src/HOL/IntDiv.thy
 author huffman Wed Feb 18 15:01:53 2009 -0800 (2009-02-18) changeset 29981 7d0ed261b712 parent 29951 a70bc5190534 child 30031 bd786c37af84 permissions -rw-r--r--
generalize int_dvd_cancel_factor simproc to idom class
 wenzelm@23164  1 (* Title: HOL/IntDiv.thy  wenzelm@23164  2  Author: Lawrence C Paulson, Cambridge University Computer Laboratory  wenzelm@23164  3  Copyright 1999 University of Cambridge  wenzelm@23164  4 wenzelm@23164  5 *)  wenzelm@23164  6 haftmann@29651  7 header{* The Division Operators div and mod *}  wenzelm@23164  8 wenzelm@23164  9 theory IntDiv  haftmann@25919  10 imports Int Divides FunDef  wenzelm@23164  11 begin  wenzelm@23164  12 haftmann@29651  13 definition divmod_rel :: "int \ int \ int \ int \ bool" where  wenzelm@23164  14  --{*definition of quotient and remainder*}  haftmann@29651  15  [code]: "divmod_rel a b = ($$q, r). a = b * q + r \  haftmann@29651  16  (if 0 < b then 0 \ r \ r < b else b < r \ r \ 0))"  wenzelm@23164  17 haftmann@29651  18 definition adjust :: "int \ int \ int \ int \ int" where  wenzelm@23164  19  --{*for the division algorithm*}  haftmann@29651  20  [code]: "adjust b = (\(q, r). if 0 \ r - b then (2 * q + 1, r - b)  haftmann@29651  21  else (2 * q, r))"  wenzelm@23164  22 wenzelm@23164  23 text{*algorithm for the case @{text "a\0, b>0"}*}  haftmann@29651  24 function posDivAlg :: "int \ int \ int \ int" where  haftmann@29651  25  "posDivAlg a b = (if a < b \ b \ 0 then (0, a)  haftmann@29651  26  else adjust b (posDivAlg a (2 * b)))"  wenzelm@23164  27 by auto  haftmann@29651  28 termination by (relation "measure (\(a, b). nat (a - b + 1))") auto  wenzelm@23164  29 wenzelm@23164  30 text{*algorithm for the case @{text "a<0, b>0"}*}  haftmann@29651  31 function negDivAlg :: "int \ int \ int \ int" where  haftmann@29651  32  "negDivAlg a b = (if 0 \a + b \ b \ 0 then (-1, a + b)  haftmann@29651  33  else adjust b (negDivAlg a (2 * b)))"  wenzelm@23164  34 by auto  haftmann@29651  35 termination by (relation "measure (\(a, b). nat (- a - b))") auto  wenzelm@23164  36 wenzelm@23164  37 text{*algorithm for the general case @{term "b\0"}*}  haftmann@29651  38 definition negateSnd :: "int \ int \ int \ int" where  haftmann@29651  39  [code inline]: "negateSnd = apsnd uminus"  wenzelm@23164  40 haftmann@29651  41 definition divmod :: "int \ int \ int \ int" where  wenzelm@23164  42  --{*The full division algorithm considers all possible signs for a, b  wenzelm@23164  43  including the special case @{text "a=0, b<0"} because  wenzelm@23164  44  @{term negDivAlg} requires @{term "a<0"}.*}  haftmann@29651  45  "divmod a b = (if 0 \ a then if 0 \ b then posDivAlg a b  haftmann@29651  46  else if a = 0 then (0, 0)  wenzelm@23164  47  else negateSnd (negDivAlg (-a) (-b))  wenzelm@23164  48  else  haftmann@29651  49  if 0 < b then negDivAlg a b  haftmann@29651  50  else negateSnd (posDivAlg (-a) (-b)))"  wenzelm@23164  51 haftmann@25571  52 instantiation int :: Divides.div  haftmann@25571  53 begin  haftmann@25571  54 haftmann@25571  55 definition  haftmann@29651  56  div_def: "a div b = fst (divmod a b)"  haftmann@25571  57 haftmann@25571  58 definition  haftmann@29651  59  mod_def: "a mod b = snd (divmod a b)"  haftmann@25571  60 haftmann@25571  61 instance ..  haftmann@25571  62 haftmann@25571  63 end  wenzelm@23164  64 haftmann@29651  65 lemma divmod_mod_div:  haftmann@29651  66  "divmod p q = (p div q, p mod q)"  wenzelm@23164  67  by (auto simp add: div_def mod_def)  wenzelm@23164  68 wenzelm@23164  69 text{*  wenzelm@23164  70 Here is the division algorithm in ML:  wenzelm@23164  71 wenzelm@23164  72 \begin{verbatim}  wenzelm@23164  73  fun posDivAlg (a,b) =  wenzelm@23164  74  if ar-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  77  end  wenzelm@23164  78 wenzelm@23164  79  fun negDivAlg (a,b) =  wenzelm@23164  80  if 0\a+b then (~1,a+b)  wenzelm@23164  81  else let val (q,r) = negDivAlg(a, 2*b)  wenzelm@23164  82  in if 0\r-b then (2*q+1, r-b) else (2*q, r)  wenzelm@23164  83  end;  wenzelm@23164  84 wenzelm@23164  85  fun negateSnd (q,r:int) = (q,~r);  wenzelm@23164  86 haftmann@29651  87  fun divmod (a,b) = if 0\a then  wenzelm@23164  88  if b>0 then posDivAlg (a,b)  wenzelm@23164  89  else if a=0 then (0,0)  wenzelm@23164  90  else negateSnd (negDivAlg (~a,~b))  wenzelm@23164  91  else  wenzelm@23164  92  if 0 b*q + r; 0 \ r'; r' < b; r < b |]  wenzelm@23164  103  ==> q' \ (q::int)"  wenzelm@23164  104 apply (subgoal_tac "r' + b * (q'-q) \ r")  wenzelm@23164  105  prefer 2 apply (simp add: right_diff_distrib)  wenzelm@23164  106 apply (subgoal_tac "0 < b * (1 + q - q') ")  wenzelm@23164  107 apply (erule_tac [2] order_le_less_trans)  wenzelm@23164  108  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  109 apply (subgoal_tac "b * q' < b * (1 + q) ")  wenzelm@23164  110  prefer 2 apply (simp add: right_diff_distrib right_distrib)  wenzelm@23164  111 apply (simp add: mult_less_cancel_left)  wenzelm@23164  112 done  wenzelm@23164  113 wenzelm@23164  114 lemma unique_quotient_lemma_neg:  wenzelm@23164  115  "[| b*q' + r' \ b*q + r; r \ 0; b < r; b < r' |]  wenzelm@23164  116  ==> q \ (q'::int)"  wenzelm@23164  117 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,  wenzelm@23164  118  auto)  wenzelm@23164  119 wenzelm@23164  120 lemma unique_quotient:  haftmann@29651  121  "[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \ 0 |]  wenzelm@23164  122  ==> q = q'"  haftmann@29651  123 apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)  wenzelm@23164  124 apply (blast intro: order_antisym  wenzelm@23164  125  dest: order_eq_refl [THEN unique_quotient_lemma]  wenzelm@23164  126  order_eq_refl [THEN unique_quotient_lemma_neg] sym)+  wenzelm@23164  127 done  wenzelm@23164  128 wenzelm@23164  129 wenzelm@23164  130 lemma unique_remainder:  haftmann@29651  131  "[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \ 0 |]  wenzelm@23164  132  ==> r = r'"  wenzelm@23164  133 apply (subgoal_tac "q = q'")  haftmann@29651  134  apply (simp add: divmod_rel_def)  wenzelm@23164  135 apply (blast intro: unique_quotient)  wenzelm@23164  136 done  wenzelm@23164  137 wenzelm@23164  138 wenzelm@23164  139 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}  wenzelm@23164  140 wenzelm@23164  141 text{*And positive divisors*}  wenzelm@23164  142 wenzelm@23164  143 lemma adjust_eq [simp]:  wenzelm@23164  144  "adjust b (q,r) =  wenzelm@23164  145  (let diff = r-b in  wenzelm@23164  146  if 0 \ diff then (2*q + 1, diff)  wenzelm@23164  147  else (2*q, r))"  wenzelm@23164  148 by (simp add: Let_def adjust_def)  wenzelm@23164  149 wenzelm@23164  150 declare posDivAlg.simps [simp del]  wenzelm@23164  151 wenzelm@23164  152 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  153 lemma posDivAlg_eqn:  wenzelm@23164  154  "0 < b ==>  wenzelm@23164  155  posDivAlg a b = (if a a" and "0 < b"  haftmann@29651  161  shows "divmod_rel a b (posDivAlg a b)"  wenzelm@23164  162 using prems apply (induct a b rule: posDivAlg.induct)  wenzelm@23164  163 apply auto  haftmann@29651  164 apply (simp add: divmod_rel_def)  wenzelm@23164  165 apply (subst posDivAlg_eqn, simp add: right_distrib)  wenzelm@23164  166 apply (case_tac "a < b")  wenzelm@23164  167 apply simp_all  wenzelm@23164  168 apply (erule splitE)  wenzelm@23164  169 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  170 done  wenzelm@23164  171 wenzelm@23164  172 wenzelm@23164  173 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}  wenzelm@23164  174 wenzelm@23164  175 text{*And positive divisors*}  wenzelm@23164  176 wenzelm@23164  177 declare negDivAlg.simps [simp del]  wenzelm@23164  178 wenzelm@23164  179 text{*use with a simproc to avoid repeatedly proving the premise*}  wenzelm@23164  180 lemma negDivAlg_eqn:  wenzelm@23164  181  "0 < b ==>  wenzelm@23164  182  negDivAlg a b =  wenzelm@23164  183  (if 0\a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"  wenzelm@23164  184 by (rule negDivAlg.simps [THEN trans], simp)  wenzelm@23164  185 wenzelm@23164  186 (*Correctness of negDivAlg: it computes quotients correctly  wenzelm@23164  187  It doesn't work if a=0 because the 0/b equals 0, not -1*)  wenzelm@23164  188 lemma negDivAlg_correct:  wenzelm@23164  189  assumes "a < 0" and "b > 0"  haftmann@29651  190  shows "divmod_rel a b (negDivAlg a b)"  wenzelm@23164  191 using prems apply (induct a b rule: negDivAlg.induct)  wenzelm@23164  192 apply (auto simp add: linorder_not_le)  haftmann@29651  193 apply (simp add: divmod_rel_def)  wenzelm@23164  194 apply (subst negDivAlg_eqn, assumption)  wenzelm@23164  195 apply (case_tac "a + b < (0\int)")  wenzelm@23164  196 apply simp_all  wenzelm@23164  197 apply (erule splitE)  wenzelm@23164  198 apply (auto simp add: right_distrib Let_def)  wenzelm@23164  199 done  wenzelm@23164  200 wenzelm@23164  201 wenzelm@23164  202 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}  wenzelm@23164  203 wenzelm@23164  204 (*the case a=0*)  haftmann@29651  205 lemma divmod_rel_0: "b \ 0 ==> divmod_rel 0 b (0, 0)"  haftmann@29651  206 by (auto simp add: divmod_rel_def linorder_neq_iff)  wenzelm@23164  207 wenzelm@23164  208 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"  wenzelm@23164  209 by (subst posDivAlg.simps, auto)  wenzelm@23164  210 wenzelm@23164  211 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"  wenzelm@23164  212 by (subst negDivAlg.simps, auto)  wenzelm@23164  213 wenzelm@23164  214 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"  wenzelm@23164  215 by (simp add: negateSnd_def)  wenzelm@23164  216 haftmann@29651  217 lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"  haftmann@29651  218 by (auto simp add: split_ifs divmod_rel_def)  wenzelm@23164  219 haftmann@29651  220 lemma divmod_correct: "b \ 0 ==> divmod_rel a b (divmod a b)"  haftmann@29651  221 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg  wenzelm@23164  222  posDivAlg_correct negDivAlg_correct)  wenzelm@23164  223 wenzelm@23164  224 text{*Arbitrary definitions for division by zero. Useful to simplify  wenzelm@23164  225  certain equations.