src/HOL/Rings.thy
 author wenzelm Mon Jun 20 21:40:48 2016 +0200 (2016-06-20) changeset 63325 1086d56cde86 parent 63040 eb4ddd18d635 child 63359 99b51ba8da1c permissions -rw-r--r--
misc tuning and modernization;
```     1 (*  Title:      HOL/Rings.thy
```
```     2     Author:     Gertrud Bauer
```
```     3     Author:     Steven Obua
```
```     4     Author:     Tobias Nipkow
```
```     5     Author:     Lawrence C Paulson
```
```     6     Author:     Markus Wenzel
```
```     7     Author:     Jeremy Avigad
```
```     8 *)
```
```     9
```
```    10 section \<open>Rings\<close>
```
```    11
```
```    12 theory Rings
```
```    13 imports Groups Set
```
```    14 begin
```
```    15
```
```    16 class semiring = ab_semigroup_add + semigroup_mult +
```
```    17   assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
```
```    18   assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
```
```    19 begin
```
```    20
```
```    21 text \<open>For the \<open>combine_numerals\<close> simproc\<close>
```
```    22 lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
```
```    23   by (simp add: distrib_right ac_simps)
```
```    24
```
```    25 end
```
```    26
```
```    27 class mult_zero = times + zero +
```
```    28   assumes mult_zero_left [simp]: "0 * a = 0"
```
```    29   assumes mult_zero_right [simp]: "a * 0 = 0"
```
```    30 begin
```
```    31
```
```    32 lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
```
```    33   by auto
```
```    34
```
```    35 end
```
```    36
```
```    37 class semiring_0 = semiring + comm_monoid_add + mult_zero
```
```    38
```
```    39 class semiring_0_cancel = semiring + cancel_comm_monoid_add
```
```    40 begin
```
```    41
```
```    42 subclass semiring_0
```
```    43 proof
```
```    44   fix a :: 'a
```
```    45   have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
```
```    46   then show "0 * a = 0" by (simp only: add_left_cancel)
```
```    47   have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
```
```    48   then show "a * 0 = 0" by (simp only: add_left_cancel)
```
```    49 qed
```
```    50
```
```    51 end
```
```    52
```
```    53 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
```
```    54   assumes distrib: "(a + b) * c = a * c + b * c"
```
```    55 begin
```
```    56
```
```    57 subclass semiring
```
```    58 proof
```
```    59   fix a b c :: 'a
```
```    60   show "(a + b) * c = a * c + b * c" by (simp add: distrib)
```
```    61   have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
```
```    62   also have "\<dots> = b * a + c * a" by (simp only: distrib)
```
```    63   also have "\<dots> = a * b + a * c" by (simp add: ac_simps)
```
```    64   finally show "a * (b + c) = a * b + a * c" by blast
```
```    65 qed
```
```    66
```
```    67 end
```
```    68
```
```    69 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
```
```    70 begin
```
```    71
```
```    72 subclass semiring_0 ..
```
```    73
```
```    74 end
```
```    75
```
```    76 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
```
```    77 begin
```
```    78
```
```    79 subclass semiring_0_cancel ..
```
```    80
```
```    81 subclass comm_semiring_0 ..
```
```    82
```
```    83 end
```
```    84
```
```    85 class zero_neq_one = zero + one +
```
```    86   assumes zero_neq_one [simp]: "0 \<noteq> 1"
```
```    87 begin
```
```    88
```
```    89 lemma one_neq_zero [simp]: "1 \<noteq> 0"
```
```    90   by (rule not_sym) (rule zero_neq_one)
```
```    91
```
```    92 definition of_bool :: "bool \<Rightarrow> 'a"
```
```    93   where "of_bool p = (if p then 1 else 0)"
```
```    94
```
```    95 lemma of_bool_eq [simp, code]:
```
```    96   "of_bool False = 0"
```
```    97   "of_bool True = 1"
```
```    98   by (simp_all add: of_bool_def)
```
```    99
```
```   100 lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
```
```   101   by (simp add: of_bool_def)
```
```   102
```
```   103 lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
```
```   104   by (cases p) simp_all
```
```   105
```
```   106 lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
```
```   107   by (cases p) simp_all
```
```   108
```
```   109 end
```
```   110
```
```   111 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
```
```   112
```
```   113 text \<open>Abstract divisibility\<close>
```
```   114
```
```   115 class dvd = times
```
```   116 begin
```
```   117
```
```   118 definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
```
```   119   where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
```
```   120
```
```   121 lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
```
```   122   unfolding dvd_def ..
```
```   123
```
```   124 lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
```
```   125   unfolding dvd_def by blast
```
```   126
```
```   127 end
```
```   128
```
```   129 context comm_monoid_mult
```
```   130 begin
```
```   131
```
```   132 subclass dvd .
```
```   133
```
```   134 lemma dvd_refl [simp]: "a dvd a"
```
```   135 proof
```
```   136   show "a = a * 1" by simp
```
```   137 qed
```
```   138
```
```   139 lemma dvd_trans [trans]:
```
```   140   assumes "a dvd b" and "b dvd c"
```
```   141   shows "a dvd c"
```
```   142 proof -
```
```   143   from assms obtain v where "b = a * v" by (auto elim!: dvdE)
```
```   144   moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
```
```   145   ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
```
```   146   then show ?thesis ..
```
```   147 qed
```
```   148
```
```   149 lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
```
```   150   by (auto simp add: subset_iff intro: dvd_trans)
```
```   151
```
```   152 lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
```
```   153   by (auto simp add: subset_iff intro: dvd_trans)
```
```   154
```
```   155 lemma one_dvd [simp]: "1 dvd a"
```
```   156   by (auto intro!: dvdI)
```
```   157
```
```   158 lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
```
```   159   by (auto intro!: mult.left_commute dvdI elim!: dvdE)
```
```   160
```
```   161 lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
```
```   162   using dvd_mult [of a b c] by (simp add: ac_simps)
```
```   163
```
```   164 lemma dvd_triv_right [simp]: "a dvd b * a"
```
```   165   by (rule dvd_mult) (rule dvd_refl)
```
```   166
```
```   167 lemma dvd_triv_left [simp]: "a dvd a * b"
```
```   168   by (rule dvd_mult2) (rule dvd_refl)
```
```   169
```
```   170 lemma mult_dvd_mono:
```
```   171   assumes "a dvd b"
```
```   172     and "c dvd d"
```
```   173   shows "a * c dvd b * d"
```
```   174 proof -
```
```   175   from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
```
```   176   moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
```
```   177   ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
```
```   178   then show ?thesis ..
```
```   179 qed
```
```   180
```
```   181 lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
```
```   182   by (simp add: dvd_def mult.assoc) blast
```
```   183
```
```   184 lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
```
```   185   using dvd_mult_left [of b a c] by (simp add: ac_simps)
```
```   186
```
```   187 end
```
```   188
```
```   189 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
```
```   190 begin
```
```   191
```
```   192 subclass semiring_1 ..
```
```   193
```
```   194 lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
```
```   195   by (auto intro: dvd_refl elim!: dvdE)
```
```   196
```
```   197 lemma dvd_0_right [iff]: "a dvd 0"
```
```   198 proof
```
```   199   show "0 = a * 0" by simp
```
```   200 qed
```
```   201
```
```   202 lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
```
```   203   by simp
```
```   204
```
```   205 lemma dvd_add [simp]:
```
```   206   assumes "a dvd b" and "a dvd c"
```
```   207   shows "a dvd (b + c)"
```
```   208 proof -
```
```   209   from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
```
```   210   moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
```
```   211   ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
```
```   212   then show ?thesis ..
```
```   213 qed
```
```   214
```
```   215 end
```
```   216
```
```   217 class semiring_1_cancel = semiring + cancel_comm_monoid_add
```
```   218   + zero_neq_one + monoid_mult
```
```   219 begin
```
```   220
```
```   221 subclass semiring_0_cancel ..
```
```   222
```
```   223 subclass semiring_1 ..
```
```   224
```
```   225 end
```
```   226
```
```   227 class comm_semiring_1_cancel =
```
```   228   comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
```
```   229   assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
```
```   230 begin
```
```   231
```
```   232 subclass semiring_1_cancel ..
```
```   233 subclass comm_semiring_0_cancel ..
```
```   234 subclass comm_semiring_1 ..
```
```   235
```
```   236 lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
```
```   237   by (simp add: algebra_simps)
```
```   238
```
```   239 lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
```
```   240 proof -
```
```   241   have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
```
```   242   proof
```
```   243     assume ?Q
```
```   244     then show ?P by simp
```
```   245   next
```
```   246     assume ?P
```
```   247     then obtain d where "a * c + b = a * d" ..
```
```   248     then have "a * c + b - a * c = a * d - a * c" by simp
```
```   249     then have "b = a * d - a * c" by simp
```
```   250     then have "b = a * (d - c)" by (simp add: algebra_simps)
```
```   251     then show ?Q ..
```
```   252   qed
```
```   253   then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
```
```   254 qed
```
```   255
```
```   256 lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
```
```   257   using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
```
```   258
```
```   259 lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
```
```   260   using dvd_add_times_triv_left_iff [of a 1 b] by simp
```
```   261
```
```   262 lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
```
```   263   using dvd_add_times_triv_right_iff [of a b 1] by simp
```
```   264
```
```   265 lemma dvd_add_right_iff:
```
```   266   assumes "a dvd b"
```
```   267   shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
```
```   268 proof
```
```   269   assume ?P
```
```   270   then obtain d where "b + c = a * d" ..
```
```   271   moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
```
```   272   ultimately have "a * e + c = a * d" by simp
```
```   273   then have "a * e + c - a * e = a * d - a * e" by simp
```
```   274   then have "c = a * d - a * e" by simp
```
```   275   then have "c = a * (d - e)" by (simp add: algebra_simps)
```
```   276   then show ?Q ..
```
```   277 next
```
```   278   assume ?Q
```
```   279   with assms show ?P by simp
```
```   280 qed
```
```   281
```
```   282 lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
```
```   283   using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
```
```   284
```
```   285 end
```
```   286
```
```   287 class ring = semiring + ab_group_add
```
```   288 begin
```
```   289
```
```   290 subclass semiring_0_cancel ..
```
```   291
```
```   292 text \<open>Distribution rules\<close>
```
```   293
```
```   294 lemma minus_mult_left: "- (a * b) = - a * b"
```
```   295   by (rule minus_unique) (simp add: distrib_right [symmetric])
```
```   296
```
```   297 lemma minus_mult_right: "- (a * b) = a * - b"
```
```   298   by (rule minus_unique) (simp add: distrib_left [symmetric])
```
```   299
```
```   300 text \<open>Extract signs from products\<close>
```
```   301 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
```
```   302 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
```
```   303
```
```   304 lemma minus_mult_minus [simp]: "- a * - b = a * b"
```
```   305   by simp
```
```   306
```
```   307 lemma minus_mult_commute: "- a * b = a * - b"
```
```   308   by simp
```
```   309
```
```   310 lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
```
```   311   using distrib_left [of a b "-c "] by simp
```
```   312
```
```   313 lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
```
```   314   using distrib_right [of a "- b" c] by simp
```
```   315
```
```   316 lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
```
```   317
```
```   318 lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
```
```   319   by (simp add: algebra_simps)
```
```   320
```
```   321 lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
```
```   322   by (simp add: algebra_simps)
```
```   323
```
```   324 end
```
```   325
```
```   326 lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
```
```   327
```
```   328 class comm_ring = comm_semiring + ab_group_add
```
```   329 begin
```
```   330
```
```   331 subclass ring ..
```
```   332 subclass comm_semiring_0_cancel ..
```
```   333
```
```   334 lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
```
```   335   by (simp add: algebra_simps)
```
```   336
```
```   337 end
```
```   338
```
```   339 class ring_1 = ring + zero_neq_one + monoid_mult
```
```   340 begin
```
```   341
```
```   342 subclass semiring_1_cancel ..
```
```   343
```
```   344 lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
```
```   345   by (simp add: algebra_simps)
```
```   346
```
```   347 end
```
```   348
```
```   349 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
```
```   350 begin
```
```   351
```
```   352 subclass ring_1 ..
```
```   353 subclass comm_semiring_1_cancel
```
```   354   by unfold_locales (simp add: algebra_simps)
```
```   355
```
```   356 lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
```
```   357 proof
```
```   358   assume "x dvd - y"
```
```   359   then have "x dvd - 1 * - y" by (rule dvd_mult)
```
```   360   then show "x dvd y" by simp
```
```   361 next
```
```   362   assume "x dvd y"
```
```   363   then have "x dvd - 1 * y" by (rule dvd_mult)
```
```   364   then show "x dvd - y" by simp
```
```   365 qed
```
```   366
```
```   367 lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
```
```   368 proof
```
```   369   assume "- x dvd y"
```
```   370   then obtain k where "y = - x * k" ..
```
```   371   then have "y = x * - k" by simp
```
```   372   then show "x dvd y" ..
```
```   373 next
```
```   374   assume "x dvd y"
```
```   375   then obtain k where "y = x * k" ..
```
```   376   then have "y = - x * - k" by simp
```
```   377   then show "- x dvd y" ..
```
```   378 qed
```
```   379
```
```   380 lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
```
```   381   using dvd_add [of x y "- z"] by simp
```
```   382
```
```   383 end
```
```   384
```
```   385 class semiring_no_zero_divisors = semiring_0 +
```
```   386   assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
```
```   387 begin
```
```   388
```
```   389 lemma divisors_zero:
```
```   390   assumes "a * b = 0"
```
```   391   shows "a = 0 \<or> b = 0"
```
```   392 proof (rule classical)
```
```   393   assume "\<not> ?thesis"
```
```   394   then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   395   with no_zero_divisors have "a * b \<noteq> 0" by blast
```
```   396   with assms show ?thesis by simp
```
```   397 qed
```
```   398
```
```   399 lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   400 proof (cases "a = 0 \<or> b = 0")
```
```   401   case False
```
```   402   then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   403     then show ?thesis using no_zero_divisors by simp
```
```   404 next
```
```   405   case True
```
```   406   then show ?thesis by auto
```
```   407 qed
```
```   408
```
```   409 end
```
```   410
```
```   411 class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
```
```   412
```
```   413 class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
```
```   414   assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
```
```   415     and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
```
```   416 begin
```
```   417
```
```   418 lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
```
```   419   by simp
```
```   420
```
```   421 lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
```
```   422   by simp
```
```   423
```
```   424 end
```
```   425
```
```   426 class ring_no_zero_divisors = ring + semiring_no_zero_divisors
```
```   427 begin
```
```   428
```
```   429 subclass semiring_no_zero_divisors_cancel
```
```   430 proof
```
```   431   fix a b c
```
```   432   have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
```
```   433     by (simp add: algebra_simps)
```
```   434   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
```
```   435     by auto
```
```   436   finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
```
```   437   have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
```
```   438     by (simp add: algebra_simps)
```
```   439   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
```
```   440     by auto
```
```   441   finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
```
```   442 qed
```
```   443
```
```   444 end
```
```   445
```
```   446 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
```
```   447 begin
```
```   448
```
```   449 subclass semiring_1_no_zero_divisors ..
```
```   450
```
```   451 lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
```
```   452 proof -
```
```   453   have "(x - 1) * (x + 1) = x * x - 1"
```
```   454     by (simp add: algebra_simps)
```
```   455   then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
```
```   456     by simp
```
```   457   then show ?thesis
```
```   458     by (simp add: eq_neg_iff_add_eq_0)
```
```   459 qed
```
```   460
```
```   461 lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
```
```   462   using mult_cancel_right [of 1 c b] by auto
```
```   463
```
```   464 lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
```
```   465   using mult_cancel_right [of a c 1] by simp
```
```   466
```
```   467 lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
```
```   468   using mult_cancel_left [of c 1 b] by force
```
```   469
```
```   470 lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
```
```   471   using mult_cancel_left [of c a 1] by simp
```
```   472
```
```   473 end
```
```   474
```
```   475 class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
```
```   476 begin
```
```   477
```
```   478 subclass semiring_1_no_zero_divisors ..
```
```   479
```
```   480 end
```
```   481
```
```   482 class idom = comm_ring_1 + semiring_no_zero_divisors
```
```   483 begin
```
```   484
```
```   485 subclass semidom ..
```
```   486
```
```   487 subclass ring_1_no_zero_divisors ..
```
```   488
```
```   489 lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   490 proof -
```
```   491   have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
```
```   492     unfolding dvd_def by (simp add: ac_simps)
```
```   493   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   494     unfolding dvd_def by simp
```
```   495   finally show ?thesis .
```
```   496 qed
```
```   497
```
```   498 lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   499 proof -
```
```   500   have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
```
```   501     unfolding dvd_def by (simp add: ac_simps)
```
```   502   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   503     unfolding dvd_def by simp
```
```   504   finally show ?thesis .
```
```   505 qed
```
```   506
```
```   507 lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
```
```   508 proof
```
```   509   assume "a * a = b * b"
```
```   510   then have "(a - b) * (a + b) = 0"
```
```   511     by (simp add: algebra_simps)
```
```   512   then show "a = b \<or> a = - b"
```
```   513     by (simp add: eq_neg_iff_add_eq_0)
```
```   514 next
```
```   515   assume "a = b \<or> a = - b"
```
```   516   then show "a * a = b * b" by auto
```
```   517 qed
```
```   518
```
```   519 end
```
```   520
```
```   521 text \<open>
```
```   522   The theory of partially ordered rings is taken from the books:
```
```   523     \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
```
```   524     \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
```
```   525
```
```   526   Most of the used notions can also be looked up in
```
```   527     \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
```
```   528     \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
```
```   529 \<close>
```
```   530
```
```   531 class divide =
```
```   532   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
```
```   533
```
```   534 setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
```
```   535
```
```   536 context semiring
```
```   537 begin
```
```   538
```
```   539 lemma [field_simps]:
```
```   540   shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
```
```   541     and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
```
```   542   by (rule distrib_left distrib_right)+
```
```   543
```
```   544 end
```
```   545
```
```   546 context ring
```
```   547 begin
```
```   548
```
```   549 lemma [field_simps]:
```
```   550   shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
```
```   551     and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
```
```   552   by (rule left_diff_distrib right_diff_distrib)+
```
```   553
```
```   554 end
```
```   555
```
```   556 setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
```
```   557
```
```   558 class semidom_divide = semidom + divide +
```
```   559   assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
```
```   560   assumes divide_zero [simp]: "a div 0 = 0"
```
```   561 begin
```
```   562
```
```   563 lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
```
```   564   using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
```
```   565
```
```   566 subclass semiring_no_zero_divisors_cancel
```
```   567 proof
```
```   568   show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
```
```   569   proof (cases "c = 0")
```
```   570     case True
```
```   571     then show ?thesis by simp
```
```   572   next
```
```   573     case False
```
```   574     {
```
```   575       assume "a * c = b * c"
```
```   576       then have "a * c div c = b * c div c"
```
```   577         by simp
```
```   578       with False have "a = b"
```
```   579         by simp
```
```   580     }
```
```   581     then show ?thesis by auto
```
```   582   qed
```
```   583   show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
```
```   584     using * [of a c b] by (simp add: ac_simps)
```
```   585 qed
```
```   586
```
```   587 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
```
```   588   using nonzero_mult_divide_cancel_left [of a 1] by simp
```
```   589
```
```   590 lemma divide_zero_left [simp]: "0 div a = 0"
```
```   591 proof (cases "a = 0")
```
```   592   case True
```
```   593   then show ?thesis by simp
```
```   594 next
```
```   595   case False
```
```   596   then have "a * 0 div a = 0"
```
```   597     by (rule nonzero_mult_divide_cancel_left)
```
```   598   then show ?thesis by simp
```
```   599 qed
```
```   600
```
```   601 lemma divide_1 [simp]: "a div 1 = a"
```
```   602   using nonzero_mult_divide_cancel_left [of 1 a] by simp
```
```   603
```
```   604 end
```
```   605
```
```   606 class idom_divide = idom + semidom_divide
```
```   607
```
```   608 class algebraic_semidom = semidom_divide
```
```   609 begin
```
```   610
```
```   611 text \<open>
```
```   612   Class @{class algebraic_semidom} enriches a integral domain
```
```   613   by notions from algebra, like units in a ring.
```
```   614   It is a separate class to avoid spoiling fields with notions
```
```   615   which are degenerated there.
```
```   616 \<close>
```
```   617
```
```   618 lemma dvd_times_left_cancel_iff [simp]:
```
```   619   assumes "a \<noteq> 0"
```
```   620   shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
```
```   621 proof
```
```   622   assume ?P
```
```   623   then obtain d where "a * c = a * b * d" ..
```
```   624   with assms have "c = b * d" by (simp add: ac_simps)
```
```   625   then show ?Q ..
```
```   626 next
```
```   627   assume ?Q
```
```   628   then obtain d where "c = b * d" ..
```
```   629   then have "a * c = a * b * d" by (simp add: ac_simps)
```
```   630   then show ?P ..
```
```   631 qed
```
```   632
```
```   633 lemma dvd_times_right_cancel_iff [simp]:
```
```   634   assumes "a \<noteq> 0"
```
```   635   shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
```
```   636   using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
```
```   637
```
```   638 lemma div_dvd_iff_mult:
```
```   639   assumes "b \<noteq> 0" and "b dvd a"
```
```   640   shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
```
```   641 proof -
```
```   642   from \<open>b dvd a\<close> obtain d where "a = b * d" ..
```
```   643   with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
```
```   644 qed
```
```   645
```
```   646 lemma dvd_div_iff_mult:
```
```   647   assumes "c \<noteq> 0" and "c dvd b"
```
```   648   shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
```
```   649 proof -
```
```   650   from \<open>c dvd b\<close> obtain d where "b = c * d" ..
```
```   651   with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
```
```   652 qed
```
```   653
```
```   654 lemma div_dvd_div [simp]:
```
```   655   assumes "a dvd b" and "a dvd c"
```
```   656   shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
```
```   657 proof (cases "a = 0")
```
```   658   case True
```
```   659   with assms show ?thesis by simp
```
```   660 next
```
```   661   case False
```
```   662   moreover from assms obtain k l where "b = a * k" and "c = a * l"
```
```   663     by (auto elim!: dvdE)
```
```   664   ultimately show ?thesis by simp
```
```   665 qed
```
```   666
```
```   667 lemma div_add [simp]:
```
```   668   assumes "c dvd a" and "c dvd b"
```
```   669   shows "(a + b) div c = a div c + b div c"
```
```   670 proof (cases "c = 0")
```
```   671   case True
```
```   672   then show ?thesis by simp
```
```   673 next
```
```   674   case False
```
```   675   moreover from assms obtain k l where "a = c * k" and "b = c * l"
```
```   676     by (auto elim!: dvdE)
```
```   677   moreover have "c * k + c * l = c * (k + l)"
```
```   678     by (simp add: algebra_simps)
```
```   679   ultimately show ?thesis
```
```   680     by simp
```
```   681 qed
```
```   682
```
```   683 lemma div_mult_div_if_dvd:
```
```   684   assumes "b dvd a" and "d dvd c"
```
```   685   shows "(a div b) * (c div d) = (a * c) div (b * d)"
```
```   686 proof (cases "b = 0 \<or> c = 0")
```
```   687   case True
```
```   688   with assms show ?thesis by auto
```
```   689 next
```
```   690   case False
```
```   691   moreover from assms obtain k l where "a = b * k" and "c = d * l"
```
```   692     by (auto elim!: dvdE)
```
```   693   moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)"
```
```   694     by (simp add: ac_simps)
```
```   695   ultimately show ?thesis by simp
```
```   696 qed
```
```   697
```
```   698 lemma dvd_div_eq_mult:
```
```   699   assumes "a \<noteq> 0" and "a dvd b"
```
```   700   shows "b div a = c \<longleftrightarrow> b = c * a"
```
```   701 proof
```
```   702   assume "b = c * a"
```
```   703   then show "b div a = c" by (simp add: assms)
```
```   704 next
```
```   705   assume "b div a = c"
```
```   706   then have "b div a * a = c * a" by simp
```
```   707   moreover from assms have "b div a * a = b"
```
```   708     by (auto elim!: dvdE simp add: ac_simps)
```
```   709   ultimately show "b = c * a" by simp
```
```   710 qed
```
```   711
```
```   712 lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
```
```   713   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
```
```   714
```
```   715 lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
```
```   716   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
```
```   717
```
```   718 lemma div_mult_swap:
```
```   719   assumes "c dvd b"
```
```   720   shows "a * (b div c) = (a * b) div c"
```
```   721 proof (cases "c = 0")
```
```   722   case True
```
```   723   then show ?thesis by simp
```
```   724 next
```
```   725   case False
```
```   726   from assms obtain d where "b = c * d" ..
```
```   727   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
```
```   728     by simp
```
```   729   ultimately show ?thesis by (simp add: ac_simps)
```
```   730 qed
```
```   731
```
```   732 lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
```
```   733   using div_mult_swap [of c b a] by (simp add: ac_simps)
```
```   734
```
```   735 lemma dvd_div_mult2_eq:
```
```   736   assumes "b * c dvd a"
```
```   737   shows "a div (b * c) = a div b div c"
```
```   738 proof -
```
```   739   from assms obtain k where "a = b * c * k" ..
```
```   740   then show ?thesis
```
```   741     by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
```
```   742 qed
```
```   743
```
```   744 lemma dvd_div_div_eq_mult:
```
```   745   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
```
```   746   shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q")
```
```   747 proof -
```
```   748   from assms have "a * c \<noteq> 0" by simp
```
```   749   then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
```
```   750     by simp
```
```   751   also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
```
```   752     by (simp add: ac_simps)
```
```   753   also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
```
```   754     using assms by (simp add: div_mult_swap)
```
```   755   also have "\<dots> \<longleftrightarrow> ?Q"
```
```   756     using assms by (simp add: ac_simps)
```
```   757   finally show ?thesis .
```
```   758 qed
```
```   759
```
```   760
```
```   761 text \<open>Units: invertible elements in a ring\<close>
```
```   762
```
```   763 abbreviation is_unit :: "'a \<Rightarrow> bool"
```
```   764   where "is_unit a \<equiv> a dvd 1"
```
```   765
```
```   766 lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
```
```   767   by simp
```
```   768
```
```   769 lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
```
```   770   by (rule dvd_trans [of _ 1]) simp_all
```
```   771
```
```   772 lemma unit_dvdE:
```
```   773   assumes "is_unit a"
```
```   774   obtains c where "a \<noteq> 0" and "b = a * c"
```
```   775 proof -
```
```   776   from assms have "a dvd b" by auto
```
```   777   then obtain c where "b = a * c" ..
```
```   778   moreover from assms have "a \<noteq> 0" by auto
```
```   779   ultimately show thesis using that by blast
```
```   780 qed
```
```   781
```
```   782 lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
```
```   783   by (rule dvd_trans)
```
```   784
```
```   785 lemma unit_div_1_unit [simp, intro]:
```
```   786   assumes "is_unit a"
```
```   787   shows "is_unit (1 div a)"
```
```   788 proof -
```
```   789   from assms have "1 = 1 div a * a" by simp
```
```   790   then show "is_unit (1 div a)" by (rule dvdI)
```
```   791 qed
```
```   792
```
```   793 lemma is_unitE [elim?]:
```
```   794   assumes "is_unit a"
```
```   795   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
```
```   796     and "is_unit b" and "1 div a = b" and "1 div b = a"
```
```   797     and "a * b = 1" and "c div a = c * b"
```
```   798 proof (rule that)
```
```   799   define b where "b = 1 div a"
```
```   800   then show "1 div a = b" by simp
```
```   801   from assms b_def show "is_unit b" by simp
```
```   802   with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   803   from assms b_def show "a * b = 1" by simp
```
```   804   then have "1 = a * b" ..
```
```   805   with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
```
```   806   from assms have "a dvd c" ..
```
```   807   then obtain d where "c = a * d" ..
```
```   808   with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
```
```   809     by (simp add: mult.assoc mult.left_commute [of a])
```
```   810 qed
```
```   811
```
```   812 lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
```
```   813   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
```
```   814
```
```   815 lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
```
```   816   by (auto dest: dvd_mult_left dvd_mult_right)
```
```   817
```
```   818 lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
```
```   819   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
```
```   820
```
```   821 lemma mult_unit_dvd_iff:
```
```   822   assumes "is_unit b"
```
```   823   shows "a * b dvd c \<longleftrightarrow> a dvd c"
```
```   824 proof
```
```   825   assume "a * b dvd c"
```
```   826   with assms show "a dvd c"
```
```   827     by (simp add: dvd_mult_left)
```
```   828 next
```
```   829   assume "a dvd c"
```
```   830   then obtain k where "c = a * k" ..
```
```   831   with assms have "c = (a * b) * (1 div b * k)"
```
```   832     by (simp add: mult_ac)
```
```   833   then show "a * b dvd c" by (rule dvdI)
```
```   834 qed
```
```   835
```
```   836 lemma dvd_mult_unit_iff:
```
```   837   assumes "is_unit b"
```
```   838   shows "a dvd c * b \<longleftrightarrow> a dvd c"
```
```   839 proof
```
```   840   assume "a dvd c * b"
```
```   841   with assms have "c * b dvd c * (b * (1 div b))"
```
```   842     by (subst mult_assoc [symmetric]) simp
```
```   843   also from assms have "b * (1 div b) = 1"
```
```   844     by (rule is_unitE) simp
```
```   845   finally have "c * b dvd c" by simp
```
```   846   with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
```
```   847 next
```
```   848   assume "a dvd c"
```
```   849   then show "a dvd c * b" by simp
```
```   850 qed
```
```   851
```
```   852 lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
```
```   853   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
```
```   854
```
```   855 lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
```
```   856   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
```
```   857
```
```   858 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
```
```   859   dvd_mult_unit_iff dvd_div_unit_iff  (* FIXME consider named_theorems *)
```
```   860
```
```   861 lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
```
```   862   by (erule is_unitE [of _ b]) simp
```
```   863
```
```   864 lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
```
```   865   by (rule dvd_div_mult_self) auto
```
```   866
```
```   867 lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
```
```   868   by (erule is_unitE) simp
```
```   869
```
```   870 lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
```
```   871   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
```
```   872
```
```   873 lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
```
```   874   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
```
```   875
```
```   876 lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
```
```   877   by (auto elim: is_unitE)
```
```   878
```
```   879 lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
```
```   880   using unit_eq_div1 [of b c a] by auto
```
```   881
```
```   882 lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
```
```   883   using mult_cancel_left [of a b c] by auto
```
```   884
```
```   885 lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
```
```   886   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
```
```   887
```
```   888 lemma unit_div_cancel:
```
```   889   assumes "is_unit a"
```
```   890   shows "b div a = c div a \<longleftrightarrow> b = c"
```
```   891 proof -
```
```   892   from assms have "is_unit (1 div a)" by simp
```
```   893   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
```
```   894     by (rule unit_mult_right_cancel)
```
```   895   with assms show ?thesis by simp
```
```   896 qed
```
```   897
```
```   898 lemma is_unit_div_mult2_eq:
```
```   899   assumes "is_unit b" and "is_unit c"
```
```   900   shows "a div (b * c) = a div b div c"
```
```   901 proof -
```
```   902   from assms have "is_unit (b * c)"
```
```   903     by (simp add: unit_prod)
```
```   904   then have "b * c dvd a"
```
```   905     by (rule unit_imp_dvd)
```
```   906   then show ?thesis
```
```   907     by (rule dvd_div_mult2_eq)
```
```   908 qed
```
```   909
```
```   910 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
```
```   911   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
```
```   912   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
```
```   913   unit_eq_div1 unit_eq_div2
```
```   914
```
```   915 lemma is_unit_divide_mult_cancel_left:
```
```   916   assumes "a \<noteq> 0" and "is_unit b"
```
```   917   shows "a div (a * b) = 1 div b"
```
```   918 proof -
```
```   919   from assms have "a div (a * b) = a div a div b"
```
```   920     by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
```
```   921   with assms show ?thesis by simp
```
```   922 qed
```
```   923
```
```   924 lemma is_unit_divide_mult_cancel_right:
```
```   925   assumes "a \<noteq> 0" and "is_unit b"
```
```   926   shows "a div (b * a) = 1 div b"
```
```   927   using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps)
```
```   928
```
```   929 end
```
```   930
```
```   931 class normalization_semidom = algebraic_semidom +
```
```   932   fixes normalize :: "'a \<Rightarrow> 'a"
```
```   933     and unit_factor :: "'a \<Rightarrow> 'a"
```
```   934   assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
```
```   935   assumes normalize_0 [simp]: "normalize 0 = 0"
```
```   936     and unit_factor_0 [simp]: "unit_factor 0 = 0"
```
```   937   assumes is_unit_normalize:
```
```   938     "is_unit a  \<Longrightarrow> normalize a = 1"
```
```   939   assumes unit_factor_is_unit [iff]:
```
```   940     "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
```
```   941   assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
```
```   942 begin
```
```   943
```
```   944 text \<open>
```
```   945   Class @{class normalization_semidom} cultivates the idea that
```
```   946   each integral domain can be split into equivalence classes
```
```   947   whose representants are associated, i.e. divide each other.
```
```   948   @{const normalize} specifies a canonical representant for each equivalence
```
```   949   class.  The rationale behind this is that it is easier to reason about equality
```
```   950   than equivalences, hence we prefer to think about equality of normalized
```
```   951   values rather than associated elements.
```
```   952 \<close>
```
```   953
```
```   954 lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
```
```   955   by (rule unit_imp_dvd) simp
```
```   956
```
```   957 lemma unit_factor_self [simp]: "unit_factor a dvd a"
```
```   958   by (cases "a = 0") simp_all
```
```   959
```
```   960 lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
```
```   961   using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
```
```   962
```
```   963 lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
```
```   964   (is "?P \<longleftrightarrow> ?Q")
```
```   965 proof
```
```   966   assume ?P
```
```   967   moreover have "unit_factor a * normalize a = a" by simp
```
```   968   ultimately show ?Q by simp
```
```   969 next
```
```   970   assume ?Q
```
```   971   then show ?P by simp
```
```   972 qed
```
```   973
```
```   974 lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
```
```   975   (is "?P \<longleftrightarrow> ?Q")
```
```   976 proof
```
```   977   assume ?P
```
```   978   moreover have "unit_factor a * normalize a = a" by simp
```
```   979   ultimately show ?Q by simp
```
```   980 next
```
```   981   assume ?Q
```
```   982   then show ?P by simp
```
```   983 qed
```
```   984
```
```   985 lemma is_unit_unit_factor:
```
```   986   assumes "is_unit a"
```
```   987   shows "unit_factor a = a"
```
```   988 proof -
```
```   989   from assms have "normalize a = 1" by (rule is_unit_normalize)
```
```   990   moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
```
```   991   ultimately show ?thesis by simp
```
```   992 qed
```
```   993
```
```   994 lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
```
```   995   by (rule is_unit_unit_factor) simp
```
```   996
```
```   997 lemma normalize_1 [simp]: "normalize 1 = 1"
```
```   998   by (rule is_unit_normalize) simp
```
```   999
```
```  1000 lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
```
```  1001   (is "?P \<longleftrightarrow> ?Q")
```
```  1002 proof
```
```  1003   assume ?Q
```
```  1004   then show ?P by (rule is_unit_normalize)
```
```  1005 next
```
```  1006   assume ?P
```
```  1007   then have "a \<noteq> 0" by auto
```
```  1008   from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
```
```  1009     by simp
```
```  1010   then have "unit_factor a = a"
```
```  1011     by simp
```
```  1012   moreover have "is_unit (unit_factor a)"
```
```  1013     using \<open>a \<noteq> 0\<close> by simp
```
```  1014   ultimately show ?Q by simp
```
```  1015 qed
```
```  1016
```
```  1017 lemma div_normalize [simp]: "a div normalize a = unit_factor a"
```
```  1018 proof (cases "a = 0")
```
```  1019   case True
```
```  1020   then show ?thesis by simp
```
```  1021 next
```
```  1022   case False
```
```  1023   then have "normalize a \<noteq> 0" by simp
```
```  1024   with nonzero_mult_divide_cancel_right
```
```  1025   have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
```
```  1026   then show ?thesis by simp
```
```  1027 qed
```
```  1028
```
```  1029 lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
```
```  1030 proof (cases "a = 0")
```
```  1031   case True
```
```  1032   then show ?thesis by simp
```
```  1033 next
```
```  1034   case False
```
```  1035   then have "unit_factor a \<noteq> 0" by simp
```
```  1036   with nonzero_mult_divide_cancel_left
```
```  1037   have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
```
```  1038   then show ?thesis by simp
```
```  1039 qed
```
```  1040
```
```  1041 lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
```
```  1042 proof (cases "a = 0")
```
```  1043   case True
```
```  1044   then show ?thesis by simp
```
```  1045 next
```
```  1046   case False
```
```  1047   have "normalize a div a = normalize a div (unit_factor a * normalize a)"
```
```  1048     by simp
```
```  1049   also have "\<dots> = 1 div unit_factor a"
```
```  1050     using False by (subst is_unit_divide_mult_cancel_right) simp_all
```
```  1051   finally show ?thesis .
