author  traytel 
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permissions  rwrr 
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(* Title: HOL/BNF/Examples/Stream.thy 
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Author: Dmitriy Traytel, TU Muenchen 

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Author: Andrei Popescu, TU Muenchen 

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Copyright 2012 

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Infinite streams. 

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*) 

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header {* Infinite Streams *} 

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theory Stream 

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imports "../BNF" 

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begin 

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codata (sset: 'a) stream (map: smap rel: stream_all2) = 
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Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65) 
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declaration {* 
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Nitpick_HOL.register_codatatype 

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@{typ "'stream_element_type stream"} @{const_name stream_case} [dest_Const @{term Stream}] 

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(*FIXME: long type variable name required to reduce the probability of 

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a name clash of Nitpick in context. E.g.: 

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context 

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fixes x :: 'stream_element_type 

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begin 

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lemma "sset s = {}" 
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nitpick 
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oops 

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end 

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*) 

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*} 

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code_datatype Stream 

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lemmas [code] = stream.sels stream.sets stream.case 

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lemma stream_case_cert: 

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assumes "CASE \<equiv> stream_case c" 

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

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using assms by simp_all 

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setup {* 

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Code.add_case @{thm stream_case_cert} 

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*} 

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(*for code generation only*) 
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definition smember :: "'a \<Rightarrow> 'a stream \<Rightarrow> bool" where 

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[code_abbrev]: "smember x s \<longleftrightarrow> x \<in> sset s" 
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lemma smember_code[code, simp]: "smember x (Stream y s) = (if x = y then True else smember x s)" 

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unfolding smember_def by auto 

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hide_const (open) smember 

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(* TODO: Provide by the package*) 
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theorem sset_induct: 
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"\<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> 
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\<forall>y \<in> sset s. P y s" 
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by (rule stream.dtor_set_induct) 
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(auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta) 

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lemma smap_simps[simp]: 
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"shd (smap f s) = f (shd s)" "stl (smap f s) = smap f (stl s)" 
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unfolding shd_def stl_def stream_case_def smap_def stream.dtor_unfold 
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by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor) 
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declare stream.map[code] 
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theorem shd_sset: "shd s \<in> sset s" 
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by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta) 
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(metis UnCI fsts_def insertI1 stream.dtor_set) 

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theorem stl_sset: "y \<in> sset (stl s) \<Longrightarrow> y \<in> sset s" 
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by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta) 
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(metis insertI1 set_mp snds_def stream.dtor_set_set_incl) 

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(* only for the nonmutual case: *) 

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theorem sset_induct1[consumes 1, case_names shd stl, induct set: "sset"]: 
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assumes "y \<in> sset s" and "\<And>s. P (shd s) s" 
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and "\<And>s y. \<lbrakk>y \<in> sset (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s" 
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shows "P y s" 
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using assms sset_induct by blast 
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(* end TODO *) 
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subsection {* prepend list to stream *} 

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

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"shift [] s = s" 

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 "shift (x # xs) s = x ## shift xs s" 
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lemma smap_shift[simp]: "smap f (xs @ s) = map f xs @ smap f s" 
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by (induct xs) auto 
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lemma shift_append[simp]: "(xs @ ys) @ s = xs @ ys @ s" 
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by (induct xs) auto 
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lemma shift_simps[simp]: 

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

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

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by (induct xs) auto 
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lemma sset_shift[simp]: "sset (xs @ s) = set xs \<union> sset s" 
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by (induct xs) auto 
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lemma shift_left_inj[simp]: "xs @ s1 = xs @ s2 \<longleftrightarrow> s1 = s2" 
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by (induct xs) auto 

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subsection {* set of streams with elements in some fixes set *} 
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coinductive_set 

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streams :: "'a set => 'a stream set" 

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for A :: "'a set" 

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where 

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

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by (induct w) auto 
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lemma sset_streams: 
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assumes "sset s \<subseteq> A" 
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shows "s \<in> streams A" 
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proof (coinduct rule: streams.coinduct[of "\<lambda>s'. \<exists>a s. s' = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A"]) 
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case streams from assms show ?case by (cases s) auto 
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next 

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fix s' assume "\<exists>a s. s' = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A" 
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then guess a s by (elim exE) 
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with assms show "\<exists>a l. s' = a ## l \<and> 
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a \<in> A \<and> ((\<exists>a s. l = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A) \<or> l \<in> streams A)" 
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by (cases s) auto 
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qed 

