src/HOL/IntDiv.thy
author nipkow
Sat Jun 16 16:27:35 2007 +0200 (2007-06-16)
changeset 23401 8c5532263ba9
parent 23365 f31794033ae1
child 23431 25ca91279a9b
permissions -rw-r--r--
tuned
     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 IntArith 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 instance int :: Divides.div
    66   div_def: "a div b == fst (divAlg (a, b))"
    67   mod_def: "a mod b == snd (divAlg (a, b))" ..
    68 
    69 lemma divAlg_mod_div:
    70   "divAlg (p, q) = (p div q, p mod q)"
    71   by (auto simp add: div_def mod_def)
    72 
    73 text{*
    74 Here is the division algorithm in ML:
    75 
    76 \begin{verbatim}
    77     fun posDivAlg (a,b) =
    78       if a<b then (0,a)
    79       else let val (q,r) = posDivAlg(a, 2*b)
    80 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
    81 	   end
    82 
    83     fun negDivAlg (a,b) =
    84       if 0\<le>a+b then (~1,a+b)
    85       else let val (q,r) = negDivAlg(a, 2*b)
    86 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
    87 	   end;
    88 
    89     fun negateSnd (q,r:int) = (q,~r);
    90 
    91     fun divAlg (a,b) = if 0\<le>a then 
    92 			  if b>0 then posDivAlg (a,b) 
    93 			   else if a=0 then (0,0)
    94 				else negateSnd (negDivAlg (~a,~b))
    95 		       else 
    96 			  if 0<b then negDivAlg (a,b)
    97 			  else        negateSnd (posDivAlg (~a,~b));
    98 \end{verbatim}
    99 *}
   100 
   101 
   102 
   103 subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
   104 
   105 lemma unique_quotient_lemma:
   106      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
   107       ==> q' \<le> (q::int)"
   108 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
   109  prefer 2 apply (simp add: right_diff_distrib)
   110 apply (subgoal_tac "0 < b * (1 + q - q') ")
   111 apply (erule_tac [2] order_le_less_trans)
   112  prefer 2 apply (simp add: right_diff_distrib right_distrib)
   113 apply (subgoal_tac "b * q' < b * (1 + q) ")
   114  prefer 2 apply (simp add: right_diff_distrib right_distrib)
   115 apply (simp add: mult_less_cancel_left)
   116 done
   117 
   118 lemma unique_quotient_lemma_neg:
   119      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
   120       ==> q \<le> (q'::int)"
   121 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
   122     auto)
   123 
   124 lemma unique_quotient:
   125      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b \<noteq> 0 |]  
   126       ==> q = q'"
   127 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)
   128 apply (blast intro: order_antisym
   129              dest: order_eq_refl [THEN unique_quotient_lemma] 
   130              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
   131 done
   132 
   133 
   134 lemma unique_remainder:
   135      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b \<noteq> 0 |]  
   136       ==> r = r'"
   137 apply (subgoal_tac "q = q'")
   138  apply (simp add: quorem_def)
   139 apply (blast intro: unique_quotient)
   140 done
   141 
   142 
   143 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
   144 
   145 text{*And positive divisors*}
   146 
   147 lemma adjust_eq [simp]:
   148      "adjust b (q,r) = 
   149       (let diff = r-b in  
   150 	if 0 \<le> diff then (2*q + 1, diff)   
   151                      else (2*q, r))"
   152 by (simp add: Let_def adjust_def)
   153 
   154 declare posDivAlg.simps [simp del]
   155 
   156 text{*use with a simproc to avoid repeatedly proving the premise*}
   157 lemma posDivAlg_eqn:
   158      "0 < b ==>  
   159       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
   160 by (rule posDivAlg.simps [THEN trans], simp)
   161 
   162 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
   163 theorem posDivAlg_correct:
   164   assumes "0 \<le> a" and "0 < b"
   165   shows "quorem ((a, b), posDivAlg a b)"
   166 using prems apply (induct a b rule: posDivAlg.induct)
   167 apply auto
   168 apply (simp add: quorem_def)
   169 apply (subst posDivAlg_eqn, simp add: right_distrib)
   170 apply (case_tac "a < b")
   171 apply simp_all
   172 apply (erule splitE)
   173 apply (auto simp add: right_distrib Let_def)
   174 done
   175 
   176 
   177 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
   178 
   179 text{*And positive divisors*}
   180 
   181 declare negDivAlg.simps [simp del]
   182 
   183 text{*use with a simproc to avoid repeatedly proving the premise*}
   184 lemma negDivAlg_eqn:
   185      "0 < b ==>  
   186       negDivAlg a b =       
   187        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
   188 by (rule negDivAlg.simps [THEN trans], simp)
   189 
   190 (*Correctness of negDivAlg: it computes quotients correctly
   191   It doesn't work if a=0 because the 0/b equals 0, not -1*)
   192 lemma negDivAlg_correct:
   193   assumes "a < 0" and "b > 0"
   194   shows "quorem ((a, b), negDivAlg a b)"
   195 using prems apply (induct a b rule: negDivAlg.induct)
   196 apply (auto simp add: linorder_not_le)
   197 apply (simp add: quorem_def)
   198 apply (subst negDivAlg_eqn, assumption)
   199 apply (case_tac "a + b < (0\<Colon>int)")
   200 apply simp_all
   201 apply (erule splitE)
   202 apply (auto simp add: right_distrib Let_def)
   203 done
   204 
   205 
   206 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
   207 
   208 (*the case a=0*)
   209 lemma quorem_0: "b \<noteq> 0 ==> quorem ((0,b), (0,0))"
   210 by (auto simp add: quorem_def linorder_neq_iff)
   211 
   212 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
   213 by (subst posDivAlg.simps, auto)
   214 
   215 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
   216 by (subst negDivAlg.simps, auto)
   217 
   218 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
   219 by (simp add: negateSnd_def)
   220 
   221 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"
   222 by (auto simp add: split_ifs quorem_def)
   223 
   224 lemma divAlg_correct: "b \<noteq> 0 ==> quorem ((a,b), divAlg (a, b))"
   225 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg
   226                     posDivAlg_correct negDivAlg_correct)
   227 
   228 text{*Arbitrary definitions for division by zero.  Useful to simplify 
