--- a/src/HOL/BNF/Examples/Koenig.thy Thu Apr 25 09:25:50 2013 +0200
+++ b/src/HOL/BNF/Examples/Koenig.thy Thu Apr 25 10:31:10 2013 +0200
@@ -110,8 +110,8 @@
(* some more stream theorems *)
-lemma stream_map[simp]: "stream_map f = stream_dtor_unfold (f o shd \<odot> stl)"
-unfolding stream_map_def pair_fun_def shd_def'[abs_def] stl_def'[abs_def]
+lemma stream_map[simp]: "smap f = stream_dtor_unfold (f o shd \<odot> stl)"
+unfolding smap_def pair_fun_def shd_def'[abs_def] stl_def'[abs_def]
map_pair_def o_def prod_case_beta by simp
definition plus :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<oplus>" 66) where
@@ -119,7 +119,7 @@
stream_dtor_unfold ((%(xs, ys). shd xs + shd ys) \<odot> (%(xs, ys). (stl xs, stl ys))) (xs, ys)"
definition scalar :: "nat \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<cdot>" 68) where
- [simp]: "scalar n = stream_map (\<lambda>x. n * x)"
+ [simp]: "scalar n = smap (\<lambda>x. n * x)"
definition ones :: "nat stream" where [simp]: "ones = stream_dtor_unfold ((%x. 1) \<odot> id) ()"
definition twos :: "nat stream" where [simp]: "twos = stream_dtor_unfold ((%x. 2) \<odot> id) ()"--- a/src/HOL/BNF/Examples/Stream.thy Thu Apr 25 09:25:50 2013 +0200
+++ b/src/HOL/BNF/Examples/Stream.thy Thu Apr 25 10:31:10 2013 +0200
@@ -12,7 +12,8 @@
imports "../BNF"
begin
-codata 'a stream = Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65)
+codata (sset: 'a) stream (map: smap rel: stream_all2) =
+ Stream (shd: 'a) (stl: "'a stream") (infixr "##" 65)
declaration {*
Nitpick_HOL.register_codatatype
@@ -23,7 +24,7 @@
fixes x :: 'stream_element_type
begin
- lemma "stream_set s = {}"
+ lemma "sset s = {}"
nitpick
oops
@@ -45,7 +46,7 @@
(*for code generation only*)
definition smember :: "'a \<Rightarrow> 'a stream \<Rightarrow> bool" where
- [code_abbrev]: "smember x s \<longleftrightarrow> x \<in> stream_set s"
+ [code_abbrev]: "smember x s \<longleftrightarrow> x \<in> sset s"
lemma smember_code[code, simp]: "smember x (Stream y s) = (if x = y then True else smember x s)"
unfolding smember_def by auto
@@ -53,33 +54,33 @@
hide_const (open) smember
(* 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"
+theorem sset_induct:
+ "\<lbrakk>\<And>s. P (shd s) s; \<And>s y. \<lbrakk>y \<in> sset (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s\<rbrakk> \<Longrightarrow>
+ \<forall>y \<in> sset 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
+lemma smap_simps[simp]:
+ "shd (smap f s) = f (shd s)" "stl (smap f s) = smap f (stl s)"
+ unfolding shd_def stl_def stream_case_def smap_def stream.dtor_unfold
by (case_tac [!] s) (auto simp: Stream_def stream.dtor_ctor)
declare stream.map[code]
-theorem shd_stream_set: "shd s \<in> stream_set s"
+theorem shd_sset: "shd s \<in> sset 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)
-theorem stl_stream_set: "y \<in> stream_set (stl s) \<Longrightarrow> y \<in> stream_set s"
+theorem stl_sset: "y \<in> sset (stl s) \<Longrightarrow> y \<in> sset 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)
(* 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"
+theorem sset_induct1[consumes 1, case_names shd stl, induct set: "sset"]:
+ 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"
shows "P y s"
- using assms stream_set_induct by blast
+ using assms sset_induct by blast
(* end TODO *)
@@ -89,7 +90,7 @@
"shift [] s = s"
| "shift (x # xs) s = x ## shift xs s"
-lemma stream_map_shift[simp]: "stream_map f (xs @- s) = map f xs @- stream_map f s"
+lemma smap_shift[simp]: "smap f (xs @- s) = map f xs @- smap f s"
by (induct xs) auto
lemma shift_append[simp]: "(xs @ ys) @- s = xs @- ys @- s"
@@ -100,7 +101,7 @@
"stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"
by (induct xs) auto
-lemma stream_set_shift[simp]: "stream_set (xs @- s) = set xs \<union> stream_set s"
+lemma sset_shift[simp]: "sset (xs @- s) = set xs \<union> sset s"
by (induct xs) auto
lemma shift_left_inj[simp]: "xs @- s1 = xs @- s2 \<longleftrightarrow> s1 = s2"
@@ -118,16 +119,16 @@
lemma shift_streams: "\<lbrakk>w \<in> lists A; s \<in> streams A\<rbrakk> \<Longrightarrow> w @- s \<in> streams A"
by (induct w) auto
-lemma stream_set_streams:
- assumes "stream_set s \<subseteq> A"
+lemma sset_streams:
+ assumes "sset 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"])
+proof (coinduct rule: streams.coinduct[of "\<lambda>s'. \<exists>a s. s' = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A"])
case streams from assms show ?case by (cases s) auto
next
- fix s' assume "\<exists>a s. s' = a ## s \<and> a \<in> A \<and> stream_set s \<subseteq> A"
+ fix s' assume "\<exists>a s. s' = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A"
then guess a s by (elim exE)
with assms show "\<exists>a l. s' = a ## l \<and>
- 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)"
+ a \<in> A \<and> ((\<exists>a s. l = a ## s \<and> a \<in> A \<and> sset s \<subseteq> A) \<or> l \<in> streams A)"
by (cases s) auto
qed
@@ -138,7 +139,7 @@
"s !! 0 = shd s"
| "s !! Suc n = stl s !! n"
-lemma snth_stream_map[simp]: "stream_map f s !! n = f (s !! n)"
+lemma snth_smap[simp]: "smap 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"
@@ -147,12 +148,12 @@
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 snth_sset[simp]: "s !! n \<in> sset s"
+ by (induct n arbitrary: s) (auto intro: shd_sset stl_sset)
-lemma stream_set_range: "stream_set s = range (snth s)"
+lemma sset_range: "sset s = range (snth s)"
proof (intro equalityI subsetI)
- fix x assume "x \<in> stream_set s"
+ fix x assume "x \<in> sset s"
thus "x \<in> range (snth s)"
proof (induct s)
case (stl s x)
@@ -168,7 +169,7 @@
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)"
+lemma stake_smap[simp]: "stake n (smap f s) = map f (stake n s)"
by (induct n arbitrary: s) auto
primrec sdrop :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
@@ -179,7 +180,7 @@
"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)"
+lemma sdrop_smap[simp]: "sdrop n (smap f s) = smap f (sdrop n s)"
by (induct n arbitrary: s) auto
lemma sdrop_stl: "sdrop n (stl s) = stl (sdrop n s)"
@@ -192,11 +193,11 @@
"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")
+lemma smap_alt: "smap 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'"])
+ by (coinduct rule: stream.coinduct[of "\<lambda>s1 s2. \<exists>n. s1 = smap f (sdrop n s) \<and> s2 = sdrop n s'"])
(auto intro: exI[of _ 0] simp del: sdrop.simps(2))
qed auto
@@ -243,8 +244,8 @@
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_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P"
+ unfolding stream_all_def sset_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
@@ -359,16 +360,16 @@
lemma sdrop_fromN[simp]: "sdrop m (fromN n) = fromN (n + m)"
unfolding fromN_def by (induct m arbitrary: n) auto
-lemma stream_set_fromN[simp]: "stream_set (fromN n) = {n ..}" (is "?L = ?R")
+lemma sset_fromN[simp]: "sset (fromN n) = {n ..}" (is "?L = ?R")
proof safe
fix m assume "m : ?L"
moreover
- { fix s assume "m \<in> stream_set s" "\<exists>n'\<ge>n. s = fromN n'"
- hence "n \<le> m" by (induct arbitrary: n rule: stream_set_induct1) fastforce+
+ { fix s assume "m \<in> sset s" "\<exists>n'\<ge>n. s = fromN n'"
+ hence "n \<le> m" by (induct arbitrary: n rule: sset_induct1) fastforce+
}
ultimately show "n \<le> m" by blast
next
- fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_stream_set)
+ fix m assume "n \<le> m" thus "m \<in> ?L" by (metis le_iff_add snth_fromN snth_sset)
qed
abbreviation "nats \<equiv> fromN 0"
@@ -393,15 +394,15 @@
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)"
by (cases ws) auto
-lemma flat_snth: "\<forall>xs \<in> stream_set s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then
+lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then
shd s ! n else flat (stl s) !! (n - length (shd s)))"
- by (metis flat_unfold not_less shd_stream_set shift_snth_ge shift_snth_less)
+ by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less)
-lemma stream_set_flat[simp]: "\<forall>xs \<in> stream_set s. xs \<noteq> [] \<Longrightarrow>
- stream_set (flat s) = (\<Union>xs \<in> stream_set s. set xs)" (is "?P \<Longrightarrow> ?L = ?R")
+lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow>
+ sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R")
proof safe
fix x assume ?P "x : ?L"
- then obtain m where "x = flat s !! m" by (metis image_iff stream_set_range)
+ then obtain m where "x = flat s !! m" by (metis image_iff sset_range)
with `?P` obtain n m' where "x = s !! n ! m'" "m' < length (s !! n)"
proof (atomize_elim, induct m arbitrary: s rule: less_induct)
case (less y)
@@ -416,60 +417,60 @@
moreover with False have "y > 0" by (cases y) simp_all
ultimately have "y - length (shd s) < y" by simp
}
- moreover have "\<forall>xs \<in> stream_set (stl s). xs \<noteq> []" using less(2) by (cases s) auto
+ moreover have "\<forall>xs \<in> sset (stl s). xs \<noteq> []" using less(2) by (cases s) auto
ultimately have "\<exists>n m'. x = stl s !! n ! m' \<and> m' < length (stl s !! n)" by (intro less(1)) auto
thus ?thesis by (metis snth.simps(2))
qed
qed
- thus "x \<in> ?R" by (auto simp: stream_set_range dest!: nth_mem)
+ thus "x \<in> ?R" by (auto simp: sset_range dest!: nth_mem)
next
- fix x xs assume "xs \<in> stream_set s" ?P "x \<in> set xs" thus "x \<in> ?L"
- by (induct rule: stream_set_induct1)
- (metis UnI1 flat_unfold shift.simps(1) stream_set_shift,
- metis UnI2 flat_unfold shd_stream_set stl_stream_set stream_set_shift)
+ fix x xs assume "xs \<in> sset s" ?P "x \<in> set xs" thus "x \<in> ?L"
+ by (induct rule: sset_induct1)
+ (metis UnI1 flat_unfold shift.simps(1) sset_shift,
+ metis UnI2 flat_unfold shd_sset stl_sset sset_shift)
qed
subsection {* merge a stream of streams *}
definition smerge :: "'a stream stream \<Rightarrow> 'a stream" where
- "smerge ss = flat (stream_map (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)"
+ "smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)"
lemma stake_nth[simp]: "m < n \<Longrightarrow> stake n s ! m = s !! m"
by (induct n arbitrary: s m) (auto simp: nth_Cons', metis Suc_pred snth.simps(2))
-lemma snth_stream_set_smerge: "ss !! n !! m \<in> stream_set (smerge ss)"
+lemma snth_sset_smerge: "ss !! n !! m \<in> sset (smerge ss)"
proof (cases "n \<le> m")
case False thus ?thesis unfolding smerge_def
- by (subst stream_set_flat)
+ by (subst sset_flat)
(auto simp: stream.set_map' in_set_conv_nth simp del: stake.simps
intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp])
next
case True thus ?thesis unfolding smerge_def
- by (subst stream_set_flat)
+ by (subst sset_flat)
(auto simp: stream.set_map' in_set_conv_nth image_iff simp del: stake.simps snth.simps
intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp])
qed
-lemma stream_set_smerge: "stream_set (smerge ss) = UNION (stream_set ss) stream_set"
+lemma sset_smerge: "sset (smerge ss) = UNION (sset ss) sset"
proof safe
- fix x assume "x \<in> stream_set (smerge ss)"
- thus "x \<in> UNION (stream_set ss) stream_set"
- unfolding smerge_def by (subst (asm) stream_set_flat)
- (auto simp: stream.set_map' in_set_conv_nth stream_set_range simp del: stake.simps, fast+)
+ fix x assume "x \<in> sset (smerge ss)"
+ thus "x \<in> UNION (sset ss) sset"
+ unfolding smerge_def by (subst (asm) sset_flat)
+ (auto simp: stream.set_map' in_set_conv_nth sset_range simp del: stake.simps, fast+)
next
- fix s x assume "s \<in> stream_set ss" "x \<in> stream_set s"
- thus "x \<in> stream_set (smerge ss)" using snth_stream_set_smerge by (auto simp: stream_set_range)
