Theory Module

theory Module
imports Ring
(*  Title:      HOL/Algebra/Module.thy
    Author:     Clemens Ballarin, started 15 April 2003
    Copyright:  Clemens Ballarin
*)

theory Module
imports Ring
begin

section {* Modules over an Abelian Group *}

subsection {* Definitions *}

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

locale module = R: cring + M: abelian_group M for M (structure) +
  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 + 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:
  fixes R (structure) and M (structure)
  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 assms)

lemma algebraI:
  fixes R (structure) and M (structure)
  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"
apply intro_locales
apply (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms assms)+
apply (rule module_axioms.intro)
 apply (simp add: smult_closed)
 apply (simp add: smult_l_distr)
 apply (simp add: smult_r_distr)
 apply (simp add: smult_assoc1)
 apply (simp add: smult_one)
apply (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms assms)+
apply (rule algebra_axioms.intro)
 apply (simp add: smult_assoc2)
done

lemma (in algebra) R_cring:
  "cring R"
  ..

lemma (in algebra) M_cring:
  "cring M"
  ..

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 [OF R.zero_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" apply algebra apply algebra done
  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"
  from RM have a_smult: "a \<odot>M x ∈ carrier M" by simp
  from RM have ma_smult: "\<ominus>a \<odot>M x ∈ carrier M" by simp
  note facts = RM a_smult ma_smult
  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)"
    apply algebra apply algebra done
  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