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(* Title: HOL/Library/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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section \<open>Infinite Streams\<close> 
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theory Stream 

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imports "~~/src/HOL/Library/Nat_Bijection" 
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begin 
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codatatype (sset: 'a) stream = 
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SCons (shd: 'a) (stl: "'a stream") (infixr "##" 65) 
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for 
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map: smap 
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rel: stream_all2 
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context 
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begin 

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(*for code generation only*) 
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qualified 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 (y ## s) = (if x = y then True else smember x s)" 
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unfolding smember_def by auto 
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end 
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lemmas smap_simps[simp] = stream.map_sel 
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lemmas shd_sset = stream.set_sel(1) 
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lemmas stl_sset = stream.set_sel(2) 
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theorem sset_induct[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" 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 by induct (metis stream.sel(1), auto) 
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lemma smap_ctr: "smap f s = x ## s' \<longleftrightarrow> f (shd s) = x \<and> smap f (stl s) = s'" 
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by (cases s) simp 

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subsection \<open>prepend list to stream\<close> 
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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 \<open>set of streams with elements in some fixed set\<close> 
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context 
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notes [[inductive_internals]] 
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begin 
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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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end 
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lemma in_streams: "stl s \<in> streams S \<Longrightarrow> shd s \<in> S \<Longrightarrow> s \<in> streams S" 
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by (cases s) auto 

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lemma streamsE: "s \<in> streams A \<Longrightarrow> (shd s \<in> A \<Longrightarrow> stl s \<in> streams A \<Longrightarrow> P) \<Longrightarrow> P" 

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by (erule streams.cases) simp_all 

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lemma Stream_image: "x ## y \<in> (op ## x') ` Y \<longleftrightarrow> x = x' \<and> y \<in> Y" 

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by auto 

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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 \<open>s \<in> streams A\<close> 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 streams_mono2: "S \<subseteq> T \<Longrightarrow> streams S \<subseteq> streams T" 
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by (auto intro: streams_mono) 

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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 \<open>nth, take, drop for streams\<close> 
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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_Stream: "(x ## s) !! Suc i = s !! i" 
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by simp 

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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 shift_snth: "(xs @ s) !! n = (if n < length xs then xs ! n else s !! (n  length xs))" 
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by auto 

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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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lemma streams_iff_snth: "s \<in> streams X \<longleftrightarrow> (\<forall>n. s !! n \<in> X)" 
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by (force simp: streams_iff_sset sset_range) 

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lemma snth_in: "s \<in> streams X \<Longrightarrow> s !! n \<in> X" 

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by (simp add: streams_iff_snth) 

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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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lemma take_stake: "take n (stake m s) = stake (min n m) s" 
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proof (induct m arbitrary: s n) 

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

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qed simp 

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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 drop_stake: "drop n (stake m s) = stake (m  n) (sdrop n s)" 
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proof (induct m arbitrary: s n) 

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

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qed simp 

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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: "sdrop i (w @ s) = drop i w @ sdrop (i  length w) s" 
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by (induct i arbitrary: w s) (auto simp: drop_tl drop_Suc neq_Nil_conv) 

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lemma stake_shift: "stake i (w @ s) = take i w @ stake (i  length w) s" 
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by (induct i arbitrary: w s) (auto simp: neq_Nil_conv) 

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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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lemma sdrop_snth: "sdrop n s !! m = s !! (n + m)" 
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by (induct n arbitrary: m 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_SCons[code]: 
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"sdrop_while P (a ## s) = (if P a then sdrop_while P s else 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_SCons) 
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next 
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case False thus ?thesis by (subst (1 2) sfilter.ctr) (simp add: sdrop_while_SCons) 
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qed 
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subsection \<open>unary predicates lifted to streams\<close> 
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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 \<open>recurring stream out of a list\<close> 
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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" 

51141  310 
by (auto dest: arg_cong[of _ _ stl]) 
50518  311 

312 
lemma stake_append: "stake n (u @ s) = take (min (length u) n) u @ stake (n  length u) s" 

313 
proof (induct n arbitrary: u) 

314 
case (Suc n) thus ?case by (cases u) auto 

315 
qed auto 

316 

317 
lemma stake_cycle_le[simp]: 

318 
assumes "u \<noteq> []" "n < length u" 

319 
shows "stake n (cycle u) = take n u" 

320 
using min_absorb2[OF less_imp_le_nat[OF assms(2)]] 

51141  321 
by (subst cycle_decomp[OF assms(1)], subst stake_append) auto 
50518  322 

