(*  Title:      HOL/Option.thy

     Author:     Folklore

 *)

 header {* Datatype option *}

 theory Option

 imports Datatype Finite_Set

 begin

 datatype 'a option = None | Some 'a

 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 option_caseE:

   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 UNIV_option_conv: "UNIV = insert None (range Some)"

 by(auto intro: classical)

 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 proof

 qed (simp add: UNIV_option_conv)

 subsubsection {* Operations *}

 primrec the :: "'a option => 'a" where

 "the (Some x) = x"

 primrec set :: "'a option => 'a set" where

 "set None = {}" |

 "set (Some x) = {x}"

 lemma ospec [dest]: "(ALL x:set A. P x) ==> A = Some x ==> P x"

   by simp

 declaration {* fn _ =>

   Classical.map_cs (fn cs => cs addSD2 ("ospec", thm "ospec"))

 *}

 lemma elem_set [iff]: "(x : set xo) = (xo = Some x)"

   by (cases xo) auto

 lemma set_empty_eq [simp]: "(set xo = {}) = (xo = None)"

   by (cases xo) auto

 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option" where

   "map = (%f y. case y of None => None | Some x => Some (f x))"

 lemma option_map_None [simp, code]: "map f None = None"

   by (simp add: map_def)

 lemma option_map_Some [simp, code]: "map f (Some x) = Some (f x)"

   by (simp add: map_def)

 lemma option_map_is_None [iff]:

     "(map f opt = None) = (opt = None)"

   by (simp add: map_def split add: option.split)

 lemma option_map_eq_Some [iff]:

     "(map f xo = Some y) = (EX z. xo = Some z & f z = y)"

   by (simp add: map_def split add: option.split)

 lemma option_map_comp:

     "map f (map g opt) = map (f o g) opt"

   by (simp add: map_def split add: option.split)

 lemma option_map_o_sum_case [simp]:

     "map f o sum_case g h = sum_case (map f o g) (map f o h)"

   by (rule ext) (simp split: sum.split)

 hide (open) const set map

 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 is_none_none:

   "is_none x \<longleftrightarrow> x = None"

   by (simp add: is_none_def)

 lemma [code_unfold]:

   "eq_class.eq x None \<longleftrightarrow> is_none x"

   by (simp add: eq is_none_none)

 hide (open) const is_none

 code_type option

   (SML "_ option")

   (OCaml "_ option")

   (Haskell "Maybe _")

 code_const None and Some

   (SML "NONE" and "SOME")

   (OCaml "None" and "Some _")

   (Haskell "Nothing" and "Just")

 code_instance option :: eq

   (Haskell -)

 code_const "eq_class.eq \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"

   (Haskell infixl 4 "==")

 code_reserved SML

   option NONE SOME

 code_reserved OCaml

   option None Some

 end