src/HOL/Algebra/Ring.thy
author wenzelm
Thu Mar 26 20:08:55 2009 +0100 (2009-03-26)
changeset 30729 461ee3e49ad3
parent 29237 e90d9d51106b
child 35054 a5db9779b026
permissions -rw-r--r--
interpretation/interpret: prefixes are mandatory by default;
     1 (*
     2   Title:     The algebraic hierarchy of rings
     3   Author:    Clemens Ballarin, started 9 December 1996
     4   Copyright: Clemens Ballarin
     5 *)
     6 
     7 theory Ring
     8 imports FiniteProduct
     9 uses ("ringsimp.ML") begin
    10 
    11 
    12 section {* The Algebraic Hierarchy of Rings *}
    13 
    14 subsection {* Abelian Groups *}
    15 
    16 record 'a ring = "'a monoid" +
    17   zero :: 'a ("\<zero>\<index>")
    18   add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65)
    19 
    20 text {* Derived operations. *}
    21 
    22 constdefs (structure R)
    23   a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
    24   "a_inv R == m_inv (| carrier = carrier R, mult = add R, one = zero R |)"
    25 
    26   a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
    27   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y == x \<oplus> (\<ominus> y)"
    28 
    29 locale abelian_monoid =
    30   fixes G (structure)
    31   assumes a_comm_monoid:
    32      "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"
    33 
    34 
    35 text {*
    36   The following definition is redundant but simple to use.
    37 *}
    38 
    39 locale abelian_group = abelian_monoid +
    40   assumes a_comm_group:
    41      "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"
    42 
    43 
    44 subsection {* Basic Properties *}
    45 
    46 lemma abelian_monoidI:
    47   fixes R (structure)
    48   assumes a_closed:
    49       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
    50     and zero_closed: "\<zero> \<in> carrier R"
    51     and a_assoc:
    52       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
    53       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
    54     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
    55     and a_comm:
    56       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
    57   shows "abelian_monoid R"
    58   by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)
    59 
    60 lemma abelian_groupI:
    61   fixes R (structure)
    62   assumes a_closed:
    63       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
    64     and zero_closed: "zero R \<in> carrier R"
    65     and a_assoc:
    66       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
    67       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
    68     and a_comm:
    69       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
    70     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
    71     and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>"
    72   shows "abelian_group R"
    73   by (auto intro!: abelian_group.intro abelian_monoidI
    74       abelian_group_axioms.intro comm_monoidI comm_groupI
    75     intro: assms)
    76 
    77 lemma (in abelian_monoid) a_monoid:
    78   "monoid (| carrier = carrier G, mult = add G, one = zero G |)"
    79 by (rule comm_monoid.axioms, rule a_comm_monoid) 
    80 
    81 lemma (in abelian_group) a_group:
    82   "group (| carrier = carrier G, mult = add G, one = zero G |)"
    83   by (simp add: group_def a_monoid)
    84     (simp add: comm_group.axioms group.axioms a_comm_group)
    85 
    86 lemmas monoid_record_simps = partial_object.simps monoid.simps
    87 
    88 lemma (in abelian_monoid) a_closed [intro, simp]:
    89   "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G"
    90   by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps]) 
    91 
    92 lemma (in abelian_monoid) zero_closed [intro, simp]:
    93   "\<zero> \<in> carrier G"
    94   by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps])
    95 
    96 lemma (in abelian_group) a_inv_closed [intro, simp]:
    97   "x \<in> carrier G ==> \<ominus> x \<in> carrier G"
    98   by (simp add: a_inv_def
    99     group.inv_closed [OF a_group, simplified monoid_record_simps])
   100 
   101 lemma (in abelian_group) minus_closed [intro, simp]:
   102   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
   103   by (simp add: a_minus_def)
   104 
   105 lemma (in abelian_group) a_l_cancel [simp]:
   106   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   107    (x \<oplus> y = x \<oplus> z) = (y = z)"
   108   by (rule group.l_cancel [OF a_group, simplified monoid_record_simps])
   109 
   110 lemma (in abelian_group) a_r_cancel [simp]:
   111   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   112    (y \<oplus> x = z \<oplus> x) = (y = z)"
