src/HOL/Rings.thy
 author wenzelm Tue Sep 26 20:54:40 2017 +0200 (21 months ago) changeset 66695 91500c024c7f parent 65811 2653f1cd8775 child 66793 deabce3ccf1f permissions -rw-r--r--
tuned;
```     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"
```
```    46     by (simp add: distrib_right [symmetric])
```
```    47   then show "0 * a = 0"
```
```    48     by (simp only: add_left_cancel)
```
```    49   have "a * 0 + a * 0 = a * 0 + 0"
```
```    50     by (simp add: distrib_left [symmetric])
```
```    51   then show "a * 0 = 0"
```
```    52     by (simp only: add_left_cancel)
```
```    53 qed
```
```    54
```
```    55 end
```
```    56
```
```    57 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
```
```    58   assumes distrib: "(a + b) * c = a * c + b * c"
```
```    59 begin
```
```    60
```
```    61 subclass semiring
```
```    62 proof
```
```    63   fix a b c :: 'a
```
```    64   show "(a + b) * c = a * c + b * c"
```
```    65     by (simp add: distrib)
```
```    66   have "a * (b + c) = (b + c) * a"
```
```    67     by (simp add: ac_simps)
```
```    68   also have "\<dots> = b * a + c * a"
```
```    69     by (simp only: distrib)
```
```    70   also have "\<dots> = a * b + a * c"
```
```    71     by (simp add: ac_simps)
```
```    72   finally show "a * (b + c) = a * b + a * c"
```
```    73     by blast
```
```    74 qed
```
```    75
```
```    76 end
```
```    77
```
```    78 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
```
```    79 begin
```
```    80
```
```    81 subclass semiring_0 ..
```
```    82
```
```    83 end
```
```    84
```
```    85 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
```
```    86 begin
```
```    87
```
```    88 subclass semiring_0_cancel ..
```
```    89
```
```    90 subclass comm_semiring_0 ..
```
```    91
```
```    92 end
```
```    93
```
```    94 class zero_neq_one = zero + one +
```
```    95   assumes zero_neq_one [simp]: "0 \<noteq> 1"
```
```    96 begin
```
```    97
```
```    98 lemma one_neq_zero [simp]: "1 \<noteq> 0"
```
```    99   by (rule not_sym) (rule zero_neq_one)
```
```   100
```
```   101 definition of_bool :: "bool \<Rightarrow> 'a"
```
```   102   where "of_bool p = (if p then 1 else 0)"
```
```   103
```
```   104 lemma of_bool_eq [simp, code]:
```
```   105   "of_bool False = 0"
```
```   106   "of_bool True = 1"
```
```   107   by (simp_all add: of_bool_def)
```
```   108
```
```   109 lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
```
```   110   by (simp add: of_bool_def)
```
```   111
```
```   112 lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
```
```   113   by (cases p) simp_all
```
```   114
```
```   115 lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
```
```   116   by (cases p) simp_all
```
```   117
```
```   118 end
```
```   119
```
```   120 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
```
```   121
```
```   122 text \<open>Abstract divisibility\<close>
```
```   123
```
```   124 class dvd = times
```
```   125 begin
```
```   126
```
```   127 definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
```
```   128   where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
```
```   129
```
```   130 lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
```
```   131   unfolding dvd_def ..
```
```   132
```
```   133 lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
```
```   134   unfolding dvd_def by blast
```
```   135
```
```   136 end
```
```   137
```
```   138 context comm_monoid_mult
```
```   139 begin
```
```   140
```
```   141 subclass dvd .
```
```   142
```
```   143 lemma dvd_refl [simp]: "a dvd a"
```
```   144 proof
```
```   145   show "a = a * 1" by simp
```
```   146 qed
```
```   147
```
```   148 lemma dvd_trans [trans]:
```
```   149   assumes "a dvd b" and "b dvd c"
```
```   150   shows "a dvd c"
```
```   151 proof -
```
```   152   from assms obtain v where "b = a * v"
```
```   153     by (auto elim!: dvdE)
```
```   154   moreover from assms obtain w where "c = b * w"
```
```   155     by (auto elim!: dvdE)
```
```   156   ultimately have "c = a * (v * w)"
```
```   157     by (simp add: mult.assoc)
```
```   158   then show ?thesis ..
```
```   159 qed
```
```   160
```
```   161 lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
```
```   162   by (auto simp add: subset_iff intro: dvd_trans)
```
```   163
```
```   164 lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
```
```   165   by (auto simp add: subset_iff intro: dvd_trans)
```
```   166
```
```   167 lemma one_dvd [simp]: "1 dvd a"
```
```   168   by (auto intro!: dvdI)
```
```   169
```
```   170 lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
```
```   171   by (auto intro!: mult.left_commute dvdI elim!: dvdE)
```
```   172
```
```   173 lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
```
```   174   using dvd_mult [of a b c] by (simp add: ac_simps)
```
```   175
```
```   176 lemma dvd_triv_right [simp]: "a dvd b * a"
```
```   177   by (rule dvd_mult) (rule dvd_refl)
```
```   178
```
```   179 lemma dvd_triv_left [simp]: "a dvd a * b"
```
```   180   by (rule dvd_mult2) (rule dvd_refl)
```
```   181
```
```   182 lemma mult_dvd_mono:
```
```   183   assumes "a dvd b"
```
```   184     and "c dvd d"
```
```   185   shows "a * c dvd b * d"
```
```   186 proof -
```
```   187   from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
```
```   188   moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
```
```   189   ultimately have "b * d = (a * c) * (b' * d')"
```
```   190     by (simp add: ac_simps)
```
```   191   then show ?thesis ..
```
```   192 qed
```
```   193
```
```   194 lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
```
```   195   by (simp add: dvd_def mult.assoc) blast
```
```   196
```
```   197 lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
```
```   198   using dvd_mult_left [of b a c] by (simp add: ac_simps)
```
```   199
```
```   200 end
```
```   201
```
```   202 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
```
```   203 begin
```
```   204
```
```   205 subclass semiring_1 ..
```
```   206
```
```   207 lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
```
```   208   by (auto intro: dvd_refl elim!: dvdE)
```
```   209
```
```   210 lemma dvd_0_right [iff]: "a dvd 0"
```
```   211 proof
```
```   212   show "0 = a * 0" by simp
```
```   213 qed
```
```   214
```
```   215 lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
```
```   216   by simp
```
```   217
```
```   218 lemma dvd_add [simp]:
```
```   219   assumes "a dvd b" and "a dvd c"
```
```   220   shows "a dvd (b + c)"
```
```   221 proof -
```
```   222   from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
```
```   223   moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
```
```   224   ultimately have "b + c = a * (b' + c')"
```
```   225     by (simp add: distrib_left)
```
```   226   then show ?thesis ..
```
```   227 qed
```
```   228
```
```   229 end
```
```   230
```
```   231 class semiring_1_cancel = semiring + cancel_comm_monoid_add
```
```   232   + zero_neq_one + monoid_mult
```
```   233 begin
```
```   234
```
```   235 subclass semiring_0_cancel ..
```
```   236
```
```   237 subclass semiring_1 ..
```
```   238
```
```   239 end
```
```   240
```
```   241 class comm_semiring_1_cancel =
```
```   242   comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
```
```   243   assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
```
```   244 begin
```
```   245
```
```   246 subclass semiring_1_cancel ..
```
```   247 subclass comm_semiring_0_cancel ..
```
```   248 subclass comm_semiring_1 ..
```
```   249
```
```   250 lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
```
```   251   by (simp add: algebra_simps)
```
```   252
```
```   253 lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
```
```   254 proof -
```
```   255   have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
```
```   256   proof
```
```   257     assume ?Q
```
```   258     then show ?P by simp
```
```   259   next
```
```   260     assume ?P
```
```   261     then obtain d where "a * c + b = a * d" ..
```
```   262     then have "a * c + b - a * c = a * d - a * c" by simp
```
```   263     then have "b = a * d - a * c" by simp
```
```   264     then have "b = a * (d - c)" by (simp add: algebra_simps)
```
```   265     then show ?Q ..
```
```   266   qed
```
```   267   then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
```
```   268 qed
```
```   269
```
```   270 lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
```
```   271   using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
```
```   272
```
```   273 lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
```
```   274   using dvd_add_times_triv_left_iff [of a 1 b] by simp
```
```   275
```
```   276 lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
```
```   277   using dvd_add_times_triv_right_iff [of a b 1] by simp
```
```   278
```
```   279 lemma dvd_add_right_iff:
```
```   280   assumes "a dvd b"
```
```   281   shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
```
```   282 proof
```
```   283   assume ?P
```
```   284   then obtain d where "b + c = a * d" ..
```
```   285   moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
```
```   286   ultimately have "a * e + c = a * d" by simp
```
```   287   then have "a * e + c - a * e = a * d - a * e" by simp
```
```   288   then have "c = a * d - a * e" by simp
```
```   289   then have "c = a * (d - e)" by (simp add: algebra_simps)
```
```   290   then show ?Q ..
```
```   291 next
```
```   292   assume ?Q
```
```   293   with assms show ?P by simp
```
```   294 qed
```
```   295
```
```   296 lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
```
```   297   using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
```
```   298
```
```   299 end
```
```   300
```
```   301 class ring = semiring + ab_group_add
```
```   302 begin
```
```   303
```
```   304 subclass semiring_0_cancel ..
```
```   305
```
```   306 text \<open>Distribution rules\<close>
```
```   307
```
```   308 lemma minus_mult_left: "- (a * b) = - a * b"
```
```   309   by (rule minus_unique) (simp add: distrib_right [symmetric])
```
```   310
```
```   311 lemma minus_mult_right: "- (a * b) = a * - b"
```
```   312   by (rule minus_unique) (simp add: distrib_left [symmetric])
```
```   313
```
```   314 text \<open>Extract signs from products\<close>
```
```   315 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
```
```   316 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
```
```   317
```
```   318 lemma minus_mult_minus [simp]: "- a * - b = a * b"
```
```   319   by simp
```
```   320
```
```   321 lemma minus_mult_commute: "- a * b = a * - b"
```
```   322   by simp
```
```   323
```
```   324 lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
```
```   325   using distrib_left [of a b "-c "] by simp
```
```   326
```
```   327 lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
```
```   328   using distrib_right [of a "- b" c] by simp
```
```   329
```
```   330 lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
```
```   331
```
```   332 lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
```
```   333   by (simp add: algebra_simps)
```
```   334
```
```   335 lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
```
```   336   by (simp add: algebra_simps)
```
```   337
```
```   338 end
```
```   339
```
```   340 lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
```
```   341
```
```   342 class comm_ring = comm_semiring + ab_group_add
```
```   343 begin
```
```   344
```
```   345 subclass ring ..
```
```   346 subclass comm_semiring_0_cancel ..
```
```   347
```
```   348 lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
```
```   349   by (simp add: algebra_simps)
```
```   350
```
```   351 end
```
```   352
```
```   353 class ring_1 = ring + zero_neq_one + monoid_mult
```
```   354 begin
```
```   355
```
```   356 subclass semiring_1_cancel ..
```
```   357
```
```   358 lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
```
```   359   by (simp add: algebra_simps)
```
```   360
```
```   361 end
```
```   362
```
```   363 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
```
```   364 begin
```
```   365
```
```   366 subclass ring_1 ..
