src/HOL/Rings.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60516 0826b7025d07 child 60529 24c2ef12318b permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 (*  Title:      HOL/Rings.thy

     2     Author:     Gertrud Bauer

     3     Author:     Steven Obua

     4     Author:     Tobias Nipkow

     5     Author:     Lawrence C Paulson

     6     Author:     Markus Wenzel

     7     Author:     Jeremy Avigad

     8 *)

     9

    10 section {* Rings *}

    11

    12 theory Rings

    13 imports Groups

    14 begin

    15

    16 class semiring = ab_semigroup_add + semigroup_mult +

    17   assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"

    18   assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"

    19 begin

    20

    21 text{*For the @{text combine_numerals} simproc*}

    22 lemma combine_common_factor:

    23   "a * e + (b * e + c) = (a + b) * e + c"

    24 by (simp add: distrib_right ac_simps)

    25

    26 end

    27

    28 class mult_zero = times + zero +

    29   assumes mult_zero_left [simp]: "0 * a = 0"

    30   assumes mult_zero_right [simp]: "a * 0 = 0"

    31 begin

    32

    33 lemma mult_not_zero:

    34   "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"

    35   by auto

    36

    37 end

    38

    39 class semiring_0 = semiring + comm_monoid_add + mult_zero

    40

    41 class semiring_0_cancel = semiring + cancel_comm_monoid_add

    42 begin

    43

    44 subclass semiring_0

    45 proof

    46   fix a :: 'a

    47   have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])

    48   thus "0 * a = 0" by (simp only: add_left_cancel)

    49 next

    50   fix a :: 'a

    51   have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])

    52   thus "a * 0 = 0" by (simp only: add_left_cancel)

    53 qed

    54

    55 end

    56

    57 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +

    58   assumes distrib: "(a + b) * c = a * c + b * c"

    59 begin

    60

    61 subclass semiring

    62 proof

    63   fix a b c :: 'a

    64   show "(a + b) * c = a * c + b * c" by (simp add: distrib)

    65   have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)

    66   also have "... = b * a + c * a" by (simp only: distrib)

    67   also have "... = a * b + a * c" by (simp add: ac_simps)

    68   finally show "a * (b + c) = a * b + a * c" by blast

    69 qed

    70

    71 end

    72

    73 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero

    74 begin

    75

    76 subclass semiring_0 ..

    77

    78 end

    79

    80 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add

    81 begin

    82

    83 subclass semiring_0_cancel ..

    84

    85 subclass comm_semiring_0 ..

    86

    87 end

    88

    89 class zero_neq_one = zero + one +

    90   assumes zero_neq_one [simp]: "0 \<noteq> 1"

    91 begin

    92

    93 lemma one_neq_zero [simp]: "1 \<noteq> 0"

    94 by (rule not_sym) (rule zero_neq_one)

    95

    96 definition of_bool :: "bool \<Rightarrow> 'a"

    97 where

    98   "of_bool p = (if p then 1 else 0)"

    99

   100 lemma of_bool_eq [simp, code]:

   101   "of_bool False = 0"

   102   "of_bool True = 1"

   103   by (simp_all add: of_bool_def)

   104

   105 lemma of_bool_eq_iff:

   106   "of_bool p = of_bool q \<longleftrightarrow> p = q"

   107   by (simp add: of_bool_def)

   108

   109 lemma split_of_bool [split]:

   110   "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"

   111   by (cases p) simp_all

   112

   113 lemma split_of_bool_asm:

   114   "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"

   115   by (cases p) simp_all

   116

   117 end

   118

   119 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

   120

   121 text {* Abstract divisibility *}

   122

   123 class dvd = times

   124 begin

   125

   126 definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where

   127   "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"

   128

   129 lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"

   130   unfolding dvd_def ..

   131

   132 lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"

   133   unfolding dvd_def by blast

   134

   135 end

   136

   137 context comm_monoid_mult

   138 begin

   139

   140 subclass dvd .

   141

   142 lemma dvd_refl [simp]:

   143   "a dvd a"

   144 proof

   145   show "a = a * 1" by simp

   146 qed

   147

   148 lemma dvd_trans:

   149   assumes "a dvd b" and "b dvd c"

   150   shows "a dvd c"

   151 proof -

   152   from assms obtain v where "b = a * v" by (auto elim!: dvdE)

   153   moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)

   154   ultimately have "c = a * (v * w)" by (simp add: mult.assoc)

   155   then show ?thesis ..

   156 qed

   157

   158 lemma one_dvd [simp]:

   159   "1 dvd a"

   160   by (auto intro!: dvdI)

   161

   162 lemma dvd_mult [simp]:

   163   "a dvd c \<Longrightarrow> a dvd (b * c)"

   164   by (auto intro!: mult.left_commute dvdI elim!: dvdE)

   165

   166 lemma dvd_mult2 [simp]:

   167   "a dvd b \<Longrightarrow> a dvd (b * c)"

   168   using dvd_mult [of a b c] by (simp add: ac_simps)

   169

   170 lemma dvd_triv_right [simp]:

   171   "a dvd b * a"

   172   by (rule dvd_mult) (rule dvd_refl)

   173

   174 lemma dvd_triv_left [simp]:

   175   "a dvd a * b"

   176   by (rule dvd_mult2) (rule dvd_refl)

   177

   178 lemma mult_dvd_mono:

   179   assumes "a dvd b"

   180     and "c dvd d"

   181   shows "a * c dvd b * d"

   182 proof -

   183   from a dvd b obtain b' where "b = a * b'" ..

   184   moreover from c dvd d obtain d' where "d = c * d'" ..

   185   ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)

   186   then show ?thesis ..

   187 qed

   188

   189 lemma dvd_mult_left:

   190   "a * b dvd c \<Longrightarrow> a dvd c"

   191   by (simp add: dvd_def mult.assoc) blast

   192

   193 lemma dvd_mult_right:

   194   "a * b dvd c \<Longrightarrow> b dvd c"

   195   using dvd_mult_left [of b a c] by (simp add: ac_simps)

   196

   197 end

   198

   199 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult

   200 begin

   201

   202 subclass semiring_1 ..