323 
lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u" 

57175  324 
by (subst cycle_decomp) (auto simp: stake_shift) 
50518  325 

326 
lemma sdrop_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> sdrop (length u) (cycle u) = cycle u" 

57175  327 
by (subst cycle_decomp) (auto simp: sdrop_shift) 
50518  328 

329 
lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> 

330 
stake n (cycle u) = concat (replicate (n div length u) u)" 

51141  331 
by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric]) 
50518  332 

333 
lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> 

334 
sdrop n (cycle u) = cycle u" 

51141  335 
by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric]) 
50518  336 

337 
lemma stake_cycle: "u \<noteq> [] \<Longrightarrow> 

338 
stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u" 

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by (subst div_mult_mod_eq[of n "length u", symmetric], unfold stake_add[symmetric]) auto 
50518  340 

341 
lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)" 

51141  342 
by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric]) 
343 

63192  344 
lemma sset_cycle[simp]: 
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assumes "xs \<noteq> []" 
63192  346 
shows "sset (cycle xs) = set xs" 
347 
proof (intro set_eqI iffI) 

348 
fix x 

349 
assume "x \<in> sset (cycle xs)" 

350 
then show "x \<in> set xs" using assms 

351 
by (induction "cycle xs" arbitrary: xs rule: sset_induct) (fastforce simp: neq_Nil_conv)+ 

352 
qed (metis assms UnI1 cycle_decomp sset_shift) 

353 

51141  354 

60500  355 
subsection \<open>iterated application of a function\<close> 
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356 

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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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360 

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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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363 

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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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366 

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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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369 

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370 
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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372 

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373 
lemma sset_siterate: "sset (siterate f x) = {(f^^n) x  n. True}" 
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374 
by (auto simp: sset_range) 
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375 

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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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378 

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379 

60500  380 
subsection \<open>stream repeating a single element\<close> 
51141  381 

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abbreviation "sconst \<equiv> siterate id" 
51141  383 

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384 
lemma shift_replicate_sconst[simp]: "replicate n x @ sconst x = sconst x" 
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385 
by (subst (3) stake_sdrop[symmetric]) (simp add: map_replicate_trivial) 
51141  386 

57175  387 
lemma sset_sconst[simp]: "sset (sconst x) = {x}" 
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388 
by (simp add: sset_siterate) 
51141  389 

57175  390 
lemma sconst_alt: "s = sconst x \<longleftrightarrow> sset s = {x}" 
391 
proof 

392 
assume "sset s = {x}" 

393 
then show "s = sconst x" 

394 
proof (coinduction arbitrary: s) 

395 
case Eq_stream 

63649  396 
then have "shd s = x" "sset (stl s) \<subseteq> {x}" by (cases s; auto)+ 
57175  397 
then have "sset (stl s) = {x}" by (cases "stl s") auto 
60500  398 
with \<open>shd s = x\<close> show ?case by auto 
57175  399 
qed 
400 
qed simp 

401 

59016  402 
lemma sconst_cycle: "sconst x = cycle [x]" 
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403 
by coinduction auto 
51141  404 

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lemma smap_sconst: "smap f (sconst x) = sconst (f x)" 
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406 
by coinduction auto 
51141  407 

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408 
lemma sconst_streams: "x \<in> A \<Longrightarrow> sconst x \<in> streams A" 
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409 
by (simp add: streams_iff_sset) 
51141  410 

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411 
lemma streams_empty_iff: "streams S = {} \<longleftrightarrow> S = {}" 
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412 
proof safe 
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413 
fix x assume "x \<in> S" "streams S = {}" 
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414 
then have "sconst x \<in> streams S" 
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415 
by (intro sconst_streams) 
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416 
then show "x \<in> {}" 
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417 
unfolding \<open>streams S = {}\<close> by simp 
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418 
qed (auto simp: streams_empty) 
51141  419 

60500  420 
subsection \<open>stream of natural numbers\<close> 
51141  421 

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422 
abbreviation "fromN \<equiv> siterate Suc" 
54469  423 

51141  424 
abbreviation "nats \<equiv> fromN 0" 
425 

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426 
lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" 
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427 
by (auto simp add: sset_siterate le_iff_add) 
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428 

57175  429 
lemma stream_smap_fromN: "s = smap (\<lambda>j. let i = j  n in s !! i) (fromN n)" 
430 
by (coinduction arbitrary: s n) 

431 
(force simp: neq_Nil_conv Let_def snth.simps(2)[symmetric] Suc_diff_Suc 

432 
intro: stream.map_cong split: if_splits simp del: snth.simps(2)) 