   229     certain equations.*}
   230 
   231 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
   232 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)  
   233 
   234 
   235 text{*Basic laws about division and remainder*}
   236 
   237 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
   238 apply (case_tac "b = 0", simp)
   239 apply (cut_tac a = a and b = b in divAlg_correct)
   240 apply (auto simp add: quorem_def div_def mod_def)
   241 done
   242 
   243 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
   244 by(simp add: zmod_zdiv_equality[symmetric])
   245 
   246 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
   247 by(simp add: mult_commute zmod_zdiv_equality[symmetric])
   248 
   249 text {* Tool setup *}
   250 
   251 ML_setup {*
   252 local 
   253 
   254 structure CancelDivMod = CancelDivModFun(
   255 struct
   256   val div_name = @{const_name Divides.div};
   257   val mod_name = @{const_name Divides.mod};
   258   val mk_binop = HOLogic.mk_binop;
   259   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
   260   val dest_sum = Int_Numeral_Simprocs.dest_sum;
   261   val div_mod_eqs =
   262     map mk_meta_eq [@{thm zdiv_zmod_equality},
   263       @{thm zdiv_zmod_equality2}];
   264   val trans = trans;
   265   val prove_eq_sums =
   266     let
   267       val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
   268     in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
   269 end)
   270 
   271 in
   272 
   273 val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc
   274   ("cancel_zdiv_zmod", ["(m::int) + n"], K CancelDivMod.proc)
   275 
   276 end;
   277 
   278 Addsimprocs [cancel_zdiv_zmod_proc]
   279 *}
   280 
   281 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
   282 apply (cut_tac a = a and b = b in divAlg_correct)
   283 apply (auto simp add: quorem_def mod_def)
   284 done
   285 
   286 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
   287    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
   288 
   289 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
   290 apply (cut_tac a = a and b = b in divAlg_correct)
   291 apply (auto simp add: quorem_def div_def mod_def)
   292 done
   293 
   294 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
   295    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
   296 
   297 
   298 
   299 subsection{*General Properties of div and mod*}
   300 
   301 lemma quorem_div_mod: "b \<noteq> 0 ==> quorem ((a, b), (a div b, a mod b))"
   302 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   303 apply (force simp add: quorem_def linorder_neq_iff)
   304 done
   305 
   306 lemma quorem_div: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a div b = q"
   307 by (simp add: quorem_div_mod [THEN unique_quotient])
   308 
   309 lemma quorem_mod: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a mod b = r"
   310 by (simp add: quorem_div_mod [THEN unique_remainder])
   311 
   312 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
   313 apply (rule quorem_div)
   314 apply (auto simp add: quorem_def)
   315 done
   316 
   317 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
   318 apply (rule quorem_div)
   319 apply (auto simp add: quorem_def)
   320 done
   321 
   322 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
   323 apply (rule quorem_div)
   324 apply (auto simp add: quorem_def)
   325 done
   326 
   327 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
   328 
   329 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
   330 apply (rule_tac q = 0 in quorem_mod)
   331 apply (auto simp add: quorem_def)
   332 done
   333 
   334 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
   335 apply (rule_tac q = 0 in quorem_mod)
   336 apply (auto simp add: quorem_def)
   337 done
   338 
   339 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
   340 apply (rule_tac q = "-1" in quorem_mod)
   341 apply (auto simp add: quorem_def)
   342 done
   343 
   344 text{*There is no @{text mod_neg_pos_trivial}.*}
   345 
   346 
   347 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
   348 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
   349 apply (case_tac "b = 0", simp)
   350 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, 
   351                                  THEN quorem_div, THEN sym])
   352 
   353 done
   354 
   355 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
   356 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
   357 apply (case_tac "b = 0", simp)
   358 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
   359        auto)
   360 done
   361 
   362 
   363 subsection{*Laws for div and mod with Unary Minus*}
   364 
   365 lemma zminus1_lemma:
   366      "quorem((a,b),(q,r))  
   367       ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
   368                           (if r=0 then 0 else b-r))"
   369 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
   370 
   371 
   372 lemma zdiv_zminus1_eq_if:
   373      "b \<noteq> (0::int)  
   374       ==> (-a) div b =  
   375           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   376 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
   377 
   378 lemma zmod_zminus1_eq_if:
   379      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
   380 apply (case_tac "b = 0", simp)
   381 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
   382 done
   383 
   384 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
   385 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
   386 
   387 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
   388 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
   389 
   390 lemma zdiv_zminus2_eq_if:
   391      "b \<noteq> (0::int)  
   392       ==> a div (-b) =  
   393           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   394 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
   395 
   396 lemma zmod_zminus2_eq_if:
   397      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
   398 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
   399 
   400 
   401 subsection{*Division of a Number by Itself*}
   402 