   203

   204 lemma dvd_0_left_iff [simp]:

   205   "0 dvd a \<longleftrightarrow> a = 0"

   206   by (auto intro: dvd_refl elim!: dvdE)

   207

   208 lemma dvd_0_right [iff]:

   209   "a dvd 0"

   210 proof

   211   show "0 = a * 0" by simp

   212 qed

   213

   214 lemma dvd_0_left:

   215   "0 dvd a \<Longrightarrow> a = 0"

   216   by simp

   217

   218 lemma dvd_add [simp]:

   219   assumes "a dvd b" and "a dvd c"

   220   shows "a dvd (b + c)"

   221 proof -

   222   from a dvd b obtain b' where "b = a * b'" ..

   223   moreover from a dvd c obtain c' where "c = a * c'" ..

   224   ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)

   225   then show ?thesis ..

   226 qed

   227

   228 end

   229

   230 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   231   + zero_neq_one + monoid_mult

   232 begin

   233

   234 subclass semiring_0_cancel ..

   235

   236 subclass semiring_1 ..

   237

   238 end

   239

   240 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   241   + zero_neq_one + comm_monoid_mult

   242 begin

   243

   244 subclass semiring_1_cancel ..

   245 subclass comm_semiring_0_cancel ..

   246 subclass comm_semiring_1 ..

   247

   248 end

   249

   250 class comm_semiring_1_diff_distrib = comm_semiring_1_cancel +

   251   assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"

   252 begin

   253

   254 lemma left_diff_distrib' [algebra_simps]:

   255   "(b - c) * a = b * a - c * a"

   256   by (simp add: algebra_simps)

   257

   258 lemma dvd_add_times_triv_left_iff [simp]:

   259   "a dvd c * a + b \<longleftrightarrow> a dvd b"

   260 proof -

   261   have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")

   262   proof

   263     assume ?Q then show ?P by simp

   264   next

   265     assume ?P

   266     then obtain d where "a * c + b = a * d" ..

   267     then have "a * c + b - a * c = a * d - a * c" by simp

   268     then have "b = a * d - a * c" by simp

   269     then have "b = a * (d - c)" by (simp add: algebra_simps)

   270     then show ?Q ..

   271   qed

   272   then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)

   273 qed

   274

   275 lemma dvd_add_times_triv_right_iff [simp]:

   276   "a dvd b + c * a \<longleftrightarrow> a dvd b"

   277   using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)

   278

   279 lemma dvd_add_triv_left_iff [simp]:

   280   "a dvd a + b \<longleftrightarrow> a dvd b"

   281   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   282

   283 lemma dvd_add_triv_right_iff [simp]:

   284   "a dvd b + a \<longleftrightarrow> a dvd b"

   285   using dvd_add_times_triv_right_iff [of a b 1] by simp

   286

   287 lemma dvd_add_right_iff:

   288   assumes "a dvd b"

   289   shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")

   290 proof

   291   assume ?P then obtain d where "b + c = a * d" ..

   292   moreover from a dvd b obtain e where "b = a * e" ..

   293   ultimately have "a * e + c = a * d" by simp

   294   then have "a * e + c - a * e = a * d - a * e" by simp

   295   then have "c = a * d - a * e" by simp

   296   then have "c = a * (d - e)" by (simp add: algebra_simps)

   297   then show ?Q ..

   298 next

   299   assume ?Q with assms show ?P by simp

   300 qed

   301

   302 lemma dvd_add_left_iff:

   303   assumes "a dvd c"

   304   shows "a dvd b + c \<longleftrightarrow> a dvd b"

   305   using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)

   306

   307 end

   308

   309 class ring = semiring + ab_group_add

   310 begin

   311

   312 subclass semiring_0_cancel ..

   313

   314 text {* Distribution rules *}

   315

   316 lemma minus_mult_left: "- (a * b) = - a * b"

   317 by (rule minus_unique) (simp add: distrib_right [symmetric])

   318

   319 lemma minus_mult_right: "- (a * b) = a * - b"

   320 by (rule minus_unique) (simp add: distrib_left [symmetric])

   321

   322 text{*Extract signs from products*}

   323 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   324 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   325

   326 lemma minus_mult_minus [simp]: "- a * - b = a * b"

   327 by simp

   328

   329 lemma minus_mult_commute: "- a * b = a * - b"

   330 by simp

   331

   332 lemma right_diff_distrib [algebra_simps]:

   333   "a * (b - c) = a * b - a * c"

   334   using distrib_left [of a b "-c "] by simp

   335

   336 lemma left_diff_distrib [algebra_simps]:

   337   "(a - b) * c = a * c - b * c"

   338   using distrib_right [of a "- b" c] by simp

   339

   340 lemmas ring_distribs =

   341   distrib_left distrib_right left_diff_distrib right_diff_distrib

   342

   343 lemma eq_add_iff1:

   344   "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"

   345 by (simp add: algebra_simps)

   346

   347 lemma eq_add_iff2:

   348   "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"

   349 by (simp add: algebra_simps)

   350

   351 end

   352

   353 lemmas ring_distribs =

   354   distrib_left distrib_right left_diff_distrib right_diff_distrib

   355

   356 class comm_ring = comm_semiring + ab_group_add

   357 begin

   358

   359 subclass ring ..

   360 subclass comm_semiring_0_cancel ..

   361

   362 lemma square_diff_square_factored:

   363   "x * x - y * y = (x + y) * (x - y)"

   364   by (simp add: algebra_simps)

   365

   366 end

   367

   368 class ring_1 = ring + zero_neq_one + monoid_mult

   369 begin

   370

   371 subclass semiring_1_cancel ..

   372

   373 lemma square_diff_one_factored:

   374   "x * x - 1 = (x + 1) * (x - 1)"

   375   by (simp add: algebra_simps)

   376

   377 end

   378

   379 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   380 begin

   381

   382 subclass ring_1 ..

   383 subclass comm_semiring_1_cancel ..

   384

   385 subclass comm_semiring_1_diff_distrib

   386   by unfold_locales (simp add: algebra_simps)

   387

   388 lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"

   389 proof

   390   assume "x dvd - y"

   391   then have "x dvd - 1 * - y" by (rule dvd_mult)

   392   then show "x dvd y" by simp

   393 next

   394   assume "x dvd y"

   395   then have "x dvd - 1 * y" by (rule dvd_mult)

   396   then show "x dvd - y" by simp

   397 qed

   398

   399 lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"

   400 proof

   401   assume "- x dvd y"

   402   then obtain k where "y = - x * k" ..

   403   then have "y = x * - k" by simp

   404   then show "x dvd y" ..

