src/HOL/IntDiv.thy
author wenzelm
Fri Mar 28 19:43:54 2008 +0100 (2008-03-28)
changeset 26462 dac4e2bce00d
parent 26101 a657683e902a
child 26480 544cef16045b
permissions -rw-r--r--
avoid rebinding of existing facts;
     1 (*  Title:      HOL/IntDiv.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1999  University of Cambridge
     5 
     6 *)
     7 
     8 header{*The Division Operators div and mod; the Divides Relation dvd*}
     9 
    10 theory IntDiv
    11 imports Int Divides FunDef
    12 begin
    13 
    14 constdefs
    15   quorem :: "(int*int) * (int*int) => bool"
    16     --{*definition of quotient and remainder*}
    17     [code func]: "quorem == %((a,b), (q,r)).
    18                       a = b*q + r &
    19                       (if 0 < b then 0\<le>r & r<b else b<r & r \<le> 0)"
    20 
    21   adjust :: "[int, int*int] => int*int"
    22     --{*for the division algorithm*}
    23     [code func]: "adjust b == %(q,r). if 0 \<le> r-b then (2*q + 1, r-b)
    24                          else (2*q, r)"
    25 
    26 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
    27 function
    28   posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
    29 where
    30   "posDivAlg a b =
    31      (if (a<b | b\<le>0) then (0,a)
    32         else adjust b (posDivAlg a (2*b)))"
    33 by auto
    34 termination by (relation "measure (%(a,b). nat(a - b + 1))") auto
    35 
    36 text{*algorithm for the case @{text "a<0, b>0"}*}
    37 function
    38   negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
    39 where
    40   "negDivAlg a b  =
    41      (if (0\<le>a+b | b\<le>0) then (-1,a+b)
    42       else adjust b (negDivAlg a (2*b)))"
    43 by auto
    44 termination by (relation "measure (%(a,b). nat(- a - b))") auto
    45 
    46 text{*algorithm for the general case @{term "b\<noteq>0"}*}
    47 constdefs
    48   negateSnd :: "int*int => int*int"
    49     [code func]: "negateSnd == %(q,r). (q,-r)"
    50 
    51 definition
    52   divAlg :: "int \<times> int \<Rightarrow> int \<times> int"
    53     --{*The full division algorithm considers all possible signs for a, b
    54        including the special case @{text "a=0, b<0"} because 
    55        @{term negDivAlg} requires @{term "a<0"}.*}
    56 where
    57   "divAlg = (\<lambda>(a, b). (if 0\<le>a then
    58                   if 0\<le>b then posDivAlg a b
    59                   else if a=0 then (0, 0)
    60                        else negateSnd (negDivAlg (-a) (-b))
    61                else 
    62                   if 0<b then negDivAlg a b
    63                   else negateSnd (posDivAlg (-a) (-b))))"
    64 
    65 instantiation int :: Divides.div
    66 begin
    67 
    68 definition
    69   div_def: "a div b = fst (divAlg (a, b))"
    70 
    71 definition
    72   mod_def: "a mod b = snd (divAlg (a, b))"
    73 
    74 instance ..
    75 
    76 end
    77 
    78 lemma divAlg_mod_div:
    79   "divAlg (p, q) = (p div q, p mod q)"
    80   by (auto simp add: div_def mod_def)
    81 
    82 text{*
    83 Here is the division algorithm in ML:
    84 
    85 \begin{verbatim}
    86     fun posDivAlg (a,b) =
    87       if a<b then (0,a)
    88       else let val (q,r) = posDivAlg(a, 2*b)
    89 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
    90 	   end
    91 
    92     fun negDivAlg (a,b) =
    93       if 0\<le>a+b then (~1,a+b)
    94       else let val (q,r) = negDivAlg(a, 2*b)
    95 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
    96 	   end;
    97 
    98     fun negateSnd (q,r:int) = (q,~r);
    99 
   100     fun divAlg (a,b) = if 0\<le>a then 
   101 			  if b>0 then posDivAlg (a,b) 
   102 			   else if a=0 then (0,0)
   103 				else negateSnd (negDivAlg (~a,~b))
   104 		       else 
   105 			  if 0<b then negDivAlg (a,b)
   106 			  else        negateSnd (posDivAlg (~a,~b));
   107 \end{verbatim}
   108 *}
   109 
   110 
   111 
   112 subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
   113 
   114 lemma unique_quotient_lemma:
   115      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
   116       ==> q' \<le> (q::int)"
   117 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
   118  prefer 2 apply (simp add: right_diff_distrib)
   119 apply (subgoal_tac "0 < b * (1 + q - q') ")
   120 apply (erule_tac [2] order_le_less_trans)
   121  prefer 2 apply (simp add: right_diff_distrib right_distrib)
   122 apply (subgoal_tac "b * q' < b * (1 + q) ")
   123  prefer 2 apply (simp add: right_diff_distrib right_distrib)
   124 apply (simp add: mult_less_cancel_left)
   125 done
   126 
   127 lemma unique_quotient_lemma_neg:
   128      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
   129       ==> q \<le> (q'::int)"
   130 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
   131     auto)
   132 
   133 lemma unique_quotient:
   134      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b \<noteq> 0 |]  
   135       ==> q = q'"
   136 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)
   137 apply (blast intro: order_antisym
   138              dest: order_eq_refl [THEN unique_quotient_lemma] 
   139              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
   140 done
   141 
   142 
   143 lemma unique_remainder:
   144      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b \<noteq> 0 |]  
   145       ==> r = r'"
   146 apply (subgoal_tac "q = q'")
   147  apply (simp add: quorem_def)
   148 apply (blast intro: unique_quotient)
   149 done
   150 
   151 
   152 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
   153 
   154 text{*And positive divisors*}
   155 
   156 lemma adjust_eq [simp]:
   157      "adjust b (q,r) = 
   158       (let diff = r-b in  
   159 	if 0 \<le> diff then (2*q + 1, diff)   
   160                      else (2*q, r))"
   161 by (simp add: Let_def adjust_def)
   162 
   163 declare posDivAlg.simps [simp del]
   164 
   165 text{*use with a simproc to avoid repeatedly proving the premise*}
   166 lemma posDivAlg_eqn:
   167      "0 < b ==>  
   168       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
   169 by (rule posDivAlg.simps [THEN trans], simp)
   170 
   171 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
   172 theorem posDivAlg_correct:
   173   assumes "0 \<le> a" and "0 < b"
   174   shows "quorem ((a, b), posDivAlg a b)"
   175 using prems apply (induct a b rule: posDivAlg.induct)
   176 apply auto
   177 apply (simp add: quorem_def)
   178 apply (subst posDivAlg_eqn, simp add: right_distrib)
   179 apply (case_tac "a < b")
   180 apply simp_all
   181 apply (erule splitE)
   182 apply (auto simp add: right_distrib Let_def)
   183 done
   184 
   185 
   186 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
   187 
   188 text{*And positive divisors*}
   189 
   190 declare negDivAlg.simps [simp del]
   191 
   192 text{*use with a simproc to avoid repeatedly proving the premise*}