   113   by (rule group.r_cancel [OF a_group, simplified monoid_record_simps])
   114 
   115 lemma (in abelian_monoid) a_assoc:
   116   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
   117   (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
   118   by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps])
   119 
   120 lemma (in abelian_monoid) l_zero [simp]:
   121   "x \<in> carrier G ==> \<zero> \<oplus> x = x"
   122   by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps])
   123 
   124 lemma (in abelian_group) l_neg:
   125   "x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>"
   126   by (simp add: a_inv_def
   127     group.l_inv [OF a_group, simplified monoid_record_simps])
   128 
   129 lemma (in abelian_monoid) a_comm:
   130   "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
   131   by (rule comm_monoid.m_comm [OF a_comm_monoid,
   132     simplified monoid_record_simps])
   133 
   134 lemma (in abelian_monoid) a_lcomm:
   135   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
   136    x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
   137   by (rule comm_monoid.m_lcomm [OF a_comm_monoid,
   138                                 simplified monoid_record_simps])
   139 
   140 lemma (in abelian_monoid) r_zero [simp]:
   141   "x \<in> carrier G ==> x \<oplus> \<zero> = x"
   142   using monoid.r_one [OF a_monoid]
   143   by simp
   144 
   145 lemma (in abelian_group) r_neg:
   146   "x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>"
   147   using group.r_inv [OF a_group]
   148   by (simp add: a_inv_def)
   149 
   150 lemma (in abelian_group) minus_zero [simp]:
   151   "\<ominus> \<zero> = \<zero>"
   152   by (simp add: a_inv_def
   153     group.inv_one [OF a_group, simplified monoid_record_simps])
   154 
   155 lemma (in abelian_group) minus_minus [simp]:
   156   "x \<in> carrier G ==> \<ominus> (\<ominus> x) = x"
   157   using group.inv_inv [OF a_group, simplified monoid_record_simps]
   158   by (simp add: a_inv_def)
   159 
   160 lemma (in abelian_group) a_inv_inj:
   161   "inj_on (a_inv G) (carrier G)"
   162   using group.inv_inj [OF a_group, simplified monoid_record_simps]
   163   by (simp add: a_inv_def)
   164 
   165 lemma (in abelian_group) minus_add:
   166   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
   167   using comm_group.inv_mult [OF a_comm_group]
   168   by (simp add: a_inv_def)
   169 
   170 lemma (in abelian_group) minus_equality: 
   171   "[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y" 
   172   using group.inv_equality [OF a_group] 
   173   by (auto simp add: a_inv_def) 
   174  
   175 lemma (in abelian_monoid) minus_unique: 
   176   "[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G;
   177       y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'" 
   178   using monoid.inv_unique [OF a_monoid] 
   179   by (simp add: a_inv_def) 
   180 
   181 lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm
   182 
   183 text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
   184 lemma comm_group_abelian_groupI:
   185   fixes G (structure)
   186   assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   187   shows "abelian_group G"
   188 proof -
   189   interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   190     by (rule cg)
   191   show "abelian_group G" ..
   192 qed
   193 
   194 
   195 subsection {* Sums over Finite Sets *}
   196 
   197 text {*
   198   This definition makes it easy to lift lemmas from @{term finprod}.
   199 *}
   200 
   201 constdefs
   202   finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b"
   203   "finsum G f A == finprod (| carrier = carrier G,
   204      mult = add G, one = zero G |) f A"
   205 
   206 syntax
   207   "_finsum" :: "index => idt => 'a set => 'b => 'b"
   208       ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
   209 syntax (xsymbols)
   210   "_finsum" :: "index => idt => 'a set => 'b => 'b"
   211       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
   212 syntax (HTML output)
   213   "_finsum" :: "index => idt => 'a set => 'b => 'b"
   214       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
   215 translations
   216   "\<Oplus>\<index>i:A. b" == "finsum \<struct>\<index> (%i. b) A"
   217   -- {* Beware of argument permutation! *}
   218 
   219 context abelian_monoid begin
   220 
   221 (*
   222   lemmas finsum_empty [simp] =
   223     comm_monoid.finprod_empty [OF a_comm_monoid, simplified]
   224   is dangeous, because attributes (like simplified) are applied upon opening
   225   the locale, simplified refers to the simpset at that time!!!
   226 
   227   lemmas finsum_empty [simp] =
   228     abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def,
   229       simplified monoid_record_simps]
   230   makes the locale slow, because proofs are repeated for every
   231   "lemma (in abelian_monoid)" command.
   232   When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down
   233   from 110 secs to 60 secs.
   234 *)
   235 
   236 lemma finsum_empty [simp]:
   237   "finsum G f {} = \<zero>"
   238   by (rule comm_monoid.finprod_empty [OF a_comm_monoid,
   239     folded finsum_def, simplified monoid_record_simps])
   240 
   241 lemma finsum_insert [simp]:
   242   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |]
   243   ==> finsum G f (insert a F) = f a \<oplus> finsum G f F"
   244   by (rule comm_monoid.finprod_insert [OF a_comm_monoid,
   245     folded finsum_def, simplified monoid_record_simps])
   246 
   247 lemma finsum_zero [simp]:
   248   "finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>"
   249   by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def,
   250     simplified monoid_record_simps])
   251 
   252 lemma finsum_closed [simp]:
   253   fixes A
   254   assumes fin: "finite A" and f: "f \<in> A -> carrier G" 
   255   shows "finsum G f A \<in> carrier G"
   256   apply (rule comm_monoid.finprod_closed [OF a_comm_monoid,
   257     folded finsum_def, simplified monoid_record_simps])
   258    apply (rule fin)
   259   apply (rule f)
   260   done
   261 
   262 lemma finsum_Un_Int:
   263   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
   264      finsum G g (A Un B) \<oplus> finsum G g (A Int B) =
   265      finsum G g A \<oplus> finsum G g B"
   266   by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid,
   267     folded finsum_def, simplified monoid_record_simps])
   268 
   269 lemma finsum_Un_disjoint:
   270   "[| finite A; finite B; A Int B = {};
   271       g \<in> A -> carrier G; g \<in> B -> carrier G |]
   272    ==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B"
   273   by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid,
   274     folded finsum_def, simplified monoid_record_simps])
   275 
   276 lemma finsum_addf:
   277   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
   278    finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)"
   279   by (rule comm_monoid.finprod_multf [OF a_comm_monoid,
   280     folded finsum_def, simplified monoid_record_simps])
   281 
   282 lemma finsum_cong':
   283   "[| A = B; g : B -> carrier G;
   284       !!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B"
   285   by (rule comm_monoid.finprod_cong' [OF a_comm_monoid,
   286     folded finsum_def, simplified monoid_record_simps]) auto
   287 
   288 lemma finsum_0 [simp]:
   289   "f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0"
   290   by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def,
   291     simplified monoid_record_simps])
   292 
   293 lemma finsum_Suc [simp]:
   294   "f : {..Suc n} -> carrier G ==>
   295    finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})"
   296   by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def,
   297     simplified monoid_record_simps])
   298 
   299 lemma finsum_Suc2:
   300   "f : {..Suc n} -> carrier G ==>
   301    finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
   302   by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def,
   303     simplified monoid_record_simps])
   304 
   305 lemma finsum_add [simp]:
   306   "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
   307      finsum G (%i. f i \<oplus> g i) {..n::nat} =
   308      finsum G f {..n} \<oplus> finsum G g {..n}"
   309   by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
   310     simplified monoid_record_simps])
   311 
   312 lemma finsum_cong:
   313   "[| A = B; f : B -> carrier G;
   314       !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
   315   by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
   316     simplified monoid_record_simps]) (auto simp add: simp_implies_def)
   317 
   318 text {*Usually, if this rule causes a failed congruence proof error,
   319    the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
   320    Adding @{thm [source] Pi_def} to the simpset is often useful. *}
   321 
   322 lemma finsum_reindex:
   323   assumes fin: "finite A"
   324     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow> 
   325         inj_on h A ==> finsum G f (h ` A) = finsum G (%x. f (h x)) A"
   326   using fin apply induct
   327   apply (auto simp add: finsum_insert Pi_def)
   328 done
   329 
   330 (* The following is wrong.  Needed is the equivalent of (^) for addition,
   331   or indeed the canonical embedding from Nat into the monoid.
   332 
   333 lemma finsum_const:
   334   assumes fin [simp]: "finite A"
   335       and a [simp]: "a : carrier G"
   336     shows "finsum G (%x. a) A = a (^) card A"
   337   using fin apply induct
   338   apply force
   339   apply (subst finsum_insert)
   340   apply auto
   341   apply (force simp add: Pi_def)
   342   apply (subst m_comm)
   343   apply auto
   344 done
   345 *)
   346 
   347 (* By Jesus Aransay. *)
   348 
   349 lemma finsum_singleton:
   350   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
   351   shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
   352   using i_in_A finsum_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<zero>)"]
   353     fin_A f_Pi finsum_zero [of "A - {i}"]
   354     finsum_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<zero>)" "(\<lambda>i. \<zero>)"] 
   355   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
   356 
   357 end
   358 
   359 
   360 subsection {* Rings: Basic Definitions *}
   361 
   362 locale ring = abelian_group R + monoid R for R (structure) +
   363   assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
   364       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
   365     and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
   366       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
   367 
   368 locale cring = ring + comm_monoid R
   369 
   370 locale "domain" = cring +
   371   assumes one_not_zero [simp]: "\<one> ~= \<zero>"
   372     and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
   373                   a = \<zero> | b = \<zero>"
   374 
   375 locale field = "domain" +
   376   assumes field_Units: "Units R = carrier R - {\<zero>}"
   377 
   378 
   379 subsection {* Rings *}
   380 
   381 lemma ringI:
   382   fixes R (structure)
   383   assumes abelian_group: "abelian_group R"
   384     and monoid: "monoid R"
   385     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
   386       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
   387     and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
   388       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
   389   shows "ring R"
   390   by (auto intro: ring.intro
   391     abelian_group.axioms ring_axioms.intro assms)
   392 
   393 lemma (in ring) is_abelian_group:
   394   "abelian_group R"
   395   ..
   396 
   397 lemma (in ring) is_monoid:
   398   "monoid R"
   399   by (auto intro!: monoidI m_assoc)
   400 
   401 lemma (in ring) is_ring:
   402   "ring R"
   403   by (rule ring_axioms)
   404 
   405 lemmas ring_record_simps = monoid_record_simps ring.simps
   406 
   407 lemma cringI:
   408   fixes R (structure)
   409   assumes abelian_group: "abelian_group R"
   410     and comm_monoid: "comm_monoid R"
   411     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
   412       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
   413   shows "cring R"
   414 proof (intro cring.intro ring.intro)
   415   show "ring_axioms R"
   416     -- {* Right-distributivity follows from left-distributivity and
   417           commutativity. *}
   418   proof (rule ring_axioms.intro)
   419     fix x y z
   420     assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
   421     note [simp] = comm_monoid.axioms [OF comm_monoid]
   422       abelian_group.axioms [OF abelian_group]
   423       abelian_monoid.a_closed
   424         
   425     from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
   426       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
   427     also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
   428     also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
   429       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
   430     finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
   431   qed (rule l_distr)
   432 qed (auto intro: cring.intro
   433   abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
   434 
   435 (*
   436 lemma (in cring) is_comm_monoid:
   437   "comm_monoid R"
   438   by (auto intro!: comm_monoidI m_assoc m_comm)
   439 *)
   440 
   441 lemma (in cring) is_cring:
   442   "cring R" by (rule cring_axioms)
   443 
   444 
   445 subsubsection {* Normaliser for Rings *}
   446 
   447 lemma (in abelian_group) r_neg2:
   448   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
   449 proof -
   450   assume G: "x \<in> carrier G" "y \<in> carrier G"
   451   then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
   452     by (simp only: r_neg l_zero)
   453   with G show ?thesis 
   454     by (simp add: a_ac)
   455 qed
   456 
   457 lemma (in abelian_group) r_neg1:
