(* Title: HOL/Option.thy
Author: Folklore
*)
header {* Datatype option *}
theory Option
imports BNF_Least_Fixpoint Finite_Set
begin
datatype 'a option =
None
| Some (the: 'a)
datatype_compat option
lemma [case_names None Some, cases type: option]:
-- {* for backward compatibility -- names of variables differ *}
"(y = None \<Longrightarrow> P) \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> P) \<Longrightarrow> P"
by (rule option.exhaust)
lemma [case_names None Some, induct type: option]:
-- {* for backward compatibility -- names of variables differ *}
"P None \<Longrightarrow> (\<And>option. P (Some option)) \<Longrightarrow> P option"
by (rule option.induct)
text {* Compatibility: *}
setup {* Sign.mandatory_path "option" *}
lemmas inducts = option.induct
lemmas cases = option.case
setup {* Sign.parent_path *}
lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
by (induct x) auto
lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
by (induct x) auto
text{*Although it may appear that both of these equalities are helpful
only when applied to assumptions, in practice it seems better to give
them the uniform iff attribute. *}
lemma inj_Some [simp]: "inj_on Some A"
by (rule inj_onI) simp
lemma case_optionE:
assumes c: "(case x of None => P | Some y => Q y)"
obtains
(None) "x = None" and P
| (Some) y where "x = Some y" and "Q y"
using c by (cases x) simp_all
lemma split_option_all: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P (Some x))"
by (auto intro: option.induct)
lemma split_option_ex: "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P (Some x))"
using split_option_all[of "\<lambda>x. \<not>P x"] by blast
lemma UNIV_option_conv: "UNIV = insert None (range Some)"
by(auto intro: classical)
subsubsection {* Operations *}
lemma ospec [dest]: "(ALL x:set_option A. P x) ==> A = Some x ==> P x"
by simp
setup {* map_theory_claset (fn ctxt => ctxt addSD2 ("ospec", @{thm ospec})) *}
lemma elem_set [iff]: "(x : set_option xo) = (xo = Some x)"
by (cases xo) auto
lemma set_empty_eq [simp]: "(set_option xo = {}) = (xo = None)"
by (cases xo) auto
lemma map_option_case: "map_option f y = (case y of None => None | Some x => Some (f x))"
by (auto split: option.split)
lemma map_option_is_None [iff]:
"(map_option f opt = None) = (opt = None)"
by (simp add: map_option_case split add: option.split)
lemma map_option_eq_Some [iff]:
"(map_option f xo = Some y) = (EX z. xo = Some z & f z = y)"
by (simp add: map_option_case split add: option.split)
lemma map_option_o_case_sum [simp]:
"map_option f o case_sum g h = case_sum (map_option f o g) (map_option f o h)"
by (rule o_case_sum)
lemma map_option_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> map_option f x = map_option g y"
by (cases x) auto
functor map_option: map_option proof -
fix f g
show "map_option f \<circ> map_option g = map_option (f \<circ> g)"
proof
fix x
show "(map_option f \<circ> map_option g) x= map_option (f \<circ> g) x"
by (cases x) simp_all
qed
next
show "map_option id = id"
proof
fix x
show "map_option id x = id x"
by (cases x) simp_all
qed
qed
lemma case_map_option [simp]:
"case_option g h (map_option f x) = case_option g (h \<circ> f) x"
by (cases x) simp_all
primrec bind :: "'a option \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'b option" where
bind_lzero: "bind None f = None" |
bind_lunit: "bind (Some x) f = f x"
lemma bind_runit[simp]: "bind x Some = x"
by (cases x) auto
lemma bind_assoc[simp]: "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
by (cases x) auto
lemma bind_rzero[simp]: "bind x (\<lambda>x. None) = None"
by (cases x) auto
lemma bind_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> bind x f = bind y g"
by (cases x) auto
definition these :: "'a option set \<Rightarrow> 'a set"
where
"these A = the ` {x \<in> A. x \<noteq> None}"
lemma these_empty [simp]:
"these {} = {}"
by (simp add: these_def)
lemma these_insert_None [simp]:
"these (insert None A) = these A"
by (auto simp add: these_def)
lemma these_insert_Some [simp]:
"these (insert (Some x) A) = insert x (these A)"
proof -
have "{y \<in> insert (Some x) A. y \<noteq> None} = insert (Some x) {y \<in> A. y \<noteq> None}"
by auto
then show ?thesis by (simp add: these_def)
qed
lemma in_these_eq:
"x \<in> these A \<longleftrightarrow> Some x \<in> A"
proof
assume "Some x \<in> A"
then obtain B where "A = insert (Some x) B" by auto
then show "x \<in> these A" by (auto simp add: these_def intro!: image_eqI)
next
assume "x \<in> these A"
then show "Some x \<in> A" by (auto simp add: these_def)
qed
lemma these_image_Some_eq [simp]:
"these (Some ` A) = A"
by (auto simp add: these_def intro!: image_eqI)
lemma Some_image_these_eq:
"Some ` these A = {x\<in>A. x \<noteq> None}"
by (auto simp add: these_def image_image intro!: image_eqI)
lemma these_empty_eq:
"these B = {} \<longleftrightarrow> B = {} \<or> B = {None}"
by (auto simp add: these_def)
lemma these_not_empty_eq:
"these B \<noteq> {} \<longleftrightarrow> B \<noteq> {} \<and> B \<noteq> {None}"
by (auto simp add: these_empty_eq)
hide_const (open) bind these
hide_fact (open) bind_cong
subsubsection {* Interaction with finite sets *}
lemma finite_option_UNIV [simp]:
"finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
instance option :: (finite) finite
by default (simp add: UNIV_option_conv)
subsubsection {* Code generator setup *}
definition is_none :: "'a option \<Rightarrow> bool" where
[code_post]: "is_none x \<longleftrightarrow> x = None"
lemma is_none_code [code]:
shows "is_none None \<longleftrightarrow> True"
and "is_none (Some x) \<longleftrightarrow> False"
unfolding is_none_def by simp_all
lemma [code_unfold]:
"HOL.equal x None \<longleftrightarrow> is_none x"
"HOL.equal None = is_none"
by (auto simp add: equal is_none_def)
hide_const (open) is_none
code_printing
type_constructor option \<rightharpoonup>
(SML) "_ option"
and (OCaml) "_ option"
and (Haskell) "Maybe _"
and (Scala) "!Option[(_)]"
| constant None \<rightharpoonup>
(SML) "NONE"
and (OCaml) "None"
and (Haskell) "Nothing"
and (Scala) "!None"
| constant Some \<rightharpoonup>
(SML) "SOME"
and (OCaml) "Some _"
and (Haskell) "Just"
and (Scala) "Some"
| class_instance option :: equal \<rightharpoonup>
(Haskell) -
| constant "HOL.equal :: 'a option \<Rightarrow> 'a option \<Rightarrow> bool" \<rightharpoonup>
(Haskell) infix 4 "=="
code_reserved SML
option NONE SOME
code_reserved OCaml
option None Some
code_reserved Scala
Option None Some
end