src/HOL/Algebra/Module.thy
 changeset 13936 d3671b878828 child 13940 c67798653056
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Algebra/Module.thy	Wed Apr 30 10:01:35 2003 +0200
1.3 @@ -0,0 +1,161 @@
1.4 +(*  Title:      HOL/Algebra/Module
1.5 +    ID:         \$Id\$
1.6 +    Author:     Clemens Ballarin, started 15 April 2003
1.8 +*)
1.9 +
1.10 +theory Module = CRing:
1.11 +
1.12 +section {* Modules over an Abelian Group *}
1.13 +
1.14 +record ('a, 'b) module = "'b ring" +
1.15 +  smult :: "['a, 'b] => 'b" (infixl "\<odot>\<index>" 70)
1.16 +
1.17 +locale module = cring R + abelian_group M +
1.18 +  assumes smult_closed [simp, intro]:
1.19 +      "[| a \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^sub>2 x \<in> carrier M"
1.20 +    and smult_l_distr:
1.21 +      "[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.22 +      (a \<oplus> b) \<odot>\<^sub>2 x = a \<odot>\<^sub>2 x \<oplus>\<^sub>2 b \<odot>\<^sub>2 x"
1.23 +    and smult_r_distr:
1.24 +      "[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
1.25 +      a \<odot>\<^sub>2 (x \<oplus>\<^sub>2 y) = a \<odot>\<^sub>2 x \<oplus>\<^sub>2 a \<odot>\<^sub>2 y"
1.26 +    and smult_assoc1:
1.27 +      "[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.28 +      (a \<otimes> b) \<odot>\<^sub>2 x = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 x)"
1.29 +    and smult_one [simp]:
1.30 +      "x \<in> carrier M ==> \<one> \<odot>\<^sub>2 x = x"
1.31 +
1.32 +locale algebra = module R M + cring M +
1.33 +  assumes smult_assoc2:
1.34 +      "[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
1.35 +      (a \<odot>\<^sub>2 x) \<otimes>\<^sub>2 y = a \<odot>\<^sub>2 (x \<otimes>\<^sub>2 y)"
1.36 +
1.37 +lemma moduleI:
1.38 +  assumes cring: "cring R"
1.39 +    and abelian_group: "abelian_group M"
1.40 +    and smult_closed:
1.41 +      "!!a x. [| a \<in> carrier R; x \<in> carrier M |] ==> smult M a x \<in> carrier M"
1.42 +    and smult_l_distr:
1.43 +      "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.44 +      smult M (add R a b) x = add M (smult M a x) (smult M b x)"
1.45 +    and smult_r_distr:
1.46 +      "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
1.47 +      smult M a (add M x y) = add M (smult M a x) (smult M a y)"
1.48 +    and smult_assoc1:
1.49 +      "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.50 +      smult M (mult R a b) x = smult M a (smult M b x)"
1.51 +    and smult_one:
1.52 +      "!!x. x \<in> carrier M ==> smult M (one R) x = x"
1.53 +  shows "module R M"
1.54 +  by (auto intro: module.intro cring.axioms abelian_group.axioms
1.55 +    module_axioms.intro prems)
1.56 +
1.57 +lemma algebraI:
1.58 +  assumes R_cring: "cring R"
1.59 +    and M_cring: "cring M"
1.60 +    and smult_closed:
1.61 +      "!!a x. [| a \<in> carrier R; x \<in> carrier M |] ==> smult M a x \<in> carrier M"
1.62 +    and smult_l_distr:
1.63 +      "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.64 +      smult M (add R a b) x = add M (smult M a x) (smult M b x)"
1.65 +    and smult_r_distr:
1.66 +      "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
1.67 +      smult M a (add M x y) = add M (smult M a x) (smult M a y)"
1.68 +    and smult_assoc1:
1.69 +      "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
1.70 +      smult M (mult R a b) x = smult M a (smult M b x)"
1.71 +    and smult_one:
1.72 +      "!!x. x \<in> carrier M ==> smult M (one R) x = x"
1.73 +    and smult_assoc2:
1.74 +      "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
1.75 +      mult M (smult M a x) y = smult M a (mult M x y)"
1.76 +  shows "algebra R M"
1.77 +  by (auto intro!: algebra.intro algebra_axioms.intro cring.axioms
1.78 +    module_axioms.intro prems)
1.79 +
1.80 +lemma (in algebra) R_cring:
1.81 +  "cring R"
1.82 +  by (rule cring.intro) assumption+
1.83 +
1.84 +lemma (in algebra) M_cring:
1.85 +  "cring M"
1.86 +  by (rule cring.intro) assumption+
1.87 +
1.88 +lemma (in algebra) module:
1.89 +  "module R M"
1.90 +  by (auto intro: moduleI R_cring is_abelian_group
1.91 +    smult_l_distr smult_r_distr smult_assoc1)
1.92 +
1.93 +subsection {* Basic Properties of Algebras *}
1.94 +
1.95 +lemma (in algebra) smult_l_null [simp]:
1.96 +  "x \<in> carrier M ==> \<zero> \<odot>\<^sub>2 x = \<zero>\<^sub>2"
1.97 +proof -
1.98 +  assume M: "x \<in> carrier M"
1.99 +  note facts = M smult_closed
1.100 +  from facts have "\<zero> \<odot>\<^sub>2 x = (\<zero> \<odot>\<^sub>2 x \<oplus>\<^sub>2 \<zero> \<odot>\<^sub>2 x) \<oplus>\<^sub>2 \<ominus>\<^sub>2 (\<zero> \<odot>\<^sub>2 x)" by algebra
1.101 +  also from M have "... = (\<zero> \<oplus> \<zero>) \<odot>\<^sub>2 x \<oplus>\<^sub>2 \<ominus>\<^sub>2 (\<zero> \<odot>\<^sub>2 x)"
1.102 +    by (simp add: smult_l_distr del: R.l_zero R.r_zero)
1.103 +  also from facts have "... = \<zero>\<^sub>2" by algebra
1.104 +  finally show ?thesis .
