src/HOL/Algebra/More_Finite_Product.thy
author wenzelm
Sat Nov 04 15:24:40 2017 +0100 (20 months ago)
changeset 67003 49850a679c2c
parent 66760 d44ea023ac09
child 67341 df79ef3b3a41
permissions -rw-r--r--
more robust sorted_entries;
     1 (*  Title:      HOL/Algebra/More_Finite_Product.thy
     2     Author:     Jeremy Avigad
     3 *)
     4 
     5 section \<open>More on finite products\<close>
     6 
     7 theory More_Finite_Product
     8   imports More_Group
     9 begin
    10 
    11 lemma (in comm_monoid) finprod_UN_disjoint:
    12   "finite I \<Longrightarrow> (\<forall>i\<in>I. finite (A i)) \<longrightarrow> (\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}) \<longrightarrow>
    13     (\<forall>i\<in>I. \<forall>x \<in> A i. g x \<in> carrier G) \<longrightarrow>
    14     finprod G g (UNION I A) = finprod G (\<lambda>i. finprod G g (A i)) I"
    15   apply (induct set: finite)
    16    apply force
    17   apply clarsimp
    18   apply (subst finprod_Un_disjoint)
    19        apply blast
    20       apply (erule finite_UN_I)
    21       apply blast
    22      apply (fastforce)
    23     apply (auto intro!: funcsetI finprod_closed)
    24   done
    25 
    26 lemma (in comm_monoid) finprod_Union_disjoint:
    27   "finite C \<Longrightarrow>
    28     \<forall>A\<in>C. finite A \<and> (\<forall>x\<in>A. f x \<in> carrier G) \<Longrightarrow>
    29     \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
    30     finprod G f (\<Union>C) = finprod G (finprod G f) C"
    31   apply (frule finprod_UN_disjoint [of C id f])
    32   apply auto
    33   done
    34 
    35 lemma (in comm_monoid) finprod_one: "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x = \<one>) \<Longrightarrow> finprod G f A = \<one>"
    36   by (induct set: finite) auto
    37 
    38 
    39 (* need better simplification rules for rings *)
    40 (* the next one holds more generally for abelian groups *)
    41 
    42 lemma (in cring) sum_zero_eq_neg: "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y"
    43   by (metis minus_equality)
    44 
    45 lemma (in domain) square_eq_one:
    46   fixes x
    47   assumes [simp]: "x \<in> carrier R"
    48     and "x \<otimes> x = \<one>"
    49   shows "x = \<one> \<or> x = \<ominus>\<one>"
    50 proof -
    51   have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>"
    52     by (simp add: ring_simprules)
    53   also from \<open>x \<otimes> x = \<one>\<close> have "\<dots> = \<zero>"
    54     by (simp add: ring_simprules)
    55   finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" .
    56   then have "(x \<oplus> \<one>) = \<zero> \<or> (x \<oplus> \<ominus> \<one>) = \<zero>"
    57     by (intro integral) auto
    58   then show ?thesis
    59     apply auto
    60      apply (erule notE)
    61      apply (rule sum_zero_eq_neg)
    62        apply auto
    63     apply (subgoal_tac "x = \<ominus> (\<ominus> \<one>)")
    64      apply (simp add: ring_simprules)
    65     apply (rule sum_zero_eq_neg)
    66       apply auto
    67     done
    68 qed
    69 
    70 lemma (in domain) inv_eq_self: "x \<in> Units R \<Longrightarrow> x = inv x \<Longrightarrow> x = \<one> \<or> x = \<ominus>\<one>"
    71   by (metis Units_closed Units_l_inv square_eq_one)
    72 
    73 
    74 text \<open>
    75   The following translates theorems about groups to the facts about
    76   the units of a ring. (The list should be expanded as more things are
    77   needed.)
    78 \<close>
    79 
    80 lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow> finite (Units R)"
    81   by (rule finite_subset) auto
    82 
    83 lemma (in monoid) units_of_pow:
    84   fixes n :: nat
    85   shows "x \<in> Units G \<Longrightarrow> x (^)\<^bsub>units_of G\<^esub> n = x (^)\<^bsub>G\<^esub> n"
    86   apply (induct n)
    87   apply (auto simp add: units_group group.is_monoid
    88     monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult)
    89   done
    90 
    91 lemma (in cring) units_power_order_eq_one:
    92   "finite (Units R) \<Longrightarrow> a \<in> Units R \<Longrightarrow> a (^) card(Units R) = \<one>"
    93   apply (subst units_of_carrier [symmetric])
    94   apply (subst units_of_one [symmetric])
    95   apply (subst units_of_pow [symmetric])
    96    apply assumption
    97   apply (rule comm_group.power_order_eq_one)
    98     apply (rule units_comm_group)
    99    apply (unfold units_of_def, auto)
   100   done
   101 
   102 end