src/HOL/Rings.thy
 author haftmann Sat Nov 08 16:53:26 2014 +0100 (2014-11-08) changeset 58952 5d82cdef6c1b parent 58889 5b7a9633cfa8 child 59000 6eb0725503fc permissions -rw-r--r--
equivalence rules for structures without zero divisors
     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 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd

   138   (*previously almost_semiring*)

   139 begin

   140

   141 subclass semiring_1 ..

   142

   143 lemma dvd_refl[simp]: "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 dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"

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

   160

   161 lemma dvd_0_right [iff]: "a dvd 0"

   162 proof

   163   show "0 = a * 0" by simp

   164 qed

   165

   166 lemma one_dvd [simp]: "1 dvd a"

   167 by (auto intro!: dvdI)

   168

   169 lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"

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

   171

   172 lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"

   173   apply (subst mult.commute)

   174   apply (erule dvd_mult)

   175   done

   176

   177 lemma dvd_triv_right [simp]: "a dvd b * a"

   178 by (rule dvd_mult) (rule dvd_refl)

   179

   180 lemma dvd_triv_left [simp]: "a dvd a * b"

   181 by (rule dvd_mult2) (rule dvd_refl)

   182

   183 lemma mult_dvd_mono:

   184   assumes "a dvd b"

   185     and "c dvd d"

   186   shows "a * c dvd b * d"

   187 proof -

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

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

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

   191   then show ?thesis ..

   192 qed

   193

   194 lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"

   195 by (simp add: dvd_def mult.assoc, blast)

   196

   197 lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"

   198   unfolding mult.commute [of a] by (rule dvd_mult_left)

   199

   200 lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"

   201 by simp

   202

   203 lemma dvd_add[simp]:

   204   assumes "a dvd b" and "a dvd c" shows "a dvd (b + c)"

   205 proof -

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

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

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

   209   then show ?thesis ..

   210 qed

   211

   212 end

   213

   214 class semiring_dvd = comm_semiring_1 +

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

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

   217 begin

   218

   219 lemma dvd_add_times_triv_right_iff [simp]:

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

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

   222

   223 lemma dvd_add_triv_left_iff [simp]:

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

   225   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   226

   227 lemma dvd_add_triv_right_iff [simp]:

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

   229   using dvd_add_times_triv_right_iff [of a b 1] by simp

   230

   231 lemma dvd_add_right_iff:

   232   assumes "a dvd b"

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

   234   using assms by (auto dest: dvd_addD)

   235

   236 lemma dvd_add_left_iff:

   237   assumes "a dvd c"

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

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

   240

   241 end

   242

   243 class no_zero_divisors = zero + times +

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

   245 begin

   246

   247 lemma divisors_zero:

   248   assumes "a * b = 0"

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

   250 proof (rule classical)

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

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

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

   254   with assms show ?thesis by simp

   255 qed

   256

   257 end

   258

   259 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   260   + zero_neq_one + monoid_mult

   261 begin

   262

   263 subclass semiring_0_cancel ..

   264

   265 subclass semiring_1 ..

   266

   267 end

   268

   269 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   270   + zero_neq_one + comm_monoid_mult

   271 begin

   272

   273 subclass semiring_1_cancel ..

   274 subclass comm_semiring_0_cancel ..

   275 subclass comm_semiring_1 ..

   276

   277 end

   278

   279 class ring = semiring + ab_group_add

   280 begin

   281

   282 subclass semiring_0_cancel ..

   283

   284 text {* Distribution rules *}

   285

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

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

   288

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

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

   291

   292 text{*Extract signs from products*}

   293 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   294 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   295

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

   297 by simp

   298

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

   300 by simp

   301

   302 lemma right_diff_distrib [algebra_simps]:

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

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

   305

   306 lemma left_diff_distrib [algebra_simps]:

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

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

   309

   310 lemmas ring_distribs =

   311   distrib_left distrib_right left_diff_distrib right_diff_distrib

   312

   313 lemma eq_add_iff1:

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

   315 by (simp add: algebra_simps)

   316

   317 lemma eq_add_iff2:

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

   319 by (simp add: algebra_simps)

   320

   321 end

   322

   323 lemmas ring_distribs =

   324   distrib_left distrib_right left_diff_distrib right_diff_distrib

   325

   326 class comm_ring = comm_semiring + ab_group_add

   327 begin

   328

   329 subclass ring ..

   330 subclass comm_semiring_0_cancel ..

   331

   332 lemma square_diff_square_factored:

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

   334   by (simp add: algebra_simps)

   335

   336 end

   337

   338 class ring_1 = ring + zero_neq_one + monoid_mult

   339 begin

   340

   341 subclass semiring_1_cancel ..

   342

   343 lemma square_diff_one_factored:

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

   345   by (simp add: algebra_simps)

   346

   347 end

   348

   349 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   350   (*previously ring*)

   351 begin

   352

   353 subclass ring_1 ..

   354 subclass comm_semiring_1_cancel ..

   355

   356 subclass semiring_dvd

   357 proof

   358   fix a b c

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

   360   proof

   361     assume ?Q then show ?P by simp

   362   next

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

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

   365     then show ?Q ..

   366   qed

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

   368   show "a dvd c"

   369   proof -

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

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

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

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

   374     then show "a dvd c" ..

   375   qed

   376 qed

   377

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

   379 proof

   380   assume "x dvd - y"

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

   382   then show "x dvd y" by simp

   383 next

   384   assume "x dvd y"

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

   386   then show "x dvd - y" by simp

   387 qed

   388

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

   390 proof

   391   assume "- x dvd y"

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

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

   394   then show "x dvd y" ..

   395 next

   396   assume "x dvd y"

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

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

   399   then show "- x dvd y" ..

   400 qed

   401

   402 lemma dvd_diff [simp]:

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

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

   405

   406 end

   407

   408 class semiring_no_zero_divisors = semiring_0 + no_zero_divisors

   409 begin

   410

   411 lemma mult_eq_0_iff [simp]:

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

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

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

   415     then show ?thesis using no_zero_divisors by simp

   416 next

   417   case True then show ?thesis by auto

   418 qed

   419

   420 end

   421

   422 class ring_no_zero_divisors = ring + semiring_no_zero_divisors

   423 begin

   424

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

   426 lemma mult_cancel_right [simp]:

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

   428 proof -

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

   430     by (simp add: algebra_simps)

   431   thus ?thesis by (simp add: disj_commute)

   432 qed

   433

   434 lemma mult_cancel_left [simp]:

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

   436 proof -

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

   438     by (simp add: algebra_simps)

   439   thus ?thesis by simp

   440 qed

   441

   442 lemma mult_left_cancel:

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

   444   by simp

   445

   446 lemma mult_right_cancel:

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

   448   by simp

   449

   450 end

   451

   452 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   453 begin

   454

   455 lemma square_eq_1_iff:

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

   457 proof -

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

   459     by (simp add: algebra_simps)

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

   461     by simp

   462   thus ?thesis

   463     by (simp add: eq_neg_iff_add_eq_0)

   464 qed

   465

   466 lemma mult_cancel_right1 [simp]:

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

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

   469

   470 lemma mult_cancel_right2 [simp]:

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

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

   473

   474 lemma mult_cancel_left1 [simp]:

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

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

   477

   478 lemma mult_cancel_left2 [simp]:

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

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

   481

   482 end

   483

   484 class idom = comm_ring_1 + no_zero_divisors

   485 begin

   486

   487 subclass ring_1_no_zero_divisors ..

   488

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

   490 proof

   491   assume "a * a = b * b"

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

   493     by (simp add: algebra_simps)

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

   495     by (simp add: eq_neg_iff_add_eq_0)

   496 next

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

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

   499 qed

   500

   501 lemma dvd_mult_cancel_right [simp]:

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

   503 proof -

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

   505     unfolding dvd_def by (simp add: ac_simps)

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

   507     unfolding dvd_def by simp

   508   finally show ?thesis .

   509 qed

   510

   511 lemma dvd_mult_cancel_left [simp]:

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

   513 proof -

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

   515     unfolding dvd_def by (simp add: ac_simps)

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

   517     unfolding dvd_def by simp

   518   finally show ?thesis .

   519 qed

   520

   521 end

   522

   523 text {*

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

   525   \begin{itemize}

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

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

   528   \end{itemize}

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

   530   \begin{itemize}

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

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

   533   \end{itemize}

   534 *}

   535

   536 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   539 begin

   540

   541 lemma mult_mono:

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

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

   544 apply (erule mult_left_mono, assumption)

   545 done

   546

   547 lemma mult_mono':

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

   549 apply (rule mult_mono)

   550 apply (fast intro: order_trans)+

   551 done

   552

   553 end

   554

   555 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   556 begin

   557

   558 subclass semiring_0_cancel ..

   559

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

   561 using mult_left_mono [of 0 b a] by simp

   562

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

   564 using mult_left_mono [of b 0 a] by simp

   565

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

   567 using mult_right_mono [of a 0 b] by simp

   568

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

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

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

   572

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

   574 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   575

   576 end

   577

   578 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   579 begin

   580

   581 subclass ordered_cancel_semiring ..

   582

   583 subclass ordered_comm_monoid_add ..

   584

   585 lemma mult_left_less_imp_less:

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

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

   588

   589 lemma mult_right_less_imp_less:

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

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

   592

   593 end

   594

   595 class linordered_semiring_1 = linordered_semiring + semiring_1

   596 begin

   597

   598 lemma convex_bound_le:

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

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

   601 proof-

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

   603     by (simp add: add_mono mult_left_mono)

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

   605 qed

   606

   607 end

   608

   609 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   612 begin

   613

   614 subclass semiring_0_cancel ..

   615

   616 subclass linordered_semiring

   617 proof

   618   fix a b c :: 'a

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

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

   621     unfolding le_less

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

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

   624     unfolding le_less

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

   626 qed

   627

   628 lemma mult_left_le_imp_le:

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

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

   631

   632 lemma mult_right_le_imp_le:

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

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

   635

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

   637 using mult_strict_left_mono [of 0 b a] by simp

   638

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

   640 using mult_strict_left_mono [of b 0 a] by simp

   641

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

   643 using mult_strict_right_mono [of a 0 b] by simp

   644

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

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

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

   648

   649 lemma zero_less_mult_pos:

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

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

   652  apply (auto simp add: le_less not_less)

   653 apply (drule_tac mult_pos_neg [of a b])

   654  apply (auto dest: less_not_sym)

   655 done

   656

   657 lemma zero_less_mult_pos2:

   658   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   660  apply (auto simp add: le_less not_less)

   661 apply (drule_tac mult_pos_neg2 [of a b])

   662  apply (auto dest: less_not_sym)

   663 done

   664

   665 text{*Strict monotonicity in both arguments*}

   666 lemma mult_strict_mono:

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

   668   shows "a * c < b * d"

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

   670   apply (simp)

   671   apply (erule mult_strict_right_mono [THEN less_trans])

   672   apply (force simp add: le_less)

   673   apply (erule mult_strict_left_mono, assumption)

   674   done

   675

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

   677 lemma mult_strict_mono':

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

   679   shows "a * c < b * d"

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

   681

   682 lemma mult_less_le_imp_less:

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

   684   shows "a * c < b * d"

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

   686   apply (erule less_le_trans)

   687   apply (erule mult_left_mono)

   688   apply simp

   689   apply (erule mult_strict_right_mono)

   690   apply assumption

   691   done

   692

   693 lemma mult_le_less_imp_less:

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

   695   shows "a * c < b * d"

