src/HOL/IntDiv.thy
 author haftmann Wed Jan 28 11:02:11 2009 +0100 (2009-01-28) changeset 29651 16a19466bf81 parent 29410 97916a925a69 child 29668 33ba3faeaa0e permissions -rw-r--r--
slightly adapted towards more uniformity with div/mod on nat
 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 wenzelm@23164  380 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"  wenzelm@23164  381 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)  wenzelm@23164  382 wenzelm@23164  383 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"  wenzelm@23164  384 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)  wenzelm@23164  385 wenzelm@23164  386 lemma zdiv_zminus2_eq_if:  wenzelm@23164  387  "b \ (0::int)  wenzelm@23164  388  ==> a div (-b) =  wenzelm@23164  389  (if a mod b = 0 then - (a div b) else - (a div b) - 1)"  wenzelm@23164  390 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)  wenzelm@23164  391 wenzelm@23164  392 lemma zmod_zminus2_eq_if:  wenzelm@23164  393  "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"  wenzelm@23164  394 by (simp add: zmod_zminus1_eq_if zmod_zminus2)  wenzelm@23164  395 wenzelm@23164  396 wenzelm@23164  397 subsection{*Division of a Number by Itself*}  wenzelm@23164  398 wenzelm@23164  399 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q"  wenzelm@23164  400 apply (subgoal_tac "0 < a*q")  wenzelm@23164  401  apply (simp add: zero_less_mult_iff, arith)  wenzelm@23164  402 done  wenzelm@23164  403 wenzelm@23164  404 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1"  wenzelm@23164  405 apply (subgoal_tac "0 \ a* (1-q) ")  wenzelm@23164  406  apply (simp add: zero_le_mult_iff)  wenzelm@23164  407 apply (simp add: right_diff_distrib)  wenzelm@23164  408 done  wenzelm@23164  409 haftmann@29651  410 lemma self_quotient: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> q = 1"  haftmann@29651  411 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)  wenzelm@23164  412 apply (rule order_antisym, safe, simp_all)  wenzelm@23164  413 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)  wenzelm@23164  414 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)  wenzelm@23164  415 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+  wenzelm@23164  416 done  wenzelm@23164  417 haftmann@29651  418 lemma self_remainder: "[| divmod_rel a a (q, r); a \ (0::int) |] ==> r = 0"  wenzelm@23164  419 apply (frule self_quotient, assumption)  haftmann@29651  420 apply (simp add: divmod_rel_def)  wenzelm@23164  421 done  wenzelm@23164  422 wenzelm@23164  423 lemma zdiv_self [simp]: "a \ 0 ==> a div a = (1::int)"  haftmann@29651  424 by (simp add: divmod_rel_div_mod [THEN self_quotient])  wenzelm@23164  425 wenzelm@23164  426 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)  wenzelm@23164  427 lemma zmod_self [simp]: "a mod a = (0::int)"  wenzelm@23164  428 apply (case_tac "a = 0", simp)  haftmann@29651  429 apply (simp add: divmod_rel_div_mod [THEN self_remainder])  wenzelm@23164  430 done  wenzelm@23164  431 wenzelm@23164  432 wenzelm@23164  433 subsection{*Computation of Division and Remainder*}  wenzelm@23164  434 wenzelm@23164  435 lemma zdiv_zero [simp]: "(0::int) div b = 0"  haftmann@29651  436 by (simp add: div_def divmod_def)  wenzelm@23164  437 wenzelm@23164  438 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@29651  439 by (simp add: div_def divmod_def)  wenzelm@23164  440 wenzelm@23164  441 lemma zmod_zero [simp]: "(0::int) mod b = 0"  haftmann@29651  442 by (simp add: mod_def divmod_def)  wenzelm@23164  443 wenzelm@23164  444 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"  haftmann@29651  445 by (simp add: div_def divmod_def)  wenzelm@23164  446 wenzelm@23164  447 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"  haftmann@29651  448 by (simp add: mod_def divmod_def)  wenzelm@23164  449 wenzelm@23164  450 text{*a positive, b positive *}  wenzelm@23164  451 wenzelm@23164  452 lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg a b)"  haftmann@29651  453 by (simp add: div_def divmod_def)  wenzelm@23164  454 wenzelm@23164  455 lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg a b)"  haftmann@29651  456 by (simp add: mod_def divmod_def)  wenzelm@23164  457 wenzelm@23164  458 text{*a negative, b positive *}  wenzelm@23164  459 wenzelm@23164  460 lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"  haftmann@29651  461 by (simp add: div_def divmod_def)  wenzelm@23164  462 wenzelm@23164  463 lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"  haftmann@29651  464 by (simp add: mod_def divmod_def)  wenzelm@23164  465 wenzelm@23164  466 text{*a positive, b negative *}  wenzelm@23164  467 wenzelm@23164  468 lemma div_pos_neg:  wenzelm@23164  469  "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  470 by (simp add: div_def divmod_def)  wenzelm@23164  471 wenzelm@23164  472 lemma mod_pos_neg:  wenzelm@23164  473  "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"  haftmann@29651  474 by (simp add: mod_def divmod_def)  wenzelm@23164  475 wenzelm@23164  476 text{*a negative, b negative *}  wenzelm@23164  477 wenzelm@23164  478 lemma div_neg_neg:  wenzelm@23164  479  "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  480 by (simp add: div_def divmod_def)  wenzelm@23164  481 wenzelm@23164  482 lemma mod_neg_neg:  wenzelm@23164  483  "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"  haftmann@29651  484 by (simp add: mod_def divmod_def)  wenzelm@23164  485 wenzelm@23164  486 text {*Simplify expresions in which div and mod combine numerical constants*}  wenzelm@23164  487 haftmann@29651  488 lemma divmod_relI:  huffman@24481  489  "\a == b * q + r; if 0 < b then 0 \ r \ r < b else b < r \ r \ 0\  haftmann@29651  490  \ divmod_rel a b (q, r)"  haftmann@29651  491  unfolding divmod_rel_def by simp  huffman@24481  492 haftmann@29651  493 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]  haftmann@29651  494 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]  huffman@24481  495 lemmas arithmetic_simps =  huffman@24481  496  arith_simps  huffman@24481  497  add_special  huffman@24481  498  OrderedGroup.add_0_left  huffman@24481  499  OrderedGroup.add_0_right  huffman@24481  500  mult_zero_left  huffman@24481  501  mult_zero_right  huffman@24481  502  mult_1_left  huffman@24481  503  mult_1_right  huffman@24481  504 huffman@24481  505 (* simprocs adapted from HOL/ex/Binary.thy *)  huffman@24481  506 ML {*  huffman@24481  507 local  huffman@24481  508  infix ==;  huffman@24481  509  val op == = Logic.mk_equals;  huffman@24481  510  fun plus m n = @{term "plus :: int \ int \ int"}  m  n;  huffman@24481  511  fun mult m n = @{term "times :: int \ int \ int"}  m  n;  huffman@24481  512 huffman@24481  513  val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};  huffman@24481  514  fun prove ctxt prop =  huffman@24481  515  Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));  huffman@24481  516 huffman@24481  517  fun binary_proc proc ss ct =  huffman@24481  518  (case Thm.term_of ct of  huffman@24481  519  _  t  u =>  huffman@24481  520  (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of  huffman@24481  521  SOME args => proc (Simplifier.the_context ss) args  huffman@24481  522  | NONE => NONE)  