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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, 2013 
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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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codatatype (sset: 'a) stream (map: smap rel: stream_all2) = 
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Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65) 
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code_datatype Stream 
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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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apply (rule stream.dtor_set_induct) 
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apply (auto simp add: shd_def stl_def fsts_def snds_def split_beta) 

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

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

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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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by (case_tac [!] s) auto 
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theorem shd_sset: "shd s \<in> sset s" 
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by (case_tac s) auto 
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theorem stl_sset: "y \<in> sset (stl s) \<Longrightarrow> y \<in> sset s" 
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by (case_tac s) auto 
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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 fixed set *} 
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coinductive_set 

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streams :: "'a set \<Rightarrow> '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 streams_Stream: "x ## s \<in> streams A \<longleftrightarrow> x \<in> A \<and> s \<in> streams A" 
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by (auto elim: streams.cases) 

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lemma streams_stl: "s \<in> streams A \<Longrightarrow> stl s \<in> streams A" 

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by (cases s) (auto simp: streams_Stream) 

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lemma streams_shd: "s \<in> streams A \<Longrightarrow> shd s \<in> A" 

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by (cases s) (auto simp: streams_Stream) 

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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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using assms proof (coinduction arbitrary: s) 
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case streams then show ?case by (cases s) simp 
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qed 
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lemma streams_sset: 
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assumes "s \<in> streams A" 

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shows "sset s \<subseteq> A" 

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proof 

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fix x assume "x \<in> sset s" from this `s \<in> streams A` show "x \<in> A" 

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by (induct s) (auto intro: streams_shd streams_stl) 

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qed 

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lemma streams_iff_sset: "s \<in> streams A \<longleftrightarrow> sset s \<subseteq> A" 

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by (metis sset_streams streams_sset) 

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lemma streams_mono: "s \<in> streams A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> s \<in> streams B" 

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unfolding streams_iff_sset by auto 

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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" 

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unfolding streams_iff_sset stream.set_map by auto 

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lemma streams_empty: "streams {} = {}" 

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by (auto elim: streams.cases) 

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lemma streams_UNIV[simp]: "streams UNIV = UNIV" 

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by (auto simp: streams_iff_sset) 

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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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then have "\<And>n. smap f (sdrop n s) = sdrop n s'" 
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by coinduction (auto intro: exI[of _ 0] simp del: sdrop.simps(2)) 
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then show ?L using sdrop.simps(1) by metis 
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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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primcorec sfilter where 
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"shd (sfilter P s) = shd (sdrop_while (Not o P) s)" 
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 "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not o P) s))" 
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lemma sfilter_Stream: "sfilter P (x ## s) = (if P x then x ## sfilter P s else sfilter P s)" 

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proof (cases "P x") 

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case True thus ?thesis by (subst sfilter.ctr) (simp add: sdrop_while_Stream) 
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next 
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case False thus ?thesis by (subst (1 2) sfilter.ctr) (simp add: sdrop_while_Stream) 
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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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lemma stream_all_Stream: "stream_all P (x ## X) \<longleftrightarrow> P x \<and> stream_all P X" 
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by simp 

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subsection {* recurring stream out of a list *} 

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primcorec cycle :: "'a list \<Rightarrow> 'a stream" where 
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"shd (cycle xs) = hd xs" 
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 "stl (cycle xs) = cycle (tl xs @ [hd xs])" 
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lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @ cycle u" 
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proof (coinduction arbitrary: u) 
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case Eq_stream then show ?case using stream.collapse[of "cycle u"] 
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by (auto intro!: exI[of _ "tl u @ [hd u]"]) 
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qed 
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lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])" 
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by (subst cycle.ctr) simp 
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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)" 

51141  319 
by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric]) 
320 

321 

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subsection {* iterated application of a function *} 
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323 

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primcorec siterate where 
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"shd (siterate f x) = x" 
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 "stl (siterate f x) = siterate f (f x)" 
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327 

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lemma stake_Suc: "stake (Suc n) s = stake n s @ [s !! n]" 
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by (induct n arbitrary: s) auto 
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330 

