src/HOL/Algebra/Module.thy
author ballarin
Fri May 02 20:02:50 2003 +0200 (2003-05-02)
changeset 13949 0ce528cd6f19
parent 13940 c67798653056
child 13952 6206d3e7672a
permissions -rw-r--r--
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
     1 (*  Title:      HOL/Algebra/Module
     2     ID:         $Id$
     3     Author:     Clemens Ballarin, started 15 April 2003
     4     Copyright:  Clemens Ballarin
     5 *)
     6 
     7 theory Module = CRing:
     8 
     9 section {* Modules over an Abelian Group *}
    10 
    11 record ('a, 'b) module = "'b ring" +
    12   smult :: "['a, 'b] => 'b" (infixl "\<odot>\<index>" 70)
    13 
    14 locale module = cring R + abelian_group M +
    15   assumes smult_closed [simp, intro]:
    16       "[| a \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^sub>2 x \<in> carrier M"
    17     and smult_l_distr:
    18       "[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    19       (a \<oplus> b) \<odot>\<^sub>2 x = a \<odot>\<^sub>2 x \<oplus>\<^sub>2 b \<odot>\<^sub>2 x"
    20     and smult_r_distr:
    21       "[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
    22       a \<odot>\<^sub>2 (x \<oplus>\<^sub>2 y) = a \<odot>\<^sub>2 x \<oplus>\<^sub>2 a \<odot>\<^sub>2 y"
    23     and smult_assoc1:
    24       "[| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    25       (a \<otimes> b) \<odot>\<^sub>2 x = a \<odot>\<^sub>2 (b \<odot>\<^sub>2 x)"
    26     and smult_one [simp]:
    27       "x \<in> carrier M ==> \<one> \<odot>\<^sub>2 x = x"
    28 
    29 locale algebra = module R M + cring M +
    30   assumes smult_assoc2:
    31       "[| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
    32       (a \<odot>\<^sub>2 x) \<otimes>\<^sub>2 y = a \<odot>\<^sub>2 (x \<otimes>\<^sub>2 y)"
    33 
    34 lemma moduleI:
    35   assumes cring: "cring R"
    36     and abelian_group: "abelian_group M"
    37     and smult_closed:
    38       "!!a x. [| a \<in> carrier R; x \<in> carrier M |] ==> smult M a x \<in> carrier M"
    39     and smult_l_distr:
    40       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    41       smult M (add R a b) x = add M (smult M a x) (smult M b x)"
    42     and smult_r_distr:
    43       "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
    44       smult M a (add M x y) = add M (smult M a x) (smult M a y)"
    45     and smult_assoc1:
    46       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    47       smult M (mult R a b) x = smult M a (smult M b x)"
    48     and smult_one:
    49       "!!x. x \<in> carrier M ==> smult M (one R) x = x"
    50   shows "module R M"
    51   by (auto intro: module.intro cring.axioms abelian_group.axioms
    52     module_axioms.intro prems)
    53 
    54 lemma algebraI:
    55   assumes R_cring: "cring R"
    56     and M_cring: "cring M"
    57     and smult_closed:
    58       "!!a x. [| a \<in> carrier R; x \<in> carrier M |] ==> smult M a x \<in> carrier M"
    59     and smult_l_distr:
    60       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    61       smult M (add R a b) x = add M (smult M a x) (smult M b x)"
    62     and smult_r_distr:
    63       "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
    64       smult M a (add M x y) = add M (smult M a x) (smult M a y)"
    65     and smult_assoc1:
    66       "!!a b x. [| a \<in> carrier R; b \<in> carrier R; x \<in> carrier M |] ==>
    67       smult M (mult R a b) x = smult M a (smult M b x)"
    68     and smult_one:
    69       "!!x. x \<in> carrier M ==> smult M (one R) x = x"
    70     and smult_assoc2:
    71       "!!a x y. [| a \<in> carrier R; x \<in> carrier M; y \<in> carrier M |] ==>
    72       mult M (smult M a x) y = smult M a (mult M x y)"
    73   shows "algebra R M"
    74   by (auto intro!: algebra.intro algebra_axioms.intro cring.axioms 
    75     module_axioms.intro prems)
    76 
    77 lemma (in algebra) R_cring:
    78   "cring R"
    79   by (rule cring.intro) assumption+
    80 
    81 lemma (in algebra) M_cring:
    82   "cring M"
    83   by (rule cring.intro) assumption+
    84 
    85 lemma (in algebra) module:
    86   "module R M"
    87   by (auto intro: moduleI R_cring is_abelian_group
    88     smult_l_distr smult_r_distr smult_assoc1)
    89 
    90 subsection {* Basic Properties of Algebras *}
    91 
    92 lemma (in algebra) smult_l_null [simp]:
    93   "x \<in> carrier M ==> \<zero> \<odot>\<^sub>2 x = \<zero>\<^sub>2"
    94 proof -
    95   assume M: "x \<in> carrier M"
    96   note facts = M smult_closed
    97   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
    98   also from M have "... = (\<zero> \<oplus> \<zero>) \<odot>\<^sub>2 x \<oplus>\<^sub>2 \<ominus>\<^sub>2 (\<zero> \<odot>\<^sub>2 x)"
    99     by (simp add: smult_l_distr del: R.l_zero R.r_zero)
   100   also from facts have "... = \<zero>\<^sub>2" by algebra
   101   finally show ?thesis .
   102 qed
   103 
   104 lemma (in algebra) smult_r_null [simp]:
   105   "a \<in> carrier R ==> a \<odot>\<^sub>2 \<zero>\<^sub>2 = \<zero>\<^sub>2";
   106 proof -
   107   assume R: "a \<in> carrier R"
   108   note facts = R smult_closed
   109   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)"
   110     by algebra
   111   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)"
   112     by (simp add: smult_r_distr del: M.l_zero M.r_zero)
   113   also from facts have "... = \<zero>\<^sub>2" by algebra
   114   finally show ?thesis .
   115 qed
   116 
   117 lemma (in algebra) smult_l_minus:
   118   "[| a \<in> carrier R; x \<in> carrier M |] ==> (\<ominus>a) \<odot>\<^sub>2 x = \<ominus>\<^sub>2 (a \<odot>\<^sub>2 x)"
   119 proof -
   120   assume RM: "a \<in> carrier R" "x \<in> carrier M"
   121   note facts = RM smult_closed
   122   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
   123   also from RM have "... = (\<ominus>a \<oplus> a) \<odot>\<^sub>2 x \<oplus>\<^sub>2 \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)"
   124     by (simp add: smult_l_distr)
   125   also from facts smult_l_null have "... = \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)" by algebra
   126   finally show ?thesis .
   127 qed
   128 
   129 lemma (in algebra) smult_r_minus:
   130   "[| a \<in> carrier R; x \<in> carrier M |] ==> a \<odot>\<^sub>2 (\<ominus>\<^sub>2x) = \<ominus>\<^sub>2 (a \<odot>\<^sub>2 x)"
   131 proof -
   132   assume RM: "a \<in> carrier R" "x \<in> carrier M"
   133   note facts = RM smult_closed
   134   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)"
   135     by algebra
   136   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)"
   137     by (simp add: smult_r_distr)
   138   also from facts smult_r_null have "... = \<ominus>\<^sub>2(a \<odot>\<^sub>2 x)" by algebra
   139   finally show ?thesis .
   140 qed
   141 
   142 subsection {* Every Abelian Group is a $\mathbb{Z}$-module *}
   143 
   144 text {* Not finished. *}
   145 
   146 constdefs
   147   INTEG :: "int ring"
   148   "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
   149 
   150 (*
   151   INTEG_MODULE :: "('a, 'm) ring_scheme => (int, 'a) module"
   152   "INTEG_MODULE R == (| carrier = carrier R, mult = mult R, one = one R,
   153     zero = zero R, add = add R, smult = (%n. %x:carrier R. ... ) |)"
   154 *)
   155 
   156 lemma cring_INTEG:
   157   "cring INTEG"
   158   by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
   159     zadd_zminus_inverse2 zadd_zmult_distrib)
   160 
   161 end