src/HOL/Rings.thy
 author haftmann Fri Apr 23 15:17:59 2010 +0200 (2010-04-23) changeset 36304 6984744e6b34 parent 36301 72f4d079ebf8 child 36348 89c54f51f55a permissions -rw-r--r--
less special treatment of times_divide_eq [simp]
     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 left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c"

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

    19 begin

    20

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

    22 lemma combine_common_factor:

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

    24 by (simp add: left_distrib add_ac)

    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

    32 class semiring_0 = semiring + comm_monoid_add + mult_zero

    33

    34 class semiring_0_cancel = semiring + cancel_comm_monoid_add

    35 begin

    36

    37 subclass semiring_0

    38 proof

    39   fix a :: 'a

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

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

    42 next

    43   fix a :: 'a

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

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

    46 qed

    47

    48 end

    49

    50 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +

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

    52 begin

    53

    54 subclass semiring

    55 proof

    56   fix a b c :: 'a

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

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

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

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

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

    62 qed

    63

    64 end

    65

    66 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero

    67 begin

    68

    69 subclass semiring_0 ..

    70

    71 end

    72

    73 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add

    74 begin

    75

    76 subclass semiring_0_cancel ..

    77

    78 subclass comm_semiring_0 ..

    79

    80 end

    81

    82 class zero_neq_one = zero + one +

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

    84 begin

    85

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

    87 by (rule not_sym) (rule zero_neq_one)

    88

    89 end

    90

    91 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

    92

    93 text {* Abstract divisibility *}

    94

    95 class dvd = times

    96 begin

    97

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

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

   100

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

   102   unfolding dvd_def ..

   103

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

   105   unfolding dvd_def by blast

   106

   107 end

   108

   109 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd

   110   (*previously almost_semiring*)

   111 begin

   112

   113 subclass semiring_1 ..

   114

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

   116 proof

   117   show "a = a * 1" by simp

   118 qed

   119

   120 lemma dvd_trans:

   121   assumes "a dvd b" and "b dvd c"

   122   shows "a dvd c"

   123 proof -

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

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

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

   127   then show ?thesis ..

   128 qed

   129

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

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

   132

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

   134 proof

   135   show "0 = a * 0" by simp

   136 qed

   137

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

   139 by (auto intro!: dvdI)

   140

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

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

   143

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

   145   apply (subst mult_commute)

   146   apply (erule dvd_mult)

   147   done

   148

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

   150 by (rule dvd_mult) (rule dvd_refl)

   151

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

   153 by (rule dvd_mult2) (rule dvd_refl)

   154

   155 lemma mult_dvd_mono:

   156   assumes "a dvd b"

   157     and "c dvd d"

   158   shows "a * c dvd b * d"

   159 proof -

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

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

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

   163   then show ?thesis ..

   164 qed

   165

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

   167 by (simp add: dvd_def mult_assoc, blast)

   168

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

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

   171

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

   173 by simp

   174

   175 lemma dvd_add[simp]:

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

   177 proof -

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

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

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

   181   then show ?thesis ..

   182 qed

   183

   184 end

   185

   186

   187 class no_zero_divisors = zero + times +

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

   189

   190 class semiring_1_cancel = semiring + cancel_comm_monoid_add

   191   + zero_neq_one + monoid_mult

   192 begin

   193

   194 subclass semiring_0_cancel ..

   195

   196 subclass semiring_1 ..

   197

   198 end

   199

   200 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add

   201   + zero_neq_one + comm_monoid_mult

   202 begin

   203

   204 subclass semiring_1_cancel ..

   205 subclass comm_semiring_0_cancel ..

   206 subclass comm_semiring_1 ..

   207

   208 end

   209

   210 class ring = semiring + ab_group_add

   211 begin

   212

   213 subclass semiring_0_cancel ..

   214

   215 text {* Distribution rules *}

   216

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

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

   219

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

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

   222

   223 text{*Extract signs from products*}

   224 lemmas mult_minus_left [simp, no_atp] = minus_mult_left [symmetric]

   225 lemmas mult_minus_right [simp,no_atp] = minus_mult_right [symmetric]

   226

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

   228 by simp

   229

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

   231 by simp

   232

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

   234 by (simp add: right_distrib diff_minus)

   235

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

   237 by (simp add: left_distrib diff_minus)

   238

   239 lemmas ring_distribs[no_atp] =

   240   right_distrib left_distrib left_diff_distrib right_diff_distrib

   241

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

   243 lemmas ring_simps[no_atp] = algebra_simps

   244

   245 lemma eq_add_iff1:

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

   247 by (simp add: algebra_simps)

   248

   249 lemma eq_add_iff2:

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

   251 by (simp add: algebra_simps)

   252

   253 end

   254

   255 lemmas ring_distribs[no_atp] =

   256   right_distrib left_distrib left_diff_distrib right_diff_distrib

   257

   258 class comm_ring = comm_semiring + ab_group_add

   259 begin

   260

   261 subclass ring ..

   262 subclass comm_semiring_0_cancel ..

   263

   264 end

   265

   266 class ring_1 = ring + zero_neq_one + monoid_mult

   267 begin

   268

   269 subclass semiring_1_cancel ..

   270

   271 end

   272

   273 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult

   274   (*previously ring*)

   275 begin

   276

   277 subclass ring_1 ..

   278 subclass comm_semiring_1_cancel ..

   279

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

   281 proof

   282   assume "x dvd - y"

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

   284   then show "x dvd y" by simp

   285 next

   286   assume "x dvd y"

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

   288   then show "x dvd - y" by simp

   289 qed

   290

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

   292 proof

   293   assume "- x dvd y"

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

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

   296   then show "x dvd y" ..

   297 next

   298   assume "x dvd y"

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

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

   301   then show "- x dvd y" ..

   302 qed

   303

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

   305 by (simp only: diff_minus dvd_add dvd_minus_iff)

   306

   307 end

   308

   309 class ring_no_zero_divisors = ring + no_zero_divisors

   310 begin

   311

   312 lemma mult_eq_0_iff [simp]:

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

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

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

   316     then show ?thesis using no_zero_divisors by simp

   317 next

   318   case True then show ?thesis by auto

   319 qed

   320

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

   322 lemma mult_cancel_right [simp, no_atp]:

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

   324 proof -

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

   326     by (simp add: algebra_simps)

   327   thus ?thesis by (simp add: disj_commute)

   328 qed

   329

   330 lemma mult_cancel_left [simp, no_atp]:

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

   332 proof -

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

   334     by (simp add: algebra_simps)

   335   thus ?thesis by simp

   336 qed

   337

   338 end

   339

   340 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

   341 begin

   342

   343 lemma mult_cancel_right1 [simp]:

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

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

   346

   347 lemma mult_cancel_right2 [simp]:

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

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

   350

   351 lemma mult_cancel_left1 [simp]:

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

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

   354

   355 lemma mult_cancel_left2 [simp]:

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

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

   358

   359 end

   360

   361 class idom = comm_ring_1 + no_zero_divisors

   362 begin

   363

   364 subclass ring_1_no_zero_divisors ..

   365

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

   367 proof

   368   assume "a * a = b * b"

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

   370     by (simp add: algebra_simps)

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

   372     by (simp add: eq_neg_iff_add_eq_0)

   373 next

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

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

   376 qed

   377

   378 lemma dvd_mult_cancel_right [simp]:

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

   380 proof -

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

   382     unfolding dvd_def by (simp add: mult_ac)

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

   384     unfolding dvd_def by simp

   385   finally show ?thesis .

   386 qed

   387

   388 lemma dvd_mult_cancel_left [simp]:

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

   390 proof -

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

   392     unfolding dvd_def by (simp add: mult_ac)

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

   394     unfolding dvd_def by simp

   395   finally show ?thesis .

   396 qed

   397

   398 end

   399

   400 class inverse =

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

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

   403

   404 class division_ring = ring_1 + inverse +

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

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

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

   408 begin

   409

   410 subclass ring_1_no_zero_divisors

   411 proof

   412   fix a b :: 'a

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

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

   415   proof

   416     assume ab: "a * b = 0"

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

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

   419       by (simp only: mult_assoc)

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

   421     finally show False by simp

   422   qed

   423 qed

   424

   425 lemma nonzero_imp_inverse_nonzero:

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

   427 proof

   428   assume ianz: "inverse a = 0"

   429   assume "a \<noteq> 0"

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

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

   432   finally have "1 = 0" .

