(* Title: HOL/Library/Quotient_Option.thy
Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
*)
header {* Quotient infrastructure for the option type *}
theory Quotient_Option
imports Main Quotient_Syntax
begin
subsection {* Relator for option type *}
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)+
fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
where
"option_pred R None = True"
| "option_pred R (Some x) = R x"
lemma option_pred_unfold:
"option_pred P x = (case x of None \<Rightarrow> True
| Some x \<Rightarrow> P x)"
by (cases x) 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, relator_eq]:
"option_rel (op =) = (op =)"
by (simp add: option_rel_unfold fun_eq_iff split: option.split)
lemma split_option_all: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P (Some x))"
by (metis option.exhaust) (* TODO: move to Option.thy *)
lemma split_option_ex: "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P (Some x))"
by (metis option.exhaust) (* TODO: move to Option.thy *)
lemma option_rel_mono[relator_mono]:
assumes "A \<le> B"
shows "(option_rel A) \<le> (option_rel B)"
using assms by (auto simp: option_rel_unfold split: option.splits)
lemma option_rel_OO[relator_distr]:
"(option_rel A) OO (option_rel B) = option_rel (A OO B)"
by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
lemma Domainp_option[relator_domain]:
assumes "Domainp A = P"
shows "Domainp (option_rel A) = (option_pred P)"
using assms unfolding Domainp_iff[abs_def] option_rel_unfold[abs_def] option_pred_unfold[abs_def]
by (auto iff: fun_eq_iff split: option.split)
lemma reflp_option_rel[reflexivity_rule]:
"reflp R \<Longrightarrow> reflp (option_rel R)"
unfolding reflp_def split_option_all by simp
lemma left_total_option_rel[reflexivity_rule]:
"left_total R \<Longrightarrow> left_total (option_rel R)"
unfolding left_total_def split_option_all split_option_ex by simp
lemma left_unique_option_rel [reflexivity_rule]:
"left_unique R \<Longrightarrow> left_unique (option_rel R)"
unfolding left_unique_def split_option_all by simp
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 right_total_option_rel [transfer_rule]:
"right_total R \<Longrightarrow> right_total (option_rel R)"
unfolding right_total_def split_option_all split_option_ex by simp
lemma right_unique_option_rel [transfer_rule]:
"right_unique R \<Longrightarrow> right_unique (option_rel R)"
unfolding right_unique_def split_option_all by simp
lemma bi_total_option_rel [transfer_rule]:
"bi_total R \<Longrightarrow> bi_total (option_rel R)"
unfolding bi_total_def split_option_all split_option_ex by simp
lemma bi_unique_option_rel [transfer_rule]:
"bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
unfolding bi_unique_def split_option_all by simp
subsection {* Transfer rules for transfer package *}
lemma None_transfer [transfer_rule]: "(option_rel A) None None"
by simp
lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
unfolding fun_rel_def by simp
lemma option_case_transfer [transfer_rule]:
"(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
unfolding fun_rel_def split_option_all by simp
lemma option_map_transfer [transfer_rule]:
"((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
unfolding Option.map_def by transfer_prover
lemma option_bind_transfer [transfer_rule]:
"(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
Option.bind Option.bind"
unfolding fun_rel_def split_option_all by simp
subsection {* Setup for lifting package *}
lemma Quotient_option[quot_map]:
assumes "Quotient R Abs Rep T"
shows "Quotient (option_rel R) (Option.map Abs)
(Option.map Rep) (option_rel T)"
using assms unfolding Quotient_alt_def option_rel_unfold
by (simp split: option.split)
lemma option_invariant_commute [invariant_commute]:
"option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
apply (simp add: fun_eq_iff Lifting.invariant_def)
apply (intro allI)
apply (case_tac x rule: option.exhaust)
apply (case_tac xa rule: option.exhaust)
apply auto[2]
apply (case_tac xa rule: option.exhaust)
apply auto
done
subsection {* Rules for quotient package *}
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 (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