```
```   367 subclass comm_semiring_1_cancel
```
```   368   by unfold_locales (simp add: algebra_simps)
```
```   369
```
```   370 lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
```
```   371 proof
```
```   372   assume "x dvd - y"
```
```   373   then have "x dvd - 1 * - y" by (rule dvd_mult)
```
```   374   then show "x dvd y" by simp
```
```   375 next
```
```   376   assume "x dvd y"
```
```   377   then have "x dvd - 1 * y" by (rule dvd_mult)
```
```   378   then show "x dvd - y" by simp
```
```   379 qed
```
```   380
```
```   381 lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
```
```   382 proof
```
```   383   assume "- x dvd y"
```
```   384   then obtain k where "y = - x * k" ..
```
```   385   then have "y = x * - k" by simp
```
```   386   then show "x dvd y" ..
```
```   387 next
```
```   388   assume "x dvd y"
```
```   389   then obtain k where "y = x * k" ..
```
```   390   then have "y = - x * - k" by simp
```
```   391   then show "- x dvd y" ..
```
```   392 qed
```
```   393
```
```   394 lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
```
```   395   using dvd_add [of x y "- z"] by simp
```
```   396
```
```   397 end
```
```   398
```
```   399 class semiring_no_zero_divisors = semiring_0 +
```
```   400   assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
```
```   401 begin
```
```   402
```
```   403 lemma divisors_zero:
```
```   404   assumes "a * b = 0"
```
```   405   shows "a = 0 \<or> b = 0"
```
```   406 proof (rule classical)
```
```   407   assume "\<not> ?thesis"
```
```   408   then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   409   with no_zero_divisors have "a * b \<noteq> 0" by blast
```
```   410   with assms show ?thesis by simp
```
```   411 qed
```
```   412
```
```   413 lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   414 proof (cases "a = 0 \<or> b = 0")
```
```   415   case False
```
```   416   then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   417     then show ?thesis using no_zero_divisors by simp
```
```   418 next
```
```   419   case True
```
```   420   then show ?thesis by auto
```
```   421 qed
```
```   422
```
```   423 end
```
```   424
```
```   425 class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
```
```   426
```
```   427 class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
```
```   428   assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
```
```   429     and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
```
```   430 begin
```
```   431
```
```   432 lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
```
```   433   by simp
```
```   434
```
```   435 lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
```
```   436   by simp
```
```   437
```
```   438 end
```
```   439
```
```   440 class ring_no_zero_divisors = ring + semiring_no_zero_divisors
```
```   441 begin
```
```   442
```
```   443 subclass semiring_no_zero_divisors_cancel
```
```   444 proof
```
```   445   fix a b c
```
```   446   have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
```
```   447     by (simp add: algebra_simps)
```
```   448   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
```
```   449     by auto
```
```   450   finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
```
```   451   have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
```
```   452     by (simp add: algebra_simps)
```
```   453   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
```
```   454     by auto
```
```   455   finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
```
```   456 qed
```
```   457
```
```   458 end
```
```   459
```
```   460 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
```
```   461 begin
```
```   462
```
```   463 subclass semiring_1_no_zero_divisors ..
```
```   464
```
```   465 lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
```
```   466 proof -
```
```   467   have "(x - 1) * (x + 1) = x * x - 1"
```
```   468     by (simp add: algebra_simps)
```
```   469   then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
```
```   470     by simp
```
```   471   then show ?thesis
```
```   472     by (simp add: eq_neg_iff_add_eq_0)
```
```   473 qed
```
```   474
```
```   475 lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
```
```   476   using mult_cancel_right [of 1 c b] by auto
```
```   477
```
```   478 lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
```
```   479   using mult_cancel_right [of a c 1] by simp
```
```   480
```
```   481 lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
```
```   482   using mult_cancel_left [of c 1 b] by force
```
```   483
```
```   484 lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
```
```   485   using mult_cancel_left [of c a 1] by simp
```
```   486
```
```   487 end
```
```   488
```
```   489 class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
```
```   490 begin
```
```   491
```
```   492 subclass semiring_1_no_zero_divisors ..
```
```   493
```
```   494 end
```
```   495
```
```   496 class idom = comm_ring_1 + semiring_no_zero_divisors
```
```   497 begin
```
```   498
```
```   499 subclass semidom ..
```
```   500
```
```   501 subclass ring_1_no_zero_divisors ..
```
```   502
```
```   503 lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   504 proof -
```
```   505   have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
```
```   506     unfolding dvd_def by (simp add: ac_simps)
```
```   507   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   508     unfolding dvd_def by simp
```
```   509   finally show ?thesis .
```
```   510 qed
```
```   511
```
```   512 lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   513 proof -
```
```   514   have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
```
```   515     unfolding dvd_def by (simp add: ac_simps)
```
```   516   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
```
```   517     unfolding dvd_def by simp
```
```   518   finally show ?thesis .
```
```   519 qed
```
```   520
```
```   521 lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
```
```   522 proof
```
```   523   assume "a * a = b * b"
```
```   524   then have "(a - b) * (a + b) = 0"
```
```   525     by (simp add: algebra_simps)
```
```   526   then show "a = b \<or> a = - b"
```
```   527     by (simp add: eq_neg_iff_add_eq_0)
```
```   528 next
```
```   529   assume "a = b \<or> a = - b"
```
```   530   then show "a * a = b * b" by auto
```
```   531 qed
```
```   532
```
```   533 end
```
```   534
```
```   535 class idom_abs_sgn = idom + abs + sgn +
```
```   536   assumes sgn_mult_abs: "sgn a * \<bar>a\<bar> = a"
```
```   537     and sgn_sgn [simp]: "sgn (sgn a) = sgn a"
```
```   538     and abs_abs [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
```
```   539     and abs_0 [simp]: "\<bar>0\<bar> = 0"
```
```   540     and sgn_0 [simp]: "sgn 0 = 0"
```
```   541     and sgn_1 [simp]: "sgn 1 = 1"
```
```   542     and sgn_minus_1: "sgn (- 1) = - 1"
```
```   543     and sgn_mult: "sgn (a * b) = sgn a * sgn b"
```
```   544 begin
```
```   545
```
```   546 lemma sgn_eq_0_iff:
```
```   547   "sgn a = 0 \<longleftrightarrow> a = 0"
```
```   548 proof -
```
```   549   { assume "sgn a = 0"
```
```   550     then have "sgn a * \<bar>a\<bar> = 0"
```
```   551       by simp
```
```   552     then have "a = 0"
```
```   553       by (simp add: sgn_mult_abs)
```
```   554   } then show ?thesis
```
```   555     by auto
```
```   556 qed
```
```   557
```
```   558 lemma abs_eq_0_iff:
```
```   559   "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
```
```   560 proof -
```
```   561   { assume "\<bar>a\<bar> = 0"
```
```   562     then have "sgn a * \<bar>a\<bar> = 0"
```
```   563       by simp
```
```   564     then have "a = 0"
```
```   565       by (simp add: sgn_mult_abs)
```
```   566   } then show ?thesis
```
```   567     by auto
```
```   568 qed
```
```   569
```
```   570 lemma abs_mult_sgn:
```
```   571   "\<bar>a\<bar> * sgn a = a"
```
```   572   using sgn_mult_abs [of a] by (simp add: ac_simps)
```
```   573
```
```   574 lemma abs_1 [simp]:
```
```   575   "\<bar>1\<bar> = 1"
```
```   576   using sgn_mult_abs [of 1] by simp
```
```   577
```
```   578 lemma sgn_abs [simp]:
```
```   579   "\<bar>sgn a\<bar> = of_bool (a \<noteq> 0)"
```
```   580   using sgn_mult_abs [of "sgn a"] mult_cancel_left [of "sgn a" "\<bar>sgn a\<bar>" 1]
```
```   581   by (auto simp add: sgn_eq_0_iff)
```
```   582
```
```   583 lemma abs_sgn [simp]:
```
```   584   "sgn \<bar>a\<bar> = of_bool (a \<noteq> 0)"
```
```   585   using sgn_mult_abs [of "\<bar>a\<bar>"] mult_cancel_right [of "sgn \<bar>a\<bar>" "\<bar>a\<bar>" 1]
```
```   586   by (auto simp add: abs_eq_0_iff)
```
```   587
```
```   588 lemma abs_mult:
```
```   589   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
```
```   590 proof (cases "a = 0 \<or> b = 0")
```
```   591   case True
```
```   592   then show ?thesis
```
```   593     by auto
```
```   594 next
```
```   595   case False
```
```   596   then have *: "sgn (a * b) \<noteq> 0"
```
```   597     by (simp add: sgn_eq_0_iff)
```
```   598   from abs_mult_sgn [of "a * b"] abs_mult_sgn [of a] abs_mult_sgn [of b]
```
```   599   have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * sgn a * \<bar>b\<bar> * sgn b"
```
```   600     by (simp add: ac_simps)
```
```   601   then have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * \<bar>b\<bar> * sgn (a * b)"
```
```   602     by (simp add: sgn_mult ac_simps)
```
```   603   with * show ?thesis
```
```   604     by simp
```
```   605 qed
```
```   606
```
```   607 lemma sgn_minus [simp]:
```
```   608   "sgn (- a) = - sgn a"
```
```   609 proof -
```
```   610   from sgn_minus_1 have "sgn (- 1 * a) = - 1 * sgn a"
```
```   611     by (simp only: sgn_mult)
```
```   612   then show ?thesis
```
```   613     by simp
```
```   614 qed
```
```   615
```
```   616 lemma abs_minus [simp]:
```
```   617   "\<bar>- a\<bar> = \<bar>a\<bar>"
```
```   618 proof -
```
```   619   have [simp]: "\<bar>- 1\<bar> = 1"
```
```   620     using sgn_mult_abs [of "- 1"] by simp
```
```   621   then have "\<bar>- 1 * a\<bar> = 1 * \<bar>a\<bar>"
```
```   622     by (simp only: abs_mult)
```
```   623   then show ?thesis
```
```   624     by simp
```
```   625 qed
```
```   626
```
```   627 end
```
```   628
```
```   629 text \<open>
```
```   630   The theory of partially ordered rings is taken from the books:
```
```   631     \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
```
```   632     \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
```
```   633
```
```   634   Most of the used notions can also be looked up in
```
```   635     \<^item> \<^url>\<open>http://www.mathworld.com\<close> by Eric Weisstein et. al.