   193 lemma negDivAlg_eqn:
   194      "0 < b ==>  
   195       negDivAlg a b =       
   196        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
   197 by (rule negDivAlg.simps [THEN trans], simp)
   198 
   199 (*Correctness of negDivAlg: it computes quotients correctly
   200   It doesn't work if a=0 because the 0/b equals 0, not -1*)
   201 lemma negDivAlg_correct:
   202   assumes "a < 0" and "b > 0"
   203   shows "quorem ((a, b), negDivAlg a b)"
   204 using prems apply (induct a b rule: negDivAlg.induct)
   205 apply (auto simp add: linorder_not_le)
   206 apply (simp add: quorem_def)
   207 apply (subst negDivAlg_eqn, assumption)
   208 apply (case_tac "a + b < (0\<Colon>int)")
   209 apply simp_all
   210 apply (erule splitE)
   211 apply (auto simp add: right_distrib Let_def)
   212 done
   213 
   214 
   215 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
   216 
   217 (*the case a=0*)
   218 lemma quorem_0: "b \<noteq> 0 ==> quorem ((0,b), (0,0))"
   219 by (auto simp add: quorem_def linorder_neq_iff)
   220 
   221 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
   222 by (subst posDivAlg.simps, auto)
   223 
   224 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
   225 by (subst negDivAlg.simps, auto)
   226 
   227 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
   228 by (simp add: negateSnd_def)
   229 
   230 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"
   231 by (auto simp add: split_ifs quorem_def)
   232 
   233 lemma divAlg_correct: "b \<noteq> 0 ==> quorem ((a,b), divAlg (a, b))"
   234 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg
   235                     posDivAlg_correct negDivAlg_correct)
   236 
   237 text{*Arbitrary definitions for division by zero.  Useful to simplify 
   238     certain equations.*}
   239 
   240 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
   241 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)  
   242 
   243 
   244 text{*Basic laws about division and remainder*}
   245 
   246 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
   247 apply (case_tac "b = 0", simp)
   248 apply (cut_tac a = a and b = b in divAlg_correct)
   249 apply (auto simp add: quorem_def div_def mod_def)
   250 done
   251 
   252 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
   253 by(simp add: zmod_zdiv_equality[symmetric])
   254 
   255 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
   256 by(simp add: mult_commute zmod_zdiv_equality[symmetric])
   257 
   258 text {* Tool setup *}
   259 
   260 ML_setup {*
   261 local 
   262 
   263 structure CancelDivMod = CancelDivModFun(
   264 struct
   265   val div_name = @{const_name Divides.div};
   266   val mod_name = @{const_name Divides.mod};
   267   val mk_binop = HOLogic.mk_binop;
   268   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
   269   val dest_sum = Int_Numeral_Simprocs.dest_sum;
   270   val div_mod_eqs =
   271     map mk_meta_eq [@{thm zdiv_zmod_equality},
   272       @{thm zdiv_zmod_equality2}];
   273   val trans = trans;
   274   val prove_eq_sums =
   275     let
   276       val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
   277     in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
   278 end)
   279 
   280 in
   281 
   282 val cancel_zdiv_zmod_proc = Simplifier.simproc @{theory}
   283   "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
   284 
   285 end;
   286 
   287 Addsimprocs [cancel_zdiv_zmod_proc]
   288 *}
   289 
   290 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
   291 apply (cut_tac a = a and b = b in divAlg_correct)
   292 apply (auto simp add: quorem_def mod_def)
   293 done
   294 
   295 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
   296    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
   297 
   298 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
   299 apply (cut_tac a = a and b = b in divAlg_correct)
   300 apply (auto simp add: quorem_def div_def mod_def)
   301 done
   302 
   303 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
   304    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
   305 
   306 
   307 
   308 subsection{*General Properties of div and mod*}
   309 
   310 lemma quorem_div_mod: "b \<noteq> 0 ==> quorem ((a, b), (a div b, a mod b))"
   311 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   312 apply (force simp add: quorem_def linorder_neq_iff)
   313 done
   314 
   315 lemma quorem_div: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a div b = q"
   316 by (simp add: quorem_div_mod [THEN unique_quotient])
   317 
   318 lemma quorem_mod: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a mod b = r"
   319 by (simp add: quorem_div_mod [THEN unique_remainder])
   320 
   321 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
   322 apply (rule quorem_div)
   323 apply (auto simp add: quorem_def)
   324 done
   325 
   326 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
   327 apply (rule quorem_div)
   328 apply (auto simp add: quorem_def)
   329 done
   330 
   331 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
   332 apply (rule quorem_div)
   333 apply (auto simp add: quorem_def)
   334 done
   335 
   336 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
   337 
   338 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
   339 apply (rule_tac q = 0 in quorem_mod)
   340 apply (auto simp add: quorem_def)
   341 done
   342 
   343 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
   344 apply (rule_tac q = 0 in quorem_mod)
   345 apply (auto simp add: quorem_def)
   346 done
   347 
   348 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
   349 apply (rule_tac q = "-1" in quorem_mod)
   350 apply (auto simp add: quorem_def)
   351 done
   352 
   353 text{*There is no @{text mod_neg_pos_trivial}.*}
   354 
   355 
   356 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
   357 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
   358 apply (case_tac "b = 0", simp)
   359 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, 
   360                                  THEN quorem_div, THEN sym])
   361 
   362 done
   363 
   364 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
   365 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
   366 apply (case_tac "b = 0", simp)
   367 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
   368        auto)
   369 done
   370 
   371 
   372 subsection{*Laws for div and mod with Unary Minus*}
   373 
   374 lemma zminus1_lemma:
   375      "quorem((a,b),(q,r))  
   376       ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
   377                           (if r=0 then 0 else b-r))"
   378 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
   379 
   380 
   381 lemma zdiv_zminus1_eq_if:
   382      "b \<noteq> (0::int)  
   383       ==> (-a) div b =  
   384           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   385 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
   386 
   387 lemma zmod_zminus1_eq_if:
   388      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
   389 apply (case_tac "b = 0", simp)
   390 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
   391 done
   392 
   393 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
   394 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
   395 
   396 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
   397 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
   398 
   399 lemma zdiv_zminus2_eq_if:
   400      "b \<noteq> (0::int)  
   401       ==> a div (-b) =  
   402           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   403 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
   404 
   405 lemma zmod_zminus2_eq_if:
   406      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
   407 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
   408 
   409 
   410 subsection{*Division of a Number by Itself*}
   411 
   412 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
   413 apply (subgoal_tac "0 < a*q")
   414  apply (simp add: zero_less_mult_iff, arith)
   415 done
   416 
   417 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
   418 apply (subgoal_tac "0 \<le> a* (1-q) ")
   419  apply (simp add: zero_le_mult_iff)
   420 apply (simp add: right_diff_distrib)
   421 done
   422 
   423 lemma self_quotient: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> q = 1"
   424 apply (simp add: split_ifs quorem_def linorder_neq_iff)
   425 apply (rule order_antisym, safe, simp_all)
   426 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
   427 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
   428 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
   429 done
   430 
   431 lemma self_remainder: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> r = 0"
   432 apply (frule self_quotient, assumption)
   433 apply (simp add: quorem_def)
   434 done
   435 
   436 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
   437 by (simp add: quorem_div_mod [THEN self_quotient])
   438 
   439 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
   440 lemma zmod_self [simp]: "a mod a = (0::int)"
   441 apply (case_tac "a = 0", simp)
   442 apply (simp add: quorem_div_mod [THEN self_remainder])
   443 done
   444 
   445 
   446 subsection{*Computation of Division and Remainder*}
   447 
   448 lemma zdiv_zero [simp]: "(0::int) div b = 0"
   449 by (simp add: div_def divAlg_def)
   450 
   451 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
   452 by (simp add: div_def divAlg_def)
   453 
   454 lemma zmod_zero [simp]: "(0::int) mod b = 0"
   455 by (simp add: mod_def divAlg_def)
   456 
   457 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
   458 by (simp add: div_def divAlg_def)
   459 
   460 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
   461 by (simp add: mod_def divAlg_def)
   462 
   463 text{*a positive, b positive *}
   464 
   465 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
   466 by (simp add: div_def divAlg_def)
   467 
   468 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
   469 by (simp add: mod_def divAlg_def)
   470 
   471 text{*a negative, b positive *}
   472 
   473 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
   474 by (simp add: div_def divAlg_def)
   475 
   476 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
   477 by (simp add: mod_def divAlg_def)
   478 
   479 text{*a positive, b negative *}
   480 
   481 lemma div_pos_neg:
   482      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
   483 by (simp add: div_def divAlg_def)
   484 
   485 lemma mod_pos_neg:
   486      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
   487 by (simp add: mod_def divAlg_def)
   488 
   489 text{*a negative, b negative *}
   490 
   491 lemma div_neg_neg:
   492      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
   493 by (simp add: div_def divAlg_def)
   494 
   495 lemma mod_neg_neg:
   496      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
   497 by (simp add: mod_def divAlg_def)
   498 
   499 text {*Simplify expresions in which div and mod combine numerical constants*}
   500 
   501 lemma quoremI:
   502   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
   503     \<Longrightarrow> quorem ((a, b), (q, r))"
   504   unfolding quorem_def by simp
   505 
   506 lemmas quorem_div_eq = quoremI [THEN quorem_div, THEN eq_reflection]
   507 lemmas quorem_mod_eq = quoremI [THEN quorem_mod, THEN eq_reflection]
   508 lemmas arithmetic_simps =
   509   arith_simps
   510   add_special
   511   OrderedGroup.add_0_left
   512   OrderedGroup.add_0_right
   513   mult_zero_left
   514   mult_zero_right
   515   mult_1_left
   516   mult_1_right
   517 
   518 (* simprocs adapted from HOL/ex/Binary.thy *)
   519 ML {*
   520 local
   521   infix ==;
   522   val op == = Logic.mk_equals;
   523   fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
   524   fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
   525 
   526   val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};
   527   fun prove ctxt prop =
   528     Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
   529 
   530   fun binary_proc proc ss ct =
   531     (case Thm.term_of ct of
   532       _ $ t $ u =>
   533       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
   534         SOME args => proc (Simplifier.the_context ss) args
   535       | NONE => NONE)
   536     | _ => NONE);
   537 in
   538 
   539 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
   540   if n = 0 then NONE
   541   else
   542     let val (k, l) = Integer.div_mod m n;
   543         fun mk_num x = HOLogic.mk_number HOLogic.intT x;
   544     in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])
   545     end);
   546 
   547 end;
   548 *}
   549 
   550 simproc_setup binary_int_div ("number_of m div number_of n :: int") =
   551   {* K (divmod_proc (@{thm quorem_div_eq})) *}
   552 
   553 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
   554   {* K (divmod_proc (@{thm quorem_mod_eq})) *}
   555 
   556 (* The following 8 lemmas are made unnecessary by the above simprocs: *)
   557 
   558 lemmas div_pos_pos_number_of =
   559     div_pos_pos [of "number_of v" "number_of w", standard]
   560 
   561 lemmas div_neg_pos_number_of =
   562     div_neg_pos [of "number_of v" "number_of w", standard]
   563 
   564 lemmas div_pos_neg_number_of =
   565     div_pos_neg [of "number_of v" "number_of w", standard]
   566 
   567 lemmas div_neg_neg_number_of =
   568     div_neg_neg [of "number_of v" "number_of w", standard]
   569 
   570 
   571 lemmas mod_pos_pos_number_of =
   572     mod_pos_pos [of "number_of v" "number_of w", standard]
   573 
   574 lemmas mod_neg_pos_number_of =
   575     mod_neg_pos [of "number_of v" "number_of w", standard]
   576 
   577 lemmas mod_pos_neg_number_of =
   578     mod_pos_neg [of "number_of v" "number_of w", standard]
   579 
   580 lemmas mod_neg_neg_number_of =
   581     mod_neg_neg [of "number_of v" "number_of w", standard]
   582 
   583 
   584 lemmas posDivAlg_eqn_number_of [simp] =
   585     posDivAlg_eqn [of "number_of v" "number_of w", standard]
   586 
   587 lemmas negDivAlg_eqn_number_of [simp] =
   588     negDivAlg_eqn [of "number_of v" "number_of w", standard]
   589 
   590 
   591 text{*Special-case simplification *}
   592 
   593 lemma zmod_1 [simp]: "a mod (1::int) = 0"
   594 apply (cut_tac a = a and b = 1 in pos_mod_sign)
   595 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
   596 apply (auto simp del:pos_mod_bound pos_mod_sign)
   597 done
   598 
   599 lemma zdiv_1 [simp]: "a div (1::int) = a"
   600 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
   601 
   602 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
   603 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
   604 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
   605 apply (auto simp del: neg_mod_sign neg_mod_bound)
   606 done
   607 
   608 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
   609 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
   610 
   611 (** The last remaining special cases for constant arithmetic:
   612     1 div z and 1 mod z **)
   613 
   614 lemmas div_pos_pos_1_number_of [simp] =
   615     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
   616 
   617 lemmas div_pos_neg_1_number_of [simp] =
   618     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
   619 
   620 lemmas mod_pos_pos_1_number_of [simp] =
   621     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
   622 
   623 lemmas mod_pos_neg_1_number_of [simp] =
   624     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
   625 
   626 
   627 lemmas posDivAlg_eqn_1_number_of [simp] =
   628     posDivAlg_eqn [of concl: 1 "number_of w", standard]
   629 
   630 lemmas negDivAlg_eqn_1_number_of [simp] =
   631     negDivAlg_eqn [of concl: 1 "number_of w", standard]
   632 
   633 
   634 
   635 subsection{*Monotonicity in the First Argument (Dividend)*}
   636 
   637 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
   638 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   639 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   640 apply (rule unique_quotient_lemma)
   641 apply (erule subst)
   642 apply (erule subst, simp_all)
   643 done
   644 
   645 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
   646 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   647 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   648 apply (rule unique_quotient_lemma_neg)
   649 apply (erule subst)
   650 apply (erule subst, simp_all)
   651 done
   652 
   653 
   654 subsection{*Monotonicity in the Second Argument (Divisor)*}
   655 
   656 lemma q_pos_lemma:
   657      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
   658 apply (subgoal_tac "0 < b'* (q' + 1) ")
   659  apply (simp add: zero_less_mult_iff)
   660 apply (simp add: right_distrib)
   661 done
   662 
   663 lemma zdiv_mono2_lemma:
   664      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
   665          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
   666       ==> q \<le> (q'::int)"
   667 apply (frule q_pos_lemma, assumption+) 
   668 apply (subgoal_tac "b*q < b* (q' + 1) ")
   669  apply (simp add: mult_less_cancel_left)
   670 apply (subgoal_tac "b*q = r' - r + b'*q'")
   671  prefer 2 apply simp
   672 apply (simp (no_asm_simp) add: right_distrib)
   673 apply (subst add_commute, rule zadd_zless_mono, arith)
   674 apply (rule mult_right_mono, auto)
   675 done
   676 
   677 lemma zdiv_mono2:
   678      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
   679 apply (subgoal_tac "b \<noteq> 0")
   680  prefer 2 apply arith
   681 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   682 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   683 apply (rule zdiv_mono2_lemma)
   684 apply (erule subst)
   685 apply (erule subst, simp_all)
   686 done
   687 
   688 lemma q_neg_lemma:
   689      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
   690 apply (subgoal_tac "b'*q' < 0")
   691  apply (simp add: mult_less_0_iff, arith)
   692 done
   693 
   694 lemma zdiv_mono2_neg_lemma:
   695      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
   696          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
   697       ==> q' \<le> (q::int)"
   698 apply (frule q_neg_lemma, assumption+) 
   699 apply (subgoal_tac "b*q' < b* (q + 1) ")
   700  apply (simp add: mult_less_cancel_left)
   701 apply (simp add: right_distrib)
   702 apply (subgoal_tac "b*q' \<le> b'*q'")
   703  prefer 2 apply (simp add: mult_right_mono_neg, arith)
   704 done
   705 
   706 lemma zdiv_mono2_neg:
   707      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
   708 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   709 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   710 apply (rule zdiv_mono2_neg_lemma)
   711 apply (erule subst)
   712 apply (erule subst, simp_all)
   713 done
   714 
   715 
   716 subsection{*More Algebraic Laws for div and mod*}
   717 
   718 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
   719 
   720 lemma zmult1_lemma:
   721      "[| quorem((b,c),(q,r));  c \<noteq> 0 |]  
   722       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   723 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
   724 
   725 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
   726 apply (case_tac "c = 0", simp)
   727 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
   728 done
   729 
   730 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
   731 apply (case_tac "c = 0", simp)
   732 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
   733 done
   734 
   735 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
   736 apply (rule trans)
   737 apply (rule_tac s = "b*a mod c" in trans)
   738 apply (rule_tac [2] zmod_zmult1_eq)
   739 apply (simp_all add: mult_commute)
   740 done
   741 
   742 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
   743 apply (rule zmod_zmult1_eq' [THEN trans])
   744 apply (rule zmod_zmult1_eq)
   745 done
   746 
   747 lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
   748 by (simp add: zdiv_zmult1_eq)
   749 
   750 instance int :: semiring_div
   751   by intro_classes auto
   752 
   753 lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
   754 by (subst mult_commute, erule zdiv_zmult_self1)
   755 
   756 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
   757 by (simp add: zmod_zmult1_eq)
   758 
   759 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
   760 by (simp add: mult_commute zmod_zmult1_eq)
   761 
   762 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
   763 proof
   764   assume "m mod d = 0"
