diff --git a/thys/Transition_Systems_and_Automata/Automata/Deterministic.thy b/thys/Transition_Systems_and_Automata/Automata/Deterministic.thy
--- a/thys/Transition_Systems_and_Automata/Automata/Deterministic.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/Deterministic.thy
@@ -1,564 +1,560 @@
 theory Deterministic
 imports
   "../Transition_Systems/Transition_System"
   "../Transition_Systems/Transition_System_Extra"
   "../Transition_Systems/Transition_System_Construction"
   "../Basic/Degeneralization"
 begin
 
   type_synonym ('label, 'state) trans = "'label \<Rightarrow> 'state \<Rightarrow> 'state"
 
   (* TODO: is there a way to be less verbose in these locales? do we need to specify all the types? *)
   locale automaton =
     fixes automaton :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition \<Rightarrow> 'automaton"
     fixes alphabet :: "'automaton \<Rightarrow> 'label set"
     fixes initial :: "'automaton \<Rightarrow> 'state"
     fixes transition :: "'automaton \<Rightarrow> ('label, 'state) trans"
     fixes condition :: "'automaton \<Rightarrow> 'condition"
     assumes automaton[simp]: "automaton (alphabet A) (initial A) (transition A) (condition A) = A"
     assumes alphabet[simp]: "alphabet (automaton a i t c) = a"
     assumes initial[simp]: "initial (automaton a i t c) = i"
     assumes transition[simp]: "transition (automaton a i t c) = t"
     assumes condition[simp]: "condition (automaton a i t c) = c"
   begin
 
     (* TODO: is there a way to use defines without renaming the constants? *)
     sublocale transition_system_initial
       "transition A" "\<lambda> a p. a \<in> alphabet A" "\<lambda> p. p = initial A"
       for A
       defines path' = path and run' = run and reachable' = reachable and nodes' = nodes
       by this
 
-    (* TODO: maybe this should be stated in terms of the lists and streams constants (adjust language_alphabet proof) *)
-    lemma path_alt_def: "path A w p \<longleftrightarrow> set w \<subseteq> alphabet A"
-    unfolding lists_iff_set[symmetric]
+    lemma path_alt_def: "path A w p \<longleftrightarrow> w \<in> lists (alphabet A)"
     proof
       show "w \<in> lists (alphabet A)" if "path A w p" using that by (induct arbitrary: p) (auto)
       show "path A w p" if "w \<in> lists (alphabet A)" using that by (induct arbitrary: p) (auto)
     qed
-    lemma run_alt_def: "run A w p \<longleftrightarrow> sset w \<subseteq> alphabet A"
-    unfolding streams_iff_sset[symmetric]
+    lemma run_alt_def: "run A w p \<longleftrightarrow> w \<in> streams (alphabet A)"
     proof
       show "w \<in> streams (alphabet A)" if "run A w p"
         using that by (coinduction arbitrary: w p) (force elim: run.cases)
       show "run A w p" if "w \<in> streams (alphabet A)"
         using that by (coinduction arbitrary: w p) (force elim: streams.cases)
     qed
 
   end
 
   (* TODO: create analogous locale for DFAs (automaton_path) *)
   locale automaton_run =
     automaton automaton alphabet initial transition condition
     for automaton :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition \<Rightarrow> 'automaton"
     and alphabet :: "'automaton \<Rightarrow> 'label set"
     and initial :: "'automaton \<Rightarrow> 'state"
     and transition :: "'automaton \<Rightarrow> ('label, 'state) trans"
     and condition :: "'automaton \<Rightarrow> 'condition"
     +
     fixes test :: "'condition \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
   begin
 
     definition language :: "'automaton \<Rightarrow> 'label stream set" where
       "language A \<equiv> {w. run A w (initial A) \<and> test (condition A) w (trace A w (initial A)) (initial A)}"
   
     lemma language[intro]:
       assumes "run A w (initial A)" "test (condition A) w (trace A w (initial A)) (initial A)"
       shows "w \<in> language A"
       using assms unfolding language_def by auto
     lemma language_elim[elim]:
       assumes "w \<in> language A"
       obtains "run A w (initial A)" "test (condition A) w (trace A w (initial A)) (initial A)"
       using assms unfolding language_def by auto
   
-    lemma language_alphabet: "language A \<subseteq> streams (alphabet A)"
-      unfolding language_def run_alt_def using sset_streams by auto
+    lemma language_alphabet: "language A \<subseteq> streams (alphabet A)" using run_alt_def by auto
 
   end
 
   locale automaton_degeneralization =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'item pred gen \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'item pred gen"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'item_degen pred \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state degen"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state degen) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'item_degen pred"
     +
     fixes item :: "'state \<times> 'label \<times> 'state \<Rightarrow> 'item"
     fixes translate :: "'item_degen \<Rightarrow> 'item degen"
   begin
 
     definition degeneralize :: "'automaton\<^sub>1 \<Rightarrow> 'automaton\<^sub>2" where
       "degeneralize A \<equiv> automaton\<^sub>2
         (alphabet\<^sub>1 A)
         (initial\<^sub>1 A, 0)
         (\<lambda> a (p, k). (transition\<^sub>1 A a p, count (condition\<^sub>1 A) (item (p, a, transition\<^sub>1 A a p)) k))
         (degen (condition\<^sub>1 A) \<circ> translate)"
 
     lemma degeneralize_simps[simp]:
       "alphabet\<^sub>2 (degeneralize A) = alphabet\<^sub>1 A"
       "initial\<^sub>2 (degeneralize A) = (initial\<^sub>1 A, 0)"
       "transition\<^sub>2 (degeneralize A) a (p, k) =
         (transition\<^sub>1 A a p, count (condition\<^sub>1 A) (item (p, a, transition\<^sub>1 A a p)) k)"
       "condition\<^sub>2 (degeneralize A) = degen (condition\<^sub>1 A) \<circ> translate"
       unfolding degeneralize_def by auto
 
     lemma degeneralize_target[simp]: "b.target (degeneralize A) w (p, k) =
       (a.target A w p, fold (count (condition\<^sub>1 A) \<circ> item) (p # a.states A w p || w || a.states A w p) k)"
       by (induct w arbitrary: p k) (auto)
     lemma degeneralize_states[simp]: "b.states (degeneralize A) w (p, k) =
       a.states A w p || scan (count (condition\<^sub>1 A) \<circ> item) (p # a.states A w p || w || a.states A w p) k"
       by (induct w arbitrary: p k) (auto)
     lemma degeneralize_trace[simp]: "b.trace (degeneralize A) w (p, k) =
       a.trace A w p ||| sscan (count (condition\<^sub>1 A) \<circ> item) (p ## a.trace A w p ||| w ||| a.trace A w p) k"
       by (coinduction arbitrary: w p k) (auto, metis sscan.code)
 
     lemma degeneralize_path[iff]: "b.path (degeneralize A) w (p, k) \<longleftrightarrow> a.path A w p"
       unfolding a.path_alt_def b.path_alt_def by simp
     lemma degeneralize_run[iff]: "b.run (degeneralize A) w (p, k) \<longleftrightarrow> a.run A w p"
       unfolding a.run_alt_def b.run_alt_def by simp
 
     lemma degeneralize_reachable_fst[simp]: "fst ` b.reachable (degeneralize A) (p, k) = a.reachable A p"
       unfolding a.reachable_alt_def b.reachable_alt_def image_def by simp
     lemma degeneralize_reachable_snd_empty[simp]:
       assumes "condition\<^sub>1 A = []"
       shows "snd ` b.reachable (degeneralize A) (p, k) = {k}"
     proof -
       have "snd ql = k" if "ql \<in> b.reachable (degeneralize A) (p, k)" for ql
         using that assms by induct auto
       then show ?thesis by auto
     qed
     lemma degeneralize_reachable_empty[simp]:
       assumes "condition\<^sub>1 A = []"
       shows "b.reachable (degeneralize A) (p, k) = a.reachable A p \<times> {k}"
       using degeneralize_reachable_fst degeneralize_reachable_snd_empty assms
       by (metis prod_singleton(2))
     lemma degeneralize_reachable_snd:
       "snd ` b.reachable (degeneralize A) (p, k) \<subseteq> insert k {0 ..< length (condition\<^sub>1 A)}"
       by (cases "condition\<^sub>1 A = []") (auto)
     lemma degeneralize_reachable:
       "b.reachable (degeneralize A) (p, k) \<subseteq> a.reachable A p \<times> insert k {0 ..< length (condition\<^sub>1 A)}"
       by (cases "condition\<^sub>1 A = []") (auto 0 3)
 
     lemma degeneralize_nodes_fst[simp]: "fst ` b.nodes (degeneralize A) = a.nodes A"
       unfolding a.nodes_alt_def b.nodes_alt_def by simp
     lemma degeneralize_nodes_snd_empty:
       assumes "condition\<^sub>1 A = []"
       shows "snd ` b.nodes (degeneralize A) = {0}"
       using assms unfolding b.nodes_alt_def by auto
     lemma degeneralize_nodes_empty:
       assumes "condition\<^sub>1 A = []"
       shows "b.nodes (degeneralize A) = a.nodes A \<times> {0}"
       using assms unfolding b.nodes_alt_def by auto
     lemma degeneralize_nodes_snd:
       "snd ` b.nodes (degeneralize A) \<subseteq> insert 0 {0 ..< length (condition\<^sub>1 A)}"
       using degeneralize_reachable_snd unfolding b.nodes_alt_def by auto
     lemma degeneralize_nodes:
       "b.nodes (degeneralize A) \<subseteq> a.nodes A \<times> insert 0 {0 ..< length (condition\<^sub>1 A)}"
       using degeneralize_reachable unfolding a.nodes_alt_def b.nodes_alt_def by simp
   
     lemma degeneralize_nodes_finite[iff]: "finite (b.nodes (degeneralize A)) \<longleftrightarrow> finite (a.nodes A)"
     proof
       show "finite (a.nodes A)" if "finite (b.nodes (degeneralize A))"
         using that by (auto simp flip: degeneralize_nodes_fst)
       show "finite (b.nodes (degeneralize A))" if "finite (a.nodes A)"
         using finite_subset degeneralize_nodes that by blast
     qed
     lemma degeneralize_nodes_card: "card (b.nodes (degeneralize A)) \<le>
       max 1 (length (condition\<^sub>1 A)) * card (a.nodes A)"
     proof (cases "finite (a.nodes A)")
       case True
       have "card (b.nodes (degeneralize A)) \<le> card (a.nodes A \<times> insert 0 {0 ..< length (condition\<^sub>1 A)})"
         using degeneralize_nodes True by (blast intro: card_mono)
       also have "\<dots> = card (insert 0 {0 ..< length (condition\<^sub>1 A)}) * card (a.nodes A)"
         unfolding card_cartesian_product by simp
       also have "card (insert 0 {0 ..< length (condition\<^sub>1 A)}) = max 1 (length (condition\<^sub>1 A))"
         by (simp add: card_insert_if Suc_leI max_absorb2)
       finally show ?thesis by this
     next
       case False
       then have "card (a.nodes A) = 0" "card (b.nodes (degeneralize A)) = 0" by auto
       then show ?thesis by simp
     qed
 
   end
 
   locale automaton_degeneralization_run =
     automaton_degeneralization
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       item translate +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'item pred gen \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'item pred gen"
     and test\<^sub>1 :: "'item pred gen \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'item_degen pred \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state degen"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state degen) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'item_degen pred"
     and test\<^sub>2 :: "'item_degen pred \<Rightarrow> 'label stream \<Rightarrow> 'state degen stream \<Rightarrow> 'state degen \<Rightarrow> bool"
     and item :: "'state \<times> 'label \<times> 'state \<Rightarrow> 'item"
     and translate :: "'item_degen \<Rightarrow> 'item degen"
     +
     assumes test[iff]: "test\<^sub>2 (degen cs \<circ> translate) w
       (r ||| sscan (count cs \<circ> item) (p ## r ||| w ||| r) k) (p, k) \<longleftrightarrow> test\<^sub>1 cs w r p"
   begin
 
     lemma degeneralize_language[simp]: "b.language (degeneralize A) = a.language A" by force
 
   end
 
   locale automaton_combination =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 +
     c: automaton automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     +
     fixes condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
   begin
 
     definition combine :: "'automaton\<^sub>1 \<Rightarrow> 'automaton\<^sub>2 \<Rightarrow> 'automaton\<^sub>3" where
       "combine A B \<equiv> automaton\<^sub>3
         (alphabet\<^sub>1 A \<inter> alphabet\<^sub>2 B)
         (initial\<^sub>1 A, initial\<^sub>2 B)
         (\<lambda> a (p, q). (transition\<^sub>1 A a p, transition\<^sub>2 B a q))
         (condition (condition\<^sub>1 A) (condition\<^sub>2 B))"
 
     lemma combine_simps[simp]:
       "alphabet\<^sub>3 (combine A B) = alphabet\<^sub>1 A \<inter> alphabet\<^sub>2 B"
       "initial\<^sub>3 (combine A B) = (initial\<^sub>1 A, initial\<^sub>2 B)"
       "transition\<^sub>3 (combine A B) a (p, q) = (transition\<^sub>1 A a p, transition\<^sub>2 B a q)"
       "condition\<^sub>3 (combine A B) = condition (condition\<^sub>1 A) (condition\<^sub>2 B)"
       unfolding combine_def by auto
 
     lemma combine_target[simp]: "c.target (combine A B) w (p, q) = (a.target A w p, b.target B w q)"
       by (induct w arbitrary: p q) (auto)
     lemma combine_states[simp]: "c.states (combine A B) w (p, q) = a.states A w p || b.states B w q"
       by (induct w arbitrary: p q) (auto)
     lemma combine_trace[simp]: "c.trace (combine A B) w (p, q) = a.trace A w p ||| b.trace B w q"
       by (coinduction arbitrary: w p q) (auto)
 
     lemma combine_path[iff]: "c.path (combine A B) w (p, q) \<longleftrightarrow> a.path A w p \<and> b.path B w q"
       unfolding a.path_alt_def b.path_alt_def c.path_alt_def by simp
     lemma combine_run[iff]: "c.run (combine A B) w (p, q) \<longleftrightarrow> a.run A w p \<and> b.run B w q"
       unfolding a.run_alt_def b.run_alt_def c.run_alt_def by simp
 
     lemma combine_reachable[simp]: "c.reachable (combine A B) (p, q) \<subseteq> a.reachable A p \<times> b.reachable B q"
       unfolding c.reachable_alt_def by auto
     lemma combine_nodes[simp]: "c.nodes (combine A B) \<subseteq> a.nodes A \<times> b.nodes B"
       unfolding a.nodes_alt_def b.nodes_alt_def c.nodes_alt_def by auto
     lemma combine_reachable_fst[simp]:
       assumes "alphabet\<^sub>1 A \<subseteq> alphabet\<^sub>2 B"
       shows "fst ` c.reachable (combine A B) (p, q) = a.reachable A p"
       using assms
       unfolding a.reachable_alt_def a.path_alt_def
       unfolding b.reachable_alt_def b.path_alt_def
       unfolding c.reachable_alt_def c.path_alt_def
       by auto force
     lemma combine_reachable_snd[simp]:
       assumes "alphabet\<^sub>1 A \<supseteq> alphabet\<^sub>2 B"
       shows "snd ` c.reachable (combine A B) (p, q) = b.reachable B q"
       using assms
       unfolding a.reachable_alt_def a.path_alt_def
       unfolding b.reachable_alt_def b.path_alt_def
       unfolding c.reachable_alt_def c.path_alt_def
       by auto force
     lemma combine_nodes_fst[simp]:
       assumes "alphabet\<^sub>1 A \<subseteq> alphabet\<^sub>2 B"
       shows "fst ` c.nodes (combine A B) = a.nodes A"
       using assms combine_reachable_fst
       unfolding a.nodes_alt_def b.nodes_alt_def c.nodes_alt_def
       by fastforce
     lemma combine_nodes_snd[simp]:
       assumes "alphabet\<^sub>1 A \<supseteq> alphabet\<^sub>2 B"
       shows "snd ` c.nodes (combine A B) = b.nodes B"
       using assms combine_reachable_snd
       unfolding a.nodes_alt_def b.nodes_alt_def c.nodes_alt_def
       by fastforce
 
     lemma combine_nodes_finite[intro]:
       assumes "finite (a.nodes A)" "finite (b.nodes B)"
       shows "finite (c.nodes (combine A B))"
     proof (rule finite_subset)
       show "c.nodes (combine A B) \<subseteq> a.nodes A \<times> b.nodes B" using combine_nodes by this
       show "finite (a.nodes A \<times> b.nodes B)" using assms by simp
     qed
     lemma combine_nodes_finite_strong[iff]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "finite (c.nodes (combine A B)) \<longleftrightarrow> finite (a.nodes A) \<and> finite (b.nodes B)"
     proof safe
       show "finite (a.nodes A)" if "finite (c.nodes (combine A B))"
         using combine_nodes_fst assms that by (metis finite_imageI equalityD1)
       show "finite (b.nodes B)" if "finite (c.nodes (combine A B))"
         using combine_nodes_snd assms that by (metis finite_imageI equalityD2)
       show "finite (c.nodes (combine A B))" if "finite (a.nodes A)" "finite (b.nodes B)"
         using that by rule
     qed
     lemma combine_nodes_card[intro]:
       assumes "finite (a.nodes A)" "finite (b.nodes B)"
       shows "card (c.nodes (combine A B)) \<le> card (a.nodes A) * card (b.nodes B)"
     proof -
       have "card (c.nodes (combine A B)) \<le> card (a.nodes A \<times> b.nodes B)"
       proof (rule card_mono)
         show "finite (a.nodes A \<times> b.nodes B)" using assms by simp
         show "c.nodes (combine A B) \<subseteq> a.nodes A \<times> b.nodes B" using combine_nodes by this
       qed
       also have "\<dots> = card (a.nodes A) * card (b.nodes B)" using card_cartesian_product by this
       finally show ?thesis by this
     qed
     lemma combine_nodes_card_strong[intro]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "card (c.nodes (combine A B)) \<le> card (a.nodes A) * card (b.nodes B)"
     proof (cases "finite (a.nodes A) \<and> finite (b.nodes B)")
       case True
       show ?thesis using True by auto
     next
       case False
       have 1: "card (c.nodes (combine A B)) = 0" using False assms by simp
       have 2: "card (a.nodes A) * card (b.nodes B) = 0" using False by auto
       show ?thesis using 1 2 by simp
     qed
 
   end
 
   locale automaton_intersection_run =
     automaton_combination
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_run automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>1 stream \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>2 stream \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label stream \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) stream \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test[iff]: "test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (r ||| s) (p, q) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w r p \<and> test\<^sub>2 c\<^sub>2 w s q"
   begin
 
     lemma combine_language[simp]: "c.language (combine A B) = a.language A \<inter> b.language B" by force
 
   end
 
   locale automaton_union_run =
     automaton_combination
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_run automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>1 stream \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>2 stream \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label stream \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) stream \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test[iff]: "test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (r ||| s) (p, q) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w r p \<or> test\<^sub>2 c\<^sub>2 w s q"
   begin
 
     lemma combine_language[simp]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "c.language (combine A B) = a.language A \<union> b.language B"
       using assms by (force simp: a.run_alt_def b.run_alt_def)
 
   end
 
   (* TODO: complete this in analogy to the pair case *)
   locale automaton_combination_list =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list \<Rightarrow> ('label, 'state list) trans \<Rightarrow>
       'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     +
     fixes condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
   begin
 
     definition combine :: "'automaton\<^sub>1 list \<Rightarrow> 'automaton\<^sub>2" where
       "combine AA \<equiv> automaton\<^sub>2
         (\<Inter> (alphabet\<^sub>1 ` set AA))
         (map initial\<^sub>1 AA)
-        (\<lambda> a pp. map2 (\<lambda> A p. transition\<^sub>1 A a p) AA pp)
+        (\<lambda> a ps. map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps)
         (condition (map condition\<^sub>1 AA))"
 
     lemma combine_simps[simp]:
       "alphabet\<^sub>2 (combine AA) = \<Inter> (alphabet\<^sub>1 ` set AA)"
       "initial\<^sub>2 (combine AA) = map initial\<^sub>1 AA"
-      "transition\<^sub>2 (combine AA) a pp = map2 (\<lambda> A p. transition\<^sub>1 A a p) AA pp"
+      "transition\<^sub>2 (combine AA) a ps = map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps"
       "condition\<^sub>2 (combine AA) = condition (map condition\<^sub>1 AA)"
       unfolding combine_def by auto
 
     (* TODO: get rid of indices, express this using stranspose and listset *)
     lemma combine_trace_smap:
-      assumes "length pp = length AA" "k < length AA"
-      shows "smap (\<lambda> pp. pp ! k) (b.trace (combine AA) w pp) = a.trace (AA ! k) w (pp ! k)"
-      using assms by (coinduction arbitrary: w pp) (force)
+      assumes "length ps = length AA" "k < length AA"
+      shows "smap (\<lambda> ps. ps ! k) (b.trace (combine AA) w ps) = a.trace (AA ! k) w (ps ! k)"
+      using assms by (coinduction arbitrary: w ps) (force)
     lemma combine_nodes_length:
-      assumes "pp \<in> b.nodes (combine AA)"
-      shows "length pp = length AA"
+      assumes "ps \<in> b.nodes (combine AA)"
+      shows "length ps = length AA"
       using assms by induct auto
     lemma combine_nodes[intro]:
-      assumes "pp \<in> b.nodes (combine AA)" "k < length pp"
-      shows "pp ! k \<in> a.nodes (AA ! k)"
+      assumes "ps \<in> b.nodes (combine AA)" "k < length ps"
+      shows "ps ! k \<in> a.nodes (AA ! k)"
       using assms by induct auto
 
     lemma combine_nodes_finite[intro]:
       assumes "list_all (finite \<circ> a.nodes) AA"
       shows "finite (b.nodes (combine AA))"
     proof (rule finite_subset)
       (* TODO: this is used more than once, make top level theorem *)
       show "b.nodes (combine AA) \<subseteq> listset (map a.nodes AA)"
         by (force simp: listset_member list_all2_conv_all_nth combine_nodes_length)
       show "finite (listset (map a.nodes AA))" using list.pred_map assms by auto
     qed
     lemma combine_nodes_card:
       assumes "list_all (finite \<circ> a.nodes) AA"
       shows "card (b.nodes (combine AA)) \<le> prod_list (map (card \<circ> a.nodes) AA)"
     proof -
       have "card (b.nodes (combine AA)) \<le> card (listset (map a.nodes AA))"
       proof (rule card_mono)
         show "finite (listset (map a.nodes AA))" using list.pred_map assms by auto
         show "b.nodes (combine AA) \<subseteq> listset (map a.nodes AA)"
           by (force simp: listset_member list_all2_conv_all_nth combine_nodes_length)
       qed
       also have "\<dots> = prod_list (map (card \<circ> a.nodes) AA)" by simp
       finally show ?thesis by this
     qed
 
   end
 
   locale automaton_intersection_list_run =
     automaton_combination_list
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list \<Rightarrow> ('label, 'state list) trans \<Rightarrow>
       'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state list stream \<Rightarrow> 'state list \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
     +
     assumes test[iff]: "test\<^sub>2 (condition cs) w rs ps \<longleftrightarrow>
       (\<forall> k < length cs. test\<^sub>1 (cs ! k) w (smap (\<lambda> ps. ps ! k) rs) (ps ! k))"
   begin
 
     lemma combine_language[simp]: "b.language (combine AA) = \<Inter> (a.language ` set AA)"
       unfolding a.language_def b.language_def
-      unfolding a.run_alt_def b.run_alt_def
+      unfolding a.run_alt_def b.run_alt_def streams_iff_sset
       by (fastforce simp: set_conv_nth combine_trace_smap)
 
   end
 
   locale automaton_union_list_run =
     automaton_combination_list
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list \<Rightarrow> ('label, 'state list) trans \<Rightarrow>
       'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state list stream \<Rightarrow> 'state list \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
     +
     assumes test[iff]: "test\<^sub>2 (condition cs) w rs ps \<longleftrightarrow>
       (\<exists> k < length cs. test\<^sub>1 (cs ! k) w (smap (\<lambda> ps. ps ! k) rs) (ps ! k))"
   begin
 
     lemma combine_language[simp]:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       shows "b.language (combine AA) = \<Union> (a.language ` set AA)"
       using assms
       unfolding a.language_def b.language_def
-      unfolding a.run_alt_def b.run_alt_def
+      unfolding a.run_alt_def b.run_alt_def streams_iff_sset
       by (fastforce simp: set_conv_nth combine_trace_smap)
 
   end
 
 end
\ No newline at end of file
diff --git a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Combine.thy b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Combine.thy
--- a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Combine.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Combine.thy
@@ -1,90 +1,90 @@
 section \<open>Nondeterministic Büchi Automata Combinations\<close>
 
 theory NBA_Combine
 imports NBA NGBA
 begin
 
   global_interpretation degeneralization: automaton_degeneralization_run
     ngba ngba.alphabet ngba.initial ngba.transition ngba.accepting "\<lambda> P w r p. gen infs P r"
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     fst id
     defines degeneralize = degeneralization.degeneralize
-    by unfold_locales auto
+    by (unfold_locales) (auto simp flip: sscan_smap)
 
   lemmas degeneralize_language[simp] = degeneralization.degeneralize_language[folded NBA.language_def]
   lemmas degeneralize_nodes_finite[iff] = degeneralization.degeneralize_nodes_finite[folded NBA.nodes_def]
 
   global_interpretation intersection: automaton_intersection_run
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     ngba ngba.alphabet ngba.initial ngba.transition ngba.accepting "\<lambda> P w r p. gen infs P r"
     "\<lambda> c\<^sub>1 c\<^sub>2. [c\<^sub>1 \<circ> fst, c\<^sub>2 \<circ> snd]"
     defines intersect' = intersection.intersect
     by unfold_locales auto
 
   lemmas intersect'_language[simp] = intersection.intersect_language[folded NGBA.language_def]
   lemmas intersect'_nodes_finite[intro] = intersection.intersect_nodes_finite[folded NGBA.nodes_def]
 
