src/HOL/Algebra/Ring.thy
author paulson <lp15@cam.ac.uk>
Wed Jun 06 14:25:53 2018 +0100 (12 months ago)
changeset 68399 0b71d08528f0
parent 67613 ce654b0e6d69
child 68443 43055b016688
permissions -rw-r--r--
resolution of name clashes in Algebra
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(*  Title:      HOL/Algebra/Ring.thy
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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
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imports FiniteProduct
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begin
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section \<open>The Algebraic Hierarchy of Rings\<close>
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subsection \<open>Abelian Groups\<close>
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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 \<open>Derived operations.\<close>
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definition
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  a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
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  where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
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definition
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  a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ \<ominus>\<index> _)" [65,66] 65)
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  where "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)"
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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 \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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definition
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  finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
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  "finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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syntax
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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>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (%i. b) A"
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  \<comment> \<open>Beware of argument permutation!\<close>
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locale abelian_group = abelian_monoid +
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  assumes a_comm_group:
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     "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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subsection \<open>Basic Properties\<close>
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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: assms)
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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: "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> 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: assms)
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lemma (in abelian_monoid) a_monoid:
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  "monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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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 \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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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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text \<open>Transfer facts from multiplicative structures via interpretation.\<close>
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sublocale abelian_monoid <
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  add: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
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    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
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    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
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  by (rule a_monoid) auto
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context abelian_monoid begin
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lemmas a_closed = add.m_closed 
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lemmas zero_closed = add.one_closed
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lemmas a_assoc = add.m_assoc
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lemmas l_zero = add.l_one
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lemmas r_zero = add.r_one
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lemmas minus_unique = add.inv_unique
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end
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sublocale abelian_monoid <
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  add: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
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    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
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    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
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    and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
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  by (rule a_comm_monoid) (auto simp: finsum_def)
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context abelian_monoid begin
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lemmas a_comm = add.m_comm
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lemmas a_lcomm = add.m_lcomm
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lemmas a_ac = a_assoc a_comm a_lcomm
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lemmas finsum_empty = add.finprod_empty
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lemmas finsum_insert = add.finprod_insert
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lemmas finsum_zero = add.finprod_one
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lemmas finsum_closed = add.finprod_closed
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lemmas finsum_Un_Int = add.finprod_Un_Int
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lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
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lemmas finsum_addf = add.finprod_multf
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lemmas finsum_cong' = add.finprod_cong'
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lemmas finsum_0 = add.finprod_0
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lemmas finsum_Suc = add.finprod_Suc
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lemmas finsum_Suc2 = add.finprod_Suc2
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lemmas finsum_add = add.finprod_mult
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lemmas finsum_infinite = add.finprod_infinite
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lemmas finsum_cong = add.finprod_cong
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text \<open>Usually, if this rule causes a failed congruence proof error,
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   the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
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   Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>
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lemmas finsum_reindex = add.finprod_reindex
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(* The following would be wrong.  Needed is the equivalent of [^] for addition,
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  or indeed the canonical embedding from Nat into the monoid.
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lemma finsum_const:
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  assumes fin [simp]: "finite A"
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      and a [simp]: "a : carrier G"
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    shows "finsum G (%x. a) A = a [^] card A"
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  using fin apply induct
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  apply force
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  apply (subst finsum_insert)
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  apply auto
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  apply (force simp add: Pi_def)
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  apply (subst m_comm)
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  apply auto
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done
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*)
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lemmas finsum_singleton = add.finprod_singleton
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end
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sublocale abelian_group <
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  add: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
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    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
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    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
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    and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
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  by (rule a_group) (auto simp: m_inv_def a_inv_def)
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context abelian_group
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begin
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lemmas a_inv_closed = add.inv_closed
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lemma 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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lemmas l_neg = add.l_inv [simp del]
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lemmas r_neg = add.r_inv [simp del]
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lemmas minus_zero = add.inv_one
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lemmas minus_minus = add.inv_inv
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lemmas a_inv_inj = add.inv_inj
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lemmas minus_equality = add.inv_equality
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end
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sublocale abelian_group <
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  add: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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  rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
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    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
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    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
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    and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
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    and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
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  by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)
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lemmas (in abelian_group) minus_add = add.inv_mult
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text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>
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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" ..
