  569  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  570 qed  haftmann@30923  571 huffman@47135  572 lemma divmod_nat_unique:  haftmann@33340  573  assumes "divmod_nat_rel m n qr"  haftmann@33340  574  shows "divmod_nat m n = qr"  haftmann@33340  575  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  576 huffman@46551  577 instantiation nat :: semiring_div  huffman@46551  578 begin  huffman@46551  579 haftmann@26100  580 definition div_nat where  haftmann@33340  581  "m div n = fst (divmod_nat m n)"  haftmann@26100  582 huffman@46551  583 lemma fst_divmod_nat [simp]:  huffman@46551  584  "fst (divmod_nat m n) = m div n"  huffman@46551  585  by (simp add: div_nat_def)  huffman@46551  586 haftmann@26100  587 definition mod_nat where  haftmann@33340  588  "m mod n = snd (divmod_nat m n)"  haftmann@25571  589 huffman@46551  590 lemma snd_divmod_nat [simp]:  huffman@46551  591  "snd (divmod_nat m n) = m mod n"  huffman@46551  592  by (simp add: mod_nat_def)  huffman@46551  593 haftmann@33340  594 lemma divmod_nat_div_mod:  haftmann@33340  595  "divmod_nat m n = (m div n, m mod n)"  huffman@46551  596  by (simp add: prod_eq_iff)  haftmann@26100  597 huffman@47135  598 lemma div_nat_unique:  haftmann@33340  599  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  600  shows "m div n = q"  huffman@47135  601  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  huffman@47135  602 huffman@47135  603 lemma mod_nat_unique:  haftmann@33340  604  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  605  shows "m mod n = r"  huffman@47135  606  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)  haftmann@25571  607 haftmann@33340  608 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  huffman@46551  609  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)  paulson@14267  610 huffman@47136  611 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  huffman@47136  612  by (simp add: divmod_nat_unique divmod_nat_rel_def)  huffman@47136  613 huffman@47136  614 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  huffman@47136  615  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  616 huffman@47137  617 lemma divmod_nat_base: "m < n \ divmod_nat m n = (0, m)"  huffman@47137  618  by (simp add: divmod_nat_unique divmod_nat_rel_def)  haftmann@25942  619 haftmann@33340  620 lemma divmod_nat_step:  haftmann@26100  621  assumes "0 < n" and "n \ m"  haftmann@33340  622  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  huffman@47135  623 proof (rule divmod_nat_unique)  huffman@47134  624  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"  huffman@47134  625  by (rule divmod_nat_rel)  huffman@47134  626  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"  huffman@47134  627  unfolding divmod_nat_rel_def using assms by auto  haftmann@26100  628 qed  haftmann@25942  629 wenzelm@26300  630 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  631 haftmann@26100  632 lemma div_less [simp]:  haftmann@26100  633  fixes m n :: nat  haftmann@26100  634  assumes "m < n"  haftmann@26100  635  shows "m div n = 0"  huffman@46551  636  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@25942  637 haftmann@26100  638 lemma le_div_geq:  haftmann@26100  639  fixes m n :: nat  haftmann@26100  640  assumes "0 < n" and "n \ m"  haftmann@26100  641  shows "m div n = Suc ((m - n) div n)"  huffman@46551  642  using assms divmod_nat_step by (simp add: prod_eq_iff)  paulson@14267  643 haftmann@26100  644 lemma mod_less [simp]:  haftmann@26100  645  fixes m n :: nat  haftmann@26100  646  assumes "m < n"  haftmann@26100  647  shows "m mod n = m"  huffman@46551  648  using assms divmod_nat_base by (simp add: prod_eq_iff)  haftmann@26100  649 haftmann@26100  650 lemma le_mod_geq:  haftmann@26100  651  fixes m n :: nat  haftmann@26100  652  assumes "n \ m"  haftmann@26100  653  shows "m mod n = (m - n) mod n"  huffman@46551  654  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)  paulson@14267  655 huffman@47136  656 instance proof  huffman@47136  657  fix m n :: nat  huffman@47136  658  show "m div n * n + m mod n = m"  huffman@47136  659  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  huffman@47136  660 next  huffman@47136  661  fix m n q :: nat  huffman@47136  662  assume "n \ 0"  huffman@47136  663  then show "(q + m * n) div n = m + q div n"  huffman@47136  664  by (induct m) (simp_all add: le_div_geq)  huffman@47136  665 next  huffman@47136  666  fix m n q :: nat  huffman@47136  667  assume "m \ 0"  huffman@47136  668  hence "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  huffman@47136  669  unfolding divmod_nat_rel_def  huffman@47136  670  by (auto split: split_if_asm, simp_all add: algebra_simps)  huffman@47136  671  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  huffman@47136  672  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  huffman@47136  673  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)  huffman@47136  674 next  huffman@47136  675  fix n :: nat show "n div 0 = 0"  haftmann@33340  676  by (simp add: div_nat_def divmod_nat_zero)  huffman@47136  677 next  huffman@47136  678  fix n :: nat show "0 div n = 0"  huffman@47136  679  by (simp add: div_nat_def divmod_nat_zero_left)  haftmann@25942  680 qed  haftmann@26100  681 haftmann@25942  682 end  paulson@14267  683 haftmann@33361  684 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  685  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  huffman@46551  686  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)  haftmann@33361  687 haftmann@26100  688 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  689 haftmann@30934  690 ML {*  wenzelm@43594  691 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  692 (  haftmann@30934  693  val div_name = @{const_name div};  haftmann@30934  694  val mod_name = @{const_name mod};  haftmann@30934  695  val mk_binop = HOLogic.mk_binop;  haftmann@30934  696  val mk_sum = Nat_Arith.mk_sum;  haftmann@30934  697  val dest_sum = Nat_Arith.dest_sum;  haftmann@25942  698 haftmann@30934  699  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  700 haftmann@30934  701  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@35050  702  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))  wenzelm@41550  703 )  haftmann@25942  704 *}  haftmann@25942  705 wenzelm@43594  706 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  707 haftmann@26100  708 haftmann@26100  709 subsubsection {* Quotient *}  haftmann@26100  710 haftmann@26100  711 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  712 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  713 haftmann@26100  714 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  715 by (simp add: div_geq)  haftmann@26100  716 haftmann@26100  717 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  718 by simp  haftmann@26100  719 haftmann@26100  720 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  721 by simp  haftmann@26100  722 haftmann@25942  723 haftmann@25942  724 subsubsection {* Remainder *}  haftmann@25942  725 haftmann@26100  726 lemma mod_less_divisor [simp]:  haftmann@26100  727  fixes m n :: nat  haftmann@26100  728  assumes "n > 0"  haftmann@26100  729  shows "m mod n < (n::nat)"  haftmann@33340  730  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  731 haftmann@26100  732 lemma mod_less_eq_dividend [simp]:  haftmann@26100  733  fixes m n :: nat  haftmann@26100  734  shows "m mod n \ m"  haftmann@26100  735 proof (rule add_leD2)  haftmann@26100  736  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  737  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  738 qed  haftmann@26100  739 haftmann@26100  740 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  741 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  742 haftmann@26100  743 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  744 by (simp add: le_mod_geq)  haftmann@26100  745 paulson@14267  746 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  747 by (induct m) (simp_all add: mod_geq)  paulson@14267  748 paulson@14267  749 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  750 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  huffman@47138  751  using mod_div_equality2 [of n m] by arith  paulson@14267  752 nipkow@15439  753 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  754  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  755  apply simp  wenzelm@22718  756  done  paulson@14267  757 haftmann@26100  758 subsubsection {* Quotient and Remainder *}  paulson@14267  759 haftmann@33340  760 lemma divmod_nat_rel_mult1_eq:  bulwahn@46552  761  "divmod_nat_rel b c (q, r)  haftmann@33340  762  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  763 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  764 haftmann@30923  765 lemma div_mult1_eq:  haftmann@30923  766  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  huffman@47135  767 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  768 haftmann@33340  769 lemma divmod_nat_rel_add1_eq:  bulwahn@46552  770  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br)  haftmann@33340  771  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  772 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  773 paulson@14267  774 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  775 lemma div_add1_eq:  nipkow@25134  776  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  huffman@47135  777 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)  paulson@14267  778 paulson@14267  779 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@22718  780  apply (cut_tac m = q and n = c in mod_less_divisor)  wenzelm@22718  781  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  wenzelm@22718  782  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)  wenzelm@22718  783  apply (simp add: add_mult_distrib2)  wenzelm@22718  784  done  paulson@10559  785 haftmann@33340  786 lemma divmod_nat_rel_mult2_eq:  bulwahn@46552  787  "divmod_nat_rel a b (q, r)  haftmann@33340  788  \ divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@33340  789 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)  