   405 next

   406   assume "x dvd y"

   407   then obtain k where "y = x * k" ..

   408   then have "y = - x * - k" by simp

   409   then show "- x dvd y" ..

   410 qed

   411

   412 lemma dvd_diff [simp]:

   413   "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"

   414   using dvd_add [of x y "- z"] by simp

   415

   416 end

   417

   418 class semiring_no_zero_divisors = semiring_0 +

   419   assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"

   420 begin

   421

   422 lemma divisors_zero:

   423   assumes "a * b = 0"

   424   shows "a = 0 \<or> b = 0"

   425 proof (rule classical)

   426   assume "\<not> (a = 0 \<or> b = 0)"

   427   then have "a \<noteq> 0" and "b \<noteq> 0" by auto

   428   with no_zero_divisors have "a * b \<noteq> 0" by blast

   429   with assms show ?thesis by simp

   430 qed

   431

   432 lemma mult_eq_0_iff [simp]:

   433   shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"

   434 proof (cases "a = 0 \<or> b = 0")

   435   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto

   436     then show ?thesis using no_zero_divisors by simp

   437 next

   438   case True then show ?thesis by auto

   439 qed

   440

   441 end

   442

   443 class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +

   444   assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"

   445     and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"

   446 begin

   447

   448 lemma mult_left_cancel:

   449   "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"

   450   by simp

   451

   452 lemma mult_right_cancel:

   453   "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"

   454   by simp

   455

   456 end

   457

   458 class ring_no_zero_divisors = ring + semiring_no_zero_divisors

   459 begin

   460

   461 subclass semiring_no_zero_divisors_cancel

   462 proof

   463   fix a b c

   464   have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"

   465     by (simp add: algebra_simps)

   466   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"

   467     by auto

   468   finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .

   469   have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"

   470     by (simp add: algebra_simps)

   471   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"

   472     by auto

   473   finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .

   474 qed

   475

   476 end

   477

   478 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   479 begin

   480

   481 lemma square_eq_1_iff:

   482   "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"

   483 proof -

   484   have "(x - 1) * (x + 1) = x * x - 1"

   485     by (simp add: algebra_simps)

   486   hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"

   487     by simp

   488   thus ?thesis

   489     by (simp add: eq_neg_iff_add_eq_0)

   490 qed

   491

   492 lemma mult_cancel_right1 [simp]:

   493   "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"

   494 by (insert mult_cancel_right [of 1 c b], force)

   495

   496 lemma mult_cancel_right2 [simp]:

   497   "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"

   498 by (insert mult_cancel_right [of a c 1], simp)

   499

   500 lemma mult_cancel_left1 [simp]:

   501   "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"

   502 by (insert mult_cancel_left [of c 1 b], force)

   503

   504 lemma mult_cancel_left2 [simp]:

   505   "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"

   506 by (insert mult_cancel_left [of c a 1], simp)

   507

   508 end

   509

   510 class semidom = comm_semiring_1_diff_distrib + semiring_no_zero_divisors

   511

   512 class idom = comm_ring_1 + semiring_no_zero_divisors

   513 begin

   514

   515 subclass semidom ..

   516

   517 subclass ring_1_no_zero_divisors ..

   518

   519 lemma dvd_mult_cancel_right [simp]:

   520   "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"

   521 proof -

   522   have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"

   523     unfolding dvd_def by (simp add: ac_simps)

   524   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"

   525     unfolding dvd_def by simp

   526   finally show ?thesis .

   527 qed

   528

   529 lemma dvd_mult_cancel_left [simp]:

   530   "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"

   531 proof -

   532   have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"

   533     unfolding dvd_def by (simp add: ac_simps)

   534   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"

   535     unfolding dvd_def by simp

   536   finally show ?thesis .

   537 qed

   538

   539 lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"

   540 proof

   541   assume "a * a = b * b"

   542   then have "(a - b) * (a + b) = 0"

   543     by (simp add: algebra_simps)

   544   then show "a = b \<or> a = - b"

   545     by (simp add: eq_neg_iff_add_eq_0)

   546 next

   547   assume "a = b \<or> a = - b"

   548   then show "a * a = b * b" by auto

   549 qed

   550

   551 end

   552

   553 text {*

   554   The theory of partially ordered rings is taken from the books:

   555   \begin{itemize}

   556   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979

   557   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963

   558   \end{itemize}

   559   Most of the used notions can also be looked up in

   560   \begin{itemize}

   561   \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.

   562   \item \emph{Algebra I} by van der Waerden, Springer.

   563   \end{itemize}

   564 *}

   565

   566 class divide =

   567   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)

   568

   569 setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}

   570

   571 context semiring

   572 begin

   573

   574 lemma [field_simps]:

   575   shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"

   576     and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"

   577   by (rule distrib_left distrib_right)+

   578

   579 end

   580

   581 context ring

   582 begin

   583

   584 lemma [field_simps]:

   585   shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"

   586     and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"

   587   by (rule left_diff_distrib right_diff_distrib)+

   588

   589 end

   590

   591 setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}

   592

   593 class semidom_divide = semidom + divide +

   594   assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"

   595   assumes divide_zero [simp]: "a div 0 = 0"

   596 begin

   597

   598 lemma nonzero_mult_divide_cancel_left [simp]:

   599   "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"

   600   using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)

   601

   602 subclass semiring_no_zero_divisors_cancel

   603 proof

   604   fix a b c

   605   { fix a b c

   606     show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"

   607     proof (cases "c = 0")

   608       case True then show ?thesis by simp

   609     next

   610       case False

   611       { assume "a * c = b * c"

   612         then have "a * c div c = b * c div c"

   613           by simp

   614         with False have "a = b"

   615           by simp

   616       } then show ?thesis by auto

   617     qed

   618   }

   619   from this [of a c b]

   620   show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"

   621     by (simp add: ac_simps)

   622 qed

   623

   624 lemma div_self [simp]:

   625   assumes "a \<noteq> 0"

   626   shows "a div a = 1"

   627   using assms nonzero_mult_divide_cancel_left [of a 1] by simp

   628

   629 end

   630

   631 class idom_divide = idom + semidom_divide

   632

   633 class algebraic_semidom = semidom_divide

   634 begin

   635

   636 lemma dvd_div_mult_self [simp]:

   637   "a dvd b \<Longrightarrow> b div a * a = b"

   638   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)

   639

   640 lemma dvd_mult_div_cancel [simp]:

   641   "a dvd b \<Longrightarrow> a * (b div a) = b"

   642   using dvd_div_mult_self [of a b] by (simp add: ac_simps)

   643

   644 lemma div_mult_swap:

   645   assumes "c dvd b"

   646   shows "a * (b div c) = (a * b) div c"

   647 proof (cases "c = 0")

   648   case True then show ?thesis by simp

   649 next

   650   case False from assms obtain d where "b = c * d" ..