   433   thus False by (simp add: eq_commute)

   434 qed

   435

   436 lemma inverse_zero_imp_zero:

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

   438 apply (rule classical)

   439 apply (drule nonzero_imp_inverse_nonzero)

   440 apply auto

   441 done

   442

   443 lemma inverse_unique:

   444   assumes ab: "a * b = 1"

   445   shows "inverse a = b"

   446 proof -

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

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

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

   450 qed

   451

   452 lemma nonzero_inverse_minus_eq:

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

   454 by (rule inverse_unique) simp

   455

   456 lemma nonzero_inverse_inverse_eq:

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

   458 by (rule inverse_unique) simp

   459

   460 lemma nonzero_inverse_eq_imp_eq:

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

   462   shows "a = b"

   463 proof -

   464   from inverse a = inverse b

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

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

   467     by (simp add: nonzero_inverse_inverse_eq)

   468 qed

   469

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

   471 by (rule inverse_unique) simp

   472

   473 lemma nonzero_inverse_mult_distrib:

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

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

   476 proof -

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

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

   479   thus ?thesis by (rule inverse_unique)

   480 qed

   481

   482 lemma division_ring_inverse_add:

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

   484 by (simp add: algebra_simps)

   485

   486 lemma division_ring_inverse_diff:

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

   488 by (simp add: algebra_simps)

   489

   490 lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"

   491 proof

   492   assume neq: "b \<noteq> 0"

   493   {

   494     hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc)

   495     also assume "a / b = 1"

   496     finally show "a = b" by simp

   497   next

   498     assume "a = b"

   499     with neq show "a / b = 1" by (simp add: divide_inverse)

   500   }

   501 qed

   502

   503 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"

   504 by (simp add: divide_inverse)

   505

   506 lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"

   507 by (simp add: divide_inverse)

   508

   509 lemma divide_zero_left [simp]: "0 / a = 0"

   510 by (simp add: divide_inverse)

   511

   512 lemma inverse_eq_divide: "inverse a = 1 / a"

   513 by (simp add: divide_inverse)

   514

   515 lemma add_divide_distrib: "(a+b) / c = a/c + b/c"

   516 by (simp add: divide_inverse algebra_simps)

   517

   518 lemma divide_1 [simp]: "a / 1 = a"

   519   by (simp add: divide_inverse)

   520

   521 lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"

   522   by (simp add: divide_inverse mult_assoc)

   523

   524 lemma minus_divide_left: "- (a / b) = (-a) / b"

   525   by (simp add: divide_inverse)

   526

   527 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"

   528   by (simp add: divide_inverse nonzero_inverse_minus_eq)

   529

   530 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"

   531   by (simp add: divide_inverse nonzero_inverse_minus_eq)

   532

   533 lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"

   534   by (simp add: divide_inverse)

   535

   536 lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"

   537   by (simp add: diff_minus add_divide_distrib)

   538

   539 lemma nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"

   540 proof -

   541   assume [simp]: "c \<noteq> 0"

   542   have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp

   543   also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)

   544   finally show ?thesis .

   545 qed

   546

   547 lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"

   548 proof -

   549   assume [simp]: "c \<noteq> 0"

   550   have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp

   551   also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc)

   552   finally show ?thesis .

   553 qed

   554

   555 lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"

   556   by (simp add: divide_inverse mult_assoc)

   557

   558 lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"

   559   by (drule sym) (simp add: divide_inverse mult_assoc)

   560

   561 end

   562

   563 class division_by_zero = division_ring +

   564   assumes inverse_zero [simp]: "inverse 0 = 0"

   565 begin

   566

   567 lemma divide_zero [simp]:

   568   "a / 0 = 0"

   569   by (simp add: divide_inverse)

   570

   571 lemma divide_self_if [simp]:

   572   "a / a = (if a = 0 then 0 else 1)"

   573   by simp

   574

   575 lemma inverse_nonzero_iff_nonzero [simp]:

   576   "inverse a = 0 \<longleftrightarrow> a = 0"

   577   by rule (fact inverse_zero_imp_zero, simp)

   578

   579 lemma inverse_minus_eq [simp]:

   580   "inverse (- a) = - inverse a"

