src/HOL/Algebra/Ring.thy
 author paulson Thu Jun 14 14:23:38 2018 +0100 (10 months ago) changeset 68445 c183a6a69f2d parent 68443 43055b016688 child 68517 6b5f15387353 permissions -rw-r--r--
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
```     1 (*  Title:      HOL/Algebra/Ring.thy
```
```     2     Author:     Clemens Ballarin, started 9 December 1996
```
```     3     Copyright:  Clemens Ballarin
```
```     4 *)
```
```     5
```
```     6 theory Ring
```
```     7 imports FiniteProduct
```
```     8 begin
```
```     9
```
```    10 section \<open>The Algebraic Hierarchy of Rings\<close>
```
```    11
```
```    12 subsection \<open>Abelian Groups\<close>
```
```    13
```
```    14 record 'a ring = "'a monoid" +
```
```    15   zero :: 'a ("\<zero>\<index>")
```
```    16   add :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<oplus>\<index>" 65)
```
```    17
```
```    18 abbreviation
```
```    19   add_monoid :: "('a, 'm) ring_scheme \<Rightarrow> ('a, 'm) monoid_scheme"
```
```    20   where "add_monoid R \<equiv> \<lparr> carrier = carrier R, mult = add R, one = zero R, \<dots> = (undefined :: 'm) \<rparr>"
```
```    21
```
```    22 text \<open>Derived operations.\<close>
```
```    23
```
```    24 definition
```
```    25   a_inv :: "[('a, 'm) ring_scheme, 'a ] \<Rightarrow> 'a" ("\<ominus>\<index> _" [81] 80)
```
```    26   where "a_inv R = m_inv (add_monoid R)"
```
```    27
```
```    28 definition
```
```    29   a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ \<ominus>\<index> _)" [65,66] 65)
```
```    30   where "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)"
```
```    31
```
```    32 definition
```
```    33   add_pow :: "[_, ('b :: semiring_1), 'a] \<Rightarrow> 'a" ("[_] \<cdot>\<index> _" [81, 81] 80)
```
```    34   where "add_pow R k a = pow (add_monoid R) a k"
```
```    35
```
```    36 locale abelian_monoid =
```
```    37   fixes G (structure)
```
```    38   assumes a_comm_monoid:
```
```    39      "comm_monoid (add_monoid G)"
```
```    40
```
```    41 definition
```
```    42   finsum :: "[('b, 'm) ring_scheme, 'a \<Rightarrow> 'b, 'a set] \<Rightarrow> 'b" where
```
```    43   "finsum G = finprod (add_monoid G)"
```
```    44
```
```    45 syntax
```
```    46   "_finsum" :: "index \<Rightarrow> idt \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"
```
```    47       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```    48 translations
```
```    49   "\<Oplus>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (\<lambda>i. b) A"
```
```    50   \<comment> \<open>Beware of argument permutation!\<close>
```
```    51
```
```    52
```
```    53 locale abelian_group = abelian_monoid +
```
```    54   assumes a_comm_group:
```
```    55      "comm_group (add_monoid G)"
```
```    56
```
```    57
```
```    58 subsection \<open>Basic Properties\<close>
```
```    59
```
```    60 lemma abelian_monoidI:
```
```    61   fixes R (structure)
```
```    62   assumes "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
```
```    63       and "\<zero> \<in> carrier R"
```
```    64       and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    65       and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
```
```    66       and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
```
```    67   shows "abelian_monoid R"
```
```    68   by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)
```
```    69
```
```    70 lemma abelian_monoidE:
```
```    71   fixes R (structure)
```
```    72   assumes "abelian_monoid R"
```
```    73   shows "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
```
```    74     and "\<zero> \<in> carrier R"
```
```    75     and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    76     and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
```
```    77     and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
```
```    78   using assms unfolding abelian_monoid_def comm_monoid_def comm_monoid_axioms_def monoid_def by auto
```
```    79
```
```    80 lemma abelian_groupI:
```
```    81   fixes R (structure)
```
```    82   assumes "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
```
```    83       and "\<zero> \<in> carrier R"
```
```    84       and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    85       and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
```
```    86       and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
```
```    87       and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> carrier R. y \<oplus> x = \<zero>"
```
```    88   shows "abelian_group R"
```
```    89   by (auto intro!: abelian_group.intro abelian_monoidI
```
```    90       abelian_group_axioms.intro comm_monoidI comm_groupI
```
```    91     intro: assms)
```
```    92
```
```    93 lemma abelian_groupE:
```
```    94   fixes R (structure)
```
```    95   assumes "abelian_group R"
```
```    96   shows "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
```
```    97     and "\<zero> \<in> carrier R"
```
```    98     and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    99     and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
```
```   100     and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
```
```   101     and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> carrier R. y \<oplus> x = \<zero>"
```
```   102   using abelian_group.a_comm_group assms comm_groupE by fastforce+
```
```   103
```
```   104 lemma (in abelian_monoid) a_monoid:
```
```   105   "monoid (add_monoid G)"
```
```   106 by (rule comm_monoid.axioms, rule a_comm_monoid)
```
```   107
```
```   108 lemma (in abelian_group) a_group:
```
```   109   "group (add_monoid G)"
```
```   110   by (simp add: group_def a_monoid)
```
```   111     (simp add: comm_group.axioms group.axioms a_comm_group)
