src/HOL/IntDiv.thy
 author haftmann Sun Mar 22 20:46:10 2009 +0100 (2009-03-22) changeset 30652 752329615264 parent 30517 51a39ed24c0f child 30930 11010e5f18f0 permissions -rw-r--r--
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
     1 (*  Title:      HOL/IntDiv.thy

     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     3     Copyright   1999  University of Cambridge

     4

     5 *)

     6

     7 header{* The Division Operators div and mod *}

     8

     9 theory IntDiv

    10 imports Int Divides FunDef

    11 uses

    12   "~~/src/Provers/Arith/cancel_numeral_factor.ML"

    13   "~~/src/Provers/Arith/extract_common_term.ML"

    14   ("Tools/int_factor_simprocs.ML")

    15 begin

    16

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

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

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

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

    21

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

    23     --{*for the division algorithm*}

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

    25                          else (2 * q, r))"

    26

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

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

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

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

    31 by auto

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

    33

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

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

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

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

    38 by auto

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

    40

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

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

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

    44

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

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

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

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

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

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

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

    52                else

    53                   if 0 < b then negDivAlg a b

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

    55

    56 instantiation int :: Divides.div

    57 begin

    58

    59 definition

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

    61

    62 definition

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

    64

    65 instance ..

    66

    67 end

    68

    69 lemma divmod_mod_div:

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

    71   by (auto simp add: div_def mod_def)

    72

    73 text{*

    74 Here is the division algorithm in ML:

    75

    76 \begin{verbatim}

    77     fun posDivAlg (a,b) =

    78       if a<b then (0,a)

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

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

    81 	   end

    82

    83     fun negDivAlg (a,b) =

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

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

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

    87 	   end;

    88

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

    90

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

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

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

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

    95 		       else

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

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

    98 \end{verbatim}

    99 *}

   100

   101

   102

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

   104

   105 lemma unique_quotient_lemma:

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

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

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

   109  prefer 2 apply (simp add: right_diff_distrib)

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

   111 apply (erule_tac [2] order_le_less_trans)

   112  prefer 2 apply (simp add: right_diff_distrib right_distrib)

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

   114  prefer 2 apply (simp add: right_diff_distrib right_distrib)

   115 apply (simp add: mult_less_cancel_left)

   116 done

   117

   118 lemma unique_quotient_lemma_neg:

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

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

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

   122     auto)

   123

   124 lemma unique_quotient:

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

   126       ==> q = q'"

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

   128 apply (blast intro: order_antisym

   129              dest: order_eq_refl [THEN unique_quotient_lemma]

   130              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

   131 done

   132

   133

   134 lemma unique_remainder:

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

   136       ==> r = r'"

   137 apply (subgoal_tac "q = q'")

   138  apply (simp add: divmod_rel_def)

   139 apply (blast intro: unique_quotient)

   140 done

   141

   142

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

   144

   145 text{*And positive divisors*}

   146

   147 lemma adjust_eq [simp]:

   148      "adjust b (q,r) =

   149       (let diff = r-b in

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

   151                      else (2*q, r))"

   152 by (simp add: Let_def adjust_def)

   153

   154 declare posDivAlg.simps [simp del]

   155

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

   157 lemma posDivAlg_eqn:

   158      "0 < b ==>

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

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

   161

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

   163 theorem posDivAlg_correct:

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

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

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

   167 apply auto

   168 apply (simp add: divmod_rel_def)

   169 apply (subst posDivAlg_eqn, simp add: right_distrib)

   170 apply (case_tac "a < b")

   171 apply simp_all

   172 apply (erule splitE)

   173 apply (auto simp add: right_distrib Let_def)

   174 done

   175

   176

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

   178

   179 text{*And positive divisors*}

   180

   181 declare negDivAlg.simps [simp del]

   182

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

   184 lemma negDivAlg_eqn:

   185      "0 < b ==>

   186       negDivAlg a b =

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

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

   189

   190 (*Correctness of negDivAlg: it computes quotients correctly

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

   192 lemma negDivAlg_correct:

