src/HOL/Option.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 49189 3f85cd15a0cc child 51096 60e4b75fefe1 permissions -rw-r--r--
introduce order topology
```     1 (*  Title:      HOL/Option.thy
```
```     2     Author:     Folklore
```
```     3 *)
```
```     4
```
```     5 header {* Datatype option *}
```
```     6
```
```     7 theory Option
```
```     8 imports Datatype
```
```     9 begin
```
```    10
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```    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
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```    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
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```    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 UNIV_option_conv: "UNIV = insert None (range Some)"
```
```    34 by(auto intro: classical)
```
```    35
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```    36
```
```    37 subsubsection {* Operations *}
```
```    38
```
```    39 primrec the :: "'a option => 'a" where
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```    40 "the (Some x) = x"
```
```    41
```
```    42 primrec set :: "'a option => 'a set" where
```
```    43 "set None = {}" |
```
```    44 "set (Some x) = {x}"
```
```    45
```
```    46 lemma ospec [dest]: "(ALL x:set A. P x) ==> A = Some x ==> P x"
```
```    47   by simp
```
```    48
```
```    49 declaration {* fn _ =>
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```    50   Classical.map_cs (fn cs => cs addSD2 ("ospec", @{thm ospec}))
```
```    51 *}
```
```    52
```
```    53 lemma elem_set [iff]: "(x : set xo) = (xo = Some x)"
```
```    54   by (cases xo) auto
```
```    55
```
```    56 lemma set_empty_eq [simp]: "(set xo = {}) = (xo = None)"
```
```    57   by (cases xo) auto
```
```    58
```
```    59 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option" where
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```    60   "map = (%f y. case y of None => None | Some x => Some (f x))"
```
```    61
```
```    62 lemma option_map_None [simp, code]: "map f None = None"
```
```    63   by (simp add: map_def)
```
```    64
```
```    65 lemma option_map_Some [simp, code]: "map f (Some x) = Some (f x)"
```
```    66   by (simp add: map_def)
```
```    67
```
```    68 lemma option_map_is_None [iff]:
```
```    69     "(map f opt = None) = (opt = None)"
```
```    70   by (simp add: map_def split add: option.split)
```
```    71
```
```    72 lemma option_map_eq_Some [iff]:
```
```    73     "(map f xo = Some y) = (EX z. xo = Some z & f z = y)"
```
```    74   by (simp add: map_def split add: option.split)
```
```    75
```
```    76 lemma option_map_comp:
```
```    77     "map f (map g opt) = map (f o g) opt"
```
```    78   by (simp add: map_def split add: option.split)
```
```    79
```
```    80 lemma option_map_o_sum_case [simp]:
```
```    81     "map f o sum_case g h = sum_case (map f o g) (map f o h)"
```
```    82   by (rule ext) (simp split: sum.split)
```
```    83
```
```    84 lemma map_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> map f x = map g y"
```
```    85 by (cases x) auto
```
```    86
```
```    87 enriched_type map: Option.map proof -
```
```    88   fix f g
```
```    89   show "Option.map f \<circ> Option.map g = Option.map (f \<circ> g)"
```
```    90   proof
```
```    91     fix x
```
```    92     show "(Option.map f \<circ> Option.map g) x= Option.map (f \<circ> g) x"
```
```    93       by (cases x) simp_all
```
```    94   qed
```
```    95 next
```
```    96   show "Option.map id = id"
```
```    97   proof
```
```    98     fix x
```
```    99     show "Option.map id x = id x"
```
```   100       by (cases x) simp_all
```
```   101   qed
```
```   102 qed
```
```   103
```
```   104 primrec bind :: "'a option \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'b option" where
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```   105 bind_lzero: "bind None f = None" |
```
```   106 bind_lunit: "bind (Some x) f = f x"
```
```   107
```
```   108 lemma bind_runit[simp]: "bind x Some = x"
```
```   109 by (cases x) auto
```
```   110
```
```   111 lemma bind_assoc[simp]: "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
```
```   112 by (cases x) auto
```
```   113
```