```
```   112
```
```   113 lemmas monoid_record_simps = partial_object.simps monoid.simps
```
```   114
```
```   115 text \<open>Transfer facts from multiplicative structures via interpretation.\<close>
```
```   116
```
```   117 sublocale abelian_monoid <
```
```   118        add: monoid "(add_monoid G)"
```
```   119   rewrites "carrier (add_monoid G) = carrier G"
```
```   120        and "mult    (add_monoid G) = add G"
```
```   121        and "one     (add_monoid G) = zero G"
```
```   122        and "(\<lambda>a k. pow (add_monoid G) a k) = (\<lambda>a k. add_pow G k a)"
```
```   123   by (rule a_monoid) (auto simp add: add_pow_def)
```
```   124
```
```   125 context abelian_monoid
```
```   126 begin
```
```   127
```
```   128 lemmas a_closed = add.m_closed
```
```   129 lemmas zero_closed = add.one_closed
```
```   130 lemmas a_assoc = add.m_assoc
```
```   131 lemmas l_zero = add.l_one
```
```   132 lemmas r_zero = add.r_one
```
```   133 lemmas minus_unique = add.inv_unique
```
```   134
```
```   135 end
```
```   136
```
```   137 sublocale abelian_monoid <
```
```   138   add: comm_monoid "(add_monoid G)"
```
```   139   rewrites "carrier (add_monoid G) = carrier G"
```
```   140        and "mult    (add_monoid G) = add G"
```
```   141        and "one     (add_monoid G) = zero G"
```
```   142        and "finprod (add_monoid G) = finsum G"
```
```   143        and "pow     (add_monoid G) = (\<lambda>a k. add_pow G k a)"
```
```   144   by (rule a_comm_monoid) (auto simp: finsum_def add_pow_def)
```
```   145
```
```   146 context abelian_monoid begin
```
```   147
```
```   148 lemmas a_comm = add.m_comm
```
```   149 lemmas a_lcomm = add.m_lcomm
```
```   150 lemmas a_ac = a_assoc a_comm a_lcomm
```
```   151
```
```   152 lemmas finsum_empty = add.finprod_empty
```
```   153 lemmas finsum_insert = add.finprod_insert
```
```   154 lemmas finsum_zero = add.finprod_one
```
```   155 lemmas finsum_closed = add.finprod_closed
```
```   156 lemmas finsum_Un_Int = add.finprod_Un_Int
```
```   157 lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
```
```   158 lemmas finsum_addf = add.finprod_multf
```
```   159 lemmas finsum_cong' = add.finprod_cong'
```
```   160 lemmas finsum_0 = add.finprod_0
```
```   161 lemmas finsum_Suc = add.finprod_Suc
```
```   162 lemmas finsum_Suc2 = add.finprod_Suc2
```
```   163 lemmas finsum_add = add.finprod_mult
```
```   164 lemmas finsum_infinite = add.finprod_infinite
```
```   165
```
```   166 lemmas finsum_cong = add.finprod_cong
```
```   167 text \<open>Usually, if this rule causes a failed congruence proof error,
```
```   168    the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
```
```   169    Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>
```
```   170
```
```   171 lemmas finsum_reindex = add.finprod_reindex
```
```   172
```
```   173 (* The following would be wrong.  Needed is the equivalent of [^] for addition,
```
```   174   or indeed the canonical embedding from Nat into the monoid.
```
```   175
```
```   176 lemma finsum_const:
```
```   177   assumes fin [simp]: "finite A"
```
```   178       and a [simp]: "a : carrier G"
```
```   179     shows "finsum G (%x. a) A = a [^] card A"
```
```   180   using fin apply induct
```
```   181   apply force
```
```   182   apply (subst finsum_insert)
```
```   183   apply auto
```
```   184   apply (force simp add: Pi_def)
```
```   185   apply (subst m_comm)
```
```   186   apply auto
```
```   187 done
```
```   188 *)
```
```   189
```
```   190 lemmas finsum_singleton = add.finprod_singleton
```
```   191
```
```   192 end
```
```   193
```
```   194 sublocale abelian_group <
```
```   195         add: group "(add_monoid G)"
```
```   196   rewrites "carrier (add_monoid G) = carrier G"
```
```   197        and "mult    (add_monoid G) = add G"
```
```   198        and "one     (add_monoid G) = zero G"
```
```   199        and "m_inv   (add_monoid G) = a_inv G"
```
```   200        and "pow     (add_monoid G) = (\<lambda>a k. add_pow G k a)"
```
```   201   by (rule a_group) (auto simp: m_inv_def a_inv_def add_pow_def)
```
```   202
```
```   203 context abelian_group
```
```   204 begin
```
```   205
```
```   206 lemmas a_inv_closed = add.inv_closed
```
```   207
```
```   208 lemma minus_closed [intro, simp]:
```
```   209   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
```
```   210   by (simp add: a_minus_def)
```
```   211
```
```   212 lemmas l_neg = add.l_inv [simp del]
```
```   213 lemmas r_neg = add.r_inv [simp del]
```
```   214 lemmas minus_minus = add.inv_inv
```
```   215 lemmas a_inv_inj = add.inv_inj
```
```   216 lemmas minus_equality = add.inv_equality
```
```   217
```
```   218 end
```
```   219
```
```   220 sublocale abelian_group <
```
```   221    add: comm_group "(add_monoid G)"
```
```   222   rewrites "carrier (add_monoid G) = carrier G"
```
```   223        and "mult    (add_monoid G) = add G"
```
```   224        and "one     (add_monoid G) = zero G"
```
```   225        and "m_inv   (add_monoid G) = a_inv G"
```
```   226        and "finprod (add_monoid G) = finsum G"
```
```   227        and "pow     (add_monoid G) = (\<lambda>a k. add_pow G k a)"
```
```   228   by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def add_pow_def)
```
```   229
```
```   230 lemmas (in abelian_group) minus_add = add.inv_mult
```
```   231
```
```   232 text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>
```
```   233
```
```   234 lemma comm_group_abelian_groupI:
```
```   235   fixes G (structure)
```
```   236   assumes cg: "comm_group (add_monoid G)"
```
```   237   shows "abelian_group G"
```
```   238 proof -
```
```   239   interpret comm_group "(add_monoid G)"
```
```   240     by (rule cg)
```
```   241   show "abelian_group G" ..