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subsection {* nth, take, drop for streams *} 
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primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!!" 100) where 

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"s !! 0 = shd s" 

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

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lemma snth_smap[simp]: "smap f s !! n = f (s !! n)" 
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by (induct n arbitrary: s) auto 
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lemma shift_snth_less[simp]: "p < length xs \<Longrightarrow> (xs @ s) !! p = xs ! p" 

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

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

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

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lemma snth_sset[simp]: "s !! n \<in> sset s" 
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by (induct n arbitrary: s) (auto intro: shd_sset stl_sset) 
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lemma sset_range: "sset s = range (snth s)" 
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proof (intro equalityI subsetI) 
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fix x assume "x \<in> sset s" 
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thus "x \<in> range (snth s)" 
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proof (induct s) 

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case (stl s x) 

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

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

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

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qed auto 

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

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"stake 0 s = []" 

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

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lemma length_stake[simp]: "length (stake n s) = n" 
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by (induct n arbitrary: s) auto 

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lemma stake_smap[simp]: "stake n (smap f s) = map f (stake n s)" 
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by (induct n arbitrary: s) auto 
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primrec sdrop :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where 
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"sdrop 0 s = s" 

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

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lemma sdrop_simps[simp]: 
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"shd (sdrop n s) = s !! n" "stl (sdrop n s) = sdrop (Suc n) s" 

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by (induct n arbitrary: s) auto 

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lemma sdrop_smap[simp]: "sdrop n (smap f s) = smap f (sdrop n s)" 
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by (induct n arbitrary: s) auto 
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lemma sdrop_stl: "sdrop n (stl s) = stl (sdrop n s)" 
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by (induct n) auto 

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lemma stake_sdrop: "stake n s @ sdrop n s = s" 
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by (induct n arbitrary: s) auto 
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lemma id_stake_snth_sdrop: 

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

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

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

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thus ?L 

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by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>n. s1 = smap f (sdrop n s) \<and> s2 = sdrop n s'"]) 
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(auto intro: exI[of _ 0] simp del: sdrop.simps(2)) 
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qed auto 

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

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by (induct n) auto 

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

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by (induct n arbitrary: w s) auto 
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lemma stake_shift: "\<lbrakk>s = w @ s'; length w = n\<rbrakk> \<Longrightarrow> stake n s = w" 

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by (induct n arbitrary: w s) auto 
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lemma stake_add[simp]: "stake m s @ stake n (sdrop m s) = stake (m + n) s" 

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by (induct m arbitrary: s) auto 
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lemma sdrop_add[simp]: "sdrop n (sdrop m s) = sdrop (m + n) s" 

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by (induct m arbitrary: s) auto 
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partial_function (tailrec) sdrop_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where 
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"sdrop_while P s = (if P (shd s) then sdrop_while P (stl s) else s)" 

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lemma sdrop_while_Stream[code]: 

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

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by (subst sdrop_while.simps) simp 

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lemma sdrop_while_sdrop_LEAST: 

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

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

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proof  

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

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

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thus ?thesis unfolding * 

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proof (induct m arbitrary: s) 

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case (Suc m) 

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

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

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

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

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

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qed 

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subsection {* unary predicates lifted to streams *} 

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

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lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P" 
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unfolding stream_all_def sset_range by auto 
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lemma stream_all_shift[simp]: "stream_all P (xs @ s) = (list_all P xs \<and> stream_all P s)" 

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unfolding stream_all_iff list_all_iff by auto 

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subsection {* recurring stream out of a list *} 

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

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"cycle = stream_unfold hd (\<lambda>xs. tl xs @ [hd xs])" 

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lemma cycle_simps[simp]: 

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"shd (cycle u) = hd u" 

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"stl (cycle u) = cycle (tl u @ [hd u])" 

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by (auto simp: cycle_def) 

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

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proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>u. s1 = cycle u \<and> s2 = u @ cycle u \<and> u \<noteq> []"]) 

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case (2 s1 s2) 

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then obtain u where "s1 = cycle u \<and> s2 = u @ cycle u \<and> u \<noteq> []" by blast 

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thus ?case using stream.unfold[of hd "\<lambda>xs. tl xs @ [hd xs]" u] by (auto simp: cycle_def) 

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qed auto 

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lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])" 
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proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>x xs. s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])"]) 
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case (2 s1 s2) 

