src/HOL/Rings.thy
 author haftmann Mon Jun 01 18:59:22 2015 +0200 (2015-06-01) changeset 60353 838025c6e278 parent 60352 d46de31a50c4 child 60429 d3d1e185cd63 permissions -rw-r--r--
implicit partial divison operation in integral domains
     1 (*  Title:      HOL/Rings.thy

     2     Author:     Gertrud Bauer

     3     Author:     Steven Obua

     4     Author:     Tobias Nipkow

     5     Author:     Lawrence C Paulson

     6     Author:     Markus Wenzel

     7     Author:     Jeremy Avigad

     8 *)

     9

    10 section {* Rings *}

    11

    12 theory Rings

    13 imports Groups

    14 begin

    15

    16 class semiring = ab_semigroup_add + semigroup_mult +

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

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

    19 begin

    20

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

    22 lemma combine_common_factor:

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

    24 by (simp add: distrib_right ac_simps)

    25

    26 end

    27

    28 class mult_zero = times + zero +

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

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

    31 begin

    32

    33 lemma mult_not_zero:

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

    35   by auto

    36

    37 end

    38

    39 class semiring_0 = semiring + comm_monoid_add + mult_zero

    40

    41 class semiring_0_cancel = semiring + cancel_comm_monoid_add

    42 begin

    43

    44 subclass semiring_0

    45 proof

    46   fix a :: 'a

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

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

    49 next

    50   fix a :: 'a

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

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

    53 qed

    54

    55 end

    56

    57 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +

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

    59 begin

    60

    61 subclass semiring

    62 proof

    63   fix a b c :: 'a

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

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

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

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

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

    69 qed

    70

    71 end

    72

    73 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero

    74 begin

    75

    76 subclass semiring_0 ..

    77

    78 end

    79

    80 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add

    81 begin

    82

    83 subclass semiring_0_cancel ..

    84

    85 subclass comm_semiring_0 ..

    86

    87 end

    88

    89 class zero_neq_one = zero + one +

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

    91 begin

    92

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

    94 by (rule not_sym) (rule zero_neq_one)

    95

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

    97 where

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

    99

   100 lemma of_bool_eq [simp, code]:

   101   "of_bool False = 0"

   102   "of_bool True = 1"

   103   by (simp_all add: of_bool_def)

   104

   105 lemma of_bool_eq_iff:

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

   107   by (simp add: of_bool_def)

   108

   109 lemma split_of_bool [split]:

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

   111   by (cases p) simp_all

   112

   113 lemma split_of_bool_asm:

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

   115   by (cases p) simp_all

   116

   117 end

   118

   119 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

   120

   121 text {* Abstract divisibility *}

   122

   123 class dvd = times

   124 begin

   125

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

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

   128

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

   130   unfolding dvd_def ..

   131

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

   133   unfolding dvd_def by blast

   134

   135 end

   136

   137 context comm_monoid_mult

   138 begin

   139

   140 subclass dvd .

   141

   142 lemma dvd_refl [simp]:

   143   "a dvd a"

   144 proof

   145   show "a = a * 1" by simp

   146 qed

   147

   148 lemma dvd_trans:

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

   150   shows "a dvd c"

   151 proof -

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

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

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

   155   then show ?thesis ..

   156 qed

   157

   158 lemma one_dvd [simp]:

   159   "1 dvd a"

   160   by (auto intro!: dvdI)

   161

   162 lemma dvd_mult [simp]:

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

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

   165

   166 lemma dvd_mult2 [simp]:

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

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

   169

   170 lemma dvd_triv_right [simp]:

   171   "a dvd b * a"

   172   by (rule dvd_mult) (rule dvd_refl)

   173

   174 lemma dvd_triv_left [simp]:

   175   "a dvd a * b"

   176   by (rule dvd_mult2) (rule dvd_refl)

   177

   178 lemma mult_dvd_mono:

   179   assumes "a dvd b"

   180     and "c dvd d"

   181   shows "a * c dvd b * d"

   182 proof -

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

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

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

   186   then show ?thesis ..

