src/HOL/BNF/Examples/Stream.thy
author traytel
Fri, 15 Mar 2013 10:08:23 +0100
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extended stream library (sdrop_while)
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(*  Title:      HOL/BNF/Examples/Stream.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Infinite streams.
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*)
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header {* Infinite Streams *}
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theory Stream
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imports "../BNF"
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begin
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codata 'a stream = Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65)
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declaration {*
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  Nitpick_HOL.register_codatatype
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    @{typ "'stream_element_type stream"} @{const_name stream_case} [dest_Const @{term Stream}]
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    (*FIXME: long type variable name required to reduce the probability of
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        a name clash of Nitpick in context. E.g.:
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        context
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        fixes x :: 'stream_element_type
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        begin
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        lemma "stream_set s = {}"
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        nitpick
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        oops
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        end
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    *)
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*}
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code_datatype Stream
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lemmas [code] = stream.sels stream.sets stream.case
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lemma stream_case_cert:
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  assumes "CASE \<equiv> stream_case c"
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  shows "CASE (a ## s) \<equiv> c a s"
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  using assms by simp_all
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setup {*
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  Code.add_case @{thm stream_case_cert}
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*}
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(* TODO: Provide by the package*)
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theorem stream_set_induct:
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  "\<lbrakk>\<And>s. P (shd s) s; \<And>s y. \<lbrakk>y \<in> stream_set (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s\<rbrakk> \<Longrightarrow>
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    \<forall>y \<in> stream_set s. P y s"
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  by (rule stream.dtor_set_induct)
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    (auto simp add:  shd_def stl_def stream_case_def fsts_def snds_def split_beta)
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lemma stream_map_simps[simp]:
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  "shd (stream_map f s) = f (shd s)" "stl (stream_map f s) = stream_map f (stl s)"
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  unfolding shd_def stl_def stream_case_def stream_map_def stream.dtor_unfold
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  by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor)
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lemma stream_map_Stream[simp, code]: "stream_map f (x ## s) = f x ## stream_map f s"
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  by (metis stream.exhaust stream.sels stream_map_simps)
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theorem shd_stream_set: "shd s \<in> stream_set s"
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  by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
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    (metis UnCI fsts_def insertI1 stream.dtor_set)
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theorem stl_stream_set: "y \<in> stream_set (stl s) \<Longrightarrow> y \<in> stream_set s"
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  by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
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    (metis insertI1 set_mp snds_def stream.dtor_set_set_incl)
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(* only for the non-mutual case: *)
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theorem stream_set_induct1[consumes 1, case_names shd stl, induct set: "stream_set"]:
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  assumes "y \<in> stream_set s" and "\<And>s. P (shd s) s"
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  and "\<And>s y. \<lbrakk>y \<in> stream_set (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s"
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  shows "P y s"
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  using assms stream_set_induct by blast
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(* end TODO *)
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subsection {* prepend list to stream *}
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primrec shift :: "'a list \<Rightarrow> 'a stream \<Rightarrow> 'a stream" (infixr "@-" 65) where
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  "shift [] s = s"
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| "shift (x # xs) s = x ## shift xs s"
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lemma stream_map_shift[simp]: "stream_map f (xs @- s) = map f xs @- stream_map f s"
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  by (induct xs) auto
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lemma shift_append[simp]: "(xs @ ys) @- s = xs @- ys @- s"
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  by (induct xs) auto
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lemma shift_simps[simp]:
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   "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
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   "stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"
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  by (induct xs) auto
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lemma stream_set_shift[simp]: "stream_set (xs @- s) = set xs \<union> stream_set s"
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  by (induct xs) auto
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lemma shift_left_inj[simp]: "xs @- s1 = xs @- s2 \<longleftrightarrow> s1 = s2"
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  by (induct xs) auto
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subsection {* set of streams with elements in some fixes set *}
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coinductive_set