```
```   242 qed
```
```   243
```
```   244
```
```   245 subsection \<open>Rings: Basic Definitions\<close>
```
```   246
```
```   247 locale semiring = abelian_monoid (* for add *) R + monoid (* for mult *) R for R (structure) +
```
```   248   assumes l_distr: "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   249       and r_distr: "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   250       and l_null[simp]: "x \<in> carrier R \<Longrightarrow> \<zero> \<otimes> x = \<zero>"
```
```   251       and r_null[simp]: "x \<in> carrier R \<Longrightarrow> x \<otimes> \<zero> = \<zero>"
```
```   252
```
```   253 locale ring = abelian_group (* for add *) R + monoid (* for mult *) R for R (structure) +
```
```   254   assumes "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   255       and "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   256
```
```   257 locale cring = ring + comm_monoid (* for mult *) R
```
```   258
```
```   259 locale "domain" = cring +
```
```   260   assumes one_not_zero [simp]: "\<one> \<noteq> \<zero>"
```
```   261       and integral: "\<lbrakk> a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R \<rbrakk> \<Longrightarrow> a = \<zero> \<or> b = \<zero>"
```
```   262
```
```   263 locale field = "domain" +
```
```   264   assumes field_Units: "Units R = carrier R - {\<zero>}"
```
```   265
```
```   266
```
```   267 subsection \<open>Rings\<close>
```
```   268
```
```   269 lemma ringI:
```
```   270   fixes R (structure)
```
```   271   assumes "abelian_group R"
```
```   272       and "monoid R"
```
```   273       and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   274       and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   275   shows "ring R"
```
```   276   by (auto intro: ring.intro
```
```   277     abelian_group.axioms ring_axioms.intro assms)
```
```   278
```
```   279 lemma ringE:
```
```   280   fixes R (structure)
```
```   281   assumes "ring R"
```
```   282   shows "abelian_group R"
```
```   283     and "monoid R"
```
```   284     and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   285     and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   286   using assms unfolding ring_def ring_axioms_def by auto
```
```   287
```
```   288 context ring begin
```
```   289
```
```   290 lemma is_abelian_group: "abelian_group R" ..
```
```   291
```
```   292 lemma is_monoid: "monoid R"
```
```   293   by (auto intro!: monoidI m_assoc)
```
```   294
```
```   295 lemma is_ring: "ring R"
```
```   296   by (rule ring_axioms)
```
```   297
```
```   298 end
```
```   299 thm monoid_record_simps
```
```   300 lemmas ring_record_simps = monoid_record_simps ring.simps
```
```   301
```
```   302 lemma cringI:
```
```   303   fixes R (structure)
```
```   304   assumes abelian_group: "abelian_group R"
```
```   305     and comm_monoid: "comm_monoid R"
```
```   306     and l_distr: "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow>
```
```   307                             (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   308   shows "cring R"
```
```   309 proof (intro cring.intro ring.intro)
```
```   310   show "ring_axioms R"
```
```   311     \<comment> \<open>Right-distributivity follows from left-distributivity and
```
```   312           commutativity.\<close>
```
```   313   proof (rule ring_axioms.intro)
```
```   314     fix x y z
```
```   315     assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
```
```   316     note [simp] = comm_monoid.axioms [OF comm_monoid]
```
```   317       abelian_group.axioms [OF abelian_group]
```
```   318       abelian_monoid.a_closed
```
```   319
```
```   320     from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
```
```   321       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
```
```   322     also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
```
```   323     also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
```
```   324       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
```
```   325     finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
```
```   326   qed (rule l_distr)
```
```   327 qed (auto intro: cring.intro
```
```   328   abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
```
```   329
```
```   330 lemma cringE:
```
```   331   fixes R (structure)
```
```   332   assumes "cring R"
```
```   333   shows "comm_monoid R"
```
```   334     and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   335   using assms cring_def apply auto by (simp add: assms cring.axioms(1) ringE(3))
```
```   336
```
```   337 (*
```
```   338 lemma (in cring) is_comm_monoid:
```
```   339   "comm_monoid R"
```
```   340   by (auto intro!: comm_monoidI m_assoc m_comm)
```
```   341 *)
```
```   342 lemma (in cring) is_cring:
```
```   343   "cring R" by (rule cring_axioms)
```
```   344
```
```   345 lemma (in ring) minus_zero [simp]: "\<ominus> \<zero> = \<zero>"
```
```   346   by (simp add: a_inv_def)
```
```   347
```
```   348 subsubsection \<open>Normaliser for Rings\<close>
```
```   349
```
```   350 lemma (in abelian_group) r_neg1:
```
```   351   "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> (x \<oplus> y) = y"
```
```   352 proof -
```
```   353   assume G: "x \<in> carrier G" "y \<in> carrier G"
```
```   354   then have "(\<ominus> x \<oplus> x) \<oplus> y = y"
```
```   355     by (simp only: l_neg l_zero)
```