   187 qed

   188

   189 lemma dvd_mult_left:

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

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

   192

   193 lemma dvd_mult_right:

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

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

   196

   197 end

   198

   199 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult

   200 begin

   201

   202 subclass semiring_1 ..

   203

   204 lemma dvd_0_left_iff [simp]:

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

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

   207

   208 lemma dvd_0_right [iff]:

   209   "a dvd 0"

   210 proof

   211   show "0 = a * 0" by simp

   212 qed

   213

   214 lemma dvd_0_left:

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

   216   by simp

   217

   218 lemma dvd_add [simp]:

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

   220   shows "a dvd (b + c)"

   221 proof -

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

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

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

   225   then show ?thesis ..

   226 qed

   227

   228 end

   229

   230 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   231   + zero_neq_one + monoid_mult

   232 begin

   233

   234 subclass semiring_0_cancel ..

   235

   236 subclass semiring_1 ..

   237

   238 end

   239

   240 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   241   + zero_neq_one + comm_monoid_mult

   242 begin

   243

   244 subclass semiring_1_cancel ..

   245 subclass comm_semiring_0_cancel ..

   246 subclass comm_semiring_1 ..

   247

   248 end

   249

   250 class comm_semiring_1_diff_distrib = comm_semiring_1_cancel +

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

   252 begin

   253

   254 lemma left_diff_distrib' [algebra_simps]:

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

   256   by (simp add: algebra_simps)

   257

   258 lemma dvd_add_times_triv_left_iff [simp]:

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

   260 proof -

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

   262   proof

   263     assume ?Q then show ?P by simp

   264   next

   265     assume ?P

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

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

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

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

   270     then show ?Q ..

   271   qed

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

   273 qed

   274

   275 lemma dvd_add_times_triv_right_iff [simp]:

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

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

   278

   279 lemma dvd_add_triv_left_iff [simp]:

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

   281   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   282

   283 lemma dvd_add_triv_right_iff [simp]:

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

   285   using dvd_add_times_triv_right_iff [of a b 1] by simp

   286

   287 lemma dvd_add_right_iff:

   288   assumes "a dvd b"

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

   290 proof

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

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

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

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

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

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

   297   then show ?Q ..

   298 next

   299   assume ?Q with assms show ?P by simp

   300 qed

   301

   302 lemma dvd_add_left_iff:

   303   assumes "a dvd c"

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

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

   306

   307 end

   308

   309 class ring = semiring + ab_group_add

   310 begin

   311

   312 subclass semiring_0_cancel ..

   313

   314 text {* Distribution rules *}

   315

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

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

   318

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

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

   321

   322 text{*Extract signs from products*}

   323 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   324 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   325

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

   327 by simp

   328

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

   330 by simp

   331

   332 lemma right_diff_distrib [algebra_simps]:

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

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

   335

   336 lemma left_diff_distrib [algebra_simps]:

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

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

   339

   340 lemmas ring_distribs =

   341   distrib_left distrib_right left_diff_distrib right_diff_distrib

   342

   343 lemma eq_add_iff1:

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

   345 by (simp add: algebra_simps)

   346

   347 lemma eq_add_iff2:

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

   349 by (simp add: algebra_simps)

   350

   351 end

   352

   353 lemmas ring_distribs =

   354   distrib_left distrib_right left_diff_distrib right_diff_distrib

   355

   356 class comm_ring = comm_semiring + ab_group_add

   357 begin

   358

   359 subclass ring ..

   360 subclass comm_semiring_0_cancel ..

   361

   362 lemma square_diff_square_factored:

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

   364   by (simp add: algebra_simps)

   365

   366 end

   367

   368 class ring_1 = ring + zero_neq_one + monoid_mult

   369 begin

   370

   371 subclass semiring_1_cancel ..

   372

   373 lemma square_diff_one_factored:

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

   375   by (simp add: algebra_simps)

   376

   377 end

   378

   379 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   380 begin

   381

   382 subclass ring_1 ..

   383 subclass comm_semiring_1_cancel ..

   384

   385 subclass comm_semiring_1_diff_distrib

   386   by unfold_locales (simp add: algebra_simps)

   387

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

   389 proof

   390   assume "x dvd - y"

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

   392   then show "x dvd y" by simp

   393 next

   394   assume "x dvd y"

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

   396   then show "x dvd - y" by simp

   397 qed

   398

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

   400 proof

   401   assume "- x dvd y"

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

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

   404   then show "x dvd y" ..

   405 next

   406   assume "x dvd y"

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

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

   409   then show "- x dvd y" ..

   410 qed

   411

   412 lemma dvd_diff [simp]:

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

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

   415

   416 end

   417

   418 class semiring_no_zero_divisors = semiring_0 +

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

   420 begin

   421

   422 lemma divisors_zero:

   423   assumes "a * b = 0"

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

   425 proof (rule classical)

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

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

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

   429   with assms show ?thesis by simp

   430 qed

   431

   432 lemma mult_eq_0_iff [simp]:

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

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

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

   436     then show ?thesis using no_zero_divisors by simp

   437 next

   438   case True then show ?thesis by auto

   439 qed

   440

   441 end

   442

   443 class ring_no_zero_divisors = ring + semiring_no_zero_divisors

   444 begin

   445

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

   447 lemma mult_cancel_right [simp]:

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

   449 proof -

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

   451     by (simp add: algebra_simps)

   452   thus ?thesis by (simp add: disj_commute)

   453 qed

   454

   455 lemma mult_cancel_left [simp]:

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

   457 proof -

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

   459     by (simp add: algebra_simps)

   460   thus ?thesis by simp

   461 qed

   462

   463 lemma mult_left_cancel:

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

   465   by simp

   466

   467 lemma mult_right_cancel:

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

   469   by simp

   470

   471 end

   472

   473 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   474 begin

   475

   476 lemma square_eq_1_iff:

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

   478 proof -

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

   480     by (simp add: algebra_simps)

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

   482     by simp

   483   thus ?thesis

   484     by (simp add: eq_neg_iff_add_eq_0)

   485 qed

   486

   487 lemma mult_cancel_right1 [simp]:

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

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

   490

   491 lemma mult_cancel_right2 [simp]:

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

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

   494

   495 lemma mult_cancel_left1 [simp]:

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

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

   498

   499 lemma mult_cancel_left2 [simp]:

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

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

   502

   503 end

   504

   505 class semidom = comm_semiring_1_diff_distrib + semiring_no_zero_divisors

   506

   507 class idom = comm_ring_1 + semiring_no_zero_divisors

   508 begin

   509

   510 subclass semidom ..

   511

   512 subclass ring_1_no_zero_divisors ..

   513

   514 lemma dvd_mult_cancel_right [simp]:

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

   516 proof -

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

   518     unfolding dvd_def by (simp add: ac_simps)

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

   520     unfolding dvd_def by simp

   521   finally show ?thesis .

   522 qed

   523

   524 lemma dvd_mult_cancel_left [simp]:

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

   526 proof -

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

   528     unfolding dvd_def by (simp add: ac_simps)

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

   530     unfolding dvd_def by simp

   531   finally show ?thesis .

   532 qed

   533

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

   535 proof

   536   assume "a * a = b * b"

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

   538     by (simp add: algebra_simps)

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

   540     by (simp add: eq_neg_iff_add_eq_0)

   541 next

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

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

   544 qed

   545

   546 end

   547

   548 text {*

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

   550   \begin{itemize}

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

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

   553   \end{itemize}

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

   555   \begin{itemize}

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

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

   558   \end{itemize}

   559 *}

   560

   561 class divide =

   562   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

   563

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

   565

   566 context semiring

   567 begin

   568

   569 lemma [field_simps]:

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

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

   572   by (rule distrib_left distrib_right)+

   573

   574 end

   575

   576 context ring

   577 begin

   578

   579 lemma [field_simps]:

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

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

   582   by (rule left_diff_distrib right_diff_distrib)+

   583

   584 end

   585

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

   587

   588 class semidom_divide = semidom + divide +

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

   590   assumes divide_zero [simp]: "divide a 0 = 0"

   591 begin

   592

   593 lemma nonzero_mult_divide_cancel_left [simp]:

   594   "a \<noteq> 0 \<Longrightarrow> divide (a * b) a = b"

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

   596

   597 end

   598

   599 class idom_divide = idom + semidom_divide

   600

   601 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   604 begin

   605

   606 lemma mult_mono:

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

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

   609 apply (erule mult_left_mono, assumption)

   610 done

   611

   612 lemma mult_mono':

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

   614 apply (rule mult_mono)

   615 apply (fast intro: order_trans)+

   616 done

   617

   618 end

   619

   620 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   621 begin

   622

   623 subclass semiring_0_cancel ..

   624

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

   626 using mult_left_mono [of 0 b a] by simp

   627

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

   629 using mult_left_mono [of b 0 a] by simp

   630

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

   632 using mult_right_mono [of a 0 b] by simp

   633

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

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

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

   637

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

   639 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   640

   641 end

   642

   643 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   644 begin

   645

   646 subclass ordered_cancel_semiring ..

   647

   648 subclass ordered_comm_monoid_add ..

   649

   650 lemma mult_left_less_imp_less:

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

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

   653

   654 lemma mult_right_less_imp_less:

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

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

   657

   658 end

   659

   660 class linordered_semiring_1 = linordered_semiring + semiring_1

   661 begin

   662

   663 lemma convex_bound_le:

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

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

   666 proof-

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

   668     by (simp add: add_mono mult_left_mono)

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

   670 qed

   671

   672 end

   673

   674 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   677 begin

   678

   679 subclass semiring_0_cancel ..

   680

   681 subclass linordered_semiring

   682 proof

   683   fix a b c :: 'a

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

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

   686     unfolding le_less

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

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

   689     unfolding le_less

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

   691 qed

   692

   693 lemma mult_left_le_imp_le:

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

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

   696

   697 lemma mult_right_le_imp_le:

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

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

   700

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

   702 using mult_strict_left_mono [of 0 b a] by simp

   703

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

   705 using mult_strict_left_mono [of b 0 a] by simp

   706

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

   708 using mult_strict_right_mono [of a 0 b] by simp

   709

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

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

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

   713

   714 lemma zero_less_mult_pos:

   715   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   717  apply (auto simp add: le_less not_less)

   718 apply (drule_tac mult_pos_neg [of a b])

   719  apply (auto dest: less_not_sym)

   720 done

   721

   722 lemma zero_less_mult_pos2:

   723   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   725  apply (auto simp add: le_less not_less)

   726 apply (drule_tac mult_pos_neg2 [of a b])

   727  apply (auto dest: less_not_sym)

   728 done

   729

   730 text{*Strict monotonicity in both arguments*}

   731 lemma mult_strict_mono:

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

   733   shows "a * c < b * d"

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

   735   apply (simp)

   736   apply (erule mult_strict_right_mono [THEN less_trans])

   737   apply (force simp add: le_less)

   738   apply (erule mult_strict_left_mono, assumption)

   739   done

   740

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

   742 lemma mult_strict_mono':

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

   744   shows "a * c < b * d"

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

   746

   747 lemma mult_less_le_imp_less:

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

   749   shows "a * c < b * d"

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

   751   apply (erule less_le_trans)

   752   apply (erule mult_left_mono)

   753   apply simp

   754   apply (erule mult_strict_right_mono)

   755   apply assumption

   756   done

   757

   758 lemma mult_le_less_imp_less:

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

   760   shows "a * c < b * d"

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

   762   apply (erule le_less_trans)

   763   apply (erule mult_strict_left_mono)

   764   apply simp

   765   apply (erule mult_right_mono)

   766   apply simp

   767   done

   768

   769 end

   770

   771 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   772 begin

   773

   774 subclass linordered_semiring_1 ..