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

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

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

   196 apply (auto simp add: linorder_not_le)

   197 apply (simp add: divmod_rel_def)

   198 apply (subst negDivAlg_eqn, assumption)

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

   200 apply simp_all

   201 apply (erule splitE)

   202 apply (auto simp add: right_distrib Let_def)

   203 done

   204

   205

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

   207

   208 (*the case a=0*)

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

   210 by (auto simp add: divmod_rel_def linorder_neq_iff)

   211

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

   213 by (subst posDivAlg.simps, auto)

   214

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

   216 by (subst negDivAlg.simps, auto)

   217

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

   219 by (simp add: negateSnd_def)

   220

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

   222 by (auto simp add: split_ifs divmod_rel_def)

   223

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

   225 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg

   226                     posDivAlg_correct negDivAlg_correct)

   227

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

   229     certain equations.*}

   230

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

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

   233

   234

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

   236

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

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

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

   240 apply (auto simp add: divmod_rel_def div_def mod_def)

   241 done

   242

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

   244 by(simp add: zmod_zdiv_equality[symmetric])

   245

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

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

   248

   249 text {* Tool setup *}

   250

   251 ML {*

   252 local

   253

   254 structure CancelDivMod = CancelDivModFun(

   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 Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac simps) end;

   269 end)

   270

   271 in

   272

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

   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 divmod_correct)

   283 apply (auto simp add: divmod_rel_def mod_def)

   284 done

   285

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

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

   288

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

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

   291 apply (auto simp add: divmod_rel_def div_def mod_def)

   292 done

   293

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

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

   296

   297

   298

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

   300

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

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

   303 apply (force simp add: divmod_rel_def linorder_neq_iff)

   304 done

   305

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

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

   308

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

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

   311

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

   313 apply (rule divmod_rel_div)

   314 apply (auto simp add: divmod_rel_def)

   315 done

   316

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

   318 apply (rule divmod_rel_div)

   319 apply (auto simp add: divmod_rel_def)

   320 done

   321

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

   323 apply (rule divmod_rel_div)

   324 apply (auto simp add: divmod_rel_def)

   325 done

   326

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

   328

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

   330 apply (rule_tac q = 0 in divmod_rel_mod)

   331 apply (auto simp add: divmod_rel_def)

   332 done

   333

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

   335 apply (rule_tac q = 0 in divmod_rel_mod)

   336 apply (auto simp add: divmod_rel_def)

   337 done

   338

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

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

   341 apply (auto simp add: divmod_rel_def)

   342 done

   343

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

   345

   346

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

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

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

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

   351                                  THEN divmod_rel_div, THEN sym])

   352

   353 done

   354

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

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

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

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

   359        auto)

   360 done

   361

   362

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

   364

   365 lemma zminus1_lemma:

   366      "divmod_rel a b (q, r)

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

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

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

   370

   371

   372 lemma zdiv_zminus1_eq_if:

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

   374       ==> (-a) div b =

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

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

   377

   378 lemma zmod_zminus1_eq_if:

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

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

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

   382 done

   383

   384 lemma zmod_zminus1_not_zero:

   385   fixes k l :: int

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

   387   unfolding zmod_zminus1_eq_if by auto

   388

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

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

   391

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

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

   394

   395 lemma zdiv_zminus2_eq_if:

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

   397       ==> a div (-b) =

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

   399 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

   400

   401 lemma zmod_zminus2_eq_if:

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

   403 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

   404

   405 lemma zmod_zminus2_not_zero:

   406   fixes k l :: int

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

   408   unfolding zmod_zminus2_eq_if by auto

   409

   410

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

   412

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

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

   415  apply (simp add: zero_less_mult_iff, arith)

   416 done

   417

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

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

   420  apply (simp add: zero_le_mult_iff)

   421 apply (simp add: right_diff_distrib)

   422 done

   423

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

   425 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)

   426 apply (rule order_antisym, safe, simp_all)

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

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

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

   430 done

   431

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

   433 apply (frule self_quotient, assumption)

