(*  Title:      HOL/Option.thy

     Author:     Folklore

 *)

 header {* Datatype option *}

 theory Option

 imports BNF_LFP Datatype Finite_Set

 begin

 datatype_new '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