src/HOL/Rings.thy
 author haftmann Sun Oct 12 16:31:28 2014 +0200 (2014-10-12) changeset 58647 fce800afeec7 parent 58198 099ca49b5231 child 58649 a62065b5e1e2 permissions -rw-r--r--
     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 header {* 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, field_simps]: "(a + b) * c = a * c + b * c"

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

    19 begin

    20

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

    22 lemma combine_common_factor:

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

    24 by (simp add: distrib_right ac_simps)

    25

    26 end

    27

    28 class mult_zero = times + zero +

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

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

    31 begin

    32

    33 lemma mult_not_zero:

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

    35   by auto

    36

    37 end

    38

    39 class semiring_0 = semiring + comm_monoid_add + mult_zero

    40

    41 class semiring_0_cancel = semiring + cancel_comm_monoid_add

    42 begin

    43

    44 subclass semiring_0

    45 proof

    46   fix a :: 'a

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

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

    49 next

    50   fix a :: 'a

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

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

    53 qed

    54

    55 end

    56

    57 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +

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

    59 begin

    60

    61 subclass semiring

    62 proof

    63   fix a b c :: 'a

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

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

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

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

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

    69 qed

    70

    71 end

    72

    73 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero

    74 begin

    75

    76 subclass semiring_0 ..

    77

    78 end

    79

    80 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add

    81 begin

    82

    83 subclass semiring_0_cancel ..

    84

    85 subclass comm_semiring_0 ..

    86

    87 end

    88

    89 class zero_neq_one = zero + one +

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

    91 begin

    92

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

    94 by (rule not_sym) (rule zero_neq_one)

    95

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

    97 where

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

    99

   100 lemma of_bool_eq [simp, code]:

   101   "of_bool False = 0"

   102   "of_bool True = 1"

   103   by (simp_all add: of_bool_def)

   104

   105 lemma of_bool_eq_iff:

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

   107   by (simp add: of_bool_def)

   108

   109 lemma split_of_bool [split]:

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

   111   by (cases p) simp_all

   112

   113 lemma split_of_bool_asm:

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

   115   by (cases p) simp_all

   116

   117 end

   118

   119 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

   120

   121 text {* Abstract divisibility *}

   122

   123 class dvd = times

   124 begin

   125

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

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

   128

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

   130   unfolding dvd_def ..

   131

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

   133   unfolding dvd_def by blast

   134

   135 end

   136

   137 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd

   138   (*previously almost_semiring*)

   139 begin

   140

   141 subclass semiring_1 ..

   142

   143 lemma dvd_refl[simp]: "a dvd a"

   144 proof

   145   show "a = a * 1" by simp

   146 qed

   147

   148 lemma dvd_trans:

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

   150   shows "a dvd c"

   151 proof -

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

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

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

   155   then show ?thesis ..

   156 qed

   157

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

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

   160

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

   162 proof

   163   show "0 = a * 0" by simp

   164 qed

   165

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

   167 by (auto intro!: dvdI)

   168

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

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

   171

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

   173   apply (subst mult.commute)

   174   apply (erule dvd_mult)

   175   done

   176

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

   178 by (rule dvd_mult) (rule dvd_refl)

   179

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

   181 by (rule dvd_mult2) (rule dvd_refl)

   182

   183 lemma mult_dvd_mono:

   184   assumes "a dvd b"

   185     and "c dvd d"

   186   shows "a * c dvd b * d"

   187 proof -

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

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

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

   191   then show ?thesis ..

   192 qed

   193

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

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

   196

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

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

   199

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

   201 by simp

   202

   203 lemma dvd_add[simp]:

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

   205 proof -

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

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

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

   209   then show ?thesis ..

