Theory Quotient_Option

theory Quotient_Option
imports Quotient_Syntax
(*  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 (Option.map f x) y <-> option_rel (λx. R (f x)) x y"
by (simp add: option_rel_def split: option.split)

lemma option_rel_map2:
"option_rel R x (Option.map f y) <-> option_rel (λx y. R x (f y)) x y"
by (simp add: option_rel_def split: option.split)

lemma option_map_id [id_simps]:
"Option.map id = id"
by (simp add: id_def Option.map.identity fun_eq_iff)

lemma option_rel_eq [id_simps]:
"option_rel (op =) = (op =)"
by (simp add: option_rel_def fun_eq_iff split: option.split)

lemma option_symp:
"symp R ==> symp (option_rel R)"
unfolding symp_def split_option_all option_rel_simps by fast

lemma option_transp:
"transp R ==> transp (option_rel R)"
unfolding transp_def split_option_all option_rel_simps by fast

lemma option_equivp [quot_equiv]:
"equivp R ==> 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) (Option.map Abs) (Option.map Rep)"
apply (rule Quotient3I)
apply (simp_all add: Option.map.compositionality comp_def Option.map.identity 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 "Option.map Abs None = None"
by simp

lemma option_Some_prs [quot_preserve]:
assumes q: "Quotient3 R Abs Rep"
shows "(Rep ---> Option.map Abs) Some = Some"
apply(simp add: fun_eq_iff)
apply(simp add: Quotient3_abs_rep[OF q])
done

end