src/HOL/Equiv_Relations.thy
changeset 23705 315c638d5856
parent 21749 3f0e86c92ff3
child 24728 e2b3a1065676
     1.1 --- a/src/HOL/Equiv_Relations.thy	Tue Jul 10 16:46:37 2007 +0200
     1.2 +++ b/src/HOL/Equiv_Relations.thy	Tue Jul 10 17:30:43 2007 +0200
     1.3 @@ -6,7 +6,7 @@
     1.4  header {* Equivalence Relations in Higher-Order Set Theory *}
     1.5  
     1.6  theory Equiv_Relations
     1.7 -imports Relation Finite_Set
     1.8 +imports Relation
     1.9  begin
    1.10  
    1.11  subsection {* Equivalence relations *}
    1.12 @@ -292,83 +292,4 @@
    1.13           erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
    1.14    done
    1.15  
    1.16 -
    1.17 -subsection {* Cardinality results *}
    1.18 -
    1.19 -text {*Suggested by Florian Kammüller*}
    1.20 -
    1.21 -lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
    1.22 -  -- {* recall @{thm equiv_type} *}
    1.23 -  apply (rule finite_subset)
    1.24 -   apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
    1.25 -  apply (unfold quotient_def)
    1.26 -  apply blast
    1.27 -  done
    1.28 -
    1.29 -lemma finite_equiv_class:
    1.30 -  "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
    1.31 -  apply (unfold quotient_def)
    1.32 -  apply (rule finite_subset)
    1.33 -   prefer 2 apply assumption
    1.34 -  apply blast
    1.35 -  done
    1.36 -
    1.37 -lemma equiv_imp_dvd_card:
    1.38 -  "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
    1.39 -    ==> k dvd card A"
    1.40 -  apply (rule Union_quotient [THEN subst])
    1.41 -   apply assumption
    1.42 -  apply (rule dvd_partition)
    1.43 -     prefer 3 apply (blast dest: quotient_disj)
    1.44 -    apply (simp_all add: Union_quotient equiv_type)
    1.45 -  done
    1.46 -
    1.47 -lemma card_quotient_disjoint:
    1.48 - "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
    1.49 -apply(simp add:quotient_def)
    1.50 -apply(subst card_UN_disjoint)
    1.51 -   apply assumption
    1.52 -  apply simp
    1.53 - apply(fastsimp simp add:inj_on_def)
    1.54 -apply (simp add:setsum_constant)
    1.55 -done
    1.56 -(*
    1.57 -ML
    1.58 -{*
    1.59 -val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
    1.60 -val UN_constant_eq = thm "UN_constant_eq";
    1.61 -val UN_equiv_class = thm "UN_equiv_class";
    1.62 -val UN_equiv_class2 = thm "UN_equiv_class2";
    1.63 -val UN_equiv_class_inject = thm "UN_equiv_class_inject";
    1.64 -val UN_equiv_class_type = thm "UN_equiv_class_type";
    1.65 -val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
    1.66 -val Union_quotient = thm "Union_quotient";
    1.67 -val comp_equivI = thm "comp_equivI";
    1.68 -val congruent2I = thm "congruent2I";
    1.69 -val congruent2_commuteI = thm "congruent2_commuteI";
    1.70 -val congruent2_def = thm "congruent2_def";
    1.71 -val congruent2_implies_congruent = thm "congruent2_implies_congruent";
    1.72 -val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
    1.73 -val congruent_def = thm "congruent_def";
    1.74 -val eq_equiv_class = thm "eq_equiv_class";
    1.75 -val eq_equiv_class_iff = thm "eq_equiv_class_iff";
    1.76 -val equiv_class_eq = thm "equiv_class_eq";
    1.77 -val equiv_class_eq_iff = thm "equiv_class_eq_iff";
    1.78 -val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
    1.79 -val equiv_class_self = thm "equiv_class_self";
    1.80 -val equiv_comp_eq = thm "equiv_comp_eq";
    1.81 -val equiv_def = thm "equiv_def";
    1.82 -val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
    1.83 -val equiv_type = thm "equiv_type";
    1.84 -val finite_equiv_class = thm "finite_equiv_class";
    1.85 -val finite_quotient = thm "finite_quotient";
    1.86 -val quotientE = thm "quotientE";
    1.87 -val quotientI = thm "quotientI";
    1.88 -val quotient_def = thm "quotient_def";
    1.89 -val quotient_disj = thm "quotient_disj";
    1.90 -val refl_comp_subset = thm "refl_comp_subset";
    1.91 -val subset_equiv_class = thm "subset_equiv_class";
    1.92 -val sym_trans_comp_subset = thm "sym_trans_comp_subset";
    1.93 -*}
    1.94 -*)
    1.95  end