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qed
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subsection \<open>Rings: Basic Definitions\<close>
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locale semiring = abelian_monoid R + monoid R for R (structure) +
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  assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
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    and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
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    and l_null[simp]: "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
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    and r_null[simp]: "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
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locale ring = abelian_group R + monoid R for R (structure) +
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  assumes "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
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    and "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
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locale cring = ring + comm_monoid R
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locale "domain" = cring +
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  assumes one_not_zero [simp]: "\<one> \<noteq> \<zero>"
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    and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
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                  a = \<zero> \<or> b = \<zero>"
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locale field = "domain" +
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  assumes field_Units: "Units R = carrier R - {\<zero>}"
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subsection \<open>Rings\<close>
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lemma ringI:
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  fixes R (structure)
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  assumes abelian_group: "abelian_group R"
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    and monoid: "monoid R"
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    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
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    and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
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  shows "ring R"
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  by (auto intro: ring.intro
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    abelian_group.axioms ring_axioms.intro assms)
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context ring begin
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lemma is_abelian_group: "abelian_group R" ..
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lemma is_monoid: "monoid R"
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  by (auto intro!: monoidI m_assoc)
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lemma is_ring: "ring R"
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  by (rule ring_axioms)
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end
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lemmas ring_record_simps = monoid_record_simps ring.simps
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lemma cringI:
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  fixes R (structure)
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  assumes abelian_group: "abelian_group R"
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    and comm_monoid: "comm_monoid R"
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    and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
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      ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
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  shows "cring R"
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proof (intro cring.intro ring.intro)
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  show "ring_axioms R"
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    \<comment> \<open>Right-distributivity follows from left-distributivity and
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          commutativity.\<close>
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  proof (rule ring_axioms.intro)
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    fix x y z
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    assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
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    note [simp] = comm_monoid.axioms [OF comm_monoid]
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      abelian_group.axioms [OF abelian_group]
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      abelian_monoid.a_closed
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    from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
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      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
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    also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
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    also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
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      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
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    finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
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  qed (rule l_distr)
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qed (auto intro: cring.intro
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  abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
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(*
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lemma (in cring) is_comm_monoid:
ballarin@20318
   305
  "comm_monoid R"
ballarin@20318
   306
  by (auto intro!: comm_monoidI m_assoc m_comm)
ballarin@27699
   307
*)
ballarin@20318
   308
ballarin@20318
   309
lemma (in cring) is_cring:
wenzelm@26202
   310
  "cring R" by (rule cring_axioms)
wenzelm@23350
   311
ballarin@20318
   312
wenzelm@61382
   313
subsubsection \<open>Normaliser for Rings\<close>
ballarin@20318
   314
ballarin@20318
   315
lemma (in abelian_group) r_neg2:
ballarin@20318
   316
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
ballarin@20318
   317
proof -
ballarin@20318
   318
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   319
  then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
ballarin@20318
   320
    by (simp only: r_neg l_zero)
ballarin@41433
   321
  with G show ?thesis
ballarin@20318
   322
    by (simp add: a_ac)
ballarin@20318
   323
qed
ballarin@20318
   324
ballarin@20318
   325
lemma (in abelian_group) r_neg1:
ballarin@20318
   326
  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
ballarin@20318
   327
proof -
ballarin@20318
   328
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   329
  then have "(\<ominus> x \<oplus> x) \<oplus> y = y" 
ballarin@20318
   330
    by (simp only: l_neg l_zero)
ballarin@20318
   331
  with G show ?thesis by (simp add: a_ac)
ballarin@20318
   332
qed
ballarin@20318
   333
ballarin@41433
   334
context ring begin
ballarin@41433
   335
wenzelm@61382
   336
text \<open>
ballarin@41433
   337
  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
wenzelm@61382
   338
\<close>
ballarin@20318
   339
rene@59851
   340
sublocale semiring
ballarin@20318
   341
proof -
rene@59851
   342
  note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
rene@59851
   343
  show "semiring R"
rene@59851
   344
  proof (unfold_locales)
rene@59851
   345
    fix x
rene@59851
   346
    assume R: "x \<in> carrier R"
rene@59851
   347
    then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
rene@59851
   348
      by (simp del: l_zero r_zero)
rene@59851
   349
    also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
rene@59851
   350
    finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
rene@59851
   351
    with R show "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero)
rene@59851
   352
    from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
rene@59851
   353
      by (simp del: l_zero r_zero)
rene@59851
   354
    also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
rene@59851
   355
    finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
rene@59851
   356
    with R show "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero)
rene@59851
   357
  qed auto
ballarin@20318
   358
qed
ballarin@20318
   359
ballarin@41433
   360
lemma l_minus:
ballarin@20318
   361
  "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   362
proof -
ballarin@20318
   363
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   364
  then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
wenzelm@44677
   365
  also from R have "... = \<zero>" by (simp add: l_neg)
ballarin@20318
   366
  finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   367
  with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
ballarin@21896
   368
  with R show ?thesis by (simp add: a_assoc r_neg)
ballarin@20318
   369
qed
ballarin@20318
   370
ballarin@41433
   371
lemma r_minus:
ballarin@20318
   372
  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   373
proof -
ballarin@20318
   374
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   375
  then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
wenzelm@44677
   376
  also from R have "... = \<zero>" by (simp add: l_neg)
ballarin@20318
   377
  finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   378
  with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
ballarin@20318
   379
  with R show ?thesis by (simp add: a_assoc r_neg )
ballarin@20318
   380
qed
ballarin@20318
   381
ballarin@41433
   382
end
ballarin@41433
   383
ballarin@23957
   384
lemma (in abelian_group) minus_eq:
ballarin@23957
   385
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
ballarin@20318
   386
  by (simp only: a_minus_def)
ballarin@20318
   387
wenzelm@61382
   388
text \<open>Setup algebra method:
wenzelm@61382
   389
  compute distributive normal form in locale contexts\<close>
ballarin@20318
   390
wenzelm@48891
   391
ML_file "ringsimp.ML"
ballarin@20318
   392
wenzelm@61382
   393
attribute_setup algebra = \<open>
wenzelm@58811
   394
  Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
wenzelm@58811
   395
    -- Scan.lift Args.name -- Scan.repeat Args.term
wenzelm@58811
   396
    >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
wenzelm@61382
   397
\<close> "theorems controlling algebra method"
wenzelm@47701
   398
wenzelm@61382
   399
method_setup algebra = \<open>
wenzelm@58811
   400
  Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
wenzelm@61382
   401
\<close> "normalisation of algebraic structure"
ballarin@20318
   402
rene@59851
   403
lemmas (in semiring) semiring_simprules
rene@59851
   404
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
rene@59851
   405
  a_closed zero_closed  m_closed one_closed
rene@59851
   406
  a_assoc l_zero  a_comm m_assoc l_one l_distr r_zero
rene@59851
   407
  a_lcomm r_distr l_null r_null 
rene@59851
   408
ballarin@20318
   409
lemmas (in ring) ring_simprules
ballarin@20318
   410
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   411
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   412
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
ballarin@20318
   413
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   414
  a_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   415
ballarin@20318
   416
lemmas (in cring)
ballarin@20318
   417
  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   418
  _
ballarin@20318
   419
ballarin@20318
   420
lemmas (in cring) cring_simprules