```   356   with G show ?thesis by (simp add: a_ac)
```
```   357 qed
```
```   358
```
```   359 lemma (in abelian_group) r_neg2:
```
```   360   "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> ((\<ominus> x) \<oplus> y) = y"
```
```   361 proof -
```
```   362   assume G: "x \<in> carrier G" "y \<in> carrier G"
```
```   363   then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
```
```   364     by (simp only: r_neg l_zero)
```
```   365   with G show ?thesis
```
```   366     by (simp add: a_ac)
```
```   367 qed
```
```   368
```
```   369 context ring begin
```
```   370
```
```   371 text \<open>
```
```   372   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
```
```   373 \<close>
```
```   374
```
```   375 sublocale semiring
```
```   376 proof -
```
```   377   note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
```
```   378   show "semiring R"
```
```   379   proof (unfold_locales)
```
```   380     fix x
```
```   381     assume R: "x \<in> carrier R"
```
```   382     then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
```
```   383       by (simp del: l_zero r_zero)
```
```   384     also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
```
```   385     finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
```
```   386     with R show "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero)
```
```   387     from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
```
```   388       by (simp del: l_zero r_zero)
```
```   389     also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
```
```   390     finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
```
```   391     with R show "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero)
```
```   392   qed auto
```
```   393 qed
```
```   394
```
```   395 lemma l_minus:
```
```   396   "\<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> (\<ominus> x) \<otimes> y = \<ominus> (x \<otimes> y)"
```
```   397 proof -
```
```   398   assume R: "x \<in> carrier R" "y \<in> carrier R"
```
```   399   then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
```
```   400   also from R have "... = \<zero>" by (simp add: l_neg)
```
```   401   finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
```
```   402   with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
```
```   403   with R show ?thesis by (simp add: a_assoc r_neg)
```
```   404 qed
```
```   405
```
```   406 lemma r_minus:
```
```   407   "\<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<otimes> (\<ominus> y) = \<ominus> (x \<otimes> y)"
```
```   408 proof -
```
```   409   assume R: "x \<in> carrier R" "y \<in> carrier R"
```
```   410   then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
```
```   411   also from R have "... = \<zero>" by (simp add: l_neg)
```
```   412   finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
```
```   413   with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
```
```   414   with R show ?thesis by (simp add: a_assoc r_neg )
```
```   415 qed
```
```   416
```
```   417 end
```
```   418
```
```   419 lemma (in abelian_group) minus_eq: "x \<ominus> y = x \<oplus> (\<ominus> y)"
```
```   420   by (rule a_minus_def)
```
```   421
```
```   422 text \<open>Setup algebra method:
```
```   423   compute distributive normal form in locale contexts\<close>
```
```   424
```
```   425
```
```   426 ML_file "ringsimp.ML"
```
```   427
```
```   428 attribute_setup algebra = \<open>
```
```   429   Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
```
```   430     -- Scan.lift Args.name -- Scan.repeat Args.term
```
```   431     >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
```
```   432 \<close> "theorems controlling algebra method"
```
```   433
```
```   434 method_setup algebra = \<open>
```
```   435   Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
```
```   436 \<close> "normalisation of algebraic structure"
```
```   437
```
```   438 lemmas (in semiring) semiring_simprules
```
```   439   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   440   a_closed zero_closed  m_closed one_closed
```
```   441   a_assoc l_zero  a_comm m_assoc l_one l_distr r_zero
```
```   442   a_lcomm r_distr l_null r_null
```
```   443
```
```   444 lemmas (in ring) ring_simprules
```
```   445   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   446   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
```
```   447   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
```
```   448   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
```
```   449   a_lcomm r_distr l_null r_null l_minus r_minus
```
```   450
```
```   451 lemmas (in cring)
```
```   452   [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   453   _
```
```   454
```
```   455 lemmas (in cring) cring_simprules
```
```   456   [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   457   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
```
```   458   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
```
```   459   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
```
```   460   a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
```
```   461
```
```   462 lemma (in semiring) nat_pow_zero:
```
```   463   "(n::nat) \<noteq> 0 \<Longrightarrow> \<zero> [^] n = \<zero>"
```
```   464   by (induct n) simp_all
```
```   465
```
```   466 context semiring begin
```
```   467
```
```   468 lemma one_zeroD:
```
