src/HOL/Divides.thy
 author kuncar Fri Dec 09 18:07:04 2011 +0100 (2011-12-09) changeset 45802 b16f976db515 parent 45607 16b4f5774621 child 46026 83caa4f4bd56 permissions -rw-r--r--
Quotient_Info stores only relation maps
 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@29404  286 lemma mod_mod_cancel:  huffman@29404  287  assumes "c dvd b"  huffman@29404  288  shows "a mod b mod c = a mod c"  huffman@29404  289 proof -  huffman@29404  290  from c dvd b obtain k where "b = c * k"  huffman@29404  291  by (rule dvdE)  huffman@29404  292  have "a mod b mod c = a mod (c * k) mod c"  huffman@29404  293  by (simp only: b = c * k)  huffman@29404  294  also have "\ = (a mod (c * k) + a div (c * k) * k * c) mod c"  huffman@29404  295  by (simp only: mod_mult_self1)  huffman@29404  296  also have "\ = (a div (c * k) * (c * k) + a mod (c * k)) mod c"  huffman@29404  297  by (simp only: add_ac mult_ac)  huffman@29404  298  also have "\ = a mod c"  huffman@29404  299  by (simp only: mod_div_equality)  huffman@29404  300  finally show ?thesis .  huffman@29404  301 qed  huffman@29404  302 haftmann@30930  303 lemma div_mult_div_if_dvd:  haftmann@30930  304  "y dvd x \ z dvd w \ (x div y) * (w div z) = (x * w) div (y * z)"  haftmann@30930  305  apply (cases "y = 0", simp)  haftmann@30930  306  apply (cases "z = 0", simp)  haftmann@30930  307  apply (auto elim!: dvdE simp add: algebra_simps)  nipkow@30476  308  apply (subst mult_assoc [symmetric])  nipkow@30476  309  apply (simp add: no_zero_divisors)  haftmann@30930  310  done  haftmann@30930  311 haftmann@35367  312 lemma div_mult_swap:  haftmann@35367  313  assumes "c dvd b"  haftmann@35367  314  shows "a * (b div c) = (a * b) div c"  haftmann@35367  315 proof -  haftmann@35367  316  from assms have "b div c * (a div 1) = b * a div (c * 1)"  haftmann@35367  317  by (simp only: div_mult_div_if_dvd one_dvd)  haftmann@35367  318  then show ?thesis by (simp add: mult_commute)  haftmann@35367  319 qed  haftmann@35367  320   haftmann@30930  321 lemma div_mult_mult2 [simp]:  haftmann@30930  322  "c \ 0 \ (a * c) div (b * c) = a div b"  haftmann@30930  323  by (drule div_mult_mult1) (simp add: mult_commute)  haftmann@30930  324 haftmann@30930  325 lemma div_mult_mult1_if [simp]:  haftmann@30930  326  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"  haftmann@30930  327  by simp_all  nipkow@30476  328 haftmann@30930  329 lemma mod_mult_mult1:  haftmann@30930  330  "(c * a) mod (c * b) = c * (a mod b)"  haftmann@30930  331 proof (cases "c = 0")  haftmann@30930  332  case True then show ?thesis by simp  haftmann@30930  333 next  haftmann@30930  334  case False  haftmann@30930  335  from mod_div_equality  haftmann@30930  336  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .  haftmann@30930  337  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)  haftmann@30930  338  = c * a + c * (a mod b)" by (simp add: algebra_simps)  haftmann@30930  339  with mod_div_equality show ?thesis by simp  haftmann@30930  340 qed  haftmann@30930  341   haftmann@30930  342 lemma mod_mult_mult2:  haftmann@30930  343  "(a * c) mod (b * c) = (a mod b) * c"  haftmann@30930  344  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)  haftmann@30930  345 huffman@31662  346 lemma dvd_mod: "k dvd m \ k dvd n \ k dvd (m mod n)"  huffman@31662  347  unfolding dvd_def by (auto simp add: mod_mult_mult1)  huffman@31662  348 huffman@31662  349 lemma dvd_mod_iff: "k dvd n \ k dvd (m mod n) \ k dvd m"  huffman@31662  350 by (blast intro: dvd_mod_imp_dvd dvd_mod)  huffman@31662  351 haftmann@31009  352 lemma div_power:  huffman@31661  353  "y dvd x \ (x div y) ^ n = x ^ n div y ^ n"  nipkow@30476  354 apply (induct n)  nipkow@30476  355  apply simp  nipkow@30476  356 apply(simp add: div_mult_div_if_dvd dvd_power_same)  nipkow@30476  357 done  nipkow@30476  358 haftmann@35367  359 lemma dvd_div_eq_mult:  haftmann@35367  360  assumes "a \ 0" and "a dvd b"  haftmann@35367  361  shows "b div a = c \ b = c * a"  haftmann@35367  362 proof  haftmann@35367  363  assume "b = c * a"  haftmann@35367  364  then show "b div a = c" by (simp add: assms)  haftmann@35367  365 next  haftmann@35367  366  assume "b div a = c"  haftmann@35367  367  then have "b div a * a = c * a" by simp  haftmann@35367  368  moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)  haftmann@35367  369  ultimately show "b = c * a" by simp  haftmann@35367  370 qed  haftmann@35367  371   haftmann@35367  372 lemma dvd_div_div_eq_mult:  haftmann@35367  373  assumes "a \ 0" "c \ 0" and "a dvd b" "c dvd d"  haftmann@35367  374  shows "b div a = d div c \ b * c = a * d"  haftmann@35367  375  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  376 huffman@31661  377 end  huffman@31661  378 haftmann@35673  379 class ring_div = semiring_div + comm_ring_1  huffman@29405  380 begin  huffman@29405  381 haftmann@36634  382 subclass ring_1_no_zero_divisors ..  haftmann@36634  383 huffman@29405  384 text {* Negation respects modular equivalence. *}  huffman@29405  385 huffman@29405  386 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"  huffman@29405  387 proof -  huffman@29405  388  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"  huffman@29405  389  by (simp only: mod_div_equality)  huffman@29405  390  also have "\ = (- (a mod b) + - (a div b) * b) mod b"  huffman@29405  391  by (simp only: minus_add_distrib minus_mult_left add_ac)  huffman@29405  392  also have "\ = (- (a mod b)) mod b"  huffman@29405  393  by (rule mod_mult_self1)  huffman@29405  394  finally show ?thesis .  huffman@29405  395 qed  huffman@29405  396 huffman@29405  397 lemma mod_minus_cong:  huffman@29405  398  assumes "a mod b = a' mod b"  huffman@29405  399  shows "(- a) mod b = (- a') mod b"  huffman@29405  400 proof -  huffman@29405  401  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"  huffman@29405  402  unfolding assms ..  huffman@29405  403  thus ?thesis  huffman@29405  404  by (simp only: mod_minus_eq [symmetric])  huffman@29405  405 qed  huffman@29405  406 huffman@29405  407 text {* Subtraction respects modular equivalence. *}  huffman@29405  408 huffman@29405  409 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"  huffman@29405  410  unfolding diff_minus  huffman@29405  411  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  412 huffman@29405  413 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"  huffman@29405  414  unfolding diff_minus  huffman@29405  415  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  416 huffman@29405  417 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"  huffman@29405  418  unfolding diff_minus  huffman@29405  419  by (intro mod_add_cong mod_minus_cong) simp_all  huffman@29405  420 huffman@29405  421 lemma mod_diff_cong:  huffman@29405  422  assumes "a mod c = a' mod c"  huffman@29405  423  assumes "b mod c = b' mod c"  huffman@29405  424  shows "(a - b) mod c = (a' - b') mod c"  huffman@29405  425  unfolding diff_minus using assms  huffman@29405  426  by (intro mod_add_cong mod_minus_cong)  huffman@29405  427 nipkow@30180  428 lemma dvd_neg_div: "y dvd x \ -x div y = - (x div y)"  nipkow@30180  429 apply (case_tac "y = 0") apply simp  nipkow@30180  430 apply (auto simp add: dvd_def)  nipkow@30180  431 apply (subgoal_tac "-(y * k) = y * - k")  nipkow@30180  432  apply (erule ssubst)  nipkow@30180  433  apply (erule div_mult_self1_is_id)  nipkow@30180  434 apply simp  nipkow@30180  435 done  nipkow@30180  436 nipkow@30180  437 lemma dvd_div_neg: "y dvd x \ x div -y = - (x div y)"  nipkow@30180  438 apply (case_tac "y = 0") apply simp  nipkow@30180  439 apply (auto simp add: dvd_def)  nipkow@30180  440 apply (subgoal_tac "y * k = -y * -k")  nipkow@30180  441  apply (erule ssubst)  nipkow@30180  442  apply (rule div_mult_self1_is_id)  nipkow@30180  443  apply simp  nipkow@30180  444 apply simp  nipkow@30180  445 done  nipkow@30180  446 huffman@29405  447 end  huffman@29405  448 haftmann@25942  449 haftmann@26100  450 subsection {* Division on @{typ nat} *}  haftmann@26100  451 haftmann@26100  452 text {*  haftmann@26100  453  We define @{const div} and @{const mod} on @{typ nat} by means  haftmann@26100  454  of a characteristic relation with two input arguments  haftmann@26100  455  @{term "m\nat"}, @{term "n\nat"} and two output arguments  haftmann@26100  456  @{term "q\nat"}(uotient) and @{term "r\nat"}(emainder).  haftmann@26100  457 *}  haftmann@26100  458 haftmann@33340  459 definition divmod_nat_rel :: "nat \ nat \ nat \ nat \ bool" where  haftmann@33340  460  "divmod_nat_rel m n qr \  haftmann@30923  461  m = fst qr * n + snd qr \  haftmann@30923  462  (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  463 haftmann@33340  464 text {* @{const divmod_nat_rel} is total: *}  haftmann@26100  465 haftmann@33340  466 lemma divmod_nat_rel_ex:  haftmann@33340  467  obtains q r where "divmod_nat_rel m n (q, r)"  haftmann@26100  468 proof (cases "n = 0")  haftmann@30923  469  case True with that show thesis  haftmann@33340  470  by (auto simp add: divmod_nat_rel_def)  haftmann@26100  471 next  haftmann@26100  472  case False  haftmann@26100  473  have "\q r. m = q * n + r \ r < n"  haftmann@26100  474  proof (induct m)  haftmann@26100  475  case 0 with n \ 0  haftmann@26100  476  have "(0\nat) = 0 * n + 0 \ 0 < n" by simp  haftmann@26100  477  then show ?case by blast  haftmann@26100  478  next  haftmann@26100  479  case (Suc m) then obtain q' r'  haftmann@26100  480  where m: "m = q' * n + r'" and n: "r' < n" by auto  haftmann@26100  481  then show ?case proof (cases "Suc r' < n")  haftmann@26100  482  case True  haftmann@26100  483  from m n have "Suc m = q' * n + Suc r'" by simp  haftmann@26100  484  with True show ?thesis by blast  haftmann@26100  485  next  haftmann@26100  486  case False then have "n \ Suc r'" by auto  haftmann@26100  487  moreover from n have "Suc r' \ n" by auto  haftmann@26100  488  ultimately have "n = Suc r'" by auto  haftmann@26100  489  with m have "Suc m = Suc q' * n + 0" by simp  haftmann@26100  490  with n \ 0 show ?thesis by blast  haftmann@26100  491  qed  haftmann@26100  492  qed  haftmann@26100  493  with that show thesis  haftmann@33340  494  using n \ 0 by (auto simp add: divmod_nat_rel_def)  haftmann@26100  495 qed  haftmann@26100  496 haftmann@33340  497 text {* @{const divmod_nat_rel} is injective: *}  haftmann@26100  498 haftmann@33340  499 lemma divmod_nat_rel_unique:  haftmann@33340  500  assumes "divmod_nat_rel m n qr"  haftmann@33340  501  and "divmod_nat_rel m n qr'"  haftmann@30923  502  shows "qr = qr'"  haftmann@26100  503 proof (cases "n = 0")  haftmann@26100  504  case True with assms show ?thesis  haftmann@30923  505  by (cases qr, cases qr')  haftmann@33340  506  (simp add: divmod_nat_rel_def)  haftmann@26100  507 next  haftmann@26100  508  case False  haftmann@26100  509  have aux: "\q r q' r'. q' * n + r' = q * n + r \ r < n \ q' \ (q\nat)"  haftmann@26100  510  apply (rule leI)  haftmann@26100  511  apply (subst less_iff_Suc_add)  haftmann@26100  512  apply (auto simp add: add_mult_distrib)  haftmann@26100  513  done  haftmann@30923  514  from n \ 0 assms have "fst qr = fst qr'"  haftmann@33340  515  by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)  haftmann@30923  516  moreover from this assms have "snd qr = snd qr'"  haftmann@33340  517  by (simp add: divmod_nat_rel_def)  haftmann@30923  518  ultimately show ?thesis by (cases qr, cases qr') simp  haftmann@26100  519 qed  haftmann@26100  520 haftmann@26100  521 text {*  haftmann@26100  522  We instantiate divisibility on the natural numbers by  haftmann@33340  523  means of @{const divmod_nat_rel}:  haftmann@26100  524 *}  haftmann@25942  525 haftmann@25942  526 instantiation nat :: semiring_div  haftmann@25571  527 begin  haftmann@25571  528 haftmann@33340  529 definition divmod_nat :: "nat \ nat \ nat \ nat" where  haftmann@37767  530  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"  haftmann@30923  531 haftmann@33340  532 lemma divmod_nat_rel_divmod_nat:  haftmann@33340  533  "divmod_nat_rel m n (divmod_nat m n)"  haftmann@30923  534 proof -  haftmann@33340  535  from divmod_nat_rel_ex  haftmann@33340  536  obtain qr where rel: "divmod_nat_rel m n qr" .  haftmann@30923  537  then show ?thesis  haftmann@33340  538  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)  haftmann@30923  539 qed  haftmann@30923  540 haftmann@33340  541 lemma divmod_nat_eq:  haftmann@33340  542  assumes "divmod_nat_rel m n qr"  haftmann@33340  543  shows "divmod_nat m n = qr"  haftmann@33340  544  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)  haftmann@26100  545 haftmann@26100  546 definition div_nat where  haftmann@33340  547  "m div n = fst (divmod_nat m n)"  haftmann@26100  548 haftmann@26100  549 definition mod_nat where  haftmann@33340  550  "m mod n = snd (divmod_nat m n)"  haftmann@25571  551 haftmann@33340  552 lemma divmod_nat_div_mod:  haftmann@33340  553  "divmod_nat m n = (m div n, m mod n)"  haftmann@26100  554  unfolding div_nat_def mod_nat_def by simp  haftmann@26100  555 haftmann@26100  556 lemma div_eq:  haftmann@33340  557  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  558  shows "m div n = q"  haftmann@33340  559  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)  haftmann@26100  560 haftmann@26100  561 lemma mod_eq:  haftmann@33340  562  assumes "divmod_nat_rel m n (q, r)"  haftmann@26100  563  shows "m mod n = r"  haftmann@33340  564  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)  haftmann@25571  565 haftmann@33340  566 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  haftmann@33340  567  by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)  paulson@14267  568 haftmann@33340  569 lemma divmod_nat_zero:  haftmann@33340  570  "divmod_nat m 0 = (0, m)"  haftmann@26100  571 proof -  haftmann@33340  572  from divmod_nat_rel [of m 0] show ?thesis  haftmann@33340  573  unfolding divmod_nat_div_mod divmod_nat_rel_def by simp  haftmann@26100  574 qed  haftmann@25942  575 haftmann@33340  576 lemma divmod_nat_base:  haftmann@26100  577  assumes "m < n"  haftmann@33340  578  shows "divmod_nat m n = (0, m)"  haftmann@26100  579 proof -  haftmann@33340  580  from divmod_nat_rel [of m n] show ?thesis  haftmann@33340  581  unfolding divmod_nat_div_mod divmod_nat_rel_def  haftmann@26100  582  using assms by (cases "m div n = 0")  haftmann@26100  583  (auto simp add: gr0_conv_Suc [of "m div n"])  haftmann@26100  584 qed  haftmann@25942  585 haftmann@33340  586 lemma divmod_nat_step:  haftmann@26100  587  assumes "0 < n" and "n \ m"  haftmann@33340  588  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"  haftmann@26100  589 proof -  haftmann@33340  590  from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .  haftmann@26100  591  with assms have m_div_n: "m div n \ 1"  haftmann@33340  592  by (cases "m div n") (auto simp add: divmod_nat_rel_def)  boehmes@35815  593  have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"  boehmes@35815  594  proof -  boehmes@35815  595  from assms have  boehmes@35815  596  "n \ 0"  boehmes@35815  597  "\k. m = Suc k * n + m mod n ==> m - n = (Suc k - Suc 0) * n + m mod n"  boehmes@35815  598  by simp_all  boehmes@35815  599  then show ?thesis using assms divmod_nat_m_n  boehmes@35815  600  by (cases "m div n")  boehmes@35815  601  (simp_all only: divmod_nat_rel_def fst_conv snd_conv, simp_all)  boehmes@35815  602  qed  haftmann@33340  603  with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp  haftmann@33340  604  moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .  haftmann@26100  605  ultimately have "m div n = Suc ((m - n) div n)"  haftmann@26100  606  and "m mod n = (m - n) mod n" using m_div_n by simp_all  haftmann@33340  607  then show ?thesis using divmod_nat_div_mod by simp  haftmann@26100  608 qed  haftmann@25942  609 wenzelm@26300  610 text {* The ''recursion'' equations for @{const div} and @{const mod} *}  haftmann@26100  611 haftmann@26100  612 lemma div_less [simp]:  haftmann@26100  613  fixes m n :: nat  haftmann@26100  614  assumes "m < n"  haftmann@26100  615  shows "m div n = 0"  haftmann@33340  616  using assms divmod_nat_base divmod_nat_div_mod by simp  haftmann@25942  617 haftmann@26100  618 lemma le_div_geq:  haftmann@26100  619  fixes m n :: nat  haftmann@26100  620  assumes "0 < n" and "n \ m"  haftmann@26100  621  shows "m div n = Suc ((m - n) div n)"  haftmann@33340  622  using assms divmod_nat_step divmod_nat_div_mod by simp  paulson@14267  623 haftmann@26100  624 lemma mod_less [simp]:  haftmann@26100  625  fixes m n :: nat  haftmann@26100  626  assumes "m < n"  haftmann@26100  627  shows "m mod n = m"  haftmann@33340  628  using assms divmod_nat_base divmod_nat_div_mod by simp  haftmann@26100  629 haftmann@26100  630 lemma le_mod_geq:  haftmann@26100  631  fixes m n :: nat  haftmann@26100  632  assumes "n \ m"  haftmann@26100  633  shows "m mod n = (m - n) mod n"  haftmann@33340  634  using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all  paulson@14267  635 haftmann@30930  636 instance proof -  haftmann@30930  637  have [simp]: "\n::nat. n div 0 = 0"  haftmann@33340  638  by (simp add: div_nat_def divmod_nat_zero)  haftmann@30930  639  have [simp]: "\n::nat. 0 div n = 0"  haftmann@30930  640  proof -  haftmann@30930  641  fix n :: nat  haftmann@30930  642  show "0 div n = 0"  haftmann@30930  643  by (cases "n = 0") simp_all  haftmann@30930  644  qed  haftmann@30930  645  show "OFCLASS(nat, semiring_div_class)" proof  haftmann@30930  646  fix m n :: nat  haftmann@30930  647  show "m div n * n + m mod n = m"  haftmann@33340  648  using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)  haftmann@30930  649  next  haftmann@30930  650  fix m n q :: nat  haftmann@30930  651  assume "n \ 0"  haftmann@30930  652  then show "(q + m * n) div n = m + q div n"  haftmann@30930  653  by (induct m) (simp_all add: le_div_geq)  haftmann@30930  654  next  haftmann@30930  655  fix m n q :: nat  haftmann@30930  656  assume "m \ 0"  haftmann@30930  657  then show "(m * n) div (m * q) = n div q"  haftmann@30930  658  proof (cases "n \ 0 \ q \ 0")  haftmann@30930  659  case False then show ?thesis by auto  haftmann@30930  660  next  haftmann@30930  661  case True with m \ 0  haftmann@30930  662  have "m > 0" and "n > 0" and "q > 0" by auto  haftmann@33340  663  then have "\a b. divmod_nat_rel n q (a, b) \ divmod_nat_rel (m * n) (m * q) (a, m * b)"  haftmann@33340  664  by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)  haftmann@33340  665  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .  haftmann@33340  666  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .  haftmann@30930  667  then show ?thesis by (simp add: div_eq)  haftmann@30930  668  qed  haftmann@30930  669  qed simp_all  haftmann@25942  670 qed  haftmann@26100  671 haftmann@25942  672 end  paulson@14267  673 haftmann@33361  674 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \ m < n then (0, m) else  haftmann@33361  675  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  haftmann@33361  676 by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)  haftmann@33361  677  (simp add: divmod_nat_div_mod)  haftmann@33361  678 haftmann@26100  679 text {* Simproc for cancelling @{const div} and @{const mod} *}  haftmann@25942  680 haftmann@30934  681 ML {*  wenzelm@43594  682 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod  wenzelm@41550  683 (  haftmann@30934  684  val div_name = @{const_name div};  haftmann@30934  685  val mod_name = @{const_name mod};  haftmann@30934  686  val mk_binop = HOLogic.mk_binop;  haftmann@30934  687  val mk_sum = Nat_Arith.mk_sum;  haftmann@30934  688  val dest_sum = Nat_Arith.dest_sum;  haftmann@25942  689 haftmann@30934  690  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  paulson@14267  691 haftmann@30934  692  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@35050  693  (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))  wenzelm@41550  694 )  haftmann@25942  695 *}  haftmann@25942  696 wenzelm@43594  697 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}  wenzelm@43594  698 haftmann@26100  699 haftmann@26100  700 subsubsection {* Quotient *}  haftmann@26100  701 haftmann@26100  702 lemma div_geq: "0 < n \ \ m < n \ m div n = Suc ((m - n) div n)"  nipkow@29667  703 by (simp add: le_div_geq linorder_not_less)  haftmann@26100  704 haftmann@26100  705 lemma div_if: "0 < n \ m div n = (if m < n then 0 else Suc ((m - n) div n))"  nipkow@29667  706 by (simp add: div_geq)  haftmann@26100  707 haftmann@26100  708 lemma div_mult_self_is_m [simp]: "0 (m*n) div n = (m::nat)"  nipkow@29667  709 by simp  haftmann@26100  710 haftmann@26100  711 lemma div_mult_self1_is_m [simp]: "0 (n*m) div n = (m::nat)"  nipkow@29667  712 by simp  haftmann@26100  713 haftmann@25942  714 haftmann@25942  715 subsubsection {* Remainder *}  haftmann@25942  716 haftmann@26100  717 lemma mod_less_divisor [simp]:  haftmann@26100  718  fixes m n :: nat  haftmann@26100  719  assumes "n > 0"  haftmann@26100  720  shows "m mod n < (n::nat)"  haftmann@33340  721  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto  paulson@14267  722 haftmann@26100  723 lemma mod_less_eq_dividend [simp]:  haftmann@26100  724  fixes m n :: nat  haftmann@26100  725  shows "m mod n \ m"  haftmann@26100  726 proof (rule add_leD2)  haftmann@26100  727  from mod_div_equality have "m div n * n + m mod n = m" .  haftmann@26100  728  then show "m div n * n + m mod n \ m" by auto  haftmann@26100  729 qed  haftmann@26100  730 haftmann@26100  731 lemma mod_geq: "\ m < (n\nat) \ m mod n = (m - n) mod n"  nipkow@29667  732 by (simp add: le_mod_geq linorder_not_less)  paulson@14267  733 haftmann@26100  734 lemma mod_if: "m mod (n\nat) = (if m < n then m else (m - n) mod n)"  nipkow@29667  735 by (simp add: le_mod_geq)  haftmann@26100  736 paulson@14267  737 lemma mod_1 [simp]: "m mod Suc 0 = 0"  nipkow@29667  738 by (induct m) (simp_all add: mod_geq)  paulson@14267  739 haftmann@26100  740 lemma mod_mult_distrib: "(m mod n) * (k\nat) = (m * k) mod (n * k)"  wenzelm@22718  741  apply (cases "n = 0", simp)  wenzelm@22718  742  apply (cases "k = 0", simp)  wenzelm@22718  743  apply (induct m rule: nat_less_induct)  wenzelm@22718  744  apply (subst mod_if, simp)  wenzelm@22718  745  apply (simp add: mod_geq diff_mult_distrib)  wenzelm@22718  746  done  paulson@14267  747 paulson@14267  748 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"  nipkow@29667  749 by (simp add: mult_commute [of k] mod_mult_distrib)  paulson@14267  750 paulson@14267  751 (* a simple rearrangement of mod_div_equality: *)  paulson@14267  752 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  nipkow@29667  753 by (cut_tac a = m and b = n in mod_div_equality2, arith)  paulson@14267  754 nipkow@15439  755 lemma mod_le_divisor[simp]: "0 < n \ m mod n \ (n::nat)"  wenzelm@22718  756  apply (drule mod_less_divisor [where m = m])  wenzelm@22718  757  apply simp  wenzelm@22718  758  done  paulson@14267  759 haftmann@26100  760 subsubsection {* Quotient and Remainder *}  paulson@14267  761 haftmann@33340  762 lemma divmod_nat_rel_mult1_eq:  haftmann@33340  763  "divmod_nat_rel b c (q, r) \ c > 0  haftmann@33340  764  \ divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"  haftmann@33340  765 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  766 haftmann@30923  767 lemma div_mult1_eq:  haftmann@30923  768  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"  nipkow@25134  769 apply (cases "c = 0", simp)  haftmann@33340  770 apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])  nipkow@25134  771 done  paulson@14267  772 haftmann@33340  773 lemma divmod_nat_rel_add1_eq:  haftmann@33340  774  "divmod_nat_rel a c (aq, ar) \ divmod_nat_rel b c (bq, br) \ c > 0  haftmann@33340  775  \ divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"  haftmann@33340  776 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)  paulson@14267  777 paulson@14267  778 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  paulson@14267  779 lemma div_add1_eq:  nipkow@25134  780  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"  nipkow@25134  781 apply (cases "c = 0", simp)  haftmann@33340  782 apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)  nipkow@25134  783 done  paulson@14267  784 paulson@14267  785 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@22718  786  apply (cut_tac m = q and n = c in mod_less_divisor)  wenzelm@22718  787  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)  wenzelm@22718  788  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)  wenzelm@22718  789  apply (simp add: add_mult_distrib2)  wenzelm@22718  790  done  paulson@10559  791 haftmann@33340  792 lemma divmod_nat_rel_mult2_eq:  haftmann@33340  793  "divmod_nat_rel a b (q, r) \ 0 < b \ 0 < c  haftmann@33340  794  \ divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"  haftmann@33340  795 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)  paulson@14267  796 paulson@14267  797 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"  wenzelm@22718  798  apply (cases "b = 0", simp)  wenzelm@22718  799  apply (cases "c = 0", simp)  haftmann@33340  800  apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])  wenzelm@22718  801  done  paulson@14267  802 paulson@14267  803 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"  wenzelm@22718  804  apply (cases "b = 0", simp)  wenzelm@22718  805  apply (cases "c = 0", simp)  haftmann@33340  806  apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])  wenzelm@22718  807  done  paulson@14267  808 paulson@14267  809 haftmann@25942  810 subsubsection{*Further Facts about Quotient and Remainder*}  paulson@14267  811 paulson@14267  812 lemma div_1 [simp]: "m div Suc 0 = m"  nipkow@29667  813 by (induct m) (simp_all add: div_geq)  paulson@14267  814 paulson@14267  815 paulson@14267  816 (* Monotonicity of div in first argument *)  haftmann@30923  817 lemma div_le_mono [rule_format (no_asm)]:  wenzelm@22718  818  "\m::nat. m \ n --> (m div k) \ (n div k)"  paulson@14267  819 apply (case_tac "k=0", simp)  paulson@15251  820 apply (induct "n" rule: nat_less_induct, clarify)  paulson@14267  821 apply (case_tac "n= k *)  paulson@14267  825 apply (case_tac "m=k *)  nipkow@15439  829 apply (simp add: div_geq diff_le_mono)  paulson@14267  830 done  paulson@14267  831 paulson@14267  832 (* Antimonotonicity of div in second argument *)  paulson@14267  833 lemma div_le_mono2: "!!m::nat. [| 0n |] ==> (k div n) \ (k div m)"  paulson@14267  834 apply (subgoal_tac "0 (k-m) div n")  paulson@14267  843  prefer 2  paulson@14267  844  apply (blast intro: div_le_mono diff_le_mono2)  paulson@14267  845 apply (rule le_trans, simp)  nipkow@15439  846 apply (simp)  paulson@14267  847 done  paulson@14267  848 paulson@14267  849 lemma div_le_dividend [simp]: "m div n \ (m::nat)"  paulson@14267  850 apply (case_tac "n=0", simp)  paulson@14267  851 apply (subgoal_tac "m div n \ m div 1", simp)  paulson@14267  852 apply (rule div_le_mono2)  paulson@14267  853 apply (simp_all (no_asm_simp))  paulson@14267  854 done  paulson@14267  855 wenzelm@22718  856 (* Similar for "less than" *)  paulson@17085  857 lemma div_less_dividend [rule_format]:  paulson@14267  858  "!!n::nat. 1 0 < m --> m div n < m"  paulson@15251  859 apply (induct_tac m rule: nat_less_induct)  paulson@14267  860 apply (rename_tac "m")  paulson@14267  861 apply (case_tac "m Suc(na) *)  paulson@16796  882 apply (simp add: linorder_not_less le_Suc_eq mod_geq)  nipkow@15439  883 apply (auto simp add: Suc_diff_le le_mod_geq)  paulson@14267  884 done  paulson@14267  885 paulson@14267  886 lemma mod_eq_0_iff: "(m mod d = 0) = (\q::nat. m = d*q)"  nipkow@29667  887 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  paulson@17084  888 wenzelm@22718  889 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]  paulson@14267  890 paulson@14267  891 (*Loses information, namely we also have r \q::nat. m = r + q*d"  haftmann@27651  893  apply (cut_tac a = m in mod_div_equality)  wenzelm@22718  894  apply (simp only: add_ac)  wenzelm@22718  895  apply (blast intro: sym)  wenzelm@22718  896  done  paulson@14267  897 nipkow@13152  898 lemma split_div:  nipkow@13189  899  "P(n div k :: nat) =  nipkow@13189  900  ((k = 0 \ P 0) \ (k \ 0 \ (!i. !j P i)))"  nipkow@13189  901  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  902 proof  nipkow@13189  903  assume P: ?P  nipkow@13189  904  show ?Q  nipkow@13189  905  proof (cases)  nipkow@13189  906  assume "k = 0"  haftmann@27651  907  with P show ?Q by simp  nipkow@13189  908  next  nipkow@13189  909  assume not0: "k \ 0"  nipkow@13189  910  thus ?Q  nipkow@13189  911  proof (simp, intro allI impI)  nipkow@13189  912  fix i j  nipkow@13189  913  assume n: "n = k*i + j" and j: "j < k"  nipkow@13189  914  show "P i"  nipkow@13189  915  proof (cases)  wenzelm@22718  916  assume "i = 0"  wenzelm@22718  917  with n j P show "P i" by simp  nipkow@13189  918  next  wenzelm@22718  919  assume "i \ 0"  wenzelm@22718  920  with not0 n j P show "P i" by(simp add:add_ac)  nipkow@13189  921  qed  nipkow@13189  922  qed  nipkow@13189  923  qed  nipkow@13189  924 next  nipkow@13189  925  assume Q: ?Q  nipkow@13189  926  show ?P  nipkow@13189  927  proof (cases)  nipkow@13189  928  assume "k = 0"  haftmann@27651  929  with Q show ?P by simp  nipkow@13189  930  next  nipkow@13189  931  assume not0: "k \ 0"  nipkow@13189  932  with Q have R: ?R by simp  nipkow@13189  933  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  934  show ?P by simp  nipkow@13189  935  qed  nipkow@13189  936 qed  nipkow@13189  937 berghofe@13882  938 lemma split_div_lemma:  haftmann@26100  939  assumes "0 < n"  haftmann@26100  940  shows "n * q \ m \ m < n * Suc q \ q = ((m\nat) div n)" (is "?lhs \ ?rhs")  haftmann@26100  941 proof  haftmann@26100  942  assume ?rhs  haftmann@26100  943  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  haftmann@26100  944  then have A: "n * q \ m" by simp  haftmann@26100  945  have "n - (m mod n) > 0" using mod_less_divisor assms by auto  haftmann@26100  946  then have "m < m + (n - (m mod n))" by simp  haftmann@26100  947  then have "m < n + (m - (m mod n))" by simp  haftmann@26100  948  with nq have "m < n + n * q" by simp  haftmann@26100  949  then have B: "m < n * Suc q" by simp  haftmann@26100  950  from A B show ?lhs ..  haftmann@26100  951 next  haftmann@26100  952  assume P: ?lhs  haftmann@33340  953  then have "divmod_nat_rel m n (q, m - n * q)"  haftmann@33340  954  unfolding divmod_nat_rel_def by (auto simp add: mult_ac)  haftmann@33340  955  with divmod_nat_rel_unique divmod_nat_rel [of m n]  haftmann@30923  956  have "(q, m - n * q) = (m div n, m mod n)" by auto  haftmann@30923  957  then show ?rhs by simp  haftmann@26100  958 qed  berghofe@13882  959 berghofe@13882  960 theorem split_div':  berghofe@13882  961  "P ((m::nat) div n) = ((n = 0 \ P 0) \  paulson@14267  962  (\q. (n * q \ m \ m < n * (Suc q)) \ P q))"  berghofe@13882  963  apply (case_tac "0 < n")  berghofe@13882  964  apply (simp only: add: split_div_lemma)  haftmann@27651  965  apply simp_all  berghofe@13882  966  done  berghofe@13882  967 nipkow@13189  968 lemma split_mod:  nipkow@13189  969  "P(n mod k :: nat) =  nipkow@13189  970  ((k = 0 \ P n) \ (k \ 0 \ (!i. !j P j)))"  nipkow@13189  971  (is "?P = ?Q" is "_ = (_ \ (_ \ ?R))")  nipkow@13189  972 proof  nipkow@13189  973  assume P: ?P  nipkow@13189  974  show ?Q  nipkow@13189  975  proof (cases)  nipkow@13189  976  assume "k = 0"  haftmann@27651  977  with P show ?Q by simp  nipkow@13189  978  next  nipkow@13189  979  assume not0: "k \ 0"  nipkow@13189  980  thus ?Q  nipkow@13189  981  proof (simp, intro allI impI)  nipkow@13189  982  fix i j  nipkow@13189  983  assume "n = k*i + j" "j < k"  nipkow@13189  984  thus "P j" using not0 P by(simp add:add_ac mult_ac)  nipkow@13189  985  qed  nipkow@13189  986  qed  nipkow@13189  987 next  nipkow@13189  988  assume Q: ?Q  nipkow@13189  989  show ?P  nipkow@13189  990  proof (cases)  nipkow@13189  991  assume "k = 0"  haftmann@27651  992  with Q show ?P by simp  nipkow@13189  993  next  nipkow@13189  994  assume not0: "k \ 0"  nipkow@13189  995  with Q have R: ?R by simp  nipkow@13189  996  from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]  nipkow@13517  997  show ?P by simp  nipkow@13189  998  qed  nipkow@13189  999 qed  nipkow@13189  1000 berghofe@13882  1001 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  berghofe@13882  1002  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in  berghofe@13882  1003  subst [OF mod_div_equality [of _ n]])  berghofe@13882  1004  apply arith  berghofe@13882  1005  done  berghofe@13882  1006 haftmann@22800  1007 lemma div_mod_equality':  haftmann@22800  1008  fixes m n :: nat  haftmann@22800  1009  shows "m div n * n = m - m mod n"  haftmann@22800  1010 proof -  haftmann@22800  1011  have "m mod n \ m mod n" ..  haftmann@22800  1012  from div_mod_equality have  haftmann@22800  1013  "m div n * n + m mod n - m mod n = m - m mod n" by simp  haftmann@22800  1014  with diff_add_assoc [OF m mod n \ m mod n, of "m div n * n"] have  haftmann@22800  1015  "m div n * n + (m mod n - m mod n) = m - m mod n"  haftmann@22800  1016  by simp  haftmann@22800  1017  then show ?thesis by simp  haftmann@22800  1018 qed  haftmann@22800  1019 haftmann@22800  1020 haftmann@25942  1021 subsubsection {*An induction'' law for modulus arithmetic.*}  paulson@14640  1022 paulson@14640  1023 lemma mod_induct_0:  paulson@14640  1024  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1025  and base: "P i" and i: "i(P 0)"  paulson@14640  1029  from i have p: "0k. 0 \ P (p-k)" (is "\k. ?A k")  paulson@14640  1031  proof  paulson@14640  1032  fix k  paulson@14640  1033  show "?A k"  paulson@14640  1034  proof (induct k)  paulson@14640  1035  show "?A 0" by simp -- "by contradiction"  paulson@14640  1036  next  paulson@14640  1037  fix n  paulson@14640  1038  assume ih: "?A n"  paulson@14640  1039  show "?A (Suc n)"  paulson@14640  1040  proof (clarsimp)  wenzelm@22718  1041  assume y: "P (p - Suc n)"  wenzelm@22718  1042  have n: "Suc n < p"  wenzelm@22718  1043  proof (rule ccontr)  wenzelm@22718  1044  assume "$$Suc n < p)"  wenzelm@22718  1045  hence "p - Suc n = 0"  wenzelm@22718  1046  by simp  wenzelm@22718  1047  with y contra show "False"  wenzelm@22718  1048  by simp  wenzelm@22718  1049  qed  wenzelm@22718  1050  hence n2: "Suc (p - Suc n) = p-n" by arith  wenzelm@22718  1051  from p have "p - Suc n < p" by arith  wenzelm@22718  1052  with y step have z: "P ((Suc (p - Suc n)) mod p)"  wenzelm@22718  1053  by blast  wenzelm@22718  1054  show "False"  wenzelm@22718  1055  proof (cases "n=0")  wenzelm@22718  1056  case True  wenzelm@22718  1057  with z n2 contra show ?thesis by simp  wenzelm@22718  1058  next  wenzelm@22718  1059  case False  wenzelm@22718  1060  with p have "p-n < p" by arith  wenzelm@22718  1061  with z n2 False ih show ?thesis by simp  wenzelm@22718  1062  qed  paulson@14640  1063  qed  paulson@14640  1064  qed  paulson@14640  1065  qed  paulson@14640  1066  moreover  paulson@14640  1067  from i obtain k where "0 i+k=p"  paulson@14640  1068  by (blast dest: less_imp_add_positive)  paulson@14640  1069  hence "0 i=p-k" by auto  paulson@14640  1070  moreover  paulson@14640  1071  note base  paulson@14640  1072  ultimately  paulson@14640  1073  show "False" by blast  paulson@14640  1074 qed  paulson@14640  1075 paulson@14640  1076 lemma mod_induct:  paulson@14640  1077  assumes step: "\i P ((Suc i) mod p)"  paulson@14640  1078  and base: "P i" and i: "ij P j" (is "?A j")  paulson@14640  1085  proof (induct j)  paulson@14640  1086  from step base i show "?A 0"  wenzelm@22718  1087  by (auto elim: mod_induct_0)  paulson@14640  1088  next  paulson@14640  1089  fix k  paulson@14640  1090  assume ih: "?A k"  paulson@14640  1091  show "?A (Suc k)"  paulson@14640  1092  proof  wenzelm@22718  1093  assume suc: "Suc k < p"  wenzelm@22718  1094  hence k: "knat) mod 2 \ m mod 2 = 1"  haftmann@33296  1122 proof -  boehmes@35815  1123  { fix n :: nat have "(n::nat) < 2 \ n = 0 \ n = 1" by (cases n) simp_all }  haftmann@33296  1124  moreover have "m mod 2 < 2" by simp  haftmann@33296  1125  ultimately have "m mod 2 = 0 \ m mod 2 = 1" .  haftmann@33296  1126  then show ?thesis by auto  haftmann@33296  1127 qed  haftmann@33296  1128 haftmann@33296  1129 text{*These lemmas collapse some needless occurrences of Suc:  haftmann@33296  1130  at least three Sucs, since two and fewer are rewritten back to Suc again!  haftmann@33296  1131  We already have some rules to simplify operands smaller than 3.*}  haftmann@33296  1132 haftmann@33296  1133 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"  haftmann@33296  1134 by (simp add: Suc3_eq_add_3)  haftmann@33296  1135 haftmann@33296  1136 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"  haftmann@33296  1137 by (simp add: Suc3_eq_add_3)  haftmann@33296  1138 haftmann@33296  1139 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"  haftmann@33296  1140 by (simp add: Suc3_eq_add_3)  haftmann@33296  1141 haftmann@33296  1142 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"  haftmann@33296  1143 by (simp add: Suc3_eq_add_3)  haftmann@33296  1144 wenzelm@45607  1145 lemmas Suc_div_eq_add3_div_number_of [simp] = Suc_div_eq_add3_div [of _ "number_of v"] for v  wenzelm@45607  1146 lemmas Suc_mod_eq_add3_mod_number_of [simp] = Suc_mod_eq_add3_mod [of _ "number_of v"] for v  haftmann@33296  1147 haftmann@33361  1148 haftmann@33361  1149 lemma Suc_times_mod_eq: "1 Suc (k * m) mod k = 1"  haftmann@33361  1150 apply (induct "m")  haftmann@33361  1151 apply (simp_all add: mod_Suc)  haftmann@33361  1152 done  haftmann@33361  1153 wenzelm@45607  1154 declare Suc_times_mod_eq [of "number_of w", simp] for w  haftmann@33361  1155 haftmann@33361  1156 lemma [simp]: "n div k \ (Suc n) div k"  haftmann@33361  1157 by (simp add: div_le_mono)  haftmann@33361  1158 haftmann@33361  1159 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"  haftmann@33361  1160 by (cases n) simp_all  haftmann@33361  1161 boehmes@35815  1162 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"  boehmes@35815  1163 proof -  boehmes@35815  1164  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  boehmes@35815  1165  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp  boehmes@35815  1166 qed  haftmann@33361  1167 haftmann@33361  1168  (* Potential use of algebra : Equality modulo n*)  haftmann@33361  1169 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"  haftmann@33361  1170 by (simp add: mult_ac add_ac)  haftmann@33361  1171 haftmann@33361  1172 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"  haftmann@33361  1173 proof -  haftmann@33361  1174  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp  haftmann@33361  1175  also have "... = Suc m mod n" by (rule mod_mult_self3)  haftmann@33361  1176  finally show ?thesis .  haftmann@33361  1177 qed  haftmann@33361  1178 haftmann@33361  1179 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"  haftmann@33361  1180 apply (subst mod_Suc [of m])  haftmann@33361  1181 apply (subst mod_Suc [of "m mod n"], simp)  haftmann@33361  1182 done  haftmann@33361  1183 haftmann@33361  1184 haftmann@33361  1185 subsection {* Division on @{typ int} *}  haftmann@33361  1186 haftmann@33361  1187 definition divmod_int_rel :: "int \ int \ int \ int \ bool" where  haftmann@33361  1188  --{*definition of quotient and remainder*}  haftmann@33361  1189  [code]: "divmod_int_rel a b = (\(q, r). a = b * q + r \  haftmann@33361  1190  (if 0 < b then 0 \ r \ r < b else b < r \ r \ 0))"  haftmann@33361  1191 haftmann@33361  1192 definition adjust :: "int \ int \ int \ int \ int" where  haftmann@33361  1193  --{*for the division algorithm*}  haftmann@33361  1194  [code]: "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@33361  1195  else (2 * q, r))"  haftmann@33361  1196 haftmann@33361  1197 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@33361  1198 function posDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1199  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@33361  1200  else adjust b (posDivAlg a (2 * b)))"  haftmann@33361  1201 by auto  haftmann@33361  1202 termination by (relation "measure (\(a, b). nat (a - b + 1))")  haftmann@33361  1203  (auto simp add: mult_2)  haftmann@33361  1204 haftmann@33361  1205 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@33361  1206 function negDivAlg :: "int \ int \ int \ int" where  haftmann@33361  1207  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@33361  1208  else adjust b (negDivAlg a (2 * b)))"  haftmann@33361  1209 by auto  haftmann@33361  1210 termination by (relation "measure (\(a, b). nat (- a - b))")  haftmann@33361  1211  (auto simp add: mult_2)  haftmann@33361  1212 haftmann@33361  1213 text{*algorithm for the general case @{term "b\0"}*}  haftmann@33361  1214 definition negateSnd :: "int \ int \ int \ int" where  haftmann@33361  1215  [code_unfold]: "negateSnd = apsnd uminus"  haftmann@33361  1216 haftmann@33361  1217 definition divmod_int :: "int \ int \ int \ int" where  haftmann@33361  1218  --{*The full division algorithm considers all possible signs for a, b  haftmann@33361  1219  including the special case @{text "a=0, b<0"} because  haftmann@33361  1220  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@33361  1221  "divmod_int a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@33361  1222  else if a = 0 then (0, 0)  haftmann@33361  1223  else negateSnd (negDivAlg (-a) (-b))  haftmann@33361  1224  else  haftmann@33361  1225  if 0 < b then negDivAlg a b  haftmann@33361  1226  else negateSnd (posDivAlg (-a) (-b)))"  haftmann@33361  1227 haftmann@33361  1228 instantiation int :: Divides.div  haftmann@33361  1229 begin  haftmann@33361  1230 haftmann@33361  1231 definition  haftmann@33361  1232  "a div b = fst (divmod_int a b)"  haftmann@33361  1233 haftmann@33361  1234 definition  haftmann@33361  1235  "a mod b = snd (divmod_int a b)"  haftmann@33361  1236 haftmann@33361  1237 instance ..  haftmann@33361  1238 paulson@3366  1239 end  haftmann@33361  1240 haftmann@33361  1241 lemma divmod_int_mod_div:  haftmann@33361  1242  "divmod_int p q = (p div q, p mod q)"  haftmann@33361  1243  by (auto simp add: div_int_def mod_int_def)  haftmann@33361  1244 haftmann@33361  1245 text{*  haftmann@33361  1246 Here is the division algorithm in ML:  haftmann@33361  1247 haftmann@33361  1248 \begin{verbatim}  haftmann@33361  1249  fun posDivAlg (a,b) =  haftmann@33361  1250  if ar-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1253  end  haftmann@33361  1254 haftmann@33361  1255  fun negDivAlg (a,b) =  haftmann@33361  1256  if 0\a+b then (~1,a+b)  haftmann@33361  1257  else let val (q,r) = negDivAlg(a, 2*b)  haftmann@33361  1258  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  haftmann@33361  1259  end;  haftmann@33361  1260 haftmann@33361  1261  fun negateSnd (q,r:int) = (q,~r);  haftmann@33361  1262 haftmann@33361  1263  fun divmod (a,b) = if 0\a then  haftmann@33361  1264  if b>0 then posDivAlg (a,b)  haftmann@33361  1265  else if a=0 then (0,0)  haftmann@33361  1266  else negateSnd (negDivAlg (~a,~b))  haftmann@33361  1267  else  haftmann@33361  1268  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  haftmann@33361  1279  ==> q' \ (q::int)"  haftmann@33361  1280 apply (subgoal_tac "r' + b * (q'-q) \ r")  haftmann@33361  1281  prefer 2 apply (simp add: right_diff_distrib)  haftmann@33361  1282 apply (subgoal_tac "0 < b * (1 + q - q') ")  haftmann@33361  1283 apply (erule_tac [2] order_le_less_trans)  haftmann@33361  1284  prefer 2 apply (simp add: right_diff_distrib right_distrib)  haftmann@33361  1285 apply (subgoal_tac "b * q' < b * (1 + q) ")  haftmann@33361  1286  prefer 2 apply (simp add: right_diff_distrib right_distrib)  haftmann@33361  1287 apply (simp add: mult_less_cancel_left)  haftmann@33361  1288 done  haftmann@33361  1289 haftmann@33361  1290 lemma unique_quotient_lemma_neg:  haftmann@33361  1291  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  haftmann@33361  1292  ==> q \ (q'::int)"  haftmann@33361  1293 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  haftmann@33361  1294  auto)  haftmann@33361  1295 haftmann@33361  1296 lemma unique_quotient:  haftmann@33361  1297  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r'); b \ 0 |]  haftmann@33361  1298  ==> q = q'"  haftmann@33361  1299 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)  haftmann@33361  1300 apply (blast intro: order_antisym  haftmann@33361  1301  dest: order_eq_refl [THEN unique_quotient_lemma]  haftmann@33361  1302  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  haftmann@33361  1303 done  haftmann@33361  1304 haftmann@33361  1305 haftmann@33361  1306 lemma unique_remainder:  haftmann@33361  1307  "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r'); b \ 0 |]  haftmann@33361  1308  ==> r = r'"  haftmann@33361  1309 apply (subgoal_tac "q = q'")  haftmann@33361  1310  apply (simp add: divmod_int_rel_def)  haftmann@33361  1311 apply (blast intro: unique_quotient)  haftmann@33361  1312 done  haftmann@33361  1313 haftmann@33361  1314 haftmann@33361  1315 subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}  haftmann@33361  1316 haftmann@33361  1317 text{*And positive divisors*}  haftmann@33361  1318 haftmann@33361  1319 lemma adjust_eq [simp]:  haftmann@33361  1320  "adjust b (q,r) =  haftmann@33361  1321  (let diff = r-b in  haftmann@33361  1322  if 0 \ diff then (2*q + 1, diff)  haftmann@33361  1323  else (2*q, r))"  haftmann@33361  1324 by (simp add: Let_def adjust_def)  haftmann@33361  1325 haftmann@33361  1326 declare posDivAlg.simps [simp del]  haftmann@33361  1327 haftmann@33361  1328 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1329 lemma posDivAlg_eqn:  haftmann@33361  1330  "0 < b ==>  haftmann@33361  1331  posDivAlg a b = (if a a" and "0 < b"  haftmann@33361  1337  shows "divmod_int_rel a b (posDivAlg a b)"  wenzelm@41550  1338  using assms  wenzelm@41550  1339  apply (induct a b rule: posDivAlg.induct)  wenzelm@41550  1340  apply auto  wenzelm@41550  1341  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1342  apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@41550  1343  apply (case_tac "a < b")  wenzelm@41550  1344  apply simp_all  wenzelm@41550  1345  apply (erule splitE)  wenzelm@41550  1346  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)  wenzelm@41550  1347  done  haftmann@33361  1348 haftmann@33361  1349 haftmann@33361  1350 subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}  haftmann@33361  1351 haftmann@33361  1352 text{*And positive divisors*}  haftmann@33361  1353 haftmann@33361  1354 declare negDivAlg.simps [simp del]  haftmann@33361  1355 haftmann@33361  1356 text{*use with a simproc to avoid repeatedly proving the premise*}  haftmann@33361  1357 lemma negDivAlg_eqn:  haftmann@33361  1358  "0 < b ==>  haftmann@33361  1359  negDivAlg a b =  haftmann@33361  1360  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  haftmann@33361  1361 by (rule negDivAlg.simps [THEN trans], simp)  haftmann@33361  1362 haftmann@33361  1363 (*Correctness of negDivAlg: it computes quotients correctly  haftmann@33361  1364  It doesn't work if a=0 because the 0/b equals 0, not -1*)  haftmann@33361  1365 lemma negDivAlg_correct:  haftmann@33361  1366  assumes "a < 0" and "b > 0"  haftmann@33361  1367  shows "divmod_int_rel a b (negDivAlg a b)"  wenzelm@41550  1368  using assms  wenzelm@41550  1369  apply (induct a b rule: negDivAlg.induct)  wenzelm@41550  1370  apply (auto simp add: linorder_not_le)  wenzelm@41550  1371  apply (simp add: divmod_int_rel_def)  wenzelm@41550  1372  apply (subst negDivAlg_eqn, assumption)  wenzelm@41550  1373  apply (case_tac "a + b < (0\int)")  wenzelm@41550  1374  apply simp_all  wenzelm@41550  1375  apply (erule splitE)  wenzelm@41550  1376  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)  wenzelm@41550  1377  done  haftmann@33361  1378 haftmann@33361  1379 haftmann@33361  1380 subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}  haftmann@33361  1381 haftmann@33361  1382 (*the case a=0*)  haftmann@33361  1383 lemma divmod_int_rel_0: "b \ 0 ==> divmod_int_rel 0 b (0, 0)"  haftmann@33361  1384 by (auto simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1385 haftmann@33361  1386 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  haftmann@33361  1387 by (subst posDivAlg.simps, auto)  haftmann@33361  1388 haftmann@33361  1389 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  haftmann@33361  1390 by (subst negDivAlg.simps, auto)  haftmann@33361  1391 haftmann@33361  1392 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"  haftmann@33361  1393 by (simp add: negateSnd_def)  haftmann@33361  1394 haftmann@33361  1395 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"  haftmann@33361  1396 by (auto simp add: split_ifs divmod_int_rel_def)  haftmann@33361  1397 haftmann@33361  1398 lemma divmod_int_correct: "b \ 0 ==> divmod_int_rel a b (divmod_int a b)"  haftmann@33361  1399 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg  haftmann@33361  1400  posDivAlg_correct negDivAlg_correct)  haftmann@33361  1401 haftmann@33361  1402 text{*Arbitrary definitions for division by zero. Useful to simplify  haftmann@33361  1403  certain equations.*}  haftmann@33361  1404 haftmann@33361  1405 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"  haftmann@33361  1406 by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  haftmann@33361  1407 haftmann@33361  1408 haftmann@33361  1409 text{*Basic laws about division and remainder*}  haftmann@33361  1410 haftmann@33361  1411 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  haftmann@33361  1412 apply (case_tac "b = 0", simp)  haftmann@33361  1413 apply (cut_tac a = a and b = b in divmod_int_correct)  haftmann@33361  1414 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)  haftmann@33361  1415 done  haftmann@33361  1416 haftmann@33361  1417 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"  haftmann@33361  1418 by(simp add: zmod_zdiv_equality[symmetric])  haftmann@33361  1419 haftmann@33361  1420 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"  haftmann@33361  1421 by(simp add: mult_commute zmod_zdiv_equality[symmetric])  haftmann@33361  1422 haftmann@33361  1423 text {* Tool setup *}  haftmann@33361  1424 haftmann@33361  1425 ML {*  wenzelm@43594  1426 structure Cancel_Div_Mod_Int = Cancel_Div_Mod  wenzelm@41550  1427 (  haftmann@33361  1428  val div_name = @{const_name div};  haftmann@33361  1429  val mod_name = @{const_name mod};  haftmann@33361  1430  val mk_binop = HOLogic.mk_binop;  haftmann@33361  1431  val mk_sum = Arith_Data.mk_sum HOLogic.intT;  haftmann@33361  1432  val dest_sum = Arith_Data.dest_sum;  haftmann@33361  1433 haftmann@33361  1434  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];  haftmann@33361  1435 haftmann@33361  1436  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@33361  1437  (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))  wenzelm@41550  1438 )  haftmann@33361  1439 *}  haftmann@33361  1440 wenzelm@43594  1441 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}  wenzelm@43594  1442 haftmann@33361  1443 lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b"  haftmann@33361  1444 apply (cut_tac a = a and b = b in divmod_int_correct)  haftmann@33361  1445 apply (auto simp add: divmod_int_rel_def mod_int_def)  haftmann@33361  1446 done  haftmann@33361  1447 wenzelm@45607  1448 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]  wenzelm@45607  1449  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]  haftmann@33361  1450 haftmann@33361  1451 lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b"  haftmann@33361  1452 apply (cut_tac a = a and b = b in divmod_int_correct)  haftmann@33361  1453 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)  haftmann@33361  1454 done  haftmann@33361  1455 wenzelm@45607  1456 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]  wenzelm@45607  1457  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]  haftmann@33361  1458 haftmann@33361  1459 haftmann@33361  1460 subsubsection{*General Properties of div and mod*}  haftmann@33361  1461 haftmann@33361  1462 lemma divmod_int_rel_div_mod: "b \ 0 ==> divmod_int_rel a b (a div b, a mod b)"  haftmann@33361  1463 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1464 apply (force simp add: divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1465 done  haftmann@33361  1466 haftmann@33361  1467 lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r); b \ 0 |] ==> a div b = q"  haftmann@33361  1468 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])  haftmann@33361  1469 haftmann@33361  1470 lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r); b \ 0 |] ==> a mod b = r"  haftmann@33361  1471 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])  haftmann@33361  1472 haftmann@33361  1473 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  haftmann@33361  1474 apply (rule divmod_int_rel_div)  haftmann@33361  1475 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1476 done  haftmann@33361  1477 haftmann@33361  1478 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  haftmann@33361  1479 apply (rule divmod_int_rel_div)  haftmann@33361  1480 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1481 done  haftmann@33361  1482 haftmann@33361  1483 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  haftmann@33361  1484 apply (rule divmod_int_rel_div)  haftmann@33361  1485 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1486 done  haftmann@33361  1487 haftmann@33361  1488 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  haftmann@33361  1489 haftmann@33361  1490 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  haftmann@33361  1491 apply (rule_tac q = 0 in divmod_int_rel_mod)  haftmann@33361  1492 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1493 done  haftmann@33361  1494 haftmann@33361  1495 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  haftmann@33361  1496 apply (rule_tac q = 0 in divmod_int_rel_mod)  haftmann@33361  1497 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1498 done  haftmann@33361  1499 haftmann@33361  1500 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  haftmann@33361  1501 apply (rule_tac q = "-1" in divmod_int_rel_mod)  haftmann@33361  1502 apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1503 done  haftmann@33361  1504 haftmann@33361  1505 text{*There is no @{text mod_neg_pos_trivial}.*}  haftmann@33361  1506 haftmann@33361  1507 haftmann@33361  1508 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)  haftmann@33361  1509 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"  haftmann@33361  1510 apply (case_tac "b = 0", simp)  haftmann@33361  1511 apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified,  haftmann@33361  1512  THEN divmod_int_rel_div, THEN sym])  haftmann@33361  1513 haftmann@33361  1514 done  haftmann@33361  1515 haftmann@33361  1516 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)  haftmann@33361  1517 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"  haftmann@33361  1518 apply (case_tac "b = 0", simp)  haftmann@33361  1519 apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],  haftmann@33361  1520  auto)  haftmann@33361  1521 done  haftmann@33361  1522 haftmann@33361  1523 haftmann@33361  1524 subsubsection{*Laws for div and mod with Unary Minus*}  haftmann@33361  1525 haftmann@33361  1526 lemma zminus1_lemma:  haftmann@33361  1527  "divmod_int_rel a b (q, r)  haftmann@33361  1528  ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@33361  1529  if r=0 then 0 else b-r)"  haftmann@33361  1530 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)  haftmann@33361  1531 haftmann@33361  1532 haftmann@33361  1533 lemma zdiv_zminus1_eq_if:  haftmann@33361  1534  "b \ (0::int)  haftmann@33361  1535  ==> (-a) div b =  haftmann@33361  1536  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  haftmann@33361  1537 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])  haftmann@33361  1538 haftmann@33361  1539 lemma zmod_zminus1_eq_if:  haftmann@33361  1540  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  haftmann@33361  1541 apply (case_tac "b = 0", simp)  haftmann@33361  1542 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])  haftmann@33361  1543 done  haftmann@33361  1544 haftmann@33361  1545 lemma zmod_zminus1_not_zero:  haftmann@33361  1546  fixes k l :: int  haftmann@33361  1547  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@33361  1548  unfolding zmod_zminus1_eq_if by auto  haftmann@33361  1549 haftmann@33361  1550 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  haftmann@33361  1551 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  haftmann@33361  1552 haftmann@33361  1553 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  haftmann@33361  1554 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  haftmann@33361  1555 haftmann@33361  1556 lemma zdiv_zminus2_eq_if:  haftmann@33361  1557  "b \ (0::int)  haftmann@33361  1558  ==> a div (-b) =  haftmann@33361  1559  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  haftmann@33361  1560 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  haftmann@33361  1561 haftmann@33361  1562 lemma zmod_zminus2_eq_if:  haftmann@33361  1563  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  haftmann@33361  1564 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  haftmann@33361  1565 haftmann@33361  1566 lemma zmod_zminus2_not_zero:  haftmann@33361  1567  fixes k l :: int  haftmann@33361  1568  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@33361  1569  unfolding zmod_zminus2_eq_if by auto  haftmann@33361  1570 haftmann@33361  1571 haftmann@33361  1572 subsubsection{*Division of a Number by Itself*}  haftmann@33361  1573 haftmann@33361  1574 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  haftmann@33361  1575 apply (subgoal_tac "0 < a*q")  haftmann@33361  1576  apply (simp add: zero_less_mult_iff, arith)  haftmann@33361  1577 done  haftmann@33361  1578 haftmann@33361  1579 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  haftmann@33361  1580 apply (subgoal_tac "0 \ a* (1-q) ")  haftmann@33361  1581  apply (simp add: zero_le_mult_iff)  haftmann@33361  1582 apply (simp add: right_diff_distrib)  haftmann@33361  1583 done  haftmann@33361  1584 haftmann@33361  1585 lemma self_quotient: "[| divmod_int_rel a a (q, r); a \ (0::int) |] ==> q = 1"  haftmann@33361  1586 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)  haftmann@33361  1587 apply (rule order_antisym, safe, simp_all)  haftmann@33361  1588 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  haftmann@33361  1589 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  haftmann@33361  1590 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  haftmann@33361  1591 done  haftmann@33361  1592 haftmann@33361  1593 lemma self_remainder: "[| divmod_int_rel a a (q, r); a \ (0::int) |] ==> r = 0"  haftmann@33361  1594 apply (frule self_quotient, assumption)  haftmann@33361  1595 apply (simp add: divmod_int_rel_def)  haftmann@33361  1596 done  haftmann@33361  1597 haftmann@33361  1598 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  haftmann@33361  1599 by (simp add: divmod_int_rel_div_mod [THEN self_quotient])  haftmann@33361  1600 haftmann@33361  1601 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  haftmann@33361  1602 lemma zmod_self [simp]: "a mod a = (0::int)"  haftmann@33361  1603 apply (case_tac "a = 0", simp)  haftmann@33361  1604 apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])  haftmann@33361  1605 done  haftmann@33361  1606 haftmann@33361  1607 haftmann@33361  1608 subsubsection{*Computation of Division and Remainder*}  haftmann@33361  1609 haftmann@33361  1610 lemma zdiv_zero [simp]: "(0::int) div b = 0"  haftmann@33361  1611 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1612 haftmann@33361  1613 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@33361  1614 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1615 haftmann@33361  1616 lemma zmod_zero [simp]: "(0::int) mod b = 0"  haftmann@33361  1617 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1618 haftmann@33361  1619 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@33361  1620 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1621 haftmann@33361  1622 text{*a positive, b positive *}  haftmann@33361  1623 haftmann@33361  1624 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@33361  1625 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1626 haftmann@33361  1627 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@33361  1628 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1629 haftmann@33361  1630 text{*a negative, b positive *}  haftmann@33361  1631 haftmann@33361  1632 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@33361  1633 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1634 haftmann@33361  1635 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@33361  1636 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1637 haftmann@33361  1638 text{*a positive, b negative *}  haftmann@33361  1639 haftmann@33361  1640 lemma div_pos_neg:  haftmann@33361  1641  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  haftmann@33361  1642 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1643 haftmann@33361  1644 lemma mod_pos_neg:  haftmann@33361  1645  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  haftmann@33361  1646 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1647 haftmann@33361  1648 text{*a negative, b negative *}  haftmann@33361  1649 haftmann@33361  1650 lemma div_neg_neg:  haftmann@33361  1651  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  haftmann@33361  1652 by (simp add: div_int_def divmod_int_def)  haftmann@33361  1653 haftmann@33361  1654 lemma mod_neg_neg:  haftmann@33361  1655  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  haftmann@33361  1656 by (simp add: mod_int_def divmod_int_def)  haftmann@33361  1657 haftmann@33361  1658 text {*Simplify expresions in which div and mod combine numerical constants*}  haftmann@33361  1659 huffman@45530  1660 lemma int_div_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a div b = q"  huffman@45530  1661  by (rule divmod_int_rel_div [of a b q r],  huffman@45530  1662  simp add: divmod_int_rel_def, simp)  huffman@45530  1663 huffman@45530  1664 lemma int_div_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a div b = q"  huffman@45530  1665  by (rule divmod_int_rel_div [of a b q r],  huffman@45530  1666  simp add: divmod_int_rel_def, simp)  huffman@45530  1667 huffman@45530  1668 lemma int_mod_pos_eq: "\(a::int) = b * q + r; 0 \ r; r < b\ \ a mod b = r"  huffman@45530  1669  by (rule divmod_int_rel_mod [of a b q r],  huffman@45530  1670  simp add: divmod_int_rel_def, simp)  huffman@45530  1671 huffman@45530  1672 lemma int_mod_neg_eq: "\(a::int) = b * q + r; r \ 0; b < r\ \ a mod b = r"  huffman@45530  1673  by (rule divmod_int_rel_mod [of a b q r],  huffman@45530  1674  simp add: divmod_int_rel_def, simp)  huffman@45530  1675 haftmann@33361  1676 lemmas arithmetic_simps =  haftmann@33361  1677  arith_simps  haftmann@33361  1678  add_special  haftmann@35050  1679  add_0_left  haftmann@35050  1680  add_0_right  haftmann@33361  1681  mult_zero_left  haftmann@33361  1682  mult_zero_right  haftmann@33361  1683  mult_1_left  haftmann@33361  1684  mult_1_right  haftmann@33361  1685 haftmann@33361  1686 (* simprocs adapted from HOL/ex/Binary.thy *)  haftmann@33361  1687 ML {*  haftmann@33361  1688 local  huffman@45530  1689  val mk_number = HOLogic.mk_number HOLogic.intT  huffman@45530  1690  val plus = @{term "plus :: int \ int \ int"}  huffman@45530  1691  val times = @{term "times :: int \ int \ int"}  huffman@45530  1692  val zero = @{term "0 :: int"}  huffman@45530  1693  val less = @{term "op < :: int \ int \ bool"}  huffman@45530  1694  val le = @{term "op \ :: int \ int \ bool"}  huffman@45530  1695  val simps = @{thms arith_simps} @ @{thms rel_simps} @  huffman@45530  1696  map (fn th => th RS sym) [@{thm numeral_0_eq_0}, @{thm numeral_1_eq_1}]  huffman@45530  1697  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)  huffman@45530  1698  (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));  haftmann@33361  1699  fun binary_proc proc ss ct =  haftmann@33361  1700  (case Thm.term_of ct of  haftmann@33361  1701  _  t  u =>  haftmann@33361  1702  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  haftmann@33361  1703  SOME args => proc (Simplifier.the_context ss) args  haftmann@33361  1704  | NONE => NONE)  haftmann@33361  1705  | _ => NONE);  haftmann@33361  1706 in  huffman@45530  1707  fun divmod_proc posrule negrule =  huffman@45530  1708  binary_proc (fn ctxt => fn ((a, t), (b, u)) =>  huffman@45530  1709  if b = 0 then NONE else let  huffman@45530  1710  val (q, r) = pairself mk_number (Integer.div_mod a b)  huffman@45530  1711  val goal1 = HOLogic.mk_eq (t, plus  (times  u  q)  r)  huffman@45530  1712  val (goal2, goal3, rule) = if b > 0  huffman@45530  1713  then (le  zero  r, less  r  u, posrule RS eq_reflection)  huffman@45530  1714  else (le  r  zero, less  u  r, negrule RS eq_reflection)  huffman@45530  1715  in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)  haftmann@33361  1716 end  haftmann@33361  1717 *}  haftmann@33361  1718 haftmann@33361  1719 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  huffman@45530  1720  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}  haftmann@33361  1721 haftmann@33361  1722 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  huffman@45530  1723  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}  haftmann@33361  1724 wenzelm@45607  1725 lemmas posDivAlg_eqn_number_of [simp] = posDivAlg_eqn [of "number_of v" "number_of w"] for v w  wenzelm@45607  1726 lemmas negDivAlg_eqn_number_of [simp] = negDivAlg_eqn [of "number_of v" "number_of w"] for v w  haftmann@33361  1727 haftmann@33361  1728 haftmann@33361  1729 text{*Special-case simplification *}  haftmann@33361  1730 haftmann@33361  1731 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  haftmann@33361  1732 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  haftmann@33361  1733 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  haftmann@33361  1734 apply (auto simp del: neg_mod_sign neg_mod_bound)  haftmann@33361  1735 done  haftmann@33361  1736 haftmann@33361  1737 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"  haftmann@33361  1738 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)  haftmann@33361  1739 haftmann@33361  1740 (** The last remaining special cases for constant arithmetic:  haftmann@33361  1741  1 div z and 1 mod z **)  haftmann@33361  1742 wenzelm@45607  1743 lemmas div_pos_pos_1_number_of [simp] = div_pos_pos [OF zero_less_one, of "number_of w"] for w  wenzelm@45607  1744 lemmas div_pos_neg_1_number_of [simp] = div_pos_neg [OF zero_less_one, of "number_of w"] for w  wenzelm@45607  1745 lemmas mod_pos_pos_1_number_of [simp] = mod_pos_pos [OF zero_less_one, of "number_of w"] for w  wenzelm@45607  1746 lemmas mod_pos_neg_1_number_of [simp] = mod_pos_neg [OF zero_less_one, of "number_of w"] for w  wenzelm@45607  1747 lemmas posDivAlg_eqn_1_number_of [simp] = posDivAlg_eqn [of concl: 1 "number_of w"] for w  wenzelm@45607  1748 lemmas negDivAlg_eqn_1_number_of [simp] = negDivAlg_eqn [of concl: 1 "number_of w"] for w  haftmann@33361  1749 haftmann@33361  1750 haftmann@33361  1751 subsubsection{*Monotonicity in the First Argument (Dividend)*}  haftmann@33361  1752 haftmann@33361  1753 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  haftmann@33361  1754 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1755 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1756 apply (rule unique_quotient_lemma)  haftmann@33361  1757 apply (erule subst)  haftmann@33361  1758 apply (erule subst, simp_all)  haftmann@33361  1759 done  haftmann@33361  1760 haftmann@33361  1761 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  haftmann@33361  1762 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1763 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  haftmann@33361  1764 apply (rule unique_quotient_lemma_neg)  haftmann@33361  1765 apply (erule subst)  haftmann@33361  1766 apply (erule subst, simp_all)  haftmann@33361  1767 done  haftmann@33361  1768 haftmann@33361  1769 haftmann@33361  1770 subsubsection{*Monotonicity in the Second Argument (Divisor)*}  haftmann@33361  1771 haftmann@33361  1772 lemma q_pos_lemma:  haftmann@33361  1773  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  haftmann@33361  1774 apply (subgoal_tac "0 < b'* (q' + 1) ")  haftmann@33361  1775  apply (simp add: zero_less_mult_iff)  haftmann@33361  1776 apply (simp add: right_distrib)  haftmann@33361  1777 done  haftmann@33361  1778 haftmann@33361  1779 lemma zdiv_mono2_lemma:  haftmann@33361  1780  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  haftmann@33361  1781  r' < b'; 0 \ r; 0 < b'; b' \ b |]  haftmann@33361  1782  ==> q \ (q'::int)"  haftmann@33361  1783 apply (frule q_pos_lemma, assumption+)  haftmann@33361  1784 apply (subgoal_tac "b*q < b* (q' + 1) ")  haftmann@33361  1785  apply (simp add: mult_less_cancel_left)  haftmann@33361  1786 apply (subgoal_tac "b*q = r' - r + b'*q'")  haftmann@33361  1787  prefer 2 apply simp  haftmann@33361  1788 apply (simp (no_asm_simp) add: right_distrib)  huffman@44766  1789 apply (subst add_commute, rule add_less_le_mono, arith)  haftmann@33361  1790 apply (rule mult_right_mono, auto)  haftmann@33361  1791 done  haftmann@33361  1792 haftmann@33361  1793 lemma zdiv_mono2:  haftmann@33361  1794  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  haftmann@33361  1795 apply (subgoal_tac "b \ 0")  haftmann@33361  1796  prefer 2 apply arith  haftmann@33361  1797 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1798 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  1799 apply (rule zdiv_mono2_lemma)  haftmann@33361  1800 apply (erule subst)  haftmann@33361  1801 apply (erule subst, simp_all)  haftmann@33361  1802 done  haftmann@33361  1803 haftmann@33361  1804 lemma q_neg_lemma:  haftmann@33361  1805  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  haftmann@33361  1806 apply (subgoal_tac "b'*q' < 0")  haftmann@33361  1807  apply (simp add: mult_less_0_iff, arith)  haftmann@33361  1808 done  haftmann@33361  1809 haftmann@33361  1810 lemma zdiv_mono2_neg_lemma:  haftmann@33361  1811  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  haftmann@33361  1812  r < b; 0 \ r'; 0 < b'; b' \ b |]  haftmann@33361  1813  ==> q' \ (q::int)"  haftmann@33361  1814 apply (frule q_neg_lemma, assumption+)  haftmann@33361  1815 apply (subgoal_tac "b*q' < b* (q + 1) ")  haftmann@33361  1816  apply (simp add: mult_less_cancel_left)  haftmann@33361  1817 apply (simp add: right_distrib)  haftmann@33361  1818 apply (subgoal_tac "b*q' \ b'*q'")  haftmann@33361  1819  prefer 2 apply (simp add: mult_right_mono_neg, arith)  haftmann@33361  1820 done  haftmann@33361  1821 haftmann@33361  1822 lemma zdiv_mono2_neg:  haftmann@33361  1823  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  haftmann@33361  1824 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@33361  1825 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  haftmann@33361  1826 apply (rule zdiv_mono2_neg_lemma)  haftmann@33361  1827 apply (erule subst)  haftmann@33361  1828 apply (erule subst, simp_all)  haftmann@33361  1829 done  haftmann@33361  1830 haftmann@33361  1831 haftmann@33361  1832 subsubsection{*More Algebraic Laws for div and mod*}  haftmann@33361  1833 haftmann@33361  1834 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  haftmann@33361  1835 haftmann@33361  1836 lemma zmult1_lemma:  haftmann@33361  1837  "[| divmod_int_rel b c (q, r); c \ 0 |]  haftmann@33361  1838  ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@33361  1839 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)  haftmann@33361  1840 haftmann@33361  1841 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  haftmann@33361  1842 apply (case_tac "c = 0", simp)  haftmann@33361  1843 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])  haftmann@33361  1844 done  haftmann@33361  1845 haftmann@33361  1846 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  haftmann@33361  1847 apply (case_tac "c = 0", simp)  haftmann@33361  1848 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])  haftmann@33361  1849 done  haftmann@33361  1850 haftmann@33361  1851 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"  haftmann@33361  1852 apply (case_tac "b = 0", simp)  haftmann@33361  1853 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  haftmann@33361  1854 done  haftmann@33361  1855 haftmann@33361  1856 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@33361  1857 haftmann@33361  1858 lemma zadd1_lemma:  haftmann@33361  1859  "[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br); c \ 0 |]  haftmann@33361  1860  ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  haftmann@33361  1861 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)  haftmann@33361  1862 haftmann@33361  1863 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@33361  1864 lemma zdiv_zadd1_eq:  haftmann@33361  1865  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@33361  1866 apply (case_tac "c = 0", simp)  haftmann@33361  1867 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)  haftmann@33361  1868 done  haftmann@33361  1869 haftmann@33361  1870 instance int :: ring_div  haftmann@33361  1871 proof  haftmann@33361  1872  fix a b c :: int  haftmann@33361  1873  assume not0: "b \ 0"  haftmann@33361  1874  show "(a + c * b) div b = c + a div b"  haftmann@33361  1875  unfolding zdiv_zadd1_eq [of a "c * b"] using not0  haftmann@33361  1876  by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)  haftmann@33361  1877 next  haftmann@33361  1878  fix a b c :: int  haftmann@33361  1879  assume "a \ 0"  haftmann@33361  1880  then show "(a * b) div (a * c) = b div c"  haftmann@33361  1881  proof (cases "b \ 0 \ c \ 0")  haftmann@33361  1882  case False then show ?thesis by auto  haftmann@33361  1883  next  haftmann@33361  1884  case True then have "b \ 0" and "c \ 0" by auto  haftmann@33361  1885  with a \ 0  haftmann@33361  1886  have "\q r. divmod_int_rel b c (q, r) \ divmod_int_rel (a * b) (a * c) (q, a * r)"  haftmann@33361  1887  apply (auto simp add: divmod_int_rel_def)  haftmann@33361  1888  apply (auto simp add: algebra_simps)  haftmann@33361  1889  apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)  haftmann@33361  1890  done  haftmann@33361  1891  moreover with c \ 0 divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto  haftmann@33361  1892  ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .  haftmann@33361  1893  moreover from a \ 0 c \ 0 have "a * c \ 0" by simp  haftmann@33361  1894  ultimately show ?thesis by (rule divmod_int_rel_div)  haftmann@33361  1895  qed  haftmann@33361  1896 qed auto  haftmann@33361  1897 haftmann@33361  1898 lemma posDivAlg_div_mod:  haftmann@33361  1899  assumes "k \ 0"  haftmann@33361  1900  and "l \ 0"  haftmann@33361  1901  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@33361  1902 proof (cases "l = 0")  haftmann@33361  1903  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@33361  1904 next  haftmann@33361  1905  case False with assms posDivAlg_correct  haftmann@33361  1906  have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@33361  1907  by simp  haftmann@33361  1908  from divmod_int_rel_div [OF this l \ 0] divmod_int_rel_mod [OF this l \ 0]  haftmann@33361  1909  show ?thesis by simp  haftmann@33361  1910 qed  haftmann@33361  1911 haftmann@33361  1912 lemma negDivAlg_div_mod:  haftmann@33361  1913  assumes "k < 0"  haftmann@33361  1914  and "l > 0"  haftmann@33361  1915  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@33361  1916 proof -  haftmann@33361  1917  from assms have "l \ 0" by simp  haftmann@33361  1918  from assms negDivAlg_correct  haftmann@33361  1919  have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@33361  1920  by simp  haftmann@33361  1921  from divmod_int_rel_div [OF this l \ 0] divmod_int_rel_mod [OF this l \ 0]  haftmann@33361  1922  show ?thesis by simp  haftmann@33361  1923 qed  haftmann@33361  1924 haftmann@33361  1925 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  haftmann@33361  1926 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  haftmann@33361  1927 haftmann@33361  1928 (* REVISIT: should this be generalized to all semiring_div types? *)  haftmann@33361  1929 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  haftmann@33361  1930 haftmann@33361  1931 haftmann@33361  1932 subsubsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  haftmann@33361  1933 haftmann@33361  1934 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  haftmann@33361  1935  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  haftmann@33361  1936  to cause particular problems.*)  haftmann@33361  1937 haftmann@33361  1938 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  haftmann@33361  1939 haftmann@33361  1940 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  haftmann@33361  1941 apply (subgoal_tac "b * (c - q mod c) < r * 1")  haftmann@33361  1942  apply (simp add: algebra_simps)  haftmann@33361  1943 apply (rule order_le_less_trans)  haftmann@33361  1944  apply (erule_tac [2] mult_strict_right_mono)  haftmann@33361  1945  apply (rule mult_left_mono_neg)  huffman@35216  1946  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)  haftmann@33361  1947  apply (simp)  haftmann@33361  1948 apply (simp)  haftmann@33361  1949 done  haftmann@33361  1950 haftmann@33361  1951 lemma zmult2_lemma_aux2:  haftmann@33361  1952  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  haftmann@33361  1953 apply (subgoal_tac "b * (q mod c) \ 0")  haftmann@33361  1954  apply arith  haftmann@33361  1955 apply (simp add: mult_le_0_iff)  haftmann@33361  1956 done  haftmann@33361  1957 haftmann@33361  1958 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  haftmann@33361  1959 apply (subgoal_tac "0 \ b * (q mod c) ")  haftmann@33361  1960 apply arith  haftmann@33361  1961 apply (simp add: zero_le_mult_iff)  haftmann@33361  1962 done  haftmann@33361  1963 haftmann@33361  1964 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  haftmann@33361  1965 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  haftmann@33361  1966  apply (simp add: right_diff_distrib)  haftmann@33361  1967 apply (rule order_less_le_trans)  haftmann@33361  1968  apply (erule mult_strict_right_mono)  haftmann@33361  1969  apply (rule_tac [2] mult_left_mono)  haftmann@33361  1970  apply simp  huffman@35216  1971  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)  haftmann@33361  1972 apply simp  haftmann@33361  1973 done  haftmann@33361  1974 haftmann@33361  1975 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); b \ 0; 0 < c |]  haftmann@33361  1976  ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@33361  1977 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff  haftmann@33361  1978  zero_less_mult_iff right_distrib [symmetric]  haftmann@33361  1979  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  haftmann@33361  1980 haftmann@33361  1981 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  haftmann@33361  1982 apply (case_tac "b = 0", simp)  haftmann@33361  1983 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])  haftmann@33361  1984 done  haftmann@33361  1985 haftmann@33361  1986 lemma zmod_zmult2_eq:  haftmann@33361  1987  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  haftmann@33361  1988 apply (case_tac "b = 0", simp)  haftmann@33361  1989 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])  haftmann@33361  1990 done  haftmann@33361  1991 haftmann@33361  1992 haftmann@33361  1993 subsubsection {*Splitting Rules for div and mod*}  haftmann@33361  1994 haftmann@33361  1995 text{*The proofs of the two lemmas below are essentially identical*}  haftmann@33361  1996 haftmann@33361  1997 lemma split_pos_lemma:  haftmann@33361  1998  "0  haftmann@33361  1999  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  haftmann@33361  2000 apply (rule iffI, clarify)  haftmann@33361  2001  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2002  apply (subst mod_add_eq)  haftmann@33361  2003  apply (subst zdiv_zadd1_eq)  haftmann@33361  2004  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  haftmann@33361  2005 txt{*converse direction*}  haftmann@33361  2006 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2007 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2008 done  haftmann@33361  2009 haftmann@33361  2010 lemma split_neg_lemma:  haftmann@33361  2011  "k<0 ==>  haftmann@33361  2012  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  haftmann@33361  2013 apply (rule iffI, clarify)  haftmann@33361  2014  apply (erule_tac P="P ?x ?y" in rev_mp)  haftmann@33361  2015  apply (subst mod_add_eq)  haftmann@33361  2016  apply (subst zdiv_zadd1_eq)  haftmann@33361  2017  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  haftmann@33361  2018 txt{*converse direction*}  haftmann@33361  2019 apply (drule_tac x = "n div k" in spec)  haftmann@33361  2020 apply (drule_tac x = "n mod k" in spec, simp)  haftmann@33361  2021 done  haftmann@33361  2022 haftmann@33361  2023 lemma split_zdiv:  haftmann@33361  2024  "P(n div k :: int) =  haftmann@33361  2025  ((k = 0 --> P 0) &  haftmann@33361  2026  (0 (\i j. 0\j & j P i)) &  haftmann@33361  2027  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  haftmann@33361  2028 apply (case_tac "k=0", simp)  haftmann@33361  2029 apply (simp only: linorder_neq_iff)  haftmann@33361  2030 apply (erule disjE)  haftmann@33361  2031  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  haftmann@33361  2032  split_neg_lemma [of concl: "%x y. P x"])  haftmann@33361  2033 done  haftmann@33361  2034 haftmann@33361  2035 lemma split_zmod:  haftmann@33361  2036  "P(n mod k :: int) =  haftmann@33361  2037  ((k = 0 --> P n) &  haftmann@33361  2038  (0 (\i j. 0\j & j P j)) &  haftmann@33361  2039  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  haftmann@33361  2040 apply (case_tac "k=0", simp)  haftmann@33361  2041 apply (simp only: linorder_neq_iff)  haftmann@33361  2042 apply (erule disjE)  haftmann@33361  2043  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  haftmann@33361  2044  split_neg_lemma [of concl: "%x y. P y"])  haftmann@33361  2045 done  haftmann@33361  2046 webertj@33730  2047 text {* Enable (lin)arith to deal with @{const div} and @{const mod}  webertj@33730  2048  when these are applied to some constant that is of the form  webertj@33730  2049  @{term "number_of k"}: *}  wenzelm@45607  2050 declare split_zdiv [of _ _ "number_of k", arith_split] for k  wenzelm@45607  2051 declare split_zmod [of _ _ "number_of k", arith_split] for k  haftmann@33361  2052 haftmann@33361  2053 haftmann@33361  2054 subsubsection{*Speeding up the Division Algorithm with Shifting*}  haftmann@33361  2055 haftmann@33361  2056 text{*computing div by shifting *}  haftmann@33361  2057 haftmann@33361  2058 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  haftmann@33361  2059 proof cases  haftmann@33361  2060  assume "a=0"  haftmann@33361  2061  thus ?thesis by simp  haftmann@33361  2062 next  haftmann@33361  2063  assume "a\0" and le_a: "0\a"  haftmann@33361  2064  hence a_pos: "1 \ a" by arith  haftmann@33361  2065  hence one_less_a2: "1 < 2 * a" by arith  haftmann@33361  2066  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  haftmann@33361  2067  unfolding mult_le_cancel_left  haftmann@33361  2068  by (simp add: add1_zle_eq add_commute [of 1])  haftmann@33361  2069  with a_pos have "0 \ b mod a" by simp  haftmann@33361  2070  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  haftmann@33361  2071  by (simp add: mod_pos_pos_trivial one_less_a2)  haftmann@33361  2072  with le_2a  haftmann@33361  2073  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  haftmann@33361  2074  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  haftmann@33361  2075  right_distrib)  haftmann@33361  2076  thus ?thesis  haftmann@33361  2077  by (subst zdiv_zadd1_eq,  haftmann@33361  2078  simp add: mod_mult_mult1 one_less_a2  haftmann@33361  2079  div_pos_pos_trivial)  haftmann@33361  2080 qed  haftmann@33361  2081 boehmes@35815  2082 lemma neg_zdiv_mult_2:  boehmes@35815  2083  assumes A: "a \ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  boehmes@35815  2084 proof -  boehmes@35815  2085  have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp  boehmes@35815  2086  have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"  boehmes@35815  2087  by (rule pos_zdiv_mult_2, simp add: A)  boehmes@35815  2088  thus ?thesis  boehmes@35815  2089  by (simp only: R zdiv_zminus_zminus diff_minus  boehmes@35815  2090  minus_add_distrib [symmetric] mult_minus_right)  boehmes@35815  2091 qed  haftmann@33361  2092 haftmann@33361  2093 lemma zdiv_number_of_Bit0 [simp]:  haftmann@33361  2094  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  haftmann@33361  2095  number_of v div (number_of w :: int)"  haftmann@33361  2096 by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])  haftmann@33361  2097 haftmann@33361  2098 lemma zdiv_number_of_Bit1 [simp]:  haftmann@33361  2099  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  haftmann@33361  2100  (if (0::int) \ number_of w  haftmann@33361  2101  then number_of v div (number_of w)  haftmann@33361  2102  else (number_of v + (1::int)) div (number_of w))"  haftmann@33361  2103 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  haftmann@33361  2104 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])  haftmann@33361  2105 done  haftmann@33361  2106 haftmann@33361  2107 haftmann@33361  2108 subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}  haftmann@33361  2109 haftmann@33361  2110 lemma pos_zmod_mult_2:  haftmann@33361  2111  fixes a b :: int  haftmann@33361  2112  assumes "0 \ a"  haftmann@33361  2113  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  haftmann@33361  2114 proof (cases "0 < a")  haftmann@33361  2115  case False with assms show ?thesis by simp  haftmann@33361  2116 next  haftmann@33361  2117  case True  haftmann@33361  2118  then have "b mod a < a" by (rule pos_mod_bound)  haftmann@33361  2119  then have "1 + b mod a \ a" by simp  haftmann@33361  2120  then have A: "2 * (1 + b mod a) \ 2 * a" by simp  haftmann@33361  2121  from 0 < a have "0 \ b mod a" by (rule pos_mod_sign)  haftmann@33361  2122  then have B: "0 \ 1 + 2 * (b mod a)" by simp  haftmann@33361  2123  have "((1\int) mod ((2\int) * a) + (2\int) * b mod ((2\int) * a)) mod ((2\int) * a) = (1\int) + (2\int) * (b mod a)"  haftmann@33361  2124  using 0 < a and A  haftmann@33361  2125  by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)  haftmann@33361  2126  then show ?thesis by (subst mod_add_eq)  haftmann@33361  2127 qed  haftmann@33361  2128 haftmann@33361  2129 lemma neg_zmod_mult_2:  haftmann@33361  2130  fixes a b :: int  haftmann@33361  2131  assumes "a \ 0"  haftmann@33361  2132  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  haftmann@33361  2133 proof -  haftmann@33361  2134  from assms have "0 \ - a" by auto  haftmann@33361  2135  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"  haftmann@33361  2136  by (rule pos_zmod_mult_2)  haftmann@33361  2137  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)  haftmann@33361  2138  (simp add: diff_minus add_ac)  haftmann@33361  2139 qed  haftmann@33361  2140 haftmann@33361  2141 lemma zmod_number_of_Bit0 [simp]:  haftmann@33361  2142  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  haftmann@33361  2143  (2::int) * (number_of v mod number_of w)"  haftmann@33361  2144 apply (simp only: number_of_eq numeral_simps)  haftmann@33361  2145 apply (simp add: mod_mult_mult1 pos_zmod_mult_2  haftmann@33361  2146  neg_zmod_mult_2 add_ac mult_2 [symmetric])  haftmann@33361  2147 done  haftmann@33361  2148 haftmann@33361  2149 lemma zmod_number_of_Bit1 [simp]:  haftmann@33361  2150  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  haftmann@33361  2151  (if (0::int) \ number_of w  haftmann@33361  2152  then 2 * (number_of v mod number_of w) + 1  haftmann@33361  2153  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  haftmann@33361  2154 apply (simp only: number_of_eq numeral_simps)  haftmann@33361  2155 apply (simp add: mod_mult_mult1 pos_zmod_mult_2  haftmann@33361  2156  neg_zmod_mult_2 add_ac mult_2 [symmetric])  haftmann@33361  2157 done  haftmann@33361  2158 haftmann@33361  2159 nipkow@39489  2160 lemma zdiv_eq_0_iff:  nipkow@39489  2161  "(i::int) div k = 0 \ k=0 \ 0\i \ i i\0 \ k ?R" by (rule split_zdiv[THEN iffD2]) simp  nipkow@39489  2165  with ?L show ?R by blast  nipkow@39489  2166 next  nipkow@39489  2167  assume ?R thus ?L  nipkow@39489  2168  by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)  nipkow@39489  2169 qed  nipkow@39489  2170 nipkow@39489  2171 haftmann@33361  2172 subsubsection{*Quotients of Signs*}  haftmann@33361  2173 haftmann@33361  2174 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  haftmann@33361  2175 apply (subgoal_tac "a div b \ -1", force)  haftmann@33361  2176 apply (rule order_trans)  haftmann@33361  2177 apply (rule_tac a' = "-1" in zdiv_mono1)  haftmann@33361  2178 apply (auto simp add: div_eq_minus1)  haftmann@33361  2179 done  haftmann@33361  2180 haftmann@33361  2181 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  haftmann@33361  2182 by (drule zdiv_mono1_neg, auto)  haftmann@33361  2183 haftmann@33361  2184 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  haftmann@33361  2185 by (drule zdiv_mono1, auto)  haftmann@33361  2186 nipkow@33804  2187 text{* Now for some equivalences of the form @{text"a div b >=< 0 \ \"}  nipkow@33804  2188 conditional upon the sign of @{text a} or @{text b}. There are many more.  nipkow@33804  2189 They should all be simp rules unless that causes too much search. *}  nipkow@33804  2190 haftmann@33361  2191 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  haftmann@33361  2192 apply auto  haftmann@33361  2193 apply (drule_tac [2] zdiv_mono1)  haftmann@33361  2194 apply (auto simp add: linorder_neq_iff)  haftmann@33361  2195 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  haftmann@33361  2196 apply (blast intro: div_neg_pos_less0)  haftmann@33361  2197 done  haftmann@33361  2198 haftmann@33361  2199 lemma neg_imp_zdiv_nonneg_iff:  nipkow@33804  2200  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  haftmann@33361  2201 apply (subst zdiv_zminus_zminus [symmetric])  haftmann@33361  2202 apply (subst pos_imp_zdiv_nonneg_iff, auto)  haftmann@33361  2203 done  haftmann@33361  2204 haftmann@33361  2205 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  haftmann@33361  2206 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  haftmann@33361  2207 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  haftmann@33361  2208 nipkow@39489  2209 lemma pos_imp_zdiv_pos_iff:  nipkow@39489  2210  "0 0 < (i::int) div k \ k \ i"  nipkow@39489  2211 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]  nipkow@39489  2212 by arith  nipkow@39489  2213 haftmann@33361  2214 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  haftmann@33361  2215 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  haftmann@33361  2216 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  haftmann@33361  2217 nipkow@33804  2218 lemma nonneg1_imp_zdiv_pos_iff:  nipkow@33804  2219  "(0::int) <= a \ (a div b > 0) = (a >= b & b>0)"  nipkow@33804  2220 apply rule  nipkow@33804  2221  apply rule  nipkow@33804  2222  using div_pos_pos_trivial[of a b]apply arith  nipkow@33804  2223  apply(cases "b=0")apply simp  nipkow@33804  2224  using div_nonneg_neg_le0[of a b]apply arith  nipkow@33804  2225 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp  nipkow@33804  2226 done  nipkow@33804  2227 haftmann@33361  2228 nipkow@39489  2229 lemma zmod_le_nonneg_dividend: "(m::int) \ 0 ==> m mod k \ m"  nipkow@39489  2230 apply (rule split_zmod[THEN iffD2])  nipkow@44890  2231 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)  nipkow@39489  2232 done  nipkow@39489  2233 nipkow@39489  2234 haftmann@33361  2235 subsubsection {* The Divides Relation *}  haftmann@33361  2236 haftmann@33361  2237 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@45607  2238  dvd_eq_mod_eq_0 [of "number_of x" "number_of y"] for x y :: int  haftmann@33361  2239 haftmann@33361  2240 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  haftmann@33361  2241  by (rule dvd_mod) (* TODO: remove *)  haftmann@33361  2242 haftmann@33361  2243 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  haftmann@33361  2244  by (rule dvd_mod_imp_dvd) (* TODO: remove *)  haftmann@33361  2245 haftmann@33361  2246 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  haftmann@33361  2247  using zmod_zdiv_equality[where a="m" and b="n"]  haftmann@33361  2248  by (simp add: algebra_simps)  haftmann@33361  2249 haftmann@33361  2250 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  haftmann@33361  2251 apply (induct "y", auto)  haftmann@33361  2252 apply (rule zmod_zmult1_eq [THEN trans])  haftmann@33361  2253 apply (simp (no_asm_simp))  haftmann@33361  2254 apply (rule mod_mult_eq [symmetric])  haftmann@33361  2255 done  haftmann@33361  2256 haftmann@33361  2257 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  haftmann@33361  2258 apply (subst split_div, auto)  haftmann@33361  2259 apply (subst split_zdiv, auto)  haftmann@33361  2260 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  2261 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2262 done  haftmann@33361  2263 haftmann@33361  2264 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  haftmann@33361  2265 apply (subst split_mod, auto)  haftmann@33361  2266 apply (subst split_zmod, auto)  haftmann@33361  2267 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  haftmann@33361  2268  in unique_remainder)  haftmann@33361  2269 apply (auto simp add: divmod_int_rel_def of_nat_mult)  haftmann@33361  2270 done  haftmann@33361  2271 haftmann@33361  2272 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  haftmann@33361  2273 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  haftmann@33361  2274 haftmann@33361  2275 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  haftmann@33361  2276 apply (subgoal_tac "m mod n = 0")  haftmann@33361  2277  apply (simp add: zmult_div_cancel)  haftmann@33361  2278 apply (simp only: dvd_eq_mod_eq_0)  haftmann@33361  2279 done  haftmann@33361  2280 haftmann@33361  2281 text{*Suggested by Matthias Daum*}  haftmann@33361  2282 lemma int_power_div_base:  haftmann@33361  2283  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  haftmann@33361  2284 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  haftmann@33361  2285  apply (erule ssubst)  haftmann@33361  2286  apply (simp only: power_add)  haftmann@33361  2287  apply simp_all  haftmann@33361  2288 done  haftmann@33361  2289 haftmann@33361  2290 text {* by Brian Huffman *}  haftmann@33361  2291 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@33361  2292 by (rule mod_minus_eq [symmetric])  haftmann@33361  2293 haftmann@33361  2294 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@33361  2295 by (rule mod_diff_left_eq [symmetric])  haftmann@33361  2296 haftmann@33361  2297 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@33361  2298 by (rule mod_diff_right_eq [symmetric])  haftmann@33361  2299 haftmann@33361  2300 lemmas zmod_simps =  haftmann@33361  2301  mod_add_left_eq [symmetric]  haftmann@33361  2302  mod_add_right_eq [symmetric]  haftmann@33361  2303  zmod_zmult1_eq [symmetric]  haftmann@33361  2304  mod_mult_left_eq [symmetric]  haftmann@33361  2305  zpower_zmod  haftmann@33361  2306  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@33361  2307 haftmann@33361  2308 text {* Distributive laws for function @{text nat}. *}  haftmann@33361  2309 haftmann@33361  2310 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  haftmann@33361  2311 apply (rule linorder_cases [of y 0])  haftmann@33361  2312 apply (simp add: div_nonneg_neg_le0)  haftmann@33361  2313 apply simp  haftmann@33361  2314 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  haftmann@33361  2315 done  haftmann@33361  2316 haftmann@33361  2317 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  haftmann@33361  2318 lemma nat_mod_distrib:  haftmann@33361  2319  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  haftmann@33361  2320 apply (case_tac "y = 0", simp)  haftmann@33361  2321 apply (simp add: nat_eq_iff zmod_int)  haftmann@33361  2322 done  haftmann@33361  2323 haftmann@33361  2324 text {* transfer setup *}  haftmann@33361  2325 haftmann@33361  2326 lemma transfer_nat_int_functions:  haftmann@33361  2327  "(x::int) >= 0 \ y >= 0 \ (nat x) div (nat y) = nat (x div y)"  haftmann@33361  2328  "(x::int) >= 0 \ y >= 0 \ (nat x) mod (nat y) = nat (x mod y)"  haftmann@33361  2329  by (auto simp add: nat_div_distrib nat_mod_distrib)  haftmann@33361  2330 haftmann@33361  2331 lemma transfer_nat_int_function_closures:  haftmann@33361  2332  "(x::int) >= 0 \ y >= 0 \ x div y >= 0"  haftmann@33361  2333  "(x::int) >= 0 \ y >= 0 \ x mod y >= 0"  haftmann@33361  2334  apply (cases "y = 0")  haftmann@33361  2335  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  haftmann@33361  2336  apply (cases "y = 0")  haftmann@33361  2337  apply auto  haftmann@33361  2338 done  haftmann@33361  2339 haftmann@35644  2340 declare transfer_morphism_nat_int [transfer add return:  haftmann@33361  2341  transfer_nat_int_functions  haftmann@33361  2342  transfer_nat_int_function_closures  haftmann@33361  2343 ]  haftmann@33361  2344 haftmann@33361  2345 lemma transfer_int_nat_functions:  haftmann@33361  2346  "(int x) div (int y) = int (x div y)"  haftmann@33361  2347  "(int x) mod (int y) = int (x mod y)"  haftmann@33361  2348  by (auto simp add: zdiv_int zmod_int)  haftmann@33361  2349 haftmann@33361  2350 lemma transfer_int_nat_function_closures:  haftmann@33361  2351  "is_nat x \ is_nat y \ is_nat (x div y)"  haftmann@33361  2352  "is_nat x \ is_nat y \ is_nat (x mod y)"  haftmann@33361  2353  by (simp_all only: is_nat_def transfer_nat_int_function_closures)  haftmann@33361  2354 haftmann@35644  2355 declare transfer_morphism_int_nat [transfer add return:  haftmann@33361  2356  transfer_int_nat_functions  haftmann@33361  2357  transfer_int_nat_function_closures  haftmann@33361  2358 ]  haftmann@33361  2359 haftmann@33361  2360 text{*Suggested by Matthias Daum*}  haftmann@33361  2361 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  haftmann@33361  2362 apply (subgoal_tac "nat x div nat k < nat x")  nipkow@34225  2363  apply (simp add: nat_div_distrib [symmetric])  haftmann@33361  2364 apply (rule Divides.div_less_dividend, simp_all)  haftmann@33361  2365 done  haftmann@33361  2366 haftmann@33361  2367 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  haftmann@33361  2368 proof  haftmann@33361  2369  assume H: "x mod n = y mod n"  haftmann@33361  2370  hence "x mod n - y mod n = 0" by simp  haftmann@33361  2371  hence "(x mod n - y mod n) mod n = 0" by simp  haftmann@33361  2372  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])  haftmann@33361  2373  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)  haftmann@33361  2374 next  haftmann@33361  2375  assume H: "n dvd x - y"  haftmann@33361  2376  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  haftmann@33361  2377  hence "x = n*k + y" by simp  haftmann@33361  2378  hence "x mod n = (n*k + y) mod n" by simp  haftmann@33361  2379  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)  haftmann@33361  2380 qed  haftmann@33361  2381 haftmann@33361  2382 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  haftmann@33361  2383  shows "\q. x = y + n * q"  haftmann@33361  2384 proof-  haftmann@33361  2385  from xy have th: "int x - int y = int (x - y)" by simp  haftmann@33361  2386  from xyn have "int x mod int n = int y mod int n"  haftmann@33361  2387  by (simp add: zmod_int[symmetric])  haftmann@33361  2388  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  haftmann@33361  2389  hence "n dvd x - y" by (simp add: th zdvd_int)  haftmann@33361  2390  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  haftmann@33361  2391 qed  haftmann@33361  2392 haftmann@33361  2393 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  haftmann@33361  2394  (is "?lhs = ?rhs")  haftmann@33361  2395 proof  haftmann@33361  2396  assume H: "x mod n = y mod n"  haftmann@33361  2397  {assume xy: "x \ y"  haftmann@33361  2398  from H have th: "y mod n = x mod n" by simp  haftmann@33361  2399  from nat_mod_eq_lemma[OF th xy] have ?rhs  haftmann@33361  2400  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  haftmann@33361  2401  moreover  haftmann@33361  2402  {assume xy: "y \ x"  haftmann@33361  2403  from nat_mod_eq_lemma[OF H xy] have ?rhs  haftmann@33361  2404  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  haftmann@33361  2405  ultimately show ?rhs using linear[of x y] by blast  haftmann@33361  2406 next  haftmann@33361  2407  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  haftmann@33361  2408  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  haftmann@33361  2409  thus ?lhs by simp  haftmann@33361  2410 qed  haftmann@33361  2411 haftmann@33361  2412 lemma div_nat_number_of [simp]:  haftmann@33361  2413  "(number_of v :: nat) div number_of v' =  haftmann@33361  2414  (if neg (number_of v :: int) then 0  haftmann@33361  2415  else nat (number_of v div number_of v'))"  haftmann@33361  2416  unfolding nat_number_of_def number_of_is_id neg_def  haftmann@33361  2417  by (simp add: nat_div_distrib)  haftmann@33361  2418 haftmann@33361  2419 lemma one_div_nat_number_of [simp]:  haftmann@33361  2420  "Suc 0 div number_of v' = nat (1 div number_of v')"  haftmann@33361  2421 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])  haftmann@33361  2422 haftmann@33361  2423 lemma mod_nat_number_of [simp]:  haftmann@33361  2424  "(number_of v :: nat) mod number_of v' =  haftmann@33361  2425  (if neg (number_of v :: int) then 0  haftmann@33361  2426  else if neg (number_of v' :: int) then number_of v  haftmann@33361  2427  else nat (number_of v mod number_of v'))"  haftmann@33361  2428  unfolding nat_number_of_def number_of_is_id neg_def  haftmann@33361  2429  by (simp add: nat_mod_distrib)  haftmann@33361  2430 haftmann@33361  2431 lemma one_mod_nat_number_of [simp]:  haftmann@33361  2432  "Suc 0 mod number_of v' =  haftmann@33361  2433  (if neg (number_of v' :: int) then Suc 0  haftmann@33361  2434  else nat (1 mod number_of v'))"  haftmann@33361  2435 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])  haftmann@33361  2436 wenzelm@45607  2437 lemmas dvd_eq_mod_eq_0_number_of [simp] =  wenzelm@45607  2438  dvd_eq_mod_eq_0 [of "number_of x" "number_of y"] for x y  haftmann@33361  2439 haftmann@33361  2440 blanchet@34126  2441 subsubsection {* Nitpick *}  blanchet@34126  2442 blanchet@34126  2443 lemma zmod_zdiv_equality':  blanchet@34126  2444 "(m\int) mod n = m - (m div n) * n"  blanchet@34126  2445 by (rule_tac P="%x. m mod n = x - (m div n) * n"  blanchet@34126  2446  in subst [OF mod_div_equality [of _ n]])  blanchet@34126  2447  arith  blanchet@34126  2448 blanchet@41792  2449 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'  blanchet@34126  2450 haftmann@35673  2451 haftmann@33361  2452 subsubsection {* Code generation *}  haftmann@33361  2453 haftmann@33361  2454 definition pdivmod :: "int \ int \ int \ int" where  haftmann@33361  2455  "pdivmod k l = (\k\ div \l\, \k\ mod \l$$"  haftmann@33361  2456 haftmann@33361  2457 lemma pdivmod_posDivAlg [code]:  haftmann@33361  2458  "pdivmod k l = (if l = 0 then (0, \k\) else posDivAlg \k\ \l\)"  haftmann@33361  2459 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)  haftmann@33361  2460 haftmann@33361  2461 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@33361  2462  apsnd ((op *) (sgn l)) (if 0 < l \ 0 \ k \ l < 0 \ k < 0  haftmann@33361  2463  then pdivmod k l  haftmann@33361  2464  else (let (r, s) = pdivmod k l in  haftmann@33361  2465  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@33361  2466 proof -  haftmann@33361  2467  have aux: "\q::int. - k = l * q \ k = l * - q" by auto  haftmann@33361  2468  show ?thesis  haftmann@33361  2469  by (simp add: divmod_int_mod_div pdivmod_def)  haftmann@33361  2470  (auto simp add: aux not_less not_le zdiv_zminus1_eq_if  haftmann@33361  2471  zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)  haftmann@33361  2472 qed  haftmann@33361  2473 haftmann@33361  2474 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  2475  apsnd ((op *) (sgn l)) (if sgn k = sgn l  haftmann@33361  2476  then pdivmod k l  haftmann@33361  2477  else (let (r, s) = pdivmod k l in  haftmann@33361  2478  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@33361  2479 proof -  haftmann@33361  2480  have "k \ 0 \ l \ 0 \ 0 < l \ 0 \ k \ l < 0 \ k < 0 \ sgn k = sgn l"  haftmann@33361  2481  by (auto simp add: not_less sgn_if)  haftmann@33361  2482  then show ?thesis by (simp add: divmod_int_pdivmod)  haftmann@33361  2483 qed  haftmann@33361  2484 haftmann@35673  2485 context ring_1  haftmann@35673  2486 begin  haftmann@35673  2487 haftmann@35673  2488 lemma of_int_num [code]:  haftmann@35673  2489  "of_int k = (if k = 0 then 0 else if k < 0 then  haftmann@35673  2490  - of_int (- k) else let  haftmann@35673  2491  (l, m) = divmod_int k 2;  haftmann@35673  2492  l' = of_int l  haftmann@35673  2493  in if m = 0 then l' + l' else l' + l' + 1)"  haftmann@35673  2494 proof -  haftmann@35673  2495  have aux1: "k mod (2\int) \ (0\int) \  haftmann@35673  2496  of_int k = of_int (k div 2 * 2 + 1)"  haftmann@35673  2497  proof -  haftmann@35673  2498  have "k mod 2 < 2" by (auto intro: pos_mod_bound)  haftmann@35673  2499  moreover have "0 \ k mod 2" by (auto intro: pos_mod_sign)  haftmann@35673  2500  moreover assume "k mod 2 \ 0"  haftmann@35673  2501  ultimately have "k mod 2 = 1" by arith  haftmann@35673  2502  moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp  haftmann@35673  2503  ultimately show ?thesis by auto `