   210 qed

   211

   212 end

   213

   214 class semiring_dvd = comm_semiring_1 +

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

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

   217 begin

   218

   219 lemma dvd_add_times_triv_right_iff [simp]:

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

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

   222

   223 lemma dvd_add_triv_left_iff [simp]:

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

   225   using dvd_add_times_triv_left_iff [of a 1 b] by simp

   226

   227 lemma dvd_add_triv_right_iff [simp]:

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

   229   using dvd_add_times_triv_right_iff [of a b 1] by simp

   230

   231 lemma dvd_add_eq_right:

   232   assumes "a dvd b"

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

   234   using assms by (auto dest: dvd_addD)

   235

   236 lemma dvd_add_eq_left:

   237   assumes "a dvd c"

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

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

   240

   241 end

   242

   243 class no_zero_divisors = zero + times +

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

   245 begin

   246

   247 lemma divisors_zero:

   248   assumes "a * b = 0"

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

   250 proof (rule classical)

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

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

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

   254   with assms show ?thesis by simp

   255 qed

   256

   257 end

   258

   259 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   260   + zero_neq_one + monoid_mult

   261 begin

   262

   263 subclass semiring_0_cancel ..

   264

   265 subclass semiring_1 ..

   266

   267 end

   268

   269 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   270   + zero_neq_one + comm_monoid_mult

   271 begin

   272

   273 subclass semiring_1_cancel ..

   274 subclass comm_semiring_0_cancel ..

   275 subclass comm_semiring_1 ..

   276

   277 end

   278

   279 class ring = semiring + ab_group_add

   280 begin

   281

   282 subclass semiring_0_cancel ..

   283

   284 text {* Distribution rules *}

   285

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

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

   288

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

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

   291

   292 text{*Extract signs from products*}

   293 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]

   294 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

   295

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

   297 by simp

   298

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

   300 by simp

   301

   302 lemma right_diff_distrib [algebra_simps, field_simps]:

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

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

   305

   306 lemma left_diff_distrib [algebra_simps, field_simps]:

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

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

   309

   310 lemmas ring_distribs =

   311   distrib_left distrib_right left_diff_distrib right_diff_distrib

   312

   313 lemma eq_add_iff1:

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

   315 by (simp add: algebra_simps)

   316

   317 lemma eq_add_iff2:

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

   319 by (simp add: algebra_simps)

   320

   321 end

   322

   323 lemmas ring_distribs =

   324   distrib_left distrib_right left_diff_distrib right_diff_distrib

   325

   326 class comm_ring = comm_semiring + ab_group_add

   327 begin

   328

   329 subclass ring ..

   330 subclass comm_semiring_0_cancel ..

   331

   332 lemma square_diff_square_factored:

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

   334   by (simp add: algebra_simps)

   335

   336 end

   337

   338 class ring_1 = ring + zero_neq_one + monoid_mult

   339 begin

   340

   341 subclass semiring_1_cancel ..

   342

   343 lemma square_diff_one_factored:

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

   345   by (simp add: algebra_simps)

   346

   347 end

   348

   349 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   350   (*previously ring*)

   351 begin

   352

   353 subclass ring_1 ..

   354 subclass comm_semiring_1_cancel ..

   355

   356 subclass semiring_dvd

   357 proof

   358   fix a b c

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

   360   proof

   361     assume ?Q then show ?P by simp

   362   next

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

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

   365     then show ?Q ..

   366   qed

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

   368   show "a dvd c"

   369   proof -

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

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

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

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

   374     then show "a dvd c" ..

   375   qed

   376 qed

   377

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

   379 proof

   380   assume "x dvd - y"

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

   382   then show "x dvd y" by simp

   383 next

   384   assume "x dvd y"

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

   386   then show "x dvd - y" by simp

   387 qed

   388

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

   390 proof

   391   assume "- x dvd y"

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

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

   394   then show "x dvd y" ..

   395 next

   396   assume "x dvd y"

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

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

   399   then show "- x dvd y" ..