```
```   636     \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
```
```   637 \<close>
```
```   638
```
```   639 text \<open>Syntactic division operator\<close>
```
```   640
```
```   641 class divide =
```
```   642   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
```
```   643
```
```   644 setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
```
```   645
```
```   646 context semiring
```
```   647 begin
```
```   648
```
```   649 lemma [field_simps]:
```
```   650   shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
```
```   651     and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
```
```   652   by (rule distrib_left distrib_right)+
```
```   653
```
```   654 end
```
```   655
```
```   656 context ring
```
```   657 begin
```
```   658
```
```   659 lemma [field_simps]:
```
```   660   shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
```
```   661     and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
```
```   662   by (rule left_diff_distrib right_diff_distrib)+
```
```   663
```
```   664 end
```
```   665
```
```   666 setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
```
```   667
```
```   668 text \<open>Algebraic classes with division\<close>
```
```   669
```
```   670 class semidom_divide = semidom + divide +
```
```   671   assumes nonzero_mult_div_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
```
```   672   assumes div_by_0 [simp]: "a div 0 = 0"
```
```   673 begin
```
```   674
```
```   675 lemma nonzero_mult_div_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
```
```   676   using nonzero_mult_div_cancel_right [of a b] by (simp add: ac_simps)
```
```   677
```
```   678 subclass semiring_no_zero_divisors_cancel
```
```   679 proof
```
```   680   show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
```
```   681   proof (cases "c = 0")
```
```   682     case True
```
```   683     then show ?thesis by simp
```
```   684   next
```
```   685     case False
```
```   686     have "a = b" if "a * c = b * c"
```
```   687     proof -
```
```   688       from that have "a * c div c = b * c div c"
```
```   689         by simp
```
```   690       with False show ?thesis
```
```   691         by simp
```
```   692     qed
```
```   693     then show ?thesis by auto
```
```   694   qed
```
```   695   show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
```
```   696     using * [of a c b] by (simp add: ac_simps)
```
```   697 qed
```
```   698
```
```   699 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
```
```   700   using nonzero_mult_div_cancel_left [of a 1] by simp
```
```   701
```
```   702 lemma div_0 [simp]: "0 div a = 0"
```
```   703 proof (cases "a = 0")
```
```   704   case True
```
```   705   then show ?thesis by simp
```
```   706 next
```
```   707   case False
```
```   708   then have "a * 0 div a = 0"
```
```   709     by (rule nonzero_mult_div_cancel_left)
```
```   710   then show ?thesis by simp
```
```   711 qed
```
```   712
```
```   713 lemma div_by_1 [simp]: "a div 1 = a"
```
```   714   using nonzero_mult_div_cancel_left [of 1 a] by simp
```
```   715
```
```   716 lemma dvd_div_eq_0_iff:
```
```   717   assumes "b dvd a"
```
```   718   shows "a div b = 0 \<longleftrightarrow> a = 0"
```
```   719   using assms by (elim dvdE, cases "b = 0") simp_all
```
```   720
```
```   721 lemma dvd_div_eq_cancel:
```
```   722   "a div c = b div c \<Longrightarrow> c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a = b"
```
```   723   by (elim dvdE, cases "c = 0") simp_all
```
```   724
```
```   725 lemma dvd_div_eq_iff:
```
```   726   "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a div c = b div c \<longleftrightarrow> a = b"
```
```   727   by (elim dvdE, cases "c = 0") simp_all
```
```   728
```
```   729 end
```
```   730
```
```   731 class idom_divide = idom + semidom_divide
```
```   732 begin
```
```   733
```
```   734 lemma dvd_neg_div:
```
```   735   assumes "b dvd a"
```
```   736   shows "- a div b = - (a div b)"
```
```   737 proof (cases "b = 0")
```
```   738   case True
```
```   739   then show ?thesis by simp
```
```   740 next
```
```   741   case False
```
```   742   from assms obtain c where "a = b * c" ..
```
```   743   then have "- a div b = (b * - c) div b"
```
```   744     by simp
```
```   745   from False also have "\<dots> = - c"
```
```   746     by (rule nonzero_mult_div_cancel_left)
```
```   747   with False \<open>a = b * c\<close> show ?thesis
```
```   748     by simp
```
```   749 qed
```
```   750
```
```   751 lemma dvd_div_neg:
```
```   752   assumes "b dvd a"
```
```   753   shows "a div - b = - (a div b)"
```
```   754 proof (cases "b = 0")
```
```   755   case True
```
```   756   then show ?thesis by simp
```
```   757 next
```
```   758   case False
```
```   759   then have "- b \<noteq> 0"
```
```   760     by simp
```
```   761   from assms obtain c where "a = b * c" ..
```
```   762   then have "a div - b = (- b * - c) div - b"
```
```   763     by simp
```
```   764   from \<open>- b \<noteq> 0\<close> also have "\<dots> = - c"
```
```   765     by (rule nonzero_mult_div_cancel_left)
```
```   766   with False \<open>a = b * c\<close> show ?thesis
```
```   767     by simp
```
```   768 qed
```
```   769
```
```   770 end
```
```   771
```
```   772 class algebraic_semidom = semidom_divide
```
```   773 begin
```
```   774
```
```   775 text \<open>
```
```   776   Class @{class algebraic_semidom} enriches a integral domain
```
```   777   by notions from algebra, like units in a ring.
```
```   778   It is a separate class to avoid spoiling fields with notions
```
```   779   which are degenerated there.
```
```   780 \<close>
```
```   781
```
```   782 lemma dvd_times_left_cancel_iff [simp]:
```
```   783   assumes "a \<noteq> 0"
```
```   784   shows "a * b dvd a * c \<longleftrightarrow> b dvd c"
```
```   785     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   786 proof
```
```   787   assume ?lhs
```
```   788   then obtain d where "a * c = a * b * d" ..
```
```   789   with assms have "c = b * d" by (simp add: ac_simps)
```
```   790   then show ?rhs ..
```
```   791 next
```
```   792   assume ?rhs
```
```   793   then obtain d where "c = b * d" ..
```
```   794   then have "a * c = a * b * d" by (simp add: ac_simps)
```
```   795   then show ?lhs ..
```
```   796 qed
```
```   797
```
```   798 lemma dvd_times_right_cancel_iff [simp]:
```
```   799   assumes "a \<noteq> 0"
```
```   800   shows "b * a dvd c * a \<longleftrightarrow> b dvd c"
```
```   801   using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
```
```   802
```
```   803 lemma div_dvd_iff_mult:
```
```   804   assumes "b \<noteq> 0" and "b dvd a"
```
```   805   shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
```
```   806 proof -
```
```   807   from \<open>b dvd a\<close> obtain d where "a = b * d" ..
```
```   808   with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
```
```   809 qed
```
```   810
```
```   811 lemma dvd_div_iff_mult:
```
```   812   assumes "c \<noteq> 0" and "c dvd b"
```
```   813   shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
```
```   814 proof -
```
```   815   from \<open>c dvd b\<close> obtain d where "b = c * d" ..
```
```   816   with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
```
```   817 qed
```
```   818
```
```   819 lemma div_dvd_div [simp]:
```
```   820   assumes "a dvd b" and "a dvd c"
```
```   821   shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
```
```   822 proof (cases "a = 0")
```
```   823   case True
```
```   824   with assms show ?thesis by simp
```
```   825 next
```
```   826   case False
```
```   827   moreover from assms obtain k l where "b = a * k" and "c = a * l"
```
```   828     by (auto elim!: dvdE)
```
```   829   ultimately show ?thesis by simp
```
```   830 qed
```
```   831
```
```   832 lemma div_add [simp]:
```
```   833   assumes "c dvd a" and "c dvd b"
```
```   834   shows "(a + b) div c = a div c + b div c"
```
```   835 proof (cases "c = 0")
```
```   836   case True
```
```   837   then show ?thesis by simp
```
```   838 next
```
```   839   case False
```
```   840   moreover from assms obtain k l where "a = c * k" and "b = c * l"
```
```   841     by (auto elim!: dvdE)
```
```   842   moreover have "c * k + c * l = c * (k + l)"
```
```   843     by (simp add: algebra_simps)
```
```   844   ultimately show ?thesis
```
```   845     by simp
```
```   846 qed
```
```   847
```
```   848 lemma div_mult_div_if_dvd:
```
```   849   assumes "b dvd a" and "d dvd c"
```
```   850   shows "(a div b) * (c div d) = (a * c) div (b * d)"
```
```   851 proof (cases "b = 0 \<or> c = 0")
```
```   852   case True
```
```   853   with assms show ?thesis by auto
```
```   854 next
```
```   855   case False
```
```   856   moreover from assms obtain k l where "a = b * k" and "c = d * l"
```
```   857     by (auto elim!: dvdE)
```
```   858   moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)"
```
```   859     by (simp add: ac_simps)
```
```   860   ultimately show ?thesis by simp
```
```   861 qed
```
```   862
```
```   863 lemma dvd_div_eq_mult:
```
```   864   assumes "a \<noteq> 0" and "a dvd b"
```
```   865   shows "b div a = c \<longleftrightarrow> b = c * a"
```
```   866     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   867 proof
```
```   868   assume ?rhs
```
```   869   then show ?lhs by (simp add: assms)
```
```   870 next
```
```   871   assume ?lhs
```
```   872   then have "b div a * a = c * a" by simp
```
```   873   moreover from assms have "b div a * a = b"
```
```   874     by (auto elim!: dvdE simp add: ac_simps)
```
```   875   ultimately show ?rhs by simp
```
```   876 qed
```
```   877
```
```   878 lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
```
```   879   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
```
```   880
```
```   881 lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
```
```   882   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
```
```   883
```
```   884 lemma div_mult_swap:
```
```   885   assumes "c dvd b"
```
```   886   shows "a * (b div c) = (a * b) div c"
```
```   887 proof (cases "c = 0")
```
```   888   case True
```
```   889   then show ?thesis by simp
```
```   890 next
```
```   891   case False
```
```   892   from assms obtain d where "b = c * d" ..
```
```   893   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
```
```   894     by simp
```
```   895   ultimately show ?thesis by (simp add: ac_simps)
```
```   896 qed
```
```   897
```
```   898 lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
```
```   899   using div_mult_swap [of c b a] by (simp add: ac_simps)
```
```   900
```
```   901 lemma dvd_div_mult2_eq:
```
```   902   assumes "b * c dvd a"
```
```   903   shows "a div (b * c) = a div b div c"
```
```   904 proof -
```
```   905   from assms obtain k where "a = b * c * k" ..
```
```   906   then show ?thesis
```
```   907     by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
```
```   908 qed
```
```   909
```
```   910 lemma dvd_div_div_eq_mult:
```
```   911   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
```
```   912   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
```
```   913     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   914 proof -
```
```   915   from assms have "a * c \<noteq> 0" by simp
```
```   916   then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
```
```   917     by simp
```
```   918   also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
```
```   919     by (simp add: ac_simps)
```
```   920   also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
```
```   921     using assms by (simp add: div_mult_swap)
```
```   922   also have "\<dots> \<longleftrightarrow> ?rhs"
```
```   923     using assms by (simp add: ac_simps)
```
```   924   finally show ?thesis .
