src/HOL/Rings.thy
 author haftmann Wed Feb 10 14:12:04 2010 +0100 (2010-02-10) changeset 35092 cfe605c54e50 parent 35083 3246e66b0874 child 35097 4554bb2abfa3 permissions -rw-r--r--
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
     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 text {*

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

    18   \begin{itemize}

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

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

    21   \end{itemize}

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

    23   \begin{itemize}

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

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

    26   \end{itemize}

    27 *}

    28

    29 class semiring = ab_semigroup_add + semigroup_mult +

    30   assumes left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c"

    31   assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c"

    32 begin

    33

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

    35 lemma combine_common_factor:

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

    37 by (simp add: left_distrib add_ac)

    38

    39 end

    40

    41 class mult_zero = times + zero +

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

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

    44

    45 class semiring_0 = semiring + comm_monoid_add + mult_zero

    46

    47 class semiring_0_cancel = semiring + cancel_comm_monoid_add

    48 begin

    49

    50 subclass semiring_0

    51 proof

    52   fix a :: 'a

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

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

    55 next

    56   fix a :: 'a

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

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

    59 qed

    60

    61 end

    62

    63 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +

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

    65 begin

    66

    67 subclass semiring

    68 proof

    69   fix a b c :: 'a

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

    71   have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)

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

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

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

    75 qed

    76

    77 end

    78

    79 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero

    80 begin

    81

    82 subclass semiring_0 ..

    83

    84 end

    85

    86 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add

    87 begin

    88

    89 subclass semiring_0_cancel ..

    90

    91 subclass comm_semiring_0 ..

    92

    93 end

    94

    95 class zero_neq_one = zero + one +

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

    97 begin

    98

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

   100 by (rule not_sym) (rule zero_neq_one)

   101

   102 end

   103

   104 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

   105

   106 text {* Abstract divisibility *}

   107

   108 class dvd = times

   109 begin

   110

   111 definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where

   112   [code del]: "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"

   113

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

   115   unfolding dvd_def ..

   116

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

   118   unfolding dvd_def by blast

   119

   120 end

   121

   122 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd

   123   (*previously almost_semiring*)

   124 begin

   125

   126 subclass semiring_1 ..

   127

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

   129 proof

   130   show "a = a * 1" by simp

   131 qed

   132

   133 lemma dvd_trans:

   134   assumes "a dvd b" and "b dvd c"

   135   shows "a dvd c"

   136 proof -

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

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

   139   ultimately have "c = a * (v * w)" by (simp add: mult_assoc)

   140   then show ?thesis ..

   141 qed

   142

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

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

   145

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

   147 proof

   148   show "0 = a * 0" by simp

   149 qed

   150

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

   152 by (auto intro!: dvdI)

   153

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

   155 by (auto intro!: mult_left_commute dvdI elim!: dvdE)

   156

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

   158   apply (subst mult_commute)

   159   apply (erule dvd_mult)

   160   done

   161

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

   163 by (rule dvd_mult) (rule dvd_refl)

   164

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

   166 by (rule dvd_mult2) (rule dvd_refl)

   167

   168 lemma mult_dvd_mono:

   169   assumes "a dvd b"

   170     and "c dvd d"

   171   shows "a * c dvd b * d"

   172 proof -

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

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

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

   176   then show ?thesis ..

   177 qed

   178

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

   180 by (simp add: dvd_def mult_assoc, blast)

   181

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

   183   unfolding mult_ac [of a] by (rule dvd_mult_left)

   184

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

   186 by simp

   187

   188 lemma dvd_add[simp]:

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

   190 proof -

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

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

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

   194   then show ?thesis ..

   195 qed

   196

   197 end

   198

   199

   200 class no_zero_divisors = zero + times +

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

   202

   203 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   204   + zero_neq_one + monoid_mult

   205 begin

   206

   207 subclass semiring_0_cancel ..

   208

   209 subclass semiring_1 ..

   210

   211 end

   212

   213 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   214   + zero_neq_one + comm_monoid_mult

   215 begin

   216

   217 subclass semiring_1_cancel ..

