src/HOL/IntDiv.thy
 author blanchet Wed Mar 04 10:45:52 2009 +0100 (2009-03-04) changeset 30240 5b25fee0362c parent 29951 a70bc5190534 child 30242 aea5d7fa7ef5 permissions -rw-r--r--
Merge.
     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 lemmas posDivAlg_eqn_number_of [simp] =

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

   552

   553 lemmas negDivAlg_eqn_number_of [simp] =

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

   555

   556

   557 text{*Special-case simplification *}

   558

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

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

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

   562 apply (auto simp del: neg_mod_sign neg_mod_bound)

   563 done

   564

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

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

   567

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

   569     1 div z and 1 mod z **)

   570

   571 lemmas div_pos_pos_1_number_of [simp] =

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

   573

   574 lemmas div_pos_neg_1_number_of [simp] =

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

   576

   577 lemmas mod_pos_pos_1_number_of [simp] =

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

   579

   580 lemmas mod_pos_neg_1_number_of [simp] =

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

   582

   583

   584 lemmas posDivAlg_eqn_1_number_of [simp] =

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

   586

   587 lemmas negDivAlg_eqn_1_number_of [simp] =

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

   589

   590

   591

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

   593

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

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

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

   597 apply (rule unique_quotient_lemma)

   598 apply (erule subst)

   599 apply (erule subst, simp_all)

   600 done

   601

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

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

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

   605 apply (rule unique_quotient_lemma_neg)

   606 apply (erule subst)

   607 apply (erule subst, simp_all)

   608 done

   609

   610

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

   612

   613 lemma q_pos_lemma:

   614      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

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

   616  apply (simp add: zero_less_mult_iff)

   617 apply (simp add: right_distrib)

   618 done

   619

   620 lemma zdiv_mono2_lemma:

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

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

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

   624 apply (frule q_pos_lemma, assumption+)

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

   626  apply (simp add: mult_less_cancel_left)

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

   628  prefer 2 apply simp

   629 apply (simp (no_asm_simp) add: right_distrib)

   630 apply (subst add_commute, rule zadd_zless_mono, arith)

   631 apply (rule mult_right_mono, auto)

   632 done

   633

   634 lemma zdiv_mono2:

   635      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

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

   637  prefer 2 apply arith

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

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

   640 apply (rule zdiv_mono2_lemma)

   641 apply (erule subst)

   642 apply (erule subst, simp_all)

   643 done

   644

   645 lemma q_neg_lemma:

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

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

   648  apply (simp add: mult_less_0_iff, arith)

   649 done

   650

   651 lemma zdiv_mono2_neg_lemma:

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

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

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

   655 apply (frule q_neg_lemma, assumption+)

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

   657  apply (simp add: mult_less_cancel_left)

   658 apply (simp add: right_distrib)

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

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

   661 done

   662

   663 lemma zdiv_mono2_neg:

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

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

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

   667 apply (rule zdiv_mono2_neg_lemma)

   668 apply (erule subst)

   669 apply (erule subst, simp_all)

   670 done

   671

   672

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

   674

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

   676

   677 lemma zmult1_lemma:

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

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

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

   681

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

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

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

   685 done

   686

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

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

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

   690 done

   691

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

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

   694 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

   695 done

   696

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

   698

   699 lemma zadd1_lemma:

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

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

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

   703

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

   705 lemma zdiv_zadd1_eq:

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

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

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

   709 done

   710

   711 instance int :: ring_div

   712 proof

   713   fix a b c :: int

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

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

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

   717       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)

   718 qed auto

   719

   720 lemma posDivAlg_div_mod:

   721   assumes "k \<ge> 0"

   722   and "l \<ge> 0"

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

   724 proof (cases "l = 0")

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

   726 next

   727   case False with assms posDivAlg_correct

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

   729     by simp

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

   731   show ?thesis by simp

   732 qed

   733

   734 lemma negDivAlg_div_mod:

   735   assumes "k < 0"

   736   and "l > 0"

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

   738 proof -

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

   740   from assms negDivAlg_correct

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

   742     by simp

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

   744   show ?thesis by simp

   745 qed

   746

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

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

   749

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

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

   752

   753

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

   755

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

   757   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

   758   to cause particular problems.*)

   759

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

   761

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

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

   764  apply (simp add: algebra_simps)

   765 apply (rule order_le_less_trans)

   766  apply (erule_tac [2] mult_strict_right_mono)

