src/HOL/IntDiv.thy
 author huffman Wed Feb 18 15:01:53 2009 -0800 (2009-02-18) changeset 29981 7d0ed261b712 parent 29951 a70bc5190534 child 30031 bd786c37af84 permissions -rw-r--r--
generalize int_dvd_cancel_factor simproc to idom class
     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 begin

    12

    13 definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where

    14     --{*definition of quotient and remainder*}

    15     [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>

    16                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"

    17

    18 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where

    19     --{*for the division algorithm*}

    20     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)

    21                          else (2 * q, r))"

    22

    23 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}

    24 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

    25   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)

    26      else adjust b (posDivAlg a (2 * b)))"

    27 by auto

    28 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto

    29

    30 text{*algorithm for the case @{text "a<0, b>0"}*}

    31 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

    32   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)

    33      else adjust b (negDivAlg a (2 * b)))"

    34 by auto

    35 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto

    36

    37 text{*algorithm for the general case @{term "b\<noteq>0"}*}

    38 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where

    39   [code inline]: "negateSnd = apsnd uminus"

    40

    41 definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

    42     --{*The full division algorithm considers all possible signs for a, b

    43        including the special case @{text "a=0, b<0"} because

    44        @{term negDivAlg} requires @{term "a<0"}.*}

    45   "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b

    46                   else if a = 0 then (0, 0)

    47                        else negateSnd (negDivAlg (-a) (-b))

    48                else

    49                   if 0 < b then negDivAlg a b

    50                   else negateSnd (posDivAlg (-a) (-b)))"

    51

    52 instantiation int :: Divides.div

    53 begin

    54

    55 definition

    56   div_def: "a div b = fst (divmod a b)"

    57

    58 definition

    59   mod_def: "a mod b = snd (divmod a b)"

    60

    61 instance ..

    62

    63 end

    64

    65 lemma divmod_mod_div:

    66   "divmod p q = (p div q, p mod q)"

    67   by (auto simp add: div_def mod_def)

    68

    69 text{*

    70 Here is the division algorithm in ML:

    71

    72 \begin{verbatim}

    73     fun posDivAlg (a,b) =

    74       if a<b then (0,a)

    75       else let val (q,r) = posDivAlg(a, 2*b)

    76 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

    77 	   end

    78

    79     fun negDivAlg (a,b) =

    80       if 0\<le>a+b then (~1,a+b)

    81       else let val (q,r) = negDivAlg(a, 2*b)

    82 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

    83 	   end;

    84

    85     fun negateSnd (q,r:int) = (q,~r);

    86

    87     fun divmod (a,b) = if 0\<le>a then

    88 			  if b>0 then posDivAlg (a,b)

    89 			   else if a=0 then (0,0)

    90 				else negateSnd (negDivAlg (~a,~b))

    91 		       else

    92 			  if 0<b then negDivAlg (a,b)

    93 			  else        negateSnd (posDivAlg (~a,~b));

    94 \end{verbatim}

    95 *}

    96

    97

    98

    99 subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}

   100

   101 lemma unique_quotient_lemma:

   102      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]

   103       ==> q' \<le> (q::int)"

   104 apply (subgoal_tac "r' + b * (q'-q) \<le> r")

   105  prefer 2 apply (simp add: right_diff_distrib)

   106 apply (subgoal_tac "0 < b * (1 + q - q') ")

   107 apply (erule_tac [2] order_le_less_trans)

   108  prefer 2 apply (simp add: right_diff_distrib right_distrib)

   109 apply (subgoal_tac "b * q' < b * (1 + q) ")

   110  prefer 2 apply (simp add: right_diff_distrib right_distrib)

   111 apply (simp add: mult_less_cancel_left)

   112 done

   113

   114 lemma unique_quotient_lemma_neg:

   115      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]

   116       ==> q \<le> (q'::int)"

   117 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,

   118     auto)

   119

   120 lemma unique_quotient:

   121      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]

   122       ==> q = q'"

   123 apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)

   124 apply (blast intro: order_antisym

   125              dest: order_eq_refl [THEN unique_quotient_lemma]

   126              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

   127 done

   128

   129

   130 lemma unique_remainder:

   131      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]

   132       ==> r = r'"

   133 apply (subgoal_tac "q = q'")

   134  apply (simp add: divmod_rel_def)

   135 apply (blast intro: unique_quotient)

   136 done

   137

   138

   139 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}

   140

   141 text{*And positive divisors*}

   142

   143 lemma adjust_eq [simp]:

   144      "adjust b (q,r) =

   145       (let diff = r-b in

   146 	if 0 \<le> diff then (2*q + 1, diff)

   147                      else (2*q, r))"

   148 by (simp add: Let_def adjust_def)

   149

   150 declare posDivAlg.simps [simp del]

   151

   152 text{*use with a simproc to avoid repeatedly proving the premise*}

   153 lemma posDivAlg_eqn:

   154      "0 < b ==>

   155       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"

   156 by (rule posDivAlg.simps [THEN trans], simp)

   157

   158 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}

   159 theorem posDivAlg_correct:

   160   assumes "0 \<le> a" and "0 < b"

   161   shows "divmod_rel a b (posDivAlg a b)"

   162 using prems apply (induct a b rule: posDivAlg.induct)

   163 apply auto

   164 apply (simp add: divmod_rel_def)

   165 apply (subst posDivAlg_eqn, simp add: right_distrib)

   166 apply (case_tac "a < b")

   167 apply simp_all

   168 apply (erule splitE)

   169 apply (auto simp add: right_distrib Let_def)

   170 done

   171

   172

   173 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}

   174

   175 text{*And positive divisors*}

   176

   177 declare negDivAlg.simps [simp del]

   178

   179 text{*use with a simproc to avoid repeatedly proving the premise*}

   180 lemma negDivAlg_eqn:

   181      "0 < b ==>

   182       negDivAlg a b =

   183        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"

   184 by (rule negDivAlg.simps [THEN trans], simp)

   185

   186 (*Correctness of negDivAlg: it computes quotients correctly

   187   It doesn't work if a=0 because the 0/b equals 0, not -1*)

   188 lemma negDivAlg_correct:

   189   assumes "a < 0" and "b > 0"

   190   shows "divmod_rel a b (negDivAlg a b)"

   191 using prems apply (induct a b rule: negDivAlg.induct)

   192 apply (auto simp add: linorder_not_le)

   193 apply (simp add: divmod_rel_def)

   194 apply (subst negDivAlg_eqn, assumption)

   195 apply (case_tac "a + b < (0\<Colon>int)")

   196 apply simp_all

   197 apply (erule splitE)

   198 apply (auto simp add: right_distrib Let_def)

   199 done

   200

   201

   202 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}

   203

   204 (*the case a=0*)

   205 lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"

   206 by (auto simp add: divmod_rel_def linorder_neq_iff)

   207

   208 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"

   209 by (subst posDivAlg.simps, auto)

   210

   211 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"

   212 by (subst negDivAlg.simps, auto)

   213

   214 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"

   215 by (simp add: negateSnd_def)

   216

   217 lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"

   218 by (auto simp add: split_ifs divmod_rel_def)

   219

   220 lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"

   221 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg

   222                     posDivAlg_correct negDivAlg_correct)

   223

   224 text{*Arbitrary definitions for division by zero.  Useful to simplify

   225     certain equations.*}

   226

   227 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"

   228 by (simp add: div_def mod_def divmod_def posDivAlg.simps)

   229

   230

   231 text{*Basic laws about division and remainder*}

   232

   233 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"

   234 apply (case_tac "b = 0", simp)

   235 apply (cut_tac a = a and b = b in divmod_correct)

   236 apply (auto simp add: divmod_rel_def div_def mod_def)

   237 done

   238

   239 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"

   240 by(simp add: zmod_zdiv_equality[symmetric])

   241

   242 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"

   243 by(simp add: mult_commute zmod_zdiv_equality[symmetric])

   244

   245 text {* Tool setup *}

   246

   247 ML {*

   248 local

   249

   250 structure CancelDivMod = CancelDivModFun(

   251 struct

   252   val div_name = @{const_name Divides.div};

   253   val mod_name = @{const_name Divides.mod};

   254   val mk_binop = HOLogic.mk_binop;

   255   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;

   256   val dest_sum = Int_Numeral_Simprocs.dest_sum;

   257   val div_mod_eqs =

   258     map mk_meta_eq [@{thm zdiv_zmod_equality},

   259       @{thm zdiv_zmod_equality2}];

   260   val trans = trans;

   261   val prove_eq_sums =

   262     let

   263       val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}

   264     in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;

   265 end)

   266

   267 in

   268

   269 val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())

   270   "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)

   271

   272 end;

   273

   274 Addsimprocs [cancel_zdiv_zmod_proc]

   275 *}

   276

   277 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"

   278 apply (cut_tac a = a and b = b in divmod_correct)

   279 apply (auto simp add: divmod_rel_def mod_def)

   280 done

   281

   282 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]

   283    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]

   284

   285 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"

   286 apply (cut_tac a = a and b = b in divmod_correct)

   287 apply (auto simp add: divmod_rel_def div_def mod_def)

   288 done

   289

   290 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]

   291    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]

   292

   293

   294

   295 subsection{*General Properties of div and mod*}

   296

   297 lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"

   298 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

   299 apply (force simp add: divmod_rel_def linorder_neq_iff)

   300 done

   301

   302 lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"

   303 by (simp add: divmod_rel_div_mod [THEN unique_quotient])

   304

   305 lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"

   306 by (simp add: divmod_rel_div_mod [THEN unique_remainder])

   307

   308 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"

   309 apply (rule divmod_rel_div)

   310 apply (auto simp add: divmod_rel_def)

   311 done

   312

   313 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"

   314 apply (rule divmod_rel_div)

   315 apply (auto simp add: divmod_rel_def)

   316 done

   317

   318 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"

   319 apply (rule divmod_rel_div)

   320 apply (auto simp add: divmod_rel_def)

   321 done

   322

   323 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

   324

   325 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"

   326 apply (rule_tac q = 0 in divmod_rel_mod)

   327 apply (auto simp add: divmod_rel_def)

   328 done

   329

   330 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"

   331 apply (rule_tac q = 0 in divmod_rel_mod)

   332 apply (auto simp add: divmod_rel_def)

   333 done

   334

   335 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"

   336 apply (rule_tac q = "-1" in divmod_rel_mod)

   337 apply (auto simp add: divmod_rel_def)

   338 done

   339

   340 text{*There is no @{text mod_neg_pos_trivial}.*}

   341

   342

   343 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)

   344 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"

   345 apply (case_tac "b = 0", simp)

   346 apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified,

   347                                  THEN divmod_rel_div, THEN sym])

   348

   349 done

   350

   351 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)

   352 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"

   353 apply (case_tac "b = 0", simp)

   354 apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],

   355        auto)

   356 done

   357

   358

   359 subsection{*Laws for div and mod with Unary Minus*}

   360

   361 lemma zminus1_lemma:

   362      "divmod_rel a b (q, r)

   363       ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,

   364                           if r=0 then 0 else b-r)"

   365 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)

   366

   367

   368 lemma zdiv_zminus1_eq_if:

   369      "b \<noteq> (0::int)

   370       ==> (-a) div b =

   371           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

   372 by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])

   373

   374 lemma zmod_zminus1_eq_if:

   375      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"

   376 apply (case_tac "b = 0", simp)

   377 apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])

   378 done

   379

   380 lemma zmod_zminus1_not_zero:

   381   fixes k l :: int

   382   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

   383   unfolding zmod_zminus1_eq_if by auto

   384

   385 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"

   386 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)

   387

   388 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"

   389 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)

   390

   391 lemma zdiv_zminus2_eq_if:

   392      "b \<noteq> (0::int)

   393       ==> a div (-b) =

   394           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

   395 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

   396

   397 lemma zmod_zminus2_eq_if:

   398      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"

   399 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

   400

   401 lemma zmod_zminus2_not_zero:

   402   fixes k l :: int

   403   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

   404   unfolding zmod_zminus2_eq_if by auto

   405

   406

   407 subsection{*Division of a Number by Itself*}

   408

   409 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"

   410 apply (subgoal_tac "0 < a*q")

   411  apply (simp add: zero_less_mult_iff, arith)

   412 done

   413

   414 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"

   415 apply (subgoal_tac "0 \<le> a* (1-q) ")

   416  apply (simp add: zero_le_mult_iff)

   417 apply (simp add: right_diff_distrib)

   418 done

   419

   420 lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"

   421 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)

   422 apply (rule order_antisym, safe, simp_all)

   423 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)

   424 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)

   425 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+

   426 done

   427

   428 lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"

   429 apply (frule self_quotient, assumption)

   430 apply (simp add: divmod_rel_def)

   431 done

   432

   433 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"

   434 by (simp add: divmod_rel_div_mod [THEN self_quotient])

   435

   436 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)

   437 lemma zmod_self [simp]: "a mod a = (0::int)"

   438 apply (case_tac "a = 0", simp)

   439 apply (simp add: divmod_rel_div_mod [THEN self_remainder])

   440 done

   441

   442

   443 subsection{*Computation of Division and Remainder*}

   444

   445 lemma zdiv_zero [simp]: "(0::int) div b = 0"

   446 by (simp add: div_def divmod_def)

   447

   448 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"

   449 by (simp add: div_def divmod_def)

   450

   451 lemma zmod_zero [simp]: "(0::int) mod b = 0"

   452 by (simp add: mod_def divmod_def)

   453

   454 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"

   455 by (simp add: mod_def divmod_def)

   456

   457 text{*a positive, b positive *}

   458

   459 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"

   460 by (simp add: div_def divmod_def)

   461

   462 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"

   463 by (simp add: mod_def divmod_def)

   464

   465 text{*a negative, b positive *}

   466

   467 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"

   468 by (simp add: div_def divmod_def)

   469

   470 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"

   471 by (simp add: mod_def divmod_def)

   472

   473 text{*a positive, b negative *}

   474

   475 lemma div_pos_neg:

   476      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"

   477 by (simp add: div_def divmod_def)

   478

   479 lemma mod_pos_neg:

   480      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"

   481 by (simp add: mod_def divmod_def)

   482

   483 text{*a negative, b negative *}

   484

   485 lemma div_neg_neg:

   486      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"

   487 by (simp add: div_def divmod_def)

   488

   489 lemma mod_neg_neg:

   490      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"

   491 by (simp add: mod_def divmod_def)

   492

   493 text {*Simplify expresions in which div and mod combine numerical constants*}

   494

   495 lemma divmod_relI:

   496   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>

   497     \<Longrightarrow> divmod_rel a b (q, r)"

   498   unfolding divmod_rel_def by simp

   499

   500 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]

   501 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]

   502 lemmas arithmetic_simps =

   503   arith_simps

   504   add_special

   505   OrderedGroup.add_0_left

   506   OrderedGroup.add_0_right

   507   mult_zero_left

   508   mult_zero_right

   509   mult_1_left

   510   mult_1_right

   511

   512 (* simprocs adapted from HOL/ex/Binary.thy *)

   513 ML {*

   514 local

   515   infix ==;

   516   val op == = Logic.mk_equals;

   517   fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $m$ n;

   518   fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $m$ n;

   519

   520   val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};

   521   fun prove ctxt prop =

   522     Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));

   523

   524   fun binary_proc proc ss ct =

   525     (case Thm.term_of ct of

   526       _ $t$ u =>

   527       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of

   528         SOME args => proc (Simplifier.the_context ss) args

   529       | NONE => NONE)

   530     | _ => NONE);

   531 in

   532

   533 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>

   534   if n = 0 then NONE

   535   else

   536     let val (k, l) = Integer.div_mod m n;

   537         fun mk_num x = HOLogic.mk_number HOLogic.intT x;

   538     in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])

   539     end);

   540

   541 end;

   542 *}

   543

   544 simproc_setup binary_int_div ("number_of m div number_of n :: int") =

   545   {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}

   546

   547 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =

   548   {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}

   549

   550 (* The following 8 lemmas are made unnecessary by the above simprocs: *)

   551

   552 lemmas div_pos_pos_number_of =

   553     div_pos_pos [of "number_of v" "number_of w", standard]

   554

   555 lemmas div_neg_pos_number_of =

   556     div_neg_pos [of "number_of v" "number_of w", standard]

   557

   558 lemmas div_pos_neg_number_of =

   559     div_pos_neg [of "number_of v" "number_of w", standard]

   560

   561 lemmas div_neg_neg_number_of =

   562     div_neg_neg [of "number_of v" "number_of w", standard]

   563

   564

   565 lemmas mod_pos_pos_number_of =

   566     mod_pos_pos [of "number_of v" "number_of w", standard]

   567

   568 lemmas mod_neg_pos_number_of =

   569     mod_neg_pos [of "number_of v" "number_of w", standard]

   570

   571 lemmas mod_pos_neg_number_of =

   572     mod_pos_neg [of "number_of v" "number_of w", standard]

   573

   574 lemmas mod_neg_neg_number_of =

   575     mod_neg_neg [of "number_of v" "number_of w", standard]

   576

   577

   578 lemmas posDivAlg_eqn_number_of [simp] =

   579     posDivAlg_eqn [of "number_of v" "number_of w", standard]

   580

   581 lemmas negDivAlg_eqn_number_of [simp] =

   582     negDivAlg_eqn [of "number_of v" "number_of w", standard]

   583

   584

   585 text{*Special-case simplification *}

   586

   587 lemma zmod_1 [simp]: "a mod (1::int) = 0"

   588 apply (cut_tac a = a and b = 1 in pos_mod_sign)

   589 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)

   590 apply (auto simp del:pos_mod_bound pos_mod_sign)

   591 done

   592

   593 lemma zdiv_1 [simp]: "a div (1::int) = a"

   594 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)

   595

   596 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"

   597 apply (cut_tac a = a and b = "-1" in neg_mod_sign)

   598 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)

   599 apply (auto simp del: neg_mod_sign neg_mod_bound)

   600 done

   601

   602 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"

   603 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)

   604

   605 (** The last remaining special cases for constant arithmetic:

   606     1 div z and 1 mod z **)

   607

   608 lemmas div_pos_pos_1_number_of [simp] =

   609     div_pos_pos [OF int_0_less_1, of "number_of w", standard]

   610

   611 lemmas div_pos_neg_1_number_of [simp] =

   612     div_pos_neg [OF int_0_less_1, of "number_of w", standard]

   613

   614 lemmas mod_pos_pos_1_number_of [simp] =

   615     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]

   616

   617 lemmas mod_pos_neg_1_number_of [simp] =

   618     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]

   619

   620

   621 lemmas posDivAlg_eqn_1_number_of [simp] =

   622     posDivAlg_eqn [of concl: 1 "number_of w", standard]

   623

   624 lemmas negDivAlg_eqn_1_number_of [simp] =

   625     negDivAlg_eqn [of concl: 1 "number_of w", standard]

   626

   627

   628

   629 subsection{*Monotonicity in the First Argument (Dividend)*}

   630

   631 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"

   632 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

   633 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

   634 apply (rule unique_quotient_lemma)

   635 apply (erule subst)

   636 apply (erule subst, simp_all)

   637 done

   638

   639 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"

   640 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

   641 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

   642 apply (rule unique_quotient_lemma_neg)

   643 apply (erule subst)

   644 apply (erule subst, simp_all)

   645 done

   646

   647

   648 subsection{*Monotonicity in the Second Argument (Divisor)*}

   649

   650 lemma q_pos_lemma:

   651      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

   652 apply (subgoal_tac "0 < b'* (q' + 1) ")

   653  apply (simp add: zero_less_mult_iff)

   654 apply (simp add: right_distrib)

   655 done

   656

   657 lemma zdiv_mono2_lemma:

   658      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';

   659          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]

   660       ==> q \<le> (q'::int)"

   661 apply (frule q_pos_lemma, assumption+)

   662 apply (subgoal_tac "b*q < b* (q' + 1) ")

   663  apply (simp add: mult_less_cancel_left)

   664 apply (subgoal_tac "b*q = r' - r + b'*q'")

   665  prefer 2 apply simp

   666 apply (simp (no_asm_simp) add: right_distrib)

   667 apply (subst add_commute, rule zadd_zless_mono, arith)

   668 apply (rule mult_right_mono, auto)

   669 done

   670

   671 lemma zdiv_mono2:

   672      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

   673 apply (subgoal_tac "b \<noteq> 0")

   674  prefer 2 apply arith

   675 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

   676 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

   677 apply (rule zdiv_mono2_lemma)

   678 apply (erule subst)

   679 apply (erule subst, simp_all)

   680 done

   681

   682 lemma q_neg_lemma:

   683      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"

   684 apply (subgoal_tac "b'*q' < 0")

   685  apply (simp add: mult_less_0_iff, arith)

   686 done

   687

   688 lemma zdiv_mono2_neg_lemma:

   689      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;

   690          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]

   691       ==> q' \<le> (q::int)"

   692 apply (frule q_neg_lemma, assumption+)

   693 apply (subgoal_tac "b*q' < b* (q + 1) ")

   694  apply (simp add: mult_less_cancel_left)

   695 apply (simp add: right_distrib)

   696 apply (subgoal_tac "b*q' \<le> b'*q'")

   697  prefer 2 apply (simp add: mult_right_mono_neg, arith)

   698 done

   699

   700 lemma zdiv_mono2_neg:

   701      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"

   702 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

   703 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

   704 apply (rule zdiv_mono2_neg_lemma)

   705 apply (erule subst)

   706 apply (erule subst, simp_all)

   707 done

   708

   709

   710 subsection{*More Algebraic Laws for div and mod*}

   711

   712 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

   713

   714 lemma zmult1_lemma:

   715      "[| divmod_rel b c (q, r);  c \<noteq> 0 |]

   716       ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"

   717 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)

   718

   719 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"

   720 apply (case_tac "c = 0", simp)

   721 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])

   722 done

   723

   724 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"

   725 apply (case_tac "c = 0", simp)

   726 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])

   727 done

   728

   729 lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"

   730 by (simp add: zdiv_zmult1_eq)

   731

   732 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"

   733 apply (case_tac "b = 0", simp)

   734 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

   735 done

   736

   737 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

   738

   739 lemma zadd1_lemma:

   740      "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]

   741       ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"

   742 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)

   743

   744 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

   745 lemma zdiv_zadd1_eq:

   746      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"

   747 apply (case_tac "c = 0", simp)

   748 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)

   749 done

   750

   751 instance int :: ring_div

   752 proof

   753   fix a b c :: int

   754   assume not0: "b \<noteq> 0"

   755   show "(a + c * b) div b = c + a div b"

   756     unfolding zdiv_zadd1_eq [of a "c * b"] using not0

   757       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)

   758 qed auto

   759

   760 lemma posDivAlg_div_mod:

   761   assumes "k \<ge> 0"

   762   and "l \<ge> 0"

   763   shows "posDivAlg k l = (k div l, k mod l)"

   764 proof (cases "l = 0")

   765   case True then show ?thesis by (simp add: posDivAlg.simps)

   766 next

   767   case False with assms posDivAlg_correct

   768     have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"

   769     by simp

   770   from divmod_rel_div [OF this l \<noteq> 0] divmod_rel_mod [OF this l \<noteq> 0]

   771   show ?thesis by simp

   772 qed

   773

   774 lemma negDivAlg_div_mod:

   775   assumes "k < 0"

   776   and "l > 0"

   777   shows "negDivAlg k l = (k div l, k mod l)"

   778 proof -

   779   from assms have "l \<noteq> 0" by simp

   780   from assms negDivAlg_correct

   781     have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"

   782     by simp

   783   from divmod_rel_div [OF this l \<noteq> 0] divmod_rel_mod [OF this l \<noteq> 0]

   784   show ?thesis by simp

   785 qed

   786

   787 lemma zdiv_zadd_self1: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"