huffman@24481  523  | _ => NONE);  huffman@24481  524 in  huffman@24481  525 huffman@24481  526 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>  huffman@24481  527  if n = 0 then NONE  huffman@24481  528  else  wenzelm@24630  529  let val (k, l) = Integer.div_mod m n;  huffman@24481  530  fun mk_num x = HOLogic.mk_number HOLogic.intT x;  huffman@24481  531  in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])  huffman@24481  532  end);  huffman@24481  533 huffman@24481  534 end;  huffman@24481  535 *}  huffman@24481  536 huffman@24481  537 simproc_setup binary_int_div ("number_of m div number_of n :: int") =  haftmann@29651  538  {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}  huffman@24481  539 huffman@24481  540 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =  haftmann@29651  541  {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}  huffman@24481  542 huffman@24481  543 (* The following 8 lemmas are made unnecessary by the above simprocs: *)  huffman@24481  544 huffman@24481  545 lemmas div_pos_pos_number_of =  wenzelm@23164  546  div_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  547 huffman@24481  548 lemmas div_neg_pos_number_of =  wenzelm@23164  549  div_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  550 huffman@24481  551 lemmas div_pos_neg_number_of =  wenzelm@23164  552  div_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  553 huffman@24481  554 lemmas div_neg_neg_number_of =  wenzelm@23164  555  div_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  556 wenzelm@23164  557 huffman@24481  558 lemmas mod_pos_pos_number_of =  wenzelm@23164  559  mod_pos_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  560 huffman@24481  561 lemmas mod_neg_pos_number_of =  wenzelm@23164  562  mod_neg_pos [of "number_of v" "number_of w", standard]  wenzelm@23164  563 huffman@24481  564 lemmas mod_pos_neg_number_of =  wenzelm@23164  565  mod_pos_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  566 huffman@24481  567 lemmas mod_neg_neg_number_of =  wenzelm@23164  568  mod_neg_neg [of "number_of v" "number_of w", standard]  wenzelm@23164  569 wenzelm@23164  570 wenzelm@23164  571 lemmas posDivAlg_eqn_number_of [simp] =  wenzelm@23164  572  posDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  573 wenzelm@23164  574 lemmas negDivAlg_eqn_number_of [simp] =  wenzelm@23164  575  negDivAlg_eqn [of "number_of v" "number_of w", standard]  wenzelm@23164  576 wenzelm@23164  577 wenzelm@23164  578 text{*Special-case simplification *}  wenzelm@23164  579 wenzelm@23164  580 lemma zmod_1 [simp]: "a mod (1::int) = 0"  wenzelm@23164  581 apply (cut_tac a = a and b = 1 in pos_mod_sign)  wenzelm@23164  582 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)  wenzelm@23164  583 apply (auto simp del:pos_mod_bound pos_mod_sign)  wenzelm@23164  584 done  wenzelm@23164  585 wenzelm@23164  586 lemma zdiv_1 [simp]: "a div (1::int) = a"  wenzelm@23164  587 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)  wenzelm@23164  588 wenzelm@23164  589 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"  wenzelm@23164  590 apply (cut_tac a = a and b = "-1" in neg_mod_sign)  wenzelm@23164  591 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)  wenzelm@23164  592 apply (auto simp del: neg_mod_sign neg_mod_bound)  wenzelm@23164  593 done  wenzelm@23164  594 wenzelm@23164  595 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"  wenzelm@23164  596 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)  wenzelm@23164  597 wenzelm@23164  598 (** The last remaining special cases for constant arithmetic:  wenzelm@23164  599  1 div z and 1 mod z **)  wenzelm@23164  600 wenzelm@23164  601 lemmas div_pos_pos_1_number_of [simp] =  wenzelm@23164  602  div_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  603 wenzelm@23164  604 lemmas div_pos_neg_1_number_of [simp] =  wenzelm@23164  605  div_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  606 wenzelm@23164  607 lemmas mod_pos_pos_1_number_of [simp] =  wenzelm@23164  608  mod_pos_pos [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  609 wenzelm@23164  610 lemmas mod_pos_neg_1_number_of [simp] =  wenzelm@23164  611  mod_pos_neg [OF int_0_less_1, of "number_of w", standard]  wenzelm@23164  612 wenzelm@23164  613 wenzelm@23164  614 lemmas posDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  615  posDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  616 wenzelm@23164  617 lemmas negDivAlg_eqn_1_number_of [simp] =  wenzelm@23164  618  negDivAlg_eqn [of concl: 1 "number_of w", standard]  wenzelm@23164  619 wenzelm@23164  620 wenzelm@23164  621 wenzelm@23164  622 subsection{*Monotonicity in the First Argument (Dividend)*}  wenzelm@23164  623 wenzelm@23164  624 lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b"  wenzelm@23164  625 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  626 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  627 apply (rule unique_quotient_lemma)  wenzelm@23164  628 apply (erule subst)  wenzelm@23164  629 apply (erule subst, simp_all)  wenzelm@23164  630 done  wenzelm@23164  631 wenzelm@23164  632 lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b"  wenzelm@23164  633 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  634 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)  wenzelm@23164  635 apply (rule unique_quotient_lemma_neg)  wenzelm@23164  636 apply (erule subst)  wenzelm@23164  637 apply (erule subst, simp_all)  wenzelm@23164  638 done  wenzelm@23164  639 wenzelm@23164  640 wenzelm@23164  641 subsection{*Monotonicity in the Second Argument (Divisor)*}  wenzelm@23164  642 wenzelm@23164  643 lemma q_pos_lemma:  wenzelm@23164  644  "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)"  wenzelm@23164  645 apply (subgoal_tac "0 < b'* (q' + 1) ")  wenzelm@23164  646  apply (simp add: zero_less_mult_iff)  wenzelm@23164  647 apply (simp add: right_distrib)  wenzelm@23164  648 done  wenzelm@23164  649 wenzelm@23164  650 lemma zdiv_mono2_lemma:  wenzelm@23164  651  "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r';  wenzelm@23164  652  r' < b'; 0 \ r; 0 < b'; b' \ b |]  wenzelm@23164  653  ==> q \ (q'::int)"  wenzelm@23164  654 apply (frule q_pos_lemma, assumption+)  wenzelm@23164  655 apply (subgoal_tac "b*q < b* (q' + 1) ")  wenzelm@23164  656  apply (simp add: mult_less_cancel_left)  wenzelm@23164  657 apply (subgoal_tac "b*q = r' - r + b'*q'")  wenzelm@23164  658  prefer 2 apply simp  wenzelm@23164  659 apply (simp (no_asm_simp) add: right_distrib)  wenzelm@23164  660 apply (subst add_commute, rule zadd_zless_mono, arith)  wenzelm@23164  661 apply (rule mult_right_mono, auto)  wenzelm@23164  662 done  wenzelm@23164  663 wenzelm@23164  664 lemma zdiv_mono2:  wenzelm@23164  665  "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'"  wenzelm@23164  666 apply (subgoal_tac "b \ 0")  wenzelm@23164  667  prefer 2 apply arith  wenzelm@23164  668 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  669 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  670 apply (rule zdiv_mono2_lemma)  wenzelm@23164  671 apply (erule subst)  wenzelm@23164  672 apply (erule subst, simp_all)  wenzelm@23164  673 done  wenzelm@23164  674 wenzelm@23164  675 lemma