src/HOL/IntDiv.thy
 author huffman Sun Feb 17 06:49:53 2008 +0100 (2008-02-17) changeset 26086 3c243098b64a parent 25961 ec39d7e40554 child 26101 a657683e902a permissions -rw-r--r--
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
 wenzelm@23164  1 (* Title: HOL/IntDiv.thy  wenzelm@23164  2  ID: $Id$  wenzelm@23164  3  Author: Lawrence C Paulson, Cambridge University Computer Laboratory  wenzelm@23164  4  Copyright 1999 University of Cambridge  wenzelm@23164  5 wenzelm@23164  6 *)  wenzelm@23164  7 wenzelm@23164  8 header{*The Division Operators div and mod; the Divides Relation dvd*}  wenzelm@23164  9 wenzelm@23164  10 theory IntDiv  haftmann@25919  11 imports Int Divides FunDef  wenzelm@23164  12 begin  wenzelm@23164  13 wenzelm@23164  14 constdefs  wenzelm@23164  15  quorem :: "(int*int) * (int*int) => bool"  wenzelm@23164  16  --{*definition of quotient and remainder*}  wenzelm@23164  17  [code func]: "quorem == %((a,b), (q,r)).  wenzelm@23164  18  a = b*q + r &  wenzelm@23164  19  (if 0 < b then 0\r & r 0)"  wenzelm@23164  20 wenzelm@23164  21  adjust :: "[int, int*int] => int*int"  wenzelm@23164  22  --{*for the division algorithm*}  wenzelm@23164  23  [code func]: "adjust b == %(q,r). if 0 \ r-b then (2*q + 1, r-b)  wenzelm@23164  24  else (2*q, r)"  wenzelm@23164  25 wenzelm@23164  26 text{*algorithm for the case @{text "a\0, b>0"}*}  wenzelm@23164  27 function  wenzelm@23164  28  posDivAlg :: "int \ int \ int \ int"  wenzelm@23164  29 where  wenzelm@23164  30  "posDivAlg a b =  wenzelm@23164  31  (if (a0) then (0,a)  wenzelm@23164  32  else adjust b (posDivAlg a (2*b)))"  wenzelm@23164  33 by auto  wenzelm@23164  34 termination by (relation "measure (%(a,b). nat(a - b + 1))") auto  wenzelm@23164  35 wenzelm@23164  36 text{*algorithm for the case @{text "a<0, b>0"}*}  wenzelm@23164  37 function  wenzelm@23164  38  negDivAlg :: "int \ int \ int \ int"  wenzelm@23164  39 where  wenzelm@23164  40  "negDivAlg a b =  wenzelm@23164  41  (if (0\a+b | b\0) then (-1,a+b)  wenzelm@23164  42  else adjust b (negDivAlg a (2*b)))"  wenzelm@23164  43 by auto  wenzelm@23164  44 termination by (relation "measure (%(a,b). nat(- a - b))") auto  wenzelm@23164  45 wenzelm@23164  46 text{*algorithm for the general case @{term "b\0"}*}  wenzelm@23164  47 constdefs  wenzelm@23164  48  negateSnd :: "int*int => int*int"  wenzelm@23164  49  [code func]: "negateSnd == %(q,r). (q,-r)"  wenzelm@23164  50 wenzelm@23164  51 definition  wenzelm@23164  52  divAlg :: "int \ int \ int \ int"  wenzelm@23164  53  --{*The full division algorithm considers all possible signs for a, b  wenzelm@23164  54  including the special case @{text "a=0, b<0"} because  wenzelm@23164  55  @{term negDivAlg} requires @{term "a<0"}.*}  wenzelm@23164  56 where  wenzelm@23164  57  "divAlg = ($$a, b). (if 0\a then  wenzelm@23164  58  if 0\b then posDivAlg a b  wenzelm@23164  59  else if a=0 then (0, 0)  wenzelm@23164  60  else negateSnd (negDivAlg (-a) (-b))  wenzelm@23164  61  else  wenzelm@23164  62  if 0r-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  90  end  wenzelm@23164  91 wenzelm@23164  92  fun negDivAlg (a,b) =  wenzelm@23164  93  if 0\a+b then (~1,a+b)  wenzelm@23164  94  else let val (q,r) = negDivAlg(a, 2*b)  wenzelm@23164  95  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  96  end;  wenzelm@23164  97 wenzelm@23164  98  fun negateSnd (q,r:int) = (q,~r);  wenzelm@23164  99 wenzelm@23164  100  fun divAlg (a,b) = if 0\a then  wenzelm@23164  101  if b>0 then posDivAlg (a,b)  wenzelm@23164  102  else if a=0 then (0,0)  wenzelm@23164  103  else negateSnd (negDivAlg (~a,~b))  wenzelm@23164  104  else  wenzelm@23164  105  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  wenzelm@23164  116  ==> q' \ (q::int)"  wenzelm@23164  117 apply (subgoal_tac "r' + b * (q'-q) \ r")  wenzelm@23164  118  prefer 2 apply (simp add: right_diff_distrib)  wenzelm@23164  119 apply (subgoal_tac "0 < b * (1 + q - q') ")  wenzelm@23164  120 apply (erule_tac [2] order_le_less_trans)  wenzelm@23164  121  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  122 apply (subgoal_tac "b * q' < b * (1 + q) ")  wenzelm@23164  123  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  124 apply (simp add: mult_less_cancel_left)  wenzelm@23164  125 done  wenzelm@23164  126 wenzelm@23164  127 lemma unique_quotient_lemma_neg:  wenzelm@23164  128  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  wenzelm@23164  129  ==> q \ (q'::int)"  wenzelm@23164  130 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  wenzelm@23164  131  auto)  wenzelm@23164  132 wenzelm@23164  133 lemma unique_quotient:  wenzelm@23164  134  "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \ 0 |]  wenzelm@23164  135  ==> q = q'"  wenzelm@23164  136 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)  wenzelm@23164  137 apply (blast intro: order_antisym  wenzelm@23164  138  dest: order_eq_refl [THEN unique_quotient_lemma]  wenzelm@23164  139  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  wenzelm@23164  140 done  wenzelm@23164  141 wenzelm@23164  142 wenzelm@23164  143 lemma unique_remainder:  wenzelm@23164  144  "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \ 0 |]  wenzelm@23164  145  ==> r = r'"  wenzelm@23164  146 apply (subgoal_tac "q = q'")  wenzelm@23164  147  apply (simp add: quorem_def)  wenzelm@23164  148 apply (blast intro: unique_quotient)  wenzelm@23164  149 done  wenzelm@23164  150 wenzelm@23164  151 wenzelm@23164  152 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}  wenzelm@23164  153 wenzelm@23164  154 text{*And positive divisors*}  wenzelm@23164  155 wenzelm@23164  156 lemma adjust_eq [simp]:  wenzelm@23164  157  "adjust b (q,r) =  wenzelm@23164  158  (let diff = r-b in  wenzelm@23164  159  if 0 \ diff then (2*q + 1, diff)  wenzelm@23164  160  else (2*q, r))"  wenzelm@23164  161 by (simp add: Let_def adjust_def)  wenzelm@23164  162 wenzelm@23164  163 declare posDivAlg.simps [simp del]  wenzelm@23164  164 wenzelm@23164  165 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  166 lemma posDivAlg_eqn:  wenzelm@23164  167  "0 < b ==>  wenzelm@23164  168  posDivAlg a b = (if a a" and "0 < b"  wenzelm@23164  174  shows "quorem ((a, b), posDivAlg a b)"  wenzelm@23164  175 using prems apply (induct a b rule: posDivAlg.induct)  wenzelm@23164  176 apply auto  wenzelm@23164  177 apply (simp add: quorem_def)  wenzelm@23164  178 apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@23164  179 apply (case_tac "a < b")  wenzelm@23164  180 apply simp_all  wenzelm@23164  181 apply (erule splitE)  wenzelm@23164  182 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  183 done  wenzelm@23164  184 wenzelm@23164  185 wenzelm@23164  186 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}  wenzelm@23164  187 wenzelm@23164  188 text{*And positive divisors*}  wenzelm@23164  189 wenzelm@23164  190 declare negDivAlg.simps [simp del]  wenzelm@23164  191 wenzelm@23164  192 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  193 lemma negDivAlg_eqn:  wenzelm@23164  194  "0 < b ==>  wenzelm@23164  195  negDivAlg a b =  wenzelm@23164  196  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  wenzelm@23164  197 by (rule negDivAlg.simps [THEN trans], simp)  wenzelm@23164  198 wenzelm@23164  199 (*Correctness of negDivAlg: it computes quotients correctly  wenzelm@23164  200  It doesn't work if a=0 because the 0/b equals 0, not -1*)  wenzelm@23164  201 lemma negDivAlg_correct:  wenzelm@23164  202  assumes "a < 0" and "b > 0"  wenzelm@23164  203  shows "quorem ((a, b), negDivAlg a b)"  wenzelm@23164  204 using prems apply (induct a b rule: negDivAlg.induct)  wenzelm@23164  205 apply (auto simp add: linorder_not_le)  wenzelm@23164  206 apply (simp add: quorem_def)  wenzelm@23164  207 apply (subst negDivAlg_eqn, assumption)  wenzelm@23164  208 apply (case_tac "a + b < (0\int)")  wenzelm@23164  209 apply simp_all  wenzelm@23164  210 apply (erule splitE)  wenzelm@23164  211 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  212 done  wenzelm@23164  213 wenzelm@23164  214 wenzelm@23164  215 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}  wenzelm@23164  216 wenzelm@23164  217 (*the case a=0*)  wenzelm@23164  218 lemma quorem_0: "b \ 0 ==> quorem ((0,b), (0,0))"  wenzelm@23164  219 by (auto simp add: quorem_def linorder_neq_iff)  wenzelm@23164  220 wenzelm@23164  221 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  wenzelm@23164  222 by (subst posDivAlg.simps, auto)  wenzelm@23164  223 wenzelm@23164  224 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  wenzelm@23164  225 by (subst negDivAlg.simps, auto)  wenzelm@23164  226 wenzelm@23164  227 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"  wenzelm@23164  228 by (simp add: negateSnd_def)  wenzelm@23164  229 wenzelm@23164  230 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"  wenzelm@23164  231 by (auto simp add: split_ifs quorem_def)  wenzelm@23164  232 wenzelm@23164  233 lemma divAlg_correct: "b \ 0 ==> quorem ((a,b), divAlg (a, b))"  wenzelm@23164  234 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg  wenzelm@23164  235  posDivAlg_correct negDivAlg_correct)  wenzelm@23164  236 wenzelm@23164  237 text{*Arbitrary definitions for division by zero. Useful to simplify  wenzelm@23164  238  certain equations.