src/HOL/IntDiv.thy
author haftmann
Mon Apr 27 10:11:44 2009 +0200 (2009-04-27)
changeset 31001 7e6ffd8f51a9
parent 30934 ed5377c2b0a3
child 31065 d87465cbfc9e
permissions -rw-r--r--
cleaned up theory power further
     1 (*  Title:      HOL/IntDiv.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1999  University of Cambridge
     4 
     5 *)
     6 
     7 header{* The Division Operators div and mod *}
     8 
     9 theory IntDiv
    10 imports Int Divides FunDef
    11 uses
    12   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
    13   "~~/src/Provers/Arith/extract_common_term.ML"
    14   ("Tools/int_factor_simprocs.ML")
    15 begin
    16 
    17 definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
    18     --{*definition of quotient and remainder*}
    19     [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
    20                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
    21 
    22 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
    23     --{*for the division algorithm*}
    24     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
    25                          else (2 * q, r))"
    26 
    27 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
    28 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
    29   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
    30      else adjust b (posDivAlg a (2 * b)))"
    31 by auto
    32 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
    33 
    34 text{*algorithm for the case @{text "a<0, b>0"}*}
    35 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
    36   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
    37      else adjust b (negDivAlg a (2 * b)))"
    38 by auto
    39 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
    40 
    41 text{*algorithm for the general case @{term "b\<noteq>0"}*}
    42 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
    43   [code inline]: "negateSnd = apsnd uminus"
    44 
    45 definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
    46     --{*The full division algorithm considers all possible signs for a, b
    47        including the special case @{text "a=0, b<0"} because 
    48        @{term negDivAlg} requires @{term "a<0"}.*}
    49   "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
    50                   else if a = 0 then (0, 0)
    51                        else negateSnd (negDivAlg (-a) (-b))
    52                else 
    53                   if 0 < b then negDivAlg a b
    54                   else negateSnd (posDivAlg (-a) (-b)))"
    55 
    56 instantiation int :: Divides.div
    57 begin
    58 
    59 definition
    60   div_def: "a div b = fst (divmod a b)"
    61 
    62 definition
    63   mod_def: "a mod b = snd (divmod a b)"
    64 
    65 instance ..
    66 
    67 end
    68 
    69 lemma divmod_mod_div:
    70   "divmod 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 divmod (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      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
   126       ==> q = q'"
   127 apply (simp add: divmod_rel_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      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
   136       ==> r = r'"
   137 apply (subgoal_tac "q = q'")
   138  apply (simp add: divmod_rel_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 "divmod_rel a b (posDivAlg a b)"
   166 using prems apply (induct a b rule: posDivAlg.induct)
   167 apply auto
   168 apply (simp add: divmod_rel_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 "divmod_rel 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: divmod_rel_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 divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
   210 by (auto simp add: divmod_rel_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 divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
   222 by (auto simp add: split_ifs divmod_rel_def)
   223 
   224 lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
   225 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_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 divmod_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 divmod_correct)
   240 apply (auto simp add: divmod_rel_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 {*
   252 local
   253 
   254 structure CancelDivMod = CancelDivModFun(struct
   255 
   256   val div_name = @{const_name div};
   257   val mod_name = @{const_name 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 
   262   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
   263 
   264   val trans = trans;
   265 
   266   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
   267     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
   268 
   269 end)
   270 
   271 in
   272 
   273 val cancel_div_mod_int_proc = Simplifier.simproc (the_context ())
   274   "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
   275 
   276 val _ = Addsimprocs [cancel_div_mod_int_proc];
   277 
   278 end
   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 divmod_correct)
   283 apply (auto simp add: divmod_rel_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 divmod_correct)
   291 apply (auto simp add: divmod_rel_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 divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel 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: divmod_rel_def linorder_neq_iff)
   304 done
   305 
   306 lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
   307 by (simp add: divmod_rel_div_mod [THEN unique_quotient])
   308 
   309 lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
   310 by (simp add: divmod_rel_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 divmod_rel_div)
   314 apply (auto simp add: divmod_rel_def)
   315 done
   316 
   317 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
   318 apply (rule divmod_rel_div)