+ fix s x assume "s \<in> sset ss" "x \<in> sset s"
+ thus "x \<in> sset (smerge ss)" using snth_sset_smerge by (auto simp: sset_range)
qed
subsection {* product of two streams *}
definition sproduct :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> ('a \<times> 'b) stream" where
- "sproduct s1 s2 = smerge (stream_map (\<lambda>x. stream_map (Pair x) s2) s1)"
+ "sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)"
-lemma stream_set_sproduct: "stream_set (sproduct s1 s2) = stream_set s1 \<times> stream_set s2"
- unfolding sproduct_def stream_set_smerge by (auto simp: stream.set_map')
+lemma sset_sproduct: "sset (sproduct s1 s2) = sset s1 \<times> sset s2"
+ unfolding sproduct_def sset_smerge by (auto simp: stream.set_map')
subsection {* interleave two streams *}
@@ -492,24 +493,24 @@
by (induct n arbitrary: s1 s2)
(auto dest: even_nat_Suc_div_2 odd_nat_plus_one_div_two[folded nat_2])
-lemma stream_set_sinterleave: "stream_set (sinterleave s1 s2) = stream_set s1 \<union> stream_set s2"
+lemma sset_sinterleave: "sset (sinterleave s1 s2) = sset s1 \<union> sset s2"
proof (intro equalityI subsetI)
- fix x assume "x \<in> stream_set (sinterleave s1 s2)"
- then obtain n where "x = sinterleave s1 s2 !! n" unfolding stream_set_range by blast
- thus "x \<in> stream_set s1 \<union> stream_set s2" by (cases "even n") auto
+ fix x assume "x \<in> sset (sinterleave s1 s2)"
+ then obtain n where "x = sinterleave s1 s2 !! n" unfolding sset_range by blast
+ thus "x \<in> sset s1 \<union> sset s2" by (cases "even n") auto
next
- fix x assume "x \<in> stream_set s1 \<union> stream_set s2"
- thus "x \<in> stream_set (sinterleave s1 s2)"
+ fix x assume "x \<in> sset s1 \<union> sset s2"
+ thus "x \<in> sset (sinterleave s1 s2)"
proof
- assume "x \<in> stream_set s1"
- then obtain n where "x = s1 !! n" unfolding stream_set_range by blast
+ assume "x \<in> sset s1"
+ then obtain n where "x = s1 !! n" unfolding sset_range by blast
hence "sinterleave s1 s2 !! (2 * n) = x" by simp
- thus ?thesis unfolding stream_set_range by blast
+ thus ?thesis unfolding sset_range by blast
next
- assume "x \<in> stream_set s2"
- then obtain n where "x = s2 !! n" unfolding stream_set_range by blast
+ assume "x \<in> sset s2"
+ then obtain n where "x = s2 !! n" unfolding sset_range by blast
hence "sinterleave s1 s2 !! (2 * n + 1) = x" by simp
- thus ?thesis unfolding stream_set_range by blast
+ thus ?thesis unfolding sset_range by blast
qed
qed
@@ -532,22 +533,22 @@
subsection {* zip via function *}
-definition "stream_map2 f s1 s2 =
+definition "smap2 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 smap2_simps[simp]:
+ "shd (smap2 f s1 s2) = f (shd s1) (shd s2)"
+ "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)"
+ unfolding smap2_def by auto
-lemma stream_map2_unfold[code]:
- "stream_map2 f (Stream a s1) (Stream b s2) = Stream (f a b) (stream_map2 f s1 s2)"
- unfolding stream_map2_def by (subst stream.unfold) simp
+lemma smap2_unfold[code]:
+ "smap2 f (Stream a s1) (Stream b s2) = Stream (f a b) (smap2 f s1 s2)"
+ unfolding smap2_def by (subst stream.unfold) simp
-lemma stream_map2_szip:
- "stream_map2 f s1 s2 = stream_map (split f) (szip s1 s2)"
+lemma smap2_szip:
+ "smap2 f s1 s2 = smap (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')"])
+ "\<lambda>s1 s2. \<exists>s1' s2'. s1 = smap2 f s1' s2' \<and> s2 = smap (split f) (szip s1' s2')"])
fastforce+
@@ -574,7 +575,7 @@
lemma stake_siterate[simp]: "stake n (siterate f x) = map (\<lambda>n. (f^^n) x) [0 ..< n]"
by (induct n arbitrary: x) (auto simp del: stake.simps(2) simp: stake_Suc)
-lemma stream_set_siterate: "stream_set (siterate f x) = {(f^^n) x | n. True}"
- by (auto simp: stream_set_range)
+lemma sset_siterate: "sset (siterate f x) = {(f^^n) x | n. True}"
+ by (auto simp: sset_range)
end