author | wenzelm |
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parent 59016 | be4a911aca71 |
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permissions | -rw-r--r-- |
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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 {* Infinite Streams *} |
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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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(*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 (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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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 {* 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 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 `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 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 {* 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: "(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 {* 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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||
54469 | 281 |
lemma stream_all_Stream: "stream_all P (x ## X) \<longleftrightarrow> P x \<and> stream_all P X" |
282 |
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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|
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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 |
51141 | 296 |
|
51409 | 297 |
lemma cycle_Cons[code]: "cycle (x # xs) = x ## cycle (xs @ [x])" |
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by (subst cycle.ctr) simp |
50518 | 299 |
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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 | 301 |
by (auto dest: arg_cong[of _ _ stl]) |
50518 | 302 |
|
303 |
lemma stake_append: "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s" |
|
304 |
proof (induct n arbitrary: u) |
|
305 |
case (Suc n) thus ?case by (cases u) auto |
|
306 |
qed auto |
|
307 |
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308 |
lemma stake_cycle_le[simp]: |
|
309 |
assumes "u \<noteq> []" "n < length u" |
|
310 |
shows "stake n (cycle u) = take n u" |
|
311 |
using min_absorb2[OF less_imp_le_nat[OF assms(2)]] |
|
51141 | 312 |
by (subst cycle_decomp[OF assms(1)], subst stake_append) auto |
50518 | 313 |
|
314 |
lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u" |
|
57175 | 315 |
by (subst cycle_decomp) (auto simp: stake_shift) |
50518 | 316 |
|
317 |
lemma sdrop_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> sdrop (length u) (cycle u) = cycle u" |
|
57175 | 318 |
by (subst cycle_decomp) (auto simp: sdrop_shift) |
50518 | 319 |
|
320 |
lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> |
|
321 |
stake n (cycle u) = concat (replicate (n div length u) u)" |
|
51141 | 322 |
by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric]) |
50518 | 323 |
|
324 |
lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow> |
|
325 |
sdrop n (cycle u) = cycle u" |
|
51141 | 326 |
by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric]) |
50518 | 327 |
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328 |
lemma stake_cycle: "u \<noteq> [] \<Longrightarrow> |
|
329 |
stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u" |
|
51141 | 330 |
by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto |
50518 | 331 |
|
332 |
lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)" |
|
51141 | 333 |
by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric]) |
334 |
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subsection {* iterated application of a function *} |
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337 |
|
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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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341 |
|
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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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344 |
|
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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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347 |
|
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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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350 |
|
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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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353 |
|
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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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356 |
|
