(* Title: HOL/BNF/Examples/Stream.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Infinite streams.
*)
header {* Infinite Streams *}
theory Stream
imports "../BNF"
begin
codata 'a stream = Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65)
declaration {*
Nitpick_HOL.register_codatatype
@{typ "'stream_element_type stream"} @{const_name stream_case} [dest_Const @{term Stream}]
(*FIXME: long type variable name required to reduce the probability of
a name clash of Nitpick in context. E.g.:
context
fixes x :: 'stream_element_type
begin
lemma "stream_set s = {}"
nitpick
oops
end
*)
*}
code_datatype Stream
lemmas [code] = stream.sels stream.sets stream.case
lemma stream_case_cert:
assumes "CASE \<equiv> stream_case c"
shows "CASE (a ## s) \<equiv> c a s"
using assms by simp_all
setup {*
Code.add_case @{thm stream_case_cert}
*}
(* TODO: Provide by the package*)
theorem stream_set_induct:
"\<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>
\<forall>y \<in> stream_set s. P y s"
by (rule stream.dtor_set_induct)
(auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
lemma stream_map_simps[simp]:
"shd (stream_map f s) = f (shd s)" "stl (stream_map f s) = stream_map f (stl s)"
unfolding shd_def stl_def stream_case_def stream_map_def stream.dtor_unfold
by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor)
lemma stream_map_Stream[simp, code]: "stream_map f (x ## s) = f x ## stream_map f s"
by (metis stream.exhaust stream.sels stream_map_simps)
theorem shd_stream_set: "shd s \<in> stream_set s"
by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
(metis UnCI fsts_def insertI1 stream.dtor_set)
theorem stl_stream_set: "y \<in> stream_set (stl s) \<Longrightarrow> y \<in> stream_set s"
by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
(metis insertI1 set_mp snds_def stream.dtor_set_set_incl)
(* only for the non-mutual case: *)
theorem stream_set_induct1[consumes 1, case_names shd stl, induct set: "stream_set"]:
assumes "y \<in> stream_set s" and "\<And>s. P (shd s) s"
and "\<And>s y. \<lbrakk>y \<in> stream_set (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s"
shows "P y s"
using assms stream_set_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 stream_map_shift[simp]: "stream_map f (xs @- s) = map f xs @- stream_map 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 stream_set_shift[simp]: "stream_set (xs @- s) = set xs \<union> stream_set 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 fixes set *}
coinductive_set
streams :: "'a set => '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 stream_set_streams:
assumes "stream_set s \<subseteq> A"
shows "s \<in> streams A"
proof (coinduct rule: streams.coinduct[of "\<lambda>s'. \<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"])
case streams from assms show ?case by (cases s) auto
next
fix s' assume "\<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"
then guess a s by (elim exE)
with assms show "\<exists>a l. s' = a ## l \<and>
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)"
by (cases s) auto
qed
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_stream_map[simp]: "stream_map 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_stream_set[simp]: "s !! n \<in> stream_set s"
by (induct n arbitrary: s) (auto intro: shd_stream_set stl_stream_set)
lemma stream_set_range: "stream_set s = range (snth s)"
proof (intro equalityI subsetI)
fix x assume "x \<in> stream_set 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_stream_map[simp]: "stake n (stream_map 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_stream_map[simp]: "sdrop n (stream_map f s) = stream_map 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 stream_map_alt: "stream_map f s = s' \<longleftrightarrow> (\<forall>n. f (s !! n) = s' !! n)" (is "?L = ?R")
proof
assume ?R
thus ?L
by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>n. s1 = stream_map f (sdrop n s) \<and> s2 = sdrop n s'"])
(auto intro: exI[of _ 0] simp del: sdrop.simps(2))
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
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 (stream_set s) P"
unfolding stream_all_def stream_set_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
subsection {* flatten a stream of lists *}
definition flat where
"flat \<equiv> stream_unfold (hd o shd) (\<lambda>s. if tl (shd s) = [] then stl s else tl (shd s) ## stl s)"
lemma flat_simps[simp]:
"shd (flat ws) = hd (shd ws)"
"stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
unfolding flat_def by auto
lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)"
unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto
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> stream_set 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_stream_set shift_snth_ge shift_snth_less)
lemma stream_set_flat[simp]: "\<forall>xs \<in> stream_set s. xs \<noteq> [] \<Longrightarrow>
stream_set (flat s) = (\<Union>xs \<in> stream_set 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 stream_set_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
moreover with False have "y > 0" by (cases y) simp_all
ultimately have "y - length (shd s) < y" by simp
}
moreover have "\<forall>xs \<in> stream_set (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: stream_set_range dest!: nth_mem)
next
fix x xs assume "xs \<in> stream_set s" ?P "x \<in> set xs" thus "x \<in> ?L"
by (induct rule: stream_set_induct1)
(metis UnI1 flat_unfold shift.simps(1) stream_set_shift,
metis UnI2 flat_unfold shd_stream_set stl_stream_set stream_set_shift)
qed
subsection {* recurring stream out of a list *}
definition cycle :: "'a list \<Rightarrow> 'a stream" where
"cycle = stream_unfold hd (\<lambda>xs. tl xs @ [hd xs])"
lemma cycle_simps[simp]:
"shd (cycle u) = hd u"
"stl (cycle u) = cycle (tl u @ [hd u])"
by (auto simp: cycle_def)
lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @- cycle u"
proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>u. s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []"])
case (2 s1 s2)
then obtain u where "s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []" by blast
thus ?case using stream.unfold[of hd "\<lambda>xs. tl xs @ [hd xs]" u] by (auto simp: cycle_def)
qed auto
lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])"
proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>x xs. s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])"])
case (2 s1 s2)
then obtain x xs where "s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])" by blast
thus ?case
by (auto simp: cycle_def intro!: exI[of _ "hd (xs @ [x])"] exI[of _ "tl (xs @ [x])"] stream.unfold)
qed auto
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 *}
definition "same x = stream_unfold (\<lambda>_. x) id ()"
lemma same_simps[simp]: "shd (same x) = x" "stl (same x) = same x"
unfolding same_def by auto
lemma same_unfold[code]: "same x = x ## same x"
by (metis same_simps stream.collapse)
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 (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. s1 = same x \<and> s2 = cycle [x]"]) auto
subsection {* stream of natural numbers *}
definition "fromN n = stream_unfold id Suc n"
lemma fromN_simps[simp]: "shd (fromN n) = n" "stl (fromN n) = fromN (Suc n)"
unfolding fromN_def by auto
lemma fromN_unfold[code]: "fromN n = n ## fromN (Suc n)"
unfolding fromN_def by (metis id_def stream.unfold)
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 stream_set_fromN[simp]: "stream_set (fromN n) = {n ..}" (is "?L = ?R")
proof safe
fix m assume "m : ?L"
moreover
{ fix s assume "m \<in> stream_set s" "\<exists>n'\<ge>n. s = fromN n'"
hence "n \<le> m" by (induct arbitrary: n rule: stream_set_induct1) fastforce+
}
ultimately show "n \<le> m" by blast
next
fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_stream_set)
qed
abbreviation "nats \<equiv> fromN 0"
subsection {* zip *}
definition "szip s1 s2 =
stream_unfold (map_pair shd shd) (map_pair stl stl) (s1, s2)"
lemma szip_simps[simp]:
"shd (szip s1 s2) = (shd s1, shd s2)" "stl (szip s1 s2) = szip (stl s1) (stl s2)"
unfolding szip_def by auto
lemma szip_unfold[code]: "szip (Stream a s1) (Stream b s2) = Stream (a, b) (szip s1 s2)"
unfolding szip_def by (subst stream.unfold) simp
lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
by (induct n arbitrary: s1 s2) auto
subsection {* zip via function *}
definition "stream_map2 f s1 s2 =
stream_unfold (\<lambda>(s1,s2). f (shd s1) (shd s2)) (map_pair stl stl) (s1, s2)"
lemma stream_map2_simps[simp]:
"shd (stream_map2 f s1 s2) = f (shd s1) (shd s2)"
"stl (stream_map2 f s1 s2) = stream_map2 f (stl s1) (stl s2)"
unfolding stream_map2_def by auto
lemma stream_map2_unfold[code]:
"stream_map2 f (Stream a s1) (Stream b s2) = Stream (f a b) (stream_map2 f s1 s2)"
unfolding stream_map2_def by (subst stream.unfold) simp
lemma stream_map2_szip:
"stream_map2 f s1 s2 = stream_map (split f) (szip s1 s2)"
by (coinduct rule: stream.coinduct[of
"\<lambda>s1 s2. \<exists>s1' s2'. s1 = stream_map2 f s1' s2' \<and> s2 = stream_map (split f) (szip s1' s2')"])
fastforce+
end