src/HOL/Limited_Sequence.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 60758 d8d85a8172b5 child 67091 1393c2340eec permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1
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```     2 (* Author: Lukas Bulwahn, TU Muenchen *)
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```     3
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```     4 section \<open>Depth-Limited Sequences with failure element\<close>
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```     5
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```     6 theory Limited_Sequence
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```     7 imports Lazy_Sequence
```
```     8 begin
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```     9
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```    10 subsection \<open>Depth-Limited Sequence\<close>
```
```    11
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```    12 type_synonym 'a dseq = "natural \<Rightarrow> bool \<Rightarrow> 'a lazy_sequence option"
```
```    13
```
```    14 definition empty :: "'a dseq"
```
```    15 where
```
```    16   "empty = (\<lambda>_ _. Some Lazy_Sequence.empty)"
```
```    17
```
```    18 definition single :: "'a \<Rightarrow> 'a dseq"
```
```    19 where
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```    20   "single x = (\<lambda>_ _. Some (Lazy_Sequence.single x))"
```
```    21
```
```    22 definition eval :: "'a dseq \<Rightarrow> natural \<Rightarrow> bool \<Rightarrow> 'a lazy_sequence option"
```
```    23 where
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```    24   [simp]: "eval f i pol = f i pol"
```
```    25
```
```    26 definition yield :: "'a dseq \<Rightarrow> natural \<Rightarrow> bool \<Rightarrow> ('a \<times> 'a dseq) option"
```
```    27 where
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```    28   "yield f i pol = (case eval f i pol of
```
```    29     None \<Rightarrow> None
```
```    30   | Some s \<Rightarrow> (map_option \<circ> apsnd) (\<lambda>r _ _. Some r) (Lazy_Sequence.yield s))"
```
```    31
```
```    32 definition map_seq :: "('a \<Rightarrow> 'b dseq) \<Rightarrow> 'a lazy_sequence \<Rightarrow> 'b dseq"
```
```    33 where
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```    34   "map_seq f xq i pol = map_option Lazy_Sequence.flat
```
```    35     (Lazy_Sequence.those (Lazy_Sequence.map (\<lambda>x. f x i pol) xq))"
```
```    36
```
```    37 lemma map_seq_code [code]:
```
```    38   "map_seq f xq i pol = (case Lazy_Sequence.yield xq of
```
```    39     None \<Rightarrow> Some Lazy_Sequence.empty
```
```    40   | Some (x, xq') \<Rightarrow> (case eval (f x) i pol of
```
```    41       None \<Rightarrow> None
```
```    42     | Some yq \<Rightarrow> (case map_seq f xq' i pol of
```
```    43         None \<Rightarrow> None
```
```    44       | Some zq \<Rightarrow> Some (Lazy_Sequence.append yq zq))))"
```
```    45   by (cases xq)
```
```    46     (auto simp add: map_seq_def Lazy_Sequence.those_def lazy_sequence_eq_iff split: list.splits option.splits)
```
```    47
```
```    48 definition bind :: "'a dseq \<Rightarrow> ('a \<Rightarrow> 'b dseq) \<Rightarrow> 'b dseq"
```
```    49 where
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```    50   "bind x f = (\<lambda>i pol.