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

   697   apply (erule le_less_trans)

   698   apply (erule mult_strict_left_mono)

   699   apply simp

   700   apply (erule mult_right_mono)

   701   apply simp

   702   done

   703

   704 lemma mult_less_imp_less_left:

   705   assumes less: "c * a < c * b" and nonneg: "0 \<le> c"

   706   shows "a < b"

   707 proof (rule ccontr)

   708   assume "\<not>  a < b"

   709   hence "b \<le> a" by (simp add: linorder_not_less)

   710   hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)

   711   with this and less show False by (simp add: not_less [symmetric])

   712 qed

   713

   714 lemma mult_less_imp_less_right:

   715   assumes less: "a * c < b * c" and nonneg: "0 \<le> c"

   716   shows "a < b"

   717 proof (rule ccontr)

   718   assume "\<not> a < b"

   719   hence "b \<le> a" by (simp add: linorder_not_less)

   720   hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)

   721   with this and less show False by (simp add: not_less [symmetric])

   722 qed

   723

   724 end

   725

   726 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   727 begin

   728

   729 subclass linordered_semiring_1 ..

   730

   731 lemma convex_bound_lt:

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

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

   734 proof -

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

   736     by (cases "u = 0")

   737        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

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

   739 qed

   740

   741 end

   742

   743 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

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

   745 begin

   746

   747 subclass ordered_semiring

   748 proof

   749   fix a b c :: 'a

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

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

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

   753 qed

   754

   755 end

   756

   757 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

   758 begin

   759

   760 subclass comm_semiring_0_cancel ..

   761 subclass ordered_comm_semiring ..

   762 subclass ordered_cancel_semiring ..

   763

   764 end

   765

   766 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   768 begin

   769

   770 subclass linordered_semiring_strict

   771 proof

   772   fix a b c :: 'a

   773   assume "a < b" "0 < c"

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

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

   776 qed

   777

   778 subclass ordered_cancel_comm_semiring

   779 proof

   780   fix a b c :: 'a

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

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

   783     unfolding le_less

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

   785 qed

   786

   787 end

   788

   789 class ordered_ring = ring + ordered_cancel_semiring

   790 begin

   791

   792 subclass ordered_ab_group_add ..

   793

   794 lemma less_add_iff1:

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

   796 by (simp add: algebra_simps)

   797

   798 lemma less_add_iff2:

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

   800 by (simp add: algebra_simps)

   801

   802 lemma le_add_iff1:

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

   804 by (simp add: algebra_simps)

   805

   806 lemma le_add_iff2:

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

   808 by (simp add: algebra_simps)

   809

   810 lemma mult_left_mono_neg:

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

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

   813   apply simp_all

   814   done

   815

   816 lemma mult_right_mono_neg:

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

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

   819   apply simp_all

   820   done

   821

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

   823 using mult_right_mono_neg [of a 0 b] by simp

   824

   825 lemma split_mult_pos_le:

   826   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   827 by (auto simp add: mult_nonpos_nonpos)

   828

   829 end

   830

   831 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   832 begin

   833

   834 subclass ordered_ring ..

   835

   836 subclass ordered_ab_group_add_abs

   837 proof

   838   fix a b

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

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

   841 qed (auto simp add: abs_if)

   842

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

   844   using linear [of 0 a]

   845   by (auto simp add: mult_nonpos_nonpos)

   846

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

   848   by (simp add: not_less)

   849

   850 end

   851

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

   853    Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.

   854  *)

   855 class linordered_ring_strict = ring + linordered_semiring_strict

   856   + ordered_ab_group_add + abs_if

   857 begin

   858

   859 subclass linordered_ring ..