paulson@14267  790 paulson@14267  791 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"  huffman@47135  792 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])  paulson@14267  793 paulson@14267  794 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"  huffman@47135  795 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])  paulson@14267  796 paulson@14267  797 huffman@46551  798 subsubsection {* Further Facts about Quotient and Remainder *}  paulson@14267  799 paulson@14267  800 lemma div_1 [simp]: "m div Suc 0 = m"  nipkow@29667  801 by (induct m) (simp_all add: div_geq)  paulson@14267  802 paulson@14267  803 (* Monotonicity of div in first argument *)  haftmann@30923  804 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  805  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  806 apply (case_tac "k=0", simp)  paulson@15251  807 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  808 apply (case_tac "n= k *)  paulson@14267  812 apply (case_tac "m=k *)  nipkow@15439  816 apply (simp add: div_geq diff_le_mono)  paulson@14267  817 done  paulson@14267  818 paulson@14267  819 (* Antimonotonicity of div in second argument *)  paulson@14267  820 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  821 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  830  prefer 2  paulson@14267  831  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  832 apply (rule le_trans, simp)  nipkow@15439  833 apply (simp)  paulson@14267  834 done  paulson@14267  835 paulson@14267  836 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  837 apply (case_tac "n=0", simp)  paulson@14267  838 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  839 apply (rule div_le_mono2)  paulson@14267  840 apply (simp_all (no_asm_simp))  paulson@14267  841 done  paulson@14267  842 wenzelm@22718  843 (* Similar for "less than" *)  huffman@47138  844 lemma div_less_dividend [simp]:  huffman@47138  845  "$$1::nat) < n; 0 < m\ \ m div n < m"  huffman@47138  846 apply (induct m rule: nat_less_induct)  paulson@14267  847 apply (rename_tac "m")  paulson@14267  848 apply (case_tac "m Suc(na) *)  paulson@16796  867 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  868 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  869 done  paulson@14267  870 paulson@14267  871 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  872 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  873 wenzelm@22718  874 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  875 paulson@14267  876 (*Loses information, namely we also have r \q::nat. m = r + q*d"  haftmann@27651  878  apply (cut_tac a = m in mod_div_equality)  wenzelm@22718  879  apply (simp only: add_ac)  wenzelm@22718  880  apply (blast intro: sym)  wenzelm@22718  881  done  paulson@14267  882 nipkow@13152  883 lemma split_div:  nipkow@13189  884  "P(n div k :: nat) =  nipkow@13189  885  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  886  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  887 proof  nipkow@13189  888  assume P: ?P  nipkow@13189  889  show ?Q  nipkow@13189  890  proof (cases)  nipkow@13189  891  assume "k = 0"  haftmann@27651  892  with P show ?Q by simp  nipkow@13189  893  next  nipkow@13189  894  assume not0: "k \ 0"  nipkow@13189  895  thus ?Q  nipkow@13189  896  proof (simp, intro allI impI)  nipkow@13189  897  fix i j  nipkow@13189  898  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  899  show "P i"  nipkow@13189  900  proof (cases)  wenzelm@22718  901  assume "i = 0"  wenzelm@22718  902  with n j P show "P i" by simp  nipkow@13189  903  next  wenzelm@22718  904  assume "i \ 0"  wenzelm@22718  905  with not0 n j P show "P i" by(simp add:add_ac)  nipkow@13189  906  qed  nipkow@13189  907  qed  nipkow@13189  908  qed  nipkow@13189  909 next  nipkow@13189  910  assume Q: ?Q  nipkow@13189  911  show ?P  nipkow@13189  912  proof (cases)  nipkow@13189  913  assume "k = 0"  haftmann@27651  914  with Q show ?P by simp  nipkow@13189  915  next  nipkow@13189  916  assume not0: "k \ 0"  nipkow@13189  917  with Q have R: ?R by simp  nipkow@13189  918  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  919  show ?P by simp  nipkow@13189  920  qed  nipkow@13189  921 qed  nipkow@13189  922 berghofe@13882  923 lemma split_div_lemma:  haftmann@26100  924  assumes "0 < n"  haftmann@26100  925  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  926 proof  haftmann@26100  927  assume ?rhs  haftmann@26100  928  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  929  then have A: "n * q \ m" by simp  haftmann@26100  930  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  931  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  932  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  933  with nq have "m < n + n * q" by simp  haftmann@26100  934  then have B: "m < n * Suc q" by simp  haftmann@26100  935  from A B show ?lhs ..  haftmann@26100  936 next  haftmann@26100  937  assume P: ?lhs  haftmann@33340  938  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@33340  939  unfolding divmod_nat_rel_def by (auto simp add: mult_ac)  haftmann@33340  940  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  941  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  942  then show ?rhs by simp  haftmann@26100  943 qed  berghofe@13882  944 berghofe@13882  945 theorem split_div':  berghofe@13882  946  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  947  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  948  apply (case_tac "0 < n")  berghofe@13882  949  apply (simp only: add: split_div_lemma)  haftmann@27651  950  apply simp_all  berghofe@13882  951  done  berghofe@13882  952 nipkow@13189  953 lemma split_mod:  nipkow@13189  954  "P(n mod k :: nat) =  nipkow@13189  955  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  956  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  957 proof  nipkow@13189  958  assume P: ?P  nipkow@13189  959  show ?Q  nipkow@13189  960  proof (cases)  nipkow@13189  961  assume "k = 0"  haftmann@27651  962  with P show ?Q by simp  nipkow@13189  963  next  nipkow@13189  964  assume not0: "k \ 0"  nipkow@13189  965  thus ?Q  nipkow@13189  966  proof (simp, intro allI impI)  nipkow@13189  967  fix i j  nipkow@13189  968  assume "n = k*i + j" "j < k"  nipkow@13189  969  thus "P j" using not0 P by(simp add:add_ac mult_ac)  nipkow@13189  970  qed  nipkow@13189  971  qed  nipkow@13189  972 next  nipkow@13189  973  assume Q: ?Q  nipkow@13189  974  show ?P  nipkow@13189  975  proof (cases)  nipkow@13189  976  assume "k = 0"  haftmann@27651  977  with Q show ?P by simp  nipkow@13189  978  next  nipkow@13189  979  assume not0: "k \ 0"  nipkow@13189  980  with Q have R: ?R by simp  nipkow@13189  981  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  982  show ?P by simp  nipkow@13189  983  qed  nipkow@13189  984 qed  nipkow@13189  985 berghofe@13882  986 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  huffman@47138  987  using mod_div_equality [of m n] by arith  huffman@47138  988 huffman@47138  989 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  huffman@47138  990  using mod_div_equality [of m n] by arith  huffman@47138  991 (* FIXME: very similar to mult_div_cancel *)  haftmann@22800  992 haftmann@22800  993 huffman@46551  994 subsubsection {* An induction'' law for modulus arithmetic. *}  paulson@14640  995 paulson@14640  996 lemma mod_induct_0:  paulson@14640  997  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  998  and base: "P i" and i: "i(P 0)"  paulson@14640  1002  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1004  proof  paulson@14640  1005  fix k  paulson@14640  1006  show "?A k"  paulson@14640  1007  proof (induct k)  paulson@14640  1008  show "?A 0" by simp -- "by contradiction"  paulson@14640  1009  next  paulson@14640  1010  fix n  paulson@14640  1011  assume ih: "?A n"  paulson@14640  1012  show "?A (Suc n)"  paulson@14640  1013  proof (clarsimp)  wenzelm@22718  1014  assume y: "P (p - Suc n)"  wenzelm@22718  1015  have n: "Suc n < p"  wenzelm@22718  1016  proof (rule ccontr)  wenzelm@22718  1017  assume "\(Suc n < p)"  wenzelm@22718  1018  hence "p - Suc n = 0"  wenzelm@22718  1019  by simp  wenzelm@22718  1020  with y contra show "False"  wenzelm@22718  1021  by simp  wenzelm@22718  1022  qed  wenzelm@22718  1023  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1024  from p have "p - Suc n < p" by arith  wenzelm@22718  1025  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1026  by blast  wenzelm@22718  1027  show "False"  wenzelm@22718  1028  proof (cases "n=0")  wenzelm@22718  1029  case True  wenzelm@22718  1030  with z n2 contra show ?thesis by simp  wenzelm@22718  1031  next  wenzelm@22718  1032  case False  wenzelm@22718  1033  with p have "p-n < p" by arith  wenzelm@22718  1034  with z n2 False ih show ?thesis by simp  wenzelm@22718  1035  qed  paulson@14640  1036  qed  paulson@14640  1037  qed  paulson@14640  1038  qed  paulson@14640  1039  moreover  paulson@14640  1040  from i obtain k where "0 i+k=p"  paulson@14640  1041  by (blast dest: less_imp_add_positive)  paulson@14640  1042  hence "0 i=p-k" by auto  paulson@14640  1043  moreover  paulson@14640  1044  note base  paulson@14640  1045  ultimately  paulson@14640  1046  show "False" by blast  paulson@14640  1047 qed  paulson@14640  1048 paulson@14640  1049 lemma mod_induct:  paulson@14640  1050  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1051  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1058  proof (induct j)  paulson@14640  1059  from step base i show "?A 0"  wenzelm@22718  1060  by (auto elim: mod_induct_0)  paulson@14640  1061  next  paulson@14640  1062  fix k  paulson@14640  1063  assume ih: "?A k"  paulson@14640  1064  show "?A (Suc k)"  paulson@14640  1065  proof  wenzelm@22718  1066  assume suc: "Suc k < p"  wenzelm@22718  1067  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1092 proof -  boehmes@35815  1093  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1094  moreover have "m mod 2 < 2" by simp  haftmann@33296  1095  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1096  then show ?thesis by auto  