```
```   925 qed
```
```   926
```
```   927 lemma dvd_mult_imp_div:
```
```   928   assumes "a * c dvd b"
```
```   929   shows "a dvd b div c"
```
```   930 proof (cases "c = 0")
```
```   931   case True then show ?thesis by simp
```
```   932 next
```
```   933   case False
```
```   934   from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
```
```   935   with False show ?thesis
```
```   936     by (simp add: mult.commute [of a] mult.assoc)
```
```   937 qed
```
```   938
```
```   939 lemma div_div_eq_right:
```
```   940   assumes "c dvd b" "b dvd a"
```
```   941   shows   "a div (b div c) = a div b * c"
```
```   942 proof (cases "c = 0 \<or> b = 0")
```
```   943   case True
```
```   944   then show ?thesis
```
```   945     by auto
```
```   946 next
```
```   947   case False
```
```   948   from assms obtain r s where "b = c * r" and "a = c * r * s"
```
```   949     by (blast elim: dvdE)
```
```   950   moreover with False have "r \<noteq> 0"
```
```   951     by auto
```
```   952   ultimately show ?thesis using False
```
```   953     by simp (simp add: mult.commute [of _ r] mult.assoc mult.commute [of c])
```
```   954 qed
```
```   955
```
```   956 lemma div_div_div_same:
```
```   957   assumes "d dvd b" "b dvd a"
```
```   958   shows   "(a div d) div (b div d) = a div b"
```
```   959 proof (cases "b = 0 \<or> d = 0")
```
```   960   case True
```
```   961   with assms show ?thesis
```
```   962     by auto
```
```   963 next
```
```   964   case False
```
```   965   from assms obtain r s
```
```   966     where "a = d * r * s" and "b = d * r"
```
```   967     by (blast elim: dvdE)
```
```   968   with False show ?thesis
```
```   969     by simp (simp add: ac_simps)
```
```   970 qed
```
```   971
```
```   972
```
```   973 text \<open>Units: invertible elements in a ring\<close>
```
```   974
```
```   975 abbreviation is_unit :: "'a \<Rightarrow> bool"
```
```   976   where "is_unit a \<equiv> a dvd 1"
```
```   977
```
```   978 lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
```
```   979   by simp
```
```   980
```
```   981 lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
```
```   982   by (rule dvd_trans [of _ 1]) simp_all
```
```   983
```
```   984 lemma unit_dvdE:
```
```   985   assumes "is_unit a"
```
```   986   obtains c where "a \<noteq> 0" and "b = a * c"
```
```   987 proof -
```
```   988   from assms have "a dvd b" by auto
```
```   989   then obtain c where "b = a * c" ..
```
```   990   moreover from assms have "a \<noteq> 0" by auto
```
```   991   ultimately show thesis using that by blast
```
```   992 qed
```
```   993
```
```   994 lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
```
```   995   by (rule dvd_trans)
```
```   996
```
```   997 lemma unit_div_1_unit [simp, intro]:
```
```   998   assumes "is_unit a"
```
```   999   shows "is_unit (1 div a)"
```
```  1000 proof -
```
```  1001   from assms have "1 = 1 div a * a" by simp
```
```  1002   then show "is_unit (1 div a)" by (rule dvdI)
```
```  1003 qed
```
```  1004
```
```  1005 lemma is_unitE [elim?]:
```
```  1006   assumes "is_unit a"
```
```  1007   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
```
```  1008     and "is_unit b" and "1 div a = b" and "1 div b = a"
```
```  1009     and "a * b = 1" and "c div a = c * b"
```
```  1010 proof (rule that)
```
```  1011   define b where "b = 1 div a"
```
```  1012   then show "1 div a = b" by simp
```
```  1013   from assms b_def show "is_unit b" by simp
```
```  1014   with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```  1015   from assms b_def show "a * b = 1" by simp
```
```  1016   then have "1 = a * b" ..
```
```  1017   with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
```
```  1018   from assms have "a dvd c" ..
```
```  1019   then obtain d where "c = a * d" ..
```
```  1020   with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
```
```  1021     by (simp add: mult.assoc mult.left_commute [of a])
```
```  1022 qed
```
```  1023
```
```  1024 lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
```
```  1025   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
```
```  1026
```
```  1027 lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
```
```  1028   by (auto dest: dvd_mult_left dvd_mult_right)
```
```  1029
```
```  1030 lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
```
```  1031   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
```
```  1032
```
```  1033 lemma mult_unit_dvd_iff:
```
```  1034   assumes "is_unit b"
```
```  1035   shows "a * b dvd c \<longleftrightarrow> a dvd c"
```
```  1036 proof
```
```  1037   assume "a * b dvd c"
```
```  1038   with assms show "a dvd c"
```
```  1039     by (simp add: dvd_mult_left)
```
```  1040 next
```
```  1041   assume "a dvd c"
```
```  1042   then obtain k where "c = a * k" ..
```
```  1043   with assms have "c = (a * b) * (1 div b * k)"
```
```  1044     by (simp add: mult_ac)
```
```  1045   then show "a * b dvd c" by (rule dvdI)
```
```  1046 qed
```
```  1047
```
```  1048 lemma mult_unit_dvd_iff': "is_unit a \<Longrightarrow> (a * b) dvd c \<longleftrightarrow> b dvd c"
```
```  1049   using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps)
```
```  1050
```
```  1051 lemma dvd_mult_unit_iff:
```
```  1052   assumes "is_unit b"
```
```  1053   shows "a dvd c * b \<longleftrightarrow> a dvd c"
```
```  1054 proof
```
```  1055   assume "a dvd c * b"
```
```  1056   with assms have "c * b dvd c * (b * (1 div b))"
```
```  1057     by (subst mult_assoc [symmetric]) simp
```
```  1058   also from assms have "b * (1 div b) = 1"
```
```  1059     by (rule is_unitE) simp
```
```  1060   finally have "c * b dvd c" by simp
```
```  1061   with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
```
```  1062 next
```
```  1063   assume "a dvd c"
```
```  1064   then show "a dvd c * b" by simp
```
```  1065 qed
```
```  1066
```
```  1067 lemma dvd_mult_unit_iff': "is_unit b \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd c"
```
```  1068   using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps)
```
```  1069
```
```  1070 lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
```
```  1071   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
```
```  1072
```
```  1073 lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
```
```  1074   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
```
```  1075
```
```  1076 lemmas unit_dvd_iff = mult_unit_dvd_iff mult_unit_dvd_iff'
```
```  1077   dvd_mult_unit_iff dvd_mult_unit_iff'
```
```  1078   div_unit_dvd_iff dvd_div_unit_iff (* FIXME consider named_theorems *)
```
```  1079
```
```  1080 lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
```
```  1081   by (erule is_unitE [of _ b]) simp
```
```  1082
```
```  1083 lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
```
```  1084   by (rule dvd_div_mult_self) auto
```
```  1085
```
```  1086 lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
```
```  1087   by (erule is_unitE) simp
```
```  1088
```
```  1089 lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
```
```  1090   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
```
```  1091
```
```  1092 lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
```
```  1093   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
```
```  1094
```
```  1095 lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
```
```  1096   by (auto elim: is_unitE)
```
```  1097
```
```  1098 lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
```
```  1099   using unit_eq_div1 [of b c a] by auto
```
```  1100
```
```  1101 lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
```
```  1102   using mult_cancel_left [of a b c] by auto
```
```  1103
```
```  1104 lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
```
```  1105   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
```
```  1106
```
```  1107 lemma unit_div_cancel:
```
```  1108   assumes "is_unit a"
```
```  1109   shows "b div a = c div a \<longleftrightarrow> b = c"
```
```  1110 proof -
```
```  1111   from assms have "is_unit (1 div a)" by simp
```
```  1112   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
```
```  1113     by (rule unit_mult_right_cancel)
```
```  1114   with assms show ?thesis by simp
```
```  1115 qed
```
```  1116
```
```  1117 lemma is_unit_div_mult2_eq:
```
```  1118   assumes "is_unit b" and "is_unit c"
```
```  1119   shows "a div (b * c) = a div b div c"
```
```  1120 proof -
```
```  1121   from assms have "is_unit (b * c)"
```
```  1122     by (simp add: unit_prod)
```
```  1123   then have "b * c dvd a"
```
```  1124     by (rule unit_imp_dvd)
```
```  1125   then show ?thesis
```
```  1126     by (rule dvd_div_mult2_eq)
```
```  1127 qed
```
```  1128
```
```  1129 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
```
```  1130   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
```
```  1131   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
```
```  1132   unit_eq_div1 unit_eq_div2
```
```  1133
```
```  1134 lemma is_unit_div_mult_cancel_left:
```
```  1135   assumes "a \<noteq> 0" and "is_unit b"
```
```  1136   shows "a div (a * b) = 1 div b"
```
```  1137 proof -
```
```  1138   from assms have "a div (a * b) = a div a div b"
```
```  1139     by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
```
```  1140   with assms show ?thesis by simp
```
```  1141 qed
```
```  1142
```
```  1143 lemma is_unit_div_mult_cancel_right:
```
```  1144   assumes "a \<noteq> 0" and "is_unit b"
```
```  1145   shows "a div (b * a) = 1 div b"
```
```  1146   using assms is_unit_div_mult_cancel_left [of a b] by (simp add: ac_simps)
```
```  1147
```
```  1148 lemma unit_div_eq_0_iff:
```
```  1149   assumes "is_unit b"
```
```  1150   shows "a div b = 0 \<longleftrightarrow> a = 0"
```
```  1151   by (rule dvd_div_eq_0_iff) (insert assms, auto)
```
```  1152
```
```  1153 lemma div_mult_unit2:
```
```  1154   "is_unit c \<Longrightarrow> b dvd a \<Longrightarrow> a div (b * c) = a div b div c"
```
```  1155   by (rule dvd_div_mult2_eq) (simp_all add: mult_unit_dvd_iff)
```
```  1156
```
```  1157 end
```
```  1158
```
```  1159 class unit_factor =
```
```  1160   fixes unit_factor :: "'a \<Rightarrow> 'a"
```
```  1161
```
```  1162 class semidom_divide_unit_factor = semidom_divide + unit_factor +
```
```  1163   assumes unit_factor_0 [simp]: "unit_factor 0 = 0"
```
```  1164     and is_unit_unit_factor: "a dvd 1 \<Longrightarrow> unit_factor a = a"
```
```  1165     and unit_factor_is_unit: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd 1"
```
```  1166     and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
```
```  1167   -- \<open>This fine-grained hierarchy will later on allow lean normalization of polynomials\<close>
```
```  1168
```
```  1169 class normalization_semidom = algebraic_semidom + semidom_divide_unit_factor +
```
```  1170   fixes normalize :: "'a \<Rightarrow> 'a"
```
```  1171   assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
```
```  1172     and normalize_0 [simp]: "normalize 0 = 0"
```
```  1173 begin
```
```  1174
```
```  1175 text \<open>
```
```  1176   Class @{class normalization_semidom} cultivates the idea that each integral
```
```  1177   domain can be split into equivalence classes whose representants are
```
```  1178   associated, i.e. divide each other. @{const normalize} specifies a canonical
```
```  1179   representant for each equivalence class. The rationale behind this is that
```
```  1180   it is easier to reason about equality than equivalences, hence we prefer to
```
```  1181   think about equality of normalized values rather than associated elements.
```
```  1182 \<close>
```
```  1183
```
```  1184 declare unit_factor_is_unit [iff]
```
```  1185
```
```  1186 lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
```
```  1187   by (rule unit_imp_dvd) simp
```
```  1188
```
```  1189 lemma unit_factor_self [simp]: "unit_factor a dvd a"
```
```  1190   by (cases "a = 0") simp_all
```
```  1191
```
```  1192 lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
```
```  1193   using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
```
```  1194
```
```  1195 lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
```
```  1196   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1197 proof
```
```  1198   assume ?lhs
```
```  1199   moreover have "unit_factor a * normalize a = a" by simp
```
```  1200   ultimately show ?rhs by simp
```
```  1201 next
```
```  1202   assume ?rhs
```
```  1203   then show ?lhs by simp
```
```  1204 qed
```
```  1205
```
```  1206 lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
```
```  1207   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1208 proof
```
```  1209   assume ?lhs
```
```  1210   moreover have "unit_factor a * normalize a = a" by simp
```
```  1211   ultimately show ?rhs by simp
```
```  1212 next
```
```  1213   assume ?rhs
```
```  1214   then show ?lhs by simp
```
```  1215 qed
```
```  1216
```
```  1217 lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
```
```  1218 proof (cases "a = 0")
```
```  1219   case True
```
```  1220   then show ?thesis by simp
```
```  1221 next
```
```  1222   case False
```
```  1223   then have "unit_factor a \<noteq> 0"
```
```  1224     by simp
```
```  1225   with nonzero_mult_div_cancel_left
```
```  1226   have "unit_factor a * normalize a div unit_factor a = normalize a"
```
```  1227     by blast
```
```  1228   then show ?thesis by simp
```
```  1229 qed
```
```  1230
```
```  1231 lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
```
```  1232 proof (cases "a = 0")
```
```  1233   case True
```
```  1234   then show ?thesis by simp
```
```  1235 next
```
```  1236   case False
```
```  1237   have "normalize a div a = normalize a div (unit_factor a * normalize a)"
```
```  1238     by simp
```
```  1239   also have "\<dots> = 1 div unit_factor a"
```
```  1240     using False by (subst is_unit_div_mult_cancel_right) simp_all
```
```  1241   finally show ?thesis .