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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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359 |
|
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360 |
|
51141 | 361 |
subsection {* stream repeating a single element *} |
362 |
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abbreviation "sconst \<equiv> siterate id" |
51141 | 364 |
|
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365 |
lemma shift_replicate_sconst[simp]: "replicate n x @- sconst x = sconst x" |
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366 |
by (subst (3) stake_sdrop[symmetric]) (simp add: map_replicate_trivial) |
51141 | 367 |
|
57175 | 368 |
lemma sset_sconst[simp]: "sset (sconst x) = {x}" |
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369 |
by (simp add: sset_siterate) |
51141 | 370 |
|
57175 | 371 |
lemma sconst_alt: "s = sconst x \<longleftrightarrow> sset s = {x}" |
372 |
proof |
|
373 |
assume "sset s = {x}" |
|
374 |
then show "s = sconst x" |
|
375 |
proof (coinduction arbitrary: s) |
|
376 |
case Eq_stream |
|
377 |
then have "shd s = x" "sset (stl s) \<subseteq> {x}" by (case_tac [!] s) auto |
|
378 |
then have "sset (stl s) = {x}" by (cases "stl s") auto |
|
379 |
with `shd s = x` show ?case by auto |
|
380 |
qed |
|
381 |
qed simp |
|
382 |
||
59016 | 383 |
lemma sconst_cycle: "sconst x = cycle [x]" |
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384 |
by coinduction auto |
51141 | 385 |
|
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386 |
lemma smap_sconst: "smap f (sconst x) = sconst (f x)" |
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387 |
by coinduction auto |
51141 | 388 |
|
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389 |
lemma sconst_streams: "x \<in> A \<Longrightarrow> sconst x \<in> streams A" |
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|
390 |
by (simp add: streams_iff_sset) |
51141 | 391 |
|
392 |
||
393 |
subsection {* stream of natural numbers *} |
|
394 |
||
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395 |
abbreviation "fromN \<equiv> siterate Suc" |
54469 | 396 |
|
51141 | 397 |
abbreviation "nats \<equiv> fromN 0" |
398 |
||
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399 |
lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" |
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400 |
by (auto simp add: sset_siterate le_iff_add) |
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401 |
|
57175 | 402 |
lemma stream_smap_fromN: "s = smap (\<lambda>j. let i = j - n in s !! i) (fromN n)" |
403 |
by (coinduction arbitrary: s n) |
|
404 |
(force simp: neq_Nil_conv Let_def snth.simps(2)[symmetric] Suc_diff_Suc |
|
405 |
intro: stream.map_cong split: if_splits simp del: snth.simps(2)) |
|
406 |
||
407 |
lemma stream_smap_nats: "s = smap (snth s) nats" |
|
408 |
using stream_smap_fromN[where n = 0] by simp |
|
409 |
||
51141 | 410 |
|
51462 | 411 |
subsection {* flatten a stream of lists *} |
412 |
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413 |
primcorec flat where |
51462 | 414 |
"shd (flat ws) = hd (shd ws)" |
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415 |
| "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)" |
51462 | 416 |
|
417 |
lemma flat_Cons[simp, code]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)" |
|
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418 |
by (subst flat.ctr) simp |
51462 | 419 |
|
420 |
lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @- flat ws" |
|
421 |
by (induct xs) auto |
|
422 |
||
423 |
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)" |
|
424 |
by (cases ws) auto |
|
425 |
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426 |
lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then |
51462 | 427 |
shd s ! n else flat (stl s) !! (n - length (shd s)))" |
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|
428 |
by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less) |
51462 | 429 |
|
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|
430 |
lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> |
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|
431 |
sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R") |
51462 | 432 |
proof safe |
433 |
fix x assume ?P "x : ?L" |
|
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434 |
then obtain m where "x = flat s !! m" by (metis image_iff sset_range) |
51462 | 435 |
with `?P` obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)" |
436 |
proof (atomize_elim, induct m arbitrary: s rule: less_induct) |
|
437 |
case (less y) |
|
438 |
thus ?case |
|
439 |
proof (cases "y < length (shd s)") |
|
440 |
case True thus ?thesis by (metis flat_snth less(2,3) snth.simps(1)) |
|
441 |
next |
|
442 |
case False |
|
443 |
hence "x = flat (stl s) !! (y - length (shd s))" by (metis less(2,3) flat_snth) |
