src/HOL/Integ/IntDiv.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16417 9bc16273c2d4 child 16733 236dfafbeb63 permissions -rw-r--r--
Constant "If" is now local
     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

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

    10

    11 theory IntDiv

    12 imports IntArith Recdef

    13 uses ("IntDiv_setup.ML")

    14 begin

    15

    16 declare zless_nat_conj [simp]

    17

    18 constdefs

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

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

    21     "quorem == %((a,b), (q,r)).

    22                       a = b*q + r &

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

    24

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

    26     --{*for the division algorithm*}

    27     "adjust b == %(q,r). if 0 \<le> r-b then (2*q + 1, r-b)

    28                          else (2*q, r)"

    29

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

    31 consts posDivAlg :: "int*int => int*int"

    32 recdef posDivAlg "measure (%(a,b). nat(a - b + 1))"

    33     "posDivAlg (a,b) =

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

    35         else adjust b (posDivAlg(a, 2*b)))"

    36

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

    38 consts negDivAlg :: "int*int => int*int"

    39 recdef negDivAlg "measure (%(a,b). nat(- a - b))"

    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

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

    45 constdefs

    46   negateSnd :: "int*int => int*int"

    47     "negateSnd == %(q,r). (q,-r)"

    48

    49   divAlg :: "int*int => int*int"

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

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

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

    53     "divAlg ==

    54        %(a,b). if 0\<le>a then

    55                   if 0\<le>b then posDivAlg (a,b)

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

    57                        else negateSnd (negDivAlg (-a,-b))

    58                else

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

    60                   else         negateSnd (posDivAlg (-a,-b))"

    61

    62 instance

    63   int :: "Divides.div" ..       --{*avoid clash with 'div' token*}

    64

    65 text{*The operators are defined with reference to the algorithm, which is

    66 proved to satisfy the specification.*}

    67 defs

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

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

    70

    71

    72 text{*

    73 Here is the division algorithm in ML:

    74

    75 \begin{verbatim}

    76     fun posDivAlg (a,b) =

    77       if a<b then (0,a)

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

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

    80 	   end

    81

    82     fun negDivAlg (a,b) =

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

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

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

    86 	   end;

    87

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

    89

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

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

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

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

    94 		       else

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

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

    97 \end{verbatim}

    98 *}

    99

   100

   101

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

   103

   104 lemma unique_quotient_lemma:

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

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

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

   108  prefer 2 apply (simp add: right_diff_distrib)

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

   110 apply (erule_tac [2] order_le_less_trans)

   111  prefer 2 apply (simp add: right_diff_distrib right_distrib)

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

   113  prefer 2 apply (simp add: right_diff_distrib right_distrib)

   114 apply (simp add: mult_less_cancel_left)

   115 done

   116

   117 lemma unique_quotient_lemma_neg:

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

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

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

   121     auto)

   122

   123 lemma unique_quotient:

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

   125       ==> q = q'"

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

   127 apply (blast intro: order_antisym

   128              dest: order_eq_refl [THEN unique_quotient_lemma]

   129              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

   130 done

   131

   132

   133 lemma unique_remainder:

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

   135       ==> r = r'"

   136 apply (subgoal_tac "q = q'")

   137  apply (simp add: quorem_def)

   138 apply (blast intro: unique_quotient)

   139 done

   140

   141

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

   143

   144 text{*And positive divisors*}

   145

   146 lemma adjust_eq [simp]:

   147      "adjust b (q,r) =

   148       (let diff = r-b in

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

   150                      else (2*q, r))"

   151 by (simp add: Let_def adjust_def)

   152

   153 declare posDivAlg.simps [simp del]

   154

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

   156 lemma posDivAlg_eqn:

   157      "0 < b ==>

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

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

   160

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

   162 theorem posDivAlg_correct [rule_format]:

   163      "0 \<le> a --> 0 < b --> quorem ((a, b), posDivAlg (a, b))"

   164 apply (induct_tac a b rule: posDivAlg.induct, auto)

   165  apply (simp_all add: quorem_def)

   166  (*base case: a<b*)

   167  apply (simp add: posDivAlg_eqn)

   168 (*main argument*)

   169 apply (subst posDivAlg_eqn, simp_all)

   170 apply (erule splitE)

   171 apply (auto simp add: right_distrib Let_def)

   172 done

   173

   174

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

   176

   177 text{*And positive divisors*}

   178

   179 declare negDivAlg.simps [simp del]

   180

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

   182 lemma negDivAlg_eqn:

   183      "0 < b ==>

   184       negDivAlg (a,b) =

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

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

   187

   188 (*Correctness of negDivAlg: it computes quotients correctly

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

   190 lemma negDivAlg_correct [rule_format]:

   191      "a < 0 --> 0 < b --> quorem ((a, b), negDivAlg (a, b))"

   192 apply (induct_tac a b rule: negDivAlg.induct, auto)

   193  apply (simp_all add: quorem_def)

   194  (*base case: 0\<le>a+b*)

   195  apply (simp add: negDivAlg_eqn)

   196 (*main argument*)

   197 apply (subst negDivAlg_eqn, assumption)

   198 apply (erule splitE)

   199 apply (auto simp add: right_distrib Let_def)

   200 done

   201

   202

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

   204

   205 (*the case a=0*)

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

   207 by (auto simp add: quorem_def linorder_neq_iff)

   208

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

   210 by (subst posDivAlg.simps, auto)

   211

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

   213 by (subst negDivAlg.simps, auto)

   214

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

   216 by (simp add: negateSnd_def)

   217

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

   219 by (auto simp add: split_ifs quorem_def)

   220

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

   222 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg

   223                     posDivAlg_correct negDivAlg_correct)

   224

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

   226     certain equations.*}

   227

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

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

   230

   231

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

   233

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

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

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

   237 apply (auto simp add: quorem_def div_def mod_def)

   238 done

   239

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

   241 by(simp add: zmod_zdiv_equality[symmetric])

   242

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

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

   245

   246 use "IntDiv_setup.ML"

   247

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

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

   250 apply (auto simp add: quorem_def mod_def)

   251 done

   252

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

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

   255

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

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

   258 apply (auto simp add: quorem_def div_def mod_def)

   259 done

   260

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

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

   263

   264

   265

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

   267

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

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

   270 apply (force simp add: quorem_def linorder_neq_iff)

   271 done

   272

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

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

   275

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

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

   278

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

   280 apply (rule quorem_div)

   281 apply (auto simp add: quorem_def)

   282 done

   283

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

   285 apply (rule quorem_div)

   286 apply (auto simp add: quorem_def)

   287 done

   288

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

   290 apply (rule quorem_div)

   291 apply (auto simp add: quorem_def)

   292 done

   293

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

   295

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

   297 apply (rule_tac q = 0 in quorem_mod)

   298 apply (auto simp add: quorem_def)

   299 done

   300

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

   302 apply (rule_tac q = 0 in quorem_mod)

   303 apply (auto simp add: quorem_def)

   304 done

   305

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

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

   308 apply (auto simp add: quorem_def)

   309 done

   310

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

   312

   313

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

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

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

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

   318                                  THEN quorem_div, THEN sym])

   319

   320 done

   321

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

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

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

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

   326        auto)

   327 done

   328

   329

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

   331

   332 lemma zminus1_lemma:

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

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

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

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

   337

   338

   339 lemma zdiv_zminus1_eq_if:

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

   341       ==> (-a) div b =

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

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

   344

   345 lemma zmod_zminus1_eq_if:

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

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

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

   349 done

   350

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

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

   353

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

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

   356

   357 lemma zdiv_zminus2_eq_if:

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

   359       ==> a div (-b) =

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

   361 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

   362

   363 lemma zmod_zminus2_eq_if:

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

   365 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

   366

   367

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

   369

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

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

   372  apply (simp add: zero_less_mult_iff, arith)

   373 done

   374

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

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

   377  apply (simp add: zero_le_mult_iff)

   378 apply (simp add: right_diff_distrib)

   379 done

   380

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

   382 apply (simp add: split_ifs quorem_def linorder_neq_iff)

   383 apply (rule order_antisym, safe, simp_all)

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

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

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

   387 done

   388

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

   390 apply (frule self_quotient, assumption)

   391 apply (simp add: quorem_def)

   392 done

   393

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

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

   396

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

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

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

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

   401 done

   402

   403

   404 subsection{*Computation of Division and Remainder*}

   405

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

   407 by (simp add: div_def divAlg_def)

   408

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

   410 by (simp add: div_def divAlg_def)

   411

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

   413 by (simp add: mod_def divAlg_def)

   414

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

   416 by (simp add: div_def divAlg_def)

   417

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

   419 by (simp add: mod_def divAlg_def)

   420

   421 text{*a positive, b positive *}

   422

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

   424 by (simp add: div_def divAlg_def)

   425

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

   427 by (simp add: mod_def divAlg_def)

   428

   429 text{*a negative, b positive *}

   430

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

   432 by (simp add: div_def divAlg_def)

   433

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

   435 by (simp add: mod_def divAlg_def)

   436

   437 text{*a positive, b negative *}

   438

   439 lemma div_pos_neg:

   440      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))"

   441 by (simp add: div_def divAlg_def)

   442

   443 lemma mod_pos_neg:

   444      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))"

   445 by (simp add: mod_def divAlg_def)

   446

   447 text{*a negative, b negative *}

   448

   449 lemma div_neg_neg:

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

   451 by (simp add: div_def divAlg_def)

   452

   453 lemma mod_neg_neg:

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

   455 by (simp add: mod_def divAlg_def)

   456

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

   458

   459 declare div_pos_pos [of "number_of v" "number_of w", standard, simp]

   460 declare div_neg_pos [of "number_of v" "number_of w", standard, simp]

   461 declare div_pos_neg [of "number_of v" "number_of w", standard, simp]

   462 declare div_neg_neg [of "number_of v" "number_of w", standard, simp]

   463

   464 declare mod_pos_pos [of "number_of v" "number_of w", standard, simp]

   465 declare mod_neg_pos [of "number_of v" "number_of w", standard, simp]

   466 declare mod_pos_neg [of "number_of v" "number_of w", standard, simp]

   467 declare mod_neg_neg [of "number_of v" "number_of w", standard, simp]

   468

   469 declare posDivAlg_eqn [of "number_of v" "number_of w", standard, simp]

   470 declare negDivAlg_eqn [of "number_of v" "number_of w", standard, simp]

   471

   472

   473 text{*Special-case simplification *}

   474

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

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

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

   478 apply (auto simp del:pos_mod_bound pos_mod_sign)

   479 done

   480

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

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

   483

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

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

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

   487 apply (auto simp del: neg_mod_sign neg_mod_bound)

   488 done

   489

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

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

   492

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

   494     1 div z and 1 mod z **)

   495

   496 declare div_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]

   497 declare div_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]

   498 declare mod_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]

   499 declare mod_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]

   500

   501 declare posDivAlg_eqn [of concl: 1 "number_of w", standard, simp]

   502 declare negDivAlg_eqn [of concl: 1 "number_of w", standard, simp]

   503

   504

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

   506

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

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

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

   510 apply (rule unique_quotient_lemma)

   511 apply (erule subst)

   512 apply (erule subst, simp_all)

   513 done

   514

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

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

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

   518 apply (rule unique_quotient_lemma_neg)

   519 apply (erule subst)

   520 apply (erule subst, simp_all)

   521 done

   522

   523

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

   525

   526 lemma q_pos_lemma:

   527      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

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

   529  apply (simp add: zero_less_mult_iff)

   530 apply (simp add: right_distrib)

   531 done

   532

   533 lemma zdiv_mono2_lemma:

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

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

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

   537 apply (frule q_pos_lemma, assumption+)

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

   539  apply (simp add: mult_less_cancel_left)

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

   541  prefer 2 apply simp

   542 apply (simp (no_asm_simp) add: right_distrib)

   543 apply (subst add_commute, rule zadd_zless_mono, arith)

   544 apply (rule mult_right_mono, auto)

   545 done

   546

   547 lemma zdiv_mono2:

   548      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

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

   550  prefer 2 apply arith

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

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

   553 apply (rule zdiv_mono2_lemma)

   554 apply (erule subst)

   555 apply (erule subst, simp_all)

   556 done

   557

   558 lemma q_neg_lemma:

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

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

   561  apply (simp add: mult_less_0_iff, arith)

   562 done

   563

   564 lemma zdiv_mono2_neg_lemma:

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

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

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

   568 apply (frule q_neg_lemma, assumption+)

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

   570  apply (simp add: mult_less_cancel_left)

   571 apply (simp add: right_distrib)

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

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

   574 done

   575

   576 lemma zdiv_mono2_neg:

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

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

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

   580 apply (rule zdiv_mono2_neg_lemma)

   581 apply (erule subst)

   582 apply (erule subst, simp_all)

   583 done

   584

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

   586

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

   588

   589 lemma zmult1_lemma:

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

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

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

   593

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

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

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

   597 done

   598

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

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

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

   602 done

   603

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

   605 apply (rule trans)

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

   607 apply (rule_tac [2] zmod_zmult1_eq)

   608 apply (simp_all add: mult_commute)

   609 done

   610

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

   612 apply (rule zmod_zmult1_eq' [THEN trans])

   613 apply (rule zmod_zmult1_eq)

   614 done

   615

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

   617 by (simp add: zdiv_zmult1_eq)

   618

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

   620 by (subst mult_commute, erule zdiv_zmult_self1)

   621

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

   623 by (simp add: zmod_zmult1_eq)

   624

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

   626 by (simp add: mult_commute zmod_zmult1_eq)

   627

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

   629 proof

   630   assume "m mod d = 0"

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

   632 next

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

   634   thus "m mod d = 0" by auto

   635 qed

   636

   637 declare zmod_eq_0_iff [THEN iffD1, dest!]

   638

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

   640

   641 lemma zadd1_lemma:

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

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

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

   645

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

   647 lemma zdiv_zadd1_eq:

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

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

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

   651 done

   652

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

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

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

   656 done

   657

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

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

   660 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

   661 done

   662

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

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

   665 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)

   666 done

   667

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

   669 apply (rule trans [symmetric])

   670 apply (rule zmod_zadd1_eq, simp)

   671 apply (rule zmod_zadd1_eq [symmetric])

   672 done

   673

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

   675 apply (rule trans [symmetric])

   676 apply (rule zmod_zadd1_eq, simp)

   677 apply (rule zmod_zadd1_eq [symmetric])

   678 done

   679

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

   681 by (simp add: zdiv_zadd1_eq)

   682

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

   684 by (simp add: zdiv_zadd1_eq)

   685

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

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

   688 apply (simp add: zmod_zadd1_eq)

   689 done

   690

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

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

   693 apply (simp add: zmod_zadd1_eq)

   694 done

   695

   696

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

   698

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

   700   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

   701   to cause particular problems.*)

   702

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

   704

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

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

   707 apply (simp add: right_diff_distrib)

   708 apply (rule order_le_less_trans)

   709 apply (erule_tac [2] mult_strict_right_mono)

   710 apply (rule mult_left_mono_neg)

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

   712                       add1_zle_eq pos_mod_bound)

   713 done

   714

   715 lemma zmult2_lemma_aux2:

   716      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

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

   718  apply arith

   719 apply (simp add: mult_le_0_iff)

   720 done

   721

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

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

   724 apply arith

   725 apply (simp add: zero_le_mult_iff)

   726 done

   727

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

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

   730 apply (simp add: right_diff_distrib)

   731 apply (rule order_less_le_trans)

   732 apply (erule mult_strict_right_mono)

   733 apply (rule_tac [2] mult_left_mono)

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

   735                       add1_zle_eq pos_mod_bound)

   736 done

   737

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

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

   740 by (auto simp add: mult_ac quorem_def linorder_neq_iff

   741                    zero_less_mult_iff right_distrib [symmetric]

   742                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

   743

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

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

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

   747 done

   748

   749 lemma zmod_zmult2_eq:

   750      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

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

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

   753 done

   754

   755

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

   757

   758 lemma zdiv_zmult_zmult1_aux1:

   759      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   760 by (subst zdiv_zmult2_eq, auto)

   761

   762 lemma zdiv_zmult_zmult1_aux2:

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

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

   765 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)

   766 done

   767

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

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

   770 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)

   771 done

   772

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

   774 apply (drule zdiv_zmult_zmult1)

   775 apply (auto simp add: mult_commute)

   776 done

   777

   778

   779

   780 subsection{*Distribution of Factors over mod*}

   781

   782 lemma zmod_zmult_zmult1_aux1:

   783      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"

   784 by (subst zmod_zmult2_eq, auto)

   785

   786 lemma zmod_zmult_zmult1_aux2:

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

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

   789 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)

   790 done

   791

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

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

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

   795 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)

   796 done

   797

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

   799 apply (cut_tac c = c in zmod_zmult_zmult1)

   800 apply (auto simp add: mult_commute)

   801 done

   802

   803

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

   805

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

   807

   808 lemma split_pos_lemma:

   809  "0<k ==>

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

   811 apply (rule iffI, clarify)

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

   813  apply (subst zmod_zadd1_eq)

   814  apply (subst zdiv_zadd1_eq)

   815  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

   816 txt{*converse direction*}

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

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

   819 done

   820

   821 lemma split_neg_lemma:

   822  "k<0 ==>

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

   824 apply (rule iffI, clarify)

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

   826  apply (subst zmod_zadd1_eq)

   827  apply (subst zdiv_zadd1_eq)

   828  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

   829 txt{*converse direction*}

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

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

   832 done

   833

   834 lemma split_zdiv:

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

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

   837    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

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

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

   840 apply (simp only: linorder_neq_iff)

   841 apply (erule disjE)

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

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

   844 done

   845

   846 lemma split_zmod:

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

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

   849    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

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

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

   852 apply (simp only: linorder_neq_iff)

   853 apply (erule disjE)

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

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

   856 done

   857

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

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

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

   861

   862

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

   864

   865 text{*computing div by shifting *}

   866

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

   868 proof cases

   869   assume "a=0"

   870     thus ?thesis by simp

   871 next

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

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

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

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

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

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

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

   879     by (simp add: mod_pos_pos_trivial one_less_a2)

   880   with  le_2a

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

   882     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

   883                   right_distrib)

   884   thus ?thesis

   885     by (subst zdiv_zadd1_eq,

   886         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2

   887                   div_pos_pos_trivial)

   888 qed

   889

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

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

   892 apply (rule_tac [2] pos_zdiv_mult_2)

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

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

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

   896        simp)

   897 done

   898

   899

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

   901   simplification problems*)

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

   903 by auto

   904

   905 lemma zdiv_number_of_BIT[simp]:

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

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

   908            then number_of v div (number_of w)

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

   910 apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if)

   911 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac

   912             split: bit.split)

   913 done

   914

   915

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

   917

   918 lemma pos_zmod_mult_2:

   919      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"

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

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

   922  prefer 2 apply arith

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

   924  apply (rule_tac [2] mult_left_mono)

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

   926                       pos_mod_bound)

   927 apply (subst zmod_zadd1_eq)

   928 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)

   929 apply (rule mod_pos_pos_trivial)

   930 apply (auto simp add: mod_pos_pos_trivial left_distrib)

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

   932 done

   933

   934 lemma neg_zmod_mult_2:

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

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

   937                     1 + 2* ((-b - 1) mod (-a))")

   938 apply (rule_tac [2] pos_zmod_mult_2)

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

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

   941  prefer 2 apply simp

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

   943 done

   944

   945 lemma zmod_number_of_BIT [simp]:

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

   947       (case b of

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

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

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

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

   952 apply (simp only: number_of_eq Bin_simps UNIV_I split: bit.split)

   953 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

   954                  not_0_le_lemma neg_zmod_mult_2 add_ac)

   955 done

   956

   957

   958 subsection{*Quotients of Signs*}

   959

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

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

   962 apply (rule order_trans)

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

   964 apply (auto simp add: zdiv_minus1)

   965 done

   966

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

   968 by (drule zdiv_mono1_neg, auto)

   969

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

   971 apply auto

   972 apply (drule_tac [2] zdiv_mono1)

   973 apply (auto simp add: linorder_neq_iff)

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

   975 apply (blast intro: div_neg_pos_less0)

   976 done

   977

   978 lemma neg_imp_zdiv_nonneg_iff:

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

   980 apply (subst zdiv_zminus_zminus [symmetric])

   981 apply (subst pos_imp_zdiv_nonneg_iff, auto)

   982 done

   983

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

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

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

   987

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

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

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

   991

   992

   993 subsection {* The Divides Relation *}

   994

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

   996 by(simp add:dvd_def zmod_eq_0_iff)

   997

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

   999 by (simp add: dvd_def)

  1000

  1001 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"

  1002   by (simp add: dvd_def)

  1003

  1004 lemma zdvd_1_left [iff]: "1 dvd (m::int)"

  1005   by (simp add: dvd_def)

  1006

  1007 lemma zdvd_refl [simp]: "m dvd (m::int)"

  1008 by (auto simp add: dvd_def intro: zmult_1_right [symmetric])

  1009

  1010 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"

  1011 by (auto simp add: dvd_def intro: mult_assoc)

  1012

  1013 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"

  1014   apply (simp add: dvd_def, auto)

  1015    apply (rule_tac [!] x = "-k" in exI, auto)

  1016   done

  1017

  1018 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"

  1019   apply (simp add: dvd_def, auto)

  1020    apply (rule_tac [!] x = "-k" in exI, auto)

  1021   done

  1022

  1023 lemma zdvd_anti_sym:

  1024     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"

  1025   apply (simp add: dvd_def, auto)

  1026   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)

  1027   done

  1028

  1029 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"

  1030   apply (simp add: dvd_def)

  1031   apply (blast intro: right_distrib [symmetric])

  1032   done

  1033

  1034 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"

  1035   apply (simp add: dvd_def)

  1036   apply (blast intro: right_diff_distrib [symmetric])

  1037   done

  1038

  1039 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"

  1040   apply (subgoal_tac "m = n + (m - n)")

  1041    apply (erule ssubst)

  1042    apply (blast intro: zdvd_zadd, simp)

  1043   done

  1044

  1045 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"

  1046   apply (simp add: dvd_def)

  1047   apply (blast intro: mult_left_commute)

  1048   done

  1049

  1050 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"

  1051   apply (subst mult_commute)

  1052   apply (erule zdvd_zmult)

  1053   done

  1054

  1055 lemma [iff]: "(k::int) dvd m * k"

  1056   apply (rule zdvd_zmult)

  1057   apply (rule zdvd_refl)

  1058   done

  1059

  1060 lemma [iff]: "(k::int) dvd k * m"

  1061   apply (rule zdvd_zmult2)