```
```  1242 qed
```
```  1243
```
```  1244 lemma is_unit_normalize:
```
```  1245   assumes "is_unit a"
```
```  1246   shows "normalize a = 1"
```
```  1247 proof -
```
```  1248   from assms have "unit_factor a = a"
```
```  1249     by (rule is_unit_unit_factor)
```
```  1250   moreover from assms have "a \<noteq> 0"
```
```  1251     by auto
```
```  1252   moreover have "normalize a = a div unit_factor a"
```
```  1253     by simp
```
```  1254   ultimately show ?thesis
```
```  1255     by simp
```
```  1256 qed
```
```  1257
```
```  1258 lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
```
```  1259   by (rule is_unit_unit_factor) simp
```
```  1260
```
```  1261 lemma normalize_1 [simp]: "normalize 1 = 1"
```
```  1262   by (rule is_unit_normalize) simp
```
```  1263
```
```  1264 lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
```
```  1265   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1266 proof
```
```  1267   assume ?rhs
```
```  1268   then show ?lhs by (rule is_unit_normalize)
```
```  1269 next
```
```  1270   assume ?lhs
```
```  1271   then have "unit_factor a * normalize a = unit_factor a * 1"
```
```  1272     by simp
```
```  1273   then have "unit_factor a = a"
```
```  1274     by simp
```
```  1275   moreover
```
```  1276   from \<open>?lhs\<close> have "a \<noteq> 0" by auto
```
```  1277   then have "is_unit (unit_factor a)" by simp
```
```  1278   ultimately show ?rhs by simp
```
```  1279 qed
```
```  1280
```
```  1281 lemma div_normalize [simp]: "a div normalize a = unit_factor a"
```
```  1282 proof (cases "a = 0")
```
```  1283   case True
```
```  1284   then show ?thesis by simp
```
```  1285 next
```
```  1286   case False
```
```  1287   then have "normalize a \<noteq> 0" by simp
```
```  1288   with nonzero_mult_div_cancel_right
```
```  1289   have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
```
```  1290   then show ?thesis by simp
```
```  1291 qed
```
```  1292
```
```  1293 lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
```
```  1294   by (cases "b = 0") simp_all
```
```  1295
```
```  1296 lemma inv_unit_factor_eq_0_iff [simp]:
```
```  1297   "1 div unit_factor a = 0 \<longleftrightarrow> a = 0"
```
```  1298   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1299 proof
```
```  1300   assume ?lhs
```
```  1301   then have "a * (1 div unit_factor a) = a * 0"
```
```  1302     by simp
```
```  1303   then show ?rhs
```
```  1304     by simp
```
```  1305 next
```
```  1306   assume ?rhs
```
```  1307   then show ?lhs by simp
```
```  1308 qed
```
```  1309
```
```  1310 lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
```
```  1311 proof (cases "a = 0 \<or> b = 0")
```
```  1312   case True
```
```  1313   then show ?thesis by auto
```
```  1314 next
```
```  1315   case False
```
```  1316   have "unit_factor (a * b) * normalize (a * b) = a * b"
```
```  1317     by (rule unit_factor_mult_normalize)
```
```  1318   then have "normalize (a * b) = a * b div unit_factor (a * b)"
```
```  1319     by simp
```
```  1320   also have "\<dots> = a * b div unit_factor (b * a)"
```
```  1321     by (simp add: ac_simps)
```
```  1322   also have "\<dots> = a * b div unit_factor b div unit_factor a"
```
```  1323     using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
```
```  1324   also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
```
```  1325     using False by (subst unit_div_mult_swap) simp_all
```
```  1326   also have "\<dots> = normalize a * normalize b"
```
```  1327     using False
```
```  1328     by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
```
```  1329   finally show ?thesis .
```
```  1330 qed
```
```  1331
```
```  1332 lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
```
```  1333   by (cases "a = 0") (auto intro: is_unit_unit_factor)
```
```  1334
```
```  1335 lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
```
```  1336   by (rule is_unit_normalize) simp
```
```  1337
```
```  1338 lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
```
```  1339 proof (cases "a = 0")
```
```  1340   case True
```
```  1341   then show ?thesis by simp
```
```  1342 next
```
```  1343   case False
```
```  1344   have "normalize a = normalize (unit_factor a * normalize a)"
```
```  1345     by simp
```
```  1346   also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
```
```  1347     by (simp only: normalize_mult)
```
```  1348   finally show ?thesis
```
```  1349     using False by simp_all
```
```  1350 qed
```
```  1351
```
```  1352 lemma unit_factor_normalize [simp]:
```
```  1353   assumes "a \<noteq> 0"
```
```  1354   shows "unit_factor (normalize a) = 1"
```
```  1355 proof -
```
```  1356   from assms have *: "normalize a \<noteq> 0"
```
```  1357     by simp
```
```  1358   have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
```
```  1359     by (simp only: unit_factor_mult_normalize)
```
```  1360   then have "unit_factor (normalize a) * normalize a = normalize a"
```
```  1361     by simp
```
```  1362   with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
```
```  1363     by simp
```
```  1364   with * show ?thesis
```
```  1365     by simp
```
```  1366 qed
```
```  1367
```
```  1368 lemma dvd_unit_factor_div:
```
```  1369   assumes "b dvd a"
```
```  1370   shows "unit_factor (a div b) = unit_factor a div unit_factor b"
```
```  1371 proof -
```
```  1372   from assms have "a = a div b * b"
```
```  1373     by simp
```
```  1374   then have "unit_factor a = unit_factor (a div b * b)"
```
```  1375     by simp
```
```  1376   then show ?thesis
```
```  1377     by (cases "b = 0") (simp_all add: unit_factor_mult)
```
```  1378 qed
```
```  1379
```
```  1380 lemma dvd_normalize_div:
```
```  1381   assumes "b dvd a"
```
```  1382   shows "normalize (a div b) = normalize a div normalize b"
```
```  1383 proof -
```
```  1384   from assms have "a = a div b * b"
```
```  1385     by simp
```
```  1386   then have "normalize a = normalize (a div b * b)"
```
```  1387     by simp
```
```  1388   then show ?thesis
```
```  1389     by (cases "b = 0") (simp_all add: normalize_mult)
```
```  1390 qed
```
```  1391
```
```  1392 lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
```
```  1393 proof -
```
```  1394   have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
```
```  1395     using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
```
```  1396       by (cases "a = 0") simp_all
```
```  1397   then show ?thesis by simp
```
```  1398 qed
```
```  1399
```
```  1400 lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
```
```  1401 proof -
```
```  1402   have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
```
```  1403     using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
```
```  1404       by (cases "b = 0") simp_all
```
```  1405   then show ?thesis by simp
```
```  1406 qed
```
```  1407
```
```  1408 lemma normalize_idem_imp_unit_factor_eq:
```
```  1409   assumes "normalize a = a"
```
```  1410   shows "unit_factor a = of_bool (a \<noteq> 0)"
```
```  1411 proof (cases "a = 0")
```
```  1412   case True
```
```  1413   then show ?thesis
```
```  1414     by simp
```
```  1415 next
```
```  1416   case False
```
```  1417   then show ?thesis
```
```  1418     using assms unit_factor_normalize [of a] by simp
```
```  1419 qed
```
```  1420
```
```  1421 lemma normalize_idem_imp_is_unit_iff:
```
```  1422   assumes "normalize a = a"
```
```  1423   shows "is_unit a \<longleftrightarrow> a = 1"
```
```  1424   using assms by (cases "a = 0") (auto dest: is_unit_normalize)
```
```  1425
```
```  1426 text \<open>
```
```  1427   We avoid an explicit definition of associated elements but prefer explicit
```
```  1428   normalisation instead. In theory we could define an abbreviation like @{prop
```
```  1429   "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is counterproductive
```
```  1430   without suggestive infix syntax, which we do not want to sacrifice for this
```
```  1431   purpose here.
```
```  1432 \<close>
```
```  1433
```
```  1434 lemma associatedI:
```
```  1435   assumes "a dvd b" and "b dvd a"
```
```  1436   shows "normalize a = normalize b"
```
```  1437 proof (cases "a = 0 \<or> b = 0")
```
```  1438   case True
```
```  1439   with assms show ?thesis by auto
```
```  1440 next
```
```  1441   case False
```
```  1442   from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
```
```  1443   moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
```
```  1444   ultimately have "b * 1 = b * (c * d)"
```
```  1445     by (simp add: ac_simps)
```
```  1446   with False have "1 = c * d"
```
```  1447     unfolding mult_cancel_left by simp
```
```  1448   then have "is_unit c" and "is_unit d"
```
```  1449     by auto
```
```  1450   with a b show ?thesis
```
```  1451     by (simp add: normalize_mult is_unit_normalize)
```
```  1452 qed
```
```  1453
```
```  1454 lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
```
```  1455   using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
```
```  1456   by simp
```
```  1457
```
```  1458 lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
```
```  1459   using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
```
```  1460   by simp
```
```  1461
```
```  1462 lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
```
```  1463   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
```
```  1464
```
```  1465 lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
```
```  1466   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1467 proof
```
```  1468   assume ?rhs
```
```  1469   then show ?lhs by (auto intro!: associatedI)
```
```  1470 next
```
```  1471   assume ?lhs
```
```  1472   then have "unit_factor a * normalize a = unit_factor a * normalize b"
```
```  1473     by simp
```
```  1474   then have *: "normalize b * unit_factor a = a"
```
```  1475     by (simp add: ac_simps)
```
```  1476   show ?rhs
```
```  1477   proof (cases "a = 0 \<or> b = 0")
```
```  1478     case True
```
```  1479     with \<open>?lhs\<close> show ?thesis by auto
```
```  1480   next
```
```  1481     case False
```
```  1482     then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
```
```  1483       by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
```
```  1484     with * show ?thesis by simp
```
```  1485   qed
```
```  1486 qed
```
```  1487
```
```  1488 lemma associated_eqI:
```
```  1489   assumes "a dvd b" and "b dvd a"
```
```  1490   assumes "normalize a = a" and "normalize b = b"
```
```  1491   shows "a = b"
```
```  1492 proof -
```
```  1493   from assms have "normalize a = normalize b"
```
```  1494     unfolding associated_iff_dvd by simp
```
```  1495   with \<open>normalize a = a\<close> have "a = normalize b"
```
```  1496     by simp
```
```  1497   with \<open>normalize b = b\<close> show "a = b"
```
```  1498     by simp
```
```  1499 qed
```
```  1500
```
```  1501 lemma normalize_unit_factor_eqI:
```
```  1502   assumes "normalize a = normalize b"
```
```  1503     and "unit_factor a = unit_factor b"
```
```  1504   shows "a = b"
```
```  1505 proof -
```
```  1506   from assms have "unit_factor a * normalize a = unit_factor b * normalize b"
```
```  1507     by simp
```
```  1508   then show ?thesis
```
```  1509     by simp
```
```  1510 qed
```
```  1511
```
```  1512 end
```
```  1513
```
```  1514
```
```  1515 text \<open>Syntactic division remainder operator\<close>
```
```  1516
```
```  1517 class modulo = dvd + divide +
```
```  1518   fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "mod" 70)
```
```  1519
```
```  1520 text \<open>Arbitrary quotient and remainder partitions\<close>
```
```  1521
```
```  1522 class semiring_modulo = comm_semiring_1_cancel + divide + modulo +
```
```  1523   assumes div_mult_mod_eq: "a div b * b + a mod b = a"
```
```  1524 begin
```
```  1525
```
```  1526 lemma mod_div_decomp:
```
```  1527   fixes a b
```
```  1528   obtains q r where "q = a div b" and "r = a mod b"
```
```  1529     and "a = q * b + r"
```
```  1530 proof -
```
```  1531   from div_mult_mod_eq have "a = a div b * b + a mod b" by simp
```
```  1532   moreover have "a div b = a div b" ..
