src/HOL/Rings.thy
 author paulson Tue Jun 30 13:56:16 2015 +0100 (2015-06-30) changeset 60615 e5fa1d5d3952 parent 60570 7ed2cde6806d child 60685 cb21b7022b00 permissions -rw-r--r--
Useful lemmas. The theorem concerning swapping the variables in a double integral.
     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   assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"

   243 begin

   244

   245 subclass semiring_1_cancel ..

   246 subclass comm_semiring_0_cancel ..

   247 subclass comm_semiring_1 ..

   248

   249 lemma left_diff_distrib' [algebra_simps]:

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

   251   by (simp add: algebra_simps)

   252

   253 lemma dvd_add_times_triv_left_iff [simp]:

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

   255 proof -

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

   257   proof

   258     assume ?Q then show ?P by simp

   259   next

   260     assume ?P

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

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

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

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

   265     then show ?Q ..

   266   qed

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

   268 qed

   269

   270 lemma dvd_add_times_triv_right_iff [simp]:

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

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

   273

   274 lemma dvd_add_triv_left_iff [simp]:

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

   276   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   277

   278 lemma dvd_add_triv_right_iff [simp]:

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

   280   using dvd_add_times_triv_right_iff [of a b 1] by simp

   281

   282 lemma dvd_add_right_iff:

   283   assumes "a dvd b"

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

   285 proof

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

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

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

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

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

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

   292   then show ?Q ..

   293 next

   294   assume ?Q with assms show ?P by simp

   295 qed

   296

   297 lemma dvd_add_left_iff:

   298   assumes "a dvd c"

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

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

   301

   302 end

   303

   304 class ring = semiring + ab_group_add

   305 begin

   306

   307 subclass semiring_0_cancel ..

   308

   309 text {* Distribution rules *}

   310

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

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

   313

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

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

   316

   317 text{*Extract signs from products*}

   318 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   319 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   320

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

   322 by simp

   323

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

   325 by simp

   326

   327 lemma right_diff_distrib [algebra_simps]:

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

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

   330

   331 lemma left_diff_distrib [algebra_simps]:

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

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

   334

   335 lemmas ring_distribs =

   336   distrib_left distrib_right left_diff_distrib right_diff_distrib

   337

   338 lemma eq_add_iff1:

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

   340 by (simp add: algebra_simps)

   341

   342 lemma eq_add_iff2:

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

   344 by (simp add: algebra_simps)

   345

   346 end

   347

   348 lemmas ring_distribs =

   349   distrib_left distrib_right left_diff_distrib right_diff_distrib

   350

   351 class comm_ring = comm_semiring + ab_group_add

   352 begin

   353

   354 subclass ring ..

   355 subclass comm_semiring_0_cancel ..

   356

   357 lemma square_diff_square_factored:

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

   359   by (simp add: algebra_simps)

   360

   361 end

   362

   363 class ring_1 = ring + zero_neq_one + monoid_mult

   364 begin

   365

   366 subclass semiring_1_cancel ..

   367

   368 lemma square_diff_one_factored:

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

   370   by (simp add: algebra_simps)

   371

   372 end

   373

   374 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   375 begin

   376

   377 subclass ring_1 ..

   378 subclass comm_semiring_1_cancel

   379   by unfold_locales (simp add: algebra_simps)

   380

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

   382 proof

   383   assume "x dvd - y"

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

   385   then show "x dvd y" by simp

   386 next

   387   assume "x dvd y"

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

   389   then show "x dvd - y" by simp

   390 qed

   391

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

   393 proof

   394   assume "- x dvd y"

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

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

   397   then show "x dvd y" ..

   398 next

   399   assume "x dvd y"

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

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

   402   then show "- x dvd y" ..

   403 qed

   404

   405 lemma dvd_diff [simp]:

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

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

   408

   409 end

   410

   411 class semiring_no_zero_divisors = semiring_0 +

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

   413 begin

   414

   415 lemma divisors_zero:

   416   assumes "a * b = 0"

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

   418 proof (rule classical)

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

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

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

   422   with assms show ?thesis by simp

   423 qed

   424

   425 lemma mult_eq_0_iff [simp]:

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

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

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

   429     then show ?thesis using no_zero_divisors by simp

   430 next

   431   case True then show ?thesis by auto

   432 qed

   433

   434 end

   435

   436 class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +

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

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

   439 begin

   440

   441 lemma mult_left_cancel:

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

   443   by simp

   444

   445 lemma mult_right_cancel:

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

   447   by simp

   448

   449 end

   450

   451 class ring_no_zero_divisors = ring + semiring_no_zero_divisors

   452 begin

   453

   454 subclass semiring_no_zero_divisors_cancel

   455 proof

   456   fix a b c

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

   458     by (simp add: algebra_simps)

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

   460     by auto

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

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

   463     by (simp add: algebra_simps)

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

   465     by auto

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

   467 qed

   468

   469 end

   470

   471 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   472 begin

   473

   474 lemma square_eq_1_iff:

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

   476 proof -

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

   478     by (simp add: algebra_simps)

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

   480     by simp

   481   thus ?thesis

   482     by (simp add: eq_neg_iff_add_eq_0)

   483 qed

   484

   485 lemma mult_cancel_right1 [simp]:

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

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

   488

   489 lemma mult_cancel_right2 [simp]:

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

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

   492

   493 lemma mult_cancel_left1 [simp]:

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

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

   496

   497 lemma mult_cancel_left2 [simp]:

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

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

   500

   501 end

   502

   503 class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors

   504

   505 class idom = comm_ring_1 + semiring_no_zero_divisors

   506 begin

   507

   508 subclass semidom ..

   509

   510 subclass ring_1_no_zero_divisors ..

   511

   512 lemma dvd_mult_cancel_right [simp]:

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

   514 proof -

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

   516     unfolding dvd_def by (simp add: ac_simps)

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

   518     unfolding dvd_def by simp

   519   finally show ?thesis .

   520 qed

   521

   522 lemma dvd_mult_cancel_left [simp]:

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

   524 proof -

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

   526     unfolding dvd_def by (simp add: ac_simps)

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

   528     unfolding dvd_def by simp

   529   finally show ?thesis .

   530 qed

   531

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

   533 proof

   534   assume "a * a = b * b"

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

   536     by (simp add: algebra_simps)

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

   538     by (simp add: eq_neg_iff_add_eq_0)

   539 next

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

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

   542 qed

   543

   544 end

   545

   546 text {*

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

   548   \begin{itemize}

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

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

   551   \end{itemize}

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

   553   \begin{itemize}

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

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

   556   \end{itemize}

   557 *}

   558

   559 class divide =

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

   561

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

   563

   564 context semiring

   565 begin

   566

   567 lemma [field_simps]:

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

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

   570   by (rule distrib_left distrib_right)+

   571

   572 end

   573

   574 context ring

   575 begin

   576

   577 lemma [field_simps]:

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

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

   580   by (rule left_diff_distrib right_diff_distrib)+

   581

   582 end

   583

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

   585

   586 class semidom_divide = semidom + divide +

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

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

   589 begin

   590

   591 lemma nonzero_mult_divide_cancel_left [simp]:

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

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

   594

   595 subclass semiring_no_zero_divisors_cancel

   596 proof

   597   fix a b c

   598   { fix a b c

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

   600     proof (cases "c = 0")

   601       case True then show ?thesis by simp

   602     next

   603       case False

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

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

   606           by simp

   607         with False have "a = b"

   608           by simp

   609       } then show ?thesis by auto

   610     qed

   611   }

   612   from this [of a c b]

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

   614     by (simp add: ac_simps)

   615 qed

   616

   617 lemma div_self [simp]:

   618   assumes "a \<noteq> 0"

   619   shows "a div a = 1"

   620   using assms nonzero_mult_divide_cancel_left [of a 1] by simp

   621

   622 lemma divide_zero_left [simp]:

   623   "0 div a = 0"

   624 proof (cases "a = 0")

   625   case True then show ?thesis by simp

   626 next

   627   case False then have "a * 0 div a = 0"

   628     by (rule nonzero_mult_divide_cancel_left)

   629   then show ?thesis by simp

   630 qed

   631

   632 end

   633

   634 class idom_divide = idom + semidom_divide

   635

   636 class algebraic_semidom = semidom_divide

   637 begin

   638

   639 lemma dvd_div_mult_self [simp]:

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

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

   642

   643 lemma dvd_mult_div_cancel [simp]:

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

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

   646

   647 lemma div_mult_swap:

   648   assumes "c dvd b"

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

   650 proof (cases "c = 0")

   651   case True then show ?thesis by simp

   652 next

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

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

   655     by simp

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

   657 qed

   658

   659 lemma dvd_div_mult:

   660   assumes "c dvd b"

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

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

   663

   664 lemma dvd_div_mult2_eq:

   665   assumes "b * c dvd a"

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

   667 using assms proof

   668   fix k

   669   assume "a = b * c * k"

   670   then show ?thesis

   671     by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)

   672 qed

   673

   674

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

   676

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

   678 where

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

   680

   681 lemma not_is_unit_0 [simp]:

   682   "\<not> is_unit 0"

   683   by simp

   684

   685 lemma unit_imp_dvd [dest]:

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

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

   688

   689 lemma unit_dvdE:

   690   assumes "is_unit a"

