src/HOL/Rings.thy
 author paulson Tue Mar 31 16:48:48 2015 +0100 (2015-03-31) changeset 59865 8a20dd967385 parent 59833 ab828c2c5d67 child 59910 815de5506080 permissions -rw-r--r--
rationalised and generalised some theorems concerning abs and x^2.
     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_cancel + 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 ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   564 begin

   565

   566 lemma mult_mono:

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

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

   569 apply (erule mult_left_mono, assumption)

   570 done

   571

   572 lemma mult_mono':

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

   574 apply (rule mult_mono)

   575 apply (fast intro: order_trans)+

   576 done

   577

   578 end

   579

   580 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   581 begin

   582

   583 subclass semiring_0_cancel ..

   584

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

   586 using mult_left_mono [of 0 b a] by simp

   587

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

   589 using mult_left_mono [of b 0 a] by simp

   590

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

   592 using mult_right_mono [of a 0 b] by simp

   593

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

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

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

   597

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

   599 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   600

   601 end

   602

   603 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   604 begin

   605

   606 subclass ordered_cancel_semiring ..

   607

   608 subclass ordered_comm_monoid_add ..

   609

   610 lemma mult_left_less_imp_less:

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

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

   613

   614 lemma mult_right_less_imp_less:

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

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

   617

   618 end

   619

   620 class linordered_semiring_1 = linordered_semiring + semiring_1

   621 begin

   622

   623 lemma convex_bound_le:

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

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

   626 proof-

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

   628     by (simp add: add_mono mult_left_mono)

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

   630 qed

   631

   632 end

   633

   634 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   637 begin

   638

   639 subclass semiring_0_cancel ..

   640

   641 subclass linordered_semiring

   642 proof

   643   fix a b c :: 'a

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

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

   646     unfolding le_less

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

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

   649     unfolding le_less

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

   651 qed

   652

   653 lemma mult_left_le_imp_le:

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

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

   656

   657 lemma mult_right_le_imp_le:

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

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

   660

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

   662 using mult_strict_left_mono [of 0 b a] by simp

   663

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

   665 using mult_strict_left_mono [of b 0 a] by simp

   666

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

   668 using mult_strict_right_mono [of a 0 b] by simp

   669

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

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

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

   673

   674 lemma zero_less_mult_pos:

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

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

   677  apply (auto simp add: le_less not_less)

   678 apply (drule_tac mult_pos_neg [of a b])

   679  apply (auto dest: less_not_sym)

   680 done

   681

   682 lemma zero_less_mult_pos2:

   683   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   685  apply (auto simp add: le_less not_less)

   686 apply (drule_tac mult_pos_neg2 [of a b])

   687  apply (auto dest: less_not_sym)

   688 done

   689

   690 text{*Strict monotonicity in both arguments*}

   691 lemma mult_strict_mono:

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

   693   shows "a * c < b * d"

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

   695   apply (simp)

   696   apply (erule mult_strict_right_mono [THEN less_trans])

   697   apply (force simp add: le_less)

   698   apply (erule mult_strict_left_mono, assumption)

   699   done

   700

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

   702 lemma mult_strict_mono':

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

   704   shows "a * c < b * d"

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

   706

   707 lemma mult_less_le_imp_less:

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

   709   shows "a * c < b * d"

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

   711   apply (erule less_le_trans)

   712   apply (erule mult_left_mono)

   713   apply simp

   714   apply (erule mult_strict_right_mono)

   715   apply assumption

   716   done

   717

   718 lemma mult_le_less_imp_less:

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

   720   shows "a * c < b * d"

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

   722   apply (erule le_less_trans)

   723   apply (erule mult_strict_left_mono)

   724   apply simp

   725   apply (erule mult_right_mono)

   726   apply simp

   727   done

   728

   729 end

   730

   731 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   732 begin

   733

   734 subclass linordered_semiring_1 ..

   735

   736 lemma convex_bound_lt:

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

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

   739 proof -

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

   741     by (cases "u = 0")

   742        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

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

   744 qed

   745

   746 end

   747

   748 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

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

   750 begin

   751

   752 subclass ordered_semiring

   753 proof

   754   fix a b c :: 'a

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

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

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

   758 qed

   759

   760 end

   761

   762 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

   763 begin

   764

   765 subclass comm_semiring_0_cancel ..

   766 subclass ordered_comm_semiring ..

   767 subclass ordered_cancel_semiring ..

