src/HOL/Option.thy
author blanchet
Fri Jan 24 11:51:45 2014 +0100 (2014-01-24)
changeset 55129 26bd1cba3ab5
parent 55089 181751ad852f
child 55404 5cb95b79a51f
permissions -rw-r--r--
killed 'More_BNFs' by moving its various bits where they (now) belong
     1 (*  Title:      HOL/Option.thy
     2     Author:     Folklore
     3 *)
     4 
     5 header {* Datatype option *}
     6 
     7 theory Option
     8 imports Datatype Finite_Set
     9 begin
    10 
    11 datatype 'a option = None | Some 'a
    12 
    13 lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
    14   by (induct x) auto
    15 
    16 lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
    17   by (induct x) auto
    18 
    19 text{*Although it may appear that both of these equalities are helpful
    20 only when applied to assumptions, in practice it seems better to give
    21 them the uniform iff attribute. *}
    22 
    23 lemma inj_Some [simp]: "inj_on Some A"
    24 by (rule inj_onI) simp
    25 
    26 lemma option_caseE:
    27   assumes c: "(case x of None => P | Some y => Q y)"
    28   obtains
    29     (None) "x = None" and P
    30   | (Some) y where "x = Some y" and "Q y"
    31   using c by (cases x) simp_all
    32 
    33 lemma split_option_all: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P (Some x))"
    34 by (auto intro: option.induct)
    35 
    36 lemma split_option_ex: "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P (Some x))"
    37 using split_option_all[of "\<lambda>x. \<not>P x"] by blast
    38 
    39 lemma UNIV_option_conv: "UNIV = insert None (range Some)"
    40 by(auto intro: classical)
    41 
    42 subsubsection {* Operations *}
    43 
    44 primrec the :: "'a option => 'a" where
    45 "the (Some x) = x"
    46 
    47 primrec set :: "'a option => 'a set" where
    48 "set None = {}" |
    49 "set (Some x) = {x}"
    50 
    51 lemma ospec [dest]: "(ALL x:set A. P x) ==> A = Some x ==> P x"
    52   by simp
    53 
    54 setup {* map_theory_claset (fn ctxt => ctxt addSD2 ("ospec", @{thm ospec})) *}
    55 
    56 lemma elem_set [iff]: "(x : set xo) = (xo = Some x)"
    57   by (cases xo) auto
    58 
    59 lemma set_empty_eq [simp]: "(set xo = {}) = (xo = None)"
    60   by (cases xo) auto
    61 
    62 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option" where
    63   "map = (%f y. case y of None => None | Some x => Some (f x))"
    64 
    65 lemma option_map_None [simp, code]: "map f None = None"
    66   by (simp add: map_def)
    67 
    68 lemma option_map_Some [simp, code]: "map f (Some x) = Some (f x)"
    69   by (simp add: map_def)
    70 
    71 lemma option_map_is_None [iff]:
    72     "(map f opt = None) = (opt = None)"
    73   by (simp add: map_def split add: option.split)
    74 
    75 lemma option_map_eq_Some [iff]:
    76     "(map f xo = Some y) = (EX z. xo = Some z & f z = y)"
    77   by (simp add: map_def split add: option.split)
    78 
    79 lemma option_map_comp:
    80     "map f (map g opt) = map (f o g) opt"
    81   by (simp add: map_def split add: option.split)
    82 
    83 lemma option_map_o_sum_case [simp]:
    84     "map f o sum_case g h = sum_case (map f o g) (map f o h)"
    85   by (rule ext) (simp split: sum.split)
    86 
    87 lemma map_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> map f x = map g y"
    88 by (cases x) auto
    89 
    90 enriched_type map: Option.map proof -
    91   fix f g
    92   show "Option.map f \<circ> Option.map g = Option.map (f \<circ> g)"
    93   proof
    94     fix x
    95     show "(Option.map f \<circ> Option.map g) x= Option.map (f \<circ> g) x"
    96       by (cases x) simp_all
    97   qed
    98 next
    99   show "Option.map id = id"
   100   proof
   101     fix x
   102     show "Option.map id x = id x"
   103       by (cases x) simp_all
   104   qed
   105 qed
   106 
   107 lemma option_case_map [simp]:
   108   "option_case g h (Option.map f x) = option_case g (h \<circ> f) x"
   109   by (cases x) simp_all
   110 
   111 primrec bind :: "'a option \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'b option" where
   112 bind_lzero: "bind None f = None" |