   434 apply (simp add: divmod_rel_def)

   435 done

   436

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

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

   439

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

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

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

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

   444 done

   445

   446

   447 subsection{*Computation of Division and Remainder*}

   448

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

   450 by (simp add: div_def divmod_def)

   451

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

   453 by (simp add: div_def divmod_def)

   454

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

   456 by (simp add: mod_def divmod_def)

   457

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

   459 by (simp add: mod_def divmod_def)

   460

   461 text{*a positive, b positive *}

   462

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

   464 by (simp add: div_def divmod_def)

   465

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

   467 by (simp add: mod_def divmod_def)

   468

   469 text{*a negative, b positive *}

   470

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

   472 by (simp add: div_def divmod_def)

   473

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

   475 by (simp add: mod_def divmod_def)

   476

   477 text{*a positive, b negative *}

   478

   479 lemma div_pos_neg:

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

   481 by (simp add: div_def divmod_def)

   482

   483 lemma mod_pos_neg:

   484      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"

   485 by (simp add: mod_def divmod_def)

   486

   487 text{*a negative, b negative *}

   488

   489 lemma div_neg_neg:

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

   491 by (simp add: div_def divmod_def)

   492

   493 lemma mod_neg_neg:

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

   495 by (simp add: mod_def divmod_def)

   496

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

   498

   499 lemma divmod_relI:

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

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

   502   unfolding divmod_rel_def by simp

   503

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

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

   506 lemmas arithmetic_simps =

   507   arith_simps

   508   add_special

   509   OrderedGroup.add_0_left

   510   OrderedGroup.add_0_right

   511   mult_zero_left

   512   mult_zero_right

   513   mult_1_left

   514   mult_1_right

   515

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

   517 ML {*

   518 local

   519   val mk_number = HOLogic.mk_number HOLogic.intT;

   520   fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $  521 (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"}$ u $mk_number k)$

   522       mk_number l;

   523   fun prove ctxt prop = Goal.prove ctxt [] [] prop

   524     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));

   525   fun binary_proc proc ss ct =

   526     (case Thm.term_of ct of

   527       _ $t$ u =>

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

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

   530       | NONE => NONE)

   531     | _ => NONE);

   532 in

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

   534     if n = 0 then NONE

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

   536     in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);

   537 end

   538 *}

   539

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

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

   542

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

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

   545

   546 lemmas posDivAlg_eqn_number_of [simp] =

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

   548

   549 lemmas negDivAlg_eqn_number_of [simp] =

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

   551

   552

   553 text{*Special-case simplification *}

   554

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

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

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

   558 apply (auto simp del: neg_mod_sign neg_mod_bound)

   559 done

   560

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

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

   563

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

   565     1 div z and 1 mod z **)

   566

   567 lemmas div_pos_pos_1_number_of [simp] =

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

   569

   570 lemmas div_pos_neg_1_number_of [simp] =

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

   572

   573 lemmas mod_pos_pos_1_number_of [simp] =

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

   575

   576 lemmas mod_pos_neg_1_number_of [simp] =

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

   578

   579

   580 lemmas posDivAlg_eqn_1_number_of [simp] =

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

   582

   583 lemmas negDivAlg_eqn_1_number_of [simp] =

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

   585

   586

   587

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

   589

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

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

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

   593 apply (rule unique_quotient_lemma)

   594 apply (erule subst)

   595 apply (erule subst, simp_all)

   596 done

   597

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

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

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

   601 apply (rule unique_quotient_lemma_neg)

   602 apply (erule subst)

   603 apply (erule subst, simp_all)

   604 done

   605

   606

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

   608

   609 lemma q_pos_lemma:

   610      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

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

   612  apply (simp add: zero_less_mult_iff)

   613 apply (simp add: right_distrib)

   614 done

   615

   616 lemma zdiv_mono2_lemma:

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

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

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

   620 apply (frule q_pos_lemma, assumption+)

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

   622  apply (simp add: mult_less_cancel_left)

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

   624  prefer 2 apply simp

   625 apply (simp (no_asm_simp) add: right_distrib)