*}  wenzelm@23164  239 wenzelm@23164  240 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"  wenzelm@23164  241 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)  wenzelm@23164  242 wenzelm@23164  243 wenzelm@23164  244 text{*Basic laws about division and remainder*}  wenzelm@23164  245 wenzelm@23164  246 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  wenzelm@23164  247 apply (case_tac "b = 0", simp)  wenzelm@23164  248 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  249 apply (auto simp add: quorem_def div_def mod_def)  wenzelm@23164  250 done  wenzelm@23164  251 wenzelm@23164  252 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"  wenzelm@23164  253 by(simp add: zmod_zdiv_equality[symmetric])  wenzelm@23164  254 wenzelm@23164  255 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"  wenzelm@23164  256 by(simp add: mult_commute zmod_zdiv_equality[symmetric])  wenzelm@23164  257 wenzelm@23164  258 text {* Tool setup *}  wenzelm@23164  259 wenzelm@23164  260 ML_setup {*  wenzelm@23164  261 local  wenzelm@23164  262 wenzelm@23164  263 structure CancelDivMod = CancelDivModFun(  wenzelm@23164  264 struct  wenzelm@23164  265  val div_name = @{const_name Divides.div};  wenzelm@23164  266  val mod_name = @{const_name Divides.mod};  wenzelm@23164  267  val mk_binop = HOLogic.mk_binop;  wenzelm@23164  268  val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;  wenzelm@23164  269  val dest_sum = Int_Numeral_Simprocs.dest_sum;  wenzelm@23164  270  val div_mod_eqs =  wenzelm@23164  271  map mk_meta_eq [@{thm zdiv_zmod_equality},  wenzelm@23164  272  @{thm zdiv_zmod_equality2}];  wenzelm@23164  273  val trans = trans;  wenzelm@23164  274  val prove_eq_sums =  wenzelm@23164  275  let  huffman@23365  276  val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}  wenzelm@23164  277  in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;  wenzelm@23164  278 end)  wenzelm@23164  279 wenzelm@23164  280 in  wenzelm@23164  281 wenzelm@23164  282 val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc  wenzelm@23164  283  ("cancel_zdiv_zmod", ["(m::int) + n"], K CancelDivMod.proc)  wenzelm@23164  284 wenzelm@23164  285 end;  wenzelm@23164  286 wenzelm@23164  287 Addsimprocs [cancel_zdiv_zmod_proc]  wenzelm@23164  288 *}  wenzelm@23164  289 wenzelm@23164  290 lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b"  wenzelm@23164  291 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  292 apply (auto simp add: quorem_def mod_def)  wenzelm@23164  293 done  wenzelm@23164  294 wenzelm@23164  295 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]  wenzelm@23164  296  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]  wenzelm@23164  297 wenzelm@23164  298 lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b"  wenzelm@23164  299 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  300 apply (auto simp add: quorem_def div_def mod_def)  wenzelm@23164  301 done  wenzelm@23164  302 wenzelm@23164  303 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]  wenzelm@23164  304  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]  wenzelm@23164  305 wenzelm@23164  306 wenzelm@23164  307 wenzelm@23164  308 subsection{*General Properties of div and mod*}  wenzelm@23164  309 wenzelm@23164  310 lemma quorem_div_mod: "b \ 0 ==> quorem ((a, b), (a div b, a mod b))"  wenzelm@23164  311 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  312 apply (force simp add: quorem_def linorder_neq_iff)  wenzelm@23164  313 done  wenzelm@23164  314 wenzelm@23164  315 lemma quorem_div: "[| quorem((a,b),(q,r)); b \ 0 |] ==> a div b = q"  wenzelm@23164  316 by (simp add: quorem_div_mod [THEN unique_quotient])  wenzelm@23164  317 wenzelm@23164  318 lemma quorem_mod: "[| quorem((a,b),(q,r)); b \ 0 |] ==> a mod b = r"  wenzelm@23164  319 by (simp add: quorem_div_mod [THEN unique_remainder])  wenzelm@23164  320 wenzelm@23164  321 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  wenzelm@23164  322 apply (rule quorem_div)  wenzelm@23164  323 apply (auto simp add: quorem_def)  wenzelm@23164  324 done  wenzelm@23164  325 wenzelm@23164  326 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  wenzelm@23164  327 apply (rule quorem_div)  wenzelm@23164  328 apply (auto simp add: quorem_def)  wenzelm@23164  329 done  wenzelm@23164  330 wenzelm@23164  331 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  wenzelm@23164  332 apply (rule quorem_div)  wenzelm@23164  333 apply (auto simp add: quorem_def)  wenzelm@23164  334 done  wenzelm@23164  335 wenzelm@23164  336 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  wenzelm@23164  337 wenzelm@23164  338 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  wenzelm@23164  339 apply (rule_tac q = 0 in quorem_mod)  wenzelm@23164  340 apply (auto simp add: quorem_def)  wenzelm@23164  341 done  wenzelm@23164  342 wenzelm@23164  343 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  wenzelm@23164  344 apply (rule_tac q = 0 in quorem_mod)  wenzelm@23164  345 apply (auto simp add: quorem_def)  wenzelm@23164  346 done  wenzelm@23164  347 wenzelm@23164  348 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  wenzelm@23164  349 apply (rule_tac q = "-1" in quorem_mod)  wenzelm@23164  350 apply (auto simp add: quorem_def)  wenzelm@23164  351 done  wenzelm@23164  352 wenzelm@23164  353 text{*There is no @{text mod_neg_pos_trivial}.*}  wenzelm@23164  354 wenzelm@23164  355 wenzelm@23164  356 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)  wenzelm@23164  357 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"  wenzelm@23164  358 apply (case_tac "b = 0", simp)  wenzelm@23164  359 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified,  wenzelm@23164  360  THEN quorem_div, THEN sym])  wenzelm@23164  361 wenzelm@23164  362 done  wenzelm@23164  363 wenzelm@23164  364 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)  wenzelm@23164  365 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"  wenzelm@23164  366 apply (case_tac "b = 0", simp)  wenzelm@23164  367 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],  wenzelm@23164  368  auto)  wenzelm@23164  369 done  wenzelm@23164  370 wenzelm@23164  371 wenzelm@23164  372 subsection{*Laws for div and mod with Unary Minus*}  wenzelm@23164  373 wenzelm@23164  374 lemma zminus1_lemma:  wenzelm@23164  375  "quorem((a,b),(q,r))  wenzelm@23164  376  ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  wenzelm@23164  377  (if r=0 then 0 else b-r))"  wenzelm@23164  378 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)  wenzelm@23164  379 wenzelm@23164  380 wenzelm@23164  381 lemma zdiv_zminus1_eq_if:  wenzelm@23164  382  "b \ (0::int)  wenzelm@23164  383  ==> (-a) div b =  wenzelm@23164  384  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  385 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])  wenzelm@23164  386 wenzelm@23164  387 lemma zmod_zminus1_eq_if:  wenzelm@23164  388  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  wenzelm@23164  389 apply (case_tac "b = 0", simp)  wenzelm@23164  390 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])  wenzelm@23164  391 done  wenzelm@23164  392 wenzelm@23164  393 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  wenzelm@23164  394 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  wenzelm@23164  395 wenzelm@23164  396 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  wenzelm@23164  397 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  wenzelm@23164  398 wenzelm@23164  399 lemma zdiv_zminus2_eq_if:  wenzelm@23164  400  "b \ (0::int)  wenzelm@23164  401  ==> a div (-b) =  wenzelm@23164  402  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  403 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  wenzelm@23164  404 wenzelm@23164  405 lemma zmod_zminus2_eq_if:  wenzelm@23164  406  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  wenzelm@23164  407 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  wenzelm@23164  408 wenzelm@23164  409 wenzelm@23164  410 subsection{*Division of a Number by Itself*}  wenzelm@23164  411 wenzelm@23164  412 