  1062   apply (rule zdvd_refl)

  1063   done

  1064

  1065 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"

  1066   apply (simp add: dvd_def)

  1067   apply (simp add: mult_assoc, blast)

  1068   done

  1069

  1070 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"

  1071   apply (rule zdvd_zmultD2)

  1072   apply (subst mult_commute, assumption)

  1073   done

  1074

  1075 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"

  1076   apply (simp add: dvd_def, clarify)

  1077   apply (rule_tac x = "k * ka" in exI)

  1078   apply (simp add: mult_ac)

  1079   done

  1080

  1081 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

  1082   apply (rule iffI)

  1083    apply (erule_tac [2] zdvd_zadd)

  1084    apply (subgoal_tac "n = (n + k * m) - k * m")

  1085     apply (erule ssubst)

  1086     apply (erule zdvd_zdiff, simp_all)

  1087   done

  1088

  1089 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  1090   apply (simp add: dvd_def)

  1091   apply (auto simp add: zmod_zmult_zmult1)

  1092   done

  1093

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

  1095   apply (subgoal_tac "k dvd n * (m div n) + m mod n")

  1096    apply (simp add: zmod_zdiv_equality [symmetric])

  1097   apply (simp only: zdvd_zadd zdvd_zmult2)

  1098   done

  1099

  1100 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"

  1101   apply (simp add: dvd_def, auto)

  1102   apply (subgoal_tac "0 < n")

  1103    prefer 2

  1104    apply (blast intro: order_less_trans)

  1105   apply (simp add: zero_less_mult_iff)

  1106   apply (subgoal_tac "n * k < n * 1")

  1107    apply (drule mult_less_cancel_left [THEN iffD1], auto)

  1108   done

  1109

  1110 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

  1111   apply (auto simp add: dvd_def nat_abs_mult_distrib)

  1112   apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)

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

  1114   apply (auto simp add: int_mult)

  1115   done

  1116

  1117 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

  1118   apply (auto simp add: dvd_def abs_if int_mult)

  1119     apply (rule_tac [3] x = "nat k" in exI)

  1120     apply (rule_tac [2] x = "-(int k)" in exI)

  1121     apply (rule_tac x = "nat (-k)" in exI)

  1122     apply (cut_tac [3] k = m in int_less_0_conv)

  1123     apply (cut_tac k = m in int_less_0_conv)

  1124     apply (auto simp add: zero_le_mult_iff mult_less_0_iff

  1125       nat_mult_distrib [symmetric] nat_eq_iff2)

  1126   done

  1127

  1128 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

  1129   apply (auto simp add: dvd_def int_mult)

  1130   apply (rule_tac x = "nat k" in exI)

  1131   apply (cut_tac k = m in int_less_0_conv)

  1132   apply (auto simp add: zero_le_mult_iff mult_less_0_iff

  1133     nat_mult_distrib [symmetric] nat_eq_iff2)

  1134   done

  1135

  1136 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"

  1137   apply (auto simp add: dvd_def)

  1138    apply (rule_tac [!] x = "-k" in exI, auto)

  1139   done

  1140

  1141 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"

  1142   apply (auto simp add: dvd_def)

  1143    apply (drule minus_equation_iff [THEN iffD1])

  1144    apply (rule_tac [!] x = "-k" in exI, auto)

  1145   done

  1146

  1147 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"

  1148   apply (rule_tac z=n in int_cases)

  1149   apply (auto simp add: dvd_int_iff)

  1150   apply (rule_tac z=z in int_cases)

  1151   apply (auto simp add: dvd_imp_le)

  1152   done

  1153

  1154

  1155 subsection{*Integer Powers*}

  1156

  1157 instance int :: power ..

  1158

  1159 primrec

  1160   "p ^ 0 = 1"

  1161   "p ^ (Suc n) = (p::int) * (p ^ n)"

  1162

  1163

  1164 instance int :: recpower

  1165 proof

  1166   fix z :: int

  1167   fix n :: nat

  1168   show "z^0 = 1" by simp

  1169   show "z^(Suc n) = z * (z^n)" by simp

  1170 qed

  1171

  1172

  1173 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  1174 apply (induct "y", auto)

  1175 apply (rule zmod_zmult1_eq [THEN trans])

  1176 apply (simp (no_asm_simp))

  1177 apply (rule zmod_zmult_distrib [symmetric])

  1178 done

  1179

  1180 lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"

  1181   by (rule Power.power_add)

  1182

  1183 lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"

  1184   by (rule Power.power_mult [symmetric])

  1185

  1186 lemma zero_less_zpower_abs_iff [simp]:

  1187      "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"

  1188 apply (induct "n")

  1189 apply (auto simp add: zero_less_mult_iff)

  1190 done

  1191

  1192 lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"

  1193 apply (induct "n")

  1194 apply (auto simp add: zero_le_mult_iff)

  1195 done

  1196

  1197 lemma int_power: "int (m^n) = (int m) ^ n"

  1198   by (induct n, simp_all add: int_mult)

  1199

  1200 text{*Compatibility binding*}

  1201 lemmas zpower_int = int_power [symmetric]

  1202

  1203 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  1204 apply (subst split_div, auto)

  1205 apply (subst split_zdiv, auto)

  1206 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)

  1207 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)

  1208 done

  1209

  1210 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  1211 apply (subst split_mod, auto)

  1212 apply (subst split_zmod, auto)

  1213 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  1214        in unique_remainder)

  1215 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)

  1216 done

  1217

  1218 text{*Suggested by Matthias Daum*}

  1219 lemma int_power_div_base:

  1220      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  1221 apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")

  1222  apply (erule ssubst)

  1223  apply (simp only: power_add)

  1224  apply simp_all

  1225 done

  1226

  1227 ML

  1228 {*

  1229 val quorem_def = thm "quorem_def";

  1230

  1231 val unique_quotient = thm "unique_quotient";

  1232 val unique_remainder = thm "unique_remainder";

  1233 val adjust_eq = thm "adjust_eq";

  1234 val posDivAlg_eqn = thm "posDivAlg_eqn";

  1235 val posDivAlg_correct = thm "posDivAlg_correct";

  1236 val negDivAlg_eqn = thm "negDivAlg_eqn";

  1237 val negDivAlg_correct = thm "negDivAlg_correct";

  1238 val quorem_0 = thm "quorem_0";

  1239 val posDivAlg_0 = thm "posDivAlg_0";

  1240 val negDivAlg_minus1 = thm "negDivAlg_minus1";

  1241 val negateSnd_eq = thm "negateSnd_eq";

  1242 val quorem_neg = thm "quorem_neg";

  1243 val divAlg_correct = thm "divAlg_correct";

  1244 val DIVISION_BY_ZERO = thm "DIVISION_BY_ZERO";

  1245 val zmod_zdiv_equality = thm "zmod_zdiv_equality";

  1246 val pos_mod_conj = thm "pos_mod_conj";

  1247 val pos_mod_sign = thm "pos_mod_sign";

  1248 val neg_mod_conj = thm "neg_mod_conj";

  1249 val neg_mod_sign = thm "neg_mod_sign";

  1250 val quorem_div_mod = thm "quorem_div_mod";

  1251 val quorem_div = thm "quorem_div";

  1252 val quorem_mod = thm "quorem_mod";

  1253 val div_pos_pos_trivial = thm "div_pos_pos_trivial";

  1254 val div_neg_neg_trivial = thm "div_neg_neg_trivial";

  1255 val div_pos_neg_trivial = thm "div_pos_neg_trivial";

  1256 val mod_pos_pos_trivial = thm "mod_pos_pos_trivial";

  1257 val mod_neg_neg_trivial = thm "mod_neg_neg_trivial";

  1258 val mod_pos_neg_trivial = thm "mod_pos_neg_trivial";

  1259 val zdiv_zminus_zminus = thm "zdiv_zminus_zminus";

  1260 val zmod_zminus_zminus = thm "zmod_zminus_zminus";

  1261 val zdiv_zminus1_eq_if = thm "zdiv_zminus1_eq_if";

  1262 val zmod_zminus1_eq_if = thm "zmod_zminus1_eq_if";

  1263 val zdiv_zminus2 = thm "zdiv_zminus2";

  1264 val zmod_zminus2 = thm "zmod_zminus2";

  1265 val zdiv_zminus2_eq_if = thm "zdiv_zminus2_eq_if";

  1266 val zmod_zminus2_eq_if = thm "zmod_zminus2_eq_if";

  1267 val self_quotient = thm "self_quotient";

  1268 val self_remainder = thm "self_remainder";

  1269 val zdiv_self = thm "zdiv_self";

  1270 val zmod_self = thm "zmod_self";

  1271 val zdiv_zero = thm "zdiv_zero";

  1272 val div_eq_minus1 = thm "div_eq_minus1";

  1273 val zmod_zero = thm "zmod_zero";

  1274 val zdiv_minus1 = thm "zdiv_minus1";

  1275 val zmod_minus1 = thm "zmod_minus1";

  1276 val div_pos_pos = thm "div_pos_pos";

  1277 val mod_pos_pos = thm "mod_pos_pos";

  1278 val div_neg_pos = thm "div_neg_pos";