   768

   769 end

   770

   771 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   773 begin

   774

   775 subclass linordered_semiring_strict

   776 proof

   777   fix a b c :: 'a

   778   assume "a < b" "0 < c"

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

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

   781 qed

   782

   783 subclass ordered_cancel_comm_semiring

   784 proof

   785   fix a b c :: 'a

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

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

   788     unfolding le_less

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

   790 qed

   791

   792 end

   793

   794 class ordered_ring = ring + ordered_cancel_semiring

   795 begin

   796

   797 subclass ordered_ab_group_add ..

   798

   799 lemma less_add_iff1:

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

   801 by (simp add: algebra_simps)

   802

   803 lemma less_add_iff2:

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

   805 by (simp add: algebra_simps)

   806

   807 lemma le_add_iff1:

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

   809 by (simp add: algebra_simps)

   810

   811 lemma le_add_iff2:

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

   813 by (simp add: algebra_simps)

   814

   815 lemma mult_left_mono_neg:

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

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

   818   apply simp_all

   819   done

   820

   821 lemma mult_right_mono_neg:

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

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

   824   apply simp_all

   825   done

   826

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

   828 using mult_right_mono_neg [of a 0 b] by simp

   829

   830 lemma split_mult_pos_le:

   831   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   832 by (auto simp add: mult_nonpos_nonpos)

   833

   834 end

   835

   836 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   837 begin

   838

   839 subclass ordered_ring ..

   840

   841 subclass ordered_ab_group_add_abs

   842 proof

   843   fix a b

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

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

   846 qed (auto simp add: abs_if)

   847

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

   849   using linear [of 0 a]

   850   by (auto simp add: mult_nonpos_nonpos)

   851

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

   853   by (simp add: not_less)

   854

   855 end

   856

   857 class linordered_ring_strict = ring + linordered_semiring_strict

   858   + ordered_ab_group_add + abs_if

   859 begin

   860

   861 subclass linordered_ring ..

   862

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

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

   865

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

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

   868

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

   870 using mult_strict_right_mono_neg [of a 0 b] by simp

   871

   872 subclass ring_no_zero_divisors

   873 proof

   874   fix a b

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

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

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

   878   proof (cases "a < 0")

   879     case True note A' = this

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

   881       case True with A'

   882       show ?thesis by (auto dest: mult_neg_neg)

   883     next

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

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

   886     qed

   887   next

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

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

   890       case True with A'

   891       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   892     next

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

   894       with A' show ?thesis by auto

   895     qed

   896   qed

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

   898 qed

   899

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

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

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

   903

   904 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"

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

   906

   907 lemma mult_less_0_iff:

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

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

   910   apply force

   911   done

   912

   913 lemma mult_le_0_iff:

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

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

   916   apply force

   917   done

   918

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

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

   921

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

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

   924

   925 lemma mult_less_cancel_right_disj:

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

   927   apply (cases "c = 0")

   928   apply (auto simp add: neq_iff mult_strict_right_mono

   929                       mult_strict_right_mono_neg)

   930   apply (auto simp add: not_less

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

   932                       not_le [symmetric, of a])

   933   apply (erule_tac [!] notE)

   934   apply (auto simp add: less_imp_le mult_right_mono

   935                       mult_right_mono_neg)

   936   done

   937

   938 lemma mult_less_cancel_left_disj:

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

   940   apply (cases "c = 0")

   941   apply (auto simp add: neq_iff mult_strict_left_mono

   942                       mult_strict_left_mono_neg)

   943   apply (auto simp add: not_less

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

   945                       not_le [symmetric, of a])

   946   apply (erule_tac [!] notE)

   947   apply (auto simp add: less_imp_le mult_left_mono

   948                       mult_left_mono_neg)

   949   done

   950

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

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

   953

   954 lemma mult_less_cancel_right:

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

   956   using mult_less_cancel_right_disj [of a c b] by auto

   957

   958 lemma mult_less_cancel_left:

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

   960   using mult_less_cancel_left_disj [of c a b] by auto

   961

   962 lemma mult_le_cancel_right:

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

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

   965

   966 lemma mult_le_cancel_left:

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

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

   969

   970 lemma mult_le_cancel_left_pos:

   971   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

   972 by (auto simp: mult_le_cancel_left)

   973

   974 lemma mult_le_cancel_left_neg:

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

   976 by (auto simp: mult_le_cancel_left)

   977

   978 lemma mult_less_cancel_left_pos:

   979   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

   980 by (auto simp: mult_less_cancel_left)

   981

   982 lemma mult_less_cancel_left_neg:

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

   984 by (auto simp: mult_less_cancel_left)

   985

   986 end

   987

   988 lemmas mult_sign_intros =

   989   mult_nonneg_nonneg mult_nonneg_nonpos

   990   mult_nonpos_nonneg mult_nonpos_nonpos

   991   mult_pos_pos mult_pos_neg

   992   mult_neg_pos mult_neg_neg

   993

   994 class ordered_comm_ring = comm_ring + ordered_comm_semiring

   995 begin

   996

   997 subclass ordered_ring ..