```
```  1052 qed
```
```  1053
```
```  1054 lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
```
```  1055   by (cases "b = 0") simp_all
```
```  1056
```
```  1057 lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
```
```  1058 proof (cases "a = 0 \<or> b = 0")
```
```  1059   case True
```
```  1060   then show ?thesis by auto
```
```  1061 next
```
```  1062   case False
```
```  1063   from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
```
```  1064   then have "normalize (a * b) = a * b div unit_factor (a * b)"
```
```  1065     by simp
```
```  1066   also have "\<dots> = a * b div unit_factor (b * a)"
```
```  1067     by (simp add: ac_simps)
```
```  1068   also have "\<dots> = a * b div unit_factor b div unit_factor a"
```
```  1069     using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
```
```  1070   also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
```
```  1071     using False by (subst unit_div_mult_swap) simp_all
```
```  1072   also have "\<dots> = normalize a * normalize b"
```
```  1073     using False
```
```  1074     by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
```
```  1075   finally show ?thesis .
```
```  1076 qed
```
```  1077
```
```  1078 lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
```
```  1079   by (cases "a = 0") (auto intro: is_unit_unit_factor)
```
```  1080
```
```  1081 lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
```
```  1082   by (rule is_unit_normalize) simp
```
```  1083
```
```  1084 lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
```
```  1085 proof (cases "a = 0")
```
```  1086   case True
```
```  1087   then show ?thesis by simp
```
```  1088 next
```
```  1089   case False
```
```  1090   have "normalize a = normalize (unit_factor a * normalize a)"
```
```  1091     by simp
```
```  1092   also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
```
```  1093     by (simp only: normalize_mult)
```
```  1094   finally show ?thesis
```
```  1095     using False by simp_all
```
```  1096 qed
```
```  1097
```
```  1098 lemma unit_factor_normalize [simp]:
```
```  1099   assumes "a \<noteq> 0"
```
```  1100   shows "unit_factor (normalize a) = 1"
```
```  1101 proof -
```
```  1102   from assms have *: "normalize a \<noteq> 0"
```
```  1103     by simp
```
```  1104   have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
```
```  1105     by (simp only: unit_factor_mult_normalize)
```
```  1106   then have "unit_factor (normalize a) * normalize a = normalize a"
```
```  1107     by simp
```
```  1108   with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
```
```  1109     by simp
```
```  1110   with * show ?thesis
```
```  1111     by simp
```
```  1112 qed
```
```  1113
```
```  1114 lemma dvd_unit_factor_div:
```
```  1115   assumes "b dvd a"
```
```  1116   shows "unit_factor (a div b) = unit_factor a div unit_factor b"
```
```  1117 proof -
```
```  1118   from assms have "a = a div b * b"
```
```  1119     by simp
```
```  1120   then have "unit_factor a = unit_factor (a div b * b)"
```
```  1121     by simp
```
```  1122   then show ?thesis
```
```  1123     by (cases "b = 0") (simp_all add: unit_factor_mult)
```
```  1124 qed
```
```  1125
```
```  1126 lemma dvd_normalize_div:
```
```  1127   assumes "b dvd a"
```
```  1128   shows "normalize (a div b) = normalize a div normalize b"
```
```  1129 proof -
```
```  1130   from assms have "a = a div b * b"
```
```  1131     by simp
```
```  1132   then have "normalize a = normalize (a div b * b)"
```
```  1133     by simp
```
```  1134   then show ?thesis
```
```  1135     by (cases "b = 0") (simp_all add: normalize_mult)
```
```  1136 qed
```
```  1137
```
```  1138 lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
```
```  1139 proof -
```
```  1140   have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
```
```  1141     using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
```
```  1142       by (cases "a = 0") simp_all
```
```  1143   then show ?thesis by simp
```
```  1144 qed
```
```  1145
```
```  1146 lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
```
```  1147 proof -
```
```  1148   have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
```
```  1149     using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
```
```  1150       by (cases "b = 0") simp_all
```
```  1151   then show ?thesis by simp
```
```  1152 qed
```
```  1153
```
```  1154 text \<open>
```
```  1155   We avoid an explicit definition of associated elements but prefer
```
```  1156   explicit normalisation instead.  In theory we could define an abbreviation
```
```  1157   like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
```
```  1158   counterproductive without suggestive infix syntax, which we do not want
```
```  1159   to sacrifice for this purpose here.
```
```  1160 \<close>
```
```  1161
```
```  1162 lemma associatedI:
```
```  1163   assumes "a dvd b" and "b dvd a"
```
```  1164   shows "normalize a = normalize b"
```
```  1165 proof (cases "a = 0 \<or> b = 0")
```
```  1166   case True
```
```  1167   with assms show ?thesis by auto
```
```  1168 next
```
```  1169   case False
```
```  1170   from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
```
```  1171   moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
```
```  1172   ultimately have "b * 1 = b * (c * d)"
```
```  1173     by (simp add: ac_simps)
```
```  1174   with False have "1 = c * d"
```
```  1175     unfolding mult_cancel_left by simp
```
```  1176   then have "is_unit c" and "is_unit d"
```
```  1177     by auto
```
```  1178   with a b show ?thesis
```
```  1179     by (simp add: normalize_mult is_unit_normalize)
```
```  1180 qed
```
```  1181
```
```  1182 lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
```
```  1183   using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
```
```  1184   by simp
```
```  1185
```
```  1186 lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
```
```  1187   using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
```
```  1188   by simp
```
```  1189
```
```  1190 lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
```
```  1191   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
```
```  1192
```
```  1193 lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
```
```  1194   (is "?P \<longleftrightarrow> ?Q")
```
```  1195 proof
```
```  1196   assume ?Q
```
```  1197   then show ?P by (auto intro!: associatedI)
```
```  1198 next
```
```  1199   assume ?P
```
```  1200   then have "unit_factor a * normalize a = unit_factor a * normalize b"
```
```  1201     by simp
```
```  1202   then have *: "normalize b * unit_factor a = a"
```
```  1203     by (simp add: ac_simps)
```
```  1204   show ?Q
```
```  1205   proof (cases "a = 0 \<or> b = 0")
```
```  1206     case True
```
```  1207     with \<open>?P\<close> show ?thesis by auto
```
```  1208   next
```
```  1209     case False
```
```  1210     then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
```
```  1211       by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
```
```  1212     with * show ?thesis by simp
```
```  1213   qed
```
```  1214 qed
```
```  1215
```
```  1216 lemma associated_eqI:
```
```  1217   assumes "a dvd b" and "b dvd a"
```
```  1218   assumes "normalize a = a" and "normalize b = b"
```
```  1219   shows "a = b"
```
```  1220 proof -
```
```  1221   from assms have "normalize a = normalize b"
```
```  1222     unfolding associated_iff_dvd by simp
```
```  1223   with \<open>normalize a = a\<close> have "a = normalize b" by simp
```
```  1224   with \<open>normalize b = b\<close> show "a = b" by simp
```
```  1225 qed
```
```  1226
```
```  1227 end
```
```  1228
```
```  1229 class ordered_semiring = semiring + ordered_comm_monoid_add +
```
```  1230   assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
```
```  1231   assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
```
```  1232 begin
```
```  1233
```
```  1234 lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
```
```  1235   apply (erule (1) mult_right_mono [THEN order_trans])
```
```  1236   apply (erule (1) mult_left_mono)
```
```  1237   done
```
```  1238
```
```  1239 lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
```
```  1240   apply (rule mult_mono)
```
```  1241   apply (fast intro: order_trans)+
```
```  1242   done
```
```  1243
```
```  1244 end
```
```  1245
```
```  1246 class ordered_semiring_0 = semiring_0 + ordered_semiring
```
```  1247 begin
```
```  1248
```
```  1249 lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
```
```  1250   using mult_left_mono [of 0 b a] by simp
```
```  1251
```
```  1252 lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
```
```  1253   using mult_left_mono [of b 0 a] by simp
```
```  1254
```
```  1255 lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
```
```  1256   using mult_right_mono [of a 0 b] by simp
```
```  1257
```
```  1258 text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
```
```  1259 lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
```
```  1260   apply (drule mult_right_mono [of b 0])
```
```  1261   apply auto
```
```  1262   done
```
```  1263
```
```  1264 lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
```
```  1265   by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
```
```  1266
```
```  1267 end
```
```  1268
```
```  1269 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
```
```  1270 begin
```
```  1271
```
```  1272 subclass semiring_0_cancel ..
```
```  1273 subclass ordered_semiring_0 ..
```
```  1274
```
```  1275 end
```
```  1276
```
```  1277 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
```
```  1278 begin
```
```  1279
```
```  1280 subclass ordered_cancel_semiring ..
```
```  1281
```
```  1282 subclass ordered_cancel_comm_monoid_add ..
```
```  1283
```
```  1284 lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
```
```  1285   by (force simp add: mult_left_mono not_le [symmetric])
```
```  1286
```
```  1287 lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
```
```  1288   by (force simp add: mult_right_mono not_le [symmetric])
```
```  1289
```
```  1290 lemma less_eq_add_cancel_left_greater_eq_zero [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
```
```  1291   using add_le_cancel_left [of a 0 b] by simp
```
```  1292
```
```  1293 lemma less_eq_add_cancel_left_less_eq_zero [simp]: "a + b \<le> a \<longleftrightarrow> b \<le> 0"
```
```  1294   using add_le_cancel_left [of a b 0] by simp
```
```  1295
```
```  1296 lemma less_eq_add_cancel_right_greater_eq_zero [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
```
```  1297   using add_le_cancel_right [of 0 a b] by simp
```
```  1298
```
```  1299 lemma less_eq_add_cancel_right_less_eq_zero [simp]: "b + a \<le> a \<longleftrightarrow> b \<le> 0"
```
```  1300   using add_le_cancel_right [of b a 0] by simp
```
```  1301
```
```  1302 lemma less_add_cancel_left_greater_zero [simp]: "a < a + b \<longleftrightarrow> 0 < b"
```
```  1303   using add_less_cancel_left [of a 0 b] by simp
```
```  1304
```
```  1305 lemma less_add_cancel_left_less_zero [simp]: "a + b < a \<longleftrightarrow> b < 0"
```
```  1306   using add_less_cancel_left [of a b 0] by simp
```
```  1307
```
```  1308 lemma less_add_cancel_right_greater_zero [simp]: "a < b + a \<longleftrightarrow> 0 < b"
```
```  1309   using add_less_cancel_right [of 0 a b] by simp
```
```  1310
```
```  1311 lemma less_add_cancel_right_less_zero [simp]: "b + a < a \<longleftrightarrow> b < 0"
```
```  1312   using add_less_cancel_right [of b a 0] by simp
```
```  1313
```
```  1314 end
```
```  1315
```
```  1316 class linordered_semiring_1 = linordered_semiring + semiring_1
```
```  1317 begin
```
```  1318
```
```  1319 lemma convex_bound_le:
```
```  1320   assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  1321   shows "u * x + v * y \<le> a"
```
```  1322 proof-
```
```  1323   from assms have "u * x + v * y \<le> u * a + v * a"
```
```  1324     by (simp add: add_mono mult_left_mono)
```
```  1325   with assms show ?thesis
```
```  1326     unfolding distrib_right[symmetric] by simp
```
```  1327 qed
```
```  1328
```
```  1329 end
```
```  1330
```
```  1331 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
```
```  1332   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
```
```  1333   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
```
```  1334 begin
```
```  1335
```
```  1336 subclass semiring_0_cancel ..
```
```  1337
```
```  1338 subclass linordered_semiring
```
```  1339 proof
```
```  1340   fix a b c :: 'a
```
```  1341   assume A: "a \<le> b" "0 \<le> c"
```
```  1342   from A show "c * a \<le> c * b"
```
```  1343     unfolding le_less
```
```  1344     using mult_strict_left_mono by (cases "c = 0") auto
```
```  1345   from A show "a * c \<le> b * c"
```
```  1346     unfolding le_less
```
```  1347     using mult_strict_right_mono by (cases "c = 0") auto
```
```  1348 qed
```
```  1349
```
```  1350 lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
```
```  1351   by (auto simp add: mult_strict_left_mono _not_less [symmetric])
```
```  1352
```
```  1353 lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
```
```  1354   by (auto simp add: mult_strict_right_mono not_less [symmetric])
```
```  1355
```
```  1356 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
```
```  1357   using mult_strict_left_mono [of 0 b a] by simp
```
```  1358
```
```  1359 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
```
```  1360   using mult_strict_left_mono [of b 0 a] by simp
```
```  1361
```
```  1362 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
```
```  1363   using mult_strict_right_mono [of a 0 b] by simp
```
```  1364
```
```  1365 text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
```
```  1366 lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
```
```  1367   apply (drule mult_strict_right_mono [of b 0])
```
```  1368   apply auto
```
```  1369   done
```
```  1370
```
```  1371 lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
```
```  1372   apply (cases "b \<le> 0")
```
```  1373    apply (auto simp add: le_less not_less)
```
```  1374   apply (drule_tac mult_pos_neg [of a b])
```
```  1375    apply (auto dest: less_not_sym)
```
```  1376   done
```
```  1377
```
```  1378 lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
```
```  1379   apply (cases "b \<le> 0")
```
```  1380    apply (auto simp add: le_less not_less)
```
```  1381   apply (drule_tac mult_pos_neg2 [of a b])
```
```  1382    apply (auto dest: less_not_sym)
```
```  1383   done
```
```  1384
```
```  1385 text \<open>Strict monotonicity in both arguments\<close>
```
```  1386 lemma mult_strict_mono:
```
```  1387   assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
```
```  1388   shows "a * c < b * d"
```
```  1389   using assms
```
```  1390   apply (cases "c = 0")
```
```  1391   apply simp
```
```  1392   apply (erule mult_strict_right_mono [THEN less_trans])
```
```  1393   apply (auto simp add: le_less)
```
```  1394   apply (erule (1) mult_strict_left_mono)
```
```  1395   done
```
```  1396
```
```  1397 text \<open>This weaker variant has more natural premises\<close>
```
```  1398 lemma mult_strict_mono':
```
```  1399   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
```
```  1400   shows "a * c < b * d"
```
```  1401   by (rule mult_strict_mono) (insert assms, auto)
```
```  1402
```
```  1403 lemma mult_less_le_imp_less:
```
```  1404   assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
```
```  1405   shows "a * c < b * d"
```
```  1406   using assms
```
```  1407   apply (subgoal_tac "a * c < b * c")
```
```  1408   apply (erule less_le_trans)
```
```  1409   apply (erule mult_left_mono)
```
```  1410   apply simp
```
```  1411   apply (erule (1) mult_strict_right_mono)
```
```  1412   done
```
```  1413
```
```  1414 lemma mult_le_less_imp_less:
```
```  1415   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
```
```  1416   shows "a * c < b * d"
```
```  1417   using assms
```
```  1418   apply (subgoal_tac "a * c \<le> b * c")
```
```  1419   apply (erule le_less_trans)
```
```  1420   apply (erule mult_strict_left_mono)
```
```  1421   apply simp
```
```  1422   apply (erule (1) mult_right_mono)
```
```  1423   done
```
```  1424
```
```  1425 end
```
```  1426
```
```  1427 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
```
```  1428 begin
```
```  1429
```
```  1430 subclass linordered_semiring_1 ..
```
```  1431
```
```  1432 lemma convex_bound_lt:
```
```  1433   assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  1434   shows "u * x + v * y < a"
```
```  1435 proof -
```
```  1436   from assms have "u * x + v * y < u * a + v * a"
```
```  1437     by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
```
```  1438   with assms show ?thesis
```
```  1439     unfolding distrib_right[symmetric] by simp
```
```  1440 qed
```
```  1441
```
```  1442 end
```
```  1443
```
```  1444 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
```
```  1445   assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
```
```  1446 begin
```
```  1447
```
```  1448 subclass ordered_semiring
```
```  1449 proof
```
```  1450   fix a b c :: 'a
```
```  1451   assume "a \<le> b" "0 \<le> c"
```
```  1452   then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
```
```  1453   then show "a * c \<le> b * c" by (simp only: mult.commute)
```
```  1454 qed
```
```  1455
```
```  1456 end
```
```  1457
```
```  1458 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
```
```  1459 begin
```
```  1460
```
```  1461 subclass comm_semiring_0_cancel ..
```
```  1462 subclass ordered_comm_semiring ..
```
```  1463 subclass ordered_cancel_semiring ..
```
```  1464
```
```  1465 end
```
```  1466
```
```  1467 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
```
```  1468   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
```
```  1469 begin
```
```  1470
```
```  1471 subclass linordered_semiring_strict
```
```  1472 proof
```
```  1473   fix a b c :: 'a
```
```  1474   assume "a < b" "0 < c"
```
```  1475   then show "c * a < c * b" by (rule comm_mult_strict_left_mono)
```
```  1476   then show "a * c < b * c" by (simp only: mult.commute)
```
```  1477 qed
```
```  1478
```
```  1479 subclass ordered_cancel_comm_semiring
```
```  1480 proof
```
```  1481   fix a b c :: 'a
```
```  1482   assume "a \<le> b" "0 \<le> c"
```
```  1483   then show "c * a \<le> c * b"
```
```  1484     unfolding le_less
```
```  1485     using mult_strict_left_mono by (cases "c = 0") auto
```
```  1486 qed
```
```  1487
```
```  1488 end
```
```  1489
```
```  1490 class ordered_ring = ring + ordered_cancel_semiring
```
```  1491 begin
```
```  1492
```
```  1493 subclass ordered_ab_group_add ..