   218 subclass comm_semiring_0_cancel ..

   219 subclass comm_semiring_1 ..

   220

   221 end

   222

   223 class ring = semiring + ab_group_add

   224 begin

   225

   226 subclass semiring_0_cancel ..

   227

   228 text {* Distribution rules *}

   229

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

   231 by (rule minus_unique) (simp add: left_distrib [symmetric])

   232

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

   234 by (rule minus_unique) (simp add: right_distrib [symmetric])

   235

   236 text{*Extract signs from products*}

   237 lemmas mult_minus_left [simp, noatp] = minus_mult_left [symmetric]

   238 lemmas mult_minus_right [simp,noatp] = minus_mult_right [symmetric]

   239

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

   241 by simp

   242

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

   244 by simp

   245

   246 lemma right_diff_distrib[algebra_simps]: "a * (b - c) = a * b - a * c"

   247 by (simp add: right_distrib diff_minus)

   248

   249 lemma left_diff_distrib[algebra_simps]: "(a - b) * c = a * c - b * c"

   250 by (simp add: left_distrib diff_minus)

   251

   252 lemmas ring_distribs[noatp] =

   253   right_distrib left_distrib left_diff_distrib right_diff_distrib

   254

   255 text{*Legacy - use @{text algebra_simps} *}

   256 lemmas ring_simps[noatp] = algebra_simps

   257

   258 lemma eq_add_iff1:

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

   260 by (simp add: algebra_simps)

   261

   262 lemma eq_add_iff2:

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

   264 by (simp add: algebra_simps)

   265

   266 end

   267

   268 lemmas ring_distribs[noatp] =

   269   right_distrib left_distrib left_diff_distrib right_diff_distrib

   270

   271 class comm_ring = comm_semiring + ab_group_add

   272 begin

   273

   274 subclass ring ..

   275 subclass comm_semiring_0_cancel ..

   276

   277 end

   278

   279 class ring_1 = ring + zero_neq_one + monoid_mult

   280 begin

   281

   282 subclass semiring_1_cancel ..

   283

   284 end

   285

   286 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   287   (*previously ring*)

   288 begin

   289

   290 subclass ring_1 ..

   291 subclass comm_semiring_1_cancel ..

   292

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

   294 proof

   295   assume "x dvd - y"

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

   297   then show "x dvd y" by simp

   298 next

   299   assume "x dvd y"

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

   301   then show "x dvd - y" by simp

   302 qed

   303

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

   305 proof

   306   assume "- x dvd y"

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

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

   309   then show "x dvd y" ..

   310 next

   311   assume "x dvd y"

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

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

   314   then show "- x dvd y" ..

   315 qed

   316

   317 lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"

   318 by (simp add: diff_minus dvd_minus_iff)

   319

   320 end

   321

   322 class ring_no_zero_divisors = ring + no_zero_divisors

   323 begin

   324

   325 lemma mult_eq_0_iff [simp]:

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

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

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

   329     then show ?thesis using no_zero_divisors by simp

   330 next

   331   case True then show ?thesis by auto

   332 qed

   333

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

   335 lemma mult_cancel_right [simp, noatp]:

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

   337 proof -

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

   339     by (simp add: algebra_simps right_minus_eq)

   340   thus ?thesis by (simp add: disj_commute right_minus_eq)

   341 qed

   342

   343 lemma mult_cancel_left [simp, noatp]:

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

   345 proof -

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

   347     by (simp add: algebra_simps right_minus_eq)

   348   thus ?thesis by (simp add: right_minus_eq)

   349 qed

   350

   351 end

   352

   353 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   354 begin

   355

   356 lemma mult_cancel_right1 [simp]:

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

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

   359

   360 lemma mult_cancel_right2 [simp]:

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

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

   363

   364 lemma mult_cancel_left1 [simp]:

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

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

   367

   368 lemma mult_cancel_left2 [simp]:

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

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

   371

   372 end

   373

   374 class idom = comm_ring_1 + no_zero_divisors

   375 begin

   376

   377 subclass ring_1_no_zero_divisors ..

   378

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

   380 proof

   381   assume "a * a = b * b"

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

   383     by (simp add: algebra_simps)

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

   385     by (simp add: right_minus_eq eq_neg_iff_add_eq_0)

   386 next

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

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

   389 qed

   390

   391 lemma dvd_mult_cancel_right [simp]:

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

   393 proof -

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

   395     unfolding dvd_def by (simp add: mult_ac)

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

   397     unfolding dvd_def by simp

   398   finally show ?thesis .

   399 qed

   400

   401 lemma dvd_mult_cancel_left [simp]:

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

   403 proof -

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

   405     unfolding dvd_def by (simp add: mult_ac)

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

   407     unfolding dvd_def by simp

   408   finally show ?thesis .

   409 qed

   410

   411 end

   412

   413 class inverse =

   414   fixes inverse :: "'a \<Rightarrow> 'a"

   415     and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)

   416

   417 class division_ring = ring_1 + inverse +

   418   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"

   419   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"

   420   assumes divide_inverse: "a / b = a * inverse b"

   421 begin

   422

   423 subclass ring_1_no_zero_divisors

   424 proof

   425   fix a b :: 'a

   426   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"

   427   show "a * b \<noteq> 0"

   428   proof

   429     assume ab: "a * b = 0"

   430     hence "0 = inverse a * (a * b) * inverse b" by simp

   431     also have "\<dots> = (inverse a * a) * (b * inverse b)"

   432       by (simp only: mult_assoc)

   433     also have "\<dots> = 1" using a b by simp

   434     finally show False by simp

   435   qed

   436 qed

   437

   438 lemma nonzero_imp_inverse_nonzero:

   439   "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"

   440 proof

   441   assume ianz: "inverse a = 0"

   442   assume "a \<noteq> 0"

   443   hence "1 = a * inverse a" by simp

   444   also have "... = 0" by (simp add: ianz)

   445   finally have "1 = 0" .

   446   thus False by (simp add: eq_commute)

   447 qed

   448

   449 lemma inverse_zero_imp_zero:

   450   "inverse a = 0 \<Longrightarrow> a = 0"

   451 apply (rule classical)

   452 apply (drule nonzero_imp_inverse_nonzero)

   453 apply auto

   454 done

   455

   456 lemma inverse_unique:

   457   assumes ab: "a * b = 1"

   458   shows "inverse a = b"

   459 proof -

   460   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all

   461   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)

   462   ultimately show ?thesis by (simp add: mult_assoc [symmetric])

   463 qed

   464

   465 lemma nonzero_inverse_minus_eq:

   466   "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"

   467 by (rule inverse_unique) simp

   468

   469 lemma nonzero_inverse_inverse_eq:

   470   "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"

   471 by (rule inverse_unique) simp

   472

   473 lemma nonzero_inverse_eq_imp_eq:

   474   assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"

   475   shows "a = b"

   476 proof -

   477   from inverse a = inverse b

   478   have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)

   479   with a \<noteq> 0 and b \<noteq> 0 show "a = b"

   480     by (simp add: nonzero_inverse_inverse_eq)

   481 qed

   482

   483 lemma inverse_1 [simp]: "inverse 1 = 1"

   484 by (rule inverse_unique) simp

   485

   486 lemma nonzero_inverse_mult_distrib:

   487   assumes "a \<noteq> 0" and "b \<noteq> 0"

   488   shows "inverse (a * b) = inverse b * inverse a"

   489 proof -

   490   have "a * (b * inverse b) * inverse a = 1" using assms by simp

   491   hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)

   492   thus ?thesis by (rule inverse_unique)

   493 qed

   494

   495 lemma division_ring_inverse_add:

   496   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"

   497 by (simp add: algebra_simps)

   498

   499 lemma division_ring_inverse_diff:

   500   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"

   501 by (simp add: algebra_simps)