   765   with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
   766 next
   767   assume "EX q::int. m = d*q"
   768   thus "m mod d = 0" by auto
   769 qed
   770 
   771 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
   772 
   773 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
   774 
   775 lemma zadd1_lemma:
   776      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c \<noteq> 0 |]  
   777       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   778 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
   779 
   780 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   781 lemma zdiv_zadd1_eq:
   782      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
   783 apply (case_tac "c = 0", simp)
   784 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
   785 done
   786 
   787 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
   788 apply (case_tac "c = 0", simp)
   789 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
   790 done
   791 
   792 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
   793 apply (case_tac "b = 0", simp)
   794 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
   795 done
   796 
   797 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
   798 apply (case_tac "b = 0", simp)
   799 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
   800 done
   801 
   802 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
   803 apply (rule trans [symmetric])
   804 apply (rule zmod_zadd1_eq, simp)
   805 apply (rule zmod_zadd1_eq [symmetric])
   806 done
   807 
   808 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
   809 apply (rule trans [symmetric])
   810 apply (rule zmod_zadd1_eq, simp)
   811 apply (rule zmod_zadd1_eq [symmetric])
   812 done
   813 
   814 lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
   815 by (simp add: zdiv_zadd1_eq)
   816 
   817 lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
   818 by (simp add: zdiv_zadd1_eq)
   819 
   820 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
   821 apply (case_tac "a = 0", simp)
   822 apply (simp add: zmod_zadd1_eq)
   823 done
   824 
   825 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
   826 apply (case_tac "a = 0", simp)
   827 apply (simp add: zmod_zadd1_eq)
   828 done
   829 
   830 
   831 lemma zmod_zdiff1_eq: fixes a::int
   832   shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r")
   833 proof -
   834   have "?l = (c + (a mod c - b mod c)) mod c"
   835     using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if)
   836   also have "\<dots> = ?r" by simp
   837   finally show ?thesis .
   838 qed
   839 
   840 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
   841 
   842 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
   843   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
   844   to cause particular problems.*)
   845 
   846 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
   847 
   848 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
   849 apply (subgoal_tac "b * (c - q mod c) < r * 1")
   850 apply (simp add: right_diff_distrib)
   851 apply (rule order_le_less_trans)
   852 apply (erule_tac [2] mult_strict_right_mono)
   853 apply (rule mult_left_mono_neg)
   854 apply (auto simp add: compare_rls add_commute [of 1]
   855                       add1_zle_eq pos_mod_bound)
   856 done
   857 
   858 lemma zmult2_lemma_aux2:
   859      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
   860 apply (subgoal_tac "b * (q mod c) \<le> 0")
   861  apply arith
   862 apply (simp add: mult_le_0_iff)
   863 done
   864 
   865 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
   866 apply (subgoal_tac "0 \<le> b * (q mod c) ")
   867 apply arith
   868 apply (simp add: zero_le_mult_iff)
   869 done
   870 
   871 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
   872 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
   873 apply (simp add: right_diff_distrib)
   874 apply (rule order_less_le_trans)
   875 apply (erule mult_strict_right_mono)
   876 apply (rule_tac [2] mult_left_mono)
   877 apply (auto simp add: compare_rls add_commute [of 1]
   878                       add1_zle_eq pos_mod_bound)
   879 done
   880 
   881 lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b \<noteq> 0;  0 < c |]  
   882       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   883 by (auto simp add: mult_ac quorem_def linorder_neq_iff
   884                    zero_less_mult_iff right_distrib [symmetric] 
   885                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
   886 
   887 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
   888 apply (case_tac "b = 0", simp)
   889 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
   890 done
   891 
   892 lemma zmod_zmult2_eq:
   893      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
   894 apply (case_tac "b = 0", simp)
   895 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
   896 done
   897 
   898 
   899 subsection{*Cancellation of Common Factors in div*}
   900 
   901 lemma zdiv_zmult_zmult1_aux1:
   902      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
   903 by (subst zdiv_zmult2_eq, auto)
   904 
   905 lemma zdiv_zmult_zmult1_aux2:
   906      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
   907 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
   908 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
   909 done
   910 
   911 lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
   912 apply (case_tac "b = 0", simp)
   913 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
   914 done
   915 
   916 lemma zdiv_zmult_zmult1_if[simp]:
   917   "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
   918 by (simp add:zdiv_zmult_zmult1)
   919 
   920 (*
   921 lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
   922 apply (drule zdiv_zmult_zmult1)
   923 apply (auto simp add: mult_commute)
   924 done
   925 *)
   926 
   927 
   928 subsection{*Distribution of Factors over mod*}
   929 
   930 lemma zmod_zmult_zmult1_aux1:
   931      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
   932 by (subst zmod_zmult2_eq, auto)
   933 
   934 lemma zmod_zmult_zmult1_aux2:
   935      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
   936 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
   937 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
   938 done
   939 
   940 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
   941 apply (case_tac "b = 0", simp)
   942 apply (case_tac "c = 0", simp)
   943 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
   944 done
   945 
   946 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
   947 apply (cut_tac c = c in zmod_zmult_zmult1)
   948 apply (auto simp add: mult_commute)
   949 done
   950 
   951 lemma zmod_zmod_cancel:
   952 assumes "n dvd m" shows "(k::int) mod m mod n = k mod n"
   953 proof -
