author | haftmann |
Fri, 01 Nov 2013 18:51:14 +0100 | |
changeset 54230 | b1d955791529 |
parent 53026 | e1a548c11845 |
child 55089 | 181751ad852f |
permissions | -rw-r--r-- |
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(* Title: HOL/Lifting_Option.thy |
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Author: Brian Huffman and Ondrej Kuncar |
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*) |
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header {* Setup for Lifting/Transfer for the option type *} |
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theory Lifting_Option |
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imports Lifting |
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begin |
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|
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subsection {* Relator and predicator properties *} |
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|
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definition |
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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 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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|
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lemma option_rel_simps[simp]: |
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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 y) = False" |
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"option_rel R (Some x) (Some y) = R x y" |
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unfolding option_rel_def by simp_all |
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|
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abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where |
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"option_pred \<equiv> option_case True" |
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|
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lemma option_rel_eq [relator_eq]: |
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"option_rel (op =) = (op =)" |
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by (simp add: option_rel_def fun_eq_iff split: option.split) |
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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_def split: option.splits) |
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|
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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_def OO_def split: option.split) |
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|
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lemma Domainp_option[relator_domain]: |
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assumes "Domainp A = P" |
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shows "Domainp (option_rel A) = (option_pred P)" |
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using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def] |
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by (auto iff: fun_eq_iff split: option.split) |
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lemma reflp_option_rel[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 left_total_option_rel[reflexivity_rule]: |
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"left_total R \<Longrightarrow> left_total (option_rel R)" |
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unfolding left_total_def split_option_all split_option_ex by simp |
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lemma left_unique_option_rel [reflexivity_rule]: |
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"left_unique R \<Longrightarrow> left_unique (option_rel R)" |
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unfolding left_unique_def split_option_all by simp |
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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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|
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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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|
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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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by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all) |
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|
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subsection {* Quotient theorem for the Lifting package *} |
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|
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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_def |
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by (simp split: option.split) |
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subsection {* Transfer rules for the Transfer package *} |
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context |
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begin |
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interpretation lifting_syntax . |
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|
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lemma None_transfer [transfer_rule]: "(option_rel A) None None" |
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97 |
by simp |
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|
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lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some" |
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100 |
unfolding fun_rel_def by simp |
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101 |
|
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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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104 |
unfolding fun_rel_def split_option_all by simp |
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105 |
|
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lemma option_map_transfer [transfer_rule]: |
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107 |
"((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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109 |
|
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lemma option_bind_transfer [transfer_rule]: |
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111 |
"(option_rel A ===> (A ===> option_rel B) ===> option_rel B) |
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112 |
Option.bind Option.bind" |
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113 |
unfolding fun_rel_def split_option_all by simp |
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114 |
|
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115 |
end |
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117 |
end |
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118 |