src/HOL/IntDiv.thy
 author wenzelm Wed Jul 15 23:48:21 2009 +0200 (2009-07-15) changeset 32010 cb1a1c94b4cd parent 31998 2c7a24f74db9 child 32075 e8e0fb5da77a permissions -rw-r--r--
more antiquotations;
 wenzelm@23164  1 (* Title: HOL/IntDiv.thy  wenzelm@23164  2  Author: Lawrence C Paulson, Cambridge University Computer Laboratory  wenzelm@23164  3  Copyright 1999 University of Cambridge  wenzelm@23164  4 wenzelm@23164  5 *)  wenzelm@23164  6 haftmann@29651  7 header{* The Division Operators div and mod *}  wenzelm@23164  8 wenzelm@23164  9 theory IntDiv  haftmann@25919  10 imports Int Divides FunDef  wenzelm@23164  11 begin  wenzelm@23164  12 haftmann@29651  13 definition divmod_rel :: "int \ int \ int \ int \ bool" where  wenzelm@23164  14  --{*definition of quotient and remainder*}  haftmann@29651  15  [code]: "divmod_rel a b = ($$q, r). a = b * q + r \  haftmann@29651  16  (if 0 < b then 0 \ r \ r < b else b < r \ r \ 0))"  wenzelm@23164  17 haftmann@29651  18 definition adjust :: "int \ int \ int \ int \ int" where  wenzelm@23164  19  --{*for the division algorithm*}  haftmann@29651  20  [code]: "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@29651  21  else (2 * q, r))"  wenzelm@23164  22 wenzelm@23164  23 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@29651  24 function posDivAlg :: "int \ int \ int \ int" where  haftmann@29651  25  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@29651  26  else adjust b (posDivAlg a (2 * b)))"  wenzelm@23164  27 by auto  haftmann@29651  28 termination by (relation "measure (\(a, b). nat (a - b + 1))") auto  wenzelm@23164  29 wenzelm@23164  30 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@29651  31 function negDivAlg :: "int \ int \ int \ int" where  haftmann@29651  32  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@29651  33  else adjust b (negDivAlg a (2 * b)))"  wenzelm@23164  34 by auto  haftmann@29651  35 termination by (relation "measure (\(a, b). nat (- a - b))") auto  wenzelm@23164  36 wenzelm@23164  37 text{*algorithm for the general case @{term "b\0"}*}  haftmann@29651  38 definition negateSnd :: "int \ int \ int \ int" where  haftmann@31998  39  [code_inline]: "negateSnd = apsnd uminus"  wenzelm@23164  40 haftmann@29651  41 definition divmod :: "int \ int \ int \ int" where  wenzelm@23164  42  --{*The full division algorithm considers all possible signs for a, b  wenzelm@23164  43  including the special case @{text "a=0, b<0"} because  wenzelm@23164  44  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@29651  45  "divmod a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@29651  46  else if a = 0 then (0, 0)  wenzelm@23164  47  else negateSnd (negDivAlg (-a) (-b))  wenzelm@23164  48  else  haftmann@29651  49  if 0 < b then negDivAlg a b  haftmann@29651  50  else negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  51 haftmann@25571  52 instantiation int :: Divides.div  haftmann@25571  53 begin  haftmann@25571  54 haftmann@25571  55 definition  haftmann@29651  56  div_def: "a div b = fst (divmod a b)"  haftmann@25571  57 haftmann@25571  58 definition  haftmann@29651  59  mod_def: "a mod b = snd (divmod a b)"  haftmann@25571  60 haftmann@25571  61 instance ..  haftmann@25571  62 haftmann@25571  63 end  wenzelm@23164  64 haftmann@29651  65 lemma divmod_mod_div:  haftmann@29651  66  "divmod p q = (p div q, p mod q)"  wenzelm@23164  67  by (auto simp add: div_def mod_def)  wenzelm@23164  68 wenzelm@23164  69 text{*  wenzelm@23164  70 Here is the division algorithm in ML:  wenzelm@23164  71 wenzelm@23164  72 \begin{verbatim}  wenzelm@23164  73  fun posDivAlg (a,b) =  wenzelm@23164  74  if ar-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  77  end  wenzelm@23164  78 wenzelm@23164  79  fun negDivAlg (a,b) =  wenzelm@23164  80  if 0\a+b then (~1,a+b)  wenzelm@23164  81  else let val (q,r) = negDivAlg(a, 2*b)  wenzelm@23164  82  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  83  end;  wenzelm@23164  84 wenzelm@23164  85  fun negateSnd (q,r:int) = (q,~r);  wenzelm@23164  86 haftmann@29651  87  fun divmod (a,b) = if 0\a then  wenzelm@23164  88  if b>0 then posDivAlg (a,b)  wenzelm@23164  89  else if a=0 then (0,0)  wenzelm@23164  90  else negateSnd (negDivAlg (~a,~b))  wenzelm@23164  91  else  wenzelm@23164  92  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  wenzelm@23164  103  ==> q' \ (q::int)"  wenzelm@23164  104 apply (subgoal_tac "r' + b * (q'-q) \ r")  wenzelm@23164  105  prefer 2 apply (simp add: right_diff_distrib)  wenzelm@23164  106 apply (subgoal_tac "0 < b * (1 + q - q') ")  wenzelm@23164  107 apply (erule_tac [2] order_le_less_trans)  wenzelm@23164  108  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  109 apply (subgoal_tac "b * q' < b * (1 + q) ")  wenzelm@23164  110  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  111 apply (simp add: mult_less_cancel_left)  wenzelm@23164  112 done  wenzelm@23164  113 wenzelm@23164  114 lemma unique_quotient_lemma_neg:  wenzelm@23164  115  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  wenzelm@23164  116  ==> q \ (q'::int)"  wenzelm@23164  117 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  wenzelm@23164  118  auto)  wenzelm@23164  119 wenzelm@23164  120 lemma unique_quotient:  haftmann@29651  121  "[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \ 0 |]  wenzelm@23164  122  ==> q = q'"  haftmann@29651  123 apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)  wenzelm@23164  124 apply (blast intro: order_antisym  wenzelm@23164  125  dest: order_eq_refl [THEN unique_quotient_lemma]  wenzelm@23164  126  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  wenzelm@23164  127 done  wenzelm@23164  128 wenzelm@23164  129 wenzelm@23164  130 lemma unique_remainder:  haftmann@29651  131  "[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \ 0 |]  wenzelm@23164  132  ==> r = r'"  wenzelm@23164  133 apply (subgoal_tac "q = q'")  haftmann@29651  134  apply (simp add: divmod_rel_def)  wenzelm@23164  135 apply (blast intro: unique_quotient)  wenzelm@23164  136 done  wenzelm@23164  137 wenzelm@23164  138 wenzelm@23164  139 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}  wenzelm@23164  140 wenzelm@23164  141 text{*And positive divisors*}  wenzelm@23164  142 wenzelm@23164  143 lemma adjust_eq [simp]:  wenzelm@23164  144  "adjust b (q,r) =  wenzelm@23164  145  (let diff = r-b in  wenzelm@23164  146  if 0 \ diff then (2*q + 1, diff)  wenzelm@23164  147  else (2*q, r))"  wenzelm@23164  148 by (simp add: Let_def adjust_def)  wenzelm@23164  149 wenzelm@23164  150 declare posDivAlg.simps [simp del]  wenzelm@23164  151 wenzelm@23164  152 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  153 lemma posDivAlg_eqn:  wenzelm@23164  154  "0 < b ==>  wenzelm@23164  155  posDivAlg a b = (if a a" and "0 < b"  haftmann@29651  161  shows "divmod_rel a b (posDivAlg a b)"  wenzelm@23164  162 using prems apply (induct a b rule: posDivAlg.induct)  wenzelm@23164  163 apply auto  haftmann@29651  164 apply (simp add: divmod_rel_def)  wenzelm@23164  165 apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@23164  166 apply (case_tac "a < b")  wenzelm@23164  167 apply simp_all  wenzelm@23164  168 apply (erule splitE)  wenzelm@23164  169 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  170 done  wenzelm@23164  171 wenzelm@23164  172 wenzelm@23164  173 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}  wenzelm@23164  174 wenzelm@23164  175 text{*And positive divisors*}  wenzelm@23164  176 wenzelm@23164  177 declare negDivAlg.simps [simp del]  wenzelm@23164  178 wenzelm@23164  179 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  180 lemma negDivAlg_eqn:  wenzelm@23164  181  "0 < b ==>  wenzelm@23164  182  negDivAlg a b =  wenzelm@23164  183  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  wenzelm@23164  184 by (rule negDivAlg.simps [THEN trans], simp)  wenzelm@23164  185 wenzelm@23164  186 (*Correctness of negDivAlg: it computes quotients correctly  wenzelm@23164  187  It doesn't work if a=0 because the 0/b equals 0, not -1*)  wenzelm@23164  188 lemma negDivAlg_correct:  wenzelm@23164  189  assumes "a < 0" and "b > 0"  haftmann@29651  190  shows "divmod_rel a b (negDivAlg a b)"  wenzelm@23164  191 using prems apply (induct a b rule: negDivAlg.induct)  wenzelm@23164  192 apply (auto simp add: linorder_not_le)  haftmann@29651  193 apply (simp add: divmod_rel_def)  wenzelm@23164  194 apply (subst negDivAlg_eqn, assumption)  wenzelm@23164  195 apply (case_tac "a + b < (0\int)")  wenzelm@23164  196 apply simp_all  wenzelm@23164  197 apply (erule splitE)  wenzelm@23164  198 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  199 done  wenzelm@23164  200 wenzelm@23164  201 wenzelm@23164  202 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}  wenzelm@23164  203 wenzelm@23164  204 (*the case a=0*)  haftmann@29651  205 lemma divmod_rel_0: "b \ 0 ==> divmod_rel 0 b (0, 0)"  haftmann@29651  206 by (auto simp add: divmod_rel_def linorder_neq_iff)  wenzelm@23164  207 wenzelm@23164  208 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  wenzelm@23164  209 by (subst posDivAlg.simps, auto)  wenzelm@23164  210 wenzelm@23164  211 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  wenzelm@23164  212 by (subst negDivAlg.simps, auto)  wenzelm@23164  213 wenzelm@23164  214 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"  wenzelm@23164  215 by (simp add: negateSnd_def)  wenzelm@23164  216 haftmann@29651  217 lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"  haftmann@29651  218 by (auto simp add: split_ifs divmod_rel_def)  wenzelm@23164  219 haftmann@29651  220 lemma divmod_correct: "b \ 0 ==> divmod_rel a b (divmod a b)"  haftmann@29651  221 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg  wenzelm@23164  222  posDivAlg_correct negDivAlg_correct)  wenzelm@23164  223 wenzelm@23164  224 text{*Arbitrary definitions for division by zero. Useful to simplify  wenzelm@23164  225  certain equations.*}  wenzelm@23164  226 wenzelm@23164  227 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"  haftmann@29651  228 by (simp add: div_def mod_def divmod_def posDivAlg.simps)  wenzelm@23164  229 wenzelm@23164  230 wenzelm@23164  231 text{*Basic laws about division and remainder*}  wenzelm@23164  232 wenzelm@23164  233 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  wenzelm@23164  234 apply (case_tac "b = 0", simp)  haftmann@29651  235 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  236 apply (auto simp add: divmod_rel_def div_def mod_def)  wenzelm@23164  237 done  wenzelm@23164  238 wenzelm@23164  239 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"  wenzelm@23164  240 by(simp add: zmod_zdiv_equality[symmetric])  wenzelm@23164  241 wenzelm@23164  242 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"  wenzelm@23164  243 by(simp add: mult_commute zmod_zdiv_equality[symmetric])  wenzelm@23164  244 wenzelm@23164  245 text {* Tool setup *}  wenzelm@23164  246 wenzelm@26480  247 ML {*  haftmann@30934  248 local  wenzelm@23164  249 haftmann@30934  250 structure CancelDivMod = CancelDivModFun(struct  haftmann@30934  251 haftmann@30934  252  val div_name = @{const_name div};  haftmann@30934  253  val mod_name = @{const_name mod};  wenzelm@23164  254  val mk_binop = HOLogic.mk_binop;  haftmann@31068  255  val mk_sum = Numeral_Simprocs.mk_sum HOLogic.intT;  haftmann@31068  256  val dest_sum = Numeral_Simprocs.dest_sum;  haftmann@30934  257 haftmann@30934  258  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];  haftmann@30934  259 wenzelm@23164  260  val trans = trans;  haftmann@30934  261 haftmann@30934  262  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac  haftmann@30934  263  (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))  haftmann@30934  264 wenzelm@23164  265 end)  wenzelm@23164  266 wenzelm@23164  267 in  wenzelm@23164  268 wenzelm@32010  269 val cancel_div_mod_int_proc = Simplifier.simproc @{theory}  haftmann@30934  270  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);  wenzelm@23164  271 haftmann@30934  272 val _ = Addsimprocs [cancel_div_mod_int_proc];  wenzelm@23164  273 haftmann@30934  274 end  wenzelm@23164  275 *}  wenzelm@23164  276 wenzelm@23164  277 lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b"  haftmann@29651  278 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  279 apply (auto simp add: divmod_rel_def mod_def)  wenzelm@23164  280 done  wenzelm@23164  281 wenzelm@23164  282 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]  wenzelm@23164  283  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]  wenzelm@23164  284 wenzelm@23164  285 lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b"  haftmann@29651  286 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  287 apply (auto simp add: divmod_rel_def div_def mod_def)  wenzelm@23164  288 done  wenzelm@23164  289 wenzelm@23164  290 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]  wenzelm@23164  291  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]  wenzelm@23164  292 wenzelm@23164  293 wenzelm@23164  294 wenzelm@23164  295 subsection{*General Properties of div and mod*}  wenzelm@23164  296 haftmann@29651  297 lemma divmod_rel_div_mod: "b \ 0 ==> divmod_rel a b (a div b, a mod b)"  wenzelm@23164  298 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@29651  299 apply (force simp add: divmod_rel_def linorder_neq_iff)  wenzelm@23164  300 done  wenzelm@23164  301 haftmann@29651  302 lemma divmod_rel_div: "[| divmod_rel a b (q, r); b \ 0 |] ==> a div b = q"  haftmann@29651  303 by (simp add: divmod_rel_div_mod [THEN unique_quotient])  wenzelm@23164  304 haftmann@29651  305 lemma divmod_rel_mod: "[| divmod_rel a b (q, r); b \ 0 |] ==> a mod b = r"  haftmann@29651  306 by (simp add: divmod_rel_div_mod [THEN unique_remainder])  wenzelm@23164  307 wenzelm@23164  308 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  haftmann@29651  309 apply (rule divmod_rel_div)  haftmann@29651  310 apply (auto simp add: divmod_rel_def)  wenzelm@23164  311 done  wenzelm@23164  312 wenzelm@23164  313 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  haftmann@29651  314 apply (rule divmod_rel_div)  haftmann@29651  315 apply (auto simp add: divmod_rel_def)  wenzelm@23164  316 done  wenzelm@23164  317 wenzelm@23164  318 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  haftmann@29651  319 apply (rule divmod_rel_div)  haftmann@29651  320 apply (auto simp add: divmod_rel_def)  wenzelm@23164  321 done  wenzelm@23164  322 wenzelm@23164  323 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  wenzelm@23164  324 wenzelm@23164  325 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  haftmann@29651  326 apply (rule_tac q = 0 in divmod_rel_mod)  haftmann@29651  327 apply (auto simp add: divmod_rel_def)  wenzelm@23164  328 done  wenzelm@23164  329 wenzelm@23164  330 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  haftmann@29651  331 apply (rule_tac q = 0 in divmod_rel_mod)  haftmann@29651  332 apply (auto simp add: divmod_rel_def)  wenzelm@23164  333 done  wenzelm@23164  334 wenzelm@23164  335 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  haftmann@29651  336 apply (rule_tac q = "-1" in divmod_rel_mod)  haftmann@29651  337 apply (auto simp add: divmod_rel_def)  wenzelm@23164  338 done  wenzelm@23164  339 wenzelm@23164  340 text{*There is no @{text mod_neg_pos_trivial}.*}  wenzelm@23164  341 wenzelm@23164  342 wenzelm@23164  343 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)  wenzelm@23164  344 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"  wenzelm@23164  345 apply (case_tac "b = 0", simp)  haftmann@29651  346 apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified,  haftmann@29651  347  THEN divmod_rel_div, THEN sym])  wenzelm@23164  348 wenzelm@23164  349 done  wenzelm@23164  350 wenzelm@23164  351 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)  wenzelm@23164  352 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"  wenzelm@23164  353 apply (case_tac "b = 0", simp)  haftmann@29651  354 apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],  wenzelm@23164  355  auto)  wenzelm@23164  356 done  wenzelm@23164  357 wenzelm@23164  358 wenzelm@23164  359 subsection{*Laws for div and mod with Unary Minus*}  wenzelm@23164  360 wenzelm@23164  361 lemma zminus1_lemma:  haftmann@29651  362  "divmod_rel a b (q, r)  haftmann@29651  363  ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@29651  364  if r=0 then 0 else b-r)"  haftmann@29651  365 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)  wenzelm@23164  366 wenzelm@23164  367 wenzelm@23164  368 lemma zdiv_zminus1_eq_if:  wenzelm@23164  369  "b \ (0::int)  wenzelm@23164  370  ==> (-a) div b =  wenzelm@23164  371  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  haftmann@29651  372 by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])  wenzelm@23164  373 wenzelm@23164  374 lemma zmod_zminus1_eq_if:  wenzelm@23164  375  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  wenzelm@23164  376 apply (case_tac "b = 0", simp)  haftmann@29651  377 apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])  wenzelm@23164  378 done  wenzelm@23164  379 haftmann@29936  380 lemma zmod_zminus1_not_zero:  haftmann@29936  381  fixes k l :: int  haftmann@29936  382  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@29936  383  unfolding zmod_zminus1_eq_if by auto  haftmann@29936  384 wenzelm@23164  385 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  wenzelm@23164  386 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  wenzelm@23164  387 wenzelm@23164  388 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  wenzelm@23164  389 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  wenzelm@23164  390 wenzelm@23164  391 lemma zdiv_zminus2_eq_if:  wenzelm@23164  392  "b \ (0::int)  wenzelm@23164  393  ==> a div (-b) =  wenzelm@23164  394  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  395 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  wenzelm@23164  396 wenzelm@23164  397 lemma zmod_zminus2_eq_if:  wenzelm@23164  398  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  wenzelm@23164  399 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  wenzelm@23164  400 haftmann@29936  401 lemma zmod_zminus2_not_zero:  haftmann@29936  402  fixes k l :: int  haftmann@29936  403  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@29936  404  unfolding zmod_zminus2_eq_if by auto  haftmann@29936  405 wenzelm@23164  406 wenzelm@23164  407 subsection{*Division of a Number by Itself*}  wenzelm@23164  408 wenzelm@23164  409 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  wenzelm@23164  410 apply (subgoal_tac "0 < a*q")  wenzelm@23164  411  apply (simp add: zero_less_mult_iff, arith)  wenzelm@23164  412 done  wenzelm@23164  413 wenzelm@23164  414 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  wenzelm@23164  415 apply (subgoal_tac "0 \ a* (1-q) ")  wenzelm@23164  416  apply (simp add: zero_le_mult_iff)  wenzelm@23164  417 apply (simp add: right_diff_distrib)  wenzelm@23164  418 done  wenzelm@23164  419 haftmann@29651  420 lemma self_quotient: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> q = 1"  haftmann@29651  421 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)  wenzelm@23164  422 apply (rule order_antisym, safe, simp_all)  wenzelm@23164  423 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  wenzelm@23164  424 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  wenzelm@23164  425 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  wenzelm@23164  426 done  wenzelm@23164  427 haftmann@29651  428 lemma self_remainder: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> r = 0"  wenzelm@23164  429 apply (frule self_quotient, assumption)  haftmann@29651  430 apply (simp add: divmod_rel_def)  wenzelm@23164  431 done  wenzelm@23164  432 wenzelm@23164  433 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  haftmann@29651  434 by (simp add: divmod_rel_div_mod [THEN self_quotient])  wenzelm@23164  435 wenzelm@23164  436 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  wenzelm@23164  437 lemma zmod_self [simp]: "a mod a = (0::int)"  wenzelm@23164  438 apply (case_tac "a = 0", simp)  haftmann@29651  439 apply (simp add: divmod_rel_div_mod [THEN self_remainder])  wenzelm@23164  440 done  wenzelm@23164  441 wenzelm@23164  442 wenzelm@23164  443 subsection{*Computation of Division and Remainder*}  wenzelm@23164  444 wenzelm@23164  445 lemma zdiv_zero [simp]: "(0::int) div b = 0"  haftmann@29651  446 by (simp add: div_def divmod_def)  wenzelm@23164  447 wenzelm@23164  448 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@29651  449 by (simp add: div_def divmod_def)  wenzelm@23164  450 wenzelm@23164  451 lemma zmod_zero [simp]: "(0::int) mod b = 0"  haftmann@29651  452 by (simp add: mod_def divmod_def)  wenzelm@23164  453 wenzelm@23164  454 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@29651  455 by (simp add: mod_def divmod_def)  wenzelm@23164  456 wenzelm@23164  457 text{*a positive, b positive *}  wenzelm@23164  458 wenzelm@23164  459 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@29651  460 by (simp add: div_def divmod_def)  wenzelm@23164  461 wenzelm@23164  462 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@29651  463 by (simp add: mod_def divmod_def)  wenzelm@23164  464 wenzelm@23164  465 text{*a negative, b positive *}  wenzelm@23164  466 wenzelm@23164  467 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@29651  468 by (simp add: div_def divmod_def)  wenzelm@23164  469 wenzelm@23164  470 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@29651  471 by (simp add: mod_def divmod_def)  wenzelm@23164  472 wenzelm@23164  473 text{*a positive, b negative *}  wenzelm@23164  474 wenzelm@23164  475 lemma div_pos_neg:  wenzelm@23164  476  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  477 by (simp add: div_def divmod_def)  wenzelm@23164  478 wenzelm@23164  479 lemma mod_pos_neg:  wenzelm@23164  480  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  481 by (simp add: mod_def divmod_def)  wenzelm@23164  482 wenzelm@23164  483 text{*a negative, b negative *}  wenzelm@23164  484 wenzelm@23164  485 lemma div_neg_neg:  wenzelm@23164  486  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  487 by (simp add: div_def divmod_def)  wenzelm@23164  488 wenzelm@23164  489 lemma mod_neg_neg:  wenzelm@23164  490  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  491 by (simp add: mod_def divmod_def)  wenzelm@23164  492 wenzelm@23164  493 text {*Simplify expresions in which div and mod combine numerical constants*}  wenzelm@23164  494 haftmann@29651  495 lemma divmod_relI:  huffman@24481  496  "\a == b * q + r; if 0 < b then 0 \ r \ r < b else b < r \ r \ 0\  haftmann@29651  497  \ divmod_rel a b (q, r)"  haftmann@29651  498  unfolding divmod_rel_def by simp  huffman@24481  499 haftmann@29651  500 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]  haftmann@29651  501 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]  huffman@24481  502 lemmas arithmetic_simps =  huffman@24481  503  arith_simps  huffman@24481  504  add_special  huffman@24481  505  OrderedGroup.add_0_left  huffman@24481  506  OrderedGroup.add_0_right  huffman@24481  507  mult_zero_left  huffman@24481  508  mult_zero_right  huffman@24481  509  mult_1_left  huffman@24481  510  mult_1_right  huffman@24481  511 huffman@24481  512 (* simprocs adapted from HOL/ex/Binary.thy *)  huffman@24481  513 ML {*  huffman@24481  514 local  haftmann@30517  515  val mk_number = HOLogic.mk_number HOLogic.intT;  haftmann@30517  516  fun mk_cert u k l = @{term "plus :: int \ int \ int"}   haftmann@30517  517  (@{term "times :: int \ int \ int"}  u  mk_number k)   haftmann@30517  518  mk_number l;  haftmann@30517  519  fun prove ctxt prop = Goal.prove ctxt [] [] prop  haftmann@30517  520  (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));  huffman@24481  521  fun binary_proc proc ss ct =  huffman@24481  522  (case Thm.term_of ct of  huffman@24481  523  _  t  u =>  huffman@24481  524  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  huffman@24481  525  SOME args => proc (Simplifier.the_context ss) args  huffman@24481  526  | NONE => NONE)  huffman@24481  527  | _ => NONE);  huffman@24481  528 in  haftmann@30517  529  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>  haftmann@30517  530  if n = 0 then NONE  haftmann@30517  531  else let val (k, l) = Integer.div_mod m n;  haftmann@30517  532  in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);  haftmann@30517  533 end  huffman@24481  534 *}  