   998 subclass ordered_cancel_comm_semiring ..

   999

  1000 end

  1001

  1002 class linordered_semidom = semidom + linordered_comm_semiring_strict +

  1003   assumes zero_less_one [simp]: "0 < 1"

  1004 begin

  1005

  1006 lemma pos_add_strict:

  1007   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1008   using add_strict_mono [of 0 a b c] by simp

  1009

  1010 lemma zero_le_one [simp]: "0 \<le> 1"

  1011 by (rule zero_less_one [THEN less_imp_le])

  1012

  1013 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1014 by (simp add: not_le)

  1015

  1016 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1017 by (simp add: not_less)

  1018

  1019 lemma less_1_mult:

  1020   assumes "1 < m" and "1 < n"

  1021   shows "1 < m * n"

  1022   using assms mult_strict_mono [of 1 m 1 n]

  1023     by (simp add:  less_trans [OF zero_less_one])

  1024

  1025 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"

  1026   using mult_left_mono[of c 1 a] by simp

  1027

  1028 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"

  1029   using mult_mono[of a 1 b 1] by simp

  1030

  1031 end

  1032

  1033 class linordered_idom = comm_ring_1 +

  1034   linordered_comm_semiring_strict + ordered_ab_group_add +

  1035   abs_if + sgn_if

  1036 begin

  1037

  1038 subclass linordered_semiring_1_strict ..

  1039 subclass linordered_ring_strict ..

  1040 subclass ordered_comm_ring ..

  1041 subclass idom ..

  1042

  1043 subclass linordered_semidom

  1044 proof

  1045   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1046   thus "0 < 1" by (simp add: le_less)

  1047 qed

  1048

  1049 lemma linorder_neqE_linordered_idom:

  1050   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1051   using assms by (rule neqE)

  1052

  1053 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1054

  1055 lemma mult_le_cancel_right1:

  1056   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1057 by (insert mult_le_cancel_right [of 1 c b], simp)

  1058

  1059 lemma mult_le_cancel_right2:

  1060   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1061 by (insert mult_le_cancel_right [of a c 1], simp)

  1062

  1063 lemma mult_le_cancel_left1:

  1064   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1065 by (insert mult_le_cancel_left [of c 1 b], simp)

  1066

  1067 lemma mult_le_cancel_left2:

  1068   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1069 by (insert mult_le_cancel_left [of c a 1], simp)

  1070

  1071 lemma mult_less_cancel_right1:

  1072   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1073 by (insert mult_less_cancel_right [of 1 c b], simp)

  1074

  1075 lemma mult_less_cancel_right2:

  1076   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1077 by (insert mult_less_cancel_right [of a c 1], simp)

  1078

  1079 lemma mult_less_cancel_left1:

  1080   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1081 by (insert mult_less_cancel_left [of c 1 b], simp)

  1082

  1083 lemma mult_less_cancel_left2:

  1084   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1085 by (insert mult_less_cancel_left [of c a 1], simp)

  1086

  1087 lemma sgn_sgn [simp]:

  1088   "sgn (sgn a) = sgn a"

  1089 unfolding sgn_if by simp

  1090

  1091 lemma sgn_0_0:

  1092   "sgn a = 0 \<longleftrightarrow> a = 0"

  1093 unfolding sgn_if by simp

  1094

  1095 lemma sgn_1_pos:

  1096   "sgn a = 1 \<longleftrightarrow> a > 0"

  1097 unfolding sgn_if by simp

  1098

  1099 lemma sgn_1_neg:

  1100   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1101 unfolding sgn_if by auto

  1102

  1103 lemma sgn_pos [simp]:

  1104   "0 < a \<Longrightarrow> sgn a = 1"

  1105 unfolding sgn_1_pos .

  1106

  1107 lemma sgn_neg [simp]:

  1108   "a < 0 \<Longrightarrow> sgn a = - 1"

  1109 unfolding sgn_1_neg .

