src/HOL/Algebra/Ring.thy
 author paulson Wed Jun 06 14:25:53 2018 +0100 (12 months ago) changeset 68399 0b71d08528f0 parent 67613 ce654b0e6d69 child 68443 43055b016688 permissions -rw-r--r--
resolution of name clashes in Algebra
```     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] => 'a" (infixl "\<oplus>\<index>" 65)
```
```    17
```
```    18 text \<open>Derived operations.\<close>
```
```    19
```
```    20 definition
```
```    21   a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _"  80)
```
```    22   where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
```
```    23
```
```    24 definition
```
```    25   a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ \<ominus>\<index> _)" [65,66] 65)
```
```    26   where "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)"
```
```    27
```
```    28 locale abelian_monoid =
```
```    29   fixes G (structure)
```
```    30   assumes a_comm_monoid:
```
```    31      "comm_monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    32
```
```    33 definition
```
```    34   finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
```
```    35   "finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    36
```
```    37 syntax
```
```    38   "_finsum" :: "index => idt => 'a set => 'b => 'b"
```
```    39       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
```
```    40 translations
```
```    41   "\<Oplus>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (%i. b) A"
```
```    42   \<comment> \<open>Beware of argument permutation!\<close>
```
```    43
```
```    44
```
```    45 locale abelian_group = abelian_monoid +
```
```    46   assumes a_comm_group:
```
```    47      "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    48
```
```    49
```
```    50 subsection \<open>Basic Properties\<close>
```
```    51
```
```    52 lemma abelian_monoidI:
```
```    53   fixes R (structure)
```
```    54   assumes a_closed:
```
```    55       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
```
```    56     and zero_closed: "\<zero> \<in> carrier R"
```
```    57     and a_assoc:
```
```    58       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
```
```    59       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    60     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
```
```    61     and a_comm:
```
```    62       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
```
```    63   shows "abelian_monoid R"
```
```    64   by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)
```
```    65
```
```    66 lemma abelian_groupI:
```
```    67   fixes R (structure)
```
```    68   assumes a_closed:
```
```    69       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
```
```    70     and zero_closed: "zero R \<in> carrier R"
```
```    71     and a_assoc:
```
```    72       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
```
```    73       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
```
```    74     and a_comm:
```
```    75       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
```
```    76     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
```
```    77     and l_inv_ex: "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> carrier R. y \<oplus> x = \<zero>"
```
```    78   shows "abelian_group R"
```
```    79   by (auto intro!: abelian_group.intro abelian_monoidI
```
```    80       abelian_group_axioms.intro comm_monoidI comm_groupI
```
```    81     intro: assms)
```
```    82
```
```    83 lemma (in abelian_monoid) a_monoid:
```
```    84   "monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    85 by (rule comm_monoid.axioms, rule a_comm_monoid)
```
```    86
```
```    87 lemma (in abelian_group) a_group:
```
```    88   "group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    89   by (simp add: group_def a_monoid)
```
```    90     (simp add: comm_group.axioms group.axioms a_comm_group)
```
```    91
```
```    92 lemmas monoid_record_simps = partial_object.simps monoid.simps
```
```    93
```
```    94 text \<open>Transfer facts from multiplicative structures via interpretation.\<close>
```
```    95
```
```    96 sublocale abelian_monoid <
```
```    97   add: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```    98   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
```
```    99     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
```
```   100     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
```
```   101   by (rule a_monoid) auto
```
```   102
```
```   103 context abelian_monoid begin
```
```   104
```
```   105 lemmas a_closed = add.m_closed
```
```   106 lemmas zero_closed = add.one_closed
```
```   107 lemmas a_assoc = add.m_assoc
```
```   108 lemmas l_zero = add.l_one
```
```   109 lemmas r_zero = add.r_one
```
```   110 lemmas minus_unique = add.inv_unique
```
```   111
```
```   112 end
```
```   113
```
```   114 sublocale abelian_monoid <
```
```   115   add: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```   116   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
```
```   117     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
```
```   118     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
```
```   119     and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
```
```   120   by (rule a_comm_monoid) (auto simp: finsum_def)
```
```   121
```
```   122 context abelian_monoid begin
```
```   123
```
```   124 lemmas a_comm = add.m_comm
```
```   125 lemmas a_lcomm = add.m_lcomm
```
```   126 lemmas a_ac = a_assoc a_comm a_lcomm
```
```   127
```
```   128 lemmas finsum_empty = add.finprod_empty
```
```   129 lemmas finsum_insert = add.finprod_insert
```
```   130 lemmas finsum_zero = add.finprod_one
```
```   131 lemmas finsum_closed = add.finprod_closed
```
```   132 lemmas finsum_Un_Int = add.finprod_Un_Int
```
```   133 lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
```
```   134 lemmas finsum_addf = add.finprod_multf
```
```   135 lemmas finsum_cong' = add.finprod_cong'
```
```   136 lemmas finsum_0 = add.finprod_0
```
```   137 lemmas finsum_Suc = add.finprod_Suc
```
```   138 lemmas finsum_Suc2 = add.finprod_Suc2
```
```   139 lemmas finsum_add = add.finprod_mult
```
```   140 lemmas finsum_infinite = add.finprod_infinite
```
```   141
```
```   142 lemmas finsum_cong = add.finprod_cong
```
```   143 text \<open>Usually, if this rule causes a failed congruence proof error,
```
```   144    the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
```
```   145    Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>
```
```   146
```
```   147 lemmas finsum_reindex = add.finprod_reindex
```
```   148
```
```   149 (* The following would be wrong.  Needed is the equivalent of [^] for addition,
```
```   150   or indeed the canonical embedding from Nat into the monoid.