|
444 |
moreover |
|
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|
445 |
{ from less(2) have *: "length (shd s) > 0" by (cases s) simp_all |
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446 |
with False have "y > 0" by (cases y) simp_all |
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|
447 |
with * have "y - length (shd s) < y" by simp |
51462 | 448 |
} |
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|
449 |
moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto |
51462 | 450 |
ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto |
451 |
thus ?thesis by (metis snth.simps(2)) |
|
452 |
qed |
|
453 |
qed |
|
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454 |
thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem) |
51462 | 455 |
next |
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456 |
fix x xs assume "xs \<in> sset s" ?P "x \<in> set xs" thus "x \<in> ?L" |
57986 | 457 |
by (induct rule: sset_induct) |
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458 |
(metis UnI1 flat_unfold shift.simps(1) sset_shift, |
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|
459 |
metis UnI2 flat_unfold shd_sset stl_sset sset_shift) |
51462 | 460 |
qed |
461 |
||
462 |
||
463 |
subsection {* merge a stream of streams *} |
|
464 |
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465 |
definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where |
|
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466 |
"smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)" |
51462 | 467 |
|
468 |
lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m" |
|
469 |
by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2)) |
|
470 |
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471 |
lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)" |
51462 | 472 |
proof (cases "n \<le> m") |
473 |
case False thus ?thesis unfolding smerge_def |
|
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|
474 |
by (subst sset_flat) |
53290 | 475 |
(auto simp: stream.set_map in_set_conv_nth simp del: stake.simps |
51462 | 476 |
intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp]) |
477 |
next |
|
478 |
case True thus ?thesis unfolding smerge_def |
|
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|
479 |
by (subst sset_flat) |
53290 | 480 |
(auto simp: stream.set_map in_set_conv_nth image_iff simp del: stake.simps snth.simps |
51462 | 481 |
intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp]) |
482 |
qed |
|
483 |
||
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|
484 |
lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset" |
51462 | 485 |
proof safe |
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486 |
fix x assume "x \<in> sset (smerge ss)" |
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|
487 |
thus "x \<in> UNION (sset ss) sset" |
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|
488 |
unfolding smerge_def by (subst (asm) sset_flat) |
53290 | 489 |
(auto simp: stream.set_map in_set_conv_nth sset_range simp del: stake.simps, fast+) |
51462 | 490 |
next |
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|
491 |
fix s x assume "s \<in> sset ss" "x \<in> sset s" |
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|
492 |
thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range) |
51462 | 493 |
qed |
494 |
||
495 |
||
496 |
subsection {* product of two streams *} |
|
497 |
||
498 |
definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where |
|
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499 |
"sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)" |
51462 | 500 |
|
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501 |
lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2" |
53290 | 502 |
unfolding sproduct_def sset_smerge by (auto simp: stream.set_map) |
51462 | 503 |
|
504 |
||
505 |
subsection {* interleave two streams *} |
|
506 |
||
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507 |
primcorec sinterleave where |
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508 |
"shd (sinterleave s1 s2) = shd s1" |
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509 |
| "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)" |
51462 | 510 |
|
511 |
lemma sinterleave_code[code]: |
|
512 |
"sinterleave (x ## s1) s2 = x ## sinterleave s2 s1" |
|
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513 |
by (subst sinterleave.ctr) simp |
51462 | 514 |
|
515 |
lemma sinterleave_snth[simp]: |
|
516 |
"even n \<Longrightarrow> sinterleave s1 s2 !! n = s1 !! (n div 2)" |
|
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|
517 |
"odd n \<Longrightarrow> sinterleave s1 s2 !! n = s2 !! (n div 2)" |
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|
518 |
by (induct n arbitrary: s1 s2) simp_all |
51462 | 519 |
|
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|
520 |
lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2" |
51462 | 521 |
proof (intro equalityI subsetI) |
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522 |
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|
523 |
then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast |
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|
524 |
thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto |
51462 | 525 |
next |
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|
526 |
fix x assume "x \<in> sset s1 \<union> sset s2" |
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|
527 |
thus "x \<in> sset (sinterleave s1 s2)" |
51462 | 528 |
proof |
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|
529 |
assume "x \<in> sset s1" |
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|
530 |
then obtain n where "x = s1 !! n" unfolding sset_range by blast |
51462 | 531 |
hence "sinterleave s1 s2 !! (2 * n) = x" by simp |
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|
532 |
thus ?thesis unfolding sset_range by blast |
51462 | 533 |
next |
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|
534 |
assume "x \<in> sset s2" |
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|
535 |
then obtain n where "x = s2 !! n" unfolding sset_range by blast |
51462 | 536 |
hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp |
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|
537 |
thus ?thesis unfolding sset_range by blast |
51462 | 538 |
qed |
539 |
qed |
|
540 |
||
541 |
||
51141 | 542 |
subsection {* zip *} |
543 |
||
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|
544 |
primcorec szip where |
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|
545 |
"shd (szip s1 s2) = (shd s1, shd s2)" |
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546 |
| "stl (szip s1 s2) = szip (stl s1) (stl s2)" |
51141 | 547 |
|
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|
548 |
lemma szip_unfold[code]: "szip (a ## s1) (b ## s2) = (a, b) ## (szip s1 s2)" |
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|
549 |
by (subst szip.ctr) simp |
51409 | 550 |
|
51141 | 551 |
lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)" |
552 |
by (induct n arbitrary: s1 s2) auto |
|
553 |
||
57175 | 554 |
lemma stake_szip[simp]: |
555 |
"stake n (szip s1 s2) = zip (stake n s1) (stake n s2)" |
|
556 |
by (induct n arbitrary: s1 s2) auto |
|
557 |
||
558 |
lemma sdrop_szip[simp]: "sdrop n (szip s1 s2) = szip (sdrop n s1) (sdrop n s2)" |
|
559 |
by (induct n arbitrary: s1 s2) auto |
|
560 |
||
561 |
lemma smap_szip_fst: |
|
562 |
"smap (\<lambda>x. f (fst x)) (szip s1 s2) = smap f s1" |
|
563 |
by (coinduction arbitrary: s1 s2) auto |
|
564 |
||
565 |
lemma smap_szip_snd: |
|
566 |
"smap (\<lambda>x. g (snd x)) (szip s1 s2) = smap g s2" |
|
567 |
by (coinduction arbitrary: s1 s2) auto |
|
568 |
||
51141 | 569 |
|
570 |
subsection {* zip via function *} |
|
571 |
||
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|
572 |
primcorec smap2 where |
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|
573 |
"shd (smap2 f s1 s2) = f (shd s1) (shd s2)" |
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|
574 |
| "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)" |
51141 | 575 |
|
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|
576 |
lemma smap2_unfold[code]: |
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|
577 |
"smap2 f (a ## s1) (b ## s2) = f a b ## (smap2 f s1 s2)" |
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|
578 |
by (subst smap2.ctr) simp |
51409 | 579 |
|
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|
580 |
lemma smap2_szip: |
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|
581 |
"smap2 f s1 s2 = smap (split f) (szip s1 s2)" |
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|
582 |
by (coinduction arbitrary: s1 s2) auto |
50518 | 583 |
|
57175 | 584 |
lemma smap_smap2[simp]: |
585 |
"smap f (smap2 g s1 s2) = smap2 (\<lambda>x y. f (g x y)) s1 s2" |
|
586 |
unfolding smap2_szip stream.map_comp o_def split_def .. |
|
587 |
||
588 |
lemma smap2_alt: |
|
589 |
"(smap2 f s1 s2 = s) = (\<forall>n. f (s1 !! n) (s2 !! n) = s !! n)" |
|
590 |
unfolding smap2_szip smap_alt by auto |
|
591 |
||
592 |
lemma snth_smap2[simp]: |
|
593 |
"smap2 f s1 s2 !! n = f (s1 !! n) (s2 !! n)" |
|
594 |
by (induct n arbitrary: s1 s2) auto |
|
595 |
||
596 |
lemma stake_smap2[simp]: |
|
597 |
"stake n (smap2 f s1 s2) = map (split f) (zip (stake n s1) (stake n s2))" |
|
598 |
by (induct n arbitrary: s1 s2) auto |
|
599 |
||
600 |
lemma sdrop_smap2[simp]: |
|
601 |
"sdrop n (smap2 f s1 s2) = smap2 f (sdrop n s1) (sdrop n s2)" |
|
602 |
by (induct n arbitrary: s1 s2) auto |
|
603 |
||
50518 | 604 |
end |