src/HOL/IntDiv.thy
 author wenzelm Wed Sep 17 21:27:14 2008 +0200 (2008-09-17) changeset 28263 69eaa97e7e96 parent 28262 aa7ca36d67fd child 28562 4e74209f113e permissions -rw-r--r--
moved global ML bindings to global place;
 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@26480  260 ML {*  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}  haftmann@26101  277  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;  wenzelm@23164  278 end)  wenzelm@23164  279 wenzelm@23164  280 in  wenzelm@23164  281 wenzelm@28262  282 val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())  haftmann@26101  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@27651  750 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"  haftmann@27651  751 apply (case_tac "b = 0", simp)  haftmann@27651  752 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  haftmann@27651  753 done  haftmann@27651  754 haftmann@27651  755 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"  haftmann@27651  756 apply (case_tac "b = 0", simp)  haftmann@27651  757 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)  haftmann@27651  758 done  haftmann@27651  759 haftmann@27651  760 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@27651  761 haftmann@27651  762 lemma zadd1_lemma:  haftmann@27651  763  "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c \ 0 |]  haftmann@27651  764  ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"  haftmann@27651  765 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)  haftmann@27651  766 haftmann@27651  767 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@27651  768 lemma zdiv_zadd1_eq:  haftmann@27651  769  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@27651  770 apply (case_tac "c = 0", simp)  haftmann@27651  771 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)  haftmann@27651  772 done  haftmann@27651  773 haftmann@27651  774 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"  haftmann@27651  775 apply (case_tac "c = 0", simp)  haftmann@27651  776 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)  haftmann@27651  777 done  haftmann@27651  778 haftmann@27651  779 lemma zdiv_zadd_self1[simp]: "a \ (0::int) ==> (a+b) div a = b div a + 1"  haftmann@27651  780 by (simp add: zdiv_zadd1_eq)  haftmann@27651  781 haftmann@27651  782 lemma zdiv_zadd_self2[simp]: "a \ (0::int) ==> (b+a) div a = b div a + 1"  haftmann@27651  783 by (simp add: zdiv_zadd1_eq)  haftmann@27651  784 haftmann@25942  785 instance int :: semiring_div  haftmann@27651  786 proof  haftmann@27651  787  fix a b c :: int  haftmann@27651  788  assume not0: "b \ 0"  haftmann@27651  789  show "(a + c * b) div b = c + a div b"  haftmann@27651  790  unfolding zdiv_zadd1_eq [of a "c * b"] using not0  haftmann@27651  791  by (simp add: zmod_zmult1_eq)  haftmann@27651  792 qed auto  haftmann@25942  793 wenzelm@23164  794 lemma zdiv_zmult_self2 [simp]: "b \ (0::int) ==> (b*a) div b = a"  wenzelm@23164  795 by (subst mult_commute, erule zdiv_zmult_self1)  wenzelm@23164  796 wenzelm@23164  797 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"  wenzelm@23164  798 by (simp add: zmod_zmult1_eq)  wenzelm@23164  799 wenzelm@23164  800 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"  wenzelm@23164  801 by (simp add: mult_commute zmod_zmult1_eq)  wenzelm@23164  802 wenzelm@23164  803 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  wenzelm@23164  804 proof  wenzelm@23164  805  assume "m mod d = 0"  wenzelm@23164  806  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto  wenzelm@23164  807 next  wenzelm@23164  808  assume "EX q::int. m = d*q"  wenzelm@23164  809  thus "m mod d = 0" by auto  wenzelm@23164  810 qed  wenzelm@23164  811 wenzelm@23164  812 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  813 wenzelm@23164  814 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"  wenzelm@23164  815 apply (rule trans [symmetric])  wenzelm@23164  816 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  817 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  818 done  wenzelm@23164  819 wenzelm@23164  820 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"  wenzelm@23164  821 apply (rule trans [symmetric])  wenzelm@23164  822 apply (rule zmod_zadd1_eq, simp)  wenzelm@23164  823 apply (rule zmod_zadd1_eq [symmetric])  wenzelm@23164  824 done  wenzelm@23164  825 wenzelm@23164  826 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"  wenzelm@23164  827 apply (case_tac "a = 0", simp)  wenzelm@23164  828 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  829 done  wenzelm@23164  830 wenzelm@23164  831 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"  wenzelm@23164  832 apply (case_tac "a = 0", simp)  wenzelm@23164  833 apply (simp add: zmod_zadd1_eq)  wenzelm@23164  834 done  wenzelm@23164  835 wenzelm@23164  836 nipkow@23983  837 lemma zmod_zdiff1_eq: fixes a::int  nipkow@23983  838  shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r")  nipkow@23983  839 proof -  nipkow@23983  840  have "?l = (c + (a mod c - b mod c)) mod c"  nipkow@23983  841  using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if)  nipkow@23983  842  also have "\ = ?r" by simp  nipkow@23983  843  finally show ?thesis .  