   860

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

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

   863

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

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

   866

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

   868 using mult_strict_right_mono_neg [of a 0 b] by simp

   869

   870 subclass ring_no_zero_divisors

   871 proof

   872   fix a b

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

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

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

   876   proof (cases "a < 0")

   877     case True note A' = this

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

   879       case True with A'

   880       show ?thesis by (auto dest: mult_neg_neg)

   881     next

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

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

   884     qed

   885   next

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

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

   888       case True with A'

   889       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   890     next

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

   892       with A' show ?thesis by auto

   893     qed

   894   qed

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

   896 qed

   897

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

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

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

   901

   902 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"

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

   904

   905 lemma mult_less_0_iff:

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

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

   908   apply force

   909   done

   910

   911 lemma mult_le_0_iff:

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

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

   914   apply force

   915   done

   916

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

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

   919

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

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

   922

   923 lemma mult_less_cancel_right_disj:

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

   925   apply (cases "c = 0")

   926   apply (auto simp add: neq_iff mult_strict_right_mono

   927                       mult_strict_right_mono_neg)

   928   apply (auto simp add: not_less

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

   930                       not_le [symmetric, of a])

   931   apply (erule_tac [!] notE)

   932   apply (auto simp add: less_imp_le mult_right_mono

   933                       mult_right_mono_neg)

   934   done

   935

   936 lemma mult_less_cancel_left_disj:

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

   938   apply (cases "c = 0")

   939   apply (auto simp add: neq_iff mult_strict_left_mono

   940                       mult_strict_left_mono_neg)

   941   apply (auto simp add: not_less

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

   943                       not_le [symmetric, of a])

   944   apply (erule_tac [!] notE)

   945   apply (auto simp add: less_imp_le mult_left_mono

   946                       mult_left_mono_neg)

   947   done

   948

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

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

   951

   952 lemma mult_less_cancel_right:

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

   954   using mult_less_cancel_right_disj [of a c b] by auto

   955

   956 lemma mult_less_cancel_left:

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

   958   using mult_less_cancel_left_disj [of c a b] by auto

   959

   960 lemma mult_le_cancel_right:

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

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

   963

   964 lemma mult_le_cancel_left:

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

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

   967

   968 lemma mult_le_cancel_left_pos:

   969   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

   970 by (auto simp: mult_le_cancel_left)

   971

   972 lemma mult_le_cancel_left_neg:

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

   974 by (auto simp: mult_le_cancel_left)

   975

   976 lemma mult_less_cancel_left_pos:

   977   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

   978 by (auto simp: mult_less_cancel_left)

   979

   980 lemma mult_less_cancel_left_neg:

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

   982 by (auto simp: mult_less_cancel_left)

   983

   984 end

   985

   986 lemmas mult_sign_intros =

   987   mult_nonneg_nonneg mult_nonneg_nonpos

   988   mult_nonpos_nonneg mult_nonpos_nonpos

   989   mult_pos_pos mult_pos_neg

   990   mult_neg_pos mult_neg_neg

   991

   992 class ordered_comm_ring = comm_ring + ordered_comm_semiring

   993 begin

   994

   995 subclass ordered_ring ..

   996 subclass ordered_cancel_comm_semiring ..

   997

   998 end

   999

  1000 class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +

  1001   (*previously linordered_semiring*)

  1002   assumes zero_less_one [simp]: "0 < 1"

  1003 begin

  1004

  1005 lemma pos_add_strict:

  1006   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1007   using add_strict_mono [of 0 a b c] by simp

  1008

  1009 lemma zero_le_one [simp]: "0 \<le> 1"

  1010 by (rule zero_less_one [THEN less_imp_le])

  1011

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

  1013 by (simp add: not_le)

  1014

  1015 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1016 by (simp add: not_less)

  1017

  1018 lemma less_1_mult:

  1019   assumes "1 < m" and "1 < n"

  1020   shows "1 < m * n"

  1021   using assms mult_strict_mono [of 1 m 1 n]

  1022     by (simp add:  less_trans [OF zero_less_one])

  1023

  1024 end

  1025

  1026 class linordered_idom = comm_ring_1 +

  1027   linordered_comm_semiring_strict + ordered_ab_group_add +

  1028   abs_if + sgn_if

  1029   (*previously linordered_ring*)

  1030 begin

  1031

  1032 subclass linordered_semiring_1_strict ..

  1033 subclass linordered_ring_strict ..

  1034 subclass ordered_comm_ring ..

  1035 subclass idom ..

  1036

  1037 subclass linordered_semidom

  1038 proof

  1039   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1040   thus "0 < 1" by (simp add: le_less)

  1041 qed

  1042

  1043 lemma linorder_neqE_linordered_idom:

  1044   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1045   using assms by (rule neqE)

  1046

  1047 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1048

  1049 lemma mult_le_cancel_right1:

  1050   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1051 by (insert mult_le_cancel_right [of 1 c b], simp)

  1052

  1053 lemma mult_le_cancel_right2:

  1054   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1055 by (insert mult_le_cancel_right [of a c 1], simp)

  1056

  1057 lemma mult_le_cancel_left1:

  1058   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1059 by (insert mult_le_cancel_left [of c 1 b], simp)

  1060

  1061 lemma mult_le_cancel_left2:

  1062   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1063 by (insert mult_le_cancel_left [of c a 1], simp)

  1064

  1065 lemma mult_less_cancel_right1:

  1066   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1067 by (insert mult_less_cancel_right [of 1 c b], simp)

  1068

  1069 lemma mult_less_cancel_right2:

  1070   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1071 by (insert mult_less_cancel_right [of a c 1], simp)

  1072

  1073 lemma mult_less_cancel_left1:

  1074   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1075 by (insert mult_less_cancel_left [of c 1 b], simp)

  1076

  1077 lemma mult_less_cancel_left2:

  1078   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1079 by (insert mult_less_cancel_left [of c a 1], simp)

  1080

  1081 lemma sgn_sgn [simp]:

  1082   "sgn (sgn a) = sgn a"

  1083 unfolding sgn_if by simp

  1084

  1085 lemma sgn_0_0:

  1086   "sgn a = 0 \<longleftrightarrow> a = 0"

  1087 unfolding sgn_if by simp

  1088

  1089 lemma sgn_1_pos:

  1090   "sgn a = 1 \<longleftrightarrow> a > 0"

  1091 unfolding sgn_if by simp

  1092

  1093 lemma sgn_1_neg:

  1094   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1095 unfolding sgn_if by auto

  1096

  1097 lemma sgn_pos [simp]:

  1098   "0 < a \<Longrightarrow> sgn a = 1"

  1099 unfolding sgn_1_pos .

  1100

  1101 lemma sgn_neg [simp]:

  1102   "a < 0 \<Longrightarrow> sgn a = - 1"

  1103 unfolding sgn_1_neg .

  1104

  1105 lemma sgn_times:

  1106   "sgn (a * b) = sgn a * sgn b"

  1107 by (auto simp add: sgn_if zero_less_mult_iff)

  1108

  1109 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1110 unfolding sgn_if abs_if by auto

  1111

  1112 lemma sgn_greater [simp]:

  1113   "0 < sgn a \<longleftrightarrow> 0 < a"

  1114   unfolding sgn_if by auto

  1115

  1116 lemma sgn_less [simp]:

  1117   "sgn a < 0 \<longleftrightarrow> a < 0"

  1118   unfolding sgn_if by auto

  1119

  1120 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1121   by (simp add: abs_if)

  1122

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

  1124   by (simp add: abs_if)

  1125

  1126 lemma dvd_if_abs_eq:

  1127   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1128 by(subst abs_dvd_iff[symmetric]) simp

  1129

  1130 text {* The following lemmas can be proven in more general structures, but

  1131 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1132 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1133

  1134 lemma equation_minus_iff_1 [simp, no_atp]:

  1135   "1 = - a \<longleftrightarrow> a = - 1"

  1136   by (fact equation_minus_iff)

  1137

  1138 lemma minus_equation_iff_1 [simp, no_atp]:

  1139   "- a = 1 \<longleftrightarrow> a = - 1"

  1140   by (subst minus_equation_iff, auto)

  1141

  1142 lemma le_minus_iff_1 [simp, no_atp]:

  1143   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1144   by (fact le_minus_iff)

  1145

  1146 lemma minus_le_iff_1 [simp, no_atp]:

  1147   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1148   by (fact minus_le_iff)

  1149

  1150 lemma less_minus_iff_1 [simp, no_atp]:

  1151   "1 < - b \<longleftrightarrow> b < - 1"

  1152   by (fact less_minus_iff)

  1153

  1154 lemma minus_less_iff_1 [simp, no_atp]:

  1155   "- a < 1 \<longleftrightarrow> - 1 < a"

  1156   by (fact minus_less_iff)

  1157

  1158 end

  1159

  1160 text {* Simprules for comparisons where common factors can be cancelled. *}

  1161

  1162 lemmas mult_compare_simps =

  1163     mult_le_cancel_right mult_le_cancel_left

  1164     mult_le_cancel_right1 mult_le_cancel_right2

  1165     mult_le_cancel_left1 mult_le_cancel_left2

  1166     mult_less_cancel_right mult_less_cancel_left

  1167     mult_less_cancel_right1 mult_less_cancel_right2

  1168     mult_less_cancel_left1 mult_less_cancel_left2

  1169     mult_cancel_right mult_cancel_left

  1170     mult_cancel_right1 mult_cancel_right2

  1171     mult_cancel_left1 mult_cancel_left2

  1172

  1173 text {* Reasoning about inequalities with division *}

  1174

  1175 context linordered_semidom

  1176 begin

  1177

  1178 lemma less_add_one: "a < a + 1"

  1179 proof -

  1180   have "a + 0 < a + 1"

  1181     by (blast intro: zero_less_one add_strict_left_mono)

  1182   thus ?thesis by simp

  1183 qed

  1184

  1185 lemma zero_less_two: "0 < 1 + 1"

  1186 by (blast intro: less_trans zero_less_one less_add_one)

  1187

  1188 end

  1189

  1190 context linordered_idom

  1191 begin

  1192

  1193 lemma mult_right_le_one_le:

  1194   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1195   by (auto simp add: mult_le_cancel_left2)

  1196

  1197 lemma mult_left_le_one_le:

  1198   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1199   by (auto simp add: mult_le_cancel_right2)

  1200

  1201 end

  1202

  1203 text {* Absolute Value *}

  1204

  1205 context linordered_idom

  1206 begin

  1207

  1208 lemma mult_sgn_abs:

  1209   "sgn x * \<bar>x\<bar> = x"

  1210   unfolding abs_if sgn_if by auto

  1211

  1212 lemma abs_one [simp]:

  1213   "\<bar>1\<bar> = 1"

  1214   by (simp add: abs_if)

  1215

  1216 end

  1217

  1218 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1219   assumes abs_eq_mult:

  1220     "(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>"

  1221

  1222 context linordered_idom

  1223 begin

  1224

  1225 subclass ordered_ring_abs proof

  1226 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1227

  1228 lemma abs_mult:

  1229   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1230   by (rule abs_eq_mult) auto

  1231

  1232 lemma abs_mult_self:

  1233   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1234   by (simp add: abs_if)

  1235

  1236 lemma abs_mult_less:

  1237   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1238 proof -

  1239   assume ac: "\<bar>a\<bar> < c"

  1240   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1241   assume "\<bar>b\<bar> < d"

  1242   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1243 qed

  1244

  1245 lemma abs_less_iff:

  1246   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1247   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1248

  1249 lemma abs_mult_pos:

  1250   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1251   by (simp add: abs_mult)

  1252

  1253 lemma abs_diff_less_iff:

  1254   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1255   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1256

  1257 end

  1258

  1259 code_identifier

  1260   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1261

  1262 end

  1263