src/HOL/IntDiv.thy
 author haftmann Thu Aug 09 15:52:45 2007 +0200 (2007-08-09) changeset 24195 7d1a16c77f7c parent 23983 79dc793bec43 child 24286 7619080e49f0 permissions -rw-r--r--
tuned
     1 (*  Title:      HOL/IntDiv.thy

     2     ID:         $Id$

     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     4     Copyright   1999  University of Cambridge

     5

     6 *)

     7

     8 header{*The Division Operators div and mod; the Divides Relation dvd*}

     9

    10 theory IntDiv

    11 imports IntArith Divides FunDef

    12 begin

    13

    14 constdefs

    15   quorem :: "(int*int) * (int*int) => bool"

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

    17     [code func]: "quorem == %((a,b), (q,r)).

    18                       a = b*q + r &

    19                       (if 0 < b then 0\<le>r & r<b else b<r & r \<le> 0)"

    20

    21   adjust :: "[int, int*int] => int*int"

    22     --{*for the division algorithm*}

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

    24                          else (2*q, r)"

    25

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

    27 function

    28   posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"

    29 where

    30   "posDivAlg a b =

    31      (if (a<b | b\<le>0) then (0,a)

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

    33 by auto

    34 termination by (relation "measure (%(a,b). nat(a - b + 1))") auto

    35

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

    37 function

    38   negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"

    39 where

    40   "negDivAlg a b  =

    41      (if (0\<le>a+b | b\<le>0) then (-1,a+b)

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

    43 by auto

    44 termination by (relation "measure (%(a,b). nat(- a - b))") auto

    45

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

    47 constdefs

    48   negateSnd :: "int*int => int*int"

    49     [code func]: "negateSnd == %(q,r). (q,-r)"

    50

    51 definition

    52   divAlg :: "int \<times> int \<Rightarrow> int \<times> int"

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

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

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

    56 where

    57   "divAlg = (\<lambda>(a, b). (if 0\<le>a then

    58                   if 0\<le>b then posDivAlg a b

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

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

    61                else

    62                   if 0<b then negDivAlg a b

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

    64

    65 instance int :: Divides.div

    66   div_def: "a div b == fst (divAlg (a, b))"

    67   mod_def: "a mod b == snd (divAlg (a, b))" ..

    68

    69 lemma divAlg_mod_div:

    70   "divAlg (p, q) = (p div q, p mod q)"

    71   by (auto simp add: div_def mod_def)

    72

    73 text{*

    74 Here is the division algorithm in ML:

    75

    76 \begin{verbatim}

    77     fun posDivAlg (a,b) =

    78       if a<b then (0,a)

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

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

    81 	   end

    82

    83     fun negDivAlg (a,b) =

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

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

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

    87 	   end;

    88

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

    90

    91     fun divAlg (a,b) = if 0\<le>a then

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

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

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

    95 		       else

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

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

    98 \end{verbatim}

    99 *}

   100

   101

   102

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

   104

   105 lemma unique_quotient_lemma:

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

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

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

   109  prefer 2 apply (simp add: right_diff_distrib)

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

   111 apply (erule_tac [2] order_le_less_trans)

   112  prefer 2 apply (simp add: right_diff_distrib right_distrib)

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

   114  prefer 2 apply (simp add: right_diff_distrib right_distrib)

   115 apply (simp add: mult_less_cancel_left)

   116 done

   117

   118 lemma unique_quotient_lemma_neg:

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

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

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

   122     auto)

   123

   124 lemma unique_quotient:

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

   126       ==> q = q'"

   127 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)

   128 apply (blast intro: order_antisym

   129              dest: order_eq_refl [THEN unique_quotient_lemma]

   130              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

   131 done

   132

   133

   134 lemma unique_remainder:

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

   136       ==> r = r'"

   137 apply (subgoal_tac "q = q'")

   138  apply (simp add: quorem_def)

   139 apply (blast intro: unique_quotient)

   140 done

   141

   142

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

   144

   145 text{*And positive divisors*}

   146

   147 lemma adjust_eq [simp]:

   148      "adjust b (q,r) =

   149       (let diff = r-b in

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

   151                      else (2*q, r))"

   152 by (simp add: Let_def adjust_def)

   153

   154 declare posDivAlg.simps [simp del]

   155

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

   157 lemma posDivAlg_eqn:

   158      "0 < b ==>

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

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

   161

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

   163 theorem posDivAlg_correct:

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

   165   shows "quorem ((a, b), posDivAlg a b)"

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

   167 apply auto

   168 apply (simp add: quorem_def)

   169 apply (subst posDivAlg_eqn, simp add: right_distrib)

   170 apply (case_tac "a < b")

   171 apply simp_all

   172 apply (erule splitE)

   173 apply (auto simp add: right_distrib Let_def)

   174 done

   175

   176

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

   178

   179 text{*And positive divisors*}

   180

   181 declare negDivAlg.simps [simp del]

   182

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

   184 lemma negDivAlg_eqn:

   185      "0 < b ==>

   186       negDivAlg a b =

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

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

   189

   190 (*Correctness of negDivAlg: it computes quotients correctly

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

   192 lemma negDivAlg_correct:

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

   194   shows "quorem ((a, b), negDivAlg a b)"

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

   196 apply (auto simp add: linorder_not_le)

   197 apply (simp add: quorem_def)

   198 apply (subst negDivAlg_eqn, assumption)

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

   200 apply simp_all

   201 apply (erule splitE)

   202 apply (auto simp add: right_distrib Let_def)

   203 done

   204

   205

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

   207

   208 (*the case a=0*)

   209 lemma quorem_0: "b \<noteq> 0 ==> quorem ((0,b), (0,0))"

   210 by (auto simp add: quorem_def linorder_neq_iff)

   211

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

   213 by (subst posDivAlg.simps, auto)

   214

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

   216 by (subst negDivAlg.simps, auto)

   217

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

   219 by (simp add: negateSnd_def)

   220

   221 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"

   222 by (auto simp add: split_ifs quorem_def)

   223

   224 lemma divAlg_correct: "b \<noteq> 0 ==> quorem ((a,b), divAlg (a, b))"

   225 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg

   226                     posDivAlg_correct negDivAlg_correct)

   227

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

   229     certain equations.*}

   230

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

   232 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)

   233

   234

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

   236

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

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

   239 apply (cut_tac a = a and b = b in divAlg_correct)

   240 apply (auto simp add: quorem_def div_def mod_def)

   241 done

   242

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

   244 by(simp add: zmod_zdiv_equality[symmetric])

   245

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

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

   248

   249 text {* Tool setup *}

   250

   251 ML_setup {*

   252 local

   253

   254 structure CancelDivMod = CancelDivModFun(

   255 struct

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

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

   258   val mk_binop = HOLogic.mk_binop;

   259   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;

   260   val dest_sum = Int_Numeral_Simprocs.dest_sum;

   261   val div_mod_eqs =

   262     map mk_meta_eq [@{thm zdiv_zmod_equality},

   263       @{thm zdiv_zmod_equality2}];

   264   val trans = trans;

   265   val prove_eq_sums =

   266     let

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

   268     in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;

   269 end)

   270

   271 in

   272

   273 val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc

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

   275

   276 end;

   277

   278 Addsimprocs [cancel_zdiv_zmod_proc]

   279 *}

   280

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

   282 apply (cut_tac a = a and b = b in divAlg_correct)

   283 apply (auto simp add: quorem_def mod_def)

   284 done

   285

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

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

   288

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

   290 apply (cut_tac a = a and b = b in divAlg_correct)

   291 apply (auto simp add: quorem_def div_def mod_def)

   292 done

   293

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

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

   296

   297

   298

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

   300

   301 lemma quorem_div_mod: "b \<noteq> 0 ==> quorem ((a, b), (a div b, a mod b))"

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

   303 apply (force simp add: quorem_def linorder_neq_iff)

   304 done

   305

   306 lemma quorem_div: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a div b = q"

   307 by (simp add: quorem_div_mod [THEN unique_quotient])

   308

   309 lemma quorem_mod: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a mod b = r"

   310 by (simp add: quorem_div_mod [THEN unique_remainder])

   311

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

   313 apply (rule quorem_div)

   314 apply (auto simp add: quorem_def)

   315 done

   316

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

   318 apply (rule quorem_div)

   319 apply (auto simp add: quorem_def)

   320 done

   321

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

   323 apply (rule quorem_div)

   324 apply (auto simp add: quorem_def)

   325 done

   326

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

   328

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

   330 apply (rule_tac q = 0 in quorem_mod)

   331 apply (auto simp add: quorem_def)

   332 done

   333

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

   335 apply (rule_tac q = 0 in quorem_mod)

   336 apply (auto simp add: quorem_def)

   337 done

   338

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

   340 apply (rule_tac q = "-1" in quorem_mod)

   341 apply (auto simp add: quorem_def)

   342 done

   343

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

   345

   346

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

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

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

   350 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified,

   351                                  THEN quorem_div, THEN sym])

   352

   353 done

   354

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

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

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

   358 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],

   359        auto)

   360 done

   361

   362

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

   364

   365 lemma zminus1_lemma:

   366      "quorem((a,b),(q,r))

   367       ==> quorem ((-a,b), (if r=0 then -q else -q - 1),

   368                           (if r=0 then 0 else b-r))"

   369 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)

   370

   371

   372 lemma zdiv_zminus1_eq_if:

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

   374       ==> (-a) div b =

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

   376 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])