   319 apply (auto simp add: divmod_rel_def)
   320 done
   321 
   322 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
   323 apply (rule divmod_rel_div)
   324 apply (auto simp add: divmod_rel_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 divmod_rel_mod)
   331 apply (auto simp add: divmod_rel_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 divmod_rel_mod)
   336 apply (auto simp add: divmod_rel_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 divmod_rel_mod)
   341 apply (auto simp add: divmod_rel_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: divmod_rel_div_mod [THEN divmod_rel_neg, simplified, 
   351                                  THEN divmod_rel_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 divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
   359        auto)
   360 done
   361 
   362 
   363 subsection{*Laws for div and mod with Unary Minus*}
   364 
   365 lemma zminus1_lemma:
   366      "divmod_rel a b (q, r)
   367       ==> divmod_rel (-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 divmod_rel_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: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_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: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
   382 done
   383 
   384 lemma zmod_zminus1_not_zero:
   385   fixes k l :: int
   386   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
   387   unfolding zmod_zminus1_eq_if by auto
   388 
   389 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
   390 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
   391 
   392 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
   393 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
   394 
   395 lemma zdiv_zminus2_eq_if:
   396      "b \<noteq> (0::int)  
   397       ==> a div (-b) =  
   398           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   399 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
   400 
   401 lemma zmod_zminus2_eq_if:
   402      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
   403 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
   404 
   405 lemma zmod_zminus2_not_zero:
   406   fixes k l :: int
   407   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
   408   unfolding zmod_zminus2_eq_if by auto 
   409 
   410 
   411 subsection{*Division of a Number by Itself*}
   412 
   413 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
   414 apply (subgoal_tac "0 < a*q")
   415  apply (simp add: zero_less_mult_iff, arith)
   416 done
   417 
   418 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
   419 apply (subgoal_tac "0 \<le> a* (1-q) ")
   420  apply (simp add: zero_le_mult_iff)
   421 apply (simp add: right_diff_distrib)
   422 done
   423 
   424 lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
   425 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
   426 apply (rule order_antisym, safe, simp_all)
   427 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
   428 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
   429 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
   430 done
   431 
   432 lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
   433 apply (frule self_quotient, assumption)
   434 apply (simp add: divmod_rel_def)
   435 done
   436 
   437 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
   438 by (simp add: divmod_rel_div_mod [THEN self_quotient])
   439 
   440 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
   441 lemma zmod_self [simp]: "a mod a = (0::int)"
   442 apply (case_tac "a = 0", simp)
   443 apply (simp add: divmod_rel_div_mod [THEN self_remainder])
   444 done
   445 
   446 
   447 subsection{*Computation of Division and Remainder*}
   448 
   449 lemma zdiv_zero [simp]: "(0::int) div b = 0"
   450 by (simp add: div_def divmod_def)
   451 
   452 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
   453 by (simp add: div_def divmod_def)
   454 
   455 lemma zmod_zero [simp]: "(0::int) mod b = 0"
   456 by (simp add: mod_def divmod_def)
   457 
   458 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
   459 by (simp add: mod_def divmod_def)
   460 
   461 text{*a positive, b positive *}
   462 
   463 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
   464 by (simp add: div_def divmod_def)
   465 
   466 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
   467 by (simp add: mod_def divmod_def)
   468 
   469 text{*a negative, b positive *}
   470 
   471 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
   472 by (simp add: div_def divmod_def)
   473 
   474 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
   475 by (simp add: mod_def divmod_def)
   476 
   477 text{*a positive, b negative *}
   478 
   479 lemma div_pos_neg:
   480      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
   481 by (simp add: div_def divmod_def)
   482 
   483 lemma mod_pos_neg:
   484      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
   485 by (simp add: mod_def divmod_def)
   486 
   487 text{*a negative, b negative *}
   488 
   489 lemma div_neg_neg:
   490      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
   491 by (simp add: div_def divmod_def)
   492 
   493 lemma mod_neg_neg:
   494      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
   495 by (simp add: mod_def divmod_def)
   496 
   497 text {*Simplify expresions in which div and mod combine numerical constants*}
   498 
   499 lemma divmod_relI:
   500   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
   501     \<Longrightarrow> divmod_rel a b (q, r)"
   502   unfolding divmod_rel_def by simp
   503 
   504 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
   505 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
   506 lemmas arithmetic_simps =
   507   arith_simps
   508   add_special
   509   OrderedGroup.add_0_left
   510   OrderedGroup.add_0_right
   511   mult_zero_left
   512   mult_zero_right
   513   mult_1_left