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then obtain x xs where "s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])" by blast 

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thus ?case 

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by (auto simp: cycle_def intro!: exI[of _ "hd (xs @ [x])"] exI[of _ "tl (xs @ [x])"] stream.unfold) 

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qed auto 

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

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by (auto dest: arg_cong[of _ _ stl]) 
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lemma stake_append: "stake n (u @ s) = take (min (length u) n) u @ stake (n  length u) s" 

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proof (induct n arbitrary: u) 

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

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qed auto 

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lemma stake_cycle_le[simp]: 

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

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

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

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by (subst cycle_decomp[OF assms(1)], subst stake_append) auto 
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lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u" 

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

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by (metis cycle_decomp sdrop_shift) 
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lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> 

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

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by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric]) 
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lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> 

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sdrop n (cycle u) = cycle u" 

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by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric]) 
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lemma stake_cycle: "u \<noteq> [] \<Longrightarrow> 

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

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by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto 
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lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)" 

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by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric]) 
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subsection {* stream repeating a single element *} 

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definition "same x = stream_unfold (\<lambda>_. x) id ()" 

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lemma same_simps[simp]: "shd (same x) = x" "stl (same x) = same x" 

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unfolding same_def by auto 

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lemma same_unfold[code]: "same x = x ## same x" 
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by (metis same_simps stream.collapse) 
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lemma snth_same[simp]: "same x !! n = x" 

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unfolding same_def by (induct n) auto 

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lemma stake_same[simp]: "stake n (same x) = replicate n x" 

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unfolding same_def by (induct n) (auto simp: upt_rec) 

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lemma sdrop_same[simp]: "sdrop n (same x) = same x" 

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unfolding same_def by (induct n) auto 

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lemma shift_replicate_same[simp]: "replicate n x @ same x = same x" 

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by (metis sdrop_same stake_same stake_sdrop) 

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lemma stream_all_same[simp]: "stream_all P (same x) \<longleftrightarrow> P x" 

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unfolding stream_all_def by auto 

339 

340 
lemma same_cycle: "same x = cycle [x]" 

341 
by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. s1 = same x \<and> s2 = cycle [x]"]) auto 

342 

343 

344 
subsection {* stream of natural numbers *} 

345 

346 
definition "fromN n = stream_unfold id Suc n" 

347 

348 
lemma fromN_simps[simp]: "shd (fromN n) = n" "stl (fromN n) = fromN (Suc n)" 

349 
unfolding fromN_def by auto 

350 

51409  351 
lemma fromN_unfold[code]: "fromN n = n ## fromN (Suc n)" 
352 
unfolding fromN_def by (metis id_def stream.unfold) 

353 

51141  354 
lemma snth_fromN[simp]: "fromN n !! m = n + m" 
355 
unfolding fromN_def by (induct m arbitrary: n) auto 

356 

357 
lemma stake_fromN[simp]: "stake m (fromN n) = [n ..< m + n]" 

358 
unfolding fromN_def by (induct m arbitrary: n) (auto simp: upt_rec) 

359 

360 
lemma sdrop_fromN[simp]: "sdrop m (fromN n) = fromN (n + m)" 

361 
unfolding fromN_def by (induct m arbitrary: n) auto 

362 

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363 
lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" (is "?L = ?R") 
51352  364 
proof safe 
365 
fix m assume "m : ?L" 

366 
moreover 

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367 
{ fix s assume "m \<in> sset s" "\<exists>n'\<ge>n. s = fromN n'" 
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368 
hence "n \<le> m" by (induct arbitrary: n rule: sset_induct1) fastforce+ 
51352  369 
} 
370 
ultimately show "n \<le> m" by blast 

371 
next 

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372 
fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_sset) 
51352  373 
qed 
374 

51141  375 
abbreviation "nats \<equiv> fromN 0" 
376 

377 

51462  378 
subsection {* flatten a stream of lists *} 
379 

380 
definition flat where 

381 
"flat \<equiv> stream_unfold (hd o shd) (\<lambda>s. if tl (shd s) = [] then stl s else tl (shd s) ## stl s)" 

382 

383 
lemma flat_simps[simp]: 

384 
"shd (flat ws) = hd (shd ws)" 

385 
"stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)" 

386 
unfolding flat_def by auto 

387 

388 
lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)" 

