src/HOL/Rings.thy
 author haftmann Wed Feb 18 22:46:48 2015 +0100 (2015-02-18) changeset 59555 05573e5504a9 parent 59537 7f2b60cb5190 child 59557 ebd8ecacfba6 permissions -rw-r--r--
eliminated fact duplicates
     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   (*previously almost_semiring*)

   201 begin

   202

   203 subclass semiring_1 ..

   204

   205 lemma dvd_0_left_iff [simp]:

   206   "0 dvd a \<longleftrightarrow> a = 0"

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

   208

   209 lemma dvd_0_right [iff]:

   210   "a dvd 0"

   211 proof

   212   show "0 = a * 0" by simp

   213 qed

   214

   215 lemma dvd_0_left:

   216   "0 dvd a \<Longrightarrow> a = 0"

   217   by simp

   218

   219 lemma dvd_add [simp]:

   220   assumes "a dvd b" and "a dvd c"

   221   shows "a dvd (b + c)"

   222 proof -

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

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

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

   226   then show ?thesis ..

   227 qed

   228

   229 end

   230

   231 class semiring_dvd = comm_semiring_1 +

   232   assumes dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"

   233   assumes dvd_addD: "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"

   234 begin

   235

   236 lemma dvd_add_times_triv_right_iff [simp]:

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

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

   239

   240 lemma dvd_add_triv_left_iff [simp]:

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

   242   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   243

   244 lemma dvd_add_triv_right_iff [simp]:

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

   246   using dvd_add_times_triv_right_iff [of a b 1] by simp

   247

   248 lemma dvd_add_right_iff:

   249   assumes "a dvd b"

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

   251   using assms by (auto dest: dvd_addD)

   252

   253 lemma dvd_add_left_iff:

   254   assumes "a dvd c"

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

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

   257

   258 end

   259

   260 class no_zero_divisors = zero + times +

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

   262 begin

   263

   264 lemma divisors_zero:

   265   assumes "a * b = 0"

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

   267 proof (rule classical)

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

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

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

   271   with assms show ?thesis by simp

   272 qed

   273

   274 end

   275

   276 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   277   + zero_neq_one + monoid_mult

   278 begin

   279

   280 subclass semiring_0_cancel ..

   281

   282 subclass semiring_1 ..

   283

   284 end

   285

   286 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   287   + zero_neq_one + comm_monoid_mult

   288 begin

   289

   290 subclass semiring_1_cancel ..

   291 subclass comm_semiring_0_cancel ..

   292 subclass comm_semiring_1 ..

   293

   294 end

   295

   296 class ring = semiring + ab_group_add

   297 begin

   298

   299 subclass semiring_0_cancel ..

   300

   301 text {* Distribution rules *}

   302

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

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

   305

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

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

   308

   309 text{*Extract signs from products*}

   310 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   311 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   312

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

   314 by simp

   315

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

   317 by simp

   318

   319 lemma right_diff_distrib [algebra_simps]:

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

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

   322

   323 lemma left_diff_distrib [algebra_simps]:

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

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

   326

   327 lemmas ring_distribs =

   328   distrib_left distrib_right left_diff_distrib right_diff_distrib

   329

   330 lemma eq_add_iff1:

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

   332 by (simp add: algebra_simps)

   333

   334 lemma eq_add_iff2:

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

   336 by (simp add: algebra_simps)

   337

   338 end

   339

   340 lemmas ring_distribs =

   341   distrib_left distrib_right left_diff_distrib right_diff_distrib

   342

   343 class comm_ring = comm_semiring + ab_group_add

   344 begin

   345

   346 subclass ring ..

   347 subclass comm_semiring_0_cancel ..

   348

   349 lemma square_diff_square_factored:

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

   351   by (simp add: algebra_simps)

   352

   353 end

   354

   355 class ring_1 = ring + zero_neq_one + monoid_mult

   356 begin

   357

   358 subclass semiring_1_cancel ..

   359

   360 lemma square_diff_one_factored:

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

   362   by (simp add: algebra_simps)

   363

   364 end

   365

   366 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   367   (*previously ring*)

   368 begin

   369

   370 subclass ring_1 ..

   371 subclass comm_semiring_1_cancel ..

   372

   373 subclass semiring_dvd

   374 proof

   375   fix a b c

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

   377   proof

   378     assume ?Q then show ?P by simp

   379   next

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

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

   382     then show ?Q ..