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  streams :: "'a set => 'a stream set"
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  for A :: "'a set"
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where
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  Stream[intro!, simp, no_atp]: "\<lbrakk>a \<in> A; s \<in> streams A\<rbrakk> \<Longrightarrow> a ## s \<in> streams A"
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lemma shift_streams: "\<lbrakk>w \<in> lists A; s \<in> streams A\<rbrakk> \<Longrightarrow> w @- s \<in> streams A"
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  by (induct w) auto
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lemma stream_set_streams:
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  assumes "stream_set s \<subseteq> A"
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  shows "s \<in> streams A"
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proof (coinduct rule: streams.coinduct[of "\<lambda>s'. \<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"])
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  case streams from assms show ?case by (cases s) auto
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next
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  fix s' assume "\<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"
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  then guess a s by (elim exE)
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  with assms show "\<exists>a l. s' = a ## l \<and>
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    a \<in> A \<and> ((\<exists>a s. l = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A) \<or> l \<in> streams A)"
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    by (cases s) auto
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qed
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subsection {* nth, take, drop for streams *}
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primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!!" 100) where
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  "s !! 0 = shd s"
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| "s !! Suc n = stl s !! n"
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lemma snth_stream_map[simp]: "stream_map f s !! n = f (s !! n)"
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  by (induct n arbitrary: s) auto
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lemma shift_snth_less[simp]: "p < length xs \<Longrightarrow> (xs @- s) !! p = xs ! p"
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  by (induct p arbitrary: xs) (auto simp: hd_conv_nth nth_tl)
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lemma shift_snth_ge[simp]: "p \<ge> length xs \<Longrightarrow> (xs @- s) !! p = s !! (p - length xs)"
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  by (induct p arbitrary: xs) (auto simp: Suc_diff_eq_diff_pred)
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lemma snth_stream_set[simp]: "s !! n \<in> stream_set s"
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  by (induct n arbitrary: s) (auto intro: shd_stream_set stl_stream_set)
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lemma stream_set_range: "stream_set s = range (snth s)"
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proof (intro equalityI subsetI)
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  fix x assume "x \<in> stream_set s"
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  thus "x \<in> range (snth s)"
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  proof (induct s)
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    case (stl s x)
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    then obtain n where "x = stl s !! n" by auto
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    thus ?case by (auto intro: range_eqI[of _ _ "Suc n"])
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  qed (auto intro: range_eqI[of _ _ 0])
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qed auto
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primrec stake :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a list" where
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  "stake 0 s = []"
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| "stake (Suc n) s = shd s # stake n (stl s)"
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lemma length_stake[simp]: "length (stake n s) = n"
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  by (induct n arbitrary: s) auto
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lemma stake_stream_map[simp]: "stake n (stream_map f s) = map f (stake n s)"
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  by (induct n arbitrary: s) auto
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primrec sdrop :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
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  "sdrop 0 s = s"
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| "sdrop (Suc n) s = sdrop n (stl s)"
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lemma sdrop_simps[simp]:
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  "shd (sdrop n s) = s !! n" "stl (sdrop n s) = sdrop (Suc n) s"
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  by (induct n arbitrary: s)  auto
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lemma sdrop_stream_map[simp]: "sdrop n (stream_map f s) = stream_map f (sdrop n s)"
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  by (induct n arbitrary: s) auto
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lemma sdrop_stl: "sdrop n (stl s) = stl (sdrop n s)"
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  by (induct n) auto
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lemma stake_sdrop: "stake n s @- sdrop n s = s"
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  by (induct n arbitrary: s) auto
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lemma id_stake_snth_sdrop:
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  "s = stake i s @- s !! i ## sdrop (Suc i) s"
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  by (subst stake_sdrop[symmetric, of _ i]) (metis sdrop_simps stream.collapse)
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lemma stream_map_alt: "stream_map f s = s' \<longleftrightarrow> (\<forall>n. f (s !! n) = s' !! n)" (is "?L = ?R")
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proof
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  assume ?R
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  thus ?L 
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    by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>n. s1 = stream_map f (sdrop n s) \<and> s2 = sdrop n s'"])
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      (auto intro: exI[of _ 0] simp del: sdrop.simps(2))
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qed auto
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lemma stake_invert_Nil[iff]: "stake n s = [] \<longleftrightarrow> n = 0"
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  by (induct n) auto
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lemma sdrop_shift: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> sdrop n s = s'"