nipkow@23983  844 qed  nipkow@23983  845 wenzelm@23164  846 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  847 wenzelm@23164  848 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  849  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  850  to cause particular problems.*)  wenzelm@23164  851 wenzelm@23164  852 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  853 wenzelm@23164  854 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  855 apply (subgoal_tac "b * (c - q mod c) < r * 1")  wenzelm@23164  856 apply (simp add: right_diff_distrib)  wenzelm@23164  857 apply (rule order_le_less_trans)  wenzelm@23164  858 apply (erule_tac [2] mult_strict_right_mono)  wenzelm@23164  859 apply (rule mult_left_mono_neg)  wenzelm@23164  860 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  861  add1_zle_eq pos_mod_bound)  wenzelm@23164  862 done  wenzelm@23164  863 wenzelm@23164  864 lemma zmult2_lemma_aux2:  wenzelm@23164  865  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  866 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  867  apply arith  wenzelm@23164  868 apply (simp add: mult_le_0_iff)  wenzelm@23164  869 done  wenzelm@23164  870 wenzelm@23164  871 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  872 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  873 apply arith  wenzelm@23164  874 apply (simp add: zero_le_mult_iff)  wenzelm@23164  875 done  wenzelm@23164  876 wenzelm@23164  877 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  878 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  wenzelm@23164  879 apply (simp add: right_diff_distrib)  wenzelm@23164  880 apply (rule order_less_le_trans)  wenzelm@23164  881 apply (erule mult_strict_right_mono)  wenzelm@23164  882 apply (rule_tac [2] mult_left_mono)  wenzelm@23164  883 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  884  add1_zle_eq pos_mod_bound)  wenzelm@23164  885 done  wenzelm@23164  886 wenzelm@23164  887 lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b \ 0; 0 < c |]  wenzelm@23164  888  ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"  wenzelm@23164  889 by (auto simp add: mult_ac quorem_def linorder_neq_iff  wenzelm@23164  890  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  891  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  892 wenzelm@23164  893 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  894 apply (case_tac "b = 0", simp)  wenzelm@23164  895 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])  wenzelm@23164  896 done  wenzelm@23164  897 wenzelm@23164  898 lemma zmod_zmult2_eq:  wenzelm@23164  899  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  900 apply (case_tac "b = 0", simp)  wenzelm@23164  901 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])  wenzelm@23164  902 done  wenzelm@23164  903 wenzelm@23164  904 wenzelm@23164  905 subsection{*Cancellation of Common Factors in div*}  wenzelm@23164  906 wenzelm@23164  907 lemma zdiv_zmult_zmult1_aux1:  wenzelm@23164  908  "[| (0::int) < b; c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  909 by (subst zdiv_zmult2_eq, auto)  wenzelm@23164  910 wenzelm@23164  911 lemma zdiv_zmult_zmult1_aux2:  wenzelm@23164  912  "[| b < (0::int); c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  913 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")  wenzelm@23164  914 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)  wenzelm@23164  915 done  wenzelm@23164  916 wenzelm@23164  917 lemma zdiv_zmult_zmult1: "c \ (0::int) ==> (c*a) div (c*b) = a div b"  wenzelm@23164  918 apply (case_tac "b = 0", simp)  wenzelm@23164  919 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)  wenzelm@23164  920 done  wenzelm@23164  921 nipkow@23401  922 lemma zdiv_zmult_zmult1_if[simp]:  nipkow@23401  923  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"  nipkow@23401  924 by (simp add:zdiv_zmult_zmult1)  nipkow@23401  925 nipkow@23401  926 (*  wenzelm@23164  927 lemma zdiv_zmult_zmult2: "c \ (0::int) ==> (a*c) div (b*c) = a div b"  wenzelm@23164  928 apply (drule zdiv_zmult_zmult1)  wenzelm@23164  929 apply (auto simp add: mult_commute)  wenzelm@23164  930 done  nipkow@23401  931 *)  wenzelm@23164  932 wenzelm@23164  933 wenzelm@23164  934 subsection{*Distribution of Factors over mod*}  wenzelm@23164  935 wenzelm@23164  936 lemma zmod_zmult_zmult1_aux1:  wenzelm@23164  937  "[| (0::int) < b; c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  938 by (subst zmod_zmult2_eq, auto)  wenzelm@23164  939 wenzelm@23164  940 lemma zmod_zmult_zmult1_aux2:  wenzelm@23164  941  "[| b < (0::int); c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  942 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")  wenzelm@23164  943 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)  wenzelm@23164  944 done  wenzelm@23164  945 wenzelm@23164  946 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"  