   383   qed

   384   assume "a dvd b + c" and "a dvd b"

   385   show "a dvd c"

   386   proof -

   387     from a dvd b obtain d where "b = a * d" ..

   388     moreover from a dvd b + c obtain e where "b + c = a * e" ..

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

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

   391     then show "a dvd c" ..

   392   qed

   393 qed

   394

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

   396 proof

   397   assume "x dvd - y"

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

   399   then show "x dvd y" by simp

   400 next

   401   assume "x dvd y"

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

   403   then show "x dvd - y" by simp

   404 qed

   405

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

   407 proof

   408   assume "- x dvd y"

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

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

   411   then show "x dvd y" ..

   412 next

   413   assume "x dvd y"

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

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

   416   then show "- x dvd y" ..

   417 qed

   418

   419 lemma dvd_diff [simp]:

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

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

   422

   423 end

   424

   425 class semiring_no_zero_divisors = semiring_0 + no_zero_divisors

   426 begin

   427

   428 lemma mult_eq_0_iff [simp]:

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

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

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

   432     then show ?thesis using no_zero_divisors by simp

   433 next

   434   case True then show ?thesis by auto

   435 qed

   436

   437 end

   438

   439 class ring_no_zero_divisors = ring + semiring_no_zero_divisors

   440 begin

   441

   442 text{*Cancellation of equalities with a common factor*}

   443 lemma mult_cancel_right [simp]:

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

   445 proof -

   446   have "(a * c = b * c) = ((a - b) * c = 0)"

   447     by (simp add: algebra_simps)

   448   thus ?thesis by (simp add: disj_commute)

   449 qed

   450

   451 lemma mult_cancel_left [simp]:

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

   453 proof -

   454   have "(c * a = c * b) = (c * (a - b) = 0)"

   455     by (simp add: algebra_simps)

   456   thus ?thesis by simp

   457 qed

   458

   459 lemma mult_left_cancel:

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

   461   by simp

   462

   463 lemma mult_right_cancel:

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

   465   by simp

   466

   467 end

   468

   469 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   470 begin

   471

   472 lemma square_eq_1_iff:

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

   474 proof -

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

   476     by (simp add: algebra_simps)

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

   478     by simp

   479   thus ?thesis

   480     by (simp add: eq_neg_iff_add_eq_0)

   481 qed

   482

   483 lemma mult_cancel_right1 [simp]:

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

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

   486

   487 lemma mult_cancel_right2 [simp]:

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

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

   490

   491 lemma mult_cancel_left1 [simp]:

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

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

   494

   495 lemma mult_cancel_left2 [simp]:

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

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

   498

   499 end

   500

   501 class idom = comm_ring_1 + no_zero_divisors

   502 begin

   503

   504 subclass ring_1_no_zero_divisors ..

   505

   506 lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"

   507 proof

   508   assume "a * a = b * b"

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

   510     by (simp add: algebra_simps)

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

   512     by (simp add: eq_neg_iff_add_eq_0)

   513 next

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

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

   516 qed

   517

   518 lemma dvd_mult_cancel_right [simp]:

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

   520 proof -

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

   522     unfolding dvd_def by (simp add: ac_simps)

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

   524     unfolding dvd_def by simp

   525   finally show ?thesis .

   526 qed

   527

   528 lemma dvd_mult_cancel_left [simp]:

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

   530 proof -

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

   532     unfolding dvd_def by (simp add: ac_simps)

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

   534     unfolding dvd_def by simp

   535   finally show ?thesis .

   536 qed

   537

   538 end

   539

   540 text {*

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

   542   \begin{itemize}

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

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

   545   \end{itemize}

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

   547   \begin{itemize}

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

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

   550   \end{itemize}

   551 *}

   552

   553 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   556 begin

   557

   558 lemma mult_mono:

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

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

   561 apply (erule mult_left_mono, assumption)

   562 done

   563

   564 lemma mult_mono':

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

   566 apply (rule mult_mono)

   567 apply (fast intro: order_trans)+

   568 done

   569

   570 end

   571

   572 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   573 begin

   574

   575 subclass semiring_0_cancel ..

   576

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

   578 using mult_left_mono [of 0 b a] by simp

   579

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

   581 using mult_left_mono [of b 0 a] by simp

   582

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

   584 using mult_right_mono [of a 0 b] by simp

   585

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

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

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

   589

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

   591 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   592

   593 end

   594

   595 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   596 begin

   597

   598 subclass ordered_cancel_semiring ..

