# Theory Quotient_Set

(*  Title:      HOL/Library/Quotient_Set.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

section ‹Quotient infrastructure for the set type›

theory Quotient_Set
imports Quotient_Syntax
begin

subsection ‹Contravariant set map (vimage) and set relator, rules for the Quotient package›

definition "rel_vset R xs ys  x y. R x y  x  xs  y  ys"

lemma rel_vset_eq [id_simps]:

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  xs = ys  (x y. x  xs  R x y  y  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 "x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
then show "xs. Rep -` (Abs -` xs) = xs"
unfolding vimage_def by auto
next
show "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  rel_vset R s s  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
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

lemma union_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows
by (intro rel_funI) (simp add: rel_vset_def)

lemma union_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]])

lemma diff_rsp[quot_respect]:
assumes "Quotient3 R Abs Rep"
shows
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
by (intro rel_funI) (auto simp add: rel_vset_def)

lemma inter_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]])

lemma mem_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(Rep ---> (-`) Abs ---> id) (∈) = (∈)"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms])

lemma mem_rsp[quot_respect]:
shows
by (intro rel_funI) (simp add: rel_vset_def)

end