   400 qed

   401

   402 lemma dvd_diff [simp]:

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

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

   405

   406 end

   407

   408 class ring_no_zero_divisors = ring + no_zero_divisors

   409 begin

   410

   411 lemma mult_eq_0_iff [simp]:

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

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

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

   415     then show ?thesis using no_zero_divisors by simp

   416 next

   417   case True then show ?thesis by auto

   418 qed

   419

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

   421 lemma mult_cancel_right [simp]:

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

   423 proof -

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

   425     by (simp add: algebra_simps)

   426   thus ?thesis by (simp add: disj_commute)

   427 qed

   428

   429 lemma mult_cancel_left [simp]:

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

   431 proof -

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

   433     by (simp add: algebra_simps)

   434   thus ?thesis by simp

   435 qed

   436

   437 lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> (c*a=c*b) = (a=b)"

   438 by simp

   439

   440 lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> (a*c=b*c) = (a=b)"

   441 by simp

   442

   443 end

   444

   445 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   446 begin

   447

   448 lemma square_eq_1_iff:

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

   450 proof -

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

   452     by (simp add: algebra_simps)

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

   454     by simp

   455   thus ?thesis

   456     by (simp add: eq_neg_iff_add_eq_0)

   457 qed

   458

   459 lemma mult_cancel_right1 [simp]:

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

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

   462

   463 lemma mult_cancel_right2 [simp]:

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

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

   466

   467 lemma mult_cancel_left1 [simp]:

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

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

   470

   471 lemma mult_cancel_left2 [simp]:

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

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

   474

   475 end

   476

   477 class idom = comm_ring_1 + no_zero_divisors

   478 begin

   479

   480 subclass ring_1_no_zero_divisors ..

   481

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

   483 proof

   484   assume "a * a = b * b"

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

   486     by (simp add: algebra_simps)

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

   488     by (simp add: eq_neg_iff_add_eq_0)

   489 next

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

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

   492 qed

   493

   494 lemma dvd_mult_cancel_right [simp]:

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

   496 proof -

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

   498     unfolding dvd_def by (simp add: ac_simps)

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

   500     unfolding dvd_def by simp

   501   finally show ?thesis .

   502 qed

   503

   504 lemma dvd_mult_cancel_left [simp]:

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

   506 proof -

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

   508     unfolding dvd_def by (simp add: ac_simps)

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

   510     unfolding dvd_def by simp

   511   finally show ?thesis .

   512 qed

   513

   514 end

   515

   516 text {*

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

   518   \begin{itemize}

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

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

   521   \end{itemize}

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

   523   \begin{itemize}

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

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

   526   \end{itemize}

   527 *}

   528

   529 class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +

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

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

   532 begin

   533

   534 lemma mult_mono:

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

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

   537 apply (erule mult_left_mono, assumption)

   538 done

   539

   540 lemma mult_mono':

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

   542 apply (rule mult_mono)

   543 apply (fast intro: order_trans)+

   544 done

   545

   546 end

   547

   548 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add

   549 begin

   550

   551 subclass semiring_0_cancel ..

   552

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

   554 using mult_left_mono [of 0 b a] by simp

   555

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

   557 using mult_left_mono [of b 0 a] by simp

   558

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

   560 using mult_right_mono [of a 0 b] by simp

   561

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

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

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

   565

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

   567 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   568

   569 end

   570

   571 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add

   572 begin

   573

   574 subclass ordered_cancel_semiring ..

   575

   576 subclass ordered_comm_monoid_add ..

   577

   578 lemma mult_left_less_imp_less:

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

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

   581

   582 lemma mult_right_less_imp_less:

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

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

   585

   586 end

   587

   588 class linordered_semiring_1 = linordered_semiring + semiring_1

   589 begin

   590

   591 lemma convex_bound_le:

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

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

   594 proof-

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

   596     by (simp add: add_mono mult_left_mono)

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

   598 qed

   599

   600 end

   601

   602 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   605 begin

   606

   607 subclass semiring_0_cancel ..

   608

   609 subclass linordered_semiring

   610 proof

   611   fix a b c :: 'a

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

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

   614     unfolding le_less

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

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

   617     unfolding le_less

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

   619 qed

   620

   621 lemma mult_left_le_imp_le:

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

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

   624

   625 lemma mult_right_le_imp_le:

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

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

   628

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

   630 using mult_strict_left_mono [of 0 b a] by simp

   631

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

   633 using mult_strict_left_mono [of b 0 a] by simp

   634

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

   636 using mult_strict_right_mono [of a 0 b] by simp

   637

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

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

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

   641

   642 lemma zero_less_mult_pos:

   643   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   645  apply (auto simp add: le_less not_less)

   646 apply (drule_tac mult_pos_neg [of a b])