wenzelm@23164  947 apply (case_tac "b = 0", simp)  wenzelm@23164  948 apply (case_tac "c = 0", simp)  wenzelm@23164  949 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)  wenzelm@23164  950 done  wenzelm@23164  951 wenzelm@23164  952 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"  wenzelm@23164  953 apply (cut_tac c = c in zmod_zmult_zmult1)  wenzelm@23164  954 apply (auto simp add: mult_commute)  wenzelm@23164  955 done  wenzelm@23164  956 nipkow@24490  957 lemma zmod_zmod_cancel:  nipkow@24490  958 assumes "n dvd m" shows "(k::int) mod m mod n = k mod n"  nipkow@24490  959 proof -  nipkow@24490  960  from n dvd m obtain r where "m = n*r" by(auto simp:dvd_def)  nipkow@24490  961  have "k mod n = (m * (k div m) + k mod m) mod n"  nipkow@24490  962  using zmod_zdiv_equality[of k m] by simp  nipkow@24490  963  also have "\ = (m * (k div m) mod n + k mod m mod n) mod n"  nipkow@24490  964  by(subst zmod_zadd1_eq, rule refl)  nipkow@24490  965  also have "m * (k div m) mod n = 0" using m = n*r  nipkow@24490  966  by(simp add:mult_ac)  nipkow@24490  967  finally show ?thesis by simp  nipkow@24490  968 qed  nipkow@24490  969 wenzelm@23164  970 wenzelm@23164  971 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  972 wenzelm@23164  973 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  974 wenzelm@23164  975 lemma split_pos_lemma:  wenzelm@23164  976  "0  wenzelm@23164  977  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  978 apply (rule iffI, clarify)  wenzelm@23164  979  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  980  apply (subst zmod_zadd1_eq)  wenzelm@23164  981  apply (subst zdiv_zadd1_eq)  wenzelm@23164  982  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@23164  983 txt{*converse direction*}  wenzelm@23164  984 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  985 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  986 done  wenzelm@23164  987 wenzelm@23164  988 lemma split_neg_lemma:  wenzelm@23164  989  "k<0 ==>  wenzelm@23164  990  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  991 apply (rule iffI, clarify)  wenzelm@23164  992  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  993  apply (subst zmod_zadd1_eq)  wenzelm@23164  994  apply (subst zdiv_zadd1_eq)  wenzelm@23164  995  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  996 txt{*converse direction*}  wenzelm@23164  997 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  998 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  999 done  wenzelm@23164  1000 wenzelm@23164  1001 lemma split_zdiv:  wenzelm@23164  1002  "P(n div k :: int) =  wenzelm@23164  1003  ((k = 0 --> P 0) &  wenzelm@23164  1004  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  1005  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  1006 apply (case_tac "k=0", simp)  wenzelm@23164  1007 apply (simp only: linorder_neq_iff)  wenzelm@23164  1008 apply (erule disjE)  wenzelm@23164  1009  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  1010  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  1011 done  wenzelm@23164  1012 wenzelm@23164  1013 lemma split_zmod:  wenzelm@23164  1014  "P(n mod k :: int) =  wenzelm@23164  1015  ((k = 0 --> P n) &  wenzelm@23164  1016  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  1017  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  1018 apply (case_tac "k=0", simp)  wenzelm@23164  1019 apply (simp only: linorder_neq_iff)  wenzelm@23164  1020 apply (erule disjE)  wenzelm@23164  1021  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  1022  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  1023 done  wenzelm@23164  1024 wenzelm@23164  1025 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  1026 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1027 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1028 wenzelm@23164  1029 wenzelm@23164  1030 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  1031 wenzelm@23164  1032 text{*computing div by shifting *}  wenzelm@23164  1033 wenzelm@23164  1034 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  1035 proof cases  wenzelm@23164  1036  assume "a=0"  wenzelm@23164  1037  thus ?thesis by simp  wenzelm@23164  1038 next  wenzelm@23164  1039  assume "a\0" and le_a: "0\a"  wenzelm@23164  1040  hence a_pos: "1 \ a" by arith  wenzelm@23164  1041  hence one_less_a2: "1 < 2*a" by arith  wenzelm@23164  1042  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  wenzelm@23164  1043  by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)  wenzelm@23164  1044  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  1045  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  1046  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  1047  with le_2a  wenzelm@23164  1048  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  1049  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  1050  right_distrib)  wenzelm@23164  1051  thus ?thesis  wenzelm@23164  1052  by (subst zdiv_zadd1_eq,  wenzelm@23164  1053  simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2  