# Theory Quotient_Set

theory Quotient_Set
imports Quotient_Syntax
```(*  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]:
"rel_vset op = = op ="
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 "((R ===> op =) ===> 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) ---> op -` Rep) Collect = Collect"
unfolding fun_eq_iff

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

lemma union_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op ∪ = op ∪"
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) op - op -"
by (intro rel_funI) (simp add: rel_vset_def)

lemma diff_prs[quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -"
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) op ∩ op ∩"
by (intro rel_funI) (auto simp add: rel_vset_def)

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

lemma mem_rsp[quot_respect]:
shows "(R ===> rel_vset R ===> op =) op ∈ op ∈"
by (intro rel_funI) (simp add: rel_vset_def)

end
```