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products/sources/formale Sprachen/Isabelle/HOL/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 65 kB

SSL Option.thy Interaktion und
PortierbarkeitIsabelle

    Authorof
*)

section

Option intro  simpxjava.lang.NullPointerException
Lifting
begin( x)simp_all

datatype java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    None
  | Some (the: 'a)

datatype_compat option

lemma [case_names None Some, cases type: optioninj_onI) simp
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
  "(y = None \ P) \ (\a. y = Some a \ P) \ P"
  by (rule option.exhaust)

lemma [case_names None Some, induct type: option]:
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
  "P None \ (\option. P (Some option)) \ P option"
  by (rule option.induct)

text \<open>Compatibility:\<close>
setup \<open>Sign.mandatory_path "option"\<close>
lemmas inducts = option.induct
lemmas cases = option.case
setup \<open>Sign.parent_path\<close>

lemma not_None_eq [iff]: "x \ None \ (\y. x = Some y)"
  by (induct x) auto

lemma not_Some_eq [iff]: "(\y. x \ Some y) \ x = None"
  by (induct x) auto

lemma comp_the_Some[simp]: "the o Some = id"
by auto

text \<open>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.\<close>

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: "(\x. P x) \ P None \ (\x. P (Some x))"
  by (auto intro: option.induct)

lemma split_option_ex: "(\x. P x) \ P None \ (\x. P (Some x))"
  using split_option_all[of "\x. \ P x"] by blast

lemma UNIV_option_conv: "UNIV = insert None (range Some)"
  by (auto intro: classical)

lemma rel_option_None1 [simp]: "rel_option P None x \ x = None"
  by (cases x) simp_all

lemma rel_option_None2 [simp]: "rel_option P x None \ x = None"
  by (cases x) simp_all

lemma option_rel_Some1: "rel_option A (Some x) y \ (\y'. y = Some y' \ A x y')" (* Option *)
by(cases y) simp_all

lemma option_rel_Some2: "rel_option A x (Some y) \ (\x'. x = Some x' \ A x' y)" (* Option *)
by(cases x) simp_all

lemma rel_option_inf: "inf (rel_option A) (rel_option B) = rel_option (inf A B)"
  (is "?lhs = ?rhs")
proof (rule antisym)
  show "?lhs \ ?rhs" by (auto elim: option.rel_cases)
  show "?rhs \ ?lhs" by (auto elim: option.rel_mono_strong)
qed

lemma rel_option_reflI:
  "(\x. x \ set_option y \ P x x) \ rel_option P y y"
  by (cases y) auto

subsubsection \<open>Operations\<close>

lemma ospec [dest]: "(\x\set_option A. P x) \ A = Some x \ P x"
  by simp

setup \<open>map_theory_claset (fn ctxt => ctxt addSD2 ("ospec", @{thm ospec}))\<close>

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: option.split)

lemma None_eq_map_option_iff [iff]: "None = map_option f x \ x = None"
by(cases x) simp_all

lemma map_option_eq_Some [iff]: "(map_option f xo = Some y) = (\z. xo = Some z \ f z = y)"
  by (simp add: map_option_case split: option.split)

lemma map_option_o_case_sum [simp]:
    "map_option f \ case_sum g h = case_sum (map_option f \ g) (map_option f \ h)"
  by (rule o_case_sum)

lemma map_option_cong: "x = y \ (\a. y = Some a \ f a = g a) \ map_option f x = map_option g y"
  by (cases x) auto

lemma map_option_idI: "(\y. y \ set_option x \ f y = y) \ map_option f x = x"
by(cases x)(simp_all)

functor map_option: map_option
  by (simp_all add: option.map_comp fun_eq_iff option.map_id)

lemma case_map_option [simp]: "case_option g h (map_option f x) = case_option g (h \ f) x"
  by (cases x) simp_all

lemma None_notin_image_Some [simp]: "None \ Some ` A"
by auto

lemma notin_range_Some: "x \ range Some \ x = None"
by(cases x) auto

lemma rel_option_iff:
  "rel_option R x y = (case (x, y) of (None, None) \ True
    | (Some x, Some y) \<Rightarrow> R x y
    | _ \<Rightarrow> False)"
  by (auto split: prod.split option.split)

definition combine_options :: "('a \ 'a \ 'a) \ 'a option \ 'a option \ 'a option"
  where "combine_options f x y =
           (case x of None \<Rightarrow> y | Some x \<Rightarrow> (case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y)))"

lemma combine_options_simps [simp]:
  "combine_options f None y = y"
  "combine_options f x None = x"
  "combine_options f (Some a) (Some b) = Some (f a b)"
  by (simp_all add: combine_options_def split: option.splits)

lemma combine_options_cases [case_names None1 None2 Some]:
  "(x = None \ P x y) \ (y = None \ P x y) \
     (\<And>a b. x = Some a \<Longrightarrow> y = Some b \<Longrightarrow> P x y) \<Longrightarrow> P x y"
  by (cases x; cases y) simp_all

lemma combine_options_commute:
  "(\x y. f x y = f y x) \ combine_options f x y = combine_options f y x"
  using combine_options_cases[of x ]
  by (induction x y rule: combine_options_cases) simp_all

lemma combine_options_assoc:
  "(\x y z. f (f x y) z = f x (f y z)) \
     combine_options f (combine_options f x y) z =
     combine_options f x (combine_options f y z)"
  by (auto simp: combine_options_def split: option.splits)

lemma combine_options_left_commute:
  "(\x y. f x y = f y x) \ (\x y z. f (f x y) z = f x (f y z)) \
     combine_options f y (combine_options f x z) =
     combine_options f x (combine_options f y z)"
  by (auto simp: combine_options_def split: option.splits)

