src/HOL/Algebra/Ring.thy
author ballarin
Thu Aug 03 14:57:26 2006 +0200 (2006-08-03)
changeset 20318 0e0ea63fe768
child 21896 9a7949815a84
permissions -rw-r--r--
Restructured algebra library, added ideals and quotient rings.
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(*
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  Title:     The algebraic hierarchy of rings
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  Id:        $Id$
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  Author:    Clemens Ballarin, started 9 December 1996
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  Copyright: Clemens Ballarin
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*)
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theory Ring imports FiniteProduct
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uses ("ringsimp.ML") begin
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section {* Abelian Groups *}
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record 'a ring = "'a monoid" +
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  zero :: 'a ("\<zero>\<index>")
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  add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65)
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text {* Derived operations. *}
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constdefs (structure R)
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  a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
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  "a_inv R == m_inv (| carrier = carrier R, mult = add R, one = zero R |)"
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  a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
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  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y == x \<oplus> (\<ominus> y)"
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locale abelian_monoid =
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  fixes G (structure)
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  assumes a_comm_monoid:
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     "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"
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text {*
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  The following definition is redundant but simple to use.
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*}
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locale abelian_group = abelian_monoid +
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  assumes a_comm_group:
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     "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"
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subsection {* Basic Properties *}
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lemma abelian_monoidI:
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  fixes R (structure)
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  assumes a_closed:
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      "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
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    and zero_closed: "\<zero> \<in> carrier R"
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    and a_assoc:
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      "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
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      (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
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    and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
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    and a_comm:
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      "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
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  shows "abelian_monoid R"
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  by (auto intro!: abelian_monoid.intro comm_monoidI intro: prems)
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lemma abelian_groupI:
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  fixes R (structure)
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  assumes a_closed:
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      "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
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    and zero_closed: "zero R \<in> carrier R"
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    and a_assoc:
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      "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
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      (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
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    and a_comm:
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      "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
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    and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
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    and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>"
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  shows "abelian_group R"
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  by (auto intro!: abelian_group.intro abelian_monoidI
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      abelian_group_axioms.intro comm_monoidI comm_groupI
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    intro: prems)
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lemma (in abelian_monoid) a_monoid:
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  "monoid (| carrier = carrier G, mult = add G, one = zero G |)"
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by (rule comm_monoid.axioms, rule a_comm_monoid) 
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lemma (in abelian_group) a_group:
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  "group (| carrier = carrier G, mult = add G, one = zero G |)"
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  by (simp add: group_def a_monoid)
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    (simp add: comm_group.axioms group.axioms a_comm_group)
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lemmas monoid_record_simps = partial_object.simps monoid.simps
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lemma (in abelian_monoid) a_closed [intro, simp]:
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  "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G"
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  by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps]) 
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lemma (in abelian_monoid) zero_closed [intro, simp]:
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  "\<zero> \<in> carrier G"
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  by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps])
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lemma (in abelian_group) a_inv_closed [intro, simp]:
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  "x \<in> carrier G ==> \<ominus> x \<in> carrier G"
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  by (simp add: a_inv_def
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    group.inv_closed [OF a_group, simplified monoid_record_simps])
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lemma (in abelian_group) minus_closed [intro, simp]:
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  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
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  by (simp add: a_minus_def)
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lemma (in abelian_group) a_l_cancel [simp]:
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  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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   (x \<oplus> y = x \<oplus> z) = (y = z)"
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  by (rule group.l_cancel [OF a_group, simplified monoid_record_simps])
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lemma (in abelian_group) a_r_cancel [simp]:
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  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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   (y \<oplus> x = z \<oplus> x) = (y = z)"
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  by (rule group.r_cancel [OF a_group, simplified monoid_record_simps])
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lemma (in abelian_monoid) a_assoc:
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  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
