src/HOL/Library/Quotient_Option.thy
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(*  Title:      HOL/Library/Quotient_Option.thy
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    Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
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*)
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header {* Quotient infrastructure for the option type *}
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theory Quotient_Option
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imports Main Quotient_Syntax
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begin
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subsection {* Relator for option type *}
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fun
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  option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
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where
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  "option_rel R None None = True"
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| "option_rel R (Some x) None = False"
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| "option_rel R None (Some x) = False"
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| "option_rel R (Some x) (Some y) = R x y"
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lemma option_rel_unfold:
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  "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
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    | (Some x, Some y) \<Rightarrow> R x y
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    | _ \<Rightarrow> False)"
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  by (cases x) (cases y, simp_all)+
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lemma option_rel_map1:
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  "option_rel R (Option.map f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
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  by (simp add: option_rel_unfold split: option.split)
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lemma option_rel_map2:
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  "option_rel R x (Option.map f y) \<longleftrightarrow> option_rel (\<lambda>x y. R x (f y)) x y"
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  by (simp add: option_rel_unfold split: option.split)
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lemma option_map_id [id_simps]:
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  "Option.map id = id"
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  by (simp add: id_def Option.map.identity fun_eq_iff)
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lemma option_rel_eq [id_simps, relator_eq]:
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  "option_rel (op =) = (op =)"
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  by (simp add: option_rel_unfold fun_eq_iff split: option.split)
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lemma split_option_all: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P (Some x))"
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  by (metis option.exhaust) (* TODO: move to Option.thy *)
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lemma split_option_ex: "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P (Some x))"
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  by (metis option.exhaust) (* TODO: move to Option.thy *)
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lemma option_rel_mono[relator_mono]:
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  assumes "A \<le> B"
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  shows "(option_rel A) \<le> (option_rel B)"
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using assms by (auto simp: option_rel_unfold split: option.splits)
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lemma option_rel_OO[relator_distr]:
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  "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
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by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
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lemma option_reflp[reflexivity_rule]:
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  "reflp R \<Longrightarrow> reflp (option_rel R)"
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  unfolding reflp_def split_option_all by simp
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lemma option_left_total[reflexivity_rule]:
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  "left_total R \<Longrightarrow> left_total (option_rel R)"
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  apply (intro left_totalI)
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  unfolding split_option_ex
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  by (case_tac x) (auto elim: left_totalE)
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lemma option_symp:
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  "symp R \<Longrightarrow> symp (option_rel R)"
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  unfolding symp_def split_option_all option_rel.simps by fast
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lemma option_transp:
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  "transp R \<Longrightarrow> transp (option_rel R)"
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  unfolding transp_def split_option_all option_rel.simps by fast
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lemma option_equivp [quot_equiv]:
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  "equivp R \<Longrightarrow> equivp (option_rel R)"
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  by (blast intro: equivpI option_reflp option_symp option_transp elim: equivpE)
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lemma right_total_option_rel [transfer_rule]:
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  "right_total R \<Longrightarrow> right_total (option_rel R)"
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  unfolding right_total_def split_option_all split_option_ex by simp
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lemma right_unique_option_rel [transfer_rule]:
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  "right_unique R \<Longrightarrow> right_unique (option_rel R)"
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  unfolding right_unique_def split_option_all by simp
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lemma bi_total_option_rel [transfer_rule]:
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  "bi_total R \<Longrightarrow> bi_total (option_rel R)"
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  unfolding bi_total_def split_option_all split_option_ex by simp
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lemma bi_unique_option_rel [transfer_rule]:
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  "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
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  unfolding bi_unique_def split_option_all by simp
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subsection {* Transfer rules for transfer package *}
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lemma None_transfer [transfer_rule]: "(option_rel A) None None"
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  by simp
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lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
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  unfolding fun_rel_def by simp
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lemma option_case_transfer [transfer_rule]:
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  "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
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  unfolding fun_rel_def split_option_all by simp
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lemma option_map_transfer [transfer_rule]:
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  "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
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  unfolding Option.map_def by transfer_prover
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lemma option_bind_transfer [transfer_rule]:
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  "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
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    Option.bind Option.bind"
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  unfolding fun_rel_def split_option_all by simp
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subsection {* Setup for lifting package *}
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lemma Quotient_option[quot_map]:
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  assumes "Quotient R Abs Rep T"
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  shows "Quotient (option_rel R) (Option.map Abs)
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    (Option.map Rep) (option_rel T)"
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  using assms unfolding Quotient_alt_def option_rel_unfold
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  by (simp split: option.split)
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fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
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where
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  "option_pred R None = True"
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| "option_pred R (Some x) = R x"
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lemma option_invariant_commute [invariant_commute]:
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  "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
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  apply (simp add: fun_eq_iff Lifting.invariant_def)
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  apply (intro allI) 
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  apply (case_tac x rule: option.exhaust)
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  apply (case_tac xa rule: option.exhaust)
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  apply auto[2]
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  apply (case_tac xa rule: option.exhaust)
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  apply auto
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done
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subsection {* Rules for quotient package *}
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lemma option_quotient [quot_thm]:
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  assumes "Quotient3 R Abs Rep"
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  shows "Quotient3 (option_rel R) (Option.map Abs) (Option.map Rep)"
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  apply (rule Quotient3I)
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  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])
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  using Quotient3_rel [OF assms]
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  apply (simp add: option_rel_unfold split: option.split)
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  done
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declare [[mapQ3 option = (option_rel, option_quotient)]]
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lemma option_None_rsp [quot_respect]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "option_rel R None None"
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  by (rule None_transfer)
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lemma option_Some_rsp [quot_respect]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "(R ===> option_rel R) Some Some"
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  by (rule Some_transfer)
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lemma option_None_prs [quot_preserve]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "Option.map Abs None = None"
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  by simp
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lemma option_Some_prs [quot_preserve]:
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  assumes q: "Quotient3 R Abs Rep"
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  shows "(Rep ---> Option.map Abs) Some = Some"
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  apply(simp add: fun_eq_iff)
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  apply(simp add: Quotient3_abs_rep[OF q])
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  done
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end