kuncar@53012: (*  Title:      HOL/Lifting_Option.thy
kuncar@53012:     Author:     Brian Huffman and Ondrej Kuncar
kuncar@53012: *)
kuncar@53012: 
kuncar@53012: header {* Setup for Lifting/Transfer for the option type *}
kuncar@53012: 
kuncar@53012: theory Lifting_Option
traytel@53026: imports Lifting
kuncar@53012: begin
kuncar@53012: 
kuncar@53012: subsection {* Relator and predicator properties *}
kuncar@53012: 
traytel@53026: definition
kuncar@53012:   option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
kuncar@53012: where
kuncar@53012:   "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
kuncar@53012:     | (Some x, Some y) \<Rightarrow> R x y
kuncar@53012:     | _ \<Rightarrow> False)"
kuncar@53012: 
traytel@53026: lemma option_rel_simps[simp]:
traytel@53026:   "option_rel R None None = True"
traytel@53026:   "option_rel R (Some x) None = False"
traytel@53026:   "option_rel R None (Some y) = False"
traytel@53026:   "option_rel R (Some x) (Some y) = R x y"
traytel@53026:   unfolding option_rel_def by simp_all
kuncar@53012: 
traytel@53026: abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
traytel@53026:   "option_pred \<equiv> option_case True"
kuncar@53012: 
kuncar@53012: lemma option_rel_eq [relator_eq]:
kuncar@53012:   "option_rel (op =) = (op =)"
traytel@53026:   by (simp add: option_rel_def fun_eq_iff split: option.split)
kuncar@53012: 
kuncar@53012: lemma option_rel_mono[relator_mono]:
kuncar@53012:   assumes "A \<le> B"
kuncar@53012:   shows "(option_rel A) \<le> (option_rel B)"
traytel@53026: using assms by (auto simp: option_rel_def split: option.splits)
kuncar@53012: 
kuncar@53012: lemma option_rel_OO[relator_distr]:
kuncar@53012:   "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
traytel@53026: by (rule ext)+ (auto simp: option_rel_def OO_def split: option.split)
kuncar@53012: 
kuncar@53012: lemma Domainp_option[relator_domain]:
kuncar@53012:   assumes "Domainp A = P"
kuncar@53012:   shows "Domainp (option_rel A) = (option_pred P)"
traytel@53026: using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def]
kuncar@53012: by (auto iff: fun_eq_iff split: option.split)
kuncar@53012: 
kuncar@53012: lemma reflp_option_rel[reflexivity_rule]:
kuncar@53012:   "reflp R \<Longrightarrow> reflp (option_rel R)"
kuncar@53012:   unfolding reflp_def split_option_all by simp
kuncar@53012: 
kuncar@53012: lemma left_total_option_rel[reflexivity_rule]:
kuncar@53012:   "left_total R \<Longrightarrow> left_total (option_rel R)"
kuncar@53012:   unfolding left_total_def split_option_all split_option_ex by simp
kuncar@53012: 
kuncar@53012: lemma left_unique_option_rel [reflexivity_rule]:
kuncar@53012:   "left_unique R \<Longrightarrow> left_unique (option_rel R)"
kuncar@53012:   unfolding left_unique_def split_option_all by simp
kuncar@53012: 
kuncar@53012: lemma right_total_option_rel [transfer_rule]:
kuncar@53012:   "right_total R \<Longrightarrow> right_total (option_rel R)"
kuncar@53012:   unfolding right_total_def split_option_all split_option_ex by simp
kuncar@53012: 
kuncar@53012: lemma right_unique_option_rel [transfer_rule]:
kuncar@53012:   "right_unique R \<Longrightarrow> right_unique (option_rel R)"
kuncar@53012:   unfolding right_unique_def split_option_all by simp
kuncar@53012: 
kuncar@53012: lemma bi_total_option_rel [transfer_rule]:
kuncar@53012:   "bi_total R \<Longrightarrow> bi_total (option_rel R)"
kuncar@53012:   unfolding bi_total_def split_option_all split_option_ex by simp
kuncar@53012: 
kuncar@53012: lemma bi_unique_option_rel [transfer_rule]:
kuncar@53012:   "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
kuncar@53012:   unfolding bi_unique_def split_option_all by simp
kuncar@53012: 
kuncar@53012: lemma option_invariant_commute [invariant_commute]:
kuncar@53012:   "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
kuncar@53012:   by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
kuncar@53012: 
kuncar@53012: subsection {* Quotient theorem for the Lifting package *}
kuncar@53012: 
kuncar@53012: lemma Quotient_option[quot_map]:
kuncar@53012:   assumes "Quotient R Abs Rep T"
kuncar@53012:   shows "Quotient (option_rel R) (Option.map Abs)
kuncar@53012:     (Option.map Rep) (option_rel T)"
traytel@53026:   using assms unfolding Quotient_alt_def option_rel_def
kuncar@53012:   by (simp split: option.split)
kuncar@53012: 
kuncar@53012: subsection {* Transfer rules for the Transfer package *}
kuncar@53012: 
kuncar@53012: context
kuncar@53012: begin
kuncar@53012: interpretation lifting_syntax .
kuncar@53012: 
kuncar@53012: lemma None_transfer [transfer_rule]: "(option_rel A) None None"
kuncar@53012:   by simp
kuncar@53012: 
kuncar@53012: lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
kuncar@53012:   unfolding fun_rel_def by simp
kuncar@53012: 
kuncar@53012: lemma option_case_transfer [transfer_rule]:
kuncar@53012:   "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
kuncar@53012:   unfolding fun_rel_def split_option_all by simp
kuncar@53012: 
kuncar@53012: lemma option_map_transfer [transfer_rule]:
kuncar@53012:   "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
kuncar@53012:   unfolding Option.map_def by transfer_prover
kuncar@53012: 
kuncar@53012: lemma option_bind_transfer [transfer_rule]:
kuncar@53012:   "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
kuncar@53012:     Option.bind Option.bind"
kuncar@53012:   unfolding fun_rel_def split_option_all by simp
kuncar@53012: 
kuncar@53012: end
kuncar@53012: 
kuncar@53012: end
kuncar@53012: