src/HOL/Option.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30327 4d1185c77f4a child 31080 21ffc770ebc0 permissions -rw-r--r--
simplified method setup;
```     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 option_caseE:
```
```    24   assumes c: "(case x of None => P | Some y => Q y)"
```
```    25   obtains
```
```    26     (None) "x = None" and P
```
```    27   | (Some) y where "x = Some y" and "Q y"
```
```    28   using c by (cases x) simp_all
```
```    29
```
```    30 lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
```
```    31   by (rule set_ext, case_tac x) auto
```
```    32
```
```    33 instance option :: (finite) finite proof
```
```    34 qed (simp add: insert_None_conv_UNIV [symmetric])
```
```    35
```
```    36 lemma inj_Some [simp]: "inj_on Some A"
```
```    37   by (rule inj_onI) simp
```
```    38
```
```    39
```
```    40 subsubsection {* Operations *}
```
```    41
```
```    42 primrec the :: "'a option => 'a" where
```
```    43 "the (Some x) = x"
```
```    44
```
```    45 primrec set :: "'a option => 'a set" where
```
```    46 "set None = {}" |
```
```    47 "set (Some x) = {x}"
```
```    48
```
```    49 lemma ospec [dest]: "(ALL x:set A. P x) ==> A = Some x ==> P x"
```
```    50   by simp
```
```    51
```
```    52 declaration {* fn _ =>
```
```    53   Classical.map_cs (fn cs => cs addSD2 ("ospec", thm "ospec"))
```
```    54 *}
```
```    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
```
```    63   map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option"
```
```    64 where
```
```    65   [code del]: "map = (%f y. case y of None => None | Some x => Some (f x))"
```
```    66
```
```    67 lemma option_map_None [simp, code]: "map f None = None"
```
```    68   by (simp add: map_def)
```
```    69
```
```    70 lemma option_map_Some [simp, code]: "map f (Some x) = Some (f x)"
```
```    71   by (simp add: map_def)
```
```    72
```
```    73 lemma option_map_is_None [iff]:
```
```    74     "(map f opt = None) = (opt = None)"
```
```    75   by (simp add: map_def split add: option.split)
```
```    76
```
```    77 lemma option_map_eq_Some [iff]:
```
```    78     "(map f xo = Some y) = (EX z. xo = Some z & f z = y)"
```
```    79   by (simp add: map_def split add: option.split)
```
```    80
```
```    81 lemma option_map_comp:
```
```    82     "map f (map g opt) = map (f o g) opt"
```
```    83   by (simp add: map_def split add: option.split)
```
```    84
```
```    85 lemma option_map_o_sum_case [simp]:
```
```    86     "map f o sum_case g h = sum_case (map f o g) (map f o h)"
```
```    87   by (rule ext) (simp split: sum.split)
```
```    88
```
```    89
```
```    90 hide (open) const set map
```
```    91
```
```    92 subsubsection {* Code generator setup *}
```
```    93
```
```    94 definition
```
```    95   is_none :: "'a option \<Rightarrow> bool" where
```
```    96   is_none_none [code post, symmetric, code inline]: "is_none x \<longleftrightarrow> x = None"
```
```    97
```
```    98 lemma is_none_code [code]:
```
```    99   shows "is_none None \<longleftrightarrow> True"
```
```   100     and "is_none (Some x) \<longleftrightarrow> False"
```
```   101   unfolding is_none_none [symmetric] by simp_all
```
```   102
```
```   103 hide (open) const is_none
```
```   104
```
```   105 code_type option
```
```   106   (SML "_ option")
```
```   107   (OCaml "_ option")
```
```   108   (Haskell "Maybe _")
```
```   109
```
```   110 code_const None and Some
```
```   111   (SML "NONE" and "SOME")
```
```   112   (OCaml "None" and "Some _")
```
```   113   (Haskell "Nothing" and "Just")
```
```   114
```
```   115 code_instance option :: eq
```
```   116   (Haskell -)
```
```   117
```
```   118 code_const "eq_class.eq \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"
```
```   119   (Haskell infixl 4 "==")
```
```   120
```
```   121 code_reserved SML
```
```   122   option NONE SOME
```
```   123
```
```   124 code_reserved OCaml
```
```   125   option None Some
```
```   126
```
```   127 end
```