   377

   378 lemma zmod_zminus1_eq_if:

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

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

   381 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])

   382 done

   383

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

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

   386

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

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

   389

   390 lemma zdiv_zminus2_eq_if:

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

   392       ==> a div (-b) =

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

   394 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

   395

   396 lemma zmod_zminus2_eq_if:

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

   398 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

   399

   400

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

   402

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

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

   405  apply (simp add: zero_less_mult_iff, arith)

   406 done

   407

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

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

   410  apply (simp add: zero_le_mult_iff)

   411 apply (simp add: right_diff_distrib)

   412 done

   413

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

   415 apply (simp add: split_ifs quorem_def linorder_neq_iff)

   416 apply (rule order_antisym, safe, simp_all)

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

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

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

   420 done

   421

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

   423 apply (frule self_quotient, assumption)

   424 apply (simp add: quorem_def)

   425 done

   426

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

   428 by (simp add: quorem_div_mod [THEN self_quotient])

   429

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

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

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

   433 apply (simp add: quorem_div_mod [THEN self_remainder])

   434 done

   435

   436

   437 subsection{*Computation of Division and Remainder*}

   438

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

   440 by (simp add: div_def divAlg_def)

   441

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

   443 by (simp add: div_def divAlg_def)

   444

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

   446 by (simp add: mod_def divAlg_def)

   447

   448 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"

   449 by (simp add: div_def divAlg_def)

   450

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

   452 by (simp add: mod_def divAlg_def)

   453

   454 text{*a positive, b positive *}

   455

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

   457 by (simp add: div_def divAlg_def)

   458

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

   460 by (simp add: mod_def divAlg_def)

   461

   462 text{*a negative, b positive *}

   463

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

   465 by (simp add: div_def divAlg_def)

   466

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

   468 by (simp add: mod_def divAlg_def)

   469

   470 text{*a positive, b negative *}

   471

   472 lemma div_pos_neg:

   473      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"

   474 by (simp add: div_def divAlg_def)

   475

   476 lemma mod_pos_neg:

   477      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"

   478 by (simp add: mod_def divAlg_def)

   479

   480 text{*a negative, b negative *}

   481

   482 lemma div_neg_neg:

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

   484 by (simp add: div_def divAlg_def)

   485

   486 lemma mod_neg_neg:

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

   488 by (simp add: mod_def divAlg_def)

   489

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

   491

   492 lemmas div_pos_pos_number_of [simp] =

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

   494

   495 lemmas div_neg_pos_number_of [simp] =

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

   497

   498 lemmas div_pos_neg_number_of [simp] =

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

   500

   501 lemmas div_neg_neg_number_of [simp] =

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

   503

   504

   505 lemmas mod_pos_pos_number_of [simp] =

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

   507

   508 lemmas mod_neg_pos_number_of [simp] =

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

   510

   511 lemmas mod_pos_neg_number_of [simp] =

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

   513

   514 lemmas mod_neg_neg_number_of [simp] =

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

   516

   517

   518 lemmas posDivAlg_eqn_number_of [simp] =

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

   520

   521 lemmas negDivAlg_eqn_number_of [simp] =

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

   523

   524

   525 text{*Special-case simplification *}

   526

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

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

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

   530 apply (auto simp del:pos_mod_bound pos_mod_sign)

   531 done

   532

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

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

   535

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

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

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

   539 apply (auto simp del: neg_mod_sign neg_mod_bound)

   540 done

   541

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

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

   544

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

   546     1 div z and 1 mod z **)

   547

   548 lemmas div_pos_pos_1_number_of [simp] =

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

   550

   551 lemmas div_pos_neg_1_number_of [simp] =

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

   553

   554 lemmas mod_pos_pos_1_number_of [simp] =

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

   556

   557 lemmas mod_pos_neg_1_number_of [simp] =

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

   559

   560

   561 lemmas posDivAlg_eqn_1_number_of [simp] =

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

   563

   564 lemmas negDivAlg_eqn_1_number_of [simp] =

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

   566

   567

   568

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

   570

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

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

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

   574 apply (rule unique_quotient_lemma)

   575 apply (erule subst)

   576 apply (erule subst, simp_all)

   577 done

   578

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

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

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

   582 apply (rule unique_quotient_lemma_neg)

   583 apply (erule subst)

   584 apply (erule subst, simp_all)

   585 done

   586

   587

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

   589

   590 lemma q_pos_lemma:

   591      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

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

   593  apply (simp add: zero_less_mult_iff)

   594 apply (simp add: right_distrib)

   595 done

   596

   597 lemma zdiv_mono2_lemma:

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

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

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

   601 apply (frule q_pos_lemma, assumption+)

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

   603  apply (simp add: mult_less_cancel_left)