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  (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
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  by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps])
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lemma (in abelian_monoid) l_zero [simp]:
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  "x \<in> carrier G ==> \<zero> \<oplus> x = x"
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  by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps])
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lemma (in abelian_group) l_neg:
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  "x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>"
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  by (simp add: a_inv_def
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    group.l_inv [OF a_group, simplified monoid_record_simps])
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lemma (in abelian_monoid) a_comm:
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  "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
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  by (rule comm_monoid.m_comm [OF a_comm_monoid,
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    simplified monoid_record_simps])
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lemma (in abelian_monoid) a_lcomm:
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  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
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   x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
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  by (rule comm_monoid.m_lcomm [OF a_comm_monoid,
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                                simplified monoid_record_simps])
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lemma (in abelian_monoid) r_zero [simp]:
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  "x \<in> carrier G ==> x \<oplus> \<zero> = x"
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  using monoid.r_one [OF a_monoid]
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  by simp
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lemma (in abelian_group) r_neg:
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  "x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>"
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  using group.r_inv [OF a_group]
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  by (simp add: a_inv_def)
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lemma (in abelian_group) minus_zero [simp]:
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  "\<ominus> \<zero> = \<zero>"
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  by (simp add: a_inv_def
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    group.inv_one [OF a_group, simplified monoid_record_simps])
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lemma (in abelian_group) minus_minus [simp]:
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  "x \<in> carrier G ==> \<ominus> (\<ominus> x) = x"
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  using group.inv_inv [OF a_group, simplified monoid_record_simps]
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  by (simp add: a_inv_def)
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lemma (in abelian_group) a_inv_inj:
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  "inj_on (a_inv G) (carrier G)"
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  using group.inv_inj [OF a_group, simplified monoid_record_simps]
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  by (simp add: a_inv_def)
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lemma (in abelian_group) minus_add:
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  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
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  using comm_group.inv_mult [OF a_comm_group]
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  by (simp add: a_inv_def)
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lemma (in abelian_group) minus_equality: 
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  "[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y" 
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  using group.inv_equality [OF a_group] 
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  by (auto simp add: a_inv_def) 
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lemma (in abelian_monoid) minus_unique: 
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  "[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G;
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      y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'" 
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  using monoid.inv_unique [OF a_monoid] 
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  by (simp add: a_inv_def) 
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lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm
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text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
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lemma comm_group_abelian_groupI:
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  fixes G (structure)
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  assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  shows "abelian_group G"
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proof -
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  interpret comm_group ["\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"]
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    by (rule cg)
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  show "abelian_group G" by (unfold_locales)
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qed
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subsection {* Sums over Finite Sets *}
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text {*
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  This definition makes it easy to lift lemmas from @{term finprod}.
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*}
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constdefs
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  finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b"
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  "finsum G f A == finprod (| carrier = carrier G,
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     mult = add G, one = zero G |) f A"
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syntax
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  "_finsum" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
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syntax (xsymbols)
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  "_finsum" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
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syntax (HTML output)
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  "_finsum" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
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translations
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  "\<Oplus>\<index>i:A. b" == "finsum \<struct>\<index> (%i. b) A"
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  -- {* Beware of argument permutation! *}
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(*
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  lemmas (in abelian_monoid) finsum_empty [simp] =
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    comm_monoid.finprod_empty [OF a_comm_monoid, simplified]
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  is dangeous, because attributes (like simplified) are applied upon opening
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  the locale, simplified refers to the simpset at that time!!!
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  lemmas (in abelian_monoid) finsum_empty [simp] =
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    abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def,
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      simplified monoid_record_simps]
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  makes the locale slow, because proofs are repeated for every
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  "lemma (in abelian_monoid)" command.
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  When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down
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  from 110 secs to 60 secs.