huffman@24481  535 huffman@24481  536 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  haftmann@29651  537  {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}  huffman@24481  538 huffman@24481  539 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  haftmann@29651  540  {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}  huffman@24481  541 wenzelm@23164  542 lemmas posDivAlg_eqn_number_of [simp] =  wenzelm@23164  543  posDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  544 wenzelm@23164  545 lemmas negDivAlg_eqn_number_of [simp] =  wenzelm@23164  546  negDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  547 wenzelm@23164  548 wenzelm@23164  549 text{*Special-case simplification *}  wenzelm@23164  550 wenzelm@23164  551 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  wenzelm@23164  552 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  wenzelm@23164  553 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  wenzelm@23164  554 apply (auto simp del: neg_mod_sign neg_mod_bound)  wenzelm@23164  555 done  wenzelm@23164  556 wenzelm@23164  557 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"  wenzelm@23164  558 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)  wenzelm@23164  559 wenzelm@23164  560 (** The last remaining special cases for constant arithmetic:  wenzelm@23164  561  1 div z and 1 mod z **)  wenzelm@23164  562 wenzelm@23164  563 lemmas div_pos_pos_1_number_of [simp] =  wenzelm@23164  564  div_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  565 wenzelm@23164  566 lemmas div_pos_neg_1_number_of [simp] =  wenzelm@23164  567  div_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  568 wenzelm@23164  569 lemmas mod_pos_pos_1_number_of [simp] =  wenzelm@23164  570  mod_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  571 wenzelm@23164  572 lemmas mod_pos_neg_1_number_of [simp] =  wenzelm@23164  573  mod_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  574 wenzelm@23164  575 wenzelm@23164  576 lemmas posDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  577  posDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  578 wenzelm@23164  579 lemmas negDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  580  negDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  581 wenzelm@23164  582 wenzelm@23164  583 wenzelm@23164  584 subsection{*Monotonicity in the First Argument (Dividend)*}  wenzelm@23164  585 wenzelm@23164  586 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  wenzelm@23164  587 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  588 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  589 apply (rule unique_quotient_lemma)  wenzelm@23164  590 apply (erule subst)  wenzelm@23164  591 apply (erule subst, simp_all)  wenzelm@23164  592 done  wenzelm@23164  593 wenzelm@23164  594 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  wenzelm@23164  595 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  596 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  597 apply (rule unique_quotient_lemma_neg)  wenzelm@23164  598 apply (erule subst)  wenzelm@23164  599 apply (erule subst, simp_all)  wenzelm@23164  600 done  wenzelm@23164  601 wenzelm@23164  602 wenzelm@23164  603 subsection{*Monotonicity in the Second Argument (Divisor)*}  wenzelm@23164  604 wenzelm@23164  605 lemma q_pos_lemma:  wenzelm@23164  606  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  wenzelm@23164  607 apply (subgoal_tac "0 < b'* (q' + 1) ")  wenzelm@23164  608  apply (simp add: zero_less_mult_iff)  wenzelm@23164  609 apply (simp add: right_distrib)  wenzelm@23164  610 done  wenzelm@23164  611 wenzelm@23164  612 lemma zdiv_mono2_lemma:  wenzelm@23164  613  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  wenzelm@23164  614  r' < b'; 0 \ r; 0 < b'; b' \ b |]  wenzelm@23164  615  ==> q \ (q'::int)"  wenzelm@23164  616 apply (frule q_pos_lemma, assumption+)  wenzelm@23164  617 apply (subgoal_tac "b*q < b* (q' + 1) ")  wenzelm@23164  618  apply (simp add: mult_less_cancel_left)  wenzelm@23164  619 apply (subgoal_tac "b*q = r' - r + b'*q'")  wenzelm@23164  620  prefer 2 apply simp  wenzelm@23164  621 apply (simp (no_asm_simp) add: right_distrib)  wenzelm@23164  622 apply (subst add_commute, rule zadd_zless_mono, arith)  wenzelm@23164  623 apply (rule mult_right_mono, auto)  wenzelm@23164  624 done  wenzelm@23164  625 wenzelm@23164  626 lemma zdiv_mono2:  wenzelm@23164  627  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  wenzelm@23164  628 apply (subgoal_tac "b \ 0")  wenzelm@23164  629  prefer 2 apply arith  wenzelm@23164  630 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  631 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  632 apply (rule zdiv_mono2_lemma)  wenzelm@23164  633 apply (erule subst)  wenzelm@23164  634 apply (erule subst, simp_all)  wenzelm@23164  635 done  wenzelm@23164  636 wenzelm@23164  637 lemma q_neg_lemma:  wenzelm@23164  638  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  wenzelm@23164  639 apply (subgoal_tac "b'*q' < 0")  wenzelm@23164  640  apply (simp add: mult_less_0_iff, arith)  wenzelm@23164  641 done  wenzelm@23164  642 wenzelm@23164  643 lemma zdiv_mono2_neg_lemma:  wenzelm@23164  644  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  wenzelm@23164  645  r < b; 0 \ r'; 0 < b'; b' \ b |]  wenzelm@23164  646  ==> q' \ (q::int)"  wenzelm@23164  647 apply (frule q_neg_lemma, assumption+)  wenzelm@23164  648 apply (subgoal_tac "b*q' < b* (q + 1) ")  wenzelm@23164  649  apply (simp add: mult_less_cancel_left)  wenzelm@23164  650 apply (simp add: right_distrib)  wenzelm@23164  651 apply (subgoal_tac "b*q' \ b'*q'")  wenzelm@23164  652  prefer 2 apply (simp add: mult_right_mono_neg, arith)  wenzelm@23164  653 done  wenzelm@23164  654 wenzelm@23164  655 lemma zdiv_mono2_neg:  wenzelm@23164  656  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  wenzelm@23164  657 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  658 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  659 apply (rule zdiv_mono2_neg_lemma)  wenzelm@23164  660 apply (erule subst)  wenzelm@23164  661 apply (erule subst, simp_all)  wenzelm@23164  662 done  wenzelm@23164  663 haftmann@25942  664 wenzelm@23164  665 subsection{*More Algebraic Laws for div and mod*}  wenzelm@23164  666 wenzelm@23164  667 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  wenzelm@23164  668 wenzelm@23164  669 lemma zmult1_lemma:  haftmann@29651  670  "[| divmod_rel b c (q, r); c \ 0 |]  haftmann@29651  671  ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@29651  672 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  wenzelm@23164  673 wenzelm@23164  674 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  wenzelm@23164  675 apply (case_tac "c = 0", simp)  haftmann@29651  676 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])  wenzelm@23164  677 done  wenzelm@23164  678 wenzelm@23164  679 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  wenzelm@23164  680 apply (case_tac "c = 0", simp)  haftmann@29651  681 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])  wenzelm@23164  682 done  wenzelm@23164  683 huffman@29403  684 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"  haftmann@27651  685 apply (case_tac "b = 0", simp)  haftmann@27651  686 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  haftmann@27651  687 done  haftmann@27651  688 haftmann@27651  689 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@27651  690 haftmann@27651  691 lemma zadd1_lemma:  haftmann@29651  692  "[| divmod_rel a c (aq, ar); divmod_rel b c (bq, br); c \ 0 |]  haftmann@29651  693  ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  haftmann@29651  694 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  haftmann@27651  695 haftmann@27651  696 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@27651  697 lemma zdiv_zadd1_eq:  haftmann@27651  698  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@27651  699 apply (case_tac "c = 0", simp)  haftmann@29651  700 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)  haftmann@27651  701 done  haftmann@27651  702 huffman@29405  703 instance int :: ring_div  haftmann@27651  704 proof  haftmann@27651  705  fix a b c :: int  haftmann@27651  706  assume not0: "b \ 0"  haftmann@27651  707  show "(a + c * b) div b = c + a div b"  haftmann@27651  708  unfolding zdiv_zadd1_eq [of a "c * b"] using not0  nipkow@30181  709  by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)  haftmann@30930  710 next  haftmann@30930  711  fix a b c :: int  haftmann@30930  712  assume "a \ 0"  haftmann@30930  713  then show "(a * b) div (a * c) = b div c"  haftmann@30930  714  proof (cases "b \ 0 \ c \ 0")  haftmann@30930  715  case False then show ?thesis by auto  haftmann@30930  716  next  haftmann@30930  717  case True then have "b \ 0" and "c \ 0" by auto  haftmann@30930  718  with a \ 0  haftmann@30930  719  have "\q r. divmod_rel b c (q, r) \ divmod_rel (a * b) (a * c) (q, a * r)"  haftmann@30930  720  apply (auto simp add: divmod_rel_def)  haftmann@30930  721  apply (auto simp add: algebra_simps)  haftmann@30930  722  apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff)  haftmann@30930  723  done  haftmann@30930  724  moreover with c \ 0 divmod_rel_div_mod have "divmod_rel b c (b div c, b mod c)" by auto  haftmann@30930  725  ultimately have "divmod_rel (a * b) (a * c) (b div c, a * (b mod c))" .  