Theory Lazy_Sequence

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theory Lazy_Sequence
imports List Code_Numeral


(* Author: Lukas Bulwahn, TU Muenchen *)

header {* Lazy sequences *}

theory Lazy_Sequence
imports List Code_Numeral
begin


datatype 'a lazy_sequence = Empty | Insert 'a "'a lazy_sequence"

definition Lazy_Sequence :: "(unit => ('a * 'a lazy_sequence) option) => 'a lazy_sequence"
where
"Lazy_Sequence f = (case f () of None => Empty | Some (x, xq) => Insert x xq)"


code_datatype Lazy_Sequence

primrec yield :: "'a lazy_sequence => ('a * 'a lazy_sequence) option"
where
"yield Empty = None"
| "yield (Insert x xq) = Some (x, xq)"


lemma [simp]: "yield xq = Some (x, xq') ==> size xq' < size xq"
by (cases xq) auto

lemma yield_Seq [code]:
"yield (Lazy_Sequence f) = f ()"

unfolding Lazy_Sequence_def by (cases "f ()") auto

lemma Seq_yield:
"Lazy_Sequence (%u. yield f) = f"

unfolding Lazy_Sequence_def by (cases f) auto

lemma lazy_sequence_size_code [code]:
"lazy_sequence_size s xq = (case yield xq of None => 0 | Some (x, xq') => s x + lazy_sequence_size s xq' + 1)"

by (cases xq) auto

lemma size_code [code]:
"size xq = (case yield xq of None => 0 | Some (x, xq') => size xq' + 1)"

by (cases xq) auto

lemma [code]: "eq_class.eq xq yq = (case (yield xq, yield yq) of
(None, None) => True | (Some (x, xq'), Some (y, yq')) => (HOL.eq x y) ∧ (eq_class.eq xq yq) | _ => False)"

apply (cases xq) apply (cases yq) apply (auto simp add: eq_class.eq_equals)
apply (cases yq) apply (auto simp add: eq_class.eq_equals) done

lemma seq_case [code]:
"lazy_sequence_case f g xq = (case (yield xq) of None => f | Some (x, xq') => g x xq')"

by (cases xq) auto

lemma [code]: "lazy_sequence_rec f g xq = (case (yield xq) of None => f | Some (x, xq') => g x xq' (lazy_sequence_rec f g xq'))"
by (cases xq) auto

definition empty :: "'a lazy_sequence"
where
[code]: "empty = Lazy_Sequence (%u. None)"


definition single :: "'a => 'a lazy_sequence"
where
[code]: "single x = Lazy_Sequence (%u. Some (x, empty))"


primrec append :: "'a lazy_sequence => 'a lazy_sequence => 'a lazy_sequence"
where
"append Empty yq = yq"
| "append (Insert x xq) yq = Insert x (append xq yq)"


lemma [code]:
"append xq yq = Lazy_Sequence (%u. case yield xq of
None => yield yq
| Some (x, xq') => Some (x, append xq' yq))"

unfolding Lazy_Sequence_def
apply (cases "xq")
apply auto
apply (cases "yq")
apply auto
done

primrec flat :: "'a lazy_sequence lazy_sequence => 'a lazy_sequence"
where
"flat Empty = Empty"
| "flat (Insert xq xqq) = append xq (flat xqq)"


lemma [code]:
"flat xqq = Lazy_Sequence (%u. case yield xqq of
None => None
| Some (xq, xqq') => yield (append xq (flat xqq')))"

apply (cases "xqq")
apply (auto simp add: Seq_yield)
unfolding Lazy_Sequence_def
by auto

primrec map :: "('a => 'b) => 'a lazy_sequence => 'b lazy_sequence"
where
"map f Empty = Empty"
| "map f (Insert x xq) = Insert (f x) (map f xq)"


lemma [code]:
"map f xq = Lazy_Sequence (%u. Option.map (%(x, xq'). (f x, map f xq')) (yield xq))"

apply (cases xq)
apply (auto simp add: Seq_yield)
unfolding Lazy_Sequence_def
apply auto
done

definition bind :: "'a lazy_sequence => ('a => 'b lazy_sequence) => 'b lazy_sequence"
where
[code]: "bind xq f = flat (map f xq)"


definition if_seq :: "bool => unit lazy_sequence"
where
"if_seq b = (if b then single () else empty)"


function iterate_upto :: "(code_numeral => 'a) => code_numeral => code_numeral => 'a Lazy_Sequence.lazy_sequence"
where
"iterate_upto f n m = Lazy_Sequence.Lazy_Sequence (%u. 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). Code_Numeral.nat_of (m + 1 - n))") auto

definition not_seq :: "unit lazy_sequence => unit lazy_sequence"
where
"not_seq xq = (case yield xq of None => single () | Some ((), xq) => empty)"


subsection {* Code setup *}

fun anamorph :: "('a => ('b × 'a) option) => code_numeral => 'a => 'b list × 'a" where
"anamorph f k x = (if k = 0 then ([], x)
else case f x of None => ([], x) | Some (v, y) =>
let (vs, z) = anamorph f (k - 1) y
in (v # vs, z))"


definition yieldn :: "code_numeral => 'a lazy_sequence => 'a list × 'a lazy_sequence" where
"yieldn = anamorph yield"


code_reflect Lazy_Sequence
datatypes lazy_sequence = Lazy_Sequence
functions map yield yieldn


subsection {* With Hit Bound Value *}
text {* assuming in negative context *}

types 'a hit_bound_lazy_sequence = "'a option lazy_sequence"

definition hit_bound :: "'a hit_bound_lazy_sequence"
where
[code]: "hit_bound = Lazy_Sequence (%u. Some (None, empty))"



definition hb_single :: "'a => 'a hit_bound_lazy_sequence"
where
[code]: "hb_single x = Lazy_Sequence (%u. Some (Some x, empty))"


primrec hb_flat :: "'a hit_bound_lazy_sequence hit_bound_lazy_sequence => 'a hit_bound_lazy_sequence"
where
"hb_flat Empty = Empty"
| "hb_flat (Insert xq xqq) = append (case xq of None => hit_bound | Some xq => xq) (hb_flat xqq)"


lemma [code]:
"hb_flat xqq = Lazy_Sequence (%u. case yield xqq of
None => None
| Some (xq, xqq') => yield (append (case xq of None => hit_bound | Some xq => xq) (hb_flat xqq')))"

apply (cases "xqq")
apply (auto simp add: Seq_yield)
unfolding Lazy_Sequence_def
by auto

primrec hb_map :: "('a => 'b) => 'a hit_bound_lazy_sequence => 'b hit_bound_lazy_sequence"
where
"hb_map f Empty = Empty"
| "hb_map f (Insert x xq) = Insert (Option.map f x) (hb_map f xq)"


lemma [code]:
"hb_map f xq = Lazy_Sequence (%u. Option.map (%(x, xq'). (Option.map f x, hb_map f xq')) (yield xq))"

apply (cases xq)
apply (auto simp add: Seq_yield)
unfolding Lazy_Sequence_def
apply auto
done

definition hb_bind :: "'a hit_bound_lazy_sequence => ('a => 'b hit_bound_lazy_sequence) => 'b hit_bound_lazy_sequence"
where
[code]: "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 => single () | Some (x, xq) => empty)"


hide_type (open) lazy_sequence
hide_const (open) Empty Insert Lazy_Sequence yield empty single append flat map bind if_seq iterate_upto not_seq
hide_fact yield.simps empty_def single_def append.simps flat.simps map.simps bind_def iterate_upto.simps if_seq_def not_seq_def

end