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(* Author: Lukas Bulwahn, TU Muenchen *)
section \<open>Lazy sequences\<close>
theory Lazy_Sequence
imports Predicate
begin
subsection \<open>Type of lazy sequences\<close>
datatype (plugins only: code extraction) (dead 'a) lazy_sequence =
lazy_sequence_of_list "'a list"
primrec list_of_lazy_sequence :: "'a lazy_sequence \ 'a list"
where
"list_of_lazy_sequence (lazy_sequence_of_list xs) = xs"
lemma lazy_sequence_of_list_of_lazy_sequence [simp]:
"lazy_sequence_of_list (list_of_lazy_sequence xq) = xq"
by (cases xq) simp_all
lemma lazy_sequence_eqI:
"list_of_lazy_sequence xq = list_of_lazy_sequence yq \ xq = yq"
by (cases xq, cases yq) simp
lemma lazy_sequence_eq_iff:
"xq = yq \ list_of_lazy_sequence xq = list_of_lazy_sequence yq"
by (auto intro: lazy_sequence_eqI)
lemma case_lazy_sequence [simp]:
"case_lazy_sequence f xq = f (list_of_lazy_sequence xq)"
by (cases xq) auto
lemma rec_lazy_sequence [simp]:
"rec_lazy_sequence f xq = f (list_of_lazy_sequence xq)"
by (cases xq) auto
definition Lazy_Sequence :: "(unit \ ('a \ 'a lazy_sequence) option) \ 'a lazy_sequence"
where
"Lazy_Sequence f = lazy_sequence_of_list (case f () of
None \<Rightarrow> []
| Some (x, xq) \<Rightarrow> x # list_of_lazy_sequence xq)"
code_datatype Lazy_Sequence
declare list_of_lazy_sequence.simps [code del]
declare lazy_sequence.case [code del]
declare lazy_sequence.rec [code del]
lemma list_of_Lazy_Sequence [simp]:
"list_of_lazy_sequence (Lazy_Sequence f) = (case f () of
None \<Rightarrow> []
| Some (x, xq) \<Rightarrow> x # list_of_lazy_sequence xq)"
by (simp add: Lazy_Sequence_def)
definition yield :: "'a lazy_sequence \ ('a \ 'a lazy_sequence) option"
where
"yield xq = (case list_of_lazy_sequence xq of
[] \<Rightarrow> None
| x # xs \<Rightarrow> Some (x, lazy_sequence_of_list xs))"
lemma yield_Seq [simp, code]:
"yield (Lazy_Sequence f) = f ()"
by (cases "f ()") (simp_all add: yield_def split_def)
lemma case_yield_eq [simp]: "case_option g h (yield xq) =
case_list g (\<lambda>x. curry h x \<circ> lazy_sequence_of_list) (list_of_lazy_sequence xq)"
by (cases "list_of_lazy_sequence xq") (simp_all add: yield_def)
lemma equal_lazy_sequence_code [code]:
"HOL.equal xq yq = (case (yield xq, yield yq) of
(None, None) \<Rightarrow> True
| (Some (x, xq'), Some (y, yq')) \<Rightarrow> HOL.equal x y \<and> HOL.equal xq yq
| _ \<Rightarrow> False)"
by (simp_all add: lazy_sequence_eq_iff equal_eq split: list.splits)
lemma [code nbe]:
"HOL.equal (x :: 'a lazy_sequence) x \ True"
by (fact equal_refl)
definition empty :: "'a lazy_sequence"
where
"empty = lazy_sequence_of_list []"
lemma list_of_lazy_sequence_empty [simp]:
"list_of_lazy_sequence empty = []"
by (simp add: empty_def)
lemma empty_code [code]:
"empty = Lazy_Sequence (\_. None)"
by (simp add: lazy_sequence_eq_iff)
definition single :: "'a \ 'a lazy_sequence"
where
"single x = lazy_sequence_of_list [x]"
lemma list_of_lazy_sequence_single [simp]:
"list_of_lazy_sequence (single x) = [x]"
by (simp add: single_def)
lemma single_code [code]:
"single x = Lazy_Sequence (\_. Some (x, empty))"
by (simp add: lazy_sequence_eq_iff)
definition append :: "'a lazy_sequence \ 'a lazy_sequence \ 'a lazy_sequence"
where
"append xq yq = lazy_sequence_of_list (list_of_lazy_sequence xq @ list_of_lazy_sequence yq)"
lemma list_of_lazy_sequence_append [simp]:
"list_of_lazy_sequence (append xq yq) = list_of_lazy_sequence xq @ list_of_lazy_sequence yq"
by (simp add: append_def)
lemma append_code [code]:
"append xq yq = Lazy_Sequence (\_. case yield xq of
None \<Rightarrow> yield yq
| Some (x, xq') \ Some (x, append xq' yq))"
by (simp_all add: lazy_sequence_eq_iff split: list.splits)
definition map :: "('a \ 'b) \ 'a lazy_sequence \ 'b lazy_sequence"
where
"map f xq = lazy_sequence_of_list (List.map f (list_of_lazy_sequence xq))"
lemma list_of_lazy_sequence_map [simp]:
"list_of_lazy_sequence (map f xq) = List.map f (list_of_lazy_sequence xq)"
by (simp add: map_def)
lemma map_code [code]:
"map f xq =
Lazy_Sequence (\<lambda>_. map_option (\<lambda>(x, xq'). (f x, map f xq')) (yield xq))"
by (simp_all add: lazy_sequence_eq_iff split: list.splits)
definition flat :: "'a lazy_sequence lazy_sequence \ 'a lazy_sequence"
where
"flat xqq = lazy_sequence_of_list (concat (List.map list_of_lazy_sequence (list_of_lazy_sequence xqq)))"
lemma list_of_lazy_sequence_flat [simp]:
"list_of_lazy_sequence (flat xqq) = concat (List.map list_of_lazy_sequence (list_of_lazy_sequence xqq))"
by (simp add: flat_def)
lemma flat_code [code]:
"flat xqq = Lazy_Sequence (\_. case yield xqq of
None \<Rightarrow> None
| Some (xq, xqq') \ yield (append xq (flat xqq')))"
by (simp add: lazy_sequence_eq_iff split: list.splits)
definition bind :: "'a lazy_sequence \ ('a \ 'b lazy_sequence) \ 'b lazy_sequence"
where
"bind xq f = flat (map f xq)"
definition if_seq :: "bool \ unit lazy_sequence"
where
"if_seq b = (if b then single () else empty)"
definition those :: "'a option lazy_sequence \ 'a lazy_sequence option"
where
"those xq = map_option lazy_sequence_of_list (List.those (list_of_lazy_sequence xq))"
function iterate_upto :: "(natural \ 'a) \ natural \ natural \ 'a lazy_sequence"
where
"iterate_upto f n m =
Lazy_Sequence (\<lambda>_. if n > m then None else Some (f n, iterate_upto f (n + 1) m))"
by pat_completeness auto
termination by (relation "measure (\(f, n, m). nat_of_natural (m + 1 - n))")
(auto simp add: less_natural_def)
definition not_seq :: "unit lazy_sequence \ unit lazy_sequence"
where
"not_seq xq = (case yield xq of
None \<Rightarrow> single ()
| Some ((), xq) \<Rightarrow> empty)"
subsection \<open>Code setup\<close>
code_reflect Lazy_Sequence
datatypes lazy_sequence = Lazy_Sequence
ML \<open>
signature LAZY_SEQUENCE =
sig
datatype 'a lazy_sequence = Lazy_Sequence of (unit -> ('a * 'a Lazy_Sequence.lazy_sequence) option)
val map: ('a -> 'b) -> 'a lazy_sequence -> 'b lazy_sequence
val yield: 'a lazy_sequence -> ('a * 'a lazy_sequence) option
val yieldn: int -> 'a lazy_sequence -> 'a list * 'a lazy_sequence
end;
structure Lazy_Sequence : LAZY_SEQUENCE =
struct
datatype lazy_sequence = datatype Lazy_Sequence.lazy_sequence;
fun map f = @{code Lazy_Sequence.map} f;
fun yield P = @{code Lazy_Sequence.yield} P;
fun yieldn k = Predicate.anamorph yield k;
end;
\<close>
subsection \<open>Generator Sequences\<close>
subsubsection \<open>General lazy sequence operation\<close>
definition product :: "'a lazy_sequence \ 'b lazy_sequence \ ('a \ 'b) lazy_sequence"
where
"product s1 s2 = bind s1 (\a. bind s2 (\b. single (a, b)))"
subsubsection \<open>Small lazy typeclasses\<close>
class small_lazy =
fixes small_lazy :: "natural \ 'a lazy_sequence"
instantiation unit :: small_lazy
begin
definition "small_lazy d = single ()"
instance ..