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

   692 proof -

   693   from assms have "a dvd b" by auto

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

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

   696   ultimately show thesis using that by blast

   697 qed

   698

   699 lemma dvd_unit_imp_unit:

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

   701   by (rule dvd_trans)

   702

   703 lemma unit_div_1_unit [simp, intro]:

   704   assumes "is_unit a"

   705   shows "is_unit (1 div a)"

   706 proof -

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

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

   709 qed

   710

   711 lemma is_unitE [elim?]:

   712   assumes "is_unit a"

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

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

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

   716 proof (rule that)

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

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

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

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

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

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

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

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

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

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

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

   728 qed

   729

   730 lemma unit_prod [intro]:

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

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

   733

   734 lemma unit_div [intro]:

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

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

   737

   738 lemma mult_unit_dvd_iff:

   739   assumes "is_unit b"

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

   741 proof

   742   assume "a * b dvd c"

   743   with assms show "a dvd c"

   744     by (simp add: dvd_mult_left)

   745 next

   746   assume "a dvd c"

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

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

   749     by (simp add: mult_ac)

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

   751 qed

   752

   753 lemma dvd_mult_unit_iff:

   754   assumes "is_unit b"

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

   756 proof

   757   assume "a dvd c * b"

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

   759     by (subst mult_assoc [symmetric]) simp

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

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

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

   763 next

   764   assume "a dvd c"

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

   766 qed

   767

   768 lemma div_unit_dvd_iff:

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

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

   771

   772 lemma dvd_div_unit_iff:

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

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

   775

   776 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff

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

   778

   779 lemma unit_mult_div_div [simp]:

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

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

   782

   783 lemma unit_div_mult_self [simp]:

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

   785   by (rule dvd_div_mult_self) auto

   786

   787 lemma unit_div_1_div_1 [simp]:

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

   789   by (erule is_unitE) simp

   790

   791 lemma unit_div_mult_swap:

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

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

   794

   795 lemma unit_div_commute:

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

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

   798

   799 lemma unit_eq_div1:

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

   801   by (auto elim: is_unitE)

   802

   803 lemma unit_eq_div2:

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

   805   using unit_eq_div1 [of b c a] by auto

   806

   807 lemma unit_mult_left_cancel:

   808   assumes "is_unit a"

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

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

   811

   812 lemma unit_mult_right_cancel:

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

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

   815

   816 lemma unit_div_cancel:

   817   assumes "is_unit a"

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

   819 proof -

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

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

   822     by (rule unit_mult_right_cancel)

   823   with assms show ?thesis by simp

   824 qed

   825

   826 lemma is_unit_div_mult2_eq:

   827   assumes "is_unit b" and "is_unit c"

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

   829 proof -

   830   from assms have "is_unit (b * c)" by (simp add: unit_prod)

   831   then have "b * c dvd a"

   832     by (rule unit_imp_dvd)

   833   then show ?thesis

   834     by (rule dvd_div_mult2_eq)

   835 qed

   836

   837

   838 text \<open>Associated elements in a ring --- an equivalence relation induced

   839   by the quasi-order divisibility.\<close>

   840

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

   842 where

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

   844

   845 lemma associatedI:

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

   847   by (simp add: associated_def)

   848

   849 lemma associatedD1:

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

   851   by (simp add: associated_def)

   852

   853 lemma associatedD2:

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

   855   by (simp add: associated_def)

   856

   857 lemma associated_refl [simp]:

   858   "associated a a"

   859   by (auto intro: associatedI)

   860

   861 lemma associated_sym:

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

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

   864

   865 lemma associated_trans:

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

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

   868

   869 lemma associated_0 [simp]:

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

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

   872   by (auto dest: associatedD1 associatedD2)

   873

   874 lemma associated_unit:

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

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

   877

   878 lemma is_unit_associatedI:

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

   880   shows "associated a b"

   881 proof (rule associatedI)

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

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

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

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

   886   then show "a dvd b" ..

   887 qed

   888

   889 lemma associated_is_unitE:

   890   assumes "associated a b"

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

   892 proof (cases "b = 0")

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

   894   with that show thesis .

   895 next

   896   case False

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

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

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

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

   901   then have "is_unit c" by auto

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

   903 qed

   904

   905 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff

   906   dvd_div_unit_iff unit_div_mult_swap unit_div_commute

   907   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel

   908   unit_eq_div1 unit_eq_div2

   909

   910 end

   911

   912 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   915 begin

   916

   917 lemma mult_mono:

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

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

   920 apply (erule mult_left_mono, assumption)

   921 done

   922

   923 lemma mult_mono':

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

   925 apply (rule mult_mono)

   926 apply (fast intro: order_trans)+

   927 done

   928

   929 end

   930

   931 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   932 begin

   933

   934 subclass semiring_0_cancel ..

