(* 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 {* stream repeating a single element *}
primcorec same where
"shd (same x) = x"
| "stl (same x) = same x"
lemma snth_same[simp]: "same x !! n = x"
unfolding same_def by (induct n) auto
lemma stake_same[simp]: "stake n (same x) = replicate n x"
unfolding same_def by (induct n) (auto simp: upt_rec)
lemma sdrop_same[simp]: "sdrop n (same x) = same x"
unfolding same_def by (induct n) auto
lemma shift_replicate_same[simp]: "replicate n x @- same x = same x"
by (metis sdrop_same stake_same stake_sdrop)
lemma stream_all_same[simp]: "stream_all P (same x) \<longleftrightarrow> P x"
unfolding stream_all_def by auto
lemma same_cycle: "same x = cycle [x]"
by coinduction auto
subsection {* stream of natural numbers *}
primcorec fromN :: "nat \<Rightarrow> nat stream" where
"fromN n = n ## fromN (n + 1)"
lemma snth_fromN[simp]: "fromN n !! m = n + m"
unfolding fromN_def by (induct m arbitrary: n) auto
lemma stake_fromN[simp]: "stake m (fromN n) = [n ..< m + n]"
unfolding fromN_def by (induct m arbitrary: n) (auto simp: upt_rec)
lemma sdrop_fromN[simp]: "sdrop m (fromN n) = fromN (n + m)"
unfolding fromN_def by (induct m arbitrary: n) auto
lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" (is "?L = ?R")
proof safe
fix m assume "m \<in> ?L"
moreover
{ fix s assume "m \<in> sset s" "\<exists>n'\<ge>n. s = fromN n'"
hence "n \<le> m" by (induct arbitrary: n rule: sset_induct1) fastforce+
}
ultimately show "n \<le> m" by auto
next
fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_sset)
qed
lemma fromN_Suc_eq_smap: "fromN (Suc n) = smap Suc (fromN n)"
by (coinduction arbitrary: n) auto
abbreviation "nats \<equiv> fromN 0"
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
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)
end