end
instantiation int :: small_lazy
begin
text \<open>maybe optimise this expression -> append (single x) xs == cons x xs
Performance difference?\<close>
function small_lazy' :: "int \ int \ int lazy_sequence"
where
"small_lazy' d i = (if d < i then empty
else append (single i) (small_lazy' d (i + 1)))"
by pat_completeness auto
termination
by (relation "measure (%(d, i). nat (d + 1 - i))") auto
definition
"small_lazy d = small_lazy' (int (nat_of_natural d)) (- (int (nat_of_natural d)))"
instance ..
end
instantiation prod :: (small_lazy, small_lazy) small_lazy
begin
definition
"small_lazy d = product (small_lazy d) (small_lazy d)"
instance ..
end
instantiation list :: (small_lazy) small_lazy
begin
fun small_lazy_list :: "natural \ 'a list lazy_sequence"
where
"small_lazy_list d = append (single [])
(if d > 0 then bind (product (small_lazy (d - 1))
(small_lazy (d - 1))) (\<lambda>(x, xs). single (x # xs)) else empty)"
instance ..
end
subsection \<open>With Hit Bound Value\<close>
text \<open>assuming in negative context\<close>
type_synonym 'a hit_bound_lazy_sequence = "'a option lazy_sequence"
definition hit_bound :: "'a hit_bound_lazy_sequence"
where
"hit_bound = Lazy_Sequence (\_. Some (None, empty))"
lemma list_of_lazy_sequence_hit_bound [simp]:
"list_of_lazy_sequence hit_bound = [None]"
by (simp add: hit_bound_def)
definition hb_single :: "'a \ 'a hit_bound_lazy_sequence"
where
"hb_single x = Lazy_Sequence (\_. Some (Some x, empty))"
definition hb_map :: "('a \ 'b) \ 'a hit_bound_lazy_sequence \ 'b hit_bound_lazy_sequence"
where
"hb_map f xq = map (map_option f) xq"
lemma hb_map_code [code]:
"hb_map f xq =
Lazy_Sequence (\<lambda>_. map_option (\<lambda>(x, xq'). (map_option f x, hb_map f xq')) (yield xq))"
using map_code [of "map_option f" xq] by (simp add: hb_map_def)
definition hb_flat :: "'a hit_bound_lazy_sequence hit_bound_lazy_sequence \ 'a hit_bound_lazy_sequence"
where
"hb_flat xqq = lazy_sequence_of_list (concat
(List.map ((\<lambda>x. case x of None \<Rightarrow> [None] | Some xs \<Rightarrow> xs) \<circ> map_option list_of_lazy_sequence) (list_of_lazy_sequence xqq)))"
lemma list_of_lazy_sequence_hb_flat [simp]:
"list_of_lazy_sequence (hb_flat xqq) =
concat (List.map ((\<lambda>x. case x of None \<Rightarrow> [None] | Some xs \<Rightarrow> xs) \<circ> map_option list_of_lazy_sequence) (list_of_lazy_sequence xqq))"
by (simp add: hb_flat_def)
lemma hb_flat_code [code]:
"hb_flat xqq = Lazy_Sequence (\_. case yield xqq of
None \<Rightarrow> None
| Some (xq, xqq') \ yield
(append (case xq of None \<Rightarrow> hit_bound | Some xq \<Rightarrow> xq) (hb_flat xqq')))"
by (simp add: lazy_sequence_eq_iff split: list.splits option.splits)
definition hb_bind :: "'a hit_bound_lazy_sequence \ ('a \ 'b hit_bound_lazy_sequence) \ 'b hit_bound_lazy_sequence"
where
"hb_bind xq f = hb_flat (hb_map f xq)"
definition hb_if_seq :: "bool \ unit hit_bound_lazy_sequence"
where
"hb_if_seq b = (if b then hb_single () else empty)"
definition hb_not_seq :: "unit hit_bound_lazy_sequence \ unit lazy_sequence"
where
"hb_not_seq xq = (case yield xq of
None \<Rightarrow> single ()
| Some (x, xq) \<Rightarrow> empty)"
hide_const (open) yield empty single append flat map bind
if_seq those iterate_upto not_seq product
hide_fact (open) yield_def empty_def single_def append_def flat_def map_def bind_def
if_seq_def those_def not_seq_def product_def
end
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