haftmann@33296  1097 qed  haftmann@33296  1098 haftmann@33296  1099 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1100  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1101  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1102 haftmann@33296  1103 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1104 by (simp add: Suc3_eq_add_3)  haftmann@33296  1105 haftmann@33296  1106 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1107 by (simp add: Suc3_eq_add_3)  haftmann@33296  1108 haftmann@33296  1109 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1110 by (simp add: Suc3_eq_add_3)  haftmann@33296  1111 haftmann@33296  1112 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1113 by (simp add: Suc3_eq_add_3)  haftmann@33296  1114 huffman@47108  1115 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v  huffman@47108  1116 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v  haftmann@33296  1117 haftmann@33361  1118 haftmann@33361  1119 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1120 apply (induct "m")  haftmann@33361  1121 apply (simp_all add: mod_Suc)  haftmann@33361  1122 done  haftmann@33361  1123 huffman@47108  1124 declare Suc_times_mod_eq [of "numeral w", simp] for w  haftmann@33361  1125 huffman@47138  1126 lemma Suc_div_le_mono [simp]: "n div k \ (Suc n) div k"  huffman@47138  1127 by (simp add: div_le_mono)  haftmann@33361  1128 haftmann@33361  1129 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1130 by (cases n) simp_all  haftmann@33361  1131 boehmes@35815  1132 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1133 proof -  boehmes@35815  1134  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1135  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1136 qed  haftmann@33361  1137 haftmann@33361  1138  (* Potential use of algebra : Equality modulo n*)  haftmann@33361  1139 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"  haftmann@33361  1140 by (simp add: mult_ac add_ac)  haftmann@33361  1141 haftmann@33361  1142 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1143 proof -  haftmann@33361  1144  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1145  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1146  finally show ?thesis .  haftmann@33361  1147 qed  haftmann@33361  1148 haftmann@33361  1149 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1150 apply (subst mod_Suc [of m])  haftmann@33361  1151 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1152 done  haftmann@33361  1153 huffman@47108  1154 lemma mod_2_not_eq_zero_eq_one_nat:  huffman@47108  1155  fixes n :: nat  huffman@47108  1156  shows "n mod 2 \ 0 \ n mod 2 = 1"  huffman@47108  1157  by simp  huffman@47108  1158 haftmann@33361  1159 haftmann@33361  1160 subsection {* Division on @{typ int} *}  haftmann@33361  1161 haftmann@33361  1162 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1163  --{*definition of quotient and remainder*}  huffman@47139  1164  "divmod_int_rel a b = (\(q, r). a = b * q + r \  huffman@47139  1165  (if 0 < b then 0 \ r \ r < b else if b < 0 then b < r \ r \ 0 else q = 0))"  haftmann@33361  1166 haftmann@33361  1167 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1168  --{*for the division algorithm*}  huffman@47108  1169  "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1170  else (2 * q, r))"  haftmann@33361  1171 haftmann@33361  1172 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1173 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1174  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1175  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1176 by auto  haftmann@33361  1177 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1178  (auto simp add: mult_2)  haftmann@33361  1179 haftmann@33361  1180 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1181 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1182  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1183  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1184 by auto  haftmann@33361  1185 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1186  (auto simp add: mult_2)  haftmann@33361  1187 haftmann@33361  1188 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1189 haftmann@33361  1190 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1191  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1192  including the special case @{text "a=0, b<0"} because  haftmann@33361  1193  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1194  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1195  else if a = 0 then (0, 0)  huffman@46560  1196  else apsnd uminus (negDivAlg (-a) (-b))  haftmann@33361  1197  else  haftmann@33361  1198  if 0 < b then negDivAlg a b  huffman@46560  1199  else apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1200 haftmann@33361  1201 instantiation int :: Divides.div  haftmann@33361  1202 begin  haftmann@33361  1203 huffman@46551  1204 definition div_int where  haftmann@33361  1205  "a div b = fst (divmod_int a b)"  haftmann@33361  1206 huffman@46551  1207 lemma fst_divmod_int [simp]:  huffman@46551  1208  "fst (divmod_int a b) = a div b"  huffman@46551  1209  by (simp add: div_int_def)  huffman@46551  1210 huffman@46551  1211 definition mod_int where  huffman@46560  1212  "a mod b = snd (divmod_int a b)"  haftmann@33361  1213 huffman@46551  1214 lemma snd_divmod_int [simp]:  huffman@46551  1215  "snd (divmod_int a b) = a mod b"  huffman@46551  1216  by (simp add: mod_int_def)  huffman@46551  1217 haftmann@33361  1218 instance ..  haftmann@33361  1219 paulson@3366  1220 end  haftmann@33361  1221 haftmann@33361  1222 lemma divmod_int_mod_div:  haftmann@33361  1223  "divmod_int p q = (p div q, p mod q)"  huffman@46551  1224  by (simp add: prod_eq_iff)  haftmann@33361  1225 haftmann@33361  1226 text{*  haftmann@33361  1227 Here is the division algorithm in ML:  haftmann@33361  1228 haftmann@33361  1229 \begin{verbatim}  haftmann@33361  1230  fun posDivAlg (a,b) =  haftmann@33361  1231  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1234  end  haftmann@33361  1235 haftmann@33361  1236  fun negDivAlg (a,b) =  haftmann@33361  1237  if 0\a+b then (~1,a+b)  haftmann@33361  1238  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1239  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1240  end;  haftmann@33361  1241 haftmann@33361  1242  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1243 haftmann@33361  1244  fun divmod (a,b) = if 0\a then  haftmann@33361  1245  if b>0 then posDivAlg (a,b)  haftmann@33361  1246  else if a=0 then (0,0)  haftmann@33361  1247  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1248  else  haftmann@33361  1249  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1259  ==> q' \ (q::int)"  haftmann@33361  1260 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1261  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1262 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1263 apply (erule_tac [2] order_le_less_trans)  haftmann@33361  1264  prefer 2 apply (simp add: right_diff_distrib right_distrib)  haftmann@33361  1265 apply (subgoal_tac "b * q' < b * (1 + q) ")  haftmann@33361  1266  prefer 2 apply (simp add: right_diff_distrib right_distrib)  haftmann@33361  1267 apply (simp add: mult_less_cancel_left)  haftmann@33361  1268 done  haftmann@33361  1269 haftmann@33361  1270 lemma unique_quotient_lemma_neg:  haftmann@33361  1271  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1272  ==> q \ (q'::int)"  haftmann@33361  1273 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1274  auto)  haftmann@33361  1275 haftmann@33361  1276 lemma unique_quotient:  bulwahn@46552  1277  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1278  ==> q = q'"  haftmann@33361  1279 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1280 apply (blast intro: order_antisym  haftmann@33361  1281  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1282  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1283 done  haftmann@33361  1284 haftmann@33361  1285 haftmann@33361  1286 lemma unique_remainder:  bulwahn@46552  1287  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  haftmann@33361  1288  ==> r = r'"  haftmann@33361  1289 apply (subgoal_tac "q = q'")  haftmann@33361  1290  apply (simp add: divmod_int_rel_def)  haftmann@33361  1291 apply (blast intro: unique_quotient)  haftmann@33361  1292 done  haftmann@33361  1293 haftmann@33361  1294 huffman@46551  1295 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}  haftmann@33361  1296 haftmann@33361  1297 text{*And positive divisors*}  haftmann@33361  1298 haftmann@33361  1299 lemma adjust_eq [simp]:  huffman@47108  1300  "adjust b (q, r) =  huffman@47108  1301  (let diff = r - b in  huffman@47108  1302  if 0 \ diff then (2 * q + 1, diff)  haftmann@33361  1303  else (2*q, r))"  huffman@47108  1304  by (simp add: Let_def adjust_def)  haftmann@33361  1305 haftmann@33361  1306 declare posDivAlg.simps [simp del]  haftmann@33361  1307 haftmann@33361  1308 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1309 lemma posDivAlg_eqn:  haftmann@33361  1310  "0 < b ==>  haftmann@33361  1311  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1317  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1318  using assms  wenzelm@41550  1319  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1320  apply auto  wenzelm@41550  1321  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1322  apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@41550  1323  apply (case_tac "a < b")  wenzelm@41550  1324  apply simp_all  wenzelm@41550  1325  apply (erule splitE)  wenzelm@41550  1326  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)  wenzelm@41550  1327  done  haftmann@33361  1328 haftmann@33361  1329 huffman@46551  1330 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}  haftmann@33361  1331 haftmann@33361  1332 text{*And positive divisors*}  haftmann@33361  1333 haftmann@33361  1334 declare negDivAlg.simps [simp del]  haftmann@33361  1335 haftmann@33361  1336 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1337 lemma negDivAlg_eqn:  haftmann@33361  1338  "0 < b ==>  haftmann@33361  1339  negDivAlg a b =  haftmann@33361  1340  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1341 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1342 haftmann@33361  1343 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1344  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1345 lemma negDivAlg_correct:  haftmann@33361  1346  assumes "a < 0" and "b > 0"  haftmann@33361  1347  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1348  using assms  wenzelm@41550  1349  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1350  apply (auto simp add: linorder_not_le)  wenzelm@41550  1351  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1352  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1353  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1354  apply simp_all  wenzelm@41550  1355  apply (erule splitE)  wenzelm@41550  1356  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)  wenzelm@41550  1357  done  haftmann@33361  1358 haftmann@33361  1359 huffman@46551  1360 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}  haftmann@33361  1361 haftmann@33361  1362 (*the case a=0*)  huffman@47139  1363 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"  haftmann@33361  1364 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1365 haftmann@33361  1366 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1367 by (subst posDivAlg.simps, auto)  haftmann@33361  1368 huffman@47139  1369 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"  huffman@47139  1370 by (subst posDivAlg.simps, auto)  huffman@47139  1371 haftmann@33361  1372 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  haftmann@33361  1373 by (subst negDivAlg.simps, auto)  haftmann@33361  1374 huffman@46560  1375 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"  huffman@47139  1376 by (auto simp add: divmod_int_rel_def)  huffman@47139  1377 huffman@47139  1378 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"  huffman@47139  1379 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)  haftmann@33361  1380 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1381  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1382 huffman@47141  1383 lemma divmod_int_unique:  huffman@47141  1384  assumes "divmod_int_rel a b qr"  huffman@47141  1385  shows "divmod_int a b = qr"  huffman@47141  1386  using assms divmod_int_correct [of a b]  huffman@47141  1387  using unique_quotient [of a b] unique_remainder [of a b]  huffman@47141  1388  by (metis pair_collapse)  huffman@47141  1389 huffman@47141  1390 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  huffman@47141  1391  using divmod_int_correct by (simp add: divmod_int_mod_div)  huffman@47141  1392 huffman@47141  1393 lemma div_int_unique: "divmod_int_rel a b (q, r) \ a div b = q"  huffman@47141  1394  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  huffman@47141  1395 huffman@47141  1396 lemma mod_int_unique: "divmod_int_rel a b (q, r) \ a mod b = r"  huffman@47141  1397  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  huffman@47141  1398 huffman@47141  1399 instance int :: ring_div  huffman@47141  1400 proof  huffman@47141  1401  fix a b :: int  huffman@47141  1402  show "a div b * b + a mod b = a"  huffman@47141  1403  using divmod_int_rel_div_mod [of a b]  huffman@47141  1404  unfolding divmod_int_rel_def by (simp add: mult_commute)  huffman@47141  1405 next  huffman@47141  1406  fix a b c :: int  huffman@47141  1407  assume "b \ 0"  huffman@47141  1408  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"  huffman@47141  1409  using divmod_int_rel_div_mod [of a b]  huffman@47141  1410  unfolding divmod_int_rel_def by (auto simp: algebra_simps)  huffman@47141  1411  thus "(a + c * b) div b = c + a div b"  huffman@47141  1412  by (rule div_int_unique)  huffman@47141  1413 next  huffman@47141  1414  fix a b c :: int  huffman@47141  1415  assume "c \ 0"  huffman@47141  1416  hence "\q r. divmod_int_rel a b (q, r)  huffman@47141  1417  \ divmod_int_rel (c * a) (c * b) (q, c * r)"  huffman@47141  1418  unfolding divmod_int_rel_def  huffman@47141  1419  by - (rule linorder_cases [of 0 b], auto simp: algebra_simps  huffman@47141  1420  mult_less_0_iff zero_less_mult_iff mult_strict_right_mono  huffman@47141  1421  mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  huffman@47141  1422  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"  huffman@47141  1423  using divmod_int_rel_div_mod [of a b] .  huffman@47141  1424  thus "(c * a) div (c * b) = a div b"  huffman@47141  1425  by (rule div_int_unique)  huffman@47141  1426 next  huffman@47141  1427  fix a :: int show "a div 0 = 0"  huffman@47141  1428  by (rule div_int_unique, simp add: divmod_int_rel_def)  huffman@47141  1429 next  huffman@47141  1430  fix a :: int show "0 div a = 0"  huffman@47141  1431  by (rule div_int_unique, auto simp add: divmod_int_rel_def)  huffman@47141  1432 qed  huffman@47141  1433 haftmann@33361  1434 text{*Basic laws about division and remainder*}  haftmann@33361  1435 haftmann@33361  1436 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  huffman@47141  1437  by (fact mod_div_equality2 [symmetric])  haftmann@33361  1438 haftmann@33361  1439 text {* Tool setup *}  haftmann@33361  1440 huffman@47108  1441 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)  huffman@47108  1442 lemmas add_0s = add_0_left add_0_right  huffman@47108  1443 haftmann@33361  1444 ML {*  wenzelm@43594  1445 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1446 (  haftmann@33361  1447  val div_name = @{const_name div};  haftmann@33361  1448  val mod_name = @{const_name mod};  haftmann@33361  1449  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1450  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1451  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1452 huffman@47165  1453  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  haftmann@33361  1454 haftmann@33361  1455  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@33361  1456  (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))  wenzelm@41550  1457 )  haftmann@33361  1458 *}  haftmann@33361  1459 wenzelm@43594  1460 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1461 huffman@47141  1462 lemma pos_mod_conj: "(0::int) < b \ 0 \ a mod b \ a mod b < b"  huffman@47141  1463  using divmod_int_correct [of a b]  huffman@47141  1464  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1465 wenzelm@45607  1466 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1467  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1468 huffman@47141  1469 lemma neg_mod_conj: "b < (0::int) \ a mod b \ 0 \ b < a mod b"  huffman@47141  1470  using divmod_int_correct [of a b]  huffman@47141  1471  by (auto simp add: divmod_int_rel_def prod_eq_iff)  haftmann@33361  1472 wenzelm@45607  1473 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1474  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1475 haftmann@33361  1476 huffman@46551  1477 subsubsection {* General Properties of div and mod *}  haftmann@33361  1478 haftmann@33361  1479 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  huffman@47140  1480 apply (rule div_int_unique)  haftmann@33361  1481 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1482 done  haftmann@33361  1483 haftmann@33361  1484 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  huffman@47140  1485 apply (rule div_int_unique)  haftmann@33361  1486 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1487 done  haftmann@33361  1488 haftmann@33361  1489 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  huffman@47140  1490 apply (rule div_int_unique)  haftmann@33361  1491 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1492 done  haftmann@33361  1493 haftmann@33361  1494 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1495 haftmann@33361  1496 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  huffman@47140  1497 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1498 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1499 done  haftmann@33361  1500 haftmann@33361  1501 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  huffman@47140  1502 apply (rule_tac q = 0 in mod_int_unique)  haftmann@33361  1503 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1504 done  haftmann@33361  1505 haftmann@33361  1506 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  huffman@47140  1507 apply (rule_tac q = "-1" in mod_int_unique)  haftmann@33361  1508 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1509 done  haftmann@33361  1510 haftmann@33361  1511 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  1512 haftmann@33361  1513 huffman@46551  1514 subsubsection {* Laws for div and mod with Unary Minus *}  haftmann@33361  1515 haftmann@33361  1516 lemma zminus1_lemma:  huffman@47139  1517  "divmod_int_rel a b (q, r) ==> b \ 0  haftmann@33361  1518  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1519  if r=0 then 0 else b-r)"  haftmann@33361  1520 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1521 haftmann@33361  1522 haftmann@33361  1523 lemma zdiv_zminus1_eq_if:  haftmann@33361  1524  "b \ (0::int)  haftmann@33361  1525  ==> (-a) div b =  haftmann@33361  1526  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47140  1527 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])  haftmann@33361  1528 haftmann@33361  1529 lemma zmod_zminus1_eq_if:  haftmann@33361  1530  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1531 apply (case_tac "b = 0", simp)  huffman@47140  1532 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])  haftmann@33361  1533 done  haftmann@33361  1534 haftmann@33361  1535 lemma zmod_zminus1_not_zero:  haftmann@33361  1536  fixes k l :: int  haftmann@33361  1537  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  1538  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  1539 haftmann@33361  1540 lemma zdiv_zminus2_eq_if:  haftmann@33361  1541  "b \ (0::int)  haftmann@33361  1542  ==> a div (-b) =  haftmann@33361  1543  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  huffman@47159  1544 by (simp add: zdiv_zminus1_eq_if div_minus_right)  haftmann@33361  1545 