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*)
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lemma (in abelian_monoid) finsum_empty [simp]:
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  "finsum G f {} = \<zero>"
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  by (rule comm_monoid.finprod_empty [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_insert [simp]:
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  "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |]
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  ==> finsum G f (insert a F) = f a \<oplus> finsum G f F"
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  by (rule comm_monoid.finprod_insert [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_zero [simp]:
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  "finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>"
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  by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def,
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    simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_closed [simp]:
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  fixes A
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  assumes fin: "finite A" and f: "f \<in> A -> carrier G" 
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  shows "finsum G f A \<in> carrier G"
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  by (rule comm_monoid.finprod_closed [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_Un_Int:
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  "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
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     finsum G g (A Un B) \<oplus> finsum G g (A Int B) =
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     finsum G g A \<oplus> finsum G g B"
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  by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_Un_disjoint:
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  "[| finite A; finite B; A Int B = {};
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      g \<in> A -> carrier G; g \<in> B -> carrier G |]
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   ==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B"
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  by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_addf:
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  "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
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   finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)"
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  by (rule comm_monoid.finprod_multf [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_cong':
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  "[| A = B; g : B -> carrier G;
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      !!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B"
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  by (rule comm_monoid.finprod_cong' [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps]) auto
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lemma (in abelian_monoid) finsum_0 [simp]:
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  "f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0"
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  by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def,
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    simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_Suc [simp]:
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  "f : {..Suc n} -> carrier G ==>
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   finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})"
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  by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def,
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    simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_Suc2:
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  "f : {..Suc n} -> carrier G ==>
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   finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
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  by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def,
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    simplified monoid_record_simps])
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lemma (in abelian_monoid) finsum_add [simp]:
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  "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
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     finsum G (%i. f i \<oplus> g i) {..n::nat} =
ballarin@20318
   301
     finsum G f {..n} \<oplus> finsum G g {..n}"
ballarin@20318
   302
  by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
ballarin@20318
   303
    simplified monoid_record_simps])
ballarin@20318
   304
ballarin@20318
   305
lemma (in abelian_monoid) finsum_cong:
ballarin@20318
   306
  "[| A = B; f : B -> carrier G;
ballarin@20318
   307
      !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
ballarin@20318
   308
  by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
ballarin@20318
   309
    simplified monoid_record_simps]) (auto simp add: simp_implies_def)
ballarin@20318
   310
ballarin@20318
   311
text {*Usually, if this rule causes a failed congruence proof error,
ballarin@20318
   312
   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
ballarin@20318
   313
   Adding @{thm [source] Pi_def} to the simpset is often useful. *}
ballarin@20318
   314
ballarin@20318
   315
ballarin@20318
   316
section {* The Algebraic Hierarchy of Rings *}
ballarin@20318
   317
ballarin@20318
   318
ballarin@20318
   319
subsection {* Basic Definitions *}
ballarin@20318
   320
ballarin@20318
   321
locale ring = abelian_group R + monoid R +
ballarin@20318
   322
  assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   323
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   324
    and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   325
      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
ballarin@20318
   326
ballarin@20318
   327
locale cring = ring + comm_monoid R
ballarin@20318
   328
ballarin@20318
   329
locale "domain" = cring +
ballarin@20318
   330
  assumes one_not_zero [simp]: "\<one> ~= \<zero>"
ballarin@20318
   331
    and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@20318
   332
                  a = \<zero> | b = \<zero>"
ballarin@20318
   333
ballarin@20318
   334
locale field = "domain" +
ballarin@20318
   335
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   336
ballarin@20318
   337
ballarin@20318
   338
subsection {* Rings *}
ballarin@20318
   339
ballarin@20318
   340
lemma ringI:
ballarin@20318
   341
  fixes R (structure)
ballarin@20318
   342
  assumes abelian_group: "abelian_group R"
ballarin@20318
   343
    and monoid: "monoid R"
ballarin@20318
   344
    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   345
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   346
    and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   347
      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
ballarin@20318
   348
  shows "ring R"
ballarin@20318
   349
  by (auto intro: ring.intro
ballarin@20318
   350
    abelian_group.axioms ring_axioms.intro prems)
ballarin@20318
   351
ballarin@20318
   352
lemma (in ring) is_abelian_group:
ballarin@20318
   353
  "abelian_group R"
ballarin@20318
   354
  by (auto intro!: abelian_groupI a_assoc a_comm l_neg)
ballarin@20318
   355
ballarin@20318
   356
lemma (in ring) is_monoid:
ballarin@20318
   357
  "monoid R"
ballarin@20318
   358
  by (auto intro!: monoidI m_assoc)
ballarin@20318
   359
ballarin@20318
   360
lemma (in ring) is_ring:
ballarin@20318
   361
  "ring R"
ballarin@20318
   362
  .
ballarin@20318
   363
ballarin@20318
   364
lemmas ring_record_simps = monoid_record_simps ring.simps
ballarin@20318
   365
ballarin@20318
   366
lemma cringI:
ballarin@20318
   367
  fixes R (structure)
ballarin@20318
   368
  assumes abelian_group: "abelian_group R"
ballarin@20318
   369
    and comm_monoid: "comm_monoid R"
ballarin@20318
   370
    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   371
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   372
  shows "cring R"
ballarin@20318
   373
  proof (intro cring.intro ring.intro)
ballarin@20318
   374
    show "ring_axioms R"
ballarin@20318
   375
    -- {* Right-distributivity follows from left-distributivity and
ballarin@20318
   376
          commutativity. *}
ballarin@20318
   377
    proof (rule ring_axioms.intro)
ballarin@20318
   378
      fix x y z
ballarin@20318
   379
      assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
ballarin@20318
   380
      note [simp]= comm_monoid.axioms [OF comm_monoid]
ballarin@20318
   381
        abelian_group.axioms [OF abelian_group]
ballarin@20318
   382
        abelian_monoid.a_closed
ballarin@20318
   383
        
ballarin@20318
   384
      from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
ballarin@20318
   385
        by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
ballarin@20318
   386
      also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
ballarin@20318
   387
      also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
ballarin@20318
   388
        by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
ballarin@20318
   389
      finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
ballarin@20318
   390
    qed
ballarin@20318
   391
  qed (auto intro: cring.intro
ballarin@20318
   392
      abelian_group.axioms comm_monoid.axioms ring_axioms.intro prems)
ballarin@20318
   393
ballarin@20318
   394
lemma (in cring) is_comm_monoid:
ballarin@20318
   395
  "comm_monoid R"
ballarin@20318
   396
  by (auto intro!: comm_monoidI m_assoc m_comm)
ballarin@20318
   397
ballarin@20318
   398
lemma (in cring) is_cring:
ballarin@20318
   399
  "cring R"
ballarin@20318
   400
  .
ballarin@20318
   401
ballarin@20318
   402
subsubsection {* Normaliser for Rings *}
ballarin@20318
   403
ballarin@20318
   404
lemma (in abelian_group) r_neg2:
ballarin@20318
   405
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
ballarin@20318
   406
proof -
ballarin@20318
   407
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   408
  then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
ballarin@20318
   409
    by (simp only: r_neg l_zero)
ballarin@20318
   410
  with G show ?thesis 
ballarin@20318
   411
    by (simp add: a_ac)
ballarin@20318
   412
qed
ballarin@20318
   413
ballarin@20318
   414
lemma (in abelian_group) r_neg1:
ballarin@20318
   415
  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
ballarin@20318
   416
proof -
ballarin@20318
   417
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   418
  then have "(\<ominus> x \<oplus> x) \<oplus> y = y" 
ballarin@20318
   419
    by (simp only: l_neg l_zero)
ballarin@20318
   420
  with G show ?thesis by (simp add: a_ac)
ballarin@20318
   421
qed
ballarin@20318
   422
ballarin@20318
   423
text {* 
ballarin@20318
   424
  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89
ballarin@20318
   425
*}
ballarin@20318
   426
ballarin@20318
   427
lemma (in ring) l_null [simp]:
ballarin@20318
   428
  "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
ballarin@20318
   429
proof -
ballarin@20318
   430
  assume R: "x \<in> carrier R"
ballarin@20318
   431
  then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
ballarin@20318
   432
    by (simp add: l_distr del: l_zero r_zero)
ballarin@20318
   433
  also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
ballarin@20318
   434
  finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
ballarin@20318
   435
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   436
qed
ballarin@20318
   437
ballarin@20318
   438
lemma (in ring) r_null [simp]:
ballarin@20318
   439
  "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
