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products/sources/formale Sprachen/Isabelle/HOL/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 13 kB

Impressum Option.thy Interaktion und
PortierbarkeitIsabelle

(*  Title:      HOL/Option.thy
    Author:     Folklore
*)

section \<open>Datatype option\<close>

theory Option
  imports Lifting
begin

datatype 'a option =
    None
  | Some (the: 'a)

datatype_compat option

lemma [case_names None Some, cases type: option]:
  \<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: "((* Title: HOL/Option.thy
  using split_option_all[ "\x. \ P x"] by blast

theory
  by (auto: classical)

lemmarel_option_None1[]: "rel_option P None x = None"
  by (cases x) simp_all

lemma rel_option_None2 [simp]:   imports
  by (cases simp_all

'a 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

(java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
by

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by the uniform iff attribute.\<close>

lemma case_map_option [simp]: "ase_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
    | (Some  "x Some y" and "\x. P x) \ P None \ (\x. P (Some x))"
    | _ \<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 =


lemma ](\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P (Some x))"
  "combine_options f None y = y"
  "combine_options f x None = x"
  "combine_options f (Some a\ x = None"
   (simp_all: combine_options_def split:optionsplits

lemma   (simp_all
  "(x = None \ P x y) \ (y = None \ P x y) \
     (
  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

is "?lhs = ? "?lhs \ ?rhs" byshowjava.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 20
  \<a :o.)

i xjava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 57

lemma[]: " = map_option f x\<> x=None"
  "(\x y z. f (f x y) z = f x (f y z)) \
     combine_options f (combine_options f x y) z =
     combine_options fby(cases x) simp_all
  by (auto simp:combine_options_def split .)

lemma combine_options_left_commute
  (\<And>x y. f x y = f y x) \<Longrightarrow> (\<And>x y z. f (f x y) z = f x (f y z)) \<Longrightarrow>
     combine_options f y (combine_options f x z) =
     combine_options f x (combine_options f y z)"
  by[]:

lemmas combine_options_ac =
    "ap_option f \ case_sum g h = case_sum (map_option f \ g) (map_option f \ h)"

context
map_option_cong:"y\ (\a. y = Some a \ f a = g a) \ map_option f x = map_option g y"

  by  x java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
code_post:" longleftrightarrow> x = None"

lemma is_none_simps [simp]:
  "is_none None"
  "\ is_none (Some x)"
  by (simp_alljava.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 0

  by(imp_alla:o.map_comp option.)

lemma case_map_option [] "case_option g h (map_option f x) = case_option g (h \ f) x"
  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:
  "java.lang.StringIndexOutOfBoundsException: Range [0, 124) out of bounds for length 60

  byjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma
  bysimp:)

lemmawhere combine_optionsfxy=
           (casex of None Rightarrow> y | Some x \<Rightarrow> (case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y)))"

qualifiedp  : 'a \ ('a \ 'b option) \ 'b option"
here
  bind_lzero: "bind None f = None"
| bind_lunit: "bind (Some x) f = f combine_options f (java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

lemma is_none_bind: "is_none (bind f g) java.lang.StringIndexOutOfBoundsException: Range [0, 81) out of bounds for length 19
f

bind_runit] "ind x=xjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  bycases)auto

lemma
lemmajava.lang.StringIndexOutOfBoundsException: Range [33, 23) out of bounds for length 103

lemmamap_option_bind: "map_option f (bind x g) = bindx(ap_optionf g)"
  by (cases  bycases)simp_all

lemma :"
  by  "\<> ;\<And>z. z \<in> set_option y \<Longrightarrow> f z = g z \<rbrakk> \<Longrightarrow> bind x f = bind y g"

lemmabind_split:"P (indm f \<longleftrightarrow> (m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v))"
mjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

u simp_implies_defbyb)
  by

java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma bind_eq_Some_conv: "bind f gbycasesx imp_all
( f) simp_all

mmabind_eq_None_conv:"Option. 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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

by

setupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

context
begin

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
     tx

qualified lemma these_eq [code  \<open>these A = the ` (Set.filter (Not \<circ> Option.is_none) A)\<close>

( add: these_def Option.is_none_def)

qualified lemma these_unfold:
  \<open>these A = {x. \<exists>y \<in> A. y = Some x}\<close>
  y(auto  dd

  simp these= {"
  by (simp add: these_def)

lemma java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
  by autosimpadd these_defjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

lemma simp:"hese ( )=these A"
proofby (auto simp add: these_def)
  have "{y java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 0
auto
    ? by add java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
qed

lemma java.lang.StringIndexOutOfBoundsException: Range [3, 4) out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Range [15, 9) out of bounds for length 25
f thesef  )
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
next
  assume "x \ these A"
  by (auto dest: finite_imageD intro: inj_Some)
qed

lemma these_image_Some_eq [simp]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by

lemma Some_image_these_eq: " ` 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 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
simpadd these_empty_eq)

qualified definition image_filter :: "('a java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 62
=fjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

end(A ==>() =>r A== =)pred_optionpred_option

finite_range_Some "inite( Some: '


subsection  finite (UNIV :: 'a set)"

contextjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

  t:
  "(rel_option A =l equal_None_code_unfold [code_unfold]:
    Option.bind Option.bind"
  unfoldingsplit_option_all java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

lemma pred_option_parametric [by (auto simp add: equal Option.is_none_def
code_printing
   java.lang.StringIndexOutOfBoundsException: Range [4, 38) out of bounds for length 22

end

subsubsection and (Scala) "!None"

lemma "
  "finite (UNIV :: 'a Some
   ption:equal java.lang.StringIndexOutOfBoundsException: Range [14, 36) out of bounds for length 15

instance option :
  SML)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

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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