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