ballarin@20318
   421
  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   422
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   423
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
ballarin@20318
   424
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   425
  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   426
rene@59851
   427
lemma (in semiring) nat_pow_zero:
nipkow@67341
   428
  "(n::nat) \<noteq> 0 \<Longrightarrow> \<zero> [^] n = \<zero>"
ballarin@20318
   429
  by (induct n) simp_all
ballarin@20318
   430
rene@59851
   431
context semiring begin
ballarin@41433
   432
ballarin@41433
   433
lemma one_zeroD:
ballarin@20318
   434
  assumes onezero: "\<one> = \<zero>"
ballarin@20318
   435
  shows "carrier R = {\<zero>}"
ballarin@20318
   436
proof (rule, rule)
ballarin@20318
   437
  fix x
ballarin@20318
   438
  assume xcarr: "x \<in> carrier R"
wenzelm@47409
   439
  from xcarr have "x = x \<otimes> \<one>" by simp
wenzelm@47409
   440
  with onezero have "x = x \<otimes> \<zero>" by simp
wenzelm@47409
   441
  with xcarr have "x = \<zero>" by simp
wenzelm@47409
   442
  then show "x \<in> {\<zero>}" by fast
ballarin@20318
   443
qed fast
ballarin@20318
   444
ballarin@41433
   445
lemma one_zeroI:
ballarin@20318
   446
  assumes carrzero: "carrier R = {\<zero>}"
ballarin@20318
   447
  shows "\<one> = \<zero>"
ballarin@20318
   448
proof -
ballarin@20318
   449
  from one_closed and carrzero
ballarin@20318
   450
      show "\<one> = \<zero>" by simp
ballarin@20318
   451
qed
ballarin@20318
   452
wenzelm@46721
   453
lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
wenzelm@46721
   454
  apply rule
wenzelm@46721
   455
   apply (erule one_zeroI)
wenzelm@46721
   456
  apply (erule one_zeroD)
wenzelm@46721
   457
  done
ballarin@20318
   458
wenzelm@46721
   459
lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
ballarin@27717
   460
  by (simp add: carrier_one_zero)
ballarin@20318
   461
ballarin@41433
   462
end
ballarin@41433
   463
wenzelm@61382
   464
text \<open>Two examples for use of method algebra\<close>
ballarin@20318
   465
ballarin@20318
   466
lemma
ballarin@27611
   467
  fixes R (structure) and S (structure)
ballarin@27611
   468
  assumes "ring R" "cring S"
ballarin@27611
   469
  assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
wenzelm@67091
   470
  shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b \<and> c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
ballarin@27611
   471
proof -
ballarin@29237
   472
  interpret ring R by fact
ballarin@29237
   473
  interpret cring S by fact
ballarin@27611
   474
  from RS show ?thesis by algebra
ballarin@27611
   475
qed
ballarin@20318
   476
ballarin@20318
   477
lemma
ballarin@27611
   478
  fixes R (structure)
ballarin@27611
   479
  assumes "ring R"
ballarin@27611
   480
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@27611
   481
  shows "a \<ominus> (a \<ominus> b) = b"
ballarin@27611
   482
proof -
ballarin@29237
   483
  interpret ring R by fact
ballarin@27611
   484
  from R show ?thesis by algebra
ballarin@27611
   485
qed
ballarin@20318
   486
wenzelm@35849
   487
wenzelm@61382
   488
subsubsection \<open>Sums over Finite Sets\<close>
ballarin@20318
   489
rene@59851
   490
lemma (in semiring) finsum_ldistr:
wenzelm@61384
   491
  "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
ballarin@20318
   492
   finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
berghofe@22265
   493
proof (induct set: finite)
ballarin@20318
   494
  case empty then show ?case by simp
ballarin@20318
   495
next
ballarin@20318
   496
  case (insert x F) then show ?case by (simp add: Pi_def l_distr)
ballarin@20318
   497
qed
ballarin@20318
   498
rene@59851
   499
lemma (in semiring) finsum_rdistr:
wenzelm@61384
   500
  "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
ballarin@20318
   501
   a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
berghofe@22265
   502
proof (induct set: finite)
ballarin@20318
   503
  case empty then show ?case by simp
ballarin@20318
   504
next
ballarin@20318
   505
  case (insert x F) then show ?case by (simp add: Pi_def r_distr)
ballarin@20318
   506
qed
ballarin@20318
   507
ballarin@20318
   508
wenzelm@61382
   509
subsection \<open>Integral Domains\<close>
ballarin@20318
   510
ballarin@41433
   511
context "domain" begin
ballarin@41433
   512
ballarin@41433
   513
lemma zero_not_one [simp]:
wenzelm@67091
   514
  "\<zero> \<noteq> \<one>"
ballarin@20318
   515
  by (rule not_sym) simp
ballarin@20318
   516
ballarin@41433
   517
lemma integral_iff: (* not by default a simp rule! *)
wenzelm@67091
   518
  "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
ballarin@20318
   519
proof
ballarin@20318
   520
  assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
wenzelm@67091
   521
  then show "a = \<zero> \<or> b = \<zero>" by (simp add: integral)
ballarin@20318
   522
next
wenzelm@67091
   523
  assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> \<or> b = \<zero>"
ballarin@20318
   524
  then show "a \<otimes> b = \<zero>" by auto
ballarin@20318
   525
qed
ballarin@20318
   526
ballarin@41433
   527
lemma m_lcancel:
wenzelm@67091
   528
  assumes prem: "a \<noteq> \<zero>"
ballarin@20318
   529
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   530
  shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
ballarin@20318
   531
proof
ballarin@20318
   532
  assume eq: "a \<otimes> b = a \<otimes> c"
ballarin@20318
   533
  with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
wenzelm@67091
   534
  with R have "a = \<zero> \<or> (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
ballarin@20318
   535
  with prem and R have "b \<ominus> c = \<zero>" by auto 
ballarin@20318
   536
  with R have "b = b \<ominus> (b \<ominus> c)" by algebra 
ballarin@20318
   537
  also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
ballarin@20318
   538
  finally show "b = c" .