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  by (induct n arbitrary: w s) auto
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lemma stake_shift: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> stake n s = w"
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  by (induct n arbitrary: w s) auto
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lemma stake_add[simp]: "stake m s @ stake n (sdrop m s) = stake (m + n) s"
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  by (induct m arbitrary: s) auto
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lemma sdrop_add[simp]: "sdrop n (sdrop m s) = sdrop (m + n) s"
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  by (induct m arbitrary: s) auto
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partial_function (tailrec) sdrop_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where 
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  "sdrop_while P s = (if P (shd s) then sdrop_while P (stl s) else s)"
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lemma sdrop_while_Stream[code]:
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  "sdrop_while P (Stream a s) = (if P a then sdrop_while P s else Stream a s)"
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  by (subst sdrop_while.simps) simp
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lemma sdrop_while_sdrop_LEAST:
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  assumes "\<exists>n. P (s !! n)"
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  shows "sdrop_while (Not o P) s = sdrop (LEAST n. P (s !! n)) s"
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proof -
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  from assms obtain m where "P (s !! m)" "\<And>n. P (s !! n) \<Longrightarrow> m \<le> n"
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    and *: "(LEAST n. P (s !! n)) = m" by atomize_elim (auto intro: LeastI Least_le)
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  thus ?thesis unfolding *
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  proof (induct m arbitrary: s)
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    case (Suc m)
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    hence "sdrop_while (Not \<circ> P) (stl s) = sdrop m (stl s)"
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      by (metis (full_types) not_less_eq_eq snth.simps(2))
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    moreover from Suc(3) have "\<not> (P (s !! 0))" by blast
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    ultimately show ?case by (subst sdrop_while.simps) simp
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  qed (metis comp_apply sdrop.simps(1) sdrop_while.simps snth.simps(1))
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qed
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subsection {* unary predicates lifted to streams *}
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definition "stream_all P s = (\<forall>p. P (s !! p))"
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lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (stream_set s) P"
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  unfolding stream_all_def stream_set_range by auto
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lemma stream_all_shift[simp]: "stream_all P (xs @- s) = (list_all P xs \<and> stream_all P s)"
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  unfolding stream_all_iff list_all_iff by auto
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subsection {* flatten a stream of lists *}
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definition flat where
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  "flat \<equiv> stream_unfold (hd o shd) (\<lambda>s. if tl (shd s) = [] then stl s else tl (shd s) ## stl s)"
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lemma flat_simps[simp]:
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  "shd (flat ws) = hd (shd ws)"
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  "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
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  unfolding flat_def by auto
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lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)"
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  unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto
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lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @- flat ws"
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  by (induct xs) auto
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lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)"
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  by (cases ws) auto
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lemma flat_snth: "\<forall>xs \<in> stream_set s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then 
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  shd s ! n else flat (stl s) !! (n - length (shd s)))"
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  by (metis flat_unfold not_less shd_stream_set shift_snth_ge shift_snth_less)
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lemma stream_set_flat[simp]: "\<forall>xs \<in> stream_set s. xs \<noteq> [] \<Longrightarrow> 
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  stream_set (flat s) = (\<Union>xs \<in> stream_set s. set xs)" (is "?P \<Longrightarrow> ?L = ?R")
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proof safe
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  fix x assume ?P "x : ?L"
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  then obtain m where "x = flat s !! m" by (metis image_iff stream_set_range)
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  with `?P` obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)"
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  proof (atomize_elim, induct m arbitrary: s rule: less_induct)
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    case (less y)
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    thus ?case
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    proof (cases "y < length (shd s)")
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      case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1))
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    next
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      case False
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      hence "x = flat (stl s) !! (y - length (shd s))" by (metis less(2,3) flat_snth)
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      moreover
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      { from less(2) have "length (shd s) > 0" by (cases s) simp_all
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        moreover with False have "y > 0" by (cases y) simp_all
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        ultimately have "y - length (shd s) < y" by simp
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      }
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      moreover have "\<forall>xs \<in> stream_set (stl s). xs \<noteq> []" using less(2) by (cases s) auto
