# Theory Stream

```(*  Title:      HOL/Library/Stream.thy
Author:     Dmitriy Traytel, TU Muenchen
Author:     Andrei Popescu, TU Muenchen

Infinite streams.
*)

section ‹Infinite Streams›

theory Stream
imports 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 ⇒ 'a stream ⇒ bool" where
[code_abbrev]: "smember x s ⟷ x ∈ 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 ∈ sset s" and "⋀s. P (shd s) s" and "⋀s y. ⟦y ∈ sset (stl s); P y (stl s)⟧ ⟹ P y s"
shows "P y s"
using assms by induct (metis stream.sel(1), auto)

lemma smap_ctr: "smap f s = x ## s' ⟷ f (shd s) = x ∧ smap f (stl s) = s'"
by (cases s) simp

subsection ‹prepend list to stream›

primrec shift :: "'a list ⇒ 'a stream ⇒ '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 ∪ sset s"
by (induct xs) auto

lemma shift_left_inj[simp]: "xs @- s1 = xs @- s2 ⟷ s1 = s2"
by (induct xs) auto

subsection ‹set of streams with elements in some fixed set›

context
notes [[inductive_internals]]
begin

coinductive_set
streams :: "'a set ⇒ 'a stream set"
for A :: "'a set"
where
Stream[intro!, simp, no_atp]: "⟦a ∈ A; s ∈ streams A⟧ ⟹ a ## s ∈ streams A"

end

lemma in_streams: "stl s ∈ streams S ⟹ shd s ∈ S ⟹ s ∈ streams S"
by (cases s) auto

lemma streamsE: "s ∈ streams A ⟹ (shd s ∈ A ⟹ stl s ∈ streams A ⟹ P) ⟹ P"
by (erule streams.cases) simp_all

lemma Stream_image: "x ## y ∈ ((##) x') ` Y ⟷ x = x' ∧ y ∈ Y"
by auto

lemma shift_streams: "⟦w ∈ lists A; s ∈ streams A⟧ ⟹ w @- s ∈ streams A"
by (induct w) auto

lemma streams_Stream: "x ## s ∈ streams A ⟷ x ∈ A ∧ s ∈ streams A"
by (auto elim: streams.cases)

lemma streams_stl: "s ∈ streams A ⟹ stl s ∈ streams A"
by (cases s) (auto simp: streams_Stream)

lemma streams_shd: "s ∈ streams A ⟹ shd s ∈ A"
by (cases s) (auto simp: streams_Stream)

lemma sset_streams:
assumes "sset s ⊆ A"
shows "s ∈ streams A"
using assms proof (coinduction arbitrary: s)
case streams then show ?case by (cases s) simp
qed

lemma streams_sset:
assumes "s ∈ streams A"
shows "sset s ⊆ A"
proof
fix x assume "x ∈ sset s" from this ‹s ∈ streams A› show "x ∈ A"
by (induct s) (auto intro: streams_shd streams_stl)
qed

lemma streams_iff_sset: "s ∈ streams A ⟷ sset s ⊆ A"
by (metis sset_streams streams_sset)

lemma streams_mono:  "s ∈ streams A ⟹ A ⊆ B ⟹ s ∈ streams B"
unfolding streams_iff_sset by auto

lemma streams_mono2: "S ⊆ T ⟹ streams S ⊆ streams T"
by (auto intro: streams_mono)

lemma smap_streams: "s ∈ streams A ⟹ (⋀x. x ∈ A ⟹ f x ∈ B) ⟹ smap f s ∈ 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 ⇒ nat ⇒ '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 ⟹ (xs @- s) !! p = xs ! p"
by (induct p arbitrary: xs) (auto simp: hd_conv_nth nth_tl)

lemma shift_snth_ge[simp]: "p ≥ length xs ⟹ (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 ∈ 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 ∈ sset s"
thus "x ∈ 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 ∈ streams X ⟷ (∀n. s !! n ∈ X)"
by (force simp: streams_iff_sset sset_range)

lemma snth_in: "s ∈ streams X ⟹ s !! n ∈ X"

primrec stake :: "nat ⇒ 'a stream ⇒ '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 ⇒ 'a stream ⇒ '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' ⟷ (∀n. f (s !! n) = s' !! n)" (is "?L = ?R")
proof
assume ?R
then have "⋀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 = [] ⟷ 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 ⇒ bool) ⇒ 'a stream ⇒ '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 "∃n. P (s !! n)"
shows "sdrop_while (Not ∘ P) s = sdrop (LEAST n. P (s !! n)) s"
proof -
from assms obtain m where "P (s !! m)" "⋀n. P (s !! n) ⟹ m ≤ 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 ∘ P) (stl s) = sdrop m (stl s)"
by (metis (full_types) not_less_eq_eq snth.simps(2))
moreover from Suc(3) have "¬ (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 ∘ P) s)"
| "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not ∘ 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 ‹unary predicates lifted to streams›

definition "stream_all P s = (∀p. P (s !! p))"

lemma stream_all_iff[iff]: "stream_all P s ⟷ 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 ∧ stream_all P s)"
unfolding stream_all_iff list_all_iff by auto

lemma stream_all_Stream: "stream_all P (x ## X) ⟷ P x ∧ stream_all P X"
by simp

subsection ‹recurring stream out of a list›

primcorec cycle :: "'a list ⇒ 'a stream" where
"shd (cycle xs) = hd xs"
| "stl (cycle xs) = cycle (tl xs @ [hd xs])"

lemma cycle_decomp: "u ≠ [] ⟹ 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: "⟦v ≠ []; cycle u = v @- s⟧ ⟹ 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 ≠ []" "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 ≠ [] ⟹ stake (length u) (cycle u) = u"
by (subst cycle_decomp) (auto simp: stake_shift)

lemma sdrop_cycle_eq[simp]: "u ≠ [] ⟹ sdrop (length u) (cycle u) = cycle u"