389 
unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto 

390 

391 
lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @ flat ws" 

392 
by (induct xs) auto 

393 

394 
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @ flat (stl ws)" 

395 
by (cases ws) auto 

396 

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397 
lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then 
51462  398 
shd s ! n else flat (stl s) !! (n  length (shd s)))" 
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399 
by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less) 
51462  400 

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401 
lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> 
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402 
sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R") 
51462  403 
proof safe 
404 
fix x assume ?P "x : ?L" 

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405 
then obtain m where "x = flat s !! m" by (metis image_iff sset_range) 
51462  406 
with `?P` obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)" 
407 
proof (atomize_elim, induct m arbitrary: s rule: less_induct) 

408 
case (less y) 

409 
thus ?case 

410 
proof (cases "y < length (shd s)") 

411 
case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1)) 

412 
next 

413 
case False 

414 
hence "x = flat (stl s) !! (y  length (shd s))" by (metis less(2,3) flat_snth) 

415 
moreover 

416 
{ from less(2) have "length (shd s) > 0" by (cases s) simp_all 

417 
moreover with False have "y > 0" by (cases y) simp_all 

418 
ultimately have "y  length (shd s) < y" by simp 

419 
} 

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420 
moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto 
51462  421 
ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto 
422 
thus ?thesis by (metis snth.simps(2)) 

423 
qed 

424 
qed 

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425 
thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem) 
51462  426 
next 
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427 
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428 
by (induct rule: sset_induct1) 
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429 
(metis UnI1 flat_unfold shift.simps(1) sset_shift, 
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430 
metis UnI2 flat_unfold shd_sset stl_sset sset_shift) 
51462  431 
qed 
432 

433 

434 
subsection {* merge a stream of streams *} 

435 

436 
definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where 

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437 
"smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)" 
51462  438 

439 
lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m" 

440 
by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2)) 

441 

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442 
lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)" 
51462  443 
proof (cases "n \<le> m") 
444 
case False thus ?thesis unfolding smerge_def 

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445 
by (subst sset_flat) 
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446 
(auto simp: stream.set_map' in_set_conv_nth simp del: stake.simps 
51462  447 
intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp]) 
448 
next 

449 
case True thus ?thesis unfolding smerge_def 

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450 
by (subst sset_flat) 
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451 
(auto simp: stream.set_map' in_set_conv_nth image_iff simp del: stake.simps snth.simps 
51462  452 
intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp]) 
453 
qed 

454 

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455 
lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset" 
51462  456 
proof safe 
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457 
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458 
thus "x \<in> UNION (sset ss) sset" 
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459 
unfolding smerge_def by (subst (asm) sset_flat) 
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460 
(auto simp: stream.set_map' in_set_conv_nth sset_range simp del: stake.simps, fast+) 
51462  461 
next 
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462 
fix s x assume "s \<in> sset ss" "x \<in> sset s" 
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463 
thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range) 
51462  464 
qed 
465 

466 

467 
subsection {* product of two streams *} 

468 

469 
definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where 

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470 
"sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)" 
51462  471 

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472 
lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2" 
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473 
unfolding sproduct_def sset_smerge by (auto simp: stream.set_map') 
51462  474 

475 

476 
subsection {* interleave two streams *} 

477 

478 
definition sinterleave :: "'a stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where 

479 
[code del]: "sinterleave s1 s2 = 

480 
stream_unfold (\<lambda>(s1, s2). shd s1) (\<lambda>(s1, s2). (s2, stl s1)) (s1, s2)" 

481 

482 
lemma sinterleave_simps[simp]: 

483 
"shd (sinterleave s1 s2) = shd s1" "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)" 

484 
unfolding sinterleave_def by auto 

485 

486 
lemma sinterleave_code[code]: 

487 
"sinterleave (x ## s1) s2 = x ## sinterleave s2 s1" 

488 
by (metis sinterleave_simps stream.exhaust stream.sels) 

489 

490 
lemma sinterleave_snth[simp]: 

491 
"even n \<Longrightarrow> sinterleave s1 s2 !! n = s1 !! (n div 2)" 

492 
"odd n \<Longrightarrow> sinterleave s1 s2 !! n = s2 !! (n div 2)" 

493 
by (induct n arbitrary: s1 s2) 

494 
(auto dest: even_nat_Suc_div_2 odd_nat_plus_one_div_two[folded nat_2]) 