```
```   151
```
```   152 lemma finsum_const:
```
```   153   assumes fin [simp]: "finite A"
```
```   154       and a [simp]: "a : carrier G"
```
```   155     shows "finsum G (%x. a) A = a [^] card A"
```
```   156   using fin apply induct
```
```   157   apply force
```
```   158   apply (subst finsum_insert)
```
```   159   apply auto
```
```   160   apply (force simp add: Pi_def)
```
```   161   apply (subst m_comm)
```
```   162   apply auto
```
```   163 done
```
```   164 *)
```
```   165
```
```   166 lemmas finsum_singleton = add.finprod_singleton
```
```   167
```
```   168 end
```
```   169
```
```   170 sublocale abelian_group <
```
```   171   add: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```   172   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
```
```   173     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
```
```   174     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
```
```   175     and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
```
```   176   by (rule a_group) (auto simp: m_inv_def a_inv_def)
```
```   177
```
```   178 context abelian_group
```
```   179 begin
```
```   180
```
```   181 lemmas a_inv_closed = add.inv_closed
```
```   182
```
```   183 lemma minus_closed [intro, simp]:
```
```   184   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
```
```   185   by (simp add: a_minus_def)
```
```   186
```
```   187 lemmas l_neg = add.l_inv [simp del]
```
```   188 lemmas r_neg = add.r_inv [simp del]
```
```   189 lemmas minus_zero = add.inv_one
```
```   190 lemmas minus_minus = add.inv_inv
```
```   191 lemmas a_inv_inj = add.inv_inj
```
```   192 lemmas minus_equality = add.inv_equality
```
```   193
```
```   194 end
```
```   195
```
```   196 sublocale abelian_group <
```
```   197   add: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```   198   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
```
```   199     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
```
```   200     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
```
```   201     and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
```
```   202     and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
```
```   203   by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)
```
```   204
```
```   205 lemmas (in abelian_group) minus_add = add.inv_mult
```
```   206
```
```   207 text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>
```
```   208
```
```   209 lemma comm_group_abelian_groupI:
```
```   210   fixes G (structure)
```
```   211   assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```   212   shows "abelian_group G"
```
```   213 proof -
```
```   214   interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
```
```   215     by (rule cg)
```
```   216   show "abelian_group G" ..
```
```   217 qed
```
```   218
```
```   219
```
```   220 subsection \<open>Rings: Basic Definitions\<close>
```
```   221
```
```   222 locale semiring = abelian_monoid R + monoid R for R (structure) +
```
```   223   assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   224       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   225     and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   226       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   227     and l_null[simp]: "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
```
```   228     and r_null[simp]: "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
```
```   229
```
```   230 locale ring = abelian_group R + monoid R for R (structure) +
```
```   231   assumes "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   232       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   233     and "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   234       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   235
```
```   236 locale cring = ring + comm_monoid R
```
```   237
```
```   238 locale "domain" = cring +
```
```   239   assumes one_not_zero [simp]: "\<one> \<noteq> \<zero>"
```
```   240     and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
```
```   241                   a = \<zero> \<or> b = \<zero>"
```
```   242
```
```   243 locale field = "domain" +
```
```   244   assumes field_Units: "Units R = carrier R - {\<zero>}"
```
```   245
```
```   246
```
```   247 subsection \<open>Rings\<close>
```
```   248
```
```   249 lemma ringI:
```
```   250   fixes R (structure)
```
```   251   assumes abelian_group: "abelian_group R"
```
```   252     and monoid: "monoid R"
```
```   253     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   254       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   255     and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   256       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
```
```   257   shows "ring R"
```
```   258   by (auto intro: ring.intro
```
```   259     abelian_group.axioms ring_axioms.intro assms)
```
```   260
```
```   261 context ring begin
```
```   262
```
```   263 lemma is_abelian_group: "abelian_group R" ..