```   469   assumes onezero: "\<one> = \<zero>"
```
```   470   shows "carrier R = {\<zero>}"
```
```   471 proof (rule, rule)
```
```   472   fix x
```
```   473   assume xcarr: "x \<in> carrier R"
```
```   474   from xcarr have "x = x \<otimes> \<one>" by simp
```
```   475   with onezero have "x = x \<otimes> \<zero>" by simp
```
```   476   with xcarr have "x = \<zero>" by simp
```
```   477   then show "x \<in> {\<zero>}" by fast
```
```   478 qed fast
```
```   479
```
```   480 lemma one_zeroI:
```
```   481   assumes carrzero: "carrier R = {\<zero>}"
```
```   482   shows "\<one> = \<zero>"
```
```   483 proof -
```
```   484   from one_closed and carrzero
```
```   485       show "\<one> = \<zero>" by simp
```
```   486 qed
```
```   487
```
```   488 lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
```
```   489   apply rule
```
```   490    apply (erule one_zeroI)
```
```   491   apply (erule one_zeroD)
```
```   492   done
```
```   493
```
```   494 lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
```
```   495   by (simp add: carrier_one_zero)
```
```   496
```
```   497 end
```
```   498
```
```   499 text \<open>Two examples for use of method algebra\<close>
```
```   500
```
```   501 lemma
```
```   502   fixes R (structure) and S (structure)
```
```   503   assumes "ring R" "cring S"
```
```   504   assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
```
```   505   shows "a \<oplus> (\<ominus> (a \<oplus> (\<ominus> b))) = b \<and> c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
```
```   506 proof -
```
```   507   interpret ring R by fact
```
```   508   interpret cring S by fact
```
```   509   from RS show ?thesis by algebra
```
```   510 qed
```
```   511
```
```   512 lemma
```
```   513   fixes R (structure)
```
```   514   assumes "ring R"
```
```   515   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   516   shows "a \<ominus> (a \<ominus> b) = b"
```
```   517 proof -
```
```   518   interpret ring R by fact
```
```   519   from R show ?thesis by algebra
```
```   520 qed
```
```   521
```
```   522
```
```   523 subsubsection \<open>Sums over Finite Sets\<close>
```
```   524
```
```   525 lemma (in semiring) finsum_ldistr:
```
```   526   "\<lbrakk> finite A; a \<in> carrier R; f: A \<rightarrow> carrier R \<rbrakk> \<Longrightarrow>
```
```   527     (\<Oplus> i \<in> A. (f i)) \<otimes> a = (\<Oplus> i \<in> A. ((f i) \<otimes> a))"
```
```   528 proof (induct set: finite)
```
```   529   case empty then show ?case by simp
```
```   530 next
```
```   531   case (insert x F) then show ?case by (simp add: Pi_def l_distr)
```
```   532 qed
```
```   533
```
```   534 lemma (in semiring) finsum_rdistr:
```
```   535   "\<lbrakk> finite A; a \<in> carrier R; f: A \<rightarrow> carrier R \<rbrakk> \<Longrightarrow>
```
```   536    a \<otimes> (\<Oplus> i \<in> A. (f i)) = (\<Oplus> i \<in> A. (a \<otimes> (f i)))"
```
```   537 proof (induct set: finite)
```
```   538   case empty then show ?case by simp
```
```   539 next
```
```   540   case (insert x F) then show ?case by (simp add: Pi_def r_distr)
```
```   541 qed
```
```   542
```
```   543 (* ************************************************************************** *)
```
```   544 (* Contributed by Paulo E. de Vilhena.                                        *)
```
```   545
```
```   546 text \<open>A quick detour\<close>
```
```   547
```
```   548 lemma add_pow_int_ge: "(k :: int) \<ge> 0 \<Longrightarrow> [ k ] \<cdot>\<^bsub>R\<^esub> a = [ nat k ] \<cdot>\<^bsub>R\<^esub> a"
```
```   549   by (simp add: add_pow_def int_pow_def nat_pow_def)
```
```   550
```
```   551 lemma add_pow_int_lt: "(k :: int) < 0 \<Longrightarrow> [ k ] \<cdot>\<^bsub>R\<^esub> a = \<ominus>\<^bsub>R\<^esub> ([ nat (- k) ] \<cdot>\<^bsub>R\<^esub> a)"
```
```   552   by (simp add: int_pow_def nat_pow_def a_inv_def add_pow_def)
```
```   553
```
```   554 corollary (in semiring) add_pow_ldistr:
```
```   555   assumes "a \<in> carrier R" "b \<in> carrier R"
```
```   556   shows "([(k :: nat)] \<cdot> a) \<otimes> b = [k] \<cdot> (a \<otimes> b)"
```
```   557 proof -
```
```   558   have "([k] \<cdot> a) \<otimes> b = (\<Oplus> i \<in> {..< k}. a) \<otimes> b"
```
```   559     using add.finprod_const[OF assms(1), of "{..<k}"] by simp
```
```   560   also have " ... = (\<Oplus> i \<in> {..< k}. (a \<otimes> b))"
```
```   561     using finsum_ldistr[of "{..<k}" b "\<lambda>x. a"] assms by simp
```
```   562   also have " ... = [k] \<cdot> (a \<otimes> b)"
```
```   563     using add.finprod_const[of "a \<otimes> b" "{..<k}"] assms by simp
```
```   564   finally show ?thesis .
```
```   565 qed
```
```   566
```
```   567 corollary (in semiring) add_pow_rdistr:
```
```   568   assumes "a \<in> carrier R" "b \<in> carrier R"
```
```   569   shows "a \<otimes> ([(k :: nat)] \<cdot> b) = [k] \<cdot> (a \<otimes> b)"
```
```   570 proof -
```
```   571   have "a \<otimes> ([k] \<cdot> b) = a \<otimes> (\<Oplus> i \<in> {..< k}. b)"
```
```   572     using add.finprod_const[OF assms(2), of "{..<k}"] by simp
```
```   573   also have " ... = (\<Oplus> i \<in> {..< k}. (a \<otimes> b))"
```
```   574     using finsum_rdistr[of "{..<k}" a "\<lambda>x. b"] assms by simp
```
```   575   also have " ... = [k] \<cdot> (a \<otimes> b)"
```
```   576     using add.finprod_const[of "a \<otimes> b" "{..<k}"] assms by simp
```
```   577   finally show ?thesis .
