(* Title: HOL/Option.thy
Author: Folklore
*)
section \<open>Datatype option\<close>
theory Option
imports Lifting
begin
datatype 'a option =
None
| Some (the: 'a)
datatype_compat option
lemma [case_names None Some, cases type: option]:
\<comment> \<open>for backward compatibility -- names of variables differ\<close>
"(y = None \ P) \ (\a. y = Some a \ P) \ P"
by (rule option.exhaust)
lemma [case_names None Some, induct type: option]:
\<comment> \<open>for backward compatibility -- names of variables differ\<close>
"P None \ (\option. P (Some option)) \ P option"
by (rule option.induct)
text \<open>Compatibility:\<close>
setup \<open>Sign.mandatory_path "option"\<close>
lemmas inducts = option.induct
lemmas cases = option.case
setup \<open>Sign.parent_path\<close>
lemma not_None_eq [iff]: "x \ None \ (\y. x = Some y)"
by (induct x) auto
lemma not_Some_eq [iff]: "(\y. x \ Some y) \ x = None"
by (induct x) auto
lemma comp_the_Some[simp]: "the o Some = id"
by auto
text \<open>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.\<close>
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: "(\x. P x) \ P None \ (\x. P (Some x))"
by (auto intro: option.induct)
lemma split_option_ex: "(\x. P x) \ P None \ (\x. P (Some x))"
using split_option_all[of "\x. \ P x"] by blast
lemma UNIV_option_conv: "UNIV = insert None (range Some)"
by (auto intro: classical)
lemma rel_option_None1 [simp]: "rel_option P None x \ x = None"
by (cases x) simp_all
lemma rel_option_None2 [simp]: "rel_option P x None \ x = None"
by (cases x) simp_all
lemma option_rel_Some1: "rel_option A (Some x) y \ (\y'. y = Some y' \ A x y')" (* Option *)
by(cases y) simp_all
lemma option_rel_Some2: "rel_option A x (Some y) \ (\x'. x = Some x' \ A x' y)" (* Option *)
by(cases x) simp_all
lemma rel_option_inf: "inf (rel_option A) (rel_option B) = rel_option (inf A B)"
(is "?lhs = ?rhs")
proof (rule antisym)
show "?lhs \ ?rhs" by (auto elim: option.rel_cases)
show "?rhs \ ?lhs" by (auto elim: option.rel_mono_strong)
qed
lemma rel_option_reflI:
"(\x. x \ set_option y \ P x x) \ rel_option P y y"
by (cases y) auto
subsubsection \<open>Operations\<close>
lemma ospec [dest]: "(\x\set_option A. P x) \ A = Some x \ P x"
by simp
setup \<open>map_theory_claset (fn ctxt => ctxt addSD2 ("ospec", @{thm ospec}))\<close>
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: option.split)
lemma None_eq_map_option_iff [iff]: "None = map_option f x \ x = None"
by(cases x) simp_all
lemma map_option_eq_Some [iff]: "(map_option f xo = Some y) = (\z. xo = Some z \ f z = y)"
by (simp add: map_option_case split: option.split)
lemma map_option_o_case_sum [simp]:
"map_option f \ case_sum g h = case_sum (map_option f \ g) (map_option f \ h)"
by (rule o_case_sum)
lemma map_option_cong: "x = y \ (\a. y = Some a \ f a = g a) \ map_option f x = map_option g y"
by (cases x) auto
lemma map_option_idI: "(\y. y \ set_option x \ f y = y) \ map_option f x = x"
by(cases x)(simp_all)
functor map_option: map_option
by (simp_all add: option.map_comp fun_eq_iff option.map_id)
lemma case_map_option [simp]: "case_option g h (map_option f x) = case_option g (h \ f) x"
by (cases x) simp_all
lemma None_notin_image_Some [simp]: "None \ Some ` A"
by auto
lemma notin_range_Some: "x \ range Some \ x = None"
by(cases x) auto
lemma rel_option_iff:
"rel_option R x y = (case (x, y) of (None, None) \ True
| (Some x, Some y) \<Rightarrow> R x y
| _ \<Rightarrow> False)"
by (auto split: prod.split option.split)
definition combine_options :: "('a \ 'a \ 'a) \ 'a option \ 'a option \ 'a option"
where "combine_options f x y =
(case x of None \<Rightarrow> y | Some x \<Rightarrow> (case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y)))"
lemma combine_options_simps [simp]:
"combine_options f None y = y"
"combine_options f x None = x"
"combine_options f (Some a) (Some b) = Some (f a b)"
by (simp_all add: combine_options_def split: option.splits)
lemma combine_options_cases [case_names None1 None2 Some]:
"(x = None \ P x y) \ (y = None \ P x y) \
(\<And>a b. x = Some a \<Longrightarrow> y = Some b \<Longrightarrow> P x y) \<Longrightarrow> P x y"
by (cases x; cases y) simp_all
lemma combine_options_commute:
"(\x y. f x y = f y x) \ combine_options f x y = combine_options f y x"
using combine_options_cases[of x ]
by (induction x y rule: combine_options_cases) simp_all
lemma combine_options_assoc:
"(\x y z. f (f x y) z = f x (f y z)) \
combine_options f (combine_options f x y) z =
combine_options f x (combine_options f y z)"
by (auto simp: combine_options_def split: option.splits)
lemma combine_options_left_commute:
"(\x y. f x y = f y x) \ (\x y z. f (f x y) z = f x (f y z)) \
combine_options f y (combine_options f x z) =
combine_options f x (combine_options f y z)"
by (auto simp: combine_options_def split: option.splits)
lemmas combine_options_ac =
combine_options_commute combine_options_assoc combine_options_left_commute
