src/HOL/Random_Sequence.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 58889 5b7a9633cfa8
child 60758 d8d85a8172b5
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 
     2 (* Author: Lukas Bulwahn, TU Muenchen *)
     3 
     4 section {* Various kind of sequences inside the random monad *}
     5 
     6 theory Random_Sequence
     7 imports Random_Pred
     8 begin
     9 
    10 type_synonym 'a random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> ('a Limited_Sequence.dseq \<times> Random.seed)"
    11 
    12 definition empty :: "'a random_dseq"
    13 where
    14   "empty = (%nrandom size. Pair (Limited_Sequence.empty))"
    15 
    16 definition single :: "'a => 'a random_dseq"
    17 where
    18   "single x = (%nrandom size. Pair (Limited_Sequence.single x))"
    19 
    20 definition bind :: "'a random_dseq => ('a \<Rightarrow> 'b random_dseq) \<Rightarrow> 'b random_dseq"
    21 where
    22   "bind R f = (\<lambda>nrandom size s. let
    23      (P, s') = R nrandom size s;
    24      (s1, s2) = Random.split_seed s'
    25   in (Limited_Sequence.bind P (%a. fst (f a nrandom size s1)), s2))"
    26 
    27 definition union :: "'a random_dseq => 'a random_dseq => 'a random_dseq"
    28 where
    29   "union R1 R2 = (\<lambda>nrandom size s. let
    30      (S1, s') = R1 nrandom size s; (S2, s'') = R2 nrandom size s'
    31   in (Limited_Sequence.union S1 S2, s''))"
    32 
    33 definition if_random_dseq :: "bool => unit random_dseq"
    34 where
    35   "if_random_dseq b = (if b then single () else empty)"
    36 
    37 definition not_random_dseq :: "unit random_dseq => unit random_dseq"
    38 where
    39   "not_random_dseq R = (\<lambda>nrandom size s. let
    40      (S, s') = R nrandom size s
    41    in (Limited_Sequence.not_seq S, s'))"
    42 
    43 definition map :: "('a => 'b) => 'a random_dseq => 'b random_dseq"
    44 where
    45   "map f P = bind P (single o f)"
    46 
    47 fun Random :: "(natural \<Rightarrow> Random.seed \<Rightarrow> (('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)) \<Rightarrow> 'a random_dseq"
    48 where
    49   "Random g nrandom = (%size. if nrandom <= 0 then (Pair Limited_Sequence.empty) else
    50      (scomp (g size) (%r. scomp (Random g (nrandom - 1) size) (%rs. Pair (Limited_Sequence.union (Limited_Sequence.single (fst r)) rs)))))"
    51 
    52 
    53 type_synonym 'a pos_random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Limited_Sequence.pos_dseq"
    54 
    55 definition pos_empty :: "'a pos_random_dseq"
    56 where
    57   "pos_empty = (%nrandom size seed. Limited_Sequence.pos_empty)"
    58 
    59 definition pos_single :: "'a => 'a pos_random_dseq"
    60 where
    61   "pos_single x = (%nrandom size seed. Limited_Sequence.pos_single x)"
    62 
    63 definition pos_bind :: "'a pos_random_dseq => ('a \<Rightarrow> 'b pos_random_dseq) \<Rightarrow> 'b pos_random_dseq"
    64 where
    65   "pos_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.pos_bind (R nrandom size seed) (%a. f a nrandom size seed))"
    66 
    67 definition pos_decr_bind :: "'a pos_random_dseq => ('a \<Rightarrow> 'b pos_random_dseq) \<Rightarrow> 'b pos_random_dseq"
    68 where
    69   "pos_decr_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.pos_decr_bind (R nrandom size seed) (%a. f a nrandom size seed))"
    70 
    71 definition pos_union :: "'a pos_random_dseq => 'a pos_random_dseq => 'a pos_random_dseq"
    72 where
    73   "pos_union R1 R2 = (\<lambda>nrandom size seed. Limited_Sequence.pos_union (R1 nrandom size seed) (R2 nrandom size seed))"
    74 
    75 definition pos_if_random_dseq :: "bool => unit pos_random_dseq"
    76 where
    77   "pos_if_random_dseq b = (if b then pos_single () else pos_empty)"
    78 
    79 definition pos_iterate_upto :: "(natural => 'a) => natural => natural => 'a pos_random_dseq"
    80 where
    81   "pos_iterate_upto f n m = (\<lambda>nrandom size seed. Limited_Sequence.pos_iterate_upto f n m)"
