src/HOL/Algebra/Ring.thy
author ballarin
Tue Jul 29 16:17:45 2008 +0200 (2008-07-29)
changeset 27699 489e3f33af0e
parent 27611 2c01c0bdb385
child 27714 27b4d7c01f8b
permissions -rw-r--r--
New theorems on summation.
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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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  apply (rule comm_monoid.finprod_closed [OF a_comm_monoid,
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    folded finsum_def, simplified monoid_record_simps])
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   apply (rule fin)
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  apply (rule f)
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  done
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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]:
ballarin@20318
   302
  "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
ballarin@20318
   303
     finsum G (%i. f i \<oplus> g i) {..n::nat} =
ballarin@20318
   304
     finsum G f {..n} \<oplus> finsum G g {..n}"
ballarin@20318
   305
  by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
ballarin@20318
   306
    simplified monoid_record_simps])
ballarin@20318
   307
ballarin@20318
   308
lemma (in abelian_monoid) finsum_cong:
ballarin@20318
   309
  "[| A = B; f : B -> carrier G;
ballarin@20318
   310
      !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
ballarin@20318
   311
  by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
ballarin@20318
   312
    simplified monoid_record_simps]) (auto simp add: simp_implies_def)
ballarin@20318
   313
ballarin@20318
   314
text {*Usually, if this rule causes a failed congruence proof error,
ballarin@20318
   315
   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
ballarin@20318
   316
   Adding @{thm [source] Pi_def} to the simpset is often useful. *}
ballarin@20318
   317
ballarin@27699
   318
lemma (in abelian_monoid) finsum_reindex:
ballarin@27699
   319
  assumes fin: "finite A"
ballarin@27699
   320
    shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow> 
ballarin@27699
   321
        inj_on h A ==> finsum G f (h ` A) = finsum G (%x. f (h x)) A"
ballarin@27699
   322
  using fin apply induct
ballarin@27699
   323
  apply (auto simp add: finsum_insert Pi_def)
ballarin@27699
   324
done
ballarin@27699
   325
ballarin@27699
   326
(* The following is wrong.  Needed is the equivalent of (^) for addition,
ballarin@27699
   327
  or indeed the canonical embedding from Nat into the monoid.
ballarin@27699
   328
ballarin@27699
   329
lemma (in abelian_monoid) finsum_const:
ballarin@27699
   330
  assumes fin [simp]: "finite A"
ballarin@27699
   331
      and a [simp]: "a : carrier G"
ballarin@27699
   332
    shows "finsum G (%x. a) A = a (^) card A"
ballarin@27699
   333
  using fin apply induct
ballarin@27699
   334
  apply force
ballarin@27699
   335
  apply (subst finsum_insert)
ballarin@27699
   336
  apply auto
ballarin@27699
   337
  apply (force simp add: Pi_def)
ballarin@27699
   338
  apply (subst m_comm)
ballarin@27699
   339
  apply auto
ballarin@27699
   340
done
ballarin@27699
   341
*)
ballarin@27699
   342
ballarin@20318
   343
ballarin@20318
   344
section {* The Algebraic Hierarchy of Rings *}
ballarin@20318
   345
ballarin@20318
   346
subsection {* Basic Definitions *}
ballarin@20318
   347
ballarin@20318
   348
locale ring = abelian_group R + monoid R +
ballarin@20318
   349
  assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   350
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   351
    and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   352
      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
ballarin@20318
   353
ballarin@20318
   354
locale cring = ring + comm_monoid R
ballarin@20318
   355
ballarin@20318
   356
locale "domain" = cring +
ballarin@20318
   357
  assumes one_not_zero [simp]: "\<one> ~= \<zero>"
ballarin@20318
   358
    and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
ballarin@20318
   359
                  a = \<zero> | b = \<zero>"
ballarin@20318
   360
ballarin@20318
   361
locale field = "domain" +
ballarin@20318
   362
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   363
ballarin@20318
   364
ballarin@20318
   365
subsection {* Rings *}
ballarin@20318
   366
ballarin@20318
   367
lemma ringI:
ballarin@20318
   368
  fixes R (structure)
ballarin@20318
   369
  assumes abelian_group: "abelian_group R"
ballarin@20318
   370
    and monoid: "monoid R"
ballarin@20318
   371
    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   372