   514   mult_1_right
   515 
   516 (* simprocs adapted from HOL/ex/Binary.thy *)
   517 ML {*
   518 local
   519   val mk_number = HOLogic.mk_number HOLogic.intT;
   520   fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
   521     (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
   522       mk_number l;
   523   fun prove ctxt prop = Goal.prove ctxt [] [] prop
   524     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
   525   fun binary_proc proc ss ct =
   526     (case Thm.term_of ct of
   527       _ $ t $ u =>
   528       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
   529         SOME args => proc (Simplifier.the_context ss) args
   530       | NONE => NONE)
   531     | _ => NONE);
   532 in
   533   fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
   534     if n = 0 then NONE
   535     else let val (k, l) = Integer.div_mod m n;
   536     in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
   537 end
   538 *}
   539 
   540 simproc_setup binary_int_div ("number_of m div number_of n :: int") =
   541   {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
   542 
   543 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
   544   {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
   545 
   546 lemmas posDivAlg_eqn_number_of [simp] =
   547     posDivAlg_eqn [of "number_of v" "number_of w", standard]
   548 
   549 lemmas negDivAlg_eqn_number_of [simp] =
   550     negDivAlg_eqn [of "number_of v" "number_of w", standard]
   551 
   552 
   553 text{*Special-case simplification *}
   554 
   555 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
   556 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
   557 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
   558 apply (auto simp del: neg_mod_sign neg_mod_bound)
   559 done
   560 
   561 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
   562 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
   563 
   564 (** The last remaining special cases for constant arithmetic:
   565     1 div z and 1 mod z **)
   566 
   567 lemmas div_pos_pos_1_number_of [simp] =
   568     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
   569 
   570 lemmas div_pos_neg_1_number_of [simp] =
   571     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
   572 
   573 lemmas mod_pos_pos_1_number_of [simp] =
   574     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
   575 
   576 lemmas mod_pos_neg_1_number_of [simp] =
   577     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
   578 
   579 
   580 lemmas posDivAlg_eqn_1_number_of [simp] =
   581     posDivAlg_eqn [of concl: 1 "number_of w", standard]
   582 
   583 lemmas negDivAlg_eqn_1_number_of [simp] =
   584     negDivAlg_eqn [of concl: 1 "number_of w", standard]
   585 
   586 
   587 
   588 subsection{*Monotonicity in the First Argument (Dividend)*}
   589 
   590 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
   591 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   592 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   593 apply (rule unique_quotient_lemma)
   594 apply (erule subst)
   595 apply (erule subst, simp_all)
   596 done
   597 
   598 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
   599 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   600 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
   601 apply (rule unique_quotient_lemma_neg)
   602 apply (erule subst)
   603 apply (erule subst, simp_all)
   604 done
   605 
   606 
   607 subsection{*Monotonicity in the Second Argument (Divisor)*}
   608 
   609 lemma q_pos_lemma:
   610      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
   611 apply (subgoal_tac "0 < b'* (q' + 1) ")
   612  apply (simp add: zero_less_mult_iff)
   613 apply (simp add: right_distrib)
   614 done
   615 
   616 lemma zdiv_mono2_lemma:
   617      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
   618          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
   619       ==> q \<le> (q'::int)"
   620 apply (frule q_pos_lemma, assumption+) 
   621 apply (subgoal_tac "b*q < b* (q' + 1) ")
   622  apply (simp add: mult_less_cancel_left)
   623 apply (subgoal_tac "b*q = r' - r + b'*q'")
   624  prefer 2 apply simp
   625 apply (simp (no_asm_simp) add: right_distrib)
   626 apply (subst add_commute, rule zadd_zless_mono, arith)
   627 apply (rule mult_right_mono, auto)
   628 done
   629 
   630 lemma zdiv_mono2:
   631      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
   632 apply (subgoal_tac "b \<noteq> 0")
   633  prefer 2 apply arith
   634 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   635 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   636 apply (rule zdiv_mono2_lemma)
   637 apply (erule subst)
   638 apply (erule subst, simp_all)
   639 done
   640 
   641 lemma q_neg_lemma:
   642      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
   643 apply (subgoal_tac "b'*q' < 0")
   644  apply (simp add: mult_less_0_iff, arith)
   645 done
   646 
   647 lemma zdiv_mono2_neg_lemma:
   648      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
   649          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
   650       ==> q' \<le> (q::int)"
   651 apply (frule q_neg_lemma, assumption+) 
   652 apply (subgoal_tac "b*q' < b* (q + 1) ")
   653  apply (simp add: mult_less_cancel_left)
   654 apply (simp add: right_distrib)
   655 apply (subgoal_tac "b*q' \<le> b'*q'")
   656  prefer 2 apply (simp add: mult_right_mono_neg, arith)
   657 done
   658 
   659 lemma zdiv_mono2_neg:
   660      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
   661 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
   662 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
   663 apply (rule zdiv_mono2_neg_lemma)
   664 apply (erule subst)
   665 apply (erule subst, simp_all)
   666 done
   667 
   668 
   669 subsection{*More Algebraic Laws for div and mod*}
   670 