    82 
    83 definition pos_map :: "('a => 'b) => 'a pos_random_dseq => 'b pos_random_dseq"
    84 where
    85   "pos_map f P = pos_bind P (pos_single o f)"
    86 
    87 fun iter :: "(Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
    88   \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Lazy_Sequence.lazy_sequence"
    89 where
    90   "iter random nrandom seed =
    91     (if nrandom = 0 then Lazy_Sequence.empty else Lazy_Sequence.Lazy_Sequence (%u. let ((x, _), seed') = random seed in Some (x, iter random (nrandom - 1) seed')))"
    92 
    93 definition pos_Random :: "(natural \<Rightarrow> Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
    94   \<Rightarrow> 'a pos_random_dseq"
    95 where
    96   "pos_Random g = (%nrandom size seed depth. iter (g size) nrandom seed)"
    97 
    98 
    99 type_synonym 'a neg_random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Limited_Sequence.neg_dseq"
   100 
   101 definition neg_empty :: "'a neg_random_dseq"
   102 where
   103   "neg_empty = (%nrandom size seed. Limited_Sequence.neg_empty)"
   104 
   105 definition neg_single :: "'a => 'a neg_random_dseq"
   106 where
   107   "neg_single x = (%nrandom size seed. Limited_Sequence.neg_single x)"
   108 
   109 definition neg_bind :: "'a neg_random_dseq => ('a \<Rightarrow> 'b neg_random_dseq) \<Rightarrow> 'b neg_random_dseq"
   110 where
   111   "neg_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.neg_bind (R nrandom size seed) (%a. f a nrandom size seed))"
   112 
   113 definition neg_decr_bind :: "'a neg_random_dseq => ('a \<Rightarrow> 'b neg_random_dseq) \<Rightarrow> 'b neg_random_dseq"
   114 where
   115   "neg_decr_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.neg_decr_bind (R nrandom size seed) (%a. f a nrandom size seed))"
   116 
   117 definition neg_union :: "'a neg_random_dseq => 'a neg_random_dseq => 'a neg_random_dseq"
   118 where
   119   "neg_union R1 R2 = (\<lambda>nrandom size seed. Limited_Sequence.neg_union (R1 nrandom size seed) (R2 nrandom size seed))"
   120 
   121 definition neg_if_random_dseq :: "bool => unit neg_random_dseq"
   122 where
   123   "neg_if_random_dseq b = (if b then neg_single () else neg_empty)"
   124 
   125 definition neg_iterate_upto :: "(natural => 'a) => natural => natural => 'a neg_random_dseq"
   126 where
   127   "neg_iterate_upto f n m = (\<lambda>nrandom size seed. Limited_Sequence.neg_iterate_upto f n m)"
   128 
   129 definition neg_not_random_dseq :: "unit pos_random_dseq => unit neg_random_dseq"
   130 where
   131   "neg_not_random_dseq R = (\<lambda>nrandom size seed. Limited_Sequence.neg_not_seq (R nrandom size seed))"
   132 
   133 definition neg_map :: "('a => 'b) => 'a neg_random_dseq => 'b neg_random_dseq"
   134 where
   135   "neg_map f P = neg_bind P (neg_single o f)"
   136 
   137 definition pos_not_random_dseq :: "unit neg_random_dseq => unit pos_random_dseq"
   138 where
   139   "pos_not_random_dseq R = (\<lambda>nrandom size seed. Limited_Sequence.pos_not_seq (R nrandom size seed))"
   140 
   141 
   142 hide_const (open)
   143   empty single bind union if_random_dseq not_random_dseq map Random
   144   pos_empty pos_single pos_bind pos_decr_bind pos_union pos_if_random_dseq pos_iterate_upto
   145   pos_not_random_dseq pos_map iter pos_Random
   146   neg_empty neg_single neg_bind neg_decr_bind neg_union neg_if_random_dseq neg_iterate_upto
   147   neg_not_random_dseq neg_map
   148 
   149 hide_fact (open) empty_def single_def bind_def union_def if_random_dseq_def not_random_dseq_def
   150   map_def Random.simps
   151   pos_empty_def pos_single_def pos_bind_def pos_decr_bind_def pos_union_def pos_if_random_dseq_def
   152   pos_iterate_upto_def pos_not_random_dseq_def pos_map_def iter.simps pos_Random_def
   153   neg_empty_def neg_single_def neg_bind_def neg_decr_bind_def neg_union_def neg_if_random_dseq_def
   154   neg_iterate_upto_def neg_not_random_dseq_def neg_map_def
   155 
   156 end
   157