   775

   776 lemma convex_bound_lt:

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

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

   779 proof -

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

   781     by (cases "u = 0")

   782        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

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

   784 qed

   785

   786 end

   787

   788 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

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

   790 begin

   791

   792 subclass ordered_semiring

   793 proof

   794   fix a b c :: 'a

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

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

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

   798 qed

   799

   800 end

   801

   802 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

   803 begin

   804

   805 subclass comm_semiring_0_cancel ..

   806 subclass ordered_comm_semiring ..

   807 subclass ordered_cancel_semiring ..

   808

   809 end

   810

   811 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   813 begin

   814

   815 subclass linordered_semiring_strict

   816 proof

   817   fix a b c :: 'a

   818   assume "a < b" "0 < c"

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

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

   821 qed

   822

   823 subclass ordered_cancel_comm_semiring

   824 proof

   825   fix a b c :: 'a

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

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

   828     unfolding le_less

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

   830 qed

   831

   832 end

   833

   834 class ordered_ring = ring + ordered_cancel_semiring

   835 begin

   836

   837 subclass ordered_ab_group_add ..

   838

   839 lemma less_add_iff1:

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

   841 by (simp add: algebra_simps)

   842

   843 lemma less_add_iff2:

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

   845 by (simp add: algebra_simps)

   846

   847 lemma le_add_iff1:

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

   849 by (simp add: algebra_simps)

   850

   851 lemma le_add_iff2:

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

   853 by (simp add: algebra_simps)

   854

   855 lemma mult_left_mono_neg:

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

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

   858   apply simp_all

   859   done

   860

   861 lemma mult_right_mono_neg:

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

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

   864   apply simp_all

   865   done

   866

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

   868 using mult_right_mono_neg [of a 0 b] by simp

   869

   870 lemma split_mult_pos_le:

   871   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   872 by (auto simp add: mult_nonpos_nonpos)

   873

   874 end

   875

   876 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   877 begin

   878

   879 subclass ordered_ring ..

   880

   881 subclass ordered_ab_group_add_abs

   882 proof

   883   fix a b

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

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

   886 qed (auto simp add: abs_if)

   887

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

   889   using linear [of 0 a]

   890   by (auto simp add: mult_nonpos_nonpos)

   891

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

   893   by (simp add: not_less)

   894

   895 end

   896

   897 class linordered_ring_strict = ring + linordered_semiring_strict

   898   + ordered_ab_group_add + abs_if

   899 begin

   900

   901 subclass linordered_ring ..

   902

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

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

   905

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

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

   908

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

   910 using mult_strict_right_mono_neg [of a 0 b] by simp

   911

   912 subclass ring_no_zero_divisors

   913 proof

   914   fix a b

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

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

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

   918   proof (cases "a < 0")

   919     case True note A' = this

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

   921       case True with A'

   922       show ?thesis by (auto dest: mult_neg_neg)

   923     next

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

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

   926     qed

   927   next

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

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

   930       case True with A'

   931       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   932     next

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

   934       with A' show ?thesis by auto

   935     qed

   936   qed

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

   938 qed

   939

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

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

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

   943

   944 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"

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

   946

   947 lemma mult_less_0_iff:

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

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

   950   apply force

   951   done

   952

   953 lemma mult_le_0_iff:

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

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

   956   apply force

   957   done

   958

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

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

   961

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

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

   964

   965 lemma mult_less_cancel_right_disj:

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

   967   apply (cases "c = 0")

