src/HOL/IntDiv.thy
 author huffman Thu Aug 30 05:01:38 2007 +0200 (2007-08-30) changeset 24481 c3a4a289decc parent 24391 b57c48f7e2d4 child 24490 a4c2a0ffa5be permissions -rw-r--r--
ported div/mod simprocs from HOL/ex/Binary.thy
 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  wenzelm@23164  11 imports IntArith 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  81  end  wenzelm@23164  82 wenzelm@23164  83  fun negDivAlg (a,b) =  wenzelm@23164  84  if 0\a+b then (~1,a+b)  wenzelm@23164  85  else let val (q,r) = negDivAlg(a, 2*b)  wenzelm@23164  86  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  87  end;  wenzelm@23164  88 wenzelm@23164  89  fun negateSnd (q,r:int) = (q,~r);  wenzelm@23164  90 wenzelm@23164  91  fun divAlg (a,b) = if 0\a then  wenzelm@23164  92  if b>0 then posDivAlg (a,b)  wenzelm@23164  93  else if a=0 then (0,0)  wenzelm@23164  94  else negateSnd (negDivAlg (~a,~b))  wenzelm@23164  95  else  wenzelm@23164  96  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  wenzelm@23164  107  ==> q' \ (q::int)"  wenzelm@23164  108 apply (subgoal_tac "r' + b * (q'-q) \ r")  wenzelm@23164  109  prefer 2 apply (simp add: right_diff_distrib)  wenzelm@23164  110 apply (subgoal_tac "0 < b * (1 + q - q') ")  wenzelm@23164  111 apply (erule_tac [2] order_le_less_trans)  wenzelm@23164  112  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  113 apply (subgoal_tac "b * q' < b * (1 + q) ")  wenzelm@23164  114  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  115 apply (simp add: mult_less_cancel_left)  wenzelm@23164  116 done  wenzelm@23164  117 wenzelm@23164  118 lemma unique_quotient_lemma_neg:  wenzelm@23164  119  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  wenzelm@23164  120  ==> q \ (q'::int)"  wenzelm@23164  121 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  wenzelm@23164  122  auto)  wenzelm@23164  123 wenzelm@23164  124 lemma unique_quotient:  wenzelm@23164  125  "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \ 0 |]  wenzelm@23164  126  ==> q = q'"  wenzelm@23164  127 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)  wenzelm@23164  128 apply (blast intro: order_antisym  wenzelm@23164  129  dest: order_eq_refl [THEN unique_quotient_lemma]  wenzelm@23164  130  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  wenzelm@23164  131 done  wenzelm@23164  132 wenzelm@23164  133 wenzelm@23164  134 lemma unique_remainder:  wenzelm@23164  135  "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \ 0 |]  wenzelm@23164  136  ==> r = r'"  wenzelm@23164  137 apply (subgoal_tac "q = q'")  wenzelm@23164  138  apply (simp add: quorem_def)  wenzelm@23164  139 apply (blast intro: unique_quotient)  wenzelm@23164  140 done  wenzelm@23164  141 wenzelm@23164  142 wenzelm@23164  143 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}  wenzelm@23164  144 wenzelm@23164  145 text{*And positive divisors*}  wenzelm@23164  146 wenzelm@23164  147 lemma adjust_eq [simp]:  wenzelm@23164  148  "adjust b (q,r) =  wenzelm@23164  149  (let diff = r-b in  wenzelm@23164  150  if 0 \ diff then (2*q + 1, diff)  wenzelm@23164  151  else (2*q, r))"  wenzelm@23164  152 by (simp add: Let_def adjust_def)  wenzelm@23164  153 wenzelm@23164  154 declare posDivAlg.simps [simp del]  wenzelm@23164  155 wenzelm@23164  156 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  157 lemma posDivAlg_eqn:  wenzelm@23164  158  "0 < b ==>  wenzelm@23164  159  posDivAlg a b = (if a a" and "0 < b"  wenzelm@23164  165  shows "quorem ((a, b), posDivAlg a b)"  wenzelm@23164  166 using prems apply (induct a b rule: posDivAlg.induct)  wenzelm@23164  167 apply auto  wenzelm@23164  168 apply (simp add: quorem_def)  wenzelm@23164  169 apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@23164  170 apply (case_tac "a < b")  wenzelm@23164  171 apply simp_all  wenzelm@23164  172 apply (erule splitE)  wenzelm@23164  173 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  174 done  wenzelm@23164  175 wenzelm@23164  176 wenzelm@23164  177 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}  wenzelm@23164  178 wenzelm@23164  179 text{*And positive divisors*}  wenzelm@23164  180 wenzelm@23164  181 declare negDivAlg.simps [simp del]  wenzelm@23164  182 wenzelm@23164  183 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  184 lemma negDivAlg_eqn:  wenzelm@23164  185  "0 < b ==>  wenzelm@23164  186  negDivAlg a b =  wenzelm@23164  187  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  wenzelm@23164  188 by (rule negDivAlg.simps [THEN trans], simp)  wenzelm@23164  189 wenzelm@23164  190 (*Correctness of negDivAlg: it computes quotients correctly  wenzelm@23164  191  It doesn't work if a=0 because the 0/b equals 0, not -1*)  wenzelm@23164  192 lemma negDivAlg_correct:  wenzelm@23164  193  assumes "a < 0" and "b > 0"  wenzelm@23164  194  shows "quorem ((a, b), negDivAlg a b)"  wenzelm@23164  195 using prems apply (induct a b rule: negDivAlg.induct)  wenzelm@23164  196 apply (auto simp add: linorder_not_le)  wenzelm@23164  197 apply (simp add: quorem_def)  wenzelm@23164  198 apply (subst negDivAlg_eqn, assumption)  wenzelm@23164  199 apply (case_tac "a + b < (0\int)")  wenzelm@23164  200 apply simp_all  wenzelm@23164  201 apply (erule splitE)  wenzelm@23164  202 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  203 done  wenzelm@23164  204 wenzelm@23164  205 wenzelm@23164  206 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}  wenzelm@23164  207 wenzelm@23164  208 (*the case a=0*)  wenzelm@23164  209 lemma quorem_0: "b \ 0 ==> quorem ((0,b), (0,0))"  wenzelm@23164  210 by (auto simp add: quorem_def linorder_neq_iff)  wenzelm@23164  211 wenzelm@23164  212 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  wenzelm@23164  213 by (subst posDivAlg.simps, auto)  wenzelm@23164  214 wenzelm@23164  215 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  wenzelm@23164  216 by (subst negDivAlg.simps, auto)  wenzelm@23164  217 wenzelm@23164  218 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"  wenzelm@23164  219 by (simp add: negateSnd_def)  wenzelm@23164  220 wenzelm@23164  221 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"  wenzelm@23164  222 by (auto simp add: split_ifs quorem_def)  wenzelm@23164  223 wenzelm@23164  224 lemma divAlg_correct: "b \ 0 ==> quorem ((a,b), divAlg (a, b))"  wenzelm@23164  225 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg  wenzelm@23164  226  posDivAlg_correct negDivAlg_correct)  wenzelm@23164  227 wenzelm@23164  228 text{*Arbitrary definitions for division by zero. Useful to simplify  wenzelm@23164  229  certain equations.*}  wenzelm@23164  230 wenzelm@23164  231 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"  wenzelm@23164  232 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)  wenzelm@23164  233 wenzelm@23164  234 wenzelm@23164  235 text{*Basic laws about division and remainder*}  wenzelm@23164  236 wenzelm@23164  237 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  wenzelm@23164  238 apply (case_tac "b = 0", simp)  wenzelm@23164  239 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  240 apply (auto simp add: quorem_def div_def mod_def)  wenzelm@23164  241 done  wenzelm@23164  242 wenzelm@23164  243 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"  wenzelm@23164  244 by(simp add: zmod_zdiv_equality[symmetric])  wenzelm@23164  245 wenzelm@23164  246 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"  wenzelm@23164  247 by(simp add: mult_commute zmod_zdiv_equality[symmetric])  wenzelm@23164  248 wenzelm@23164  249 text {* Tool setup *}  wenzelm@23164  250 wenzelm@23164  251 ML_setup {*  wenzelm@23164  252 local  wenzelm@23164  253 wenzelm@23164  254 structure CancelDivMod = CancelDivModFun(  wenzelm@23164  255 struct  wenzelm@23164  256  val div_name = @{const_name Divides.div};  wenzelm@23164  257  val mod_name = @{const_name Divides.mod};  wenzelm@23164  258  val mk_binop = HOLogic.mk_binop;  wenzelm@23164  259  val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;  wenzelm@23164  260  val dest_sum = Int_Numeral_Simprocs.dest_sum;  wenzelm@23164  261  val div_mod_eqs =  wenzelm@23164  262  map mk_meta_eq [@{thm zdiv_zmod_equality},  wenzelm@23164  263  @{thm zdiv_zmod_equality2}];  wenzelm@23164  264  val trans = trans;  wenzelm@23164  265  val prove_eq_sums =  wenzelm@23164  266  let  huffman@23365  267  val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}  wenzelm@23164  268  in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;  wenzelm@23164  269 end)  wenzelm@23164  270 wenzelm@23164  271 in  wenzelm@23164  272 wenzelm@23164  273 val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc  wenzelm@23164  274  ("cancel_zdiv_zmod", ["(m::int) + n"], K CancelDivMod.proc)  wenzelm@23164  275 wenzelm@23164  276 end;  wenzelm@23164  277 wenzelm@23164  278 Addsimprocs [cancel_zdiv_zmod_proc]  wenzelm@23164  279 *}  wenzelm@23164  280 wenzelm@23164  281 lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b"  wenzelm@23164  282 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  283 apply (auto simp add: quorem_def mod_def)  wenzelm@23164  284 done  wenzelm@23164  285 wenzelm@23164  286 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]  wenzelm@23164  287  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]  wenzelm@23164  288 wenzelm@23164  289 lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b"  wenzelm@23164  290 apply (cut_tac a = a and b = b in divAlg_correct)  wenzelm@23164  291 apply (auto simp add: quorem_def div_def mod_def)  wenzelm@23164  292 done  wenzelm@23164  293 wenzelm@23164  294 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]  wenzelm@23164  295  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]  wenzelm@23164  296 wenzelm@23164  297 wenzelm@23164  298 wenzelm@23164  299 subsection{*General Properties of div and mod*}  wenzelm@23164  300 wenzelm@23164  301 lemma quorem_div_mod: "b \ 0 ==> quorem ((a, b), (a div b, a mod b))"  wenzelm@23164  302 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  303 apply (force simp add: quorem_def linorder_neq_iff)  wenzelm@23164  304 done  wenzelm@23164  305 wenzelm@23164  306 lemma quorem_div: "[| quorem((a,b),(q,r)); b \ 0 |] ==> a div b = q"  wenzelm@23164  307 by (simp add: quorem_div_mod [THEN unique_quotient])  wenzelm@23164  308 wenzelm@23164  309 lemma quorem_mod: "[| quorem((a,b),(q,r)); b \ 0 |] ==> a mod b = r"  wenzelm@23164  310 by (simp add: quorem_div_mod [THEN unique_remainder])  wenzelm@23164  311 wenzelm@23164  312 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  wenzelm@23164  313 apply (rule quorem_div)  wenzelm@23164  314 apply (auto simp add: quorem_def)  wenzelm@23164  315 done  wenzelm@23164  316 wenzelm@23164  317 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  wenzelm@23164  318 apply (rule quorem_div)  wenzelm@23164  319 apply (auto simp add: quorem_def)  wenzelm@23164  320 done  wenzelm@23164  321 wenzelm@23164  322 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  wenzelm@23164  323 apply (rule quorem_div)  wenzelm@23164  324 apply (auto simp add: quorem_def)  wenzelm@23164  325 done  wenzelm@23164  326 wenzelm@23164  327 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  wenzelm@23164  328 wenzelm@23164  329 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  wenzelm@23164  330 apply (rule_tac q = 0 in quorem_mod)  wenzelm@23164  331 apply (auto simp add: quorem_def)  wenzelm@23164  332 done  wenzelm@23164  333 wenzelm@23164  334 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  wenzelm@23164  335 apply (rule_tac q = 0 in quorem_mod)  wenzelm@23164  336 apply (auto simp add: quorem_def)  wenzelm@23164  337 done  wenzelm@23164  338 wenzelm@23164  339 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  wenzelm@23164  340 apply (rule_tac q = "-1" in quorem_mod)  wenzelm@23164  341 apply (auto simp add: quorem_def)  wenzelm@23164  342 done  wenzelm@23164  343 wenzelm@23164  344 text{*There is no @{text mod_neg_pos_trivial}.*}  wenzelm@23164  345 wenzelm@23164  346 wenzelm@23164  347 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)  wenzelm@23164  348 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"  wenzelm@23164  349 apply (case_tac "b = 0", simp)  wenzelm@23164  350 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified,  wenzelm@23164  351  THEN quorem_div, THEN sym])  wenzelm@23164  352 wenzelm@23164  353 done  wenzelm@23164  354 wenzelm@23164  355 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)  wenzelm@23164  356 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"  wenzelm@23164  357 apply (case_tac "b = 0", simp)  wenzelm@23164  358 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],  wenzelm@23164  359  auto)  wenzelm@23164  360 done  wenzelm@23164  361 wenzelm@23164  362 wenzelm@23164  363 subsection{*Laws for div and mod with Unary Minus*}  wenzelm@23164  364 wenzelm@23164  365 lemma zminus1_lemma:  wenzelm@23164  366  "quorem((a,b),(q,r))  wenzelm@23164  367  ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  wenzelm@23164  368  (if r=0 then 0 else b-r))"  wenzelm@23164  369 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)  wenzelm@23164  370 wenzelm@23164  371 wenzelm@23164  372 lemma zdiv_zminus1_eq_if:  wenzelm@23164  373  "b \ (0::int)  wenzelm@23164  374  ==> (-a) div b =  wenzelm@23164  375  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  376 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])  wenzelm@23164  377 wenzelm@23164  378 lemma zmod_zminus1_eq_if:  wenzelm@23164  379  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  wenzelm@23164  380 apply (case_tac "b = 0", simp)  wenzelm@23164  381 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])  wenzelm@23164  382 done  wenzelm@23164  383 wenzelm@23164  384 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  wenzelm@23164  385 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  wenzelm@23164  386 wenzelm@23164  387 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  wenzelm@23164  388 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  wenzelm@23164  389 wenzelm@23164  390 lemma zdiv_zminus2_eq_if:  wenzelm@23164  391  "b \ (0::int)  wenzelm@23164  392  ==> a div (-b) =  wenzelm@23164  393  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  394 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  wenzelm@23164  395 wenzelm@23164  396 lemma zmod_zminus2_eq_if:  wenzelm@23164  397  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  wenzelm@23164  398 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  wenzelm@23164  399 wenzelm@23164  400 wenzelm@23164  401 subsection{*Division of a Number by Itself*}  wenzelm@23164  402 wenzelm@23164  403 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  wenzelm@23164  404 apply (subgoal_tac "0 < a*q")  wenzelm@23164  405  apply (simp add: zero_less_mult_iff, arith)  wenzelm@23164  406 done  wenzelm@23164  407 wenzelm@23164  408 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  wenzelm@23164  409 apply (subgoal_tac "0 \ a* (1-q) ")  wenzelm@23164  410  apply (simp add: zero_le_mult_iff)  wenzelm@23164  411 apply (simp add: right_diff_distrib)  wenzelm@23164  412 done  wenzelm@23164  413 wenzelm@23164  414 lemma self_quotient: "[| quorem((a,a),(q,r)); a \ (0::int) |] ==> q = 1"  wenzelm@23164  415 apply (simp add: split_ifs quorem_def linorder_neq_iff)  wenzelm@23164  416 apply (rule order_antisym, safe, simp_all)  wenzelm@23164  417 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  wenzelm@23164  418 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  wenzelm@23164  419 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  wenzelm@23164  420 done  wenzelm@23164  421 wenzelm@23164  422 lemma self_remainder: "[| quorem((a,a),(q,r)); a \ (0::int) |] ==> r = 0"  wenzelm@23164  423 apply (frule self_quotient, assumption)  wenzelm@23164  424 apply (simp add: quorem_def)  wenzelm@23164  425 done  wenzelm@23164  426 wenzelm@23164  427 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  wenzelm@23164  428 by (simp add: quorem_div_mod [THEN self_quotient])  wenzelm@23164  429 wenzelm@23164  430 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  wenzelm@23164  431 lemma zmod_self [simp]: "a mod a = (0::int)"  wenzelm@23164  432 apply (case_tac "a = 0", simp)  wenzelm@23164  433 apply (simp add: quorem_div_mod [THEN self_remainder])  wenzelm@23164  434 done  wenzelm@23164  435 wenzelm@23164  436 wenzelm@23164  437 subsection{*Computation of Division and Remainder*}  wenzelm@23164  438 wenzelm@23164  439 lemma zdiv_zero [simp]: "(0::int) div b = 0"  wenzelm@23164  440 by (simp add: div_def divAlg_def)  wenzelm@23164  441 wenzelm@23164  442 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  wenzelm@23164  443 by (simp add: div_def divAlg_def)  wenzelm@23164  444 wenzelm@23164  445 lemma zmod_zero [simp]: "(0::int) mod b = 0"  wenzelm@23164  446 by (simp add: mod_def divAlg_def)  wenzelm@23164  447 wenzelm@23164  448 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"  wenzelm@23164  449 by (simp add: div_def divAlg_def)  wenzelm@23164  450 wenzelm@23164  451 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  wenzelm@23164  452 by (simp add: mod_def divAlg_def)  wenzelm@23164  453 wenzelm@23164  454 text{*a positive, b positive *}  wenzelm@23164  455 wenzelm@23164  456 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  wenzelm@23164  457 by (simp add: div_def divAlg_def)  wenzelm@23164  458 wenzelm@23164  459 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  wenzelm@23164  460 by (simp add: mod_def divAlg_def)  wenzelm@23164  461 wenzelm@23164  462 text{*a negative, b positive *}  wenzelm@23164  463 wenzelm@23164  464 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  wenzelm@23164  465 by (simp add: div_def divAlg_def)  wenzelm@23164  466 wenzelm@23164  467 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  wenzelm@23164  468 by (simp add: mod_def divAlg_def)  wenzelm@23164  469 wenzelm@23164  470 text{*a positive, b negative *}  wenzelm@23164  471 wenzelm@23164  472 lemma div_pos_neg:  wenzelm@23164  473  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  wenzelm@23164  474 by (simp add: div_def divAlg_def)  wenzelm@23164  475 wenzelm@23164  476 lemma mod_pos_neg:  wenzelm@23164  477  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  wenzelm@23164  478 by (simp add: mod_def divAlg_def)  wenzelm@23164  479 wenzelm@23164  480 text{*a negative, b negative *}  wenzelm@23164  481 wenzelm@23164  482 lemma div_neg_neg:  wenzelm@23164  483  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  484 by (simp add: div_def divAlg_def)  wenzelm@23164  485 wenzelm@23164  486 lemma mod_neg_neg:  wenzelm@23164  487  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  488 by (simp add: mod_def divAlg_def)  wenzelm@23164  489 wenzelm@23164  490 text {*Simplify expresions in which div and mod combine numerical constants*}  wenzelm@23164  491 huffman@24481  492 lemma quoremI:  huffman@24481  493  "\a == b * q + r; if 0 < b then 0 \ r \ r < b else b < r \ r \ 0\  huffman@24481  494  \ quorem ((a, b), (q, r))"  huffman@24481  495  unfolding quorem_def by simp  huffman@24481  496 huffman@24481  497 lemmas quorem_div_eq = quoremI [THEN quorem_div, THEN eq_reflection]  huffman@24481  498 lemmas quorem_mod_eq = quoremI [THEN quorem_mod, THEN eq_reflection]  huffman@24481  499 lemmas arithmetic_simps =  huffman@24481  500  arith_simps  huffman@24481  501  add_special  huffman@24481  502  OrderedGroup.add_0_left  huffman@24481  503  OrderedGroup.add_0_right  huffman@24481  504  mult_zero_left  huffman@24481  505  mult_zero_right  huffman@24481  506  mult_1_left  huffman@24481  507  mult_1_right  huffman@24481  508 huffman@24481  509 (* simprocs adapted from HOL/ex/Binary.thy *)  huffman@24481  510 ML {*  huffman@24481  511 local  huffman@24481  512  infix ==;  huffman@24481  513  val op == = Logic.mk_equals;  huffman@24481  514  fun plus m n = @{term "plus :: int \ int \ int"}  m  n;  huffman@24481  515  fun mult m n = @{term "times :: int \ int \ int"}  m  n;  huffman@24481  516 huffman@24481  517  val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};  huffman@24481  518  fun prove ctxt prop =  huffman@24481  519  Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));  huffman@24481  520 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  huffman@24481  529 huffman@24481  530 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>  huffman@24481  531  if n = 0 then NONE  huffman@24481  532  else  huffman@24481  533  let val (k, l) = IntInf.divMod (m, n);  huffman@24481  534  fun mk_num x = HOLogic.mk_number HOLogic.intT x;  huffman@24481  535  in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])  huffman@24481  536  end);  huffman@24481  537 huffman@24481  538 end;  huffman@24481  539 *}  huffman@24481  540 huffman@24481  541 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  huffman@24481  542  {* K (divmod_proc (@{thm quorem_div_eq})) *}  huffman@24481  543 huffman@24481  544 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  huffman@24481  545  {* K (divmod_proc (@{thm quorem_mod_eq})) *}  huffman@24481  546 huffman@24481  547 (* The following 8 lemmas are made unnecessary by the above simprocs: *)  huffman@24481  548 huffman@24481  549 lemmas div_pos_pos_number_of =  wenzelm@23164  550  div_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  551 huffman@24481  552 lemmas div_neg_pos_number_of =  wenzelm@23164  553  div_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  554 huffman@24481  555 lemmas div_pos_neg_number_of =  wenzelm@23164  556  div_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  557 huffman@24481  558 lemmas div_neg_neg_number_of =  wenzelm@23164  559  div_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  560 wenzelm@23164  561 huffman@24481  562 lemmas mod_pos_pos_number_of =  wenzelm@23164  563  mod_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  564 huffman@24481  565 lemmas mod_neg_pos_number_of =  wenzelm@23164  566  mod_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  567 huffman@24481  568 lemmas mod_pos_neg_number_of =  wenzelm@23164  569  mod_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  570 huffman@24481  571 lemmas mod_neg_neg_number_of =  wenzelm@23164  572  mod_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  573 wenzelm@23164  574 wenzelm@23164  575 lemmas posDivAlg_eqn_number_of [simp] =  wenzelm@23164  576  posDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  577 wenzelm@23164  578 lemmas negDivAlg_eqn_number_of [simp] =  wenzelm@23164  579  negDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  580 wenzelm@23164  581 wenzelm@23164  582 text{*Special-case simplification *}  wenzelm@23164  583 wenzelm@23164  584 lemma zmod_1 [simp]: "a mod (1::int) = 0"  wenzelm@23164  585 apply (cut_tac a = a and b = 1 in pos_mod_sign)  wenzelm@23164  586 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)  wenzelm@23164  587 apply (auto simp del:pos_mod_bound pos_mod_sign)  wenzelm@23164  588 done  wenzelm@23164  589 wenzelm@23164  590 lemma zdiv_1 [simp]: "a div (1::int) = a"  wenzelm@23164  591 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)  wenzelm@23164  592 wenzelm@23164  593 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  wenzelm@23164  594 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  wenzelm@23164  595 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  wenzelm@23164  596 apply (auto simp del: neg_mod_sign neg_mod_bound)  wenzelm@23164  597 done  wenzelm@23164  598 wenzelm@23164  599 lemma zdiv_minus1_right [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 (** The last remaining special cases for constant arithmetic:  wenzelm@23164  603  1 div z and 1 mod z **)  wenzelm@23164  604 wenzelm@23164  605 lemmas div_pos_pos_1_number_of [simp] =  wenzelm@23164  606  div_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  607 wenzelm@23164  608 lemmas div_pos_neg_1_number_of [simp] =  wenzelm@23164  609  div_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  610 wenzelm@23164  611 lemmas mod_pos_pos_1_number_of [simp] =  wenzelm@23164  612  mod_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  613 wenzelm@23164  614 lemmas mod_pos_neg_1_number_of [simp] =  wenzelm@23164  615  mod_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  616 wenzelm@23164  617 wenzelm@23164  618 lemmas posDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  619  posDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  620 wenzelm@23164  621 lemmas negDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  622  negDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  623 wenzelm@23164  624 wenzelm@23164  625 wenzelm@23164  626 subsection{*Monotonicity in the First Argument (Dividend)*}  wenzelm@23164  627 wenzelm@23164  628 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  wenzelm@23164  629 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  630 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  631 apply (rule unique_quotient_lemma)  wenzelm@23164  632 apply (erule subst)  wenzelm@23164  633 apply (erule subst, simp_all)  wenzelm@23164  