   651   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"

   652     by simp

   653   ultimately show ?thesis by (simp add: ac_simps)

   654 qed

   655

   656 lemma dvd_div_mult:

   657   assumes "c dvd b"

   658   shows "b div c * a = (b * a) div c"

   659   using assms div_mult_swap [of c b a] by (simp add: ac_simps)

   660

   661

   662 text \<open>Units: invertible elements in a ring\<close>

   663

   664 abbreviation is_unit :: "'a \<Rightarrow> bool"

   665 where

   666   "is_unit a \<equiv> a dvd 1"

   667

   668 lemma not_is_unit_0 [simp]:

   669   "\<not> is_unit 0"

   670   by simp

   671

   672 lemma unit_imp_dvd [dest]:

   673   "is_unit b \<Longrightarrow> b dvd a"

   674   by (rule dvd_trans [of _ 1]) simp_all

   675

   676 lemma unit_dvdE:

   677   assumes "is_unit a"

   678   obtains c where "a \<noteq> 0" and "b = a * c"

   679 proof -

   680   from assms have "a dvd b" by auto

   681   then obtain c where "b = a * c" ..

   682   moreover from assms have "a \<noteq> 0" by auto

   683   ultimately show thesis using that by blast

   684 qed

   685

   686 lemma dvd_unit_imp_unit:

   687   "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"

   688   by (rule dvd_trans)

   689

   690 lemma unit_div_1_unit [simp, intro]:

   691   assumes "is_unit a"

   692   shows "is_unit (1 div a)"

   693 proof -

   694   from assms have "1 = 1 div a * a" by simp

   695   then show "is_unit (1 div a)" by (rule dvdI)

   696 qed

   697

   698 lemma is_unitE [elim?]:

   699   assumes "is_unit a"

   700   obtains b where "a \<noteq> 0" and "b \<noteq> 0"

   701     and "is_unit b" and "1 div a = b" and "1 div b = a"

   702     and "a * b = 1" and "c div a = c * b"

   703 proof (rule that)

   704   def b \<equiv> "1 div a"

   705   then show "1 div a = b" by simp

   706   from b_def is_unit a show "is_unit b" by simp

   707   from is_unit a and is_unit b show "a \<noteq> 0" and "b \<noteq> 0" by auto

   708   from b_def is_unit a show "a * b = 1" by simp

   709   then have "1 = a * b" ..

   710   with b_def b \<noteq> 0 show "1 div b = a" by simp

   711   from is_unit a have "a dvd c" ..

   712   then obtain d where "c = a * d" ..

   713   with a \<noteq> 0 a * b = 1 show "c div a = c * b"

   714     by (simp add: mult.assoc mult.left_commute [of a])

   715 qed

   716

   717 lemma unit_prod [intro]:

   718   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"

   719   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)

   720

   721 lemma unit_div [intro]:

   722   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"

   723   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)

   724

   725 lemma mult_unit_dvd_iff:

   726   assumes "is_unit b"

   727   shows "a * b dvd c \<longleftrightarrow> a dvd c"

   728 proof

   729   assume "a * b dvd c"

   730   with assms show "a dvd c"

   731     by (simp add: dvd_mult_left)

   732 next

   733   assume "a dvd c"

   734   then obtain k where "c = a * k" ..

   735   with assms have "c = (a * b) * (1 div b * k)"

   736     by (simp add: mult_ac)

   737   then show "a * b dvd c" by (rule dvdI)

   738 qed

   739

   740 lemma dvd_mult_unit_iff:

   741   assumes "is_unit b"

   742   shows "a dvd c * b \<longleftrightarrow> a dvd c"

   743 proof

   744   assume "a dvd c * b"

   745   with assms have "c * b dvd c * (b * (1 div b))"

   746     by (subst mult_assoc [symmetric]) simp

   747   also from is_unit b have "b * (1 div b) = 1" by (rule is_unitE) simp

   748   finally have "c * b dvd c" by simp

   749   with a dvd c * b show "a dvd c" by (rule dvd_trans)

   750 next

   751   assume "a dvd c"

   752   then show "a dvd c * b" by simp

   753 qed

   754

   755 lemma div_unit_dvd_iff:

   756   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"

   757   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)

   758

   759 lemma dvd_div_unit_iff:

   760   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"

   761   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)

   762

   763 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff

   764   dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>

   765

   766 lemma unit_mult_div_div [simp]:

   767   "is_unit a \<Longrightarrow> b * (1 div a) = b div a"

   768   by (erule is_unitE [of _ b]) simp

   769

   770 lemma unit_div_mult_self [simp]:

   771   "is_unit a \<Longrightarrow> b div a * a = b"

   772   by (rule dvd_div_mult_self) auto

   773

   774 lemma unit_div_1_div_1 [simp]:

   775   "is_unit a \<Longrightarrow> 1 div (1 div a) = a"

   776   by (erule is_unitE) simp

   777

   778 lemma unit_div_mult_swap:

   779   "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"

   780   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])

   781

   782 lemma unit_div_commute:

   783   "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"

   784   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)

   785

   786 lemma unit_eq_div1:

   787   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"

   788   by (auto elim: is_unitE)

   789

   790 lemma unit_eq_div2:

   791   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"

   792   using unit_eq_div1 [of b c a] by auto

   793

   794 lemma unit_mult_left_cancel:

   795   assumes "is_unit a"

   796   shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")

   797   using assms mult_cancel_left [of a b c] by auto

   798

   799 lemma unit_mult_right_cancel:

   800   "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"

   801   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)

   802

   803 lemma unit_div_cancel:

   804   assumes "is_unit a"

   805   shows "b div a = c div a \<longleftrightarrow> b = c"

   806 proof -

   807   from assms have "is_unit (1 div a)" by simp

   808   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"

   809     by (rule unit_mult_right_cancel)

   810   with assms show ?thesis by simp

   811 qed

   812

   813

   814 text \<open>Associated elements in a ring â€“ an equivalence relation induced by the quasi-order divisibility \<close>

