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
```