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  wenzelm@23164  413 apply (subgoal_tac "0 < a*q")  wenzelm@23164  414  apply (simp add: zero_less_mult_iff, arith)  wenzelm@23164  415 done  wenzelm@23164  416 wenzelm@23164  417 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  wenzelm@23164  418 apply (subgoal_tac "0 \ a* (1-q) ")  wenzelm@23164  419  apply (simp add: zero_le_mult_iff)  wenzelm@23164  420 apply (simp add: right_diff_distrib)  wenzelm@23164  421 done  wenzelm@23164  422 wenzelm@23164  423 lemma self_quotient: "[| quorem((a,a),(q,r)); a \ (0::int) |] ==> q = 1"  wenzelm@23164  424 apply (simp add: split_ifs quorem_def linorder_neq_iff)  wenzelm@23164  425 apply (rule order_antisym, safe, simp_all)  wenzelm@23164  426 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  wenzelm@23164  427 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  wenzelm@23164  428 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  wenzelm@23164  429 done  wenzelm@23164  430 wenzelm@23164  431 lemma self_remainder: "[| quorem((a,a),(q,r)); a \ (0::int) |] ==> r = 0"  wenzelm@23164  432 apply (frule self_quotient, assumption)  wenzelm@23164  433 apply (simp add: quorem_def)  wenzelm@23164  434 done  wenzelm@23164  435 wenzelm@23164  436 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  wenzelm@23164  437 by (simp add: quorem_div_mod [THEN self_quotient])  wenzelm@23164  438 wenzelm@23164  439 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  wenzelm@23164  440 lemma zmod_self [simp]: "a mod a = (0::int)"  wenzelm@23164  441 apply (case_tac "a = 0", simp)  wenzelm@23164  442 apply (simp add: quorem_div_mod [THEN self_remainder])  wenzelm@23164  443 done  wenzelm@23164  444 wenzelm@23164  445 wenzelm@23164  446 subsection{*Computation of Division and Remainder*}  wenzelm@23164  447 wenzelm@23164  448 lemma zdiv_zero [simp]: "(0::int) div b = 0"  wenzelm@23164  449 by (simp add: div_def divAlg_def)  wenzelm@23164  450 wenzelm@23164  451 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  wenzelm@23164  452 by (simp add: div_def divAlg_def)  wenzelm@23164  453 wenzelm@23164  454 lemma zmod_zero [simp]: "(0::int) mod b = 0"  wenzelm@23164  455 by (simp add: mod_def divAlg_def)  wenzelm@23164  456 wenzelm@23164  457 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"  wenzelm@23164  458 by (simp add: div_def divAlg_def)  wenzelm@23164  459 wenzelm@23164  460 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  wenzelm@23164  461 by (simp add: mod_def divAlg_def)  wenzelm@23164  462 wenzelm@23164  463 text{*a positive, b positive *}  wenzelm@23164  464 wenzelm@23164  465 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  wenzelm@23164  466 by (simp add: div_def divAlg_def)  wenzelm@23164  467 wenzelm@23164  468 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  wenzelm@23164  469 by (simp add: mod_def divAlg_def)  wenzelm@23164  470 wenzelm@23164  471 text{*a negative, b positive *}  wenzelm@23164  472 wenzelm@23164  473 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  wenzelm@23164  474 by (simp add: div_def divAlg_def)  wenzelm@23164  475 wenzelm@23164  476 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  wenzelm@23164  477 by (simp add: mod_def divAlg_def)  wenzelm@23164  478 wenzelm@23164  479 text{*a positive, b negative *}  wenzelm@23164  480 wenzelm@23164  481 lemma div_pos_neg:  wenzelm@23164  482  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  wenzelm@23164  483 by (simp add: div_def divAlg_def)  wenzelm@23164  484 wenzelm@23164  485 lemma mod_pos_neg:  wenzelm@23164  486  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  wenzelm@23164  487 by (simp add: mod_def divAlg_def)  wenzelm@23164  488 wenzelm@23164  489 text{*a negative, b negative *}  wenzelm@23164  490 wenzelm@23164  491 lemma div_neg_neg:  wenzelm@23164  492  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  493 by (simp add: div_def divAlg_def)  wenzelm@23164  494 wenzelm@23164  495 lemma mod_neg_neg:  wenzelm@23164  496  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  497 by (simp add: mod_def divAlg_def)  wenzelm@23164  498 wenzelm@23164  499 text {*Simplify expresions in which div and mod combine numerical constants*}  wenzelm@23164  500 huffman@24481  501 lemma quoremI:  huffman@24481  502  "\a == b * q + r; if 0 < b then 0 \ r \ r < b else b < r \ r \ 0\  huffman@24481  503  \ quorem ((a, b), (q, r))"  huffman@24481  504  unfolding quorem_def by simp  huffman@24481  505 huffman@24481  506 lemmas quorem_div_eq = quoremI [THEN quorem_div, THEN eq_reflection]  huffman@24481  507 lemmas quorem_mod_eq = quoremI [THEN quorem_mod, THEN eq_reflection]  huffman@24481  508 lemmas arithmetic_simps =  huffman@24481  509  arith_simps  huffman@24481  510  add_special  huffman@24481  511  OrderedGroup.add_0_left  huffman@24481  512  OrderedGroup.add_0_right  huffman@24481  513  mult_zero_left  huffman@24481  514  mult_zero_right  huffman@24481  515  mult_1_left  huffman@24481  516  mult_1_right  huffman@24481  517 huffman@24481  518 (* simprocs adapted from HOL/ex/Binary.thy *)  huffman@24481  519 ML {*  huffman@24481  520 local  huffman@24481  521  infix ==;  huffman@24481  522  val op == = Logic.mk_equals;  huffman@24481  523  fun plus m n = @{term "plus :: int \ int \ int"}  m  n;  huffman@24481  524  fun mult m n = @{term "times :: int \ int \ int"}  m  n;  huffman@24481  525 huffman@24481  526  val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};  huffman@24481  527  fun prove ctxt prop =  huffman@24481  528  Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));  huffman@24481  529 huffman@24481  530  fun binary_proc proc ss ct =  huffman@24481  531  (case Thm.term_of ct of  huffman@24481  532  _  t  u =>  huffman@24481  533  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  huffman@24481  534  SOME args => proc (Simplifier.the_context ss) args  huffman@24481  535  | NONE => NONE)  huffman@24481  536  | _ => NONE);  huffman@24481  537 in  huffman@24481  538 huffman@24481  539 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>  huffman@24481  540  if n = 0 then NONE  huffman@24481  541  else  wenzelm@24630  542  let val (k, l) = Integer.div_mod m n;  huffman@24481  543  fun mk_num x = HOLogic.mk_number HOLogic.intT x;  huffman@24481  544  in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])  huffman@24481  545  end);  huffman@24481  546 huffman@24481  547 end;  huffman@24481  548 *}  huffman@24481  549 huffman@24481  550 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  huffman@24481  551  {* K (divmod_proc (@{thm quorem_div_eq})) *}  huffman@24481  552 huffman@24481  553 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  huffman@24481  554  {* K (divmod_proc (@{thm quorem_mod_eq})) *}  huffman@24481  555 huffman@24481  556 (* The following 8 lemmas are made unnecessary by the above simprocs: *)  huffman@24481  557 huffman@24481  558 lemmas div_pos_pos_number_of =  wenzelm@23164  559  div_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  560 huffman@24481  561 lemmas div_neg_pos_number_of =  wenzelm@23164  562  div_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  563 huffman@24481  564 lemmas div_pos_neg_number_of =  wenzelm@23164  565  div_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  566 huffman@24481  567 lemmas div_neg_neg_number_of =  wenzelm@23164  568  div_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  569 wenzelm@23164  570 huffman@24481  571 lemmas mod_pos_pos_number_of =  wenzelm@23164  572  mod_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  573 huffman@24481  574 lemmas mod_neg_pos_number_of =  wenzelm@23164  575  mod_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  576 huffman@24481  577 lemmas mod_pos_neg_number_of =  wenzelm@23164  578  mod_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  579 huffman@24481  580 lemmas mod_neg_neg_number_of =  wenzelm@23164  581  mod_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  582 wenzelm@23164  583 wenzelm@23164  584 lemmas posDivAlg_eqn_number_of [simp] =  wenzelm@23164  585  posDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  586 wenzelm@23164  587 lemmas negDivAlg_eqn_number_of [simp] =  wenzelm@23164  588  negDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  589 wenzelm@23164  590 wenzelm@23164  591 text{*Special-case simplification *}  wenzelm@23164  592 wenzelm@23164  593 lemma zmod_1 [simp]: "a mod (1::int) = 0"  wenzelm@23164  594 apply (cut_tac a = a and b = 1 in pos_mod_sign)  wenzelm@23164  595 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)  wenzelm@23164  596 apply (auto simp del:pos_mod_bound pos_mod_sign)  wenzelm@23164  597 done  wenzelm@23164  