haftmann@30930  726  moreover from a \ 0 c \ 0 have "a * c \ 0" by simp  haftmann@30930  727  ultimately show ?thesis by (rule divmod_rel_div)  haftmann@30930  728  qed  haftmann@27651  729 qed auto  haftmann@25942  730 haftmann@29651  731 lemma posDivAlg_div_mod:  haftmann@29651  732  assumes "k \ 0"  haftmann@29651  733  and "l \ 0"  haftmann@29651  734  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@29651  735 proof (cases "l = 0")  haftmann@29651  736  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@29651  737 next  haftmann@29651  738  case False with assms posDivAlg_correct  haftmann@29651  739  have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@29651  740  by simp  haftmann@29651  741  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  742  show ?thesis by simp  haftmann@29651  743 qed  haftmann@29651  744 haftmann@29651  745 lemma negDivAlg_div_mod:  haftmann@29651  746  assumes "k < 0"  haftmann@29651  747  and "l > 0"  haftmann@29651  748  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@29651  749 proof -  haftmann@29651  750  from assms have "l \ 0" by simp  haftmann@29651  751  from assms negDivAlg_correct  haftmann@29651  752  have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@29651  753  by simp  haftmann@29651  754  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  755  show ?thesis by simp  haftmann@29651  756 qed  haftmann@29651  757 wenzelm@23164  758 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  huffman@29403  759 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  wenzelm@23164  760 huffman@29403  761 (* REVISIT: should this be generalized to all semiring_div types? *)  wenzelm@23164  762 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  763 nipkow@23983  764 wenzelm@23164  765 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  766 wenzelm@23164  767 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  768  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  769  to cause particular problems.*)  wenzelm@23164  770 wenzelm@23164  771 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  772 wenzelm@23164  773 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  774 apply (subgoal_tac "b * (c - q mod c) < r * 1")  nipkow@29667  775  apply (simp add: algebra_simps)  wenzelm@23164  776 apply (rule order_le_less_trans)  nipkow@29667  777  apply (erule_tac [2] mult_strict_right_mono)  nipkow@29667  778  apply (rule mult_left_mono_neg)  nipkow@29667  779  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)  nipkow@29667  780  apply (simp)  nipkow@29667  781 apply (simp)  wenzelm@23164  782 done  wenzelm@23164  783 wenzelm@23164  784 lemma zmult2_lemma_aux2:  wenzelm@23164  785  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  786 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  787  apply arith  wenzelm@23164  788 apply (simp add: mult_le_0_iff)  wenzelm@23164  789 done  wenzelm@23164  790 wenzelm@23164  791 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  792 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  793 apply arith  wenzelm@23164  794 apply (simp add: zero_le_mult_iff)  wenzelm@23164  795 done  wenzelm@23164  796 wenzelm@23164  797 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  798 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  nipkow@29667  799  apply (simp add: right_diff_distrib)  wenzelm@23164  800 apply (rule order_less_le_trans)  nipkow@29667  801  apply (erule mult_strict_right_mono)  nipkow@29667  802  apply (rule_tac [2] mult_left_mono)  nipkow@29667  803  apply simp  nipkow@29667  804  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)  nipkow@29667  805 apply simp  wenzelm@23164  806 done  wenzelm@23164  807 haftmann@29651  808 lemma zmult2_lemma: "[| divmod_rel a b (q, r); b \ 0; 0 < c |]  haftmann@29651  809  ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@29651  810 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff  wenzelm@23164  811  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  812  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  813 wenzelm@23164  814 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  815 apply (case_tac "b = 0", simp)  haftmann@29651  816 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])  wenzelm@23164  817 done  wenzelm@23164  818 wenzelm@23164  819 lemma zmod_zmult2_eq:  wenzelm@23164  820  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  821 apply (case_tac "b = 0", simp)  haftmann@29651  822 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])  wenzelm@23164  823 done  wenzelm@23164  824 wenzelm@23164  825 wenzelm@23164  826 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  827 wenzelm@23164  828 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  829 wenzelm@23164  830 lemma split_pos_lemma:  wenzelm@23164  831  "0  wenzelm@23164  832  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  833 apply (rule iffI, clarify)  wenzelm@23164  834  apply (erule_tac P="P ?x ?y" in rev_mp)  nipkow@29948  835  apply (subst mod_add_eq)  wenzelm@23164  836  apply (subst zdiv_zadd1_eq)  wenzelm@23164  837  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@23164  838 txt{*converse direction*}  wenzelm@23164  839 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  840 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  841 done  wenzelm@23164  842 wenzelm@23164  843 lemma split_neg_lemma:  wenzelm@23164  844  "k<0 ==>  wenzelm@23164  845  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  846 apply (rule iffI, clarify)  wenzelm@23164  847  apply (erule_tac P="P ?x ?y" in rev_mp)  nipkow@29948  848  apply (subst mod_add_eq)  wenzelm@23164  849  apply (subst zdiv_zadd1_eq)  wenzelm@23164  850  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  851 txt{*converse direction*}  wenzelm@23164  852 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  853 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  854 done  wenzelm@23164  855 wenzelm@23164  856 lemma split_zdiv:  wenzelm@23164  857  "P(n div k :: int) =  wenzelm@23164  858  ((k = 0 --> P 0) &  wenzelm@23164  859  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  860  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  861 apply (case_tac "k=0", simp)  wenzelm@23164  862 apply (simp only: linorder_neq_iff)  wenzelm@23164  863 apply (erule disjE)  wenzelm@23164  864  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  865  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  866 done  wenzelm@23164  867 wenzelm@23164  868 lemma split_zmod:  wenzelm@23164  869  "P(n mod k :: int) =  wenzelm@23164  870  ((k = 0 --> P n) &  wenzelm@23164  871  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  872  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  873 apply (case_tac "k=0", simp)  wenzelm@23164  874 apply (simp only: linorder_neq_iff)  wenzelm@23164  875 apply (erule disjE)  wenzelm@23164  876  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  877  