   458   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
   459 proof -
   460   assume G: "x \<in> carrier G" "y \<in> carrier G"
   461   then have "(\<ominus> x \<oplus> x) \<oplus> y = y" 
   462     by (simp only: l_neg l_zero)
   463   with G show ?thesis by (simp add: a_ac)
   464 qed
   465 
   466 text {* 
   467   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89
   468 *}
   469 
   470 lemma (in ring) l_null [simp]:
   471   "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
   472 proof -
   473   assume R: "x \<in> carrier R"
   474   then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
   475     by (simp add: l_distr del: l_zero r_zero)
   476   also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
   477   finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
   478   with R show ?thesis by (simp del: r_zero)
   479 qed
   480 
   481 lemma (in ring) r_null [simp]:
   482   "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
   483 proof -
   484   assume R: "x \<in> carrier R"
   485   then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
   486     by (simp add: r_distr del: l_zero r_zero)
   487   also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
   488   finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
   489   with R show ?thesis by (simp del: r_zero)
   490 qed
   491 
   492 lemma (in ring) l_minus:
   493   "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
   494 proof -
   495   assume R: "x \<in> carrier R" "y \<in> carrier R"
   496   then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
   497   also from R have "... = \<zero>" by (simp add: l_neg l_null)
   498   finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
   499   with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
   500   with R show ?thesis by (simp add: a_assoc r_neg)
   501 qed
   502 
   503 lemma (in ring) r_minus:
   504   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
   505 proof -
   506   assume R: "x \<in> carrier R" "y \<in> carrier R"
   507   then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
   508   also from R have "... = \<zero>" by (simp add: l_neg r_null)
   509   finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
   510   with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
   511   with R show ?thesis by (simp add: a_assoc r_neg )
   512 qed
   513 
   514 lemma (in abelian_group) minus_eq:
   515   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
   516   by (simp only: a_minus_def)
   517 
   518 text {* Setup algebra method:
   519   compute distributive normal form in locale contexts *}
   520 
   521 use "ringsimp.ML"
   522 
   523 setup Algebra.setup
   524 
   525 lemmas (in ring) ring_simprules
   526   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
   527   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
   528   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
   529   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
   530   a_lcomm r_distr l_null r_null l_minus r_minus
   531 
   532 lemmas (in cring)
   533   [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
   534   _
   535 
   536 lemmas (in cring) cring_simprules
   537   [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
   538   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
   539   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
   540   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
   541   a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
   542 
   543 
   544 lemma (in cring) nat_pow_zero:
   545   "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
   546   by (induct n) simp_all
   547 
   548 lemma (in ring) one_zeroD:
   549   assumes onezero: "\<one> = \<zero>"
   550   shows "carrier R = {\<zero>}"
   551 proof (rule, rule)
   552   fix x
   553   assume xcarr: "x \<in> carrier R"
   554   from xcarr
   555       have "x = x \<otimes> \<one>" by simp
   556   from this and onezero
   557       have "x = x \<otimes> \<zero>" by simp
   558   from this and xcarr
   559       have "x = \<zero>" by simp
   560   thus "x \<in> {\<zero>}" by fast
   561 qed fast
   562 
   563 lemma (in ring) one_zeroI:
   564   assumes carrzero: "carrier R = {\<zero>}"
   565   shows "\<one> = \<zero>"
   566 proof -
   567   from one_closed and carrzero
   568       show "\<one> = \<zero>" by simp
   569 qed
   570 
   571 lemma (in ring) carrier_one_zero:
   572   shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
   573   by (rule, erule one_zeroI, erule one_zeroD)
   574 
   575 lemma (in ring) carrier_one_not_zero:
   576   shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
   577   by (simp add: carrier_one_zero)