   935

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

   937 using mult_left_mono [of 0 b a] by simp

   938

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

   940 using mult_left_mono [of b 0 a] by simp

   941

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

   943 using mult_right_mono [of a 0 b] by simp

   944

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

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

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

   948

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

   950 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   951

   952 end

   953

   954 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   955 begin

   956

   957 subclass ordered_cancel_semiring ..

   958

   959 subclass ordered_comm_monoid_add ..

   960

   961 lemma mult_left_less_imp_less:

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

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

   964

   965 lemma mult_right_less_imp_less:

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

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

   968

   969 end

   970

   971 class linordered_semiring_1 = linordered_semiring + semiring_1

   972 begin

   973

   974 lemma convex_bound_le:

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

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

   977 proof-

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

   979     by (simp add: add_mono mult_left_mono)

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

   981 qed

   982

   983 end

   984

   985 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   988 begin

   989

   990 subclass semiring_0_cancel ..

   991

   992 subclass linordered_semiring

   993 proof

   994   fix a b c :: 'a

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

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

   997     unfolding le_less

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

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

  1000     unfolding le_less

  1001     using mult_strict_right_mono by (cases "c = 0") auto

  1002 qed

  1003

  1004 lemma mult_left_le_imp_le:

  1005   "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"

  1006 by (force simp add: mult_strict_left_mono _not_less [symmetric])

  1007

  1008 lemma mult_right_le_imp_le:

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

  1010 by (force simp add: mult_strict_right_mono not_less [symmetric])

  1011

  1012 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"

  1013 using mult_strict_left_mono [of 0 b a] by simp

  1014

  1015 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"

  1016 using mult_strict_left_mono [of b 0 a] by simp

  1017

  1018 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"

  1019 using mult_strict_right_mono [of a 0 b] by simp

  1020

  1021 text {* Legacy - use @{text mult_neg_pos} *}

  1022 lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"

  1023 by (drule mult_strict_right_mono [of b 0], auto)

  1024

  1025 lemma zero_less_mult_pos:

  1026   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

  1027 apply (cases "b\<le>0")

  1028  apply (auto simp add: le_less not_less)

  1029 apply (drule_tac mult_pos_neg [of a b])

  1030  apply (auto dest: less_not_sym)

  1031 done

  1032

  1033 lemma zero_less_mult_pos2:

  1034   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

  1035 apply (cases "b\<le>0")

  1036  apply (auto simp add: le_less not_less)

  1037 apply (drule_tac mult_pos_neg2 [of a b])

  1038  apply (auto dest: less_not_sym)

  1039 done

  1040

  1041 text{*Strict monotonicity in both arguments*}

  1042 lemma mult_strict_mono:

  1043   assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"

  1044   shows "a * c < b * d"

  1045   using assms apply (cases "c=0")

  1046   apply (simp)

  1047   apply (erule mult_strict_right_mono [THEN less_trans])

  1048   apply (force simp add: le_less)

  1049   apply (erule mult_strict_left_mono, assumption)

  1050   done

  1051

  1052 text{*This weaker variant has more natural premises*}

  1053 lemma mult_strict_mono':

  1054   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"

  1055   shows "a * c < b * d"

  1056 by (rule mult_strict_mono) (insert assms, auto)

  1057

  1058 lemma mult_less_le_imp_less:

  1059   assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"

  1060   shows "a * c < b * d"

  1061   using assms apply (subgoal_tac "a * c < b * c")

  1062   apply (erule less_le_trans)

  1063   apply (erule mult_left_mono)

  1064   apply simp

  1065   apply (erule mult_strict_right_mono)

  1066   apply assumption

  1067   done

  1068

  1069 lemma mult_le_less_imp_less:

  1070   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"

  1071   shows "a * c < b * d"

  1072   using assms apply (subgoal_tac "a * c \<le> b * c")

  1073   apply (erule le_less_trans)

  1074   apply (erule mult_strict_left_mono)

  1075   apply simp

  1076   apply (erule mult_right_mono)

  1077   apply simp

  1078   done

  1079

  1080 end

  1081

  1082 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

  1083 begin

  1084

  1085 subclass linordered_semiring_1 ..