   599

   600 subclass ordered_comm_monoid_add ..

   601

   602 lemma mult_left_less_imp_less:

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

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

   605

   606 lemma mult_right_less_imp_less:

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

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

   609

   610 end

   611

   612 class linordered_semiring_1 = linordered_semiring + semiring_1

   613 begin

   614

   615 lemma convex_bound_le:

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

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

   618 proof-

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

   620     by (simp add: add_mono mult_left_mono)

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

   622 qed

   623

   624 end

   625

   626 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   629 begin

   630

   631 subclass semiring_0_cancel ..

   632

   633 subclass linordered_semiring

   634 proof

   635   fix a b c :: 'a

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

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

   638     unfolding le_less

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

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

   641     unfolding le_less

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

   643 qed

   644

   645 lemma mult_left_le_imp_le:

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

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

   648

   649 lemma mult_right_le_imp_le:

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

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

   652

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

   654 using mult_strict_left_mono [of 0 b a] by simp

   655

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

   657 using mult_strict_left_mono [of b 0 a] by simp

   658

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

   660 using mult_strict_right_mono [of a 0 b] by simp

   661

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

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

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

   665

   666 lemma zero_less_mult_pos:

   667   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   669  apply (auto simp add: le_less not_less)

   670 apply (drule_tac mult_pos_neg [of a b])

   671  apply (auto dest: less_not_sym)

   672 done

   673

   674 lemma zero_less_mult_pos2:

   675   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   677  apply (auto simp add: le_less not_less)

   678 apply (drule_tac mult_pos_neg2 [of a b])

   679  apply (auto dest: less_not_sym)

   680 done

   681

   682 text{*Strict monotonicity in both arguments*}

   683 lemma mult_strict_mono:

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

   685   shows "a * c < b * d"

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

   687   apply (simp)

   688   apply (erule mult_strict_right_mono [THEN less_trans])

   689   apply (force simp add: le_less)

   690   apply (erule mult_strict_left_mono, assumption)

   691   done

   692

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

   694 lemma mult_strict_mono':

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

   696   shows "a * c < b * d"

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

   698

   699 lemma mult_less_le_imp_less:

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

   701   shows "a * c < b * d"

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

   703   apply (erule less_le_trans)

   704   apply (erule mult_left_mono)

   705   apply simp

   706   apply (erule mult_strict_right_mono)

   707   apply assumption

   708   done

   709

   710 lemma mult_le_less_imp_less:

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

   712   shows "a * c < b * d"

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

   714   apply (erule le_less_trans)

   715   apply (erule mult_strict_left_mono)

   716   apply simp

   717   apply (erule mult_right_mono)

   718   apply simp

   719   done

   720

   721 end

   722

   723 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   724 begin

   725

   726 subclass linordered_semiring_1 ..

   727

   728 lemma convex_bound_lt:

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

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

   731 proof -

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

   733     by (cases "u = 0")

   734        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

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

   736 qed

   737

   738 end

   739

   740 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

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

   742 begin

   743

   744 subclass ordered_semiring

   745 proof

   746   fix a b c :: 'a

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

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

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

   750 qed

   751

   752 end

   753

   754 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

   755 begin

   756

   757 subclass comm_semiring_0_cancel ..

   758 subclass ordered_comm_semiring ..

   759 subclass ordered_cancel_semiring ..

   760

   761 end

   762

   763 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   765 begin

   766

   767 subclass linordered_semiring_strict

   768 proof

   769   fix a b c :: 'a

   770   assume "a < b" "0 < c"

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

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

   773 qed

   774

   775 subclass ordered_cancel_comm_semiring

   776 proof

   777   fix a b c :: 'a

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

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

   780     unfolding le_less

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

   782 qed

   783

   784 end

   785

   786 class ordered_ring = ring + ordered_cancel_semiring

   787 begin

   788

   789 subclass ordered_ab_group_add ..