```
```   578 qed
```
```   579
```
```   580 (* For integers, we need the uniqueness of the additive inverse *)
```
```   581 lemma (in ring) add_pow_ldistr_int:
```
```   582   assumes "a \<in> carrier R" "b \<in> carrier R"
```
```   583   shows "([(k :: int)] \<cdot> a) \<otimes> b = [k] \<cdot> (a \<otimes> b)"
```
```   584 proof (cases "k \<ge> 0")
```
```   585   case True thus ?thesis
```
```   586     using add_pow_int_ge[of k R] add_pow_ldistr[OF assms] by auto
```
```   587 next
```
```   588   case False thus ?thesis
```
```   589     using add_pow_int_lt[of k R a] add_pow_int_lt[of k R "a \<otimes> b"]
```
```   590           add_pow_ldistr[OF assms, of "nat (- k)"] assms l_minus by auto
```
```   591 qed
```
```   592
```
```   593 lemma (in ring) add_pow_rdistr_int:
```
```   594   assumes "a \<in> carrier R" "b \<in> carrier R"
```
```   595   shows "a \<otimes> ([(k :: int)] \<cdot> b) = [k] \<cdot> (a \<otimes> b)"
```
```   596 proof (cases "k \<ge> 0")
```
```   597   case True thus ?thesis
```
```   598     using add_pow_int_ge[of k R] add_pow_rdistr[OF assms] by auto
```
```   599 next
```
```   600   case False thus ?thesis
```
```   601     using add_pow_int_lt[of k R b] add_pow_int_lt[of k R "a \<otimes> b"]
```
```   602           add_pow_rdistr[OF assms, of "nat (- k)"] assms r_minus by auto
```
```   603 qed
```
```   604
```
```   605
```
```   606 subsection \<open>Integral Domains\<close>
```
```   607
```
```   608 context "domain" begin
```
```   609
```
```   610 lemma zero_not_one [simp]: "\<zero> \<noteq> \<one>"
```
```   611   by (rule not_sym) simp
```
```   612
```
```   613 lemma integral_iff: (* not by default a simp rule! *)
```
```   614   "\<lbrakk> a \<in> carrier R; b \<in> carrier R \<rbrakk> \<Longrightarrow> (a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
```
```   615 proof
```
```   616   assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
```
```   617   then show "a = \<zero> \<or> b = \<zero>" by (simp add: integral)
```
```   618 next
```
```   619   assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> \<or> b = \<zero>"
```
```   620   then show "a \<otimes> b = \<zero>" by auto
```
```   621 qed
```
```   622
```
```   623 lemma m_lcancel:
```
```   624   assumes prem: "a \<noteq> \<zero>"
```
```   625     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
```
```   626   shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
```
```   627 proof
```
```   628   assume eq: "a \<otimes> b = a \<otimes> c"
```
```   629   with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
```
```   630   with R have "a = \<zero> \<or> (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
```
```   631   with prem and R have "b \<ominus> c = \<zero>" by auto
```
```   632   with R have "b = b \<ominus> (b \<ominus> c)" by algebra
```
```   633   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
```
```   634   finally show "b = c" .
```
```   635 next
```
```   636   assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
```
```   637 qed
```
```   638
```
```   639 lemma m_rcancel:
```
```   640   assumes prem: "a \<noteq> \<zero>"
```
```   641     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
```
```   642   shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
```
```   643 proof -
```
```   644   from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
```
```   645   with R show ?thesis by algebra
```
```   646 qed
```
```   647
```
```   648 end
```
```   649
```
```   650
```
```   651 subsection \<open>Fields\<close>
```
```   652
```
```   653 text \<open>Field would not need to be derived from domain, the properties
```
```   654   for domain follow from the assumptions of field\<close>
```
```   655
```
```   656 lemma (in cring) cring_fieldI:
```
```   657   assumes field_Units: "Units R = carrier R - {\<zero>}"
```
```   658   shows "field R"
```
```   659 proof
```
```   660   from field_Units have "\<zero> \<notin> Units R" by fast
```
```   661   moreover have "\<one> \<in> Units R" by fast
```
```   662   ultimately show "\<one> \<noteq> \<zero>" by force
```
```   663 next
```
```   664   fix a b
```
```   665   assume acarr: "a \<in> carrier R"
```
```   666     and bcarr: "b \<in> carrier R"
```
```   667     and ab: "a \<otimes> b = \<zero>"
```
```   668   show "a = \<zero> \<or> b = \<zero>"
```
```   669   proof (cases "a = \<zero>", simp)
```
```   670     assume "a \<noteq> \<zero>"
```
```   671     with field_Units and acarr have aUnit: "a \<in> Units R" by fast
```
```   672     from bcarr have "b = \<one> \<otimes> b" by algebra
```
```   673     also from aUnit acarr have "... = (inv a \<otimes> a) \<otimes> b" by simp
```
```   674     also from acarr bcarr aUnit[THEN Units_inv_closed]
```
```   675     have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
```
```   676     also from ab and acarr bcarr aUnit have "... = (inv a) \<otimes> \<zero>" by simp
```
```   677     also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra
```
```   678     finally have "b = \<zero>" .
