src/HOL/BNF/Examples/Stream.thy

 
 codata 'a stream = Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65)
 
 (* TODO: Provide by the package*)
 theorem stream_set_induct:
-   "\<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>
-   \<forall>y \<in> stream_set s. P y s"
-by (rule stream.dtor_set_induct)
-   (auto simp add:  shd_def stl_def stream_case_def fsts_def snds_def split_beta)
+  "\<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>
+    \<forall>y \<in> stream_set s. P y s"
+  by (rule stream.dtor_set_induct)
+    (auto simp add:  shd_def stl_def stream_case_def fsts_def snds_def split_beta)
+
+lemma stream_map_simps[simp]:
+  "shd (stream_map f s) = f (shd s)" "stl (stream_map f s) = stream_map f (stl s)"
+  unfolding shd_def stl_def stream_case_def stream_map_def stream.dtor_unfold
+  by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor)
+
+lemma stream_map_Stream[simp]: "stream_map f (x ## s) = f x ## stream_map f s"
+  by (metis stream.exhaust stream.sels stream_map_simps)
 
 theorem shd_stream_set: "shd s \<in> stream_set s"
-by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
-   (metis UnCI fsts_def insertI1 stream.dtor_set)
+  by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
+    (metis UnCI fsts_def insertI1 stream.dtor_set)
 
 theorem stl_stream_set: "y \<in> stream_set (stl s) \<Longrightarrow> y \<in> stream_set s"
-by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
-   (metis insertI1 set_mp snds_def stream.dtor_set_set_incl)
+  by (auto simp add: shd_def stl_def stream_case_def fsts_def snds_def split_beta)
+    (metis insertI1 set_mp snds_def stream.dtor_set_set_incl)
 
 (* only for the non-mutual case: *)
 theorem stream_set_induct1[consumes 1, case_names shd stl, induct set: "stream_set"]:
   assumes "y \<in> stream_set s" and "\<And>s. P (shd s) s"
   and "\<And>s y. \<lbrakk>y \<in> stream_set (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s"
   shows "P y s"
-using assms stream_set_induct by blast
+  using assms stream_set_induct by blast
 (* end TODO *)
 
 
 subsection {* prepend list to stream *}
 
 primrec shift :: "'a list \<Rightarrow> 'a stream \<Rightarrow> 'a stream" (infixr "@-" 65) where
   "shift [] s = s"
 | "shift (x # xs) s = x ## shift xs s"
 
 lemma shift_append[simp]: "(xs @ ys) @- s = xs @- ys @- s"
-by (induct xs) auto
+  by (induct xs) auto
 
 lemma shift_simps[simp]:
    "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
    "stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"
-by (induct xs) auto
-
-
-subsection {* recurring stream out of a list *}
-
-definition cycle :: "'a list \<Rightarrow> 'a stream" where
-  "cycle = stream_unfold hd (\<lambda>xs. tl xs @ [hd xs])"
-
-lemma cycle_simps[simp]:
-  "shd (cycle u) = hd u"
-  "stl (cycle u) = cycle (tl u @ [hd u])"
-by (auto simp: cycle_def)
-
-
-lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @- cycle u"
-proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>u. s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []"])
-  case (2 s1 s2)
-  then obtain u where "s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []" by blast
-  thus ?case using stream.unfold[of hd "\<lambda>xs. tl xs @ [hd xs]" u] by (auto simp: cycle_def)
-qed auto
-
-lemma cycle_Cons: "cycle (x # xs) = x ## cycle (xs @ [x])"
-proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>x xs. s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])"])
-  case (2 s1 s2)
-  then obtain x xs where "s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])" by blast
-  thus ?case
-    by (auto simp: cycle_def intro!: exI[of _ "hd (xs @ [x])"] exI[of _ "tl (xs @ [x])"] stream.unfold)
-qed auto
+  by (induct xs) auto
+
+lemma stream_set_shift[simp]: "stream_set (xs @- s) = set xs \<union> stream_set s"
+  by (induct xs) auto
+
+
+subsection {* set of streams with elements in some fixes set *}
 
 coinductive_set
   streams :: "'a set => 'a stream set"
   for A :: "'a set"
 where
   Stream[intro!, simp, no_atp]: "\<lbrakk>a \<in> A; s \<in> streams A\<rbrakk> \<Longrightarrow> a ## s \<in> streams A"
 
 lemma shift_streams: "\<lbrakk>w \<in> lists A; s \<in> streams A\<rbrakk> \<Longrightarrow> w @- s \<in> streams A"
-by (induct w) auto
+  by (induct w) auto
 
 lemma stream_set_streams:
   assumes "stream_set s \<subseteq> A"
   shows "s \<in> streams A"
 proof (coinduct rule: streams.coinduct[of "\<lambda>s'. \<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"])