   788 by (rule div_add_self1) (* already declared [simp] *)

   789

   790 lemma zdiv_zadd_self2: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"

   791 by (rule div_add_self2) (* already declared [simp] *)

   792

   793 lemma zdiv_zmult_self2: "b \<noteq> (0::int) ==> (b*a) div b = a"

   794 by (rule div_mult_self1_is_id) (* already declared [simp] *)

   795

   796 lemma zmod_zmult_self1: "(a*b) mod b = (0::int)"

   797 by (rule mod_mult_self2_is_0) (* already declared [simp] *)

   798

   799 lemma zmod_zmult_self2: "(b*a) mod b = (0::int)"

   800 by (rule mod_mult_self1_is_0) (* already declared [simp] *)

   801

   802 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"

   803 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

   804

   805 (* REVISIT: should this be generalized to all semiring_div types? *)

   806 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]

   807

   808 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"

   809 by (rule mod_add_left_eq)

   810

   811 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"

   812 by (rule mod_add_right_eq)

   813

   814 lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)"

   815 by (rule mod_add_self1) (* already declared [simp] *)

   816

   817 lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)"

   818 by (rule mod_add_self2) (* already declared [simp] *)

   819

   820 lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)"

   821 by (rule mod_diff_eq)

   822

   823 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}

   824

   825 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but

   826   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

   827   to cause particular problems.*)

   828

   829 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

   830

   831 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"

   832 apply (subgoal_tac "b * (c - q mod c) < r * 1")

   833  apply (simp add: algebra_simps)

   834 apply (rule order_le_less_trans)

   835  apply (erule_tac [2] mult_strict_right_mono)

   836  apply (rule mult_left_mono_neg)

   837   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)

   838  apply (simp)

   839 apply (simp)

   840 done

   841

   842 lemma zmult2_lemma_aux2:

   843      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

   844 apply (subgoal_tac "b * (q mod c) \<le> 0")

   845  apply arith

   846 apply (simp add: mult_le_0_iff)

   847 done

   848

   849 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"

   850 apply (subgoal_tac "0 \<le> b * (q mod c) ")

   851 apply arith

   852 apply (simp add: zero_le_mult_iff)

   853 done

   854

   855 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"

   856 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")

   857  apply (simp add: right_diff_distrib)

   858 apply (rule order_less_le_trans)

   859  apply (erule mult_strict_right_mono)

   860  apply (rule_tac [2] mult_left_mono)

   861   apply simp

   862  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)

   863 apply simp

   864 done

   865

   866 lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]

   867       ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"

   868 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff

   869                    zero_less_mult_iff right_distrib [symmetric]

   870                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

   871

   872 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"

   873 apply (case_tac "b = 0", simp)

   874 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])

   875 done

   876

   877 lemma zmod_zmult2_eq:

   878      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

   879 apply (case_tac "b = 0", simp)

   880 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])

   881 done

   882

   883

   884 subsection{*Cancellation of Common Factors in div*}

   885

   886 lemma zdiv_zmult_zmult1_aux1:

   887      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   888 by (subst zdiv_zmult2_eq, auto)

   889

   890 lemma zdiv_zmult_zmult1_aux2:

   891      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   892 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")

   893 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)

   894 done

   895

   896 lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"

   897 apply (case_tac "b = 0", simp)

   898 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)

   899 done

   900

   901 lemma zdiv_zmult_zmult1_if[simp]:

   902   "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"

   903 by (simp add:zdiv_zmult_zmult1)

   904

   905 (*

   906 lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"

   907 apply (drule zdiv_zmult_zmult1)

   908 apply (auto simp add: mult_commute)

   909 done

   910 *)

   911

   912

   913 subsection{*Distribution of Factors over mod*}

   914

   915 lemma zmod_zmult_zmult1_aux1:

   916      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"

   917 by (subst zmod_zmult2_eq, auto)

   918

   919 lemma zmod_zmult_zmult1_aux2:

   920      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"

   921 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")

   922 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)

   923 done

   924

   925 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"

   926 apply (case_tac "b = 0", simp)

   927 apply (case_tac "c = 0", simp)

   928 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)

   929 done

   930

   931 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"

   932 apply (cut_tac c = c in zmod_zmult_zmult1)

   933 apply (auto simp add: mult_commute)

   934 done

   935

   936 lemma zmod_zmod_cancel: "n dvd m \<Longrightarrow> (k::int) mod m mod n = k mod n"

   937 by (rule mod_mod_cancel)

   938

   939

   940 subsection {*Splitting Rules for div and mod*}

   941

   942 text{*The proofs of the two lemmas below are essentially identical*}

   943

   944 lemma split_pos_lemma:

   945  "0<k ==>

   946     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"

   947 apply (rule iffI, clarify)

   948  apply (erule_tac P="P ?x ?y" in rev_mp)

   949  apply (subst mod_add_eq)

   950  apply (subst zdiv_zadd1_eq)

   951  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

   952 txt{*converse direction*}

   953 apply (drule_tac x = "n div k" in spec)

   954 apply (drule_tac x = "n mod k" in spec, simp)

   955 done

   956

   957 lemma split_neg_lemma:

   958  "k<0 ==>

   959     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"

   960 apply (rule iffI, clarify)

   961  apply (erule_tac P="P ?x ?y" in rev_mp)

   962  apply (subst mod_add_eq)

   963  apply (subst zdiv_zadd1_eq)

   964  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

   965 txt{*converse direction*}

   966 apply (drule_tac x = "n div k" in spec)

   967 apply (drule_tac x = "n mod k" in spec, simp)

   968 done

   969

   970 lemma split_zdiv:

   971  "P(n div k :: int) =

   972   ((k = 0 --> P 0) &

   973    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

   974    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"

   975 apply (case_tac "k=0", simp)

   976 apply (simp only: linorder_neq_iff)

   977 apply (erule disjE)

   978  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]

   979                       split_neg_lemma [of concl: "%x y. P x"])

   980 done

   981

   982 lemma split_zmod:

   983  "P(n mod k :: int) =

   984   ((k = 0 --> P n) &

   985    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

   986    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"

   987 apply (case_tac "k=0", simp)

   988 apply (simp only: linorder_neq_iff)

   989 apply (erule disjE)

   990  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]

   991                       split_neg_lemma [of concl: "%x y. P y"])

   992 done

   993

   994 (* Enable arith to deal with div 2 and mod 2: *)

   995 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]

   996 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]

   997

   998

   999 subsection{*Speeding up the Division Algorithm with Shifting*}

  1000

  1001 text{*computing div by shifting *}

  1002

  1003 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"

  1004 proof cases

  1005   assume "a=0"

  1006     thus ?thesis by simp

  1007 next

  1008   assume "a\<noteq>0" and le_a: "0\<le>a"

  1009   hence a_pos: "1 \<le> a" by arith

  1010   hence one_less_a2: "1 < 2*a" by arith

  1011   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"

  1012     by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)

  1013   with a_pos have "0 \<le> b mod a" by simp

  1014   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"

  1015     by (simp add: mod_pos_pos_trivial one_less_a2)

  1016   with  le_2a

  1017   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"

  1018     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

  1019                   right_distrib)

  1020   thus ?thesis

  1021     by (subst zdiv_zadd1_eq,

  1022         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2

  1023                   div_pos_pos_trivial)

  1024 qed

  1025

  1026 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"

  1027 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")

  1028 apply (rule_tac [2] pos_zdiv_mult_2)

  1029 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)

  1030 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")

  1031 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],

  1032        simp)

  1033 done

  1034

  1035 lemma zdiv_number_of_Bit0 [simp]:

  1036      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =

  1037           number_of v div (number_of w :: int)"

  1038 by (simp only: number_of_eq numeral_simps) simp

  1039

  1040 lemma zdiv_number_of_Bit1 [simp]:

  1041      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =

  1042           (if (0::int) \<le> number_of w

  1043            then number_of v div (number_of w)

  1044            else (number_of v + (1::int)) div (number_of w))"

  1045 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)

  1046 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)

  1047 done

  1048

  1049

  1050 subsection{*Computing mod by Shifting (proofs resemble those for div)*}

  1051

  1052 lemma pos_zmod_mult_2:

  1053      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"

  1054 apply (case_tac "a = 0", simp)

  1055 apply (subgoal_tac "1 < a * 2")

  1056  prefer 2 apply arith

  1057 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")

  1058  apply (rule_tac [2] mult_left_mono)

  1059 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq

  1060                       pos_mod_bound)

  1061 apply (subst mod_add_eq)

  1062 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)

  1063 apply (rule mod_pos_pos_trivial)

  1064 apply (auto simp add: mod_pos_pos_trivial ring_distribs)

  1065 apply (subgoal_tac "0 \<le> b mod a", arith, simp)

  1066 done

  1067

  1068 lemma neg_zmod_mult_2:

  1069      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"

  1070 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =

  1071                     1 + 2* ((-b - 1) mod (-a))")

  1072 apply (rule_tac [2] pos_zmod_mult_2)

  1073 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)

  1074 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")

  1075  prefer 2 apply simp

  1076 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])

  1077 done

  1078

  1079 lemma zmod_number_of_Bit0 [simp]:

  1080      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =

  1081       (2::int) * (number_of v mod number_of w)"

  1082 apply (simp only: number_of_eq numeral_simps)

  1083 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1084                  neg_zmod_mult_2 add_ac)

  1085 done

  1086

  1087 lemma zmod_number_of_Bit1 [simp]:

  1088      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =

  1089       (if (0::int) \<le> number_of w

  1090                 then 2 * (number_of v mod number_of w) + 1

  1091                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  1092 apply (simp only: number_of_eq numeral_simps)

  1093 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1094                  neg_zmod_mult_2 add_ac)

  1095 done

  1096

  1097

  1098 subsection{*Quotients of Signs*}

  1099

  1100 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  1101 apply (subgoal_tac "a div b \<le> -1", force)

  1102 apply (rule order_trans)

  1103 apply (rule_tac a' = "-1" in zdiv_mono1)

  1104 apply (auto simp add: div_eq_minus1)

  1105 done

  1106

  1107 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"

  1108 by (drule zdiv_mono1_neg, auto)

  1109

  1110 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  1111 apply auto

  1112 apply (drule_tac [2] zdiv_mono1)

  1113 apply (auto simp add: linorder_neq_iff)

  1114 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  1115 apply (blast intro: div_neg_pos_less0)

  1116 done

  1117

  1118 lemma neg_imp_zdiv_nonneg_iff:

  1119      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  1120 apply (subst zdiv_zminus_zminus [symmetric])

  1121 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  1122 done

  1123

  1124 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  1125 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  1126 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  1127

  1128 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  1129 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  1130 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  1131

  1132

  1133 subsection {* The Divides Relation *}

  1134

  1135 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"

  1136   by (rule dvd_eq_mod_eq_0)

  1137

  1138 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

  1139   zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]

  1140

  1141 lemma zdvd_0_right: "(m::int) dvd 0"

  1142   by (rule dvd_0_right) (* already declared [iff] *)

  1143

  1144 lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"

  1145   by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)

  1146

  1147 lemma zdvd_1_left: "1 dvd (m::int)"

  1148   by (rule one_dvd) (* already declared [simp] *)

  1149

  1150 lemma zdvd_refl: "m dvd (m::int)"

  1151   by (rule dvd_refl) (* already declared [simp] *)

  1152

  1153 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"

  1154   by (rule dvd_trans)

  1155

  1156 lemma zdvd_zminus_iff: "m dvd -n \<longleftrightarrow> m dvd (n::int)"

  1157   by (rule dvd_minus_iff) (* already declared [simp] *)

  1158

  1159 lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)"

  1160   by (rule minus_dvd_iff) (* already declared [simp] *)

  1161

  1162 lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"

  1163   by (rule abs_dvd_iff) (* already declared [simp] *)

  1164

  1165 lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"

  1166   by (rule dvd_abs_iff) (* already declared [simp] *)

  1167

  1168 lemma zdvd_anti_sym:

  1169     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"

  1170   apply (simp add: dvd_def, auto)

  1171   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)

  1172   done

  1173

  1174 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"

  1175   by (rule dvd_add)

  1176

  1177 lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a"

  1178   shows "\<bar>a\<bar> = \<bar>b\<bar>"

  1179 proof-

  1180   from ab obtain k where k:"b = a*k" unfolding dvd_def by blast

  1181   from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast

  1182   from k k' have "a = a*k*k'" by simp

  1183   with mult_cancel_left1[where c="a" and b="k*k'"]

  1184   have kk':"k*k' = 1" using anz by (simp add: mult_assoc)

  1185   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)

  1186   thus ?thesis using k k' by auto

  1187 qed

  1188

  1189 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"

  1190   by (rule Ring_and_Field.dvd_diff)

  1191

  1192 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"

  1193   apply (subgoal_tac "m = n + (m - n)")

  1194    apply (erule ssubst)

  1195    apply (blast intro: zdvd_zadd, simp)

  1196   done

  1197

  1198 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"

  1199   by (rule dvd_mult)

  1200

  1201 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"

  1202   by (rule dvd_mult2)

  1203

  1204 lemma zdvd_triv_right: "(k::int) dvd m * k"

  1205   by (rule dvd_triv_right) (* already declared [simp] *)

  1206

  1207 lemma zdvd_triv_left: "(k::int) dvd k * m"

  1208   by (rule dvd_triv_left) (* already declared [simp] *)

  1209

  1210 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"

  1211   by (rule dvd_mult_left)

  1212

  1213 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"

  1214   by (rule dvd_mult_right)

  1215

  1216 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"

  1217   by (rule mult_dvd_mono)

  1218

  1219 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

  1220   apply (rule iffI)

  1221    apply (erule_tac [2] zdvd_zadd)

  1222    apply (subgoal_tac "n = (n + k * m) - k * m")

  1223     apply (erule ssubst)

  1224     apply (erule zdvd_zdiff, simp_all)

  1225   done

  1226

  1227 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  1228   apply (simp add: dvd_def)

  1229   apply (auto simp add: zmod_zmult_zmult1)

  1230   done

  1231

  1232 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  1233   apply (subgoal_tac "k dvd n * (m div n) + m mod n")

  1234    apply (simp add: zmod_zdiv_equality [symmetric])

  1235   apply (simp only: zdvd_zadd zdvd_zmult2)

  1236   done

  1237

  1238 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"

  1239   apply (auto elim!: dvdE)

  1240   apply (subgoal_tac "0 < n")

  1241    prefer 2

  1242    apply (blast intro: order_less_trans)

  1243   apply (simp add: zero_less_mult_iff)

  1244   apply (subgoal_tac "n * k < n * 1")

  1245    apply (drule mult_less_cancel_left [THEN iffD1], auto)

  1246   done

  1247

  1248 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  1249   using zmod_zdiv_equality[where a="m" and b="n"]

  1250   by (simp add: algebra_simps)

  1251

  1252 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  1253 apply (subgoal_tac "m mod n = 0")

  1254  apply (simp add: zmult_div_cancel)

  1255 apply (simp only: zdvd_iff_zmod_eq_0)

  1256 done

  1257

  1258 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"

  1259   shows "m dvd n"

  1260 proof-

  1261   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast

  1262   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp

  1263     with h have False by (simp add: mult_assoc)}

  1264   hence "n = m * h" by blast

  1265   thus ?thesis by simp

  1266 qed

  1267

  1268 lemma zdvd_zmult_cancel_disj:

  1269   "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"

  1270 by (rule dvd_mult_cancel_left) (* already declared [simp] *)

  1271

  1272

  1273 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"

  1274 apply (simp split add: split_nat)

  1275 apply (rule iffI)

  1276 apply (erule exE)

  1277 apply (rule_tac x = "int x" in exI)

  1278 apply simp

  1279 apply (erule exE)

  1280 apply (rule_tac x = "nat x" in exI)

  1281 apply (erule conjE)

  1282 apply (erule_tac x = "nat x" in allE)

  1283 apply simp

  1284 done

  1285

  1286 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"

  1287 proof -

  1288   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"

  1289   proof -

  1290     fix k

  1291     assume A: "int y = int x * k"

  1292     then show "x dvd y" proof (cases k)

  1293       case (1 n) with A have "y = x * n" by (simp add: zmult_int)

  1294       then show ?thesis ..

  1295     next

  1296       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp

  1297       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)

  1298       also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)

  1299       finally have "- int (x * Suc n) = int y" ..

  1300       then show ?thesis by (simp only: negative_eq_positive) auto

  1301     qed

  1302   qed

  1303   then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)

  1304 qed

  1305

  1306 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"

  1307 proof

  1308   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)

  1309   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)

  1310   hence "nat \<bar>x\<bar> = 1"  by simp

  1311   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)

  1312 next

  1313   assume "\<bar>x\<bar>=1" thus "x dvd 1"

  1314     by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)

  1315 qed

  1316 lemma zdvd_mult_cancel1:

  1317   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"

  1318 proof

  1319   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"

  1320     by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)

  1321 next

  1322   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp

  1323   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)

  1324 qed

  1325

  1326 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

  1327   unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus_iff)

  1328

  1329 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

  1330   unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus2_iff)

  1331

  1332 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

  1333   by (auto simp add: dvd_int_iff)

  1334

  1335 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"

  1336   by (rule minus_dvd_iff)

  1337

  1338 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"

  1339   by (rule dvd_minus_iff)

  1340

  1341 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"

  1342   apply (rule_tac z=n in int_cases)

  1343   apply (auto simp add: dvd_int_iff)

  1344   apply (rule_tac z=z in int_cases)

  1345   apply (auto simp add: dvd_imp_le)

  1346   done

  1347

  1348 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  1349 apply (induct "y", auto)

  1350 apply (rule zmod_zmult1_eq [THEN trans])

  1351 apply (simp (no_asm_simp))

  1352 apply (rule mod_mult_eq [symmetric])

  1353 done

  1354

  1355 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  1356 apply (subst split_div, auto)

  1357 apply (subst split_zdiv, auto)

  1358 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)

  1359 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)

  1360 done

  1361

  1362 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  1363 apply (subst split_mod, auto)

  1364 apply (subst split_zmod, auto)

  1365 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  1366        in unique_remainder)

  1367 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)

  1368 done

  1369

  1370 text{*Suggested by Matthias Daum*}

  1371 lemma int_power_div_base:

  1372      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  1373 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")

  1374  apply (erule ssubst)

  1375  apply (simp only: power_add)

  1376  apply simp_all

  1377 done

  1378

  1379 text {* by Brian Huffman *}

  1380 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  1381 by (rule mod_minus_eq [symmetric])

  1382

  1383 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  1384 by (rule mod_diff_left_eq [symmetric])

  1385

  1386 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  1387 by (rule mod_diff_right_eq [symmetric])

  1388

  1389 lemmas zmod_simps =

  1390   IntDiv.zmod_zadd_left_eq  [symmetric]

  1391   IntDiv.zmod_zadd_right_eq [symmetric]

  1392   IntDiv.zmod_zmult1_eq     [symmetric]

  1393   mod_mult_left_eq          [symmetric]

  1394   IntDiv.zpower_zmod

  1395   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  1396

  1397 text {* Distributive laws for function @{text nat}. *}

  1398

  1399 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  1400 apply (rule linorder_cases [of y 0])

  1401 apply (simp add: div_nonneg_neg_le0)

  1402 apply simp

  1403 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  1404 done

  1405

  1406 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  1407 lemma nat_mod_distrib:

  1408   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  1409 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)

  1410 apply (simp add: nat_eq_iff zmod_int)

  1411 done

  1412

  1413 text{*Suggested by Matthias Daum*}

  1414 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  1415 apply (subgoal_tac "nat x div nat k < nat x")

  1416  apply (simp (asm_lr) add: nat_div_distrib [symmetric])

  1417 apply (rule Divides.div_less_dividend, simp_all)

  1418 done

  1419

  1420 text {* code generator setup *}

  1421

  1422 context ring_1

  1423 begin

  1424

  1425 lemma of_int_num [code]:

  1426   "of_int k = (if k = 0 then 0 else if k < 0 then

  1427      - of_int (- k) else let

  1428        (l, m) = divmod k 2;

  1429        l' = of_int l

  1430      in if m = 0 then l' + l' else l' + l' + 1)"

  1431 proof -

  1432   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>

  1433     of_int k = of_int (k div 2 * 2 + 1)"

  1434   proof -

  1435     have "k mod 2 < 2" by (auto intro: pos_mod_bound)

  1436     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)

  1437     moreover assume "k mod 2 \<noteq> 0"

  1438     ultimately have "k mod 2 = 1" by arith

  1439     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp

  1440     ultimately show ?thesis by auto

  1441   qed

  1442   have aux2: "\<And>x. of_int 2 * x = x + x"

  1443   proof -

  1444     fix x

  1445     have int2: "(2::int) = 1 + 1" by arith

  1446     show "of_int 2 * x = x + x"

  1447     unfolding int2 of_int_add left_distrib by simp

  1448   qed

  1449   have aux3: "\<And>x. x * of_int 2 = x + x"

  1450   proof -

  1451     fix x

  1452     have int2: "(2::int) = 1 + 1" by arith

  1453     show "x * of_int 2 = x + x"

  1454     unfolding int2 of_int_add right_distrib by simp

  1455   qed

  1456   from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)

  1457 qed

  1458

  1459 end

  1460

  1461 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  1462 proof

  1463   assume H: "x mod n = y mod n"

  1464   hence "x mod n - y mod n = 0" by simp

  1465   hence "(x mod n - y mod n) mod n = 0" by simp

  1466   hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])

  1467   thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)

  1468 next

  1469   assume H: "n dvd x - y"

  1470   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  1471   hence "x = n*k + y" by simp

  1472   hence "x mod n = (n*k + y) mod n" by simp

  1473   thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)

  1474 qed

  1475

  1476 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  1477   shows "\<exists>q. x = y + n * q"

  1478 proof-

  1479   from xy have th: "int x - int y = int (x - y)" by simp

  1480   from xyn have "int x mod int n = int y mod int n"

  1481     by (simp add: zmod_int[symmetric])

  1482   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  1483   hence "n dvd x - y" by (simp add: th zdvd_int)

  1484   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  1485 qed

  1486

  1487 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  1488   (is "?lhs = ?rhs")

  1489 proof

  1490   assume H: "x mod n = y mod n"

  1491   {assume xy: "x \<le> y"

  1492     from H have th: "y mod n = x mod n" by simp

  1493     from nat_mod_eq_lemma[OF th xy] have ?rhs

  1494       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  1495   moreover

  1496   {assume xy: "y \<le> x"

  1497     from nat_mod_eq_lemma[OF H xy] have ?rhs

  1498       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  1499   ultimately  show ?rhs using linear[of x y] by blast

  1500 next

  1501   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  1502   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  1503   thus  ?lhs by simp

  1504 qed

  1505

  1506

  1507 subsection {* Code generation *}

  1508

  1509 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1510   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  1511

  1512 lemma pdivmod_posDivAlg [code]:

  1513   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  1514 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  1515

  1516 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1517   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  1518     then pdivmod k l

  1519     else (let (r, s) = pdivmod k l in

  1520       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1521 proof -

  1522   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  1523   show ?thesis

  1524     by (simp add: divmod_mod_div pdivmod_def)

  1525       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  1526       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  1527 qed

  1528

  1529 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1530   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  1531     then pdivmod k l

  1532     else (let (r, s) = pdivmod k l in

  1533       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1534 proof -

  1535   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"

  1536     by (auto simp add: not_less sgn_if)

  1537   then show ?thesis by (simp add: divmod_pdivmod)

  1538 qed

  1539

  1540 code_modulename SML

  1541   IntDiv Integer

  1542

  1543 code_modulename OCaml

  1544   IntDiv Integer

  1545

  1546 code_modulename Haskell

  1547   IntDiv Integer

  1548

  1549 end
`