by (subst cycle_decomp) (auto simp: sdrop_shift)

lemma stake_cycle_eq_mod_0[simp]: "⟦u ≠ []; n mod length u = 0⟧ ⟹
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] mod_eq_0_iff_dvd elim!: dvdE)

lemma sdrop_cycle_eq_mod_0[simp]: "⟦u ≠ []; n mod length u = 0⟧ ⟹
sdrop n (cycle u) = cycle u"
by (induct "n div length u" arbitrary: n u)
(auto simp: sdrop_add [symmetric] mod_eq_0_iff_dvd elim!: dvdE)

lemma stake_cycle: "u ≠ [] ⟹
stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u"
by (subst div_mult_mod_eq[of n "length u", symmetric], unfold stake_add[symmetric]) auto

lemma sdrop_cycle: "u ≠ [] ⟹ 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])

lemma sset_cycle[simp]:
assumes "xs ≠ []"
shows "sset (cycle xs) = set xs"
proof (intro set_eqI iffI)
fix x
assume "x ∈ sset (cycle xs)"
then show "x ∈ set xs" using assms
by (induction "cycle xs" arbitrary: xs rule: sset_induct) (fastforce simp: neq_Nil_conv)+
qed (metis assms UnI1 cycle_decomp sset_shift)

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 (λn. (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 ≡ 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}"

lemma sconst_alt: "s = sconst x ⟷ 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) ⊆ {x}" by (cases s; auto)+
then have "sset (stl s) = {x}" by (cases "stl s") auto
with ‹shd s = x› 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 ∈ A ⟹ sconst x ∈ streams A"

lemma streams_empty_iff: "streams S = {} ⟷ S = {}"
proof safe
fix x assume "x ∈ S" "streams S = {}"
then have "sconst x ∈ streams S"
by (intro sconst_streams)
then show "x ∈ {}"
unfolding ‹streams S = {}› by simp
qed (auto simp: streams_empty)

subsection ‹stream of natural numbers›

abbreviation "fromN ≡ siterate Suc"

abbreviation "nats ≡ fromN 0"

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

lemma stream_smap_fromN: "s = smap (λj. let i = j - n in s !! i) (fromN n)"
by (coinduction arbitrary: s n)
(force simp: neq_Nil_conv Let_def Suc_diff_Suc simp flip: snth.simps(2)
intro: stream.map_cong split: if_splits)

lemma stream_smap_nats: "s = smap (snth s) nats"
using stream_smap_fromN[where n = 0] by simp

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 ≠ [] ⟹ flat (xs ## ws) = xs @- flat ws"
by (induct xs) auto

lemma flat_unfold: "shd ws ≠ [] ⟹ flat ws = shd ws @- flat (stl ws)"
by (cases ws) auto

lemma flat_snth: "∀xs ∈ sset s. xs ≠ [] ⟹ 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]: "∀xs ∈ sset s. xs ≠ [] ⟹
sset (flat s) = (⋃xs ∈ sset s. set xs)" (is "?P ⟹ ?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 "∀xs ∈ sset (stl s). xs ≠ []" using less(2) by (cases s) auto
ultimately have "∃n m'. x = stl s !! n ! m' ∧ m' < length (stl s !! n)" by (intro less(1)) auto
thus ?thesis by (metis snth.simps(2))
qed
qed
thus "x ∈ ?R" by (auto simp: sset_range dest!: nth_mem)
next
fix x xs assume "xs ∈ sset s" ?P "x ∈ set xs" thus "x ∈ ?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 ‹merge a stream of streams›

definition smerge :: "'a stream stream ⇒ 'a stream" where
"smerge ss = flat (smap (λn. map (λs. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)"

lemma stake_nth[simp]: "m < n ⟹ 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 ∈ sset (smerge ss)"
proof (cases "n ≤ 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) = ⋃(sset ` (sset ss))"
proof safe
fix x assume "x ∈ sset (smerge ss)"
thus "x ∈ ⋃(sset ` (sset ss))"
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 ∈ sset ss" "x ∈ sset s"
thus "x ∈ sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range)
qed

subsection ‹product of two streams›

definition sproduct :: "'a stream ⇒ 'b stream ⇒ ('a × 'b) stream" where
"sproduct s1 s2 = smerge (smap (λx. smap (Pair x) s2) s1)"

lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 × 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 ⟹ sinterleave s1 s2 !! n = s1 !! (n div 2)"
"odd n ⟹ 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 ∪ sset s2"
proof (intro equalityI subsetI)
fix x assume "x ∈ sset (sinterleave s1 s2)"
then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast
thus "x ∈ sset s1 ∪ sset s2" by (cases "even n") auto
next
fix x assume "x ∈ sset s1 ∪ sset s2"
thus "x ∈ sset (sinterleave s1 s2)"
proof
assume "x ∈ 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 ∈ 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 (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 (λx. f (fst x)) (szip s1 s2) = smap f s1"
by (coinduction arbitrary: s1 s2) auto

lemma smap_szip_snd:
"smap (λx. g (snd x)) (szip s1 s2) = smap g s2"
by (coinduction 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 (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 (λx 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) = (∀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
```