   403 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
   404 apply (subgoal_tac "0 < a*q")
   405  apply (simp add: zero_less_mult_iff, arith)
   406 done
   407 
   408 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
   409 apply (subgoal_tac "0 \<le> a* (1-q) ")
   410  apply (simp add: zero_le_mult_iff)
   411 apply (simp add: right_diff_distrib)
   412 done
   413 
   414 lemma self_quotient: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> q = 1"
   415 apply (simp add: split_ifs quorem_def linorder_neq_iff)
   416 apply (rule order_antisym, safe, simp_all)
   417 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
   418 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
   419 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
   420 done
   421 
   422 lemma self_remainder: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> r = 0"
   423 apply (frule self_quotient, assumption)
   424 apply (simp add: quorem_def)
   425 done
   426 
   427 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
   428 by (simp add: quorem_div_mod [THEN self_quotient])
   429 
   430 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
   431 lemma zmod_self [simp]: "a mod a = (0::int)"
   432 apply (case_tac "a = 0", simp)
   433 apply (simp add: quorem_div_mod [THEN self_remainder])
   434 done
   435 
   436 
   437 subsection{*Computation of Division and Remainder*}
   438 
   439 lemma zdiv_zero [simp]: "(0::int) div b = 0"
   440 by (simp add: div_def divAlg_def)
   441 
   442 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
   443 by (simp add: div_def divAlg_def)
   444 
   445 lemma zmod_zero [simp]: "(0::int) mod b = 0"
   446 by (simp add: mod_def divAlg_def)
   447 
   448 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
   449 by (simp add: div_def divAlg_def)
   450 
   451 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
   452 by (simp add: mod_def divAlg_def)
   453 
   454 text{*a positive, b positive *}
   455 
   456 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
   457 by (simp add: div_def divAlg_def)
   458 
   459 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
   460 by (simp add: mod_def divAlg_def)
   461 
   462 text{*a negative, b positive *}
   463 
   464 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
   465 by (simp add: div_def divAlg_def)
   466 
   467 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
   468 by (simp add: mod_def divAlg_def)
   469 
   470 text{*a positive, b negative *}
   471 
   472 lemma div_pos_neg:
   473      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
   474 by (simp add: div_def divAlg_def)
   475 
   476 lemma mod_pos_neg:
   477      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
   478 by (simp add: mod_def divAlg_def)
   479 
   480 text{*a negative, b negative *}
   481 
   482 lemma div_neg_neg:
   483      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
   484 by (simp add: div_def divAlg_def)
   485 
   486 lemma mod_neg_neg:
   487      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
   488 by (simp add: mod_def divAlg_def)
   489 
   490 text {*Simplify expresions in which div and mod combine numerical constants*}
   491 
   492 lemmas div_pos_pos_number_of [simp] =
   493     div_pos_pos [of "number_of v" "number_of w", standard]
   494 
   495 lemmas div_neg_pos_number_of [simp] =
   496     div_neg_pos [of "number_of v" "number_of w", standard]
   497 
   498 lemmas div_pos_neg_number_of [simp] =
   499     div_pos_neg [of "number_of v" "number_of w", standard]
   500 
   501 lemmas div_neg_neg_number_of [simp] =
   502     div_neg_neg [of "number_of v" "number_of w", standard]
   503 
   504 
   505 lemmas mod_pos_pos_number_of [simp] =
   506     mod_pos_pos [of "number_of v" "number_of w", standard]
   507 
   508 lemmas mod_neg_pos_number_of [simp] =
   509     mod_neg_pos [of "number_of v" "number_of w", standard]
   510 
   511 lemmas mod_pos_neg_number_of [simp] =
   512     mod_pos_neg [of "number_of v" "number_of w", standard]
   513 
   514 lemmas mod_neg_neg_number_of [simp] =
   515     mod_neg_neg [of "number_of v" "number_of w", standard]
   516 
   517 
   518 lemmas posDivAlg_eqn_number_of [simp] =
   519     posDivAlg_eqn [of "number_of v" "number_of w", standard]
   520 
   521 lemmas negDivAlg_eqn_number_of [simp] =
   522     negDivAlg_eqn [of "number_of v" "number_of w", standard]
   523 
   524 
   525 text{*Special-case simplification *}
   526 
   527 lemma zmod_1 [simp]: "a mod (1::int) = 0"
   528 apply (cut_tac a = a and b = 1 in pos_mod_sign)
   529 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
   530 apply (auto simp del:pos_mod_bound pos_mod_sign)
   531 done
   532 
   533 lemma zdiv_1 [simp]: "a div (1::int) = a"
   534 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
   535 
   536 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
   537 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
   538 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
   539 apply (auto simp del: neg_mod_sign neg_mod_bound)
   540 done
   541 
   542 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
   543 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
   544 
   545 (** The last remaining special cases for constant arithmetic:
   546     1 div z and 1 mod z **)
   547 
   548 lemmas div_pos_pos_1_number_of [simp] =
   549     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
   550 
   551 lemmas div_pos_neg_1_number_of [simp] =
   552     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
   553 
   554 lemmas mod_pos_pos_1_number_of [simp] =
   555     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
   556 
   557 lemmas mod_pos_neg_1_number_of [simp] =
   558     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
   559 
   560 
   561 lemmas posDivAlg_eqn_1_number_of [simp] =
   562     posDivAlg_eqn [of concl: 1 "number_of w", standard]
   563 
   564 lemmas negDivAlg_eqn_1_number_of [simp] =
   565     negDivAlg_eqn [of concl: 1 "number_of w", standard]
   566 
   567 
   568 
   569 subsection{*Monotonicity in the First Argument (Dividend)*}
   570 
   571 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