wenzelm@23164  1054  div_pos_pos_trivial)  wenzelm@23164  1055 qed  wenzelm@23164  1056 wenzelm@23164  1057 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  1058 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  1059 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  1060 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1061 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1062 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  1063  simp)  wenzelm@23164  1064 done  wenzelm@23164  1065 wenzelm@23164  1066 (*Not clear why this must be proved separately; probably number_of causes  wenzelm@23164  1067  simplification problems*)  wenzelm@23164  1068 lemma not_0_le_lemma: "~ 0 \ x ==> x \ (0::int)"  wenzelm@23164  1069 by auto  wenzelm@23164  1070 huffman@26086  1071 lemma zdiv_number_of_Bit0 [simp]:  huffman@26086  1072  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  huffman@26086  1073  number_of v div (number_of w :: int)"  huffman@26086  1074 by (simp only: number_of_eq numeral_simps) simp  huffman@26086  1075 huffman@26086  1076 lemma zdiv_number_of_Bit1 [simp]:  huffman@26086  1077  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  huffman@26086  1078  (if (0::int) \ number_of w  wenzelm@23164  1079  then number_of v div (number_of w)  wenzelm@23164  1080  else (number_of v + (1::int)) div (number_of w))"  wenzelm@23164  1081 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  huffman@26086  1082 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)  wenzelm@23164  1083 done  wenzelm@23164  1084 wenzelm@23164  1085 wenzelm@23164  1086 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  1087 wenzelm@23164  1088 lemma pos_zmod_mult_2:  wenzelm@23164  1089  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  1090 apply (case_tac "a = 0", simp)  wenzelm@23164  1091 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  1092  prefer 2 apply arith  wenzelm@23164  1093 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  1094  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  1095 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  1096  pos_mod_bound)  wenzelm@23164  1097 apply (subst zmod_zadd1_eq)  wenzelm@23164  1098 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)  wenzelm@23164  1099 apply (rule mod_pos_pos_trivial)  huffman@26086  1100 apply (auto simp add: mod_pos_pos_trivial ring_distribs)  wenzelm@23164  1101 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  1102 done  wenzelm@23164  1103 wenzelm@23164  1104 lemma neg_zmod_mult_2:  wenzelm@23164  1105  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  1106 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  1107  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  1108 apply (rule_tac [2] pos_zmod_mult_2)  wenzelm@23164  1109 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1110 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1111  prefer 2 apply simp  wenzelm@23164  1112 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  1113 done  wenzelm@23164  1114 huffman@26086  1115 lemma zmod_number_of_Bit0 [simp]:  huffman@26086  1116  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  huffman@26086  1117  (2::int) * (number_of v mod number_of w)"  huffman@26086  1118 apply (simp only: number_of_eq numeral_simps)  huffman@26086  1119 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  huffman@26086  1120  not_0_le_lemma neg_zmod_mult_2 add_ac)  huffman@26086  1121 done  huffman@26086  1122 huffman@26086  1123 lemma zmod_number_of_Bit1 [simp]:  huffman@26086  1124  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  huffman@26086  1125  (if (0::int) \ number_of w  wenzelm@23164  1126  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  1127  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  huffman@26086  1128 apply (simp only: number_of_eq numeral_simps)  wenzelm@23164  1129 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  wenzelm@23164  1130  not_0_le_lemma neg_zmod_mult_2 add_ac)  wenzelm@23164  1131 done  wenzelm@23164  1132 wenzelm@23164  1133 wenzelm@23164  1134 subsection{*Quotients of Signs*}  wenzelm@23164  1135 wenzelm@23164  1136 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  1137 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  1138 apply (rule order_trans)  wenzelm@23164  1139 apply (rule_tac a' = "-1" in zdiv_mono1)  wenzelm@23164  1140 apply (auto simp add: zdiv_minus1)  wenzelm@23164  1141 done  wenzelm@23164  1142 wenzelm@23164  1143 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  1144 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  1145 wenzelm@23164  1146 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1147 apply auto  wenzelm@23164  1148 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1149 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1150 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1151 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1152 done  wenzelm@23164  1153 wenzelm@23164  1154 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1155  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1156 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1157 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1158 done  wenzelm@23164  1159 wenzelm@23164  1160 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1161 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1162 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1163 wenzelm@23164  1164 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1165 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1166 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1167 wenzelm@23164  1168 wenzelm@23164  1169 subsection {* The Divides Relation *}  wenzelm@23164  1170 wenzelm@23164  1171 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"  haftmann@23512  1172  by (simp add: dvd_def zmod_eq_0_iff)  haftmann@23512  1173 wenzelm@23164  1174 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@23164  1175  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]  wenzelm@23164  1176 wenzelm@23164  1177 lemma zdvd_0_right [iff]: "(m::int) dvd 0"  haftmann@23512  1178  by (simp add: dvd_def)  wenzelm@23164  1179 paulson@24286  1180 lemma zdvd_0_left [iff,noatp]: "(0 dvd (m::int)) = (m = 0)"  wenzelm@23164  1181  by (simp add: dvd_def)  wenzelm@23164  1182 wenzelm@23164  1183 lemma zdvd_1_left [iff]: "1 dvd (m::int)"  wenzelm@23164  1184  by (simp add: dvd_def)  wenzelm@23164  1185 wenzelm@23164  1186 lemma zdvd_refl [simp]: "m dvd (m::int)"  haftmann@23512  1187  by (auto simp add: dvd_def intro: zmult_1_right [symmetric])  wenzelm@23164  1188 wenzelm@23164  1189 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"  haftmann@23512  1190  by (auto simp add: dvd_def intro: mult_assoc)  wenzelm@23164  1191 haftmann@27651  1192 lemma zdvd_zminus_iff: "m dvd -n \ m dvd (n::int)"  haftmann@27651  1193 proof  haftmann@27651  1194  assume "m dvd - n"  haftmann@27651  1195  then obtain k where "- n = m * k" ..  haftmann@27651  1196  then have "n = m * - k" by simp  haftmann@27651  1197  then show "m dvd n" ..  haftmann@27651  1198 next  haftmann@27651  1199  assume "m dvd n"  haftmann@27651  1200  then have "m dvd n * -1" by (rule dvd_mult2)  haftmann@27651  1201  then show "m dvd - n" by simp  haftmann@27651  1202 qed  wenzelm@23164  1203 haftmann@27651  1204 lemma zdvd_zminus2_iff: "-m dvd n \ m dvd (n::int)"  haftmann@27651  1205 proof  haftmann@27651  1206  assume "- m dvd n"  haftmann@27651  1207  then obtain k where "n = - m * k" ..  haftmann@27651  1208  then have "n = m * - k" by simp  haftmann@27651  1209  then show "m dvd n" ..  haftmann@27651  1210 next  haftmann@27651  1211  assume "m dvd n"  haftmann@27651  1212  then obtain k where "n = m * k" ..  haftmann@27651  1213  then have "n = - m * - k" by simp  haftmann@27651  1214  then show "- m dvd n" ..  haftmann@27651  1215 qed  haftmann@27651  1216 wenzelm@23164  1217 lemma zdvd_abs1: "( \i::int\ dvd j) = (i dvd j)"  haftmann@27651  1218  by (cases "i > 0") (simp_all add: zdvd_zminus2_iff)  haftmann@27651  1219 wenzelm@23164  1220 lemma zdvd_abs2: "( (i::int) dvd \j$$ = (i dvd j)"  haftmann@27651  1221  by (cases "j > 0") (simp_all add: zdvd_zminus_iff)  wenzelm@23164  1222 wenzelm@23164  1223 lemma zdvd_anti_sym:  wenzelm@23164  1224  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1225  apply (simp add: dvd_def, auto)  wenzelm@23164  1226  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1227  done  wenzelm@23164  1228 wenzelm@23164  1229 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"  wenzelm@23164  1230  apply (simp add: dvd_def)  wenzelm@23164  1231  apply (blast intro: right_distrib [symmetric])  wenzelm@23164  1232  done  wenzelm@23164  1233 wenzelm@23164  1234 lemma zdvd_dvd_eq: assumes anz:"a \ 0" and ab: "(a::int) dvd b" and ba:"b dvd a"  wenzelm@23164  1235  shows "\a\ = \b\"  wenzelm@23164  1236 proof-  wenzelm@23164  1237  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast  wenzelm@23164  1238  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1239  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1240  with mult_cancel_left1[where c="a" and b="k*k'"]  wenzelm@23164  1241  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)  wenzelm@23164  1242  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1243  thus ?thesis using k k' by auto  wenzelm@23164  1244 qed  wenzelm@23164  1245 wenzelm@23164  1246 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"  wenzelm@23164  1247  apply (simp add: dvd_def)  wenzelm@23164  1248  apply (blast intro: right_diff_distrib [symmetric])  wenzelm@23164  1249  done  wenzelm@23164  1250 wenzelm@23164  1251 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1252  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1253  apply (erule ssubst)  wenzelm@23164  1254  apply (blast intro: zdvd_zadd, simp)  wenzelm@23164  1255  done  wenzelm@23164  1256 wenzelm@23164  1257 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"  wenzelm@23164  1258  apply (simp add: dvd_def)  wenzelm@23164  1259  apply (blast intro: mult_left_commute)  wenzelm@23164  1260  done  wenzelm@23164  1261 wenzelm@23164  1262 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"  wenzelm@23164  1263  apply (subst mult_commute)  wenzelm@23164  1264  apply (erule zdvd_zmult)  wenzelm@23164  1265  done  wenzelm@23164  1266 wenzelm@23164  1267 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"  wenzelm@23164  1268  apply (rule zdvd_zmult)  wenzelm@23164  1269  apply (rule zdvd_refl)  wenzelm@23164  1270  done  wenzelm@23164  1271 wenzelm@23164  1272 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"  wenzelm@23164  1273  apply (rule zdvd_zmult2)  wenzelm@23164  1274  apply (rule zdvd_refl)  wenzelm@23164  1275  done  wenzelm@23164  1276 wenzelm@23164  1277 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"  wenzelm@23164  1278  apply (simp add: dvd_def)  wenzelm@23164  1279  apply (simp add: mult_assoc, blast)  wenzelm@23164  1280  done  wenzelm@23164  1281 wenzelm@23164  1282 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"  wenzelm@23164  1283  apply (rule zdvd_zmultD2)  wenzelm@23164  1284  apply (subst mult_commute, assumption)  wenzelm@23164  1285  done  wenzelm@23164  1286 wenzelm@23164  1287 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"  haftmann@27651  1288  by (rule mult_dvd_mono)  wenzelm@23164  1289 wenzelm@23164  1290 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  wenzelm@23164  1291  apply (rule iffI)  wenzelm@23164  1292  apply (erule_tac [2] zdvd_zadd)  wenzelm@23164  1293  apply (subgoal_tac "n = (n + k * m) - k * m")  wenzelm@23164  1294  apply (erule ssubst)  wenzelm@23164  1295  apply (erule zdvd_zdiff, simp_all)  wenzelm@23164  1296  done  wenzelm@23164  1297 wenzelm@23164  1298 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  wenzelm@23164  1299  apply (simp add: dvd_def)  wenzelm@23164  1300  apply (auto simp add: zmod_zmult_zmult1)  wenzelm@23164  1301  done  wenzelm@23164  1302 wenzelm@23164  1303 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1304  apply (subgoal_tac "k dvd n * (m div n) + m mod n")  wenzelm@23164  1305  apply (simp add: zmod_zdiv_equality [symmetric])  wenzelm@23164  1306  apply (simp only: zdvd_zadd zdvd_zmult2)  wenzelm@23164  1307  done  wenzelm@23164  1308 wenzelm@23164  1309 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  haftmann@27651  1310  apply (auto elim!: dvdE)  wenzelm@23164  1311  apply (subgoal_tac "0 < n")  wenzelm@23164  1312  prefer 2  wenzelm@23164  1313  apply (blast intro: order_less_trans)  wenzelm@23164  1314  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1315  apply (subgoal_tac "n * k < n * 1")  wenzelm@23164  1316  apply (drule mult_less_cancel_left [THEN iffD1], auto)  wenzelm@23164  1317  done  haftmann@27651  1318 wenzelm@23164  1319 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1320  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@23477  1321  by (simp add: ring_simps)  wenzelm@23164  1322 wenzelm@23164  1323 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1324 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1325  apply (simp add: zmult_div_cancel)  wenzelm@23164  1326 apply (simp only: zdvd_iff_zmod_eq_0)  wenzelm@23164  1327 done  wenzelm@23164  1328 wenzelm@23164  1329 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1330  shows "m dvd n"  wenzelm@23164  1331 proof-  wenzelm@23164  1332  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1333  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1334  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1335  hence "n = m * h" by blast  wenzelm@23164  1336  thus ?thesis by blast  wenzelm@23164  1337 qed  wenzelm@23164  1338 nipkow@23969  1339 lemma zdvd_zmult_cancel_disj[simp]:  nipkow@23969  1340  "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"  nipkow@23969  1341 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)  nipkow@23969  1342 nipkow@23969  1343 wenzelm@23164  1344 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  nipkow@25134  1345 apply (simp split add: split_nat)  nipkow@25134  1346 apply (rule iffI)  nipkow@25134  1347 apply (erule exE)  nipkow@25134  1348 apply (rule_tac x = "int x" in exI)  nipkow@25134  1349 apply simp  nipkow@25134  1350 apply (erule exE)  nipkow@25134  1351 apply (rule_tac x = "nat x" in exI)  nipkow@25134  1352 apply (erule conjE)  nipkow@25134  1353 apply (erule_tac x = "nat x" in allE)  nipkow@25134  1354 apply simp  nipkow@25134  1355 done  wenzelm@23164  1356 huffman@23365  1357 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  haftmann@27651  1358 proof -  haftmann@27651  1359  have "\k. int y = int x * k \ x dvd y"  haftmann@27651  1360  proof -  haftmann@27651  1361  fix k  haftmann@27651  1362  assume A: "int y = int x * k"  haftmann@27651  1363  then show "x dvd y" proof (cases k)  haftmann@27651  1364  case (1 n) with A have "y = x * n" by (simp add: zmult_int)  haftmann@27651  1365  then show ?thesis ..  haftmann@27651  1366  next  haftmann@27651  1367  case (2 n) with A have "int y = int x * (- int (Suc n))" by simp  haftmann@27651  1368  also have "\ = - (int x * int (Suc n))" by (simp only: mult_minus_right)  haftmann@27651  1369  also have "\ = - int (x * Suc n)" by (simp only: zmult_int)  haftmann@27651  1370  finally have "- int (x * Suc n) = int y" ..  haftmann@27651  1371  then show ?thesis by (simp only: negative_eq_positive) auto  haftmann@27651  1372  qed  haftmann@27651  1373  qed  haftmann@27651  1374  then show ?thesis by (auto elim!: dvdE simp only: zmult_int [symmetric])  haftmann@27651  1375 qed  wenzelm@23164  1376 wenzelm@23164  1377 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1378 proof  wenzelm@23164  1379  assume d: "x dvd 1" hence "int (nat \x\) dvd int (nat 1)" by (simp add: zdvd_abs1)  wenzelm@23164  1380  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1381  hence "nat \x\ = 1" by simp  wenzelm@23164  1382  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1383 next  wenzelm@23164  1384  assume "\x\=1" thus "x dvd 1"  wenzelm@23164  1385  by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)  wenzelm@23164  1386 qed  wenzelm@23164  1387 lemma zdvd_mult_cancel1:  wenzelm@23164  1388  assumes mp:"m \(0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1389 proof  wenzelm@23164  1390  assume n1: "\n\ = 1" thus "m * n dvd m"  wenzelm@23164  1391  by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)  wenzelm@23164  1392 next  wenzelm@23164  1393  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1394  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1395 qed  wenzelm@23164  1396 huffman@23365  1397 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  haftmann@27651  1398  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus_iff)  huffman@23306  1399 huffman@23365  1400 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  haftmann@27651  1401  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus2_iff)  wenzelm@23164  1402 wenzelm@23164  1403 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  haftmann@27651  1404  by (auto simp add: dvd_int_iff)  wenzelm@23164  1405 wenzelm@23164  1406 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"  haftmann@27651  1407  by (simp add: zdvd_zminus2_iff)  wenzelm@23164  1408 wenzelm@23164  1409 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"  haftmann@27651  1410  by (simp add: zdvd_zminus_iff)  wenzelm@23164  1411 wenzelm@23164  1412 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1413  apply (rule_tac z=n in int_cases)  huffman@23365  1414  apply (auto simp add: dvd_int_iff)  huffman@23365  1415  apply (rule_tac z=z in int_cases)  huffman@23307  1416  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1417  done  wenzelm@23164  1418 wenzelm@23164  1419 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1420 apply (induct "y", auto)  wenzelm@23164  1421 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1422 apply (simp (no_asm_simp))  wenzelm@23164  1423 apply (rule zmod_zmult_distrib [symmetric])  wenzelm@23164  1424 done  wenzelm@23164  1425 huffman@23365  1426 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1427 apply (subst split_div, auto)  wenzelm@23164  1428 apply (subst split_zdiv, auto)  huffman@23365  1429 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  1430 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  wenzelm@23164  1431 done  wenzelm@23164  1432 wenzelm@23164  1433 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1434 apply (subst split_mod, auto)  huffman@23365  1435 apply (subst split_zmod, auto)  huffman@23365  1436 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1437  in unique_remainder)  huffman@23431  1438 apply (auto simp add: IntDiv.quorem_def of_nat_mult)  huffman@23365  1439 done  wenzelm@23164  1440 wenzelm@23164  1441 text{*Suggested by Matthias Daum*}  wenzelm@23164  1442 lemma int_power_div_base:  wenzelm@23164  1443  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  wenzelm@23164  1444 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")  wenzelm@23164  1445  apply (erule ssubst)  wenzelm@23164  1446  apply (simp only: power_add)  wenzelm@23164  1447  apply simp_all  wenzelm@23164  1448 done  wenzelm@23164  1449 haftmann@23853  1450 text {* by Brian Huffman *}  haftmann@23853  1451 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  haftmann@23853  1452 by (simp only: zmod_zminus1_eq_if mod_mod_trivial)  haftmann@23853  1453 haftmann@23853  1454 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  haftmann@23853  1455 by (simp only: diff_def zmod_zadd_left_eq [symmetric])  haftmann@23853  1456 haftmann@23853  1457 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  haftmann@23853  1458 proof -  haftmann@23853  1459  have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m"  haftmann@23853  1460  by (simp only: zminus_zmod)  haftmann@23853  1461  hence "(x + - (y mod m)) mod m = (x + - y) mod m"  haftmann@23853  1462  by (simp only: zmod_zadd_right_eq [symmetric])  haftmann@23853  1463  thus "(x - y mod m) mod m = (x - y) mod m"  haftmann@23853  1464  by (simp only: diff_def)  haftmann@23853  1465 qed  