  1279 val mod_neg_pos = thm "mod_neg_pos";

  1280 val div_pos_neg = thm "div_pos_neg";

  1281 val mod_pos_neg = thm "mod_pos_neg";

  1282 val div_neg_neg = thm "div_neg_neg";

  1283 val mod_neg_neg = thm "mod_neg_neg";

  1284 val zmod_1 = thm "zmod_1";

  1285 val zdiv_1 = thm "zdiv_1";

  1286 val zmod_minus1_right = thm "zmod_minus1_right";

  1287 val zdiv_minus1_right = thm "zdiv_minus1_right";

  1288 val zdiv_mono1 = thm "zdiv_mono1";

  1289 val zdiv_mono1_neg = thm "zdiv_mono1_neg";

  1290 val zdiv_mono2 = thm "zdiv_mono2";

  1291 val zdiv_mono2_neg = thm "zdiv_mono2_neg";

  1292 val zdiv_zmult1_eq = thm "zdiv_zmult1_eq";

  1293 val zmod_zmult1_eq = thm "zmod_zmult1_eq";

  1294 val zmod_zmult1_eq' = thm "zmod_zmult1_eq'";

  1295 val zmod_zmult_distrib = thm "zmod_zmult_distrib";

  1296 val zdiv_zmult_self1 = thm "zdiv_zmult_self1";

  1297 val zdiv_zmult_self2 = thm "zdiv_zmult_self2";

  1298 val zmod_zmult_self1 = thm "zmod_zmult_self1";

  1299 val zmod_zmult_self2 = thm "zmod_zmult_self2";

  1300 val zmod_eq_0_iff = thm "zmod_eq_0_iff";

  1301 val zdiv_zadd1_eq = thm "zdiv_zadd1_eq";

  1302 val zmod_zadd1_eq = thm "zmod_zadd1_eq";

  1303 val mod_div_trivial = thm "mod_div_trivial";

  1304 val mod_mod_trivial = thm "mod_mod_trivial";

  1305 val zmod_zadd_left_eq = thm "zmod_zadd_left_eq";

  1306 val zmod_zadd_right_eq = thm "zmod_zadd_right_eq";

  1307 val zdiv_zadd_self1 = thm "zdiv_zadd_self1";

  1308 val zdiv_zadd_self2 = thm "zdiv_zadd_self2";

  1309 val zmod_zadd_self1 = thm "zmod_zadd_self1";

  1310 val zmod_zadd_self2 = thm "zmod_zadd_self2";

  1311 val zdiv_zmult2_eq = thm "zdiv_zmult2_eq";

  1312 val zmod_zmult2_eq = thm "zmod_zmult2_eq";

  1313 val zdiv_zmult_zmult1 = thm "zdiv_zmult_zmult1";

  1314 val zdiv_zmult_zmult2 = thm "zdiv_zmult_zmult2";

  1315 val zmod_zmult_zmult1 = thm "zmod_zmult_zmult1";

  1316 val zmod_zmult_zmult2 = thm "zmod_zmult_zmult2";

  1317 val pos_zdiv_mult_2 = thm "pos_zdiv_mult_2";

  1318 val neg_zdiv_mult_2 = thm "neg_zdiv_mult_2";

  1319 val zdiv_number_of_BIT = thm "zdiv_number_of_BIT";

  1320 val pos_zmod_mult_2 = thm "pos_zmod_mult_2";

  1321 val neg_zmod_mult_2 = thm "neg_zmod_mult_2";

  1322 val zmod_number_of_BIT = thm "zmod_number_of_BIT";

  1323 val div_neg_pos_less0 = thm "div_neg_pos_less0";

  1324 val div_nonneg_neg_le0 = thm "div_nonneg_neg_le0";

  1325 val pos_imp_zdiv_nonneg_iff = thm "pos_imp_zdiv_nonneg_iff";

  1326 val neg_imp_zdiv_nonneg_iff = thm "neg_imp_zdiv_nonneg_iff";

  1327 val pos_imp_zdiv_neg_iff = thm "pos_imp_zdiv_neg_iff";

  1328 val neg_imp_zdiv_neg_iff = thm "neg_imp_zdiv_neg_iff";

  1329

  1330 val zpower_zmod = thm "zpower_zmod";

  1331 val zpower_zadd_distrib = thm "zpower_zadd_distrib";

  1332 val zpower_zpower = thm "zpower_zpower";

  1333 val zero_less_zpower_abs_iff = thm "zero_less_zpower_abs_iff";

  1334 val zero_le_zpower_abs = thm "zero_le_zpower_abs";

  1335 val zpower_int = thm "zpower_int";

  1336 *}

  1337

  1338 end