(* 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
fun
option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
where
"option_rel R None None = True"
| "option_rel R (Some x) None = False"
| "option_rel R None (Some x) = False"
| "option_rel R (Some x) (Some y) = R x y"
lemma option_rel_unfold:
"option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
| (Some x, Some y) \<Rightarrow> R x y
| _ \<Rightarrow> False)"
by (cases x) (cases y, simp_all)+
lemma option_rel_map1:
"option_rel R (Option.map f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
by (simp add: option_rel_unfold split: option.split)
lemma option_rel_map2:
"option_rel R x (Option.map f y) \<longleftrightarrow> option_rel (\<lambda>x y. R x (f y)) x y"
by (simp add: option_rel_unfold 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_unfold fun_eq_iff split: option.split)
lemma option_reflp:
"reflp R \<Longrightarrow> reflp (option_rel R)"
by (auto simp add: option_rel_unfold split: option.splits intro!: reflpI elim: reflpE)
lemma option_symp:
"symp R \<Longrightarrow> symp (option_rel R)"
by (auto simp add: option_rel_unfold split: option.splits intro!: sympI elim: sympE)
lemma option_transp:
"transp R \<Longrightarrow> transp (option_rel R)"
by (auto simp add: option_rel_unfold split: option.splits intro!: transpI elim: transpE)
lemma option_equivp [quot_equiv]:
"equivp R \<Longrightarrow> equivp (option_rel R)"
by (blast intro: equivpI option_reflp 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_unfold 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 simp
lemma option_Some_rsp [quot_respect]:
assumes q: "Quotient3 R Abs Rep"
shows "(R ===> option_rel R) Some Some"
by auto
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