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

   605  prefer 2 apply simp

   606 apply (simp (no_asm_simp) add: right_distrib)

   607 apply (subst add_commute, rule zadd_zless_mono, arith)

   608 apply (rule mult_right_mono, auto)

   609 done

   610

   611 lemma zdiv_mono2:

   612      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

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

   614  prefer 2 apply arith

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

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

   617 apply (rule zdiv_mono2_lemma)

   618 apply (erule subst)

   619 apply (erule subst, simp_all)

   620 done

   621

   622 lemma q_neg_lemma:

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

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

   625  apply (simp add: mult_less_0_iff, arith)

   626 done

   627

   628 lemma zdiv_mono2_neg_lemma:

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

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

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

   632 apply (frule q_neg_lemma, assumption+)

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

   634  apply (simp add: mult_less_cancel_left)

   635 apply (simp add: right_distrib)

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

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

   638 done

   639

   640 lemma zdiv_mono2_neg:

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

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

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

   644 apply (rule zdiv_mono2_neg_lemma)

   645 apply (erule subst)

   646 apply (erule subst, simp_all)

   647 done

   648

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

   650

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

   652

   653 lemma zmult1_lemma:

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

   655       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"

   656 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)

   657

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

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

   660 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])

   661 done

   662

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

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

   665 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])

   666 done

   667

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

   669 apply (rule trans)

   670 apply (rule_tac s = "b*a mod c" in trans)

   671 apply (rule_tac [2] zmod_zmult1_eq)

   672 apply (simp_all add: mult_commute)

   673 done

   674

   675 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"

   676 apply (rule zmod_zmult1_eq' [THEN trans])

   677 apply (rule zmod_zmult1_eq)

   678 done

   679

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

   681 by (simp add: zdiv_zmult1_eq)

   682

   683 lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"

   684 by (subst mult_commute, erule zdiv_zmult_self1)

   685

   686 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"

   687 by (simp add: zmod_zmult1_eq)

   688

   689 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"

   690 by (simp add: mult_commute zmod_zmult1_eq)

   691

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

   693 proof

   694   assume "m mod d = 0"

   695   with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto

   696 next

   697   assume "EX q::int. m = d*q"

   698   thus "m mod d = 0" by auto

   699 qed

   700

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

   702

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

   704

   705 lemma zadd1_lemma:

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

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

   708 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)

   709

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

   711 lemma zdiv_zadd1_eq:

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

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

   714 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)

   715 done

   716

   717 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"

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

   719 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)

   720 done

   721

   722 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"

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

   724 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

   725 done

   726

   727 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"

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

   729 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)

   730 done

   731

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

   733 apply (rule trans [symmetric])

   734 apply (rule zmod_zadd1_eq, simp)

   735 apply (rule zmod_zadd1_eq [symmetric])

   736 done

   737

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

   739 apply (rule trans [symmetric])

   740 apply (rule zmod_zadd1_eq, simp)

   741 apply (rule zmod_zadd1_eq [symmetric])

   742 done

   743

   744 lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"

   745 by (simp add: zdiv_zadd1_eq)

   746

   747 lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"

   748 by (simp add: zdiv_zadd1_eq)

   749

   750 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"

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

   752 apply (simp add: zmod_zadd1_eq)

   753 done

   754

   755 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"

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

   757 apply (simp add: zmod_zadd1_eq)

   758 done

   759

   760

   761 lemma zmod_zdiff1_eq: fixes a::int

   762   shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r")

   763 proof -

   764   have "?l = (c + (a mod c - b mod c)) mod c"

   765     using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if)

   766   also have "\<dots> = ?r" by simp

   767   finally show ?thesis .

   768 qed

   769

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

   771

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

   773   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

   774   to cause particular problems.*)

   775

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

   777

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

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

   780 apply (simp add: right_diff_distrib)

   781 apply (rule order_le_less_trans)

   782 apply (erule_tac [2] mult_strict_right_mono)

   783 apply (rule mult_left_mono_neg)

   784 apply (auto simp add: compare_rls add_commute [of 1]

   785                       add1_zle_eq pos_mod_bound)

   786 done

   787

   788 lemma zmult2_lemma_aux2:

   789      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

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

   791  apply arith

   792 apply (simp add: mult_le_0_iff)

   793 done

   794

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

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

   797 apply arith

   798 apply (simp add: zero_le_mult_iff)

   799 done

   800

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

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

   803 apply (simp add: right_diff_distrib)

   804 apply (rule order_less_le_trans)

   805 apply (erule mult_strict_right_mono)

   806 apply (rule_tac [2] mult_left_mono)

   807 apply (auto simp add: compare_rls add_commute [of 1]

   808                       add1_zle_eq pos_mod_bound)

   809 done

   810

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

   812       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"

   813 by (auto simp add: mult_ac quorem_def linorder_neq_iff

   814                    zero_less_mult_iff right_distrib [symmetric]

   815                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

   816

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

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

   819 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])

   820 done

   821

   822 lemma zmod_zmult2_eq:

   823      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

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

   825 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])

   826 done

   827

   828

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

   830

   831 lemma zdiv_zmult_zmult1_aux1:

   832      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   833 by (subst zdiv_zmult2_eq, auto)

   834

   835 lemma zdiv_zmult_zmult1_aux2:

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

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

   838 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)

   839 done

   840

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

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

   843 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)

   844 done

   845

   846 lemma zdiv_zmult_zmult1_if[simp]:

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

   848 by (simp add:zdiv_zmult_zmult1)

   849

   850 (*

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

   852 apply (drule zdiv_zmult_zmult1)

   853 apply (auto simp add: mult_commute)

   854 done

   855 *)

   856

   857

   858 subsection{*Distribution of Factors over mod*}

   859

   860 lemma zmod_zmult_zmult1_aux1:

   861      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"

   862 by (subst zmod_zmult2_eq, auto)

   863

   864 lemma zmod_zmult_zmult1_aux2:

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

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

   867 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)

   868 done

   869

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

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

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

   873 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)

   874 done

   875

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

   877 apply (cut_tac c = c in zmod_zmult_zmult1)

   878 apply (auto simp add: mult_commute)

   879 done

   880

   881

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

   883

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

   885

   886 lemma split_pos_lemma:

   887  "0<k ==>

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

   889 apply (rule iffI, clarify)

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

   891  apply (subst zmod_zadd1_eq)

   892  apply (subst zdiv_zadd1_eq)

   893  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

   894 txt{*converse direction*}

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

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

   897 done

   898

   899 lemma split_neg_lemma:

   900  "k<0 ==>

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

   902 apply (rule iffI, clarify)

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

   904  apply (subst zmod_zadd1_eq)

   905  apply (subst zdiv_zadd1_eq)

   906  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

   907 txt{*converse direction*}

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

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

   910 done

   911

   912 lemma split_zdiv:

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

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

   915    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

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

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

   918 apply (simp only: linorder_neq_iff)

   919 apply (erule disjE)

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

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

   922 done

   923

   924 lemma split_zmod:

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

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

   927    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

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

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

   930 apply (simp only: linorder_neq_iff)

   931 apply (erule disjE)

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

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

   934 done

   935

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

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

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

   939

   940

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

   942

   943 text{*computing div by shifting *}

   944

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

   946 proof cases

   947   assume "a=0"

   948     thus ?thesis by simp

   949 next

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

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

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

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

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

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

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

   957     by (simp add: mod_pos_pos_trivial one_less_a2)

   958   with  le_2a

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

   960     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

   961                   right_distrib)

   962   thus ?thesis

   963     by (subst zdiv_zadd1_eq,

   964         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2

   965                   div_pos_pos_trivial)

   966 qed

   967

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

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

   970 apply (rule_tac [2] pos_zdiv_mult_2)

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

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

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

   974        simp)

   975 done

   976

   977

   978 (*Not clear why this must be proved separately; probably number_of causes

   979   simplification problems*)

   980 lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"

   981 by auto

   982

   983 lemma zdiv_number_of_BIT[simp]:

   984      "number_of (v BIT b) div number_of (w BIT bit.B0) =

   985           (if b=bit.B0 | (0::int) \<le> number_of w

   986            then number_of v div (number_of w)

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

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

   989 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac

   990             split: bit.split)

   991 done

   992

   993

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

   995

   996 lemma pos_zmod_mult_2:

   997      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"

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

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

  1000  prefer 2 apply arith

  1001 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")

  1002  apply (rule_tac [2] mult_left_mono)

  1003 apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq

  1004                       pos_mod_bound)

  1005 apply (subst zmod_zadd1_eq)

  1006 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)

  1007 apply (rule mod_pos_pos_trivial)

  1008 apply (auto simp add: mod_pos_pos_trivial left_distrib)

  1009 apply (subgoal_tac "0 \<le> b mod a", arith, simp)

  1010 done

  1011

  1012 lemma neg_zmod_mult_2:

  1013      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"

  1014 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =

  1015                     1 + 2* ((-b - 1) mod (-a))")

  1016 apply (rule_tac [2] pos_zmod_mult_2)

  1017 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)

  1018 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")

  1019  prefer 2 apply simp

  1020 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])

  1021 done

  1022

  1023 lemma zmod_number_of_BIT [simp]:

  1024      "number_of (v BIT b) mod number_of (w BIT bit.B0) =

  1025       (case b of

  1026           bit.B0 => 2 * (number_of v mod number_of w)

  1027         | bit.B1 => if (0::int) \<le> number_of w

  1028                 then 2 * (number_of v mod number_of w) + 1

  1029                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  1030 apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split)

  1031 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1032                  not_0_le_lemma neg_zmod_mult_2 add_ac)

  1033 done

  1034

  1035

  1036 subsection{*Quotients of Signs*}

  1037

  1038 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  1039 apply (subgoal_tac "a div b \<le> -1", force)

  1040 apply (rule order_trans)

  1041 apply (rule_tac a' = "-1" in zdiv_mono1)

  1042 apply (auto simp add: zdiv_minus1)

  1043 done

  1044

  1045 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"

  1046 by (drule zdiv_mono1_neg, auto)

  1047

  1048 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  1049 apply auto

  1050 apply (drule_tac [2] zdiv_mono1)

  1051 apply (auto simp add: linorder_neq_iff)

  1052 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  1053 apply (blast intro: div_neg_pos_less0)

  1054 done

  1055

  1056 lemma neg_imp_zdiv_nonneg_iff:

  1057      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  1058 apply (subst zdiv_zminus_zminus [symmetric])

  1059 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  1060 done

  1061

  1062 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  1063 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  1064 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  1065

  1066 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  1067 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  1068 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  1069

  1070

  1071 subsection {* The Divides Relation *}

  1072

  1073 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"

  1074   by (simp add: dvd_def zmod_eq_0_iff)

  1075

  1076 instance int :: dvd_mod

  1077   by default (simp add: times_class.dvd [symmetric] zdvd_iff_zmod_eq_0)

  1078

  1079 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

  1080   zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]

  1081

  1082 lemma zdvd_0_right [iff]: "(m::int) dvd 0"

  1083   by (simp add: dvd_def)

  1084

  1085 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"

  1086   by (simp add: dvd_def)

  1087

  1088 lemma zdvd_1_left [iff]: "1 dvd (m::int)"

  1089   by (simp add: dvd_def)

  1090

  1091 lemma zdvd_refl [simp]: "m dvd (m::int)"

  1092   by (auto simp add: dvd_def intro: zmult_1_right [symmetric])

  1093

  1094 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"

  1095   by (auto simp add: dvd_def intro: mult_assoc)

  1096

  1097 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"

  1098   apply (simp add: dvd_def, auto)

  1099    apply (rule_tac [!] x = "-k" in exI, auto)

  1100   done

  1101

  1102 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"

  1103   apply (simp add: dvd_def, auto)

  1104    apply (rule_tac [!] x = "-k" in exI, auto)

  1105   done

  1106 lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"

  1107   apply (cases "i > 0", simp)

  1108   apply (simp add: dvd_def)

  1109   apply (rule iffI)

  1110   apply (erule exE)

  1111   apply (rule_tac x="- k" in exI, simp)

  1112   apply (erule exE)

  1113   apply (rule_tac x="- k" in exI, simp)

  1114   done

  1115 lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"

  1116   apply (cases "j > 0", simp)

  1117   apply (simp add: dvd_def)

  1118   apply (rule iffI)

  1119   apply (erule exE)

  1120   apply (rule_tac x="- k" in exI, simp)

  1121   apply (erule exE)

  1122   apply (rule_tac x="- k" in exI, simp)

  1123   done

  1124

  1125 lemma zdvd_anti_sym:

  1126     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"

  1127   apply (simp add: dvd_def, auto)

  1128   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)

  1129   done

  1130

  1131 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"

  1132   apply (simp add: dvd_def)

  1133   apply (blast intro: right_distrib [symmetric])

  1134   done

  1135

  1136 lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a"

  1137   shows "\<bar>a\<bar> = \<bar>b\<bar>"

  1138 proof-

  1139   from ab obtain k where k:"b = a*k" unfolding dvd_def by blast

  1140   from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast

  1141   from k k' have "a = a*k*k'" by simp

  1142   with mult_cancel_left1[where c="a" and b="k*k'"]

  1143   have kk':"k*k' = 1" using anz by (simp add: mult_assoc)

  1144   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)

  1145   thus ?thesis using k k' by auto

  1146 qed

  1147

  1148 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"

  1149   apply (simp add: dvd_def)

  1150   apply (blast intro: right_diff_distrib [symmetric])

  1151   done

  1152

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

  1154   apply (subgoal_tac "m = n + (m - n)")

  1155    apply (erule ssubst)

  1156    apply (blast intro: zdvd_zadd, simp)

  1157   done

  1158

  1159 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"

  1160   apply (simp add: dvd_def)

  1161   apply (blast intro: mult_left_commute)

  1162   done

  1163

  1164 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"

  1165   apply (subst mult_commute)

  1166   apply (erule zdvd_zmult)

  1167   done

  1168

  1169 lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"

  1170   apply (rule zdvd_zmult)

  1171   apply (rule zdvd_refl)

  1172   done

  1173

  1174 lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"

