Theory Module

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theory Module
imports CRing
begin

(*  Title:      HOL/Algebra/Module.thy
    ID:         $Id: Module.thy,v 1.10 2005/06/17 14:13:05 haftmann Exp $
    Author:     Clemens Ballarin, started 15 April 2003
    Copyright:  Clemens Ballarin
*)

header {* Modules over an Abelian Group *}

theory Module imports CRing begin

record ('a, 'b) module = "'b ring" +
  smult :: "['a, 'b] => 'b" (infixl "\<odot>\<index>" 70)

locale module = cring R + abelian_group M +
  assumes smult_closed [simp, intro]:
      "[| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M x ∈ carrier M"
    and smult_l_distr:
      "[| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊕ b) \<odot>M x = a \<odot>M x ⊕M b \<odot>M x"
    and smult_r_distr:
      "[| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |] ==>
      a \<odot>M (x ⊕M y) = a \<odot>M x ⊕M a \<odot>M y"
    and smult_assoc1:
      "[| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊗ b) \<odot>M x = a \<odot>M (b \<odot>M x)"
    and smult_one [simp]:
      "x ∈ carrier M ==> \<one> \<odot>M x = x"

locale algebra = module R M + cring M +
  assumes smult_assoc2:
      "[| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |] ==>
      (a \<odot>M x) ⊗M y = a \<odot>M (x ⊗M y)"

lemma moduleI:
  includes struct R + struct M
  assumes cring: "cring R"
    and abelian_group: "abelian_group M"
    and smult_closed:
      "!!a x. [| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M x ∈ carrier M"
    and smult_l_distr:
      "!!a b x. [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊕ b) \<odot>M x = (a \<odot>M x) ⊕M (b \<odot>M x)"
    and smult_r_distr:
      "!!a x y. [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |] ==>
      a \<odot>M (x ⊕M y) = (a \<odot>M x) ⊕M (a \<odot>M y)"
    and smult_assoc1:
      "!!a b x. [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊗ b) \<odot>M x = a \<odot>M (b \<odot>M x)"
    and smult_one:
      "!!x. x ∈ carrier M ==> \<one> \<odot>M x = x"
  shows "module R M"
  by (auto intro: module.intro cring.axioms abelian_group.axioms
    module_axioms.intro prems)

lemma algebraI:
  includes struct R + struct M
  assumes R_cring: "cring R"
    and M_cring: "cring M"
    and smult_closed:
      "!!a x. [| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M x ∈ carrier M"
    and smult_l_distr:
      "!!a b x. [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊕ b) \<odot>M x = (a \<odot>M x) ⊕M (b \<odot>M x)"
    and smult_r_distr:
      "!!a x y. [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |] ==>
      a \<odot>M (x ⊕M y) = (a \<odot>M x) ⊕M (a \<odot>M y)"
    and smult_assoc1:
      "!!a b x. [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |] ==>
      (a ⊗ b) \<odot>M x = a \<odot>M (b \<odot>M x)"
    and smult_one:
      "!!x. x ∈ carrier M ==> (one R) \<odot>M x = x"
    and smult_assoc2:
      "!!a x y. [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |] ==>
      (a \<odot>M x) ⊗M y = a \<odot>M (x ⊗M y)"
  shows "algebra R M"
  by (auto intro!: algebra.intro algebra_axioms.intro cring.axioms 
    module_axioms.intro prems)

lemma (in algebra) R_cring:
  "cring R"
  by (rule cring.intro)

lemma (in algebra) M_cring:
  "cring M"
  by (rule cring.intro)

lemma (in algebra) module:
  "module R M"
  by (auto intro: moduleI R_cring is_abelian_group
    smult_l_distr smult_r_distr smult_assoc1)


subsection {* Basic Properties of Algebras *}

lemma (in algebra) smult_l_null [simp]:
  "x ∈ carrier M ==> \<zero> \<odot>M x = \<zero>M"
proof -
  assume M: "x ∈ carrier M"
  note facts = M smult_closed
  from facts have "\<zero> \<odot>M x = (\<zero> \<odot>M x ⊕M \<zero> \<odot>M x) ⊕M \<ominus>M (\<zero> \<odot>M x)" by algebra
  also from M have "... = (\<zero> ⊕ \<zero>) \<odot>M x ⊕M \<ominus>M (\<zero> \<odot>M x)"
    by (simp add: smult_l_distr del: R.l_zero R.r_zero)
  also from facts have "... = \<zero>M" by algebra
  finally show ?thesis .
qed