   815

   816 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

   817 where

   818   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"

   819

   820 lemma associatedI:

   821   "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"

   822   by (simp add: associated_def)

   823

   824 lemma associatedD1:

   825   "associated a b \<Longrightarrow> a dvd b"

   826   by (simp add: associated_def)

   827

   828 lemma associatedD2:

   829   "associated a b \<Longrightarrow> b dvd a"

   830   by (simp add: associated_def)

   831

   832 lemma associated_refl [simp]:

   833   "associated a a"

   834   by (auto intro: associatedI)

   835

   836 lemma associated_sym:

   837   "associated b a \<longleftrightarrow> associated a b"

   838   by (auto intro: associatedI dest: associatedD1 associatedD2)

   839

   840 lemma associated_trans:

   841   "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"

   842   by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)

   843

   844 lemma associated_0 [simp]:

   845   "associated 0 b \<longleftrightarrow> b = 0"

   846   "associated a 0 \<longleftrightarrow> a = 0"

   847   by (auto dest: associatedD1 associatedD2)

   848

   849 lemma associated_unit:

   850   "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"

   851   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)

   852

   853 lemma is_unit_associatedI:

   854   assumes "is_unit c" and "a = c * b"

   855   shows "associated a b"

   856 proof (rule associatedI)

   857   from a = c * b show "b dvd a" by auto

   858   from is_unit c obtain d where "c * d = 1" by (rule is_unitE)

   859   moreover from a = c * b have "d * a = d * (c * b)" by simp

   860   ultimately have "b = a * d" by (simp add: ac_simps)

   861   then show "a dvd b" ..

   862 qed

   863

   864 lemma associated_is_unitE:

   865   assumes "associated a b"

   866   obtains c where "is_unit c" and "a = c * b"

   867 proof (cases "b = 0")

   868   case True with assms have "is_unit 1" and "a = 1 * b" by simp_all

   869   with that show thesis .

   870 next

   871   case False

   872   from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)

   873   then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)

   874   then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)

   875   with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp

   876   then have "is_unit c" by auto

   877   with a = c * b that show thesis by blast

   878 qed

   879

   880 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff

   881   dvd_div_unit_iff unit_div_mult_swap unit_div_commute

   882   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel

   883   unit_eq_div1 unit_eq_div2

   884

   885 end

   886

   887 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

   888   assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"

   889   assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"

   890 begin

   891

   892 lemma mult_mono:

   893   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"

   894 apply (erule mult_right_mono [THEN order_trans], assumption)

   895 apply (erule mult_left_mono, assumption)

   896 done

   897

   898 lemma mult_mono':

   899   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"

   900 apply (rule mult_mono)

   901 apply (fast intro: order_trans)+

   902 done

   903

   904 end

   905

   906 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   907 begin

   908

   909 subclass semiring_0_cancel ..

   910

   911 lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"

   912 using mult_left_mono [of 0 b a] by simp

   913

   914 lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"

   915 using mult_left_mono [of b 0 a] by simp

   916

   917 lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"

   918 using mult_right_mono [of a 0 b] by simp

   919

   920 text {* Legacy - use @{text mult_nonpos_nonneg} *}

   921 lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"

   922 by (drule mult_right_mono [of b 0], auto)

   923

   924 lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"

   925 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   926

   927 end

   928

   929 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   930 begin

   931

   932 subclass ordered_cancel_semiring ..

   933

   934 subclass ordered_comm_monoid_add ..

   935

   936 lemma mult_left_less_imp_less:

   937   "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"

   938 by (force simp add: mult_left_mono not_le [symmetric])

   939

   940 lemma mult_right_less_imp_less:

   941   "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"

   942 by (force simp add: mult_right_mono not_le [symmetric])

   943

   944 end

   945

   946 class linordered_semiring_1 = linordered_semiring + semiring_1

   947 begin

   948

   949 lemma convex_bound_le:

   950   assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"

   951   shows "u * x + v * y \<le> a"

   952 proof-

   953   from assms have "u * x + v * y \<le> u * a + v * a"

   954     by (simp add: add_mono mult_left_mono)

   955   thus ?thesis using assms unfolding distrib_right[symmetric] by simp

   956 qed

   957

   958 end

   959

   960 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

   961   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   962   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"

   963 begin

   964

   965 subclass semiring_0_cancel ..

   966

   967 subclass linordered_semiring

   968 proof

   969   fix a b c :: 'a

   970   assume A: "a \<le> b" "0 \<le> c"

   971   from A show "c * a \<le> c * b"

   972     unfolding le_less

   973     using mult_strict_left_mono by (cases "c = 0") auto

   974   from A show "a * c \<le> b * c"

   975     unfolding le_less

   976     using mult_strict_right_mono by (cases "c = 0") auto

   977 qed

   978

   979 lemma mult_left_le_imp_le:

   980   "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"

   981 by (force simp add: mult_strict_left_mono _not_less [symmetric])

   982

   983 lemma mult_right_le_imp_le:

   984   "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"

   985 by (force simp add: mult_strict_right_mono not_less [symmetric])

   986

   987 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"

   988 using mult_strict_left_mono [of 0 b a] by simp

   989

   990 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"

   991 using mult_strict_left_mono [of b 0 a] by simp

   992

   993 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"

   994 using mult_strict_right_mono [of a 0 b] by simp

   995

   996 text {* Legacy - use @{text mult_neg_pos} *}

   997 lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"

   998 by (drule mult_strict_right_mono [of b 0], auto)

   999

  1000 lemma zero_less_mult_pos:

  1001   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

  1002 apply (cases "b\<le>0")

  1003  apply (auto simp add: le_less not_less)

  1004 apply (drule_tac mult_pos_neg [of a b])

  1005  apply (auto dest: less_not_sym)

  1006 done

  1007

  1008 lemma zero_less_mult_pos2:

  1009   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

  1010 apply (cases "b\<le>0")

  1011  apply (auto simp add: le_less not_less)

  1012 apply (drule_tac mult_pos_neg2 [of a b])

  1013  apply (auto dest: less_not_sym)

  1014 done

  1015

  1016 text{*Strict monotonicity in both arguments*}

  1017 lemma mult_strict_mono:

  1018   assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"

  1019   shows "a * c < b * d"

  1020   using assms apply (cases "c=0")

  1021   apply (simp)

  1022   apply (erule mult_strict_right_mono [THEN less_trans])