   767  apply (rule mult_left_mono_neg)

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

   769  apply (simp)

   770 apply (simp)

   771 done

   772

   773 lemma zmult2_lemma_aux2:

   774      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

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

   776  apply arith

   777 apply (simp add: mult_le_0_iff)

   778 done

   779

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

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

   782 apply arith

   783 apply (simp add: zero_le_mult_iff)

   784 done

   785

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

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

   788  apply (simp add: right_diff_distrib)

   789 apply (rule order_less_le_trans)

   790  apply (erule mult_strict_right_mono)

   791  apply (rule_tac [2] mult_left_mono)

   792   apply simp

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

   794 apply simp

   795 done

   796

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

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

   799 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff

   800                    zero_less_mult_iff right_distrib [symmetric]

   801                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

   802

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

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

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

   806 done

   807

   808 lemma zmod_zmult2_eq:

   809      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

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

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

   812 done

   813

   814

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

   816

   817 lemma zdiv_zmult_zmult1_aux1:

   818      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   819 by (subst zdiv_zmult2_eq, auto)

   820

   821 lemma zdiv_zmult_zmult1_aux2:

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

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

   824 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)

   825 done

   826

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

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

   829 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)

   830 done

   831

   832 lemma zdiv_zmult_zmult1_if[simp]:

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

   834 by (simp add:zdiv_zmult_zmult1)

   835

   836

   837 subsection{*Distribution of Factors over mod*}

   838

   839 lemma zmod_zmult_zmult1_aux1:

   840      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"

   841 by (subst zmod_zmult2_eq, auto)

   842

   843 lemma zmod_zmult_zmult1_aux2:

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

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

   846 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)

   847 done

   848

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

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

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

   852 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)

   853 done

   854

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

   856 apply (cut_tac c = c in zmod_zmult_zmult1)

   857 apply (auto simp add: mult_commute)

   858 done

   859

   860

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

   862

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

   864

   865 lemma split_pos_lemma:

   866  "0<k ==>

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

   868 apply (rule iffI, clarify)

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

   870  apply (subst mod_add_eq)

   871  apply (subst zdiv_zadd1_eq)

   872  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

   873 txt{*converse direction*}

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

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

   876 done

   877

   878 lemma split_neg_lemma:

   879  "k<0 ==>

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

   881 apply (rule iffI, clarify)

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

   883  apply (subst mod_add_eq)

   884  apply (subst zdiv_zadd1_eq)

   885  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

   886 txt{*converse direction*}

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

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

   889 done

   890

   891 lemma split_zdiv:

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

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

   894    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

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

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

   897 apply (simp only: linorder_neq_iff)

   898 apply (erule disjE)

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

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

   901 done

   902

   903 lemma split_zmod:

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

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

   906    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

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

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

   909 apply (simp only: linorder_neq_iff)

   910 apply (erule disjE)

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

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

   913 done

   914

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

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

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

   918

   919

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

   921

   922 text{*computing div by shifting *}

   923

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

   925 proof cases

   926   assume "a=0"

   927     thus ?thesis by simp

   928 next

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

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

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

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

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

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

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

   936     by (simp add: mod_pos_pos_trivial one_less_a2)

   937   with  le_2a

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

   939     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

   940                   right_distrib)

   941   thus ?thesis

   942     by (subst zdiv_zadd1_eq,

   943         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2

   944                   div_pos_pos_trivial)

   945 qed

   946

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

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

   949 apply (rule_tac [2] pos_zdiv_mult_2)

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

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

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

   953        simp)

   954 done

   955

   956 lemma zdiv_number_of_Bit0 [simp]:

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

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

   959 by (simp only: number_of_eq numeral_simps) simp

   960

   961 lemma zdiv_number_of_Bit1 [simp]:

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

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

   964            then number_of v div (number_of w)

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

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

   967 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)

   968 done

   969

   970

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

   972

   973 lemma pos_zmod_mult_2:

   974      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"

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

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

   977  prefer 2 apply arith

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

   979  apply (rule_tac [2] mult_left_mono)

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

   981                       pos_mod_bound)

   982 apply (subst mod_add_eq)

   983 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)

   984 apply (rule mod_pos_pos_trivial)

   985 apply (auto simp add: mod_pos_pos_trivial ring_distribs)

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

   987 done

   988

   989 lemma neg_zmod_mult_2:

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

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

   992                     1 + 2* ((-b - 1) mod (-a))")