+  global_interpretation union: automaton_union_run
+    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
+    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
+    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
+    case_sum
+    defines union = union.union
+    by (unfold_locales) (auto simp: comp_def)
+
+  lemmas union_language = union.union_language
+  lemmas union_nodes_finite = union.union_nodes_finite
+
   global_interpretation intersection_list: automaton_intersection_list_run
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     ngba ngba.alphabet ngba.initial ngba.transition ngba.accepting "\<lambda> P w r p. gen infs P r"
     "\<lambda> cs. map (\<lambda> k ps. (cs ! k) (ps ! k)) [0 ..< length cs]"
     defines intersect_list' = intersection_list.intersect
   proof unfold_locales
     fix cs :: "('b \<Rightarrow> bool) list" and rs :: "'b stream list" and w :: "'a stream" and ps :: "'b list"
     assume 1: "length rs = length cs" "length ps = length cs"
     have "gen infs (map (\<lambda> k pp. (cs ! k) (pp ! k)) [0 ..< length cs]) (stranspose rs) \<longleftrightarrow>
       (\<forall> k < length cs. infs (\<lambda> pp. (cs ! k) (pp ! k)) (stranspose rs))"
       by (auto simp: gen_def)
     also have "\<dots> \<longleftrightarrow> (\<forall> k < length cs. infs (cs ! k) (smap (\<lambda> pp. pp ! k) (stranspose rs)))"
       by (simp add: comp_def)
     also have "\<dots> \<longleftrightarrow> (\<forall> k < length cs. infs (cs ! k) (rs ! k))" using 1 by simp
     also have "\<dots> \<longleftrightarrow> list_all (\<lambda> (c, r, p). infs c r) (cs || rs || ps)"
       using 1 unfolding list_all_length by simp
     finally show "gen infs (map (\<lambda> k ps. (cs ! k) (ps ! k)) [0 ..< length cs]) (stranspose rs) \<longleftrightarrow>
       list_all (\<lambda> (c, r, p). infs c r) (cs || rs || ps)" by this
   qed
 
   lemmas intersect_list'_language[simp] = intersection_list.intersect_language[folded NGBA.language_def]
   lemmas intersect_list'_nodes_finite[intro] = intersection_list.intersect_nodes_finite[folded NGBA.nodes_def]
 
-  global_interpretation union: automaton_union_run
-    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
-    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
-    nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
-    case_sum
-    defines union = union.union
-    by (unfold_locales) (auto simp: comp_def)
-
-  lemmas union_language = union.union_language
-  lemmas union_nodes_finite = union.union_nodes_finite
-
   global_interpretation union_list: automaton_union_list_run
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     nba nba.alphabet nba.initial nba.transition nba.accepting "\<lambda> P w r p. infs P r"
     "\<lambda> cs (k, p). (cs ! k) p"
     defines union_list = union_list.union
     by (unfold_locales) (auto simp: szip_sconst_smap_fst comp_def)
 
   lemmas union_list_language = union_list.union_language
   lemmas union_list_nodes_finite = union_list.union_nodes_finite
 
   abbreviation intersect where "intersect A B \<equiv> degeneralize (intersect' A B)"
 
   lemma intersect_language[simp]: "NBA.language (intersect A B) = NBA.language A \<inter> NBA.language B"
     by simp
   lemma intersect_nodes_finite[intro]:
     assumes "finite (NBA.nodes A)" "finite (NBA.nodes B)"
     shows "finite (NBA.nodes (intersect A B))"
     using intersect'_nodes_finite assms by simp
 
   abbreviation intersect_list where "intersect_list AA \<equiv> degeneralize (intersect_list' AA)"
 
   lemma intersect_list_language[simp]: "NBA.language (intersect_list AA) = \<Inter> (NBA.language ` set AA)"
     by simp
   lemma intersect_list_nodes_finite[intro]:
     assumes "list_all (finite \<circ> NBA.nodes) AA"
     shows "finite (NBA.nodes (intersect_list AA))"
     using intersect_list'_nodes_finite assms by simp
 
 end
\ No newline at end of file
diff --git a/thys/Transition_Systems_and_Automata/Automata/Nondeterministic.thy b/thys/Transition_Systems_and_Automata/Automata/Nondeterministic.thy
--- a/thys/Transition_Systems_and_Automata/Automata/Nondeterministic.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/Nondeterministic.thy
@@ -1,895 +1,893 @@
 theory Nondeterministic
 imports
   "../Transition_Systems/Transition_System"
   "../Transition_Systems/Transition_System_Extra"
   "../Transition_Systems/Transition_System_Construction"
   "../Basic/Degeneralization"
 begin
 
   (* TODO: might not be needed anymore *)
   type_synonym ('label, 'state) trans = "'label \<Rightarrow> 'state \<Rightarrow> 'state set"
 
   (* TODO: many of these type annotations can be left out *)
   locale automaton =
     fixes automaton :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition \<Rightarrow> 'automaton"
     fixes alphabet :: "'automaton \<Rightarrow> 'label set"
     fixes initial :: "'automaton \<Rightarrow> 'state set"
     fixes transition :: "'automaton \<Rightarrow> ('label, 'state) trans"
     fixes condition :: "'automaton \<Rightarrow> 'condition"
     assumes automaton[simp]: "automaton (alphabet A) (initial A) (transition A) (condition A) = A"
     assumes alphabet[simp]: "alphabet (automaton a i t c) = a"
     assumes initial[simp]: "initial (automaton a i t c) = i"
     assumes transition[simp]: "transition (automaton a i t c) = t"
     assumes condition[simp]: "condition (automaton a i t c) = c"
   begin
 
     sublocale transition_system_initial
       "\<lambda> a p. snd a" "\<lambda> a p. fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p" "\<lambda> p. p \<in> initial A"
       for A
       defines path' = path and run' = run and reachable' = reachable and nodes' = nodes
       by this
 
     lemma states_alt_def: "states r p = map snd r" by (induct r arbitrary: p) (auto)
     lemma trace_alt_def: "trace r p = smap snd r" by (coinduction arbitrary: r p) (auto)
 
     lemma successors_alt_def: "successors A p = (\<Union> a \<in> alphabet A. transition A a p)" by auto
 
     lemma reachable_transition[intro]:
       assumes "a \<in> alphabet A" "q \<in> reachable A p" "r \<in> transition A a q"
       shows "r \<in> reachable A p"
       using reachable.execute assms by force
     lemma nodes_transition[intro]:
       assumes "a \<in> alphabet A" "p \<in> nodes A" "q \<in> transition A a p"
       shows "q \<in> nodes A"
       using nodes.execute assms by force
 
     lemma path_alphabet:
       assumes "length r = length w" "path A (w || r) p"
       shows "w \<in> lists (alphabet A)"
       using assms by (induct arbitrary: p rule: list_induct2) (auto)
     lemma run_alphabet:
       assumes "run A (w ||| r) p"
       shows "w \<in> streams (alphabet A)"
       using assms by (coinduction arbitrary: w r p) (metis run.cases stream.map szip_smap szip_smap_fst)
 
     definition restrict :: "'automaton \<Rightarrow> 'automaton" where
       "restrict A \<equiv> automaton
         (alphabet A)
         (initial A)
         (\<lambda> a p. if a \<in> alphabet A then transition A a p else {})
         (condition A)"
 
     lemma restrict_simps[simp]:
       "alphabet (restrict A) = alphabet A"
       "initial (restrict A) = initial A"
       "transition (restrict A) a p = (if a \<in> alphabet A then transition A a p else {})"
       "condition (restrict A) = condition A"
       unfolding restrict_def by auto
 
     lemma restrict_path[simp]: "path (restrict A) = path A"
     proof (intro ext iffI)
       show "path A wr p" if "path (restrict A) wr p" for wr p using that by induct auto
       show "path (restrict A) wr p" if "path A wr p" for wr p using that by induct auto
     qed
     lemma restrict_run[simp]: "run (restrict A) = run A"
     proof (intro ext iffI)
       show "run A wr p" if "run (restrict A) wr p" for wr p using that by coinduct auto
       show "run (restrict A) wr p" if "run A wr p" for wr p using that by coinduct auto
     qed
 
   end
 
   locale automaton_path =
     automaton automaton alphabet initial transition condition
     for automaton :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition \<Rightarrow> 'automaton"
     and alphabet :: "'automaton \<Rightarrow> 'label set"
     and initial :: "'automaton \<Rightarrow> 'state set"
     and transition :: "'automaton \<Rightarrow> ('label, 'state) trans"
     and condition :: "'automaton \<Rightarrow> 'condition"
     +
     fixes test :: "'condition \<Rightarrow> 'label list \<Rightarrow> 'state list \<Rightarrow> 'state \<Rightarrow> bool"
   begin
 
     definition language :: "'automaton \<Rightarrow> 'label list set" where
       "language A \<equiv> {w |w r p. length r = length w \<and> p \<in> initial A \<and> path A (w || r) p \<and> test (condition A) w r p}"
 
     lemma language[intro]:
       assumes "length r = length w" "p \<in> initial A" "path A (w || r) p" "test (condition A) w r p"
       shows "w \<in> language A"
       using assms unfolding language_def by auto
     lemma language_elim[elim]:
       assumes "w \<in> language A"
       obtains r p
       where "length r = length w" "p \<in> initial A" "path A (w || r) p" "test (condition A) w r p"
       using assms unfolding language_def by auto
 
     lemma language_alphabet: "language A \<subseteq> lists (alphabet A)" by (auto dest: path_alphabet)
 
     lemma restrict_language[simp]: "language (restrict A) = language A" by force
 
   end
 
   locale automaton_run =
     automaton automaton alphabet initial transition condition
     for automaton :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition \<Rightarrow> 'automaton"
     and alphabet :: "'automaton \<Rightarrow> 'label set"
     and initial :: "'automaton \<Rightarrow> 'state set"
     and transition :: "'automaton \<Rightarrow> ('label, 'state) trans"
     and condition :: "'automaton \<Rightarrow> 'condition"
     +
     fixes test :: "'condition \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
   begin
 
     definition language :: "'automaton \<Rightarrow> 'label stream set" where
       "language A \<equiv> {w |w r p. p \<in> initial A \<and> run A (w ||| r) p \<and> test (condition A) w r p}"
 
     lemma language[intro]:
       assumes "p \<in> initial A" "run A (w ||| r) p" "test (condition A) w r p"
       shows "w \<in> language A"
       using assms unfolding language_def by auto
     lemma language_elim[elim]:
       assumes "w \<in> language A"
       obtains r p
       where "p \<in> initial A" "run A (w ||| r) p" "test (condition A) w r p"
       using assms unfolding language_def by auto
 
     lemma language_alphabet: "language A \<subseteq> streams (alphabet A)" by (auto dest: run_alphabet)
 
     lemma restrict_language[simp]: "language (restrict A) = language A" by force
 
   end
 
   locale automaton_degeneralization =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'item pred gen \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'item pred gen"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen set \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'item_degen pred \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state degen set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state degen) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'item_degen pred"
     +
     fixes item :: "'state \<times> 'label \<times> 'state \<Rightarrow> 'item"
     fixes translate :: "'item_degen \<Rightarrow> 'item degen"
   begin
 
     definition degeneralize :: "'automaton\<^sub>1 \<Rightarrow> 'automaton\<^sub>2" where
       "degeneralize A \<equiv> automaton\<^sub>2
         (alphabet\<^sub>1 A)
         (initial\<^sub>1 A \<times> {0})
         (\<lambda> a (p, k). {(q, count (condition\<^sub>1 A) (item (p, a, q)) k) |q. q \<in> transition\<^sub>1 A a p})
         (degen (condition\<^sub>1 A) \<circ> translate)"
 
     lemma degeneralize_simps[simp]:
       "alphabet\<^sub>2 (degeneralize A) = alphabet\<^sub>1 A"
       "initial\<^sub>2 (degeneralize A) = initial\<^sub>1 A \<times> {0}"
       "transition\<^sub>2 (degeneralize A) a (p, k) =
         {(q, count (condition\<^sub>1 A) (item (p, a, q)) k) |q. q \<in> transition\<^sub>1 A a p}"
       "condition\<^sub>2 (degeneralize A) = degen (condition\<^sub>1 A) \<circ> translate"
       unfolding degeneralize_def by auto
 