ballarin@20318
   440
proof -
ballarin@20318
   441
  assume R: "x \<in> carrier R"
ballarin@20318
   442
  then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
ballarin@20318
   443
    by (simp add: r_distr del: l_zero r_zero)
ballarin@20318
   444
  also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
ballarin@20318
   445
  finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
ballarin@20318
   446
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   447
qed
ballarin@20318
   448
ballarin@20318
   449
lemma (in ring) l_minus:
ballarin@20318
   450
  "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   451
proof -
ballarin@20318
   452
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   453
  then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
ballarin@20318
   454
  also from R have "... = \<zero>" by (simp add: l_neg l_null)
ballarin@20318
   455
  finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   456
  with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
ballarin@20318
   457
  with R show ?thesis by (simp add: a_assoc r_neg )
ballarin@20318
   458
qed
ballarin@20318
   459
ballarin@20318
   460
lemma (in ring) r_minus:
ballarin@20318
   461
  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   462
proof -
ballarin@20318
   463
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   464
  then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
ballarin@20318
   465
  also from R have "... = \<zero>" by (simp add: l_neg r_null)
ballarin@20318
   466
  finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   467
  with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
ballarin@20318
   468
  with R show ?thesis by (simp add: a_assoc r_neg )
ballarin@20318
   469
qed
ballarin@20318
   470
ballarin@20318
   471
lemma (in ring) minus_eq:
ballarin@20318
   472
  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
ballarin@20318
   473
  by (simp only: a_minus_def)
ballarin@20318
   474
ballarin@20318
   475
text {* Setup algebra method:
ballarin@20318
   476
  compute distributive normal form in locale contexts *}
ballarin@20318
   477
ballarin@20318
   478
use "ringsimp.ML"
ballarin@20318
   479
ballarin@20318
   480
setup Algebra.setup
ballarin@20318
   481
ballarin@20318
   482
lemmas (in ring) ring_simprules
ballarin@20318
   483
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   484
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   485
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
ballarin@20318
   486
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   487
  a_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   488
ballarin@20318
   489
lemmas (in cring)
ballarin@20318
   490
  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   491
  _
ballarin@20318
   492
ballarin@20318
   493
lemmas (in cring) cring_simprules
ballarin@20318
   494
  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   495
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   496
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
ballarin@20318
   497
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   498
  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   499
ballarin@20318
   500
ballarin@20318
   501
lemma (in cring) nat_pow_zero:
ballarin@20318
   502
  "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
ballarin@20318
   503
  by (induct n) simp_all
ballarin@20318
   504
ballarin@20318
   505
lemma (in ring) one_zeroD:
ballarin@20318
   506
  assumes onezero: "\<one> = \<zero>"
ballarin@20318
   507
  shows "carrier R = {\<zero>}"
ballarin@20318
   508
proof (rule, rule)
ballarin@20318
   509
  fix x
ballarin@20318
   510
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   511
  from xcarr
ballarin@20318
   512
      have "x = x \<otimes> \<one>" by simp
ballarin@20318
   513
  from this and onezero
ballarin@20318
   514
      have "x = x \<otimes> \<zero>" by simp
ballarin@20318
   515
  from this and xcarr
ballarin@20318
   516
      have "x = \<zero>" by simp
ballarin@20318
   517
  thus "x \<in> {\<zero>}" by fast
ballarin@20318
   518
qed fast
ballarin@20318