   626 apply (subst add_commute, rule zadd_zless_mono, arith)

   627 apply (rule mult_right_mono, auto)

   628 done

   629

   630 lemma zdiv_mono2:

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

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

   633  prefer 2 apply arith

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

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

   636 apply (rule zdiv_mono2_lemma)

   637 apply (erule subst)

   638 apply (erule subst, simp_all)

   639 done

   640

   641 lemma q_neg_lemma:

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

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

   644  apply (simp add: mult_less_0_iff, arith)

   645 done

   646

   647 lemma zdiv_mono2_neg_lemma:

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

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

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

   651 apply (frule q_neg_lemma, assumption+)

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

   653  apply (simp add: mult_less_cancel_left)

   654 apply (simp add: right_distrib)

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

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

   657 done

   658

   659 lemma zdiv_mono2_neg:

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

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

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

   663 apply (rule zdiv_mono2_neg_lemma)

   664 apply (erule subst)

   665 apply (erule subst, simp_all)

   666 done

   667

   668

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

   670

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

   672

   673 lemma zmult1_lemma:

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

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

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

   677

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

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

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

   681 done

   682

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

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

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

   686 done

   687

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

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

   690 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

   691 done

   692

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

   694

   695 lemma zadd1_lemma:

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

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

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

   699

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

   701 lemma zdiv_zadd1_eq:

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

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

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

   705 done

   706

   707 instance int :: ring_div

   708 proof

   709   fix a b c :: int

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

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

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

   713       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)

   714 qed auto

   715

   716 lemma posDivAlg_div_mod:

   717   assumes "k \<ge> 0"

   718   and "l \<ge> 0"

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

   720 proof (cases "l = 0")

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

   722 next

   723   case False with assms posDivAlg_correct

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

   725     by simp

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

   727   show ?thesis by simp

   728 qed

   729

   730 lemma negDivAlg_div_mod:

   731   assumes "k < 0"

   732   and "l > 0"

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

   734 proof -

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

   736   from assms negDivAlg_correct

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

   738     by simp

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

   740   show ?thesis by simp

   741 qed

   742

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

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

   745

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

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

   748

   749

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

   751

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

   753   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

   754   to cause particular problems.*)

   755

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

   757

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

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

   760  apply (simp add: algebra_simps)

   761 apply (rule order_le_less_trans)

   762  apply (erule_tac [2] mult_strict_right_mono)

   763  apply (rule mult_left_mono_neg)

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

   765  apply (simp)

   766 apply (simp)

   767 done

   768

   769 lemma zmult2_lemma_aux2:

   770      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

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

   772  apply arith

   773 apply (simp add: mult_le_0_iff)

   774 done

   775

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

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

   778 apply arith

   779 apply (simp add: zero_le_mult_iff)

   780 done

   781

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

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

   784  apply (simp add: right_diff_distrib)

   785 apply (rule order_less_le_trans)

   786  apply (erule mult_strict_right_mono)

   787  apply (rule_tac [2] mult_left_mono)

   788   apply simp

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

   790 apply simp

   791 done

   792

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

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

   795 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff

   796                    zero_less_mult_iff right_distrib [symmetric]

   797                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

   798

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

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

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

   802 done

   803

   804 lemma zmod_zmult2_eq:

   805      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

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

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

   808 done

   809

   810

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

   812

   813 lemma zdiv_zmult_zmult1_aux1:

   814      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"

   815 by (subst zdiv_zmult2_eq, auto)

   816

   817 lemma zdiv_zmult_zmult1_aux2:

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

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

   820 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)

   821 done

   822

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

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

   825 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)

   826 done

   827

   828 lemma zdiv_zmult_zmult1_if[simp]:

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

   830 by (simp add:zdiv_zmult_zmult1)

   831

   832

   833 subsection{*Distribution of Factors over mod*}

   834

   835 lemma zmod_zmult_zmult1_aux1:

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

   837 by (subst zmod_zmult2_eq, auto)

   838

   839 lemma zmod_zmult_zmult1_aux2:

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

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

   842 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)

   843 done

   844

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

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

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

   848 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)

   849 done

   850

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

   852 apply (cut_tac c = c in zmod_zmult_zmult1)

   853 apply (auto simp add: mult_commute)

   854 done

   855

   856

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

   858

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

   860

   861 lemma split_pos_lemma:

   862  "0<k ==>

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

   864 apply (rule iffI, clarify)

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

   866  apply (subst mod_add_eq)

   867  apply (subst zdiv_zadd1_eq)

   868  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

   869 txt{*converse direction*}

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

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

   872 done

   873

   874 lemma split_neg_lemma:

   875  "k<0 ==>

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

   877 apply (rule iffI, clarify)

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

   879  apply (subst mod_add_eq)

   880  apply (subst zdiv_zadd1_eq)

   881  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

   882 txt{*converse direction*}

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

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

   885 done

   886

   887 lemma split_zdiv:

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

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

   890    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

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

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

   893 apply (simp only: linorder_neq_iff)

   894 apply (erule disjE)

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

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

   897 done

   898

   899 lemma split_zmod:

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

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

   902    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

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

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

   905 apply (simp only: linorder_neq_iff)

   906 apply (erule disjE)

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

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

   909 done

   910

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

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

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

   914

   915

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

   917

   918 text{*computing div by shifting *}

   919

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

   921 proof cases

   922   assume "a=0"

   923     thus ?thesis by simp

   924 next

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

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

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

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

   929     unfolding mult_le_cancel_left

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

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

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

   933     by (simp add: mod_pos_pos_trivial one_less_a2)

   934   with  le_2a

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

   936     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

   937                   right_distrib)

   938   thus ?thesis

   939     by (subst zdiv_zadd1_eq,

   940         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2

   941                   div_pos_pos_trivial)

   942 qed

   943

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

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

   946 apply (rule_tac [2] pos_zdiv_mult_2)

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

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

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

   950        simp)

   951 done

   952

   953 lemma zdiv_number_of_Bit0 [simp]:

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

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

   956 by (simp only: number_of_eq numeral_simps) simp

   957

   958 lemma zdiv_number_of_Bit1 [simp]:

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

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

   961            then number_of v div (number_of w)

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

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

   964 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)

   965 done

   966

   967

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

   969

   970 lemma pos_zmod_mult_2:

   971      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"

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

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

   974  prefer 2 apply arith

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

   976  apply (rule_tac [2] mult_left_mono)

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

   978                       pos_mod_bound)

   979 apply (subst mod_add_eq)

   980 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)

   981 apply (rule mod_pos_pos_trivial)

   982 apply (auto simp add: mod_pos_pos_trivial ring_distribs)

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

   984 done

   985

   986 lemma neg_zmod_mult_2:

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

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

   989                     1 + 2* ((-b - 1) mod (-a))")

   990 apply (rule_tac [2] pos_zmod_mult_2)

   991 apply (auto simp add: right_diff_distrib)

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

   993  prefer 2 apply simp

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

   995 done

   996

   997 lemma zmod_number_of_Bit0 [simp]:

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

   999       (2::int) * (number_of v mod number_of w)"

  1000 apply (simp only: number_of_eq numeral_simps)

  1001 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1002                  neg_zmod_mult_2 add_ac)

  1003 done

  1004

  1005 lemma zmod_number_of_Bit1 [simp]:

  1006      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =

  1007       (if (0::int) \<le> number_of w

  1008                 then 2 * (number_of v mod number_of w) + 1

  1009                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  1010 apply (simp only: number_of_eq numeral_simps)

  1011 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2

  1012                  neg_zmod_mult_2 add_ac)

  1013 done

  1014

  1015

  1016 subsection{*Quotients of Signs*}

  1017

  1018 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  1019 apply (subgoal_tac "a div b \<le> -1", force)

  1020 apply (rule order_trans)

  1021 apply (rule_tac a' = "-1" in zdiv_mono1)

  1022 apply (auto simp add: div_eq_minus1)