   647  apply (auto dest: less_not_sym)

   648 done

   649

   650 lemma zero_less_mult_pos2:

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

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

   653  apply (auto simp add: le_less not_less)

   654 apply (drule_tac mult_pos_neg2 [of a b])

   655  apply (auto dest: less_not_sym)

   656 done

   657

   658 text{*Strict monotonicity in both arguments*}

   659 lemma mult_strict_mono:

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

   661   shows "a * c < b * d"

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

   663   apply (simp)

   664   apply (erule mult_strict_right_mono [THEN less_trans])

   665   apply (force simp add: le_less)

   666   apply (erule mult_strict_left_mono, assumption)

   667   done

   668

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

   670 lemma mult_strict_mono':

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

   672   shows "a * c < b * d"

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

   674

   675 lemma mult_less_le_imp_less:

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

   677   shows "a * c < b * d"

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

   679   apply (erule less_le_trans)

   680   apply (erule mult_left_mono)

   681   apply simp

   682   apply (erule mult_strict_right_mono)

   683   apply assumption

   684   done

   685

   686 lemma mult_le_less_imp_less:

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

   688   shows "a * c < b * d"

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

   690   apply (erule le_less_trans)

   691   apply (erule mult_strict_left_mono)

   692   apply simp

   693   apply (erule mult_right_mono)

   694   apply simp

   695   done

   696

   697 lemma mult_less_imp_less_left:

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

   699   shows "a < b"

   700 proof (rule ccontr)

   701   assume "\<not>  a < b"

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

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

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

   705 qed

   706

   707 lemma mult_less_imp_less_right:

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

   709   shows "a < b"

   710 proof (rule ccontr)

   711   assume "\<not> a < b"

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

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

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

   715 qed

   716

   717 end

   718

   719 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   720 begin

   721

   722 subclass linordered_semiring_1 ..

   723

   724 lemma convex_bound_lt:

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

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

   727 proof -

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

   729     by (cases "u = 0")

   730        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)

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

   732 qed

   733

   734 end

   735

   736 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +

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

   738 begin

   739

   740 subclass ordered_semiring

   741 proof

   742   fix a b c :: 'a

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

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

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

   746 qed

   747

   748 end

   749

   750 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add

   751 begin

   752

   753 subclass comm_semiring_0_cancel ..

   754 subclass ordered_comm_semiring ..

   755 subclass ordered_cancel_semiring ..

   756

   757 end

   758

   759 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   761 begin

   762

   763 subclass linordered_semiring_strict

   764 proof

   765   fix a b c :: 'a

   766   assume "a < b" "0 < c"

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

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

   769 qed

   770

   771 subclass ordered_cancel_comm_semiring

   772 proof

   773   fix a b c :: 'a

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

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

   776     unfolding le_less

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

   778 qed

   779

   780 end

   781

   782 class ordered_ring = ring + ordered_cancel_semiring

   783 begin

   784

   785 subclass ordered_ab_group_add ..

   786

   787 lemma less_add_iff1:

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

   789 by (simp add: algebra_simps)

   790

   791 lemma less_add_iff2:

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

   793 by (simp add: algebra_simps)

   794

   795 lemma le_add_iff1:

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

   797 by (simp add: algebra_simps)

   798

   799 lemma le_add_iff2:

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

   801 by (simp add: algebra_simps)

   802

   803 lemma mult_left_mono_neg:

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

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

   806   apply simp_all

   807   done

   808

   809 lemma mult_right_mono_neg:

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

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

   812   apply simp_all

   813   done

   814

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

   816 using mult_right_mono_neg [of a 0 b] by simp

   817

   818 lemma split_mult_pos_le:

   819   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   820 by (auto simp add: mult_nonpos_nonpos)

   821

   822 end

   823

   824 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   825 begin

   826

   827 subclass ordered_ring ..

   828

   829 subclass ordered_ab_group_add_abs

   830 proof

   831   fix a b

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

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

   834 qed (auto simp add: abs_if)

   835

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

   837   using linear [of 0 a]

   838   by (auto simp add: mult_nonpos_nonpos)

   839

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

   841   by (simp add: not_less)

   842

   843 end

   844

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

   846    Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.