   519
ballarin@20318
   520
lemma (in ring) one_zeroI:
ballarin@20318
   521
  assumes carrzero: "carrier R = {\<zero>}"
ballarin@20318
   522
  shows "\<one> = \<zero>"
ballarin@20318
   523
proof -
ballarin@20318
   524
  from one_closed and carrzero
ballarin@20318
   525
      show "\<one> = \<zero>" by simp
ballarin@20318
   526
qed
ballarin@20318
   527
ballarin@20318
   528
lemma (in ring) one_zero:
ballarin@20318
   529
  shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
ballarin@20318
   530
  by (rule, erule one_zeroI, erule one_zeroD)
ballarin@20318
   531
ballarin@20318
   532
lemma (in ring) one_not_zero:
ballarin@20318
   533
  shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
ballarin@20318
   534
  by (simp add: one_zero)
ballarin@20318
   535
ballarin@20318
   536
text {* Two examples for use of method algebra *}
ballarin@20318
   537
ballarin@20318
   538
lemma
ballarin@20318
   539
  includes ring R + cring S
ballarin@20318
   540
  shows "[| a \<in> carrier R; b \<in> carrier R; c \<in> carrier S; d \<in> carrier S |] ==> 
ballarin@20318
   541
  a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
ballarin@20318
   542
  by algebra
ballarin@20318
   543
ballarin@20318
   544
lemma
ballarin@20318
   545
  includes cring
ballarin@20318
   546
  shows "[| a \<in> carrier R; b \<in> carrier R |] ==> a \<ominus> (a \<ominus> b) = b"
ballarin@20318
   547
  by algebra
ballarin@20318
   548
ballarin@20318
   549
ballarin@20318
   550
subsubsection {* Sums over Finite Sets *}
ballarin@20318
   551
ballarin@20318
   552
lemma (in cring) finsum_ldistr:
ballarin@20318
   553
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   554
   finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
ballarin@20318
   555
proof (induct set: Finites)
ballarin@20318
   556
  case empty then show ?case by simp
ballarin@20318
   557
next
ballarin@20318
   558
  case (insert x F) then show ?case by (simp add: Pi_def l_distr)
ballarin@20318
   559
qed
ballarin@20318
   560
ballarin@20318
   561
lemma (in cring) finsum_rdistr:
ballarin@20318
   562
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   563
   a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
ballarin@20318
   564
proof (induct set: Finites)
ballarin@20318
   565
  case empty then show ?case by simp
ballarin@20318
   566
next
ballarin@20318
   567
  case (insert x F) then show ?case by (simp add: Pi_def r_distr)
ballarin@20318
   568
qed
ballarin@20318
   569
ballarin@20318
   570
ballarin@20318
   571
subsection {* Integral Domains *}
ballarin@20318
   572
ballarin@20318
   573
lemma (in "domain") zero_not_one [simp]:
ballarin@20318
   574
  "\<zero> ~= \<one>"
ballarin@20318
   575
  by (rule not_sym) simp
ballarin@20318
   576
ballarin@20318
   577
lemma (in "domain") integral_iff: (* not by default a simp rule! *)
ballarin@20318
   578
  "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
ballarin@20318
   579
proof
ballarin@20318
   580
  assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
ballarin@20318
   581
  then show "a = \<zero> | b = \<zero>" by (simp add: integral)
ballarin@20318
   582
next
ballarin@20318
   583
  assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
ballarin@20318
   584
  then show "a \<otimes> b = \<zero>" by auto
ballarin@20318
   585
qed
ballarin@20318
   586
ballarin@20318
   587
lemma (in "domain") m_lcancel:
ballarin@20318
   588
  assumes prem: "a ~= \<zero>"
ballarin@20318
   589
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   590
  shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
ballarin@20318
   591
proof
ballarin@20318
   592
  assume eq: "a \<otimes> b = a \<otimes> c"
ballarin@20318
   593
  with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
ballarin@20318
   594
  with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
ballarin@20318
   595
  with prem and R have "b \<ominus> c = \<zero>" by auto 
ballarin@20318
   596
  with R have "b = b \<ominus> (b \<ominus> c)" by algebra 
ballarin@20318
   597
  also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
ballarin@20318
   598
  finally show "b = c" .