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   373
    and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   374
      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
ballarin@20318
   375
  shows "ring R"
ballarin@20318
   376
  by (auto intro: ring.intro
ballarin@20318
   377
    abelian_group.axioms ring_axioms.intro prems)
ballarin@20318
   378
ballarin@20318
   379
lemma (in ring) is_abelian_group:
ballarin@20318
   380
  "abelian_group R"
ballarin@27699
   381
  by unfold_locales
ballarin@20318
   382
ballarin@20318
   383
lemma (in ring) is_monoid:
ballarin@20318
   384
  "monoid R"
ballarin@20318
   385
  by (auto intro!: monoidI m_assoc)
ballarin@20318
   386
ballarin@20318
   387
lemma (in ring) is_ring:
ballarin@20318
   388
  "ring R"
wenzelm@26202
   389
  by (rule ring_axioms)
ballarin@20318
   390
ballarin@20318
   391
lemmas ring_record_simps = monoid_record_simps ring.simps
ballarin@20318
   392
ballarin@20318
   393
lemma cringI:
ballarin@20318
   394
  fixes R (structure)
ballarin@20318
   395
  assumes abelian_group: "abelian_group R"
ballarin@20318
   396
    and comm_monoid: "comm_monoid R"
ballarin@20318
   397
    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
ballarin@20318
   398
      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
ballarin@20318
   399
  shows "cring R"
wenzelm@23350
   400
proof (intro cring.intro ring.intro)
wenzelm@23350
   401
  show "ring_axioms R"
ballarin@20318
   402
    -- {* Right-distributivity follows from left-distributivity and
ballarin@20318
   403
          commutativity. *}
wenzelm@23350
   404
  proof (rule ring_axioms.intro)
wenzelm@23350
   405
    fix x y z
wenzelm@23350
   406
    assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
wenzelm@23350
   407
    note [simp] = comm_monoid.axioms [OF comm_monoid]
wenzelm@23350
   408
      abelian_group.axioms [OF abelian_group]
wenzelm@23350
   409
      abelian_monoid.a_closed
ballarin@20318
   410
        
wenzelm@23350
   411
    from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
wenzelm@23350
   412
      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
wenzelm@23350
   413
    also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
wenzelm@23350
   414
    also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
wenzelm@23350
   415
      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
wenzelm@23350
   416
    finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
wenzelm@23350
   417
  qed (rule l_distr)
wenzelm@23350
   418
qed (auto intro: cring.intro
wenzelm@23350
   419
  abelian_group.axioms comm_monoid.axioms ring_axioms.intro prems)
ballarin@20318
   420
ballarin@27699
   421
(*
ballarin@20318
   422
lemma (in cring) is_comm_monoid:
ballarin@20318
   423
  "comm_monoid R"
ballarin@20318
   424
  by (auto intro!: comm_monoidI m_assoc m_comm)
ballarin@27699
   425
*)
ballarin@20318
   426
ballarin@20318
   427
lemma (in cring) is_cring:
wenzelm@26202
   428
  "cring R" by (rule cring_axioms)
wenzelm@23350
   429
ballarin@20318
   430
ballarin@20318
   431
subsubsection {* Normaliser for Rings *}
ballarin@20318
   432
ballarin@20318
   433
lemma (in abelian_group) r_neg2:
ballarin@20318
   434
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
ballarin@20318
   435
proof -
ballarin@20318
   436
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   437
  then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
ballarin@20318
   438
    by (simp only: r_neg l_zero)
ballarin@20318
   439
  with G show ?thesis 
ballarin@20318
   440
    by (simp add: a_ac)
ballarin@20318
   441
qed
ballarin@20318
   442
ballarin@20318
   443
lemma (in abelian_group) r_neg1:
ballarin@20318
   444
  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
ballarin@20318
   445
proof -
ballarin@20318
   446
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   447
  then have "(\<ominus> x \<oplus> x) \<oplus> y = y" 
ballarin@20318
   448
    by (simp only: l_neg l_zero)
ballarin@20318
   449
  with G show ?thesis by (simp add: a_ac)
ballarin@20318
   450
qed
ballarin@20318
   451
ballarin@20318
   452
text {* 
ballarin@20318
   453
  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89
ballarin@20318
   454
*}
ballarin@20318
   455
ballarin@20318
   456
lemma (in ring) l_null [simp]:
ballarin@20318
   457
  "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
ballarin@20318
   458
proof -
ballarin@20318
   459
  assume R: "x \<in> carrier R"
ballarin@20318
   460
  then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