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  878 done  wenzelm@23164  879 wenzelm@23164  880 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  881 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  882 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  883 wenzelm@23164  884 wenzelm@23164  885 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  886 wenzelm@23164  887 text{*computing div by shifting *}  wenzelm@23164  888 wenzelm@23164  889 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  890 proof cases  wenzelm@23164  891  assume "a=0"  wenzelm@23164  892  thus ?thesis by simp  wenzelm@23164  893 next  wenzelm@23164  894  assume "a\0" and le_a: "0\a"  wenzelm@23164  895  hence a_pos: "1 \ a" by arith  haftmann@30652  896  hence one_less_a2: "1 < 2 * a" by arith  wenzelm@23164  897  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  haftmann@30652  898  unfolding mult_le_cancel_left  haftmann@30652  899  by (simp add: add1_zle_eq add_commute [of 1])  wenzelm@23164  900  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  901  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  902  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  903  with le_2a  wenzelm@23164  904  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  905  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  906  right_distrib)  wenzelm@23164  907  thus ?thesis  wenzelm@23164  908  by (subst zdiv_zadd1_eq,  haftmann@30930  909  simp add: mod_mult_mult1 one_less_a2  wenzelm@23164  910  div_pos_pos_trivial)  wenzelm@23164  911 qed  wenzelm@23164  912 wenzelm@23164  913 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  914 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  915 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  916 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  917 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  918 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  919  simp)  wenzelm@23164  920 done  wenzelm@23164  921 huffman@26086  922 lemma zdiv_number_of_Bit0 [simp]:  huffman@26086  923  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  huffman@26086  924  number_of v div (number_of w :: int)"  huffman@26086  925 by (simp only: number_of_eq numeral_simps) simp  huffman@26086  926 huffman@26086  927 lemma zdiv_number_of_Bit1 [simp]:  huffman@26086  928  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  huffman@26086  929  (if (0::int) \ number_of w  wenzelm@23164  930  then number_of v div (number_of w)  wenzelm@23164  931  else (number_of v + (1::int)) div (number_of w))"  wenzelm@23164  932 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  haftmann@30930  933 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)  wenzelm@23164  934 done  wenzelm@23164  935 wenzelm@23164  936 wenzelm@23164  937 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  938 wenzelm@23164  939 lemma pos_zmod_mult_2:  wenzelm@23164  940  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  941 apply (case_tac "a = 0", simp)  wenzelm@23164  942 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  943  prefer 2 apply arith  wenzelm@23164  944 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  945  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  946 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  947  pos_mod_bound)  nipkow@29948  948 apply (subst mod_add_eq)  haftmann@30930  949 apply (simp add: mod_mult_mult2 mod_pos_pos_trivial)  wenzelm@23164  950 apply (rule mod_pos_pos_trivial)  huffman@26086  951 apply (auto simp add: mod_pos_pos_trivial ring_distribs)  wenzelm@23164  952 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  953 done  wenzelm@23164  954 wenzelm@23164  955 lemma neg_zmod_mult_2:  wenzelm@23164  956  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  957 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  958  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  959 apply (rule_tac [2] pos_zmod_mult_2)  nipkow@30042  960 apply (auto simp add: right_diff_distrib)  wenzelm@23164  961 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  962  prefer 2 apply simp  wenzelm@23164  963 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  964 done  wenzelm@23164  965 huffman@26086  966 lemma zmod_number_of_Bit0 [simp]:  huffman@26086  967  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  huffman@26086  968  (2::int) * (number_of v mod number_of w)"  huffman@26086  969 apply (simp only: number_of_eq numeral_simps)  haftmann@30930  970 apply (simp add: mod_mult_mult1 pos_zmod_mult_2  nipkow@29948  971  neg_zmod_mult_2 add_ac)  huffman@26086  972 done  huffman@26086  973 huffman@26086  974 lemma zmod_number_of_Bit1 [simp]:  huffman@26086  975  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  huffman@26086  976  (if (0::int) \ number_of w  wenzelm@23164  977  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  978  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  huffman@26086  979 apply (simp only: number_of_eq numeral_simps)  haftmann@30930  980 apply (simp add: mod_mult_mult1 pos_zmod_mult_2  nipkow@29948  981  neg_zmod_mult_2 add_ac)  wenzelm@23164  982 done  wenzelm@23164  983 wenzelm@23164  984 wenzelm@23164  985 subsection{*Quotients of Signs*}  wenzelm@23164  986 wenzelm@23164  987 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  988 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  989 apply (rule order_trans)  wenzelm@23164  990 apply (rule_tac a' = "-1" in zdiv_mono1)  nipkow@29948  991 apply (auto simp add: div_eq_minus1)  wenzelm@23164  992 done  wenzelm@23164  993 nipkow@30323  994 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  995 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  996 nipkow@30323  997 lemma div_nonpos_pos_le0: "[| (a::int) \ 0; b > 0 |] ==> a div b \ 0"  nipkow@30323  998 by (drule zdiv_mono1, auto)  nipkow@30323  999 wenzelm@23164  1000 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1001 apply auto  wenzelm@23164  1002 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1003 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1004 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1005 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1006 done  wenzelm@23164  1007 wenzelm@23164  1008 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1009  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1010 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1011 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1012 done  wenzelm@23164  1013 wenzelm@23164  1014 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1015 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1016 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1017 wenzelm@23164  1018 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1019 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1020 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1021 wenzelm@23164  1022 wenzelm@23164  1023 subsection {* The Divides Relation *}  wenzelm@23164  1024 wenzelm@23164  1025 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  nipkow@30042  1026  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]  wenzelm@23164  1027 wenzelm@23164  1028 lemma zdvd_anti_sym:  wenzelm@23164  1029  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1030  apply (simp add: dvd_def, auto)  wenzelm@23164  1031  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1032  done  wenzelm@23164  1033 nipkow@30042  1034 lemma zdvd_dvd_eq: assumes "a \ 0" and "(a::int) dvd b" and "b dvd a"  wenzelm@23164  1035  shows "\a\ = \b\"  wenzelm@23164  1036 proof-  nipkow@30042  1037  from a dvd b obtain k where k:"b = a*k" unfolding dvd_def by blast  nipkow@30042  1038  from b dvd a obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1039  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1040  with mult_cancel_left1[where c="a" and b="k*k'"]  nipkow@30042  1041  have kk':"k*k' = 1" using a\0 by (simp add: mult_assoc)  wenzelm@23164  1042  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1043  thus ?thesis using k k' by auto  wenzelm@23164  1044 qed  wenzelm@23164  1045 wenzelm@23164  1046 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1047  