```
```   679     then show "a = \<zero> \<or> b = \<zero>" by simp
```
```   680   qed
```
```   681 qed (rule field_Units)
```
```   682
```
```   683 text \<open>Another variant to show that something is a field\<close>
```
```   684 lemma (in cring) cring_fieldI2:
```
```   685   assumes notzero: "\<zero> \<noteq> \<one>"
```
```   686   and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
```
```   687   shows "field R"
```
```   688   apply (rule cring_fieldI, simp add: Units_def)
```
```   689   apply (rule, clarsimp)
```
```   690   apply (simp add: notzero)
```
```   691 proof (clarsimp)
```
```   692   fix x
```
```   693   assume xcarr: "x \<in> carrier R"
```
```   694     and "x \<noteq> \<zero>"
```
```   695   then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
```
```   696   then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast
```
```   697   from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
```
```   698   with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
```
```   699 qed
```
```   700
```
```   701
```
```   702 subsection \<open>Morphisms\<close>
```
```   703
```
```   704 definition
```
```   705   ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
```
```   706   where "ring_hom R S =
```
```   707     {h. h \<in> carrier R \<rightarrow> carrier S \<and>
```
```   708       (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow>
```
```   709         h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and>
```
```   710       h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
```
```   711
```
```   712 lemma ring_hom_memI:
```
```   713   fixes R (structure) and S (structure)
```
```   714   assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"
```
```   715       and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```   716       and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```   717       and "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```   718   shows "h \<in> ring_hom R S"
```
```   719   by (auto simp add: ring_hom_def assms Pi_def)
```
```   720
```
```   721 lemma ring_hom_memE:
```
```   722   fixes R (structure) and S (structure)
```
```   723   assumes "h \<in> ring_hom R S"
```
```   724   shows "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"
```
```   725     and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```   726     and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```   727     and "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```   728   using assms unfolding ring_hom_def by auto
```
```   729
```
```   730 lemma ring_hom_closed:
```
```   731   "\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R \<rbrakk> \<Longrightarrow> h x \<in> carrier S"
```
```   732   by (auto simp add: ring_hom_def funcset_mem)
```
```   733
```
```   734 lemma ring_hom_mult:
```
```   735   fixes R (structure) and S (structure)
```
```   736   shows "\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```   737     by (simp add: ring_hom_def)
```
```   738
```
```   739 lemma ring_hom_add:
```
```   740   fixes R (structure) and S (structure)
```
```   741   shows "\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```   742     by (simp add: ring_hom_def)
```
```   743
```
```   744 lemma ring_hom_one:
```
```   745   fixes R (structure) and S (structure)
```
```   746   shows "h \<in> ring_hom R S \<Longrightarrow> h \<one> = \<one>\<^bsub>S\<^esub>"
```
```   747   by (simp add: ring_hom_def)
```
```   748
```
```   749 lemma ring_hom_zero:
```
```   750   fixes R (structure) and S (structure)
```
```   751   assumes "h \<in> ring_hom R S" "ring R" "ring S"
```
```   752   shows "h \<zero> = \<zero>\<^bsub>S\<^esub>"
```
```   753 proof -
```
```   754   have "h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero>"
```
```   755     using ring_hom_add[OF assms(1), of \<zero> \<zero>] assms(2)
```
```   756     by (simp add: ring.ring_simprules(2) ring.ring_simprules(15))
```
```   757   thus ?thesis
```
```   758     by (metis abelian_group.l_neg assms ring.is_abelian_group ring.ring_simprules(18) ring.ring_simprules(2) ring_hom_closed)
```
```   759 qed
```
```   760
```
```   761 locale ring_hom_cring =
```
```   762   R?: cring R + S?: cring S for R (structure) and S (structure) + fixes h
```
```   763   assumes homh [simp, intro]: "h \<in> ring_hom R S"
```
```   764   notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
```
```   765     and hom_mult [simp] = ring_hom_mult [OF homh]
```
```   766     and hom_add [simp] = ring_hom_add [OF homh]
```
```   767     and hom_one [simp] = ring_hom_one [OF homh]
```
```   768
```
```   769 lemma (in ring_hom_cring) hom_zero [simp]: "h \<zero> = \<zero>\<^bsub>S\<^esub>"
```
```   770 proof -
```
```   771   have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
```
```   772     by (simp add: hom_add [symmetric] del: hom_add)
```
```   773   then show ?thesis by (simp del: S.r_zero)
```
```   774 qed
```
```   775
```
```   776 lemma (in ring_hom_cring) hom_a_inv [simp]:
```
```   777   "x \<in> carrier R \<Longrightarrow> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
```
```   778 proof -
```
```   779   assume R: "x \<in> carrier R"
```
```   780   then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
```
```   781     by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
```
```   782   with R show ?thesis by simp
```
```   783 qed
```
```   784
```
```   785 lemma (in ring_hom_cring) hom_finsum [simp]:
```
```   786   assumes "f: A \<rightarrow> carrier R"
```
```   787   shows "h (\<Oplus> i \<in> A. f i) = (\<Oplus>\<^bsub>S\<^esub> i \<in> A. (h o f) i)"
```
```   788   using assms by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
```
```   789
```
```   790 lemma (in ring_hom_cring) hom_finprod:
```
```   791   assumes "f: A \<rightarrow> carrier R"
```
```   792   shows "h (\<Otimes> i \<in> A. f i) = (\<Otimes>\<^bsub>S\<^esub> i \<in> A. (h o f) i)"
```
```   793   using assms by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
```
```   794
```
```   795 declare ring_hom_cring.hom_finprod [simp]
```
```   796
```
```   797 lemma id_ring_hom [simp]: "id \<in> ring_hom R R"
```
```   798   by (auto intro!: ring_hom_memI)
```
```   799
```
```   800 (* Next lemma contributed by Paulo EmÃ­lio de Vilhena. *)
```
```   801
```
```   802 lemma ring_hom_trans:
```
```   803   "\<lbrakk> f \<in> ring_hom R S; g \<in> ring_hom S T \<rbrakk> \<Longrightarrow> g \<circ> f \<in> ring_hom R T"
```
```   804   by (rule ring_hom_memI) (auto simp add: ring_hom_closed ring_hom_mult ring_hom_add ring_hom_one)
```
```   805
```
```   806 subsection\<open>Jeremy Avigad's @{text"More_Finite_Product"} material\<close>
```
```   807
```
```   808 (* need better simplification rules for rings *)
```
```   809 (* the next one holds more generally for abelian groups *)
```
```   810
```
```   811 lemma (in cring) sum_zero_eq_neg: "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y"
```
```   812   by (metis minus_equality)
```
```   813
```
```   814 lemma (in domain) square_eq_one:
```
```   815   fixes x
```
```   816   assumes [simp]: "x \<in> carrier R"
```
```   817     and "x \<otimes> x = \<one>"
```
```   818   shows "x = \<one> \<or> x = \<ominus>\<one>"
```
```   819 proof -
```
```   820   have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>"
```
```   821     by (simp add: ring_simprules)
```
```   822   also from \<open>x \<otimes> x = \<one>\<close> have "\<dots> = \<zero>"
```
```   823     by (simp add: ring_simprules)
```
```   824   finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" .