598 wenzelm@23164  599 lemma zdiv_1 [simp]: "a div (1::int) = a"  wenzelm@23164  600 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)  wenzelm@23164  601 wenzelm@23164  602 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  wenzelm@23164  603 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  wenzelm@23164  604 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  wenzelm@23164  605 apply (auto simp del: neg_mod_sign neg_mod_bound)  wenzelm@23164  606 done  wenzelm@23164  607 wenzelm@23164  608 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"  wenzelm@23164  609 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)  wenzelm@23164  610 wenzelm@23164  611 (** The last remaining special cases for constant arithmetic:  wenzelm@23164  612  1 div z and 1 mod z **)  wenzelm@23164  613 wenzelm@23164  614 lemmas div_pos_pos_1_number_of [simp] =  wenzelm@23164  615  div_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  616 wenzelm@23164  617 lemmas div_pos_neg_1_number_of [simp] =  wenzelm@23164  618  div_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  619 wenzelm@23164  620 lemmas mod_pos_pos_1_number_of [simp] =  wenzelm@23164  621  mod_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  622 wenzelm@23164  623 lemmas mod_pos_neg_1_number_of [simp] =  wenzelm@23164  624  mod_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  625 wenzelm@23164  626 wenzelm@23164  627 lemmas posDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  628  posDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  629 wenzelm@23164  630 lemmas negDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  631  negDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  632 wenzelm@23164  633 wenzelm@23164  634 wenzelm@23164  635 subsection{*Monotonicity in the First Argument (Dividend)*}  wenzelm@23164  636 wenzelm@23164  637 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  wenzelm@23164  638 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  639 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  640 apply (rule unique_quotient_lemma)  wenzelm@23164  641 apply (erule subst)  wenzelm@23164  642 apply (erule subst, simp_all)  wenzelm@23164  643 done  wenzelm@23164  644 wenzelm@23164  645 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  wenzelm@23164  646 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  647 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  648 apply (rule unique_quotient_lemma_neg)  wenzelm@23164  649 apply (erule subst)  wenzelm@23164  650 apply (erule subst, simp_all)  wenzelm@23164  651 done  wenzelm@23164  652 wenzelm@23164  653 wenzelm@23164  654 subsection{*Monotonicity in the Second Argument (Divisor)*}  wenzelm@23164  655 wenzelm@23164  656 lemma q_pos_lemma:  wenzelm@23164  657  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  wenzelm@23164  658 apply (subgoal_tac "0 < b'* (q' + 1) ")  wenzelm@23164  659  apply (simp add: zero_less_mult_iff)  wenzelm@23164  660 apply (simp add: right_distrib)  wenzelm@23164  661 done  wenzelm@23164  662 wenzelm@23164  663 lemma zdiv_mono2_lemma:  wenzelm@23164  664  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  wenzelm@23164  665  r' < b'; 0 \ r; 0 < b'; b' \ b |]  wenzelm@23164  666  ==> q \ (q'::int)"  wenzelm@23164  667 apply (frule q_pos_lemma, assumption+)  wenzelm@23164  668 apply (subgoal_tac "b*q < b* (q' + 1) ")  wenzelm@23164  669  apply (simp add: mult_less_cancel_left)  wenzelm@23164  670 apply (subgoal_tac "b*q = r' - r + b'*q'")  wenzelm@23164  671  prefer 2 apply simp  wenzelm@23164  672 apply (simp (no_asm_simp) add: right_distrib)  wenzelm@23164  673 apply (subst add_commute, rule zadd_zless_mono, arith)  wenzelm@23164  674 apply (rule mult_right_mono, auto)  wenzelm@23164  675 done  wenzelm@23164  676 wenzelm@23164  677 lemma zdiv_mono2:  wenzelm@23164  678  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  wenzelm@23164  679 apply (subgoal_tac "b \ 0")  wenzelm@23164  680  prefer 2 apply arith  wenzelm@23164  681 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  682 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  683 apply (rule zdiv_mono2_lemma)  wenzelm@23164  684 apply (erule subst)  wenzelm@23164  685 apply (erule subst, simp_all)  wenzelm@23164  686 done  wenzelm@23164  687 wenzelm@23164  688 lemma q_neg_lemma:  wenzelm@23164  689  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  wenzelm@23164  690 apply (subgoal_tac "b'*q' < 0")  wenzelm@23164  691  apply (simp add: mult_less_0_iff, arith)  wenzelm@23164  692 done  wenzelm@23164  693 wenzelm@23164  694 lemma zdiv_mono2_neg_lemma:  wenzelm@23164  695  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  wenzelm@23164  696  r < b; 0 \ r'; 0 < b'; b' \ b |]  wenzelm@23164  697  ==> q' \ (q::int)"  wenzelm@23164  698 apply (frule q_neg_lemma, assumption+)  wenzelm@23164  699 apply (subgoal_tac "b*q' < b* (q + 1) ")  wenzelm@23164  700  apply (simp add: mult_less_cancel_left)  wenzelm@23164  701 apply (simp add: right_distrib)  wenzelm@23164  702 apply (subgoal_tac "b*q' \ b'*q'")  wenzelm@23164  703  prefer 2 apply (simp add: mult_right_mono_neg, arith)  wenzelm@23164  704 done  wenzelm@23164  705 wenzelm@23164  706 lemma zdiv_mono2_neg:  wenzelm@23164  707  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  wenzelm@23164  708 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  709 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  710 apply (rule zdiv_mono2_neg_lemma)  wenzelm@23164  711 apply (erule subst)  wenzelm@23164  712 apply (erule subst, simp_all)  wenzelm@23164  713 done  wenzelm@23164  714 haftmann@25942  715 wenzelm@23164  716 subsection{*More Algebraic Laws for div and mod*}  wenzelm@23164  717 wenzelm@23164  718 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  wenzelm@23164  719 wenzelm@23164  720 lemma zmult1_lemma:  wenzelm@23164  721  "[| quorem((b,c),(q,r)); c \ 0 |]  wenzelm@23164  722  ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"  wenzelm@23164  723 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)  wenzelm@23164  724 wenzelm@23164  725 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  wenzelm@23164  726 apply (case_tac "c = 0", simp)  wenzelm@23164  727 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])  wenzelm@23164  728 done  wenzelm@23164  729 wenzelm@23164  730 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  wenzelm@23164  731 apply (case_tac "c = 0", simp)  wenzelm@23164  732 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])  wenzelm@23164  733 done  wenzelm@23164  734 wenzelm@23164  735 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"  wenzelm@23164  736 apply (rule trans)  wenzelm@23164  737 apply (rule_tac s = "b*a mod c" in trans)  wenzelm@23164  738 apply (rule_tac [2] zmod_zmult1_eq)  wenzelm@23164  739 apply (simp_all add: mult_commute)  wenzelm@23164  740 done  wenzelm@23164  741 wenzelm@23164  742 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"  wenzelm@23164  743 apply (rule zmod_zmult1_eq' [THEN trans])  wenzelm@23164  744 apply (rule zmod_zmult1_eq)  wenzelm@23164  745 done  wenzelm@23164  746 wenzelm@23164  747 lemma zdiv_zmult_self1 [simp]: "b \ (0::int) ==> (a*b) div b = a"  wenzelm@23164  748 by (simp add: zdiv_zmult1_eq)  wenzelm@23164  749 haftmann@25942  750 instance int :: semiring_div  haftmann@25942  751  by intro_classes auto  haftmann@25942  752 wenzelm@23164  753 lemma zdiv_zmult_self2 [simp]: "b \ (0::int) ==> (b*a) div b = a"  wenzelm@23164  754 by (subst mult_commute, erule zdiv_zmult_self1)  wenzelm@23164  755 wenzelm@23164  756 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"  wenzelm@23164  757 by (simp add: zmod_zmult1_eq)  wenzelm@23164  758 wenzelm@23164  759 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"  wenzelm@23164  760 by (simp add: mult_commute zmod_zmult1_eq)  wenzelm@23164  761 wenzelm@23164  762 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  wenzelm@23164  763 proof  wenzelm@23164  764  assume "m mod d = 0"  wenzelm@23164  765  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto  wenzelm@23164  766 next  wenzelm@23164  767  assume "EX q::int. m = d*q"  wenzelm@23164  768  thus "m mod d = 0" by auto  wenzelm@23164  769 qed  wenzelm@23164  770 wenzelm@23164  771 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  772 wenzelm@23164  773 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  wenzelm@23164  774 wenzelm@23164  775 lemma zadd1_lemma:  wenzelm@23164  776  "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c \ 0 |]  wenzelm@23164  777  ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"  wenzelm@23164  778 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)  