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