  1086

  1087 lemma convex_bound_lt:

  1088   assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"

  1089   shows "u * x + v * y < a"

  1090 proof -

  1091   from assms have "u * x + v * y < u * a + v * a"

  1092     by (cases "u = 0")

  1093        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

  1094   thus ?thesis using assms unfolding distrib_right[symmetric] by simp

  1095 qed

  1096

  1097 end

  1098

  1099 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

  1100   assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"

  1101 begin

  1102

  1103 subclass ordered_semiring

  1104 proof

  1105   fix a b c :: 'a

  1106   assume "a \<le> b" "0 \<le> c"

  1107   thus "c * a \<le> c * b" by (rule comm_mult_left_mono)

  1108   thus "a * c \<le> b * c" by (simp only: mult.commute)

  1109 qed

  1110

  1111 end

  1112

  1113 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

  1114 begin

  1115

  1116 subclass comm_semiring_0_cancel ..

  1117 subclass ordered_comm_semiring ..

  1118 subclass ordered_cancel_semiring ..

  1119

  1120 end

  1121

  1122 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

  1123   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

  1124 begin

  1125

  1126 subclass linordered_semiring_strict

  1127 proof

  1128   fix a b c :: 'a

  1129   assume "a < b" "0 < c"

  1130   thus "c * a < c * b" by (rule comm_mult_strict_left_mono)

  1131   thus "a * c < b * c" by (simp only: mult.commute)

  1132 qed

  1133

  1134 subclass ordered_cancel_comm_semiring

  1135 proof

  1136   fix a b c :: 'a

  1137   assume "a \<le> b" "0 \<le> c"

  1138   thus "c * a \<le> c * b"

  1139     unfolding le_less

  1140     using mult_strict_left_mono by (cases "c = 0") auto

  1141 qed

  1142

  1143 end

  1144

  1145 class ordered_ring = ring + ordered_cancel_semiring

  1146 begin

  1147

  1148 subclass ordered_ab_group_add ..

  1149

  1150 lemma less_add_iff1:

  1151   "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"

  1152 by (simp add: algebra_simps)

  1153

  1154 lemma less_add_iff2:

  1155   "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"

  1156 by (simp add: algebra_simps)

  1157

  1158 lemma le_add_iff1:

  1159   "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"

  1160 by (simp add: algebra_simps)

  1161

  1162 lemma le_add_iff2:

  1163   "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"

  1164 by (simp add: algebra_simps)

  1165

  1166 lemma mult_left_mono_neg:

  1167   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"

  1168   apply (drule mult_left_mono [of _ _ "- c"])

  1169   apply simp_all

  1170   done

  1171

  1172 lemma mult_right_mono_neg:

  1173   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"

  1174   apply (drule mult_right_mono [of _ _ "- c"])

  1175   apply simp_all

  1176   done

  1177

  1178 lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"

  1179 using mult_right_mono_neg [of a 0 b] by simp

  1180

  1181 lemma split_mult_pos_le:

  1182   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

  1183 by (auto simp add: mult_nonpos_nonpos)

  1184

  1185 end

  1186

  1187 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

  1188 begin

  1189

  1190 subclass ordered_ring ..

  1191

  1192 subclass ordered_ab_group_add_abs

  1193 proof

  1194   fix a b

  1195   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"

  1196     by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)

  1197 qed (auto simp add: abs_if)

  1198

  1199 lemma zero_le_square [simp]: "0 \<le> a * a"

  1200   using linear [of 0 a]

  1201   by (auto simp add: mult_nonpos_nonpos)

  1202

  1203 lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"

  1204   by (simp add: not_less)

  1205

  1206 end

  1207

  1208 class linordered_ring_strict = ring + linordered_semiring_strict

  1209   + ordered_ab_group_add + abs_if

  1210 begin

  1211

  1212 subclass linordered_ring ..

  1213

  1214 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"

  1215 using mult_strict_left_mono [of b a "- c"] by simp

  1216

  1217 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"

  1218 using mult_strict_right_mono [of b a "- c"] by simp

  1219

  1220 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"

  1221 using mult_strict_right_mono_neg [of a 0 b] by simp

  1222

  1223 subclass ring_no_zero_divisors

  1224 proof

  1225   fix a b

  1226   assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)

  1227   assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)

  1228   have "a * b < 0 \<or> 0 < a * b"

  1229   proof (cases "a < 0")

  1230     case True note A' = this

  1231     show ?thesis proof (cases "b < 0")

  1232       case True with A'

  1233       show ?thesis by (auto dest: mult_neg_neg)

  1234     next

  1235       case False with B have "0 < b" by auto

  1236       with A' show ?thesis by (auto dest: mult_strict_right_mono)

  1237     qed

  1238   next

  1239     case False with A have A': "0 < a" by auto

  1240     show ?thesis proof (cases "b < 0")

  1241       case True with A'

  1242       show ?thesis by (auto dest: mult_strict_right_mono_neg)

  1243     next

  1244       case False with B have "0 < b" by auto

  1245       with A' show ?thesis by auto

  1246     qed

  1247   qed

  1248   then show "a * b \<noteq> 0" by (simp add: neq_iff)

  1249 qed

  1250

  1251 lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"

  1252   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])

  1253      (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)

  1254

  1255 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"

  1256   by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)

  1257

  1258 lemma mult_less_0_iff:

  1259   "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"

  1260   apply (insert zero_less_mult_iff [of "-a" b])

  1261   apply force

  1262   done

  1263

  1264 lemma mult_le_0_iff:

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

  1266   apply (insert zero_le_mult_iff [of "-a" b])

  1267   apply force

  1268   done

  1269

  1270 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},

  1271    also with the relations @{text "\<le>"} and equality.*}

  1272

  1273 text{*These disjunction'' versions produce two cases when the comparison is

  1274  an assumption, but effectively four when the comparison is a goal.*}

  1275

  1276 lemma mult_less_cancel_right_disj:

  1277   "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1278   apply (cases "c = 0")

  1279   apply (auto simp add: neq_iff mult_strict_right_mono

  1280                       mult_strict_right_mono_neg)

  1281   apply (auto simp add: not_less

  1282                       not_le [symmetric, of "a*c"]

  1283                       not_le [symmetric, of a])

  1284   apply (erule_tac [!] notE)

  1285   apply (auto simp add: less_imp_le mult_right_mono

  1286                       mult_right_mono_neg)

  1287   done

  1288

  1289 lemma mult_less_cancel_left_disj:

  1290   "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1291   apply (cases "c = 0")

  1292   apply (auto simp add: neq_iff mult_strict_left_mono

  1293                       mult_strict_left_mono_neg)

  1294   apply (auto simp add: not_less

  1295                       not_le [symmetric, of "c*a"]

  1296                       not_le [symmetric, of a])

  1297   apply (erule_tac [!] notE)

  1298   apply (auto simp add: less_imp_le mult_left_mono

  1299                       mult_left_mono_neg)

  1300   done

  1301

  1302 text{*The conjunction of implication'' lemmas produce two cases when the

  1303 comparison is a goal, but give four when the comparison is an assumption.*}

  1304

  1305 lemma mult_less_cancel_right:

  1306   "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"

  1307   using mult_less_cancel_right_disj [of a c b] by auto

  1308

  1309 lemma mult_less_cancel_left:

  1310   "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"

  1311   using mult_less_cancel_left_disj [of c a b] by auto

  1312

  1313 lemma mult_le_cancel_right:

  1314    "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1315 by (simp add: not_less [symmetric] mult_less_cancel_right_disj)

  1316

  1317 lemma mult_le_cancel_left:

  1318   "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1319 by (simp add: not_less [symmetric] mult_less_cancel_left_disj)

  1320

  1321 lemma mult_le_cancel_left_pos:

  1322   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

  1323 by (auto simp: mult_le_cancel_left)

  1324

  1325 lemma mult_le_cancel_left_neg:

  1326   "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"

  1327 by (auto simp: mult_le_cancel_left)

  1328

  1329 lemma mult_less_cancel_left_pos:

  1330   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

  1331 by (auto simp: mult_less_cancel_left)

  1332

  1333 lemma mult_less_cancel_left_neg:

  1334   "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"

  1335 by (auto simp: mult_less_cancel_left)

  1336

  1337 end

  1338

  1339 lemmas mult_sign_intros =

  1340   mult_nonneg_nonneg mult_nonneg_nonpos

  1341   mult_nonpos_nonneg mult_nonpos_nonpos

  1342   mult_pos_pos mult_pos_neg

  1343   mult_neg_pos mult_neg_neg

  1344

  1345 class ordered_comm_ring = comm_ring + ordered_comm_semiring

  1346 begin

  1347

  1348 subclass ordered_ring ..

  1349 subclass ordered_cancel_comm_semiring ..

  1350

  1351 end

  1352

  1353 class linordered_semidom = semidom + linordered_comm_semiring_strict +

  1354   assumes zero_less_one [simp]: "0 < 1"

  1355   assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"

  1356 begin

  1357

  1358 text {* Addition is the inverse of subtraction. *}

  1359

  1360 lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"

  1361   by (frule le_add_diff_inverse2) (simp add: add.commute)

  1362

  1363 lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"

  1364   by simp

  1365

  1366 lemma add_le_imp_le_diff:

  1367   shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"

  1368   apply (subst add_le_cancel_right [where c=k, symmetric])

  1369   apply (frule le_add_diff_inverse2)

  1370   apply (simp only: add.assoc [symmetric])

  1371   using add_implies_diff by fastforce

  1372

  1373 lemma add_le_add_imp_diff_le:

  1374   assumes a1: "i + k \<le> n"

  1375       and a2: "n \<le> j + k"

  1376   shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"