   968   apply (auto simp add: neq_iff mult_strict_right_mono

   969                       mult_strict_right_mono_neg)

   970   apply (auto simp add: not_less

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

   972                       not_le [symmetric, of a])

   973   apply (erule_tac [!] notE)

   974   apply (auto simp add: less_imp_le mult_right_mono

   975                       mult_right_mono_neg)

   976   done

   977

   978 lemma mult_less_cancel_left_disj:

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

   980   apply (cases "c = 0")

   981   apply (auto simp add: neq_iff mult_strict_left_mono

   982                       mult_strict_left_mono_neg)

   983   apply (auto simp add: not_less

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

   985                       not_le [symmetric, of a])

   986   apply (erule_tac [!] notE)

   987   apply (auto simp add: less_imp_le mult_left_mono

   988                       mult_left_mono_neg)

   989   done

   990

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

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

   993

   994 lemma mult_less_cancel_right:

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

   996   using mult_less_cancel_right_disj [of a c b] by auto

   997

   998 lemma mult_less_cancel_left:

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

  1000   using mult_less_cancel_left_disj [of c a b] by auto

  1001

  1002 lemma mult_le_cancel_right:

  1003    "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1004 by (simp add: not_less [symmetric] mult_less_cancel_right_disj)

  1005

  1006 lemma mult_le_cancel_left:

  1007   "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

  1008 by (simp add: not_less [symmetric] mult_less_cancel_left_disj)

  1009

  1010 lemma mult_le_cancel_left_pos:

  1011   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

  1012 by (auto simp: mult_le_cancel_left)

  1013

  1014 lemma mult_le_cancel_left_neg:

  1015   "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"

  1016 by (auto simp: mult_le_cancel_left)

  1017

  1018 lemma mult_less_cancel_left_pos:

  1019   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

  1020 by (auto simp: mult_less_cancel_left)

  1021

  1022 lemma mult_less_cancel_left_neg:

  1023   "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"

  1024 by (auto simp: mult_less_cancel_left)

  1025

  1026 end

  1027

  1028 lemmas mult_sign_intros =

  1029   mult_nonneg_nonneg mult_nonneg_nonpos

  1030   mult_nonpos_nonneg mult_nonpos_nonpos

  1031   mult_pos_pos mult_pos_neg

  1032   mult_neg_pos mult_neg_neg

  1033

  1034 class ordered_comm_ring = comm_ring + ordered_comm_semiring

  1035 begin

  1036

  1037 subclass ordered_ring ..

  1038 subclass ordered_cancel_comm_semiring ..

  1039

  1040 end

  1041

  1042 class linordered_semidom = semidom + linordered_comm_semiring_strict +

  1043   assumes zero_less_one [simp]: "0 < 1"

  1044 begin

  1045

  1046 lemma pos_add_strict:

  1047   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1048   using add_strict_mono [of 0 a b c] by simp

  1049

  1050 lemma zero_le_one [simp]: "0 \<le> 1"

  1051 by (rule zero_less_one [THEN less_imp_le])

  1052

  1053 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1054 by (simp add: not_le)

  1055

  1056 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1057 by (simp add: not_less)

  1058

  1059 lemma less_1_mult:

  1060   assumes "1 < m" and "1 < n"

  1061   shows "1 < m * n"

  1062   using assms mult_strict_mono [of 1 m 1 n]

  1063     by (simp add:  less_trans [OF zero_less_one])

  1064

  1065 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"

  1066   using mult_left_mono[of c 1 a] by simp

  1067

  1068 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"

  1069   using mult_mono[of a 1 b 1] by simp

  1070

  1071 end

  1072

  1073 class linordered_idom = comm_ring_1 +

  1074   linordered_comm_semiring_strict + ordered_ab_group_add +

  1075   abs_if + sgn_if

  1076 begin

  1077

  1078 subclass linordered_semiring_1_strict ..

  1079 subclass linordered_ring_strict ..