q_neg_lemma:  wenzelm@23164  676  "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)"  wenzelm@23164  677 apply (subgoal_tac "b'*q' < 0")  wenzelm@23164  678  apply (simp add: mult_less_0_iff, arith)  wenzelm@23164  679 done  wenzelm@23164  680 wenzelm@23164  681 lemma zdiv_mono2_neg_lemma:  wenzelm@23164  682  "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;  wenzelm@23164  683  r < b; 0 \ r'; 0 < b'; b' \ b |]  wenzelm@23164  684  ==> q' \ (q::int)"  wenzelm@23164  685 apply (frule q_neg_lemma, assumption+)  wenzelm@23164  686 apply (subgoal_tac "b*q' < b* (q + 1) ")  wenzelm@23164  687  apply (simp add: mult_less_cancel_left)  wenzelm@23164  688 apply (simp add: right_distrib)  wenzelm@23164  689 apply (subgoal_tac "b*q' \ b'*q'")  wenzelm@23164  690  prefer 2 apply (simp add: mult_right_mono_neg, arith)  wenzelm@23164  691 done  wenzelm@23164  692 wenzelm@23164  693 lemma zdiv_mono2_neg:  wenzelm@23164  694  "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b"  wenzelm@23164  695 apply (cut_tac a = a and b = b in zmod_zdiv_equality)  wenzelm@23164  696 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)  wenzelm@23164  697 apply (rule zdiv_mono2_neg_lemma)  wenzelm@23164  698 apply (erule subst)  wenzelm@23164  699 apply (erule subst, simp_all)  wenzelm@23164  700 done  wenzelm@23164  701 haftmann@25942  702 wenzelm@23164  703 subsection{*More Algebraic Laws for div and mod*}  wenzelm@23164  704 wenzelm@23164  705 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}  wenzelm@23164  706 wenzelm@23164  707 lemma zmult1_lemma:  haftmann@29651  708  "[| divmod_rel b c (q, r); c \ 0 |]  haftmann@29651  709  ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"  haftmann@29651  710 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  wenzelm@23164  711 wenzelm@23164  712 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"  wenzelm@23164  713 apply (case_tac "c = 0", simp)  haftmann@29651  714 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])  wenzelm@23164  715 done  wenzelm@23164  716 wenzelm@23164  717 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"  wenzelm@23164  718 apply (case_tac "c = 0", simp)  haftmann@29651  719 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])  wenzelm@23164  720 done  wenzelm@23164  721 wenzelm@23164  722 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"  wenzelm@23164  723 apply (rule trans)  wenzelm@23164  724 apply (rule_tac s = "b*a mod c" in trans)  wenzelm@23164  725 apply (rule_tac [2] zmod_zmult1_eq)  wenzelm@23164  726 apply (simp_all add: mult_commute)  wenzelm@23164  727 done  wenzelm@23164  728 wenzelm@23164  729 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"  wenzelm@23164  730 apply (rule zmod_zmult1_eq' [THEN trans])  wenzelm@23164  731 apply (rule zmod_zmult1_eq)  wenzelm@23164  732 done  wenzelm@23164  733 wenzelm@23164  734 lemma zdiv_zmult_self1 [simp]: "b \ (0::int) ==> (a*b) div b = a"  wenzelm@23164  735 by (simp add: zdiv_zmult1_eq)  wenzelm@23164  736 huffman@29403  737 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"  haftmann@27651  738 apply (case_tac "b = 0", simp)  haftmann@27651  739 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)  haftmann@27651  740 done  haftmann@27651  741 huffman@29403  742 lemma zmod_zmod_trivial: "(a mod b) mod b = a mod (b::int)"  haftmann@27651  743 apply (case_tac "b = 0", simp)  haftmann@27651  744 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)  haftmann@27651  745 done  haftmann@27651  746 haftmann@27651  747 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}  haftmann@27651  748 haftmann@27651  749 lemma zadd1_lemma:  haftmann@29651  750  "[| divmod_rel a c (aq, ar); divmod_rel b c (bq, br); c \ 0 |]  haftmann@29651  751  ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"  haftmann@29651  752 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)  haftmann@27651  753 haftmann@27651  754 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)  haftmann@27651  755 lemma zdiv_zadd1_eq:  haftmann@27651  756  "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"  haftmann@27651  757 apply (case_tac "c = 0", simp)  haftmann@29651  758 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)  haftmann@27651  759 done  haftmann@27651  760 haftmann@27651  761 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"  haftmann@27651  762 apply (case_tac "c = 0", simp)  haftmann@29651  763 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_mod)  haftmann@27651  764 done  haftmann@27651  765 huffman@29405  766 instance int :: ring_div  haftmann@27651  767 proof  haftmann@27651  768  fix a b c :: int  haftmann@27651  769  assume not0: "b \ 0"  haftmann@27651  770  show "(a + c * b) div b = c + a div b"  haftmann@27651  771  unfolding zdiv_zadd1_eq [of a "c * b"] using not0  huffman@29403  772  by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)  haftmann@27651  773 qed auto  haftmann@25942  774 haftmann@29651  775 lemma posDivAlg_div_mod:  haftmann@29651  776  assumes "k \ 0"  haftmann@29651  777  and "l \ 0"  haftmann@29651  778  shows "posDivAlg k l = (k div l, k mod l)"  haftmann@29651  779 proof (cases "l = 0")  haftmann@29651  780  case True then show ?thesis by (simp add: posDivAlg.simps)  haftmann@29651  781 next  haftmann@29651  782  case False with assms posDivAlg_correct  haftmann@29651  783  have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"  haftmann@29651  784  by simp  haftmann@29651  785  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  786  show ?thesis by simp  haftmann@29651  787 qed  haftmann@29651  788 haftmann@29651  789 lemma negDivAlg_div_mod:  haftmann@29651  790  assumes "k < 0"  haftmann@29651  791  and "l > 0"  haftmann@29651  792  shows "negDivAlg k l = (k div l, k mod l)"  haftmann@29651  793 proof -  haftmann@29651  794  from assms have "l \ 0" by simp  haftmann@29651  795  from assms negDivAlg_correct  haftmann@29651  796  have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"  haftmann@29651  797  by simp  haftmann@29651  798  from divmod_rel_div [OF this l \ 0] divmod_rel_mod [OF this l \ 0]  haftmann@29651  799  show ?thesis by simp  haftmann@29651  800 qed  haftmann@29651  801 huffman@29403  802 lemma zdiv_zadd_self1: "a \ (0::int) ==> (a+b) div a = b div a + 1"  huffman@29403  803 by (rule div_add_self1) (* already declared [simp] *)  huffman@29403  804 huffman@29403  805 lemma zdiv_zadd_self2: "a \ (0::int) ==> (b+a) div a = b div a + 1"  huffman@29403  806 by (rule div_add_self2) (* already declared [simp] *)  wenzelm@23164  807 huffman@29403  808 lemma zdiv_zmult_self2: "b \ (0::int) ==> (b*a) div b = a"  huffman@29403  809 by (rule div_mult_self1_is_id) (* already declared [simp] *)  wenzelm@23164  810 huffman@29403  811 lemma zmod_zmult_self1: "(a*b) mod b = (0::int)"  huffman@29403  812 by (rule mod_mult_self2_is_0) (* already declared [simp] *)  huffman@29403  813 huffman@29403  814 lemma zmod_zmult_self2: "(b*a) mod b = (0::int)"  huffman@29403  815 by (rule mod_mult_self1_is_0) (* already declared [simp] *)  wenzelm@23164  816 wenzelm@23164  817 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"  huffman@29403  818 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)  wenzelm@23164  819 huffman@29403  820 (* REVISIT: should this be generalized to all semiring_div types? *)  wenzelm@23164  821 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]  wenzelm@23164  822 wenzelm@23164  823 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"  huffman@29403  824 by (rule mod_add_left_eq)  wenzelm@23164  825 wenzelm@23164  826 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"  huffman@29403  827 by (rule mod_add_right_eq)  wenzelm@23164  828 huffman@29403  829 lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)"  huffman@29403  830 by (rule mod_add_self1) (* already declared [simp] *)  wenzelm@23164  831 huffman@29403  832 lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)"  huffman@29403  833 by (rule mod_add_self2) (* already declared [simp] *)  wenzelm@23164  834 huffman@29405  835 lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)"  huffman@29405  836 by (rule mod_diff_eq)  nipkow@23983  837 wenzelm@23164  838 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}  wenzelm@23164  839 wenzelm@23164  840 (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but  wenzelm@23164  841  7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems  wenzelm@23164  842  to cause particular problems.*)  wenzelm@23164  843 wenzelm@23164  844 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}  wenzelm@23164  845 wenzelm@23164  846 lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r"  wenzelm@23164  847 apply (subgoal_tac "b * (c - q mod c) < r * 1")  wenzelm@23164  848 apply (simp add: right_diff_distrib)  wenzelm@23164  849 apply (rule order_le_less_trans)  wenzelm@23164  850 apply (erule_tac [2] mult_strict_right_mono)  wenzelm@23164  851 apply (rule mult_left_mono_neg)  wenzelm@23164  852 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  853  add1_zle_eq pos_mod_bound)  wenzelm@23164  854 done  wenzelm@23164  855 wenzelm@23164  856 lemma zmult2_lemma_aux2:  wenzelm@23164  857  "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0"  wenzelm@23164  858 apply (subgoal_tac "b * (q mod c) \ 0")  wenzelm@23164  859  apply arith  wenzelm@23164  860 apply (simp add: mult_le_0_iff)  wenzelm@23164  861 done  wenzelm@23164  862 wenzelm@23164  863 lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r"  wenzelm@23164  864 apply (subgoal_tac "0 \ b * (q mod c) ")  wenzelm@23164  865 apply arith  wenzelm@23164  866 apply (simp add: zero_le_mult_iff)  wenzelm@23164  867 done  wenzelm@23164  868 wenzelm@23164  869 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c"  wenzelm@23164  870 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")  wenzelm@23164  871 apply (simp add: right_diff_distrib)  wenzelm@23164  872 apply (rule order_less_le_trans)  wenzelm@23164  873 apply (erule mult_strict_right_mono)  wenzelm@23164  874 apply (rule_tac [2] mult_left_mono)  wenzelm@23164  875 apply (auto simp add: compare_rls add_commute [of 1]  wenzelm@23164  876  add1_zle_eq pos_mod_bound)  wenzelm@23164  877 done  wenzelm@23164  878 haftmann@29651  879 lemma zmult2_lemma: "[| divmod_rel a b (q, r); b \ 0; 0 < c |]  haftmann@29651  880  ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"  haftmann@29651  881 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff  wenzelm@23164  882  zero_less_mult_iff right_distrib [symmetric]  wenzelm@23164  883  zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)  wenzelm@23164  884 wenzelm@23164  885 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"  wenzelm@23164  886 apply (case_tac "b = 0", simp)  haftmann@29651  887 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])  wenzelm@23164  888 done  wenzelm@23164  889 wenzelm@23164  890 lemma zmod_zmult2_eq:  wenzelm@23164  891  "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"  wenzelm@23164  892 apply (case_tac "b = 0", simp)  haftmann@29651  893 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])  wenzelm@23164  894 done  wenzelm@23164  895 wenzelm@23164  896 wenzelm@23164  897 subsection{*Cancellation of Common Factors in div*}  wenzelm@23164  898 wenzelm@23164  899 lemma zdiv_zmult_zmult1_aux1:  wenzelm@23164  900  "[| (0::int) < b; c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  901 by (subst zdiv_zmult2_eq, auto)  wenzelm@23164  902 wenzelm@23164  903 lemma zdiv_zmult_zmult1_aux2:  wenzelm@23164  904  "[| b < (0::int); c \ 0 |] ==> (c*a) div (c*b) = a div b"  wenzelm@23164  905 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")  wenzelm@23164  906 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)  wenzelm@23164  907 done  wenzelm@23164  908 wenzelm@23164  909 lemma zdiv_zmult_zmult1: "c \ (0::int) ==> (c*a) div (c*b) = a div b"  wenzelm@23164  910 apply (case_tac "b = 0", simp)  wenzelm@23164  911 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)  wenzelm@23164  912 done  wenzelm@23164  913 nipkow@23401  914 lemma zdiv_zmult_zmult1_if[simp]:  nipkow@23401  915  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"  nipkow@23401  916 by (simp add:zdiv_zmult_zmult1)  nipkow@23401  917 nipkow@23401  918 (*  wenzelm@23164  919 lemma zdiv_zmult_zmult2: "c \ (0::int) ==> (a*c) div (b*c) = a div b"  wenzelm@23164  920 apply (drule zdiv_zmult_zmult1)  wenzelm@23164  921 apply (auto simp add: mult_commute)  wenzelm@23164  922 done  nipkow@23401  923 *)  wenzelm@23164  924 wenzelm@23164  925 wenzelm@23164  926 subsection{*Distribution of Factors over mod*}  wenzelm@23164  927 wenzelm@23164  928 lemma zmod_zmult_zmult1_aux1:  wenzelm@23164  929  "[| (0::int) < b; c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  930 by (subst zmod_zmult2_eq, auto)  wenzelm@23164  931 wenzelm@23164  932 lemma zmod_zmult_zmult1_aux2:  wenzelm@23164  933  "[| b < (0::int); c \ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"  wenzelm@23164  934 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")  wenzelm@23164  935 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)  wenzelm@23164  936 done  wenzelm@23164  937 wenzelm@23164  938 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"  wenzelm@23164  939 apply (case_tac "b = 0", simp)  wenzelm@23164  940 apply (case_tac "c = 0", simp)  wenzelm@23164  941 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)  wenzelm@23164  942 done  wenzelm@23164  943 wenzelm@23164  944 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"  wenzelm@23164  945 apply (cut_tac c = c in zmod_zmult_zmult1)  wenzelm@23164  946 apply (auto simp add: mult_commute)  wenzelm@23164  947 done  wenzelm@23164  948 huffman@29404  949 lemma zmod_zmod_cancel: "n dvd m \ (k::int) mod m mod n = k mod n"  huffman@29404  950 by (rule mod_mod_cancel)  nipkow@24490  951 wenzelm@23164  952 wenzelm@23164  953 subsection {*Splitting Rules for div and mod*}  wenzelm@23164  954 wenzelm@23164  955 text{*The proofs of the two lemmas below are essentially identical*}  wenzelm@23164  956 wenzelm@23164  957 lemma split_pos_lemma:  wenzelm@23164  958  "0  wenzelm@23164  959  P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)"  wenzelm@23164  960 apply (rule iffI, clarify)  wenzelm@23164  961  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  962  apply (subst zmod_zadd1_eq)  wenzelm@23164  963  apply (subst zdiv_zadd1_eq)  wenzelm@23164  964  apply (simp add: div_pos_pos_trivial mod_pos_pos_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_neg_lemma:  wenzelm@23164  971  "k<0 ==>  wenzelm@23164  972  P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)"  wenzelm@23164  973 apply (rule iffI, clarify)  wenzelm@23164  974  apply (erule_tac P="P ?x ?y" in rev_mp)  wenzelm@23164  975  apply (subst zmod_zadd1_eq)  wenzelm@23164  976  apply (subst zdiv_zadd1_eq)  wenzelm@23164  977  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  wenzelm@23164  978 txt{*converse direction*}  wenzelm@23164  979 apply (drule_tac x = "n div k" in spec)  wenzelm@23164  980 apply (drule_tac x = "n mod k" in spec, simp)  wenzelm@23164  981 done  wenzelm@23164  982 wenzelm@23164  983 lemma split_zdiv:  wenzelm@23164  984  "P(n div k :: int) =  wenzelm@23164  985  ((k = 0 --> P 0) &  wenzelm@23164  986  (0 (\i j. 0\j & j P i)) &  wenzelm@23164  987  (k<0 --> (\i j. k0 & n = k*i + j --> P i)))"  wenzelm@23164  988 apply (case_tac "k=0", simp)  wenzelm@23164  989 apply (simp only: linorder_neq_iff)  wenzelm@23164  990 apply (erule disjE)  wenzelm@23164  991  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]  wenzelm@23164  992  split_neg_lemma [of concl: "%x y. P x"])  wenzelm@23164  993 done  wenzelm@23164  994 wenzelm@23164  995 lemma split_zmod:  wenzelm@23164  996  "P(n mod k :: int) =  wenzelm@23164  997  ((k = 0 --> P n) &  wenzelm@23164  998  (0 (\i j. 0\j & j P j)) &  wenzelm@23164  999  (k<0 --> (\i j. k0 & n = k*i + j --> P j)))"  wenzelm@23164  1000 apply (case_tac "k=0", simp)  wenzelm@23164  1001 apply (simp only: linorder_neq_iff)  wenzelm@23164  1002 apply (erule disjE)  wenzelm@23164  1003  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]  wenzelm@23164  1004  split_neg_lemma [of concl: "%x y. P y"])  wenzelm@23164  1005 done  wenzelm@23164  1006 wenzelm@23164  1007 (* Enable arith to deal with div 2 and mod 2: *)  wenzelm@23164  1008 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1009 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]  wenzelm@23164  1010 wenzelm@23164  1011 wenzelm@23164  1012 subsection{*Speeding up the Division Algorithm with Shifting*}  wenzelm@23164  1013 wenzelm@23164  1014 text{*computing div by shifting *}  wenzelm@23164  1015 wenzelm@23164  1016 lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a"  wenzelm@23164  1017 proof cases  wenzelm@23164  1018  assume "a=0"  wenzelm@23164  1019  thus ?thesis by simp  wenzelm@23164  1020 next  wenzelm@23164  1021  assume "a\0" and le_a: "0\a"  wenzelm@23164  1022  hence a_pos: "1 \ a" by arith  wenzelm@23164  1023  hence one_less_a2: "1 < 2*a" by arith  wenzelm@23164  1024  hence le_2a: "2 * (1 + b mod a) \ 2 * a"  wenzelm@23164  1025  by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)  wenzelm@23164  1026  with a_pos have "0 \ b mod a" by simp  wenzelm@23164  1027  hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)"  wenzelm@23164  1028  by (simp add: mod_pos_pos_trivial one_less_a2)  wenzelm@23164  1029  with le_2a  wenzelm@23164  1030  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"  wenzelm@23164  1031  by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2  wenzelm@23164  1032  right_distrib)  wenzelm@23164  1033  thus ?thesis  wenzelm@23164  1034  by (subst zdiv_zadd1_eq,  wenzelm@23164  1035  simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2  wenzelm@23164  1036  div_pos_pos_trivial)  wenzelm@23164  1037 qed  wenzelm@23164  1038 wenzelm@23164  1039 lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"  wenzelm@23164  1040 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")  wenzelm@23164  1041 apply (rule_tac [2] pos_zdiv_mult_2)  wenzelm@23164  1042 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1043 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1044 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],  wenzelm@23164  1045  simp)  wenzelm@23164  1046 done  wenzelm@23164  1047 wenzelm@23164  1048 (*Not clear why this must be proved separately; probably number_of causes  wenzelm@23164  1049  simplification problems*)  wenzelm@23164  1050 lemma not_0_le_lemma: "~ 0 \ x ==> x \ (0::int)"  wenzelm@23164  1051 by auto  wenzelm@23164  1052 huffman@26086  1053 lemma zdiv_number_of_Bit0 [simp]:  huffman@26086  1054  "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  huffman@26086  1055  number_of v div (number_of w :: int)"  huffman@26086  1056 by (simp only: number_of_eq numeral_simps) simp  huffman@26086  1057 huffman@26086  1058 lemma zdiv_number_of_Bit1 [simp]:  huffman@26086  1059  "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  huffman@26086  1060  (if (0::int) \ number_of w  wenzelm@23164  1061  then number_of v div (number_of w)  wenzelm@23164  1062  else (number_of v + (1::int)) div (number_of w))"  wenzelm@23164  1063 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)  huffman@26086  1064 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)  wenzelm@23164  1065 done  wenzelm@23164  1066 wenzelm@23164  1067 wenzelm@23164  1068 subsection{*Computing mod by Shifting (proofs resemble those for div)*}  wenzelm@23164  1069 wenzelm@23164  1070 lemma pos_zmod_mult_2:  wenzelm@23164  1071  "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"  wenzelm@23164  1072 apply (case_tac "a = 0", simp)  wenzelm@23164  1073 apply (subgoal_tac "1 < a * 2")  wenzelm@23164  1074  prefer 2 apply arith  wenzelm@23164  1075 apply (subgoal_tac "2* (1 + b mod a) \ 2*a")  wenzelm@23164  1076  apply (rule_tac [2] mult_left_mono)  wenzelm@23164  1077 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq  wenzelm@23164  1078  pos_mod_bound)  wenzelm@23164  1079 apply (subst zmod_zadd1_eq)  wenzelm@23164  1080 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)  wenzelm@23164  1081 apply (rule mod_pos_pos_trivial)  huffman@26086  1082 apply (auto simp add: mod_pos_pos_trivial ring_distribs)  wenzelm@23164  1083 apply (subgoal_tac "0 \ b mod a", arith, simp)  wenzelm@23164  1084 done  wenzelm@23164  1085 wenzelm@23164  1086 lemma neg_zmod_mult_2:  wenzelm@23164  1087  "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"  wenzelm@23164  1088 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =  wenzelm@23164  1089  1 + 2* ((-b - 1) mod (-a))")  wenzelm@23164  1090 apply (rule_tac [2] pos_zmod_mult_2)  wenzelm@23164  1091 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)  wenzelm@23164  1092 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")  wenzelm@23164  1093  prefer 2 apply simp  wenzelm@23164  1094 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])  wenzelm@23164  1095 done  wenzelm@23164  1096 huffman@26086  1097 lemma zmod_number_of_Bit0 [simp]:  huffman@26086  1098  "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  huffman@26086  1099  (2::int) * (number_of v mod number_of w)"  huffman@26086  1100 apply (simp only: number_of_eq numeral_simps)  huffman@26086  1101 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  huffman@26086  1102  not_0_le_lemma neg_zmod_mult_2 add_ac)  huffman@26086  1103 done  huffman@26086  1104 huffman@26086  1105 lemma zmod_number_of_Bit1 [simp]:  huffman@26086  1106  "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  huffman@26086  1107  (if (0::int) \ number_of w  wenzelm@23164  1108  then 2 * (number_of v mod number_of w) + 1  wenzelm@23164  1109  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"  huffman@26086  1110 apply (simp only: number_of_eq numeral_simps)  wenzelm@23164  1111 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2  wenzelm@23164  1112  not_0_le_lemma neg_zmod_mult_2 add_ac)  wenzelm@23164  1113 done  wenzelm@23164  1114 wenzelm@23164  1115 wenzelm@23164  1116 subsection{*Quotients of Signs*}  wenzelm@23164  1117 wenzelm@23164  1118 lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"  wenzelm@23164  1119 apply (subgoal_tac "a div b \ -1", force)  wenzelm@23164  1120 apply (rule order_trans)  wenzelm@23164  1121 apply (rule_tac a' = "-1" in zdiv_mono1)  wenzelm@23164  1122 apply (auto simp add: zdiv_minus1)  wenzelm@23164  1123 done  wenzelm@23164  1124 wenzelm@23164  1125 lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0"  wenzelm@23164  1126 by (drule zdiv_mono1_neg, auto)  wenzelm@23164  1127 wenzelm@23164  1128 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)"  wenzelm@23164  1129 apply auto  wenzelm@23164  1130 apply (drule_tac [2] zdiv_mono1)  wenzelm@23164  1131 apply (auto simp add: linorder_neq_iff)  wenzelm@23164  1132 apply (simp (no_asm_use) add: linorder_not_less [symmetric])  wenzelm@23164  1133 apply (blast intro: div_neg_pos_less0)  wenzelm@23164  1134 done  wenzelm@23164  1135 wenzelm@23164  1136 lemma neg_imp_zdiv_nonneg_iff:  wenzelm@23164  1137  "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))"  wenzelm@23164  1138 apply (subst zdiv_zminus_zminus [symmetric])  wenzelm@23164  1139 apply (subst pos_imp_zdiv_nonneg_iff, auto)  wenzelm@23164  1140 done  wenzelm@23164  1141 wenzelm@23164  1142 (*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*)  wenzelm@23164  1143 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"  wenzelm@23164  1144 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)  wenzelm@23164  1145 wenzelm@23164  1146 (*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*)  wenzelm@23164  1147 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"  wenzelm@23164  1148 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)  wenzelm@23164  1149 wenzelm@23164  1150 wenzelm@23164  1151 subsection {* The Divides Relation *}  wenzelm@23164  1152 wenzelm@23164  1153 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"  huffman@29410  1154  by (rule dvd_eq_mod_eq_0)  haftmann@23512  1155 wenzelm@23164  1156 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =  wenzelm@23164  1157  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]  wenzelm@23164  1158 huffman@29410  1159 lemma zdvd_0_right: "(m::int) dvd 0"  huffman@29410  1160  by (rule dvd_0_right) (* already declared [iff] *)  wenzelm@23164  1161 huffman@29410  1162 lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"  huffman@29410  1163  by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)  wenzelm@23164  1164 huffman@29410  1165 lemma zdvd_1_left: "1 dvd (m::int)"  huffman@29410  1166  by (rule one_dvd) (* already declared [simp] *)  wenzelm@23164  1167 wenzelm@23164  1168 lemma zdvd_refl [simp]: "m dvd (m::int)"  huffman@29410  1169  by (rule dvd_refl) (* TODO: declare generic dvd_refl [simp] *)  wenzelm@23164  1170 wenzelm@23164  1171 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"  huffman@29410  1172  by (rule dvd_trans)  wenzelm@23164  1173 haftmann@27651  1174 lemma zdvd_zminus_iff: "m dvd -n \ m dvd (n::int)"  huffman@29410  1175  by (rule dvd_minus_iff)  wenzelm@23164  1176 haftmann@27651  1177 lemma zdvd_zminus2_iff: "-m dvd n \ m dvd (n::int)"  huffman@29410  1178  by (rule minus_dvd_iff)  haftmann@27651  1179 wenzelm@23164  1180 lemma zdvd_abs1: "( \i::int\ dvd j) = (i dvd j)"  haftmann@27651  1181  by (cases "i > 0") (simp_all add: zdvd_zminus2_iff)  haftmann@27651  1182 wenzelm@23164  1183 lemma zdvd_abs2: "( (i::int) dvd \j$$ = (i dvd j)"  haftmann@27651  1184  by (cases "j > 0") (simp_all add: zdvd_zminus_iff)  wenzelm@23164  1185 wenzelm@23164  1186 lemma zdvd_anti_sym:  wenzelm@23164  1187  "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"  wenzelm@23164  1188  apply (simp add: dvd_def, auto)  wenzelm@23164  1189  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)  wenzelm@23164  1190  done  wenzelm@23164  1191 wenzelm@23164  1192 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"  huffman@29410  1193  by (rule dvd_add)  wenzelm@23164  1194 wenzelm@23164  1195 lemma zdvd_dvd_eq: assumes anz:"a \ 0" and ab: "(a::int) dvd b" and ba:"b dvd a"  wenzelm@23164  1196  shows "\a\ = \b\"  wenzelm@23164  1197 proof-  wenzelm@23164  1198  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast  wenzelm@23164  1199  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast  wenzelm@23164  1200  from k k' have "a = a*k*k'" by simp  wenzelm@23164  1201  with mult_cancel_left1[where c="a" and b="k*k'"]  wenzelm@23164  1202  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)  wenzelm@23164  1203  hence "k = 1 \ k' = 1 \ k = -1 \ k' = -1" by (simp add: zmult_eq_1_iff)  wenzelm@23164  1204  thus ?thesis using k k' by auto  wenzelm@23164  1205 qed  wenzelm@23164  1206 wenzelm@23164  1207 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"  huffman@29410  1208  by (rule Ring_and_Field.dvd_diff)  wenzelm@23164  1209 wenzelm@23164  1210 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1211  apply (subgoal_tac "m = n + (m - n)")  wenzelm@23164  1212  apply (erule ssubst)  wenzelm@23164  1213  apply (blast intro: zdvd_zadd, simp)  wenzelm@23164  1214  done  wenzelm@23164  1215 wenzelm@23164  1216 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"  huffman@29410  1217  by (rule dvd_mult)  wenzelm@23164  1218 wenzelm@23164  1219 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"  huffman@29410  1220  by (rule dvd_mult2)  wenzelm@23164  1221 huffman@29410  1222 lemma zdvd_triv_right: "(k::int) dvd m * k"  huffman@29410  1223  by (rule dvd_triv_right) (* already declared [simp] *)  wenzelm@23164  1224 huffman@29410  1225 lemma zdvd_triv_left: "(k::int) dvd k * m"  huffman@29410  1226  by (rule dvd_triv_left) (* already declared [simp] *)  wenzelm@23164  1227 wenzelm@23164  1228 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"  huffman@29410  1229  by (rule dvd_mult_left)  wenzelm@23164  1230 wenzelm@23164  1231 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"  huffman@29410  1232  by (rule dvd_mult_right)  wenzelm@23164  1233 wenzelm@23164  1234 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"  haftmann@27651  1235  by (rule mult_dvd_mono)  wenzelm@23164  1236 wenzelm@23164  1237 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"  wenzelm@23164  1238  apply (rule iffI)  wenzelm@23164  1239  apply (erule_tac [2] zdvd_zadd)  wenzelm@23164  1240  apply (subgoal_tac "n = (n + k * m) - k * m")  wenzelm@23164  1241  apply (erule ssubst)  wenzelm@23164  1242  apply (erule zdvd_zdiff, simp_all)  wenzelm@23164  1243  done  wenzelm@23164  1244 wenzelm@23164  1245 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"  wenzelm@23164  1246  apply (simp add: dvd_def)  wenzelm@23164  1247  apply (auto simp add: zmod_zmult_zmult1)  wenzelm@23164  1248  done  wenzelm@23164  1249 wenzelm@23164  1250 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"  wenzelm@23164  1251  apply (subgoal_tac "k dvd n * (m div n) + m mod n")  wenzelm@23164  1252  apply (simp add: zmod_zdiv_equality [symmetric])  wenzelm@23164  1253  apply (simp only: zdvd_zadd zdvd_zmult2)  wenzelm@23164  1254  done  wenzelm@23164  1255 wenzelm@23164  1256 lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)"  haftmann@27651  1257  apply (auto elim!: dvdE)  wenzelm@23164  1258  apply (subgoal_tac "0 < n")  wenzelm@23164  1259  prefer 2  wenzelm@23164  1260  apply (blast intro: order_less_trans)  wenzelm@23164  1261  apply (simp add: zero_less_mult_iff)  wenzelm@23164  1262  apply (subgoal_tac "n * k < n * 1")  wenzelm@23164  1263  apply (drule mult_less_cancel_left [THEN iffD1], auto)  wenzelm@23164  1264  done  haftmann@27651  1265 wenzelm@23164  1266 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"  wenzelm@23164  1267  using zmod_zdiv_equality[where a="m" and b="n"]  nipkow@23477  1268  by (simp add: ring_simps)  wenzelm@23164  1269 wenzelm@23164  1270 lemma zdvd_mult_div_cancel:"(n::int) dvd m \ n * (m div n) = m"  wenzelm@23164  1271 apply (subgoal_tac "m mod n = 0")  wenzelm@23164  1272  apply (simp add: zmult_div_cancel)  wenzelm@23164  1273 apply (simp only: zdvd_iff_zmod_eq_0)  wenzelm@23164  1274 done  wenzelm@23164  1275 wenzelm@23164  1276 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \ (0::int)"  wenzelm@23164  1277  shows "m dvd n"  wenzelm@23164  1278 proof-  wenzelm@23164  1279  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast  wenzelm@23164  1280  {assume "n \ m*h" hence "k* n \ k* (m*h)" using kz by simp  wenzelm@23164  1281  with h have False by (simp add: mult_assoc)}  wenzelm@23164  1282  hence "n = m * h" by blast  huffman@29410  1283  thus ?thesis by simp  wenzelm@23164  1284 qed  wenzelm@23164  1285 nipkow@23969  1286 lemma zdvd_zmult_cancel_disj[simp]:  nipkow@23969  1287  "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"  nipkow@23969  1288 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)  nipkow@23969  1289 nipkow@23969  1290 wenzelm@23164  1291 theorem ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))"  nipkow@25134  1292 apply (simp split add: split_nat)  nipkow@25134  1293 apply (rule iffI)  nipkow@25134  1294 apply (erule exE)  nipkow@25134  1295 apply (rule_tac x = "int x" in exI)  nipkow@25134  1296 apply simp  nipkow@25134  1297 apply (erule exE)  nipkow@25134  1298 apply (rule_tac x = "nat x" in exI)  nipkow@25134  1299 apply (erule conjE)  nipkow@25134  1300 apply (erule_tac x = "nat x" in allE)  nipkow@25134  1301 apply simp  nipkow@25134  1302 done  wenzelm@23164  1303 huffman@23365  1304 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"  haftmann@27651  1305 proof -  haftmann@27651  1306  have "\k. int y = int x * k \ x dvd y"  haftmann@27651  1307  proof -  haftmann@27651  1308  fix k  haftmann@27651  1309  assume A: "int y = int x * k"  haftmann@27651  1310  then show "x dvd y" proof (cases k)  haftmann@27651  1311  case (1 n) with A have "y = x * n" by (simp add: zmult_int)  haftmann@27651  1312  then show ?thesis ..  haftmann@27651  1313  next  haftmann@27651  1314  case (2 n) with A have "int y = int x * (- int (Suc n))" by simp  haftmann@27651  1315  also have "\ = - (int x * int (Suc n))" by (simp only: mult_minus_right)  haftmann@27651  1316  also have "\ = - int (x * Suc n)" by (simp only: zmult_int)  haftmann@27651  1317  finally have "- int (x * Suc n) = int y" ..  haftmann@27651  1318  then show ?thesis by (simp only: negative_eq_positive) auto  haftmann@27651  1319  qed  haftmann@27651  1320  qed  huffman@29410  1321  then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)  huffman@29410  1322 qed  wenzelm@23164  1323 wenzelm@23164  1324 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \x\ = 1)"  wenzelm@23164  1325 proof  wenzelm@23164  1326  assume d: "x dvd 1" hence "int (nat \x\) dvd int (nat 1)" by (simp add: zdvd_abs1)  wenzelm@23164  1327  hence "nat \x\ dvd 1" by (simp add: zdvd_int)  wenzelm@23164  1328  hence "nat \x\ = 1" by simp  wenzelm@23164  1329  thus "\x\ = 1" by (cases "x < 0", auto)  wenzelm@23164  1330 next  wenzelm@23164  1331  assume "\x\=1" thus "x dvd 1"  wenzelm@23164  1332  by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)  wenzelm@23164  1333 qed  wenzelm@23164  1334 lemma zdvd_mult_cancel1:  wenzelm@23164  1335  assumes mp:"m \(0::int)" shows "(m * n dvd m) = (\n\ = 1)"  wenzelm@23164  1336 proof  wenzelm@23164  1337  assume n1: "\n\ = 1" thus "m * n dvd m"  wenzelm@23164  1338  by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)  wenzelm@23164  1339 next  wenzelm@23164  1340  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp  wenzelm@23164  1341  from zdvd_mult_cancel[OF H2 mp] show "\n\ = 1" by (simp only: zdvd1_eq)  wenzelm@23164  1342 qed  wenzelm@23164  1343 huffman@23365  1344 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"  haftmann@27651  1345  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus_iff)  huffman@23306  1346 huffman@23365  1347 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"  haftmann@27651  1348  unfolding zdvd_int by (cases "z \ 0") (simp_all add: zdvd_zminus2_iff)  wenzelm@23164  1349 wenzelm@23164  1350 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)"  haftmann@27651  1351  by (auto simp add: dvd_int_iff)  wenzelm@23164  1352 wenzelm@23164  1353 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"  huffman@29410  1354  by (rule minus_dvd_iff)  wenzelm@23164  1355 wenzelm@23164  1356 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"  huffman@29410  1357  by (rule dvd_minus_iff)  wenzelm@23164  1358 wenzelm@23164  1359 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)"  huffman@23365  1360  apply (rule_tac z=n in int_cases)  huffman@23365  1361  apply (auto simp add: dvd_int_iff)  huffman@23365  1362  apply (rule_tac z=z in int_cases)  huffman@23307  1363  apply (auto simp add: dvd_imp_le)  wenzelm@23164  1364  done  wenzelm@23164  1365 wenzelm@23164  1366 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"  wenzelm@23164  1367 apply (induct "y", auto)  wenzelm@23164  1368 apply (rule zmod_zmult1_eq [THEN trans])  wenzelm@23164  1369 apply (simp (no_asm_simp))  wenzelm@23164  1370 apply (rule zmod_zmult_distrib [symmetric])  wenzelm@23164  1371 done  wenzelm@23164  1372 huffman@23365  1373 lemma zdiv_int: "int (a div b) = (int a) div (int b)"  wenzelm@23164  1374 apply (subst split_div, auto)  wenzelm@23164  1375 apply (subst split_zdiv, auto)  huffman@23365  1376 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  1377 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  wenzelm@23164  1378 done  wenzelm@23164  1379 wenzelm@23164  1380 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"  huffman@23365  1381 apply (subst split_mod, auto)  huffman@23365  1382 apply (subst split_zmod, auto)  huffman@23365  1383 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia  huffman@23365  1384  in unique_remainder)  haftmann@29651  1385 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)  huffman@23365  1386 done  wenzelm@23164  1387 wenzelm@23164  1388 text{*Suggested by Matthias Daum*}  wenzelm@23164  1389 lemma int_power_div_base:  wenzelm@23164  1390  "\0 < m; 0 < k\ \ k ^ m div k = (k::int) ^ (m - Suc 0)"  wenzelm@23164  1391 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")  wenzelm@23164  1392  apply (erule ssubst)  wenzelm@23164  1393  apply (simp only: power_add)  wenzelm@23164  1394  apply simp_all  wenzelm@23164  1395 done  wenzelm@23164  1396 haftmann@23853  1397 text {* by Brian Huffman *}  haftmann@23853  1398 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"  huffman@29405  1399 by (rule mod_minus_eq [symmetric])  haftmann@23853  1400 haftmann@23853  1401 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"  huffman@29405  1402 by (rule mod_diff_left_eq [symmetric])  haftmann@23853  1403 haftmann@23853  1404 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"  huffman@29405  1405 by (rule mod_diff_right_eq [symmetric])  haftmann@23853  1406 haftmann@23853  1407 lemmas zmod_simps =  haftmann@23853  1408  IntDiv.zmod_zadd_left_eq [symmetric]  haftmann@23853  1409  IntDiv.zmod_zadd_right_eq [symmetric]  haftmann@23853  1410  IntDiv.zmod_zmult1_eq [symmetric]  haftmann@23853  1411  IntDiv.zmod_zmult1_eq' [symmetric]  haftmann@23853  1412  IntDiv.zpower_zmod  haftmann@23853  1413  zminus_zmod zdiff_zmod_left zdiff_zmod_right  haftmann@23853  1414 huffman@29045  1415 text {* Distributive laws for function @{text nat}. *}  huffman@29045  1416 huffman@29045  1417 lemma nat_div_distrib: "0 \ x \ nat (x div y) = nat x div nat y"  huffman@29045  1418 apply (rule linorder_cases [of y 0])  huffman@29045  1419 apply (simp add: div_nonneg_neg_le0)  huffman@29045  1420 apply simp  huffman@29045  1421 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)  huffman@29045  1422 done  huffman@29045  1423 huffman@29045  1424 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)  huffman@29045  1425 lemma nat_mod_distrib:  huffman@29045  1426  "\0 \ x; 0 \ y\ \ nat (x mod y) = nat x mod nat y"  huffman@29045  1427 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)  huffman@29045  1428 apply (simp add: nat_eq_iff zmod_int)  huffman@29045  1429 done  huffman@29045  1430 huffman@29045  1431 text{*Suggested by Matthias Daum*}  huffman@29045  1432 lemma int_div_less_self: "\0 < x; 1 < k\ \ x div k < (x::int)"  huffman@29045  1433 apply (subgoal_tac "nat x div nat k < nat x")  huffman@29045  1434  apply (simp (asm_lr) add: nat_div_distrib [symmetric])  huffman@29045  1435 apply (rule Divides.div_less_dividend, simp_all)  huffman@29045  1436 done  huffman@29045  1437 haftmann@23853  1438 text {* code generator setup *}  wenzelm@23164  1439 haftmann@26507  1440 context ring_1  haftmann@26507  1441 begin  haftmann@26507  1442 haftmann@28562  1443 lemma of_int_num [code]:  haftmann@26507  1444  "of_int k = (if k = 0 then 0 else if k < 0 then  haftmann@26507  1445  - of_int (- k) else let  haftmann@29651  1446  (l, m) = divmod k 2;  haftmann@26507  1447  l' = of_int l  haftmann@26507  1448  in if m = 0 then l' + l' else l' + l' + 1)"  haftmann@26507  1449 proof -  haftmann@26507  1450  have aux1: "k mod (2\int) \ (0\int) \  haftmann@26507  1451  of_int k = of_int (k div 2 * 2 + 1)"  haftmann@26507  1452  proof -  haftmann@26507  1453  have "k mod 2 < 2" by (auto intro: pos_mod_bound)  haftmann@26507  1454  moreover have "0 \ k mod 2" by (auto intro: pos_mod_sign)  haftmann@26507  1455  moreover assume "k mod 2 \ 0"  haftmann@26507  1456  ultimately have "k mod 2 = 1" by arith  haftmann@26507  1457  moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp  haftmann@26507  1458  ultimately show ?thesis by auto  haftmann@26507  1459  qed  haftmann@26507  1460  have aux2: "\x. of_int 2 * x = x + x"  haftmann@26507  1461  proof -  haftmann@26507  1462  fix x  haftmann@26507  1463  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1464  show "of_int 2 * x = x + x"  haftmann@26507  1465  unfolding int2 of_int_add left_distrib by simp  haftmann@26507  1466  qed  haftmann@26507  1467  have aux3: "\x. x * of_int 2 = x + x"  haftmann@26507  1468  proof -  haftmann@26507  1469  fix x  haftmann@26507  1470  have int2: "(2::int) = 1 + 1" by arith  haftmann@26507  1471  show "x * of_int 2 = x + x"  haftmann@26507  1472  unfolding int2 of_int_add right_distrib by simp  haftmann@26507  1473  qed  haftmann@29651  1474  from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)  haftmann@26507  1475 qed  haftmann@26507  1476 haftmann@26507  1477 end  haftmann@26507  1478 chaieb@27667  1479 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \ n dvd x - y"  chaieb@27667  1480 proof  chaieb@27667  1481  assume H: "x mod n = y mod n"  chaieb@27667  1482  hence "x mod n - y mod n = 0" by simp  chaieb@27667  1483  hence "(x mod n - y mod n) mod n = 0" by simp  chaieb@27667  1484  hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])  chaieb@27667  1485  thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)  chaieb@27667  1486 next  chaieb@27667  1487  assume H: "n dvd x - y"  chaieb@27667  1488  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast  chaieb@27667  1489  hence "x = n*k + y" by simp  chaieb@27667  1490  hence "x mod n = (n*k + y) mod n" by simp  chaieb@27667  1491  thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)  chaieb@27667  1492 qed  chaieb@27667  1493 chaieb@27667  1494 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \ x"  chaieb@27667  1495  shows "\q. x = y + n * q"  chaieb@27667  1496 proof-  chaieb@27667  1497  from xy have th: "int x - int y = int (x - y)" by simp  chaieb@27667  1498  from xyn have "int x mod int n = int y mod int n"  chaieb@27667  1499  by (simp add: zmod_int[symmetric])  chaieb@27667  1500  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])  chaieb@27667  1501  hence "n dvd x - y" by (simp add: th zdvd_int)  chaieb@27667  1502  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith  chaieb@27667  1503 qed  chaieb@27667  1504 chaieb@27667  1505 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \ (\q1 q2. x + n * q1 = y + n * q2)"  chaieb@27667  1506  (is "?lhs = ?rhs")  chaieb@27667  1507 proof  chaieb@27667  1508  assume H: "x mod n = y mod n"  chaieb@27667  1509  {assume xy: "x \ y"  chaieb@27667  1510  from H have th: "y mod n = x mod n" by simp  chaieb@27667  1511  from nat_mod_eq_lemma[OF th xy] have ?rhs  chaieb@27667  1512  apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}  chaieb@27667  1513  moreover  chaieb@27667  1514  {assume xy: "y \ x"  chaieb@27667  1515  from nat_mod_eq_lemma[OF H xy] have ?rhs  chaieb@27667  1516  apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}  chaieb@27667  1517  ultimately show ?rhs using linear[of x y] by blast  chaieb@27667  1518 next  chaieb@27667  1519  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast  chaieb@27667  1520  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp  chaieb@27667  1521  thus ?lhs by simp  chaieb@27667  1522 qed  chaieb@27667  1523 wenzelm@23164  1524 code_modulename SML  wenzelm@23164  1525  IntDiv Integer  wenzelm@23164  1526 wenzelm@23164  1527 code_modulename OCaml  wenzelm@23164  1528  IntDiv Integer  wenzelm@23164  1529 wenzelm@23164  1530 code_modulename Haskell  haftmann@24195  1531  IntDiv Integer  wenzelm@23164  1532 wenzelm@23164  1533 end `