wenzelm@23164  779 wenzelm@23164  780 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  wenzelm@23164  781 lemma zdiv_zadd1_eq:  wenzelm@23164  782  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  wenzelm@23164  783 apply (case_tac "c = 0", simp)  wenzelm@23164  784 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)  wenzelm@23164  785 done  wenzelm@23164  786 wenzelm@23164  787 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"  wenzelm@23164  788 apply (case_tac "c = 0", simp)  wenzelm@23164  789 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)  wenzelm@23164  790 done  wenzelm@23164  791 wenzelm@23164  792 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"  wenzelm@23164  793 apply (case_tac "b = 0", simp)  wenzelm@23164  794 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  wenzelm@23164  795 done  wenzelm@23164  796 wenzelm@23164  797 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"  wenzelm@23164  798 apply (case_tac "b = 0", simp)  wenzelm@23164  799 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)  wenzelm@23164  800 done  wenzelm@23164  801 wenzelm@23164  802 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"  wenzelm@23164  803 apply (rule trans [symmetric])  wenzelm@23164  804 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  805 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  806 done  wenzelm@23164  807 wenzelm@23164  808 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"  wenzelm@23164  809 apply (rule trans [symmetric])  wenzelm@23164  810 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  811 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  812 done  wenzelm@23164  813 wenzelm@23164  814 lemma zdiv_zadd_self1[simp]: "a \ (0::int) ==> (a+b) div a = b div a + 1"  wenzelm@23164  815 by (simp add: zdiv_zadd1_eq)  wenzelm@23164  816 wenzelm@23164  817 lemma zdiv_zadd_self2[simp]: "a \ (0::int) ==> (b+a) div a = b div a + 1"  wenzelm@23164  818 by (simp add: zdiv_zadd1_eq)  wenzelm@23164  819 wenzelm@23164  820 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"  wenzelm@23164  821 apply (case_tac "a = 0", simp)  wenzelm@23164  822 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  823 done  wenzelm@23164  824 wenzelm@23164  825 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"  wenzelm@23164  826 apply (case_tac "a = 0", simp)  wenzelm@23164  827 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  828 done  wenzelm@23164  829 wenzelm@23164  830 nipkow@23983  831 lemma zmod_zdiff1_eq: fixes a::int  nipkow@23983  832  shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r")  nipkow@23983  833 proof -  nipkow@23983  834  have "?l = (c + (a mod c - b mod c)) mod c"  nipkow@23983  835  using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if)  nipkow@23983  836  also have "\ = ?r" by simp  nipkow@23983  837  finally show ?thesis .  nipkow@23983  838 qed  nipkow@23983  839 wenzelm@23164  840 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  841 wenzelm@23164  842 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  843  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  844  to cause particular problems.*)  wenzelm@23164  845 wenzelm@23164  846 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  847 wenzelm@23164  848 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  849 apply (subgoal_tac "b * (c - q mod c) < r * 1")  wenzelm@23164  850 apply (simp add: right_diff_distrib)  wenzelm@23164  851 apply (rule order_le_less_trans)  wenzelm@23164  852 apply (erule_tac [2] mult_strict_right_mono)  wenzelm@23164  853 apply (rule mult_left_mono_neg)  wenzelm@23164  854 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  855  add1_zle_eq pos_mod_bound)  wenzelm@23164  856 done  wenzelm@23164  857 wenzelm@23164  858 lemma zmult2_lemma_aux2:  wenzelm@23164  859  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  860 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  861  apply arith  wenzelm@23164  862 apply (simp add: mult_le_0_iff)  wenzelm@23164  863 done  wenzelm@23164  864 wenzelm@23164  865 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  866 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  867 apply arith  wenzelm@23164  868 apply (simp add: zero_le_mult_iff)  wenzelm@23164  869 done  wenzelm@23164  870 wenzelm@23164  871 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  872 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  wenzelm@23164  873 apply (simp add: right_diff_distrib)  wenzelm@23164  874 apply (rule order_less_le_trans)  wenzelm@23164  875 apply (erule mult_strict_right_mono)  wenzelm@23164  876 apply (rule_tac [2] mult_left_mono)  wenzelm@23164  877 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  878  add1_zle_eq pos_mod_bound)  wenzelm@23164  879 done  wenzelm@23164  880 wenzelm@23164  881 lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b \ 0; 0 < c |]  wenzelm@23164  882  ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"  wenzelm@23164  883 by (auto simp add: mult_ac quorem_def linorder_neq_iff  wenzelm@23164  884  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  885  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  886 wenzelm@23164  887 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  888 apply (case_tac "b = 0", simp)  wenzelm@23164  889 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])  wenzelm@23164  890 done  wenzelm@23164  891 wenzelm@23164  892 lemma zmod_zmult2_eq:  wenzelm@23164  893  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  894 apply (case_tac "b = 0", simp)  wenzelm@23164  895 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])  wenzelm@23164  896 done  wenzelm@23164  897 wenzelm@23164  898 wenzelm@23164  899 subsection{*Cancellation of Common Factors in div*}  wenzelm@23164  900 wenzelm@23164  901 lemma zdiv_zmult_zmult1_aux1:  wenzelm@23164  902  "[| (0::int) < b; c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  903 by (subst zdiv_zmult2_eq, auto)  wenzelm@23164  904 wenzelm@23164  905 lemma zdiv_zmult_zmult1_aux2:  wenzelm@23164  906  "[| b < (0::int); c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  907 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")  wenzelm@23164  908 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)  wenzelm@23164  909 done  wenzelm@23164  910 wenzelm@23164  911 lemma zdiv_zmult_zmult1: "c \ (0::int) ==> (c*a) div (c*b) = a div b"  wenzelm@23164  912 apply (case_tac "b = 0", simp)  wenzelm@23164  913 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)  wenzelm@23164  914 done  wenzelm@23164  915 nipkow@23401  916 lemma zdiv_zmult_zmult1_if[simp]:  nipkow@23401  917  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"  nipkow@23401  918 by (simp add:zdiv_zmult_zmult1)  nipkow@23401  919 nipkow@23401  920 (*  wenzelm@23164  921 lemma zdiv_zmult_zmult2: "c \ (0::int) ==> (a*c) div (b*c) = a div b"  wenzelm@23164  922 apply (drule zdiv_zmult_zmult1)  wenzelm@23164  923 apply (auto simp add: mult_commute)  wenzelm@23164  924 done  nipkow@23401  925 *)  wenzelm@23164  926 wenzelm@23164  927 wenzelm@23164  928 subsection{*Distribution of Factors over mod*}  wenzelm@23164  929 wenzelm@23164  930 lemma zmod_zmult_zmult1_aux1:  wenzelm@23164  931  "[| (0::int) < b; c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  932 by (subst zmod_zmult2_eq, auto)  wenzelm@23164  933 wenzelm@23164  934 lemma zmod_zmult_zmult1_aux2:  wenzelm@23164  935  "[| b < (0::int); c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  936 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")  wenzelm@23164  937 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)  wenzelm@23164  938 done  wenzelm@23164  939 wenzelm@23164  940 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"  wenzelm@23164  941 apply (case_tac "b = 0", simp)  wenzelm@23164  942 apply (case_tac "c = 0", simp)  wenzelm@23164  943 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)  wenzelm@23164  944 done  wenzelm@23164  945 wenzelm@23164  946 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"  wenzelm@23164  947 apply (cut_tac c = c in zmod_zmult_zmult1)  wenzelm@23164  948 apply (auto simp add: mult_commute)  wenzelm@23164  949 done  wenzelm@23164  950 nipkow@24490  951 lemma zmod_zmod_cancel:  nipkow@24490  952 assumes "n dvd m" shows "(k::int) mod m mod n = k mod n"  nipkow@24490  953 proof -  nipkow@24490  954  from n dvd m obtain r where "m = n*r" by(auto simp:dvd_def)  nipkow@24490  955  have "k mod n = (m * (k div m) + k mod m) mod n"  nipkow@24490  956  using zmod_zdiv_equality[of k m] by simp  nipkow@24490  957  also have "\ = (m * (k div m) mod n + k mod m mod n) mod n"  nipkow@24490  958  by(subst zmod_zadd1_eq, rule refl)  nipkow@24490  959  also have "m * (k div m) mod n = 0" using m = n*r  nipkow@24490  960  by(simp add:mult_ac)  nipkow@24490  961  finally show ?thesis by simp  nipkow@24490  962 qed  nipkow@24490  963 wenzelm@23164  964 wenzelm@23164  965 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  966 wenzelm@23164  967 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  968 wenzelm@23164  969 lemma split_pos_lemma:  wenzelm@23164  970  "0  wenzelm@23164  971  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  972 apply (rule iffI, clarify)  wenzelm@23164  973  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  974  apply (subst zmod_zadd1_eq)  wenzelm@23164  975  apply (subst zdiv_zadd1_eq)  wenzelm@23164  976  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@23164  977 txt{*converse direction*}  wenzelm@23164  978 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  979 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  980 done  wenzelm@23164  981 wenzelm@23164  982 lemma split_neg_lemma:  wenzelm@23164  983  "k<0 ==>  wenzelm@23164  984  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  985 apply (rule iffI, clarify)  wenzelm@23164  986  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  987  apply (subst zmod_zadd1_eq)  wenzelm@23164  988  apply (subst zdiv_zadd1_eq)  wenzelm@23164  989  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  990 txt{*converse direction*}  wenzelm@23164  991 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  992 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  993 done  wenzelm@23164  994 wenzelm@23164  995 lemma split_zdiv:  wenzelm@23164  996  "P(n div k :: int) =  wenzelm@23164  997  ((k = 0 --> P 0) &  wenzelm@23164  998  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  999  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  1000 apply (case_tac "k=0", simp)  wenzelm@23164  1001 apply (simp only: linorder_neq_iff)  wenzelm@23164  1002 apply (erule disjE)  wenzelm@23164  1003  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  1004  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  1005 done  wenzelm@23164  1006 wenzelm@23164  1007 lemma split_zmod:  wenzelm@23164  1008  "P(n mod k :: int) =  wenzelm@23164  1009  ((k = 0 --> P n) &  wenzelm@23164  1010  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  1011  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  1012 apply (case_tac "k=0", simp)  wenzelm@23164  1013 apply (simp only: linorder_neq_iff)  wenzelm@23164  1014 apply (erule disjE)  wenzelm@23164  1015  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  1016  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  1017 done  wenzelm@23164  1018 wenzelm@23164  1019 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  1020 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1021 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1022 wenzelm@23164  1023 wenzelm@23164  1024 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  1025 wenzelm@23164  1026 text{*computing div by shifting *}  wenzelm@23164  1027 wenzelm@23164  1028 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  1029 proof cases  wenzelm@23164  1030  assume "a=0"  wenzelm@23164  1031  thus ?thesis by simp  wenzelm@23164  1032 next  wenzelm@23164  1033  assume "a\0" and le_a: "0\a"  wenzelm@23164  1034  hence a_pos: "1 \ a" by arith  wenzelm@23164  1035  hence one_less_a2: "1 < 2*a" by arith  wenzelm@23164  1036  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  wenzelm@23164  1037  by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)  wenzelm@23164  1038  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  1039  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  1040  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  1041  with le_2a  wenzelm@23164  1042  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  1043  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  1044  right_distrib)  wenzelm@23164  1045  thus ?thesis  wenzelm@23164  1046  by (subst zdiv_zadd1_eq,  wenzelm@23164  1047  simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2  wenzelm@23164  1048  div_pos_pos_trivial)  wenzelm@23164  1049 qed  wenzelm@23164  1050 wenzelm@23164  1051 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  1052 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  1053 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  1054 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1055 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1056 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  1057  simp)  wenzelm@23164  1058 done  wenzelm@23164  1059 wenzelm@23164  1060 (*Not clear why this must be proved separately; probably number_of causes  wenzelm@23164  1061  simplification problems*)  wenzelm@23164  1062 lemma not_0_le_lemma: "~ 0 \ x ==> x \ (0::int)"  wenzelm@23164  1063 by auto  wenzelm@23164  1064 huffman@26086  1065 lemma zdiv_number_of_Bit0 [simp]:  huffman@26086  1066  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  huffman@26086  1067  number_of v div (number_of w :: int)"  huffman@26086  1068 by (simp only: number_of_eq numeral_simps) simp  huffman@26086  1069 huffman@26086  1070 lemma zdiv_number_of_Bit1 [simp]:  huffman@26086  1071  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  huffman@26086  1072  (if (0::int) \ number_of w  wenzelm@23164  1073  then number_of v div (number_of w)  wenzelm@23164  1074  else (number_of v + (1::int)) div (number_of w))"  wenzelm@23164  1075 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  huffman@26086  1076 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)  wenzelm@23164  1077 done  wenzelm@23164  1078 wenzelm@23164  1079 wenzelm@23164  1080 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  1081 wenzelm@23164  1082 lemma pos_zmod_mult_2:  wenzelm@23164  1083  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  1084 apply (case_tac "a = 0", simp)  wenzelm@23164  1085 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  1086  prefer 2 apply arith  wenzelm@23164  1087 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  1088  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  1089 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  1090  pos_mod_bound)  wenzelm@23164  1091 apply (subst zmod_zadd1_eq)  wenzelm@23164  1092 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)  wenzelm@23164  1093 apply (rule mod_pos_pos_trivial)  huffman@26086  1094 apply (auto simp add: mod_pos_pos_trivial ring_distribs)  wenzelm@23164  1095 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  1096 done  wenzelm@23164  1097 wenzelm@23164  1098 lemma neg_zmod_mult_2:  wenzelm@23164  1099  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  1100 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  1101  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  1102 apply (rule_tac [2] pos_zmod_mult_2)  wenzelm@23164  1103 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1104 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1105  prefer 2 apply simp  wenzelm@23164  1106 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  1107 done  wenzelm@23164  1108 huffman@26086  1109 lemma zmod_number_of_Bit0 [simp]:  huffman@26086  1110  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  huffman@26086  1111  (2::int) * (number_of v mod number_of w)"  huffman@26086  1112 apply (simp only: number_of_eq numeral_simps)  huffman@26086  1113 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  huffman@26086  1114  not_0_le_lemma neg_zmod_mult_2 add_ac)  huffman@26086  1115 done  huffman@26086  1116 huffman@26086  1117 lemma zmod_number_of_Bit1 [simp]:  huffman@26086  1118  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  huffman@26086  1119  (if (0::int) \ number_of w  wenzelm@23164  1120  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  1121  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  huffman@26086  1122 apply (simp only: number_of_eq numeral_simps)  wenzelm@23164  1123 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  wenzelm@23164  1124  not_0_le_lemma neg_zmod_mult_2 add_ac)  wenzelm@23164  1125 done  wenzelm@23164  1126 wenzelm@23164  1127 wenzelm@23164  1128 subsection{*Quotients of Signs*}  wenzelm@23164  1129 wenzelm@23164  1130 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  1131 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  1132 apply (rule order_trans)  wenzelm@23164  1133 apply (rule_tac a' = "-1" in zdiv_mono1)  wenzelm@23164  1134 apply (auto simp add: zdiv_minus1)  wenzelm@23164  1135 done  wenzelm@23164  1136 wenzelm@23164  1137 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  