-    abbreviation "counter cs w r p k \<equiv> sscan (count cs) (smap item (p ## r ||| w ||| r)) k"
-
     lemma run_degeneralize:
       assumes "a.run A (w ||| r) p"
-      shows "b.run (degeneralize A) (w ||| r ||| counter (condition\<^sub>1 A) w r p k) (p, k)"
+      shows "b.run (degeneralize A) (w ||| r ||| sscan (count (condition\<^sub>1 A) \<circ> item) (p ## r ||| w ||| r) k) (p, k)"
       using assms by (coinduction arbitrary: w r p k) (force elim: a.run.cases)
     lemma degeneralize_run:
       assumes "b.run (degeneralize A) (w ||| rs) pk"
       obtains r s p k
-      where "rs = r ||| s" "pk = (p, k)" "a.run A (w ||| r) p" "s = counter (condition\<^sub>1 A) w r p k"
+      where "rs = r ||| s" "pk = (p, k)" "a.run A (w ||| r) p" "s = sscan (count (condition\<^sub>1 A) \<circ> item) (p ## r ||| w ||| r) k"
     proof -
       obtain r s p k where 1: "rs = r ||| s" "pk = (p, k)" using szip_smap surjective_pairing by metis
       show ?thesis
       proof
         show "rs = r ||| s" "pk = (p, k)" using 1 by this
         show "a.run A (w ||| r) p"
           using assms unfolding 1 by (coinduction arbitrary: w r s p k) (force elim: b.run.cases)
-        show "s = counter (condition\<^sub>1 A) w r p k"
+        show "s = sscan (count (condition\<^sub>1 A) \<circ> item) (p ## r ||| w ||| r) k"
           using assms unfolding 1 by (coinduction arbitrary: w r s p k) (erule b.run.cases, force)
       qed
     qed
 
     lemma degeneralize_nodes:
       "b.nodes (degeneralize A) \<subseteq> a.nodes A \<times> insert 0 {0 ..< length (condition\<^sub>1 A)}"
     proof
       fix pk
       assume "pk \<in> b.nodes (degeneralize A)"
       then show "pk \<in> a.nodes A \<times> insert 0 {0 ..< length (condition\<^sub>1 A)}"
         by (induct) (force, cases "condition\<^sub>1 A = []", auto)
     qed
     lemma nodes_degeneralize: "a.nodes A \<subseteq> fst ` b.nodes (degeneralize A)"
     proof
       fix p
       assume "p \<in> a.nodes A"
       then show "p \<in> fst ` b.nodes (degeneralize A)"
       proof induct
         case (initial p)
         have "(p, 0) \<in> b.nodes (degeneralize A)" using initial by auto
         then show ?case using image_iff fst_conv by force
       next
         case (execute p aq)
         obtain k where "(p, k) \<in> b.nodes (degeneralize A)" using execute(2) by auto
         then have "(snd aq, count (condition\<^sub>1 A) (item (p, aq)) k) \<in> b.nodes (degeneralize A)"
           using execute(3) by auto
         then show ?case using image_iff snd_conv by force
       qed
     qed
 
     lemma degeneralize_nodes_finite[iff]: "finite (b.nodes (degeneralize A)) \<longleftrightarrow> finite (a.nodes A)"
     proof
       show "finite (a.nodes A)" if "finite (b.nodes (degeneralize A))"
         using finite_subset nodes_degeneralize that by blast
       show "finite (b.nodes (degeneralize A))" if "finite (a.nodes A)"
         using finite_subset degeneralize_nodes that by blast
     qed
 
   end
 
   locale automaton_degeneralization_run =
     automaton_degeneralization
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       item translate +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'item pred gen \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'item pred gen"
     and test\<^sub>1 :: "'item pred gen \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen set \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'item_degen pred \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state degen set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state degen) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'item_degen pred"
     and test\<^sub>2 :: "'item_degen pred \<Rightarrow> 'label stream \<Rightarrow> 'state degen stream \<Rightarrow> 'state degen \<Rightarrow> bool"
     and item :: "'state \<times> 'label \<times> 'state \<Rightarrow> 'item"
     and translate :: "'item_degen \<Rightarrow> 'item degen"
     +
-    assumes test[iff]: "test\<^sub>2 (degen cs \<circ> translate) w (r ||| counter cs w r p k) (p, k) \<longleftrightarrow> test\<^sub>1 cs w r p"
+    assumes test[iff]: "test\<^sub>2 (degen cs \<circ> translate) w (r ||| sscan (count cs \<circ> item) (p ## r ||| w ||| r) k) (p, k) \<longleftrightarrow> test\<^sub>1 cs w r p"
   begin
 
     lemma degeneralize_language[simp]: "b.language (degeneralize A) = a.language A"
       unfolding a.language_def b.language_def by (auto dest: run_degeneralize elim!: degeneralize_run)
 
   end
 
   locale automaton_intersection =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 +
     c: automaton automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     +
     fixes condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
   begin
 
     definition intersect :: "'automaton\<^sub>1 \<Rightarrow> 'automaton\<^sub>2 \<Rightarrow> 'automaton\<^sub>3" where
       "intersect A B \<equiv> automaton\<^sub>3
         (alphabet\<^sub>1 A \<inter> alphabet\<^sub>2 B)
         (initial\<^sub>1 A \<times> initial\<^sub>2 B)
         (\<lambda> a (p, q). transition\<^sub>1 A a p \<times> transition\<^sub>2 B a q)
         (condition (condition\<^sub>1 A) (condition\<^sub>2 B))"
 
     lemma intersect_simps[simp]:
       "alphabet\<^sub>3 (intersect A B) = alphabet\<^sub>1 A \<inter> alphabet\<^sub>2 B"
       "initial\<^sub>3 (intersect A B) = initial\<^sub>1 A \<times> initial\<^sub>2 B"
       "transition\<^sub>3 (intersect A B) a (p, q) = transition\<^sub>1 A a p \<times> transition\<^sub>2 B a q"
       "condition\<^sub>3 (intersect A B) = condition (condition\<^sub>1 A) (condition\<^sub>2 B)"
       unfolding intersect_def by auto
 
     lemma intersect_target[simp]:
       assumes "length w = length r" "length r = length s"
       shows "c.target (w || r || s) (p, q) = (a.target (w || r) p, b.target (w || s) q)"
       using assms by (induct arbitrary: p q rule: list_induct3) (auto)
 
     lemma intersect_path[iff]:
       assumes "length w = length r" "length r = length s"
       shows "c.path (intersect A B) (w || r || s) (p, q) \<longleftrightarrow>
         a.path A (w || r) p \<and> b.path B (w || s) q"
       using assms by (induct arbitrary: p q rule: list_induct3) (auto)
     lemma intersect_run[iff]: "c.run (intersect A B) (w ||| r ||| s) (p, q) \<longleftrightarrow>
       a.run A (w ||| r) p \<and> b.run B (w ||| s) q"
     proof safe
       show "a.run A (w ||| r) p" if "c.run (intersect A B) (w ||| r ||| s) (p, q)"
         using that by (coinduction arbitrary: w r s p q) (force elim: c.run.cases)
       show "b.run B (w ||| s) q" if "c.run (intersect A B) (w ||| r ||| s) (p, q)"
         using that by (coinduction arbitrary: w r s p q) (force elim: c.run.cases)
       show "c.run (intersect A B) (w ||| r ||| s) (p, q)" if "a.run A (w ||| r) p" "b.run B (w ||| s) q"
         using that by (coinduction arbitrary: w r s p q) (auto elim: a.run.cases b.run.cases)
     qed
 
     lemma intersect_nodes: "c.nodes (intersect A B) \<subseteq> a.nodes A \<times> b.nodes B"
     proof
       fix pq
       assume "pq \<in> c.nodes (intersect A B)"
       then show "pq \<in> a.nodes A \<times> b.nodes B" by induct auto
     qed
 
     lemma intersect_nodes_finite[intro]:
       assumes "finite (a.nodes A)" "finite (b.nodes B)"
       shows "finite (c.nodes (intersect A B))"
       using finite_subset intersect_nodes assms by blast
 
   end
 
   (* TODO: format these the same as degeneralization *)
   locale automaton_intersection_path =
     automaton_intersection
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_path automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_path automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_path automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label list \<Rightarrow> 'state\<^sub>1 list \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label list \<Rightarrow> 'state\<^sub>2 list \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label list \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) list \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test[iff]: "length u = length w \<Longrightarrow> length v = length w \<Longrightarrow>
       test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (u || v) (p, q) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w u p \<and> test\<^sub>2 c\<^sub>2 w v q"
   begin
 
     lemma intersect_language[simp]: "c.language (intersect A B) = a.language A \<inter> b.language B"
       unfolding a.language_def b.language_def c.language_def by (force iff: split_zip)
 
   end
 
   locale automaton_intersection_run =
     automaton_intersection
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_run automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>1 stream \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>2 stream \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 \<times> 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label stream \<Rightarrow> ('state\<^sub>1 \<times> 'state\<^sub>2) stream \<Rightarrow> 'state\<^sub>1 \<times> 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test[iff]: "test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (u ||| v) (p, q) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w u p \<and> test\<^sub>2 c\<^sub>2 w v q"
   begin
 
     lemma intersect_language[simp]: "c.language (intersect A B) = a.language A \<inter> b.language B"
       unfolding a.language_def b.language_def c.language_def by (fastforce iff: split_szip)
 
   end
 
-  locale automaton_intersection_list =
-    a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
-    b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
-    for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
-    and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
-    and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
-    and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
-    and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
-    and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list set \<Rightarrow> ('label, 'state list) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
-    and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
-    and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list set"
-    and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
-    and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
-    +
-    fixes condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
-  begin
-
-    definition intersect :: "'automaton\<^sub>1 list \<Rightarrow> 'automaton\<^sub>2" where
-      "intersect AA \<equiv> automaton\<^sub>2
-        (\<Inter> (alphabet\<^sub>1 ` set AA))
-        (listset (map initial\<^sub>1 AA))
-        (\<lambda> a ps. listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps))
-        (condition (map condition\<^sub>1 AA))"
-
-    lemma intersect_simps[simp]:
-      "alphabet\<^sub>2 (intersect AA) = \<Inter> (alphabet\<^sub>1 ` set AA)"
-      "initial\<^sub>2 (intersect AA) = listset (map initial\<^sub>1 AA)"
-      "transition\<^sub>2 (intersect AA) a ps = listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps)"
-      "condition\<^sub>2 (intersect AA) = condition (map condition\<^sub>1 AA)"
-      unfolding intersect_def by auto
-
-    lemma intersect_run_length:
-      assumes "length ps = length AA"
-      assumes "b.run (intersect AA) (w ||| r) ps"
-      assumes "qs \<in> sset r"
-      shows "length qs = length AA"
-    proof -
-      have "pred_stream (\<lambda> qs. length qs = length AA) r"
-        using assms(1, 2) by (coinduction arbitrary: w r ps)
-          (force elim: b.run.cases simp: listset_member list_all2_conv_all_nth)
-      then show ?thesis using assms(3) unfolding stream.pred_set by auto
-    qed
-    lemma intersect_run_stranspose:
-      assumes "length ps = length AA"
-      assumes "b.run (intersect AA) (w ||| r) ps"
-      obtains rs where "r = stranspose rs" "length rs = length AA"
-    proof
-      define rs where "rs \<equiv> map (\<lambda> k. smap (\<lambda> ps. ps ! k) r) [0 ..< length AA]"
-      have "length qs = length AA" if "qs \<in> sset r" for qs using intersect_run_length assms that by this
-      then show "r = stranspose rs"
-        unfolding rs_def by (coinduction arbitrary: r) (auto intro: nth_equalityI simp: comp_def)
-      show "length rs = length AA" unfolding rs_def by auto
-    qed
-
-    lemma run_intersect:
-      assumes "length rs = length AA" "length ps = length AA"
-      assumes "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
-      shows "b.run (intersect AA) (w ||| stranspose rs) ps"
-    using assms
-    proof (coinduction arbitrary: w rs ps)
-      case (run ap r)
-      then show ?case
-      proof (intro conjI exI)
-        show "fst ap \<in> alphabet\<^sub>2 (intersect AA)"
-          using run by (force elim: a.run.cases simp: set_conv_nth)
-        show "snd ap \<in> transition\<^sub>2 (intersect AA) (fst ap) ps"
-          using run by (force elim: a.run.cases simp: listset_member list_all2_conv_all_nth)
-        show "\<forall> k < length AA. a.run' (AA ! k) (stl w ||| map stl rs ! k) (map shd rs ! k)"
-          using run by (force elim: a.run.cases)
-      qed auto
-    qed
-    lemma intersect_run:
-      assumes "length rs = length AA" "length ps = length AA"
-      assumes "b.run (intersect AA) (w ||| stranspose rs) ps"
-      shows "k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
-    using assms
-    proof (coinduction arbitrary: w rs ps)
-      case (run ap wr)
-      then show ?case
-      proof (intro exI conjI)
-        show "fst ap \<in> alphabet\<^sub>1 (AA ! k)"
-          using run by (force elim: b.run.cases)
-        show "snd ap \<in> transition\<^sub>1 (AA ! k) (fst ap) (ps ! k)"
-          using run by (force elim: b.run.cases simp: listset_member list_all2_conv_all_nth)
-        show "b.run' (intersect AA) (stl w ||| stranspose (map stl rs)) (shd (stranspose rs))"
-          using run by (force elim: b.run.cases)
-      qed auto
-    qed
-
-    lemma intersect_nodes: "b.nodes (intersect AA) \<subseteq> listset (map a.nodes AA)"
-    proof
-      show "ps \<in> listset (map a.nodes AA)" if "ps \<in> b.nodes (intersect AA)" for ps
-        using that by (induct) (auto 0 3 simp: listset_member list_all2_conv_all_nth)
-    qed
-
-    lemma intersect_nodes_finite[intro]:
-      assumes "list_all (finite \<circ> a.nodes) AA"
-      shows "finite (b.nodes (intersect AA))"
-    proof (rule finite_subset)
-      show "b.nodes (intersect AA) \<subseteq> listset (map a.nodes AA)" using intersect_nodes by this
-      show "finite (listset (map a.nodes AA))" using list.pred_map assms by auto
-    qed
-
-  end
-
-  locale automaton_intersection_list_run =
-    automaton_intersection_list
-      automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
-      automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 condition +
-    a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
-    for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
-    and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
-    and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
-    and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
-    and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
-    and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
-    and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list set \<Rightarrow> ('label, 'state list) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
-    and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
-    and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list set"
-    and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
-    and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
-    and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state list stream \<Rightarrow> 'state list \<Rightarrow> bool"
-    and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
-    +
-    assumes test[iff]: "length rs = length cs \<Longrightarrow> length ps = length cs \<Longrightarrow>
-      test\<^sub>2 (condition cs) w (stranspose rs) ps \<longleftrightarrow> list_all (\<lambda> (c, r, p). test\<^sub>1 c w r p) (cs || rs || ps)"
-  begin
-
-    lemma intersect_language[simp]: "b.language (intersect AA) = \<Inter> (a.language ` set AA)"
-    proof safe
-      fix A w
-      assume 1: "w \<in> b.language (intersect AA)" "A \<in> set AA"
-      obtain r ps where 2:
-        "ps \<in> initial\<^sub>2 (intersect AA)"
-        "b.run (intersect AA) (w ||| r) ps"
-        "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w r ps"
-        using 1(1) by auto
-      have 3: "length ps = length AA" using 2(1) by (simp add: listset_member list_all2_conv_all_nth)
-      obtain rs where 4: "r = stranspose rs" "length rs = length AA"
-        using intersect_run_stranspose 3 2(2) by this
-      obtain k where 5: "k < length AA" "A = AA ! k" using 1(2) unfolding set_conv_nth by auto
-      show "w \<in> a.language A"
-      proof
-        show "ps ! k \<in> initial\<^sub>1 A" using 2(1) 5 by (auto simp: listset_member list_all2_conv_all_nth)
-        show "a.run A (w ||| rs ! k) (ps ! k)" using 2(2) 3 4 5 by (auto intro: intersect_run)
-        show "test\<^sub>1 (condition\<^sub>1 A) w (rs ! k) (ps ! k)" using 2(3) 3 4 5 by (simp add: list_all_length)
-      qed
-    next
-      fix w
-      assume 1: "w \<in> \<Inter> (a.language ` set AA)"
-      have 2: "\<forall> A \<in> set AA. \<exists> r p. p \<in> initial\<^sub>1 A \<and> a.run A (w ||| r) p \<and> test\<^sub>1 (condition\<^sub>1 A) w r p"
-        using 1 by blast
-      obtain rs ps where 3:
-        "length rs = length AA" "length ps = length AA"
-        "\<And> k. k < length AA \<Longrightarrow> ps ! k \<in> initial\<^sub>1 (AA ! k)"
-        "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
-        "\<And> k. k < length AA \<Longrightarrow> test\<^sub>1 (condition\<^sub>1 (AA ! k)) w (rs ! k) (ps ! k)"
-        using 2
-        unfolding Ball_set list_choice_zip list_choice_pair
-        unfolding list.pred_set set_conv_nth
-        by force
-      show "w \<in> b.language (intersect AA)"
-      proof
-        show "ps \<in> initial\<^sub>2 (intersect AA)" using 3 by (auto simp: listset_member list_all2_conv_all_nth)
-        show "b.run (intersect AA) (w ||| stranspose rs) ps" using 3 by (auto intro: run_intersect)
-        show "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w (stranspose rs) ps" using 3 by (auto simp: list_all_length)
-      qed
-    qed
-
-  end
-
   locale automaton_union =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 +
     c: automaton automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     +
     fixes condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
   begin
 