```
```   264
```
```   265 lemma is_monoid: "monoid R"
```
```   266   by (auto intro!: monoidI m_assoc)
```
```   267
```
```   268 lemma is_ring: "ring R"
```
```   269   by (rule ring_axioms)
```
```   270
```
```   271 end
```
```   272
```
```   273 lemmas ring_record_simps = monoid_record_simps ring.simps
```
```   274
```
```   275 lemma cringI:
```
```   276   fixes R (structure)
```
```   277   assumes abelian_group: "abelian_group R"
```
```   278     and comm_monoid: "comm_monoid R"
```
```   279     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
```
```   280       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
```
```   281   shows "cring R"
```
```   282 proof (intro cring.intro ring.intro)
```
```   283   show "ring_axioms R"
```
```   284     \<comment> \<open>Right-distributivity follows from left-distributivity and
```
```   285           commutativity.\<close>
```
```   286   proof (rule ring_axioms.intro)
```
```   287     fix x y z
```
```   288     assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
```
```   289     note [simp] = comm_monoid.axioms [OF comm_monoid]
```
```   290       abelian_group.axioms [OF abelian_group]
```
```   291       abelian_monoid.a_closed
```
```   292
```
```   293     from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
```
```   294       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
```
```   295     also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
```
```   296     also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
```
```   297       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
```
```   298     finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
```
```   299   qed (rule l_distr)
```
```   300 qed (auto intro: cring.intro
```
```   301   abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
```
```   302
```
```   303 (*
```
```   304 lemma (in cring) is_comm_monoid:
```
```   305   "comm_monoid R"
```
```   306   by (auto intro!: comm_monoidI m_assoc m_comm)
```
```   307 *)
```
```   308
```
```   309 lemma (in cring) is_cring:
```
```   310   "cring R" by (rule cring_axioms)
```
```   311
```
```   312
```
```   313 subsubsection \<open>Normaliser for Rings\<close>
```
```   314
```
```   315 lemma (in abelian_group) r_neg2:
```
```   316   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
```
```   317 proof -
```
```   318   assume G: "x \<in> carrier G" "y \<in> carrier G"
```
```   319   then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
```
```   320     by (simp only: r_neg l_zero)
```
```   321   with G show ?thesis
```
```   322     by (simp add: a_ac)
```
```   323 qed
```
```   324
```
```   325 lemma (in abelian_group) r_neg1:
```
```   326   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
```
```   327 proof -
```
```   328   assume G: "x \<in> carrier G" "y \<in> carrier G"
```
```   329   then have "(\<ominus> x \<oplus> x) \<oplus> y = y"
```
```   330     by (simp only: l_neg l_zero)
```
```   331   with G show ?thesis by (simp add: a_ac)
```
```   332 qed
```
```   333
```
```   334 context ring begin
```
```   335
```
```   336 text \<open>
```
```   337   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
```
```   338 \<close>
```
```   339
```
```   340 sublocale semiring
```
```   341 proof -
```
```   342   note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
```
```   343   show "semiring R"
```
```   344   proof (unfold_locales)
```
```   345     fix x
```
```   346     assume R: "x \<in> carrier R"
```
```   347     then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
```
```   348       by (simp del: l_zero r_zero)
```
```   349     also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
```
```   350     finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
```
```   351     with R show "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero)
```
```   352     from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
```
```   353       by (simp del: l_zero r_zero)
```
```   354     also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
```
```   355     finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
```
```   356     with R show "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero)
```
```   357   qed auto
```
```   358 qed
```
```   359
```
```   360 lemma l_minus:
```
```   361   "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
```
```   362 proof -
```
```   363   assume R: "x \<in> carrier R" "y \<in> carrier R"
```
```   364   then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
```
```   365   also from R have "... = \<zero>" by (simp add: l_neg)
```
```   366   finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
```
```   367   with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
```
```   368   with R show ?thesis by (simp add: a_assoc r_neg)