```
```  1494
```
```  1495 lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
```
```  1496   by (simp add: algebra_simps)
```
```  1497
```
```  1498 lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
```
```  1499   by (simp add: algebra_simps)
```
```  1500
```
```  1501 lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
```
```  1502   by (simp add: algebra_simps)
```
```  1503
```
```  1504 lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
```
```  1505   by (simp add: algebra_simps)
```
```  1506
```
```  1507 lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
```
```  1508   apply (drule mult_left_mono [of _ _ "- c"])
```
```  1509   apply simp_all
```
```  1510   done
```
```  1511
```
```  1512 lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
```
```  1513   apply (drule mult_right_mono [of _ _ "- c"])
```
```  1514   apply simp_all
```
```  1515   done
```
```  1516
```
```  1517 lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
```
```  1518   using mult_right_mono_neg [of a 0 b] by simp
```
```  1519
```
```  1520 lemma split_mult_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
```
```  1521   by (auto simp add: mult_nonpos_nonpos)
```
```  1522
```
```  1523 end
```
```  1524
```
```  1525 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
```
```  1526 begin
```
```  1527
```
```  1528 subclass ordered_ring ..
```
```  1529
```
```  1530 subclass ordered_ab_group_add_abs
```
```  1531 proof
```
```  1532   fix a b
```
```  1533   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
```
```  1534     by (auto simp add: abs_if not_le not_less algebra_simps
```
```  1535         simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
```
```  1536 qed (auto simp add: abs_if)
```
```  1537
```
```  1538 lemma zero_le_square [simp]: "0 \<le> a * a"
```
```  1539   using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
```
```  1540
```
```  1541 lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
```
```  1542   by (simp add: not_less)
```
```  1543
```
```  1544 proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
```
```  1545   by (auto simp add: abs_if split: if_split_asm)
```
```  1546
```
```  1547 lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
```
```  1548   by (intro add_nonneg_nonneg zero_le_square)
```
```  1549
```
```  1550 lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
```
```  1551   by (simp add: not_less sum_squares_ge_zero)
```
```  1552
```
```  1553 end
```
```  1554
```
```  1555 class linordered_ring_strict = ring + linordered_semiring_strict
```
```  1556   + ordered_ab_group_add + abs_if
```
```  1557 begin
```
```  1558
```
```  1559 subclass linordered_ring ..
```
```  1560
```
```  1561 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
```
```  1562   using mult_strict_left_mono [of b a "- c"] by simp
```
```  1563
```
```  1564 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
```
```  1565   using mult_strict_right_mono [of b a "- c"] by simp
```
```  1566
```
```  1567 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
```
```  1568   using mult_strict_right_mono_neg [of a 0 b] by simp
```
```  1569
```
```  1570 subclass ring_no_zero_divisors
```
```  1571 proof
```
```  1572   fix a b
```
```  1573   assume "a \<noteq> 0"
```
```  1574   then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
```
```  1575   assume "b \<noteq> 0"
```
```  1576   then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
```
```  1577   have "a * b < 0 \<or> 0 < a * b"
```
```  1578   proof (cases "a < 0")
```
```  1579     case A': True
```
```  1580     show ?thesis
```
```  1581     proof (cases "b < 0")
```
```  1582       case True
```
```  1583       with A' show ?thesis by (auto dest: mult_neg_neg)
```
```  1584     next
```
```  1585       case False
```
```  1586       with B have "0 < b" by auto
```
```  1587       with A' show ?thesis by (auto dest: mult_strict_right_mono)
```
```  1588     qed
```
```  1589   next
```
```  1590     case False
```
```  1591     with A have A': "0 < a" by auto
```
```  1592     show ?thesis
```
```  1593     proof (cases "b < 0")
```
```  1594       case True
```
```  1595       with A' show ?thesis
```
```  1596         by (auto dest: mult_strict_right_mono_neg)
```
```  1597     next
```
```  1598       case False
```
```  1599       with B have "0 < b" by auto
```
```  1600       with A' show ?thesis by auto
```
```  1601     qed
```
```  1602   qed
```
```  1603   then show "a * b \<noteq> 0"
```
```  1604     by (simp add: neq_iff)
```
```  1605 qed
```
```  1606
```
```  1607 lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
```
```  1608   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
```
```  1609      (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
```
```  1610
```
```  1611 lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
```
```  1612   by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
```
```  1613
```
```  1614 lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
```
```  1615   using zero_less_mult_iff [of "- a" b] by auto
```
```  1616
```
```  1617 lemma mult_le_0_iff: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
```
```  1618   using zero_le_mult_iff [of "- a" b] by auto
```
```  1619
```
```  1620 text \<open>
```
```  1621   Cancellation laws for @{term "c * a < c * b"} and @{term "a * c < b * c"},
```
```  1622   also with the relations \<open>\<le>\<close> and equality.
```
```  1623 \<close>
```
```  1624
```
```  1625 text \<open>
```
```  1626   These ``disjunction'' versions produce two cases when the comparison is
```
```  1627   an assumption, but effectively four when the comparison is a goal.
```
```  1628 \<close>
```
```  1629
```
```  1630 lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
```
```  1631   apply (cases "c = 0")
```
```  1632   apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
```
```  1633   apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
```
```  1634   apply (erule_tac [!] notE)
```
```  1635   apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
```
```  1636   done
```
```  1637
```
```  1638 lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
```
```  1639   apply (cases "c = 0")
```
```  1640   apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
```
```  1641   apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
```
```  1642   apply (erule_tac [!] notE)
```
```  1643   apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
```
```  1644   done
```
```  1645
```
```  1646 text \<open>
```
```  1647   The ``conjunction of implication'' lemmas produce two cases when the
```
```  1648   comparison is a goal, but give four when the comparison is an assumption.
```
```  1649 \<close>
```
```  1650
```
```  1651 lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
```
```  1652   using mult_less_cancel_right_disj [of a c b] by auto
```
```  1653
```
```  1654 lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
```
```  1655   using mult_less_cancel_left_disj [of c a b] by auto
```
```  1656
```
```  1657 lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1658   by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
```
```  1659
```
```  1660 lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1661   by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
```
```  1662
```
```  1663 lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
```
```  1664   by (auto simp: mult_le_cancel_left)
```
```  1665
```
```  1666 lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
```
```  1667   by (auto simp: mult_le_cancel_left)
```
```  1668
```
```  1669 lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
```
```  1670   by (auto simp: mult_less_cancel_left)
```
```  1671
```
```  1672 lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
```
```  1673   by (auto simp: mult_less_cancel_left)
```
```  1674
```
```  1675 end
```
```  1676
```
```  1677 lemmas mult_sign_intros =
```
```  1678   mult_nonneg_nonneg mult_nonneg_nonpos
```
```  1679   mult_nonpos_nonneg mult_nonpos_nonpos
```
```  1680   mult_pos_pos mult_pos_neg
```
```  1681   mult_neg_pos mult_neg_neg
```
```  1682
```
```  1683 class ordered_comm_ring = comm_ring + ordered_comm_semiring
```
```  1684 begin
```
```  1685
```
```  1686 subclass ordered_ring ..
```
```  1687 subclass ordered_cancel_comm_semiring ..
```
```  1688
```
```  1689 end
```
```  1690
```
```  1691 class zero_less_one = order + zero + one +
```
```  1692   assumes zero_less_one [simp]: "0 < 1"
```
```  1693
```
```  1694 class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one
```
```  1695 begin
```
```  1696
```
```  1697 subclass zero_neq_one
```
```  1698   by standard (insert zero_less_one, blast)
```
```  1699
```
```  1700 subclass comm_semiring_1
```
```  1701   by standard (rule mult_1_left)
```
```  1702
```
```  1703 lemma zero_le_one [simp]: "0 \<le> 1"
```
```  1704   by (rule zero_less_one [THEN less_imp_le])
```
```  1705
```
```  1706 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
```
```  1707   by (simp add: not_le)
```
```  1708
```
```  1709 lemma not_one_less_zero [simp]: "\<not> 1 < 0"
```
```  1710   by (simp add: not_less)
```
```  1711
```
```  1712 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
```
```  1713   using mult_left_mono[of c 1 a] by simp
```
```  1714
```
```  1715 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
```
```  1716   using mult_mono[of a 1 b 1] by simp
```
```  1717
```
```  1718 lemma zero_less_two: "0 < 1 + 1"
```
```  1719   using add_pos_pos[OF zero_less_one zero_less_one] .
```
```  1720
```
```  1721 end
```
```  1722
```
```  1723 class linordered_semidom = semidom + linordered_comm_semiring_strict + zero_less_one +
```
```  1724   assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
```
```  1725 begin
```
```  1726
```
```  1727 subclass linordered_nonzero_semiring ..