     definition union :: "'automaton\<^sub>1 \<Rightarrow> 'automaton\<^sub>2 \<Rightarrow> 'automaton\<^sub>3" where
       "union A B \<equiv> automaton\<^sub>3
         (alphabet\<^sub>1 A \<union> alphabet\<^sub>2 B)
         (initial\<^sub>1 A <+> initial\<^sub>2 B)
         (\<lambda> a. \<lambda> Inl p \<Rightarrow> Inl ` transition\<^sub>1 A a p | Inr q \<Rightarrow> Inr ` transition\<^sub>2 B a q)
         (condition (condition\<^sub>1 A) (condition\<^sub>2 B))"
 
     lemma union_simps[simp]:
       "alphabet\<^sub>3 (union A B) = alphabet\<^sub>1 A \<union> alphabet\<^sub>2 B"
       "initial\<^sub>3 (union A B) = initial\<^sub>1 A <+> initial\<^sub>2 B"
       "transition\<^sub>3 (union A B) a (Inl p) = Inl ` transition\<^sub>1 A a p"
       "transition\<^sub>3 (union A B) a (Inr q) = Inr ` transition\<^sub>2 B a q"
       "condition\<^sub>3 (union A B) = condition (condition\<^sub>1 A) (condition\<^sub>2 B)"
       unfolding union_def by auto
 
     lemma path_union_a:
       assumes "length r = length w" "a.path A (w || r) p"
       shows "c.path (union A B) (w || map Inl r) (Inl p)"
       using assms by (induct arbitrary: p rule: list_induct2) (auto)
     lemma path_union_b:
       assumes "length s = length w" "b.path B (w || s) q"
       shows "c.path (union A B) (w || map Inr s) (Inr q)"
       using assms by (induct arbitrary: q rule: list_induct2) (auto)
     lemma union_path:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       assumes "length rs = length w" "c.path (union A B) (w || rs) pq"
       obtains
         (a) r p where "rs = map Inl r" "pq = Inl p" "a.path A (w || r) p" |
         (b) s q where "rs = map Inr s" "pq = Inr q" "b.path B (w || s) q"
     proof (cases pq)
       case (Inl p)
       have 1: "rs = map Inl (map projl rs)"
         using assms(2, 3) unfolding Inl by (induct arbitrary: p rule: list_induct2) (auto)
       have 2: "a.path A (w || map projl rs) p"
         using assms(2, 1, 3) unfolding Inl by (induct arbitrary: p rule: list_induct2) (auto)
       show ?thesis using a 1 Inl 2 by this
     next
       case (Inr q)
       have 1: "rs = map Inr (map projr rs)"
         using assms(2, 3) unfolding Inr by (induct arbitrary: q rule: list_induct2) (auto)
       have 2: "b.path B (w || map projr rs) q"
         using assms(2, 1, 3) unfolding Inr by (induct arbitrary: q rule: list_induct2) (auto)
       show ?thesis using b 1 Inr 2 by this
     qed
 
     lemma run_union_a:
       assumes "a.run A (w ||| r) p"
       shows "c.run (union A B) (w ||| smap Inl r) (Inl p)"
       using assms by (coinduction arbitrary: w r p) (force elim: a.run.cases)
     lemma run_union_b:
       assumes "b.run B (w ||| s) q"
       shows "c.run (union A B) (w ||| smap Inr s) (Inr q)"
       using assms by (coinduction arbitrary: w s q) (force elim: b.run.cases)
     lemma union_run:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       assumes "c.run (union A B) (w ||| rs) pq"
       obtains
         (a) r p where "rs = smap Inl r" "pq = Inl p" "a.run A (w ||| r) p" |
         (b) s q where "rs = smap Inr s" "pq = Inr q" "b.run B (w ||| s) q"
     proof (cases pq)
       case (Inl p)
       have 1: "rs = smap Inl (smap projl rs)"
         using assms(2) unfolding Inl by (coinduction arbitrary: w rs p) (force elim: c.run.cases)
       have 2: "a.run A (w ||| smap projl rs) p"
         using assms unfolding Inl by (coinduction arbitrary: w rs p) (force elim: c.run.cases)
       show ?thesis using a 1 Inl 2 by this
     next
       case (Inr q)
       have 1: "rs = smap Inr (smap projr rs)"
         using assms(2) unfolding Inr by (coinduction arbitrary: w rs q) (force elim: c.run.cases)
       have 2: "b.run B (w ||| smap projr rs) q"
         using assms unfolding Inr by (coinduction arbitrary: w rs q) (force elim: c.run.cases)
       show ?thesis using b 1 Inr 2 by this
     qed
 
     lemma union_nodes:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "c.nodes (union A B) \<subseteq> a.nodes A <+> b.nodes B"
     proof
       fix pq
       assume "pq \<in> c.nodes (union A B)"
       then show "pq \<in> a.nodes A <+> b.nodes B" using assms by (induct) (auto 0 3)
     qed
 
     lemma union_nodes_finite[intro]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       assumes "finite (a.nodes A)" "finite (b.nodes B)"
       shows "finite (c.nodes (union A B))"
       using finite_subset union_nodes assms by (auto intro: finite_Plus)
 
   end
 
   locale automaton_union_path =
     automaton_union
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_path automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_path automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_path automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label list \<Rightarrow> 'state\<^sub>1 list \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label list \<Rightarrow> 'state\<^sub>2 list \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label list \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) list \<Rightarrow> 'state\<^sub>1 + 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test\<^sub>1[iff]: "length u = length w \<Longrightarrow> test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (map Inl u) (Inl p) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w u p"
     assumes test\<^sub>2[iff]: "length v = length w \<Longrightarrow> test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (map Inr v) (Inr q) \<longleftrightarrow> test\<^sub>2 c\<^sub>2 w v q"
   begin
 
     lemma union_language[simp]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "c.language (union A B) = a.language A \<union> b.language B"
       using assms unfolding a.language_def b.language_def c.language_def
       by (force intro: path_union_a path_union_b elim!: union_path)
 
   end
 
   locale automaton_union_run =
     automaton_union
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
       automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_run automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state\<^sub>1 set \<Rightarrow> ('label, 'state\<^sub>1) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state\<^sub>1 set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state\<^sub>1) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>1 stream \<Rightarrow> 'state\<^sub>1 \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state\<^sub>2 set \<Rightarrow> ('label, 'state\<^sub>2) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state\<^sub>2 set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state\<^sub>2) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state\<^sub>2 stream \<Rightarrow> 'state\<^sub>2 \<Rightarrow> bool"
     and automaton\<^sub>3 :: "'label set \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans \<Rightarrow>
       'condition\<^sub>3 \<Rightarrow> 'automaton\<^sub>3"
     and alphabet\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'label set"
     and initial\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) set"
     and transition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> ('label, 'state\<^sub>1 + 'state\<^sub>2) trans"
     and condition\<^sub>3 :: "'automaton\<^sub>3 \<Rightarrow> 'condition\<^sub>3"
     and test\<^sub>3 :: "'condition\<^sub>3 \<Rightarrow> 'label stream \<Rightarrow> ('state\<^sub>1 + 'state\<^sub>2) stream \<Rightarrow> 'state\<^sub>1 + 'state\<^sub>2 \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'condition\<^sub>3"
     +
     assumes test\<^sub>1[iff]: "test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (smap Inl u) (Inl p) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 w u p"
     assumes test\<^sub>2[iff]: "test\<^sub>3 (condition c\<^sub>1 c\<^sub>2) w (smap Inr v) (Inr q) \<longleftrightarrow> test\<^sub>2 c\<^sub>2 w v q"
   begin
 
     lemma union_language[simp]:
       assumes "alphabet\<^sub>1 A = alphabet\<^sub>2 B"
       shows "c.language (union A B) = a.language A \<union> b.language B"
       using assms unfolding a.language_def b.language_def c.language_def
       by (auto intro: run_union_a run_union_b elim!: union_run)
 
   end
 
+  locale automaton_intersection_list =
+    a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
+    b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
+    for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
+    and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
+    and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
+    and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
+    and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
+    and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list set \<Rightarrow> ('label, 'state list) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
+    and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
+    and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list set"
+    and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
+    and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
+    +
+    fixes condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
+  begin
+
+    definition intersect :: "'automaton\<^sub>1 list \<Rightarrow> 'automaton\<^sub>2" where
+      "intersect AA \<equiv> automaton\<^sub>2
+        (\<Inter> (alphabet\<^sub>1 ` set AA))
+        (listset (map initial\<^sub>1 AA))
+        (\<lambda> a ps. listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps))
+        (condition (map condition\<^sub>1 AA))"
+
+    lemma intersect_simps[simp]:
+      "alphabet\<^sub>2 (intersect AA) = \<Inter> (alphabet\<^sub>1 ` set AA)"
+      "initial\<^sub>2 (intersect AA) = listset (map initial\<^sub>1 AA)"
+      "transition\<^sub>2 (intersect AA) a ps = listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA ps)"
+      "condition\<^sub>2 (intersect AA) = condition (map condition\<^sub>1 AA)"
+      unfolding intersect_def by auto
+
+    lemma intersect_run_length:
+      assumes "length ps = length AA"
+      assumes "b.run (intersect AA) (w ||| r) ps"
+      assumes "qs \<in> sset r"
+      shows "length qs = length AA"
+    proof -
+      have "pred_stream (\<lambda> qs. length qs = length AA) r"
+        using assms(1, 2) by (coinduction arbitrary: w r ps)
+          (force elim: b.run.cases simp: listset_member list_all2_conv_all_nth)
+      then show ?thesis using assms(3) unfolding stream.pred_set by auto
+    qed
+    lemma intersect_run_stranspose:
+      assumes "length ps = length AA"
+      assumes "b.run (intersect AA) (w ||| r) ps"
+      obtains rs where "r = stranspose rs" "length rs = length AA"
+    proof
+      define rs where "rs \<equiv> map (\<lambda> k. smap (\<lambda> ps. ps ! k) r) [0 ..< length AA]"
+      have "length qs = length AA" if "qs \<in> sset r" for qs using intersect_run_length assms that by this
+      then show "r = stranspose rs"
+        unfolding rs_def by (coinduction arbitrary: r) (auto intro: nth_equalityI simp: comp_def)
+      show "length rs = length AA" unfolding rs_def by auto
+    qed
+
+    lemma run_intersect:
+      assumes "length rs = length AA" "length ps = length AA"
+      assumes "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
+      shows "b.run (intersect AA) (w ||| stranspose rs) ps"
+    using assms
+    proof (coinduction arbitrary: w rs ps)
+      case (run ap r)
+      then show ?case
+      proof (intro conjI exI)
+        show "fst ap \<in> alphabet\<^sub>2 (intersect AA)"
+          using run by (force elim: a.run.cases simp: set_conv_nth)
+        show "snd ap \<in> transition\<^sub>2 (intersect AA) (fst ap) ps"
+          using run by (force elim: a.run.cases simp: listset_member list_all2_conv_all_nth)
+        show "\<forall> k < length AA. a.run' (AA ! k) (stl w ||| map stl rs ! k) (map shd rs ! k)"
+          using run by (force elim: a.run.cases)
+      qed auto
+    qed
+    lemma intersect_run:
+      assumes "length rs = length AA" "length ps = length AA"
+      assumes "b.run (intersect AA) (w ||| stranspose rs) ps"
+      shows "k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
+    using assms
+    proof (coinduction arbitrary: w rs ps)
+      case (run ap wr)
+      then show ?case
+      proof (intro exI conjI)
+        show "fst ap \<in> alphabet\<^sub>1 (AA ! k)"
+          using run by (force elim: b.run.cases)
+        show "snd ap \<in> transition\<^sub>1 (AA ! k) (fst ap) (ps ! k)"
+          using run by (force elim: b.run.cases simp: listset_member list_all2_conv_all_nth)
+        show "b.run' (intersect AA) (stl w ||| stranspose (map stl rs)) (shd (stranspose rs))"
+          using run by (force elim: b.run.cases)
+      qed auto
+    qed
+
+    lemma intersect_nodes: "b.nodes (intersect AA) \<subseteq> listset (map a.nodes AA)"
+    proof
+      show "ps \<in> listset (map a.nodes AA)" if "ps \<in> b.nodes (intersect AA)" for ps
+        using that by (induct) (auto 0 3 simp: listset_member list_all2_conv_all_nth)
+    qed
+
+    lemma intersect_nodes_finite[intro]:
+      assumes "list_all (finite \<circ> a.nodes) AA"
+      shows "finite (b.nodes (intersect AA))"
+    proof (rule finite_subset)
+      show "b.nodes (intersect AA) \<subseteq> listset (map a.nodes AA)" using intersect_nodes by this
+      show "finite (listset (map a.nodes AA))" using list.pred_map assms by auto
+    qed
+
+  end
+
+  locale automaton_intersection_list_run =
+    automaton_intersection_list
+      automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
+      automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 condition +
+    a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
+    b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
+    for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
+    and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
+    and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
+    and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
+    and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
+    and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
+    and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state list set \<Rightarrow> ('label, 'state list) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
+    and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
+    and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'state list set"
+    and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, 'state list) trans"
+    and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
+    and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> 'state list stream \<Rightarrow> 'state list \<Rightarrow> bool"
+    and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
+    +
+    assumes test[iff]: "length rs = length cs \<Longrightarrow> length ps = length cs \<Longrightarrow>
+      test\<^sub>2 (condition cs) w (stranspose rs) ps \<longleftrightarrow> list_all (\<lambda> (c, r, p). test\<^sub>1 c w r p) (cs || rs || ps)"
+  begin
+
+    lemma intersect_language[simp]: "b.language (intersect AA) = \<Inter> (a.language ` set AA)"
+    proof safe
+      fix A w
+      assume 1: "w \<in> b.language (intersect AA)" "A \<in> set AA"
+      obtain r ps where 2:
+        "ps \<in> initial\<^sub>2 (intersect AA)"
+        "b.run (intersect AA) (w ||| r) ps"
+        "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w r ps"
+        using 1(1) by auto
+      have 3: "length ps = length AA" using 2(1) by (simp add: listset_member list_all2_conv_all_nth)
+      obtain rs where 4: "r = stranspose rs" "length rs = length AA"
+        using intersect_run_stranspose 3 2(2) by this
+      obtain k where 5: "k < length AA" "A = AA ! k" using 1(2) unfolding set_conv_nth by auto
+      show "w \<in> a.language A"
+      proof
+        show "ps ! k \<in> initial\<^sub>1 A" using 2(1) 5 by (auto simp: listset_member list_all2_conv_all_nth)
+        show "a.run A (w ||| rs ! k) (ps ! k)" using 2(2) 3 4 5 by (auto intro: intersect_run)
+        show "test\<^sub>1 (condition\<^sub>1 A) w (rs ! k) (ps ! k)" using 2(3) 3 4 5 by (simp add: list_all_length)
+      qed
+    next
+      fix w
+      assume 1: "w \<in> \<Inter> (a.language ` set AA)"
+      have 2: "\<forall> A \<in> set AA. \<exists> r p. p \<in> initial\<^sub>1 A \<and> a.run A (w ||| r) p \<and> test\<^sub>1 (condition\<^sub>1 A) w r p"
+        using 1 by blast
+      obtain rs ps where 3:
+        "length rs = length AA" "length ps = length AA"
+        "\<And> k. k < length AA \<Longrightarrow> ps ! k \<in> initial\<^sub>1 (AA ! k)"
+        "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"
+        "\<And> k. k < length AA \<Longrightarrow> test\<^sub>1 (condition\<^sub>1 (AA ! k)) w (rs ! k) (ps ! k)"
+        using 2
+        unfolding Ball_set list_choice_zip list_choice_pair
+        unfolding list.pred_set set_conv_nth
+        by force
+      show "w \<in> b.language (intersect AA)"
+      proof
+        show "ps \<in> initial\<^sub>2 (intersect AA)" using 3 by (auto simp: listset_member list_all2_conv_all_nth)
+        show "b.run (intersect AA) (w ||| stranspose rs) ps" using 3 by (auto intro: run_intersect)
+        show "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w (stranspose rs) ps" using 3 by (auto simp: list_all_length)
+      qed
+    qed
+
+  end
+
   locale automaton_union_list =
     a: automaton automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 +
     b: automaton automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> (nat \<times> 'state) set \<Rightarrow> ('label, nat \<times> 'state) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> (nat \<times> 'state) set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, nat \<times> 'state) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     +
     fixes condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
   begin
 
     definition union :: "'automaton\<^sub>1 list \<Rightarrow> 'automaton\<^sub>2" where
       "union AA \<equiv> automaton\<^sub>2
         (\<Union> (alphabet\<^sub>1 ` set AA))
         (\<Union> k < length AA. {k} \<times> initial\<^sub>1 (AA ! k))
         (\<lambda> a (k, p). {k} \<times> transition\<^sub>1 (AA ! k) a p)
         (condition (map condition\<^sub>1 AA))"
 
     lemma union_simps[simp]:
       "alphabet\<^sub>2 (union AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       "initial\<^sub>2 (union AA) = (\<Union> k < length AA. {k} \<times> initial\<^sub>1 (AA ! k))"
       "transition\<^sub>2 (union AA) a (k, p) = {k} \<times> transition\<^sub>1 (AA ! k) a p"
       "condition\<^sub>2 (union AA) = condition (map condition\<^sub>1 AA)"
       unfolding union_def by auto
 
     lemma run_union:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       assumes "A \<in> set AA"
       assumes "a.run A (w ||| s) p"
       obtains k where "k < length AA" "A = AA ! k" "b.run (union AA) (w ||| sconst k ||| s) (k, p)"
     proof -
       obtain k where 1: "k < length AA" "A = AA ! k" using assms(2) unfolding set_conv_nth by auto
       show ?thesis
       proof
         show "k < length AA" "A = AA ! k" using 1 by this
         show "b.run (union AA) (w ||| sconst k ||| s) (k, p)"
           using assms 1(2) by (coinduction arbitrary: w s p) (force elim: a.run.cases)
       qed
     qed
     lemma union_run:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       assumes "k < length AA"
       assumes "b.run (union AA) (w ||| r) (k, p)"
       obtains s where "r = sconst k ||| s" "a.run (AA ! k) (w ||| s) p"
     proof
       show "r = sconst k ||| smap snd r"
         using assms by (coinduction arbitrary: w r p) (force elim: b.run.cases)
       show "a.run (AA ! k) (w ||| smap snd r) p"
         using assms by (coinduction arbitrary: w r p) (force elim: b.run.cases)
     qed
 
     lemma union_nodes:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       shows "b.nodes (union AA) \<subseteq> (\<Union> k < length AA. {k} \<times> a.nodes (AA ! k))"
     proof
       show "kp \<in> (\<Union> k < length AA. {k} \<times> a.nodes (AA ! k))" if "kp \<in> b.nodes (union AA)" for kp
         using that assms by (induct) (auto 0 4)
     qed
 
     lemma union_nodes_finite[intro]:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       assumes "list_all (finite \<circ> a.nodes) AA"
       shows "finite (b.nodes (union AA))"
     proof (rule finite_subset)
       show "b.nodes (union AA) \<subseteq> (\<Union> k < length AA. {k} \<times> a.nodes (AA ! k))"
         using union_nodes assms(1) by this
       show "finite (\<Union> k < length AA. {k} \<times> a.nodes' (AA ! k))"
         using assms(2) unfolding list_all_length by auto
     qed
 
   end
 
   locale automaton_union_list_run =
     automaton_union_list
       automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1
       automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 condition +
     a: automaton_run automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_run automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
     for automaton\<^sub>1 :: "'label set \<Rightarrow> 'state set \<Rightarrow> ('label, 'state) trans \<Rightarrow> 'condition\<^sub>1 \<Rightarrow> 'automaton\<^sub>1"
     and alphabet\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'label set"
     and initial\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'state set"
     and transition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> ('label, 'state) trans"
     and condition\<^sub>1 :: "'automaton\<^sub>1 \<Rightarrow> 'condition\<^sub>1"
     and test\<^sub>1 :: "'condition\<^sub>1 \<Rightarrow> 'label stream \<Rightarrow> 'state stream \<Rightarrow> 'state \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> (nat \<times> 'state) set \<Rightarrow> ('label, nat \<times> 'state) trans \<Rightarrow> 'condition\<^sub>2 \<Rightarrow> 'automaton\<^sub>2"
     and alphabet\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'label set"
     and initial\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> (nat \<times> 'state) set"
     and transition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> ('label, nat \<times> 'state) trans"
     and condition\<^sub>2 :: "'automaton\<^sub>2 \<Rightarrow> 'condition\<^sub>2"
     and test\<^sub>2 :: "'condition\<^sub>2 \<Rightarrow> 'label stream \<Rightarrow> (nat \<times> 'state) stream \<Rightarrow> nat \<times> 'state \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
     +
     assumes test[iff]: "k < length cs \<Longrightarrow> test\<^sub>2 (condition cs) w (sconst k ||| r) (k, p) \<longleftrightarrow> test\<^sub>1 (cs ! k) w r p"
   begin
 
     lemma union_language[simp]:
       assumes "\<Inter> (alphabet\<^sub>1 ` set AA) = \<Union> (alphabet\<^sub>1 ` set AA)"
       shows "b.language (union AA) = \<Union> (a.language ` set AA)"
     proof
       show "b.language (union AA) \<subseteq> \<Union> (a.language ` set AA)"
         using assms unfolding a.language_def b.language_def by (force elim: union_run)
       show "\<Union> (a.language ` set AA) \<subseteq> b.language (union AA)"
         using assms unfolding a.language_def b.language_def by (force elim!: run_union)
     qed
 
   end
 
 end
\ No newline at end of file
diff --git a/thys/Transition_Systems_and_Automata/Basic/Sequence.thy b/thys/Transition_Systems_and_Automata/Basic/Sequence.thy
--- a/thys/Transition_Systems_and_Automata/Basic/Sequence.thy
+++ b/thys/Transition_Systems_and_Automata/Basic/Sequence.thy
@@ -1,468 +1,470 @@
 section \<open>Finite and Infinite Sequences\<close>
 
 theory Sequence
 imports
   "Basic"
   "HOL-Library.Stream"
   "HOL-Library.Monad_Syntax"
 begin
 
   subsection \<open>List Basics\<close>
 
   declare upt_Suc[simp del]
   declare last.simps[simp del]
   declare butlast.simps[simp del]
   declare Cons_nth_drop_Suc[simp]
   declare list.pred_True[simp]
 
   lemma list_pred_cases:
     assumes "list_all P xs"
     obtains (nil) "xs = []" | (cons) y ys where "xs = y # ys" "P y" "list_all P ys"
     using assms by (cases xs) (auto)
 
   lemma lists_iff_set: "w \<in> lists A \<longleftrightarrow> set w \<subseteq> A" by auto
 
   lemma fold_const: "fold const xs a = last (a # xs)"
     by (induct xs arbitrary: a) (auto simp: last.simps)
 
   lemma take_Suc: "take (Suc n) xs = (if xs = [] then [] else hd xs # take n (tl xs))"
     by (simp add: take_Suc)
 
   lemma bind_map[simp]: "map f xs \<bind> g = xs \<bind> g \<circ> f" unfolding List.bind_def by simp
 
   lemma ball_bind[iff]: "Ball (set (xs \<bind> f)) P \<longleftrightarrow> (\<forall> x \<in> set xs. \<forall> y \<in> set (f x). P y)"
     unfolding set_list_bind by simp
   lemma bex_bind[iff]: "Bex (set (xs \<bind> f)) P \<longleftrightarrow> (\<exists> x \<in> set xs. \<exists> y \<in> set (f x). P y)"
     unfolding set_list_bind by simp
 
   lemma list_choice: "list_all (\<lambda> x. \<exists> y. P x y) xs \<longleftrightarrow> (\<exists> ys. list_all2 P xs ys)"
     by (induct xs) (auto simp: list_all2_Cons1)
 
   lemma listset_member: "ys \<in> listset XS \<longleftrightarrow> list_all2 (\<in>) ys XS"
     by (induct XS arbitrary: ys) (auto simp: set_Cons_def list_all2_Cons2)
   lemma listset_empty[iff]: "listset XS = {} \<longleftrightarrow> \<not> list_all (\<lambda> A. A \<noteq> {}) XS"
     by (induct XS) (auto simp: set_Cons_def)
   lemma listset_finite[iff]:
     assumes "list_all (\<lambda> A. A \<noteq> {}) XS"
     shows "finite (listset XS) \<longleftrightarrow> list_all finite XS"
   using assms
   proof (induct XS)
     case Nil
     show ?case by simp
   next
     case (Cons A XS)
     note [simp] = finite_image_iff finite_cartesian_product_iff
     have "listset (A # XS) = case_prod Cons ` (A \<times> listset XS)" by (auto simp: set_Cons_def)
     also have "finite \<dots> \<longleftrightarrow> finite (A \<times> listset XS)" by (simp add: inj_on_def)
     also have "\<dots> \<longleftrightarrow> finite A \<and> finite (listset XS)" using Cons(2) by simp
     also have "finite (listset XS) \<longleftrightarrow> list_all finite XS" using Cons by simp
     also have "finite A \<and> \<dots> \<longleftrightarrow> list_all finite (A # XS)" by simp
     finally show ?case by this
   qed
   lemma listset_finite'[intro]:
     assumes "list_all finite XS"
     shows "finite (listset XS)"
     using infinite_imp_nonempty assms by blast
   lemma listset_card[simp]: "card (listset XS) = prod_list (map card XS)"
   proof (induct XS)
     case Nil
     show ?case by simp
   next
     case (Cons A XS)
     have 1: "inj (case_prod Cons)" unfolding inj_def by simp
     have "listset (A # XS) = case_prod Cons ` (A \<times> listset XS)" by (auto simp: set_Cons_def)
     also have "card \<dots> = card (A \<times> listset XS)" using card_image 1 by auto
     also have "\<dots> = card A * card (listset XS)" using card_cartesian_product by this
     also have "card (listset XS) = prod_list (map card XS)" using Cons by this
     also have "card A * \<dots> = prod_list (map card (A # XS))" by simp
     finally show ?case by this
   qed
 
   subsection \<open>Stream Basics\<close>
 
   declare stream.map_id[simp]
   declare stream.set_map[simp]
   declare stream.set_sel(1)[intro!, simp]
   declare stream.pred_True[simp]
   declare stream.pred_map[iff]
   declare stream.rel_map[iff]
   declare shift_simps[simp del]
   declare stake_sdrop[simp]
   declare stake_siterate[simp del]
   declare sdrop_snth[simp]
 
   lemma stream_pred_cases:
     assumes "pred_stream P xs"
     obtains (scons) y ys where "xs = y ## ys" "P y" "pred_stream P ys"
     using assms by (cases xs) (auto)
 
   lemma stream_rel_coinduct[case_names stream_rel, coinduct pred: stream_all2]:
     assumes "R u v"
     assumes "\<And> a u b v. R (a ## u) (b ## v) \<Longrightarrow> P a b \<and> R u v"
     shows "stream_all2 P u v"
     using assms by (coinduct) (metis stream.collapse)
   lemma stream_rel_coinduct_shift[case_names stream_rel, consumes 1]:
     assumes "R u v"
     assumes "\<And> u v. R u v \<Longrightarrow>
       \<exists> u\<^sub>1 u\<^sub>2 v\<^sub>1 v\<^sub>2. u = u\<^sub>1 @- u\<^sub>2 \<and> v = v\<^sub>1 @- v\<^sub>2 \<and> u\<^sub>1 \<noteq> [] \<and> v\<^sub>1 \<noteq> [] \<and> list_all2 P u\<^sub>1 v\<^sub>1 \<and> R u\<^sub>2 v\<^sub>2"
     shows "stream_all2 P u v"
   proof -
     have "\<exists> u\<^sub>1 u\<^sub>2 v\<^sub>1 v\<^sub>2. u = u\<^sub>1 @- u\<^sub>2 \<and> v = v\<^sub>1 @- v\<^sub>2 \<and> list_all2 P u\<^sub>1 v\<^sub>1 \<and> R u\<^sub>2 v\<^sub>2"
       using assms(1) by force
     then show ?thesis using assms(2) by (coinduct) (force elim: list.rel_cases)
   qed
 
   lemma stream_pred_coinduct[case_names stream_pred, coinduct pred: pred_stream]:
     assumes "R w"
     assumes "\<And> a w. R (a ## w) \<Longrightarrow> P a \<and> R w"
     shows "pred_stream P w"
     using assms unfolding stream.pred_rel eq_onp_def by (coinduction arbitrary: w) (auto)
   lemma stream_pred_coinduct_shift[case_names stream_pred, consumes 1]:
     assumes "R w"
     assumes "\<And> w. R w \<Longrightarrow> \<exists> u v. w = u @- v \<and> u \<noteq> [] \<and> list_all P u \<and> R v"
     shows "pred_stream P w"
   proof -
     have "\<exists> u v. w = u @- v \<and> list_all P u \<and> R v"
       using assms(1) by (metis list_all_simps(2) shift.simps(1))
     then show ?thesis using assms(2) by (coinduct) (force elim: list_pred_cases)
   qed
   lemma stream_pred_flat_coinduct[case_names stream_pred, consumes 1]:
     assumes "R ws"
     assumes "\<And> w ws. R (w ## ws) \<Longrightarrow> w \<noteq> [] \<and> list_all P w \<and> R ws"
     shows "pred_stream P (flat ws)"
     using assms
     by (coinduction arbitrary: ws rule: stream_pred_coinduct_shift) (metis stream.exhaust flat_Stream)
 
   lemmas stream_eq_coinduct[case_names stream_eq, coinduct pred: HOL.eq] =
     stream_rel_coinduct[where ?P = HOL.eq, unfolded stream.rel_eq]
   lemmas stream_eq_coinduct_shift[case_names stream_eq, consumes 1] =
     stream_rel_coinduct_shift[where ?P = HOL.eq, unfolded stream.rel_eq list.rel_eq]
 
   lemma stream_pred_shift[iff]: "pred_stream P (u @- v) \<longleftrightarrow> list_all P u \<and> pred_stream P v"
     by (induct u) (auto)
   lemma stream_rel_shift[iff]:
     assumes "length u\<^sub>1 = length v\<^sub>1"
     shows "stream_all2 P (u\<^sub>1 @- u\<^sub>2) (v\<^sub>1 @- v\<^sub>2) \<longleftrightarrow> list_all2 P u\<^sub>1 v\<^sub>1 \<and> stream_all2 P u\<^sub>2 v\<^sub>2"
     using assms by (induct rule: list_induct2) (auto)
 
   lemma sset_subset_stream_pred: "sset w \<subseteq> A \<longleftrightarrow> pred_stream (\<lambda> a. a \<in> A) w"
     unfolding stream.pred_set by auto
 
   lemma eq_scons: "w = a ## v \<longleftrightarrow> a = shd w \<and> v = stl w" by auto
   lemma scons_eq: "a ## v = w \<longleftrightarrow> shd w = a \<and> stl w = v" by auto
   lemma eq_shift: "w = u @- v \<longleftrightarrow> stake (length u) w = u \<and> sdrop (length u) w = v"
     by (induct u arbitrary: w) (force+)
   lemma shift_eq: "u @- v = w \<longleftrightarrow> u = stake (length u) w \<and> v = sdrop (length u) w"
     by (induct u arbitrary: w) (force+)
   lemma scons_eq_shift: "a ## w = u @- v \<longleftrightarrow> ([] = u \<and> a ## w = v) \<or> (\<exists> u'. a # u' = u \<and> w = u' @- v)"
     by (cases u) (auto)
   lemma shift_eq_scons: "u @- v = a ## w \<longleftrightarrow> (u = [] \<and> v = a ## w) \<or> (\<exists> u'. u = a # u' \<and> u' @- v = w)"
     by (cases u) (auto)
 
   lemma stream_all2_sset1:
     assumes "stream_all2 P xs ys"
     shows "\<forall> x \<in> sset xs. \<exists> y \<in> sset ys. P x y"
   proof -
     have "pred_stream (\<lambda> x. \<exists> y \<in> S. P x y) xs" if "sset ys \<subseteq> S" for S
       using assms that by (coinduction arbitrary: xs ys) (force elim: stream.rel_cases)
     then show ?thesis unfolding stream.pred_set by auto
   qed
   lemma stream_all2_sset2:
     assumes "stream_all2 P xs ys"
     shows "\<forall> y \<in> sset ys. \<exists> x \<in> sset xs. P x y"
   proof -
     have "pred_stream (\<lambda> y. \<exists> x \<in> S. P x y) ys" if "sset xs \<subseteq> S" for S
       using assms that by (coinduction arbitrary: xs ys) (force elim: stream.rel_cases)
     then show ?thesis unfolding stream.pred_set by auto
   qed
 
   lemma smap_eq_scons[iff]: "smap f xs = y ## ys \<longleftrightarrow> f (shd xs) = y \<and> smap f (stl xs) = ys"
     using smap_ctr by metis
   lemma scons_eq_smap[iff]: "y ## ys = smap f xs \<longleftrightarrow> y = f (shd xs) \<and> ys = smap f (stl xs)"
     using smap_ctr by metis
   lemma smap_eq_shift[iff]:
     "smap f w = u @- v \<longleftrightarrow> (\<exists> w\<^sub>1 w\<^sub>2. w = w\<^sub>1 @- w\<^sub>2 \<and> map f w\<^sub>1 = u \<and> smap f w\<^sub>2 = v)"
     using sdrop_smap eq_shift stake_sdrop stake_smap by metis
   lemma shift_eq_smap[iff]:
     "u @- v = smap f w \<longleftrightarrow> (\<exists> w\<^sub>1 w\<^sub>2. w = w\<^sub>1 @- w\<^sub>2 \<and> u = map f w\<^sub>1 \<and> v = smap f w\<^sub>2)"
     using sdrop_smap eq_shift stake_sdrop stake_smap by metis
 
   lemma szip_eq_scons[iff]: "szip xs ys = z ## zs \<longleftrightarrow> (shd xs, shd ys) = z \<and> szip (stl xs) (stl ys) = zs"
     using szip.ctr stream.inject by metis
   lemma scons_eq_szip[iff]: "z ## zs = szip xs ys \<longleftrightarrow> z = (shd xs, shd ys) \<and> zs = szip (stl xs) (stl ys)"
     using szip.ctr stream.inject by metis
 
   lemma siterate_eq_scons[iff]: "siterate f s = a ## w \<longleftrightarrow> s = a \<and> siterate f (f s) = w"
     using siterate.ctr stream.inject by metis
   lemma scons_eq_siterate[iff]: "a ## w = siterate f s \<longleftrightarrow> a = s \<and> w = siterate f (f s)"
     using siterate.ctr stream.inject by metis
 
   lemma snth_0: "(a ## w) !! 0 = a" by simp
   lemma eqI_snth:
     assumes "\<And> i. u !! i = v !! i"
     shows "u = v"
     using assms by (coinduction arbitrary: u v) (metis stream.sel snth.simps)
 
   lemma stream_pred_snth: "pred_stream P w \<longleftrightarrow> (\<forall> i. P (w !! i))"
     unfolding stream.pred_set sset_range by simp
   lemma stream_rel_snth: "stream_all2 P u v \<longleftrightarrow> (\<forall> i. P (u !! i) (v !! i))"
   proof safe
     show "P (u !! i) (v !! i)" if "stream_all2 P u v" for i
       using that by (induct i arbitrary: u v) (auto elim: stream.rel_cases)
     show "stream_all2 P u v" if "\<forall> i. P (u !! i) (v !! i)"
       using that by (coinduct) (metis snth_0 snth_Stream)
   qed
 
   lemma stream_rel_pred_szip: "stream_all2 P u v \<longleftrightarrow> pred_stream (case_prod P) (szip u v)"
     unfolding stream_pred_snth stream_rel_snth by simp
 
   lemma sconst_eq[iff]: "sconst x = sconst y \<longleftrightarrow> x = y" by (auto) (metis siterate.simps(1))
   lemma stream_pred__sconst[iff]: "pred_stream P (sconst x) \<longleftrightarrow> P x"
     unfolding stream_pred_snth by simp
   lemma stream_rel_sconst[iff]: "stream_all2 P (sconst x) (sconst y) \<longleftrightarrow> P x y"
     unfolding stream_rel_snth by simp
 
   lemma set_sset_stake[intro!, simp]: "set (stake n xs) \<subseteq> sset xs"
     by (metis sset_shift stake_sdrop sup_ge1)
   lemma sset_sdrop[intro!, simp]: "sset (sdrop n xs) \<subseteq> sset xs"
     by (metis sset_shift stake_sdrop sup_ge2)
 
   lemma set_stake_snth: "x \<in> set (stake n xs) \<longleftrightarrow> (\<exists> i < n. xs !! i = x)"
     unfolding in_set_conv_nth by auto
 