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      ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto
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      thus ?thesis by (metis snth.simps(2))
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    qed
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  qed
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  thus "x \<in> ?R" by (auto simp: stream_set_range dest!: nth_mem)
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next
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  fix x xs assume "xs \<in> stream_set s" ?P "x \<in> set xs" thus "x \<in> ?L"
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    by (induct rule: stream_set_induct1)
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      (metis UnI1 flat_unfold shift.simps(1) stream_set_shift,
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       metis UnI2 flat_unfold shd_stream_set stl_stream_set stream_set_shift)
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qed
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subsection {* recurring stream out of a list *}
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definition cycle :: "'a list \<Rightarrow> 'a stream" where
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  "cycle = stream_unfold hd (\<lambda>xs. tl xs @ [hd xs])"
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lemma cycle_simps[simp]:
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  "shd (cycle u) = hd u"
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  "stl (cycle u) = cycle (tl u @ [hd u])"
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  by (auto simp: cycle_def)
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lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @- cycle u"
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proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>u. s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []"])
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  case (2 s1 s2)
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  then obtain u where "s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []" by blast
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  thus ?case using stream.unfold[of hd "\<lambda>xs. tl xs @ [hd xs]" u] by (auto simp: cycle_def)
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qed auto
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lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])"
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proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>x xs. s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])"])
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  case (2 s1 s2)
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  then obtain x xs where "s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])" by blast
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  thus ?case
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    by (auto simp: cycle_def intro!: exI[of _ "hd (xs @ [x])"] exI[of _ "tl (xs @ [x])"] stream.unfold)
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qed auto
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lemma cycle_rotated: "\<lbrakk>v \<noteq> []; cycle u = v @- s\<rbrakk> \<Longrightarrow> cycle (tl u @ [hd u]) = tl v @- s"
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  by (auto dest: arg_cong[of _ _ stl])
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lemma stake_append: "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s"
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proof (induct n arbitrary: u)
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  case (Suc n) thus ?case by (cases u) auto
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qed auto
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lemma stake_cycle_le[simp]:
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  assumes "u \<noteq> []" "n < length u"
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  shows "stake n (cycle u) = take n u"
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using min_absorb2[OF less_imp_le_nat[OF assms(2)]]
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  by (subst cycle_decomp[OF assms(1)], subst stake_append) auto
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lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u"
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  by (metis cycle_decomp stake_shift)
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lemma sdrop_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> sdrop (length u) (cycle u) = cycle u"
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  by (metis cycle_decomp sdrop_shift)
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lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>
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   stake n (cycle u) = concat (replicate (n div length u) u)"
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  by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric])
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lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>
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   sdrop n (cycle u) = cycle u"
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  by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric])
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lemma stake_cycle: "u \<noteq> [] \<Longrightarrow>
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   stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u"
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  by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto
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lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)"
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  by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric])
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subsection {* stream repeating a single element *}
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definition "same x = stream_unfold (\<lambda>_. x) id ()"
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lemma same_simps[simp]: "shd (same x) = x" "stl (same x) = same x"
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  unfolding same_def by auto
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lemma same_unfold[code]: "same x = x ## same x"
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  by (metis same_simps stream.collapse)
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lemma snth_same[simp]: "same x !! n = x"
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  unfolding same_def by (induct n) auto
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lemma stake_same[simp]: "stake n (same x) = replicate n x"
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  unfolding same_def by (induct n) (auto simp: upt_rec)
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lemma sdrop_same[simp]: "sdrop n (same x) = same x"
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  unfolding same_def by (induct n) auto