context
begin
qualified definition is_none :: "'a option \ bool"
where [code_post]: "is_none x \ x = None"
lemma is_none_simps [simp]:
"is_none None"
"\ is_none (Some x)"
by (simp_all add: is_none_def)
lemma is_none_code [code]:
"is_none None = True"
"is_none (Some x) = False"
by simp_all
lemma rel_option_unfold:
"rel_option R x y \
(is_none x \<longleftrightarrow> is_none y) \<and> (\<not> is_none x \<longrightarrow> \<not> is_none y \<longrightarrow> R (the x) (the y))"
by (simp add: rel_option_iff split: option.split)
lemma rel_optionI:
"\ is_none x \ is_none y; \ \ is_none x; \ is_none y \ \ P (the x) (the y) \
\<Longrightarrow> rel_option P x y"
by (simp add: rel_option_unfold)
lemma is_none_map_option [simp]: "is_none (map_option f x) \ is_none x"
by (simp add: is_none_def)
lemma the_map_option: "\ is_none x \ the (map_option f x) = f (the x)"
by (auto simp add: is_none_def)
qualified primrec bind :: "'a option \ ('a \ 'b option) \ 'b option"
where
bind_lzero: "bind None f = None"
| bind_lunit: "bind (Some x) f = f x"
lemma is_none_bind: "is_none (bind f g) \ is_none f \ is_none (g (the f))"
by (cases f) simp_all
lemma bind_runit[simp]: "bind x Some = x"
by (cases x) auto
lemma bind_assoc[simp]: "bind (bind x f) g = bind x (\y. bind (f y) g)"
by (cases x) auto
lemma bind_rzero[simp]: "bind x (\x. None) = None"
by (cases x) auto
qualified lemma bind_cong: "x = y \ (\a. y = Some a \ f a = g a) \ bind x f = bind y g"
by (cases x) auto
lemma bind_split: "P (bind m f) \ (m = None \ P None) \ (\v. m = Some v \ P (f v))"
by (cases m) auto
lemma bind_split_asm: "P (bind m f) \ \ (m = None \ \ P None \ (\x. m = Some x \ \ P (f x)))"
by (cases m) auto
lemmas bind_splits = bind_split bind_split_asm
lemma bind_eq_Some_conv: "bind f g = Some x \ (\y. f = Some y \ g y = Some x)"
by (cases f) simp_all
lemma bind_eq_None_conv: "Option.bind a f = None \ a = None \ f (the a) = None"
by(cases a) simp_all
lemma map_option_bind: "map_option f (bind x g) = bind x (map_option f \ g)"
by (cases x) simp_all
lemma bind_option_cong:
"\ x = y; \z. z \ set_option y \ f z = g z \ \ bind x f = bind y g"
by (cases y) simp_all
lemma bind_option_cong_simp:
"\ x = y; \z. z \ set_option y =simp=> f z = g z \ \ bind x f = bind y g"
unfolding simp_implies_def by (rule bind_option_cong)
lemma bind_option_cong_code: "x = y \ bind x f = bind y f"
by simp
lemma bind_map_option: "bind (map_option f x) g = bind x (g \ f)"
by(cases x) simp_all
lemma set_bind_option [simp]: "set_option (bind x f) = (\((set_option \ f) ` set_option x))"
by(cases x) auto
lemma map_conv_bind_option: "map_option f x = Option.bind x (Some \ f)"
by(cases x) simp_all
end
setup \<open>Code_Simp.map_ss (Simplifier.add_cong @{thm bind_option_cong_code})\<close>
context
begin
qualified definition these :: "'a option set \ 'a set"
where "these A = the ` {x \ A. x \ 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 \ insert (Some x) A. y \ None} = insert (Some x) {y \ A. y \ None}"
by auto
then show ?thesis by (simp add: these_def)
qed
lemma in_these_eq: "x \ these A \ Some x \ A"
proof
assume "Some x \ A"
then obtain B where "A = insert (Some x) B" by auto
then show "x \ these A" by (auto simp add: these_def intro!: image_eqI)
next
assume "x \ these A"
then show "Some x \ 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\A. x \ None}"
by (auto simp add: these_def image_image intro!: image_eqI)
lemma these_empty_eq: "these B = {} \ B = {} \ B = {None}"
by (auto simp add: these_def)
lemma these_not_empty_eq: "these B \ {} \ B \ {} \ B \ {None}"
by (auto simp add: these_empty_eq)
end
lemma finite_range_Some: "finite (range (Some :: 'a \ 'a option)) = finite (UNIV :: 'a set)"
by (auto dest: finite_imageD intro: inj_Some)
subsection \<open>Transfer rules for the Transfer package\<close>
context includes lifting_syntax
begin
lemma option_bind_transfer [transfer_rule]:
"(rel_option A ===> (A ===> rel_option B) ===> rel_option B)
Option.bind Option.bind"
unfolding rel_fun_def split_option_all by simp
lemma pred_option_parametric [transfer_rule]:
"((A ===> (=)) ===> rel_option A ===> (=)) pred_option pred_option"
by (rule rel_funI)+ (auto simp add: rel_option_unfold Option.is_none_def dest: rel_funD)
end
subsubsection \<open>Interaction with finite sets\<close>
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 standard (simp add: UNIV_option_conv)
subsubsection \<open>Code generator setup\<close>
lemma equal_None_code_unfold [code_unfold]:
"HOL.equal x None \ Option.is_none x"
"HOL.equal None = Option.is_none"
by (auto simp add: equal Option.is_none_def)
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 \ 'a option \ bool" \
(Haskell) infix 4 "=="
code_reserved SML
option NONE SOME
code_reserved OCaml
option None Some
code_reserved Scala
Option None Some
end
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