   671 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
   672 
   673 lemma zmult1_lemma:
   674      "[| divmod_rel b c (q, r);  c \<noteq> 0 |]  
   675       ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
   676 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
   677 
   678 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
   679 apply (case_tac "c = 0", simp)
   680 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
   681 done
   682 
   683 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
   684 apply (case_tac "c = 0", simp)
   685 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
   686 done
   687 
   688 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
   689 apply (case_tac "b = 0", simp)
   690 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
   691 done
   692 
   693 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
   694 
   695 lemma zadd1_lemma:
   696      "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]  
   697       ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
   698 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
   699 
   700 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   701 lemma zdiv_zadd1_eq:
   702      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
   703 apply (case_tac "c = 0", simp)
   704 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
   705 done
   706 
   707 instance int :: ring_div
   708 proof
   709   fix a b c :: int
   710   assume not0: "b \<noteq> 0"
   711   show "(a + c * b) div b = c + a div b"
   712     unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
   713       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
   714 next
   715   fix a b c :: int
   716   assume "a \<noteq> 0"
   717   then show "(a * b) div (a * c) = b div c"
   718   proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
   719     case False then show ?thesis by auto
   720   next
   721     case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
   722     with `a \<noteq> 0`
   723     have "\<And>q r. divmod_rel b c (q, r) \<Longrightarrow> divmod_rel (a * b) (a * c) (q, a * r)"
   724       apply (auto simp add: divmod_rel_def) 
   725       apply (auto simp add: algebra_simps)
   726       apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff)
   727       apply (simp_all add: mult_less_cancel_left_disj mult_commute [of _ a])
   728       done
   729     moreover with `c \<noteq> 0` divmod_rel_div_mod have "divmod_rel b c (b div c, b mod c)" by auto
   730     ultimately have "divmod_rel (a * b) (a * c) (b div c, a * (b mod c))" .
   731     moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
   732     ultimately show ?thesis by (rule divmod_rel_div)
   733   qed
   734 qed auto
   735 
   736 lemma posDivAlg_div_mod:
   737   assumes "k \<ge> 0"
   738   and "l \<ge> 0"
   739   shows "posDivAlg k l = (k div l, k mod l)"
   740 proof (cases "l = 0")
   741   case True then show ?thesis by (simp add: posDivAlg.simps)
   742 next
   743   case False with assms posDivAlg_correct
   744     have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
   745     by simp
   746   from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
   747   show ?thesis by simp
   748 qed
   749 
   750 lemma negDivAlg_div_mod:
   751   assumes "k < 0"
   752   and "l > 0"
   753   shows "negDivAlg k l = (k div l, k mod l)"
   754 proof -
   755   from assms have "l \<noteq> 0" by simp
   756   from assms negDivAlg_correct
   757     have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
   758     by simp
   759   from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
   760   show ?thesis by simp
   761 qed
   762 
   763 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
   764 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   765 
   766 (* REVISIT: should this be generalized to all semiring_div types? *)
   767 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
   768 
   769 
   770 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
   771 
   772 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
   773   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
   774   to cause particular problems.*)
   775 
   776 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
   777 
   778 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
   779 apply (subgoal_tac "b * (c - q mod c) < r * 1")
   780  apply (simp add: algebra_simps)
   781 apply (rule order_le_less_trans)
   782  apply (erule_tac [2] mult_strict_right_mono)
   783  apply (rule mult_left_mono_neg)
   784   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
   785  apply (simp)
   786 apply (simp)
   787 done
   788 
   789 lemma zmult2_lemma_aux2:
   790      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
   791 apply (subgoal_tac "b * (q mod c) \<le> 0")
   792  apply arith
   793 apply (simp add: mult_le_0_iff)
   794 done
   795 
   796 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
   797 apply (subgoal_tac "0 \<le> b * (q mod c) ")
   798 apply arith
   799 apply (simp add: zero_le_mult_iff)
   800 done
   801 
   802 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
   803 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
   804  apply (simp add: right_diff_distrib)
   805 apply (rule order_less_le_trans)
   806  apply (erule mult_strict_right_mono)
   807  apply (rule_tac [2] mult_left_mono)
   808   apply simp
   809  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
   810 apply simp
   811 done
   812 
   813 lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
   814       ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
   815 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
   816                    zero_less_mult_iff right_distrib [symmetric] 
   817                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
   818 
   819 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
