author haftmann Tue Jul 10 17:30:43 2007 +0200 (2007-07-10) changeset 23705 315c638d5856 parent 23704 18d6ee425689 child 23706 b7abba3c230e
moved finite lemmas to Finite_Set.thy
 src/HOL/Equiv_Relations.thy file | annotate | diff | revisions src/HOL/IntDef.thy file | annotate | diff | revisions
```     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"
1.50 -apply(subst card_UN_disjoint)
1.51 -   apply assumption
1.52 -  apply simp
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
```
```     2.1 --- a/src/HOL/IntDef.thy	Tue Jul 10 16:46:37 2007 +0200
2.2 +++ b/src/HOL/IntDef.thy	Tue Jul 10 17:30:43 2007 +0200
2.3 @@ -11,6 +11,7 @@
2.4  imports Equiv_Relations Nat
2.5  begin
2.6
2.7 +
2.8  text {* the equivalence relation underlying the integers *}
2.9
2.10  definition
2.11 @@ -622,52 +623,6 @@
2.12    by (rule Ints_cases) auto
2.13
2.14
2.15 -(* int (Suc n) = 1 + int n *)
2.16 -
2.17 -
2.18 -
2.19 -subsection{*More Properties of @{term setsum} and  @{term setprod}*}
2.20 -
2.22 -
2.23 -
2.24 -lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
2.25 -  apply (cases "finite A")
2.26 -  apply (erule finite_induct, auto)
2.27 -  done
2.28 -
2.29 -lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
2.30 -  apply (cases "finite A")
2.31 -  apply (erule finite_induct, auto)
2.32 -  done
2.33 -
2.34 -lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
2.35 -  apply (cases "finite A")
2.36 -  apply (erule finite_induct, auto simp add: of_nat_mult)
2.37 -  done
2.38 -
2.39 -lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
2.40 -  apply (cases "finite A")
2.41 -  apply (erule finite_induct, auto)
2.42 -  done
2.43 -
2.44 -lemma setprod_nonzero_nat:
2.45 -    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
2.46 -  by (rule setprod_nonzero, auto)
2.47 -
2.48 -lemma setprod_zero_eq_nat:
2.49 -    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
2.50 -  by (rule setprod_zero_eq, auto)
2.51 -
2.52 -lemma setprod_nonzero_int:
2.53 -    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
2.54 -  by (rule setprod_nonzero, auto)
2.55 -
2.56 -lemma setprod_zero_eq_int:
2.57 -    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
2.58 -  by (rule setprod_zero_eq, auto)
2.59 -
2.60 -
2.61  subsection {* Further properties *}
2.62
2.63  text{*Now we replace the case analysis rule by a more conventional one:
2.64 @@ -766,8 +721,6 @@
2.65  lemmas int_Suc = of_nat_Suc [where ?'a_1.0=int]
2.66  lemmas abs_int_eq = abs_of_nat [where 'a=int and n="?m"]
2.67  lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
2.68 -lemmas int_setsum = of_nat_setsum [where 'a=int]
2.69 -lemmas int_setprod = of_nat_setprod [where 'a=int]
2.70  lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
2.71  lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq]
2.72  lemmas int_eq_of_nat = TrueI
```