  1175   apply (rule zdvd_zmult2)

  1176   apply (rule zdvd_refl)

  1177   done

  1178

  1179 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"

  1180   apply (simp add: dvd_def)

  1181   apply (simp add: mult_assoc, blast)

  1182   done

  1183

  1184 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"

  1185   apply (rule zdvd_zmultD2)

  1186   apply (subst mult_commute, assumption)

  1187   done

  1188

  1189 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"

  1190   apply (simp add: dvd_def, clarify)

  1191   apply (rule_tac x = "k * ka" in exI)

  1192   apply (simp add: mult_ac)

  1193   done

  1194

  1195 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

  1196   apply (rule iffI)

  1197    apply (erule_tac [2] zdvd_zadd)

  1198    apply (subgoal_tac "n = (n + k * m) - k * m")

  1199     apply (erule ssubst)

  1200     apply (erule zdvd_zdiff, simp_all)

  1201   done

  1202

  1203 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  1204   apply (simp add: dvd_def)

  1205   apply (auto simp add: zmod_zmult_zmult1)

  1206   done

  1207

  1208 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  1209   apply (subgoal_tac "k dvd n * (m div n) + m mod n")

  1210    apply (simp add: zmod_zdiv_equality [symmetric])

  1211   apply (simp only: zdvd_zadd zdvd_zmult2)

  1212   done

  1213

  1214 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"

  1215   apply (simp add: dvd_def, auto)

  1216   apply (subgoal_tac "0 < n")

  1217    prefer 2

  1218    apply (blast intro: order_less_trans)

  1219   apply (simp add: zero_less_mult_iff)

  1220   apply (subgoal_tac "n * k < n * 1")

  1221    apply (drule mult_less_cancel_left [THEN iffD1], auto)

  1222   done

  1223 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  1224   using zmod_zdiv_equality[where a="m" and b="n"]

  1225   by (simp add: ring_simps)

  1226

  1227 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  1228 apply (subgoal_tac "m mod n = 0")

  1229  apply (simp add: zmult_div_cancel)

  1230 apply (simp only: zdvd_iff_zmod_eq_0)

  1231 done

  1232

  1233 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"

  1234   shows "m dvd n"

  1235 proof-

  1236   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast

  1237   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp

  1238     with h have False by (simp add: mult_assoc)}

  1239   hence "n = m * h" by blast

  1240   thus ?thesis by blast

  1241 qed

  1242

  1243 lemma zdvd_zmult_cancel_disj[simp]:

  1244   "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"

  1245 by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)

  1246

  1247

  1248 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"

  1249   apply (simp split add: split_nat)

  1250   apply (rule iffI)

  1251   apply (erule exE)

  1252   apply (rule_tac x = "int x" in exI)

  1253   apply simp

  1254   apply (erule exE)

  1255   apply (rule_tac x = "nat x" in exI)

  1256   apply (erule conjE)

  1257   apply (erule_tac x = "nat x" in allE)

  1258   apply simp

  1259   done

  1260

  1261 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"

  1262   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]

  1263     nat_0_le cong add: conj_cong)

  1264   apply (rule iffI)

  1265   apply iprover

  1266   apply (erule exE)

  1267   apply (case_tac "x=0")

  1268   apply (rule_tac x=0 in exI)

  1269   apply simp

  1270   apply (case_tac "0 \<le> k")

  1271   apply iprover

  1272   apply (simp add: linorder_not_le)

  1273   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])

  1274   apply assumption

  1275   apply (simp add: mult_ac)

  1276   done

  1277

  1278 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"

  1279 proof

  1280   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)

  1281   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)

  1282   hence "nat \<bar>x\<bar> = 1"  by simp

  1283   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)

  1284 next

  1285   assume "\<bar>x\<bar>=1" thus "x dvd 1"

  1286     by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)

  1287 qed

  1288 lemma zdvd_mult_cancel1:

  1289   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"

  1290 proof

  1291   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"

  1292     by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)

  1293 next

  1294   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp

  1295   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)

  1296 qed

  1297

  1298 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

  1299   apply (auto simp add: dvd_def nat_abs_mult_distrib)

  1300   apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)

  1301    apply (rule_tac x = "-(int k)" in exI)

  1302   apply (auto simp add: int_mult)

  1303   done

  1304

  1305 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

  1306   apply (auto simp add: dvd_def abs_if int_mult)

  1307     apply (rule_tac [3] x = "nat k" in exI)

  1308     apply (rule_tac [2] x = "-(int k)" in exI)

  1309     apply (rule_tac x = "nat (-k)" in exI)

  1310     apply (cut_tac [3] k = m in int_less_0_conv)

  1311     apply (cut_tac k = m in int_less_0_conv)