   790

   791 lemma less_add_iff1:

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

   793 by (simp add: algebra_simps)

   794

   795 lemma less_add_iff2:

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

   797 by (simp add: algebra_simps)

   798

   799 lemma le_add_iff1:

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

   801 by (simp add: algebra_simps)

   802

   803 lemma le_add_iff2:

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

   805 by (simp add: algebra_simps)

   806

   807 lemma mult_left_mono_neg:

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

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

   810   apply simp_all

   811   done

   812

   813 lemma mult_right_mono_neg:

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

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

   816   apply simp_all

   817   done

   818

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

   820 using mult_right_mono_neg [of a 0 b] by simp

   821

   822 lemma split_mult_pos_le:

   823   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   824 by (auto simp add: mult_nonpos_nonpos)

   825

   826 end

   827

   828 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   829 begin

   830

   831 subclass ordered_ring ..

   832

   833 subclass ordered_ab_group_add_abs

   834 proof

   835   fix a b

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

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

   838 qed (auto simp add: abs_if)

   839

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

   841   using linear [of 0 a]

   842   by (auto simp add: mult_nonpos_nonpos)

   843

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

   845   by (simp add: not_less)

   846

   847 end

   848

   849 (* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.

   850    Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.

   851  *)

   852 class linordered_ring_strict = ring + linordered_semiring_strict

   853   + ordered_ab_group_add + abs_if

   854 begin

   855

   856 subclass linordered_ring ..

   857

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

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

   860

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

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

   863

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

   865 using mult_strict_right_mono_neg [of a 0 b] by simp

   866

   867 subclass ring_no_zero_divisors

   868 proof

   869   fix a b

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

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

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

   873   proof (cases "a < 0")

   874     case True note A' = this

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

   876       case True with A'

   877       show ?thesis by (auto dest: mult_neg_neg)

   878     next

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

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

   881     qed

   882   next

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

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

   885       case True with A'

   886       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   887     next

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

   889       with A' show ?thesis by auto

   890     qed

   891   qed

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

   893 qed

   894

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

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

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

   898

   899 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"

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

   901

   902 lemma mult_less_0_iff:

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

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

   905   apply force

   906   done

   907

   908 lemma mult_le_0_iff:

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

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

   911   apply force

   912   done

   913

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

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

   916

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

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

   919

   920 lemma mult_less_cancel_right_disj:

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

   922   apply (cases "c = 0")

   923   apply (auto simp add: neq_iff mult_strict_right_mono

   924                       mult_strict_right_mono_neg)

   925   apply (auto simp add: not_less

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

   927                       not_le [symmetric, of a])

   928   apply (erule_tac [!] notE)

   929   apply (auto simp add: less_imp_le mult_right_mono

   930                       mult_right_mono_neg)

   931   done

   932

   933 lemma mult_less_cancel_left_disj:

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

   935   apply (cases "c = 0")

   936   apply (auto simp add: neq_iff mult_strict_left_mono

   937                       mult_strict_left_mono_neg)

   938   apply (auto simp add: not_less

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

   940                       not_le [symmetric, of a])

   941   apply (erule_tac [!] notE)

   942   apply (auto simp add: less_imp_le mult_left_mono

   943                       mult_left_mono_neg)

   944   done

   945

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

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

   948

   949 lemma mult_less_cancel_right:

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

   951   using mult_less_cancel_right_disj [of a c b] by auto

   952

   953 lemma mult_less_cancel_left:

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

   955   using mult_less_cancel_left_disj [of c a b] by auto

   956

   957 lemma mult_le_cancel_right:

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

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

   960

   961 lemma mult_le_cancel_left:

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

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

   964

   965 lemma mult_le_cancel_left_pos:

   966   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

   967 by (auto simp: mult_le_cancel_left)

   968

   969 lemma mult_le_cancel_left_neg:

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

   971 by (auto simp: mult_le_cancel_left)

   972

   973 lemma mult_less_cancel_left_pos:

   974   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

   975 by (auto simp: mult_less_cancel_left)

   976

   977 lemma mult_less_cancel_left_neg:

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

   979 by (auto simp: mult_less_cancel_left)

   980

   981 end

   982

   983 lemmas mult_sign_intros =

   984   mult_nonneg_nonneg mult_nonneg_nonpos

   985   mult_nonpos_nonneg mult_nonpos_nonpos

   986   mult_pos_pos mult_pos_neg

   987   mult_neg_pos mult_neg_neg

   988

   989 class ordered_comm_ring = comm_ring + ordered_comm_semiring

   990 begin

   991

   992 subclass ordered_ring ..

   993 subclass ordered_cancel_comm_semiring ..