lemma (in algebra) smult_r_null [simp]:
  "a ∈ carrier R ==> a \<odot>M \<zero>M = \<zero>M";
proof -
  assume R: "a ∈ carrier R"
  note facts = R smult_closed
  from facts have "a \<odot>M \<zero>M = (a \<odot>M \<zero>MM a \<odot>M \<zero>M) ⊕M \<ominus>M (a \<odot>M \<zero>M)"
    by algebra
  also from R have "... = a \<odot>M (\<zero>MM \<zero>M) ⊕M \<ominus>M (a \<odot>M \<zero>M)"
    by (simp add: smult_r_distr del: M.l_zero M.r_zero)
  also from facts have "... = \<zero>M" by algebra
  finally show ?thesis .
qed

lemma (in algebra) smult_l_minus:
  "[| a ∈ carrier R; x ∈ carrier M |] ==> (\<ominus>a) \<odot>M x = \<ominus>M (a \<odot>M x)"
proof -
  assume RM: "a ∈ carrier R" "x ∈ carrier M"
  note facts = RM smult_closed
  from facts have "(\<ominus>a) \<odot>M x = (\<ominus>a \<odot>M x ⊕M a \<odot>M x) ⊕M \<ominus>M(a \<odot>M x)"
    by algebra
  also from RM have "... = (\<ominus>a ⊕ a) \<odot>M x ⊕M \<ominus>M(a \<odot>M x)"
    by (simp add: smult_l_distr)
  also from facts smult_l_null have "... = \<ominus>M(a \<odot>M x)" by algebra
  finally show ?thesis .
qed

lemma (in algebra) smult_r_minus:
  "[| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M (\<ominus>Mx) = \<ominus>M (a \<odot>M x)"
proof -
  assume RM: "a ∈ carrier R" "x ∈ carrier M"
  note facts = RM smult_closed
  from facts have "a \<odot>M (\<ominus>Mx) = (a \<odot>M \<ominus>Mx ⊕M a \<odot>M x) ⊕M \<ominus>M(a \<odot>M x)"
    by algebra
  also from RM have "... = a \<odot>M (\<ominus>Mx ⊕M x) ⊕M \<ominus>M(a \<odot>M x)"
    by (simp add: smult_r_distr)
  also from facts smult_r_null have "... = \<ominus>M(a \<odot>M x)" by algebra
  finally show ?thesis .
qed

end

lemma moduleI:

  [| cring R; abelian_group M;
     !!a x. [| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M x ∈ carrier M;
     !!a b x.
        [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
        ==> (aR b) \<odot>M x = a \<odot>M xM b \<odot>M x;
     !!a x y.
        [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |]
        ==> a \<odot>M (xM y) = a \<odot>M xM a \<odot>M y;
     !!a b x.
        [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
        ==> aR b \<odot>M x = a \<odot>M (b \<odot>M x);
     !!x. x ∈ carrier M ==> \<one>R \<odot>M x = x |]
  ==> module R M

lemma algebraI:

  [| cring R; cring M;
     !!a x. [| a ∈ carrier R; x ∈ carrier M |] ==> a \<odot>M x ∈ carrier M;
     !!a b x.
        [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
        ==> (aR b) \<odot>M x = a \<odot>M xM b \<odot>M x;
     !!a x y.
        [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |]
        ==> a \<odot>M (xM y) = a \<odot>M xM a \<odot>M y;
     !!a b x.
        [| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
        ==> aR b \<odot>M x = a \<odot>M (b \<odot>M x);
     !!x. x ∈ carrier M ==> \<one>R \<odot>M x = x;
     !!a x y.
        [| a ∈ carrier R; x ∈ carrier M; y ∈ carrier M |]
        ==> a \<odot>M xM y = a \<odot>M (xM y) |]
  ==> algebra R M

lemma R_cring:

  algebra R M ==> cring R

lemma M_cring:

  algebra R M ==> cring M

lemma module:

  algebra R M ==> module R M

Basic Properties of Algebras

lemma smult_l_null:

  [| algebra R M; x ∈ carrier M |] ==> \<zero>R \<odot>M x = \<zero>M

lemma smult_r_null:

  [| algebra R M; a ∈ carrier R |] ==> a \<odot>M \<zero>M = \<zero>M

lemma smult_l_minus:

  [| algebra R M; a ∈ carrier R; x ∈ carrier M |]
  ==> \<ominus>R a \<odot>M x = \<ominus>M (a \<odot>M x)

lemma smult_r_minus:

  [| algebra R M; a ∈ carrier R; x ∈ carrier M |]
  ==> a \<odot>M \<ominus>M x = \<ominus>M (a \<odot>M x)