   "flat \<equiv> stream_unfold (hd o shd) (\<lambda>s. if tl (shd s) = [] then stl s else tl (shd s) ## stl s)"
 
 lemma flat_simps[simp]:
   "shd (flat ws) = hd (shd ws)"
   "stl (flat ws) = flat (if tl (shd ws) = [] then stl ws else tl (shd ws) ## stl ws)"
-unfolding flat_def by auto
+  unfolding flat_def by auto
 
 lemma flat_Cons[simp]: "flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)"
-unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto
+  unfolding flat_def using stream.unfold[of "hd o shd" _ "(x # xs) ## ws"] by auto
 
 lemma flat_Stream[simp]: "xs \<noteq> [] \<Longrightarrow> flat (xs ## ws) = xs @- flat ws"
-by (induct xs) auto
+  by (induct xs) auto
 
 lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)"
-by (cases ws) auto
-
-
-subsection {* take, drop, nth for streams *}
+  by (cases ws) auto
+
+
+subsection {* nth, take, drop for streams *}
+
+primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!!" 100) where
+  "s !! 0 = shd s"
+| "s !! Suc n = stl s !! n"
+
+lemma snth_stream_map[simp]: "stream_map f s !! n = f (s !! n)"
+  by (induct n arbitrary: s) auto
+
+lemma shift_snth_less[simp]: "p < length xs \<Longrightarrow> (xs @- s) !! p = xs ! p"
+  by (induct p arbitrary: xs) (auto simp: hd_conv_nth nth_tl)
+
+lemma shift_snth_ge[simp]: "p \<ge> length xs \<Longrightarrow> (xs @- s) !! p = s !! (p - length xs)"
+  by (induct p arbitrary: xs) (auto simp: Suc_diff_eq_diff_pred)
+
+lemma snth_stream_set[simp]: "s !! n \<in> stream_set s"
+  by (induct n arbitrary: s) (auto intro: shd_stream_set stl_stream_set)
+
+lemma stream_set_range: "stream_set s = range (snth s)"
+proof (intro equalityI subsetI)
+  fix x assume "x \<in> stream_set s"
+  thus "x \<in> range (snth s)"
+  proof (induct s)
+    case (stl s x)
+    then obtain n where "x = stl s !! n" by auto
+    thus ?case by (auto intro: range_eqI[of _ _ "Suc n"])
+  qed (auto intro: range_eqI[of _ _ 0])
+qed auto
 
 primrec stake :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a list" where
   "stake 0 s = []"
 | "stake (Suc n) s = shd s # stake n (stl s)"
 
+lemma length_stake[simp]: "length (stake n s) = n"
+  by (induct n arbitrary: s) auto
+
+lemma stake_stream_map[simp]: "stake n (stream_map f s) = map f (stake n s)"
+  by (induct n arbitrary: s) auto
+
 primrec sdrop :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
   "sdrop 0 s = s"
 | "sdrop (Suc n) s = sdrop n (stl s)"
 
-primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!!" 100) where
-  "s !! 0 = shd s"
-| "s !! Suc n = stl s !! n"
+lemma sdrop_simps[simp]:
+  "shd (sdrop n s) = s !! n" "stl (sdrop n s) = sdrop (Suc n) s"
+  by (induct n arbitrary: s)  auto
+
+lemma sdrop_stream_map[simp]: "sdrop n (stream_map f s) = stream_map f (sdrop n s)"
+  by (induct n arbitrary: s) auto
 
 lemma stake_sdrop: "stake n s @- sdrop n s = s"
-by (induct n arbitrary: s) auto
-
-lemma stake_empty: "stake n s = [] \<longleftrightarrow> n = 0"
-by (cases n) auto
+  by (induct n arbitrary: s) auto
+
+lemma id_stake_snth_sdrop:
+  "s = stake i s @- s !! i ## sdrop (Suc i) s"
+  by (subst stake_sdrop[symmetric, of _ i]) (metis sdrop_simps stream.collapse)
+
+lemma stream_map_alt: "stream_map f s = s' \<longleftrightarrow> (\<forall>n. f (s !! n) = s' !! n)" (is "?L = ?R")
+proof
+  assume ?R
+  thus ?L 
+    by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>n. s1 = stream_map f (sdrop n s) \<and> s2 = sdrop n s'"])
+      (auto intro: exI[of _ 0] simp del: sdrop.simps(2))
+qed auto
+
+lemma stake_invert_Nil[iff]: "stake n s = [] \<longleftrightarrow> n = 0"
+  by (induct n) auto
 