```
```   369 qed
```
```   370
```
```   371 lemma r_minus:
```
```   372   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
```
```   373 proof -
```
```   374   assume R: "x \<in> carrier R" "y \<in> carrier R"
```
```   375   then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
```
```   376   also from R have "... = \<zero>" by (simp add: l_neg)
```
```   377   finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
```
```   378   with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
```
```   379   with R show ?thesis by (simp add: a_assoc r_neg )
```
```   380 qed
```
```   381
```
```   382 end
```
```   383
```
```   384 lemma (in abelian_group) minus_eq:
```
```   385   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
```
```   386   by (simp only: a_minus_def)
```
```   387
```
```   388 text \<open>Setup algebra method:
```
```   389   compute distributive normal form in locale contexts\<close>
```
```   390
```
```   391 ML_file "ringsimp.ML"
```
```   392
```
```   393 attribute_setup algebra = \<open>
```
```   394   Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
```
```   395     -- Scan.lift Args.name -- Scan.repeat Args.term
```
```   396     >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
```
```   397 \<close> "theorems controlling algebra method"
```
```   398
```
```   399 method_setup algebra = \<open>
```
```   400   Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
```
```   401 \<close> "normalisation of algebraic structure"
```
```   402
```
```   403 lemmas (in semiring) semiring_simprules
```
```   404   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   405   a_closed zero_closed  m_closed one_closed
```
```   406   a_assoc l_zero  a_comm m_assoc l_one l_distr r_zero
```
```   407   a_lcomm r_distr l_null r_null
```
```   408
```
```   409 lemmas (in ring) ring_simprules
```
```   410   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   411   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
```
```   412   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
```
```   413   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
```
```   414   a_lcomm r_distr l_null r_null l_minus r_minus
```
```   415
```
```   416 lemmas (in cring)
```
```   417   [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   418   _
```
```   419
```
```   420 lemmas (in cring) cring_simprules
```
```   421   [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
```
```   422   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
```
```   423   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
```
```   424   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
```
```   425   a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
```
```   426
```
```   427 lemma (in semiring) nat_pow_zero:
```
```   428   "(n::nat) \<noteq> 0 \<Longrightarrow> \<zero> [^] n = \<zero>"
```
```   429   by (induct n) simp_all
```
```   430
```
```   431 context semiring begin
```
```   432
```
```   433 lemma one_zeroD:
```
```   434   assumes onezero: "\<one> = \<zero>"
```
```   435   shows "carrier R = {\<zero>}"
```
```   436 proof (rule, rule)
```
```   437   fix x
```
```   438   assume xcarr: "x \<in> carrier R"
```
```   439   from xcarr have "x = x \<otimes> \<one>" by simp
```
```   440   with onezero have "x = x \<otimes> \<zero>" by simp
```
```   441   with xcarr have "x = \<zero>" by simp
```
```   442   then show "x \<in> {\<zero>}" by fast
```
```   443 qed fast
```
```   444
```
```   445 lemma one_zeroI:
```
```   446   assumes carrzero: "carrier R = {\<zero>}"
```
```   447   shows "\<one> = \<zero>"
```
```   448 proof -
```
```   449   from one_closed and carrzero
```
```   450       show "\<one> = \<zero>" by simp
```
```   451 qed
```
```   452
```
```   453 lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
```
```   454   apply rule
```
```   455    apply (erule one_zeroI)
```
```   456   apply (erule one_zeroD)
```
```   457   done
```
```   458
```
```   459 lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
```
```   460   by (simp add: carrier_one_zero)
```
```   461
```
```   462 end
```
```   463
```
```   464 text \<open>Two examples for use of method algebra\<close>
```
```   465
```
```   466 lemma
```
```   467   fixes R (structure) and S (structure)
```
```   468   assumes "ring R" "cring S"
```
```   469   assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
```
```   470   shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b \<and> c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
```
```   471 proof -
```
```   472   interpret ring R by fact
```
```   473   interpret cring S by fact
```
```   474   from RS show ?thesis by algebra
```
```   475 qed
```
```   476
```
```   477 lemma
```
```   478   fixes R (structure)
```