   847  *)

   848 class linordered_ring_strict = ring + linordered_semiring_strict

   849   + ordered_ab_group_add + abs_if

   850 begin

   851

   852 subclass linordered_ring ..

   853

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

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

   856

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

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

   859

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

   861 using mult_strict_right_mono_neg [of a 0 b] by simp

   862

   863 subclass ring_no_zero_divisors

   864 proof

   865   fix a b

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

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

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

   869   proof (cases "a < 0")

   870     case True note A' = this

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

   872       case True with A'

   873       show ?thesis by (auto dest: mult_neg_neg)

   874     next

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

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

   877     qed

   878   next

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

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

   881       case True with A'

   882       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   883     next

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

   885       with A' show ?thesis by auto

   886     qed

   887   qed

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

   889 qed

   890

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

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

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

   894

   895 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"

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

   897

   898 lemma mult_less_0_iff:

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

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

   901   apply force

   902   done

   903

   904 lemma mult_le_0_iff:

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

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

   907   apply force

   908   done

   909

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

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

   912

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

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

   915

   916 lemma mult_less_cancel_right_disj:

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

   918   apply (cases "c = 0")

   919   apply (auto simp add: neq_iff mult_strict_right_mono

   920                       mult_strict_right_mono_neg)

   921   apply (auto simp add: not_less

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

   923                       not_le [symmetric, of a])

   924   apply (erule_tac [!] notE)

   925   apply (auto simp add: less_imp_le mult_right_mono

   926                       mult_right_mono_neg)

   927   done

   928

   929 lemma mult_less_cancel_left_disj:

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

   931   apply (cases "c = 0")

   932   apply (auto simp add: neq_iff mult_strict_left_mono

   933                       mult_strict_left_mono_neg)

   934   apply (auto simp add: not_less

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

   936                       not_le [symmetric, of a])

   937   apply (erule_tac [!] notE)

   938   apply (auto simp add: less_imp_le mult_left_mono

   939                       mult_left_mono_neg)

   940   done

   941

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

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

   944

   945 lemma mult_less_cancel_right:

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

   947   using mult_less_cancel_right_disj [of a c b] by auto

   948

   949 lemma mult_less_cancel_left:

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

   951   using mult_less_cancel_left_disj [of c a b] by auto

   952

   953 lemma mult_le_cancel_right:

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

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

   956

   957 lemma mult_le_cancel_left:

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

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

   960

   961 lemma mult_le_cancel_left_pos:

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

   963 by (auto simp: mult_le_cancel_left)

   964

   965 lemma mult_le_cancel_left_neg:

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

   967 by (auto simp: mult_le_cancel_left)

   968

   969 lemma mult_less_cancel_left_pos:

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

   971 by (auto simp: mult_less_cancel_left)

   972

   973 lemma mult_less_cancel_left_neg:

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

   975 by (auto simp: mult_less_cancel_left)

   976

   977 end

   978

   979 lemmas mult_sign_intros =

   980   mult_nonneg_nonneg mult_nonneg_nonpos

   981   mult_nonpos_nonneg mult_nonpos_nonpos

   982   mult_pos_pos mult_pos_neg

   983   mult_neg_pos mult_neg_neg

   984

   985 class ordered_comm_ring = comm_ring + ordered_comm_semiring

   986 begin

   987

   988 subclass ordered_ring ..

   989 subclass ordered_cancel_comm_semiring ..

   990

   991 end

   992

   993 class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +

   994   (*previously linordered_semiring*)

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

   996 begin

   997

   998 lemma pos_add_strict:

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

  1000   using add_strict_mono [of 0 a b c] by simp

  1001

  1002 lemma zero_le_one [simp]: "0 \<le> 1"

  1003 by (rule zero_less_one [THEN less_imp_le])

  1004

  1005 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1006 by (simp add: not_le)

  1007

  1008 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1009 by (simp add: not_less)

  1010

  1011 lemma less_1_mult:

  1012   assumes "1 < m" and "1 < n"

  1013   shows "1 < m * n"

  1014   using assms mult_strict_mono [of 1 m 1 n]

  1015     by (simp add:  less_trans [OF zero_less_one])

  1016

  1017 end

  1018

  1019 class linordered_idom = comm_ring_1 +

  1020   linordered_comm_semiring_strict + ordered_ab_group_add +

  1021   abs_if + sgn_if

  1022   (*previously linordered_ring*)

  1023 begin

  1024

  1025 subclass linordered_semiring_1_strict ..