634 done  wenzelm@23164  635 wenzelm@23164  636 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  wenzelm@23164  637 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  638 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  639 apply (rule unique_quotient_lemma_neg)  wenzelm@23164  640 apply (erule subst)  wenzelm@23164  641 apply (erule subst, simp_all)  wenzelm@23164  642 done  wenzelm@23164  643 wenzelm@23164  644 wenzelm@23164  645 subsection{*Monotonicity in the Second Argument (Divisor)*}  wenzelm@23164  646 wenzelm@23164  647 lemma q_pos_lemma:  wenzelm@23164  648  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  wenzelm@23164  649 apply (subgoal_tac "0 < b'* (q' + 1) ")  wenzelm@23164  650  apply (simp add: zero_less_mult_iff)  wenzelm@23164  651 apply (simp add: right_distrib)  wenzelm@23164  652 done  wenzelm@23164  653 wenzelm@23164  654 lemma zdiv_mono2_lemma:  wenzelm@23164  655  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  wenzelm@23164  656  r' < b'; 0 \ r; 0 < b'; b' \ b |]  wenzelm@23164  657  ==> q \ (q'::int)"  wenzelm@23164  658 apply (frule q_pos_lemma, assumption+)  wenzelm@23164  659 apply (subgoal_tac "b*q < b* (q' + 1) ")  wenzelm@23164  660  apply (simp add: mult_less_cancel_left)  wenzelm@23164  661 apply (subgoal_tac "b*q = r' - r + b'*q'")  wenzelm@23164  662  prefer 2 apply simp  wenzelm@23164  663 apply (simp (no_asm_simp) add: right_distrib)  wenzelm@23164  664 apply (subst add_commute, rule zadd_zless_mono, arith)  wenzelm@23164  665 apply (rule mult_right_mono, auto)  wenzelm@23164  666 done  wenzelm@23164  667 wenzelm@23164  668 lemma zdiv_mono2:  wenzelm@23164  669  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  wenzelm@23164  670 apply (subgoal_tac "b \ 0")  wenzelm@23164  671  prefer 2 apply arith  wenzelm@23164  672 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  673 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  674 apply (rule zdiv_mono2_lemma)  wenzelm@23164  675 apply (erule subst)  wenzelm@23164  676 apply (erule subst, simp_all)  wenzelm@23164  677 done  wenzelm@23164  678 wenzelm@23164  679 lemma q_neg_lemma:  wenzelm@23164  680  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  wenzelm@23164  681 apply (subgoal_tac "b'*q' < 0")  wenzelm@23164  682  apply (simp add: mult_less_0_iff, arith)  wenzelm@23164  683 done  wenzelm@23164  684 wenzelm@23164  685 lemma zdiv_mono2_neg_lemma:  wenzelm@23164  686  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  wenzelm@23164  687  r < b; 0 \ r'; 0 < b'; b' \ b |]  wenzelm@23164  688  ==> q' \ (q::int)"  wenzelm@23164  689 apply (frule q_neg_lemma, assumption+)  wenzelm@23164  690 apply (subgoal_tac "b*q' < b* (q + 1) ")  wenzelm@23164  691  apply (simp add: mult_less_cancel_left)  wenzelm@23164  692 apply (simp add: right_distrib)  wenzelm@23164  693 apply (subgoal_tac "b*q' \ b'*q'")  wenzelm@23164  694  prefer 2 apply (simp add: mult_right_mono_neg, arith)  wenzelm@23164  695 done  wenzelm@23164  696 wenzelm@23164  697 lemma zdiv_mono2_neg:  wenzelm@23164  698  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  wenzelm@23164  699 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  700 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  701 apply (rule zdiv_mono2_neg_lemma)  wenzelm@23164  702 apply (erule subst)  wenzelm@23164  703 apply (erule subst, simp_all)  wenzelm@23164  704 done  wenzelm@23164  705 wenzelm@23164  706 subsection{*More Algebraic Laws for div and mod*}  wenzelm@23164  707 wenzelm@23164  708 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  wenzelm@23164  709 wenzelm@23164  710 lemma zmult1_lemma:  wenzelm@23164  711  "[| quorem((b,c),(q,r)); c \ 0 |]  wenzelm@23164  712  ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"  wenzelm@23164  713 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)  wenzelm@23164  714 wenzelm@23164  715 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  wenzelm@23164  716 apply (case_tac "c = 0", simp)  wenzelm@23164  717 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])  wenzelm@23164  718 done  wenzelm@23164  719 wenzelm@23164  720 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  wenzelm@23164  721 apply (case_tac "c = 0", simp)  wenzelm@23164  722 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])  wenzelm@23164  723 done  wenzelm@23164  724 wenzelm@23164  725 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"  wenzelm@23164  726 apply (rule trans)  wenzelm@23164  727 apply (rule_tac s = "b*a mod c" in trans)  wenzelm@23164  728 apply (rule_tac [2] zmod_zmult1_eq)  wenzelm@23164  729 apply (simp_all add: mult_commute)  wenzelm@23164  730 done  wenzelm@23164  731 wenzelm@23164  732 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"  wenzelm@23164  733 apply (rule zmod_zmult1_eq' [THEN trans])  wenzelm@23164  734 apply (rule zmod_zmult1_eq)  wenzelm@23164  735 done  wenzelm@23164  736 wenzelm@23164  737 lemma zdiv_zmult_self1 [simp]: "b \ (0::int) ==> (a*b) div b = a"  wenzelm@23164  738 by (simp add: zdiv_zmult1_eq)  wenzelm@23164  739 wenzelm@23164  740 lemma zdiv_zmult_self2 [simp]: "b \ (0::int) ==> (b*a) div b = a"  wenzelm@23164  741 by (subst mult_commute, erule zdiv_zmult_self1)  wenzelm@23164  742 wenzelm@23164  743 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"  wenzelm@23164  744 by (simp add: zmod_zmult1_eq)  wenzelm@23164  745 wenzelm@23164  746 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"  wenzelm@23164  747 by (simp add: mult_commute zmod_zmult1_eq)  wenzelm@23164  748 wenzelm@23164  749 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  wenzelm@23164  750 proof  wenzelm@23164  751  assume "m mod d = 0"  wenzelm@23164  752  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto  wenzelm@23164  753 next  wenzelm@23164  754  assume "EX q::int. m = d*q"  wenzelm@23164  755  thus "m mod d = 0" by auto  wenzelm@23164  756 qed  wenzelm@23164  757 wenzelm@23164  758 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  759 wenzelm@23164  760 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  wenzelm@23164  761 wenzelm@23164  762 lemma zadd1_lemma:  wenzelm@23164  763  "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c \ 0 |]  wenzelm@23164  764  ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"  wenzelm@23164  765 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)  wenzelm@23164  766 wenzelm@23164  767 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  wenzelm@23164  768 lemma zdiv_zadd1_eq:  wenzelm@23164  769  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  wenzelm@23164  770 apply (case_tac "c = 0", simp)  wenzelm@23164  771 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)  wenzelm@23164  772 done  wenzelm@23164  773 wenzelm@23164  774 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"  wenzelm@23164  775 apply (case_tac "c = 0", simp)  wenzelm@23164  776 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)  wenzelm@23164  777 done  wenzelm@23164  778 wenzelm@23164  779 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"  wenzelm@23164  780 apply (case_tac "b = 0", simp)  wenzelm@23164  781 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  wenzelm@23164  782 done  wenzelm@23164  783 wenzelm@23164  784 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"  wenzelm@23164  785 apply (case_tac "b = 0", simp)  wenzelm@23164  786 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)  wenzelm@23164  787 done  wenzelm@23164  788 wenzelm@23164  789 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"  wenzelm@23164  790 apply (rule trans [symmetric])  wenzelm@23164  791 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  792 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  793 done  wenzelm@23164  794 wenzelm@23164  795 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"  wenzelm@23164  796 apply (rule trans [symmetric])  wenzelm@23164  797 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  798 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  799 done  wenzelm@23164  800 wenzelm@23164  801 lemma zdiv_zadd_self1[simp]: "a \ (0::int) ==> (a+b) div a = b div a + 1"  wenzelm@23164  802 by (simp add: zdiv_zadd1_eq)  wenzelm@23164  803 wenzelm@23164  804 lemma zdiv_zadd_self2[simp]: "a \ (0::int) ==> (b+a) div a = b div a + 1"  wenzelm@23164  805 by (simp add: zdiv_zadd1_eq)  wenzelm@23164  806 wenzelm@23164  807 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"  wenzelm@23164  808 apply (case_tac "a = 0", simp)  wenzelm@23164  809 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  810 done  wenzelm@23164  811 wenzelm@23164  812 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"  wenzelm@23164  813 apply (case_tac "a = 0", simp)  wenzelm@23164  814 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  815 done  wenzelm@23164  816 wenzelm@23164  817 nipkow@23983  818 lemma zmod_zdiff1_eq: fixes a::int  nipkow@23983  819  shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r")  nipkow@23983  820 proof -  nipkow@23983  821  have "?l = (c + (a mod c - b mod c)) mod c"  nipkow@23983  822  using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if)  nipkow@23983  823  also have "\ = ?r" by simp  nipkow@23983  824  finally show ?thesis .  