   581 proof cases

   582   assume "a=0" thus ?thesis by simp

   583 next

   584   assume "a\<noteq>0"

   585   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

   586 qed

   587

   588 lemma inverse_eq_imp_eq:

   589   "inverse a = inverse b \<Longrightarrow> a = b"

   590 apply (cases "a=0 | b=0")

   591  apply (force dest!: inverse_zero_imp_zero

   592               simp add: eq_commute [of "0::'a"])

   593 apply (force dest!: nonzero_inverse_eq_imp_eq)

   594 done

   595

   596 lemma inverse_eq_iff_eq [simp]:

   597   "inverse a = inverse b \<longleftrightarrow> a = b"

   598   by (force dest!: inverse_eq_imp_eq)

   599

   600 lemma inverse_inverse_eq [simp]:

   601   "inverse (inverse a) = a"

   602 proof cases

   603   assume "a=0" thus ?thesis by simp

   604 next

   605   assume "a\<noteq>0"

   606   thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

   607 qed

   608

   609 end

   610

   611 class mult_mono = times + zero + ord +

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

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

   614

   615 text {*

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

   617   \begin{itemize}

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

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

   620   \end{itemize}

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

   622   \begin{itemize}

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

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

   625   \end{itemize}

   626 *}

   627

   628 class ordered_semiring = mult_mono + semiring_0 + ordered_ab_semigroup_add

   629 begin

   630

   631 lemma mult_mono:

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

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

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

   635 apply (erule mult_left_mono, assumption)

   636 done

   637

   638 lemma mult_mono':

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

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

   641 apply (rule mult_mono)

   642 apply (fast intro: order_trans)+

   643 done

   644

   645 end

   646

   647 class ordered_cancel_semiring = mult_mono + ordered_ab_semigroup_add

   648   + semiring + cancel_comm_monoid_add

   649 begin

   650

   651 subclass semiring_0_cancel ..

   652 subclass ordered_semiring ..

   653

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

   655 using mult_left_mono [of 0 b a] by simp

   656

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

   658 using mult_left_mono [of b 0 a] by simp

   659

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

   661 using mult_right_mono [of a 0 b] by simp

   662

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

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

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

   666

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

   668 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

   669

   670 end

   671

   672 class linordered_semiring = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + mult_mono

   673 begin

   674

   675 subclass ordered_cancel_semiring ..

   676

   677 subclass ordered_comm_monoid_add ..

   678

   679 lemma mult_left_less_imp_less:

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

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

   682

   683 lemma mult_right_less_imp_less:

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

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

   686

   687 end

   688

   689 class linordered_semiring_1 = linordered_semiring + semiring_1

   690

   691 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +

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

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

   694 begin

   695

   696 subclass semiring_0_cancel ..

   697

   698 subclass linordered_semiring

   699 proof

   700   fix a b c :: 'a

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

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

   703     unfolding le_less

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

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

   706     unfolding le_less

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

   708 qed

   709

   710 lemma mult_left_le_imp_le:

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

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

   713

   714 lemma mult_right_le_imp_le:

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

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

   717

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

   719 using mult_strict_left_mono [of 0 b a] by simp

   720

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

   722 using mult_strict_left_mono [of b 0 a] by simp

   723

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

   725 using mult_strict_right_mono [of a 0 b] by simp

   726

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

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

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

   730

   731 lemma zero_less_mult_pos:

   732   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   734  apply (auto simp add: le_less not_less)

   735 apply (drule_tac mult_pos_neg [of a b])

   736  apply (auto dest: less_not_sym)

   737 done

   738

   739 lemma zero_less_mult_pos2:

   740   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"

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

   742  apply (auto simp add: le_less not_less)

   743 apply (drule_tac mult_pos_neg2 [of a b])

   744  apply (auto dest: less_not_sym)

   745 done

   746

   747 text{*Strict monotonicity in both arguments*}

   748 lemma mult_strict_mono:

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

   750   shows "a * c < b * d"

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

   752   apply (simp add: mult_pos_pos)

   753   apply (erule mult_strict_right_mono [THEN less_trans])

   754   apply (force simp add: le_less)

   755   apply (erule mult_strict_left_mono, assumption)

   756   done

   757

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

   759 lemma mult_strict_mono':

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

   761   shows "a * c < b * d"

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

   763

   764 lemma mult_less_le_imp_less:

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

   766   shows "a * c < b * d"

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

   768   apply (erule less_le_trans)

   769   apply (erule mult_left_mono)

   770   apply simp

   771   apply (erule mult_strict_right_mono)

   772   apply assumption

   773   done

   774

   775 lemma mult_le_less_imp_less:

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

   777   shows "a * c < b * d"

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

   779   apply (erule le_less_trans)

   780   apply (erule mult_strict_left_mono)

   781   apply simp

   782   apply (erule mult_right_mono)

   783   apply simp

   784   done

   785

   786 lemma mult_less_imp_less_left:

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

   788   shows "a < b"

   789 proof (rule ccontr)

   790   assume "\<not>  a < b"

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

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

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

   794 qed

   795

   796 lemma mult_less_imp_less_right:

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

   798   shows "a < b"

   799 proof (rule ccontr)

   800   assume "\<not> a < b"

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

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

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

   804 qed

   805

   806 end

   807

   808 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1

   809

   810 class mult_mono1 = times + zero + ord +

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

   812

   813 class ordered_comm_semiring = comm_semiring_0

   814   + ordered_ab_semigroup_add + mult_mono1

   815 begin

   816

   817 subclass ordered_semiring

   818 proof

   819   fix a b c :: 'a

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

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

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

   823 qed

   824

   825 end

   826

   827 class ordered_cancel_comm_semiring = comm_semiring_0_cancel

   828   + ordered_ab_semigroup_add + mult_mono1

   829 begin

   830

   831 subclass ordered_comm_semiring ..

   832 subclass ordered_cancel_semiring ..

   833

   834 end

   835

   836 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +

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

   838 begin

   839

   840 subclass linordered_semiring_strict

   841 proof

   842   fix a b c :: 'a

   843   assume "a < b" "0 < c"

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

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

   846 qed

   847

   848 subclass ordered_cancel_comm_semiring

   849 proof

   850   fix a b c :: 'a

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

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

   853     unfolding le_less

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

   855 qed

   856

   857 end

   858

   859 class ordered_ring = ring + ordered_cancel_semiring

   860 begin

   861

   862 subclass ordered_ab_group_add ..

   863

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

   865 lemmas ring_simps[no_atp] = algebra_simps

   866

   867 lemma less_add_iff1:

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

   869 by (simp add: algebra_simps)

   870

   871 lemma less_add_iff2:

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

   873 by (simp add: algebra_simps)

   874

   875 lemma le_add_iff1:

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

   877 by (simp add: algebra_simps)

   878

   879 lemma le_add_iff2:

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

   881 by (simp add: algebra_simps)

   882

   883 lemma mult_left_mono_neg:

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

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

   886   apply simp_all

   887   done

   888

   889 lemma mult_right_mono_neg:

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

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

   892   apply simp_all

   893   done

   894

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

   896 using mult_right_mono_neg [of a 0 b] by simp

   897

   898 lemma split_mult_pos_le:

   899   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"

   900 by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)

   901

   902 end

   903

   904 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if

   905 begin

   906

   907 subclass ordered_ring ..

   908

   909 subclass ordered_ab_group_add_abs

   910 proof

   911   fix a b

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

   913     by (auto simp add: abs_if not_less)

   914     (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric],

   915      auto intro: add_nonneg_nonneg, auto intro!: less_imp_le add_neg_neg)

   916 qed (auto simp add: abs_if)

   917

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

   919   using linear [of 0 a]

   920   by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)

   921

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

   923   by (simp add: not_less)

   924

   925 end

   926

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

   928    Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.

   929  *)

   930 class linordered_ring_strict = ring + linordered_semiring_strict

   931   + ordered_ab_group_add + abs_if

   932 begin

   933

   934 subclass linordered_ring ..