   994

   995 end

   996

   997 class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +

   998   (*previously linordered_semiring*)

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

  1000 begin

  1001

  1002 lemma pos_add_strict:

  1003   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1004   using add_strict_mono [of 0 a b c] by simp

  1005

  1006 lemma zero_le_one [simp]: "0 \<le> 1"

  1007 by (rule zero_less_one [THEN less_imp_le])

  1008

  1009 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1010 by (simp add: not_le)

  1011

  1012 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1013 by (simp add: not_less)

  1014

  1015 lemma less_1_mult:

  1016   assumes "1 < m" and "1 < n"

  1017   shows "1 < m * n"

  1018   using assms mult_strict_mono [of 1 m 1 n]

  1019     by (simp add:  less_trans [OF zero_less_one])

  1020

  1021 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"

  1022   using mult_left_mono[of c 1 a] by simp

  1023

  1024 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"

  1025   using mult_mono[of a 1 b 1] by simp

  1026

  1027 end

  1028

  1029 class linordered_idom = comm_ring_1 +

  1030   linordered_comm_semiring_strict + ordered_ab_group_add +

  1031   abs_if + sgn_if

  1032   (*previously linordered_ring*)

  1033 begin

  1034

  1035 subclass linordered_semiring_1_strict ..

  1036 subclass linordered_ring_strict ..

  1037 subclass ordered_comm_ring ..

  1038 subclass idom ..

  1039

  1040 subclass linordered_semidom

  1041 proof

  1042   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1043   thus "0 < 1" by (simp add: le_less)

  1044 qed

  1045

  1046 lemma linorder_neqE_linordered_idom:

  1047   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1048   using assms by (rule neqE)

  1049

  1050 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1051

  1052 lemma mult_le_cancel_right1:

  1053   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1054 by (insert mult_le_cancel_right [of 1 c b], simp)

  1055

  1056 lemma mult_le_cancel_right2:

  1057   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1058 by (insert mult_le_cancel_right [of a c 1], simp)

  1059

  1060 lemma mult_le_cancel_left1:

  1061   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1062 by (insert mult_le_cancel_left [of c 1 b], simp)

  1063

  1064 lemma mult_le_cancel_left2:

  1065   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1066 by (insert mult_le_cancel_left [of c a 1], simp)

  1067

  1068 lemma mult_less_cancel_right1:

  1069   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1070 by (insert mult_less_cancel_right [of 1 c b], simp)

  1071

  1072 lemma mult_less_cancel_right2:

  1073   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1074 by (insert mult_less_cancel_right [of a c 1], simp)

  1075

  1076 lemma mult_less_cancel_left1:

  1077   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1078 by (insert mult_less_cancel_left [of c 1 b], simp)

  1079

  1080 lemma mult_less_cancel_left2:

  1081   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1082 by (insert mult_less_cancel_left [of c a 1], simp)

  1083

  1084 lemma sgn_sgn [simp]:

  1085   "sgn (sgn a) = sgn a"

  1086 unfolding sgn_if by simp

  1087

  1088 lemma sgn_0_0:

  1089   "sgn a = 0 \<longleftrightarrow> a = 0"

  1090 unfolding sgn_if by simp

  1091

  1092 lemma sgn_1_pos:

  1093   "sgn a = 1 \<longleftrightarrow> a > 0"

  1094 unfolding sgn_if by simp

  1095

  1096 lemma sgn_1_neg:

  1097   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1098 unfolding sgn_if by auto

  1099

  1100 lemma sgn_pos [simp]:

  1101   "0 < a \<Longrightarrow> sgn a = 1"

  1102 unfolding sgn_1_pos .

  1103

  1104 lemma sgn_neg [simp]:

  1105   "a < 0 \<Longrightarrow> sgn a = - 1"

  1106 unfolding sgn_1_neg .

  1107

  1108 lemma sgn_times:

  1109   "sgn (a * b) = sgn a * sgn b"

  1110 by (auto simp add: sgn_if zero_less_mult_iff)

  1111

  1112 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1113 unfolding sgn_if abs_if by auto

  1114

  1115 lemma sgn_greater [simp]:

  1116   "0 < sgn a \<longleftrightarrow> 0 < a"

  1117   unfolding sgn_if by auto

  1118

  1119 lemma sgn_less [simp]:

  1120   "sgn a < 0 \<longleftrightarrow> a < 0"

  1121   unfolding sgn_if by auto

  1122

  1123 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1124   by (simp add: abs_if)

  1125

  1126 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1127   by (simp add: abs_if)