ballarin@20318
   461
    by (simp add: l_distr del: l_zero r_zero)
ballarin@20318
   462
  also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
ballarin@20318
   463
  finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
ballarin@20318
   464
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   465
qed
ballarin@20318
   466
ballarin@20318
   467
lemma (in ring) r_null [simp]:
ballarin@20318
   468
  "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
ballarin@20318
   469
proof -
ballarin@20318
   470
  assume R: "x \<in> carrier R"
ballarin@20318
   471
  then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
ballarin@20318
   472
    by (simp add: r_distr del: l_zero r_zero)
ballarin@20318
   473
  also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
ballarin@20318
   474
  finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
ballarin@20318
   475
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   476
qed
ballarin@20318
   477
ballarin@20318
   478
lemma (in ring) l_minus:
ballarin@20318
   479
  "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   480
proof -
ballarin@20318
   481
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   482
  then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
ballarin@20318
   483
  also from R have "... = \<zero>" by (simp add: l_neg l_null)
ballarin@20318
   484
  finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   485
  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@21896
   486
  with R show ?thesis by (simp add: a_assoc r_neg)
ballarin@20318
   487
qed
ballarin@20318
   488
ballarin@20318
   489
lemma (in ring) r_minus:
ballarin@20318
   490
  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   491
proof -
ballarin@20318
   492
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   493
  then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
ballarin@20318
   494
  also from R have "... = \<zero>" by (simp add: l_neg r_null)
ballarin@20318
   495
  finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   496
  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
   497
  with R show ?thesis by (simp add: a_assoc r_neg )
ballarin@20318
   498
qed
ballarin@20318
   499
ballarin@23957
   500
lemma (in abelian_group) minus_eq:
ballarin@23957
   501
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
ballarin@20318
   502
  by (simp only: a_minus_def)
ballarin@20318
   503
ballarin@20318
   504
text {* Setup algebra method:
ballarin@20318
   505
  compute distributive normal form in locale contexts *}
ballarin@20318
   506
ballarin@20318
   507
use "ringsimp.ML"
ballarin@20318
   508
ballarin@20318
   509
setup Algebra.setup
ballarin@20318
   510
ballarin@20318
   511
lemmas (in ring) ring_simprules
ballarin@20318
   512
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   513
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   514
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
ballarin@20318
   515
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   516
  a_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   517
ballarin@20318
   518
lemmas (in cring)
ballarin@20318
   519
  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   520
  _
ballarin@20318
   521
ballarin@20318
   522
lemmas (in cring) cring_simprules
ballarin@20318
   523
  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   524
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   525
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
ballarin@20318
   526
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   527
  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   528
ballarin@20318
   529
ballarin@20318
   530
lemma (in cring) nat_pow_zero:
ballarin@20318
   531
  "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
ballarin@20318
   532
  by (induct n) simp_all
ballarin@20318
   533
ballarin@20318
   534
lemma (in ring) one_zeroD:
ballarin@20318
   535
  assumes onezero: "\<one> = \<zero>"
ballarin@20318
   536
  shows "carrier R = {\<zero>}"
ballarin@20318
   537
proof (rule, rule)
ballarin@20318
   538
  fix x
ballarin@20318
   539
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   540
  from xcarr
ballarin@20318
   541
      have "x = x \<otimes> \<one>" by simp
ballarin@20318
   542
  from this and onezero
ballarin@20318
   543
      have "x = x \<otimes> \<zero>" by simp
ballarin@20318
   544
  from this and xcarr
ballarin@20318
   545
      have "x = \<zero>" by simp
ballarin@20318
   546
  thus "x \<in> {\<zero>}" by fast
ballarin@20318
   547
qed fast
ballarin@20318
