src/HOL/Lazy_Sequence.thy
author blanchet
Fri Feb 14 07:53:46 2014 +0100 (2014-02-14)
changeset 55466 786edc984c98
parent 55416 dd7992d4a61a
child 55642 63beb38e9258
permissions -rw-r--r--
merged 'Option.map' and 'Option.map_option'
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(* Author: Lukas Bulwahn, TU Muenchen *)
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header {* Lazy sequences *}
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theory Lazy_Sequence
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imports Predicate
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begin
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subsection {* Type of lazy sequences *}
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datatype 'a lazy_sequence = lazy_sequence_of_list "'a list"
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primrec list_of_lazy_sequence :: "'a lazy_sequence \<Rightarrow> 'a list"
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where
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  "list_of_lazy_sequence (lazy_sequence_of_list xs) = xs"
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lemma lazy_sequence_of_list_of_lazy_sequence [simp]:
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  "lazy_sequence_of_list (list_of_lazy_sequence xq) = xq"
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  by (cases xq) simp_all
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lemma lazy_sequence_eqI:
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  "list_of_lazy_sequence xq = list_of_lazy_sequence yq \<Longrightarrow> xq = yq"
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  by (cases xq, cases yq) simp
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lemma lazy_sequence_eq_iff:
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  "xq = yq \<longleftrightarrow> list_of_lazy_sequence xq = list_of_lazy_sequence yq"
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  by (auto intro: lazy_sequence_eqI)
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lemma lazy_sequence_size_eq:
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  "lazy_sequence_size f xq = Suc (list_size f (list_of_lazy_sequence xq))"
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  by (cases xq) simp
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lemma size_lazy_sequence_eq [code]:
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  "size (xq :: 'a lazy_sequence) = 0"
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  by (cases xq) simp
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lemma case_lazy_sequence [simp]:
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  "case_lazy_sequence f xq = f (list_of_lazy_sequence xq)"
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  by (cases xq) auto
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lemma rec_lazy_sequence [simp]:
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  "rec_lazy_sequence f xq = f (list_of_lazy_sequence xq)"
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  by (cases xq) auto
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definition Lazy_Sequence :: "(unit \<Rightarrow> ('a \<times> 'a lazy_sequence) option) \<Rightarrow> 'a lazy_sequence"
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where
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  "Lazy_Sequence f = lazy_sequence_of_list (case f () of
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    None \<Rightarrow> []
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  | Some (x, xq) \<Rightarrow> x # list_of_lazy_sequence xq)"
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code_datatype Lazy_Sequence
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declare list_of_lazy_sequence.simps [code del]
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declare lazy_sequence.cases [code del]
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declare lazy_sequence.recs [code del]
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lemma list_of_Lazy_Sequence [simp]:
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  "list_of_lazy_sequence (Lazy_Sequence f) = (case f () of
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    None \<Rightarrow> []
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  | Some (x, xq) \<Rightarrow> x # list_of_lazy_sequence xq)"
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  by (simp add: Lazy_Sequence_def)
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definition yield :: "'a lazy_sequence \<Rightarrow> ('a \<times> 'a lazy_sequence) option"
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where
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  "yield xq = (case list_of_lazy_sequence xq of
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    [] \<Rightarrow> None
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  | x # xs \<Rightarrow> Some (x, lazy_sequence_of_list xs))" 
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lemma yield_Seq [simp, code]:
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  "yield (Lazy_Sequence f) = f ()"
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  by (cases "f ()") (simp_all add: yield_def split_def)
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lemma case_yield_eq [simp]: "case_option g h (yield xq) =
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  case_list g (\<lambda>x. curry h x \<circ> lazy_sequence_of_list) (list_of_lazy_sequence xq)"
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  by (cases "list_of_lazy_sequence xq") (simp_all add: yield_def)
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lemma lazy_sequence_size_code [code]:
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  "lazy_sequence_size s xq = (case yield xq of
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    None \<Rightarrow> 1
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  | Some (x, xq') \<Rightarrow> Suc (s x + lazy_sequence_size s xq'))"
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  by (cases "list_of_lazy_sequence xq") (simp_all add: lazy_sequence_size_eq)
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lemma equal_lazy_sequence_code [code]:
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  "HOL.equal xq yq = (case (yield xq, yield yq) of
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    (None, None) \<Rightarrow> True
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  | (Some (x, xq'), Some (y, yq')) \<Rightarrow> HOL.equal x y \<and> HOL.equal xq yq
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  | _ \<Rightarrow> False)"
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  by (simp_all add: lazy_sequence_eq_iff equal_eq split: list.splits)
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lemma [code nbe]:
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  "HOL.equal (x :: 'a lazy_sequence) x \<longleftrightarrow> True"
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  by (fact equal_refl)
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definition empty :: "'a lazy_sequence"
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where
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  "empty = lazy_sequence_of_list []"
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lemma list_of_lazy_sequence_empty [simp]:
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  "list_of_lazy_sequence empty = []"
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  by (simp add: empty_def)
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lemma empty_code [code]:
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  "empty = Lazy_Sequence (\<lambda>_. None)"
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  by (simp add: lazy_sequence_eq_iff)
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definition single :: "'a \<Rightarrow> 'a lazy_sequence"
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where
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  "single x = lazy_sequence_of_list [x]"
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lemma list_of_lazy_sequence_single [simp]:
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  "list_of_lazy_sequence (single x) = [x]"
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  by (simp add: single_def)
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lemma single_code [code]:
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  "single x = Lazy_Sequence (\<lambda>_. Some (x, empty))"