haftmann@23853  1466 haftmann@23853  1467 lemmas zmod_simps =  haftmann@23853  1468  IntDiv.zmod_zadd_left_eq [symmetric]  haftmann@23853  1469  IntDiv.zmod_zadd_right_eq [symmetric]  haftmann@23853  1470  IntDiv.zmod_zmult1_eq [symmetric]  haftmann@23853  1471  IntDiv.zmod_zmult1_eq' [symmetric]  haftmann@23853  1472  IntDiv.zpower_zmod  haftmann@23853  1473  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1474 haftmann@23853  1475 text {* code generator setup *}  wenzelm@23164  1476 haftmann@26507  1477 context ring_1  haftmann@26507  1478 begin  haftmann@26507  1479 haftmann@26507  1480 lemma of_int_num [code func]:  haftmann@26507  1481  "of_int k = (if k = 0 then 0 else if k < 0 then  haftmann@26507  1482  - of_int (- k) else let  haftmann@26507  1483  (l, m) = divAlg (k, 2);  haftmann@26507  1484  l' = of_int l  haftmann@26507  1485  in if m = 0 then l' + l' else l' + l' + 1)"  haftmann@26507  1486 proof -  haftmann@26507  1487  have aux1: "k mod (2\int) \ (0\int) \  haftmann@26507  1488  of_int k = of_int (k div 2 * 2 + 1)"  haftmann@26507  1489  proof -  haftmann@26507  1490  have "k mod 2 < 2" by (auto intro: pos_mod_bound)  haftmann@26507  1491  moreover have "0 \ k mod 2" by (auto intro: pos_mod_sign)  haftmann@26507  1492  moreover assume "k mod 2 \ 0"  haftmann@26507  1493  ultimately have "k mod 2 = 1" by arith  haftmann@26507  1494  moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp  haftmann@26507  1495  ultimately show ?thesis by auto  haftmann@26507  1496  qed  haftmann@26507  1497  have aux2: "\x. of_int 2 * x = x + x"  haftmann@26507  1498  proof -  haftmann@26507  1499  fix x  haftmann@26507  1500  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1501  show "of_int 2 * x = x + x"  haftmann@26507  1502  unfolding int2 of_int_add left_distrib by simp  haftmann@26507  1503  qed  haftmann@26507  1504  have aux3: "\x. x * of_int 2 = x + x"  haftmann@26507  1505  proof -  haftmann@26507  1506  fix x  haftmann@26507  1507  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1508  show "x * of_int 2 = x + x"  haftmann@26507  1509  unfolding int2 of_int_add right_distrib by simp  haftmann@26507  1510  qed  haftmann@26507  1511  from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3)  haftmann@26507  1512 qed  haftmann@26507  1513 haftmann@26507  1514 end  haftmann@26507  1515 chaieb@27667  1516 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  chaieb@27667  1517 proof  chaieb@27667  1518  assume H: "x mod n = y mod n"  chaieb@27667  1519  hence "x mod n - y mod n = 0" by simp  chaieb@27667  1520  hence "(x mod n - y mod n) mod n = 0" by simp  chaieb@27667  1521  hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])  chaieb@27667  1522  thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)  chaieb@27667  1523 next  chaieb@27667  1524  assume H: "n dvd x - y"  chaieb@27667  1525  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  chaieb@27667  1526  hence "x = n*k + y" by simp  chaieb@27667  1527  hence "x mod n = (n*k + y) mod n" by simp  chaieb@27667  1528  thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)  chaieb@27667  1529 qed  chaieb@27667  1530 chaieb@27667  1531 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  chaieb@27667  1532  shows "\q. x = y + n * q"  chaieb@27667  1533 proof-  chaieb@27667  1534  from xy have th: "int x - int y = int (x - y)" by simp  chaieb@27667  1535  from xyn have "int x mod int n = int y mod int n"  chaieb@27667  1536  by (simp add: zmod_int[symmetric])  chaieb@27667  1537  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  chaieb@27667  1538  hence "n dvd x - y" by (simp add: th zdvd_int)  chaieb@27667  1539  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  chaieb@27667  1540 qed  chaieb@27667  1541 chaieb@27667  1542 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  chaieb@27667  1543  (is "?lhs = ?rhs")  chaieb@27667  1544 proof  chaieb@27667  1545  assume H: "x mod n = y mod n"  chaieb@27667  1546  {assume xy: "x \ y"  chaieb@27667  1547  from H have th: "y mod n = x mod n" by simp  chaieb@27667  1548  from nat_mod_eq_lemma[OF th xy] have ?rhs  chaieb@27667  1549  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  chaieb@27667  1550  moreover  chaieb@27667  1551  {assume xy: "y \ x"  chaieb@27667  1552  from nat_mod_eq_lemma[OF H xy] have ?rhs  chaieb@27667  1553  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  chaieb@27667  1554  ultimately show ?rhs using linear[of x y] by blast  chaieb@27667  1555 next  chaieb@27667  1556  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  chaieb@27667  1557  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  chaieb@27667  1558  thus ?lhs by simp  chaieb@27667  1559 qed  chaieb@27667  1560 wenzelm@23164  1561 code_modulename SML  wenzelm@23164  1562  IntDiv Integer  wenzelm@23164  1563 wenzelm@23164  1564 code_modulename OCaml  wenzelm@23164  1565  IntDiv Integer  wenzelm@23164  1566 wenzelm@23164  1567 code_modulename Haskell  haftmann@24195  1568  IntDiv Integer  wenzelm@23164  1569 wenzelm@23164  1570 end `