   572 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   573 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   574 apply (rule unique_quotient_lemma)
   575 apply (erule subst)
   576 apply (erule subst, simp_all)
   577 done
   578 
   579 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
   580 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   581 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   582 apply (rule unique_quotient_lemma_neg)
   583 apply (erule subst)
   584 apply (erule subst, simp_all)
   585 done
   586 
   587 
   588 subsection{*Monotonicity in the Second Argument (Divisor)*}
   589 
   590 lemma q_pos_lemma:
   591      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
   592 apply (subgoal_tac "0 < b'* (q' + 1) ")
   593  apply (simp add: zero_less_mult_iff)
   594 apply (simp add: right_distrib)
   595 done
   596 
   597 lemma zdiv_mono2_lemma:
   598      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
   599          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
   600       ==> q \<le> (q'::int)"
   601 apply (frule q_pos_lemma, assumption+) 
   602 apply (subgoal_tac "b*q < b* (q' + 1) ")
   603  apply (simp add: mult_less_cancel_left)
   604 apply (subgoal_tac "b*q = r' - r + b'*q'")
   605  prefer 2 apply simp
   606 apply (simp (no_asm_simp) add: right_distrib)
   607 apply (subst add_commute, rule zadd_zless_mono, arith)
   608 apply (rule mult_right_mono, auto)
   609 done
   610 
   611 lemma zdiv_mono2:
   612      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
   613 apply (subgoal_tac "b \<noteq> 0")
   614  prefer 2 apply arith
   615 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   616 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   617 apply (rule zdiv_mono2_lemma)
   618 apply (erule subst)
   619 apply (erule subst, simp_all)
   620 done
   621 
   622 lemma q_neg_lemma:
   623      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
   624 apply (subgoal_tac "b'*q' < 0")
   625  apply (simp add: mult_less_0_iff, arith)
   626 done
   627 
   628 lemma zdiv_mono2_neg_lemma:
   629      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
   630          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
   631       ==> q' \<le> (q::int)"
   632 apply (frule q_neg_lemma, assumption+) 
   633 apply (subgoal_tac "b*q' < b* (q + 1) ")
   634  apply (simp add: mult_less_cancel_left)
   635 apply (simp add: right_distrib)
   636 apply (subgoal_tac "b*q' \<le> b'*q'")
   637  prefer 2 apply (simp add: mult_right_mono_neg, arith)
   638 done
   639 
   640 lemma zdiv_mono2_neg:
   641      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
   642 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   643 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   644 apply (rule zdiv_mono2_neg_lemma)
   645 apply (erule subst)
   646 apply (erule subst, simp_all)
   647 done
   648 
   649 subsection{*More Algebraic Laws for div and mod*}
   650 
   651 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
   652 
   653 lemma zmult1_lemma:
   654      "[| quorem((b,c),(q,r));  c \<noteq> 0 |]  
   655       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   656 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
   657 
   658 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
   659 apply (case_tac "c = 0", simp)
   660 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
   661 done
   662 
   663 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
   664 apply (case_tac "c = 0", simp)
   665 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
   666 done
   667 
   668 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
   669 apply (rule trans)
   670 apply (rule_tac s = "b*a mod c" in trans)
   671 apply (rule_tac [2] zmod_zmult1_eq)
   672 apply (simp_all add: mult_commute)
   673 done
   674 
   675 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
   676 apply (rule zmod_zmult1_eq' [THEN trans])
   677 apply (rule zmod_zmult1_eq)
   678 done
   679 
   680 lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
   681 by (simp add: zdiv_zmult1_eq)
   682 
   683 lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
   684 by (subst mult_commute, erule zdiv_zmult_self1)
   685 
   686 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
   687 by (simp add: zmod_zmult1_eq)
   688 
   689 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
   690 by (simp add: mult_commute zmod_zmult1_eq)
   691 
   692 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
   693 proof
   694   assume "m mod d = 0"
   695   with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
   696 next
   697   assume "EX q::int. m = d*q"
   698   thus "m mod d = 0" by auto
   699 qed
   700 
   701 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
   702 
   703 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
   704 
   705 lemma zadd1_lemma:
   706      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c \<noteq> 0 |]  
   707       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   708 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
   709 
   710 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   711 lemma zdiv_zadd1_eq:
   712      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
   713 apply (case_tac "c = 0", simp)
   714 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
   715 done
   716 
   717 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
   718 apply (case_tac "c = 0", simp)
   719 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
   720 done
   721 
   722 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
   723 apply (case_tac "b = 0", simp)
   724 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
   725 done
   726 
   727 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
   728 apply (case_tac "b = 0", simp)
   729 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
   730 done
   731 
   732 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
   733 apply (rule trans [symmetric])
   734 apply (rule zmod_zadd1_eq, simp)
   735 apply (rule zmod_zadd1_eq [symmetric])
   736 done
   737 
   738 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