495 

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496 
lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2" 
51462  497 
proof (intro equalityI subsetI) 
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498 
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499 
then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast 
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500 
thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto 
51462  501 
next 
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502 
fix x assume "x \<in> sset s1 \<union> sset s2" 
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503 
thus "x \<in> sset (sinterleave s1 s2)" 
51462  504 
proof 
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505 
assume "x \<in> sset s1" 
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506 
then obtain n where "x = s1 !! n" unfolding sset_range by blast 
51462  507 
hence "sinterleave s1 s2 !! (2 * n) = x" by simp 
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508 
thus ?thesis unfolding sset_range by blast 
51462  509 
next 
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510 
assume "x \<in> sset s2" 
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511 
then obtain n where "x = s2 !! n" unfolding sset_range by blast 
51462  512 
hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp 
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513 
thus ?thesis unfolding sset_range by blast 
51462  514 
qed 
515 
qed 

516 

517 

51141  518 
subsection {* zip *} 
519 

520 
definition "szip s1 s2 = 

521 
stream_unfold (map_pair shd shd) (map_pair stl stl) (s1, s2)" 

522 

523 
lemma szip_simps[simp]: 

524 
"shd (szip s1 s2) = (shd s1, shd s2)" "stl (szip s1 s2) = szip (stl s1) (stl s2)" 

525 
unfolding szip_def by auto 

526 

51409  527 
lemma szip_unfold[code]: "szip (Stream a s1) (Stream b s2) = Stream (a, b) (szip s1 s2)" 
528 
unfolding szip_def by (subst stream.unfold) simp 

529 

51141  530 
lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)" 
531 
by (induct n arbitrary: s1 s2) auto 

532 

533 

534 
subsection {* zip via function *} 

535 

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536 
definition "smap2 f s1 s2 = 
51141  537 
stream_unfold (\<lambda>(s1,s2). f (shd s1) (shd s2)) (map_pair stl stl) (s1, s2)" 
538 

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539 
lemma smap2_simps[simp]: 
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540 
"shd (smap2 f s1 s2) = f (shd s1) (shd s2)" 
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541 
"stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)" 
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542 
unfolding smap2_def by auto 
51141  543 

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544 
lemma smap2_unfold[code]: 
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545 
"smap2 f (Stream a s1) (Stream b s2) = Stream (f a b) (smap2 f s1 s2)" 
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546 
unfolding smap2_def by (subst stream.unfold) simp 
51409  547 

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548 
lemma smap2_szip: 
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549 
"smap2 f s1 s2 = smap (split f) (szip s1 s2)" 
51141  550 
by (coinduct rule: stream.coinduct[of 
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551 
"\<lambda>s1 s2. \<exists>s1' s2'. s1 = smap2 f s1' s2' \<and> s2 = smap (split f) (szip s1' s2')"]) 
51141  552 
fastforce+ 
50518  553 

51462  554 

555 
subsection {* iterated application of a function *} 

556 

557 
definition siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where 

558 
"siterate f x = x ## stream_unfold f f x" 

559 

560 
lemma siterate_simps[simp]: "shd (siterate f x) = x" "stl (siterate f x) = siterate f (f x)" 

561 
unfolding siterate_def by (auto intro: stream.unfold) 

562 

563 
lemma siterate_code[code]: "siterate f x = x ## siterate f (f x)" 

564 
by (metis siterate_def stream.unfold) 

565 

566 
lemma stake_Suc: "stake (Suc n) s = stake n s @ [s !! n]" 

567 
by (induct n arbitrary: s) auto 

568 

569 
lemma snth_siterate[simp]: "siterate f x !! n = (f^^n) x" 

570 
by (induct n arbitrary: x) (auto simp: funpow_swap1) 

571 

572 
lemma sdrop_siterate[simp]: "sdrop n (siterate f x) = siterate f ((f^^n) x)" 

573 
by (induct n arbitrary: x) (auto simp: funpow_swap1) 

574 

575 
lemma stake_siterate[simp]: "stake n (siterate f x) = map (\<lambda>n. (f^^n) x) [0 ..< n]" 

576 
by (induct n arbitrary: x) (auto simp del: stake.simps(2) simp: stake_Suc) 

577 

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578 
lemma sset_siterate: "sset (siterate f x) = {(f^^n) x  n. True}" 
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579 
by (auto simp: sset_range) 
51462  580 

50518  581 
end 