(* Title: HOL/Library/Quotient_Option.thy
Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the option type *}
theory Quotient_Option
imports Main Quotient_Syntax
begin
subsection {* Rules for the Quotient package *}
lemma option_rel_map1:
"option_rel R (map_option f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
by (simp add: option_rel_def split: option.split)
lemma option_rel_map2:
"option_rel R x (map_option f y) \<longleftrightarrow> option_rel (\<lambda>x y. R x (f y)) x y"
by (simp add: option_rel_def split: option.split)
declare
map_option.id [id_simps]
option_rel_eq [id_simps]
lemma option_symp:
"symp R \<Longrightarrow> symp (option_rel R)"
unfolding symp_def split_option_all option_rel_simps by fast
lemma option_transp:
"transp R \<Longrightarrow> transp (option_rel R)"
unfolding transp_def split_option_all option_rel_simps by fast
lemma option_equivp [quot_equiv]:
"equivp R \<Longrightarrow> equivp (option_rel R)"
by (blast intro: equivpI reflp_option_rel option_symp option_transp elim: equivpE)
lemma option_quotient [quot_thm]:
assumes "Quotient3 R Abs Rep"
shows "Quotient3 (option_rel R) (map_option Abs) (map_option Rep)"
apply (rule Quotient3I)
apply (simp_all add: option.map_comp comp_def option.map_id[unfolded id_def] option_rel_eq option_rel_map1 option_rel_map2 Quotient3_abs_rep [OF assms] Quotient3_rel_rep [OF assms])
using Quotient3_rel [OF assms]
apply (simp add: option_rel_def split: option.split)
done
declare [[mapQ3 option = (option_rel, option_quotient)]]
lemma option_None_rsp [quot_respect]:
assumes q: "Quotient3 R Abs Rep"
shows "option_rel R None None"
by (rule None_transfer)
lemma option_Some_rsp [quot_respect]:
assumes q: "Quotient3 R Abs Rep"
shows "(R ===> option_rel R) Some Some"
by (rule Some_transfer)
lemma option_None_prs [quot_preserve]:
assumes q: "Quotient3 R Abs Rep"
shows "map_option Abs None = None"
by simp
lemma option_Some_prs [quot_preserve]:
assumes q: "Quotient3 R Abs Rep"
shows "(Rep ---> map_option Abs) Some = Some"
apply(simp add: fun_eq_iff)
apply(simp add: Quotient3_abs_rep[OF q])
done
end