ballarin@20318
   599
next
ballarin@20318
   600
  assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
ballarin@20318
   601
qed
ballarin@20318
   602
ballarin@20318
   603
lemma (in "domain") m_rcancel:
ballarin@20318
   604
  assumes prem: "a ~= \<zero>"
ballarin@20318
   605
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   606
  shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
ballarin@20318
   607
proof -
ballarin@20318
   608
  from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
ballarin@20318
   609
  with R show ?thesis by algebra
ballarin@20318
   610
qed
ballarin@20318
   611
ballarin@20318
   612
ballarin@20318
   613
subsection {* Fields *}
ballarin@20318
   614
ballarin@20318
   615
text {* Field would not need to be derived from domain, the properties
ballarin@20318
   616
  for domain follow from the assumptions of field *}
ballarin@20318
   617
lemma (in cring) cring_fieldI:
ballarin@20318
   618
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   619
  shows "field R"
ballarin@20318
   620
proof unfold_locales
ballarin@20318
   621
  from field_Units
ballarin@20318
   622
  have a: "\<zero> \<notin> Units R" by fast
ballarin@20318
   623
  have "\<one> \<in> Units R" by fast
ballarin@20318
   624
  from this and a
ballarin@20318
   625
  show "\<one> \<noteq> \<zero>" by force
ballarin@20318
   626
next
ballarin@20318
   627
  fix a b
ballarin@20318
   628
  assume acarr: "a \<in> carrier R"
ballarin@20318
   629
    and bcarr: "b \<in> carrier R"
ballarin@20318
   630
    and ab: "a \<otimes> b = \<zero>"
ballarin@20318
   631
  show "a = \<zero> \<or> b = \<zero>"
ballarin@20318
   632
  proof (cases "a = \<zero>", simp)
ballarin@20318
   633
    assume "a \<noteq> \<zero>"
ballarin@20318
   634
    from this and field_Units and acarr
ballarin@20318
   635
    have aUnit: "a \<in> Units R" by fast
ballarin@20318
   636
    from bcarr
ballarin@20318
   637
    have "b = \<one> \<otimes> b" by algebra
ballarin@20318
   638
    also from aUnit acarr
ballarin@20318
   639
    have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
ballarin@20318
   640
    also from acarr bcarr aUnit[THEN Units_inv_closed]
ballarin@20318
   641
    have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
ballarin@20318
   642
    also from ab and acarr bcarr aUnit
ballarin@20318
   643
    have "... = (inv a) \<otimes> \<zero>" by simp
ballarin@20318
   644
    also from aUnit[THEN Units_inv_closed]
ballarin@20318
   645
    have "... = \<zero>" by algebra
ballarin@20318
   646
    finally
ballarin@20318
   647
    have "b = \<zero>" .
ballarin@20318
   648
    thus "a = \<zero> \<or> b = \<zero>" by simp
ballarin@20318
   649
  qed
ballarin@20318
   650
qed
ballarin@20318
   651
ballarin@20318
   652
text {* Another variant to show that something is a field *}
ballarin@20318
   653
lemma (in cring) cring_fieldI2:
ballarin@20318
   654
  assumes notzero: "\<zero> \<noteq> \<one>"
ballarin@20318
   655
  and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
ballarin@20318
   656
  shows "field R"
ballarin@20318
   657
  apply (rule cring_fieldI, simp add: Units_def)
ballarin@20318
   658
  apply (rule, clarsimp)
ballarin@20318
   659
  apply (simp add: notzero)
ballarin@20318
   660
proof (clarsimp)
ballarin@20318
   661
  fix x
ballarin@20318
   662
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   663
    and "x \<noteq> \<zero>"
ballarin@20318
   664
  from this
ballarin@20318
   665
  have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
ballarin@20318
   666
  from this
ballarin@20318
   667
  obtain y
ballarin@20318
   668
    where ycarr: "y \<in> carrier R"
ballarin@20318
   669
    and xy: "x \<otimes> y = \<one>"
ballarin@20318
   670
    by fast
ballarin@20318
   671
  from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
ballarin@20318
   672
  from ycarr and this and xy
ballarin@20318
   673
  show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
ballarin@20318
   674
qed
ballarin@20318
   675
ballarin@20318
   676
ballarin@20318
   677
subsection {* Morphisms *}
ballarin@20318
   678
ballarin@20318
   679
constdefs (structure R S)
ballarin@20318
   680
  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
ballarin@20318
   681
  "ring_hom R S == {h. h \<in> carrier R -> carrier S &
ballarin@20318
   682
      (ALL x y. x \<in> carrier R & y \<in> carrier R -->
ballarin@20318
   683
        h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