433 

434 
lemma stream_smap_nats: "s = smap (snth s) nats" 

435 
using stream_smap_fromN[where n = 0] by simp 

436 

51141  437 

60500  438 
subsection \<open>flatten a stream of lists\<close> 
51462  439 

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440 
primcorec flat where 
51462  441 
"shd (flat ws) = hd (shd ws)" 
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442 
 "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)" 
51462  443 

444 
lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)" 

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445 
by (subst flat.ctr) simp 
51462  446 

447 
lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @ flat ws" 

448 
by (induct xs) auto 

449 

450 
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @ flat (stl ws)" 

451 
by (cases ws) auto 

452 

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453 
lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then 
51462  454 
shd s ! n else flat (stl s) !! (n  length (shd s)))" 
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455 
by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less) 
51462  456 

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457 
lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> 
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458 
sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R") 
51462  459 
proof safe 
460 
fix x assume ?P "x : ?L" 

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461 
then obtain m where "x = flat s !! m" by (metis image_iff sset_range) 
60500  462 
with \<open>?P\<close> obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)" 
51462  463 
proof (atomize_elim, induct m arbitrary: s rule: less_induct) 
464 
case (less y) 

465 
thus ?case 

466 
proof (cases "y < length (shd s)") 

467 
case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1)) 

468 
next 

469 
case False 

470 
hence "x = flat (stl s) !! (y  length (shd s))" by (metis less(2,3) flat_snth) 

471 
moreover 

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472 
{ from less(2) have *: "length (shd s) > 0" by (cases s) simp_all 
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473 
with False have "y > 0" by (cases y) simp_all 
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474 
with * have "y  length (shd s) < y" by simp 
51462  475 
} 
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476 
moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto 
51462  477 
ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto 
478 
thus ?thesis by (metis snth.simps(2)) 

479 
qed 

480 
qed 

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481 
thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem) 
51462  482 
next 
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483 
fix x xs assume "xs \<in> sset s" ?P "x \<in> set xs" thus "x \<in> ?L" 
57986  484 
by (induct rule: sset_induct) 
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485 
(metis UnI1 flat_unfold shift.simps(1) sset_shift, 
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486 
metis UnI2 flat_unfold shd_sset stl_sset sset_shift) 
51462  487 
qed 
488 

489 

60500  490 
subsection \<open>merge a stream of streams\<close> 
51462  491 

492 
definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where 

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493 
"smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)" 
51462  494 

495 
lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m" 

496 
by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2)) 

497 

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498 
lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)" 
51462  499 
proof (cases "n \<le> m") 
500 
case False thus ?thesis unfolding smerge_def 

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501 
by (subst sset_flat) 
53290  502 
(auto simp: stream.set_map in_set_conv_nth simp del: stake.simps 
51462  503 
intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp]) 
504 
next 

505 
case True thus ?thesis unfolding smerge_def 

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506 
by (subst sset_flat) 
53290  507 
(auto simp: stream.set_map in_set_conv_nth image_iff simp del: stake.simps snth.simps 
51462  508 
intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp]) 
509 
qed 

510 

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511 
lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset" 
51462  512 
proof safe 
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513 
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514 
thus "x \<in> UNION (sset ss) sset" 
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515 
unfolding smerge_def by (subst (asm) sset_flat) 
53290  516 
(auto simp: stream.set_map in_set_conv_nth sset_range simp del: stake.simps, fast+) 
51462  517 
next 
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518 
fix s x assume "s \<in> sset ss" "x \<in> sset s" 
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519 
thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range) 
51462  520 
qed 
521 

522 

60500  523 
subsection \<open>product of two streams\<close> 
51462  524 

525 
definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where 

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526 
"sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)" 
51462  527 

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528 
lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2" 
53290  529 
unfolding sproduct_def sset_smerge by (auto simp: stream.set_map) 
51462  530 

531 

60500  532 
subsection \<open>interleave two streams\<close> 
51462  533 

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534 
primcorec sinterleave where 
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535 
"shd (sinterleave s1 s2) = shd s1" 
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536 
 "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)" 
51462  537 

538 
lemma sinterleave_code[code]: 

539 
"sinterleave (x ## s1) s2 = x ## sinterleave s2 s1" 

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540 
by (subst sinterleave.ctr) simp 
51462  541 

542 
lemma sinterleave_snth[simp]: 

543 
"even n \<Longrightarrow> sinterleave s1 s2 !! n = s1 !! (n div 2)" 

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544 
"odd n \<Longrightarrow> sinterleave s1 s2 !! n = s2 !! (n div 2)" 
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545 
by (induct n arbitrary: s1 s2) simp_all 
51462  546 

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547 
lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2" 
51462  548 
proof (intro equalityI subsetI) 
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549 
fix x assume "x \<in> sset (sinterleave s1 s2)" 
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550 
then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast 
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551 
thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto 
51462  552 
next 
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553 
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554 
thus "x \<in> sset (sinterleave s1 s2)" 
51462  555 
proof 
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556 
assume "x \<in> sset s1" 
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557 
then obtain n where "x = s1 !! n" unfolding sset_range by blast 
51462  558 
hence "sinterleave s1 s2 !! (2 * n) = x" by simp 
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559 
thus ?thesis unfolding sset_range by blast 
51462  560 
next 
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561 
assume "x \<in> sset s2" 
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562 
then obtain n where "x = s2 !! n" unfolding sset_range by blast 
51462  563 
hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp 
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564 
thus ?thesis unfolding sset_range by blast 
51462  565 
qed 
566 
qed 

567 

568 

60500  569 
subsection \<open>zip\<close> 
51141  570 

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571 
primcorec szip where 
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572 
"shd (szip s1 s2) = (shd s1, shd s2)" 
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573 
 "stl (szip s1 s2) = szip (stl s1) (stl s2)" 
51141  574 

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575 
lemma szip_unfold[code]: "szip (a ## s1) (b ## s2) = (a, b) ## (szip s1 s2)" 
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576 
by (subst szip.ctr) simp 
51409  577 

51141  578 
lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)" 
579 
by (induct n arbitrary: s1 s2) auto 

580 

57175  581 
lemma stake_szip[simp]: 
582 
"stake n (szip s1 s2) = zip (stake n s1) (stake n s2)" 

583 
by (induct n arbitrary: s1 s2) auto 

584 

585 
lemma sdrop_szip[simp]: "sdrop n (szip s1 s2) = szip (sdrop n s1) (sdrop n s2)" 

586 
by (induct n arbitrary: s1 s2) auto 

587 

588 
lemma smap_szip_fst: 

589 
"smap (\<lambda>x. f (fst x)) (szip s1 s2) = smap f s1" 

590 
by (coinduction arbitrary: s1 s2) auto 

591 

592 
lemma smap_szip_snd: 

593 
"smap (\<lambda>x. g (snd x)) (szip s1 s2) = smap g s2" 

594 
by (coinduction arbitrary: s1 s2) auto 

595 

51141  596 

60500  597 
subsection \<open>zip via function\<close> 
51141  598 

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599 
primcorec smap2 where 
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600 
"shd (smap2 f s1 s2) = f (shd s1) (shd s2)" 
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601 
 "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)" 
51141  602 

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603 
lemma smap2_unfold[code]: 
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604 
"smap2 f (a ## s1) (b ## s2) = f a b ## (smap2 f s1 s2)" 
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605 
by (subst smap2.ctr) simp 
51409  606 

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607 
lemma smap2_szip: 
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608 
"smap2 f s1 s2 = smap (case_prod f) (szip s1 s2)" 
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609 
by (coinduction arbitrary: s1 s2) auto 
50518  610 

57175  611 
lemma smap_smap2[simp]: 
612 
"smap f (smap2 g s1 s2) = smap2 (\<lambda>x y. f (g x y)) s1 s2" 

613 
unfolding smap2_szip stream.map_comp o_def split_def .. 

614 

615 
lemma smap2_alt: 

616 
"(smap2 f s1 s2 = s) = (\<forall>n. f (s1 !! n) (s2 !! n) = s !! n)" 

617 
unfolding smap2_szip smap_alt by auto 

618 

619 
lemma snth_smap2[simp]: 

620 
"smap2 f s1 s2 !! n = f (s1 !! n) (s2 !! n)" 

621 
by (induct n arbitrary: s1 s2) auto 

622 

623 
lemma stake_smap2[simp]: 

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624 
"stake n (smap2 f s1 s2) = map (case_prod f) (zip (stake n s1) (stake n s2))" 
57175  625 
by (induct n arbitrary: s1 s2) auto 
626 

627 
lemma sdrop_smap2[simp]: 

628 
"sdrop n (smap2 f s1 s2) = smap2 f (sdrop n s1) (sdrop n s2)" 

629 
by (induct n arbitrary: s1 s2) auto 

630 

50518  631 
end 