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1048  apply (erule ssubst)  nipkow@30042  1049  apply (blast intro: dvd_add, simp)  wenzelm@23164  1050  done  wenzelm@23164  1051 wenzelm@23164  1052 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  nipkow@30042  1053 apply (rule iffI)  nipkow@30042  1054  apply (erule_tac [2] dvd_add)  nipkow@30042  1055  apply (subgoal_tac "n = (n + k * m) - k * m")  nipkow@30042  1056  apply (erule ssubst)  nipkow@30042  1057  apply (erule dvd_diff)  nipkow@30042  1058  apply(simp_all)  nipkow@30042  1059 done  wenzelm@23164  1060 wenzelm@23164  1061 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  huffman@31662  1062  by (rule dvd_mod) (* TODO: remove *)  wenzelm@23164  1063 wenzelm@23164  1064 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  huffman@31662  1065  by (rule dvd_mod_imp_dvd) (* TODO: remove *)  wenzelm@23164  1066 nipkow@31734  1067 lemma dvd_imp_le_int: "(i::int) ~= 0 ==> d dvd i ==> abs d <= abs i"  nipkow@31734  1068 apply(auto simp:abs_if)  nipkow@31734  1069  apply(clarsimp simp:dvd_def mult_less_0_iff)  nipkow@31734  1070  using mult_le_cancel_left_neg[of _ "-1::int"]  nipkow@31734  1071  apply(clarsimp simp:dvd_def mult_less_0_iff)  nipkow@31734  1072  apply(clarsimp simp:dvd_def mult_less_0_iff  nipkow@31734  1073  minus_mult_right simp del: mult_minus_right)  nipkow@31734  1074 apply(clarsimp simp:dvd_def mult_less_0_iff)  nipkow@31734  1075 done  nipkow@31734  1076 wenzelm@23164  1077 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  haftmann@27651  1078  apply (auto elim!: dvdE)  wenzelm@23164  1079  apply (subgoal_tac "0 < n")  wenzelm@23164  1080  prefer 2  wenzelm@23164  1081  apply (blast intro: order_less_trans)  wenzelm@23164  1082  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1083  done  haftmann@27651  1084 wenzelm@23164  1085 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1086  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@29667  1087  by (simp add: algebra_simps)  wenzelm@23164  1088 wenzelm@23164  1089 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1090 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1091  apply (simp add: zmult_div_cancel)  nipkow@30042  1092 apply (simp only: dvd_eq_mod_eq_0)  wenzelm@23164  1093 done  wenzelm@23164  1094 wenzelm@23164  1095 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1096  shows "m dvd n"  wenzelm@23164  1097 proof-  wenzelm@23164  1098  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1099  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1100  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1101  hence "n = m * h" by blast  huffman@29410  1102  thus ?thesis by simp  wenzelm@23164  1103 qed  wenzelm@23164  1104 nipkow@23969  1105 wenzelm@23164  1106 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  nipkow@25134  1107 apply (simp split add: split_nat)  nipkow@25134  1108 apply (rule iffI)  nipkow@25134  1109 apply (erule exE)  nipkow@25134  1110 apply (rule_tac x = "int x" in exI)  nipkow@25134  1111 apply simp  nipkow@25134  1112 apply (erule exE)  nipkow@25134  1113 apply (rule_tac x = "nat x" in exI)  nipkow@25134  1114 apply (erule conjE)  nipkow@25134  1115 apply (erule_tac x = "nat x" in allE)  nipkow@25134  1116 apply simp  nipkow@25134  1117 done  wenzelm@23164  1118 huffman@23365  1119 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  haftmann@27651  1120 proof -  haftmann@27651  1121  have "\k. int y = int x * k \ x dvd y"  haftmann@27651  1122  proof -  haftmann@27651  1123  fix k  haftmann@27651  1124  assume A: "int y = int x * k"  haftmann@27651  1125  then show "x dvd y" proof (cases k)  haftmann@27651  1126  case (1 n) with A have "y = x * n" by (simp add: zmult_int)  haftmann@27651  1127  then show ?thesis ..  haftmann@27651  1128  next  haftmann@27651  1129  case (2 n) with A have "int y = int x * (- int (Suc n))" by simp  haftmann@27651  1130  also have "\ = - (int x * int (Suc n))" by (simp only: mult_minus_right)  haftmann@27651  1131  also have "\ = - int (x * Suc n)" by (simp only: zmult_int)  haftmann@27651  1132  finally have "- int (x * Suc n) = int y" ..  haftmann@27651  1133  then show ?thesis by (simp only: negative_eq_positive) auto  haftmann@27651  1134  qed  haftmann@27651  1135  qed  nipkow@30042  1136  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)  huffman@29410  1137 qed  wenzelm@23164  1138 wenzelm@23164  1139 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1140 proof  nipkow@30042  1141  assume d: "x dvd 1" hence "int (nat \x$$ dvd int (nat 1)" by simp  wenzelm@23164  1142  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1143  hence "nat \x\ = 1" by simp  wenzelm@23164  1144  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1145 next  wenzelm@23164  1146  assume "\x\=1" thus "x dvd 1"  nipkow@30042  1147  by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)  wenzelm@23164  1148 qed  wenzelm@23164  1149 lemma zdvd_mult_cancel1:  wenzelm@23164  1150  assumes mp:"m $$0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1151 proof  wenzelm@23164  1152  assume n1: "\n\ = 1" thus "m * n dvd m"  nipkow@30042  1153  by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)  wenzelm@23164  1154 next  wenzelm@23164  1155  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1156  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1157 qed  wenzelm@23164  1158 huffman@23365  1159 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  nipkow@30042  1160  unfolding zdvd_int by (cases "z \ 0") simp_all  huffman@23306  1161 huffman@23365  1162 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  nipkow@30042  1163  unfolding zdvd_int by (cases "z \ 0") simp_all  wenzelm@23164  1164 wenzelm@23164  1165 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  haftmann@27651  1166  by (auto simp add: dvd_int_iff)  wenzelm@23164  1167 wenzelm@23164  1168 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1169  apply (rule_tac z=n in int_cases)  huffman@23365  1170  apply (auto simp add: dvd_int_iff)  huffman@23365  1171  apply (rule_tac z=z in int_cases)  huffman@23307  1172  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1173  done  wenzelm@23164  1174 wenzelm@23164  1175 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1176 apply (induct "y", auto)  wenzelm@23164  1177 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1178 apply (simp (no_asm_simp))  nipkow@29948  1179 apply (rule mod_mult_eq [symmetric])  wenzelm@23164  1180 done  wenzelm@23164  1181 huffman@23365  1182 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1183 apply (subst split_div, auto)  wenzelm@23164  1184 apply (subst split_zdiv, auto)  huffman@23365  1185 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)  haftmann@29651  1186 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  wenzelm@23164  1187 done  wenzelm@23164  1188 wenzelm@23164  1189 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1190 apply (subst split_mod, auto)  huffman@23365  1191 apply (subst split_zmod, auto)  huffman@23365  1192 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1193  in unique_remainder)  haftmann@29651  1194 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  huffman@23365  1195 done  wenzelm@23164  1196 nipkow@30180  1197 lemma abs_div: "(y::int) dvd x \ abs (x div y) = abs x div abs y"  nipkow@30180  1198 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)  nipkow@30180  1199 wenzelm@23164  1200 text{*Suggested by Matthias Daum*}  wenzelm@23164  1201 lemma int_power_div_base:  wenzelm@23164  1202  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  huffman@30079  1203 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")  wenzelm@23164  1204  apply (erule ssubst)  wenzelm@23164  1205  apply (simp only: power_add)  wenzelm@23164  1206  apply simp_all  wenzelm@23164  1207 done  wenzelm@23164  1208 haftmann@23853  1209 text {* by Brian Huffman *}  haftmann@23853  1210 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  huffman@29405  1211 by (rule mod_minus_eq [symmetric])  haftmann@23853  1212 haftmann@23853  1213 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  huffman@29405  1214 by (rule mod_diff_left_eq [symmetric])  haftmann@23853  1215 haftmann@23853  1216 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  huffman@29405  1217 by (rule mod_diff_right_eq [symmetric])  haftmann@23853  1218 haftmann@23853  1219 lemmas zmod_simps =  nipkow@30034  1220  mod_add_left_eq [symmetric]  nipkow@30034  1221  mod_add_right_eq [symmetric]  haftmann@30930  1222  zmod_zmult1_eq [symmetric]  haftmann@30930  1223  mod_mult_left_eq [symmetric]  haftmann@30930  1224  zpower_zmod  haftmann@23853  1225  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1226 huffman@29045  1227 text {* Distributive laws for function @{text nat}. *}  huffman@29045  1228 huffman@29045  1229 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  huffman@29045  1230 apply (rule linorder_cases [of y 0])  huffman@29045  1231 apply (simp add: div_nonneg_neg_le0)  huffman@29045  1232 apply simp  huffman@29045  1233 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  huffman@29045  1234 done  huffman@29045  1235 huffman@29045  1236 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  huffman@29045  1237 lemma nat_mod_distrib:  huffman@29045  1238  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  huffman@29045  1239 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)  huffman@29045  1240 apply (simp add: nat_eq_iff zmod_int)  huffman@29045  1241 done  huffman@29045  1242 huffman@29045  1243 text{*Suggested by Matthias Daum*}  huffman@29045  1244 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  huffman@29045  1245 apply (subgoal_tac "nat x div nat k < nat x")  huffman@29045  1246  apply (simp (asm_lr) add: nat_div_distrib [symmetric])  huffman@29045  1247 apply (rule Divides.div_less_dividend, simp_all)  huffman@29045  1248 done  huffman@29045  1249 haftmann@23853  1250 text {* code generator setup *}  wenzelm@23164  1251 haftmann@26507  1252 context ring_1  haftmann@26507  1253 begin  haftmann@26507  1254 haftmann@28562  1255 lemma of_int_num [code]:  haftmann@26507  1256  "of_int k = (if k = 0 then 0 else if k < 0 then  haftmann@26507  1257  - of_int (- k) else let  haftmann@29651  1258  (l, m) = divmod k 2;  haftmann@26507  1259  l' = of_int l  haftmann@26507  1260  in if m = 0 then l' + l' else l' + l' + 1)"  haftmann@26507  1261 proof -  haftmann@26507  1262  have aux1: "k mod (2\int) \ (0\int) \  haftmann@26507  1263  of_int k = of_int (k div 2 * 2 + 1)"  haftmann@26507  1264  proof -  haftmann@26507  1265  have "k mod 2 < 2" by (auto intro: pos_mod_bound)  haftmann@26507  1266  moreover have "0 \ k mod 2" by (auto intro: pos_mod_sign)  haftmann@26507  1267  moreover assume "k mod 2 \ 0"  haftmann@26507  1268  ultimately have "k mod 2 = 1" by arith  haftmann@26507  1269  moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp  haftmann@26507  1270  ultimately show ?thesis by auto  haftmann@26507  1271  qed  haftmann@26507  1272  have aux2: "\x. of_int 2 * x = x + x"  haftmann@26507  1273  proof -  haftmann@26507  1274  fix x  haftmann@26507  1275  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1276  show "of_int 2 * x = x + x"  haftmann@26507  1277  unfolding int2 of_int_add left_distrib by simp  haftmann@26507  1278  qed  haftmann@26507  1279  have aux3: "\x. x * of_int 2 = x + x"  haftmann@26507  1280  proof -  haftmann@26507  1281  fix x  haftmann@26507  1282  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1283  show "x * of_int 2 = x + x"  haftmann@26507  1284  unfolding int2 of_int_add right_distrib by simp  haftmann@26507  1285  qed  haftmann@29651  1286  from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)  haftmann@26507  1287 qed  haftmann@26507  1288 haftmann@26507  1289 end  haftmann@26507  1290 chaieb@27667  1291 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  chaieb@27667  1292 proof  chaieb@27667  1293  assume H: "x mod n = y mod n"  chaieb@27667  1294  hence "x mod n - y mod n = 0" by simp  chaieb@27667  1295  hence "(x mod n - y mod n) mod n = 0" by simp  nipkow@30034  1296  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])  nipkow@30042  1297  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)  chaieb@27667  1298 next  chaieb@27667  1299  assume H: "n dvd x - y"  chaieb@27667  1300  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  chaieb@27667  1301  hence "x = n*k + y" by simp  chaieb@27667  1302  hence "x mod n = (n*k + y) mod n" by simp  nipkow@30034  1303  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)  chaieb@27667  1304 qed  chaieb@27667  1305 chaieb@27667  1306 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  chaieb@27667  1307  shows "\q. x = y + n * q"  chaieb@27667  1308 proof-  chaieb@27667  1309  from xy have th: "int x - int y = int (x - y)" by simp  chaieb@27667  1310  from xyn have "int x mod int n = int y mod int n"  chaieb@27667  1311  by (simp add: zmod_int[symmetric])  chaieb@27667  1312  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  chaieb@27667  1313  hence "n dvd x - y" by (simp add: th zdvd_int)  chaieb@27667  1314  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  chaieb@27667  1315 qed  chaieb@27667  1316 chaieb@27667  1317 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  chaieb@27667  1318  (is "?lhs = ?rhs")  chaieb@27667  1319 proof  chaieb@27667  1320  assume H: "x mod n = y mod n"  chaieb@27667  1321  {assume xy: "x \ y"  chaieb@27667  1322  from H have th: "y mod n = x mod n" by simp  chaieb@27667  1323  from nat_mod_eq_lemma[OF th xy] have ?rhs  chaieb@27667  1324  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  chaieb@27667  1325  moreover  chaieb@27667  1326  {assume xy: "y \ x"  chaieb@27667  1327  from nat_mod_eq_lemma[OF H xy] have ?rhs  chaieb@27667  1328  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  chaieb@27667  1329  ultimately show ?rhs using linear[of x y] by blast  chaieb@27667  1330 next  chaieb@27667  1331  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  chaieb@27667  1332  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  chaieb@27667  1333  thus ?lhs by simp  chaieb@27667  1334 qed  chaieb@27667  1335 haftmann@29936  1336 haftmann@29936  1337 subsection {* Code generation *}  haftmann@29936  1338 haftmann@29936  1339 definition pdivmod :: "int \ int \ int \ int" where  haftmann@29936  1340  "pdivmod k l = (\k\ div \l\, \k\ mod \l$$"  haftmann@29936  1341 haftmann@29936  1342 lemma pdivmod_posDivAlg [code]:  haftmann@29936  1343  "pdivmod k l = (if l = 0 then (0, \k\) else posDivAlg \k\ \l\)"  haftmann@29936  1344 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)  haftmann@29936  1345 haftmann@29936  1346 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@29936  1347  apsnd ((op *) (sgn l)) (if 0 < l \ 0 \ k \ l < 0 \ k < 0  haftmann@29936  1348  then pdivmod k l  haftmann@29936  1349  else (let (r, s) = pdivmod k l in  haftmann@29936  1350  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@29936  1351 proof -  haftmann@29936  1352  have aux: "\q::int. - k = l * q \ k = l * - q" by auto  haftmann@29936  1353  show ?thesis  haftmann@29936  1354  by (simp add: divmod_mod_div pdivmod_def)  haftmann@29936  1355  (auto simp add: aux not_less not_le zdiv_zminus1_eq_if  haftmann@29936  1356  zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)  haftmann@29936  1357 qed  haftmann@29936  1358 haftmann@29936  1359 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@29936  1360  apsnd ((op *) (sgn l)) (if sgn k = sgn l  haftmann@29936  1361  then pdivmod k l  haftmann@29936  1362  else (let (r, s) = pdivmod k l in  haftmann@29936  1363  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@29936  1364 proof -  haftmann@29936  1365  have "k \ 0 \ l \ 0 \ 0 < l \ 0 \ k \ l < 0 \ k < 0 \ sgn k = sgn l"  haftmann@29936  1366  by (auto simp add: not_less sgn_if)  haftmann@29936  1367  then show ?thesis by (simp add: divmod_pdivmod)  haftmann@29936  1368 qed  haftmann@29936  1369 wenzelm@23164  1370 code_modulename SML  wenzelm@23164  1371  IntDiv Integer  wenzelm@23164  1372 wenzelm@23164  1373 code_modulename OCaml  wenzelm@23164  1374  IntDiv Integer  wenzelm@23164  1375 wenzelm@23164  1376 code_modulename Haskell  haftmann@24195  1377  IntDiv Integer  wenzelm@23164  1378 wenzelm@23164  1379 end `