  1377 proof -

  1378   have "n - (i + k) + (i + k) = n"

  1379     using a1 by simp

  1380   moreover have "n - k = n - k - i + i"

  1381     using a1 by (simp add: add_le_imp_le_diff)

  1382   ultimately show ?thesis

  1383     using a2

  1384     apply (simp add: add.assoc [symmetric])

  1385     apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])

  1386     by (simp add: add.commute diff_diff_add)

  1387 qed

  1388

  1389 lemma pos_add_strict:

  1390   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1391   using add_strict_mono [of 0 a b c] by simp

  1392

  1393 lemma zero_le_one [simp]: "0 \<le> 1"

  1394 by (rule zero_less_one [THEN less_imp_le])

  1395

  1396 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1397 by (simp add: not_le)

  1398

  1399 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1400 by (simp add: not_less)

  1401

  1402 lemma less_1_mult:

  1403   assumes "1 < m" and "1 < n"

  1404   shows "1 < m * n"

  1405   using assms mult_strict_mono [of 1 m 1 n]

  1406     by (simp add:  less_trans [OF zero_less_one])

  1407

  1408 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"

  1409   using mult_left_mono[of c 1 a] by simp

  1410

  1411 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"

  1412   using mult_mono[of a 1 b 1] by simp

  1413

  1414 end

  1415

  1416 class linordered_idom = comm_ring_1 +

  1417   linordered_comm_semiring_strict + ordered_ab_group_add +

  1418   abs_if + sgn_if

  1419 begin

  1420

  1421 subclass linordered_semiring_1_strict ..

  1422 subclass linordered_ring_strict ..

  1423 subclass ordered_comm_ring ..

  1424 subclass idom ..

  1425

  1426 subclass linordered_semidom

  1427 proof

  1428   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1429   thus "0 < 1" by (simp add: le_less)

  1430   show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"

  1431     by simp

  1432 qed

  1433

  1434 lemma linorder_neqE_linordered_idom:

  1435   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1436   using assms by (rule neqE)

  1437

  1438 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1439

  1440 lemma mult_le_cancel_right1:

  1441   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1442 by (insert mult_le_cancel_right [of 1 c b], simp)

  1443

  1444 lemma mult_le_cancel_right2:

  1445   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1446 by (insert mult_le_cancel_right [of a c 1], simp)

  1447

  1448 lemma mult_le_cancel_left1:

  1449   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1450 by (insert mult_le_cancel_left [of c 1 b], simp)

  1451

  1452 lemma mult_le_cancel_left2:

  1453   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1454 by (insert mult_le_cancel_left [of c a 1], simp)

  1455

  1456 lemma mult_less_cancel_right1:

  1457   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1458 by (insert mult_less_cancel_right [of 1 c b], simp)

  1459

  1460 lemma mult_less_cancel_right2:

  1461   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1462 by (insert mult_less_cancel_right [of a c 1], simp)

  1463

  1464 lemma mult_less_cancel_left1:

  1465   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1466 by (insert mult_less_cancel_left [of c 1 b], simp)

  1467

  1468 lemma mult_less_cancel_left2:

  1469   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1470 by (insert mult_less_cancel_left [of c a 1], simp)

  1471

  1472 lemma sgn_sgn [simp]:

  1473   "sgn (sgn a) = sgn a"

  1474 unfolding sgn_if by simp

  1475

  1476 lemma sgn_0_0:

  1477   "sgn a = 0 \<longleftrightarrow> a = 0"

  1478 unfolding sgn_if by simp

  1479

  1480 lemma sgn_1_pos:

  1481   "sgn a = 1 \<longleftrightarrow> a > 0"

  1482 unfolding sgn_if by simp

  1483

  1484 lemma sgn_1_neg:

  1485   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1486 unfolding sgn_if by auto

  1487

  1488 lemma sgn_pos [simp]:

  1489   "0 < a \<Longrightarrow> sgn a = 1"

  1490 unfolding sgn_1_pos .

  1491

  1492 lemma sgn_neg [simp]:

  1493   "a < 0 \<Longrightarrow> sgn a = - 1"

  1494 unfolding sgn_1_neg .

  1495

  1496 lemma sgn_times:

  1497   "sgn (a * b) = sgn a * sgn b"

  1498 by (auto simp add: sgn_if zero_less_mult_iff)

  1499

  1500 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1501 unfolding sgn_if abs_if by auto

  1502

  1503 lemma sgn_greater [simp]:

  1504   "0 < sgn a \<longleftrightarrow> 0 < a"

  1505   unfolding sgn_if by auto

  1506

  1507 lemma sgn_less [simp]:

  1508   "sgn a < 0 \<longleftrightarrow> a < 0"

  1509   unfolding sgn_if by auto

  1510

  1511 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1512   by (simp add: abs_if)

  1513

  1514 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1515   by (simp add: abs_if)

  1516

  1517 lemma dvd_if_abs_eq:

  1518   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1519 by(subst abs_dvd_iff[symmetric]) simp

  1520

  1521 text {* The following lemmas can be proven in more general structures, but

  1522 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1523 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1524

  1525 lemma equation_minus_iff_1 [simp, no_atp]:

  1526   "1 = - a \<longleftrightarrow> a = - 1"

  1527   by (fact equation_minus_iff)

  1528

  1529 lemma minus_equation_iff_1 [simp, no_atp]:

  1530   "- a = 1 \<longleftrightarrow> a = - 1"

  1531   by (subst minus_equation_iff, auto)

  1532

  1533 lemma le_minus_iff_1 [simp, no_atp]:

  1534   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1535   by (fact le_minus_iff)

  1536

  1537 lemma minus_le_iff_1 [simp, no_atp]:

  1538   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1539   by (fact minus_le_iff)

  1540

  1541 lemma less_minus_iff_1 [simp, no_atp]:

  1542   "1 < - b \<longleftrightarrow> b < - 1"

  1543   by (fact less_minus_iff)

  1544

  1545 lemma minus_less_iff_1 [simp, no_atp]:

  1546   "- a < 1 \<longleftrightarrow> - 1 < a"

  1547   by (fact minus_less_iff)

  1548

  1549 end

  1550

  1551 text {* Simprules for comparisons where common factors can be cancelled. *}

  1552

  1553 lemmas mult_compare_simps =

  1554     mult_le_cancel_right mult_le_cancel_left

  1555     mult_le_cancel_right1 mult_le_cancel_right2

  1556     mult_le_cancel_left1 mult_le_cancel_left2

  1557     mult_less_cancel_right mult_less_cancel_left

  1558     mult_less_cancel_right1 mult_less_cancel_right2

  1559     mult_less_cancel_left1 mult_less_cancel_left2

  1560     mult_cancel_right mult_cancel_left

  1561     mult_cancel_right1 mult_cancel_right2

  1562     mult_cancel_left1 mult_cancel_left2

  1563

  1564 text {* Reasoning about inequalities with division *}

  1565

  1566 context linordered_semidom

  1567 begin

  1568

  1569 lemma less_add_one: "a < a + 1"

  1570 proof -

  1571   have "a + 0 < a + 1"

  1572     by (blast intro: zero_less_one add_strict_left_mono)

  1573   thus ?thesis by simp

  1574 qed

  1575

  1576 lemma zero_less_two: "0 < 1 + 1"

  1577 by (blast intro: less_trans zero_less_one less_add_one)

  1578

  1579 end

  1580

  1581 context linordered_idom

  1582 begin

  1583

  1584 lemma mult_right_le_one_le:

  1585   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1586   by (rule mult_left_le)

  1587

  1588 lemma mult_left_le_one_le:

  1589   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1590   by (auto simp add: mult_le_cancel_right2)

  1591

  1592 end

  1593

  1594 text {* Absolute Value *}

  1595

  1596 context linordered_idom

  1597 begin

  1598

  1599 lemma mult_sgn_abs:

  1600   "sgn x * \<bar>x\<bar> = x"

  1601   unfolding abs_if sgn_if by auto

  1602

  1603 lemma abs_one [simp]:

  1604   "\<bar>1\<bar> = 1"

  1605   by (simp add: abs_if)

  1606

  1607 end

  1608

  1609 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1610   assumes abs_eq_mult:

  1611     "(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>"

  1612

  1613 context linordered_idom

  1614 begin

  1615

  1616 subclass ordered_ring_abs proof

  1617 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1618

  1619 lemma abs_mult:

  1620   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1621   by (rule abs_eq_mult) auto

  1622

  1623 lemma abs_mult_self:

  1624   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1625   by (simp add: abs_if)

  1626

  1627 lemma abs_mult_less:

  1628   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1629 proof -

  1630   assume ac: "\<bar>a\<bar> < c"

  1631   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1632   assume "\<bar>b\<bar> < d"

  1633   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1634 qed

  1635

  1636 lemma abs_less_iff:

  1637   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1638   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1639

  1640 lemma abs_mult_pos:

  1641   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1642   by (simp add: abs_mult)

  1643

  1644 lemma abs_diff_less_iff:

  1645   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1646   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1647

  1648 lemma abs_diff_le_iff:

  1649    "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"

  1650   by (auto simp add: diff_le_eq ac_simps abs_le_iff)

  1651

  1652 end

  1653

  1654 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib

  1655

  1656 code_identifier

  1657   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1658

  1659 end