  1110

  1111 lemma sgn_times:

  1112   "sgn (a * b) = sgn a * sgn b"

  1113 by (auto simp add: sgn_if zero_less_mult_iff)

  1114

  1115 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1116 unfolding sgn_if abs_if by auto

  1117

  1118 lemma sgn_greater [simp]:

  1119   "0 < sgn a \<longleftrightarrow> 0 < a"

  1120   unfolding sgn_if by auto

  1121

  1122 lemma sgn_less [simp]:

  1123   "sgn a < 0 \<longleftrightarrow> a < 0"

  1124   unfolding sgn_if by auto

  1125

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

  1127   by (simp add: abs_if)

  1128

  1129 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1130   by (simp add: abs_if)

  1131

  1132 lemma dvd_if_abs_eq:

  1133   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1134 by(subst abs_dvd_iff[symmetric]) simp

  1135

  1136 text {* The following lemmas can be proven in more general structures, but

  1137 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1138 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1139

  1140 lemma equation_minus_iff_1 [simp, no_atp]:

  1141   "1 = - a \<longleftrightarrow> a = - 1"

  1142   by (fact equation_minus_iff)

  1143

  1144 lemma minus_equation_iff_1 [simp, no_atp]:

  1145   "- a = 1 \<longleftrightarrow> a = - 1"

  1146   by (subst minus_equation_iff, auto)

  1147

  1148 lemma le_minus_iff_1 [simp, no_atp]:

  1149   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1150   by (fact le_minus_iff)

  1151

  1152 lemma minus_le_iff_1 [simp, no_atp]:

  1153   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1154   by (fact minus_le_iff)

  1155

  1156 lemma less_minus_iff_1 [simp, no_atp]:

  1157   "1 < - b \<longleftrightarrow> b < - 1"

  1158   by (fact less_minus_iff)

  1159

  1160 lemma minus_less_iff_1 [simp, no_atp]:

  1161   "- a < 1 \<longleftrightarrow> - 1 < a"

  1162   by (fact minus_less_iff)

  1163

  1164 end

  1165

  1166 text {* Simprules for comparisons where common factors can be cancelled. *}

  1167

  1168 lemmas mult_compare_simps =

  1169     mult_le_cancel_right mult_le_cancel_left

  1170     mult_le_cancel_right1 mult_le_cancel_right2

  1171     mult_le_cancel_left1 mult_le_cancel_left2

  1172     mult_less_cancel_right mult_less_cancel_left

  1173     mult_less_cancel_right1 mult_less_cancel_right2

  1174     mult_less_cancel_left1 mult_less_cancel_left2

  1175     mult_cancel_right mult_cancel_left

  1176     mult_cancel_right1 mult_cancel_right2

  1177     mult_cancel_left1 mult_cancel_left2

  1178

  1179 text {* Reasoning about inequalities with division *}

  1180

  1181 context linordered_semidom

  1182 begin

  1183

  1184 lemma less_add_one: "a < a + 1"

  1185 proof -

  1186   have "a + 0 < a + 1"

  1187     by (blast intro: zero_less_one add_strict_left_mono)

  1188   thus ?thesis by simp

  1189 qed

  1190

  1191 lemma zero_less_two: "0 < 1 + 1"

  1192 by (blast intro: less_trans zero_less_one less_add_one)

  1193

  1194 end

  1195

  1196 context linordered_idom

  1197 begin

  1198

  1199 lemma mult_right_le_one_le:

  1200   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1201   by (rule mult_left_le)

  1202

  1203 lemma mult_left_le_one_le:

  1204   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1205   by (auto simp add: mult_le_cancel_right2)

  1206

  1207 end

  1208

  1209 text {* Absolute Value *}

  1210

  1211 context linordered_idom

  1212 begin

  1213

  1214 lemma mult_sgn_abs:

  1215   "sgn x * \<bar>x\<bar> = x"

  1216   unfolding abs_if sgn_if by auto

  1217

  1218 lemma abs_one [simp]:

  1219   "\<bar>1\<bar> = 1"

  1220   by (simp add: abs_if)

  1221

  1222 end

  1223

  1224 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1225   assumes abs_eq_mult:

  1226     "(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>"

  1227

  1228 context linordered_idom

  1229 begin

  1230

  1231 subclass ordered_ring_abs proof

  1232 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1233

  1234 lemma abs_mult:

  1235   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1236   by (rule abs_eq_mult) auto

  1237

  1238 lemma abs_mult_self:

  1239   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1240   by (simp add: abs_if)

  1241

  1242 lemma abs_mult_less:

  1243   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1244 proof -

  1245   assume ac: "\<bar>a\<bar> < c"

  1246   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1247   assume "\<bar>b\<bar> < d"

  1248   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1249 qed

  1250

  1251 lemma abs_less_iff:

  1252   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1253   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1254

  1255 lemma abs_mult_pos:

  1256   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1257   by (simp add: abs_mult)

  1258

  1259 lemma abs_diff_less_iff:

  1260   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1261   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1262

  1263 lemma abs_diff_le_iff:

  1264    "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"

  1265   by (auto simp add: diff_le_eq ac_simps abs_le_iff)

  1266

  1267 end

  1268

  1269 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib

  1270

  1271 code_identifier

  1272   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1273

  1274 end