lemmas combine_options_ac =
  combine_options_commute combine_options_assoc combine_options_left_commute

context
begin

qualified definition is_none :: "'a option \ bool"
  where [code_post]: "is_none x \ x = None"

lemma is_none_simps [simp]:
  "is_none None"
  "\ is_none (Some x)"
  by (simp_all add: is_none_def)

lemma is_none_code [code]:
  "is_none None \ True"
  "is_none (Some x) \ False"
  by simp_all

lemma rel_option_unfold:
  "rel_option R x y \
   (is_none x \<longleftrightarrow> is_none y) \<and> (\<not> is_none x \<longrightarrow> \<not> is_none y \<longrightarrow> R (the x) (the y))"
  by (simp add: rel_option_iff split: option.split)

lemma rel_optionI:
  "\ is_none x \ is_none y; \ \ is_none x; \ is_none y \ \ P (the x) (the y) \
  \<Longrightarrow> rel_option P x y"
  by (simp add: rel_option_unfold)

lemma is_none_map_option [simp]: "is_none (map_option f x) \ is_none x"
  by (simp add: is_none_def)

lemma the_map_option: "\ is_none x \ the (map_option f x) = f (the x)"
  by (auto simp add: is_none_def)

qualified 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 is_none_bind: "is_none (bind f g) \ is_none f \ is_none (g (the f))"
  by (cases f) simp_all

lemma bind_runit[simp]: "bind x Some = x"
  by (cases x) auto

lemma bind_assoc[simp]: "bind (bind x f) g = bind x (\y. bind (f y) g)"
  by (cases x) auto

lemma bind_rzero[simp]: "bind x (\x. None) = None"
  by (cases x) auto

qualified lemma bind_cong: "x = y \ (\a. y = Some a \ f a = g a) \ bind x f = bind y g"
  by (cases x) auto

lemma bind_split: "P (bind m f) \ (m = None \ P None) \ (\v. m = Some v \ P (f v))"
  by (cases m) auto

lemma bind_split_asm: "P (bind m f) \ \ (m = None \ \ P None \ (\x. m = Some x \ \ P (f x)))"
  by (cases m) auto

lemmas bind_splits = bind_split bind_split_asm

lemma bind_eq_Some_conv: "bind f g = Some x \ (\y. f = Some y \ g y = Some x)"
  by (cases f) simp_all

lemma bind_eq_None_conv: "Option.bind a f = None \ a = None \ f (the a) = None"
by(cases a) simp_all

lemma map_option_bind: "map_option f (bind x g) = bind x (map_option f \ g)"
  by (cases x) simp_all

lemma bind_option_cong:
  "\ x = y; \z. z \ set_option y \ f z = g z \ \ bind x f = bind y g"
  by (cases y) simp_all

lemma bind_option_cong_simp:
  "\ x = y; \z. z \ set_option y =simp=> f z = g z \ \ bind x f = bind y g"
  unfolding simp_implies_def by (rule bind_option_cong)

lemma bind_option_cong_code: "x = y \ bind x f = bind y f"
  by simp

lemma bind_map_option: "bind (map_option f x) g = bind x (g \ f)"
by(cases x) simp_all

lemma set_bind_option [simp]: "set_option (bind x f) = (\((set_option \ f) ` set_option x))"
by(cases x) auto

lemma map_conv_bind_option: "map_option f x = Option.bind x (Some \ f)"
by(cases x) simp_all

end

setup \<open>Code_Simp.map_ss (Simplifier.add_cong @{thm bind_option_cong_code})\<close>

context
begin

qualified definition these :: "'a option set \ 'a set"
  where "these A = the ` {x \ A. x \ None}"

qualified lemma these_eq [code]:
  \<open>these A = the ` (Set.filter (Not \<circ> Option.is_none) A)\<close>
  by (simp add: these_def Option.is_none_def)

qualified lemma these_unfold:
  \<open>these A = {x. \<exists>y \<in> A. y = Some x}\<close>
  by (auto simp add: these_def set_eq_iff image_iff)

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)

qualified definition image_filter :: "('a \ 'b option) \ 'a set \ 'b set"
  where image_filter_eq: "image_filter f A = these (f ` A)"

end

lemma finite_range_Some: "finite (range (Some :: 'a \ 'a option)) = finite (UNIV :: 'a set)"
  by (auto dest: finite_imageD intro: inj_Some)

subsection \<open>Transfer rules for the Transfer package\<close>

context includes lifting_syntax
begin

lemma option_bind_transfer [transfer_rule]:
  "(rel_option A ===> (A ===> rel_option B) ===> rel_option B)
    Option.bind Option.bind"
  unfolding rel_fun_def split_option_all by simp

lemma pred_option_parametric [transfer_rule]:
  "((A ===> (=)) ===> rel_option A ===> (=)) pred_option pred_option"
  by (rule rel_funI)+ (auto simp add: rel_option_unfold Option.is_none_def dest: rel_funD)

end

subsubsection \<open>Interaction with finite sets\<close>

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 standard (simp add: UNIV_option_conv)

subsubsection \<open>Code generator setup\<close>

lemma equal_None_code_unfold [code_unfold]:
  "HOL.equal x None \ Option.is_none x"
  "HOL.equal None = Option.is_none"
  by (auto simp add: equal Option.is_none_def)

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 \ 'a option \ bool" \
    (Haskell) infix 4 "=="

code_reserved
  (SML) option NONE SOME
  and (OCaml) option None Some
  and (Scala) Option None Some

end

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