  1023 done

  1024

  1025 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"

  1026 by (drule zdiv_mono1_neg, auto)

  1027

  1028 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"

  1029 by (drule zdiv_mono1, auto)

  1030

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

  1032 apply auto

  1033 apply (drule_tac [2] zdiv_mono1)

  1034 apply (auto simp add: linorder_neq_iff)

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

  1036 apply (blast intro: div_neg_pos_less0)

  1037 done

  1038

  1039 lemma neg_imp_zdiv_nonneg_iff:

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

  1041 apply (subst zdiv_zminus_zminus [symmetric])

  1042 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  1043 done

  1044

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

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

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

  1048

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

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

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

  1052

  1053

  1054 subsection {* The Divides Relation *}

  1055

  1056 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

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

  1058

  1059 lemma zdvd_anti_sym:

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

  1061   apply (simp add: dvd_def, auto)

  1062   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)

  1063   done

  1064

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

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

  1067 proof-

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

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

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

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

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

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

  1074   thus ?thesis using k k' by auto

  1075 qed

  1076

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

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

  1079    apply (erule ssubst)

  1080    apply (blast intro: dvd_add, simp)

  1081   done

  1082

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

  1084 apply (rule iffI)

  1085  apply (erule_tac [2] dvd_add)

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

  1087   apply (erule ssubst)

  1088   apply (erule dvd_diff)

  1089   apply(simp_all)

  1090 done

  1091

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

  1093   apply (simp add: dvd_def)

  1094   apply (auto simp add: zmod_zmult_zmult1)

  1095   done

  1096

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

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

  1099    apply (simp add: zmod_zdiv_equality [symmetric])

  1100   apply (simp only: dvd_add dvd_mult2)

  1101   done

  1102

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

  1104   apply (auto elim!: dvdE)

  1105   apply (subgoal_tac "0 < n")

  1106    prefer 2

  1107    apply (blast intro: order_less_trans)

  1108   apply (simp add: zero_less_mult_iff)

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

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

  1111   done

  1112

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

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

  1115   by (simp add: algebra_simps)

  1116

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

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

  1119  apply (simp add: zmult_div_cancel)

  1120 apply (simp only: dvd_eq_mod_eq_0)

  1121 done

  1122

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

  1124   shows "m dvd n"

  1125 proof-

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

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

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

  1129   hence "n = m * h" by blast

  1130   thus ?thesis by simp

  1131 qed

  1132

  1133

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

  1135 apply (simp split add: split_nat)

  1136 apply (rule iffI)

  1137 apply (erule exE)

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

  1139 apply simp

  1140 apply (erule exE)

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

  1142 apply (erule conjE)

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

  1144 apply simp

  1145 done

  1146

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

  1148 proof -

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

  1150   proof -

  1151     fix k

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

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

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

  1155       then show ?thesis ..

  1156     next

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

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

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

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

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

  1162     qed

  1163   qed

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

  1165 qed

  1166

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

  1168 proof

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

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

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

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

  1173 next

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

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

  1176 qed

  1177 lemma zdvd_mult_cancel1:

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

  1179 proof

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

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

  1182 next

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

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

  1185 qed

  1186

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

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

  1189

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

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

  1192

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

  1194   by (auto simp add: dvd_int_iff)

  1195

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

  1197   apply (rule_tac z=n in int_cases)

  1198   apply (auto simp add: dvd_int_iff)

  1199   apply (rule_tac z=z in int_cases)

  1200   apply (auto simp add: dvd_imp_le)

  1201   done

  1202

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

  1204 apply (induct "y", auto)

  1205 apply (rule zmod_zmult1_eq [THEN trans])

  1206 apply (simp (no_asm_simp))

  1207 apply (rule mod_mult_eq [symmetric])

  1208 done

  1209

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

  1211 apply (subst split_div, auto)

  1212 apply (subst split_zdiv, auto)

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

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

  1215 done

  1216

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

  1218 apply (subst split_mod, auto)

  1219 apply (subst split_zmod, auto)

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

  1221        in unique_remainder)