  1023   apply (force simp add: le_less)

  1024   apply (erule mult_strict_left_mono, assumption)

  1025   done

  1026

  1027 text{*This weaker variant has more natural premises*}

  1028 lemma mult_strict_mono':

  1029   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"

  1030   shows "a * c < b * d"

  1031 by (rule mult_strict_mono) (insert assms, auto)

  1032

  1033 lemma mult_less_le_imp_less:

  1034   assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"

  1035   shows "a * c < b * d"

  1036   using assms apply (subgoal_tac "a * c < b * c")

  1037   apply (erule less_le_trans)

  1038   apply (erule mult_left_mono)

  1039   apply simp

  1040   apply (erule mult_strict_right_mono)

  1041   apply assumption

  1042   done

  1043

  1044 lemma mult_le_less_imp_less:

  1045   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"

  1046   shows "a * c < b * d"

  1047   using assms apply (subgoal_tac "a * c \<le> b * c")

  1048   apply (erule le_less_trans)

  1049   apply (erule mult_strict_left_mono)

  1050   apply simp

  1051   apply (erule mult_right_mono)

  1052   apply simp

  1053   done

  1054

  1055 end

  1056

  1057 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

  1058 begin

  1059

  1060 subclass linordered_semiring_1 ..

  1061

  1062 lemma convex_bound_lt:

  1063   assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"

  1064   shows "u * x + v * y < a"

  1065 proof -

  1066   from assms have "u * x + v * y < u * a + v * a"

  1067     by (cases "u = 0")

  1068        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

  1069   thus ?thesis using assms unfolding distrib_right[symmetric] by simp

  1070 qed

  1071

  1072 end

  1073

  1074 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

  1075   assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"

  1076 begin

  1077

  1078 subclass ordered_semiring

  1079 proof

  1080   fix a b c :: 'a

  1081   assume "a \<le> b" "0 \<le> c"

  1082   thus "c * a \<le> c * b" by (rule comm_mult_left_mono)

  1083   thus "a * c \<le> b * c" by (simp only: mult.commute)

  1084 qed

  1085

  1086 end

  1087

  1088 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

  1089 begin

  1090

  1091 subclass comm_semiring_0_cancel ..

  1092 subclass ordered_comm_semiring ..

  1093 subclass ordered_cancel_semiring ..

  1094

  1095 end

  1096

  1097 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

  1098   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

  1099 begin

  1100

  1101 subclass linordered_semiring_strict

  1102 proof

  1103   fix a b c :: 'a

  1104   assume "a < b" "0 < c"

  1105   thus "c * a < c * b" by (rule comm_mult_strict_left_mono)

  1106   thus "a * c < b * c" by (simp only: mult.commute)

  1107 qed

  1108

  1109 subclass ordered_cancel_comm_semiring

  1110 proof

  1111   fix a b c :: 'a

  1112   assume "a \<le> b" "0 \<le> c"

  1113   thus "c * a \<le> c * b"

  1114     unfolding le_less

  1115     using mult_strict_left_mono by (cases "c = 0") auto

  1116 qed

  1117

  1118 end

  1119

  1120 class ordered_ring = ring + ordered_cancel_semiring

  1121 begin

  1122

  1123 subclass ordered_ab_group_add ..

  1124

  1125 lemma less_add_iff1:

  1126   "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"

  1127 by (simp add: algebra_simps)

  1128

  1129 lemma less_add_iff2:

  1130   "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"

  1131 by (simp add: algebra_simps)

  1132

  1133 lemma le_add_iff1:

  1134   "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"

  1135 by (simp add: algebra_simps)

  1136

  1137 lemma le_add_iff2:

  1138   "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"

  1139 by (simp add: algebra_simps)

  1140

  1141 lemma mult_left_mono_neg:

  1142   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"

  1143   apply (drule mult_left_mono [of _ _ "- c"])

  1144   apply simp_all

  1145   done

  1146

  1147 lemma mult_right_mono_neg:

  1148   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"

  1149   apply (drule mult_right_mono [of _ _ "- c"])

  1150   apply simp_all

  1151   done

  1152

  1153 lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"

  1154 using mult_right_mono_neg [of a 0 b] by simp

  1155

  1156 lemma split_mult_pos_le:

  1157   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

  1158 by (auto simp add: mult_nonpos_nonpos)

  1159

  1160 end

  1161

  1162 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

  1163 begin

  1164

  1165 subclass ordered_ring ..

  1166

  1167 subclass ordered_ab_group_add_abs

  1168 proof

  1169   fix a b

  1170   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"

  1171     by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)

  1172 qed (auto simp add: abs_if)

  1173

  1174 lemma zero_le_square [simp]: "0 \<le> a * a"

  1175   using linear [of 0 a]

  1176   by (auto simp add: mult_nonpos_nonpos)

  1177

  1178 lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"

  1179   by (simp add: not_less)

  1180

  1181 end

  1182

  1183 class linordered_ring_strict = ring + linordered_semiring_strict

  1184   + ordered_ab_group_add + abs_if

  1185 begin

  1186

  1187 subclass linordered_ring ..

  1188

  1189 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"

  1190 using mult_strict_left_mono [of b a "- c"] by simp

  1191

  1192 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"

  1193 using mult_strict_right_mono [of b a "- c"] by simp

  1194

  1195 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"

  1196 using mult_strict_right_mono_neg [of a 0 b] by simp

  1197

  1198 subclass ring_no_zero_divisors

  1199 proof

  1200   fix a b

  1201   assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)

  1202   assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)

  1203   have "a * b < 0 \<or> 0 < a * b"

  1204   proof (cases "a < 0")

  1205     case True note A' = this

  1206     show ?thesis proof (cases "b < 0")

  1207       case True with A'

  1208       show ?thesis by (auto dest: mult_neg_neg)

  1209     next

  1210       case False with B have "0 < b" by auto

  1211       with A' show ?thesis by (auto dest: mult_strict_right_mono)

  1212     qed

  1213   next

  1214     case False with A have A': "0 < a" by auto

  1215     show ?thesis proof (cases "b < 0")

  1216       case True with A'

  1217       show ?thesis by (auto dest: mult_strict_right_mono_neg)

  1218     next

  1219       case False with B have "0 < b" by auto

  1220       with A' show ?thesis by auto

  1221     qed

  1222   qed

  1223   then show "a * b \<noteq> 0" by (simp add: neq_iff)

  1224 qed

  1225

  1226 lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"

  1227   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])

  1228      (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)

  1229

  1230 lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"

  1231   by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)

  1232

  1233 lemma mult_less_0_iff:

  1234   "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"

  1235   apply (insert zero_less_mult_iff [of "-a" b])

  1236   apply force

  1237   done

  1238

  1239 lemma mult_le_0_iff:

  1240   "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"

  1241   apply (insert zero_le_mult_iff [of "-a" b])

  1242   apply force

  1243   done

  1244

  1245 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},

  1246    also with the relations @{text "\<le>"} and equality.*}

  1247

  1248 text{*These disjunction'' versions produce two cases when the comparison is

  1249  an assumption, but effectively four when the comparison is a goal.*}

  1250

  1251 lemma mult_less_cancel_right_disj:

  1252   "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1253   apply (cases "c = 0")

  1254   apply (auto simp add: neq_iff mult_strict_right_mono

  1255                       mult_strict_right_mono_neg)

  1256   apply (auto simp add: not_less

  1257                       not_le [symmetric, of "a*c"]

  1258                       not_le [symmetric, of a])

  1259   apply (erule_tac [!] notE)

  1260   apply (auto simp add: less_imp_le mult_right_mono

  1261                       mult_right_mono_neg)

  1262   done

  1263

  1264 lemma mult_less_cancel_left_disj:

  1265   "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1266   apply (cases "c = 0")

  1267   apply (auto simp add: neq_iff mult_strict_left_mono

  1268                       mult_strict_left_mono_neg)

  1269   apply (auto simp add: not_less

  1270                       not_le [symmetric, of "c*a"]

  1271                       not_le [symmetric, of a])

  1272   apply (erule_tac [!] notE)

  1273   apply (auto simp add: less_imp_le mult_left_mono

  1274                       mult_left_mono_neg)

  1275   done

  1276

  1277 text{*The conjunction of implication'' lemmas produce two cases when the

  1278 comparison is a goal, but give four when the comparison is an assumption.*}

  1279

  1280 lemma mult_less_cancel_right:

  1281   "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"

  1282   using mult_less_cancel_right_disj [of a c b] by auto

  1283

  1284 lemma mult_less_cancel_left:

  1285   "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"

  1286   using mult_less_cancel_left_disj [of c a b] by auto

  1287

  1288 lemma mult_le_cancel_right:

  1289    "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1290 by (simp add: not_less [symmetric] mult_less_cancel_right_disj)

  1291

  1292 lemma mult_le_cancel_left:

  1293   "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1294 by (simp add: not_less [symmetric] mult_less_cancel_left_disj)

  1295

  1296 lemma mult_le_cancel_left_pos:

  1297   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

  1298 by (auto simp: mult_le_cancel_left)

  1299

  1300 lemma mult_le_cancel_left_neg:

  1301   "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"

  1302 by (auto simp: mult_le_cancel_left)

  1303

  1304 lemma mult_less_cancel_left_pos:

  1305   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

  1306 by (auto simp: mult_less_cancel_left)

  1307

  1308 lemma mult_less_cancel_left_neg:

  1309   "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"

  1310 by (auto simp: mult_less_cancel_left)

  1311

  1312 end

  1313

  1314 lemmas mult_sign_intros =

  1315   mult_nonneg_nonneg mult_nonneg_nonpos

  1316   mult_nonpos_nonneg mult_nonpos_nonpos

  1317   mult_pos_pos mult_pos_neg

  1318   mult_neg_pos mult_neg_neg

  1319

  1320 class ordered_comm_ring = comm_ring + ordered_comm_semiring

  1321 begin

  1322

  1323 subclass ordered_ring ..

  1324 subclass ordered_cancel_comm_semiring ..

  1325

  1326 end

  1327

  1328 class linordered_semidom = semidom + linordered_comm_semiring_strict +

  1329   assumes zero_less_one [simp]: "0 < 1"

  1330 begin

  1331

  1332 lemma pos_add_strict:

  1333   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1334   using add_strict_mono [of 0 a b c] by simp

  1335

  1336 lemma zero_le_one [simp]: "0 \<le> 1"

  1337 by (rule zero_less_one [THEN less_imp_le])

  1338

  1339 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1340 by (simp add: not_le)

  1341

  1342 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1343 by (simp add: not_less)

  1344

  1345 lemma less_1_mult:

  1346   assumes "1 < m" and "1 < n"

  1347   shows "1 < m * n"

  1348   using assms mult_strict_mono [of 1 m 1 n]

  1349     by (simp add:  less_trans [OF zero_less_one])

  1350

  1351 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"

  1352   using mult_left_mono[of c 1 a] by simp

  1353

  1354 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"

  1355   using mult_mono[of a 1 b 1] by simp

  1356

  1357 end

  1358

  1359 class linordered_idom = comm_ring_1 +

  1360   linordered_comm_semiring_strict + ordered_ab_group_add +

  1361   abs_if + sgn_if

  1362 begin

  1363

  1364 subclass linordered_semiring_1_strict ..

  1365 subclass linordered_ring_strict ..

  1366 subclass ordered_comm_ring ..

  1367 subclass idom ..

  1368

  1369 subclass linordered_semidom

  1370 proof

  1371   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1372   thus "0 < 1" by (simp add: le_less)

  1373 qed

  1374

  1375 lemma linorder_neqE_linordered_idom:

  1376   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1377   using assms by (rule neqE)

  1378

  1379 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1380

  1381 lemma mult_le_cancel_right1:

  1382   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1383 by (insert mult_le_cancel_right [of 1 c b], simp)

  1384

  1385 lemma mult_le_cancel_right2:

  1386   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1387 by (insert mult_le_cancel_right [of a c 1], simp)

  1388

  1389 lemma mult_le_cancel_left1:

  1390   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1391 by (insert mult_le_cancel_left [of c 1 b], simp)

  1392

  1393 lemma mult_le_cancel_left2:

  1394   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1395 by (insert mult_le_cancel_left [of c a 1], simp)

  1396

  1397 lemma mult_less_cancel_right1:

  1398   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1399 by (insert mult_less_cancel_right [of 1 c b], simp)

  1400

  1401 lemma mult_less_cancel_right2:

  1402   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1403 by (insert mult_less_cancel_right [of a c 1], simp)

  1404

  1405 lemma mult_less_cancel_left1:

  1406   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1407 by (insert mult_less_cancel_left [of c 1 b], simp)

  1408

  1409 lemma mult_less_cancel_left2:

  1410   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1411 by (insert mult_less_cancel_left [of c a 1], simp)

  1412

  1413 lemma sgn_sgn [simp]:

  1414   "sgn (sgn a) = sgn a"

  1415 unfolding sgn_if by simp

  1416

  1417 lemma sgn_0_0:

  1418   "sgn a = 0 \<longleftrightarrow> a = 0"

  1419 unfolding sgn_if by simp

  1420

  1421 lemma sgn_1_pos:

  1422   "sgn a = 1 \<longleftrightarrow> a > 0"

  1423 unfolding sgn_if by simp

  1424

  1425 lemma sgn_1_neg:

  1426   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1427 unfolding sgn_if by auto

  1428

  1429 lemma sgn_pos [simp]:

  1430   "0 < a \<Longrightarrow> sgn a = 1"

  1431 unfolding sgn_1_pos .

  1432

  1433 lemma sgn_neg [simp]:

  1434   "a < 0 \<Longrightarrow> sgn a = - 1"

  1435 unfolding sgn_1_neg .

  1436

  1437 lemma sgn_times:

  1438   "sgn (a * b) = sgn a * sgn b"

  1439 by (auto simp add: sgn_if zero_less_mult_iff)

  1440

  1441 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1442 unfolding sgn_if abs_if by auto

  1443

  1444 lemma sgn_greater [simp]:

  1445   "0 < sgn a \<longleftrightarrow> 0 < a"

  1446   unfolding sgn_if by auto

  1447

  1448 lemma sgn_less [simp]:

  1449   "sgn a < 0 \<longleftrightarrow> a < 0"

  1450   unfolding sgn_if by auto

  1451

  1452 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1453   by (simp add: abs_if)

  1454

  1455 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1456   by (simp add: abs_if)

  1457

  1458 lemma dvd_if_abs_eq:

  1459   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1460 by(subst abs_dvd_iff[symmetric]) simp

  1461

  1462 text {* The following lemmas can be proven in more general structures, but

  1463 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1464 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1465

  1466 lemma equation_minus_iff_1 [simp, no_atp]:

  1467   "1 = - a \<longleftrightarrow> a = - 1"

  1468   by (fact equation_minus_iff)

  1469

  1470 lemma minus_equation_iff_1 [simp, no_atp]:

  1471   "- a = 1 \<longleftrightarrow> a = - 1"

  1472   by (subst minus_equation_iff, auto)

  1473

  1474 lemma le_minus_iff_1 [simp, no_atp]:

  1475   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1476   by (fact le_minus_iff)

  1477

  1478 lemma minus_le_iff_1 [simp, no_atp]:

  1479   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1480   by (fact minus_le_iff)

  1481

  1482 lemma less_minus_iff_1 [simp, no_atp]:

  1483   "1 < - b \<longleftrightarrow> b < - 1"

  1484   by (fact less_minus_iff)

  1485

  1486 lemma minus_less_iff_1 [simp, no_atp]:

  1487   "- a < 1 \<longleftrightarrow> - 1 < a"

  1488   by (fact minus_less_iff)

  1489

  1490 end

  1491

  1492 text {* Simprules for comparisons where common factors can be cancelled. *}

  1493

  1494 lemmas mult_compare_simps =

  1495     mult_le_cancel_right mult_le_cancel_left

  1496     mult_le_cancel_right1 mult_le_cancel_right2

  1497     mult_le_cancel_left1 mult_le_cancel_left2

  1498     mult_less_cancel_right mult_less_cancel_left

  1499     mult_less_cancel_right1 mult_less_cancel_right2

  1500     mult_less_cancel_left1 mult_less_cancel_left2

  1501     mult_cancel_right mult_cancel_left

  1502     mult_cancel_right1 mult_cancel_right2

  1503     mult_cancel_left1 mult_cancel_left2

  1504

  1505 text {* Reasoning about inequalities with division *}

  1506

  1507 context linordered_semidom

  1508 begin

  1509

  1510 lemma less_add_one: "a < a + 1"

  1511 proof -

  1512   have "a + 0 < a + 1"

  1513     by (blast intro: zero_less_one add_strict_left_mono)

  1514   thus ?thesis by simp

  1515 qed

  1516

  1517 lemma zero_less_two: "0 < 1 + 1"

  1518 by (blast intro: less_trans zero_less_one less_add_one)

  1519

  1520 end

  1521

  1522 context linordered_idom

  1523 begin

  1524

  1525 lemma mult_right_le_one_le:

  1526   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1527   by (rule mult_left_le)

  1528

  1529 lemma mult_left_le_one_le:

  1530   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1531   by (auto simp add: mult_le_cancel_right2)

  1532

  1533 end

  1534

  1535 text {* Absolute Value *}

  1536

  1537 context linordered_idom

  1538 begin

  1539

  1540 lemma mult_sgn_abs:

  1541   "sgn x * \<bar>x\<bar> = x"

  1542   unfolding abs_if sgn_if by auto

  1543

  1544 lemma abs_one [simp]:

  1545   "\<bar>1\<bar> = 1"

  1546   by (simp add: abs_if)

  1547

  1548 end

  1549

  1550 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1551   assumes abs_eq_mult:

  1552     "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1553

  1554 context linordered_idom

  1555 begin

  1556

  1557 subclass ordered_ring_abs proof

  1558 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1559

  1560 lemma abs_mult:

  1561   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1562   by (rule abs_eq_mult) auto

  1563

  1564 lemma abs_mult_self:

  1565   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1566   by (simp add: abs_if)

  1567

  1568 lemma abs_mult_less:

  1569   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1570 proof -

  1571   assume ac: "\<bar>a\<bar> < c"

  1572   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1573   assume "\<bar>b\<bar> < d"

  1574   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1575 qed

  1576

  1577 lemma abs_less_iff:

  1578   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1579   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1580

  1581 lemma abs_mult_pos:

  1582   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1583   by (simp add: abs_mult)

  1584

  1585 lemma abs_diff_less_iff:

  1586   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1587   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1588

  1589 lemma abs_diff_le_iff:

  1590    "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"

  1591   by (auto simp add: diff_le_eq ac_simps abs_le_iff)

  1592

  1593 end

  1594

  1595 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib

  1596

  1597 code_identifier

  1598   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1599

  1600 end