(*  Title:      HOL/Library/Stream.thy

    Author:     Dmitriy Traytel, TU Muenchen

    Author:     Andrei Popescu, TU Muenchen

    Copyright   2012, 2013

Infinite streams.

*)

section \<open>Infinite Streams\<close>

theory Stream

imports "~~/src/HOL/Library/Nat_Bijection"

begin

codatatype (sset: 'a) stream =

  SCons (shd: 'a) (stl: "'a stream") (infixr "##" 65)

for

  map: smap

  rel: stream_all2

context

begin

(*for code generation only*)

qualified definition smember :: "'a \<Rightarrow> 'a stream \<Rightarrow> bool" where

  [code_abbrev]: "smember x s \<longleftrightarrow> x \<in> sset s"

lemma smember_code[code, simp]: "smember x (y ## s) = (if x = y then True else smember x s)"

  unfolding smember_def by auto

end

lemmas smap_simps[simp] = stream.map_sel

lemmas shd_sset = stream.set_sel(1)

lemmas stl_sset = stream.set_sel(2)

theorem sset_induct[consumes 1, case_names shd stl, induct set: sset]:

  assumes "y \<in> sset s" and "\<And>s. P (shd s) s" and "\<And>s y. \<lbrakk>y \<in> sset (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s"

  shows "P y s"

using assms by induct (metis stream.sel(1), auto)

lemma smap_ctr: "smap f s = x ## s' \<longleftrightarrow> f (shd s) = x \<and> smap f (stl s) = s'"

  by (cases s) simp

subsection \<open>prepend list to stream\<close>

primrec shift :: "'a list \<Rightarrow> 'a stream \<Rightarrow> 'a stream" (infixr "@-" 65) where

  "shift [] s = s"

| "shift (x # xs) s = x ## shift xs s"

lemma smap_shift[simp]: "smap f (xs @- s) = map f xs @- smap f s"

  by (induct xs) auto

lemma shift_append[simp]: "(xs @ ys) @- s = xs @- ys @- s"

  by (induct xs) auto

lemma shift_simps[simp]:

   "shd (xs @- s) = (if xs = [] then shd s else hd xs)"

   "stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"

  by (induct xs) auto

lemma sset_shift[simp]: "sset (xs @- s) = set xs \<union> sset s"

  by (induct xs) auto

lemma shift_left_inj[simp]: "xs @- s1 = xs @- s2 \<longleftrightarrow> s1 = s2"

  by (induct xs) auto

subsection \<open>set of streams with elements in some fixed set\<close>

context

  notes [[inductive_defs]]

begin

coinductive_set

  streams :: "'a set \<Rightarrow> 'a stream set"

  for A :: "'a set"

where

  Stream[intro!, simp, no_atp]: "\<lbrakk>a \<in> A; s \<in> streams A\<rbrakk> \<Longrightarrow> a ## s \<in> streams A"

end

lemma in_streams: "stl s \<in> streams S \<Longrightarrow> shd s \<in> S \<Longrightarrow> s \<in> streams S"

  by (cases s) auto

lemma streamsE: "s \<in> streams A \<Longrightarrow> (shd s \<in> A \<Longrightarrow> stl s \<in> streams A \<Longrightarrow> P) \<Longrightarrow> P"

  by (erule streams.cases) simp_all

lemma Stream_image: "x ## y \<in> (op ## x') ` Y \<longleftrightarrow> x = x' \<and> y \<in> Y"

  by auto

lemma shift_streams: "\<lbrakk>w \<in> lists A; s \<in> streams A\<rbrakk> \<Longrightarrow> w @- s \<in> streams A"

  by (induct w) auto

lemma streams_Stream: "x ## s \<in> streams A \<longleftrightarrow> x \<in> A \<and> s \<in> streams A"

  by (auto elim: streams.cases)

lemma streams_stl: "s \<in> streams A \<Longrightarrow> stl s \<in> streams A"

  by (cases s) (auto simp: streams_Stream)

lemma streams_shd: "s \<in> streams A \<Longrightarrow> shd s \<in> A"