   502

   503 end

   504

   505 class mult_mono = times + zero + ord +

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

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

   508

   509 class ordered_semiring = mult_mono + semiring_0 + ordered_ab_semigroup_add

   510 begin

   511

   512 lemma mult_mono:

   513   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c

   514      \<Longrightarrow> a * c \<le> b * d"

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

   516 apply (erule mult_left_mono, assumption)

   517 done

   518

   519 lemma mult_mono':

   520   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c

   521      \<Longrightarrow> a * c \<le> b * d"

   522 apply (rule mult_mono)

   523 apply (fast intro: order_trans)+

   524 done

   525

   526 end

   527

   528 class ordered_cancel_semiring = mult_mono + ordered_ab_semigroup_add

   529   + semiring + cancel_comm_monoid_add

   530 begin

   531

   532 subclass semiring_0_cancel ..

   533 subclass ordered_semiring ..

   534

   535 lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"

   536 using mult_left_mono [of zero b a] by simp

   537

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

   539 using mult_left_mono [of b zero a] by simp

   540

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

   542 using mult_right_mono [of a zero b] by simp

   543

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

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

   546 by (drule mult_right_mono [of b zero], auto)

   547

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

   549 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   550

   551 end

   552

   553 class linordered_semiring = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + mult_mono

   554 begin

   555

   556 subclass ordered_cancel_semiring ..

   557

   558 subclass ordered_comm_monoid_add ..

   559

   560 lemma mult_left_less_imp_less:

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

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

   563

   564 lemma mult_right_less_imp_less:

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

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

   567

   568 end

   569

   570 class linordered_semiring_1 = linordered_semiring + semiring_1

   571

   572 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   575 begin

   576

   577 subclass semiring_0_cancel ..

   578

   579 subclass linordered_semiring

   580 proof

   581   fix a b c :: 'a

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

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

   584     unfolding le_less

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

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

   587     unfolding le_less

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

   589 qed

   590

   591 lemma mult_left_le_imp_le:

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

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

   594

   595 lemma mult_right_le_imp_le:

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

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

   598

   599 lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"

   600 using mult_strict_left_mono [of zero b a] by simp

   601

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

   603 using mult_strict_left_mono [of b zero a] by simp

   604

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

   606 using mult_strict_right_mono [of a zero b] by simp

   607

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

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

   610 by (drule mult_strict_right_mono [of b zero], auto)

   611

   612 lemma zero_less_mult_pos:

   613   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   615  apply (auto simp add: le_less not_less)

   616 apply (drule_tac mult_pos_neg [of a b])

   617  apply (auto dest: less_not_sym)

   618 done

   619

   620 lemma zero_less_mult_pos2:

   621   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   623  apply (auto simp add: le_less not_less)

   624 apply (drule_tac mult_pos_neg2 [of a b])

   625  apply (auto dest: less_not_sym)

   626 done

   627

   628 text{*Strict monotonicity in both arguments*}

   629 lemma mult_strict_mono:

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

   631   shows "a * c < b * d"

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

   633   apply (simp add: mult_pos_pos)

   634   apply (erule mult_strict_right_mono [THEN less_trans])

   635   apply (force simp add: le_less)

   636   apply (erule mult_strict_left_mono, assumption)

   637   done

   638

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

   640 lemma mult_strict_mono':

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

   642   shows "a * c < b * d"

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

   644

   645 lemma mult_less_le_imp_less:

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

   647   shows "a * c < b * d"

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

   649   apply (erule less_le_trans)

   650   apply (erule mult_left_mono)

   651   apply simp

   652   apply (erule mult_strict_right_mono)

   653   apply assumption

   654   done

   655

   656 lemma mult_le_less_imp_less:

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

   658   shows "a * c < b * d"

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

   660   apply (erule le_less_trans)

   661   apply (erule mult_strict_left_mono)

   662   apply simp

   663   apply (erule mult_right_mono)

   664   apply simp

   665   done

   666

   667 lemma mult_less_imp_less_left:

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

   669   shows "a < b"

   670 proof (rule ccontr)