  1080 subclass ordered_comm_ring ..

  1081 subclass idom ..

  1082

  1083 subclass linordered_semidom

  1084 proof

  1085   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1086   thus "0 < 1" by (simp add: le_less)

  1087 qed

  1088

  1089 lemma linorder_neqE_linordered_idom:

  1090   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1091   using assms by (rule neqE)

  1092

  1093 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1094

  1095 lemma mult_le_cancel_right1:

  1096   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1097 by (insert mult_le_cancel_right [of 1 c b], simp)

  1098

  1099 lemma mult_le_cancel_right2:

  1100   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1101 by (insert mult_le_cancel_right [of a c 1], simp)

  1102

  1103 lemma mult_le_cancel_left1:

  1104   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1105 by (insert mult_le_cancel_left [of c 1 b], simp)

  1106

  1107 lemma mult_le_cancel_left2:

  1108   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1109 by (insert mult_le_cancel_left [of c a 1], simp)

  1110

  1111 lemma mult_less_cancel_right1:

  1112   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1113 by (insert mult_less_cancel_right [of 1 c b], simp)

  1114

  1115 lemma mult_less_cancel_right2:

  1116   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1117 by (insert mult_less_cancel_right [of a c 1], simp)

  1118

  1119 lemma mult_less_cancel_left1:

  1120   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1121 by (insert mult_less_cancel_left [of c 1 b], simp)

  1122

  1123 lemma mult_less_cancel_left2:

  1124   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1125 by (insert mult_less_cancel_left [of c a 1], simp)

  1126

  1127 lemma sgn_sgn [simp]:

  1128   "sgn (sgn a) = sgn a"

  1129 unfolding sgn_if by simp

  1130

  1131 lemma sgn_0_0:

  1132   "sgn a = 0 \<longleftrightarrow> a = 0"

  1133 unfolding sgn_if by simp

  1134

  1135 lemma sgn_1_pos:

  1136   "sgn a = 1 \<longleftrightarrow> a > 0"

  1137 unfolding sgn_if by simp

  1138

  1139 lemma sgn_1_neg:

  1140   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1141 unfolding sgn_if by auto

  1142

  1143 lemma sgn_pos [simp]:

  1144   "0 < a \<Longrightarrow> sgn a = 1"

  1145 unfolding sgn_1_pos .

  1146

  1147 lemma sgn_neg [simp]:

  1148   "a < 0 \<Longrightarrow> sgn a = - 1"

  1149 unfolding sgn_1_neg .

  1150

  1151 lemma sgn_times:

  1152   "sgn (a * b) = sgn a * sgn b"

  1153 by (auto simp add: sgn_if zero_less_mult_iff)

  1154

  1155 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1156 unfolding sgn_if abs_if by auto

  1157

  1158 lemma sgn_greater [simp]:

  1159   "0 < sgn a \<longleftrightarrow> 0 < a"

  1160   unfolding sgn_if by auto

  1161

  1162 lemma sgn_less [simp]:

  1163   "sgn a < 0 \<longleftrightarrow> a < 0"

  1164   unfolding sgn_if by auto

  1165

  1166 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1167   by (simp add: abs_if)

  1168

  1169 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1170   by (simp add: abs_if)

  1171

  1172 lemma dvd_if_abs_eq:

  1173   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1174 by(subst abs_dvd_iff[symmetric]) simp

  1175

  1176 text {* The following lemmas can be proven in more general structures, but

  1177 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1178 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1179

  1180 lemma equation_minus_iff_1 [simp, no_atp]:

  1181   "1 = - a \<longleftrightarrow> a = - 1"

  1182   by (fact equation_minus_iff)

  1183

  1184 lemma minus_equation_iff_1 [simp, no_atp]:

  1185   "- a = 1 \<longleftrightarrow> a = - 1"

  1186   by (subst minus_equation_iff, auto)

