wenzelm@47455: (* Title: HOL/Library/Quotient_Option.thy huffman@47624: Author: Cezary Kaliszyk, Christian Urban and Brian Huffman kaliszyk@35222: *) wenzelm@35788: wenzelm@35788: header {* Quotient infrastructure for the option type *} wenzelm@35788: kaliszyk@35222: theory Quotient_Option kaliszyk@35222: imports Main Quotient_Syntax kaliszyk@35222: begin kaliszyk@35222: huffman@47624: subsection {* Relator for option type *} huffman@47624: kaliszyk@35222: fun haftmann@40542: option_rel :: "('a \ 'b \ bool) \ 'a option \ 'b option \ bool" kaliszyk@35222: where kaliszyk@35222: "option_rel R None None = True" kaliszyk@35222: | "option_rel R (Some x) None = False" kaliszyk@35222: | "option_rel R None (Some x) = False" kaliszyk@35222: | "option_rel R (Some x) (Some y) = R x y" kaliszyk@35222: haftmann@40820: lemma option_rel_unfold: haftmann@40820: "option_rel R x y = (case (x, y) of (None, None) \ True haftmann@40820: | (Some x, Some y) \ R x y haftmann@40820: | _ \ False)" haftmann@40820: by (cases x) (cases y, simp_all)+ haftmann@40820: haftmann@40820: lemma option_rel_map1: haftmann@40820: "option_rel R (Option.map f x) y \ option_rel (\x. R (f x)) x y" haftmann@40820: by (simp add: option_rel_unfold split: option.split) haftmann@40820: haftmann@40820: lemma option_rel_map2: haftmann@40820: "option_rel R x (Option.map f y) \ option_rel (\x y. R x (f y)) x y" haftmann@40820: by (simp add: option_rel_unfold split: option.split) haftmann@40820: haftmann@40820: lemma option_map_id [id_simps]: haftmann@40820: "Option.map id = id" haftmann@40820: by (simp add: id_def Option.map.identity fun_eq_iff) haftmann@40820: huffman@47624: lemma option_rel_eq [id_simps, relator_eq]: haftmann@40820: "option_rel (op =) = (op =)" haftmann@40820: by (simp add: option_rel_unfold fun_eq_iff split: option.split) haftmann@40820: huffman@47624: lemma split_option_all: "(\x. P x) \ P None \ (\x. P (Some x))" huffman@47624: by (metis option.exhaust) (* TODO: move to Option.thy *) huffman@47624: huffman@47624: lemma split_option_ex: "(\x. P x) \ P None \ (\x. P (Some x))" huffman@47624: by (metis option.exhaust) (* TODO: move to Option.thy *) huffman@47624: kuncar@47982: lemma option_reflp[reflexivity_rule]: haftmann@40820: "reflp R \ reflp (option_rel R)" huffman@47624: unfolding reflp_def split_option_all by simp haftmann@40820: kuncar@47982: lemma option_left_total[reflexivity_rule]: kuncar@47982: "left_total R \ left_total (option_rel R)" kuncar@47982: apply (intro left_totalI) kuncar@47982: unfolding split_option_ex kuncar@47982: by (case_tac x) (auto elim: left_totalE) kuncar@47982: haftmann@40820: lemma option_symp: haftmann@40820: "symp R \ symp (option_rel R)" huffman@47624: unfolding symp_def split_option_all option_rel.simps by fast haftmann@40820: haftmann@40820: lemma option_transp: haftmann@40820: "transp R \ transp (option_rel R)" huffman@47624: unfolding transp_def split_option_all option_rel.simps by fast haftmann@40820: haftmann@40820: lemma option_equivp [quot_equiv]: haftmann@40820: "equivp R \ equivp (option_rel R)" haftmann@40820: by (blast intro: equivpI option_reflp option_symp option_transp elim: equivpE) haftmann@40820: huffman@47624: lemma right_total_option_rel [transfer_rule]: huffman@47624: "right_total R \ right_total (option_rel R)" huffman@47624: unfolding right_total_def split_option_all split_option_ex by simp huffman@47624: huffman@47624: lemma right_unique_option_rel [transfer_rule]: huffman@47624: "right_unique R \ right_unique (option_rel R)" huffman@47624: unfolding right_unique_def split_option_all by simp huffman@47624: huffman@47624: lemma bi_total_option_rel [transfer_rule]: huffman@47624: "bi_total R \ bi_total (option_rel R)" huffman@47624: unfolding bi_total_def split_option_all split_option_ex by simp huffman@47624: huffman@47624: lemma bi_unique_option_rel [transfer_rule]: huffman@47624: "bi_unique R \ bi_unique (option_rel R)" huffman@47624: unfolding bi_unique_def split_option_all by simp huffman@47624: huffman@47635: subsection {* Transfer rules for transfer package *} huffman@47624: huffman@47624: lemma None_transfer [transfer_rule]: "(option_rel A) None None" huffman@47624: by simp huffman@47624: huffman@47624: lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some" huffman@47624: unfolding fun_rel_def by simp huffman@47624: huffman@47624: lemma option_case_transfer [transfer_rule]: huffman@47624: "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case" huffman@47624: unfolding fun_rel_def split_option_all by simp huffman@47624: huffman@47624: lemma option_map_transfer [transfer_rule]: huffman@47624: "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map" huffman@47635: unfolding Option.map_def by transfer_prover huffman@47624: huffman@47624: lemma option_bind_transfer [transfer_rule]: huffman@47624: "(option_rel A ===> (A ===> option_rel B) ===> option_rel B) huffman@47624: Option.bind Option.bind" huffman@47624: unfolding fun_rel_def split_option_all by simp huffman@47624: huffman@47624: subsection {* Setup for lifting package *} huffman@47624: kuncar@47777: lemma Quotient_option[quot_map]: huffman@47624: assumes "Quotient R Abs Rep T" huffman@47624: shows "Quotient (option_rel R) (Option.map Abs) huffman@47624: (Option.map Rep) (option_rel T)" huffman@47624: using assms unfolding Quotient_alt_def option_rel_unfold huffman@47624: by (simp split: option.split) huffman@47624: kuncar@47634: fun option_pred :: "('a \ bool) \ 'a option \ bool" kuncar@47634: where kuncar@47634: "option_pred R None = True" kuncar@47634: | "option_pred R (Some x) = R x" kuncar@47634: kuncar@47634: lemma option_invariant_commute [invariant_commute]: kuncar@47634: "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)" kuncar@47634: apply (simp add: fun_eq_iff Lifting.invariant_def) kuncar@47634: apply (intro allI) kuncar@47634: apply (case_tac x rule: option.exhaust) kuncar@47634: apply (case_tac xa rule: option.exhaust) kuncar@47634: apply auto[2] kuncar@47634: apply (case_tac xa rule: option.exhaust) kuncar@47634: apply auto kuncar@47634: done kuncar@47634: huffman@47624: subsection {* Rules for quotient package *} huffman@47624: haftmann@40820: lemma option_quotient [quot_thm]: kuncar@47308: assumes "Quotient3 R Abs Rep" kuncar@47308: shows "Quotient3 (option_rel R) (Option.map Abs) (Option.map Rep)" kuncar@47308: apply (rule Quotient3I) kuncar@47308: 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]) kuncar@47308: using Quotient3_rel [OF assms] haftmann@40820: apply (simp add: option_rel_unfold split: option.split) kaliszyk@35222: done kaliszyk@35222: kuncar@47308: declare [[mapQ3 option = (option_rel, option_quotient)]] kuncar@47094: haftmann@40820: lemma option_None_rsp [quot_respect]: kuncar@47308: assumes q: "Quotient3 R Abs Rep" kaliszyk@35222: shows "option_rel R None None" huffman@47624: by (rule None_transfer) kaliszyk@35222: haftmann@40820: lemma option_Some_rsp [quot_respect]: kuncar@47308: assumes q: "Quotient3 R Abs Rep" kaliszyk@35222: shows "(R ===> option_rel R) Some Some" huffman@47624: by (rule Some_transfer) kaliszyk@35222: haftmann@40820: lemma option_None_prs [quot_preserve]: kuncar@47308: assumes q: "Quotient3 R Abs Rep" kaliszyk@35222: shows "Option.map Abs None = None" kaliszyk@35222: by simp kaliszyk@35222: haftmann@40820: lemma option_Some_prs [quot_preserve]: kuncar@47308: assumes q: "Quotient3 R Abs Rep" kaliszyk@35222: shows "(Rep ---> Option.map Abs) Some = Some" nipkow@39302: apply(simp add: fun_eq_iff) kuncar@47308: apply(simp add: Quotient3_abs_rep[OF q]) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: end