(* Title: HOL/Library/Quotient_Set.thy
Author: Cezary Kaliszyk and Christian Urban
*)
section \<open>Quotient infrastructure for the set type\<close>
theory Quotient_Set
imports Quotient_Syntax
begin
subsection \<open>Contravariant set map (vimage) and set relator, rules for the Quotient package\<close>
definition "rel_vset R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
lemma rel_vset_eq [id_simps]:
"rel_vset (=) = (=)"
by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff rel_vset_def)
lemma rel_vset_equivp:
assumes e: "equivp R"
shows "rel_vset R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
unfolding rel_vset_def
using equivp_reflp[OF e]
by auto (metis, metis equivp_symp[OF e])
lemma set_quotient [quot_thm]:
assumes "Quotient3 R Abs Rep"
shows "Quotient3 (rel_vset R) (vimage Rep) (vimage Abs)"
proof (rule Quotient3I)
from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
then show "\<And>xs. Rep -` (Abs -` xs) = xs"
unfolding vimage_def by auto
next
show "\<And>xs. rel_vset R (Abs -` xs) (Abs -` xs)"
unfolding rel_vset_def vimage_def
by auto (metis Quotient3_rel_abs[OF assms])+
next
fix r s
show "rel_vset R r s = (rel_vset R r r \<and> rel_vset R s s \<and> Rep -` r = Rep -` s)"
unfolding rel_vset_def vimage_def set_eq_iff
by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+
qed
declare [[mapQ3 set = (rel_vset, set_quotient)]]
lemma empty_set_rsp[quot_respect]:
"rel_vset R {} {}"
unfolding rel_vset_def by simp
lemma collect_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows "((R ===> (=)) ===> rel_vset R) Collect Collect"
by (intro rel_funI) (simp add: rel_fun_def rel_vset_def)
lemma collect_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "((Abs ---> id) ---> (-`) Rep) Collect = Collect"
unfolding fun_eq_iff
by (simp add: Quotient3_abs_rep[OF assms])
lemma union_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows "(rel_vset R ===> rel_vset R ===> rel_vset R) (\<union>) (\<union>)"
by (intro rel_funI) (simp add: rel_vset_def)
lemma union_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "((-`) Abs ---> (-`) Abs ---> (-`) Rep) (\<union>) = (\<union>)"
unfolding fun_eq_iff
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]])
lemma diff_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows "(rel_vset R ===> rel_vset R ===> rel_vset R) (-) (-)"
by (intro rel_funI) (simp add: rel_vset_def)
lemma diff_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "((-`) Abs ---> (-`) Abs ---> (-`) Rep) (-) = (-)"
unfolding fun_eq_iff
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff)
lemma inter_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows "(rel_vset R ===> rel_vset R ===> rel_vset R) (\<inter>) (\<inter>)"
by (intro rel_funI) (auto simp add: rel_vset_def)
lemma inter_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "((-`) Abs ---> (-`) Abs ---> (-`) Rep) (\<inter>) = (\<inter>)"
unfolding fun_eq_iff
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]])
lemma mem_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(Rep ---> (-`) Abs ---> id) (\<in>) = (\<in>)"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms])
lemma mem_rsp[quot_respect]:
shows "(R ===> rel_vset R ===> (=)) (\<in>) (\<in>)"
by (intro rel_funI) (simp add: rel_vset_def)
end