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 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)
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 => 'b) => 'a option => '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)
lemma map_cong: "x = y ==> (!!a. y = Some a ==> f a = g a) ==> map f x = map g y"
by (cases x) auto
enriched_type map: Option.map proof -
fix f g
show "Option.map f o Option.map g = Option.map (f o g)"
proof
fix x
show "(Option.map f o Option.map g) x= Option.map (f o g) x"
by (cases x) simp_all
qed
next
show "Option.map id = id"
proof
fix x
show "Option.map id x = id x"
by (cases x) simp_all
qed
qed
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 bind_runit[simp]: "bind x Some = x"
by (cases x) auto
lemma bind_assoc[simp]: "bind (bind x f) g = bind x (λy. bind (f y) g)"
by (cases x) auto
lemma bind_rzero[simp]: "bind x (λx. None) = None"
by (cases x) auto
lemma bind_cong: "x = y ==> (!!a. y = Some a ==> f a = g a) ==> bind x f = bind y g"
by (cases x) auto
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)
hide_const (open) set map bind these
hide_fact (open) map_cong bind_cong
subsubsection {* Code generator setup *}
definition is_none :: "'a option => bool" where
[code_post]: "is_none x <-> x = None"
lemma is_none_code [code]:
shows "is_none None <-> True"
and "is_none (Some x) <-> False"
unfolding is_none_def by simp_all
lemma [code_unfold]:
"HOL.equal x None <-> is_none x"
by (simp add: equal is_none_def)
hide_const (open) is_none
code_type option
(SML "_ option")
(OCaml "_ option")
(Haskell "Maybe _")
(Scala "!Option[(_)]")
code_const None and Some
(SML "NONE" and "SOME")
(OCaml "None" and "Some _")
(Haskell "Nothing" and "Just")
(Scala "!None" and "Some")
code_instance option :: equal
(Haskell -)
code_const "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