  1128

  1129 lemma dvd_if_abs_eq:

  1130   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1131 by(subst abs_dvd_iff[symmetric]) simp

  1132

  1133 text {* The following lemmas can be proven in more general structures, but

  1134 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1135 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1136

  1137 lemma equation_minus_iff_1 [simp, no_atp]:

  1138   "1 = - a \<longleftrightarrow> a = - 1"

  1139   by (fact equation_minus_iff)

  1140

  1141 lemma minus_equation_iff_1 [simp, no_atp]:

  1142   "- a = 1 \<longleftrightarrow> a = - 1"

  1143   by (subst minus_equation_iff, auto)

  1144

  1145 lemma le_minus_iff_1 [simp, no_atp]:

  1146   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1147   by (fact le_minus_iff)

  1148

  1149 lemma minus_le_iff_1 [simp, no_atp]:

  1150   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1151   by (fact minus_le_iff)

  1152

  1153 lemma less_minus_iff_1 [simp, no_atp]:

  1154   "1 < - b \<longleftrightarrow> b < - 1"

  1155   by (fact less_minus_iff)

  1156

  1157 lemma minus_less_iff_1 [simp, no_atp]:

  1158   "- a < 1 \<longleftrightarrow> - 1 < a"

  1159   by (fact minus_less_iff)

  1160

  1161 end

  1162

  1163 text {* Simprules for comparisons where common factors can be cancelled. *}

  1164

  1165 lemmas mult_compare_simps =

  1166     mult_le_cancel_right mult_le_cancel_left

  1167     mult_le_cancel_right1 mult_le_cancel_right2

  1168     mult_le_cancel_left1 mult_le_cancel_left2

  1169     mult_less_cancel_right mult_less_cancel_left

  1170     mult_less_cancel_right1 mult_less_cancel_right2

  1171     mult_less_cancel_left1 mult_less_cancel_left2

  1172     mult_cancel_right mult_cancel_left

  1173     mult_cancel_right1 mult_cancel_right2

  1174     mult_cancel_left1 mult_cancel_left2

  1175

  1176 text {* Reasoning about inequalities with division *}

  1177

  1178 context linordered_semidom

  1179 begin

  1180

  1181 lemma less_add_one: "a < a + 1"

  1182 proof -

  1183   have "a + 0 < a + 1"

  1184     by (blast intro: zero_less_one add_strict_left_mono)

  1185   thus ?thesis by simp

  1186 qed

  1187

  1188 lemma zero_less_two: "0 < 1 + 1"

  1189 by (blast intro: less_trans zero_less_one less_add_one)

  1190

  1191 end

  1192

  1193 context linordered_idom

  1194 begin

  1195

  1196 lemma mult_right_le_one_le:

  1197   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1198   by (auto simp add: mult_le_cancel_left2)

  1199

  1200 lemma mult_left_le_one_le:

  1201   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1202   by (auto simp add: mult_le_cancel_right2)

  1203

  1204 end

  1205

  1206 text {* Absolute Value *}

  1207

  1208 context linordered_idom

  1209 begin

  1210

  1211 lemma mult_sgn_abs:

  1212   "sgn x * \<bar>x\<bar> = x"

  1213   unfolding abs_if sgn_if by auto

  1214

  1215 lemma abs_one [simp]:

  1216   "\<bar>1\<bar> = 1"

  1217   by (simp add: abs_if)

  1218

  1219 end

  1220

  1221 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1222   assumes abs_eq_mult:

  1223     "(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>"

  1224

  1225 context linordered_idom

  1226 begin

  1227

  1228 subclass ordered_ring_abs proof

  1229 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1230

  1231 lemma abs_mult:

  1232   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1233   by (rule abs_eq_mult) auto

  1234

  1235 lemma abs_mult_self:

  1236   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1237   by (simp add: abs_if)

  1238

  1239 lemma abs_mult_less:

  1240   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1241 proof -

  1242   assume ac: "\<bar>a\<bar> < c"

  1243   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1244   assume "\<bar>b\<bar> < d"

  1245   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1246 qed

  1247

  1248 lemma abs_less_iff:

  1249   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1250   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1251

  1252 lemma abs_mult_pos:

  1253   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1254   by (simp add: abs_mult)

  1255

  1256 lemma abs_diff_less_iff:

  1257   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1258   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1259

  1260 end

  1261

  1262 code_identifier

  1263   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1264

  1265 end

  1266