```
```    51      if i = 0 then
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```    52        (if pol then Some Lazy_Sequence.empty else None)
```
```    53      else
```
```    54        (case x (i - 1) pol of
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```    55          None \<Rightarrow> None
```
```    56        | Some xq \<Rightarrow> map_seq f xq i pol))"
```
```    57
```
```    58 definition union :: "'a dseq \<Rightarrow> 'a dseq \<Rightarrow> 'a dseq"
```
```    59 where
```
```    60   "union x y = (\<lambda>i pol. case (x i pol, y i pol) of
```
```    61       (Some xq, Some yq) \<Rightarrow> Some (Lazy_Sequence.append xq yq)
```
```    62     | _ \<Rightarrow> None)"
```
```    63
```
```    64 definition if_seq :: "bool \<Rightarrow> unit dseq"
```
```    65 where
```
```    66   "if_seq b = (if b then single () else empty)"
```
```    67
```
```    68 definition not_seq :: "unit dseq \<Rightarrow> unit dseq"
```
```    69 where
```
```    70   "not_seq x = (\<lambda>i pol. case x i (\<not> pol) of
```
```    71     None \<Rightarrow> Some Lazy_Sequence.empty
```
```    72   | Some xq \<Rightarrow> (case Lazy_Sequence.yield xq of
```
```    73       None \<Rightarrow> Some (Lazy_Sequence.single ())
```
```    74     | Some _ \<Rightarrow> Some (Lazy_Sequence.empty)))"
```
```    75
```
```    76 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dseq \<Rightarrow> 'b dseq"
```
```    77 where
```
```    78   "map f g = (\<lambda>i pol. case g i pol of
```
```    79      None \<Rightarrow> None
```
```    80    | Some xq \<Rightarrow> Some (Lazy_Sequence.map f xq))"
```
```    81
```
```    82
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```    83 subsection \<open>Positive Depth-Limited Sequence\<close>
```
```    84
```
```    85 type_synonym 'a pos_dseq = "natural \<Rightarrow> 'a Lazy_Sequence.lazy_sequence"
```
```    86
```
```    87 definition pos_empty :: "'a pos_dseq"
```
```    88 where
```
```    89   "pos_empty = (\<lambda>i. Lazy_Sequence.empty)"
```
```    90
```
```    91 definition pos_single :: "'a \<Rightarrow> 'a pos_dseq"
```
```    92 where
```
```    93   "pos_single x = (\<lambda>i. Lazy_Sequence.single x)"
```
```    94
```
```    95 definition pos_bind :: "'a pos_dseq \<Rightarrow> ('a \<Rightarrow> 'b pos_dseq) \<Rightarrow> 'b pos_dseq"
```
```    96 where
```
```    97   "pos_bind x f = (\<lambda>i. Lazy_Sequence.bind (x i) (\<lambda>a. f a i))"
```
```    98
```
```    99 definition pos_decr_bind :: "'a pos_dseq \<Rightarrow> ('a \<Rightarrow> 'b pos_dseq) \<Rightarrow> 'b pos_dseq"
```
```   100 where
```
```   101   "pos_decr_bind x f = (\<lambda>i.
```
```   102      if i = 0 then
```
```   103        Lazy_Sequence.empty
```
```   104      else
```
```   105        Lazy_Sequence.bind (x (i - 1)) (\<lambda>a. f a i))"
```
```   106
```
```   107 definition pos_union :: "'a pos_dseq \<Rightarrow> 'a pos_dseq \<Rightarrow> 'a pos_dseq"
```
```   108 where
```
```   109   "pos_union xq yq = (\<lambda>i. Lazy_Sequence.append (xq i) (yq i))"
```
```   110
```
```   111 definition pos_if_seq :: "bool \<Rightarrow> unit pos_dseq"
```
```   112 where
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```   113   "pos_if_seq b = (if b then pos_single () else pos_empty)"
```
```   114
```
```   115 definition pos_iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a pos_dseq"
```
```   116 where
```
```   117   "pos_iterate_upto f n m = (\<lambda>i. Lazy_Sequence.iterate_upto f n m)"
```
```   118
```
```   119 definition pos_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pos_dseq \<Rightarrow> 'b pos_dseq"
```
```   120 where
```
```   121   "pos_map f xq = (\<lambda>i. Lazy_Sequence.map f (xq i))"
```
```   122
```
```   123
```
```   124 subsection \<open>Negative Depth-Limited Sequence\<close>
```
```   125
```
```   126 type_synonym 'a neg_dseq = "natural \<Rightarrow> 'a Lazy_Sequence.hit_bound_lazy_sequence"
```
```   127
```
```   128 definition neg_empty :: "'a neg_dseq"
```
```   129 where
```
```   130   "neg_empty = (\<lambda>i. Lazy_Sequence.empty)"
```
```   131
```
```   132 definition neg_single :: "'a \<Rightarrow> 'a neg_dseq"
```
```   133 where
```
```   134   "neg_single x = (\<lambda>i. Lazy_Sequence.hb_single x)"
```
```   135
```
```   136 definition neg_bind :: "'a neg_dseq \<Rightarrow> ('a \<Rightarrow> 'b neg_dseq) \<Rightarrow> 'b neg_dseq"
```
```   137 where
```
```   138   "neg_bind x f = (\<lambda>i. hb_bind (x i) (\<lambda>a. f a i))"
```
```   139
```
```   140 definition neg_decr_bind :: "'a neg_dseq \<Rightarrow> ('a \<Rightarrow> 'b neg_dseq) \<Rightarrow> 'b neg_dseq"
```
```   141 where
```
```   142   "neg_decr_bind x f = (\<lambda>i.