   935

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

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

   938

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

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

   941

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

   943 using mult_strict_right_mono_neg [of a 0 b] by simp

   944

   945 subclass ring_no_zero_divisors

   946 proof

   947   fix a b

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

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

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

   951   proof (cases "a < 0")

   952     case True note A' = this

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

   954       case True with A'

   955       show ?thesis by (auto dest: mult_neg_neg)

   956     next

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

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

   959     qed

   960   next

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

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

   963       case True with A'

   964       show ?thesis by (auto dest: mult_strict_right_mono_neg)

   965     next

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

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

   968     qed

   969   qed

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

   971 qed

   972

   973 lemma zero_less_mult_iff:

   974   "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"

   975   apply (auto simp add: mult_pos_pos mult_neg_neg)

   976   apply (simp_all add: not_less le_less)

   977   apply (erule disjE) apply assumption defer

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

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

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

   981   apply (drule sym) apply simp

   982   apply (blast dest: zero_less_mult_pos)

   983   apply (blast dest: zero_less_mult_pos2)

   984   done

   985

   986 lemma zero_le_mult_iff:

   987   "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"

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

   989

   990 lemma mult_less_0_iff:

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

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

   993   apply force

   994   done

   995

   996 lemma mult_le_0_iff:

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

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

   999   apply force

  1000   done

  1001

  1002 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},

  1003    also with the relations @{text "\<le>"} and equality.*}

  1004

  1005 text{*These disjunction'' versions produce two cases when the comparison is

  1006  an assumption, but effectively four when the comparison is a goal.*}

  1007

  1008 lemma mult_less_cancel_right_disj:

  1009   "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1010   apply (cases "c = 0")

  1011   apply (auto simp add: neq_iff mult_strict_right_mono

  1012                       mult_strict_right_mono_neg)

  1013   apply (auto simp add: not_less

  1014                       not_le [symmetric, of "a*c"]

  1015                       not_le [symmetric, of a])

  1016   apply (erule_tac [!] notE)

  1017   apply (auto simp add: less_imp_le mult_right_mono

  1018                       mult_right_mono_neg)

  1019   done

  1020

  1021 lemma mult_less_cancel_left_disj:

  1022   "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"

  1023   apply (cases "c = 0")

  1024   apply (auto simp add: neq_iff mult_strict_left_mono

  1025                       mult_strict_left_mono_neg)

  1026   apply (auto simp add: not_less

  1027                       not_le [symmetric, of "c*a"]

  1028                       not_le [symmetric, of a])

  1029   apply (erule_tac [!] notE)

  1030   apply (auto simp add: less_imp_le mult_left_mono

  1031                       mult_left_mono_neg)

  1032   done

  1033

  1034 text{*The conjunction of implication'' lemmas produce two cases when the

  1035 comparison is a goal, but give four when the comparison is an assumption.*}

  1036

  1037 lemma mult_less_cancel_right:

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

  1039   using mult_less_cancel_right_disj [of a c b] by auto

  1040

  1041 lemma mult_less_cancel_left:

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

  1043   using mult_less_cancel_left_disj [of c a b] by auto

  1044

  1045 lemma mult_le_cancel_right:

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

  1047 by (simp add: not_less [symmetric] mult_less_cancel_right_disj)

  1048

  1049 lemma mult_le_cancel_left:

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

  1051 by (simp add: not_less [symmetric] mult_less_cancel_left_disj)

  1052

  1053 lemma mult_le_cancel_left_pos:

  1054   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"

  1055 by (auto simp: mult_le_cancel_left)

  1056

  1057 lemma mult_le_cancel_left_neg:

  1058   "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"

  1059 by (auto simp: mult_le_cancel_left)

  1060

  1061 lemma mult_less_cancel_left_pos:

  1062   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"

  1063 by (auto simp: mult_less_cancel_left)

  1064

  1065 lemma mult_less_cancel_left_neg:

  1066   "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"

  1067 by (auto simp: mult_less_cancel_left)

  1068

  1069 end

  1070

  1071 text{*Legacy - use @{text algebra_simps} *}

  1072 lemmas ring_simps[no_atp] = algebra_simps

  1073

  1074 lemmas mult_sign_intros =

  1075   mult_nonneg_nonneg mult_nonneg_nonpos

  1076   mult_nonpos_nonneg mult_nonpos_nonpos

  1077   mult_pos_pos mult_pos_neg

  1078   mult_neg_pos mult_neg_neg

  1079

  1080 class ordered_comm_ring = comm_ring + ordered_comm_semiring

  1081 begin

  1082

  1083 subclass ordered_ring ..

  1084 subclass ordered_cancel_comm_semiring ..