   954   from `n dvd m` obtain r where "m = n*r" by(auto simp:dvd_def)
   955   have "k mod n = (m * (k div m) + k mod m) mod n"
   956     using zmod_zdiv_equality[of k m] by simp
   957   also have "\<dots> = (m * (k div m) mod n + k mod m mod n) mod n"
   958     by(subst zmod_zadd1_eq, rule refl)
   959   also have "m * (k div m) mod n = 0" using `m = n*r`
   960     by(simp add:mult_ac)
   961   finally show ?thesis by simp
   962 qed
   963 
   964 
   965 subsection {*Splitting Rules for div and mod*}
   966 
   967 text{*The proofs of the two lemmas below are essentially identical*}
   968 
   969 lemma split_pos_lemma:
   970  "0<k ==> 
   971     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
   972 apply (rule iffI, clarify)
   973  apply (erule_tac P="P ?x ?y" in rev_mp)  
   974  apply (subst zmod_zadd1_eq) 
   975  apply (subst zdiv_zadd1_eq) 
   976  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
   977 txt{*converse direction*}
   978 apply (drule_tac x = "n div k" in spec) 
   979 apply (drule_tac x = "n mod k" in spec, simp)
   980 done
   981 
   982 lemma split_neg_lemma:
   983  "k<0 ==>
   984     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
   985 apply (rule iffI, clarify)
   986  apply (erule_tac P="P ?x ?y" in rev_mp)  
   987  apply (subst zmod_zadd1_eq) 
   988  apply (subst zdiv_zadd1_eq) 
   989  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
   990 txt{*converse direction*}
   991 apply (drule_tac x = "n div k" in spec) 
   992 apply (drule_tac x = "n mod k" in spec, simp)
   993 done
   994 
   995 lemma split_zdiv:
   996  "P(n div k :: int) =
   997   ((k = 0 --> P 0) & 
   998    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
   999    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
  1000 apply (case_tac "k=0", simp)
  1001 apply (simp only: linorder_neq_iff)
  1002 apply (erule disjE) 
  1003  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
  1004                       split_neg_lemma [of concl: "%x y. P x"])
  1005 done
  1006 
  1007 lemma split_zmod:
  1008  "P(n mod k :: int) =
  1009   ((k = 0 --> P n) & 
  1010    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
  1011    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
  1012 apply (case_tac "k=0", simp)
  1013 apply (simp only: linorder_neq_iff)
  1014 apply (erule disjE) 
  1015  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
  1016                       split_neg_lemma [of concl: "%x y. P y"])
  1017 done
  1018 
  1019 (* Enable arith to deal with div 2 and mod 2: *)
  1020 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
  1021 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
  1022 
  1023 
  1024 subsection{*Speeding up the Division Algorithm with Shifting*}
  1025 
  1026 text{*computing div by shifting *}
  1027 
  1028 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
  1029 proof cases
  1030   assume "a=0"
  1031     thus ?thesis by simp
  1032 next
  1033   assume "a\<noteq>0" and le_a: "0\<le>a"   
  1034   hence a_pos: "1 \<le> a" by arith
  1035   hence one_less_a2: "1 < 2*a" by arith
  1036   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
  1037     by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
  1038   with a_pos have "0 \<le> b mod a" by simp
  1039   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
  1040     by (simp add: mod_pos_pos_trivial one_less_a2)
  1041   with  le_2a
  1042   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
  1043     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
  1044                   right_distrib) 
  1045   thus ?thesis
  1046     by (subst zdiv_zadd1_eq,
  1047         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
  1048                   div_pos_pos_trivial)
  1049 qed
  1050 
  1051 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
  1052 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
  1053 apply (rule_tac [2] pos_zdiv_mult_2)
  1054 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
  1055 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
  1056 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
  1057        simp) 
  1058 done
  1059 
  1060 (*Not clear why this must be proved separately; probably number_of causes
  1061   simplification problems*)
  1062 lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
  1063 by auto
  1064 
  1065 lemma zdiv_number_of_Bit0 [simp]:
  1066      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
  1067           number_of v div (number_of w :: int)"
  1068 by (simp only: number_of_eq numeral_simps) simp
  1069 
  1070 lemma zdiv_number_of_Bit1 [simp]:
  1071      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
  1072           (if (0::int) \<le> number_of w                    
  1073            then number_of v div (number_of w)     
  1074            else (number_of v + (1::int)) div (number_of w))"
  1075 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
  1076 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
  1077 done
  1078 
  1079 
  1080 subsection{*Computing mod by Shifting (proofs resemble those for div)*}
  1081 
  1082 lemma pos_zmod_mult_2:
  1083      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
  1084 apply (case_tac "a = 0", simp)
  1085 apply (subgoal_tac "1 < a * 2")
  1086  prefer 2 apply arith
  1087 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
  1088  apply (rule_tac [2] mult_left_mono)
  1089 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
  1090                       pos_mod_bound)
  1091 apply (subst zmod_zadd1_eq)
  1092 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
  1093 apply (rule mod_pos_pos_trivial)
  1094 apply (auto simp add: mod_pos_pos_trivial ring_distribs)
  1095 apply (subgoal_tac "0 \<le> b mod a", arith, simp)
  1096 done
  1097 
  1098 lemma neg_zmod_mult_2:
  1099      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
  1100 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
  1101                     1 + 2* ((-b - 1) mod (-a))")
  1102 apply (rule_tac [2] pos_zmod_mult_2)
  1103 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
  1104 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
  1105  prefer 2 apply simp 
  1106 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
  1107 done
  1108 
  1109 lemma zmod_number_of_Bit0 [simp]:
  1110      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
  1111       (2::int) * (number_of v mod number_of w)"
  1112 apply (simp only: number_of_eq numeral_simps) 
  1113 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
  1114                  not_0_le_lemma neg_zmod_mult_2 add_ac)
  1115 done
  1116 
  1117 lemma zmod_number_of_Bit1 [simp]:
  1118      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
  1119       (if (0::int) \<le> number_of w  
  1120                 then 2 * (number_of v mod number_of w) + 1     