  by (cases s) (auto simp: streams_Stream)

lemma sset_streams:

  assumes "sset s \<subseteq> A"

  shows "s \<in> streams A"

using assms proof (coinduction arbitrary: s)

  case streams then show ?case by (cases s) simp

qed

lemma streams_sset:

  assumes "s \<in> streams A"

  shows "sset s \<subseteq> A"

proof

  fix x assume "x \<in> sset s" from this \<open>s \<in> streams A\<close> show "x \<in> A"

    by (induct s) (auto intro: streams_shd streams_stl)

qed

lemma streams_iff_sset: "s \<in> streams A \<longleftrightarrow> sset s \<subseteq> A"

  by (metis sset_streams streams_sset)

lemma streams_mono:  "s \<in> streams A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> s \<in> streams B"

  unfolding streams_iff_sset by auto

lemma streams_mono2: "S \<subseteq> T \<Longrightarrow> streams S \<subseteq> streams T"

  by (auto intro: streams_mono)

lemma smap_streams: "s \<in> streams A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> smap f s \<in> streams B"

  unfolding streams_iff_sset stream.set_map by auto

lemma streams_empty: "streams {} = {}"

  by (auto elim: streams.cases)

lemma streams_UNIV[simp]: "streams UNIV = UNIV"

  by (auto simp: streams_iff_sset)

subsection \<open>nth, take, drop for streams\<close>

primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!!" 100) where

  "s !! 0 = shd s"

| "s !! Suc n = stl s !! n"

lemma snth_Stream: "(x ## s) !! Suc i = s !! i"

  by simp

lemma snth_smap[simp]: "smap f s !! n = f (s !! n)"

  by (induct n arbitrary: s) auto

lemma shift_snth_less[simp]: "p < length xs \<Longrightarrow> (xs @- s) !! p = xs ! p"

  by (induct p arbitrary: xs) (auto simp: hd_conv_nth nth_tl)

lemma shift_snth_ge[simp]: "p \<ge> length xs \<Longrightarrow> (xs @- s) !! p = s !! (p - length xs)"

  by (induct p arbitrary: xs) (auto simp: Suc_diff_eq_diff_pred)

lemma shift_snth: "(xs @- s) !! n = (if n < length xs then xs ! n else s !! (n - length xs))"

  by auto

lemma snth_sset[simp]: "s !! n \<in> sset s"

  by (induct n arbitrary: s) (auto intro: shd_sset stl_sset)

lemma sset_range: "sset s = range (snth s)"

proof (intro equalityI subsetI)

  fix x assume "x \<in> sset s"

  thus "x \<in> range (snth s)"

  proof (induct s)

    case (stl s x)

    then obtain n where "x = stl s !! n" by auto

    thus ?case by (auto intro: range_eqI[of _ _ "Suc n"])

  qed (auto intro: range_eqI[of _ _ 0])

qed auto

lemma streams_iff_snth: "s \<in> streams X \<longleftrightarrow> (\<forall>n. s !! n \<in> X)"

  by (force simp: streams_iff_sset sset_range)

lemma snth_in: "s \<in> streams X \<Longrightarrow> s !! n \<in> X"

  by (simp add: streams_iff_snth)

primrec stake :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a list" where

  "stake 0 s = []"

| "stake (Suc n) s = shd s # stake n (stl s)"

lemma length_stake[simp]: "length (stake n s) = n"

  by (induct n arbitrary: s) auto

lemma stake_smap[simp]: "stake n (smap f s) = map f (stake n s)"

  by (induct n arbitrary: s) auto

lemma take_stake: "take n (stake m s) = stake (min n m) s"

proof (induct m arbitrary: s n)

  case (Suc m) thus ?case by (cases n) auto

qed simp

primrec sdrop :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where

  "sdrop 0 s = s"

| "sdrop (Suc n) s = sdrop n (stl s)"

lemma sdrop_simps[simp]:

  "shd (sdrop n s) = s !! n" "stl (sdrop n s) = sdrop (Suc n) s"

  by (induct n arbitrary: s)  auto

lemma sdrop_smap[simp]: "sdrop n (smap f s) = smap f (sdrop n s)"

  by (induct n arbitrary: s) auto

lemma sdrop_stl: "sdrop n (stl s) = stl (sdrop n s)"

  by (induct n) auto

lemma drop_stake: "drop n (stake m s) = stake (m - n) (sdrop n s)"

proof (induct m arbitrary: s n)

  case (Suc m) thus ?case by (cases n) auto

qed simp

lemma stake_sdrop: "stake n s @- sdrop n s = s"

  by (induct n arbitrary: s) auto

lemma id_stake_snth_sdrop:

  "s = stake i s @- s !! i ## sdrop (Suc i) s"

  by (subst stake_sdrop[symmetric, of _ i]) (metis sdrop_simps stream.collapse)

lemma smap_alt: "smap f s = s' \<longleftrightarrow> (\<forall>n. f (s !! n) = s' !! n)" (is "?L = ?R")

proof

  assume ?R

  then have "\<And>n. smap f (sdrop n s) = sdrop n s'"

    by coinduction (auto intro: exI[of _ 0] simp del: sdrop.simps(2))

  then show ?L using sdrop.simps(1) by metis

qed auto

lemma stake_invert_Nil[iff]: "stake n s = [] \<longleftrightarrow> n = 0"

  by (induct n) auto

lemma sdrop_shift: "sdrop i (w @- s) = drop i w @- sdrop (i - length w) s"

  by (induct i arbitrary: w s) (auto simp: drop_tl drop_Suc neq_Nil_conv)

lemma stake_shift: "stake i (w @- s) = take i w @ stake (i - length w) s"

  by (induct i arbitrary: w s) (auto simp: neq_Nil_conv)

lemma stake_add[simp]: "stake m s @ stake n (sdrop m s) = stake (m + n) s"

  by (induct m arbitrary: s) auto

lemma sdrop_add[simp]: "sdrop n (sdrop m s) = sdrop (m + n) s"

  by (induct m arbitrary: s) auto

lemma sdrop_snth: "sdrop n s !! m = s !! (n + m)"

  by (induct n arbitrary: m s) auto

partial_function (tailrec) sdrop_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where 

  "sdrop_while P s = (if P (shd s) then sdrop_while P (stl s) else s)"

lemma sdrop_while_SCons[code]:

  "sdrop_while P (a ## s) = (if P a then sdrop_while P s else a ## s)"

  by (subst sdrop_while.simps) simp

lemma sdrop_while_sdrop_LEAST:

  assumes "\<exists>n. P (s !! n)"

  shows "sdrop_while (Not o P) s = sdrop (LEAST n. P (s !! n)) s"

proof -

  from assms obtain m where "P (s !! m)" "\<And>n. P (s !! n) \<Longrightarrow> m \<le> n"

    and *: "(LEAST n. P (s !! n)) = m" by atomize_elim (auto intro: LeastI Least_le)

  thus ?thesis unfolding *

  proof (induct m arbitrary: s)

    case (Suc m)

    hence "sdrop_while (Not \<circ> P) (stl s) = sdrop m (stl s)"

      by (metis (full_types) not_less_eq_eq snth.simps(2))

    moreover from Suc(3) have "\<not> (P (s !! 0))" by blast

    ultimately show ?case by (subst sdrop_while.simps) simp

  qed (metis comp_apply sdrop.simps(1) sdrop_while.simps snth.simps(1))

qed

primcorec sfilter where

  "shd (sfilter P s) = shd (sdrop_while (Not o P) s)"

| "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not o P) s))"

lemma sfilter_Stream: "sfilter P (x ## s) = (if P x then x ## sfilter P s else sfilter P s)"

proof (cases "P x")

  case True thus ?thesis by (subst sfilter.ctr) (simp add: sdrop_while_SCons)

next

  case False thus ?thesis by (subst (1 2) sfilter.ctr) (simp add: sdrop_while_SCons)

qed

subsection \<open>unary predicates lifted to streams\<close>

definition "stream_all P s = (\<forall>p. P (s !! p))"

lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P"

  unfolding stream_all_def sset_range by auto

lemma stream_all_shift[simp]: "stream_all P (xs @- s) = (list_all P xs \<and> stream_all P s)"

  unfolding stream_all_iff list_all_iff by auto

lemma stream_all_Stream: "stream_all P (x ## X) \<longleftrightarrow> P x \<and> stream_all P X"

  by simp

subsection \<open>recurring stream out of a list\<close>

primcorec cycle :: "'a list \<Rightarrow> 'a stream" where

  "shd (cycle xs) = hd xs"

| "stl (cycle xs) = cycle (tl xs @ [hd xs])"

lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @- cycle u"

proof (coinduction arbitrary: u)

  case Eq_stream then show ?case using stream.collapse[of "cycle u"]

    by (auto intro!: exI[of _ "tl u @ [hd u]"])

qed

lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])"

  by (subst cycle.ctr) simp

lemma cycle_rotated: "\<lbrakk>v \<noteq> []; cycle u = v @- s\<rbrakk> \<Longrightarrow> cycle (tl u @ [hd u]) = tl v @- s"

  by (auto dest: arg_cong[of _ _ stl])

lemma stake_append: "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s"

proof (induct n arbitrary: u)

  case (Suc n) thus ?case by (cases u) auto

qed auto

lemma stake_cycle_le[simp]:

  assumes "u \<noteq> []" "n < length u"

  shows "stake n (cycle u) = take n u"

using min_absorb2[OF less_imp_le_nat[OF assms(2)]]

  by (subst cycle_decomp[OF assms(1)], subst stake_append) auto

lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u"

  by (subst cycle_decomp) (auto simp: stake_shift)

lemma sdrop_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> sdrop (length u) (cycle u) = cycle u"

  by (subst cycle_decomp) (auto simp: sdrop_shift)

lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>

   stake n (cycle u) = concat (replicate (n div length u) u)"

  by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric])

lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>

   sdrop n (cycle u) = cycle u"

  by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric])

lemma stake_cycle: "u \<noteq> [] \<Longrightarrow>

   stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u"

  by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto

lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)"

  by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric])

subsection \<open>iterated application of a function\<close>

primcorec siterate where

  "shd (siterate f x) = x"

| "stl (siterate f x) = siterate f (f x)"

lemma stake_Suc: "stake (Suc n) s = stake n s @ [s !! n]"

  by (induct n arbitrary: s) auto

lemma snth_siterate[simp]: "siterate f x !! n = (f^^n) x"

  by (induct n arbitrary: x) (auto simp: funpow_swap1)

lemma sdrop_siterate[simp]: "sdrop n (siterate f x) = siterate f ((f^^n) x)"

  by (induct n arbitrary: x) (auto simp: funpow_swap1)

lemma stake_siterate[simp]: "stake n (siterate f x) = map (\<lambda>n. (f^^n) x) [0 ..< n]"

  by (induct n arbitrary: x) (auto simp del: stake.simps(2) simp: stake_Suc)

lemma sset_siterate: "sset (siterate f x) = {(f^^n) x | n. True}"

  by (auto simp: sset_range)

lemma smap_siterate: "smap f (siterate f x) = siterate f (f x)"

  by (coinduction arbitrary: x) auto

subsection \<open>stream repeating a single element\<close>

abbreviation "sconst \<equiv> siterate id"

lemma shift_replicate_sconst[simp]: "replicate n x @- sconst x = sconst x"

  by (subst (3) stake_sdrop[symmetric]) (simp add: map_replicate_trivial)

lemma sset_sconst[simp]: "sset (sconst x) = {x}"

  by (simp add: sset_siterate)

lemma sconst_alt: "s = sconst x \<longleftrightarrow> sset s = {x}"

proof

  assume "sset s = {x}"

  then show "s = sconst x"

  proof (coinduction arbitrary: s)

    case Eq_stream

    then have "shd s = x" "sset (stl s) \<subseteq> {x}" by (case_tac [!] s) auto

    then have "sset (stl s) = {x}" by (cases "stl s") auto

    with \<open>shd s = x\<close> show ?case by auto

  qed

qed simp

lemma sconst_cycle: "sconst x = cycle [x]"

  by coinduction auto

lemma smap_sconst: "smap f (sconst x) = sconst (f x)"

  by coinduction auto

lemma sconst_streams: "x \<in> A \<Longrightarrow> sconst x \<in> streams A"

  by (simp add: streams_iff_sset)

subsection \<open>stream of natural numbers\<close>

abbreviation "fromN \<equiv> siterate Suc"

abbreviation "nats \<equiv> fromN 0"

lemma sset_fromN[simp]: "sset (fromN n) = {n ..}"

  by (auto simp add: sset_siterate le_iff_add)

lemma stream_smap_fromN: "s = smap (\<lambda>j. let i = j - n in s !! i) (fromN n)"

  by (coinduction arbitrary: s n)

    (force simp: neq_Nil_conv Let_def snth.simps(2)[symmetric] Suc_diff_Suc

      intro: stream.map_cong split: if_splits simp del: snth.simps(2))

lemma stream_smap_nats: "s = smap (snth s) nats"

  using stream_smap_fromN[where n = 0] by simp

subsection \<open>flatten a stream of lists\<close>

primcorec flat where

  "shd (flat ws) = hd (shd ws)"

| "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"

lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)"

  by (subst flat.ctr) simp

lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @- flat ws"

  by (induct xs) auto

lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)"

  by (cases ws) auto

lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then 

  shd s ! n else flat (stl s) !! (n - length (shd s)))"

  by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less)

lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> 

  sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R")

proof safe

  fix x assume ?P "x : ?L"

  then obtain m where "x = flat s !! m" by (metis image_iff sset_range)

  with \<open>?P\<close> obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)"

  proof (atomize_elim, induct m arbitrary: s rule: less_induct)

    case (less y)

    thus ?case

    proof (cases "y < length (shd s)")

      case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1))

    next

      case False

      hence "x = flat (stl s) !! (y - length (shd s))" by (metis less(2,3) flat_snth)

      moreover

      { from less(2) have *: "length (shd s) > 0" by (cases s) simp_all

        with False have "y > 0" by (cases y) simp_all

        with * have "y - length (shd s) < y" by simp

      }

      moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto

      ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto

      thus ?thesis by (metis snth.simps(2))

    qed

  qed

  thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem)

next

  fix x xs assume "xs \<in> sset s" ?P "x \<in> set xs" thus "x \<in> ?L"

    by (induct rule: sset_induct)

      (metis UnI1 flat_unfold shift.simps(1) sset_shift,

       metis UnI2 flat_unfold shd_sset stl_sset sset_shift)

qed

subsection \<open>merge a stream of streams\<close>

definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where

  "smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)"

lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m"

  by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2))

lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)"

proof (cases "n \<le> m")

  case False thus ?thesis unfolding smerge_def

    by (subst sset_flat)

      (auto simp: stream.set_map in_set_conv_nth simp del: stake.simps

        intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp])

next

  case True thus ?thesis unfolding smerge_def

    by (subst sset_flat)

      (auto simp: stream.set_map in_set_conv_nth image_iff simp del: stake.simps snth.simps

        intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp])

qed

lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset"

proof safe

  fix x assume "x \<in> sset (smerge ss)"

  thus "x \<in> UNION (sset ss) sset"

    unfolding smerge_def by (subst (asm) sset_flat)

      (auto simp: stream.set_map in_set_conv_nth sset_range simp del: stake.simps, fast+)

next

  fix s x assume "s \<in> sset ss" "x \<in> sset s"

  thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range)

qed

subsection \<open>product of two streams\<close>

definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where

  "sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)"

lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2"

  unfolding sproduct_def sset_smerge by (auto simp: stream.set_map)

subsection \<open>interleave two streams\<close>

primcorec sinterleave where

  "shd (sinterleave s1 s2) = shd s1"

| "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)"

lemma sinterleave_code[code]:

  "sinterleave (x ## s1) s2 = x ## sinterleave s2 s1"

  by (subst sinterleave.ctr) simp

lemma sinterleave_snth[simp]:

  "even n \<Longrightarrow> sinterleave s1 s2 !! n = s1 !! (n div 2)"

  "odd n \<Longrightarrow> sinterleave s1 s2 !! n = s2 !! (n div 2)"

  by (induct n arbitrary: s1 s2) simp_all

lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2"

proof (intro equalityI subsetI)

  fix x assume "x \<in> sset (sinterleave s1 s2)"

  then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast

  thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto

next

  fix x assume "x \<in> sset s1 \<union> sset s2"

  thus "x \<in> sset (sinterleave s1 s2)"

  proof

    assume "x \<in> sset s1"

    then obtain n where "x = s1 !! n" unfolding sset_range by blast

    hence "sinterleave s1 s2 !! (2 * n) = x" by simp

    thus ?thesis unfolding sset_range by blast

  next

    assume "x \<in> sset s2"

    then obtain n where "x = s2 !! n" unfolding sset_range by blast

    hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp

    thus ?thesis unfolding sset_range by blast

  qed

qed

subsection \<open>zip\<close>

primcorec szip where

  "shd (szip s1 s2) = (shd s1, shd s2)"

| "stl (szip s1 s2) = szip (stl s1) (stl s2)"

lemma szip_unfold[code]: "szip (a ## s1) (b ## s2) = (a, b) ## (szip s1 s2)"

  by (subst szip.ctr) simp

lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)"

  by (induct n arbitrary: s1 s2) auto

lemma stake_szip[simp]:

  "stake n (szip s1 s2) = zip (stake n s1) (stake n s2)"

  by (induct n arbitrary: s1 s2) auto

lemma sdrop_szip[simp]: "sdrop n (szip s1 s2) = szip (sdrop n s1) (sdrop n s2)"

  by (induct n arbitrary: s1 s2) auto

lemma smap_szip_fst:

  "smap (\<lambda>x. f (fst x)) (szip s1 s2) = smap f s1"

  by (coinduction arbitrary: s1 s2) auto

lemma smap_szip_snd:

  "smap (\<lambda>x. g (snd x)) (szip s1 s2) = smap g s2"

  by (coinduction arbitrary: s1 s2) auto

subsection \<open>zip via function\<close>

primcorec smap2 where

  "shd (smap2 f s1 s2) = f (shd s1) (shd s2)"

| "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)"

lemma smap2_unfold[code]:

  "smap2 f (a ## s1) (b ## s2) = f a b ## (smap2 f s1 s2)"

  by (subst smap2.ctr) simp

lemma smap2_szip:

  "smap2 f s1 s2 = smap (case_prod f) (szip s1 s2)"

  by (coinduction arbitrary: s1 s2) auto

lemma smap_smap2[simp]:

  "smap f (smap2 g s1 s2) = smap2 (\<lambda>x y. f (g x y)) s1 s2"

  unfolding smap2_szip stream.map_comp o_def split_def ..

lemma smap2_alt:

  "(smap2 f s1 s2 = s) = (\<forall>n. f (s1 !! n) (s2 !! n) = s !! n)"

  unfolding smap2_szip smap_alt by auto

lemma snth_smap2[simp]:

  "smap2 f s1 s2 !! n = f (s1 !! n) (s2 !! n)"

  by (induct n arbitrary: s1 s2) auto

lemma stake_smap2[simp]:

  "stake n (smap2 f s1 s2) = map (case_prod f) (zip (stake n s1) (stake n s2))"

  by (induct n arbitrary: s1 s2) auto

lemma sdrop_smap2[simp]:

  "sdrop n (smap2 f s1 s2) = smap2 f (sdrop n s1) (sdrop n s2)"

  by (induct n arbitrary: s1 s2) auto

end