```
```  1533   moreover have "a mod b = a mod b" ..
```
```  1534   note that ultimately show thesis by blast
```
```  1535 qed
```
```  1536
```
```  1537 lemma mult_div_mod_eq: "b * (a div b) + a mod b = a"
```
```  1538   using div_mult_mod_eq [of a b] by (simp add: ac_simps)
```
```  1539
```
```  1540 lemma mod_div_mult_eq: "a mod b + a div b * b = a"
```
```  1541   using div_mult_mod_eq [of a b] by (simp add: ac_simps)
```
```  1542
```
```  1543 lemma mod_mult_div_eq: "a mod b + b * (a div b) = a"
```
```  1544   using div_mult_mod_eq [of a b] by (simp add: ac_simps)
```
```  1545
```
```  1546 lemma minus_div_mult_eq_mod: "a - a div b * b = a mod b"
```
```  1547   by (rule add_implies_diff [symmetric]) (fact mod_div_mult_eq)
```
```  1548
```
```  1549 lemma minus_mult_div_eq_mod: "a - b * (a div b) = a mod b"
```
```  1550   by (rule add_implies_diff [symmetric]) (fact mod_mult_div_eq)
```
```  1551
```
```  1552 lemma minus_mod_eq_div_mult: "a - a mod b = a div b * b"
```
```  1553   by (rule add_implies_diff [symmetric]) (fact div_mult_mod_eq)
```
```  1554
```
```  1555 lemma minus_mod_eq_mult_div: "a - a mod b = b * (a div b)"
```
```  1556   by (rule add_implies_diff [symmetric]) (fact mult_div_mod_eq)
```
```  1557
```
```  1558 end
```
```  1559
```
```  1560
```
```  1561 class ordered_semiring = semiring + ordered_comm_monoid_add +
```
```  1562   assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
```
```  1563   assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
```
```  1564 begin
```
```  1565
```
```  1566 lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
```
```  1567   apply (erule (1) mult_right_mono [THEN order_trans])
```
```  1568   apply (erule (1) mult_left_mono)
```
```  1569   done
```
```  1570
```
```  1571 lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
```
```  1572   by (rule mult_mono) (fast intro: order_trans)+
```
```  1573
```
```  1574 end
```
```  1575
```
```  1576 class ordered_semiring_0 = semiring_0 + ordered_semiring
```
```  1577 begin
```
```  1578
```
```  1579 lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
```
```  1580   using mult_left_mono [of 0 b a] by simp
```
```  1581
```
```  1582 lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
```
```  1583   using mult_left_mono [of b 0 a] by simp
```
```  1584
```
```  1585 lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
```
```  1586   using mult_right_mono [of a 0 b] by simp
```
```  1587
```
```  1588 text \<open>Legacy -- use @{thm [source] mult_nonpos_nonneg}.\<close>
```
```  1589 lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
```
```  1590   by (drule mult_right_mono [of b 0]) auto
```
```  1591
```
```  1592 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"
```
```  1593   by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
```
```  1594
```
```  1595 end
```
```  1596
```
```  1597 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
```
```  1598 begin
```
```  1599
```
```  1600 subclass semiring_0_cancel ..
```
```  1601
```
```  1602 subclass ordered_semiring_0 ..
```
```  1603
```
```  1604 end
```
```  1605
```
```  1606 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
```
```  1607 begin
```
```  1608
```
```  1609 subclass ordered_cancel_semiring ..
```
```  1610
```
```  1611 subclass ordered_cancel_comm_monoid_add ..
```
```  1612
```
```  1613 subclass ordered_ab_semigroup_monoid_add_imp_le ..
```
```  1614
```
```  1615 lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
```
```  1616   by (force simp add: mult_left_mono not_le [symmetric])
```
```  1617
```
```  1618 lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
```
```  1619   by (force simp add: mult_right_mono not_le [symmetric])
```
```  1620
```
```  1621 end
```
```  1622
```
```  1623 class linordered_semiring_1 = linordered_semiring + semiring_1
```
```  1624 begin
```
```  1625
```
```  1626 lemma convex_bound_le:
```
```  1627   assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  1628   shows "u * x + v * y \<le> a"
```
```  1629 proof-
```
```  1630   from assms have "u * x + v * y \<le> u * a + v * a"
```
```  1631     by (simp add: add_mono mult_left_mono)
```
```  1632   with assms show ?thesis
```
```  1633     unfolding distrib_right[symmetric] by simp
```
```  1634 qed
```
```  1635
```
```  1636 end
```
```  1637
```
```  1638 class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
```
```  1639   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
```
```  1640   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
```
```  1641 begin
```
```  1642
```
```  1643 subclass semiring_0_cancel ..
```
```  1644
```
```  1645 subclass linordered_semiring
```
```  1646 proof
```
```  1647   fix a b c :: 'a
```
```  1648   assume *: "a \<le> b" "0 \<le> c"
```
```  1649   then show "c * a \<le> c * b"
```
```  1650     unfolding le_less
```
```  1651     using mult_strict_left_mono by (cases "c = 0") auto
```
```  1652   from * show "a * c \<le> b * c"
```
```  1653     unfolding le_less
```
```  1654     using mult_strict_right_mono by (cases "c = 0") auto
```
```  1655 qed
```
```  1656
```
```  1657 lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
```
```  1658   by (auto simp add: mult_strict_left_mono _not_less [symmetric])
```
```  1659
```
```  1660 lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
```
```  1661   by (auto simp add: mult_strict_right_mono not_less [symmetric])
```
```  1662
```
```  1663 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
```
```  1664   using mult_strict_left_mono [of 0 b a] by simp
```
```  1665
```
```  1666 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
```
```  1667   using mult_strict_left_mono [of b 0 a] by simp
```
```  1668
```
```  1669 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
```
```  1670   using mult_strict_right_mono [of a 0 b] by simp
```
```  1671
```
```  1672 text \<open>Legacy -- use @{thm [source] mult_neg_pos}.\<close>
```
```  1673 lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
```
```  1674   by (drule mult_strict_right_mono [of b 0]) auto
```
```  1675
```
```  1676 lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
```
```  1677   apply (cases "b \<le> 0")
```
```  1678    apply (auto simp add: le_less not_less)
```
```  1679   apply (drule_tac mult_pos_neg [of a b])
```
```  1680    apply (auto dest: less_not_sym)
```
```  1681   done
```
```  1682
```
```  1683 lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
```
```  1684   apply (cases "b \<le> 0")
```
```  1685    apply (auto simp add: le_less not_less)
```
```  1686   apply (drule_tac mult_pos_neg2 [of a b])
```
```  1687    apply (auto dest: less_not_sym)
```
```  1688   done
```
```  1689
```
```  1690 text \<open>Strict monotonicity in both arguments\<close>
```
```  1691 lemma mult_strict_mono:
```
```  1692   assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
```
```  1693   shows "a * c < b * d"
```
```  1694   using assms
```
```  1695   apply (cases "c = 0")
```
```  1696    apply simp
```
```  1697   apply (erule mult_strict_right_mono [THEN less_trans])
```
```  1698    apply (auto simp add: le_less)
```
```  1699   apply (erule (1) mult_strict_left_mono)
```
```  1700   done
```
```  1701
```
```  1702 text \<open>This weaker variant has more natural premises\<close>
```
```  1703 lemma mult_strict_mono':
```
```  1704   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
```
```  1705   shows "a * c < b * d"
```
```  1706   by (rule mult_strict_mono) (insert assms, auto)
```
```  1707
```
```  1708 lemma mult_less_le_imp_less:
```
```  1709   assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
```
```  1710   shows "a * c < b * d"
```
```  1711   using assms
```
```  1712   apply (subgoal_tac "a * c < b * c")
```
```  1713    apply (erule less_le_trans)
```
```  1714    apply (erule mult_left_mono)
```
```  1715    apply simp
```
```  1716   apply (erule (1) mult_strict_right_mono)
```
```  1717   done
```
```  1718
```
```  1719 lemma mult_le_less_imp_less:
```
```  1720   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
```
```  1721   shows "a * c < b * d"
```
```  1722   using assms
```
```  1723   apply (subgoal_tac "a * c \<le> b * c")
```
```  1724    apply (erule le_less_trans)
```
```  1725    apply (erule mult_strict_left_mono)
```
```  1726    apply simp
```
```  1727   apply (erule (1) mult_right_mono)
```
```  1728   done
```
```  1729
```
```  1730 end
```
```  1731
```
```  1732 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
```
```  1733 begin
```
```  1734
```
```  1735 subclass linordered_semiring_1 ..
```
```  1736
```
```  1737 lemma convex_bound_lt:
```
```  1738   assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```  1739   shows "u * x + v * y < a"
```
```  1740 proof -
```
```  1741   from assms have "u * x + v * y < u * a + v * a"
```
```  1742     by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
```
```  1743   with assms show ?thesis
```
```  1744     unfolding distrib_right[symmetric] by simp
```
```  1745 qed
```
```  1746
```
```  1747 end
```
```  1748
```
```  1749 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
```
```  1750   assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
```
```  1751 begin
```
```  1752
```
```  1753 subclass ordered_semiring
```
```  1754 proof
```
```  1755   fix a b c :: 'a
```
```  1756   assume "a \<le> b" "0 \<le> c"
```
```  1757   then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
```
```  1758   then show "a * c \<le> b * c" by (simp only: mult.commute)
```
```  1759 qed
```
```  1760
```
```  1761 end
```
```  1762
```
```  1763 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
```
```  1764 begin
```
```  1765
```
```  1766 subclass comm_semiring_0_cancel ..
```
```  1767 subclass ordered_comm_semiring ..
```
```  1768 subclass ordered_cancel_semiring ..
```
```  1769
```
```  1770 end
```
```  1771
```
```  1772 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
```
```  1773   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
```
```  1774 begin
```
```  1775
```
```  1776 subclass linordered_semiring_strict
```
```  1777 proof
```
```  1778   fix a b c :: 'a
```
```  1779   assume "a < b" "0 < c"
```
```  1780   then show "c * a < c * b"
```
```  1781     by (rule comm_mult_strict_left_mono)
```
```  1782   then show "a * c < b * c"
```
```  1783     by (simp only: mult.commute)
```
```  1784 qed
```
```  1785
```
```  1786 subclass ordered_cancel_comm_semiring
```
```  1787 proof
```
```  1788   fix a b c :: 'a
```
```  1789   assume "a \<le> b" "0 \<le> c"
```
```  1790   then show "c * a \<le> c * b"
```
```  1791     unfolding le_less
```
```  1792     using mult_strict_left_mono by (cases "c = 0") auto
```
```  1793 qed
```
```  1794
```
```  1795 end
```
```  1796
```
```  1797 class ordered_ring = ring + ordered_cancel_semiring
```
```  1798 begin
```
```  1799
```
```  1800 subclass ordered_ab_group_add ..
```
```  1801
```
```  1802 lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
```
```  1803   by (simp add: algebra_simps)
```
```  1804
```
```  1805 lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
```
```  1806   by (simp add: algebra_simps)
```
```  1807
```
```  1808 lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
```
```  1809   by (simp add: algebra_simps)
```
```  1810
```
```  1811 lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
```
```  1812   by (simp add: algebra_simps)
```
```  1813
```
```  1814 lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
```
```  1815   apply (drule mult_left_mono [of _ _ "- c"])
```
```  1816   apply simp_all
```
```  1817   done
```
```  1818
```
```  1819 lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
```
```  1820   apply (drule mult_right_mono [of _ _ "- c"])
```
```  1821   apply simp_all
```
```  1822   done
```
```  1823
```
```  1824 lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
```
```  1825   using mult_right_mono_neg [of a 0 b] by simp
```
```  1826
```
```  1827 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"
```
```  1828   by (auto simp add: mult_nonpos_nonpos)
```
```  1829
```
```  1830 end
```
```  1831
```
```  1832 class abs_if = minus + uminus + ord + zero + abs +
```
```  1833   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
```
```  1834
```
```  1835 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
```
```  1836 begin
```
```  1837
```
```  1838 subclass ordered_ring ..
```
```  1839
```
```  1840 subclass ordered_ab_group_add_abs
```
```  1841 proof
```
```  1842   fix a b
```
```  1843   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
```
```  1844     by (auto simp add: abs_if not_le not_less algebra_simps
```
```  1845         simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
```
```  1846 qed (auto simp: abs_if)
```
```  1847
```
```  1848 lemma zero_le_square [simp]: "0 \<le> a * a"
```
```  1849   using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
```
```  1850
```
```  1851 lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
```
```  1852   by (simp add: not_less)
```
```  1853
```
```  1854 proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
```
```  1855   by (auto simp add: abs_if split: if_split_asm)
```
```  1856
```
```  1857 lemma abs_eq_iff':
```
```  1858   "\<bar>a\<bar> = b \<longleftrightarrow> b \<ge> 0 \<and> (a = b \<or> a = - b)"
```
```  1859   by (cases "a \<ge> 0") auto
```
```  1860
```
```  1861 lemma eq_abs_iff':
```
```  1862   "a = \<bar>b\<bar> \<longleftrightarrow> a \<ge> 0 \<and> (b = a \<or> b = - a)"
```
```  1863   using abs_eq_iff' [of b a] by auto
```
```  1864
```
```  1865 lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
```
```  1866   by (intro add_nonneg_nonneg zero_le_square)
```
```  1867
```
```  1868 lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
```
```  1869   by (simp add: not_less sum_squares_ge_zero)
```
```  1870
```
```  1871 end
```
```  1872
```
```  1873 class linordered_ring_strict = ring + linordered_semiring_strict
```
```  1874   + ordered_ab_group_add + abs_if
```
```  1875 begin
```
```  1876
```
```  1877 subclass linordered_ring ..
```
```  1878
```
```  1879 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
```
```  1880   using mult_strict_left_mono [of b a "- c"] by simp
```
```  1881
```
```  1882 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
```
```  1883   using mult_strict_right_mono [of b a "- c"] by simp
```
```  1884
```
```  1885 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
```
```  1886   using mult_strict_right_mono_neg [of a 0 b] by simp
```
```  1887
```
```  1888 subclass ring_no_zero_divisors
```
```  1889 proof
```
```  1890   fix a b
```
```  1891   assume "a \<noteq> 0"
```
```  1892   then have a: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
```
```  1893   assume "b \<noteq> 0"
```
```  1894   then have b: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
```
```  1895   have "a * b < 0 \<or> 0 < a * b"
```
```  1896   proof (cases "a < 0")
```
```  1897     case True
```
```  1898     show ?thesis
```
```  1899     proof (cases "b < 0")
```
```  1900       case True
```
```  1901       with \<open>a < 0\<close> show ?thesis by (auto dest: mult_neg_neg)
```
```  1902     next
```
```  1903       case False
```
```  1904       with b have "0 < b" by auto
```
```  1905       with \<open>a < 0\<close> show ?thesis by (auto dest: mult_strict_right_mono)
```
```  1906     qed
```
```  1907   next
```
```  1908     case False
```
```  1909     with a have "0 < a" by auto
```
```  1910     show ?thesis
```
```  1911     proof (cases "b < 0")
```
```  1912       case True
```
```  1913       with \<open>0 < a\<close> show ?thesis
```
```  1914         by (auto dest: mult_strict_right_mono_neg)
```
```  1915     next
```
```  1916       case False
```
```  1917       with b have "0 < b" by auto
```
```  1918       with \<open>0 < a\<close> show ?thesis by auto
```
```  1919     qed
```
```  1920   qed
```
```  1921   then show "a * b \<noteq> 0"
```
```  1922     by (simp add: neq_iff)
```
```  1923 qed
```
```  1924
```
```  1925 lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
```
```  1926   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
```
```  1927      (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
```
```  1928
```
```  1929 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"
```
```  1930   by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
```
```  1931
```
```  1932 lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
```
```  1933   using zero_less_mult_iff [of "- a" b] by auto
```
```  1934
```
```  1935 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"
```
```  1936   using zero_le_mult_iff [of "- a" b] by auto
```
```  1937
```
```  1938 text \<open>
```
```  1939   Cancellation laws for @{term "c * a < c * b"} and @{term "a * c < b * c"},
```
```  1940   also with the relations \<open>\<le>\<close> and equality.
```
```  1941 \<close>
```
```  1942
```
```  1943 text \<open>
```
```  1944   These ``disjunction'' versions produce two cases when the comparison is
```
```  1945   an assumption, but effectively four when the comparison is a goal.
```
```  1946 \<close>
```
```  1947
```
```  1948 lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
```
```  1949   apply (cases "c = 0")
```
```  1950    apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
```
```  1951      apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
```
```  1952      apply (erule_tac [!] notE)
```
```  1953      apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
```
```  1954   done
```
```  1955
```
```  1956 lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
```
```  1957   apply (cases "c = 0")
```
```  1958    apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
```
```  1959      apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
```
```  1960      apply (erule_tac [!] notE)
```
```  1961      apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
```
```  1962   done
```
```  1963
```
```  1964 text \<open>
```
```  1965   The ``conjunction of implication'' lemmas produce two cases when the
```
```  1966   comparison is a goal, but give four when the comparison is an assumption.
```
```  1967 \<close>
```
```  1968
```
```  1969 lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
```
```  1970   using mult_less_cancel_right_disj [of a c b] by auto
```
```  1971
```
```  1972 lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
```
```  1973   using mult_less_cancel_left_disj [of c a b] by auto
```
```  1974
```
```  1975 lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1976   by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
```
```  1977
```
```  1978 lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```  1979   by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
```
```  1980
```
```  1981 lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
```
```  1982   by (auto simp: mult_le_cancel_left)
```
```  1983
```
```  1984 lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
```
```  1985   by (auto simp: mult_le_cancel_left)
```
```  1986
```
```  1987 lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
```
```  1988   by (auto simp: mult_less_cancel_left)
```
```  1989
```
```  1990 lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
```
```  1991   by (auto simp: mult_less_cancel_left)
```
```  1992
```
```  1993 end
```
```  1994
```
```  1995 lemmas mult_sign_intros =
```
```  1996   mult_nonneg_nonneg mult_nonneg_nonpos
```
```  1997   mult_nonpos_nonneg mult_nonpos_nonpos
```
```  1998   mult_pos_pos mult_pos_neg
```
```  1999   mult_neg_pos mult_neg_neg
```
```  2000
```
```  2001 class ordered_comm_ring = comm_ring + ordered_comm_semiring
```
```  2002 begin
```
```  2003
```
```  2004 subclass ordered_ring ..
```
```  2005 subclass ordered_cancel_comm_semiring ..
```
```  2006
```
```  2007 end
```
```  2008
```
```  2009 class zero_less_one = order + zero + one +
```
```  2010   assumes zero_less_one [simp]: "0 < 1"
```
```  2011
```
```  2012 class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one
```
```  2013 begin
```
```  2014
```
```  2015 subclass zero_neq_one
```
```  2016   by standard (insert zero_less_one, blast)
```
```  2017
```
```  2018 subclass comm_semiring_1
```
```  2019   by standard (rule mult_1_left)
```
```  2020
```
```  2021 lemma zero_le_one [simp]: "0 \<le> 1"
```
```  2022   by (rule zero_less_one [THEN less_imp_le])
```
```  2023
```
```  2024 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
```
```  2025   by (simp add: not_le)
```
```  2026
```
```  2027 lemma not_one_less_zero [simp]: "\<not> 1 < 0"
```
```  2028   by (simp add: not_less)
```
```  2029
```
```  2030 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
```
```  2031   using mult_left_mono[of c 1 a] by simp
```
```  2032
```
```  2033 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
```
```  2034   using mult_mono[of a 1 b 1] by simp
```
```  2035
```
```  2036 lemma zero_less_two: "0 < 1 + 1"
```
```  2037   using add_pos_pos[OF zero_less_one zero_less_one] .
```
```  2038
```
```  2039 end
```
```  2040
```
```  2041 class linordered_semidom = semidom + linordered_comm_semiring_strict + zero_less_one +
```
```  2042   assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
```
```  2043 begin
```
```  2044
```
```  2045 subclass linordered_nonzero_semiring ..
```
```  2046
```
```  2047 text \<open>Addition is the inverse of subtraction.\<close>
```
```  2048
```
```  2049 lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
```
```  2050   by (frule le_add_diff_inverse2) (simp add: add.commute)
```
```  2051
```
```  2052 lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
```
```  2053   by simp
```
```  2054
```
```  2055 lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
```
```  2056   apply (subst add_le_cancel_right [where c=k, symmetric])
```
```  2057   apply (frule le_add_diff_inverse2)
```
```  2058   apply (simp only: add.assoc [symmetric])
```
```  2059   using add_implies_diff
```
```  2060   apply fastforce
```
```  2061   done
```
```  2062
```
```  2063 lemma add_le_add_imp_diff_le:
```
```  2064   assumes 1: "i + k \<le> n"
```
```  2065     and 2: "n \<le> j + k"
```
```  2066   shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
```
```  2067 proof -
```
```  2068   have "n - (i + k) + (i + k) = n"
```
```  2069     using 1 by simp
```
```  2070   moreover have "n - k = n - k - i + i"
```
```  2071     using 1 by (simp add: add_le_imp_le_diff)
```
```  2072   ultimately show ?thesis
```
```  2073     using 2
```
```  2074     apply (simp add: add.assoc [symmetric])
```
```  2075     apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
```
```  2076     apply (simp add: add.commute diff_diff_add)
```
```  2077     done
```
```  2078 qed
```
```  2079
```
```  2080 lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
```
```  2081   using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
```
```  2082
```
```  2083 end
```
```  2084
```
```  2085 class linordered_idom =
```
```  2086   comm_ring_1 + linordered_comm_semiring_strict + ordered_ab_group_add + abs_if + sgn +
```
```  2087   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```  2088 begin
```
```  2089
```
```  2090 subclass linordered_semiring_1_strict ..