*}  wenzelm@23164  226 wenzelm@23164  227 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"  haftmann@29651  228 by (simp add: div_def mod_def divmod_def posDivAlg.simps)  wenzelm@23164  229 wenzelm@23164  230 wenzelm@23164  231 text{*Basic laws about division and remainder*}  wenzelm@23164  232 wenzelm@23164  233 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  wenzelm@23164  234 apply (case_tac "b = 0", simp)  haftmann@29651  235 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  236 apply (auto simp add: divmod_rel_def div_def mod_def)  wenzelm@23164  237 done  wenzelm@23164  238 wenzelm@23164  239 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"  wenzelm@23164  240 by(simp add: zmod_zdiv_equality[symmetric])  wenzelm@23164  241 wenzelm@23164  242 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"  wenzelm@23164  243 by(simp add: mult_commute zmod_zdiv_equality[symmetric])  wenzelm@23164  244 wenzelm@23164  245 text {* Tool setup *}  wenzelm@23164  246 wenzelm@26480  247 ML {*  wenzelm@23164  248 local  wenzelm@23164  249 wenzelm@23164  250 structure CancelDivMod = CancelDivModFun(  wenzelm@23164  251 struct  wenzelm@23164  252  val div_name = @{const_name Divides.div};  wenzelm@23164  253  val mod_name = @{const_name Divides.mod};  wenzelm@23164  254  val mk_binop = HOLogic.mk_binop;  wenzelm@23164  255  val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;  wenzelm@23164  256  val dest_sum = Int_Numeral_Simprocs.dest_sum;  wenzelm@23164  257  val div_mod_eqs =  wenzelm@23164  258  map mk_meta_eq [@{thm zdiv_zmod_equality},  wenzelm@23164  259  @{thm zdiv_zmod_equality2}];  wenzelm@23164  260  val trans = trans;  wenzelm@23164  261  val prove_eq_sums =  wenzelm@23164  262  let  huffman@23365  263  val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}  haftmann@26101  264  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;  wenzelm@23164  265 end)  wenzelm@23164  266 wenzelm@23164  267 in  wenzelm@23164  268 wenzelm@28262  269 val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())  haftmann@26101  270  "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)  wenzelm@23164  271 wenzelm@23164  272 end;  wenzelm@23164  273 wenzelm@23164  274 Addsimprocs [cancel_zdiv_zmod_proc]  wenzelm@23164  275 *}  wenzelm@23164  276 wenzelm@23164  277 lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b"  haftmann@29651  278 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  279 apply (auto simp add: divmod_rel_def mod_def)  wenzelm@23164  280 done  wenzelm@23164  281 wenzelm@23164  282 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]  wenzelm@23164  283  and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]  wenzelm@23164  284 wenzelm@23164  285 lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b"  haftmann@29651  286 apply (cut_tac a = a and b = b in divmod_correct)  haftmann@29651  287 apply (auto simp add: divmod_rel_def div_def mod_def)  wenzelm@23164  288 done  wenzelm@23164  289 wenzelm@23164  290 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]  wenzelm@23164  291  and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]  wenzelm@23164  292 wenzelm@23164  293 wenzelm@23164  294 wenzelm@23164  295 subsection{*General Properties of div and mod*}  wenzelm@23164  296 haftmann@29651  297 lemma divmod_rel_div_mod: "b \ 0 ==> divmod_rel a b (a div b, a mod b)"  wenzelm@23164  298 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  haftmann@29651  299 apply (force simp add: divmod_rel_def linorder_neq_iff)  wenzelm@23164  300 done  wenzelm@23164  301 haftmann@29651  302 lemma divmod_rel_div: "[| divmod_rel a b (q, r); b \ 0 |] ==> a div b = q"  haftmann@29651  303 by (simp add: divmod_rel_div_mod [THEN unique_quotient])  wenzelm@23164  304 haftmann@29651  305 lemma divmod_rel_mod: "[| divmod_rel a b (q, r); b \ 0 |] ==> a mod b = r"  haftmann@29651  306 by (simp add: divmod_rel_div_mod [THEN unique_remainder])  wenzelm@23164  307 wenzelm@23164  308 lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0"  haftmann@29651  309 apply (rule divmod_rel_div)  haftmann@29651  310 apply (auto simp add: divmod_rel_def)  wenzelm@23164  311 done  wenzelm@23164  312 wenzelm@23164  313 lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0"  haftmann@29651  314 apply (rule divmod_rel_div)  haftmann@29651  315 apply (auto simp add: divmod_rel_def)  wenzelm@23164  316 done  wenzelm@23164  317 wenzelm@23164  318 lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1"  haftmann@29651  319 apply (rule divmod_rel_div)  haftmann@29651  320 apply (auto simp add: divmod_rel_def)  wenzelm@23164  321 done  wenzelm@23164  322 wenzelm@23164  323 (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)  wenzelm@23164  324 wenzelm@23164  325 lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a"  haftmann@29651  326 apply (rule_tac q = 0 in divmod_rel_mod)  haftmann@29651  327 apply (auto simp add: divmod_rel_def)  wenzelm@23164  328 done  wenzelm@23164  329 wenzelm@23164  330 lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a"  haftmann@29651  331 apply (rule_tac q = 0 in divmod_rel_mod)  haftmann@29651  332 apply (auto simp add: divmod_rel_def)  wenzelm@23164  333 done  wenzelm@23164  334 wenzelm@23164  335 lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b"  haftmann@29651  336 apply (rule_tac q = "-1" in divmod_rel_mod)  haftmann@29651  337 apply (auto simp add: divmod_rel_def)  wenzelm@23164  338 done  wenzelm@23164  339 wenzelm@23164  340 text{*There is no @{text mod_neg_pos_trivial}.*}  wenzelm@23164  341 wenzelm@23164  342 wenzelm@23164  343 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)  wenzelm@23164  344 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"  wenzelm@23164  345 apply (case_tac "b = 0", simp)  haftmann@29651  346 apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified,  haftmann@29651  347  THEN divmod_rel_div, THEN sym])  wenzelm@23164  348 wenzelm@23164  349 done  wenzelm@23164  350 wenzelm@23164  351 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)  wenzelm@23164  352 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"  wenzelm@23164  353 apply (case_tac "b = 0", simp)  haftmann@29651  354 apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],  wenzelm@23164  355  auto)  wenzelm@23164  356 done  wenzelm@23164  357 wenzelm@23164  358 wenzelm@23164  359 subsection{*Laws for div and mod with Unary Minus*}  wenzelm@23164  360 wenzelm@23164  361 lemma zminus1_lemma:  haftmann@29651  362  "divmod_rel a b (q, r)  haftmann@29651  363  ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,  haftmann@29651  364  if r=0 then 0 else b-r)"  haftmann@29651  365 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)  wenzelm@23164  366 wenzelm@23164  367 wenzelm@23164  368 lemma zdiv_zminus1_eq_if:  wenzelm@23164  369  "b \ (0::int)  wenzelm@23164  370  ==> (-a) div b =  wenzelm@23164  371  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  haftmann@29651  372 by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])  wenzelm@23164  373 wenzelm@23164  374 lemma zmod_zminus1_eq_if:  wenzelm@23164  375  "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"  wenzelm@23164  376 apply (case_tac "b = 0", simp)  haftmann@29651  377 apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])  wenzelm@23164  378 done  wenzelm@23164  379 haftmann@29936  380 lemma zmod_zminus1_not_zero:  haftmann@29936  381  fixes k l :: int  haftmann@29936  382  shows "- k mod l \ 0 \ k mod l \ 0"  haftmann@29936  383  unfolding zmod_zminus1_eq_if by auto  haftmann@29936  384 wenzelm@23164  385 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  wenzelm@23164  386 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  wenzelm@23164  387 wenzelm@23164  388 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  wenzelm@23164  389 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  wenzelm@23164  390 wenzelm@23164  391 lemma zdiv_zminus2_eq_if:  wenzelm@23164  392  "b \ (0::int)  wenzelm@23164  393  ==> a div (-b) =  wenzelm@23164  394  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  395 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  wenzelm@23164  396 wenzelm@23164  397 lemma zmod_zminus2_eq_if:  wenzelm@23164  398  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  wenzelm@23164  399 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  wenzelm@23164  400 haftmann@29936  