  1026 subclass linordered_ring_strict ..

  1027 subclass ordered_comm_ring ..

  1028 subclass idom ..

  1029

  1030 subclass linordered_semidom

  1031 proof

  1032   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1033   thus "0 < 1" by (simp add: le_less)

  1034 qed

  1035

  1036 lemma linorder_neqE_linordered_idom:

  1037   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1038   using assms by (rule neqE)

  1039

  1040 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1041

  1042 lemma mult_le_cancel_right1:

  1043   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1044 by (insert mult_le_cancel_right [of 1 c b], simp)

  1045

  1046 lemma mult_le_cancel_right2:

  1047   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1048 by (insert mult_le_cancel_right [of a c 1], simp)

  1049

  1050 lemma mult_le_cancel_left1:

  1051   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1052 by (insert mult_le_cancel_left [of c 1 b], simp)

  1053

  1054 lemma mult_le_cancel_left2:

  1055   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1056 by (insert mult_le_cancel_left [of c a 1], simp)

  1057

  1058 lemma mult_less_cancel_right1:

  1059   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1060 by (insert mult_less_cancel_right [of 1 c b], simp)

  1061

  1062 lemma mult_less_cancel_right2:

  1063   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1064 by (insert mult_less_cancel_right [of a c 1], simp)

  1065

  1066 lemma mult_less_cancel_left1:

  1067   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1068 by (insert mult_less_cancel_left [of c 1 b], simp)

  1069

  1070 lemma mult_less_cancel_left2:

  1071   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1072 by (insert mult_less_cancel_left [of c a 1], simp)

  1073

  1074 lemma sgn_sgn [simp]:

  1075   "sgn (sgn a) = sgn a"

  1076 unfolding sgn_if by simp

  1077

  1078 lemma sgn_0_0:

  1079   "sgn a = 0 \<longleftrightarrow> a = 0"

  1080 unfolding sgn_if by simp

  1081

  1082 lemma sgn_1_pos:

  1083   "sgn a = 1 \<longleftrightarrow> a > 0"

  1084 unfolding sgn_if by simp

  1085

  1086 lemma sgn_1_neg:

  1087   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1088 unfolding sgn_if by auto

  1089

  1090 lemma sgn_pos [simp]:

  1091   "0 < a \<Longrightarrow> sgn a = 1"

  1092 unfolding sgn_1_pos .

  1093

  1094 lemma sgn_neg [simp]:

  1095   "a < 0 \<Longrightarrow> sgn a = - 1"

  1096 unfolding sgn_1_neg .

  1097

  1098 lemma sgn_times:

  1099   "sgn (a * b) = sgn a * sgn b"

  1100 by (auto simp add: sgn_if zero_less_mult_iff)

  1101

  1102 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1103 unfolding sgn_if abs_if by auto

  1104

  1105 lemma sgn_greater [simp]:

  1106   "0 < sgn a \<longleftrightarrow> 0 < a"

  1107   unfolding sgn_if by auto

  1108

  1109 lemma sgn_less [simp]:

  1110   "sgn a < 0 \<longleftrightarrow> a < 0"

  1111   unfolding sgn_if by auto

  1112

  1113 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1114   by (simp add: abs_if)

  1115

  1116 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1117   by (simp add: abs_if)

  1118

  1119 lemma dvd_if_abs_eq:

  1120   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1121 by(subst abs_dvd_iff[symmetric]) simp

  1122

  1123 text {* The following lemmas can be proven in more general structures, but

  1124 are dangerous as simp rules in absence of @{thm neg_equal_zero},

  1125 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

  1126

  1127 lemma equation_minus_iff_1 [simp, no_atp]:

  1128   "1 = - a \<longleftrightarrow> a = - 1"