   993 apply (rule_tac [2] pos_zmod_mult_2)

   994 apply (auto simp add: right_diff_distrib)

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

   996  prefer 2 apply simp

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

   998 done

   999

  1000 lemma zmod_number_of_Bit0 [simp]:

  1001      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =

  1002       (2::int) * (number_of v mod number_of w)"

  1003 apply (simp only: number_of_eq numeral_simps)

  1004 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1005                  neg_zmod_mult_2 add_ac)

  1006 done

  1007

  1008 lemma zmod_number_of_Bit1 [simp]:

  1009      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =

  1010       (if (0::int) \<le> number_of w

  1011                 then 2 * (number_of v mod number_of w) + 1

  1012                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  1013 apply (simp only: number_of_eq numeral_simps)

  1014 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1015                  neg_zmod_mult_2 add_ac)

  1016 done

  1017

  1018

  1019 subsection{*Quotients of Signs*}

  1020

  1021 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  1022 apply (subgoal_tac "a div b \<le> -1", force)

  1023 apply (rule order_trans)

  1024 apply (rule_tac a' = "-1" in zdiv_mono1)

  1025 apply (auto simp add: div_eq_minus1)

  1026 done

  1027

  1028 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"

  1029 by (drule zdiv_mono1_neg, auto)

  1030

  1031 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  1032 apply auto

  1033 apply (drule_tac [2] zdiv_mono1)

  1034 apply (auto simp add: linorder_neq_iff)

  1035 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  1036 apply (blast intro: div_neg_pos_less0)

  1037 done

  1038

  1039 lemma neg_imp_zdiv_nonneg_iff:

  1040      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  1041 apply (subst zdiv_zminus_zminus [symmetric])

  1042 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  1043 done

  1044

  1045 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  1046 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  1047 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  1048

  1049 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  1050 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  1051 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  1052

  1053

  1054 subsection {* The Divides Relation *}

  1055

  1056 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

  1057   dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]

  1058

  1059 lemma zdvd_anti_sym:

  1060     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"

  1061   apply (simp add: dvd_def, auto)

  1062   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)

  1063   done

  1064

  1065 lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a"

  1066   shows "\<bar>a\<bar> = \<bar>b\<bar>"

  1067 proof-

  1068   from a dvd b obtain k where k:"b = a*k" unfolding dvd_def by blast

  1069   from b dvd a obtain k' where k':"a = b*k'" unfolding dvd_def by blast

  1070   from k k' have "a = a*k*k'" by simp

  1071   with mult_cancel_left1[where c="a" and b="k*k'"]

  1072   have kk':"k*k' = 1" using a\<noteq>0 by (simp add: mult_assoc)

  1073   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)

  1074   thus ?thesis using k k' by auto

  1075 qed

  1076

  1077 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"

  1078   apply (subgoal_tac "m = n + (m - n)")

  1079    apply (erule ssubst)

  1080    apply (blast intro: dvd_add, simp)

  1081   done

  1082

  1083 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

  1084 apply (rule iffI)

  1085  apply (erule_tac [2] dvd_add)

  1086  apply (subgoal_tac "n = (n + k * m) - k * m")

  1087   apply (erule ssubst)

  1088   apply (erule dvd_diff)

  1089   apply(simp_all)

  1090 done

  1091

  1092 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  1093   apply (simp add: dvd_def)

  1094   apply (auto simp add: zmod_zmult_zmult1)

  1095   done

  1096

  1097 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  1098   apply (subgoal_tac "k dvd n * (m div n) + m mod n")

  1099    apply (simp add: zmod_zdiv_equality [symmetric])

  1100   apply (simp only: dvd_add dvd_mult2)

  1101   done

  1102

  1103 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"

  1104   apply (auto elim!: dvdE)

  1105   apply (subgoal_tac "0 < n")

  1106    prefer 2

  1107    apply (blast intro: order_less_trans)

  1108   apply (simp add: zero_less_mult_iff)

  1109   apply (subgoal_tac "n * k < n * 1")

  1110    apply (drule mult_less_cancel_left [THEN iffD1], auto)

  1111   done

  1112

  1113 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  1114   using zmod_zdiv_equality[where a="m" and b="n"]

  1115   by (simp add: algebra_simps)

  1116

  1117 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  1118 apply (subgoal_tac "m mod n = 0")

  1119  apply (simp add: zmult_div_cancel)

  1120 apply (simp only: dvd_eq_mod_eq_0)