  1121                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
  1122 apply (simp only: number_of_eq numeral_simps) 
  1123 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
  1124                  not_0_le_lemma neg_zmod_mult_2 add_ac)
  1125 done
  1126 
  1127 
  1128 subsection{*Quotients of Signs*}
  1129 
  1130 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
  1131 apply (subgoal_tac "a div b \<le> -1", force)
  1132 apply (rule order_trans)
  1133 apply (rule_tac a' = "-1" in zdiv_mono1)
  1134 apply (auto simp add: zdiv_minus1)
  1135 done
  1136 
  1137 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
  1138 by (drule zdiv_mono1_neg, auto)
  1139 
  1140 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
  1141 apply auto
  1142 apply (drule_tac [2] zdiv_mono1)
  1143 apply (auto simp add: linorder_neq_iff)
  1144 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
  1145 apply (blast intro: div_neg_pos_less0)
  1146 done
  1147 
  1148 lemma neg_imp_zdiv_nonneg_iff:
  1149      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
  1150 apply (subst zdiv_zminus_zminus [symmetric])
  1151 apply (subst pos_imp_zdiv_nonneg_iff, auto)
  1152 done
  1153 
  1154 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
  1155 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
  1156 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
  1157 
  1158 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
  1159 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
  1160 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
  1161 
  1162 
  1163 subsection {* The Divides Relation *}
  1164 
  1165 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
  1166   by (simp add: dvd_def zmod_eq_0_iff)
  1167 
  1168 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
  1169   zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
  1170 
  1171 lemma zdvd_0_right [iff]: "(m::int) dvd 0"
  1172   by (simp add: dvd_def)
  1173 
  1174 lemma zdvd_0_left [iff,noatp]: "(0 dvd (m::int)) = (m = 0)"
  1175   by (simp add: dvd_def)
  1176 
  1177 lemma zdvd_1_left [iff]: "1 dvd (m::int)"
  1178   by (simp add: dvd_def)
  1179 
  1180 lemma zdvd_refl [simp]: "m dvd (m::int)"
  1181   by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
  1182 
  1183 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
  1184   by (auto simp add: dvd_def intro: mult_assoc)
  1185 
  1186 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
  1187   apply (simp add: dvd_def, auto)
  1188    apply (rule_tac [!] x = "-k" in exI, auto)
  1189   done
  1190 
  1191 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
  1192   apply (simp add: dvd_def, auto)
  1193    apply (rule_tac [!] x = "-k" in exI, auto)
  1194   done
  1195 lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)" 
  1196   apply (cases "i > 0", simp)
  1197   apply (simp add: dvd_def)
  1198   apply (rule iffI)
  1199   apply (erule exE)
  1200   apply (rule_tac x="- k" in exI, simp)
  1201   apply (erule exE)
  1202   apply (rule_tac x="- k" in exI, simp)
  1203   done
  1204 lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)" 
  1205   apply (cases "j > 0", simp)
  1206   apply (simp add: dvd_def)
  1207   apply (rule iffI)
  1208   apply (erule exE)
  1209   apply (rule_tac x="- k" in exI, simp)
  1210   apply (erule exE)
  1211   apply (rule_tac x="- k" in exI, simp)
  1212   done
  1213 
  1214 lemma zdvd_anti_sym:
  1215     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
  1216   apply (simp add: dvd_def, auto)
  1217   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
  1218   done
  1219 
  1220 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
  1221   apply (simp add: dvd_def)
  1222   apply (blast intro: right_distrib [symmetric])
  1223   done
  1224 
  1225 lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a" 
  1226   shows "\<bar>a\<bar> = \<bar>b\<bar>"
  1227 proof-
  1228   from ab obtain k where k:"b = a*k" unfolding dvd_def by blast 
  1229   from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
  1230   from k k' have "a = a*k*k'" by simp
  1231   with mult_cancel_left1[where c="a" and b="k*k'"]
  1232   have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
  1233   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
  1234   thus ?thesis using k k' by auto
  1235 qed
  1236 
  1237 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
  1238   apply (simp add: dvd_def)
  1239   apply (blast intro: right_diff_distrib [symmetric])
  1240   done
  1241 
  1242 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
  1243   apply (subgoal_tac "m = n + (m - n)")
  1244    apply (erule ssubst)
  1245    apply (blast intro: zdvd_zadd, simp)
  1246   done
  1247 
  1248 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
  1249   apply (simp add: dvd_def)
  1250   apply (blast intro: mult_left_commute)
  1251   done
  1252 
  1253 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
  1254   apply (subst mult_commute)
  1255   apply (erule zdvd_zmult)
  1256   done
  1257 
  1258 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"
  1259   apply (rule zdvd_zmult)
  1260   apply (rule zdvd_refl)
  1261   done
  1262 
  1263 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"
  1264   apply (rule zdvd_zmult2)
  1265   apply (rule zdvd_refl)
  1266   done
  1267 
  1268 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
  1269   apply (simp add: dvd_def)
  1270   apply (simp add: mult_assoc, blast)
  1271   done
  1272 
  1273 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
  1274   apply (rule zdvd_zmultD2)
  1275   apply (subst mult_commute, assumption)
  1276   done
  1277 
  1278 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
  1279   apply (simp add: dvd_def, clarify)
  1280   apply (rule_tac x = "k * ka" in exI)
  1281   apply (simp add: mult_ac)
  1282   done
  1283 
  1284 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
  1285   apply (rule iffI)
  1286    apply (erule_tac [2] zdvd_zadd)
  1287    apply (subgoal_tac "n = (n + k * m) - k * m")
  1288     apply (erule ssubst)
  1289     apply (erule zdvd_zdiff, simp_all)
  1290   done
  1291 
  1292 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  1293   apply (simp add: dvd_def)
  1294   apply (auto simp add: zmod_zmult_zmult1)
  1295   done
  1296 
  1297 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  1298   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
  1299    apply (simp add: zmod_zdiv_equality [symmetric])
  1300   apply (simp only: zdvd_zadd zdvd_zmult2)
  1301   done
  1302 
  1303 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
  1304   apply (simp add: dvd_def, auto)
  1305   apply (subgoal_tac "0 < n")
  1306    prefer 2
  1307    apply (blast intro: order_less_trans)
  1308   apply (simp add: zero_less_mult_iff)
  1309   apply (subgoal_tac "n * k < n * 1")
  1310    apply (drule mult_less_cancel_left [THEN iffD1], auto)