```
```  1728
```
```  1729 text \<open>Addition is the inverse of subtraction.\<close>
```
```  1730
```
```  1731 lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
```
```  1732   by (frule le_add_diff_inverse2) (simp add: add.commute)
```
```  1733
```
```  1734 lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
```
```  1735   by simp
```
```  1736
```
```  1737 lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
```
```  1738   apply (subst add_le_cancel_right [where c=k, symmetric])
```
```  1739   apply (frule le_add_diff_inverse2)
```
```  1740   apply (simp only: add.assoc [symmetric])
```
```  1741   using add_implies_diff apply fastforce
```
```  1742   done
```
```  1743
```
```  1744 lemma add_le_add_imp_diff_le:
```
```  1745   assumes 1: "i + k \<le> n"
```
```  1746     and 2: "n \<le> j + k"
```
```  1747   shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
```
```  1748 proof -
```
```  1749   have "n - (i + k) + (i + k) = n"
```
```  1750     using 1 by simp
```
```  1751   moreover have "n - k = n - k - i + i"
```
```  1752     using 1 by (simp add: add_le_imp_le_diff)
```
```  1753   ultimately show ?thesis
```
```  1754     using 2
```
```  1755     apply (simp add: add.assoc [symmetric])
```
```  1756     apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
```
```  1757     apply (simp add: add.commute diff_diff_add)
```
```  1758     done
```
```  1759 qed
```
```  1760
```
```  1761 lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
```
```  1762   using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
```
```  1763
```
```  1764 end
```
```  1765
```
```  1766 class linordered_idom =
```
```  1767   comm_ring_1 + linordered_comm_semiring_strict + ordered_ab_group_add + abs_if + sgn_if
```
```  1768 begin
```
```  1769
```
```  1770 subclass linordered_semiring_1_strict ..
```
```  1771 subclass linordered_ring_strict ..
```
```  1772 subclass ordered_comm_ring ..
```
```  1773 subclass idom ..
```
```  1774
```
```  1775 subclass linordered_semidom
```
```  1776 proof
```
```  1777   have "0 \<le> 1 * 1" by (rule zero_le_square)
```
```  1778   then show "0 < 1" by (simp add: le_less)
```
```  1779   show "b \<le> a \<Longrightarrow> a - b + b = a" for a b
```
```  1780     by simp
```
```  1781 qed
```
```  1782
```
```  1783 lemma linorder_neqE_linordered_idom:
```
```  1784   assumes "x \<noteq> y"
```
```  1785   obtains "x < y" | "y < x"
```
```  1786   using assms by (rule neqE)
```
```  1787
```
```  1788 text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
```
```  1789
```
```  1790 lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
```
```  1791   using mult_le_cancel_right [of 1 c b] by simp
```
```  1792
```
```  1793 lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
```
```  1794   using mult_le_cancel_right [of a c 1] by simp
```
```  1795
```
```  1796 lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
```
```  1797   using mult_le_cancel_left [of c 1 b] by simp
```
```  1798
```
```  1799 lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
```
```  1800   using mult_le_cancel_left [of c a 1] by simp
```
```  1801
```
```  1802 lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
```
```  1803   using mult_less_cancel_right [of 1 c b] by simp
```
```  1804
```
```  1805 lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
```
```  1806   using mult_less_cancel_right [of a c 1] by simp
```
```  1807
```
```  1808 lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
```
```  1809   using mult_less_cancel_left [of c 1 b] by simp
```
```  1810
```
```  1811 lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
```
```  1812   using mult_less_cancel_left [of c a 1] by simp
```
```  1813
```
```  1814 lemma sgn_sgn [simp]: "sgn (sgn a) = sgn a"
```
```  1815   unfolding sgn_if by simp
```
```  1816
```
```  1817 lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
```
```  1818   unfolding sgn_if by simp
```
```  1819
```
```  1820 lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
```
```  1821   unfolding sgn_if by simp
```
```  1822
```
```  1823 lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
```
```  1824   unfolding sgn_if by auto
```
```  1825
```
```  1826 lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
```
```  1827   by (simp only: sgn_1_pos)
```
```  1828
```
```  1829 lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
```
```  1830   by (simp only: sgn_1_neg)
```
```  1831
```
```  1832 lemma sgn_times: "sgn (a * b) = sgn a * sgn b"
```
```  1833   by (auto simp add: sgn_if zero_less_mult_iff)
```
```  1834
```
```  1835 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
```
```  1836   unfolding sgn_if abs_if by auto
```
```  1837
```
```  1838 lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
```
```  1839   unfolding sgn_if by auto
```
```  1840
```
```  1841 lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
```
```  1842   unfolding sgn_if by auto
```
```  1843
```
```  1844 lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
```
```  1845   by (simp add: sgn_if)
```
```  1846
```
```  1847 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
```
```  1848   by (simp add: abs_if)
```
```  1849
```
```  1850 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
```
```  1851   by (simp add: abs_if)
```
```  1852
```
```  1853 lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
```
```  1854   by (subst abs_dvd_iff [symmetric]) simp
```
```  1855
```
```  1856 text \<open>
```
```  1857   The following lemmas can be proven in more general structures, but
```
```  1858   are dangerous as simp rules in absence of @{thm neg_equal_zero},
```
```  1859   @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
```
```  1860 \<close>
```
```  1861
```
```  1862 lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
```
```  1863   by (fact equation_minus_iff)
```
```  1864
```
```  1865 lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
```
```  1866   by (subst minus_equation_iff, auto)
```
```  1867
```
```  1868 lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
```
```  1869   by (fact le_minus_iff)
```
```  1870
```
```  1871 lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
```
```  1872   by (fact minus_le_iff)
```
```  1873
```
```  1874 lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
```
```  1875   by (fact less_minus_iff)
```
```  1876
```
```  1877 lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
```
```  1878   by (fact minus_less_iff)
```
```  1879
```
```  1880 end
```
```  1881
```
```  1882 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
```
```  1883
```
```  1884 lemmas mult_compare_simps =
```
```  1885   mult_le_cancel_right mult_le_cancel_left
```
```  1886   mult_le_cancel_right1 mult_le_cancel_right2
```
```  1887   mult_le_cancel_left1 mult_le_cancel_left2
```
```  1888   mult_less_cancel_right mult_less_cancel_left
```
```  1889   mult_less_cancel_right1 mult_less_cancel_right2
```
```  1890   mult_less_cancel_left1 mult_less_cancel_left2
```
```  1891   mult_cancel_right mult_cancel_left
```
```  1892   mult_cancel_right1 mult_cancel_right2
```
```  1893   mult_cancel_left1 mult_cancel_left2
```
```  1894
```
```  1895
```
```  1896 text \<open>Reasoning about inequalities with division\<close>
```
```  1897
```
```  1898 context linordered_semidom
```
```  1899 begin
```
```  1900
```
```  1901 lemma less_add_one: "a < a + 1"
```
```  1902 proof -
```
```  1903   have "a + 0 < a + 1"
```
```  1904     by (blast intro: zero_less_one add_strict_left_mono)
```
```  1905   then show ?thesis by simp
```
```  1906 qed
```
```  1907
```
```  1908 end
```
```  1909
```
```  1910 context linordered_idom
```
```  1911 begin
```
```  1912
```
```  1913 lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
```
```  1914   by (rule mult_left_le)
```
```  1915
```
```  1916 lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
```
```  1917   by (auto simp add: mult_le_cancel_right2)
```
```  1918
```
```  1919 end
```
```  1920
```
```  1921 text \<open>Absolute Value\<close>
```
```  1922
```
```  1923 context linordered_idom
```
```  1924 begin
```
```  1925
```
```  1926 lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
```
```  1927   unfolding abs_if sgn_if by auto
```
```  1928
```
```  1929 lemma abs_one [simp]: "\<bar>1\<bar> = 1"
```
```  1930   by (simp add: abs_if)
```
```  1931
```
```  1932 end
```
```  1933
```
```  1934 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
```
```  1935   assumes abs_eq_mult:
```
```  1936     "(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>"
```
```  1937
```
```  1938 context linordered_idom
```
```  1939 begin
```
```  1940
```
```  1941 subclass ordered_ring_abs
```
```  1942   by standard (auto simp add: abs_if not_less mult_less_0_iff)
```
```  1943
```
```  1944 lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
```
```  1945   by (rule abs_eq_mult) auto
```
```  1946
```
```  1947 lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
```
```  1948   by (simp add: abs_if)
```
```  1949
```
```  1950 lemma abs_mult_less:
```
```  1951   assumes ac: "\<bar>a\<bar> < c"
```
```  1952     and bd: "\<bar>b\<bar> < d"
```
```  1953   shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
```
```  1954 proof -
```
```  1955   from ac have "0 < c"
```
```  1956     by (blast intro: le_less_trans abs_ge_zero)
```
```  1957   with bd show ?thesis by (simp add: ac mult_strict_mono)
```
```  1958 qed
```
```  1959
```
```  1960 lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
```
```  1961   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
```
```  1962
```
```  1963 lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
```
```  1964   by (simp add: abs_mult)
```
```  1965
```
```  1966 lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
```
```  1967   by (auto simp add: diff_less_eq ac_simps abs_less_iff)
```
```  1968
```
```  1969 lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
```
```  1970   by (auto simp add: diff_le_eq ac_simps abs_le_iff)
```
```  1971
```
```  1972 lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
```
```  1973   by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
```
```  1974
```
```  1975 end
```
```  1976
```
```  1977 subsection \<open>Dioids\<close>
```
```  1978
```
```  1979 text \<open>
```
```  1980   Dioids are the alternative extensions of semirings, a semiring can
```
```  1981   either be a ring or a dioid but never both.
```
```  1982 \<close>
```
```  1983
```
```  1984 class dioid = semiring_1 + canonically_ordered_monoid_add
```
```  1985 begin
```
```  1986
```
```  1987 subclass ordered_semiring
```
```  1988   by standard (auto simp: le_iff_add distrib_left distrib_right)
```
```  1989
```
```  1990 end
```
```  1991
```
```  1992
```
```  1993 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
```
```  1994
```
```  1995 code_identifier
```
```  1996   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1997
```
```  1998 end
```