401 lemma zmod_zminus2_not_zero:  haftmann@29936  402  fixes k l :: int  haftmann@29936  403  shows "k mod - l \ 0 \ k mod l \ 0"  haftmann@29936  404  unfolding zmod_zminus2_eq_if by auto  haftmann@29936  405 wenzelm@23164  406 wenzelm@23164  407 subsection{*Division of a Number by Itself*}  wenzelm@23164  408 wenzelm@23164  409 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  wenzelm@23164  410 apply (subgoal_tac "0 < a*q")  wenzelm@23164  411  apply (simp add: zero_less_mult_iff, arith)  wenzelm@23164  412 done  wenzelm@23164  413 wenzelm@23164  414 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  wenzelm@23164  415 apply (subgoal_tac "0 \ a* (1-q) ")  wenzelm@23164  416  apply (simp add: zero_le_mult_iff)  wenzelm@23164  417 apply (simp add: right_diff_distrib)  wenzelm@23164  418 done  wenzelm@23164  419 haftmann@29651  420 lemma self_quotient: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> q = 1"  haftmann@29651  421 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)  wenzelm@23164  422 apply (rule order_antisym, safe, simp_all)  wenzelm@23164  423 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  wenzelm@23164  424 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  wenzelm@23164  425 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  wenzelm@23164  426 done  wenzelm@23164  427 haftmann@29651  428 lemma self_remainder: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> r = 0"  wenzelm@23164  429 apply (frule self_quotient, assumption)  haftmann@29651  430 apply (simp add: divmod_rel_def)  wenzelm@23164  431 done  wenzelm@23164  432 wenzelm@23164  433 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  haftmann@29651  434 by (simp add: divmod_rel_div_mod [THEN self_quotient])  wenzelm@23164  435 wenzelm@23164  436 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  wenzelm@23164  437 lemma zmod_self [simp]: "a mod a = (0::int)"  wenzelm@23164  438 apply (case_tac "a = 0", simp)  haftmann@29651  439 apply (simp add: divmod_rel_div_mod [THEN self_remainder])  wenzelm@23164  440 done  wenzelm@23164  441 wenzelm@23164  442 wenzelm@23164  443 subsection{*Computation of Division and Remainder*}  wenzelm@23164  444 wenzelm@23164  445 lemma zdiv_zero [simp]: "(0::int) div b = 0"  haftmann@29651  446 by (simp add: div_def divmod_def)  wenzelm@23164  447 wenzelm@23164  448 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@29651  449 by (simp add: div_def divmod_def)  wenzelm@23164  450 wenzelm@23164  451 lemma zmod_zero [simp]: "(0::int) mod b = 0"  haftmann@29651  452 by (simp add: mod_def divmod_def)  wenzelm@23164  453 wenzelm@23164  454 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@29651  455 by (simp add: mod_def divmod_def)  wenzelm@23164  456 wenzelm@23164  457 text{*a positive, b positive *}  wenzelm@23164  458 wenzelm@23164  459 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@29651  460 by (simp add: div_def divmod_def)  wenzelm@23164  461 wenzelm@23164  462 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@29651  463 by (simp add: mod_def divmod_def)  wenzelm@23164  464 wenzelm@23164  465 text{*a negative, b positive *}  wenzelm@23164  466 wenzelm@23164  467 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@29651  468 by (simp add: div_def divmod_def)  wenzelm@23164  469 wenzelm@23164  470 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@29651  471 by (simp add: mod_def divmod_def)  wenzelm@23164  472 wenzelm@23164  473 text{*a positive, b negative *}  wenzelm@23164  474 wenzelm@23164  475 lemma div_pos_neg:  wenzelm@23164  476  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  477 by (simp add: div_def divmod_def)  wenzelm@23164  478 wenzelm@23164  479 lemma mod_pos_neg:  wenzelm@23164  480  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  481 by (simp add: mod_def divmod_def)  wenzelm@23164  482 wenzelm@23164  483 text{*a negative, b negative *}  wenzelm@23164  484 wenzelm@23164  485 lemma div_neg_neg:  wenzelm@23164  486  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  487 by (simp add: div_def divmod_def)  wenzelm@23164  488 wenzelm@23164  489 lemma mod_neg_neg:  wenzelm@23164  490  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  491 by (simp add: mod_def divmod_def)  wenzelm@23164  492 wenzelm@23164  493 text {*Simplify expresions in which div and mod combine numerical constants*}  wenzelm@23164  494 haftmann@29651  495 lemma divmod_relI:  huffman@24481  496  "\a == b * q + r; if 0 < b then 0 \ r \ r < b else b < r \ r \ 0\  haftmann@29651  497  \ divmod_rel a b (q, r)"  haftmann@29651  498  unfolding divmod_rel_def by simp  huffman@24481  499 haftmann@29651  500 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]  haftmann@29651  501 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]  huffman@24481  502 lemmas arithmetic_simps =  huffman@24481  503  arith_simps  huffman@24481  504  add_special  huffman@24481  505  OrderedGroup.add_0_left  huffman@24481  506  OrderedGroup.add_0_right  huffman@24481  507  mult_zero_left  huffman@24481  508  mult_zero_right  huffman@24481  509  mult_1_left  huffman@24481  510  mult_1_right  huffman@24481  511 huffman@24481  512 (* simprocs adapted from HOL/ex/Binary.thy *)  huffman@24481  513 ML {*  huffman@24481  514 local  huffman@24481  515  infix ==;  huffman@24481  516  val op == = Logic.mk_equals;  huffman@24481  517  fun plus m n = @{term "plus :: int \ int \ int"}  m  n;  huffman@24481  518  fun mult m n = @{term "times :: int \ int \ int"}  m  n;  huffman@24481  519 huffman@24481  520  val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};  huffman@24481  521  fun prove ctxt prop =  huffman@24481  522  Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));  huffman@24481  523 huffman@24481  524  fun binary_proc proc ss ct =  huffman@24481  525  (case Thm.term_of ct of  huffman@24481  526  _  t  u =>  huffman@24481  527  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  huffman@24481  528  SOME args => proc (Simplifier.the_context ss) args  huffman@24481  529  | NONE => NONE)  huffman@24481  530  | _ => NONE);  huffman@24481  531 in  huffman@24481  532 huffman@24481  533 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>  huffman@24481  534  if n = 0 then NONE  huffman@24481  535  else  wenzelm@24630  536  let val (k, l) = Integer.div_mod m n;  huffman@24481  537  fun mk_num x = HOLogic.mk_number HOLogic.intT x;  huffman@24481  538  in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])  huffman@24481  539  end);  huffman@24481  540 huffman@24481  541 end;  huffman@24481  542 *}  huffman@24481  543 huffman@24481  544 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  haftmann@29651  545  {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}  huffman@24481  546 huffman@24481  547 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  haftmann@29651  548  {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}  huffman@24481  549 huffman@24481  550 (* The following 8 lemmas are made unnecessary by the above simprocs: *)  huffman@24481  551 huffman@24481  552 lemmas div_pos_pos_number_of =  wenzelm@23164  553  div_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  554 huffman@24481  555 lemmas div_neg_pos_number_of =  wenzelm@23164  556  div_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  557 huffman@24481  558 lemmas div_pos_neg_number_of =  wenzelm@23164  559  div_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  560 huffman@24481  561 lemmas div_neg_neg_number_of =  wenzelm@23164  562  div_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  563 wenzelm@23164  564 huffman@24481  565 lemmas mod_pos_pos_number_of =  wenzelm@23164  566  mod_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  567 huffman@24481  568 lemmas mod_neg_pos_number_of =  wenzelm@23164  569  mod_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  570 huffman@24481  571 lemmas mod_pos_neg_number_of =  wenzelm@23164  572  mod_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  573 huffman@24481  574 lemmas mod_neg_neg_number_of =  wenzelm@23164  575  mod_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  576 wenzelm@23164  577 wenzelm@23164  578 lemmas posDivAlg_eqn_number_of [simp] =  wenzelm@23164  579  posDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  580 wenzelm@23164  581 lemmas negDivAlg_eqn_number_of [simp] =  wenzelm@23164  582  negDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  583 wenzelm@23164  584 wenzelm@23164  585 text{*Special-case simplification *}  wenzelm@23164  586 wenzelm@23164  587 lemma zmod_1 [simp]: "a mod (1::int) = 0"  wenzelm@23164  588 apply (cut_tac a = a and b = 1 in pos_mod_sign)  wenzelm@23164  589 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)  wenzelm@23164  590 apply (auto simp del:pos_mod_bound pos_mod_sign)  wenzelm@23164  591 done  wenzelm@23164  592 wenzelm@23164  593 lemma zdiv_1 [simp]: "a div (1::int) = a"  wenzelm@23164  594 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)  wenzelm@23164  595 wenzelm@23164  596 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  wenzelm@23164  597 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  wenzelm@23164  598 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  wenzelm@23164  599 apply (auto simp del: neg_mod_sign neg_mod_bound)  wenzelm@23164  600 done  wenzelm@23164  601 wenzelm@23164  602 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"  wenzelm@23164  603 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)  wenzelm@23164  604 wenzelm@23164  605 (** The last remaining special cases for constant arithmetic:  wenzelm@23164  606  1 div z and 1 mod z **)  wenzelm@23164  607 wenzelm@23164  608 lemmas div_pos_pos_1_number_of [simp] =  wenzelm@23164  609  div_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  610 wenzelm@23164  611 lemmas div_pos_neg_1_number_of [simp] =  wenzelm@23164  612  div_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  613 wenzelm@23164  614 lemmas mod_pos_pos_1_number_of [simp] =  wenzelm@23164  615  mod_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  616 wenzelm@23164  617 lemmas mod_pos_neg_1_number_of [simp] =  wenzelm@23164  618  mod_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  619 wenzelm@23164  620 wenzelm@23164  621 lemmas posDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  622  posDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  623 wenzelm@23164  624 lemmas negDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  625  negDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  626 wenzelm@23164  627 wenzelm@23164  628 wenzelm@23164  629 subsection{*Monotonicity in the First Argument (Dividend)*}  wenzelm@23164  630 wenzelm@23164  631 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  wenzelm@23164  632 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  633 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  634 apply (rule unique_quotient_lemma)  wenzelm@23164  635 apply (erule subst)  wenzelm@23164  636 apply (erule subst, simp_all)  wenzelm@23164  637 done  wenzelm@23164  638 wenzelm@23164  639 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  wenzelm@23164  640 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  641 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  642 apply (rule unique_quotient_lemma_neg)  wenzelm@23164  643 apply (erule subst)  wenzelm@23164  644 apply (erule subst, simp_all)  wenzelm@23164  645 done  wenzelm@23164  646 wenzelm@23164  647 wenzelm@23164  648 subsection{*Monotonicity in the Second Argument (Divisor)*}  wenzelm@23164  649 wenzelm@23164  650 lemma q_pos_lemma:  wenzelm@23164  651  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  wenzelm@23164  652 apply (subgoal_tac "0 < b'* (q' + 1) ")  wenzelm@23164  653  apply (simp add: zero_less_mult_iff)  wenzelm@23164  654 apply (simp add: right_distrib)  wenzelm@23164  655 done  wenzelm@23164  656 wenzelm@23164  657 lemma zdiv_mono2_lemma:  wenzelm@23164  658  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  wenzelm@23164  659  r' < b'; 0 \ r; 0 < b'; b' \ b |]  wenzelm@23164  660  ==> q \ (q'::int)"  wenzelm@23164  661 apply (frule q_pos_lemma, assumption+)  wenzelm@23164  662 apply (subgoal_tac "b*q < b* (q' + 1) ")  wenzelm@23164  663  apply (simp add: mult_less_cancel_left)  wenzelm@23164  664 apply (subgoal_tac "b*q = r' - r + b'*q'")  wenzelm@23164  665  prefer 2 apply simp  wenzelm@23164  666 apply (simp (no_asm_simp) add: right_distrib)  wenzelm@23164  667 apply (subst add_commute, rule zadd_zless_mono, arith)  wenzelm@23164  668 apply (rule mult_right_mono, auto)  wenzelm@23164  669 done  wenzelm@23164  670 wenzelm@23164  671 lemma zdiv_mono2:  wenzelm@23164  672  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  wenzelm@23164  673 apply (subgoal_tac "b \ 0")  wenzelm@23164  674  prefer 2 apply arith  wenzelm@23164  675 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  676 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  677 apply (rule zdiv_mono2_lemma)  wenzelm@23164  678 apply (erule subst)  wenzelm@23164  679 apply (erule subst, simp_all)  wenzelm@23164  680 done  wenzelm@23164  681 wenzelm@23164  682 lemma q_neg_lemma:  wenzelm@23164  683  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  wenzelm@23164  684 apply (subgoal_tac "b'*q' < 0")  wenzelm@23164  685  apply (simp add: mult_less_0_iff, arith)  wenzelm@23164  686 done  wenzelm@23164  687 wenzelm@23164  688 lemma zdiv_mono2_neg_lemma:  wenzelm@23164  689  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  wenzelm@23164  690  r < b; 0 \ r'; 0 < b'; b' \ b |]  wenzelm@23164  691  ==> q' \ (q::int)"  wenzelm@23164  692 apply (frule q_neg_lemma, assumption+)  wenzelm@23164  693 apply (subgoal_tac "b*q' < b* (q + 1) ")  wenzelm@23164  694  apply (simp add: mult_less_cancel_left)  wenzelm@23164  695 apply (simp add: right_distrib)  wenzelm@23164  696 apply (subgoal_tac "b*q' \ b'*q'")  wenzelm@23164  697  prefer 2 apply (simp add: mult_right_mono_neg, arith)  wenzelm@23164  698 done  wenzelm@23164  699 wenzelm@23164  700 lemma zdiv_mono2_neg:  wenzelm@23164  701  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  wenzelm@23164  702 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  703 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  704 apply (rule zdiv_mono2_neg_lemma)  wenzelm@23164  705 apply (erule subst)  wenzelm@23164  706 apply (erule subst, simp_all)  wenzelm@23164  707 done  wenzelm@23164  708 haftmann@25942  709 wenzelm@23164  710 subsection{*More Algebraic Laws for div and mod*}  wenzelm@23164  711 wenzelm@23164  712 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  wenzelm@23164  713 wenzelm@23164  714 lemma zmult1_lemma:  haftmann@29651  715  "[| divmod_rel b c (q, r); c \ 0 |]  haftmann@29651  716  ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@29651  717 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  wenzelm@23164  718 wenzelm@23164  719 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  wenzelm@23164  720 apply (case_tac "c = 0", simp)  haftmann@29651  721 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])  wenzelm@23164  722 done  wenzelm@23164  723 wenzelm@23164  724 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  wenzelm@23164  725 apply (case_tac "c = 0", simp)  haftmann@29651  726 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])  wenzelm@23164  727 done  wenzelm@23164  728 wenzelm@23164  729 lemma zdiv_zmult_self1 [simp]: "b \ (0::int) ==> (a*b) div b = a"  wenzelm@23164  730 by (simp add: zdiv_zmult1_eq)  wenzelm@23164  731 huffman@29403  732 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"  haftmann@27651  733 apply (case_tac "b = 0", simp)  haftmann@27651  734 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  haftmann@27651  735 done  haftmann@27651  736 haftmann@27651  737 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@27651  738 haftmann@27651  739 lemma zadd1_lemma:  haftmann@29651  740  "[| divmod_rel a c (aq, ar); divmod_rel b c (bq, br); c \ 0 |]  haftmann@29651  741  ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  haftmann@29651  742 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  haftmann@27651  743 haftmann@27651  744 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@27651  745 lemma zdiv_zadd1_eq:  haftmann@27651  746  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@27651  747 apply (case_tac "c = 0", simp)  haftmann@29651  748 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)  haftmann@27651  749 done  haftmann@27651  750 huffman@29405  751 instance int :: ring_div  haftmann@27651  752 proof  haftmann@27651  753  fix a b c :: int  haftmann@27651  754  assume not0: "b \ 0"  haftmann@27651  755  show "(a + c * b) div b = c + a div b"  haftmann@27651  756  unfolding zdiv_zadd1_eq [of a "c * b"] using not0  huffman@29403  757  by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)  haftmann@27651  758 qed auto  haftmann@25942  759 haftmann@29651  760 lemma posDivAlg_div_mod:  haftmann@29651  761  assumes "k \ 0"  haftmann@29651  762  and "l \ 0"  haftmann@29651  763  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@29651  764 proof (cases "l = 0")  haftmann@29651  765  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@29651  766 next  haftmann@29651  767  case False with assms posDivAlg_correct  haftmann@29651  768  have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@29651  769  by simp  haftmann@29651  770  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  771  show ?thesis by simp  haftmann@29651  772 qed  haftmann@29651  773 haftmann@29651  774 lemma negDivAlg_div_mod:  haftmann@29651  775  assumes "k < 0"  haftmann@29651  776  and "l > 0"  haftmann@29651  777  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@29651  778 proof -  haftmann@29651  779  from assms have "l \ 0" by simp  haftmann@29651  780  from assms negDivAlg_correct  haftmann@29651  781  have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@29651  782  by simp  haftmann@29651  783  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  784  show ?thesis by simp  haftmann@29651  785 qed  haftmann@29651  786 huffman@29403  787 lemma zdiv_zadd_self1: "a \ (0::int) ==> (a+b) div a = b div a + 1"  huffman@29403  788 by (rule div_add_self1) (* already declared [simp] *)  huffman@29403  789 huffman@29403  790 lemma zdiv_zadd_self2: "a \ (0::int) ==> (b+a) div a = b div a + 1"  huffman@29403  791 by (rule div_add_self2) (* already declared [simp] *)  wenzelm@23164  