  1311   done
  1312 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  1313   using zmod_zdiv_equality[where a="m" and b="n"]
  1314   by (simp add: ring_simps)
  1315 
  1316 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
  1317 apply (subgoal_tac "m mod n = 0")
  1318  apply (simp add: zmult_div_cancel)
  1319 apply (simp only: zdvd_iff_zmod_eq_0)
  1320 done
  1321 
  1322 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
  1323   shows "m dvd n"
  1324 proof-
  1325   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
  1326   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
  1327     with h have False by (simp add: mult_assoc)}
  1328   hence "n = m * h" by blast
  1329   thus ?thesis by blast
  1330 qed
  1331 
  1332 lemma zdvd_zmult_cancel_disj[simp]:
  1333   "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"
  1334 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)
  1335 
  1336 
  1337 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
  1338 apply (simp split add: split_nat)
  1339 apply (rule iffI)
  1340 apply (erule exE)
  1341 apply (rule_tac x = "int x" in exI)
  1342 apply simp
  1343 apply (erule exE)
  1344 apply (rule_tac x = "nat x" in exI)
  1345 apply (erule conjE)
  1346 apply (erule_tac x = "nat x" in allE)
  1347 apply simp
  1348 done
  1349 
  1350 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
  1351 apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
  1352     nat_0_le cong add: conj_cong)
  1353 apply (rule iffI)
  1354 apply iprover
  1355 apply (erule exE)
  1356 apply (case_tac "x=0")
  1357 apply (rule_tac x=0 in exI)
  1358 apply simp
  1359 apply (case_tac "0 \<le> k")
  1360 apply iprover
  1361 apply (simp add: neq0_conv linorder_not_le)
  1362 apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
  1363 apply assumption
  1364 apply (simp add: mult_ac)
  1365 done
  1366 
  1367 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
  1368 proof
  1369   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
  1370   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
  1371   hence "nat \<bar>x\<bar> = 1"  by simp
  1372   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
  1373 next
  1374   assume "\<bar>x\<bar>=1" thus "x dvd 1" 
  1375     by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
  1376 qed
  1377 lemma zdvd_mult_cancel1: 
  1378   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
  1379 proof
  1380   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
  1381     by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
  1382 next
  1383   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
  1384   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
  1385 qed
  1386 
  1387 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
  1388   apply (auto simp add: dvd_def nat_abs_mult_distrib)
  1389   apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
  1390    apply (rule_tac x = "-(int k)" in exI)
  1391   apply (auto simp add: int_mult)
  1392   done
  1393 
  1394 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
  1395   apply (auto simp add: dvd_def abs_if int_mult)
  1396     apply (rule_tac [3] x = "nat k" in exI)
  1397     apply (rule_tac [2] x = "-(int k)" in exI)
  1398     apply (rule_tac x = "nat (-k)" in exI)
  1399     apply (cut_tac [3] k = m in int_less_0_conv)
  1400     apply (cut_tac k = m in int_less_0_conv)
  1401     apply (auto simp add: zero_le_mult_iff mult_less_0_iff
  1402       nat_mult_distrib [symmetric] nat_eq_iff2)
  1403   done
  1404 
  1405 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
  1406   apply (auto simp add: dvd_def int_mult)
  1407   apply (rule_tac x = "nat k" in exI)
  1408   apply (cut_tac k = m in int_less_0_conv)
  1409   apply (auto simp add: zero_le_mult_iff mult_less_0_iff
  1410     nat_mult_distrib [symmetric] nat_eq_iff2)
  1411   done
  1412 
  1413 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
  1414   apply (auto simp add: dvd_def)
  1415    apply (rule_tac [!] x = "-k" in exI, auto)
  1416   done
  1417 
  1418 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
  1419   apply (auto simp add: dvd_def)
  1420    apply (drule minus_equation_iff [THEN iffD1])
  1421    apply (rule_tac [!] x = "-k" in exI, auto)
  1422   done
  1423 
  1424 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
  1425   apply (rule_tac z=n in int_cases)
  1426   apply (auto simp add: dvd_int_iff)
  1427   apply (rule_tac z=z in int_cases)
  1428   apply (auto simp add: dvd_imp_le)
  1429   done
  1430 
  1431 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
  1432 apply (induct "y", auto)
  1433 apply (rule zmod_zmult1_eq [THEN trans])
  1434 apply (simp (no_asm_simp))
  1435 apply (rule zmod_zmult_distrib [symmetric])
  1436 done
  1437 
  1438 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
  1439 apply (subst split_div, auto)
  1440 apply (subst split_zdiv, auto)
  1441 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
  1442 apply (auto simp add: IntDiv.quorem_def of_nat_mult)
  1443 done
  1444 
  1445 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
  1446 apply (subst split_mod, auto)
  1447 apply (subst split_zmod, auto)
  1448 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
  1449        in unique_remainder)
  1450 apply (auto simp add: IntDiv.quorem_def of_nat_mult)
  1451 done
  1452 
  1453 text{*Suggested by Matthias Daum*}
  1454 lemma int_power_div_base:
  1455      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
  1456 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
  1457  apply (erule ssubst)
  1458  apply (simp only: power_add)
  1459  apply simp_all
  1460 done
  1461 
  1462 text {* by Brian Huffman *}
  1463 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
  1464 by (simp only: zmod_zminus1_eq_if mod_mod_trivial)
  1465 
  1466 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
  1467 by (simp only: diff_def zmod_zadd_left_eq [symmetric])
  1468 
  1469 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
  1470 proof -
  1471   have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m"
  1472     by (simp only: zminus_zmod)
  1473   hence "(x + - (y mod m)) mod m = (x + - y) mod m"
  1474     by (simp only: zmod_zadd_right_eq [symmetric])
  1475   thus "(x - y mod m) mod m = (x - y) mod m"
  1476     by (simp only: diff_def)
  1477 qed
  1478 
  1479 lemmas zmod_simps =
  1480   IntDiv.zmod_zadd_left_eq  [symmetric]
  1481   IntDiv.zmod_zadd_right_eq [symmetric]
  1482   IntDiv.zmod_zmult1_eq     [symmetric]
  1483   IntDiv.zmod_zmult1_eq'    [symmetric]
  1484   IntDiv.zpower_zmod
  1485   zminus_zmod zdiff_zmod_left zdiff_zmod_right
  1486 
  1487 text {* code generator setup *}
  1488 
  1489 code_modulename SML
  1490   IntDiv Integer
  1491 
  1492 code_modulename OCaml
  1493   IntDiv Integer
  1494 
  1495 code_modulename Haskell
  1496   IntDiv Integer
  1497 
  1498 end