   739 apply (rule trans [symmetric])
   740 apply (rule zmod_zadd1_eq, simp)
   741 apply (rule zmod_zadd1_eq [symmetric])
   742 done
   743 
   744 lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
   745 by (simp add: zdiv_zadd1_eq)
   746 
   747 lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
   748 by (simp add: zdiv_zadd1_eq)
   749 
   750 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
   751 apply (case_tac "a = 0", simp)
   752 apply (simp add: zmod_zadd1_eq)
   753 done
   754 
   755 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
   756 apply (case_tac "a = 0", simp)
   757 apply (simp add: zmod_zadd1_eq)
   758 done
   759 
   760 
   761 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
   762 
   763 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
   764   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
   765   to cause particular problems.*)
   766 
   767 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
   768 
   769 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
   770 apply (subgoal_tac "b * (c - q mod c) < r * 1")
   771 apply (simp add: right_diff_distrib)
   772 apply (rule order_le_less_trans)
   773 apply (erule_tac [2] mult_strict_right_mono)
   774 apply (rule mult_left_mono_neg)
   775 apply (auto simp add: compare_rls add_commute [of 1]
   776                       add1_zle_eq pos_mod_bound)
   777 done
   778 
   779 lemma zmult2_lemma_aux2:
   780      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
   781 apply (subgoal_tac "b * (q mod c) \<le> 0")
   782  apply arith
   783 apply (simp add: mult_le_0_iff)
   784 done
   785 
   786 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
   787 apply (subgoal_tac "0 \<le> b * (q mod c) ")
   788 apply arith
   789 apply (simp add: zero_le_mult_iff)
   790 done
   791 
   792 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
   793 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
   794 apply (simp add: right_diff_distrib)
   795 apply (rule order_less_le_trans)
   796 apply (erule mult_strict_right_mono)
   797 apply (rule_tac [2] mult_left_mono)
   798 apply (auto simp add: compare_rls add_commute [of 1]
   799                       add1_zle_eq pos_mod_bound)
   800 done
   801 
   802 lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b \<noteq> 0;  0 < c |]  
   803       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   804 by (auto simp add: mult_ac quorem_def linorder_neq_iff
   805                    zero_less_mult_iff right_distrib [symmetric] 
   806                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
   807 
   808 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
   809 apply (case_tac "b = 0", simp)
   810 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
   811 done
   812 
   813 lemma zmod_zmult2_eq:
   814      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
   815 apply (case_tac "b = 0", simp)
   816 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
   817 done
   818 
   819 
   820 subsection{*Cancellation of Common Factors in div*}
   821 
   822 lemma zdiv_zmult_zmult1_aux1:
   823      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
   824 by (subst zdiv_zmult2_eq, auto)
   825 
   826 lemma zdiv_zmult_zmult1_aux2:
   827      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
   828 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
   829 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
   830 done
   831 
   832 lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
   833 apply (case_tac "b = 0", simp)
   834 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
   835 done
   836 
   837 lemma zdiv_zmult_zmult1_if[simp]:
   838   "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
   839 by (simp add:zdiv_zmult_zmult1)
   840 
   841 (*
   842 lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
   843 apply (drule zdiv_zmult_zmult1)
   844 apply (auto simp add: mult_commute)
   845 done
   846 *)
   847 
   848 
   849 subsection{*Distribution of Factors over mod*}
   850 
   851 lemma zmod_zmult_zmult1_aux1:
   852      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
   853 by (subst zmod_zmult2_eq, auto)
   854 
   855 lemma zmod_zmult_zmult1_aux2:
   856      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
   857 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
   858 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
   859 done
   860 
   861 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
   862 apply (case_tac "b = 0", simp)
   863 apply (case_tac "c = 0", simp)
   864 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
   865 done
   866 
   867 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
   868 apply (cut_tac c = c in zmod_zmult_zmult1)
   869 apply (auto simp add: mult_commute)
   870 done
   871 
   872 
   873 subsection {*Splitting Rules for div and mod*}
   874 
   875 text{*The proofs of the two lemmas below are essentially identical*}
   876 
   877 lemma split_pos_lemma:
   878  "0<k ==> 
   879     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
   880 apply (rule iffI, clarify)
   881  apply (erule_tac P="P ?x ?y" in rev_mp)  
   882  apply (subst zmod_zadd1_eq) 
   883  apply (subst zdiv_zadd1_eq) 
   884  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
   885 txt{*converse direction*}
   886 apply (drule_tac x = "n div k" in spec) 
   887 apply (drule_tac x = "n mod k" in spec, simp)
   888 done
   889 
   890 lemma split_neg_lemma:
   891  "k<0 ==>
   892     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
   893 apply (rule iffI, clarify)
   894  apply (erule_tac P="P ?x ?y" in rev_mp)  
   895  apply (subst zmod_zadd1_eq) 
   896  apply (subst zdiv_zadd1_eq) 
   897  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
   898 txt{*converse direction*}
   899 apply (drule_tac x = "n div k" in spec) 
   900 apply (drule_tac x = "n mod k" in spec, simp)
   901 done
   902 
   903 lemma split_zdiv:
   904  "P(n div k :: int) =
   905   ((k = 0 --> P 0) & 
   906    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
   907    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