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   380
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lemma shift_replicate_same[simp]: "replicate n x @- same x = same x"
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  by (metis sdrop_same stake_same stake_sdrop)
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lemma stream_all_same[simp]: "stream_all P (same x) \<longleftrightarrow> P x"
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  unfolding stream_all_def by auto
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   386
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lemma same_cycle: "same x = cycle [x]"
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  by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. s1 = same x \<and> s2 = cycle [x]"]) auto
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   390
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subsection {* stream of natural numbers *}
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definition "fromN n = stream_unfold id Suc n"
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   394
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lemma fromN_simps[simp]: "shd (fromN n) = n" "stl (fromN n) = fromN (Suc n)"
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  unfolding fromN_def by auto
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   397
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lemma fromN_unfold[code]: "fromN n = n ## fromN (Suc n)"
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   399
  unfolding fromN_def by (metis id_def stream.unfold)
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   400
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lemma snth_fromN[simp]: "fromN n !! m = n + m"
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   402
  unfolding fromN_def by (induct m arbitrary: n) auto
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   403
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   404
lemma stake_fromN[simp]: "stake m (fromN n) = [n ..< m + n]"
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   405
  unfolding fromN_def by (induct m arbitrary: n) (auto simp: upt_rec)
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   406
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lemma sdrop_fromN[simp]: "sdrop m (fromN n) = fromN (n + m)"
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   408
  unfolding fromN_def by (induct m arbitrary: n) auto
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   409
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lemma stream_set_fromN[simp]: "stream_set (fromN n) = {n ..}" (is "?L = ?R")
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proof safe
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   412
  fix m assume "m : ?L"
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   413
  moreover
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   414
  { fix s assume "m \<in> stream_set s" "\<exists>n'\<ge>n. s = fromN n'"
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    hence "n \<le> m" by (induct arbitrary: n rule: stream_set_induct1) fastforce+
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  }
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  ultimately show "n \<le> m" by blast
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   418
next
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   419
  fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_stream_set)
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qed
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   421
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abbreviation "nats \<equiv> fromN 0"
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   423
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   424
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subsection {* zip *}
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   426
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   427
definition "szip s1 s2 =
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  stream_unfold (map_pair shd shd) (map_pair stl stl) (s1, s2)"
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   429
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   430
lemma szip_simps[simp]:
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  "shd (szip s1 s2) = (shd s1, shd s2)" "stl (szip s1 s2) = szip (stl s1) (stl s2)"
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   432
  unfolding szip_def by auto
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   433
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   434
lemma szip_unfold[code]: "szip (Stream a s1) (Stream b s2) = Stream (a, b) (szip s1 s2)"
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   435
  unfolding szip_def by (subst stream.unfold) simp
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   436
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lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
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   438
  by (induct n arbitrary: s1 s2) auto
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   439
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   440
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   441
subsection {* zip via function *}
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   442
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   443
definition "stream_map2 f s1 s2 =
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   444
  stream_unfold (\<lambda>(s1,s2). f (shd s1) (shd s2)) (map_pair stl stl) (s1, s2)"
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   445
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   446
lemma stream_map2_simps[simp]:
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  "shd (stream_map2 f s1 s2) = f (shd s1) (shd s2)"
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  "stl (stream_map2 f s1 s2) = stream_map2 f (stl s1) (stl s2)"
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   449
  unfolding stream_map2_def by auto
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   450
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   451
lemma stream_map2_unfold[code]:
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   452
  "stream_map2 f (Stream a s1) (Stream b s2) = Stream (f a b) (stream_map2 f s1 s2)"
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   453
  unfolding stream_map2_def by (subst stream.unfold) simp
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   454
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   455
lemma stream_map2_szip:
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   456
  "stream_map2 f s1 s2 = stream_map (split f) (szip s1 s2)"
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   457
  by (coinduct rule: stream.coinduct[of
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   458
    "\<lambda>s1 s2. \<exists>s1' s2'. s1 = stream_map2 f s1' s2' \<and> s2 = stream_map (split f) (szip s1' s2')"])
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   459
    fastforce+
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   460
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end