   820 apply (case_tac "b = 0", simp)
   821 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
   822 done
   823 
   824 lemma zmod_zmult2_eq:
   825      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
   826 apply (case_tac "b = 0", simp)
   827 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
   828 done
   829 
   830 
   831 subsection {*Splitting Rules for div and mod*}
   832 
   833 text{*The proofs of the two lemmas below are essentially identical*}
   834 
   835 lemma split_pos_lemma:
   836  "0<k ==> 
   837     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
   838 apply (rule iffI, clarify)
   839  apply (erule_tac P="P ?x ?y" in rev_mp)  
   840  apply (subst mod_add_eq) 
   841  apply (subst zdiv_zadd1_eq) 
   842  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
   843 txt{*converse direction*}
   844 apply (drule_tac x = "n div k" in spec) 
   845 apply (drule_tac x = "n mod k" in spec, simp)
   846 done
   847 
   848 lemma split_neg_lemma:
   849  "k<0 ==>
   850     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
   851 apply (rule iffI, clarify)
   852  apply (erule_tac P="P ?x ?y" in rev_mp)  
   853  apply (subst mod_add_eq) 
   854  apply (subst zdiv_zadd1_eq) 
   855  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
   856 txt{*converse direction*}
   857 apply (drule_tac x = "n div k" in spec) 
   858 apply (drule_tac x = "n mod k" in spec, simp)
   859 done
   860 
   861 lemma split_zdiv:
   862  "P(n div k :: int) =
   863   ((k = 0 --> P 0) & 
   864    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
   865    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
   866 apply (case_tac "k=0", simp)
   867 apply (simp only: linorder_neq_iff)
   868 apply (erule disjE) 
   869  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
   870                       split_neg_lemma [of concl: "%x y. P x"])
   871 done
   872 
   873 lemma split_zmod:
   874  "P(n mod k :: int) =
   875   ((k = 0 --> P n) & 
   876    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
   877    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
   878 apply (case_tac "k=0", simp)
   879 apply (simp only: linorder_neq_iff)
   880 apply (erule disjE) 
   881  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
   882                       split_neg_lemma [of concl: "%x y. P y"])
   883 done
   884 
   885 (* Enable arith to deal with div 2 and mod 2: *)
   886 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
   887 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
   888 
   889 
   890 subsection{*Speeding up the Division Algorithm with Shifting*}
   891 
   892 text{*computing div by shifting *}
   893 
   894 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
   895 proof cases
   896   assume "a=0"
   897     thus ?thesis by simp
   898 next
   899   assume "a\<noteq>0" and le_a: "0\<le>a"   
   900   hence a_pos: "1 \<le> a" by arith
   901   hence one_less_a2: "1 < 2 * a" by arith
   902   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
   903     unfolding mult_le_cancel_left
   904     by (simp add: add1_zle_eq add_commute [of 1])
   905   with a_pos have "0 \<le> b mod a" by simp
   906   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
   907     by (simp add: mod_pos_pos_trivial one_less_a2)
   908   with  le_2a
   909   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
   910     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
   911                   right_distrib) 
   912   thus ?thesis
   913     by (subst zdiv_zadd1_eq,
   914         simp add: mod_mult_mult1 one_less_a2
   915                   div_pos_pos_trivial)
   916 qed
   917 
   918 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
   919 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
   920 apply (rule_tac [2] pos_zdiv_mult_2)
   921 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
   922 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
   923 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
   924        simp) 
   925 done
   926 
   927 lemma zdiv_number_of_Bit0 [simp]:
   928      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
   929           number_of v div (number_of w :: int)"
   930 by (simp only: number_of_eq numeral_simps) simp
   931 
   932 lemma zdiv_number_of_Bit1 [simp]:
   933      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
   934           (if (0::int) \<le> number_of w                    
   935            then number_of v div (number_of w)     
   936            else (number_of v + (1::int)) div (number_of w))"
   937 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
   938 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
   939 done
   940 
   941 
   942 subsection{*Computing mod by Shifting (proofs resemble those for div)*}
   943 
   944 lemma pos_zmod_mult_2:
   945      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
   946 apply (case_tac "a = 0", simp)
   947 apply (subgoal_tac "1 < a * 2")
   948  prefer 2 apply arith
   949 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
   950  apply (rule_tac [2] mult_left_mono)
   951 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
   952                       pos_mod_bound)
   953 apply (subst mod_add_eq)
   954 apply (simp add: mod_mult_mult2 mod_pos_pos_trivial)
   955 apply (rule mod_pos_pos_trivial)
   956 apply (auto simp add: mod_pos_pos_trivial ring_distribs)
   957 apply (subgoal_tac "0 \<le> b mod a", arith, simp)
   958 done
   959 
   960 lemma neg_zmod_mult_2:
   961      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
   962 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
   963                     1 + 2* ((-b - 1) mod (-a))")
   964 apply (rule_tac [2] pos_zmod_mult_2)
   965 apply (auto simp add: right_diff_distrib)
   966 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
   967  prefer 2 apply simp 
   968 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
   969 done
   970 
   971 lemma zmod_number_of_Bit0 [simp]:
   972      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
   973       (2::int) * (number_of v mod number_of w)"
   974 apply (simp only: number_of_eq numeral_simps) 
   975 apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
   976                  neg_zmod_mult_2 add_ac)
   977 done
   978 
   979 lemma zmod_number_of_Bit1 [simp]:
   980      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
   981       (if (0::int) \<le> number_of w  
   982                 then 2 * (number_of v mod number_of w) + 1     
   983                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
   984 apply (simp only: number_of_eq numeral_simps) 
   985 apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
   986                  neg_zmod_mult_2 add_ac)
   987 done
   988 
   989 
   990 subsection{*Quotients of Signs*}
   991 
   992 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
   993 apply (subgoal_tac "a div b \<le> -1", force)
   994 apply (rule order_trans)
   995 apply (rule_tac a' = "-1" in zdiv_mono1)
   996 apply (auto simp add: div_eq_minus1)
   997 done
   998 
   999 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
  1000 by (drule zdiv_mono1_neg, auto)
  1001 
  1002 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
  1003 by (drule zdiv_mono1, auto)
  1004 
  1005 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
  1006 apply auto
  1007 apply (drule_tac [2] zdiv_mono1)
  1008 apply (auto simp add: linorder_neq_iff)
  1009 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
  1010 apply (blast intro: div_neg_pos_less0)
  1011 done
  1012 
  1013 lemma neg_imp_zdiv_nonneg_iff:
  1014      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
  1015 apply (subst zdiv_zminus_zminus [symmetric])
  1016 apply (subst pos_imp_zdiv_nonneg_iff, auto)
  1017 done
  1018 
  1019 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
  1020 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
  1021 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
  1022 
  1023 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
  1024 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
  1025 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
  1026 
  1027 
  1028 subsection {* The Divides Relation *}
  1029 
  1030 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
  1031   dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
  1032 
  1033 lemma zdvd_anti_sym:
  1034     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
  1035   apply (simp add: dvd_def, auto)
  1036   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
  1037   done
  1038 
  1039 lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a" 
  1040   shows "\<bar>a\<bar> = \<bar>b\<bar>"
  1041 proof-
  1042   from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
  1043   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
  1044   from k k' have "a = a*k*k'" by simp
  1045   with mult_cancel_left1[where c="a" and b="k*k'"]
  1046   have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
  1047   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
  1048   thus ?thesis using k k' by auto
  1049 qed
  1050 
  1051 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
  1052   apply (subgoal_tac "m = n + (m - n)")
  1053    apply (erule ssubst)
  1054    apply (blast intro: dvd_add, simp)
  1055   done
  1056 
  1057 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
  1058 apply (rule iffI)
  1059  apply (erule_tac [2] dvd_add)
  1060  apply (subgoal_tac "n = (n + k * m) - k * m")
  1061   apply (erule ssubst)
  1062   apply (erule dvd_diff)
  1063   apply(simp_all)
  1064 done
  1065 
  1066 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  1067   by (auto elim!: dvdE simp add: mod_mult_mult1)
  1068 
  1069 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  1070   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
  1071    apply (simp add: zmod_zdiv_equality [symmetric])
  1072   apply (simp only: dvd_add dvd_mult2)
  1073   done
  1074 
  1075 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
  1076   apply (auto elim!: dvdE)
  1077   apply (subgoal_tac "0 < n")
  1078    prefer 2
  1079    apply (blast intro: order_less_trans)
  1080   apply (simp add: zero_less_mult_iff)
  1081   apply (subgoal_tac "n * k < n * 1")
  1082    apply (drule mult_less_cancel_left [THEN iffD1], auto)
  1083   done
  1084 
  1085 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  1086   using zmod_zdiv_equality[where a="m" and b="n"]
  1087   by (simp add: algebra_simps)
  1088 
  1089 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
  1090 apply (subgoal_tac "m mod n = 0")
  1091  apply (simp add: zmult_div_cancel)
  1092 apply (simp only: dvd_eq_mod_eq_0)
  1093 done
  1094 
  1095 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
  1096   shows "m dvd n"
  1097 proof-
  1098   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
  1099   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
  1100     with h have False by (simp add: mult_assoc)}
  1101   hence "n = m * h" by blast
  1102   thus ?thesis by simp
  1103 qed
  1104 
  1105 
  1106 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
  1107 apply (simp split add: split_nat)
  1108 apply (rule iffI)
  1109 apply (erule exE)
  1110 apply (rule_tac x = "int x" in exI)
  1111 apply simp
  1112 apply (erule exE)
  1113 apply (rule_tac x = "nat x" in exI)
  1114 apply (erule conjE)
  1115 apply (erule_tac x = "nat x" in allE)
  1116 apply simp
  1117 done
  1118 
  1119 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
  1120 proof -
  1121   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
  1122   proof -
  1123     fix k
  1124     assume A: "int y = int x * k"
  1125     then show "x dvd y" proof (cases k)
  1126       case (1 n) with A have "y = x * n" by (simp add: zmult_int)
  1127       then show ?thesis ..
