src/HOL/Algebra/Ring.thy
author ballarin
Thu Jan 06 21:06:17 2011 +0100 (2011-01-06)
changeset 41433 1b8ff770f02c
parent 35849 b5522b51cb1e
child 41959 b460124855b8
permissions -rw-r--r--
Abelian group facts obtained from group facts via interpretation (sublocale).
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(*  Title:      The algebraic hierarchy of rings
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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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uses ("ringsimp.ML")
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begin
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section {* The Algebraic Hierarchy of Rings *}
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subsection {* 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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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 (| carrier = carrier R, mult = add R, one = zero R |)"
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definition
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  a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 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 (| carrier = carrier G, mult = add G, one = zero G |)"
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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 (| carrier = carrier G, mult = add G, one = zero G |)"
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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" == "CONST finsum \<struct>\<index> (%i. b) A"
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  -- {* Beware of argument permutation! *}
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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: 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: "!!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: assms)
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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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text {* Transfer facts from multiplicative structures via interpretation. *}
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sublocale abelian_monoid <
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  add!: monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
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  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
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    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
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    and "one (| carrier = carrier G, mult = add G, one = zero G |) = 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 "(| carrier = carrier G, mult = add G, one = zero G |)"
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  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
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    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
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    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
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    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = 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_cong = add.finprod_cong
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text {*Usually, if this rule causes a failed congruence proof error,
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   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
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   Adding @{thm [source] Pi_def} to the simpset is often useful. *}
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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 "(| carrier = carrier G, mult = add G, one = zero G |)"
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  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
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    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
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    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
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    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = 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 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 a_l_cancel = add.l_cancel
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lemmas a_r_cancel = add.r_cancel
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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 "(| carrier = carrier G, mult = add G, one = zero G |)"
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  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
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    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
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    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
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    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
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    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = 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 {* 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" ..
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qed
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subsection {* Rings: Basic Definitions *}
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locale ring = abelian_group 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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locale cring = ring + comm_monoid R
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locale "domain" = cring +
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  assumes one_not_zero [simp]: "\<one> ~= \<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> | 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 {* Rings *}
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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:
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  "abelian_group R"
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  ..
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lemma is_monoid:
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  "monoid R"
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  by (auto intro!: monoidI m_assoc)
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lemma is_ring:
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  "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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    -- {* Right-distributivity follows from left-distributivity and
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          commutativity. *}
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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:
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  "comm_monoid R"
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  by (auto intro!: comm_monoidI m_assoc m_comm)
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*)
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lemma (in cring) is_cring:
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  "cring R" by (rule cring_axioms)
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subsubsection {* Normaliser for Rings *}
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lemma (in abelian_group) r_neg2:
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  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
ballarin@20318
   320
proof -
ballarin@20318
   321
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   322
  then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
ballarin@20318
   323
    by (simp only: r_neg l_zero)
ballarin@41433
   324
  with G show ?thesis
ballarin@20318
   325
    by (simp add: a_ac)
ballarin@20318
   326
qed
ballarin@20318
   327
ballarin@20318
   328
lemma (in abelian_group) r_neg1:
ballarin@20318
   329
  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
ballarin@20318
   330
proof -
ballarin@20318
   331
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@20318
   332
  then have "(\<ominus> x \<oplus> x) \<oplus> y = y" 
ballarin@20318
   333
    by (simp only: l_neg l_zero)
ballarin@20318
   334
  with G show ?thesis by (simp add: a_ac)
ballarin@20318
   335
qed
ballarin@20318
   336
ballarin@41433
   337
context ring begin
ballarin@41433
   338
ballarin@20318
   339
text {* 
ballarin@41433
   340
  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
ballarin@20318
   341
*}
ballarin@20318
   342
ballarin@41433
   343
lemma l_null [simp]:
ballarin@20318
   344
  "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
ballarin@20318
   345
proof -
ballarin@20318
   346
  assume R: "x \<in> carrier R"
ballarin@20318
   347
  then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
ballarin@20318
   348
    by (simp add: l_distr del: l_zero r_zero)
ballarin@20318
   349
  also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
ballarin@20318
   350
  finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
ballarin@20318
   351
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   352
qed
ballarin@20318
   353
ballarin@41433
   354
lemma r_null [simp]:
ballarin@20318
   355
  "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
ballarin@20318
   356
proof -
ballarin@20318
   357
  assume R: "x \<in> carrier R"
ballarin@20318
   358
  then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
ballarin@20318
   359
    by (simp add: r_distr del: l_zero r_zero)
ballarin@20318
   360
  also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
ballarin@20318
   361
  finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
ballarin@20318
   362
  with R show ?thesis by (simp del: r_zero)
ballarin@20318
   363
qed
ballarin@20318
   364
ballarin@41433
   365
lemma l_minus:
ballarin@20318
   366
  "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   367
proof -
ballarin@20318
   368
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   369
  then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
ballarin@20318
   370
  also from R have "... = \<zero>" by (simp add: l_neg l_null)
ballarin@20318
   371
  finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   372
  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
   373
  with R show ?thesis by (simp add: a_assoc r_neg)
ballarin@20318
   374
qed
ballarin@20318
   375
ballarin@41433
   376
lemma r_minus:
ballarin@20318
   377
  "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
ballarin@20318
   378
proof -
ballarin@20318
   379
  assume R: "x \<in> carrier R" "y \<in> carrier R"
ballarin@20318
   380
  then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
ballarin@20318
   381
  also from R have "... = \<zero>" by (simp add: l_neg r_null)
ballarin@20318
   382
  finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
ballarin@20318
   383
  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
   384
  with R show ?thesis by (simp add: a_assoc r_neg )
ballarin@20318
   385
qed
ballarin@20318
   386
ballarin@41433
   387
end
ballarin@41433
   388
ballarin@23957
   389
lemma (in abelian_group) minus_eq:
ballarin@23957
   390
  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
ballarin@20318
   391
  by (simp only: a_minus_def)
ballarin@20318
   392
ballarin@20318
   393
text {* Setup algebra method:
ballarin@20318
   394
  compute distributive normal form in locale contexts *}
ballarin@20318
   395
ballarin@20318
   396
use "ringsimp.ML"
ballarin@20318
   397
ballarin@20318
   398
setup Algebra.setup
ballarin@20318
   399
ballarin@20318
   400
lemmas (in ring) ring_simprules
ballarin@20318
   401
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   402
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   403
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
ballarin@20318
   404
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   405
  a_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   406
ballarin@20318
   407
lemmas (in cring)
ballarin@20318
   408
  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   409
  _
ballarin@20318
   410
ballarin@20318
   411
lemmas (in cring) cring_simprules
ballarin@20318
   412
  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
ballarin@20318
   413
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
ballarin@20318
   414
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
ballarin@20318
   415
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
ballarin@20318
   416
  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
ballarin@20318
   417
ballarin@20318
   418
lemma (in cring) nat_pow_zero:
ballarin@20318
   419
  "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
ballarin@20318
   420
  by (induct n) simp_all
ballarin@20318
   421
ballarin@41433
   422
context ring begin
ballarin@41433
   423
ballarin@41433
   424
lemma one_zeroD:
ballarin@20318
   425
  assumes onezero: "\<one> = \<zero>"
ballarin@20318
   426
  shows "carrier R = {\<zero>}"
ballarin@20318
   427
proof (rule, rule)
ballarin@20318
   428
  fix x
ballarin@20318
   429
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   430
  from xcarr
ballarin@20318
   431
      have "x = x \<otimes> \<one>" by simp
ballarin@20318
   432
  from this and onezero
ballarin@20318
   433
      have "x = x \<otimes> \<zero>" by simp
ballarin@20318
   434
  from this and xcarr
ballarin@20318
   435
      have "x = \<zero>" by simp
ballarin@20318
   436
  thus "x \<in> {\<zero>}" by fast
ballarin@20318
   437
qed fast
ballarin@20318
   438
ballarin@41433
   439
lemma one_zeroI:
ballarin@20318
   440
  assumes carrzero: "carrier R = {\<zero>}"
ballarin@20318
   441
  shows "\<one> = \<zero>"
ballarin@20318
   442
proof -
ballarin@20318
   443
  from one_closed and carrzero
ballarin@20318
   444
      show "\<one> = \<zero>" by simp
ballarin@20318
   445
qed
ballarin@20318
   446
ballarin@41433
   447
lemma carrier_one_zero:
ballarin@20318
   448
  shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
ballarin@20318
   449
  by (rule, erule one_zeroI, erule one_zeroD)
ballarin@20318
   450
ballarin@41433
   451
lemma carrier_one_not_zero:
ballarin@20318
   452
  shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
ballarin@27717
   453
  by (simp add: carrier_one_zero)
ballarin@20318
   454
ballarin@41433
   455
end
ballarin@41433
   456
ballarin@20318
   457
text {* Two examples for use of method algebra *}
ballarin@20318
   458
ballarin@20318
   459
lemma
ballarin@27611
   460
  fixes R (structure) and S (structure)
ballarin@27611
   461
  assumes "ring R" "cring S"
ballarin@27611
   462
  assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
ballarin@27611
   463
  shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
ballarin@27611
   464
proof -
ballarin@29237
   465
  interpret ring R by fact
ballarin@29237
   466
  interpret cring S by fact
ballarin@27611
   467
ML_val {* Algebra.print_structures @{context} *}
ballarin@27611
   468
  from RS show ?thesis by algebra
ballarin@27611
   469
qed
ballarin@20318
   470
ballarin@20318
   471
lemma
ballarin@27611
   472
  fixes R (structure)
ballarin@27611
   473
  assumes "ring R"
ballarin@27611
   474
  assumes R: "a \<in> carrier R" "b \<in> carrier R"
ballarin@27611
   475
  shows "a \<ominus> (a \<ominus> b) = b"
ballarin@27611
   476
proof -
ballarin@29237
   477
  interpret ring R by fact
ballarin@27611
   478
  from R show ?thesis by algebra
ballarin@27611
   479
qed
ballarin@20318
   480
wenzelm@35849
   481
ballarin@20318
   482
subsubsection {* Sums over Finite Sets *}
ballarin@20318
   483
ballarin@27717
   484
lemma (in ring) finsum_ldistr:
ballarin@20318
   485
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   486
   finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
berghofe@22265
   487
proof (induct set: finite)
ballarin@20318
   488
  case empty then show ?case by simp
ballarin@20318
   489
next
ballarin@20318
   490
  case (insert x F) then show ?case by (simp add: Pi_def l_distr)
ballarin@20318
   491
qed
ballarin@20318
   492
ballarin@27717
   493
lemma (in ring) finsum_rdistr:
ballarin@20318
   494
  "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
ballarin@20318
   495
   a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
berghofe@22265
   496
proof (induct set: finite)
ballarin@20318
   497
  case empty then show ?case by simp
ballarin@20318
   498
next
ballarin@20318
   499
  case (insert x F) then show ?case by (simp add: Pi_def r_distr)
ballarin@20318
   500
qed
ballarin@20318
   501
ballarin@20318
   502
ballarin@20318
   503
subsection {* Integral Domains *}
ballarin@20318
   504
ballarin@41433
   505
context "domain" begin
ballarin@41433
   506
ballarin@41433
   507
lemma zero_not_one [simp]:
ballarin@20318
   508
  "\<zero> ~= \<one>"
ballarin@20318
   509
  by (rule not_sym) simp
ballarin@20318
   510
ballarin@41433
   511
lemma integral_iff: (* not by default a simp rule! *)
ballarin@20318
   512
  "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
ballarin@20318
   513
proof
ballarin@20318
   514
  assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
ballarin@20318
   515
  then show "a = \<zero> | b = \<zero>" by (simp add: integral)
ballarin@20318
   516
next
ballarin@20318
   517
  assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
ballarin@20318
   518
  then show "a \<otimes> b = \<zero>" by auto
ballarin@20318
   519
qed
ballarin@20318
   520
ballarin@41433
   521
lemma m_lcancel:
ballarin@20318
   522
  assumes prem: "a ~= \<zero>"
ballarin@20318
   523
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   524
  shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
ballarin@20318
   525
proof
ballarin@20318
   526
  assume eq: "a \<otimes> b = a \<otimes> c"
ballarin@20318
   527
  with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
ballarin@20318
   528
  with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
ballarin@20318
   529
  with prem and R have "b \<ominus> c = \<zero>" by auto 
ballarin@20318
   530
  with R have "b = b \<ominus> (b \<ominus> c)" by algebra 
ballarin@20318
   531
  also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
ballarin@20318
   532
  finally show "b = c" .
ballarin@20318
   533
next
ballarin@20318
   534
  assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
ballarin@20318
   535
qed
ballarin@20318
   536
ballarin@41433
   537
lemma m_rcancel:
ballarin@20318
   538
  assumes prem: "a ~= \<zero>"
ballarin@20318
   539
    and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
ballarin@20318
   540
  shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
ballarin@20318
   541
proof -
ballarin@20318
   542
  from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
ballarin@20318
   543
  with R show ?thesis by algebra
ballarin@20318
   544
qed
ballarin@20318
   545
ballarin@41433
   546
end
ballarin@41433
   547
ballarin@20318
   548
ballarin@20318
   549
subsection {* Fields *}
ballarin@20318
   550
ballarin@20318
   551
text {* Field would not need to be derived from domain, the properties
ballarin@20318
   552
  for domain follow from the assumptions of field *}
ballarin@20318
   553
lemma (in cring) cring_fieldI:
ballarin@20318
   554
  assumes field_Units: "Units R = carrier R - {\<zero>}"
ballarin@20318
   555
  shows "field R"
haftmann@28823
   556
proof
ballarin@20318
   557
  from field_Units
ballarin@20318
   558
  have a: "\<zero> \<notin> Units R" by fast
ballarin@20318
   559
  have "\<one> \<in> Units R" by fast
ballarin@20318
   560
  from this and a
ballarin@20318
   561
  show "\<one> \<noteq> \<zero>" by force
ballarin@20318
   562
next
ballarin@20318
   563
  fix a b
ballarin@20318
   564
  assume acarr: "a \<in> carrier R"
ballarin@20318
   565
    and bcarr: "b \<in> carrier R"
ballarin@20318
   566
    and ab: "a \<otimes> b = \<zero>"
ballarin@20318
   567
  show "a = \<zero> \<or> b = \<zero>"
ballarin@20318
   568
  proof (cases "a = \<zero>", simp)
ballarin@20318
   569
    assume "a \<noteq> \<zero>"
ballarin@20318
   570
    from this and field_Units and acarr
ballarin@20318
   571
    have aUnit: "a \<in> Units R" by fast
ballarin@20318
   572
    from bcarr
ballarin@20318
   573
    have "b = \<one> \<otimes> b" by algebra
ballarin@20318
   574
    also from aUnit acarr
ballarin@20318
   575
    have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
ballarin@20318
   576
    also from acarr bcarr aUnit[THEN Units_inv_closed]
ballarin@20318
   577
    have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
ballarin@20318
   578
    also from ab and acarr bcarr aUnit
ballarin@20318
   579
    have "... = (inv a) \<otimes> \<zero>" by simp
ballarin@20318
   580
    also from aUnit[THEN Units_inv_closed]
ballarin@20318
   581
    have "... = \<zero>" by algebra
ballarin@20318
   582
    finally
ballarin@20318
   583
    have "b = \<zero>" .
ballarin@20318
   584
    thus "a = \<zero> \<or> b = \<zero>" by simp
ballarin@20318
   585
  qed
wenzelm@23350
   586
qed (rule field_Units)
ballarin@20318
   587
ballarin@20318
   588
text {* Another variant to show that something is a field *}
ballarin@20318
   589
lemma (in cring) cring_fieldI2:
ballarin@20318
   590
  assumes notzero: "\<zero> \<noteq> \<one>"
ballarin@20318
   591
  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
   592
  shows "field R"
ballarin@20318
   593
  apply (rule cring_fieldI, simp add: Units_def)
ballarin@20318
   594
  apply (rule, clarsimp)
ballarin@20318
   595
  apply (simp add: notzero)
ballarin@20318
   596
proof (clarsimp)
ballarin@20318
   597
  fix x
ballarin@20318
   598
  assume xcarr: "x \<in> carrier R"
ballarin@20318
   599
    and "x \<noteq> \<zero>"
ballarin@20318
   600
  from this
ballarin@20318
   601
  have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
ballarin@20318
   602
  from this
ballarin@20318
   603
  obtain y
ballarin@20318
   604
    where ycarr: "y \<in> carrier R"
ballarin@20318
   605
    and xy: "x \<otimes> y = \<one>"
ballarin@20318
   606
    by fast
ballarin@20318
   607
  from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
ballarin@20318
   608
  from ycarr and this and xy
ballarin@20318
   609
  show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
ballarin@20318
   610
qed
ballarin@20318
   611
ballarin@20318
   612
ballarin@20318
   613
subsection {* Morphisms *}
ballarin@20318
   614
wenzelm@35847
   615
definition
ballarin@20318
   616
  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
wenzelm@35848
   617
  where "ring_hom R S =
wenzelm@35847
   618
    {h. h \<in> carrier R -> carrier S &
ballarin@20318
   619
      (ALL x y. x \<in> carrier R & y \<in> carrier R -->
wenzelm@35847
   620
        h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
wenzelm@35847
   621
      h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
ballarin@20318
   622
ballarin@20318
   623
lemma ring_hom_memI:
ballarin@20318
   624
  fixes R (structure) and S (structure)
ballarin@20318
   625
  assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
ballarin@20318
   626
    and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   627
      h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   628
    and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   629
      h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   630
    and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   631
  shows "h \<in> ring_hom R S"
ballarin@27714
   632
  by (auto simp add: ring_hom_def assms Pi_def)
ballarin@20318
   633
ballarin@20318
   634
lemma ring_hom_closed:
ballarin@20318
   635
  "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
ballarin@20318
   636
  by (auto simp add: ring_hom_def funcset_mem)
ballarin@20318
   637
ballarin@20318
   638
lemma ring_hom_mult:
ballarin@20318
   639
  fixes R (structure) and S (structure)
ballarin@20318
   640
  shows
ballarin@20318
   641
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   642
    h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
ballarin@20318
   643
    by (simp add: ring_hom_def)
ballarin@20318
   644
ballarin@20318
   645
lemma ring_hom_add:
ballarin@20318
   646
  fixes R (structure) and S (structure)
ballarin@20318
   647
  shows
ballarin@20318
   648
    "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
ballarin@20318
   649
    h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ballarin@20318
   650
    by (simp add: ring_hom_def)
ballarin@20318
   651
ballarin@20318
   652
lemma ring_hom_one:
ballarin@20318
   653
  fixes R (structure) and S (structure)
ballarin@20318
   654
  shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
ballarin@20318
   655
  by (simp add: ring_hom_def)
ballarin@20318
   656
ballarin@29237
   657
locale ring_hom_cring = R: cring R + S: cring S
ballarin@29237
   658
    for R (structure) and S (structure) +
ballarin@20318
   659
  fixes h
ballarin@20318
   660
  assumes homh [simp, intro]: "h \<in> ring_hom R S"
ballarin@20318
   661
  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
ballarin@20318
   662
    and hom_mult [simp] = ring_hom_mult [OF homh]
ballarin@20318
   663
    and hom_add [simp] = ring_hom_add [OF homh]
ballarin@20318
   664
    and hom_one [simp] = ring_hom_one [OF homh]
ballarin@20318
   665
ballarin@20318
   666
lemma (in ring_hom_cring) hom_zero [simp]:
ballarin@20318
   667
  "h \<zero> = \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   668
proof -
ballarin@20318
   669
  have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
ballarin@20318
   670
    by (simp add: hom_add [symmetric] del: hom_add)
ballarin@20318
   671
  then show ?thesis by (simp del: S.r_zero)
ballarin@20318
   672
qed
ballarin@20318
   673
ballarin@20318
   674
lemma (in ring_hom_cring) hom_a_inv [simp]:
ballarin@20318
   675
  "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
ballarin@20318
   676
proof -
ballarin@20318
   677
  assume R: "x \<in> carrier R"
ballarin@20318
   678
  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
   679
    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
ballarin@20318
   680
  with R show ?thesis by simp
ballarin@20318
   681
qed
ballarin@20318
   682
ballarin@20318
   683
lemma (in ring_hom_cring) hom_finsum [simp]:
ballarin@20318
   684
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   685
  h (finsum R f A) = finsum S (h o f) A"
berghofe@22265
   686
proof (induct set: finite)
ballarin@20318
   687
  case empty then show ?case by simp
ballarin@20318
   688
next
ballarin@20318
   689
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   690
qed
ballarin@20318
   691
ballarin@20318
   692
lemma (in ring_hom_cring) hom_finprod:
ballarin@20318
   693
  "[| finite A; f \<in> A -> carrier R |] ==>
ballarin@20318
   694
  h (finprod R f A) = finprod S (h o f) A"
berghofe@22265
   695
proof (induct set: finite)
ballarin@20318
   696
  case empty then show ?case by simp
ballarin@20318
   697
next
ballarin@20318
   698
  case insert then show ?case by (simp add: Pi_def)
ballarin@20318
   699
qed
ballarin@20318
   700
ballarin@20318
   701
declare ring_hom_cring.hom_finprod [simp]
ballarin@20318
   702
ballarin@20318
   703
lemma id_ring_hom [simp]:
ballarin@20318
   704
  "id \<in> ring_hom R R"
ballarin@20318
   705
  by (auto intro!: ring_hom_memI)
ballarin@20318
   706
ballarin@20318
   707
end