   671   assume "\<not>  a < b"

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

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

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

   675 qed

   676

   677 lemma mult_less_imp_less_right:

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

   679   shows "a < b"

   680 proof (rule ccontr)

   681   assume "\<not> a < b"

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

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

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

   685 qed

   686

   687 end

   688

   689 class linlinordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   690

   691 class mult_mono1 = times + zero + ord +

   692   assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"

   693

   694 class ordered_comm_semiring = comm_semiring_0

   695   + ordered_ab_semigroup_add + mult_mono1

   696 begin

   697

   698 subclass ordered_semiring

   699 proof

   700   fix a b c :: 'a

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

   702   thus "c * a \<le> c * b" by (rule mult_mono1)

   703   thus "a * c \<le> b * c" by (simp only: mult_commute)

   704 qed

   705

   706 end

   707

   708 class ordered_cancel_comm_semiring = comm_semiring_0_cancel

   709   + ordered_ab_semigroup_add + mult_mono1

   710 begin

   711

   712 subclass ordered_comm_semiring ..

   713 subclass ordered_cancel_semiring ..

   714

   715 end

   716

   717 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

   718   assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   719 begin

   720

   721 subclass linordered_semiring_strict

   722 proof

   723   fix a b c :: 'a

   724   assume "a < b" "0 < c"

   725   thus "c * a < c * b" by (rule mult_strict_left_mono_comm)

   726   thus "a * c < b * c" by (simp only: mult_commute)

   727 qed

   728

   729 subclass ordered_cancel_comm_semiring

   730 proof

   731   fix a b c :: 'a

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

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

   734     unfolding le_less

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

   736 qed

   737

   738 end

   739

   740 class ordered_ring = ring + ordered_cancel_semiring

   741 begin

   742

   743 subclass ordered_ab_group_add ..

   744

   745 text{*Legacy - use @{text algebra_simps} *}

   746 lemmas ring_simps[noatp] = algebra_simps

   747

   748 lemma less_add_iff1:

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

   750 by (simp add: algebra_simps)

   751

   752 lemma less_add_iff2:

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

   754 by (simp add: algebra_simps)

   755

   756 lemma le_add_iff1:

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

   758 by (simp add: algebra_simps)

   759

   760 lemma le_add_iff2:

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

   762 by (simp add: algebra_simps)

   763

   764 lemma mult_left_mono_neg:

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

   766   apply (drule mult_left_mono [of _ _ "uminus c"])

   767   apply (simp_all add: minus_mult_left [symmetric])

   768   done

   769

   770 lemma mult_right_mono_neg:

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

   772   apply (drule mult_right_mono [of _ _ "uminus c"])

   773   apply (simp_all add: minus_mult_right [symmetric])

   774   done

   775

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

   777 using mult_right_mono_neg [of a zero b] by simp

   778

   779 lemma split_mult_pos_le:

   780   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   781 by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)

   782

   783 end

   784

   785 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   786 begin

   787

   788 subclass ordered_ring ..

   789

   790 subclass ordered_ab_group_add_abs

   791 proof

   792   fix a b

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

   794     by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)

   795     (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]

   796      neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,

   797       auto intro!: less_imp_le add_neg_neg)

   798 qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)

   799

   800 end

   801

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

   803    Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.

   804  *)

   805 class linordered_ring_strict = ring + linordered_semiring_strict

   806   + ordered_ab_group_add + abs_if

   807 begin

   808

   809 subclass linordered_ring ..

   810

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

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

   813

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

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

   816

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

   818 using mult_strict_right_mono_neg [of a zero b] by simp

   819

   820 subclass ring_no_zero_divisors

   821 proof

   822   fix a b

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

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

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

   826   proof (cases "a < 0")

   827     case True note A' = this

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

   829       case True with A'

   830       show ?thesis by (auto dest: mult_neg_neg)

   831     next

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

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

   834     qed

   835   next

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

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

   838       case True with A'

   839       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   840     next

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

   842       with A' show ?thesis by (auto dest: mult_pos_pos)

   843     qed

   844   qed

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

   846 qed

   847

   848 lemma zero_less_mult_iff:

   849   "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"

   850   apply (auto simp add: mult_pos_pos mult_neg_neg)

   851   apply (simp_all add: not_less le_less)

   852   apply (erule disjE) apply assumption defer

   853   apply (erule disjE) defer apply (drule sym) apply simp

   854   apply (erule disjE) defer apply (drule sym) apply simp

   855   apply (erule disjE) apply assumption apply (drule sym) apply simp

   856   apply (drule sym) apply simp

   857   apply (blast dest: zero_less_mult_pos)

   858   apply (blast dest: zero_less_mult_pos2)

   859   done

   860

   861 lemma zero_le_mult_iff:

   862   "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"

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

   864

   865 lemma mult_less_0_iff:

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

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

   868   apply (force simp add: minus_mult_left[symmetric])

   869   done

   870

   871 lemma mult_le_0_iff:

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

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

   874   apply (force simp add: minus_mult_left[symmetric])

   875   done

   876

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

   878 by (simp add: zero_le_mult_iff linear)

   879

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

   881 by (simp add: not_less)

   882

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

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

   885

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

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

   888

   889 lemma mult_less_cancel_right_disj:

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

   891   apply (cases "c = 0")

   892   apply (auto simp add: neq_iff mult_strict_right_mono

   893                       mult_strict_right_mono_neg)

   894   apply (auto simp add: not_less

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

   896                       not_le [symmetric, of a])

   897   apply (erule_tac [!] notE)

   898   apply (auto simp add: less_imp_le mult_right_mono

   899                       mult_right_mono_neg)

   900   done

   901

   902 lemma mult_less_cancel_left_disj:

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

   904   apply (cases "c = 0")

   905   apply (auto simp add: neq_iff mult_strict_left_mono

   906                       mult_strict_left_mono_neg)

   907   apply (auto simp add: not_less

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

   909                       not_le [symmetric, of a])

   910   apply (erule_tac [!] notE)

   911   apply (auto simp add: less_imp_le mult_left_mono

   912                       mult_left_mono_neg)

   913   done

   914

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

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

   917

   918 lemma mult_less_cancel_right:

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

   920   using mult_less_cancel_right_disj [of a c b] by auto

   921

   922 lemma mult_less_cancel_left:

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

   924   using mult_less_cancel_left_disj [of c a b] by auto

   925

   926 lemma mult_le_cancel_right:

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

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

   929

   930 lemma mult_le_cancel_left:

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

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

   933

   934 lemma mult_le_cancel_left_pos:

   935   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

   936 by (auto simp: mult_le_cancel_left)

   937

   938 lemma mult_le_cancel_left_neg:

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

   940 by (auto simp: mult_le_cancel_left)

   941

   942 lemma mult_less_cancel_left_pos:

   943   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

   944 by (auto simp: mult_less_cancel_left)

   945

   946 lemma mult_less_cancel_left_neg:

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

   948 by (auto simp: mult_less_cancel_left)

   949

   950 end

   951

   952 text{*Legacy - use @{text algebra_simps} *}

   953 lemmas ring_simps[noatp] = algebra_simps

   954

   955 lemmas mult_sign_intros =

   956   mult_nonneg_nonneg mult_nonneg_nonpos

   957   mult_nonpos_nonneg mult_nonpos_nonpos

   958   mult_pos_pos mult_pos_neg

   959   mult_neg_pos mult_neg_neg

   960

   961 class ordered_comm_ring = comm_ring + ordered_comm_semiring

   962 begin

   963

   964 subclass ordered_ring ..

   965 subclass ordered_cancel_comm_semiring ..