   548
ballarin@20318
   549
lemma (in ring) one_zeroI:
ballarin@20318
   550
  assumes carrzero: "carrier R = {\<zero>}"
ballarin@20318
   551
  shows "\<one> = \<zero>"
ballarin@20318
   552
proof -
ballarin@20318
   553
  from one_closed and carrzero
ballarin@20318
   554
      show "\<one> = \<zero>" by simp
ballarin@20318
   555
qed
ballarin@20318
   556
ballarin@20318
   557
lemma (in ring) one_zero:
ballarin@20318
   558
  shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
ballarin@20318
   559
  by (rule, erule one_zeroI, erule one_zeroD)
ballarin@20318
   560
ballarin@20318
   561
lemma (in ring) one_not_zero:
ballarin@20318
   562
  shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
ballarin@20318
   563
  by (simp add: one_zero)
ballarin@20318
   564
ballarin@20318
   565
text {* Two examples for use of method algebra *}
ballarin@20318
   566
ballarin@20318
   567
lemma
ballarin@27611
   568
  fixes R (structure) and S (structure)
ballarin@27611
   569
  assumes "ring R" "cring S"
ballarin@27611
   570
  assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
ballarin@27611
   571
  shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
ballarin@27611
   572
proof -
ballarin@27611
   573
  interpret ring [R] by fact
ballarin@27611
   574
  interpret cring [S] by fact
ballarin@27611
   575
ML_val {* Algebra.print_structures @{context} *}
ballarin@27611
   576
  from RS show ?thesis by algebra
ballarin@27611
   577
qed
ballarin@20318
   578
ballarin@20318
   579
lemma
ballarin@27611
   580
  fixes R (structure)
ballarin@27611
   581
  assumes "ring R"
ballarin@27611
   582
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@27611
   583
  shows "a \<ominus> (a \<ominus> b) = b"
ballarin@27611
   584
proof -
ballarin@27611
   585
  interpret ring [R] by fact
ballarin@27611
   586
  from R show ?thesis by algebra
ballarin@27611
   587
qed
ballarin@20318
   588
ballarin@20318
   589
subsubsection {* Sums over Finite Sets *}
ballarin@20318
   590
ballarin@20318
   591
lemma (in cring) finsum_ldistr:
ballarin@20318
   592
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   593
   finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
berghofe@22265
   594
proof (induct set: finite)
ballarin@20318
   595
  case empty then show ?case by simp
ballarin@20318
   596
next
ballarin@20318
   597
  case (insert x F) then show ?case by (simp add: Pi_def l_distr)
ballarin@20318
   598
qed
ballarin@20318
   599
ballarin@20318
   600
lemma (in cring) finsum_rdistr:
ballarin@20318
   601
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   602
   a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
berghofe@22265
   603
proof (induct set: finite)
ballarin@20318
   604
  case empty then show ?case by simp
ballarin@20318
   605
next
ballarin@20318
   606
  case (insert x F) then show ?case by (simp add: Pi_def r_distr)
ballarin@20318
   607
qed
ballarin@20318
   608
ballarin@20318
   609
ballarin@20318
   610
subsection {* Integral Domains *}
ballarin@20318
   611
ballarin@20318
   612
lemma (in "domain") zero_not_one [simp]:
ballarin@20318
   613
  "\<zero> ~= \<one>"
ballarin@20318
   614
  by (rule not_sym) simp
ballarin@20318
   615
ballarin@20318
   616
lemma (in "domain") integral_iff: (* not by default a simp rule! *)
ballarin@20318
   617
  "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
ballarin@20318
   618
proof
ballarin@20318
   619
  assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
ballarin@20318
   620
  then show "a = \<zero> | b = \<zero>" by (simp add: integral)
ballarin@20318
   621
next
ballarin@20318
   622
  assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
ballarin@20318
   623
  then show "a \<otimes> b = \<zero>" by auto
ballarin@20318
   624
qed
ballarin@20318
   625
ballarin@20318
   626
lemma (in "domain") m_lcancel:
ballarin@20318
   627
  assumes prem: "a ~= \<zero>"
ballarin@20318
   628
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   629
  shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
ballarin@20318
   630
proof
ballarin@20318
   631
  assume eq: "a \<otimes> b = a \<otimes> c"
ballarin@20318
   632
  with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
ballarin@20318
   633
  with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
ballarin@20318
   634
  with prem and R have "b \<ominus> c = \<zero>" by auto 
ballarin@20318
   635
  with R have "b = b \<ominus> (b \<ominus> c)" by algebra 
ballarin@20318
   636
  also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
ballarin@20318
   637
  finally show "b = c" .
ballarin@20318
   638
next
ballarin@20318
   639
  assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
ballarin@20318
   640
qed
ballarin@20318
   641
ballarin@20318
   642
lemma (in "domain") m_rcancel:
ballarin@20318
   643
  assumes prem: "a ~= \<zero>"
ballarin@20318
   644
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   645
  shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
ballarin@20318
   646
proof -
ballarin@20318
   647
  from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
ballarin@20318
   648
  with R show ?thesis by algebra
ballarin@20318
   649
qed
ballarin@20318
   650
ballarin@20318
   651
ballarin@20318
   652
subsection {* Fields *}
ballarin@20318
   653
ballarin@20318
   654
text {* Field would not need to be derived from domain, the properties
ballarin@20318
   655
  for domain follow from the assumptions of field *}
ballarin@20318
   656
lemma (in cring) cring_fieldI:
ballarin@20318
   657
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   658
  shows "field R"
ballarin@20318
   659
proof unfold_locales
ballarin@20318
   660
  from field_Units
ballarin@20318
   661
  have a: "\<zero> \<notin> Units R" by fast
ballarin@20318
   662
  have "\<one> \<in> Units R" by fast
ballarin@20318
   663
  from this and a
ballarin@20318
   664
  show "\<one> \<noteq> \<zero>" by force
ballarin@20318
   665
next
ballarin@20318
   666
  fix a b
ballarin@20318
   667
  assume acarr: "a \<in> carrier R"
ballarin@20318
   668
    and bcarr: "b \<in> carrier R"
ballarin@20318
   669
    and ab: "a \<otimes> b = \<zero>"
ballarin@20318
   670
  show "a = \<zero> \<or> b = \<zero>"
ballarin@20318
   671
  proof (cases "a = \<zero>", simp)
ballarin@20318
   672
    assume "a \<noteq> \<zero>"
ballarin@20318
   673
    from this and field_Units and acarr
ballarin@20318
   674
    have aUnit: "a \<in> Units R" by fast
ballarin@20318
   675
    from bcarr
ballarin@20318
   676
    have "b = \<one> \<otimes> b" by algebra
ballarin@20318
   677
    also from aUnit acarr
ballarin@20318
   678
    have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
ballarin@20318
   679
    also from acarr bcarr aUnit[THEN Units_inv_closed]
ballarin@20318
   680
    have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
ballarin@20318
   681
    also from ab and acarr bcarr aUnit
ballarin@20318
   682
    have "... = (inv a) \<otimes> \<zero>" by simp
ballarin@20318
   683
    also from aUnit[THEN Units_inv_closed]
ballarin@20318
   684
    have "... = \<zero>" by algebra
ballarin@20318
   685
    finally
ballarin@20318
   686
    have "b = \<zero>" .
ballarin@20318
   687
    thus "a = \<zero> \<or> b = \<zero>" by simp
ballarin@20318
   688
  qed
wenzelm@23350
   689
qed (rule field_Units)
ballarin@20318
   690
ballarin@20318
   691
text {* Another variant to show that something is a field *}
ballarin@20318
   692
lemma (in cring) cring_fieldI2:
ballarin@20318
   693
  assumes notzero: "\<zero> \<noteq> \<one>"
ballarin@20318
   694
  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
   695
  shows "field R"
ballarin@20318
   696
  apply (rule cring_fieldI, simp add: Units_def)
ballarin@20318
   697
  apply (rule, clarsimp)
ballarin@20318
   698
  apply (simp add: notzero)
ballarin@20318
   699
proof (clarsimp)
ballarin@20318
   700
  fix x
ballarin@20318
   701
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   702
    and "x \<noteq> \<zero>"
ballarin@20318
   703
  from this
ballarin@20318
   704
  have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
ballarin@20318
   705
  from this
ballarin@20318
   706
  obtain y
ballarin@20318
   707
    where ycarr: "y \<in> carrier R"
ballarin@20318
   708
    and xy: "x \<otimes> y = \<one>"
ballarin@20318
   709
    by fast
ballarin@20318
   710
  from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
ballarin@20318
   711
  from ycarr and this and xy
ballarin@20318
   712
  show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
ballarin@20318
   713
qed
ballarin@20318
   714
ballarin@20318
   715
ballarin@20318
   716
subsection {* Morphisms *}
ballarin@20318
   717
ballarin@20318
   718
constdefs (structure R S)
ballarin@20318
   719
  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
ballarin@20318
   720
  "ring_hom R S == {h. h \<in> carrier R -> carrier S &
ballarin@20318
   721
      (ALL x y. x \<in> carrier R & y \<in> carrier R -->
ballarin@20318
   722