  1312     apply (auto simp add: zero_le_mult_iff mult_less_0_iff

  1313       nat_mult_distrib [symmetric] nat_eq_iff2)

  1314   done

  1315

  1316 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

  1317   apply (auto simp add: dvd_def int_mult)

  1318   apply (rule_tac x = "nat k" in exI)

  1319   apply (cut_tac k = m in int_less_0_conv)

  1320   apply (auto simp add: zero_le_mult_iff mult_less_0_iff

  1321     nat_mult_distrib [symmetric] nat_eq_iff2)

  1322   done

  1323

  1324 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"

  1325   apply (auto simp add: dvd_def)

  1326    apply (rule_tac [!] x = "-k" in exI, auto)

  1327   done

  1328

  1329 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"

  1330   apply (auto simp add: dvd_def)

  1331    apply (drule minus_equation_iff [THEN iffD1])

  1332    apply (rule_tac [!] x = "-k" in exI, auto)

  1333   done

  1334

  1335 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"

  1336   apply (rule_tac z=n in int_cases)

  1337   apply (auto simp add: dvd_int_iff)

  1338   apply (rule_tac z=z in int_cases)

  1339   apply (auto simp add: dvd_imp_le)

  1340   done

  1341

  1342

  1343 subsection{*Integer Powers*}

  1344

  1345 instance int :: power ..

  1346

  1347 primrec

  1348   "p ^ 0 = 1"

  1349   "p ^ (Suc n) = (p::int) * (p ^ n)"

  1350

  1351

  1352 instance int :: recpower

  1353 proof

  1354   fix z :: int

  1355   fix n :: nat

  1356   show "z^0 = 1" by simp

  1357   show "z^(Suc n) = z * (z^n)" by simp

  1358 qed

  1359

  1360

  1361 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  1362 apply (induct "y", auto)

  1363 apply (rule zmod_zmult1_eq [THEN trans])

  1364 apply (simp (no_asm_simp))

  1365 apply (rule zmod_zmult_distrib [symmetric])

  1366 done

  1367

  1368 lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"

  1369   by (rule Power.power_add)

  1370

  1371 lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"

  1372   by (rule Power.power_mult [symmetric])

  1373

  1374 lemma zero_less_zpower_abs_iff [simp]:

  1375      "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"

  1376 apply (induct "n")

  1377 apply (auto simp add: zero_less_mult_iff)

  1378 done

  1379

  1380 lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"

  1381 apply (induct "n")

  1382 apply (auto simp add: zero_le_mult_iff)

  1383 done

  1384

  1385 lemma int_power: "int (m^n) = (int m) ^ n"

  1386   by (rule of_nat_power)

  1387

  1388 text{*Compatibility binding*}

  1389 lemmas zpower_int = int_power [symmetric]

  1390

  1391 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  1392 apply (subst split_div, auto)

  1393 apply (subst split_zdiv, auto)

  1394 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)

  1395 apply (auto simp add: IntDiv.quorem_def of_nat_mult)

  1396 done

  1397

  1398 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  1399 apply (subst split_mod, auto)

  1400 apply (subst split_zmod, auto)

  1401 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  1402        in unique_remainder)

  1403 apply (auto simp add: IntDiv.quorem_def of_nat_mult)

  1404 done

  1405

  1406 text{*Suggested by Matthias Daum*}

  1407 lemma int_power_div_base:

  1408      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  1409 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")

  1410  apply (erule ssubst)

  1411  apply (simp only: power_add)

  1412  apply simp_all

  1413 done

  1414

  1415 text {* by Brian Huffman *}

  1416 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  1417 by (simp only: zmod_zminus1_eq_if mod_mod_trivial)

  1418

  1419 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  1420 by (simp only: diff_def zmod_zadd_left_eq [symmetric])

  1421

  1422 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  1423 proof -

  1424   have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m"

  1425     by (simp only: zminus_zmod)

  1426   hence "(x + - (y mod m)) mod m = (x + - y) mod m"

  1427     by (simp only: zmod_zadd_right_eq [symmetric])

  1428   thus "(x - y mod m) mod m = (x - y) mod m"

  1429     by (simp only: diff_def)

  1430 qed

  1431

  1432 lemmas zmod_simps =

  1433   IntDiv.zmod_zadd_left_eq  [symmetric]

  1434   IntDiv.zmod_zadd_right_eq [symmetric]

  1435   IntDiv.zmod_zmult1_eq     [symmetric]

  1436   IntDiv.zmod_zmult1_eq'    [symmetric]

  1437   IntDiv.zpower_zmod

  1438   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  1439

  1440 text {* code generator setup *}

  1441

  1442 code_modulename SML

  1443   IntDiv Integer

  1444

  1445 code_modulename OCaml

  1446   IntDiv Integer

  1447

  1448 code_modulename Haskell

  1449   IntDiv Integer

  1450

  1451 end