 lemma sdrop_shift: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> sdrop n s = s'"
-by (induct n arbitrary: w s) auto
+  by (induct n arbitrary: w s) auto
 
 lemma stake_shift: "\<lbrakk>s = w @- s'; length w = n\<rbrakk> \<Longrightarrow> stake n s = w"
-by (induct n arbitrary: w s) auto
+  by (induct n arbitrary: w s) auto
 
 lemma stake_add[simp]: "stake m s @ stake n (sdrop m s) = stake (m + n) s"
-by (induct m arbitrary: s) auto
+  by (induct m arbitrary: s) auto
 
 lemma sdrop_add[simp]: "sdrop n (sdrop m s) = sdrop (m + n) s"
-by (induct m arbitrary: s) auto
+  by (induct m arbitrary: s) auto
+
+
+subsection {* unary predicates lifted to streams *}
+
+definition "stream_all P s = (\<forall>p. P (s !! p))"
+
+lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (stream_set s) P"
+  unfolding stream_all_def stream_set_range by auto
+
+lemma stream_all_shift[simp]: "stream_all P (xs @- s) = (list_all P xs \<and> stream_all P s)"
+  unfolding stream_all_iff list_all_iff by auto
+
+
+subsection {* recurring stream out of a list *}
+
+definition cycle :: "'a list \<Rightarrow> 'a stream" where
+  "cycle = stream_unfold hd (\<lambda>xs. tl xs @ [hd xs])"
+
+lemma cycle_simps[simp]:
+  "shd (cycle u) = hd u"
+  "stl (cycle u) = cycle (tl u @ [hd u])"
+  by (auto simp: cycle_def)
+
+lemma cycle_decomp: "u \<noteq> [] \<Longrightarrow> cycle u = u @- cycle u"
+proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>u. s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []"])
+  case (2 s1 s2)
+  then obtain u where "s1 = cycle u \<and> s2 = u @- cycle u \<and> u \<noteq> []" by blast
+  thus ?case using stream.unfold[of hd "\<lambda>xs. tl xs @ [hd xs]" u] by (auto simp: cycle_def)
+qed auto
+
+lemma cycle_Cons: "cycle (x # xs) = x ## cycle (xs @ [x])"
+proof (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>x xs. s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])"])
+  case (2 s1 s2)
+  then obtain x xs where "s1 = cycle (x # xs) \<and> s2 = x ## cycle (xs @ [x])" by blast
+  thus ?case
+    by (auto simp: cycle_def intro!: exI[of _ "hd (xs @ [x])"] exI[of _ "tl (xs @ [x])"] stream.unfold)
+qed auto
 
 lemma cycle_rotated: "\<lbrakk>v \<noteq> []; cycle u = v @- s\<rbrakk> \<Longrightarrow> cycle (tl u @ [hd u]) = tl v @- s"
-by (auto dest: arg_cong[of _ _ stl])
+  by (auto dest: arg_cong[of _ _ stl])
 
 lemma stake_append: "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s"
 proof (induct n arbitrary: u)
   case (Suc n) thus ?case by (cases u) auto
 qed auto
 
 lemma stake_cycle_le[simp]:
   assumes "u \<noteq> []" "n < length u"
   shows "stake n (cycle u) = take n u"
 using min_absorb2[OF less_imp_le_nat[OF assms(2)]]
-by (subst cycle_decomp[OF assms(1)], subst stake_append) auto
+  by (subst cycle_decomp[OF assms(1)], subst stake_append) auto
 
 lemma stake_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> stake (length u) (cycle u) = u"
-by (metis cycle_decomp stake_shift)
+  by (metis cycle_decomp stake_shift)
 
 lemma sdrop_cycle_eq[simp]: "u \<noteq> [] \<Longrightarrow> sdrop (length u) (cycle u) = cycle u"
-by (metis cycle_decomp sdrop_shift)
+  by (metis cycle_decomp sdrop_shift)
 