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lemma snth_siterate[simp]: "siterate f x !! n = (f^^n) x" 
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by (induct n arbitrary: x) (auto simp: funpow_swap1) 
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333 

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lemma sdrop_siterate[simp]: "sdrop n (siterate f x) = siterate f ((f^^n) x)" 
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by (induct n arbitrary: x) (auto simp: funpow_swap1) 
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336 

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lemma stake_siterate[simp]: "stake n (siterate f x) = map (\<lambda>n. (f^^n) x) [0 ..< n]" 
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by (induct n arbitrary: x) (auto simp del: stake.simps(2) simp: stake_Suc) 
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339 

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lemma sset_siterate: "sset (siterate f x) = {(f^^n) x  n. True}" 
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by (auto simp: sset_range) 
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342 

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lemma smap_siterate: "smap f (siterate f x) = siterate f (f x)" 
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by (coinduction arbitrary: x) auto 
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345 

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346 

51141  347 
subsection {* stream repeating a single element *} 
348 

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abbreviation "sconst \<equiv> siterate id" 
51141  350 

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lemma shift_replicate_sconst[simp]: "replicate n x @ sconst x = sconst x" 
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352 
by (subst (3) stake_sdrop[symmetric]) (simp add: map_replicate_trivial) 
51141  353 

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lemma stream_all_same[simp]: "sset (sconst x) = {x}" 
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by (simp add: sset_siterate) 
51141  356 

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lemma same_cycle: "sconst x = cycle [x]" 
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358 
by coinduction auto 
51141  359 

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lemma smap_sconst: "smap f (sconst x) = sconst (f x)" 
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by coinduction auto 
51141  362 

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lemma sconst_streams: "x \<in> A \<Longrightarrow> sconst x \<in> streams A" 
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364 
by (simp add: streams_iff_sset) 
51141  365 

366 

367 
subsection {* stream of natural numbers *} 

368 

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abbreviation "fromN \<equiv> siterate Suc" 
54469  370 

51141  371 
abbreviation "nats \<equiv> fromN 0" 
372 

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lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" 
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374 
by (auto simp add: sset_siterate) arith 
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375 

51141  376 

51462  377 
subsection {* flatten a stream of lists *} 
378 

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379 
primcorec flat where 
51462  380 
"shd (flat ws) = hd (shd ws)" 
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 "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)" 
51462  382 

383 
lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)" 

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by (subst flat.ctr) simp 
51462  385 

386 
lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @ flat ws" 

387 
by (induct xs) auto 

388 

389 
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @ flat (stl ws)" 

390 
by (cases ws) auto 

391 

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lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then 
51462  393 
shd s ! n else flat (stl s) !! (n  length (shd s)))" 
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394 
by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less) 
51462  395 

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lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> 
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397 
sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R") 
51462  398 
proof safe 
399 
fix x assume ?P "x : ?L" 

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400 
then obtain m where "x = flat s !! m" by (metis image_iff sset_range) 
51462  401 
with `?P` obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)" 
402 
proof (atomize_elim, induct m arbitrary: s rule: less_induct) 

403 
case (less y) 

404 
thus ?case 

405 
proof (cases "y < length (shd s)") 

406 
case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1)) 

407 
next 

408 
case False 

409 
hence "x = flat (stl s) !! (y  length (shd s))" by (metis less(2,3) flat_snth) 

410 
moreover 

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{ from less(2) have *: "length (shd s) > 0" by (cases s) simp_all 
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with False have "y > 0" by (cases y) simp_all 
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413 
with * have "y  length (shd s) < y" by simp 
51462  414 
} 
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415 
moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto 
51462  416 
ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto 
417 
thus ?thesis by (metis snth.simps(2)) 

418 
qed 

419 
qed 

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420 
thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem) 
51462  421 
next 
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422 
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423 
by (induct rule: sset_induct1) 
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424 
(metis UnI1 flat_unfold shift.simps(1) sset_shift, 
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425 
metis UnI2 flat_unfold shd_sset stl_sset sset_shift) 
51462  426 
qed 
427 

428 

429 
subsection {* merge a stream of streams *} 

430 

431 
definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where 

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432 
"smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)" 
51462  433 

434 
lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m" 

435 
by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2)) 

436 

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437 
lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)" 
51462  438 
proof (cases "n \<le> m") 
439 
case False thus ?thesis unfolding smerge_def 

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440 
by (subst sset_flat) 
53290  441 
(auto simp: stream.set_map in_set_conv_nth simp del: stake.simps 
51462  442 
intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp]) 
443 
next 

444 
case True thus ?thesis unfolding smerge_def 

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445 
by (subst sset_flat) 
53290  446 
(auto simp: stream.set_map in_set_conv_nth image_iff simp del: stake.simps snth.simps 
51462  447 
intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp]) 
448 
qed 

449 

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450 
lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset" 
51462  451 
proof safe 
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452 
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453 
thus "x \<in> UNION (sset ss) sset" 
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454 
unfolding smerge_def by (subst (asm) sset_flat) 
53290  455 
(auto simp: stream.set_map in_set_conv_nth sset_range simp del: stake.simps, fast+) 
51462  456 
next 
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457 
fix s x assume "s \<in> sset ss" "x \<in> sset s" 
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458 
thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range) 
51462  459 
qed 
460 

461 

462 
subsection {* product of two streams *} 

463 

464 
definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where 

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"sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)" 
51462  466 

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lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2" 
53290  468 
unfolding sproduct_def sset_smerge by (auto simp: stream.set_map) 
51462  469 

470 

471 
subsection {* interleave two streams *} 

472 

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primcorec sinterleave where 
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"shd (sinterleave s1 s2) = shd s1" 
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475 
 "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)" 
51462  476 

477 
lemma sinterleave_code[code]: 

478 
"sinterleave (x ## s1) s2 = x ## sinterleave s2 s1" 

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479 
by (subst sinterleave.ctr) simp 
51462  480 

481 
lemma sinterleave_snth[simp]: 

482 
"even n \<Longrightarrow> sinterleave s1 s2 !! n = s1 !! (n div 2)" 

483 
"odd n \<Longrightarrow> sinterleave s1 s2 !! n = s2 !! (n div 2)" 

484 
by (induct n arbitrary: s1 s2) 

485 
(auto dest: even_nat_Suc_div_2 odd_nat_plus_one_div_two[folded nat_2]) 

486 

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487 
lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2" 
51462  488 
proof (intro equalityI subsetI) 
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489 
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490 
then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast 
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491 
thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto 
51462  492 
next 
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493 
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494 
thus "x \<in> sset (sinterleave s1 s2)" 
51462  495 
proof 
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496 
assume "x \<in> sset s1" 
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497 
then obtain n where "x = s1 !! n" unfolding sset_range by blast 
51462  498 
hence "sinterleave s1 s2 !! (2 * n) = x" by simp 
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499 
thus ?thesis unfolding sset_range by blast 
51462  500 
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501 
assume "x \<in> sset s2" 
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502 
then obtain n where "x = s2 !! n" unfolding sset_range by blast 
51462  503 
hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp 
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504 
thus ?thesis unfolding sset_range by blast 
51462  505 
qed 
506 
qed 

507 

508 

51141  509 
subsection {* zip *} 
510 

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511 
primcorec szip where 
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512 
"shd (szip s1 s2) = (shd s1, shd s2)" 
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 "stl (szip s1 s2) = szip (stl s1) (stl s2)" 
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lemma szip_unfold[code]: "szip (Stream a s1) (Stream b s2) = Stream (a, b) (szip s1 s2)" 
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by (subst szip.ctr) simp 
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lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)" 
519 
by (induct n arbitrary: s1 s2) auto 

520 

521 

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subsection {* zip via function *} 

523 

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primcorec smap2 where 
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"shd (smap2 f s1 s2) = f (shd s1) (shd s2)" 
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 "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)" 
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lemma smap2_unfold[code]: 
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"smap2 f (Stream a s1) (Stream b s2) = Stream (f a b) (smap2 f s1 s2)" 
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by (subst smap2.ctr) simp 
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lemma smap2_szip: 
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"smap2 f s1 s2 = smap (split f) (szip s1 s2)" 
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by (coinduction arbitrary: s1 s2) auto 
50518  535 

536 
end 