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  by (simp add: lazy_sequence_eq_iff)
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definition append :: "'a lazy_sequence \<Rightarrow> 'a lazy_sequence \<Rightarrow> 'a lazy_sequence"
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where
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  "append xq yq = lazy_sequence_of_list (list_of_lazy_sequence xq @ list_of_lazy_sequence yq)"
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lemma list_of_lazy_sequence_append [simp]:
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  "list_of_lazy_sequence (append xq yq) = list_of_lazy_sequence xq @ list_of_lazy_sequence yq"
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  by (simp add: append_def)
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lemma append_code [code]:
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  "append xq yq = Lazy_Sequence (\<lambda>_. case yield xq of
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    None \<Rightarrow> yield yq
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  | Some (x, xq') \<Rightarrow> Some (x, append xq' yq))"
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  by (simp_all add: lazy_sequence_eq_iff split: list.splits)
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definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a lazy_sequence \<Rightarrow> 'b lazy_sequence"
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where
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  "map f xq = lazy_sequence_of_list (List.map f (list_of_lazy_sequence xq))"
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lemma list_of_lazy_sequence_map [simp]:
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  "list_of_lazy_sequence (map f xq) = List.map f (list_of_lazy_sequence xq)"
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  by (simp add: map_def)
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lemma map_code [code]:
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  "map f xq =
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    Lazy_Sequence (\<lambda>_. map_option (\<lambda>(x, xq'). (f x, map f xq')) (yield xq))"
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  by (simp_all add: lazy_sequence_eq_iff split: list.splits)
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definition flat :: "'a lazy_sequence lazy_sequence \<Rightarrow> 'a lazy_sequence"
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where
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  "flat xqq = lazy_sequence_of_list (concat (List.map list_of_lazy_sequence (list_of_lazy_sequence xqq)))"
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lemma list_of_lazy_sequence_flat [simp]:
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  "list_of_lazy_sequence (flat xqq) = concat (List.map list_of_lazy_sequence (list_of_lazy_sequence xqq))"
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  by (simp add: flat_def)
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lemma flat_code [code]:
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  "flat xqq = Lazy_Sequence (\<lambda>_. case yield xqq of
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    None \<Rightarrow> None
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  | Some (xq, xqq') \<Rightarrow> yield (append xq (flat xqq')))"
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  by (simp add: lazy_sequence_eq_iff split: list.splits)
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definition bind :: "'a lazy_sequence \<Rightarrow> ('a \<Rightarrow> 'b lazy_sequence) \<Rightarrow> 'b lazy_sequence"
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where
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  "bind xq f = flat (map f xq)"
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definition if_seq :: "bool \<Rightarrow> unit lazy_sequence"
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where
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  "if_seq b = (if b then single () else empty)"
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definition those :: "'a option lazy_sequence \<Rightarrow> 'a lazy_sequence option"
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where
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  "those xq = map_option lazy_sequence_of_list (List.those (list_of_lazy_sequence xq))"
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function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a lazy_sequence"
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where
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  "iterate_upto f n m =
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    Lazy_Sequence (\<lambda>_. if n > m then None else Some (f n, iterate_upto f (n + 1) m))"
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  by pat_completeness auto
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termination by (relation "measure (\<lambda>(f, n, m). nat_of_natural (m + 1 - n))")
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  (auto simp add: less_natural_def)
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definition not_seq :: "unit lazy_sequence \<Rightarrow> unit lazy_sequence"
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where
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  "not_seq xq = (case yield xq of
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    None \<Rightarrow> single ()
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  | Some ((), xq) \<Rightarrow> empty)"
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subsection {* Code setup *}
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code_reflect Lazy_Sequence
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  datatypes lazy_sequence = Lazy_Sequence
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ML {*
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signature LAZY_SEQUENCE =
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sig
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  datatype 'a lazy_sequence = Lazy_Sequence of (unit -> ('a * 'a Lazy_Sequence.lazy_sequence) option)
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  val map: ('a -> 'b) -> 'a lazy_sequence -> 'b lazy_sequence
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  val yield: 'a lazy_sequence -> ('a * 'a lazy_sequence) option
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  val yieldn: int -> 'a lazy_sequence -> 'a list * 'a lazy_sequence
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end;
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structure Lazy_Sequence : LAZY_SEQUENCE =
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struct
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datatype lazy_sequence = datatype Lazy_Sequence.lazy_sequence;
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fun map f = @{code Lazy_Sequence.map} f;
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fun yield P = @{code Lazy_Sequence.yield} P;
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fun yieldn k = Predicate.anamorph yield k;
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end;
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*}
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subsection {* Generator Sequences *}
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subsubsection {* General lazy sequence operation *}
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definition product :: "'a lazy_sequence \<Rightarrow> 'b lazy_sequence \<Rightarrow> ('a \<times> 'b) lazy_sequence"
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where
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  "product s1 s2 = bind s1 (\<lambda>a. bind s2 (\<lambda>b. single (a, b)))"
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subsubsection {* Small lazy typeclasses *}
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class small_lazy =
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  fixes small_lazy :: "natural \<Rightarrow> 'a lazy_sequence"
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instantiation unit :: small_lazy
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begin
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definition "small_lazy d = single ()"
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instance ..
