author huffman Sat Apr 21 10:59:52 2012 +0200 (2012-04-21) changeset 47647 ec29cc09599d parent 47646 9460f3f22365 child 47648 6b9d20a095ae
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
```     1.1 --- a/src/HOL/Library/Quotient_Set.thy	Sat Apr 21 11:15:49 2012 +0200
1.2 +++ b/src/HOL/Library/Quotient_Set.thy	Sat Apr 21 10:59:52 2012 +0200
1.3 @@ -10,47 +10,47 @@
1.4
1.5  subsection {* set map (vimage) and set relation *}
1.6
1.7 -definition "set_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
1.8 +definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
1.9
1.10 -lemma set_rel_eq [id_simps]:
1.11 -  "set_rel op = = op ="
1.12 -  by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff set_rel_def)
1.13 +lemma vset_rel_eq [id_simps]:
1.14 +  "vset_rel op = = op ="
1.15 +  by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff vset_rel_def)
1.16
1.17 -lemma set_rel_equivp:
1.18 +lemma vset_rel_equivp:
1.19    assumes e: "equivp R"
1.20 -  shows "set_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
1.21 -  unfolding set_rel_def
1.22 +  shows "vset_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
1.23 +  unfolding vset_rel_def
1.24    using equivp_reflp[OF e]
1.25    by auto (metis, metis equivp_symp[OF e])
1.26
1.27  lemma set_quotient [quot_thm]:
1.28    assumes "Quotient3 R Abs Rep"
1.29 -  shows "Quotient3 (set_rel R) (vimage Rep) (vimage Abs)"
1.30 +  shows "Quotient3 (vset_rel R) (vimage Rep) (vimage Abs)"
1.31  proof (rule Quotient3I)
1.32    from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
1.33    then show "\<And>xs. Rep -` (Abs -` xs) = xs"
1.34      unfolding vimage_def by auto
1.35  next
1.36 -  show "\<And>xs. set_rel R (Abs -` xs) (Abs -` xs)"
1.37 -    unfolding set_rel_def vimage_def
1.38 +  show "\<And>xs. vset_rel R (Abs -` xs) (Abs -` xs)"
1.39 +    unfolding vset_rel_def vimage_def
1.40      by auto (metis Quotient3_rel_abs[OF assms])+
1.41  next
1.42    fix r s
1.43 -  show "set_rel R r s = (set_rel R r r \<and> set_rel R s s \<and> Rep -` r = Rep -` s)"
1.44 -    unfolding set_rel_def vimage_def set_eq_iff
1.45 +  show "vset_rel R r s = (vset_rel R r r \<and> vset_rel R s s \<and> Rep -` r = Rep -` s)"
1.46 +    unfolding vset_rel_def vimage_def set_eq_iff
1.47      by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+
1.48  qed
1.49
1.50 -declare [[mapQ3 set = (set_rel, set_quotient)]]
1.51 +declare [[mapQ3 set = (vset_rel, set_quotient)]]
1.52
1.53  lemma empty_set_rsp[quot_respect]:
1.54 -  "set_rel R {} {}"
1.55 -  unfolding set_rel_def by simp
1.56 +  "vset_rel R {} {}"
1.57 +  unfolding vset_rel_def by simp
1.58
1.59  lemma collect_rsp[quot_respect]:
1.60    assumes "Quotient3 R Abs Rep"
1.61 -  shows "((R ===> op =) ===> set_rel R) Collect Collect"
1.62 -  by (intro fun_relI) (simp add: fun_rel_def set_rel_def)
1.63 +  shows "((R ===> op =) ===> vset_rel R) Collect Collect"
1.64 +  by (intro fun_relI) (simp add: fun_rel_def vset_rel_def)
1.65
1.66  lemma collect_prs[quot_preserve]:
1.67    assumes "Quotient3 R Abs Rep"
1.68 @@ -60,8 +60,8 @@
1.69
1.70  lemma union_rsp[quot_respect]:
1.71    assumes "Quotient3 R Abs Rep"
1.72 -  shows "(set_rel R ===> set_rel R ===> set_rel R) op \<union> op \<union>"
1.73 -  by (intro fun_relI) (simp add: set_rel_def)
1.74 +  shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<union> op \<union>"
1.75 +  by (intro fun_relI) (simp add: vset_rel_def)
1.76
1.77  lemma union_prs[quot_preserve]:
1.78    assumes "Quotient3 R Abs Rep"
1.79 @@ -71,8 +71,8 @@
1.80
1.81  lemma diff_rsp[quot_respect]:
1.82    assumes "Quotient3 R Abs Rep"
1.83 -  shows "(set_rel R ===> set_rel R ===> set_rel R) op - op -"
1.84 -  by (intro fun_relI) (simp add: set_rel_def)
1.85 +  shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op - op -"
1.86 +  by (intro fun_relI) (simp add: vset_rel_def)
1.87
1.88  lemma diff_prs[quot_preserve]:
1.89    assumes "Quotient3 R Abs Rep"
1.90 @@ -82,8 +82,8 @@
1.91
1.92  lemma inter_rsp[quot_respect]:
1.93    assumes "Quotient3 R Abs Rep"
1.94 -  shows "(set_rel R ===> set_rel R ===> set_rel R) op \<inter> op \<inter>"
1.95 -  by (intro fun_relI) (auto simp add: set_rel_def)
1.96 +  shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<inter> op \<inter>"
1.97 +  by (intro fun_relI) (auto simp add: vset_rel_def)
1.98
1.99  lemma inter_prs[quot_preserve]:
1.100    assumes "Quotient3 R Abs Rep"
1.101 @@ -97,7 +97,7 @@
1.102    by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms])
1.103
1.104  lemma mem_rsp[quot_respect]:
1.105 -  shows "(R ===> set_rel R ===> op =) op \<in> op \<in>"
1.106 -  by (intro fun_relI) (simp add: set_rel_def)
1.107 +  shows "(R ===> vset_rel R ===> op =) op \<in> op \<in>"
1.108 +  by (intro fun_relI) (simp add: vset_rel_def)
1.109
1.110  end
```