(* Title: HOL/Algebra/Module.thy ID: $Id: Module.thy,v 1.15 2006/08/03 12:58:13 ballarin Exp $ 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 = 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: 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 prems) 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 prems)+ 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 prems)+ apply (rule algebra_axioms.intro) apply (simp add: smult_assoc2) done lemma (in algebra) R_cring: "cring R" by unfold_locales lemma (in algebra) M_cring: "cring M" by unfold_locales 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>M ⊕M a \<odot>M \<zero>M) ⊕M \<ominus>M (a \<odot>M \<zero>M)" by algebra also from R have "... = a \<odot>M (\<zero>M ⊕M \<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
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 |]
==> (a ⊕R b) \<odot>M x = a \<odot>M x ⊕M b \<odot>M x;
!!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;
!!a b x.
[| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
==> a ⊗R 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 |]
==> (a ⊕R b) \<odot>M x = a \<odot>M x ⊕M b \<odot>M x;
!!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;
!!a b x.
[| a ∈ carrier R; b ∈ carrier R; x ∈ carrier M |]
==> a ⊗R 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 x ⊗M y = a \<odot>M (x ⊗M y) |]
==> algebra R M
lemma R_cring:
cring R
lemma M_cring:
cring M
lemma module:
module R M
lemma smult_l_null:
x ∈ carrier M ==> \<zero> \<odot>M x = \<zero>M
lemma smult_r_null:
a ∈ carrier R ==> a \<odot>M \<zero>M = \<zero>M
lemma smult_l_minus:
[| a ∈ carrier R; x ∈ carrier M |]
==> \<ominus> a \<odot>M x = \<ominus>M (a \<odot>M x)
lemma smult_r_minus:
[| a ∈ carrier R; x ∈ carrier M |]
==> a \<odot>M \<ominus>M x = \<ominus>M (a \<odot>M x)