  1187

  1188 lemma le_minus_iff_1 [simp, no_atp]:

  1189   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1190   by (fact le_minus_iff)

  1191

  1192 lemma minus_le_iff_1 [simp, no_atp]:

  1193   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1194   by (fact minus_le_iff)

  1195

  1196 lemma less_minus_iff_1 [simp, no_atp]:

  1197   "1 < - b \<longleftrightarrow> b < - 1"

  1198   by (fact less_minus_iff)

  1199

  1200 lemma minus_less_iff_1 [simp, no_atp]:

  1201   "- a < 1 \<longleftrightarrow> - 1 < a"

  1202   by (fact minus_less_iff)

  1203

  1204 end

  1205

  1206 text {* Simprules for comparisons where common factors can be cancelled. *}

  1207

  1208 lemmas mult_compare_simps =

  1209     mult_le_cancel_right mult_le_cancel_left

  1210     mult_le_cancel_right1 mult_le_cancel_right2

  1211     mult_le_cancel_left1 mult_le_cancel_left2

  1212     mult_less_cancel_right mult_less_cancel_left

  1213     mult_less_cancel_right1 mult_less_cancel_right2

  1214     mult_less_cancel_left1 mult_less_cancel_left2

  1215     mult_cancel_right mult_cancel_left

  1216     mult_cancel_right1 mult_cancel_right2

  1217     mult_cancel_left1 mult_cancel_left2

  1218

  1219 text {* Reasoning about inequalities with division *}

  1220

  1221 context linordered_semidom

  1222 begin

  1223

  1224 lemma less_add_one: "a < a + 1"

  1225 proof -

  1226   have "a + 0 < a + 1"

  1227     by (blast intro: zero_less_one add_strict_left_mono)

  1228   thus ?thesis by simp

  1229 qed

  1230

  1231 lemma zero_less_two: "0 < 1 + 1"

  1232 by (blast intro: less_trans zero_less_one less_add_one)

  1233

  1234 end

  1235

  1236 context linordered_idom

  1237 begin

  1238

  1239 lemma mult_right_le_one_le:

  1240   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1241   by (rule mult_left_le)

  1242

  1243 lemma mult_left_le_one_le:

  1244   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1245   by (auto simp add: mult_le_cancel_right2)

  1246

  1247 end

  1248

  1249 text {* Absolute Value *}

  1250

  1251 context linordered_idom

  1252 begin

  1253

  1254 lemma mult_sgn_abs:

  1255   "sgn x * \<bar>x\<bar> = x"

  1256   unfolding abs_if sgn_if by auto

  1257

  1258 lemma abs_one [simp]:

  1259   "\<bar>1\<bar> = 1"

  1260   by (simp add: abs_if)

  1261

  1262 end

  1263

  1264 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1265   assumes abs_eq_mult:

  1266     "(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>"

  1267

  1268 context linordered_idom

  1269 begin

  1270

  1271 subclass ordered_ring_abs proof

  1272 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1273

  1274 lemma abs_mult:

  1275   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1276   by (rule abs_eq_mult) auto

  1277

  1278 lemma abs_mult_self:

  1279   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1280   by (simp add: abs_if)

  1281

  1282 lemma abs_mult_less:

  1283   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1284 proof -

  1285   assume ac: "\<bar>a\<bar> < c"

  1286   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1287   assume "\<bar>b\<bar> < d"

  1288   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1289 qed

  1290

  1291 lemma abs_less_iff:

  1292   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1293   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1294

  1295 lemma abs_mult_pos:

  1296   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1297   by (simp add: abs_mult)

  1298

  1299 lemma abs_diff_less_iff:

  1300   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1301   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1302

  1303 lemma abs_diff_le_iff:

  1304    "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"

  1305   by (auto simp add: diff_le_eq ac_simps abs_le_iff)

  1306

  1307 end

  1308

  1309 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib

  1310

  1311 code_identifier

  1312   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1313

  1314 end