   578 
   579 text {* Two examples for use of method algebra *}
   580 
   581 lemma
   582   fixes R (structure) and S (structure)
   583   assumes "ring R" "cring S"
   584   assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
   585   shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
   586 proof -
   587   interpret ring R by fact
   588   interpret cring S by fact
   589 ML_val {* Algebra.print_structures @{context} *}
   590   from RS show ?thesis by algebra
   591 qed
   592 
   593 lemma
   594   fixes R (structure)
   595   assumes "ring R"
   596   assumes R: "a \<in> carrier R" "b \<in> carrier R"
   597   shows "a \<ominus> (a \<ominus> b) = b"
   598 proof -
   599   interpret ring R by fact
   600   from R show ?thesis by algebra
   601 qed
   602 
   603 subsubsection {* Sums over Finite Sets *}
   604 
   605 lemma (in ring) finsum_ldistr:
   606   "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
   607    finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
   608 proof (induct set: finite)
   609   case empty then show ?case by simp
   610 next
   611   case (insert x F) then show ?case by (simp add: Pi_def l_distr)
   612 qed
   613 
   614 lemma (in ring) finsum_rdistr:
   615   "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
   616    a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
   617 proof (induct set: finite)
   618   case empty then show ?case by simp
   619 next
   620   case (insert x F) then show ?case by (simp add: Pi_def r_distr)
   621 qed
   622 
   623 
   624 subsection {* Integral Domains *}
   625 
   626 lemma (in "domain") zero_not_one [simp]:
   627   "\<zero> ~= \<one>"
   628   by (rule not_sym) simp
   629 
   630 lemma (in "domain") integral_iff: (* not by default a simp rule! *)
   631   "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
   632 proof
   633   assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
   634   then show "a = \<zero> | b = \<zero>" by (simp add: integral)
   635 next
   636   assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
   637   then show "a \<otimes> b = \<zero>" by auto
   638 qed
   639 
   640 lemma (in "domain") m_lcancel:
   641   assumes prem: "a ~= \<zero>"
   642     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
   643   shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
   644 proof
   645   assume eq: "a \<otimes> b = a \<otimes> c"
   646   with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
   647   with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
   648   with prem and R have "b \<ominus> c = \<zero>" by auto 
   649   with R have "b = b \<ominus> (b \<ominus> c)" by algebra 
   650   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
   651   finally show "b = c" .
   652 next
   653   assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
   654 qed
   655 
   656 lemma (in "domain") m_rcancel:
   657   assumes prem: "a ~= \<zero>"
   658     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
   659   shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
   660 proof -
   661   from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
   662   with R show ?thesis by algebra
   663 qed
   664 
   665 
   666 subsection {* Fields *}
   667 
   668 text {* Field would not need to be derived from domain, the properties
   669   for domain follow from the assumptions of field *}
   670 lemma (in cring) cring_fieldI:
   671   assumes field_Units: "Units R = carrier R - {\<zero>}"
   672   shows "field R"
   673 proof
   674   from field_Units
   675   have a: "\<zero> \<notin> Units R" by fast
   676   have "\<one> \<in> Units R" by fast
   677   from this and a
   678   show "\<one> \<noteq> \<zero>" by force
   679 next
   680   fix a b
   681   assume acarr: "a \<in> carrier R"
   682     and bcarr: "b \<in> carrier R"
   683     and ab: "a \<otimes> b = \<zero>"
   684   show "a = \<zero> \<or> b = \<zero>"
   685   proof (cases "a = \<zero>", simp)
   686     assume "a \<noteq> \<zero>"
   687     from this and field_Units and acarr
   688     have aUnit: "a \<in> Units R" by fast
   689     from bcarr
   690     have "b = \<one> \<otimes> b" by algebra
   691     also from aUnit acarr
   692     have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
   693     also from acarr bcarr aUnit[THEN Units_inv_closed]
   694     have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
   695     also from ab and acarr bcarr aUnit
   696     have "... = (inv a) \<otimes> \<zero>" by simp
   697     also from aUnit[THEN Units_inv_closed]
   698     have "... = \<zero>" by algebra
   699     finally
   700     have "b = \<zero>" .
   701     thus "a = \<zero> \<or> b = \<zero>" by simp
   702   qed
   703 qed (rule field_Units)
   704 
   705 text {* Another variant to show that something is a field *}
   706 lemma (in cring) cring_fieldI2:
   707   assumes notzero: "\<zero> \<noteq> \<one>"
   708   and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
   709   shows "field R"
   710   apply (rule cring_fieldI, simp add: Units_def)
   711   apply (rule, clarsimp)
   712   apply (simp add: notzero)
   713 proof (clarsimp)
   714   fix x
   715   assume xcarr: "x \<in> carrier R"
   716     and "x \<noteq> \<zero>"
   717   from this
   718   have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
   719   from this
   720   obtain y
   721     where ycarr: "y \<in> carrier R"
   722     and xy: "x \<otimes> y = \<one>"
   723     by fast
   724   from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
   725   from ycarr and this and xy
   726   show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
   727 qed
   728 
   729 
   730 subsection {* Morphisms *}
   731 
   732 constdefs (structure R S)
   733   ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
   734   "ring_hom R S == {h. h \<in> carrier R -> carrier S &
   735       (ALL x y. x \<in> carrier R & y \<in> carrier R -->
   736         h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
   737       h \<one> = \<one>\<^bsub>S\<^esub>}"
   738 
   739 lemma ring_hom_memI:
   740   fixes R (structure) and S (structure)
   741   assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
   742     and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
   743       h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
   744     and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
   745       h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
   746     and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
   747   shows "h \<in> ring_hom R S"
   748   by (auto simp add: ring_hom_def assms Pi_def)
   749 
   750 lemma ring_hom_closed:
   751   "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
   752   by (auto simp add: ring_hom_def funcset_mem)
   753 
   754 lemma ring_hom_mult:
   755   fixes R (structure) and S (structure)
   756   shows
   757     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
   758     h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
   759     by (simp add: ring_hom_def)
   760 
   761 lemma ring_hom_add:
   762   fixes R (structure) and S (structure)
   763   shows
   764     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
   765     h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
   766     by (simp add: ring_hom_def)
   767 
   768 lemma ring_hom_one:
   769   fixes R (structure) and S (structure)
   770   shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
   771   by (simp add: ring_hom_def)
   772 
   773 locale ring_hom_cring = R: cring R + S: cring S
   774     for R (structure) and S (structure) +
   775   fixes h
   776   assumes homh [simp, intro]: "h \<in> ring_hom R S"
   777   notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
   778     and hom_mult [simp] = ring_hom_mult [OF homh]
   779     and hom_add [simp] = ring_hom_add [OF homh]
   780     and hom_one [simp] = ring_hom_one [OF homh]
   781 
   782 lemma (in ring_hom_cring) hom_zero [simp]:
   783   "h \<zero> = \<zero>\<^bsub>S\<^esub>"
   784 proof -
   785   have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
   786     by (simp add: hom_add [symmetric] del: hom_add)
   787   then show ?thesis by (simp del: S.r_zero)
   788 qed
   789 
   790 lemma (in ring_hom_cring) hom_a_inv [simp]:
   791   "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
   792 proof -
   793   assume R: "x \<in> carrier R"
   794   then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
   795     by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
   796   with R show ?thesis by simp
   797 qed
   798 
   799 lemma (in ring_hom_cring) hom_finsum [simp]:
   800   "[| finite A; f \<in> A -> carrier R |] ==>
   801   h (finsum R f A) = finsum S (h o f) A"
   802 proof (induct set: finite)
   803   case empty then show ?case by simp
   804 next
   805   case insert then show ?case by (simp add: Pi_def)
   806 qed
   807 
   808 lemma (in ring_hom_cring) hom_finprod:
   809   "[| finite A; f \<in> A -> carrier R |] ==>
   810   h (finprod R f A) = finprod S (h o f) A"
   811 proof (induct set: finite)
   812   case empty then show ?case by simp
   813 next
   814   case insert then show ?case by (simp add: Pi_def)
   815 qed
   816 
   817 declare ring_hom_cring.hom_finprod [simp]
   818 
   819 lemma id_ring_hom [simp]:
   820   "id \<in> ring_hom R R"
   821   by (auto intro!: ring_hom_memI)
   822 
   823 end