haftmann@33361  1546 lemma zmod_zminus2_eq_if:  haftmann@33361  1547  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  huffman@47159  1548 by (simp add: zmod_zminus1_eq_if mod_minus_right)  haftmann@33361  1549 haftmann@33361  1550 lemma zmod_zminus2_not_zero:  haftmann@33361  1551  fixes k l :: int  haftmann@33361  1552  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  1553  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  1554 haftmann@33361  1555 huffman@46551  1556 subsubsection {* Computation of Division and Remainder *}  haftmann@33361  1557 haftmann@33361  1558 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  1559 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1560 haftmann@33361  1561 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  1562 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1563 haftmann@33361  1564 text{*a positive, b positive *}  haftmann@33361  1565 haftmann@33361  1566 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  1567 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1568 haftmann@33361  1569 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  1570 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1571 haftmann@33361  1572 text{*a negative, b positive *}  haftmann@33361  1573 haftmann@33361  1574 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  1575 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1576 haftmann@33361  1577 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  1578 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1579 haftmann@33361  1580 text{*a positive, b negative *}  haftmann@33361  1581 haftmann@33361  1582 lemma div_pos_neg:  huffman@46560  1583  "[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1584 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1585 haftmann@33361  1586 lemma mod_pos_neg:  huffman@46560  1587  "[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"  haftmann@33361  1588 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1589 haftmann@33361  1590 text{*a negative, b negative *}  haftmann@33361  1591 haftmann@33361  1592 lemma div_neg_neg:  huffman@46560  1593  "[| a < 0; b \ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1594 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1595 haftmann@33361  1596 lemma mod_neg_neg:  huffman@46560  1597  "[| a < 0; b \ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"  haftmann@33361  1598 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1599 haftmann@33361  1600 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  1601 huffman@45530  1602 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@47140  1603  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)  huffman@45530  1604 huffman@45530  1605 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@47140  1606  by (rule div_int_unique [of a b q r],  bulwahn@46552  1607  simp add: divmod_int_rel_def)  huffman@45530  1608 huffman@45530  1609 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@47140  1610  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1611  simp add: divmod_int_rel_def)  huffman@45530  1612 huffman@45530  1613 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@47140  1614  by (rule mod_int_unique [of a b q r],  bulwahn@46552  1615  simp add: divmod_int_rel_def)  huffman@45530  1616 haftmann@33361  1617 (* simprocs adapted from HOL/ex/Binary.thy *)  haftmann@33361  1618 ML {*  haftmann@33361  1619 local  huffman@45530  1620  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  1621  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  1622  val times = @{term "times :: int \ int \ int"}  huffman@45530  1623  val zero = @{term "0 :: int"}  huffman@45530  1624  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  1625  val le = @{term "op \ :: int \ int \ bool"}  huffman@45530  1626  val simps = @{thms arith_simps} @ @{thms rel_simps} @  huffman@47108  1627  map (fn th => th RS sym) [@{thm numeral_1_eq_1}]  huffman@45530  1628  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  huffman@45530  1629  (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));  haftmann@33361  1630  fun binary_proc proc ss ct =  haftmann@33361  1631  (case Thm.term_of ct of  haftmann@33361  1632  _  t  u =>  haftmann@33361  1633  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  haftmann@33361  1634  SOME args => proc (Simplifier.the_context ss) args  haftmann@33361  1635  | NONE => NONE)  haftmann@33361  1636  | _ => NONE);  haftmann@33361  1637 in  huffman@45530  1638  fun divmod_proc posrule negrule =  huffman@45530  1639  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  huffman@45530  1640  if b = 0 then NONE else let  huffman@45530  1641  val (q, r) = pairself mk_number (Integer.div_mod a b)  huffman@45530  1642  val goal1 = HOLogic.mk_eq (t, plus  (times  u  q)  r)  huffman@45530  1643  val (goal2, goal3, rule) = if b > 0  huffman@45530  1644  then (le  zero  r, less  r  u, posrule RS eq_reflection)  huffman@45530  1645  else (le  r  zero, less  u  r, negrule RS eq_reflection)  huffman@45530  1646  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  1647 end  haftmann@33361  1648 *}  haftmann@33361  1649 huffman@47108  1650 simproc_setup binary_int_div  huffman@47108  1651  ("numeral m div numeral n :: int" |  huffman@47108  1652  "numeral m div neg_numeral n :: int" |  huffman@47108  1653  "neg_numeral m div numeral n :: int" |  huffman@47108  1654  "neg_numeral m div neg_numeral n :: int") =  huffman@45530  1655  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  1656 huffman@47108  1657 simproc_setup binary_int_mod  huffman@47108  1658  ("numeral m mod numeral n :: int" |  huffman@47108  1659  "numeral m mod neg_numeral n :: int" |  huffman@47108  1660  "neg_numeral m mod numeral n :: int" |  huffman@47108  1661  "neg_numeral m mod neg_numeral n :: int") =  huffman@45530  1662  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  1663 huffman@47108  1664 lemmas posDivAlg_eqn_numeral [simp] =  huffman@47108  1665  posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w  huffman@47108  1666 huffman@47108  1667 lemmas negDivAlg_eqn_numeral [simp] =  huffman@47108  1668  negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w  haftmann@33361  1669 haftmann@33361  1670 haftmann@33361  1671 text{*Special-case simplification *}  haftmann@33361  1672 haftmann@33361  1673 (** The last remaining special cases for constant arithmetic:  haftmann@33361  1674  1 div z and 1 mod z **)  haftmann@33361  1675 huffman@47108  1676 lemmas div_pos_pos_1_numeral [simp] =  huffman@47108  1677  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1678 huffman@47108  1679 lemmas div_pos_neg_1_numeral [simp] =  huffman@47108  1680  div_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1681  OF neg_numeral_less_zero] for w  huffman@47108  1682 huffman@47108  1683 lemmas mod_pos_pos_1_numeral [simp] =  huffman@47108  1684  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w  huffman@47108  1685 huffman@47108  1686 lemmas mod_pos_neg_1_numeral [simp] =  huffman@47108  1687  mod_pos_neg [OF zero_less_one, of "neg_numeral w",  huffman@47108  1688  OF neg_numeral_less_zero] for w  huffman@47108  1689 huffman@47108  1690 lemmas posDivAlg_eqn_1_numeral [simp] =  huffman@47108  1691  posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  huffman@47108  1692 huffman@47108  1693 lemmas negDivAlg_eqn_1_numeral [simp] =  huffman@47108  1694  negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w  haftmann@33361  1695 haftmann@33361  1696 huffman@46551  1697 subsubsection {* Monotonicity in the First Argument (Dividend) *}  haftmann@33361  1698 haftmann@33361  1699 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  1700 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1701 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1702 apply (rule unique_quotient_lemma)  haftmann@33361  1703 apply (erule subst)  haftmann@33361  1704 apply (erule subst, simp_all)  haftmann@33361  1705 done  haftmann@33361  1706 haftmann@33361  1707 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  1708 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1709 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1710 apply (rule unique_quotient_lemma_neg)  haftmann@33361  1711 apply (erule subst)  haftmann@33361  1712 apply (erule subst, simp_all)  haftmann@33361  1713 done  haftmann@33361  1714 haftmann@33361  1715 huffman@46551  1716 subsubsection {* Monotonicity in the Second Argument (Divisor) *}  haftmann@33361  1717 haftmann@33361  1718 lemma q_pos_lemma:  haftmann@33361  1719  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  1720 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  1721  apply (simp add: zero_less_mult_iff)  haftmann@33361  1722 apply (simp add: right_distrib)  haftmann@33361  1723 done  haftmann@33361  1724 haftmann@33361  1725 lemma zdiv_mono2_lemma:  haftmann@33361  1726  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  1727  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  1728  ==> q \ (q'::int)"  haftmann@33361  1729 apply (frule q_pos_lemma, assumption+)  haftmann@33361  1730 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  1731  apply (simp add: mult_less_cancel_left)  haftmann@33361  1732 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  1733  prefer 2 apply simp  haftmann@33361  1734 apply (simp (no_asm_simp) add: right_distrib)  huffman@44766  1735 apply (subst add_commute, rule add_less_le_mono, arith)  haftmann@33361  1736 apply (rule mult_right_mono, auto)  haftmann@33361  1737 done  haftmann@33361  1738 haftmann@33361  1739 lemma zdiv_mono2:  haftmann@33361  1740  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  1741 apply (subgoal_tac "b \ 0")  haftmann@33361  1742  prefer 2 apply arith  haftmann@33361  1743 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1744 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  1745 apply (rule zdiv_mono2_lemma)  haftmann@33361  1746 apply (erule subst)  haftmann@33361  1747 apply (erule subst, simp_all)  haftmann@33361  1748 done  haftmann@33361  1749 haftmann@33361  1750 lemma q_neg_lemma:  