```
```   825   then have "(x \<oplus> \<one>) = \<zero> \<or> (x \<oplus> \<ominus> \<one>) = \<zero>"
```
```   826     by (intro integral) auto
```
```   827   then show ?thesis
```
```   828     by (metis add.inv_closed add.inv_solve_right assms(1) l_zero one_closed zero_closed)
```
```   829 qed
```
```   830
```
```   831 lemma (in domain) inv_eq_self: "x \<in> Units R \<Longrightarrow> x = inv x \<Longrightarrow> x = \<one> \<or> x = \<ominus>\<one>"
```
```   832   by (metis Units_closed Units_l_inv square_eq_one)
```
```   833
```
```   834
```
```   835 text \<open>
```
```   836   The following translates theorems about groups to the facts about
```
```   837   the units of a ring. (The list should be expanded as more things are
```
```   838   needed.)
```
```   839 \<close>
```
```   840
```
```   841 lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow> finite (Units R)"
```
```   842   by (rule finite_subset) auto
```
```   843
```
```   844 lemma (in monoid) units_of_pow:
```
```   845   fixes n :: nat
```
```   846   shows "x \<in> Units G \<Longrightarrow> x [^]\<^bsub>units_of G\<^esub> n = x [^]\<^bsub>G\<^esub> n"
```
```   847   apply (induct n)
```
```   848   apply (auto simp add: units_group group.is_monoid
```
```   849     monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult)
```
```   850   done
```
```   851
```
```   852 lemma (in cring) units_power_order_eq_one:
```
```   853   "finite (Units R) \<Longrightarrow> a \<in> Units R \<Longrightarrow> a [^] card(Units R) = \<one>"
```
```   854   by (metis comm_group.power_order_eq_one units_comm_group units_of_carrier units_of_one units_of_pow)
```
```   855
```
```   856 subsection\<open>Jeremy Avigad's @{text"More_Ring"} material\<close>
```
```   857
```
```   858 lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub> \<Longrightarrow> \<forall>x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}. x \<in> Units R \<Longrightarrow> field R"
```
```   859   apply (unfold_locales)
```
```   860     apply (use cring_axioms in auto)
```
```   861    apply (rule trans)
```
```   862     apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
```
```   863      apply assumption
```
```   864     apply (subst m_assoc)
```
```   865        apply auto
```
```   866   apply (unfold Units_def)
```
```   867   apply auto
```
```   868   done
```
```   869
```
```   870 lemma (in monoid) inv_char:
```
```   871   "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
```
```   872   apply (subgoal_tac "x \<in> Units G")
```
```   873    apply (subgoal_tac "y = inv x \<otimes> \<one>")
```
```   874     apply simp
```
```   875    apply (erule subst)
```
```   876    apply (subst m_assoc [symmetric])
```
```   877       apply auto
```
```   878   apply (unfold Units_def)
```
```   879   apply auto
```
```   880   done
```
```   881
```
```   882 lemma (in comm_monoid) comm_inv_char: "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
```
```   883   by (simp add: inv_char m_comm)
```
```   884
```
```   885 lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
```
```   886   apply (rule inv_char)
```
```   887      apply (auto simp add: l_minus r_minus)
```
```   888   done
```
```   889
```
```   890 lemma (in monoid) inv_eq_imp_eq: "x \<in> Units G \<Longrightarrow> y \<in> Units G \<Longrightarrow> inv x = inv y \<Longrightarrow> x = y"
```
```   891   apply (subgoal_tac "inv (inv x) = inv (inv y)")
```
```   892    apply (subst (asm) Units_inv_inv)+
```
```   893     apply auto
```
```   894   done
```
```   895
```
```   896 lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> \<in> Units R"
```
```   897   apply (unfold Units_def)
```
```   898   apply auto
```
```   899   apply (rule_tac x = "\<ominus> \<one>" in bexI)
```
```   900    apply auto
```
```   901   apply (simp add: l_minus r_minus)
```
```   902   done
```
```   903
```
```   904 lemma (in ring) inv_eq_neg_one_eq: "x \<in> Units R \<Longrightarrow> inv x = \<ominus> \<one> \<longleftrightarrow> x = \<ominus> \<one>"
```
```   905   apply auto
```
```   906   apply (subst Units_inv_inv [symmetric])
```
```   907    apply auto
```
```   908   done
```
```   909
```
```   910 lemma (in monoid) inv_eq_one_eq: "x \<in> Units G \<Longrightarrow> inv x = \<one> \<longleftrightarrow> x = \<one>"
```
```   911   by (metis Units_inv_inv inv_one)
```
```   912
```
```   913 end
```