 lemma stake_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>
    stake n (cycle u) = concat (replicate (n div length u) u)"
-by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric])
+  by (induct "n div length u" arbitrary: n u) (auto simp: stake_add[symmetric])
 
 lemma sdrop_cycle_eq_mod_0[simp]: "\<lbrakk>u \<noteq> []; n mod length u = 0\<rbrakk> \<Longrightarrow>
    sdrop n (cycle u) = cycle u"
-by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric])
+  by (induct "n div length u" arbitrary: n u) (auto simp: sdrop_add[symmetric])
 
 lemma stake_cycle: "u \<noteq> [] \<Longrightarrow>
    stake n (cycle u) = concat (replicate (n div length u) u) @ take (n mod length u) u"
-by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto
+  by (subst mod_div_equality[of n "length u", symmetric], unfold stake_add[symmetric]) auto
 
 lemma sdrop_cycle: "u \<noteq> [] \<Longrightarrow> sdrop n (cycle u) = cycle (rotate (n mod length u) u)"
-by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric])
+  by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric])
+
+
+subsection {* stream repeating a single element *}
+
+definition "same x = stream_unfold (\<lambda>_. x) id ()"
+
+lemma same_simps[simp]: "shd (same x) = x" "stl (same x) = same x"
+  unfolding same_def by auto
+
+lemma same_unfold: "same x = Stream x (same x)"
+  by (metis same_simps stream.collapse)
+
+lemma snth_same[simp]: "same x !! n = x"
+  unfolding same_def by (induct n) auto
+
+lemma stake_same[simp]: "stake n (same x) = replicate n x"
+  unfolding same_def by (induct n) (auto simp: upt_rec)
+
+lemma sdrop_same[simp]: "sdrop n (same x) = same x"
+  unfolding same_def by (induct n) auto
+
+lemma shift_replicate_same[simp]: "replicate n x @- same x = same x"
+  by (metis sdrop_same stake_same stake_sdrop)
+
+lemma stream_all_same[simp]: "stream_all P (same x) \<longleftrightarrow> P x"
+  unfolding stream_all_def by auto
+
+lemma same_cycle: "same x = cycle [x]"
+  by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. s1 = same x \<and> s2 = cycle [x]"]) auto
+
+
+subsection {* stream of natural numbers *}
+
+definition "fromN n = stream_unfold id Suc n"
+
+lemma fromN_simps[simp]: "shd (fromN n) = n" "stl (fromN n) = fromN (Suc n)"
+  unfolding fromN_def by auto
+
+lemma snth_fromN[simp]: "fromN n !! m = n + m"
+  unfolding fromN_def by (induct m arbitrary: n) auto
+
+lemma stake_fromN[simp]: "stake m (fromN n) = [n ..< m + n]"
+  unfolding fromN_def by (induct m arbitrary: n) (auto simp: upt_rec)
+
+lemma sdrop_fromN[simp]: "sdrop m (fromN n) = fromN (n + m)"
+  unfolding fromN_def by (induct m arbitrary: n) auto
+
+abbreviation "nats \<equiv> fromN 0"
+
+
+subsection {* zip *}
+
+definition "szip s1 s2 =
+  stream_unfold (map_pair shd shd) (map_pair stl stl) (s1, s2)"
+
+lemma szip_simps[simp]:
+  "shd (szip s1 s2) = (shd s1, shd s2)" "stl (szip s1 s2) = szip (stl s1) (stl s2)"
+  unfolding szip_def by auto
+
+lemma snth_szip[simp]: "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
+  by (induct n arbitrary: s1 s2) auto
+
+
+subsection {* zip via function *}
+
+definition "stream_map2 f s1 s2 =
+  stream_unfold (\<lambda>(s1,s2). f (shd s1) (shd s2)) (map_pair stl stl) (s1, s2)"
+
+lemma stream_map2_simps[simp]:
+ "shd (stream_map2 f s1 s2) = f (shd s1) (shd s2)"
+ "stl (stream_map2 f s1 s2) = stream_map2 f (stl s1) (stl s2)"
+  unfolding stream_map2_def by auto
+
+lemma stream_map2_szip:
+  "stream_map2 f s1 s2 = stream_map (split f) (szip s1 s2)"
+  by (coinduct rule: stream.coinduct[of
+    "\<lambda>s1 s2. \<exists>s1' s2'. s1 = stream_map2 f s1' s2' \<and> s2 = stream_map (split f) (szip s1' s2')"])
+    fastforce+
 
 end