nipkow@23983  825 qed  nipkow@23983  826 wenzelm@23164  827 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  828 wenzelm@23164  829 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  830  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  831  to cause particular problems.*)  wenzelm@23164  832 wenzelm@23164  833 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  834 wenzelm@23164  835 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  836 apply (subgoal_tac "b * (c - q mod c) < r * 1")  wenzelm@23164  837 apply (simp add: right_diff_distrib)  wenzelm@23164  838 apply (rule order_le_less_trans)  wenzelm@23164  839 apply (erule_tac [2] mult_strict_right_mono)  wenzelm@23164  840 apply (rule mult_left_mono_neg)  wenzelm@23164  841 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  842  add1_zle_eq pos_mod_bound)  wenzelm@23164  843 done  wenzelm@23164  844 wenzelm@23164  845 lemma zmult2_lemma_aux2:  wenzelm@23164  846  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  847 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  848  apply arith  wenzelm@23164  849 apply (simp add: mult_le_0_iff)  wenzelm@23164  850 done  wenzelm@23164  851 wenzelm@23164  852 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  853 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  854 apply arith  wenzelm@23164  855 apply (simp add: zero_le_mult_iff)  wenzelm@23164  856 done  wenzelm@23164  857 wenzelm@23164  858 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  859 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  wenzelm@23164  860 apply (simp add: right_diff_distrib)  wenzelm@23164  861 apply (rule order_less_le_trans)  wenzelm@23164  862 apply (erule mult_strict_right_mono)  wenzelm@23164  863 apply (rule_tac [2] mult_left_mono)  wenzelm@23164  864 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  865  add1_zle_eq pos_mod_bound)  wenzelm@23164  866 done  wenzelm@23164  867 wenzelm@23164  868 lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b \ 0; 0 < c |]  wenzelm@23164  869  ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"  wenzelm@23164  870 by (auto simp add: mult_ac quorem_def linorder_neq_iff  wenzelm@23164  871  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  872  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  873 wenzelm@23164  874 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  875 apply (case_tac "b = 0", simp)  wenzelm@23164  876 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])  wenzelm@23164  877 done  wenzelm@23164  878 wenzelm@23164  879 lemma zmod_zmult2_eq:  wenzelm@23164  880  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  881 apply (case_tac "b = 0", simp)  wenzelm@23164  882 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])  wenzelm@23164  883 done  wenzelm@23164  884 wenzelm@23164  885 wenzelm@23164  886 subsection{*Cancellation of Common Factors in div*}  wenzelm@23164  887 wenzelm@23164  888 lemma zdiv_zmult_zmult1_aux1:  wenzelm@23164  889  "[| (0::int) < b; c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  890 by (subst zdiv_zmult2_eq, auto)  wenzelm@23164  891 wenzelm@23164  892 lemma zdiv_zmult_zmult1_aux2:  wenzelm@23164  893  "[| b < (0::int); c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  894 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")  wenzelm@23164  895 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)  wenzelm@23164  896 done  wenzelm@23164  897 wenzelm@23164  898 lemma zdiv_zmult_zmult1: "c \ (0::int) ==> (c*a) div (c*b) = a div b"  wenzelm@23164  899 apply (case_tac "b = 0", simp)  wenzelm@23164  900 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)  wenzelm@23164  901 done  wenzelm@23164  902 nipkow@23401  903 lemma zdiv_zmult_zmult1_if[simp]:  nipkow@23401  904  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"  nipkow@23401  905 by (simp add:zdiv_zmult_zmult1)  nipkow@23401  906 nipkow@23401  907 (*  wenzelm@23164  908 lemma zdiv_zmult_zmult2: "c \ (0::int) ==> (a*c) div (b*c) = a div b"  wenzelm@23164  909 apply (drule zdiv_zmult_zmult1)  wenzelm@23164  910 apply (auto simp add: mult_commute)  wenzelm@23164  911 done  nipkow@23401  912 *)  wenzelm@23164  913 wenzelm@23164  914 wenzelm@23164  915 subsection{*Distribution of Factors over mod*}  wenzelm@23164  916 wenzelm@23164  917 lemma zmod_zmult_zmult1_aux1:  wenzelm@23164  918  "[| (0::int) < b; c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  919 by (subst zmod_zmult2_eq, auto)  wenzelm@23164  920 wenzelm@23164  921 lemma zmod_zmult_zmult1_aux2:  wenzelm@23164  922  "[| b < (0::int); c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  923 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")  wenzelm@23164  924 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)  wenzelm@23164  925 done  wenzelm@23164  926 wenzelm@23164  927 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"  wenzelm@23164  928 apply (case_tac "b = 0", simp)  wenzelm@23164  929 apply (case_tac "c = 0", simp)  wenzelm@23164  930 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)  wenzelm@23164  931 done  wenzelm@23164  932 wenzelm@23164  933 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"  wenzelm@23164  934 apply (cut_tac c = c in zmod_zmult_zmult1)  wenzelm@23164  935 apply (auto simp add: mult_commute)  wenzelm@23164  936 done  wenzelm@23164  937 wenzelm@23164  938 wenzelm@23164  939 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  940 wenzelm@23164  941 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  942 wenzelm@23164  943 lemma split_pos_lemma:  wenzelm@23164  944  "0  wenzelm@23164  945  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  946 apply (rule iffI, clarify)  wenzelm@23164  947  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  948  apply (subst zmod_zadd1_eq)  wenzelm@23164  949  apply (subst zdiv_zadd1_eq)  wenzelm@23164  950  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@23164  951 txt{*converse direction*}  wenzelm@23164  952 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  953 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  954 done  wenzelm@23164  955 wenzelm@23164  956 lemma split_neg_lemma:  wenzelm@23164  957  "k<0 ==>  wenzelm@23164  958  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  959 apply (rule iffI, clarify)  wenzelm@23164  960  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  961  apply (subst zmod_zadd1_eq)  wenzelm@23164  962  apply (subst zdiv_zadd1_eq)  wenzelm@23164  963  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  964 txt{*converse direction*}  wenzelm@23164  965 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  966 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  967 done  wenzelm@23164  968 wenzelm@23164  969 lemma split_zdiv:  wenzelm@23164  970  "P(n div k :: int) =  wenzelm@23164  971  ((k = 0 --> P 0) &  wenzelm@23164  972  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  973  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  974 apply (case_tac "k=0", simp)  wenzelm@23164  975 apply (simp only: linorder_neq_iff)  wenzelm@23164  976 apply (erule disjE)  wenzelm@23164  977  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  978  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  979 done  wenzelm@23164  980 wenzelm@23164  981 lemma split_zmod:  wenzelm@23164  982  "P(n mod k :: int) =  wenzelm@23164  983  ((k = 0 --> P n) &  wenzelm@23164  984  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  985  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  