```   479   assumes "ring R"
```
```   480   assumes R: "a \<in> carrier R" "b \<in> carrier R"
```
```   481   shows "a \<ominus> (a \<ominus> b) = b"
```
```   482 proof -
```
```   483   interpret ring R by fact
```
```   484   from R show ?thesis by algebra
```
```   485 qed
```
```   486
```
```   487
```
```   488 subsubsection \<open>Sums over Finite Sets\<close>
```
```   489
```
```   490 lemma (in semiring) finsum_ldistr:
```
```   491   "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
```
```   492    finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
```
```   493 proof (induct set: finite)
```
```   494   case empty then show ?case by simp
```
```   495 next
```
```   496   case (insert x F) then show ?case by (simp add: Pi_def l_distr)
```
```   497 qed
```
```   498
```
```   499 lemma (in semiring) finsum_rdistr:
```
```   500   "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
```
```   501    a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
```
```   502 proof (induct set: finite)
```
```   503   case empty then show ?case by simp
```
```   504 next
```
```   505   case (insert x F) then show ?case by (simp add: Pi_def r_distr)
```
```   506 qed
```
```   507
```
```   508
```
```   509 subsection \<open>Integral Domains\<close>
```
```   510
```
```   511 context "domain" begin
```
```   512
```
```   513 lemma zero_not_one [simp]:
```
```   514   "\<zero> \<noteq> \<one>"
```
```   515   by (rule not_sym) simp
```
```   516
```
```   517 lemma integral_iff: (* not by default a simp rule! *)
```
```   518   "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
```
```   519 proof
```
```   520   assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
```
```   521   then show "a = \<zero> \<or> b = \<zero>" by (simp add: integral)
```
```   522 next
```
```   523   assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> \<or> b = \<zero>"
```
```   524   then show "a \<otimes> b = \<zero>" by auto
```
```   525 qed
```
```   526
```
```   527 lemma m_lcancel:
```
```   528   assumes prem: "a \<noteq> \<zero>"
```
```   529     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
```
```   530   shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
```
```   531 proof
```
```   532   assume eq: "a \<otimes> b = a \<otimes> c"
```
```   533   with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
```
```   534   with R have "a = \<zero> \<or> (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
```
```   535   with prem and R have "b \<ominus> c = \<zero>" by auto
```
```   536   with R have "b = b \<ominus> (b \<ominus> c)" by algebra
```
```   537   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
```
```   538   finally show "b = c" .
```
```   539 next
```
```   540   assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
```
```   541 qed
```
```   542
```
```   543 lemma m_rcancel:
```
```   544   assumes prem: "a \<noteq> \<zero>"
```
```   545     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
```
```   546   shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
```
```   547 proof -
```
```   548   from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
```
```   549   with R show ?thesis by algebra
```
```   550 qed
```
```   551
```
```   552 end
```
```   553
```
```   554
```
```   555 subsection \<open>Fields\<close>
```
```   556
```
```   557 text \<open>Field would not need to be derived from domain, the properties
```
```   558   for domain follow from the assumptions of field\<close>
```
```   559 lemma (in cring) cring_fieldI:
```
```   560   assumes field_Units: "Units R = carrier R - {\<zero>}"
```
```   561   shows "field R"
```
```   562 proof
```
```   563   from field_Units have "\<zero> \<notin> Units R" by fast
```
```   564   moreover have "\<one> \<in> Units R" by fast
```
```   565   ultimately show "\<one> \<noteq> \<zero>" by force
```
```   566 next
```
```   567   fix a b
```
```   568   assume acarr: "a \<in> carrier R"
```
```   569     and bcarr: "b \<in> carrier R"
```
```   570     and ab: "a \<otimes> b = \<zero>"
```
```   571   show "a = \<zero> \<or> b = \<zero>"
```
```   572   proof (cases "a = \<zero>", simp)
```
```   573     assume "a \<noteq> \<zero>"
```
```   574     with field_Units and acarr have aUnit: "a \<in> Units R" by fast
```
```   575     from bcarr have "b = \<one> \<otimes> b" by algebra
```
```   576     also from aUnit acarr have "... = (inv a \<otimes> a) \<otimes> b" by simp
```
```   577     also from acarr bcarr aUnit[THEN Units_inv_closed]
```
```   578     have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
```
```   579     also from ab and acarr bcarr aUnit have "... = (inv a) \<otimes> \<zero>" by simp
```
```   580     also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra
```
```   581     finally have "b = \<zero>" .
```
```   582     then show "a = \<zero> \<or> b = \<zero>" by simp
```
```   583   qed
```
```   584 qed (rule field_Units)
```
```   585
```
```   586 text \<open>Another variant to show that something is a field\<close>
```
```   587 lemma (in cring) cring_fieldI2:
```
```   588   assumes notzero: "\<zero> \<noteq> \<one>"
```
```   589   and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
```
```   590   shows "field R"
```
```   591   apply (rule cring_fieldI, simp add: Units_def)
```
```   592   apply (rule, clarsimp)
```
```   593   apply (simp add: notzero)
```
```   594 proof (clarsimp)
```
```   595   fix x
```
```   596   assume xcarr: "x \<in> carrier R"
```
```   597     and "x \<noteq> \<zero>"
```
```   598   then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
```
```   599   then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast
```
```   600   from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
```
```   601   with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
```
```   602 qed
```
```   603
```
```   604
```
```   605 subsection \<open>Morphisms\<close>
```
```   606
```
```   607 definition
```
```   608   ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
```
```   609   where "ring_hom R S =
```
```   610     {h. h \<in> carrier R \<rightarrow> carrier S \<and>
```
```   611       (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow>
```
```   612         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>
```
```   613       h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
```
```   614
```
```   615 lemma ring_hom_memI:
```
```   616   fixes R (structure) and S (structure)
```
```   617   assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
```
```   618     and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
```
```   619       h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```   620     and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
```
```   621       h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```   622     and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```   623   shows "h \<in> ring_hom R S"
```
```   624   by (auto simp add: ring_hom_def assms Pi_def)
```
```   625
```
```   626 lemma ring_hom_closed:
```
```   627   "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
```
```   628   by (auto simp add: ring_hom_def funcset_mem)
```
```   629
```
```   630 lemma ring_hom_mult:
```
```   631   fixes R (structure) and S (structure)
```
```   632   shows
```
```   633     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
```
```   634     h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```   635     by (simp add: ring_hom_def)
```
```   636
```
```   637 lemma ring_hom_add:
```
```   638   fixes R (structure) and S (structure)
```
```   639   shows
```
```   640     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
```
```   641     h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```   642     by (simp add: ring_hom_def)
```
```   643
```
```   644 lemma ring_hom_one:
```
```   645   fixes R (structure) and S (structure)
```
```   646   shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
```
```   647   by (simp add: ring_hom_def)
```
```   648
```
```   649 locale ring_hom_cring = R?: cring R + S?: cring S
```
```   650     for R (structure) and S (structure) +
```
```   651   fixes h
```
```   652   assumes homh [simp, intro]: "h \<in> ring_hom R S"
```
```   653   notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
```
```   654     and hom_mult [simp] = ring_hom_mult [OF homh]
```
```   655     and hom_add [simp] = ring_hom_add [OF homh]
```
```   656     and hom_one [simp] = ring_hom_one [OF homh]
```
```   657
```
```   658 lemma (in ring_hom_cring) hom_zero [simp]:
```
```   659   "h \<zero> = \<zero>\<^bsub>S\<^esub>"
```
```   660 proof -
```
```   661   have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
```
```   662     by (simp add: hom_add [symmetric] del: hom_add)
```
```   663   then show ?thesis by (simp del: S.r_zero)
```
```   664 qed
```
```   665
```
```   666 lemma (in ring_hom_cring) hom_a_inv [simp]:
```
```   667   "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
```
```   668 proof -
```
```   669   assume R: "x \<in> carrier R"
```
```   670   then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
```
```   671     by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
```
```   672   with R show ?thesis by simp
```
```   673 qed
```
```   674
```
```   675 lemma (in ring_hom_cring) hom_finsum [simp]:
```
```   676   "f \<in> A \<rightarrow> carrier R \<Longrightarrow>
```
```   677   h (finsum R f A) = finsum S (h \<circ> f) A"
```
```   678   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
```
```   679
```
```   680 lemma (in ring_hom_cring) hom_finprod:
```
```   681   "f \<in> A \<rightarrow> carrier R \<Longrightarrow>
```
```   682   h (finprod R f A) = finprod S (h \<circ> f) A"
```
```   683   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
```
```   684
```
```   685 declare ring_hom_cring.hom_finprod [simp]
```
```   686
```
```   687 lemma id_ring_hom [simp]:
```
```   688   "id \<in> ring_hom R R"
```
```   689   by (auto intro!: ring_hom_memI)
```
```   690
```
```   691 end
```