```   114 lemma bind_rzero[simp]: "bind x (\<lambda>x. None) = None"
```
```   115 by (cases x) auto
```
```   116
```
```   117 lemma bind_cong: "x = y \<Longrightarrow> (\<And>a. y = Some a \<Longrightarrow> f a = g a) \<Longrightarrow> bind x f = bind y g"
```
```   118 by (cases x) auto
```
```   119
```
```   120 definition these :: "'a option set \<Rightarrow> 'a set"
```
```   121 where
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```   122   "these A = the ` {x \<in> A. x \<noteq> None}"
```
```   123
```
```   124 lemma these_empty [simp]:
```
```   125   "these {} = {}"
```
```   126   by (simp add: these_def)
```
```   127
```
```   128 lemma these_insert_None [simp]:
```
```   129   "these (insert None A) = these A"
```
```   130   by (auto simp add: these_def)
```
```   131
```
```   132 lemma these_insert_Some [simp]:
```
```   133   "these (insert (Some x) A) = insert x (these A)"
```
```   134 proof -
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```   135   have "{y \<in> insert (Some x) A. y \<noteq> None} = insert (Some x) {y \<in> A. y \<noteq> None}"
```
```   136     by auto
```
```   137   then show ?thesis by (simp add: these_def)
```
```   138 qed
```
```   139
```
```   140 lemma in_these_eq:
```
```   141   "x \<in> these A \<longleftrightarrow> Some x \<in> A"
```
```   142 proof
```
```   143   assume "Some x \<in> A"
```
```   144   then obtain B where "A = insert (Some x) B" by auto
```
```   145   then show "x \<in> these A" by (auto simp add: these_def intro!: image_eqI)
```
```   146 next
```
```   147   assume "x \<in> these A"
```
```   148   then show "Some x \<in> A" by (auto simp add: these_def)
```
```   149 qed
```
```   150
```
```   151 lemma these_image_Some_eq [simp]:
```
```   152   "these (Some ` A) = A"
```
```   153   by (auto simp add: these_def intro!: image_eqI)
```
```   154
```
```   155 lemma Some_image_these_eq:
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```   156   "Some ` these A = {x\<in>A. x \<noteq> None}"
```
```   157   by (auto simp add: these_def image_image intro!: image_eqI)
```
```   158
```
```   159 lemma these_empty_eq:
```
```   160   "these B = {} \<longleftrightarrow> B = {} \<or> B = {None}"
```
```   161   by (auto simp add: these_def)
```
```   162
```
```   163 lemma these_not_empty_eq:
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```   164   "these B \<noteq> {} \<longleftrightarrow> B \<noteq> {} \<and> B \<noteq> {None}"
```
```   165   by (auto simp add: these_empty_eq)
```
```   166
```
```   167 hide_const (open) set map bind these
```
```   168 hide_fact (open) map_cong bind_cong
```
```   169
```
```   170
```
```   171 subsubsection {* Code generator setup *}
```
```   172
```
```   173 definition is_none :: "'a option \<Rightarrow> bool" where
```
```   174   [code_post]: "is_none x \<longleftrightarrow> x = None"
```
```   175
```
```   176 lemma is_none_code [code]:
```
```   177   shows "is_none None \<longleftrightarrow> True"
```
```   178     and "is_none (Some x) \<longleftrightarrow> False"
```
```   179   unfolding is_none_def by simp_all
```
```   180
```
```   181 lemma [code_unfold]:
```
```   182   "HOL.equal x None \<longleftrightarrow> is_none x"
```
```   183   by (simp add: equal is_none_def)
```
```   184
```
```   185 hide_const (open) is_none
```
```   186
```
```   187 code_type option
```
```   188   (SML "_ option")
```
```   189   (OCaml "_ option")
```
```   190   (Haskell "Maybe _")
```
```   191   (Scala "!Option[(_)]")
```
```   192
```
```   193 code_const None and Some
```
```   194   (SML "NONE" and "SOME")
```
```   195   (OCaml "None" and "Some _")
```
```   196   (Haskell "Nothing" and "Just")
```
```   197   (Scala "!None" and "Some")
```
```   198
```
```   199 code_instance option :: equal
```
```   200   (Haskell -)
```
```   201
```
```   202 code_const "HOL.equal \<Colon> 'a option \<Rightarrow> 'a option \<Rightarrow> bool"
```
```   203   (Haskell infix 4 "==")
```
```   204
```
```   205 code_reserved SML
```
```   206   option NONE SOME
```
```   207
```
```   208 code_reserved OCaml
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```   209   option None Some
```
```   210
```
```   211 code_reserved Scala
```
```   212   Option None Some
```
```   213
```
```   214 end
```
```   215
```