(* Title: HOL/Lifting_Option.thy
Author: Brian Huffman and Ondrej Kuncar
Author: Andreas Lochbihler, Karlsruhe Institute of Technology
*)
header {* Setup for Lifting/Transfer for the option type *}
theory Lifting_Option
imports Lifting Partial_Function
begin
subsection {* Relator and predicator properties *}
definition
option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
where
"option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
| (Some x, Some y) \<Rightarrow> R x y
| _ \<Rightarrow> False)"
lemma option_rel_simps[simp]:
"option_rel R None None = True"
"option_rel R (Some x) None = False"
"option_rel R None (Some y) = False"
"option_rel R (Some x) (Some y) = R x y"
unfolding option_rel_def by simp_all
abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
"option_pred \<equiv> option_case True"
lemma option_rel_eq [relator_eq]:
"option_rel (op =) = (op =)"
by (simp add: option_rel_def fun_eq_iff split: option.split)
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_def 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_def 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_def[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 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
lemma option_invariant_commute [invariant_commute]:
"option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
subsection {* Quotient theorem for the 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_def
by (simp split: option.split)
subsection {* Transfer rules for the Transfer package *}
context
begin
interpretation lifting_syntax .
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
end
subsubsection {* BNF setup *}
lemma option_rec_conv_option_case: "option_rec = option_case"
by (simp add: fun_eq_iff split: option.split)
bnf "'a option"
map: Option.map
sets: Option.set
bd: natLeq
wits: None
rel: option_rel
proof -
show "Option.map id = id" by (rule Option.map.id)
next
fix f g
show "Option.map (g \<circ> f) = Option.map g \<circ> Option.map f"
by (auto simp add: fun_eq_iff Option.map_def split: option.split)
next
fix f g x
assume "\<And>z. z \<in> Option.set x \<Longrightarrow> f z = g z"
thus "Option.map f x = Option.map g x"
by (simp cong: Option.map_cong)
next
fix f
show "Option.set \<circ> Option.map f = op ` f \<circ> Option.set"
by fastforce
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix x
show "|Option.set x| \<le>o natLeq"
by (cases x) (simp_all add: ordLess_imp_ordLeq finite_iff_ordLess_natLeq[symmetric])
next
fix R S
show "option_rel R OO option_rel S \<le> option_rel (R OO S)"
by (auto simp: option_rel_def split: option.splits)
next
fix z
assume "z \<in> Option.set None"
thus False by simp
next
fix R
show "option_rel R =
(Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map fst))\<inverse>\<inverse> OO
Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map snd)"
unfolding option_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff prod.cases
by (auto simp: trans[OF eq_commute option_map_is_None] trans[OF eq_commute option_map_eq_Some]
split: option.splits)
qed
end