src/HOL/Option.thy
 author haftmann Tue, 14 Jul 2009 10:54:04 +0200 changeset 31998 2c7a24f74db9 parent 31154 f919b8e67413 child 32069 6d28bbd33e2c permissions -rw-r--r--
code attributes use common underscore convention
```
(*  Title:      HOL/Option.thy
Author:     Folklore
*)

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

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"

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

lemma option_map_is_None [iff]:
"(map f opt = None) = (opt = None)"

lemma option_map_eq_Some [iff]:
"(map f xo = Some y) = (EX z. xo = Some z & f z = y)"

lemma option_map_comp:
"map f (map g opt) = map (f o g) opt"

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"

lemma [code_inline]:
"eq_class.eq x None \<longleftrightarrow> is_none x"

hide (open) const is_none

code_type option
(SML "_ option")
(OCaml "_ option")

code_const None and Some
(SML "NONE" and "SOME")
(OCaml "None" and "Some _")

code_instance option :: eq

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

code_reserved SML
option NONE SOME

code_reserved OCaml
option None Some

end
```