1138 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  1139 wenzelm@23164  1140 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1141 apply auto  wenzelm@23164  1142 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1143 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1144 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1145 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1146 done  wenzelm@23164  1147 wenzelm@23164  1148 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1149  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1150 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1151 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1152 done  wenzelm@23164  1153 wenzelm@23164  1154 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1155 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1156 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1157 wenzelm@23164  1158 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1159 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1160 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1161 wenzelm@23164  1162 wenzelm@23164  1163 subsection {* The Divides Relation *}  wenzelm@23164  1164 wenzelm@23164  1165 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"  haftmann@23512  1166  by (simp add: dvd_def zmod_eq_0_iff)  haftmann@23512  1167 wenzelm@23164  1168 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@23164  1169  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]  wenzelm@23164  1170 wenzelm@23164  1171 lemma zdvd_0_right [iff]: "(m::int) dvd 0"  haftmann@23512  1172  by (simp add: dvd_def)  wenzelm@23164  1173 paulson@24286  1174 lemma zdvd_0_left [iff,noatp]: "(0 dvd (m::int)) = (m = 0)"  wenzelm@23164  1175  by (simp add: dvd_def)  wenzelm@23164  1176 wenzelm@23164  1177 lemma zdvd_1_left [iff]: "1 dvd (m::int)"  wenzelm@23164  1178  by (simp add: dvd_def)  wenzelm@23164  1179 wenzelm@23164  1180 lemma zdvd_refl [simp]: "m dvd (m::int)"  haftmann@23512  1181  by (auto simp add: dvd_def intro: zmult_1_right [symmetric])  wenzelm@23164  1182 wenzelm@23164  1183 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"  haftmann@23512  1184  by (auto simp add: dvd_def intro: mult_assoc)  wenzelm@23164  1185 wenzelm@23164  1186 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"  wenzelm@23164  1187  apply (simp add: dvd_def, auto)  wenzelm@23164  1188  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1189  done  wenzelm@23164  1190 wenzelm@23164  1191 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"  wenzelm@23164  1192  apply (simp add: dvd_def, auto)  wenzelm@23164  1193  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1194  done  wenzelm@23164  1195 lemma zdvd_abs1: "( \i::int\ dvd j) = (i dvd j)"  wenzelm@23164  1196  apply (cases "i > 0", simp)  wenzelm@23164  1197  apply (simp add: dvd_def)  wenzelm@23164  1198  apply (rule iffI)  wenzelm@23164  1199  apply (erule exE)  wenzelm@23164  1200  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1201  apply (erule exE)  wenzelm@23164  1202  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1203  done  wenzelm@23164  1204 lemma zdvd_abs2: "( (i::int) dvd \j$$ = (i dvd j)"  wenzelm@23164  1205  apply (cases "j > 0", simp)  wenzelm@23164  1206  apply (simp add: dvd_def)  wenzelm@23164  1207  apply (rule iffI)  wenzelm@23164  1208  apply (erule exE)  wenzelm@23164  1209  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1210  apply (erule exE)  wenzelm@23164  1211  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1212  done  wenzelm@23164  1213 wenzelm@23164  1214 lemma zdvd_anti_sym:  wenzelm@23164  1215  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1216  apply (simp add: dvd_def, auto)  wenzelm@23164  1217  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1218  done  wenzelm@23164  1219 wenzelm@23164  1220 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"  wenzelm@23164  1221  apply (simp add: dvd_def)  wenzelm@23164  1222  apply (blast intro: right_distrib [symmetric])  wenzelm@23164  1223  done  wenzelm@23164  1224 wenzelm@23164  1225 lemma zdvd_dvd_eq: assumes anz:"a \ 0" and ab: "(a::int) dvd b" and ba:"b dvd a"  wenzelm@23164  1226  shows "\a\ = \b\"  wenzelm@23164  1227 proof-  wenzelm@23164  1228  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast  wenzelm@23164  1229  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1230  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1231  with mult_cancel_left1[where c="a" and b="k*k'"]  wenzelm@23164  1232  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)  wenzelm@23164  1233  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1234  thus ?thesis using k k' by auto  wenzelm@23164  1235 qed  wenzelm@23164  1236 wenzelm@23164  1237 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"  wenzelm@23164  1238  apply (simp add: dvd_def)  wenzelm@23164  1239  apply (blast intro: right_diff_distrib [symmetric])  wenzelm@23164  1240  done  wenzelm@23164  1241 wenzelm@23164  1242 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1243  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1244  apply (erule ssubst)  wenzelm@23164  1245  apply (blast intro: zdvd_zadd, simp)  wenzelm@23164  1246  done  wenzelm@23164  1247 wenzelm@23164  1248 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"  wenzelm@23164  1249  apply (simp add: dvd_def)  wenzelm@23164  1250  apply (blast intro: mult_left_commute)  wenzelm@23164  1251  done  wenzelm@23164  1252 wenzelm@23164  1253 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"  wenzelm@23164  1254  apply (subst mult_commute)  wenzelm@23164  1255  apply (erule zdvd_zmult)  wenzelm@23164  1256  done  wenzelm@23164  1257 wenzelm@23164  1258 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"  wenzelm@23164  1259  apply (rule zdvd_zmult)  wenzelm@23164  1260  apply (rule zdvd_refl)  wenzelm@23164  1261  done  wenzelm@23164  1262 wenzelm@23164  1263 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"  wenzelm@23164  1264  apply (rule zdvd_zmult2)  wenzelm@23164  1265  apply (rule zdvd_refl)  wenzelm@23164  1266  done  wenzelm@23164  1267 wenzelm@23164  1268 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"  wenzelm@23164  1269  apply (simp add: dvd_def)  wenzelm@23164  1270  apply (simp add: mult_assoc, blast)  wenzelm@23164  1271  done  wenzelm@23164  1272 wenzelm@23164  1273 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"  wenzelm@23164  1274  apply (rule zdvd_zmultD2)  wenzelm@23164  1275  apply (subst mult_commute, assumption)  wenzelm@23164  1276  done  wenzelm@23164  1277 wenzelm@23164  1278 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"  wenzelm@23164  1279  apply (simp add: dvd_def, clarify)  wenzelm@23164  1280  apply (rule_tac x = "k * ka" in exI)  wenzelm@23164  1281  apply (simp add: mult_ac)  wenzelm@23164  1282  done  wenzelm@23164  1283 wenzelm@23164  1284 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  wenzelm@23164  1285  apply (rule iffI)  wenzelm@23164  1286  apply (erule_tac [2] zdvd_zadd)  wenzelm@23164  1287  apply (subgoal_tac "n = (n + k * m) - k * m")  wenzelm@23164  1288  apply (erule ssubst)  wenzelm@23164  1289  apply (erule zdvd_zdiff, simp_all)  wenzelm@23164  1290  done  wenzelm@23164  1291 wenzelm@23164  1292 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  wenzelm@23164  1293  apply (simp add: dvd_def)  wenzelm@23164  1294  apply (auto simp add: zmod_zmult_zmult1)  wenzelm@23164  1295  done  wenzelm@23164  1296 wenzelm@23164  1297 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1298  apply (subgoal_tac "k dvd n * (m div n) + m mod n")  wenzelm@23164  1299  apply (simp add: zmod_zdiv_equality [symmetric])  wenzelm@23164  1300  apply (simp only: zdvd_zadd zdvd_zmult2)  wenzelm@23164  1301  done  wenzelm@23164  1302 wenzelm@23164  1303 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  wenzelm@23164  1304  apply (simp add: dvd_def, auto)  wenzelm@23164  1305  apply (subgoal_tac "0 < n")  wenzelm@23164  1306  prefer 2  wenzelm@23164  1307  apply (blast intro: order_less_trans)  wenzelm@23164  1308  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1309  apply (subgoal_tac "n * k < n * 1")  wenzelm@23164  1310  apply (drule mult_less_cancel_left [THEN iffD1], auto)  wenzelm@23164  1311  done  wenzelm@23164  1312 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1313  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@23477  1314  by (simp add: ring_simps)  wenzelm@23164  1315 wenzelm@23164  1316 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1317 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1318  apply (simp add: zmult_div_cancel)  wenzelm@23164  1319 apply (simp only: zdvd_iff_zmod_eq_0)  wenzelm@23164  1320 done  wenzelm@23164  1321 wenzelm@23164  1322 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1323  shows "m dvd n"  wenzelm@23164  1324 proof-  wenzelm@23164  1325  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1326  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1327  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1328  hence "n = m * h" by blast  wenzelm@23164  1329  thus ?thesis by blast  wenzelm@23164  1330 qed  wenzelm@23164  1331 nipkow@23969  1332 lemma zdvd_zmult_cancel_disj[simp]:  nipkow@23969  1333  "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"  nipkow@23969  1334 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)  nipkow@23969  1335 nipkow@23969  1336 wenzelm@23164  1337 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  nipkow@25134  1338 apply (simp split add: split_nat)  nipkow@25134  1339 apply (rule iffI)  nipkow@25134  1340 apply (erule exE)  nipkow@25134  1341 apply (rule_tac x = "int x" in exI)  nipkow@25134  1342 apply simp  nipkow@25134  1343 apply (erule exE)  nipkow@25134  1344 apply (rule_tac x = "nat x" in exI)  nipkow@25134  1345 apply (erule conjE)  nipkow@25134  1346 apply (erule_tac x = "nat x" in allE)  nipkow@25134  1347 apply simp  nipkow@25134  1348 done  wenzelm@23164  1349 huffman@23365  1350 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  nipkow@25134  1351 apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]  huffman@23431  1352  nat_0_le cong add: conj_cong)  nipkow@25134  1353 apply (rule iffI)  nipkow@25134  1354 apply iprover  nipkow@25134  1355 apply (erule exE)  nipkow@25134  1356 apply (case_tac "x=0")  nipkow@25134  1357 apply (rule_tac x=0 in exI)  nipkow@25134  1358 apply simp  nipkow@25134  1359 apply (case_tac "0 \ k")  nipkow@25134  1360 apply iprover  nipkow@25134  1361 apply (simp add: neq0_conv linorder_not_le)  nipkow@25134  1362 apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])  nipkow@25134  1363 apply assumption  nipkow@25134  1364 apply (simp add: mult_ac)  nipkow@25134  1365 done  wenzelm@23164  1366 wenzelm@23164  1367 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1368 proof  wenzelm@23164  1369  assume d: "x dvd 1" hence "int (nat \x\) dvd int (nat 1)" by (simp add: zdvd_abs1)  wenzelm@23164  1370  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1371  hence "nat \x\ = 1" by simp  wenzelm@23164  1372  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1373 next  wenzelm@23164  1374  assume "\x\=1" thus "x dvd 1"  wenzelm@23164  1375  by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)  wenzelm@23164  1376 qed  wenzelm@23164  1377 lemma zdvd_mult_cancel1:  wenzelm@23164  1378  assumes mp:"m \(0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1379 proof  wenzelm@23164  1380  assume n1: "\n\ = 1" thus "m * n dvd m"  wenzelm@23164  1381  by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)  wenzelm@23164  1382 next  wenzelm@23164  1383  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1384  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1385 qed  wenzelm@23164  1386 huffman@23365  1387 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  wenzelm@23164  1388  apply (auto simp add: dvd_def nat_abs_mult_distrib)  huffman@23365  1389  apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)  huffman@23365  1390  apply (rule_tac x = "-(int k)" in exI)  huffman@23431  1391  apply (auto simp add: int_mult)  huffman@23306  1392  done  huffman@23306  1393 huffman@23365  1394 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  huffman@23431  1395  apply (auto simp add: dvd_def abs_if int_mult)  huffman@23306  1396  apply (rule_tac [3] x = "nat k" in exI)  huffman@23365  1397  apply (rule_tac [2] x = "-(int k)" in exI)  huffman@23306  1398  apply (rule_tac x = "nat (-k)" in exI)  huffman@23431  1399  apply (cut_tac [3] k = m in int_less_0_conv)  huffman@23431  1400  apply (cut_tac k = m in int_less_0_conv)  huffman@23306  1401  apply (auto simp add: zero_le_mult_iff mult_less_0_iff  huffman@23365  1402  nat_mult_distrib [symmetric] nat_eq_iff2)  wenzelm@23164  1403  done  wenzelm@23164  1404 wenzelm@23164  1405 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  huffman@23431  1406  apply (auto simp add: dvd_def int_mult)  huffman@23365  1407  apply (rule_tac x = "nat k" in exI)  huffman@23431  1408  apply (cut_tac k = m in int_less_0_conv)  huffman@23365  1409  apply (auto simp add: zero_le_mult_iff mult_less_0_iff  huffman@23365  1410  nat_mult_distrib [symmetric] nat_eq_iff2)  huffman@23365  1411  done  wenzelm@23164  1412 wenzelm@23164  1413 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"  wenzelm@23164  1414  apply (auto simp add: dvd_def)  wenzelm@23164  1415  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1416  done  wenzelm@23164  1417 wenzelm@23164  1418 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"  wenzelm@23164  1419  apply (auto simp add: dvd_def)  wenzelm@23164  1420  apply (drule minus_equation_iff [THEN iffD1])  wenzelm@23164  1421  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1422  done  wenzelm@23164  1423 wenzelm@23164  1424 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1425  apply (rule_tac z=n in int_cases)  huffman@23365  1426  apply (auto simp add: dvd_int_iff)  huffman@23365  1427  apply (rule_tac z=z in int_cases)  huffman@23307  1428  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1429  done  wenzelm@23164  1430 wenzelm@23164  1431 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1432 apply (induct "y", auto)  wenzelm@23164  1433 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1434 apply (simp (no_asm_simp))  wenzelm@23164  1435 apply (rule zmod_zmult_distrib [symmetric])  wenzelm@23164  1436 done  wenzelm@23164  1437 huffman@23365  1438 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1439 apply (subst split_div, auto)  wenzelm@23164  1440 apply (subst split_zdiv, auto)  huffman@23365  1441 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)  huffman@23431  1442 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  wenzelm@23164  1443 done  wenzelm@23164  1444 wenzelm@23164  1445 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1446 apply (subst split_mod, auto)  huffman@23365  1447 apply (subst split_zmod, auto)  huffman@23365  1448 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1449  in unique_remainder)  huffman@23431  1450 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  huffman@23365  1451 done  wenzelm@23164  1452 wenzelm@23164  1453 text{*Suggested by Matthias Daum*}  wenzelm@23164  1454 lemma int_power_div_base:  wenzelm@23164  1455  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  wenzelm@23164  1456 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")  wenzelm@23164  1457  apply (erule ssubst)  wenzelm@23164  1458  apply (simp only: power_add)  wenzelm@23164  1459  apply simp_all  wenzelm@23164  1460 done  wenzelm@23164  1461 haftmann@23853  1462 text {* by Brian Huffman *}  haftmann@23853  1463 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@23853  1464 by (simp only: zmod_zminus1_eq_if mod_mod_trivial)  haftmann@23853  1465 haftmann@23853  1466 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@23853  1467 by (simp only: diff_def zmod_zadd_left_eq [symmetric])  haftmann@23853  1468 haftmann@23853  1469 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@23853  1470 proof -  haftmann@23853  1471  have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m"  haftmann@23853  1472  by (simp only: zminus_zmod)  haftmann@23853  1473  hence "(x + - (y mod m)) mod m = (x + - y) mod m"  haftmann@23853  1474  by (simp only: zmod_zadd_right_eq [symmetric])  haftmann@23853  1475  thus "(x - y mod m) mod m = (x - y) mod m"  haftmann@23853  1476  by (simp only: diff_def)  haftmann@23853  1477 qed  haftmann@23853  1478 haftmann@23853  1479 lemmas zmod_simps =  haftmann@23853  1480  IntDiv.zmod_zadd_left_eq [symmetric]  haftmann@23853  1481  IntDiv.zmod_zadd_right_eq [symmetric]  haftmann@23853  1482  IntDiv.zmod_zmult1_eq [symmetric]  haftmann@23853  1483  IntDiv.zmod_zmult1_eq' [symmetric]  haftmann@23853  1484  IntDiv.zpower_zmod  haftmann@23853  1485  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1486 haftmann@23853  1487 text {* code generator setup *}  wenzelm@23164  1488 wenzelm@23164  1489 code_modulename SML  wenzelm@23164  1490  IntDiv Integer  wenzelm@23164  1491 wenzelm@23164  1492 code_modulename OCaml  wenzelm@23164  1493  IntDiv Integer  wenzelm@23164  1494 wenzelm@23164  1495 code_modulename Haskell  haftmann@24195  1496  IntDiv Integer  wenzelm@23164  1497 wenzelm@23164  1498 end `