```
```   143      if i = 0 then
```
```   144        Lazy_Sequence.hit_bound
```
```   145      else
```
```   146        hb_bind (x (i - 1)) (\<lambda>a. f a i))"
```
```   147
```
```   148 definition neg_union :: "'a neg_dseq \<Rightarrow> 'a neg_dseq \<Rightarrow> 'a neg_dseq"
```
```   149 where
```
```   150   "neg_union x y = (\<lambda>i. Lazy_Sequence.append (x i) (y i))"
```
```   151
```
```   152 definition neg_if_seq :: "bool \<Rightarrow> unit neg_dseq"
```
```   153 where
```
```   154   "neg_if_seq b = (if b then neg_single () else neg_empty)"
```
```   155
```
```   156 definition neg_iterate_upto
```
```   157 where
```
```   158   "neg_iterate_upto f n m = (\<lambda>i. Lazy_Sequence.iterate_upto (\<lambda>i. Some (f i)) n m)"
```
```   159
```
```   160 definition neg_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a neg_dseq \<Rightarrow> 'b neg_dseq"
```
```   161 where
```
```   162   "neg_map f xq = (\<lambda>i. Lazy_Sequence.hb_map f (xq i))"
```
```   163
```
```   164
```
```   165 subsection \<open>Negation\<close>
```
```   166
```
```   167 definition pos_not_seq :: "unit neg_dseq \<Rightarrow> unit pos_dseq"
```
```   168 where
```
```   169   "pos_not_seq xq = (\<lambda>i. Lazy_Sequence.hb_not_seq (xq (3 * i)))"
```
```   170
```
```   171 definition neg_not_seq :: "unit pos_dseq \<Rightarrow> unit neg_dseq"
```
```   172 where
```
```   173   "neg_not_seq x = (\<lambda>i. case Lazy_Sequence.yield (x i) of
```
```   174     None => Lazy_Sequence.hb_single ()
```
```   175   | Some ((), xq) => Lazy_Sequence.empty)"
```
```   176
```
```   177
```
```   178 ML \<open>
```
```   179 signature LIMITED_SEQUENCE =
```
```   180 sig
```
```   181   type 'a dseq = Code_Numeral.natural -> bool -> 'a Lazy_Sequence.lazy_sequence option
```
```   182   val map : ('a -> 'b) -> 'a dseq -> 'b dseq
```
```   183   val yield : 'a dseq -> Code_Numeral.natural -> bool -> ('a * 'a dseq) option
```
```   184   val yieldn : int -> 'a dseq -> Code_Numeral.natural -> bool -> 'a list * 'a dseq
```
```   185 end;
```
```   186
```
```   187 structure Limited_Sequence : LIMITED_SEQUENCE =
```
```   188 struct
```
```   189
```
```   190 type 'a dseq = Code_Numeral.natural -> bool -> 'a Lazy_Sequence.lazy_sequence option
```
```   191
```
```   192 fun map f = @{code Limited_Sequence.map} f;
```
```   193
```
```   194 fun yield f = @{code Limited_Sequence.yield} f;
```
```   195
```
```   196 fun yieldn n f i pol = (case f i pol of
```
```   197     NONE => ([], fn _ => fn _ => NONE)
```
```   198   | SOME s => let val (xs, s') = Lazy_Sequence.yieldn n s in (xs, fn _ => fn _ => SOME s') end);
```
```   199
```
```   200 end;
```
```   201 \<close>
```
```   202
```
```   203 code_reserved Eval Limited_Sequence
```
```   204
```
```   205
```
```   206 hide_const (open) yield empty single eval map_seq bind union if_seq not_seq map
```
```   207   pos_empty pos_single pos_bind pos_decr_bind pos_union pos_if_seq pos_iterate_upto pos_not_seq pos_map
```
```   208   neg_empty neg_single neg_bind neg_decr_bind neg_union neg_if_seq neg_iterate_upto neg_not_seq neg_map
```
```   209
```
```   210 hide_fact (open) yield_def empty_def single_def eval_def map_seq_def bind_def union_def
```
```   211   if_seq_def not_seq_def map_def
```
```   212   pos_empty_def pos_single_def pos_bind_def pos_union_def pos_if_seq_def pos_iterate_upto_def pos_not_seq_def pos_map_def
```
```   213   neg_empty_def neg_single_def neg_bind_def neg_union_def neg_if_seq_def neg_iterate_upto_def neg_not_seq_def neg_map_def
```
```   214
```
```   215 end
```
```   216
```