haftmann@33361  1751  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  1752 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  1753  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  1754 done  haftmann@33361  1755 haftmann@33361  1756 lemma zdiv_mono2_neg_lemma:  haftmann@33361  1757  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  1758  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  1759  ==> q' \ (q::int)"  haftmann@33361  1760 apply (frule q_neg_lemma, assumption+)  haftmann@33361  1761 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  1762  apply (simp add: mult_less_cancel_left)  haftmann@33361  1763 apply (simp add: right_distrib)  haftmann@33361  1764 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  1765  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  1766 done  haftmann@33361  1767 haftmann@33361  1768 lemma zdiv_mono2_neg:  haftmann@33361  1769  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  1770 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1771 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  1772 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  1773 apply (erule subst)  haftmann@33361  1774 apply (erule subst, simp_all)  haftmann@33361  1775 done  haftmann@33361  1776 haftmann@33361  1777 huffman@46551  1778 subsubsection {* More Algebraic Laws for div and mod *}  haftmann@33361  1779 haftmann@33361  1780 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  1781 haftmann@33361  1782 lemma zmult1_lemma:  bulwahn@46552  1783  "[| divmod_int_rel b c (q, r) |]  haftmann@33361  1784  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@33361  1785 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)  haftmann@33361  1786 haftmann@33361  1787 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  1788 apply (case_tac "c = 0", simp)  huffman@47140  1789 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])  haftmann@33361  1790 done  haftmann@33361  1791 haftmann@33361  1792 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  1793 haftmann@33361  1794 lemma zadd1_lemma:  bulwahn@46552  1795  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]  haftmann@33361  1796  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  haftmann@33361  1797 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)  haftmann@33361  1798 haftmann@33361  1799 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  1800 lemma zdiv_zadd1_eq:  haftmann@33361  1801  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  1802 apply (case_tac "c = 0", simp)  huffman@47140  1803 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)  haftmann@33361  1804 done  haftmann@33361  1805 haftmann@33361  1806 lemma posDivAlg_div_mod:  haftmann@33361  1807  assumes "k \ 0"  haftmann@33361  1808  and "l \ 0"  haftmann@33361  1809  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  1810 proof (cases "l = 0")  haftmann@33361  1811  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  1812 next  haftmann@33361  1813  case False with assms posDivAlg_correct  haftmann@33361  1814  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  1815  by simp  huffman@47140  1816  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  1817  show ?thesis by simp  haftmann@33361  1818 qed  haftmann@33361  1819 haftmann@33361  1820 lemma negDivAlg_div_mod:  haftmann@33361  1821  assumes "k < 0"  haftmann@33361  1822  and "l > 0"  haftmann@33361  1823  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  1824 proof -  haftmann@33361  1825  from assms have "l \ 0" by simp  haftmann@33361  1826  from assms negDivAlg_correct  haftmann@33361  1827  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  1828  by simp  huffman@47140  1829  from div_int_unique [OF this] mod_int_unique [OF this]  haftmann@33361  1830  show ?thesis by simp  haftmann@33361  1831 qed  haftmann@33361  1832 haftmann@33361  1833 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  1834 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  1835 haftmann@33361  1836 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  1837 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  1838 huffman@47108  1839 lemma zmod_zdiv_equality':  huffman@47108  1840  "(m\int) mod n = m - (m div n) * n"  huffman@47141  1841  using mod_div_equality [of m n] by arith  huffman@47108  1842 haftmann@33361  1843 huffman@46551  1844 subsubsection {* Proving @{term "a div (b*c) = (a div b) div c"} *}  haftmann@33361  1845 haftmann@33361  1846 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  1847  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  1848  to cause particular problems.*)  haftmann@33361  1849 haftmann@33361  1850 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  1851 haftmann@33361  1852 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  haftmann@33361  1853 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  1854  apply (simp add: algebra_simps)  haftmann@33361  1855 apply (rule order_le_less_trans)  haftmann@33361  1856  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  1857  apply (rule mult_left_mono_neg)  huffman@35216  1858  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  1859  apply (simp)  haftmann@33361  1860 apply (simp)  haftmann@33361  1861 done  haftmann@33361  1862 haftmann@33361  1863 lemma zmult2_lemma_aux2:  haftmann@33361  1864  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  1865 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  1866  apply arith  haftmann@33361  1867 apply (simp add: mult_le_0_iff)  haftmann@33361  1868 done  haftmann@33361  1869 haftmann@33361  1870 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  1871 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  1872 apply arith  haftmann@33361  1873 apply (simp add: zero_le_mult_iff)  haftmann@33361  1874 done  haftmann@33361  1875 haftmann@33361  1876 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  1877 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  1878  apply (simp add: right_diff_distrib)  haftmann@33361  1879 apply (rule order_less_le_trans)  haftmann@33361  1880  apply (erule mult_strict_right_mono)  haftmann@33361  1881  apply (rule_tac [2] mult_left_mono)  haftmann@33361  1882  apply simp  huffman@35216  1883  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  1884 apply simp  haftmann@33361  1885 done  haftmann@33361  1886 bulwahn@46552  1887 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  haftmann@33361  1888  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@33361  1889 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff  haftmann@33361  1890  zero_less_mult_iff right_distrib [symmetric]  huffman@47139  1891  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)  haftmann@33361  1892 haftmann@33361  1893 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  haftmann@33361  1894 apply (case_tac "b = 0", simp)  huffman@47140  1895 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])  haftmann@33361  1896 done  haftmann@33361  1897 haftmann@33361  1898 lemma zmod_zmult2_eq:  haftmann@33361  1899  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  haftmann@33361  1900 apply (case_tac "b = 0", simp)  huffman@47140  1901 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])  haftmann@33361  1902 done  haftmann@33361  1903 huffman@47108  1904 lemma div_pos_geq:  huffman@47108  1905  fixes k l :: int  huffman@47108  1906  assumes "0 < l" and "l \ k"  huffman@47108  1907  shows "k div l = (k - l) div l + 1"  huffman@47108  1908 proof -  huffman@47108  1909  have "k = (k - l) + l" by simp  huffman@47108  1910  then obtain j where k: "k = j + l" ..  huffman@47108  1911  with assms show ?thesis by simp  huffman@47108  1912 qed  huffman@47108  1913 huffman@47108  1914 lemma mod_pos_geq:  huffman@47108  1915  fixes k l :: int  huffman@47108  1916  assumes "0 < l" and "l \ k"  huffman@47108  1917  shows "k mod l = (k - l) mod l"  huffman@47108  1918 proof -  huffman@47108  1919  have "k = (k - l) + l" by simp  huffman@47108  1920  then obtain j where k: "k = j + l" ..  huffman@47108  1921  with assms show ?thesis by simp  huffman@47108  1922 qed  huffman@47108  1923 haftmann@33361  1924 huffman@46551  1925 subsubsection {* Splitting Rules for div and mod *}  haftmann@33361  1926 haftmann@33361  1927 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  1928 haftmann@33361  1929 lemma split_pos_lemma:  haftmann@33361  1930  "0  haftmann@33361  1931  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  1932 apply (rule iffI, clarify)  haftmann@33361  1933  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  1934  apply (subst mod_add_eq)  haftmann@33361  1935  apply (subst zdiv_zadd1_eq)  haftmann@33361  1936  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  1937 txt{*converse direction*}  haftmann@33361  1938 apply (drule_tac x = "n div k" in spec)  haftmann@33361  1939 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  1940 done  haftmann@33361  1941 haftmann@33361  1942 lemma split_neg_lemma:  haftmann@33361  1943  "k<0 ==>  haftmann@33361  1944  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  1945 apply (rule iffI, clarify)  haftmann@33361  1946  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  1947  apply (subst mod_add_eq)  haftmann@33361  1948  apply (subst zdiv_zadd1_eq)  haftmann@33361  1949  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  1950 txt{*converse direction*}  haftmann@33361  1951 apply (drule_tac x = "n div k" in spec)  haftmann@33361  1952 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  1953 done  haftmann@33361  1954 haftmann@33361  1955 lemma split_zdiv:  haftmann@33361  1956  "P(n div k :: int) =  haftmann@33361  1957  ((k = 0 --> P 0) &  haftmann@33361  1958  (0 (\i j. 0\j & j P i)) &  haftmann@33361  1959  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  1960 apply (case_tac "k=0", simp)  haftmann@33361  1961 apply (simp only: linorder_neq_iff)  haftmann@33361  1962 apply (erule disjE)  haftmann@33361  1963  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  1964  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  1965 done  