   908 apply (case_tac "k=0", simp)
   909 apply (simp only: linorder_neq_iff)
   910 apply (erule disjE) 
   911  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
   912                       split_neg_lemma [of concl: "%x y. P x"])
   913 done
   914 
   915 lemma split_zmod:
   916  "P(n mod k :: int) =
   917   ((k = 0 --> P n) & 
   918    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
   919    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
   920 apply (case_tac "k=0", simp)
   921 apply (simp only: linorder_neq_iff)
   922 apply (erule disjE) 
   923  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
   924                       split_neg_lemma [of concl: "%x y. P y"])
   925 done
   926 
   927 (* Enable arith to deal with div 2 and mod 2: *)
   928 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
   929 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
   930 
   931 
   932 subsection{*Speeding up the Division Algorithm with Shifting*}
   933 
   934 text{*computing div by shifting *}
   935 
   936 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
   937 proof cases
   938   assume "a=0"
   939     thus ?thesis by simp
   940 next
   941   assume "a\<noteq>0" and le_a: "0\<le>a"   
   942   hence a_pos: "1 \<le> a" by arith
   943   hence one_less_a2: "1 < 2*a" by arith
   944   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
   945     by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
   946   with a_pos have "0 \<le> b mod a" by simp
   947   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
   948     by (simp add: mod_pos_pos_trivial one_less_a2)
   949   with  le_2a
   950   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
   951     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
   952                   right_distrib) 
   953   thus ?thesis
   954     by (subst zdiv_zadd1_eq,
   955         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
   956                   div_pos_pos_trivial)
   957 qed
   958 
   959 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
   960 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
   961 apply (rule_tac [2] pos_zdiv_mult_2)
   962 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
   963 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
   964 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
   965        simp) 
   966 done
   967 
   968 
   969 (*Not clear why this must be proved separately; probably number_of causes
   970   simplification problems*)
   971 lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
   972 by auto
   973 
   974 lemma zdiv_number_of_BIT[simp]:
   975      "number_of (v BIT b) div number_of (w BIT bit.B0) =  
   976           (if b=bit.B0 | (0::int) \<le> number_of w                    
   977            then number_of v div (number_of w)     
   978            else (number_of v + (1::int)) div (number_of w))"
   979 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
   980 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac 
   981             split: bit.split)
   982 done
   983 
   984 
   985 subsection{*Computing mod by Shifting (proofs resemble those for div)*}
   986 
   987 lemma pos_zmod_mult_2:
   988      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
   989 apply (case_tac "a = 0", simp)
   990 apply (subgoal_tac "1 < a * 2")
   991  prefer 2 apply arith
   992 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
   993  apply (rule_tac [2] mult_left_mono)
   994 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
   995                       pos_mod_bound)
   996 apply (subst zmod_zadd1_eq)
   997 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
   998 apply (rule mod_pos_pos_trivial)
   999 apply (auto simp add: mod_pos_pos_trivial left_distrib)
  1000 apply (subgoal_tac "0 \<le> b mod a", arith, simp)
  1001 done
  1002 
  1003 lemma neg_zmod_mult_2:
  1004      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
  1005 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
  1006                     1 + 2* ((-b - 1) mod (-a))")
  1007 apply (rule_tac [2] pos_zmod_mult_2)
  1008 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
  1009 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
  1010  prefer 2 apply simp 
  1011 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
  1012 done
  1013 
  1014 lemma zmod_number_of_BIT [simp]:
  1015      "number_of (v BIT b) mod number_of (w BIT bit.B0) =  
  1016       (case b of
  1017           bit.B0 => 2 * (number_of v mod number_of w)
  1018         | bit.B1 => if (0::int) \<le> number_of w  
  1019                 then 2 * (number_of v mod number_of w) + 1     
  1020                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
  1021 apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split) 
  1022 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
  1023                  not_0_le_lemma neg_zmod_mult_2 add_ac)
  1024 done
  1025 
  1026 
  1027 subsection{*Quotients of Signs*}
  1028 
  1029 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
  1030 apply (subgoal_tac "a div b \<le> -1", force)
  1031 apply (rule order_trans)
  1032 apply (rule_tac a' = "-1" in zdiv_mono1)
  1033 apply (auto simp add: zdiv_minus1)
  1034 done
  1035 
  1036 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
  1037 by (drule zdiv_mono1_neg, auto)
  1038 
  1039 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
  1040 apply auto
  1041 apply (drule_tac [2] zdiv_mono1)
  1042 apply (auto simp add: linorder_neq_iff)
  1043 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
  1044 apply (blast intro: div_neg_pos_less0)
  1045 done
  1046 
  1047 lemma neg_imp_zdiv_nonneg_iff:
  1048      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
  1049 apply (subst zdiv_zminus_zminus [symmetric])
  1050 apply (subst pos_imp_zdiv_nonneg_iff, auto)
  1051 done
  1052 
  1053 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
  1054 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
  1055 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
  1056 
  1057 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
  1058 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
  1059 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
  1060 
  1061 
  1062 subsection {* The Divides Relation *}
  1063 
  1064 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
  1065 by(simp add:dvd_def zmod_eq_0_iff)
  1066 
  1067 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
  1068   zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
  1069 
  1070 lemma zdvd_0_right [iff]: "(m::int) dvd 0"
  1071 by (simp add: dvd_def)
  1072 
  1073 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
  1074   by (simp add: dvd_def)
  1075 
  1076 lemma zdvd_1_left [iff]: "1 dvd (m::int)"
  1077   by (simp add: dvd_def)
  1078 
  1079 lemma zdvd_refl [simp]: "m dvd (m::int)"
  1080 by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
  1081 
  1082 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
  1083 by (auto simp add: dvd_def intro: mult_assoc)
  1084 
  1085 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
  1086   apply (simp add: dvd_def, auto)
  1087    apply (rule_tac [!] x = "-k" in exI, auto)
  1088   done
  1089 
  1090 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
  1091   apply (simp add: dvd_def, auto)
  1092    apply (rule_tac [!] x = "-k" in exI, auto)
  1093   done
  1094 lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)" 
  1095   apply (cases "i > 0", simp)
  1096   apply (simp add: dvd_def)
  1097   apply (rule iffI)
  1098   apply (erule exE)
  1099   apply (rule_tac x="- k" in exI, simp)
  1100   apply (erule exE)
  1101   apply (rule_tac x="- k" in exI, simp)
  1102   done
  1103 lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)" 
  1104   apply (cases "j > 0", simp)
  1105   apply (simp add: dvd_def)
  1106   apply (rule iffI)
  1107   apply (erule exE)
  1108   apply (rule_tac x="- k" in exI, simp)
  1109   apply (erule exE)
  1110   apply (rule_tac x="- k" in exI, simp)
  1111   done
  1112 
  1113 lemma zdvd_anti_sym:
  1114     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
  1115   apply (simp add: dvd_def, auto)
  1116   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
  1117   done
  1118 
  1119 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
  1120   apply (simp add: dvd_def)
  1121   apply (blast intro: right_distrib [symmetric])
  1122   done
  1123 
  1124 lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a" 
  1125   shows "\<bar>a\<bar> = \<bar>b\<bar>"
  1126 proof-
  1127   from ab obtain k where k:"b = a*k" unfolding dvd_def by blast 
  1128   from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
  1129   from k k' have "a = a*k*k'" by simp
  1130   with mult_cancel_left1[where c="a" and b="k*k'"]
  1131   have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
  1132   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
  1133   thus ?thesis using k k' by auto
  1134 qed
  1135 
  1136 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
  1137   apply (simp add: dvd_def)
  1138   apply (blast intro: right_diff_distrib [symmetric])
  1139   done
  1140 
  1141 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
  1142   apply (subgoal_tac "m = n + (m - n)")
  1143    apply (erule ssubst)
  1144    apply (blast intro: zdvd_zadd, simp)
  1145   done
  1146 
  1147 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
  1148   apply (simp add: dvd_def)
  1149   apply (blast intro: mult_left_commute)
  1150   done
  1151 
  1152 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
  1153   apply (subst mult_commute)
  1154   apply (erule zdvd_zmult)
  1155   done
  1156 
  1157 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"
  1158   apply (rule zdvd_zmult)
  1159   apply (rule zdvd_refl)
  1160   done
  1161 
  1162 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"
  1163   apply (rule zdvd_zmult2)
  1164   apply (rule zdvd_refl)
  1165   done
  1166 
  1167 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
  1168   apply (simp add: dvd_def)
  1169   apply (simp add: mult_assoc, blast)
  1170   done
  1171 
  1172 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
  1173   apply (rule zdvd_zmultD2)
  1174   apply (subst mult_commute, assumption)
  1175   done
  1176 
  1177 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
  1178   apply (simp add: dvd_def, clarify)
  1179   apply (rule_tac x = "k * ka" in exI)
  1180   apply (simp add: mult_ac)
  1181   done
  1182 
  1183 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
  1184   apply (rule iffI)
  1185    apply (erule_tac [2] zdvd_zadd)
  1186    apply (subgoal_tac "n = (n + k * m) - k * m")
  1187     apply (erule ssubst)
  1188     apply (erule zdvd_zdiff, simp_all)
  1189   done
  1190 
  1191 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  1192   apply (simp add: dvd_def)
  1193   apply (auto simp add: zmod_zmult_zmult1)
  1194   done
  1195 
  1196 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  1197   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
  1198    apply (simp add: zmod_zdiv_equality [symmetric])
  1199   apply (simp only: zdvd_zadd zdvd_zmult2)
  1200   done
  1201 
  1202 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
  1203   apply (simp add: dvd_def, auto)
  1204   apply (subgoal_tac "0 < n")
  1205    prefer 2
  1206    apply (blast intro: order_less_trans)
  1207   apply (simp add: zero_less_mult_iff)
  1208   apply (subgoal_tac "n * k < n * 1")
  1209    apply (drule mult_less_cancel_left [THEN iffD1], auto)
  1210   done
  1211 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  1212   using zmod_zdiv_equality[where a="m" and b="n"]
  1213   by (simp add: ring_eq_simps)
  1214 
  1215 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
  1216 apply (subgoal_tac "m mod n = 0")
  1217  apply (simp add: zmult_div_cancel)
  1218 apply (simp only: zdvd_iff_zmod_eq_0)
  1219 done
  1220 
  1221 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
  1222   shows "m dvd n"
  1223 proof-
  1224   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
  1225   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
  1226     with h have False by (simp add: mult_assoc)}
  1227   hence "n = m * h" by blast
  1228   thus ?thesis by blast
  1229 qed
  1230 
  1231 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