  1121 done

  1122

  1123 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"

  1124   shows "m dvd n"

  1125 proof-

  1126   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast

  1127   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp

  1128     with h have False by (simp add: mult_assoc)}

  1129   hence "n = m * h" by blast

  1130   thus ?thesis by simp

  1131 qed

  1132

  1133

  1134 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"

  1135 apply (simp split add: split_nat)

  1136 apply (rule iffI)

  1137 apply (erule exE)

  1138 apply (rule_tac x = "int x" in exI)

  1139 apply simp

  1140 apply (erule exE)

  1141 apply (rule_tac x = "nat x" in exI)

  1142 apply (erule conjE)

  1143 apply (erule_tac x = "nat x" in allE)

  1144 apply simp

  1145 done

  1146

  1147 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"

  1148 proof -

  1149   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"

  1150   proof -

  1151     fix k

  1152     assume A: "int y = int x * k"

  1153     then show "x dvd y" proof (cases k)

  1154       case (1 n) with A have "y = x * n" by (simp add: zmult_int)

  1155       then show ?thesis ..

  1156     next

  1157       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp

  1158       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)

  1159       also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)

  1160       finally have "- int (x * Suc n) = int y" ..

  1161       then show ?thesis by (simp only: negative_eq_positive) auto

  1162     qed

  1163   qed

  1164   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)

  1165 qed

  1166

  1167 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"

  1168 proof

  1169   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp

  1170   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)

  1171   hence "nat \<bar>x\<bar> = 1"  by simp

  1172   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)

  1173 next

  1174   assume "\<bar>x\<bar>=1" thus "x dvd 1"

  1175     by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)

  1176 qed

  1177 lemma zdvd_mult_cancel1:

  1178   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"

  1179 proof

  1180   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"

  1181     by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)

  1182 next

  1183   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp

  1184   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)

  1185 qed

  1186

  1187 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

  1188   unfolding zdvd_int by (cases "z \<ge> 0") simp_all

  1189

  1190 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

  1191   unfolding zdvd_int by (cases "z \<ge> 0") simp_all

  1192

  1193 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

  1194   by (auto simp add: dvd_int_iff)

  1195

  1196 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"

  1197   apply (rule_tac z=n in int_cases)

  1198   apply (auto simp add: dvd_int_iff)

  1199   apply (rule_tac z=z in int_cases)

  1200   apply (auto simp add: dvd_imp_le)

  1201   done

  1202

  1203 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  1204 apply (induct "y", auto)

  1205 apply (rule zmod_zmult1_eq [THEN trans])

  1206 apply (simp (no_asm_simp))

  1207 apply (rule mod_mult_eq [symmetric])

  1208 done

  1209

  1210 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  1211 apply (subst split_div, auto)

  1212 apply (subst split_zdiv, auto)

  1213 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)

  1214 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)

  1215 done

  1216

  1217 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  1218 apply (subst split_mod, auto)

  1219 apply (subst split_zmod, auto)

  1220 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  1221        in unique_remainder)

  1222 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)

  1223 done

  1224

  1225 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"

  1226 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

  1227

  1228 text{*Suggested by Matthias Daum*}

  1229 lemma int_power_div_base:

  1230      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  1231 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")

  1232  apply (erule ssubst)

  1233  apply (simp only: power_add)

  1234  apply simp_all

  1235 done

  1236

  1237 text {* by Brian Huffman *}

  1238 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  1239 by (rule mod_minus_eq [symmetric])

  1240

  1241 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  1242 by (rule mod_diff_left_eq [symmetric])

  1243

  1244 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  1245 by (rule mod_diff_right_eq [symmetric])

  1246

  1247 lemmas zmod_simps =

  1248   mod_add_left_eq  [symmetric]

  1249   mod_add_right_eq [symmetric]

  1250   IntDiv.zmod_zmult1_eq     [symmetric]

  1251   mod_mult_left_eq          [symmetric]

  1252   IntDiv.zpower_zmod

  1253   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  1254

  1255 text {* Distributive laws for function @{text nat}. *}

  1256

  1257 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  1258 apply (rule linorder_cases [of y 0])

  1259 apply (simp add: div_nonneg_neg_le0)

  1260 apply simp

  1261 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  1262 done

  1263

  1264 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  1265 lemma nat_mod_distrib:

  1266   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  1267 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)

  1268 apply (simp add: nat_eq_iff zmod_int)

  1269 done

  1270

  1271 text{*Suggested by Matthias Daum*}

  1272 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  1273 apply (subgoal_tac "nat x div nat k < nat x")

  1274  apply (simp (asm_lr) add: nat_div_distrib [symmetric])

  1275 apply (rule Divides.div_less_dividend, simp_all)

  1276 done

  1277

  1278 text {* code generator setup *}

  1279

  1280 context ring_1

  1281 begin

  1282

  1283 lemma of_int_num [code]:

  1284   "of_int k = (if k = 0 then 0 else if k < 0 then

  1285      - of_int (- k) else let

  1286        (l, m) = divmod k 2;

  1287        l' = of_int l

  1288      in if m = 0 then l' + l' else l' + l' + 1)"

  1289 proof -

  1290   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>

  1291     of_int k = of_int (k div 2 * 2 + 1)"

  1292   proof -

  1293     have "k mod 2 < 2" by (auto intro: pos_mod_bound)

  1294     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)

  1295     moreover assume "k mod 2 \<noteq> 0"

  1296     ultimately have "k mod 2 = 1" by arith

  1297     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp

  1298     ultimately show ?thesis by auto

  1299   qed

  1300   have aux2: "\<And>x. of_int 2 * x = x + x"

  1301   proof -

  1302     fix x

  1303     have int2: "(2::int) = 1 + 1" by arith

  1304     show "of_int 2 * x = x + x"

  1305     unfolding int2 of_int_add left_distrib by simp

  1306   qed

  1307   have aux3: "\<And>x. x * of_int 2 = x + x"

  1308   proof -

  1309     fix x

  1310     have int2: "(2::int) = 1 + 1" by arith

  1311     show "x * of_int 2 = x + x"

  1312     unfolding int2 of_int_add right_distrib by simp

  1313   qed

  1314   from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)

  1315 qed

  1316

  1317 end

  1318

  1319 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  1320 proof

  1321   assume H: "x mod n = y mod n"

  1322   hence "x mod n - y mod n = 0" by simp

  1323   hence "(x mod n - y mod n) mod n = 0" by simp

  1324   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])

  1325   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)

  1326 next

  1327   assume H: "n dvd x - y"

  1328   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  1329   hence "x = n*k + y" by simp

  1330   hence "x mod n = (n*k + y) mod n" by simp

  1331   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)

  1332 qed

  1333

  1334 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  1335   shows "\<exists>q. x = y + n * q"

  1336 proof-

  1337   from xy have th: "int x - int y = int (x - y)" by simp

  1338   from xyn have "int x mod int n = int y mod int n"

  1339     by (simp add: zmod_int[symmetric])

  1340   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  1341   hence "n dvd x - y" by (simp add: th zdvd_int)

  1342   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  1343 qed

  1344

  1345 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  1346   (is "?lhs = ?rhs")

  1347 proof

  1348   assume H: "x mod n = y mod n"

  1349   {assume xy: "x \<le> y"

  1350     from H have th: "y mod n = x mod n" by simp

  1351     from nat_mod_eq_lemma[OF th xy] have ?rhs

  1352       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  1353   moreover

  1354   {assume xy: "y \<le> x"

  1355     from nat_mod_eq_lemma[OF H xy] have ?rhs

  1356       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  1357   ultimately  show ?rhs using linear[of x y] by blast

  1358 next

  1359   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  1360   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  1361   thus  ?lhs by simp

  1362 qed

  1363

  1364

  1365 subsection {* Code generation *}

  1366

  1367 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1368   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  1369

  1370 lemma pdivmod_posDivAlg [code]:

  1371   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  1372 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  1373

  1374 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1375   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  1376     then pdivmod k l

  1377     else (let (r, s) = pdivmod k l in

  1378       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1379 proof -

  1380   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  1381   show ?thesis

  1382     by (simp add: divmod_mod_div pdivmod_def)

  1383       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  1384       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  1385 qed

  1386

  1387 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1388   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  1389     then pdivmod k l

  1390     else (let (r, s) = pdivmod k l in

  1391       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1392 proof -

  1393   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"

  1394     by (auto simp add: not_less sgn_if)

  1395   then show ?thesis by (simp add: divmod_pdivmod)

  1396 qed

  1397

  1398 code_modulename SML

  1399   IntDiv Integer

  1400

  1401 code_modulename OCaml

  1402   IntDiv Integer

  1403

  1404 code_modulename Haskell

  1405   IntDiv Integer

  1406

  1407 end
`