(* Title: HOL/Algebra/Module.thy
Author: Clemens Ballarin, started 15 April 2003
Copyright: Clemens Ballarin
*)
theory Module
imports Ring
begin
section \<open>Modules over an Abelian Group\<close>
subsection \<open>Definitions\<close>
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 \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
and smult_l_distr:
"[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<oplus> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"
and smult_r_distr:
"[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"
and smult_assoc1:
"[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<otimes> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
and smult_one [simp]:
"x \<in> carrier M ==> \<one> \<odot>\<^bsub>M\<^esub> x = x"
locale algebra = module + cring M +
assumes smult_assoc2:
"[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
(a \<odot>\<^bsub>M\<^esub> x) \<otimes>\<^bsub>M\<^esub> y = a \<odot>\<^bsub>M\<^esub> (x \<otimes>\<^bsub>M\<^esub> 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 \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
and smult_l_distr:
"!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<oplus> b) \<odot>\<^bsub>M\<^esub> x = (a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
and smult_r_distr:
"!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = (a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> y)"
and smult_assoc1:
"!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<otimes> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
and smult_one:
"!!x. x \<in> carrier M ==> \<one> \<odot>\<^bsub>M\<^esub> 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 \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
and smult_l_distr:
"!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<oplus> b) \<odot>\<^bsub>M\<^esub> x = (a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
and smult_r_distr:
"!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = (a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> y)"
and smult_assoc1:
"!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
(a \<otimes> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
and smult_one:
"!!x. x \<in> carrier M ==> (one R) \<odot>\<^bsub>M\<^esub> x = x"
and smult_assoc2:
"!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
(a \<odot>\<^bsub>M\<^esub> x) \<otimes>\<^bsub>M\<^esub> y = a \<odot>\<^bsub>M\<^esub> (x \<otimes>\<^bsub>M\<^esub> 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 \<open>Basic Properties of Modules\<close>
lemma (in module) smult_l_null [simp]:
"x \<in> carrier M ==> \<zero> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
proof-
assume M : "x \<in> carrier M"
note facts = M smult_closed [OF R.zero_closed]
from facts have "\<zero> \<odot>\<^bsub>M\<^esub> x = (\<zero> \<oplus> \<zero>) \<odot>\<^bsub>M\<^esub> x "
using smult_l_distr by auto
also have "... = \<zero> \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> \<zero> \<odot>\<^bsub>M\<^esub> x"
using smult_l_distr[of \<zero> \<zero> x] facts by auto
finally show "\<zero> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>" using facts
by (metis M.add.r_cancel_one')
qed
lemma (in module) smult_r_null [simp]:
"a \<in> carrier R ==> a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
proof -
assume R: "a \<in> carrier R"
note facts = R smult_closed
from facts have "a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = (a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub>) \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub>)"
by (simp add: M.add.inv_solve_right)
also from R have "... = a \<odot>\<^bsub>M\<^esub> (\<zero>\<^bsub>M\<^esub> \<oplus>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub>) \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub>)"
by (simp add: smult_r_distr del: M.l_zero M.r_zero)
also from facts have "... = \<zero>\<^bsub>M\<^esub>"
by (simp add: M.r_neg)
finally show ?thesis .
qed
lemma (in module) smult_l_minus:
"\<lbrakk> a \<in> carrier R; x \<in> carrier M \<rbrakk> \<Longrightarrow> (\<ominus>a) \<odot>\<^bsub>M\<^esub> x = \<ominus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> x)"
proof-
assume RM: "a \<in> carrier R" "x \<in> carrier M"
from RM have a_smult: "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M" by simp
from RM have ma_smult: "\<ominus>a \<odot>\<^bsub>M\<^esub> x \<in> carrier M" by simp
note facts = RM a_smult ma_smult
from facts have "(\<ominus>a) \<odot>\<^bsub>M\<^esub> x = (\<ominus>a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (simp add: M.add.inv_solve_right)
also from RM have "... = (\<ominus>a \<oplus> a) \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (simp add: smult_l_distr)
also from facts smult_l_null have "... = \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (simp add: R.l_neg)
finally show ?thesis .
qed
lemma (in module) smult_r_minus:
"[| a \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^bsub>M\<^esub> (\<ominus>\<^bsub>M\<^esub>x) = \<ominus>\<^bsub>M\<^esub> (a \<odot>\<^bsub>M\<^esub> x)"
proof -
assume RM: "a \<in> carrier R" "x \<in> carrier M"
note facts = RM smult_closed
from facts have "a \<odot>\<^bsub>M\<^esub> (\<ominus>\<^bsub>M\<^esub>x) = (a \<odot>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub>x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (simp add: M.add.inv_solve_right)
also from RM have "... = a \<odot>\<^bsub>M\<^esub> (\<ominus>\<^bsub>M\<^esub>x \<oplus>\<^bsub>M\<^esub> x) \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (simp add: smult_r_distr)
also from facts smult_l_null have "... = \<ominus>\<^bsub>M\<^esub>(a \<odot>\<^bsub>M\<^esub> x)"
by (metis M.add.inv_closed M.add.inv_solve_right M.l_neg R.zero_closed r_null smult_assoc1)
finally show ?thesis .
qed
lemma (in module) finsum_smult_ldistr:
"\<lbrakk> finite A; a \<in> carrier R; f: A \<rightarrow> carrier M \<rbrakk> \<Longrightarrow>
a \<odot>\<^bsub>M\<^esub> (\<Oplus>\<^bsub>M\<^esub> i \<in> A. (f i)) = (\<Oplus>\<^bsub>M\<^esub> i \<in> A. ( a \<odot>\<^bsub>M\<^esub> (f i)))"
proof (induct set: finite)
case empty then show ?case
by (metis M.finsum_empty M.zero_closed R.zero_closed r_null smult_assoc1 smult_l_null)
next
case (insert x F) then show ?case
by (simp add: Pi_def smult_r_distr)
qed
end