  1232   apply (simp split add: split_nat)
  1233   apply (rule iffI)
  1234   apply (erule exE)
  1235   apply (rule_tac x = "int x" in exI)
  1236   apply simp
  1237   apply (erule exE)
  1238   apply (rule_tac x = "nat x" in exI)
  1239   apply (erule conjE)
  1240   apply (erule_tac x = "nat x" in allE)
  1241   apply simp
  1242   done
  1243 
  1244 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
  1245   unfolding dvd_def
  1246   apply (rule_tac s="\<exists>k. int y = int x * int k" in trans)
  1247   apply (simp only: of_nat_mult [symmetric] of_nat_eq_iff)
  1248   apply (simp add: ex_nat cong add: conj_cong)
  1249   apply (rule iffI)
  1250   apply iprover
  1251   apply (erule exE)
  1252   apply (case_tac "x=0")
  1253   apply (rule_tac x=0 in exI)
  1254   apply simp
  1255   apply (case_tac "0 \<le> k")
  1256   apply iprover
  1257   apply (simp add: linorder_not_le)
  1258   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF of_nat_0_less_iff]])
  1259   apply assumption
  1260   apply (simp add: mult_ac)
  1261   done
  1262 
  1263 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
  1264 proof
  1265   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
  1266   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
  1267   hence "nat \<bar>x\<bar> = 1"  by simp
  1268   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
  1269 next
  1270   assume "\<bar>x\<bar>=1" thus "x dvd 1" 
  1271     by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
  1272 qed
  1273 lemma zdvd_mult_cancel1: 
  1274   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
  1275 proof
  1276   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
  1277     by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
  1278 next
  1279   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
  1280   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
  1281 qed
  1282 
  1283 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
  1284   apply (auto simp add: dvd_def nat_abs_mult_distrib)
  1285   apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
  1286    apply (rule_tac x = "-(int k)" in exI)
  1287   apply auto
  1288   done
  1289 
  1290 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
  1291   apply (auto simp add: dvd_def abs_if)
  1292     apply (rule_tac [3] x = "nat k" in exI)
  1293     apply (rule_tac [2] x = "-(int k)" in exI)
  1294     apply (rule_tac x = "nat (-k)" in exI)
  1295     apply (cut_tac [3] m = m in int_less_0_conv)
  1296     apply (cut_tac m = m in int_less_0_conv)
  1297     apply (auto simp add: zero_le_mult_iff mult_less_0_iff
  1298       nat_mult_distrib [symmetric] nat_eq_iff2)
  1299   done
  1300 
  1301 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
  1302   apply (auto simp add: dvd_def)
  1303   apply (rule_tac x = "nat k" in exI)
  1304   apply (cut_tac m = m in int_less_0_conv)
  1305   apply (auto simp add: zero_le_mult_iff mult_less_0_iff
  1306     nat_mult_distrib [symmetric] nat_eq_iff2)
  1307   done
  1308 
  1309 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
  1310   apply (auto simp add: dvd_def)
  1311    apply (rule_tac [!] x = "-k" in exI, auto)
  1312   done
  1313 
  1314 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
  1315   apply (auto simp add: dvd_def)
  1316    apply (drule minus_equation_iff [THEN iffD1])
  1317    apply (rule_tac [!] x = "-k" in exI, auto)
  1318   done
  1319 
  1320 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
  1321   apply (rule_tac z=n in int_cases)
  1322   apply (auto simp add: dvd_int_iff)
  1323   apply (rule_tac z=z in int_cases)
  1324   apply (auto simp add: dvd_imp_le)
  1325   done
  1326 
  1327 
  1328 subsection{*Integer Powers*} 
  1329 
  1330 instance int :: power ..
  1331 
  1332 primrec
  1333   "p ^ 0 = 1"
  1334   "p ^ (Suc n) = (p::int) * (p ^ n)"
  1335 
  1336 
  1337 instance int :: recpower
  1338 proof
  1339   fix z :: int
  1340   fix n :: nat
  1341   show "z^0 = 1" by simp
  1342   show "z^(Suc n) = z * (z^n)" by simp
  1343 qed
  1344 
  1345 
  1346 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
  1347 apply (induct "y", auto)
  1348 apply (rule zmod_zmult1_eq [THEN trans])
  1349 apply (simp (no_asm_simp))
  1350 apply (rule zmod_zmult_distrib [symmetric])
  1351 done
  1352 
  1353 lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
  1354   by (rule Power.power_add)
  1355 
  1356 lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
  1357   by (rule Power.power_mult [symmetric])
  1358 
  1359 lemma zero_less_zpower_abs_iff [simp]:
  1360      "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
  1361 apply (induct "n")
  1362 apply (auto simp add: zero_less_mult_iff)
  1363 done
  1364 
  1365 lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
  1366 apply (induct "n")
  1367 apply (auto simp add: zero_le_mult_iff)
  1368 done
  1369 
  1370 lemma int_power: "int (m^n) = (int m) ^ n"
  1371   by (rule of_nat_power)
  1372 
  1373 text{*Compatibility binding*}
  1374 lemmas zpower_int = int_power [symmetric]
  1375 
  1376 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
  1377 apply (subst split_div, auto)
  1378 apply (subst split_zdiv, auto)
  1379 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
  1380 apply (auto simp add: IntDiv.quorem_def)
  1381 done
  1382 
  1383 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
  1384 apply (subst split_mod, auto)
  1385 apply (subst split_zmod, auto)
  1386 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
  1387        in unique_remainder)
  1388 apply (auto simp add: IntDiv.quorem_def)
  1389 done
  1390 
  1391 text{*Suggested by Matthias Daum*}
  1392 lemma int_power_div_base:
  1393      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
  1394 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
  1395  apply (erule ssubst)
  1396  apply (simp only: power_add)
  1397  apply simp_all
  1398 done
  1399 
  1400 text {* code serializer setup *}
  1401 
  1402 code_modulename SML
  1403   IntDiv Integer
  1404 
  1405 code_modulename OCaml
  1406   IntDiv Integer
  1407 
  1408 code_modulename Haskell
  1409   IntDiv Divides
  1410 
  1411 end