   113 bind_lunit: "bind (Some x) f = f x"
   114 
   115 lemma bind_runit[simp]: "bind x Some = x"
   116 by (cases x) auto
   117 
   118 lemma bind_assoc[simp]: "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
   119 by (cases x) auto
   120 
   121 lemma bind_rzero[simp]: "bind x (\<lambda>x. None) = None"
   122 by (cases x) auto
   123 
   124 lemma bind_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> bind x f = bind y g"
   125 by (cases x) auto
   126 
   127 definition these :: "'a option set \<Rightarrow> 'a set"
   128 where
   129   "these A = the ` {x \<in> A. x \<noteq> None}"
   130 
   131 lemma these_empty [simp]:
   132   "these {} = {}"
   133   by (simp add: these_def)
   134 
   135 lemma these_insert_None [simp]:
   136   "these (insert None A) = these A"
   137   by (auto simp add: these_def)
   138 
   139 lemma these_insert_Some [simp]:
   140   "these (insert (Some x) A) = insert x (these A)"
   141 proof -
   142   have "{y \<in> insert (Some x) A. y \<noteq> None} = insert (Some x) {y \<in> A. y \<noteq> None}"
   143     by auto
   144   then show ?thesis by (simp add: these_def)
   145 qed
   146 
   147 lemma in_these_eq:
   148   "x \<in> these A \<longleftrightarrow> Some x \<in> A"
   149 proof
   150   assume "Some x \<in> A"
   151   then obtain B where "A = insert (Some x) B" by auto
   152   then show "x \<in> these A" by (auto simp add: these_def intro!: image_eqI)
   153 next
   154   assume "x \<in> these A"
   155   then show "Some x \<in> A" by (auto simp add: these_def)
   156 qed
   157 
   158 lemma these_image_Some_eq [simp]:
   159   "these (Some ` A) = A"
   160   by (auto simp add: these_def intro!: image_eqI)
   161 
   162 lemma Some_image_these_eq:
   163   "Some ` these A = {x\<in>A. x \<noteq> None}"
   164   by (auto simp add: these_def image_image intro!: image_eqI)
   165 
   166 lemma these_empty_eq:
   167   "these B = {} \<longleftrightarrow> B = {} \<or> B = {None}"
   168   by (auto simp add: these_def)
   169 
   170 lemma these_not_empty_eq:
   171   "these B \<noteq> {} \<longleftrightarrow> B \<noteq> {} \<and> B \<noteq> {None}"
   172   by (auto simp add: these_empty_eq)
   173 
   174 hide_const (open) set map bind these
   175 hide_fact (open) map_cong bind_cong
   176 
   177 
   178 subsubsection {* Interaction with finite sets *}
   179 
   180 lemma finite_option_UNIV [simp]:
   181   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
   182   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
   183 
   184 instance option :: (finite) finite
   185   by default (simp add: UNIV_option_conv)
   186 
   187 
   188 subsubsection {* Code generator setup *}
   189 
   190 definition is_none :: "'a option \<Rightarrow> bool" where
   191   [code_post]: "is_none x \<longleftrightarrow> x = None"
   192 
   193 lemma is_none_code [code]:
   194   shows "is_none None \<longleftrightarrow> True"
   195     and "is_none (Some x) \<longleftrightarrow> False"
   196   unfolding is_none_def by simp_all
   197 
   198 lemma [code_unfold]:
   199   "HOL.equal x None \<longleftrightarrow> is_none x"
   200   "HOL.equal None = is_none"
   201   by (auto simp add: equal is_none_def)
   202 
   203 hide_const (open) is_none
   204 
   205 code_printing
   206   type_constructor option \<rightharpoonup>
   207     (SML) "_ option"
   208     and (OCaml) "_ option"
   209     and (Haskell) "Maybe _"
   210     and (Scala) "!Option[(_)]"
   211 | constant None \<rightharpoonup>
   212     (SML) "NONE"
   213     and (OCaml) "None"
   214     and (Haskell) "Nothing"
   215     and (Scala) "!None"
   216 | constant Some \<rightharpoonup>
   217     (SML) "SOME"
   218     and (OCaml) "Some _"
   219     and (Haskell) "Just"
   220     and (Scala) "Some"
   221 | class_instance option :: equal \<rightharpoonup>
   222     (Haskell) -
   223 | constant "HOL.equal :: 'a option \<Rightarrow> 'a option \<Rightarrow> bool" \<rightharpoonup>
   224     (Haskell) infix 4 "=="
   225 
   226 code_reserved SML
   227   option NONE SOME
   228 
   229 code_reserved OCaml
   230   option None Some
   231 
   232 code_reserved Scala
   233   Option None Some
   234 
   235 end