--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Option.thy Wed Mar 04 11:48:52 2009 +0100
@@ -0,0 +1,124 @@
+(* Title: HOL/Option.thy
+ Author: Folklore
+*)
+
+header {* Datatype option *}
+
+theory Option
+imports Datatype
+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 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 insert_None_conv_UNIV: "insert None (range Some) = UNIV"
+ by (rule set_ext, case_tac x) auto
+
+lemma inj_Some [simp]: "inj_on Some A"
+ by (rule inj_onI) simp
+
+
+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
+ [code del]: "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
+ is_none_none [code post, symmetric, code inline]: "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_none [symmetric] by simp_all
+
+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