  1128     next
  1129       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
  1130       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
  1131       also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
  1132       finally have "- int (x * Suc n) = int y" ..
  1133       then show ?thesis by (simp only: negative_eq_positive) auto
  1134     qed
  1135   qed
  1136   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)
  1137 qed
  1138 
  1139 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
  1140 proof
  1141   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
  1142   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
  1143   hence "nat \<bar>x\<bar> = 1"  by simp
  1144   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
  1145 next
  1146   assume "\<bar>x\<bar>=1" thus "x dvd 1" 
  1147     by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)
  1148 qed
  1149 lemma zdvd_mult_cancel1: 
  1150   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
  1151 proof
  1152   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
  1153     by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)
  1154 next
  1155   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
  1156   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
  1157 qed
  1158 
  1159 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
  1160   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
  1161 
  1162 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
  1163   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
  1164 
  1165 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
  1166   by (auto simp add: dvd_int_iff)
  1167 
  1168 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
  1169   apply (rule_tac z=n in int_cases)
  1170   apply (auto simp add: dvd_int_iff)
  1171   apply (rule_tac z=z in int_cases)
  1172   apply (auto simp add: dvd_imp_le)
  1173   done
  1174 
  1175 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
  1176 apply (induct "y", auto)
  1177 apply (rule zmod_zmult1_eq [THEN trans])
  1178 apply (simp (no_asm_simp))
  1179 apply (rule mod_mult_eq [symmetric])
  1180 done
  1181 
  1182 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
  1183 apply (subst split_div, auto)
  1184 apply (subst split_zdiv, auto)
  1185 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
  1186 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
  1187 done
  1188 
  1189 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
  1190 apply (subst split_mod, auto)
  1191 apply (subst split_zmod, auto)
  1192 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
  1193        in unique_remainder)
  1194 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
  1195 done
  1196 
  1197 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
  1198 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
  1199 
  1200 text{*Suggested by Matthias Daum*}
  1201 lemma int_power_div_base:
  1202      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
  1203 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
  1204  apply (erule ssubst)
  1205  apply (simp only: power_add)
  1206  apply simp_all
  1207 done
  1208 
  1209 text {* by Brian Huffman *}
  1210 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
  1211 by (rule mod_minus_eq [symmetric])
  1212 
  1213 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
  1214 by (rule mod_diff_left_eq [symmetric])
  1215 
  1216 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
  1217 by (rule mod_diff_right_eq [symmetric])
  1218 
  1219 lemmas zmod_simps =
  1220   mod_add_left_eq  [symmetric]
  1221   mod_add_right_eq [symmetric]
  1222   zmod_zmult1_eq   [symmetric]
  1223   mod_mult_left_eq [symmetric]
  1224   zpower_zmod
  1225   zminus_zmod zdiff_zmod_left zdiff_zmod_right
  1226 
  1227 text {* Distributive laws for function @{text nat}. *}
  1228 
  1229 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
  1230 apply (rule linorder_cases [of y 0])
  1231 apply (simp add: div_nonneg_neg_le0)
  1232 apply simp
  1233 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
  1234 done
  1235 
  1236 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
  1237 lemma nat_mod_distrib:
  1238   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
  1239 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
  1240 apply (simp add: nat_eq_iff zmod_int)
  1241 done
  1242 
  1243 text{*Suggested by Matthias Daum*}
  1244 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
  1245 apply (subgoal_tac "nat x div nat k < nat x")
  1246  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
  1247 apply (rule Divides.div_less_dividend, simp_all)
  1248 done
  1249 
  1250 text {* code generator setup *}
  1251 
  1252 context ring_1
  1253 begin
  1254 
  1255 lemma of_int_num [code]:
  1256   "of_int k = (if k = 0 then 0 else if k < 0 then
  1257      - of_int (- k) else let
  1258        (l, m) = divmod k 2;
  1259        l' = of_int l
  1260      in if m = 0 then l' + l' else l' + l' + 1)"
  1261 proof -
  1262   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
  1263     of_int k = of_int (k div 2 * 2 + 1)"
  1264   proof -
  1265     have "k mod 2 < 2" by (auto intro: pos_mod_bound)
  1266     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
  1267     moreover assume "k mod 2 \<noteq> 0"
  1268     ultimately have "k mod 2 = 1" by arith
  1269     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
  1270     ultimately show ?thesis by auto
  1271   qed
  1272   have aux2: "\<And>x. of_int 2 * x = x + x"
  1273   proof -
  1274     fix x
  1275     have int2: "(2::int) = 1 + 1" by arith
  1276     show "of_int 2 * x = x + x"
  1277     unfolding int2 of_int_add left_distrib by simp
  1278   qed
  1279   have aux3: "\<And>x. x * of_int 2 = x + x"
  1280   proof -
  1281     fix x
  1282     have int2: "(2::int) = 1 + 1" by arith
  1283     show "x * of_int 2 = x + x" 
  1284     unfolding int2 of_int_add right_distrib by simp
  1285   qed
  1286   from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
  1287 qed
  1288 
  1289 end
  1290 
  1291 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
  1292 proof
  1293   assume H: "x mod n = y mod n"
  1294   hence "x mod n - y mod n = 0" by simp
  1295   hence "(x mod n - y mod n) mod n = 0" by simp 
  1296   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
  1297   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
  1298 next
  1299   assume H: "n dvd x - y"
  1300   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
  1301   hence "x = n*k + y" by simp
  1302   hence "x mod n = (n*k + y) mod n" by simp
  1303   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
  1304 qed
  1305 
  1306 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
  1307   shows "\<exists>q. x = y + n * q"
  1308 proof-
  1309   from xy have th: "int x - int y = int (x - y)" by simp 
  1310   from xyn have "int x mod int n = int y mod int n" 
  1311     by (simp add: zmod_int[symmetric])
  1312   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
  1313   hence "n dvd x - y" by (simp add: th zdvd_int)
  1314   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
  1315 qed
  1316 
  1317 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
  1318   (is "?lhs = ?rhs")
  1319 proof
  1320   assume H: "x mod n = y mod n"
  1321   {assume xy: "x \<le> y"
  1322     from H have th: "y mod n = x mod n" by simp
  1323     from nat_mod_eq_lemma[OF th xy] have ?rhs 
  1324       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
  1325   moreover
  1326   {assume xy: "y \<le> x"
  1327     from nat_mod_eq_lemma[OF H xy] have ?rhs 
  1328       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
  1329   ultimately  show ?rhs using linear[of x y] by blast  
  1330 next
  1331   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
  1332   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
  1333   thus  ?lhs by simp
  1334 qed
  1335 
  1336 
  1337 subsection {* Simproc setup *}
  1338 
  1339 use "Tools/int_factor_simprocs.ML"
  1340 
  1341 
  1342 subsection {* Code generation *}
  1343 
  1344 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1345   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
  1346 
  1347 lemma pdivmod_posDivAlg [code]:
  1348   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
  1349 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
  1350 
  1351 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  1352   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
  1353     then pdivmod k l
  1354     else (let (r, s) = pdivmod k l in
  1355       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
  1356 proof -
  1357   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
  1358   show ?thesis
  1359     by (simp add: divmod_mod_div pdivmod_def)
  1360       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
  1361       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
  1362 qed
  1363 
  1364 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  1365   apsnd ((op *) (sgn l)) (if sgn k = sgn l
  1366     then pdivmod k l
  1367     else (let (r, s) = pdivmod k l in
  1368       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
  1369 proof -
  1370   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
  1371     by (auto simp add: not_less sgn_if)
  1372   then show ?thesis by (simp add: divmod_pdivmod)
  1373 qed
  1374 
  1375 code_modulename SML
  1376   IntDiv Integer
  1377 
  1378 code_modulename OCaml
  1379   IntDiv Integer
  1380 
  1381 code_modulename Haskell
  1382   IntDiv Integer
  1383 
  1384 end