792 huffman@29403  793 lemma zdiv_zmult_self2: "b \ (0::int) ==> (b*a) div b = a"  huffman@29403  794 by (rule div_mult_self1_is_id) (* already declared [simp] *)  wenzelm@23164  795 huffman@29403  796 lemma zmod_zmult_self1: "(a*b) mod b = (0::int)"  huffman@29403  797 by (rule mod_mult_self2_is_0) (* already declared [simp] *)  huffman@29403  798 huffman@29403  799 lemma zmod_zmult_self2: "(b*a) mod b = (0::int)"  huffman@29403  800 by (rule mod_mult_self1_is_0) (* already declared [simp] *)  wenzelm@23164  801 wenzelm@23164  802 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  huffman@29403  803 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  wenzelm@23164  804 huffman@29403  805 (* REVISIT: should this be generalized to all semiring_div types? *)  wenzelm@23164  806 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  807 wenzelm@23164  808 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"  huffman@29403  809 by (rule mod_add_left_eq)  wenzelm@23164  810 wenzelm@23164  811 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"  huffman@29403  812 by (rule mod_add_right_eq)  wenzelm@23164  813 huffman@29403  814 lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)"  huffman@29403  815 by (rule mod_add_self1) (* already declared [simp] *)  wenzelm@23164  816 huffman@29403  817 lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)"  huffman@29403  818 by (rule mod_add_self2) (* already declared [simp] *)  wenzelm@23164  819 huffman@29405  820 lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)"  huffman@29405  821 by (rule mod_diff_eq)  nipkow@23983  822 wenzelm@23164  823 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  824 wenzelm@23164  825 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  826  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  827  to cause particular problems.*)  wenzelm@23164  828 wenzelm@23164  829 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  830 wenzelm@23164  831 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  832 apply (subgoal_tac "b * (c - q mod c) < r * 1")  nipkow@29667  833  apply (simp add: algebra_simps)  wenzelm@23164  834 apply (rule order_le_less_trans)  nipkow@29667  835  apply (erule_tac [2] mult_strict_right_mono)  nipkow@29667  836  apply (rule mult_left_mono_neg)  nipkow@29667  837  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)  nipkow@29667  838  apply (simp)  nipkow@29667  839 apply (simp)  wenzelm@23164  840 done  wenzelm@23164  841 wenzelm@23164  842 lemma zmult2_lemma_aux2:  wenzelm@23164  843  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  844 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  845  apply arith  wenzelm@23164  846 apply (simp add: mult_le_0_iff)  wenzelm@23164  847 done  wenzelm@23164  848 wenzelm@23164  849 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  850 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  851 apply arith  wenzelm@23164  852 apply (simp add: zero_le_mult_iff)  wenzelm@23164  853 done  wenzelm@23164  854 wenzelm@23164  855 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  856 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  nipkow@29667  857  apply (simp add: right_diff_distrib)  wenzelm@23164  858 apply (rule order_less_le_trans)  nipkow@29667  859  apply (erule mult_strict_right_mono)  nipkow@29667  860  apply (rule_tac [2] mult_left_mono)  nipkow@29667  861  apply simp  nipkow@29667  862  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)  nipkow@29667  863 apply simp  wenzelm@23164  864 done  wenzelm@23164  865 haftmann@29651  866 lemma zmult2_lemma: "[| divmod_rel a b (q, r); b \ 0; 0 < c |]  haftmann@29651  867  ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@29651  868 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff  wenzelm@23164  869  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  870  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  871 wenzelm@23164  872 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  873 apply (case_tac "b = 0", simp)  haftmann@29651  874 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])  wenzelm@23164  875 done  wenzelm@23164  876 wenzelm@23164  877 lemma zmod_zmult2_eq:  wenzelm@23164  878  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  879 apply (case_tac "b = 0", simp)  haftmann@29651  880 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])  wenzelm@23164  881 done  wenzelm@23164  882 wenzelm@23164  883 wenzelm@23164  884 subsection{*Cancellation of Common Factors in div*}  wenzelm@23164  885 wenzelm@23164  886 lemma zdiv_zmult_zmult1_aux1:  wenzelm@23164  887  "[| (0::int) < b; c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  888 by (subst zdiv_zmult2_eq, auto)  wenzelm@23164  889 wenzelm@23164  890 lemma zdiv_zmult_zmult1_aux2:  wenzelm@23164  891  "[| b < (0::int); c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  892 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")  wenzelm@23164  893 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)  wenzelm@23164  894 done  wenzelm@23164  895 wenzelm@23164  896 lemma zdiv_zmult_zmult1: "c \ (0::int) ==> (c*a) div (c*b) = a div b"  wenzelm@23164  897 apply (case_tac "b = 0", simp)  wenzelm@23164  898 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)  wenzelm@23164  899 done  wenzelm@23164  900 nipkow@23401  901 lemma zdiv_zmult_zmult1_if[simp]:  nipkow@23401  902  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"  nipkow@23401  903 by (simp add:zdiv_zmult_zmult1)  nipkow@23401  904 nipkow@23401  905 (*  wenzelm@23164  906 lemma zdiv_zmult_zmult2: "c \ (0::int) ==> (a*c) div (b*c) = a div b"  wenzelm@23164  907 apply (drule zdiv_zmult_zmult1)  wenzelm@23164  908 apply (auto simp add: mult_commute)  wenzelm@23164  909 done  nipkow@23401  910 *)  wenzelm@23164  911 wenzelm@23164  912 wenzelm@23164  913 subsection{*Distribution of Factors over mod*}  wenzelm@23164  914 wenzelm@23164  915 lemma zmod_zmult_zmult1_aux1:  wenzelm@23164  916  "[| (0::int) < b; c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  917 by (subst zmod_zmult2_eq, auto)  wenzelm@23164  918 wenzelm@23164  919 lemma zmod_zmult_zmult1_aux2:  wenzelm@23164  920  "[| b < (0::int); c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  921 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")  wenzelm@23164  922 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)  wenzelm@23164  923 done  wenzelm@23164  924 wenzelm@23164  925 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"  wenzelm@23164  926 apply (case_tac "b = 0", simp)  wenzelm@23164  927 apply (case_tac "c = 0", simp)  wenzelm@23164  928 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)  wenzelm@23164  929 done  wenzelm@23164  930 wenzelm@23164  931 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"  wenzelm@23164  932 apply (cut_tac c = c in zmod_zmult_zmult1)  wenzelm@23164  933 apply (auto simp add: mult_commute)  wenzelm@23164  934 done  wenzelm@23164  935 huffman@29404  936 lemma zmod_zmod_cancel: "n dvd m \ (k::int) mod m mod n = k mod n"  huffman@29404  937 by (rule mod_mod_cancel)  nipkow@24490  938 wenzelm@23164  939 wenzelm@23164  940 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  941 wenzelm@23164  942 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  943 wenzelm@23164  944 lemma split_pos_lemma:  wenzelm@23164  945  "0  wenzelm@23164  946  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  947 apply (rule iffI, clarify)  wenzelm@23164  948  apply (erule_tac P="P ?x ?y" in rev_mp)  nipkow@29948  949  apply (subst mod_add_eq)  wenzelm@23164  950  apply (subst zdiv_zadd1_eq)  wenzelm@23164  951  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  wenzelm@23164  952 txt{*converse direction*}  wenzelm@23164  953 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  954 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  955 done  wenzelm@23164  956 wenzelm@23164  957 lemma split_neg_lemma:  wenzelm@23164  958  "k<0 ==>  wenzelm@23164  959  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  960 apply (rule iffI, clarify)  wenzelm@23164  961  apply (erule_tac P="P ?x ?y" in rev_mp)  nipkow@29948  962  apply (subst mod_add_eq)  wenzelm@23164  963  apply (subst zdiv_zadd1_eq)  wenzelm@23164  964  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  965 txt{*converse direction*}  wenzelm@23164  966 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  967 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  968 done  wenzelm@23164  969 wenzelm@23164  970 lemma split_zdiv:  wenzelm@23164  971  "P(n div k :: int) =  wenzelm@23164  972  ((k = 0 --> P 0) &  wenzelm@23164  973  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  974  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  975 apply (case_tac "k=0", simp)  wenzelm@23164  976 apply (simp only: linorder_neq_iff)  wenzelm@23164  977 apply (erule disjE)  wenzelm@23164  978  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  979  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  980 done  wenzelm@23164  981 wenzelm@23164  982 lemma split_zmod:  wenzelm@23164  983  "P(n mod k :: int) =  wenzelm@23164  984  ((k = 0 --> P n) &  wenzelm@23164  985  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  986  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  987 apply (case_tac "k=0", simp)  wenzelm@23164  988 apply (simp only: linorder_neq_iff)  wenzelm@23164  989 apply (erule disjE)  wenzelm@23164  990  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  991  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  992 done  wenzelm@23164  993 wenzelm@23164  994 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  995 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  996 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  997 wenzelm@23164  998 wenzelm@23164  999 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  1000 wenzelm@23164  1001 text{*computing div by shifting *}  wenzelm@23164  1002 wenzelm@23164  1003 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  1004 proof cases  wenzelm@23164  1005  assume "a=0"  wenzelm@23164  1006  thus ?thesis by simp  wenzelm@23164  1007 next  wenzelm@23164  1008  assume "a\0" and le_a: "0\a"  wenzelm@23164  1009  hence a_pos: "1 \ a" by arith  wenzelm@23164  1010  hence one_less_a2: "1 < 2*a" by arith  wenzelm@23164  1011  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  wenzelm@23164  1012  by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)  wenzelm@23164  1013  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  1014  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  1015  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  1016  with le_2a  wenzelm@23164  1017  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  1018  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  1019  right_distrib)  wenzelm@23164  1020  thus ?thesis  wenzelm@23164  1021  by (subst zdiv_zadd1_eq,  wenzelm@23164  1022  simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2  wenzelm@23164  1023  div_pos_pos_trivial)  wenzelm@23164  1024 qed  wenzelm@23164  1025 wenzelm@23164  1026 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  1027 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  1028 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  1029 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1030 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1031 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  1032  simp)  wenzelm@23164  1033 done  wenzelm@23164  1034 huffman@26086  1035 lemma zdiv_number_of_Bit0 [simp]:  huffman@26086  1036  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  huffman@26086  1037  number_of v div (number_of w :: int)"  huffman@26086  1038 by (simp only: number_of_eq numeral_simps) simp  huffman@26086  1039 huffman@26086  1040 lemma zdiv_number_of_Bit1 [simp]:  huffman@26086  1041  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  huffman@26086  1042  (if (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)  huffman@26086  1046 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)  wenzelm@23164  1047 done  wenzelm@23164  1048 wenzelm@23164  1049 wenzelm@23164  1050 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  1051 wenzelm@23164  1052 lemma pos_zmod_mult_2:  wenzelm@23164  1053  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  1054 apply (case_tac "a = 0", simp)  wenzelm@23164  1055 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  1056  prefer 2 apply arith  wenzelm@23164  1057 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  1058  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  1059 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  1060  pos_mod_bound)  nipkow@29948  1061 apply (subst mod_add_eq)  wenzelm@23164  1062 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)  wenzelm@23164  1063 apply (rule mod_pos_pos_trivial)  huffman@26086  1064 apply (auto simp add: mod_pos_pos_trivial ring_distribs)  wenzelm@23164  1065 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  1066 done  wenzelm@23164  1067 wenzelm@23164  1068 lemma neg_zmod_mult_2:  wenzelm@23164  1069  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  1070 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  1071  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  1072 apply (rule_tac [2] pos_zmod_mult_2)  wenzelm@23164  1073 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1074 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1075  prefer 2 apply simp  wenzelm@23164  1076 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  1077 done  wenzelm@23164  1078 huffman@26086  1079 lemma zmod_number_of_Bit0 [simp]:  huffman@26086  1080  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  huffman@26086  1081  (2::int) * (number_of v mod number_of w)"  huffman@26086  1082 apply (simp only: number_of_eq numeral_simps)  huffman@26086  1083 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  nipkow@29948  1084  neg_zmod_mult_2 add_ac)  huffman@26086  1085 done  huffman@26086  1086 huffman@26086  1087 lemma zmod_number_of_Bit1 [simp]:  huffman@26086  1088  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  huffman@26086  1089  (if (0::int) \ number_of w  wenzelm@23164  1090  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  1091  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  huffman@26086  1092 apply (simp only: number_of_eq numeral_simps)  wenzelm@23164  1093 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  nipkow@29948  1094  neg_zmod_mult_2 add_ac)  wenzelm@23164  1095 done  wenzelm@23164  1096 wenzelm@23164  1097 wenzelm@23164  1098 subsection{*Quotients of Signs*}  wenzelm@23164  1099 wenzelm@23164  1100 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  1101 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  1102 apply (rule order_trans)  wenzelm@23164  1103 apply (rule_tac a' = "-1" in zdiv_mono1)  nipkow@29948  1104 apply (auto simp add: div_eq_minus1)  wenzelm@23164  1105 done  wenzelm@23164  1106 wenzelm@23164  1107 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  1108 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  1109 wenzelm@23164  1110 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1111 apply auto  wenzelm@23164  1112 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1113 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1114 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1115 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1116 done  wenzelm@23164  1117 wenzelm@23164  1118 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1119  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1120 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1121 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1122 done  wenzelm@23164  1123 wenzelm@23164  1124 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1125 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1126 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1127 wenzelm@23164  1128 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1129 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1130 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1131 wenzelm@23164  1132 wenzelm@23164  1133 subsection {* The Divides Relation *}  wenzelm@23164  1134 wenzelm@23164  1135 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"  huffman@29410  1136  by (rule dvd_eq_mod_eq_0)  haftmann@23512  1137 wenzelm@23164  1138 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@23164  1139  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]  wenzelm@23164  1140 huffman@29410  1141 lemma zdvd_0_right: "(m::int) dvd 0"  huffman@29410  1142  by (rule dvd_0_right) (* already declared [iff] *)  wenzelm@23164  1143 huffman@29410  1144 lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"  huffman@29410  1145  by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)  wenzelm@23164  1146 huffman@29410  1147 lemma zdvd_1_left: "1 dvd (m::int)"  huffman@29410  1148  by (rule one_dvd) (* already declared [simp] *)  wenzelm@23164  1149 huffman@29950  1150 lemma zdvd_refl: "m dvd (m::int)"  huffman@29950  1151  by (rule dvd_refl) (* already declared [simp] *)  wenzelm@23164  1152 wenzelm@23164  1153 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"  huffman@29410  1154  by (rule dvd_trans)  wenzelm@23164  1155 huffman@29950  1156 lemma zdvd_zminus_iff: "m dvd -n \ m dvd (n::int)"  huffman@29950  1157  by (rule dvd_minus_iff) (* already declared [simp] *)  wenzelm@23164  1158 huffman@29950  1159 lemma zdvd_zminus2_iff: "-m dvd n \ m dvd (n::int)"  huffman@29950  1160  by (rule minus_dvd_iff) (* already declared [simp] *)  haftmann@27651  1161 huffman@29950  1162 lemma zdvd_abs1: "( \i::int\ dvd j) = (i dvd j)"  huffman@29950  1163  by (rule abs_dvd_iff) (* already declared [simp] *)  haftmann@27651  1164 huffman@29950  1165 lemma zdvd_abs2: "( (i::int) dvd \j$$ = (i dvd j)"  huffman@29950  1166  by (rule dvd_abs_iff) (* already declared [simp] *)  wenzelm@23164  1167 wenzelm@23164  1168 lemma zdvd_anti_sym:  wenzelm@23164  1169  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1170  apply (simp add: dvd_def, auto)  wenzelm@23164  1171  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1172  done  wenzelm@23164  1173 wenzelm@23164  1174 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"  huffman@29410  1175  by (rule dvd_add)  wenzelm@23164  1176 wenzelm@23164  1177 lemma zdvd_dvd_eq: assumes anz:"a \ 0" and ab: "(a::int) dvd b" and ba:"b dvd a"  wenzelm@23164  1178  shows "\a\ = \b\"  wenzelm@23164  1179 proof-  wenzelm@23164  1180  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast  wenzelm@23164  1181  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1182  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1183  with mult_cancel_left1[where c="a" and b="k*k'"]  wenzelm@23164  1184  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)  wenzelm@23164  1185  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1186  thus ?thesis using k k' by auto  wenzelm@23164  1187 qed  wenzelm@23164  1188 wenzelm@23164  1189 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"  huffman@29410  1190  by (rule Ring_and_Field.dvd_diff)  wenzelm@23164  1191 wenzelm@23164  1192 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1193  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1194  apply (erule ssubst)  wenzelm@23164  1195  apply (blast intro: zdvd_zadd, simp)  wenzelm@23164  1196  done  wenzelm@23164  1197 wenzelm@23164  1198 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"  huffman@29410  1199  by (rule dvd_mult)  wenzelm@23164  1200 wenzelm@23164  1201 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"  huffman@29410  1202  by (rule dvd_mult2)  wenzelm@23164  1203 huffman@29410  1204 lemma zdvd_triv_right: "(k::int) dvd m * k"  huffman@29410  1205  by (rule dvd_triv_right) (* already declared [simp] *)  wenzelm@23164  1206 huffman@29410  1207 lemma zdvd_triv_left: "(k::int) dvd k * m"  huffman@29410  1208  by (rule dvd_triv_left) (* already declared [simp] *)  wenzelm@23164  1209 wenzelm@23164  1210 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"  huffman@29410  1211  by (rule dvd_mult_left)  wenzelm@23164  1212 wenzelm@23164  1213 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"  huffman@29410  1214  by (rule dvd_mult_right)  wenzelm@23164  1215 wenzelm@23164  1216 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"  haftmann@27651  1217  by (rule mult_dvd_mono)  wenzelm@23164  1218 wenzelm@23164  1219 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  wenzelm@23164  1220  apply (rule iffI)  wenzelm@23164  1221  apply (erule_tac [2] zdvd_zadd)  wenzelm@23164  1222  apply (subgoal_tac "n = (n + k * m) - k * m")  wenzelm@23164  1223  apply (erule ssubst)  wenzelm@23164  1224  apply (erule zdvd_zdiff, simp_all)  wenzelm@23164  1225  done  wenzelm@23164  1226 wenzelm@23164  1227 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  wenzelm@23164  1228  apply (simp add: dvd_def)  wenzelm@23164  1229  apply (auto simp add: zmod_zmult_zmult1)  wenzelm@23164  1230  done  wenzelm@23164  1231 wenzelm@23164  1232 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1233  apply (subgoal_tac "k dvd n * (m div n) + m mod n")  wenzelm@23164  1234  apply (simp add: zmod_zdiv_equality [symmetric])  wenzelm@23164  1235  apply (simp only: zdvd_zadd zdvd_zmult2)  wenzelm@23164  1236  done  wenzelm@23164  1237 wenzelm@23164  1238 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  haftmann@27651  1239  apply (auto elim!: dvdE)  wenzelm@23164  1240  apply (subgoal_tac "0 < n")  wenzelm@23164  1241  prefer 2  wenzelm@23164  1242  apply (blast intro: order_less_trans)  wenzelm@23164  1243  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1244  apply (subgoal_tac "n * k < n * 1")  wenzelm@23164  1245  apply (drule mult_less_cancel_left [THEN iffD1], auto)  wenzelm@23164  1246  done  haftmann@27651  1247 wenzelm@23164  1248 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1249  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@29667  1250  by (simp add: algebra_simps)  wenzelm@23164  1251 wenzelm@23164  1252 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1253 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1254  apply (simp add: zmult_div_cancel)  wenzelm@23164  1255 apply (simp only: zdvd_iff_zmod_eq_0)  wenzelm@23164  1256 done  wenzelm@23164  1257 wenzelm@23164  1258 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1259  shows "m dvd n"  wenzelm@23164  1260 proof-  wenzelm@23164  1261  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1262  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1263  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1264  hence "n = m * h" by blast  huffman@29410  1265  thus ?thesis by simp  wenzelm@23164  1266 qed  wenzelm@23164  1267 huffman@29981  1268 lemma zdvd_zmult_cancel_disj:  nipkow@23969  1269  "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"  huffman@29981  1270 by (rule dvd_mult_cancel_left) (* already declared [simp] *)  nipkow@23969  1271 nipkow@23969  1272 wenzelm@23164  1273 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  nipkow@25134  1274 apply (simp split add: split_nat)  nipkow@25134  1275 apply (rule iffI)  nipkow@25134  1276 apply (erule exE)  nipkow@25134  1277 apply (rule_tac x = "int x" in exI)  nipkow@25134  1278 apply simp  nipkow@25134  1279 apply (erule exE)  nipkow@25134  1280 apply (rule_tac x = "nat x" in exI)  nipkow@25134  1281 apply (erule conjE)  nipkow@25134  1282 apply (erule_tac x = "nat x" in allE)  nipkow@25134  1283 apply simp  nipkow@25134  1284 done  wenzelm@23164  1285 huffman@23365  1286 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  haftmann@27651  1287 proof -  haftmann@27651  1288  have "\k. int y = int x * k \ x dvd y"  haftmann@27651  1289  proof -  haftmann@27651  1290  fix k  haftmann@27651  1291  assume A: "int y = int x * k"  haftmann@27651  1292  then show "x dvd y" proof (cases k)  haftmann@27651  1293  case (1 n) with A have "y = x * n" by (simp add: zmult_int)  haftmann@27651  1294  then show ?thesis ..  haftmann@27651  1295  next  haftmann@27651  1296  case (2 n) with A have "int y = int x * (- int (Suc n))" by simp  haftmann@27651  1297  also have "\ = - (int x * int (Suc n))" by (simp only: mult_minus_right)  haftmann@27651  1298  also have "\ = - int (x * Suc n)" by (simp only: zmult_int)  haftmann@27651  1299  finally have "- int (x * Suc n) = int y" ..  haftmann@27651  1300  then show ?thesis by (simp only: negative_eq_positive) auto  haftmann@27651  1301  qed  haftmann@27651  1302  qed  huffman@29410  1303  then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)  huffman@29410  1304 qed  wenzelm@23164  1305 wenzelm@23164  1306 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1307 proof  wenzelm@23164  1308  assume d: "x dvd 1" hence "int (nat \x\) dvd int (nat 1)" by (simp add: zdvd_abs1)  wenzelm@23164  1309  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1310  hence "nat \x\ = 1" by simp  wenzelm@23164  1311  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1312 next  wenzelm@23164  1313  assume "\x\=1" thus "x dvd 1"  wenzelm@23164  1314  by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)  wenzelm@23164  1315 qed  wenzelm@23164  1316 lemma zdvd_mult_cancel1:  wenzelm@23164  1317  assumes mp:"m $$0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1318 proof  wenzelm@23164  1319  assume n1: "\n\ = 1" thus "m * n dvd m"  wenzelm@23164  1320  by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)  wenzelm@23164  1321 next  wenzelm@23164  1322  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1323  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1324 qed  wenzelm@23164  1325 huffman@23365  1326 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  haftmann@27651  1327  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus_iff)  huffman@23306  1328 huffman@23365  1329 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  haftmann@27651  1330  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus2_iff)  wenzelm@23164  1331 wenzelm@23164  1332 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  haftmann@27651  1333  by (auto simp add: dvd_int_iff)  wenzelm@23164  1334 wenzelm@23164  1335 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"  huffman@29410  1336  by (rule minus_dvd_iff)  wenzelm@23164  1337 wenzelm@23164  1338 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"  huffman@29410  1339  by (rule dvd_minus_iff)  wenzelm@23164  1340 wenzelm@23164  1341 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1342  apply (rule_tac z=n in int_cases)  huffman@23365  1343  apply (auto simp add: dvd_int_iff)  huffman@23365  1344  apply (rule_tac z=z in int_cases)  huffman@23307  1345  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1346  done  wenzelm@23164  1347 wenzelm@23164  1348 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1349 apply (induct "y", auto)  wenzelm@23164  1350 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1351 apply (simp (no_asm_simp))  nipkow@29948  1352 apply (rule mod_mult_eq [symmetric])  wenzelm@23164  1353 done  wenzelm@23164  1354 huffman@23365  1355 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1356 apply (subst split_div, auto)  wenzelm@23164  1357 apply (subst split_zdiv, auto)  huffman@23365  1358 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)  haftmann@29651  1359 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  wenzelm@23164  1360 done  wenzelm@23164  1361 wenzelm@23164  1362 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1363 apply (subst split_mod, auto)  huffman@23365  1364 apply (subst split_zmod, auto)  huffman@23365  1365 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1366  in unique_remainder)  haftmann@29651  1367 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  huffman@23365  1368 done  wenzelm@23164  1369 wenzelm@23164  1370 text{*Suggested by Matthias Daum*}  wenzelm@23164  1371 lemma int_power_div_base:  wenzelm@23164  1372  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  wenzelm@23164  1373 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")  wenzelm@23164  1374  apply (erule ssubst)  wenzelm@23164  1375  apply (simp only: power_add)  wenzelm@23164  1376  apply simp_all  wenzelm@23164  1377 done  wenzelm@23164  1378 haftmann@23853  1379 text {* by Brian Huffman *}  haftmann@23853  1380 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  huffman@29405  1381 by (rule mod_minus_eq [symmetric])  haftmann@23853  1382 haftmann@23853  1383 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  huffman@29405  1384 by (rule mod_diff_left_eq [symmetric])  haftmann@23853  1385 haftmann@23853  1386 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  huffman@29405  1387 by (rule mod_diff_right_eq [symmetric])  haftmann@23853  1388 haftmann@23853  1389 lemmas zmod_simps =  haftmann@23853  1390  IntDiv.zmod_zadd_left_eq [symmetric]  haftmann@23853  1391  IntDiv.zmod_zadd_right_eq [symmetric]  haftmann@23853  1392  IntDiv.zmod_zmult1_eq [symmetric]  nipkow@29948  1393  mod_mult_left_eq [symmetric]  haftmann@23853  1394  IntDiv.zpower_zmod  haftmann@23853  1395  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1396 huffman@29045  1397 text {* Distributive laws for function @{text nat}. *}  huffman@29045  1398 huffman@29045  1399 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  huffman@29045  1400 apply (rule linorder_cases [of y 0])  huffman@29045  1401 apply (simp add: div_nonneg_neg_le0)  huffman@29045  1402 apply simp  huffman@29045  1403 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  huffman@29045  1404 done  huffman@29045  1405 huffman@29045  1406 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  huffman@29045  1407 lemma nat_mod_distrib:  huffman@29045  1408  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  huffman@29045  1409 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)  huffman@29045  1410 apply (simp add: nat_eq_iff zmod_int)  huffman@29045  1411 done  huffman@29045  1412 huffman@29045  1413 text{*Suggested by Matthias Daum*}  huffman@29045  1414 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  huffman@29045  1415 apply (subgoal_tac "nat x div nat k < nat x")  huffman@29045  1416  apply (simp (asm_lr) add: nat_div_distrib [symmetric])  huffman@29045  1417 apply (rule Divides.div_less_dividend, simp_all)  huffman@29045  1418 done  huffman@29045  1419 haftmann@23853  1420 text {* code generator setup *}  wenzelm@23164  1421 haftmann@26507  1422 context ring_1  haftmann@26507  1423 begin  haftmann@26507  1424 haftmann@28562  1425 lemma of_int_num [code]:  haftmann@26507  1426  "of_int k = (if k = 0 then 0 else if k < 0 then  haftmann@26507  1427  - of_int (- k) else let  haftmann@29651  1428  (l, m) = divmod k 2;  haftmann@26507  1429  l' = of_int l  haftmann@26507  1430  in if m = 0 then l' + l' else l' + l' + 1)"  haftmann@26507  1431 proof -  haftmann@26507  1432  have aux1: "k mod (2\int) \ (0\int) \  haftmann@26507  1433  of_int k = of_int (k div 2 * 2 + 1)"  haftmann@26507  1434  proof -  haftmann@26507  1435  have "k mod 2 < 2" by (auto intro: pos_mod_bound)  haftmann@26507  1436  moreover have "0 \ k mod 2" by (auto intro: pos_mod_sign)  haftmann@26507  1437  moreover assume "k mod 2 \ 0"  haftmann@26507  1438  ultimately have "k mod 2 = 1" by arith  haftmann@26507  1439  moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp  haftmann@26507  1440  ultimately show ?thesis by auto  haftmann@26507  1441  qed  haftmann@26507  1442  have aux2: "\x. of_int 2 * x = x + x"  haftmann@26507  1443  proof -  haftmann@26507  1444  fix x  haftmann@26507  1445  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1446  show "of_int 2 * x = x + x"  haftmann@26507  1447  unfolding int2 of_int_add left_distrib by simp  haftmann@26507  1448  qed  haftmann@26507  1449  have aux3: "\x. x * of_int 2 = x + x"  haftmann@26507  1450  proof -  haftmann@26507  1451  fix x  haftmann@26507  1452  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1453  show "x * of_int 2 = x + x"  haftmann@26507  1454  unfolding int2 of_int_add right_distrib by simp  haftmann@26507  1455  qed  haftmann@29651  1456  from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)  haftmann@26507  1457 qed  haftmann@26507  1458 haftmann@26507  1459 end  haftmann@26507  1460 chaieb@27667  1461 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  chaieb@27667  1462 proof  chaieb@27667  1463  assume H: "x mod n = y mod n"  chaieb@27667  1464  hence "x mod n - y mod n = 0" by simp  chaieb@27667  1465  hence "(x mod n - y mod n) mod n = 0" by simp  chaieb@27667  1466  hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])  chaieb@27667  1467  thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)  chaieb@27667  1468 next  chaieb@27667  1469  assume H: "n dvd x - y"  chaieb@27667  1470  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  chaieb@27667  1471  hence "x = n*k + y" by simp  chaieb@27667  1472  hence "x mod n = (n*k + y) mod n" by simp  chaieb@27667  1473  thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)  chaieb@27667  1474 qed  chaieb@27667  1475 chaieb@27667  1476 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  chaieb@27667  1477  shows "\q. x = y + n * q"  chaieb@27667  1478 proof-  chaieb@27667  1479  from xy have th: "int x - int y = int (x - y)" by simp  chaieb@27667  1480  from xyn have "int x mod int n = int y mod int n"  chaieb@27667  1481  by (simp add: zmod_int[symmetric])  chaieb@27667  1482  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  chaieb@27667  1483  hence "n dvd x - y" by (simp add: th zdvd_int)  chaieb@27667  1484  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  chaieb@27667  1485 qed  chaieb@27667  1486 chaieb@27667  1487 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  chaieb@27667  1488  (is "?lhs = ?rhs")  chaieb@27667  1489 proof  chaieb@27667  1490  assume H: "x mod n = y mod n"  chaieb@27667  1491  {assume xy: "x \ y"  chaieb@27667  1492  from H have th: "y mod n = x mod n" by simp  chaieb@27667  1493  from nat_mod_eq_lemma[OF th xy] have ?rhs  chaieb@27667  1494  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  chaieb@27667  1495  moreover  chaieb@27667  1496  {assume xy: "y \ x"  chaieb@27667  1497  from nat_mod_eq_lemma[OF H xy] have ?rhs  chaieb@27667  1498  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  chaieb@27667  1499  ultimately show ?rhs using linear[of x y] by blast  chaieb@27667  1500 next  chaieb@27667  1501  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  chaieb@27667  1502  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  chaieb@27667  1503  thus ?lhs by simp  chaieb@27667  1504 qed  chaieb@27667  1505 haftmann@29936  1506 haftmann@29936  1507 subsection {* Code generation *}  haftmann@29936  1508 haftmann@29936  1509 definition pdivmod :: "int \ int \ int \ int" where  haftmann@29936  1510  "pdivmod k l = (\k\ div \l\, \k\ mod \l$$"  haftmann@29936  1511 haftmann@29936  1512 lemma pdivmod_posDivAlg [code]:  haftmann@29936  1513  "pdivmod k l = (if l = 0 then (0, \k\) else posDivAlg \k\ \l\)"  haftmann@29936  1514 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)  haftmann@29936  1515 haftmann@29936  1516 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@29936  1517  apsnd ((op *) (sgn l)) (if 0 < l \ 0 \ k \ l < 0 \ k < 0  haftmann@29936  1518  then pdivmod k l  haftmann@29936  1519  else (let (r, s) = pdivmod k l in  haftmann@29936  1520  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@29936  1521 proof -  haftmann@29936  1522  have aux: "\q::int. - k = l * q \ k = l * - q" by auto  haftmann@29936  1523  show ?thesis  haftmann@29936  1524  by (simp add: divmod_mod_div pdivmod_def)  haftmann@29936  1525  (auto simp add: aux not_less not_le zdiv_zminus1_eq_if  haftmann@29936  1526  zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)  haftmann@29936  1527 qed  haftmann@29936  1528 haftmann@29936  1529 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else  haftmann@29936  1530  apsnd ((op *) (sgn l)) (if sgn k = sgn l  haftmann@29936  1531  then pdivmod k l  haftmann@29936  1532  else (let (r, s) = pdivmod k l in  haftmann@29936  1533  if s = 0 then (- r, 0) else (- r - 1, \l\ - s))))"  haftmann@29936  1534 proof -  haftmann@29936  1535  have "k \ 0 \ l \ 0 \ 0 < l \ 0 \ k \ l < 0 \ k < 0 \ sgn k = sgn l"  haftmann@29936  1536  by (auto simp add: not_less sgn_if)  haftmann@29936  1537  then show ?thesis by (simp add: divmod_pdivmod)  haftmann@29936  1538 qed  haftmann@29936  1539 wenzelm@23164  1540 code_modulename SML  wenzelm@23164  1541  IntDiv Integer  wenzelm@23164  1542 wenzelm@23164  1543 code_modulename OCaml  wenzelm@23164  1544  IntDiv Integer  wenzelm@23164  1545 wenzelm@23164  1546 code_modulename Haskell  haftmann@24195  1547  IntDiv Integer  wenzelm@23164  1548 wenzelm@23164  1549 end `