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end
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instantiation int :: small_lazy
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begin
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text {* maybe optimise this expression -> append (single x) xs == cons x xs 
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Performance difference? *}
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function small_lazy' :: "int \<Rightarrow> int \<Rightarrow> int lazy_sequence"
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where
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  "small_lazy' d i = (if d < i then empty
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    else append (single i) (small_lazy' d (i + 1)))"
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    by pat_completeness auto
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termination 
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  by (relation "measure (%(d, i). nat (d + 1 - i))") auto
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definition
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  "small_lazy d = small_lazy' (int (nat_of_natural d)) (- (int (nat_of_natural d)))"
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instance ..
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end
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instantiation prod :: (small_lazy, small_lazy) small_lazy
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begin
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definition
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  "small_lazy d = product (small_lazy d) (small_lazy d)"
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instance ..
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end
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instantiation list :: (small_lazy) small_lazy
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begin
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fun small_lazy_list :: "natural \<Rightarrow> 'a list lazy_sequence"
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where
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  "small_lazy_list d = append (single [])
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    (if d > 0 then bind (product (small_lazy (d - 1))
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      (small_lazy (d - 1))) (\<lambda>(x, xs). single (x # xs)) else empty)"
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instance ..
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end
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subsection {* With Hit Bound Value *}
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text {* assuming in negative context *}
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type_synonym 'a hit_bound_lazy_sequence = "'a option lazy_sequence"
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definition hit_bound :: "'a hit_bound_lazy_sequence"
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where
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  "hit_bound = Lazy_Sequence (\<lambda>_. Some (None, empty))"
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lemma list_of_lazy_sequence_hit_bound [simp]:
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  "list_of_lazy_sequence hit_bound = [None]"
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  by (simp add: hit_bound_def)
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definition hb_single :: "'a \<Rightarrow> 'a hit_bound_lazy_sequence"
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where
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  "hb_single x = Lazy_Sequence (\<lambda>_. Some (Some x, empty))"
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definition hb_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a hit_bound_lazy_sequence \<Rightarrow> 'b hit_bound_lazy_sequence"
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where
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  "hb_map f xq = map (map_option f) xq"
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lemma hb_map_code [code]:
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  "hb_map f xq =
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    Lazy_Sequence (\<lambda>_. map_option (\<lambda>(x, xq'). (map_option f x, hb_map f xq')) (yield xq))"
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  using map_code [of "map_option f" xq] by (simp add: hb_map_def)
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definition hb_flat :: "'a hit_bound_lazy_sequence hit_bound_lazy_sequence \<Rightarrow> 'a hit_bound_lazy_sequence"
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where
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  "hb_flat xqq = lazy_sequence_of_list (concat
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    (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)))"
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lemma list_of_lazy_sequence_hb_flat [simp]:
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  "list_of_lazy_sequence (hb_flat xqq) =
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    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))"
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  by (simp add: hb_flat_def)
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lemma hb_flat_code [code]:
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  "hb_flat xqq = Lazy_Sequence (\<lambda>_. case yield xqq of
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    None \<Rightarrow> None
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  | Some (xq, xqq') \<Rightarrow> yield
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     (append (case xq of None \<Rightarrow> hit_bound | Some xq \<Rightarrow> xq) (hb_flat xqq')))"
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  by (simp add: lazy_sequence_eq_iff split: list.splits option.splits)
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definition hb_bind :: "'a hit_bound_lazy_sequence \<Rightarrow> ('a \<Rightarrow> 'b hit_bound_lazy_sequence) \<Rightarrow> 'b hit_bound_lazy_sequence"
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where
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  "hb_bind xq f = hb_flat (hb_map f xq)"
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definition hb_if_seq :: "bool \<Rightarrow> unit hit_bound_lazy_sequence"
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where
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  "hb_if_seq b = (if b then hb_single () else empty)"
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definition hb_not_seq :: "unit hit_bound_lazy_sequence \<Rightarrow> unit lazy_sequence"
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where
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  "hb_not_seq xq = (case yield xq of
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    None \<Rightarrow> single ()
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  | Some (x, xq) \<Rightarrow> empty)"
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hide_const (open) yield empty single append flat map bind
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  if_seq those iterate_upto not_seq product
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hide_fact (open) yield_def empty_def single_def append_def flat_def map_def bind_def
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  if_seq_def those_def not_seq_def product_def 
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end