1.105 +qed
1.106 +
1.107 +lemma (in algebra) smult_r_null [simp]:
1.108 +  "a \<in> carrier R ==> a \<odot>\<^sub>2 \<zero>\<^sub>2 = \<zero>\<^sub>2";
1.109 +proof -
1.110 +  assume R: "a \<in> carrier R"
1.111 +  note facts = R smult_closed
1.112 +  from facts have "a \<odot>\<^sub>2 \<zero>\<^sub>2 = (a \<odot>\<^sub>2 \<zero>\<^sub>2 \<oplus>\<^sub>2 a \<odot>\<^sub>2 \<zero>\<^sub>2) \<oplus>\<^sub>2 \<ominus>\<^sub>2 (a \<odot>\<^sub>2 \<zero>\<^sub>2)"
1.113 +    by algebra
1.114 +  also from R have "... = a \<odot>\<^sub>2 (\<zero>\<^sub>2 \<oplus>\<^sub>2 \<zero>\<^sub>2) \<oplus>\<^sub>2 \<ominus>\<^sub>2 (a \<odot>\<^sub>2 \<zero>\<^sub>2)"
1.115 +    by (simp add: smult_r_distr del: M.l_zero M.r_zero)
1.116 +  also from facts have "... = \<zero>\<^sub>2" by algebra
1.117 +  finally show ?thesis .
1.118 +qed
1.119 +
1.120 +lemma (in algebra) smult_l_minus:
1.121 +  "[| a \<in> carrier R; x \<in> carrier M |] ==> (\<ominus>a) \<odot>\<^sub>2 x = \<ominus>\<^sub>2 (a \<odot>\<^sub>2 x)"
1.122 +proof -
1.123 +  assume RM: "a \<in> carrier R" "x \<in> carrier M"
1.124 +  note facts = RM smult_closed
1.125 +  from facts have "(\<ominus>a) \<odot>\<^sub>2 x = (\<ominus>a \<odot>\<^sub>2 x \<oplus>\<^sub>2 a \<odot>\<^sub>2 x) \<oplus>\<^sub>2 \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)" by algebra
1.126 +  also from RM have "... = (\<ominus>a \<oplus> a) \<odot>\<^sub>2 x \<oplus>\<^sub>2 \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)"
1.127 +    by (simp add: smult_l_distr)
1.128 +  also from facts smult_l_null have "... = \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)" by algebra
1.129 +  finally show ?thesis .
1.130 +qed
1.131 +
1.132 +lemma (in algebra) smult_r_minus:
1.133 +  "[| a \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^sub>2 (\<ominus>\<^sub>2x) = \<ominus>\<^sub>2 (a \<odot>\<^sub>2 x)"
1.134 +proof -
1.135 +  assume RM: "a \<in> carrier R" "x \<in> carrier M"
1.136 +  note facts = RM smult_closed
1.137 +  from facts have "a \<odot>\<^sub>2 (\<ominus>\<^sub>2x) = (a \<odot>\<^sub>2 \<ominus>\<^sub>2x \<oplus>\<^sub>2 a \<odot>\<^sub>2 x) \<oplus>\<^sub>2 \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)"
1.138 +    by algebra
1.139 +  also from RM have "... = a \<odot>\<^sub>2 (\<ominus>\<^sub>2x \<oplus>\<^sub>2 x) \<oplus>\<^sub>2 \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)"
1.140 +    by (simp add: smult_r_distr)
1.141 +  also from facts smult_r_null have "... = \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)" by algebra
1.142 +  finally show ?thesis .
1.143 +qed
1.144 +
1.145 +subsection {* Every Abelian Group is a \$\mathbb{Z}\$-module *}
1.146 +
1.147 +text {* Not finished. *}
1.148 +
1.149 +constdefs
1.150 +  INTEG :: "int ring"
1.151 +  "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1.152 +
1.153 +(*
1.154 +  INTEG_MODULE :: "('a, 'm) ring_scheme => (int, 'a) module"
1.155 +  "INTEG_MODULE R == (| carrier = carrier R, mult = mult R, one = one R,
1.156 +    zero = zero R, add = add R, smult = (%n. %x:carrier R. ... ) |)"
1.157 +*)
1.158 +
1.159 +lemma cring_INTEG:
1.160 +  "cring INTEG"
1.161 +  by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI