author  traytel 
Fri, 15 Feb 2013 11:31:59 +0100  
changeset 51141  cc7423ce6774 
parent 51023  550f265864e3 
child 51352  fdecc2cd5649 
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 'a stream = Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65) 
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(* TODO: Provide by the package*) 

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theorem stream_set_induct: 

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"\<lbrakk>\<And>s. P (shd s) s; \<And>s y. \<lbrakk>y \<in> stream_set (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s\<rbrakk> \<Longrightarrow> 
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\<forall>y \<in> stream_set 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 stream_map_simps[simp]: 

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"shd (stream_map f s) = f (shd s)" "stl (stream_map f s) = stream_map f (stl s)" 

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unfolding shd_def stl_def stream_case_def stream_map_def stream.dtor_unfold 

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by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor) 

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lemma stream_map_Stream[simp]: "stream_map f (x ## s) = f x ## stream_map f s" 

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by (metis stream.exhaust stream.sels stream_map_simps) 

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theorem shd_stream_set: "shd s \<in> stream_set 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_stream_set: "y \<in> stream_set (stl s) \<Longrightarrow> y \<in> stream_set 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 stream_set_induct1[consumes 1, case_names shd stl, induct set: "stream_set"]: 

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assumes "y \<in> stream_set s" and "\<And>s. P (shd s) s" 

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and "\<And>s y. \<lbrakk>y \<in> stream_set (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 stream_set_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 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 stream_set_shift[simp]: "stream_set (xs @ s) = set xs \<union> stream_set s" 
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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 stream_set_streams: 

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assumes "stream_set 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> stream_set 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> stream_set 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> stream_set 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 {* flatten a stream of lists *} 

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definition flat where 

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

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"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)" 
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unfolding flat_def by auto 
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lemma flat_Cons[simp]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)" 
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unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto 
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lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @ flat ws" 
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by (induct xs) auto 
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lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @ flat (stl ws)" 

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by (cases ws) auto 
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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_stream_map[simp]: "stream_map 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_stream_set[simp]: "s !! n \<in> stream_set s" 

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by (induct n arbitrary: s) (auto intro: shd_stream_set stl_stream_set) 

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lemma stream_set_range: "stream_set s = range (snth s)" 

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proof (intro equalityI subsetI) 

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fix x assume "x \<in> stream_set 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_stream_map[simp]: "stake n (stream_map 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_stream_map[simp]: "sdrop n (stream_map f s) = stream_map f (sdrop n s)" 

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by (induct n arbitrary: s) 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 stream_map_alt: "stream_map 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 = stream_map 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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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 (stream_set s) P" 

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unfolding stream_all_def stream_set_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: "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: "same x = Stream 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 

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

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

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subsection {* stream of natural numbers *} 

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definition "fromN n = stream_unfold id Suc n" 

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

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unfolding fromN_def by auto 

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lemma snth_fromN[simp]: "fromN n !! m = n + m" 

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unfolding fromN_def by (induct m arbitrary: n) auto 

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

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

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

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unfolding fromN_def by (induct m arbitrary: n) auto 

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abbreviation "nats \<equiv> fromN 0" 

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subsection {* zip *} 

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definition "szip s1 s2 = 

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

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lemma szip_simps[simp]: 

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

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unfolding szip_def by auto 

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lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)" 

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by (induct n arbitrary: s1 s2) auto 

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subsection {* zip via function *} 

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definition "stream_map2 f s1 s2 = 

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

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lemma stream_map2_simps[simp]: 

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"shd (stream_map2 f s1 s2) = f (shd s1) (shd s2)" 

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"stl (stream_map2 f s1 s2) = stream_map2 f (stl s1) (stl s2)" 

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unfolding stream_map2_def by auto 

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lemma stream_map2_szip: 

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"stream_map2 f s1 s2 = stream_map (split f) (szip s1 s2)" 

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by (coinduct rule: stream.coinduct[of 

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"\<lambda>s1 s2. \<exists>s1' s2'. s1 = stream_map2 f s1' s2' \<and> s2 = stream_map (split f) (szip s1' s2')"]) 

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fastforce+ 

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end 