src/HOL/Library/Quotient_Option.thy
 author kuncar Fri, 09 Dec 2011 18:07:04 +0100 changeset 45802 b16f976db515 parent 41372 551eb49a6e91 child 47094 1a7ad2601cb5 permissions -rw-r--r--
Quotient_Info stores only relation maps
```
(*  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"

declare [[map option = option_rel]]

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 "Quotient R Abs Rep"
shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
apply (rule QuotientI)
apply (simp_all add: Option.map.compositionality comp_def Option.map.identity option_rel_eq option_rel_map1 option_rel_map2 Quotient_abs_rep [OF assms] Quotient_rel_rep [OF assms])
using Quotient_rel [OF assms]
apply (simp add: option_rel_unfold split: option.split)
done

lemma option_None_rsp [quot_respect]:
assumes q: "Quotient R Abs Rep"
shows "option_rel R None None"
by simp

lemma option_Some_rsp [quot_respect]:
assumes q: "Quotient R Abs Rep"
shows "(R ===> option_rel R) Some Some"
by auto

lemma option_None_prs [quot_preserve]:
assumes q: "Quotient R Abs Rep"
shows "Option.map Abs None = None"
by simp

lemma option_Some_prs [quot_preserve]:
assumes q: "Quotient R Abs Rep"
shows "(Rep ---> Option.map Abs) Some = Some"