(*  Title:      HOL/BNF/Examples/Stream.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Andrei Popescu, TU Muenchen

     Copyright   2012, 2013

 Infinite streams.

 *)

 header {* Infinite Streams *}

 theory Stream

 imports "../BNF"

 begin

 codatatype (sset: 'a) stream (map: smap rel: stream_all2) =

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

 code_datatype Stream

 lemma stream_case_cert:

   assumes "CASE \<equiv> case_stream c"

   shows "CASE (a ## s) \<equiv> c a s"

   using assms by simp_all

 setup {*

   Code.add_case @{thm stream_case_cert}

 *}

 (*for code generation only*)

 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 (Stream y s) = (if x = y then True else smember x s)"

   unfolding smember_def by auto

 hide_const (open) smember

 (* TODO: Provide by the package*)

 theorem sset_induct:

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

     \<forall>y \<in> sset s. P y s"

   apply (rule stream.dtor_set_induct)

   apply (auto simp add: shd_def stl_def fsts_def snds_def split_beta)

   apply (metis Stream_def fst_conv stream.case stream.dtor_ctor stream.exhaust)

   by (metis Stream_def sndI stl_def stream.collapse stream.dtor_ctor)

 lemma smap_simps[simp]:

   "shd (smap f s) = f (shd s)" "stl (smap f s) = smap f (stl s)"

   by (case_tac [!] s) auto

 theorem shd_sset: "shd s \<in> sset s"

   by (case_tac s) auto

 theorem stl_sset: "y \<in> sset (stl s) \<Longrightarrow> y \<in> sset s"

   by (case_tac s) auto

 (* only for the non-mutual case: *)

 theorem sset_induct1[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 sset_induct by blast

 (* end TODO *)

 subsection {* prepend list to stream *}

 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 {* set of streams with elements in some fixed set *}

 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"

 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 `s \<in> streams A` 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 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 {* nth, take, drop for streams *}

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

   "s !! 0 = shd s"

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

 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 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

 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

 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 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: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> sdrop n s = s'"

   by (induct n arbitrary: w s) auto

 lemma stake_shift: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> stake n s = w"

   by (induct n arbitrary: w s) auto

 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

 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_Stream[code]:

   "sdrop_while P (Stream a s) = (if P a then sdrop_while P s else Stream 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_Stream)

 next

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

 qed

 subsection {* unary predicates lifted to streams *}

 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 {* recurring stream out of a list *}

 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 (metis cycle_decomp stake_shift)

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

   by (metis cycle_decomp 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 {* iterated application of a function *}

 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 {* stream repeating a single element *}

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

   by (simp add: sset_siterate)

 lemma same_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 {* stream of natural numbers *}

 abbreviation "fromN \<equiv> siterate Suc"

 abbreviation "nats \<equiv> fromN 0"

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

   by (auto simp add: sset_siterate) arith

 subsection {* flatten a stream of lists *}

 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 `?P` 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_induct1)

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

        metis UnI2 flat_unfold shd_sset stl_sset sset_shift)

 qed

 subsection {* merge a stream of streams *}

 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 {* product of two streams *}

 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 {* interleave two streams *}

 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)

     (auto dest: even_nat_Suc_div_2 odd_nat_plus_one_div_two[folded nat_2])

 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 {* zip *}

 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 (Stream a s1) (Stream b s2) = Stream (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

 subsection {* zip via function *}

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

   by (subst smap2.ctr) simp

 lemma smap2_szip:

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

   by (coinduction arbitrary: s1 s2) auto

 end