  1085

  1086 end

  1087

  1088 class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +

  1089   (*previously linordered_semiring*)

  1090   assumes zero_less_one [simp]: "0 < 1"

  1091 begin

  1092

  1093 lemma pos_add_strict:

  1094   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

  1095   using add_strict_mono [of 0 a b c] by simp

  1096

  1097 lemma zero_le_one [simp]: "0 \<le> 1"

  1098 by (rule zero_less_one [THEN less_imp_le])

  1099

  1100 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"

  1101 by (simp add: not_le)

  1102

  1103 lemma not_one_less_zero [simp]: "\<not> 1 < 0"

  1104 by (simp add: not_less)

  1105

  1106 lemma less_1_mult:

  1107   assumes "1 < m" and "1 < n"

  1108   shows "1 < m * n"

  1109   using assms mult_strict_mono [of 1 m 1 n]

  1110     by (simp add:  less_trans [OF zero_less_one])

  1111

  1112 end

  1113

  1114 class linordered_idom = comm_ring_1 +

  1115   linordered_comm_semiring_strict + ordered_ab_group_add +

  1116   abs_if + sgn_if

  1117   (*previously linordered_ring*)

  1118 begin

  1119

  1120 subclass linordered_ring_strict ..

  1121 subclass ordered_comm_ring ..

  1122 subclass idom ..

  1123

  1124 subclass linordered_semidom

  1125 proof

  1126   have "0 \<le> 1 * 1" by (rule zero_le_square)

  1127   thus "0 < 1" by (simp add: le_less)

  1128 qed

  1129

  1130 lemma linorder_neqE_linordered_idom:

  1131   assumes "x \<noteq> y" obtains "x < y" | "y < x"

  1132   using assms by (rule neqE)

  1133

  1134 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

  1135

  1136 lemma mult_le_cancel_right1:

  1137   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1138 by (insert mult_le_cancel_right [of 1 c b], simp)

  1139

  1140 lemma mult_le_cancel_right2:

  1141   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1142 by (insert mult_le_cancel_right [of a c 1], simp)

  1143

  1144 lemma mult_le_cancel_left1:

  1145   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"

  1146 by (insert mult_le_cancel_left [of c 1 b], simp)

  1147

  1148 lemma mult_le_cancel_left2:

  1149   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"

  1150 by (insert mult_le_cancel_left [of c a 1], simp)

  1151

  1152 lemma mult_less_cancel_right1:

  1153   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1154 by (insert mult_less_cancel_right [of 1 c b], simp)

  1155

  1156 lemma mult_less_cancel_right2:

  1157   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1158 by (insert mult_less_cancel_right [of a c 1], simp)

  1159

  1160 lemma mult_less_cancel_left1:

  1161   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"

  1162 by (insert mult_less_cancel_left [of c 1 b], simp)

  1163

  1164 lemma mult_less_cancel_left2:

  1165   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"

  1166 by (insert mult_less_cancel_left [of c a 1], simp)

  1167

  1168 lemma sgn_sgn [simp]:

  1169   "sgn (sgn a) = sgn a"

  1170 unfolding sgn_if by simp

  1171

  1172 lemma sgn_0_0:

  1173   "sgn a = 0 \<longleftrightarrow> a = 0"

  1174 unfolding sgn_if by simp

  1175

  1176 lemma sgn_1_pos:

  1177   "sgn a = 1 \<longleftrightarrow> a > 0"

  1178 unfolding sgn_if by simp

  1179

  1180 lemma sgn_1_neg:

  1181   "sgn a = - 1 \<longleftrightarrow> a < 0"

  1182 unfolding sgn_if by auto

  1183

  1184 lemma sgn_pos [simp]:

  1185   "0 < a \<Longrightarrow> sgn a = 1"

  1186 unfolding sgn_1_pos .

  1187

  1188 lemma sgn_neg [simp]:

  1189   "a < 0 \<Longrightarrow> sgn a = - 1"

  1190 unfolding sgn_1_neg .

  1191

  1192 lemma sgn_times:

  1193   "sgn (a * b) = sgn a * sgn b"

  1194 by (auto simp add: sgn_if zero_less_mult_iff)

  1195

  1196 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"

  1197 unfolding sgn_if abs_if by auto

  1198

  1199 lemma sgn_greater [simp]:

  1200   "0 < sgn a \<longleftrightarrow> 0 < a"

  1201   unfolding sgn_if by auto

  1202

  1203 lemma sgn_less [simp]:

  1204   "sgn a < 0 \<longleftrightarrow> a < 0"

  1205   unfolding sgn_if by auto

  1206

  1207 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"

  1208   by (simp add: abs_if)

  1209

  1210 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"

  1211   by (simp add: abs_if)

  1212

  1213 lemma dvd_if_abs_eq:

  1214   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"

  1215 by(subst abs_dvd_iff[symmetric]) simp

  1216

  1217 end

  1218

  1219 text {* Simprules for comparisons where common factors can be cancelled. *}

  1220

  1221 lemmas mult_compare_simps[no_atp] =

  1222     mult_le_cancel_right mult_le_cancel_left

  1223     mult_le_cancel_right1 mult_le_cancel_right2

  1224     mult_le_cancel_left1 mult_le_cancel_left2

  1225     mult_less_cancel_right mult_less_cancel_left

  1226     mult_less_cancel_right1 mult_less_cancel_right2

  1227     mult_less_cancel_left1 mult_less_cancel_left2

  1228     mult_cancel_right mult_cancel_left

  1229     mult_cancel_right1 mult_cancel_right2

  1230     mult_cancel_left1 mult_cancel_left2

  1231

  1232 text {* Reasoning about inequalities with division *}

  1233

  1234 context linordered_semidom

  1235 begin

  1236

  1237 lemma less_add_one: "a < a + 1"

  1238 proof -

  1239   have "a + 0 < a + 1"

  1240     by (blast intro: zero_less_one add_strict_left_mono)

  1241   thus ?thesis by simp

  1242 qed

  1243

  1244 lemma zero_less_two: "0 < 1 + 1"

  1245 by (blast intro: less_trans zero_less_one less_add_one)

  1246

  1247 end

  1248

  1249 context linordered_idom

  1250 begin

  1251

  1252 lemma mult_right_le_one_le:

  1253   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"

  1254   by (auto simp add: mult_le_cancel_left2)

  1255

  1256 lemma mult_left_le_one_le:

  1257   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"

  1258   by (auto simp add: mult_le_cancel_right2)

  1259

  1260 end

  1261

  1262 text {* Absolute Value *}

  1263

  1264 context linordered_idom

  1265 begin

  1266

  1267 lemma mult_sgn_abs:

  1268   "sgn x * \<bar>x\<bar> = x"

  1269   unfolding abs_if sgn_if by auto

  1270

  1271 lemma abs_one [simp]:

  1272   "\<bar>1\<bar> = 1"

  1273   by (simp add: abs_if zero_less_one [THEN less_not_sym])

  1274

  1275 end

  1276

  1277 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +

  1278   assumes abs_eq_mult:

  1279     "(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>"

  1280

  1281 context linordered_idom

  1282 begin

  1283

  1284 subclass ordered_ring_abs proof

  1285 qed (auto simp add: abs_if not_less mult_less_0_iff)

  1286

  1287 lemma abs_mult:

  1288   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

  1289   by (rule abs_eq_mult) auto

  1290

  1291 lemma abs_mult_self:

  1292   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"

  1293   by (simp add: abs_if)

  1294

  1295 lemma abs_mult_less:

  1296   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"

  1297 proof -

  1298   assume ac: "\<bar>a\<bar> < c"

  1299   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)

  1300   assume "\<bar>b\<bar> < d"

  1301   thus ?thesis by (simp add: ac cpos mult_strict_mono)

  1302 qed

  1303

  1304 lemma less_minus_self_iff:

  1305   "a < - a \<longleftrightarrow> a < 0"

  1306   by (simp only: less_le less_eq_neg_nonpos equal_neg_zero)

  1307

  1308 lemma abs_less_iff:

  1309   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"

  1310   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

  1311

  1312 lemma abs_mult_pos:

  1313   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"

  1314   by (simp add: abs_mult)

  1315

  1316 end

  1317

  1318 code_modulename SML

  1319   Rings Arith

  1320

  1321 code_modulename OCaml

  1322   Rings Arith

  1323

  1324 code_modulename Haskell

  1325   Rings Arith

  1326

  1327 end