src/HOL/Limited_Sequence.thy
author paulson <lp15@cam.ac.uk>
Tue Apr 25 16:39:54 2017 +0100 (2017-04-25)
changeset 65578 e4997c181cce
parent 60758 d8d85a8172b5
child 67091 1393c2340eec
permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
     1 
     2 (* Author: Lukas Bulwahn, TU Muenchen *)
     3 
     4 section \<open>Depth-Limited Sequences with failure element\<close>
     5 
     6 theory Limited_Sequence
     7 imports Lazy_Sequence
     8 begin
     9 
    10 subsection \<open>Depth-Limited Sequence\<close>
    11 
    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
    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
    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
    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
    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
    50   "bind x f = (\<lambda>i pol. 
    51      if i = 0 then
    52        (if pol then Some Lazy_Sequence.empty else None)
    53      else
    54        (case x (i - 1) pol of
    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 
    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
   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