ballarin@20318
   539
next
ballarin@20318
   540
  assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
ballarin@20318
   541
qed
ballarin@20318
   542
ballarin@41433
   543
lemma m_rcancel:
wenzelm@67091
   544
  assumes prem: "a \<noteq> \<zero>"
ballarin@20318
   545
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   546
  shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
ballarin@20318
   547
proof -
ballarin@20318
   548
  from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
ballarin@20318
   549
  with R show ?thesis by algebra
ballarin@20318
   550
qed
ballarin@20318
   551
ballarin@41433
   552
end
ballarin@41433
   553
ballarin@20318
   554
wenzelm@61382
   555
subsection \<open>Fields\<close>
ballarin@20318
   556
wenzelm@61382
   557
text \<open>Field would not need to be derived from domain, the properties
wenzelm@61382
   558
  for domain follow from the assumptions of field\<close>
ballarin@20318
   559
lemma (in cring) cring_fieldI:
ballarin@20318
   560
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   561
  shows "field R"
haftmann@28823
   562
proof
wenzelm@47409
   563
  from field_Units have "\<zero> \<notin> Units R" by fast
wenzelm@47409
   564
  moreover have "\<one> \<in> Units R" by fast
wenzelm@47409
   565
  ultimately show "\<one> \<noteq> \<zero>" by force
ballarin@20318
   566
next
ballarin@20318
   567
  fix a b
ballarin@20318
   568
  assume acarr: "a \<in> carrier R"
ballarin@20318
   569
    and bcarr: "b \<in> carrier R"
ballarin@20318
   570
    and ab: "a \<otimes> b = \<zero>"
ballarin@20318
   571
  show "a = \<zero> \<or> b = \<zero>"
ballarin@20318
   572
  proof (cases "a = \<zero>", simp)
ballarin@20318
   573
    assume "a \<noteq> \<zero>"
wenzelm@47409
   574
    with field_Units and acarr have aUnit: "a \<in> Units R" by fast
wenzelm@47409
   575
    from bcarr have "b = \<one> \<otimes> b" by algebra
wenzelm@47409
   576
    also from aUnit acarr have "... = (inv a \<otimes> a) \<otimes> b" by simp
ballarin@20318
   577
    also from acarr bcarr aUnit[THEN Units_inv_closed]
ballarin@20318
   578
    have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
wenzelm@47409
   579
    also from ab and acarr bcarr aUnit have "... = (inv a) \<otimes> \<zero>" by simp
wenzelm@47409
   580
    also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra
wenzelm@47409
   581
    finally have "b = \<zero>" .
wenzelm@47409
   582
    then show "a = \<zero> \<or> b = \<zero>" by simp
ballarin@20318
   583
  qed
wenzelm@23350
   584
qed (rule field_Units)
ballarin@20318
   585
wenzelm@61382
   586
text \<open>Another variant to show that something is a field\<close>
ballarin@20318
   587
lemma (in cring) cring_fieldI2:
ballarin@20318
   588
  assumes notzero: "\<zero> \<noteq> \<one>"
ballarin@20318
   589
  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
   590
  shows "field R"
ballarin@20318
   591
  apply (rule cring_fieldI, simp add: Units_def)
ballarin@20318
   592
  apply (rule, clarsimp)
ballarin@20318
   593
  apply (simp add: notzero)
ballarin@20318
   594
proof (clarsimp)
ballarin@20318
   595
  fix x
ballarin@20318
   596
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   597
    and "x \<noteq> \<zero>"
wenzelm@47409
   598
  then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
wenzelm@47409
   599
  then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast
ballarin@20318
   600
  from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
wenzelm@47409
   601
  with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
ballarin@20318
   602
qed
ballarin@20318
   603
ballarin@20318
   604
wenzelm@61382
   605
subsection \<open>Morphisms\<close>
ballarin@20318
   606
wenzelm@35847
   607
definition
ballarin@20318
   608
  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
wenzelm@35848
   609
  where "ring_hom R S =
wenzelm@67091
   610
    {h. h \<in> carrier R \<rightarrow> carrier S \<and>
wenzelm@67091
   611
      (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow>
wenzelm@67091
   612
        h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and>
wenzelm@35847
   613
      h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
ballarin@20318
   614
ballarin@20318
   615
lemma ring_hom_memI:
ballarin@20318
   616
  fixes R (structure) and S (structure)
ballarin@20318
   617
  assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
ballarin@20318
   618
    and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   619
      h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   620
    and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   621
      h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   622
    and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   623
  shows "h \<in> ring_hom R S"
ballarin@27714
   624
  by (auto simp add: ring_hom_def assms Pi_def)
ballarin@20318
   625
ballarin@20318
   626
lemma ring_hom_closed:
ballarin@20318
   627
  "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
ballarin@20318
   628
  by (auto simp add: ring_hom_def funcset_mem)
ballarin@20318
   629
ballarin@20318
   630
lemma ring_hom_mult:
ballarin@20318
   631
  fixes R (structure) and S (structure)
ballarin@20318
   632
  shows
ballarin@20318
   633
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   634
    h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   635
    by (simp add: ring_hom_def)
ballarin@20318
   636
ballarin@20318
   637
lemma ring_hom_add:
ballarin@20318
   638
  fixes R (structure) and S (structure)
ballarin@20318
   639
  shows
ballarin@20318
   640
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   641
    h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   642
    by (simp add: ring_hom_def)
ballarin@20318
   643
ballarin@20318
   644
lemma ring_hom_one:
ballarin@20318
   645
  fixes R (structure) and S (structure)
ballarin@20318
   646
  shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   647
  by (simp add: ring_hom_def)
ballarin@20318
   648
ballarin@61565
   649
locale ring_hom_cring = R?: cring R + S?: cring S
ballarin@29237
   650
    for R (structure) and S (structure) +
ballarin@20318
   651
  fixes h
ballarin@20318
   652
  assumes homh [simp, intro]: "h \<in> ring_hom R S"
ballarin@20318
   653
  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
ballarin@20318
   654
    and hom_mult [simp] = ring_hom_mult [OF homh]
ballarin@20318
   655
    and hom_add [simp] = ring_hom_add [OF homh]
ballarin@20318
   656
    and hom_one [simp] = ring_hom_one [OF homh]
ballarin@20318
   657
ballarin@20318
   658
lemma (in ring_hom_cring) hom_zero [simp]:
ballarin@20318
   659
  "h \<zero> = \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   660
proof -
ballarin@20318
   661
  have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   662
    by (simp add: hom_add [symmetric] del: hom_add)
ballarin@20318
   663
  then show ?thesis by (simp del: S.r_zero)
ballarin@20318
   664
qed
ballarin@20318
   665
ballarin@20318
   666
lemma (in ring_hom_cring) hom_a_inv [simp]:
ballarin@20318
   667
  "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
ballarin@20318
   668
proof -
ballarin@20318
   669
  assume R: "x \<in> carrier R"
ballarin@20318
   670
  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
   671
    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
ballarin@20318
   672
  with R show ?thesis by simp
ballarin@20318
   673
qed
ballarin@20318
   674
ballarin@20318
   675
lemma (in ring_hom_cring) hom_finsum [simp]:
wenzelm@67091
   676
  "f \<in> A \<rightarrow> carrier R \<Longrightarrow>
wenzelm@67091
   677
  h (finsum R f A) = finsum S (h \<circ> f) A"
rene@60112
   678
  by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
ballarin@20318
   679
ballarin@20318
   680
lemma (in ring_hom_cring) hom_finprod:
wenzelm@67091
   681
  "f \<in> A \<rightarrow> carrier R \<Longrightarrow>
wenzelm@67091
   682
  h (finprod R f A) = finprod S (h \<circ> f) A"
rene@60112
   683
  by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
ballarin@20318
   684
ballarin@20318
   685
declare ring_hom_cring.hom_finprod [simp]
ballarin@20318
   686
ballarin@20318
   687
lemma id_ring_hom [simp]:
ballarin@20318
   688
  "id \<in> ring_hom R R"
ballarin@20318
   689
  by (auto intro!: ring_hom_memI)
ballarin@20318
   690
ballarin@20318
   691
end