   966

   967 end

   968

   969 class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +

   970   (*previously linordered_semiring*)

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

   972 begin

   973

   974 lemma pos_add_strict:

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

   976   using add_strict_mono [of zero a b c] by simp

   977

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

   979 by (rule zero_less_one [THEN less_imp_le])

   980

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

   982 by (simp add: not_le)

   983

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

   985 by (simp add: not_less)

   986

   987 lemma less_1_mult:

   988   assumes "1 < m" and "1 < n"

   989   shows "1 < m * n"

   990   using assms mult_strict_mono [of 1 m 1 n]

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

   992

   993 end

   994

   995 class linordered_idom = comm_ring_1 +

   996   linordered_comm_semiring_strict + ordered_ab_group_add +

   997   abs_if + sgn_if

   998   (*previously linordered_ring*)

   999 begin

  1000

  1001 subclass linordered_ring_strict ..

  1002 subclass ordered_comm_ring ..

  1003 subclass idom ..

  1004

  1005 subclass linordered_semidom

  1006 proof

  1007   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1008   thus "0 < 1" by (simp add: le_less)

  1009 qed

  1010

  1011 lemma linorder_neqE_linordered_idom:

  1012   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1013   using assms by (rule neqE)

  1014

  1015 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1016

  1017 lemma mult_le_cancel_right1:

  1018   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1019 by (insert mult_le_cancel_right [of 1 c b], simp)

  1020

  1021 lemma mult_le_cancel_right2:

  1022   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1023 by (insert mult_le_cancel_right [of a c 1], simp)

  1024

  1025 lemma mult_le_cancel_left1:

  1026   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1027 by (insert mult_le_cancel_left [of c 1 b], simp)

  1028

  1029 lemma mult_le_cancel_left2:

  1030   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1031 by (insert mult_le_cancel_left [of c a 1], simp)

  1032

  1033 lemma mult_less_cancel_right1:

  1034   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1035 by (insert mult_less_cancel_right [of 1 c b], simp)

  1036

  1037 lemma mult_less_cancel_right2:

  1038   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1039 by (insert mult_less_cancel_right [of a c 1], simp)

  1040

  1041 lemma mult_less_cancel_left1:

  1042   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1043 by (insert mult_less_cancel_left [of c 1 b], simp)

  1044

  1045 lemma mult_less_cancel_left2:

  1046   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1047 by (insert mult_less_cancel_left [of c a 1], simp)

  1048

  1049 lemma sgn_sgn [simp]:

  1050   "sgn (sgn a) = sgn a"

  1051 unfolding sgn_if by simp

  1052

  1053 lemma sgn_0_0:

  1054   "sgn a = 0 \<longleftrightarrow> a = 0"

  1055 unfolding sgn_if by simp

  1056

  1057 lemma sgn_1_pos:

  1058   "sgn a = 1 \<longleftrightarrow> a > 0"

  1059 unfolding sgn_if by (simp add: neg_equal_zero)

  1060

  1061 lemma sgn_1_neg:

  1062   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1063 unfolding sgn_if by (auto simp add: equal_neg_zero)

  1064

  1065 lemma sgn_pos [simp]:

  1066   "0 < a \<Longrightarrow> sgn a = 1"

  1067 unfolding sgn_1_pos .

  1068

  1069 lemma sgn_neg [simp]:

  1070   "a < 0 \<Longrightarrow> sgn a = - 1"

  1071 unfolding sgn_1_neg .

  1072

  1073 lemma sgn_times:

  1074   "sgn (a * b) = sgn a * sgn b"

  1075 by (auto simp add: sgn_if zero_less_mult_iff)

  1076

  1077 lemma abs_sgn: "abs k = k * sgn k"

  1078 unfolding sgn_if abs_if by auto

  1079

  1080 lemma sgn_greater [simp]:

  1081   "0 < sgn a \<longleftrightarrow> 0 < a"

  1082   unfolding sgn_if by auto

  1083

  1084 lemma sgn_less [simp]:

  1085   "sgn a < 0 \<longleftrightarrow> a < 0"

  1086   unfolding sgn_if by auto

  1087

  1088 lemma abs_dvd_iff [simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"

  1089   by (simp add: abs_if)

  1090

  1091 lemma dvd_abs_iff [simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"

  1092   by (simp add: abs_if)

  1093

  1094 lemma dvd_if_abs_eq:

  1095   "abs l = abs (k) \<Longrightarrow> l dvd k"

  1096 by(subst abs_dvd_iff[symmetric]) simp

  1097

  1098 end

  1099

  1100 text {* Simprules for comparisons where common factors can be cancelled. *}

  1101

  1102 lemmas mult_compare_simps[noatp] =

  1103     mult_le_cancel_right mult_le_cancel_left

  1104     mult_le_cancel_right1 mult_le_cancel_right2

  1105     mult_le_cancel_left1 mult_le_cancel_left2

  1106     mult_less_cancel_right mult_less_cancel_left

  1107     mult_less_cancel_right1 mult_less_cancel_right2

  1108     mult_less_cancel_left1 mult_less_cancel_left2

  1109     mult_cancel_right mult_cancel_left

  1110     mult_cancel_right1 mult_cancel_right2

  1111     mult_cancel_left1 mult_cancel_left2

  1112

  1113 -- {* FIXME continue localization here *}

  1114

  1115 subsection {* Reasoning about inequalities with division *}

  1116

  1117 lemma mult_right_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1

  1118     ==> x * y <= x"

  1119 by (auto simp add: mult_compare_simps)

  1120

  1121 lemma mult_left_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1

  1122     ==> y * x <= x"

  1123 by (auto simp add: mult_compare_simps)

  1124

  1125 context linordered_semidom

  1126 begin

  1127

  1128 lemma less_add_one: "a < a + 1"

  1129 proof -

  1130   have "a + 0 < a + 1"

  1131     by (blast intro: zero_less_one add_strict_left_mono)

  1132   thus ?thesis by simp

  1133 qed

  1134

  1135 lemma zero_less_two: "0 < 1 + 1"

  1136 by (blast intro: less_trans zero_less_one less_add_one)

  1137

  1138 end

  1139

  1140

  1141 subsection {* Absolute Value *}

  1142

  1143 context linordered_idom

  1144 begin

  1145

  1146 lemma mult_sgn_abs: "sgn x * abs x = x"

  1147   unfolding abs_if sgn_if by auto

  1148

  1149 end

  1150

  1151 lemma abs_one [simp]: "abs 1 = (1::'a::linordered_idom)"

  1152 by (simp add: abs_if zero_less_one [THEN order_less_not_sym])

  1153

  1154 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1155   assumes abs_eq_mult:

  1156     "(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>"

  1157

  1158 context linordered_idom

  1159 begin

  1160

  1161 subclass ordered_ring_abs proof

  1162 qed (auto simp add: abs_if not_less equal_neg_zero neg_equal_zero mult_less_0_iff)

  1163

  1164 lemma abs_mult:

  1165   "abs (a * b) = abs a * abs b"

  1166   by (rule abs_eq_mult) auto

  1167

  1168 lemma abs_mult_self:

  1169   "abs a * abs a = a * a"

  1170   by (simp add: abs_if)

  1171

  1172 end

  1173

  1174 lemma abs_mult_less:

  1175      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::linordered_idom)"

  1176 proof -

  1177   assume ac: "abs a < c"

  1178   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)

  1179   assume "abs b < d"

  1180   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1181 qed

  1182

  1183 lemmas eq_minus_self_iff[noatp] = equal_neg_zero

  1184

  1185 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::linordered_idom))"

  1186   unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..

  1187

  1188 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::linordered_idom))"

  1189 apply (simp add: order_less_le abs_le_iff)

  1190 apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)

  1191 done

  1192

  1193 lemma abs_mult_pos: "(0::'a::linordered_idom) <= x ==>

  1194     (abs y) * x = abs (y * x)"

  1195   apply (subst abs_mult)

  1196   apply simp

  1197 done

  1198

  1199 code_modulename SML

  1200   Rings Arith

  1201

  1202 code_modulename OCaml

  1203   Rings Arith

  1204

  1205 code_modulename Haskell

  1206   Rings Arith

  1207

  1208 end