haftmann@33361  1966 haftmann@33361  1967 lemma split_zmod:  haftmann@33361  1968  "P(n mod k :: int) =  haftmann@33361  1969  ((k = 0 --> P n) &  haftmann@33361  1970  (0 (\i j. 0\j & j P j)) &  haftmann@33361  1971  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  1972 apply (case_tac "k=0", simp)  haftmann@33361  1973 apply (simp only: linorder_neq_iff)  haftmann@33361  1974 apply (erule disjE)  haftmann@33361  1975  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  1976  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  1977 done  haftmann@33361  1978 webertj@33730  1979 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  1980  when these are applied to some constant that is of the form  huffman@47108  1981  @{term "numeral k"}: *}  huffman@47108  1982 declare split_zdiv [of _ _ "numeral k", arith_split] for k  huffman@47108  1983 declare split_zmod [of _ _ "numeral k", arith_split] for k  haftmann@33361  1984 haftmann@33361  1985 huffman@47166  1986 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}  huffman@47166  1987 huffman@47166  1988 lemma pos_divmod_int_rel_mult_2:  huffman@47166  1989  assumes "0 \ b"  huffman@47166  1990  assumes "divmod_int_rel a b (q, r)"  huffman@47166  1991  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  huffman@47166  1992  using assms unfolding divmod_int_rel_def by auto  huffman@47166  1993 huffman@47166  1994 lemma neg_divmod_int_rel_mult_2:  huffman@47166  1995  assumes "b \ 0"  huffman@47166  1996  assumes "divmod_int_rel (a + 1) b (q, r)"  huffman@47166  1997  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  huffman@47166  1998  using assms unfolding divmod_int_rel_def by auto  haftmann@33361  1999 haftmann@33361  2000 text{*computing div by shifting *}  haftmann@33361  2001 haftmann@33361  2002 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  huffman@47166  2003  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  huffman@47166  2004  by (rule div_int_unique)  haftmann@33361  2005 boehmes@35815  2006 lemma neg_zdiv_mult_2:  boehmes@35815  2007  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  huffman@47166  2008  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  huffman@47166  2009  by (rule div_int_unique)  haftmann@33361  2010 huffman@47108  2011 (* FIXME: add rules for negative numerals *)  huffman@47108  2012 lemma zdiv_numeral_Bit0 [simp]:  huffman@47108  2013  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =  huffman@47108  2014  numeral v div (numeral w :: int)"  huffman@47108  2015  unfolding numeral.simps unfolding mult_2 [symmetric]  huffman@47108  2016  by (rule div_mult_mult1, simp)  huffman@47108  2017 huffman@47108  2018 lemma zdiv_numeral_Bit1 [simp]:  huffman@47108  2019  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  huffman@47108  2020  (numeral v div (numeral w :: int))"  huffman@47108  2021  unfolding numeral.simps  huffman@47108  2022  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2023  by (rule pos_zdiv_mult_2, simp)  haftmann@33361  2024 haftmann@33361  2025 lemma pos_zmod_mult_2:  haftmann@33361  2026  fixes a b :: int  haftmann@33361  2027  assumes "0 \ a"  haftmann@33361  2028  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  huffman@47166  2029  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2030  by (rule mod_int_unique)  haftmann@33361  2031 haftmann@33361  2032 lemma neg_zmod_mult_2:  haftmann@33361  2033  fixes a b :: int  haftmann@33361  2034  assumes "a \ 0"  haftmann@33361  2035  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  huffman@47166  2036  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  huffman@47166  2037  by (rule mod_int_unique)  haftmann@33361  2038 huffman@47108  2039 (* FIXME: add rules for negative numerals *)  huffman@47108  2040 lemma zmod_numeral_Bit0 [simp]:  huffman@47108  2041  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  huffman@47108  2042  (2::int) * (numeral v mod numeral w)"  huffman@47108  2043  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  huffman@47108  2044  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)  huffman@47108  2045 huffman@47108  2046 lemma zmod_numeral_Bit1 [simp]:  huffman@47108  2047  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =  huffman@47108  2048  2 * (numeral v mod numeral w) + (1::int)"  huffman@47108  2049  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]  huffman@47108  2050  unfolding mult_2 [symmetric] add_commute [of _ 1]  huffman@47108  2051  by (rule pos_zmod_mult_2, simp)  haftmann@33361  2052 nipkow@39489  2053 lemma zdiv_eq_0_iff:  nipkow@39489  2054  "(i::int) div k = 0 \ k=0 \ 0\i \ i i\0 \ k ?R" by (rule split_zdiv[THEN iffD2]) simp  nipkow@39489  2058  with ?L show ?R by blast  nipkow@39489  2059 next  nipkow@39489  2060  assume ?R thus ?L  nipkow@39489  2061  by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)  nipkow@39489  2062 qed  nipkow@39489  2063 nipkow@39489  2064 huffman@46551  2065 subsubsection {* Quotients of Signs *}  haftmann@33361  2066 haftmann@33361  2067 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  haftmann@33361  2068 apply (subgoal_tac "a div b \ -1", force)  haftmann@33361  2069 apply (rule order_trans)  haftmann@33361  2070 apply (rule_tac a' = "-1" in zdiv_mono1)  haftmann@33361  2071 apply (auto simp add: div_eq_minus1)  haftmann@33361  2072 done  haftmann@33361  2073 haftmann@33361  2074 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  haftmann@33361  2075 by (drule zdiv_mono1_neg, auto)  haftmann@33361  2076 haftmann@33361  2077 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  haftmann@33361  2078 by (drule zdiv_mono1, auto)  haftmann@33361  2079 nipkow@33804  2080 text{* Now for some equivalences of the form @{text"a div b >=< 0 \ \"}  nipkow@33804  2081 conditional upon the sign of @{text a} or @{text b}. There are many more.  nipkow@33804  2082 They should all be simp rules unless that causes too much search. *}  nipkow@33804  2083 haftmann@33361  2084 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  haftmann@33361  2085 apply auto  haftmann@33361  2086 apply (drule_tac [2] zdiv_mono1)  haftmann@33361  2087 apply (auto simp add: linorder_neq_iff)  haftmann@33361  2088 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  haftmann@33361  2089 apply (blast intro: div_neg_pos_less0)  haftmann@33361  2090 done  haftmann@33361  2091 haftmann@33361  2092 lemma neg_imp_zdiv_nonneg_iff:  nipkow@33804  2093  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  huffman@47159  2094 apply (subst div_minus_minus [symmetric])  haftmann@33361  2095 apply (subst pos_imp_zdiv_nonneg_iff, auto)  haftmann@33361  2096 done  haftmann@33361  2097 haftmann@33361  2098 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  haftmann@33361  2099 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  haftmann@33361  2100 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  haftmann@33361  2101 nipkow@39489  2102 lemma pos_imp_zdiv_pos_iff:  nipkow@39489  2103  "0 0 < (i::int) div k \ k \ i"  nipkow@39489  2104 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]  nipkow@39489  2105 by arith  nipkow@39489  2106 haftmann@33361  2107 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  haftmann@33361  2108 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  haftmann@33361  2109 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  haftmann@33361  2110 nipkow@33804  2111 lemma nonneg1_imp_zdiv_pos_iff:  nipkow@33804  2112  "(0::int) <= a \ (a div b > 0) = (a >= b & b>0)"  nipkow@33804  2113 apply rule  nipkow@33804  2114  apply rule  nipkow@33804  2115  using div_pos_pos_trivial[of a b]apply arith  nipkow@33804  2116  apply(cases "b=0")apply simp  nipkow@33804  2117  using div_nonneg_neg_le0[of a b]apply arith  nipkow@33804  2118 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp  nipkow@33804  2119 done  nipkow@33804  2120 nipkow@39489  2121 lemma zmod_le_nonneg_dividend: "(m::int) \ 0 ==> m mod k \ m"  nipkow@39489  2122 apply (rule split_zmod[THEN iffD2])  nipkow@44890  2123 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)  nipkow@39489  2124 done  nipkow@39489  2125 nipkow@39489  2126 haftmann@33361  2127 subsubsection {* The Divides Relation *}  haftmann@33361  2128 huffman@47108  2129 lemmas zdvd_iff_zmod_eq_0_numeral [simp] =  huffman@47108  2130  dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]  huffman@47108  2131  dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]  huffman@47108  2132  dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]  huffman@47108  2133  dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y  haftmann@33361  2134 huffman@47108  2135 lemmas dvd_eq_mod_eq_0_numeral [simp] =  huffman@47108  2136  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y  huffman@47108  2137 huffman@47108  2138 huffman@47108  2139 subsubsection {* Further properties *}  huffman@47108  2140 haftmann@33361  2141 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  haftmann@33361  2142  using zmod_zdiv_equality[where a="m" and b="n"]  huffman@47142  2143  by (simp add: algebra_simps) (* FIXME: generalize *)  haftmann@33361  2144 haftmann@33361  2145 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  haftmann@33361  2146 apply (subst split_div, auto)  haftmann@33361  2147 apply (subst split_zdiv, auto)  haftmann@33361  2148 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  2149 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2150 done  haftmann@33361  2151 haftmann@33361  2152 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  haftmann@33361  2153 apply (subst split_mod, auto)  haftmann@33361  2154 apply (subst split_zmod, auto)  haftmann@33361  2155 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  haftmann@33361  2156  in unique_remainder)  haftmann@33361  2157 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2158 done  haftmann@33361  2159 haftmann@33361  2160 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  haftmann@33361  2161 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  haftmann@33361  2162 haftmann@33361  2163 text{*Suggested by Matthias Daum*}  haftmann@33361  2164 lemma int_power_div_base:  haftmann@33361  2165  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  haftmann@33361  2166 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  haftmann@33361  2167  apply (erule ssubst)  haftmann@33361  2168  apply (simp only: power_add)  haftmann@33361  2169  apply simp_all  haftmann@33361  2170 done  haftmann@33361  2171 haftmann@33361  2172 text {* by Brian Huffman *}  haftmann@33361  2173 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@33361  2174 by (rule mod_minus_eq [symmetric])  haftmann@33361  2175 haftmann@33361  2176 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@33361  2177 by (rule mod_diff_left_eq [symmetric])  haftmann@33361  2178 haftmann@33361  2179 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@33361  2180 by (rule mod_diff_right_eq [symmetric])  haftmann@33361  2181 haftmann@33361  2182 lemmas zmod_simps =  haftmann@33361  2183  mod_add_left_eq [symmetric]  haftmann@33361  2184  mod_add_right_eq [symmetric]  huffman@47142  2185  mod_mult_right_eq[symmetric]  haftmann@33361  2186  mod_mult_left_eq [symmetric]  huffman@47164  2187  power_mod  haftmann@33361  2188  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@33361  2189 haftmann@33361  2190 text {* Distributive laws for function @{text nat}. *}  haftmann@33361  2191 haftmann@33361  2192 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  haftmann@33361  2193 apply (rule linorder_cases [of y 0])  haftmann@33361  2194 apply (simp add: div_nonneg_neg_le0)  haftmann@33361  2195 apply simp  haftmann@33361  2196 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  haftmann@33361  2197 done  haftmann@33361  2198 haftmann@33361  2199 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  haftmann@33361  2200 lemma nat_mod_distrib:  haftmann@33361  2201  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  haftmann@33361  2202 apply (case_tac "y = 0", simp)  haftmann@33361  2203 apply (simp add: nat_eq_iff zmod_int)  haftmann@33361  2204 done  haftmann@33361  2205 haftmann@33361  2206 text {* transfer setup *}  haftmann@33361  2207 haftmann@33361  2208 lemma transfer_nat_int_functions:  haftmann@33361  2209  "(x::int) >= 0 \ y >= 0 \ (nat x) div (nat y) = nat (x div y)"  haftmann@33361  2210  "(x::int) >= 0 \ y >= 0 \ (nat x) mod (nat y) = nat (x mod y)"  haftmann@33361  2211  by (auto simp add: nat_div_distrib nat_mod_distrib)  haftmann@33361  2212 haftmann@33361  2213 lemma transfer_nat_int_function_closures:  haftmann@33361  2214  "(x::int) >= 0 \ y >= 0 \ x div y >= 0"  haftmann@33361  2215  "(x::int) >= 0 \ y >= 0 \ x mod y >= 0"  haftmann@33361  2216  apply (cases "y = 0")  haftmann@33361  2217  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  haftmann@33361  2218  apply (cases "y = 0")  haftmann@33361  2219  apply auto  haftmann@33361  2220 done  haftmann@33361  2221 haftmann@35644  2222 declare transfer_morphism_nat_int [transfer add return:  haftmann@33361  2223  transfer_nat_int_functions  haftmann@33361  2224  transfer_nat_int_function_closures  haftmann@33361  2225 ]  haftmann@33361  2226 haftmann@33361  2227 lemma transfer_int_nat_functions:  haftmann@33361  2228  "(int x) div (int y) = int (x div y)"  haftmann@33361  2229  "(int x) mod (int y) = int (x mod y)"  haftmann@33361  2230  by (auto simp add: zdiv_int zmod_int)  haftmann@33361  2231 haftmann@33361  2232 lemma transfer_int_nat_function_closures:  haftmann@33361  2233  "is_nat x \ is_nat y \ is_nat (x div y)"  haftmann@33361  2234  "is_nat x \ is_nat y \ is_nat (x mod y)"  haftmann@33361  2235  by (simp_all only: is_nat_def transfer_nat_int_function_closures)  haftmann@33361  2236 haftmann@35644  2237 declare transfer_morphism_int_nat [transfer add return:  haftmann@33361  2238  transfer_int_nat_functions  haftmann@33361  2239  transfer_int_nat_function_closures  haftmann@33361  2240 ]  haftmann@33361  2241 haftmann@33361  2242 text{*Suggested by Matthias Daum*}  haftmann@33361  2243 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  haftmann@33361  2244 apply (subgoal_tac "nat x div nat k < nat x")  nipkow@34225  2245  apply (simp add: nat_div_distrib [symmetric])  haftmann@33361  2246 apply (rule Divides.div_less_dividend, simp_all)  haftmann@33361  2247 done  haftmann@33361  2248 haftmann@33361  2249 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  haftmann@33361  2250 proof  haftmann@33361  2251  assume H: "x mod n = y mod n"  haftmann@33361  2252  hence "x mod n - y mod n = 0" by simp  haftmann@33361  2253  hence "(x mod n - y mod n) mod n = 0" by simp  haftmann@33361  2254  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])  haftmann@33361  2255  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)  haftmann@33361  2256 next  haftmann@33361  2257  assume H: "n dvd x - y"  haftmann@33361  2258  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  haftmann@33361  2259  hence "x = n*k + y" by simp  haftmann@33361  2260  hence "x mod n = (n*k + y) mod n" by simp  haftmann@33361  2261  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)  haftmann@33361  2262 qed  haftmann@33361  2263 haftmann@33361  2264 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  haftmann@33361  2265  shows "\q. x = y + n * q"  haftmann@33361  2266 proof-  haftmann@33361  2267  from xy have th: "int x - int y = int (x - y)" by simp  haftmann@33361  2268  from xyn have "int x mod int n = int y mod int n"  huffman@46551  2269  by (simp add: zmod_int [symmetric])  haftmann@33361  2270  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  haftmann@33361  2271  hence "n dvd x - y" by (simp add: th zdvd_int)  haftmann@33361  2272  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  haftmann@33361  2273 qed  haftmann@33361  2274 haftmann@33361  2275 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  haftmann@33361  2276  (is "?lhs = ?rhs")  haftmann@33361  2277 proof  haftmann@33361  2278  assume H: "x mod n = y mod n"  haftmann@33361  2279  {assume xy: "x \ y"  haftmann@33361  2280  from H have th: "y mod n = x mod n" by simp  haftmann@33361  2281  from nat_mod_eq_lemma[OF th xy] have ?rhs  haftmann@33361  2282  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  haftmann@33361  2283  moreover  haftmann@33361  2284  {assume xy: "y \ x"  haftmann@33361  2285  from nat_mod_eq_lemma[OF H xy] have ?rhs  haftmann@33361  2286  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  haftmann@33361  2287  ultimately show ?rhs using linear[of x y] by blast  haftmann@33361  2288 next  haftmann@33361  2289  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  haftmann@33361  2290  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  haftmann@33361  2291  thus ?lhs by simp  haftmann@33361  2292 qed  haftmann@33361  2293 huffman@47108  2294 lemma div_nat_numeral [simp]:  huffman@47108  2295  "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"  haftmann@33361  2296  by (simp add: nat_div_distrib)  haftmann@33361  2297 huffman@47108  2298 lemma one_div_nat_numeral [simp]:  huffman@47108  2299  "Suc 0 div numeral v' = nat (1 div numeral v')"  huffman@47108  2300  by (subst nat_div_distrib, simp_all)  huffman@47108  2301 huffman@47108  2302 lemma mod_nat_numeral [simp]:  huffman@47108  2303  "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"  haftmann@33361  2304  by (simp add: nat_mod_distrib)  haftmann@33361  2305 huffman@47108  2306 lemma one_mod_nat_numeral [simp]:  huffman@47108  2307  "Suc 0 mod numeral v' = nat (1 mod numeral v')"  huffman@47108  2308  by (subst nat_mod_distrib) simp_all  huffman@47108  2309 huffman@47108  2310 lemma mod_2_not_eq_zero_eq_one_int:  huffman@47108  2311  fixes k :: int  huffman@47108  2312  shows "k mod 2 \ 0 \ k mod 2 = 1"  huffman@47108  2313  by auto  huffman@47108  2314 huffman@47108  2315 huffman@47108  2316 subsubsection {* Tools setup *}  huffman@47108  2317 huffman@47108  2318 text {* Nitpick *}  blanchet@34126  2319 blanchet@41792  2320 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'  blanchet@34126  2321 haftmann@35673  2322 haftmann@33361  2323 subsubsection {* Code generation *}  haftmann@33361  2324 haftmann@33361  2325 definition pdivmod :: "int \ int \ int \ int" where  haftmann@33361  2326  "pdivmod k l = (\k\ div \l\, \k\ mod \l$$"  haftmann@33361  2327 haftmann@33361  2328 lemma pdivmod_posDivAlg [code]:  haftmann@33361  2329  "pdivmod k l = (if l = 0 then (0, \k\) else posDivAlg \k\ \l\)"  haftmann@33361  2330 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)  haftmann@33361  2331 haftmann@33361  2332 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@33361  2333  apsnd ((op *) (sgn l)) (if 0 < l \ 0 \ k \ l < 0 \ k < 0  haftmann@33361  2334  then pdivmod k l  haftmann@33361  2335  else (let (r, s) = pdivmod k l in  huffman@47108  2336  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@33361  2337 proof -  haftmann@33361  2338  have aux: "\q::int. - k = l * q \ k = l * - q" by auto  haftmann@33361  2339  show ?thesis  haftmann@33361  2340  by (simp add: divmod_int_mod_div pdivmod_def)  haftmann@33361  2341  (auto simp add: aux not_less not_le zdiv_zminus1_eq_if  haftmann@33361  2342  zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)  haftmann@33361  2343 qed  haftmann@33361  2344 haftmann@33361  2345 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@33361  2346  apsnd ((op *) (sgn l)) (if sgn k = sgn l  haftmann@33361  2347  then pdivmod k l  haftmann@33361  2348  else (let (r, s) = pdivmod k l in  haftmann@33361  2349  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@33361  2350 proof -  haftmann@33361  2351  have "k \ 0 \ l \ 0 \ 0 < l \ 0 \ k \ l < 0 \ k < 0 \ sgn k = sgn l"  haftmann@33361  2352  by (auto simp add: not_less sgn_if)  haftmann@33361  2353  then show ?thesis by (simp add: divmod_int_pdivmod)  haftmann@33361  2354 qed  haftmann@33361  2355 haftmann@33364  2356 code_modulename SML  haftmann@33364  2357  Divides Arith  haftmann@33364  2358 haftmann@33364  2359 code_modulename OCaml  haftmann@33364  2360  Divides Arith  haftmann@33364  2361 haftmann@33364  2362 code_modulename Haskell  haftmann@33364  2363  Divides Arith  haftmann@33364  2364 haftmann@33361  2365 end `