
bulwahn@34948: 
bulwahn@34948: (* Author: Lukas Bulwahn, TU Muenchen *)
bulwahn@34948: 
haftmann@51126: header {* Various kind of sequences inside the random monad *}
haftmann@51126: 
bulwahn@34948: theory Random_Sequence
haftmann@51126: imports Random_Pred
bulwahn@34948: begin
bulwahn@34948: 
haftmann@51143: type_synonym 'a random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> ('a Limited_Sequence.dseq \<times> Random.seed)"
bulwahn@34948: 
bulwahn@34948: definition empty :: "'a random_dseq"
bulwahn@34948: where
haftmann@51126:   "empty = (%nrandom size. Pair (Limited_Sequence.empty))"
bulwahn@34948: 
bulwahn@34948: definition single :: "'a => 'a random_dseq"
bulwahn@34948: where
haftmann@51126:   "single x = (%nrandom size. Pair (Limited_Sequence.single x))"
bulwahn@34948: 
bulwahn@34948: definition bind :: "'a random_dseq => ('a \<Rightarrow> 'b random_dseq) \<Rightarrow> 'b random_dseq"
bulwahn@34948: where
bulwahn@34948:   "bind R f = (\<lambda>nrandom size s. let
bulwahn@34948:      (P, s') = R nrandom size s;
bulwahn@34948:      (s1, s2) = Random.split_seed s'
haftmann@51126:   in (Limited_Sequence.bind P (%a. fst (f a nrandom size s1)), s2))"
bulwahn@34948: 
bulwahn@34948: definition union :: "'a random_dseq => 'a random_dseq => 'a random_dseq"
bulwahn@34948: where
bulwahn@34948:   "union R1 R2 = (\<lambda>nrandom size s. let
bulwahn@34948:      (S1, s') = R1 nrandom size s; (S2, s'') = R2 nrandom size s'
haftmann@51126:   in (Limited_Sequence.union S1 S2, s''))"
bulwahn@34948: 
bulwahn@34948: definition if_random_dseq :: "bool => unit random_dseq"
bulwahn@34948: where
bulwahn@34948:   "if_random_dseq b = (if b then single () else empty)"
bulwahn@34948: 
bulwahn@34948: definition not_random_dseq :: "unit random_dseq => unit random_dseq"
bulwahn@34948: where
bulwahn@34948:   "not_random_dseq R = (\<lambda>nrandom size s. let
bulwahn@34948:      (S, s') = R nrandom size s
haftmann@51126:    in (Limited_Sequence.not_seq S, s'))"
bulwahn@34948: 
bulwahn@34948: definition map :: "('a => 'b) => 'a random_dseq => 'b random_dseq"
bulwahn@34948: where
bulwahn@34948:   "map f P = bind P (single o f)"
bulwahn@34948: 
haftmann@51143: fun Random :: "(natural \<Rightarrow> Random.seed \<Rightarrow> (('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)) \<Rightarrow> 'a random_dseq"
haftmann@51126: where
haftmann@51126:   "Random g nrandom = (%size. if nrandom <= 0 then (Pair Limited_Sequence.empty) else
haftmann@51126:      (scomp (g size) (%r. scomp (Random g (nrandom - 1) size) (%rs. Pair (Limited_Sequence.union (Limited_Sequence.single (fst r)) rs)))))"
haftmann@51126: 
haftmann@51126: 
haftmann@51143: type_synonym 'a pos_random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Limited_Sequence.pos_dseq"
haftmann@51126: 
haftmann@51126: definition pos_empty :: "'a pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_empty = (%nrandom size seed. Limited_Sequence.pos_empty)"
haftmann@51126: 
haftmann@51126: definition pos_single :: "'a => 'a pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_single x = (%nrandom size seed. Limited_Sequence.pos_single x)"
haftmann@51126: 
haftmann@51126: definition pos_bind :: "'a pos_random_dseq => ('a \<Rightarrow> 'b pos_random_dseq) \<Rightarrow> 'b pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.pos_bind (R nrandom size seed) (%a. f a nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition pos_decr_bind :: "'a pos_random_dseq => ('a \<Rightarrow> 'b pos_random_dseq) \<Rightarrow> 'b pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_decr_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.pos_decr_bind (R nrandom size seed) (%a. f a nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition pos_union :: "'a pos_random_dseq => 'a pos_random_dseq => 'a pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_union R1 R2 = (\<lambda>nrandom size seed. Limited_Sequence.pos_union (R1 nrandom size seed) (R2 nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition pos_if_random_dseq :: "bool => unit pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_if_random_dseq b = (if b then pos_single () else pos_empty)"
haftmann@51126: 
haftmann@51143: definition pos_iterate_upto :: "(natural => 'a) => natural => natural => 'a pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_iterate_upto f n m = (\<lambda>nrandom size seed. Limited_Sequence.pos_iterate_upto f n m)"
haftmann@51126: 
haftmann@51126: definition pos_map :: "('a => 'b) => 'a pos_random_dseq => 'b pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_map f P = pos_bind P (pos_single o f)"
haftmann@51126: 
haftmann@51126: fun iter :: "(Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
haftmann@51143:   \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Lazy_Sequence.lazy_sequence"
haftmann@51126: where
haftmann@51126:   "iter random nrandom seed =
haftmann@51126:     (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')))"
haftmann@51126: 
haftmann@51143: definition pos_Random :: "(natural \<Rightarrow> Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
haftmann@51126:   \<Rightarrow> 'a pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_Random g = (%nrandom size seed depth. iter (g size) nrandom seed)"
haftmann@51126: 
haftmann@51126: 
haftmann@51143: type_synonym 'a neg_random_dseq = "natural \<Rightarrow> natural \<Rightarrow> Random.seed \<Rightarrow> 'a Limited_Sequence.neg_dseq"
bulwahn@34948: 
haftmann@51126: definition neg_empty :: "'a neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_empty = (%nrandom size seed. Limited_Sequence.neg_empty)"
haftmann@51126: 
haftmann@51126: definition neg_single :: "'a => 'a neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_single x = (%nrandom size seed. Limited_Sequence.neg_single x)"
haftmann@51126: 
haftmann@51126: definition neg_bind :: "'a neg_random_dseq => ('a \<Rightarrow> 'b neg_random_dseq) \<Rightarrow> 'b neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.neg_bind (R nrandom size seed) (%a. f a nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition neg_decr_bind :: "'a neg_random_dseq => ('a \<Rightarrow> 'b neg_random_dseq) \<Rightarrow> 'b neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_decr_bind R f = (\<lambda>nrandom size seed. Limited_Sequence.neg_decr_bind (R nrandom size seed) (%a. f a nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition neg_union :: "'a neg_random_dseq => 'a neg_random_dseq => 'a neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_union R1 R2 = (\<lambda>nrandom size seed. Limited_Sequence.neg_union (R1 nrandom size seed) (R2 nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition neg_if_random_dseq :: "bool => unit neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_if_random_dseq b = (if b then neg_single () else neg_empty)"
haftmann@51126: 
haftmann@51143: definition neg_iterate_upto :: "(natural => 'a) => natural => natural => 'a neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_iterate_upto f n m = (\<lambda>nrandom size seed. Limited_Sequence.neg_iterate_upto f n m)"
bulwahn@34948: 
haftmann@51126: definition neg_not_random_dseq :: "unit pos_random_dseq => unit neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_not_random_dseq R = (\<lambda>nrandom size seed. Limited_Sequence.neg_not_seq (R nrandom size seed))"
haftmann@51126: 
haftmann@51126: definition neg_map :: "('a => 'b) => 'a neg_random_dseq => 'b neg_random_dseq"
haftmann@51126: where
haftmann@51126:   "neg_map f P = neg_bind P (neg_single o f)"
haftmann@51126: 
haftmann@51126: definition pos_not_random_dseq :: "unit neg_random_dseq => unit pos_random_dseq"
haftmann@51126: where
haftmann@51126:   "pos_not_random_dseq R = (\<lambda>nrandom size seed. Limited_Sequence.pos_not_seq (R nrandom size seed))"
haftmann@51126: 
bulwahn@34948: 
haftmann@51126: hide_const (open)
haftmann@51126:   empty single bind union if_random_dseq not_random_dseq map Random
haftmann@51126:   pos_empty pos_single pos_bind pos_decr_bind pos_union pos_if_random_dseq pos_iterate_upto
haftmann@51126:   pos_not_random_dseq pos_map iter pos_Random
haftmann@51126:   neg_empty neg_single neg_bind neg_decr_bind neg_union neg_if_random_dseq neg_iterate_upto
haftmann@51126:   neg_not_random_dseq neg_map
haftmann@51126: 
haftmann@51126: hide_fact (open) empty_def single_def bind_def union_def if_random_dseq_def not_random_dseq_def
haftmann@51126:   map_def Random.simps
haftmann@51126:   pos_empty_def pos_single_def pos_bind_def pos_decr_bind_def pos_union_def pos_if_random_dseq_def
haftmann@51126:   pos_iterate_upto_def pos_not_random_dseq_def pos_map_def iter.simps pos_Random_def
haftmann@51126:   neg_empty_def neg_single_def neg_bind_def neg_decr_bind_def neg_union_def neg_if_random_dseq_def
haftmann@51126:   neg_iterate_upto_def neg_not_random_dseq_def neg_map_def
haftmann@51126: 
haftmann@51126: end
haftmann@51126: 