  1129   by (fact equation_minus_iff)

  1130

  1131 lemma minus_equation_iff_1 [simp, no_atp]:

  1132   "- a = 1 \<longleftrightarrow> a = - 1"

  1133   by (subst minus_equation_iff, auto)

  1134

  1135 lemma le_minus_iff_1 [simp, no_atp]:

  1136   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"

  1137   by (fact le_minus_iff)

  1138

  1139 lemma minus_le_iff_1 [simp, no_atp]:

  1140   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"

  1141   by (fact minus_le_iff)

  1142

  1143 lemma less_minus_iff_1 [simp, no_atp]:

  1144   "1 < - b \<longleftrightarrow> b < - 1"

  1145   by (fact less_minus_iff)

  1146

  1147 lemma minus_less_iff_1 [simp, no_atp]:

  1148   "- a < 1 \<longleftrightarrow> - 1 < a"

  1149   by (fact minus_less_iff)

  1150

  1151 end

  1152

  1153 text {* Simprules for comparisons where common factors can be cancelled. *}

  1154

  1155 lemmas mult_compare_simps =

  1156     mult_le_cancel_right mult_le_cancel_left

  1157     mult_le_cancel_right1 mult_le_cancel_right2

  1158     mult_le_cancel_left1 mult_le_cancel_left2

  1159     mult_less_cancel_right mult_less_cancel_left

  1160     mult_less_cancel_right1 mult_less_cancel_right2

  1161     mult_less_cancel_left1 mult_less_cancel_left2

  1162     mult_cancel_right mult_cancel_left

  1163     mult_cancel_right1 mult_cancel_right2

  1164     mult_cancel_left1 mult_cancel_left2

  1165

  1166 text {* Reasoning about inequalities with division *}

  1167

  1168 context linordered_semidom

  1169 begin

  1170

  1171 lemma less_add_one: "a < a + 1"

  1172 proof -

  1173   have "a + 0 < a + 1"

  1174     by (blast intro: zero_less_one add_strict_left_mono)

  1175   thus ?thesis by simp

  1176 qed

  1177

  1178 lemma zero_less_two: "0 < 1 + 1"

  1179 by (blast intro: less_trans zero_less_one less_add_one)

  1180

  1181 end

  1182

  1183 context linordered_idom

  1184 begin

  1185

  1186 lemma mult_right_le_one_le:

  1187   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1188   by (auto simp add: mult_le_cancel_left2)

  1189

  1190 lemma mult_left_le_one_le:

  1191   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1192   by (auto simp add: mult_le_cancel_right2)

  1193

  1194 end

  1195

  1196 text {* Absolute Value *}

  1197

  1198 context linordered_idom

  1199 begin

  1200

  1201 lemma mult_sgn_abs:

  1202   "sgn x * \<bar>x\<bar> = x"

  1203   unfolding abs_if sgn_if by auto

  1204

  1205 lemma abs_one [simp]:

  1206   "\<bar>1\<bar> = 1"

  1207   by (simp add: abs_if)

  1208

  1209 end

  1210

  1211 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1212   assumes abs_eq_mult:

  1213     "(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>"

  1214

  1215 context linordered_idom

  1216 begin

  1217

  1218 subclass ordered_ring_abs proof

  1219 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1220

  1221 lemma abs_mult:

  1222   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1223   by (rule abs_eq_mult) auto

  1224

  1225 lemma abs_mult_self:

  1226   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1227   by (simp add: abs_if)

  1228

  1229 lemma abs_mult_less:

  1230   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1231 proof -

  1232   assume ac: "\<bar>a\<bar> < c"

  1233   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1234   assume "\<bar>b\<bar> < d"

  1235   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1236 qed

  1237

  1238 lemma abs_less_iff:

  1239   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1240   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1241

  1242 lemma abs_mult_pos:

  1243   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1244   by (simp add: abs_mult)

  1245

  1246 lemma abs_diff_less_iff:

  1247   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"

  1248   by (auto simp add: diff_less_eq ac_simps abs_less_iff)

  1249

  1250 end

  1251

  1252 code_identifier

  1253   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1254

  1255 end

  1256