986 apply (case_tac "k=0", simp)  wenzelm@23164  987 apply (simp only: linorder_neq_iff)  wenzelm@23164  988 apply (erule disjE)  wenzelm@23164  989  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  990  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  991 done  wenzelm@23164  992 wenzelm@23164  993 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  994 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  995 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  996 wenzelm@23164  997 wenzelm@23164  998 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  999 wenzelm@23164  1000 text{*computing div by shifting *}  wenzelm@23164  1001 wenzelm@23164  1002 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  1003 proof cases  wenzelm@23164  1004  assume "a=0"  wenzelm@23164  1005  thus ?thesis by simp  wenzelm@23164  1006 next  wenzelm@23164  1007  assume "a\0" and le_a: "0\a"  wenzelm@23164  1008  hence a_pos: "1 \ a" by arith  wenzelm@23164  1009  hence one_less_a2: "1 < 2*a" by arith  wenzelm@23164  1010  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  wenzelm@23164  1011  by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)  wenzelm@23164  1012  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  1013  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  1014  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  1015  with le_2a  wenzelm@23164  1016  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  1017  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  1018  right_distrib)  wenzelm@23164  1019  thus ?thesis  wenzelm@23164  1020  by (subst zdiv_zadd1_eq,  wenzelm@23164  1021  simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2  wenzelm@23164  1022  div_pos_pos_trivial)  wenzelm@23164  1023 qed  wenzelm@23164  1024 wenzelm@23164  1025 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  1026 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  1027 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  1028 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1029 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1030 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  1031  simp)  wenzelm@23164  1032 done  wenzelm@23164  1033 wenzelm@23164  1034 wenzelm@23164  1035 (*Not clear why this must be proved separately; probably number_of causes  wenzelm@23164  1036  simplification problems*)  wenzelm@23164  1037 lemma not_0_le_lemma: "~ 0 \ x ==> x \ (0::int)"  wenzelm@23164  1038 by auto  wenzelm@23164  1039 wenzelm@23164  1040 lemma zdiv_number_of_BIT[simp]:  wenzelm@23164  1041  "number_of (v BIT b) div number_of (w BIT bit.B0) =  wenzelm@23164  1042  (if b=bit.B0 | (0::int) \ number_of w  wenzelm@23164  1043  then number_of v div (number_of w)  wenzelm@23164  1044  else (number_of v + (1::int)) div (number_of w))"  wenzelm@23164  1045 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  wenzelm@23164  1046 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac  wenzelm@23164  1047  split: bit.split)  wenzelm@23164  1048 done  wenzelm@23164  1049 wenzelm@23164  1050 wenzelm@23164  1051 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  1052 wenzelm@23164  1053 lemma pos_zmod_mult_2:  wenzelm@23164  1054  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  1055 apply (case_tac "a = 0", simp)  wenzelm@23164  1056 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  1057  prefer 2 apply arith  wenzelm@23164  1058 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  1059  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  1060 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  1061  pos_mod_bound)  wenzelm@23164  1062 apply (subst zmod_zadd1_eq)  wenzelm@23164  1063 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)  wenzelm@23164  1064 apply (rule mod_pos_pos_trivial)  wenzelm@23164  1065 apply (auto simp add: mod_pos_pos_trivial left_distrib)  wenzelm@23164  1066 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  1067 done  wenzelm@23164  1068 wenzelm@23164  1069 lemma neg_zmod_mult_2:  wenzelm@23164  1070  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  1071 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  1072  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  1073 apply (rule_tac [2] pos_zmod_mult_2)  wenzelm@23164  1074 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1075 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1076  prefer 2 apply simp  wenzelm@23164  1077 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  1078 done  wenzelm@23164  1079 wenzelm@23164  1080 lemma zmod_number_of_BIT [simp]:  wenzelm@23164  1081  "number_of (v BIT b) mod number_of (w BIT bit.B0) =  wenzelm@23164  1082  (case b of  wenzelm@23164  1083  bit.B0 => 2 * (number_of v mod number_of w)  wenzelm@23164  1084  | bit.B1 => if (0::int) \ number_of w  wenzelm@23164  1085  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  1086  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  wenzelm@23164  1087 apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split)  wenzelm@23164  1088 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  wenzelm@23164  1089  not_0_le_lemma neg_zmod_mult_2 add_ac)  wenzelm@23164  1090 done  wenzelm@23164  1091 wenzelm@23164  1092 wenzelm@23164  1093 subsection{*Quotients of Signs*}  wenzelm@23164  1094 wenzelm@23164  1095 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  1096 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  1097 apply (rule order_trans)  wenzelm@23164  1098 apply (rule_tac a' = "-1" in zdiv_mono1)  wenzelm@23164  1099 apply (auto simp add: zdiv_minus1)  wenzelm@23164  1100 done  wenzelm@23164  1101 wenzelm@23164  1102 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  1103 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  1104 wenzelm@23164  1105 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1106 apply auto  wenzelm@23164  1107 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1108 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1109 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1110 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1111 done  wenzelm@23164  1112 wenzelm@23164  1113 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1114  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1115 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1116 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1117 done  wenzelm@23164  1118 wenzelm@23164  1119 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1120 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1121 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1122 wenzelm@23164  1123 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1124 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1125 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1126 wenzelm@23164  1127 wenzelm@23164  1128 subsection {* The Divides Relation *}  wenzelm@23164  1129 wenzelm@23164  1130 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"  haftmann@23512  1131  by (simp add: dvd_def zmod_eq_0_iff)  haftmann@23512  1132 haftmann@23684  1133 instance int :: dvd_mod  haftmann@23684  1134  by default (simp add: times_class.dvd [symmetric] zdvd_iff_zmod_eq_0)  wenzelm@23164  1135 wenzelm@23164  1136 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@23164  1137  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]  wenzelm@23164  1138 wenzelm@23164  1139 lemma zdvd_0_right [iff]: "(m::int) dvd 0"  haftmann@23512  1140  by (simp add: dvd_def)  wenzelm@23164  1141 paulson@24286  1142 lemma zdvd_0_left [iff,noatp]: "(0 dvd (m::int)) = (m = 0)"  wenzelm@23164  1143  by (simp add: dvd_def)  wenzelm@23164  1144 wenzelm@23164  1145 lemma zdvd_1_left [iff]: "1 dvd (m::int)"  wenzelm@23164  1146  by (simp add: dvd_def)  wenzelm@23164  1147 wenzelm@23164  1148 lemma zdvd_refl [simp]: "m dvd (m::int)"  haftmann@23512  1149  by (auto simp add: dvd_def intro: zmult_1_right [symmetric])  wenzelm@23164  1150 wenzelm@23164  1151 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"  haftmann@23512  1152  by (auto simp add: dvd_def intro: mult_assoc)  wenzelm@23164  1153 wenzelm@23164  1154 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"  wenzelm@23164  1155  apply (simp add: dvd_def, auto)  wenzelm@23164  1156  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1157  done  wenzelm@23164  1158 