        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
   723
      h \<one> = \<one>\<^bsub>S\<^esub>}"
ballarin@20318
   724
ballarin@20318
   725
lemma ring_hom_memI:
ballarin@20318
   726
  fixes R (structure) and S (structure)
ballarin@20318
   727
  assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
ballarin@20318
   728
    and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   729
      h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   730
    and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   731
      h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   732
    and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   733
  shows "h \<in> ring_hom R S"
ballarin@20318
   734
  by (auto simp add: ring_hom_def prems Pi_def)
ballarin@20318
   735
ballarin@20318
   736
lemma ring_hom_closed:
ballarin@20318
   737
  "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
ballarin@20318
   738
  by (auto simp add: ring_hom_def funcset_mem)
ballarin@20318
   739
ballarin@20318
   740
lemma ring_hom_mult:
ballarin@20318
   741
  fixes R (structure) and S (structure)
ballarin@20318
   742
  shows
ballarin@20318
   743
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   744
    h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   745
    by (simp add: ring_hom_def)
ballarin@20318
   746
ballarin@20318
   747
lemma ring_hom_add:
ballarin@20318
   748
  fixes R (structure) and S (structure)
ballarin@20318
   749
  shows
ballarin@20318
   750
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   751
    h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   752
    by (simp add: ring_hom_def)
ballarin@20318
   753
ballarin@20318
   754
lemma ring_hom_one:
ballarin@20318
   755
  fixes R (structure) and S (structure)
ballarin@20318
   756
  shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   757
  by (simp add: ring_hom_def)
ballarin@20318
   758
ballarin@20318
   759
locale ring_hom_cring = cring R + cring S +
ballarin@20318
   760
  fixes h
ballarin@20318
   761
  assumes homh [simp, intro]: "h \<in> ring_hom R S"
ballarin@20318
   762
  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
ballarin@20318
   763
    and hom_mult [simp] = ring_hom_mult [OF homh]
ballarin@20318
   764
    and hom_add [simp] = ring_hom_add [OF homh]
ballarin@20318
   765
    and hom_one [simp] = ring_hom_one [OF homh]
ballarin@20318
   766
ballarin@20318
   767
lemma (in ring_hom_cring) hom_zero [simp]:
ballarin@20318
   768
  "h \<zero> = \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   769
proof -
ballarin@20318
   770
  have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   771
    by (simp add: hom_add [symmetric] del: hom_add)
ballarin@20318
   772
  then show ?thesis by (simp del: S.r_zero)
ballarin@20318
   773
qed
ballarin@20318
   774
ballarin@20318
   775
lemma (in ring_hom_cring) hom_a_inv [simp]:
ballarin@20318
   776
  "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
ballarin@20318
   777
proof -
ballarin@20318
   778
  assume R: "x \<in> carrier R"
ballarin@20318
   779
  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
   780
    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
ballarin@20318
   781
  with R show ?thesis by simp
ballarin@20318
   782
qed
ballarin@20318
   783
ballarin@20318
   784
lemma (in ring_hom_cring) hom_finsum [simp]:
ballarin@20318
   785
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   786
  h (finsum R f A) = finsum S (h o f) A"
berghofe@22265
   787
proof (induct set: finite)
ballarin@20318
   788
  case empty then show ?case by simp
ballarin@20318
   789
next
ballarin@20318
   790
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   791
qed
ballarin@20318
   792
ballarin@20318
   793
lemma (in ring_hom_cring) hom_finprod:
ballarin@20318
   794
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   795
  h (finprod R f A) = finprod S (h o f) A"
berghofe@22265
   796
proof (induct set: finite)
ballarin@20318
   797
  case empty then show ?case by simp
ballarin@20318
   798
next
ballarin@20318
   799
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   800
qed
ballarin@20318
   801
ballarin@20318
   802
declare ring_hom_cring.hom_finprod [simp]
ballarin@20318
   803
ballarin@20318
   804
lemma id_ring_hom [simp]:
ballarin@20318
   805
  "id \<in> ring_hom R R"
ballarin@20318
   806
  by (auto intro!: ring_hom_memI)
ballarin@20318
   807
ballarin@20318
   808
end