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

  1223 done

  1224

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

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

  1227

  1228 text{*Suggested by Matthias Daum*}

  1229 lemma int_power_div_base:

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

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

  1232  apply (erule ssubst)

  1233  apply (simp only: power_add)

  1234  apply simp_all

  1235 done

  1236

  1237 text {* by Brian Huffman *}

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

  1239 by (rule mod_minus_eq [symmetric])

  1240

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

  1242 by (rule mod_diff_left_eq [symmetric])

  1243

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

  1245 by (rule mod_diff_right_eq [symmetric])

  1246

  1247 lemmas zmod_simps =

  1248   mod_add_left_eq  [symmetric]

  1249   mod_add_right_eq [symmetric]

  1250   IntDiv.zmod_zmult1_eq     [symmetric]

  1251   mod_mult_left_eq          [symmetric]

  1252   IntDiv.zpower_zmod

  1253   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  1254

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

  1256

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

  1258 apply (rule linorder_cases [of y 0])

  1259 apply (simp add: div_nonneg_neg_le0)

  1260 apply simp

  1261 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  1262 done

  1263

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

  1265 lemma nat_mod_distrib:

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

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

  1268 apply (simp add: nat_eq_iff zmod_int)

  1269 done

  1270

  1271 text{*Suggested by Matthias Daum*}

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

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

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

  1275 apply (rule Divides.div_less_dividend, simp_all)

  1276 done

  1277

  1278 text {* code generator setup *}

  1279

  1280 context ring_1

  1281 begin

  1282

  1283 lemma of_int_num [code]:

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

  1285      - of_int (- k) else let

  1286        (l, m) = divmod k 2;

  1287        l' = of_int l

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

  1289 proof -

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

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

  1292   proof -

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

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

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

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

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

  1298     ultimately show ?thesis by auto

  1299   qed

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

  1301   proof -

  1302     fix x

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

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

  1305     unfolding int2 of_int_add left_distrib by simp

  1306   qed

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

  1308   proof -

  1309     fix x

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

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

  1312     unfolding int2 of_int_add right_distrib by simp

  1313   qed

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

  1315 qed

  1316

  1317 end

  1318

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

  1320 proof

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

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

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

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

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

  1326 next

  1327   assume H: "n dvd x - y"

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

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

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

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

  1332 qed

  1333

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

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

  1336 proof-

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

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

  1339     by (simp add: zmod_int[symmetric])

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

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

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

  1343 qed

  1344

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

  1346   (is "?lhs = ?rhs")

  1347 proof

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

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

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

  1351     from nat_mod_eq_lemma[OF th xy] have ?rhs

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

  1353   moreover

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

  1355     from nat_mod_eq_lemma[OF H xy] have ?rhs

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

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

  1358 next

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

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

  1361   thus  ?lhs by simp

  1362 qed

  1363

  1364

  1365 subsection {* Simproc setup *}

  1366

  1367 use "Tools/int_factor_simprocs.ML"

  1368

  1369

  1370 subsection {* Code generation *}

  1371

  1372 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1373   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  1374

  1375 lemma pdivmod_posDivAlg [code]:

  1376   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  1377 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  1378

  1379 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1380   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  1381     then pdivmod k l

  1382     else (let (r, s) = pdivmod k l in

  1383       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1384 proof -

  1385   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  1386   show ?thesis

  1387     by (simp add: divmod_mod_div pdivmod_def)

  1388       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  1389       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  1390 qed

  1391

  1392 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  1393   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  1394     then pdivmod k l

  1395     else (let (r, s) = pdivmod k l in

  1396       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  1397 proof -

  1398   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"

  1399     by (auto simp add: not_less sgn_if)

  1400   then show ?thesis by (simp add: divmod_pdivmod)

  1401 qed

  1402

  1403 code_modulename SML

  1404   IntDiv Integer

  1405

  1406 code_modulename OCaml

  1407   IntDiv Integer

  1408

  1409 code_modulename Haskell

  1410   IntDiv Integer

  1411

  1412 end
`