wenzelm@23164  1159 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"  wenzelm@23164  1160  apply (simp add: dvd_def, auto)  wenzelm@23164  1161  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1162  done  wenzelm@23164  1163 lemma zdvd_abs1: "( \i::int\ dvd j) = (i dvd j)"  wenzelm@23164  1164  apply (cases "i > 0", simp)  wenzelm@23164  1165  apply (simp add: dvd_def)  wenzelm@23164  1166  apply (rule iffI)  wenzelm@23164  1167  apply (erule exE)  wenzelm@23164  1168  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1169  apply (erule exE)  wenzelm@23164  1170  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1171  done  wenzelm@23164  1172 lemma zdvd_abs2: "( (i::int) dvd \j$$ = (i dvd j)"  wenzelm@23164  1173  apply (cases "j > 0", simp)  wenzelm@23164  1174  apply (simp add: dvd_def)  wenzelm@23164  1175  apply (rule iffI)  wenzelm@23164  1176  apply (erule exE)  wenzelm@23164  1177  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1178  apply (erule exE)  wenzelm@23164  1179  apply (rule_tac x="- k" in exI, simp)  wenzelm@23164  1180  done  wenzelm@23164  1181 wenzelm@23164  1182 lemma zdvd_anti_sym:  wenzelm@23164  1183  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1184  apply (simp add: dvd_def, auto)  wenzelm@23164  1185  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1186  done  wenzelm@23164  1187 wenzelm@23164  1188 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"  wenzelm@23164  1189  apply (simp add: dvd_def)  wenzelm@23164  1190  apply (blast intro: right_distrib [symmetric])  wenzelm@23164  1191  done  wenzelm@23164  1192 wenzelm@23164  1193 lemma zdvd_dvd_eq: assumes anz:"a \ 0" and ab: "(a::int) dvd b" and ba:"b dvd a"  wenzelm@23164  1194  shows "\a\ = \b\"  wenzelm@23164  1195 proof-  wenzelm@23164  1196  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast  wenzelm@23164  1197  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1198  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1199  with mult_cancel_left1[where c="a" and b="k*k'"]  wenzelm@23164  1200  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)  wenzelm@23164  1201  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1202  thus ?thesis using k k' by auto  wenzelm@23164  1203 qed  wenzelm@23164  1204 wenzelm@23164  1205 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"  wenzelm@23164  1206  apply (simp add: dvd_def)  wenzelm@23164  1207  apply (blast intro: right_diff_distrib [symmetric])  wenzelm@23164  1208  done  wenzelm@23164  1209 wenzelm@23164  1210 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1211  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1212  apply (erule ssubst)  wenzelm@23164  1213  apply (blast intro: zdvd_zadd, simp)  wenzelm@23164  1214  done  wenzelm@23164  1215 wenzelm@23164  1216 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"  wenzelm@23164  1217  apply (simp add: dvd_def)  wenzelm@23164  1218  apply (blast intro: mult_left_commute)  wenzelm@23164  1219  done  wenzelm@23164  1220 wenzelm@23164  1221 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"  wenzelm@23164  1222  apply (subst mult_commute)  wenzelm@23164  1223  apply (erule zdvd_zmult)  wenzelm@23164  1224  done  wenzelm@23164  1225 wenzelm@23164  1226 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"  wenzelm@23164  1227  apply (rule zdvd_zmult)  wenzelm@23164  1228  apply (rule zdvd_refl)  wenzelm@23164  1229  done  wenzelm@23164  1230 wenzelm@23164  1231 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"  wenzelm@23164  1232  apply (rule zdvd_zmult2)  wenzelm@23164  1233  apply (rule zdvd_refl)  wenzelm@23164  1234  done  wenzelm@23164  1235 wenzelm@23164  1236 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"  wenzelm@23164  1237  apply (simp add: dvd_def)  wenzelm@23164  1238  apply (simp add: mult_assoc, blast)  wenzelm@23164  1239  done  wenzelm@23164  1240 wenzelm@23164  1241 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"  wenzelm@23164  1242  apply (rule zdvd_zmultD2)  wenzelm@23164  1243  apply (subst mult_commute, assumption)  wenzelm@23164  1244  done  wenzelm@23164  1245 wenzelm@23164  1246 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"  wenzelm@23164  1247  apply (simp add: dvd_def, clarify)  wenzelm@23164  1248  apply (rule_tac x = "k * ka" in exI)  wenzelm@23164  1249  apply (simp add: mult_ac)  wenzelm@23164  1250  done  wenzelm@23164  1251 wenzelm@23164  1252 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  wenzelm@23164  1253  apply (rule iffI)  wenzelm@23164  1254  apply (erule_tac [2] zdvd_zadd)  wenzelm@23164  1255  apply (subgoal_tac "n = (n + k * m) - k * m")  wenzelm@23164  1256  apply (erule ssubst)  wenzelm@23164  1257  apply (erule zdvd_zdiff, simp_all)  wenzelm@23164  1258  done  wenzelm@23164  1259 wenzelm@23164  1260 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  wenzelm@23164  1261  apply (simp add: dvd_def)  wenzelm@23164  1262  apply (auto simp add: zmod_zmult_zmult1)  wenzelm@23164  1263  done  wenzelm@23164  1264 wenzelm@23164  1265 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1266  apply (subgoal_tac "k dvd n * (m div n) + m mod n")  wenzelm@23164  1267  apply (simp add: zmod_zdiv_equality [symmetric])  wenzelm@23164  1268  apply (simp only: zdvd_zadd zdvd_zmult2)  wenzelm@23164  1269  done  wenzelm@23164  1270 wenzelm@23164  1271 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  wenzelm@23164  1272  apply (simp add: dvd_def, auto)  wenzelm@23164  1273  apply (subgoal_tac "0 < n")  wenzelm@23164  1274  prefer 2  wenzelm@23164  1275  apply (blast intro: order_less_trans)  wenzelm@23164  1276  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1277  apply (subgoal_tac "n * k < n * 1")  wenzelm@23164  1278  apply (drule mult_less_cancel_left [THEN iffD1], auto)  wenzelm@23164  1279  done  wenzelm@23164  1280 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1281  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@23477  1282  by (simp add: ring_simps)  wenzelm@23164  1283 wenzelm@23164  1284 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1285 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1286  apply (simp add: zmult_div_cancel)  wenzelm@23164  1287 apply (simp only: zdvd_iff_zmod_eq_0)  wenzelm@23164  1288 done  wenzelm@23164  1289 wenzelm@23164  1290 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1291  shows "m dvd n"  wenzelm@23164  1292 proof-  wenzelm@23164  1293  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1294  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1295  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1296  hence "n = m * h" by blast  wenzelm@23164  1297  thus ?thesis by blast  wenzelm@23164  1298 qed  wenzelm@23164  1299 nipkow@23969  1300 lemma zdvd_zmult_cancel_disj[simp]:  nipkow@23969  1301  "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"  nipkow@23969  1302 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)  nipkow@23969  1303 nipkow@23969  1304 wenzelm@23164  1305 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  wenzelm@23164  1306  apply (simp split add: split_nat)  wenzelm@23164  1307  apply (rule iffI)  wenzelm@23164  1308  apply (erule exE)  wenzelm@23164  1309  apply (rule_tac x = "int x" in exI)  wenzelm@23164  1310  apply simp  wenzelm@23164  1311  apply (erule exE)  wenzelm@23164  1312  apply (rule_tac x = "nat x" in exI)  wenzelm@23164  1313  apply (erule conjE)  wenzelm@23164  1314  apply (erule_tac x = "nat x" in allE)  wenzelm@23164  1315  apply simp  wenzelm@23164  1316  done  wenzelm@23164  1317 huffman@23365  1318 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  huffman@23431  1319  apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]  huffman@23431  1320  nat_0_le cong add: conj_cong)  wenzelm@23164  1321  apply (rule iffI)  wenzelm@23164  1322  apply iprover  wenzelm@23164  1323  apply (erule exE)  wenzelm@23164  1324  apply (case_tac "x=0")  wenzelm@23164  1325  apply (rule_tac x=0 in exI)  wenzelm@23164  1326  apply simp  wenzelm@23164  1327  apply (case_tac "0 \ k")  wenzelm@23164  1328  apply iprover  wenzelm@23164  1329  apply (simp add: linorder_not_le)  huffman@23431  1330  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])  wenzelm@23164  1331  apply assumption  wenzelm@23164  1332  apply (simp add: mult_ac)  wenzelm@23164  1333  done  wenzelm@23164  1334 wenzelm@23164  1335 