# HG changeset patch
# User blanchet
# Date 1360924594 -3600
# Node ID 43502299c9350167002f82ad8d1b0a3aee48bf5c
# Parent  4f0147ed8bcb35449e995576377589e4a46a9cab# Parent  cc7423ce677414aee2043166012450013fb64917
merge

diff -r 4f0147ed8bcb -r 43502299c935 src/HOL/BNF/Examples/Stream.thy
--- a/src/HOL/BNF/Examples/Stream.thy	Fri Feb 15 11:27:15 2013 +0100
+++ b/src/HOL/BNF/Examples/Stream.thy	Fri Feb 15 11:36:34 2013 +0100
@@ -16,25 +16,33 @@
 
 (* 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 *)
 
 
@@ -45,39 +53,18 @@
 | "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 *}
+  by (induct xs) auto
 
-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 stream_set_shift[simp]: "stream_set (xs @- s) = set xs \<union> stream_set s"
+  by (induct xs) auto
 
 
-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
+subsection {* set of streams with elements in some fixes set *}
 
 coinductive_set
   streams :: "'a set => 'a stream set"
@@ -86,7 +73,7 @@
   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"
@@ -110,52 +97,137 @@
 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
+  by (cases ws) auto
 
 
-subsection {* take, drop, nth for streams *}
+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
+  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 stake_empty: "stake n s = [] \<longleftrightarrow> n = 0"
-by (cases n) auto
+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)
@@ -166,27 +238,105 @@
   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
diff -r 4f0147ed8bcb -r 43502299c935 src/HOL/Codegenerator_Test/RBT_Set_Test.thy
--- a/src/HOL/Codegenerator_Test/RBT_Set_Test.thy	Fri Feb 15 11:27:15 2013 +0100
+++ b/src/HOL/Codegenerator_Test/RBT_Set_Test.thy	Fri Feb 15 11:36:34 2013 +0100
@@ -25,9 +25,17 @@
 lemma [code, code del]:
   "acc = acc" ..
 
-lemmas [code del] =
-  finite_set_code finite_coset_code 
-  equal_set_code
+lemma [code, code del]:
+  "Cardinality.card' = Cardinality.card'" ..
+
+lemma [code, code del]:
+  "Cardinality.finite' = Cardinality.finite'" ..
+
+lemma [code, code del]:
+  "Cardinality.subset' = Cardinality.subset'" ..
+
+lemma [code, code del]:
+  "Cardinality.eq_set = Cardinality.eq_set" ..
 
 (*
   If the code generation ends with an exception with the following message:
diff -r 4f0147ed8bcb -r 43502299c935 src/HOL/Library/Cardinality.thy
--- a/src/HOL/Library/Cardinality.thy	Fri Feb 15 11:27:15 2013 +0100
+++ b/src/HOL/Library/Cardinality.thy	Fri Feb 15 11:36:34 2013 +0100
@@ -388,65 +388,133 @@
 subsection {* Code setup for sets *}
 
 text {*
-  Implement operations @{term "finite"}, @{term "card"}, @{term "op \<subseteq>"}, and @{term "op ="} 
-  for sets using @{term "finite_UNIV"} and @{term "card_UNIV"}.
+  Implement @{term "CARD('a)"} via @{term card_UNIV} and provide
+  implementations for @{term "finite"}, @{term "card"}, @{term "op \<subseteq>"}, 
+  and @{term "op ="}if the calling context already provides @{class finite_UNIV}
+  and @{class card_UNIV} instances. If we implemented the latter
+  always via @{term card_UNIV}, we would require instances of essentially all 
+  element types, i.e., a lot of instantiation proofs and -- at run time --
+  possibly slow dictionary constructions.
 *}
 
+definition card_UNIV' :: "'a card_UNIV"
+where [code del]: "card_UNIV' = Phantom('a) CARD('a)"
+
+lemma CARD_code [code_unfold]:
+  "CARD('a) = of_phantom (card_UNIV' :: 'a card_UNIV)"
+by(simp add: card_UNIV'_def)
+
+lemma card_UNIV'_code [code]:
+  "card_UNIV' = card_UNIV"
+by(simp add: card_UNIV card_UNIV'_def)
+
+hide_const (open) card_UNIV'
+
 lemma card_Compl:
   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
 
-lemma card_coset_code [code]:
-  fixes xs :: "'a :: card_UNIV list" 
-  shows "card (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
-by(simp add: List.card_set card_Compl card_UNIV)
-
-lemma [code, code del]: "finite = finite" ..
+context fixes xs :: "'a :: finite_UNIV list"
+begin
 
-lemma [code]:
-  fixes xs :: "'a :: card_UNIV list" 
-  shows finite_set_code:
-  "finite (set xs) = True" 
-  and finite_coset_code:
-  "finite (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
+definition finite' :: "'a set \<Rightarrow> bool"
+where [simp, code del, code_abbrev]: "finite' = finite"
+
+lemma finite'_code [code]:
+  "finite' (set xs) \<longleftrightarrow> True"
+  "finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
 by(simp_all add: card_gt_0_iff finite_UNIV)
 
-lemma coset_subset_code [code]:
-  fixes xs :: "'a list" shows
-  "List.coset xs \<subseteq> set ys \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card (set (xs @ ys)) = n)"
+end
+
+context fixes xs :: "'a :: card_UNIV list"
+begin
+
+definition card' :: "'a set \<Rightarrow> nat" 
+where [simp, code del, code_abbrev]: "card' = card"
+ 
+lemma card'_code [code]:
+  "card' (set xs) = length (remdups xs)"
+  "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
+by(simp_all add: List.card_set card_Compl card_UNIV)
+
+
+definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
+where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"
+
+lemma subset'_code [code]:
+  "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
+  "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
+  "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
   (metis finite_compl finite_set rev_finite_subset)
 
-lemma equal_set_code [code]:
-  fixes xs ys :: "'a :: equal list"
+definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
+where [simp, code del, code_abbrev]: "eq_set = op ="
+
+lemma eq_set_code [code]:
+  fixes ys
   defines "rhs \<equiv> 
   let n = CARD('a)
   in if n = 0 then False else 
         let xs' = remdups xs; ys' = remdups ys 
         in length xs' + length ys' = n \<and> (\<forall>x \<in> set xs'. x \<notin> set ys') \<and> (\<forall>y \<in> set ys'. y \<notin> set xs')"
-  shows "equal_class.equal (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
-  and "equal_class.equal (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
-  and "equal_class.equal (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3)
-  and "equal_class.equal (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4)
+  shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
+  and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
+  and "eq_set (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3)
+  and "eq_set (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4)
 proof -
   show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs")
   proof
     assume ?lhs thus ?rhs
-      by(auto simp add: equal_eq rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
+      by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
   next
     assume ?rhs
     moreover have "\<lbrakk> \<forall>y\<in>set xs. y \<notin> set ys; \<forall>x\<in>set ys. x \<notin> set xs \<rbrakk> \<Longrightarrow> set xs \<inter> set ys = {}" by blast
     ultimately show ?lhs
-      by(auto simp add: equal_eq rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
+      by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
   qed
-  thus ?thesis2 unfolding equal_eq by blast
-  show ?thesis3 ?thesis4 unfolding equal_eq List.coset_def by blast+
+  thus ?thesis2 unfolding eq_set_def by blast
+  show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+
 qed
 
-notepad begin (* test code setup *)
-have "List.coset [True] = set [False] \<and> List.coset [] \<subseteq> List.set [True, False] \<and> finite (List.coset [True])"
+end
+
+text {* 
+  Provide more informative exceptions than Match for non-rewritten cases.
+  If generated code raises one these exceptions, then a code equation calls
+  the mentioned operator for an element type that is not an instance of
+  @{class card_UNIV} and is therefore not implemented via @{term card_UNIV}.
+  Constrain the element type with sort @{class card_UNIV} to change this.
+*}
+
+definition card_coset_requires_card_UNIV :: "'a list \<Rightarrow> nat"
+where [code del, simp]: "card_coset_requires_card_UNIV xs = card (List.coset xs)"
+
+code_abort card_coset_requires_card_UNIV
+
+lemma card_coset_error [code]:
+  "card (List.coset xs) = card_coset_requires_card_UNIV xs"
+by(simp)
+
+definition coset_subseteq_set_requires_card_UNIV :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+where [code del, simp]: "coset_subseteq_set_requires_card_UNIV xs ys \<longleftrightarrow> List.coset xs \<subseteq> set ys"
+
+code_abort coset_subseteq_set_requires_card_UNIV
+
+lemma coset_subseteq_set_code [code]:
+  "List.coset xs \<subseteq> set ys \<longleftrightarrow> 
+  (if xs = [] \<and> ys = [] then False else coset_subseteq_set_requires_card_UNIV xs ys)"
+by simp
+
+notepad begin -- "test code setup"
+have "List.coset [True] = set [False] \<and> 
+      List.coset [] \<subseteq> List.set [True, False] \<and> 
+      finite (List.coset [True])"
   by eval
 end
 
+hide_const (open) card' finite' subset' eq_set
+
 end