```
```  2091 subclass linordered_ring_strict ..
```
```  2092 subclass ordered_comm_ring ..
```
```  2093 subclass idom ..
```
```  2094
```
```  2095 subclass linordered_semidom
```
```  2096 proof
```
```  2097   have "0 \<le> 1 * 1" by (rule zero_le_square)
```
```  2098   then show "0 < 1" by (simp add: le_less)
```
```  2099   show "b \<le> a \<Longrightarrow> a - b + b = a" for a b by simp
```
```  2100 qed
```
```  2101
```
```  2102 subclass idom_abs_sgn
```
```  2103   by standard
```
```  2104     (auto simp add: sgn_if abs_if zero_less_mult_iff)
```
```  2105
```
```  2106 lemma linorder_neqE_linordered_idom:
```
```  2107   assumes "x \<noteq> y"
```
```  2108   obtains "x < y" | "y < x"
```
```  2109   using assms by (rule neqE)
```
```  2110
```
```  2111 text \<open>These cancellation simp rules also produce two cases when the comparison is a goal.\<close>
```
```  2112
```
```  2113 lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
```
```  2114   using mult_le_cancel_right [of 1 c b] by simp
```
```  2115
```
```  2116 lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
```
```  2117   using mult_le_cancel_right [of a c 1] by simp
```
```  2118
```
```  2119 lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
```
```  2120   using mult_le_cancel_left [of c 1 b] by simp
```
```  2121
```
```  2122 lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
```
```  2123   using mult_le_cancel_left [of c a 1] by simp
```
```  2124
```
```  2125 lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
```
```  2126   using mult_less_cancel_right [of 1 c b] by simp
```
```  2127
```
```  2128 lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
```
```  2129   using mult_less_cancel_right [of a c 1] by simp
```
```  2130
```
```  2131 lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
```
```  2132   using mult_less_cancel_left [of c 1 b] by simp
```
```  2133
```
```  2134 lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
```
```  2135   using mult_less_cancel_left [of c a 1] by simp
```
```  2136
```
```  2137 lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
```
```  2138   by (fact sgn_eq_0_iff)
```
```  2139
```
```  2140 lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
```
```  2141   unfolding sgn_if by simp
```
```  2142
```
```  2143 lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
```
```  2144   unfolding sgn_if by auto
```
```  2145
```
```  2146 lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
```
```  2147   by (simp only: sgn_1_pos)
```
```  2148
```
```  2149 lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
```
```  2150   by (simp only: sgn_1_neg)
```
```  2151
```
```  2152 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
```
```  2153   unfolding sgn_if abs_if by auto
```
```  2154
```
```  2155 lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
```
```  2156   unfolding sgn_if by auto
```
```  2157
```
```  2158 lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
```
```  2159   unfolding sgn_if by auto
```
```  2160
```
```  2161 lemma abs_sgn_eq_1 [simp]:
```
```  2162   "a \<noteq> 0 \<Longrightarrow> \<bar>sgn a\<bar> = 1"
```
```  2163   by simp
```
```  2164
```
```  2165 lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
```
```  2166   by (simp add: sgn_if)
```
```  2167
```
```  2168 lemma sgn_mult_self_eq [simp]:
```
```  2169   "sgn a * sgn a = of_bool (a \<noteq> 0)"
```
```  2170   by (cases "a > 0") simp_all
```
```  2171
```
```  2172 lemma abs_mult_self_eq [simp]:
```
```  2173   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
```
```  2174   by (cases "a > 0") simp_all
```
```  2175
```
```  2176 lemma same_sgn_sgn_add:
```
```  2177   "sgn (a + b) = sgn a" if "sgn b = sgn a"
```
```  2178 proof (cases a 0 rule: linorder_cases)
```
```  2179   case equal
```
```  2180   with that show ?thesis
```
```  2181     by simp
```
```  2182 next
```
```  2183   case less
```
```  2184   with that have "b < 0"
```
```  2185     by (simp add: sgn_1_neg)
```
```  2186   with \<open>a < 0\<close> have "a + b < 0"
```
```  2187     by (rule add_neg_neg)
```
```  2188   with \<open>a < 0\<close> show ?thesis
```
```  2189     by simp
```
```  2190 next
```
```  2191   case greater
```
```  2192   with that have "b > 0"
```
```  2193     by (simp add: sgn_1_pos)
```
```  2194   with \<open>a > 0\<close> have "a + b > 0"
```
```  2195     by (rule add_pos_pos)
```
```  2196   with \<open>a > 0\<close> show ?thesis
```
```  2197     by simp
```
```  2198 qed
```
```  2199
```
```  2200 lemma same_sgn_abs_add:
```
```  2201   "\<bar>a + b\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" if "sgn b = sgn a"
```
```  2202 proof -
```
```  2203   have "a + b = sgn a * \<bar>a\<bar> + sgn b * \<bar>b\<bar>"
```
```  2204     by (simp add: sgn_mult_abs)
```
```  2205   also have "\<dots> = sgn a * (\<bar>a\<bar> + \<bar>b\<bar>)"
```
```  2206     using that by (simp add: algebra_simps)
```
```  2207   finally show ?thesis
```
```  2208     by (auto simp add: abs_mult)
```
```  2209 qed
```
```  2210
```
```  2211 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
```
```  2212   by (simp add: abs_if)
```
```  2213
```
```  2214 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
```
```  2215   by (simp add: abs_if)
```
```  2216
```
```  2217 lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
```
```  2218   by (subst abs_dvd_iff [symmetric]) simp
```
```  2219
```
```  2220 text \<open>
```
```  2221   The following lemmas can be proven in more general structures, but
```
```  2222   are dangerous as simp rules in absence of @{thm neg_equal_zero},
```
```  2223   @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
```
```  2224 \<close>
```
```  2225
```
```  2226 lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
```
```  2227   by (fact equation_minus_iff)
```
```  2228
```
```  2229 lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
```
```  2230   by (subst minus_equation_iff, auto)
```
```  2231
```
```  2232 lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
```
```  2233   by (fact le_minus_iff)
```
```  2234
```
```  2235 lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
```
```  2236   by (fact minus_le_iff)
```
```  2237
```
```  2238 lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
```
```  2239   by (fact less_minus_iff)
```
```  2240
```
```  2241 lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
```
```  2242   by (fact minus_less_iff)
```
```  2243
```
```  2244 end
```
```  2245
```
```  2246 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
```
```  2247
```
```  2248 lemmas mult_compare_simps =
```
```  2249   mult_le_cancel_right mult_le_cancel_left
```
```  2250   mult_le_cancel_right1 mult_le_cancel_right2
```
```  2251   mult_le_cancel_left1 mult_le_cancel_left2
```
```  2252   mult_less_cancel_right mult_less_cancel_left
```
```  2253   mult_less_cancel_right1 mult_less_cancel_right2
```
```  2254   mult_less_cancel_left1 mult_less_cancel_left2
```
```  2255   mult_cancel_right mult_cancel_left
```
```  2256   mult_cancel_right1 mult_cancel_right2
```
```  2257   mult_cancel_left1 mult_cancel_left2
```
```  2258
```
```  2259
```
```  2260 text \<open>Reasoning about inequalities with division\<close>
```
```  2261
```
```  2262 context linordered_semidom
```
```  2263 begin
```
```  2264
```
```  2265 lemma less_add_one: "a < a + 1"
```
```  2266 proof -
```
```  2267   have "a + 0 < a + 1"
```
```  2268     by (blast intro: zero_less_one add_strict_left_mono)
```
```  2269   then show ?thesis by simp
```
```  2270 qed
```
```  2271
```
```  2272 end
```
```  2273
```
```  2274 context linordered_idom
```
```  2275 begin
```
```  2276
```
```  2277 lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
```
```  2278   by (rule mult_left_le)
```
```  2279
```
```  2280 lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
```
```  2281   by (auto simp add: mult_le_cancel_right2)
```
```  2282
```
```  2283 end
```
```  2284
```
```  2285 text \<open>Absolute Value\<close>
```
```  2286
```
```  2287 context linordered_idom
```
```  2288 begin
```
```  2289
```
```  2290 lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
```
```  2291   by (fact sgn_mult_abs)
```
```  2292
```
```  2293 lemma abs_one: "\<bar>1\<bar> = 1"
```
```  2294   by (fact abs_1)
```
```  2295
```
```  2296 end
```
```  2297
```
```  2298 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
```
```  2299   assumes abs_eq_mult:
```
```  2300     "(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>"
```
```  2301
```
```  2302 context linordered_idom
```
```  2303 begin
```
```  2304
```
```  2305 subclass ordered_ring_abs
```
```  2306   by standard (auto simp: abs_if not_less mult_less_0_iff)
```
```  2307
```
```  2308 lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
```
```  2309   by (simp add: abs_if)
```
```  2310
```
```  2311 lemma abs_mult_less:
```
```  2312   assumes ac: "\<bar>a\<bar> < c"
```
```  2313     and bd: "\<bar>b\<bar> < d"
```
```  2314   shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
```
```  2315 proof -
```
```  2316   from ac have "0 < c"
```
```  2317     by (blast intro: le_less_trans abs_ge_zero)
```
```  2318   with bd show ?thesis by (simp add: ac mult_strict_mono)
```
```  2319 qed
```
```  2320
```
```  2321 lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
```
```  2322   by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
```
```  2323
```
```  2324 lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
```
```  2325   by (simp add: abs_mult)
```
```  2326
```
```  2327 lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
```
```  2328   by (auto simp add: diff_less_eq ac_simps abs_less_iff)
```
```  2329
```
```  2330 lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
```
```  2331   by (auto simp add: diff_le_eq ac_simps abs_le_iff)
```
```  2332
```
```  2333 lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
```
```  2334   by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
```
```  2335
```
```  2336 end
```
```  2337
```
```  2338 subsection \<open>Dioids\<close>
```
```  2339
```
```  2340 text \<open>
```
```  2341   Dioids are the alternative extensions of semirings, a semiring can
```
```  2342   either be a ring or a dioid but never both.
```
```  2343 \<close>
```
```  2344
```
```  2345 class dioid = semiring_1 + canonically_ordered_monoid_add
```
```  2346 begin
```
```  2347
```
```  2348 subclass ordered_semiring
```
```  2349   by standard (auto simp: le_iff_add distrib_left distrib_right)
```
```  2350
```
```  2351 end
```
```  2352
```
```  2353
```
```  2354 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
```
```  2355
```
```  2356 code_identifier
```
```  2357   code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  2358
```
```  2359 end
```