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1336 proof  wenzelm@23164  1337  assume d: "x dvd 1" hence "int (nat \x\) dvd int (nat 1)" by (simp add: zdvd_abs1)  wenzelm@23164  1338  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1339  hence "nat \x\ = 1" by simp  wenzelm@23164  1340  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1341 next  wenzelm@23164  1342  assume "\x\=1" thus "x dvd 1"  wenzelm@23164  1343  by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)  wenzelm@23164  1344 qed  wenzelm@23164  1345 lemma zdvd_mult_cancel1:  wenzelm@23164  1346  assumes mp:"m \(0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1347 proof  wenzelm@23164  1348  assume n1: "\n\ = 1" thus "m * n dvd m"  wenzelm@23164  1349  by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)  wenzelm@23164  1350 next  wenzelm@23164  1351  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1352  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1353 qed  wenzelm@23164  1354 huffman@23365  1355 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  wenzelm@23164  1356  apply (auto simp add: dvd_def nat_abs_mult_distrib)  huffman@23365  1357  apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)  huffman@23365  1358  apply (rule_tac x = "-(int k)" in exI)  huffman@23431  1359  apply (auto simp add: int_mult)  huffman@23306  1360  done  huffman@23306  1361 huffman@23365  1362 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  huffman@23431  1363  apply (auto simp add: dvd_def abs_if int_mult)  huffman@23306  1364  apply (rule_tac [3] x = "nat k" in exI)  huffman@23365  1365  apply (rule_tac [2] x = "-(int k)" in exI)  huffman@23306  1366  apply (rule_tac x = "nat (-k)" in exI)  huffman@23431  1367  apply (cut_tac [3] k = m in int_less_0_conv)  huffman@23431  1368  apply (cut_tac k = m in int_less_0_conv)  huffman@23306  1369  apply (auto simp add: zero_le_mult_iff mult_less_0_iff  huffman@23365  1370  nat_mult_distrib [symmetric] nat_eq_iff2)  wenzelm@23164  1371  done  wenzelm@23164  1372 wenzelm@23164  1373 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  huffman@23431  1374  apply (auto simp add: dvd_def int_mult)  huffman@23365  1375  apply (rule_tac x = "nat k" in exI)  huffman@23431  1376  apply (cut_tac k = m in int_less_0_conv)  huffman@23365  1377  apply (auto simp add: zero_le_mult_iff mult_less_0_iff  huffman@23365  1378  nat_mult_distrib [symmetric] nat_eq_iff2)  huffman@23365  1379  done  wenzelm@23164  1380 wenzelm@23164  1381 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"  wenzelm@23164  1382  apply (auto simp add: dvd_def)  wenzelm@23164  1383  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1384  done  wenzelm@23164  1385 wenzelm@23164  1386 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"  wenzelm@23164  1387  apply (auto simp add: dvd_def)  wenzelm@23164  1388  apply (drule minus_equation_iff [THEN iffD1])  wenzelm@23164  1389  apply (rule_tac [!] x = "-k" in exI, auto)  wenzelm@23164  1390  done  wenzelm@23164  1391 wenzelm@23164  1392 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1393  apply (rule_tac z=n in int_cases)  huffman@23365  1394  apply (auto simp add: dvd_int_iff)  huffman@23365  1395  apply (rule_tac z=z in int_cases)  huffman@23307  1396  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1397  done  wenzelm@23164  1398 wenzelm@23164  1399 wenzelm@23164  1400 subsection{*Integer Powers*}  wenzelm@23164  1401 wenzelm@23164  1402 instance int :: power ..  wenzelm@23164  1403 wenzelm@23164  1404 primrec  wenzelm@23164  1405  "p ^ 0 = 1"  wenzelm@23164  1406  "p ^ (Suc n) = (p::int) * (p ^ n)"  wenzelm@23164  1407 wenzelm@23164  1408 wenzelm@23164  1409 instance int :: recpower  wenzelm@23164  1410 proof  wenzelm@23164  1411  fix z :: int  wenzelm@23164  1412  fix n :: nat  wenzelm@23164  1413  show "z^0 = 1" by simp  wenzelm@23164  1414  show "z^(Suc n) = z * (z^n)" by simp  wenzelm@23164  1415 qed  wenzelm@23164  1416 huffman@24391  1417 lemma of_int_power:  huffman@24391  1418  "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"  huffman@24391  1419  by (induct n) (simp_all add: power_Suc)  wenzelm@23164  1420 wenzelm@23164  1421 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1422 apply (induct "y", auto)  wenzelm@23164  1423 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1424 apply (simp (no_asm_simp))  wenzelm@23164  1425 apply (rule zmod_zmult_distrib [symmetric])  wenzelm@23164  1426 done  wenzelm@23164  1427 wenzelm@23164  1428 lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"  wenzelm@23164  1429  by (rule Power.power_add)  wenzelm@23164  1430 wenzelm@23164  1431 lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"  wenzelm@23164  1432  by (rule Power.power_mult [symmetric])  wenzelm@23164  1433 wenzelm@23164  1434 lemma zero_less_zpower_abs_iff [simp]:  wenzelm@23164  1435  "(0 < (abs x)^n) = (x \ (0::int) | n=0)"  wenzelm@23164  1436 apply (induct "n")  wenzelm@23164  1437 apply (auto simp add: zero_less_mult_iff)  wenzelm@23164  1438 done  wenzelm@23164  1439 wenzelm@23164  1440 lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"  wenzelm@23164  1441 apply (induct "n")  wenzelm@23164  1442 apply (auto simp add: zero_le_mult_iff)  wenzelm@23164  1443 done  wenzelm@23164  1444 wenzelm@23164  1445 lemma int_power: "int (m^n) = (int m) ^ n"  huffman@23365  1446  by (rule of_nat_power)  wenzelm@23164  1447 wenzelm@23164  1448 text{*Compatibility binding*}  wenzelm@23164  1449 lemmas zpower_int = int_power [symmetric]  wenzelm@23164  1450 huffman@23365  1451 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1452 apply (subst split_div, auto)  wenzelm@23164  1453 apply (subst split_zdiv, auto)  huffman@23365  1454 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  1455 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  wenzelm@23164  1456 done  wenzelm@23164  1457 wenzelm@23164  1458 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1459 apply (subst split_mod, auto)  huffman@23365  1460 apply (subst split_zmod, auto)  huffman@23365  1461 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1462  in unique_remainder)  huffman@23431  1463 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  huffman@23365  1464 done  wenzelm@23164  1465 wenzelm@23164  1466 text{*Suggested by Matthias Daum*}  wenzelm@23164  1467 lemma int_power_div_base:  wenzelm@23164  1468  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  wenzelm@23164  1469 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")  wenzelm@23164  1470  apply (erule ssubst)  wenzelm@23164  1471  apply (simp only: power_add)  wenzelm@23164  1472  apply simp_all  wenzelm@23164  1473 done  wenzelm@23164  1474 haftmann@23853  1475 text {* by Brian Huffman *}  haftmann@23853  1476 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@23853  1477 by (simp only: zmod_zminus1_eq_if mod_mod_trivial)  haftmann@23853  1478 haftmann@23853  1479 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@23853  1480 by (simp only: diff_def zmod_zadd_left_eq [symmetric])  haftmann@23853  1481 haftmann@23853  1482 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@23853  1483 proof -  haftmann@23853  1484  have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m"  haftmann@23853  1485  by (simp only: zminus_zmod)  haftmann@23853  1486  hence "(x + - (y mod m)) mod m = (x + - y) mod m"  haftmann@23853  1487  by (simp only: zmod_zadd_right_eq [symmetric])  haftmann@23853  1488  thus "(x - y mod m) mod m = (x - y) mod m"  haftmann@23853  1489  by (simp only: diff_def)  haftmann@23853  1490 qed  haftmann@23853  1491 haftmann@23853  1492 lemmas zmod_simps =  haftmann@23853  1493  IntDiv.zmod_zadd_left_eq [symmetric]  haftmann@23853  1494  IntDiv.zmod_zadd_right_eq [symmetric]  haftmann@23853  1495  IntDiv.zmod_zmult1_eq [symmetric]  haftmann@23853  1496  IntDiv.zmod_zmult1_eq' [symmetric]  haftmann@23853  1497  IntDiv.zpower_zmod  haftmann@23853  1498  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1499 haftmann@23853  1500 text {* code generator setup *}  wenzelm@23164  1501 wenzelm@23164  1502 code_modulename SML  wenzelm@23164  1503  IntDiv Integer  wenzelm@23164  1504 wenzelm@23164  1505 code_modulename OCaml  wenzelm@23164  1506  IntDiv Integer  wenzelm@23164  1507 wenzelm@23164  1508 code_modulename Haskell  haftmann@24195  1509  IntDiv Integer  wenzelm@23164  1510 wenzelm@23164  1511 end `