ballarin@20318
   684
      h \<one> = \<one>\<^bsub>S\<^esub>}"
ballarin@20318
   685
ballarin@20318
   686
lemma ring_hom_memI:
ballarin@20318
   687
  fixes R (structure) and S (structure)
ballarin@20318
   688
  assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
ballarin@20318
   689
    and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   690
      h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   691
    and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   692
      h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   693
    and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   694
  shows "h \<in> ring_hom R S"
ballarin@20318
   695
  by (auto simp add: ring_hom_def prems Pi_def)
ballarin@20318
   696
ballarin@20318
   697
lemma ring_hom_closed:
ballarin@20318
   698
  "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
ballarin@20318
   699
  by (auto simp add: ring_hom_def funcset_mem)
ballarin@20318
   700
ballarin@20318
   701
lemma ring_hom_mult:
ballarin@20318
   702
  fixes R (structure) and S (structure)
ballarin@20318
   703
  shows
ballarin@20318
   704
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   705
    h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   706
    by (simp add: ring_hom_def)
ballarin@20318
   707
ballarin@20318
   708
lemma ring_hom_add:
ballarin@20318
   709
  fixes R (structure) and S (structure)
ballarin@20318
   710
  shows
ballarin@20318
   711
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   712
    h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   713
    by (simp add: ring_hom_def)
ballarin@20318
   714
ballarin@20318
   715
lemma ring_hom_one:
ballarin@20318
   716
  fixes R (structure) and S (structure)
ballarin@20318
   717
  shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   718
  by (simp add: ring_hom_def)
ballarin@20318
   719
ballarin@20318
   720
locale ring_hom_cring = cring R + cring S +
ballarin@20318
   721
  fixes h
ballarin@20318
   722
  assumes homh [simp, intro]: "h \<in> ring_hom R S"
ballarin@20318
   723
  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
ballarin@20318
   724
    and hom_mult [simp] = ring_hom_mult [OF homh]
ballarin@20318
   725
    and hom_add [simp] = ring_hom_add [OF homh]
ballarin@20318
   726
    and hom_one [simp] = ring_hom_one [OF homh]
ballarin@20318
   727
ballarin@20318
   728
lemma (in ring_hom_cring) hom_zero [simp]:
ballarin@20318
   729
  "h \<zero> = \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   730
proof -
ballarin@20318
   731
  have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   732
    by (simp add: hom_add [symmetric] del: hom_add)
ballarin@20318
   733
  then show ?thesis by (simp del: S.r_zero)
ballarin@20318
   734
qed
ballarin@20318
   735
ballarin@20318
   736
lemma (in ring_hom_cring) hom_a_inv [simp]:
ballarin@20318
   737
  "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
ballarin@20318
   738
proof -
ballarin@20318
   739
  assume R: "x \<in> carrier R"
ballarin@20318
   740
  then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
ballarin@20318
   741
    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
ballarin@20318
   742
  with R show ?thesis by simp
ballarin@20318
   743
qed
ballarin@20318
   744
ballarin@20318
   745
lemma (in ring_hom_cring) hom_finsum [simp]:
ballarin@20318
   746
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   747
  h (finsum R f A) = finsum S (h o f) A"
ballarin@20318
   748
proof (induct set: Finites)
ballarin@20318
   749
  case empty then show ?case by simp
ballarin@20318
   750
next
ballarin@20318
   751
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   752
qed
ballarin@20318
   753
ballarin@20318
   754
lemma (in ring_hom_cring) hom_finprod:
ballarin@20318
   755
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   756
  h (finprod R f A) = finprod S (h o f) A"
ballarin@20318
   757
proof (induct set: Finites)
ballarin@20318
   758
  case empty then show ?case by simp
ballarin@20318
   759
next
ballarin@20318
   760
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   761
qed
ballarin@20318
   762
ballarin@20318
   763
declare ring_hom_cring.hom_finprod [simp]
ballarin@20318
   764
ballarin@20318
   765
lemma id_ring_hom [simp]:
ballarin@20318
   766
  "id \<in> ring_hom R R"
ballarin@20318
   767
  by (auto intro!: ring_hom_memI)
ballarin@20318
   768
ballarin@20318
   769
end