   (* TODO: these should be generated by the transfer option on the primcorec definitions *)
   (* TODO: transfer_prover cannot handle corecursive definitions due to the \<lambda>(s1, s2). undefined
     abortion term that is never used *)
   lemma szip_transfer[transfer_rule]:
     includes lifting_syntax
     shows "(stream_all2 A ===> stream_all2 B ===> stream_all2 (rel_prod A B)) szip szip"
     by (intro rel_funI, coinduction) (force elim: stream.rel_cases)
   lemma siterate_transfer[transfer_rule]:
     includes lifting_syntax
     shows "((A ===> A) ===> A ===> stream_all2 A) siterate siterate"
     by (intro rel_funI, coinduction) (force dest: rel_funD)
 
   lemma split_stream_first:
     assumes "A \<inter> sset xs \<noteq> {}"
     obtains ys a zs
     where "xs = ys @- a ## zs" "A \<inter> set ys = {}" "a \<in> A"
   proof
     let ?n = "LEAST n. xs !! n \<in> A"
     have 1: "xs !! n \<notin> A" if "n < ?n" for n using that by (metis (full_types) not_less_Least)
     show "xs = stake ?n xs @- (xs !! ?n) ## sdrop (Suc ?n) xs" using id_stake_snth_sdrop by blast
     show "A \<inter> set (stake ?n xs) = {}" using 1 by (metis (no_types, lifting) disjoint_iff_not_equal set_stake_snth)
     show "xs !! ?n \<in> A" using assms unfolding sset_range by (auto intro: LeastI)
   qed
   lemma split_stream_first':
     assumes "x \<in> sset xs"
     obtains ys zs
     where "xs = ys @- x ## zs" "x \<notin> set ys"
   proof
     let ?n = "LEAST n. xs !! n = x"
     have 1: "xs !! ?n = x" using assms unfolding sset_range by (auto intro: LeastI)
     have 2: "xs !! n \<noteq> x" if "n < ?n" for n using that by (metis (full_types) not_less_Least)
     show "xs = stake ?n xs @- x ## sdrop (Suc ?n) xs" using 1 by (metis id_stake_snth_sdrop)
     show "x \<notin> set (stake ?n xs)" using 2 by (meson set_stake_snth)
   qed
 
   lemma streams_UNIV[iff]: "streams A = UNIV \<longleftrightarrow> A = UNIV"
   proof
     show "A = UNIV \<Longrightarrow> streams A = UNIV" by simp
   next
     assume "streams A = UNIV"
     then have "w \<in> streams A" for w by simp
     then have "sset w \<subseteq> A" for w unfolding streams_iff_sset by this
     then have "sset (sconst a) \<subseteq> A" for a by blast
     then have "a \<in> A" for a by simp
     then show "A = UNIV" by auto
   qed
+  lemma streams_int[simp]: "streams (A \<inter> B) = streams A \<inter> streams B" by (auto iff: streams_iff_sset)
+  lemma streams_Int[simp]: "streams (\<Inter> S) = \<Inter> (streams ` S)" by (auto iff: streams_iff_sset)
 
   lemma pred_list_listsp[pred_set_conv]: "list_all = listsp"
     unfolding list.pred_set by auto
   lemma pred_stream_streamsp[pred_set_conv]: "pred_stream = streamsp"
     unfolding stream.pred_set streams_iff_sset[to_pred] by auto
 
   subsection \<open>The scan Function\<close>
 
   primrec (transfer) scan :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b list" where
     "scan f [] a = []" | "scan f (x # xs) a = f x a # scan f xs (f x a)"
 
   lemma scan_append[simp]: "scan f (xs @ ys) a = scan f xs a @ scan f ys (fold f xs a)"
     by (induct xs arbitrary: a) (auto)
 
   lemma scan_eq_nil[iff]: "scan f xs a = [] \<longleftrightarrow> xs = []" by (cases xs) (auto)
   lemma scan_eq_cons[iff]:
     "scan f xs a = b # w \<longleftrightarrow> (\<exists> y ys. xs = y # ys \<and> f y a = b \<and> scan f ys (f y a) = w)"
     by (cases xs) (auto)
   lemma scan_eq_append[iff]:
     "scan f xs a = u @ v \<longleftrightarrow> (\<exists> ys zs. xs = ys @ zs \<and> scan f ys a = u \<and> scan f zs (fold f ys a) = v)"
     by (induct u arbitrary: xs a) (auto, metis append_Cons fold_simps(2), blast)
 
   lemma scan_length[simp]: "length (scan f xs a) = length xs"
     by (induct xs arbitrary: a) (auto)
   (* TODO: figure out how this is used, should it be a simp rule? or flipped? also target_alt_def *)
   lemma scan_last: "last (a # scan f xs a) = fold f xs a"
     by (induct xs arbitrary: a) (auto simp: last.simps)
   lemma scan_butlast[simp]: "scan f (butlast xs) a = butlast (scan f xs a)"
     by (induct xs arbitrary: a) (auto simp: butlast.simps)
 
   lemma scan_const[simp]: "scan const xs a = xs"
     by (induct xs arbitrary: a) (auto)
   lemma scan_nth[simp]:
     assumes "i < length (scan f xs a)"
     shows "scan f xs a ! i = fold f (take (Suc i) xs) a"
     using assms by (cases xs, simp, induct i arbitrary: xs a, auto simp: take_Suc neq_Nil_conv)
   lemma scan_map[simp]: "scan f (map g xs) a = scan (f \<circ> g) xs a"
     by (induct xs arbitrary: a) (auto)
   lemma scan_take[simp]: "take k (scan f xs a) = scan f (take k xs) a"
     by (induct k arbitrary: xs a) (auto simp: take_Suc neq_Nil_conv)
   lemma scan_drop[simp]: "drop k (scan f xs a) = scan f (drop k xs) (fold f (take k xs) a)"
     by (induct k arbitrary: xs a) (auto simp: take_Suc neq_Nil_conv)
 
   primcorec (transfer) sscan :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a stream \<Rightarrow> 'b \<Rightarrow> 'b stream" where
     "sscan f xs a = f (shd xs) a ## sscan f (stl xs) (f (shd xs) a)"
 
   lemma sscan_scons[simp]: "sscan f (x ## xs) a = f x a ## sscan f xs (f x a)"
     by (simp add: stream.expand)
   lemma sscan_shift[simp]: "sscan f (xs @- ys) a = scan f xs a @- sscan f ys (fold f xs a)"
     by (induct xs arbitrary: a) (auto)
 
   lemma sscan_eq_scons[iff]:
     "sscan f xs a = b ## w \<longleftrightarrow> f (shd xs) a = b \<and> sscan f (stl xs) (f (shd xs) a) = w"
     using sscan.ctr stream.inject by metis
   lemma scons_eq_sscan[iff]:
     "b ## w = sscan f xs a \<longleftrightarrow> b = f (shd xs) a \<and> w = sscan f (stl xs) (f (shd xs) a)"
     using sscan.ctr stream.inject by metis
 
   lemma sscan_const[simp]: "sscan const xs a = xs"
     by (coinduction arbitrary: xs a) (auto)
   lemma sscan_snth[simp]: "sscan f xs a !! i = fold f (stake (Suc i) xs) a"
     by (induct i arbitrary: xs a) (auto)
   lemma sscan_scons_snth[simp]: "(a ## sscan f xs a) !! i = fold f (stake i xs) a"
     by (induct i arbitrary: xs a) (auto)
   lemma sscan_smap[simp]: "sscan f (smap g xs) a = sscan (f \<circ> g) xs a"
     by (coinduction arbitrary: xs a) (auto)
   lemma sscan_stake[simp]: "stake k (sscan f xs a) = scan f (stake k xs) a"
     by (induct k arbitrary: a xs) (auto)
   lemma sscan_sdrop[simp]: "sdrop k (sscan f xs a) = sscan f (sdrop k xs) (fold f (stake k xs) a)"
     by (induct k arbitrary: a xs) (auto)
 
   subsection \<open>Transposing Streams\<close>
 
   primcorec (transfer) stranspose :: "'a stream list \<Rightarrow> 'a list stream" where
     "stranspose ws = map shd ws ## stranspose (map stl ws)"
 
   lemma stranspose_eq_scons[iff]: "stranspose ws = a ## w \<longleftrightarrow> map shd ws = a \<and> stranspose (map stl ws) = w"
     using stranspose.ctr stream.inject by metis
   lemma scons_eq_stranspose[iff]: "a ## w = stranspose ws \<longleftrightarrow> a = map shd ws \<and> w = stranspose (map stl ws)"
     using stranspose.ctr stream.inject by metis
 
   lemma stranspose_nil[simp]: "stranspose [] = sconst []" by coinduction auto
   lemma stranspose_cons[simp]: "stranspose (w # ws) = smap2 Cons w (stranspose ws)"
     by (coinduction arbitrary: w ws) (metis list.simps(9) smap2.simps stranspose.simps stream.sel)
 
   lemma snth_stranspose[simp]: "stranspose ws !! k = map (\<lambda> w. w !! k) ws" by (induct k arbitrary: ws) (auto)
   lemma stranspose_nth[simp]:
     assumes "k < length ws"
     shows "smap (\<lambda> xs. xs ! k) (stranspose ws) = ws ! k"
     using assms by (auto intro: eqI_snth)
 
   subsection \<open>Distinct Streams\<close>
 
   coinductive sdistinct :: "'a stream \<Rightarrow> bool" where
     scons[intro!]: "x \<notin> sset xs \<Longrightarrow> sdistinct xs \<Longrightarrow> sdistinct (x ## xs)"
 
   lemma sdistinct_scons_elim[elim!]:
     assumes "sdistinct (x ## xs)"
     obtains "x \<notin> sset xs" "sdistinct xs"
     using assms by (auto elim: sdistinct.cases)
 
   lemma sdistinct_coinduct[case_names sdistinct, coinduct pred: sdistinct]:
     assumes "P xs"
     assumes "\<And> x xs. P (x ## xs) \<Longrightarrow> x \<notin> sset xs \<and> P xs"
     shows "sdistinct xs"
     using stream.collapse sdistinct.coinduct assms by metis
 
   lemma sdistinct_shift[intro!]:
     assumes "distinct xs" "sdistinct ys" "set xs \<inter> sset ys = {}"
     shows "sdistinct (xs @- ys)"
     using assms by (induct xs) (auto)
   lemma sdistinct_shift_elim[elim!]:
     assumes "sdistinct (xs @- ys)"
     obtains "distinct xs" "sdistinct ys" "set xs \<inter> sset ys = {}"
     using assms by (induct xs) (auto)
 
   lemma sdistinct_infinite_sset:
     assumes "sdistinct w"
     shows "infinite (sset w)"
     using assms by (coinduction arbitrary: w) (force elim: sdistinct.cases)
 
   lemma not_sdistinct_decomp:
     assumes "\<not> sdistinct w"
     obtains u v a w'
     where "w = u @- a ## v @- a ## w'"
   proof (rule ccontr)
     assume 1: "\<not> thesis"
     assume 2: "w = u @- a ## v @- a ## w' \<Longrightarrow> thesis" for u a v w'
     have 3: "\<forall> u v a w'. w \<noteq> u @- a ## v @- a ## w'" using 1 2 by auto
     have 4: "sdistinct w" using 3 by (coinduct) (metis id_stake_snth_sdrop imageE shift.simps sset_range)
     show False using assms 4 by auto
   qed
 
   subsection \<open>Sorted Streams\<close>
 
   coinductive (in order) sascending :: "'a stream \<Rightarrow> bool" where
     "a \<le> b \<Longrightarrow> sascending (b ## w) \<Longrightarrow> sascending (a ## b ## w)"
 
   coinductive (in order) sdescending :: "'a stream \<Rightarrow> bool" where
     "a \<ge> b \<Longrightarrow> sdescending (b ## w) \<Longrightarrow> sdescending (a ## b ## w)"
 
   lemma sdescending_coinduct[case_names sdescending, coinduct pred: sdescending]:
     assumes "P w"
     assumes "\<And> a b w. P (a ## b ## w) \<Longrightarrow> a \<ge> b \<and> P (b ## w)"
     shows "sdescending w"
     using stream.collapse sdescending.coinduct assms by (metis (no_types))
 
   lemma sdescending_sdrop:
     assumes "sdescending w"
     shows "sdescending (sdrop k w)"
     using assms by (induct k) (auto, metis sdescending.cases sdrop_stl stream.sel(2))
 
   lemma sdescending_snth_antimono:
     assumes "sdescending w"
     shows "antimono (snth w)"
   unfolding antimono_iff_le_Suc
   proof
     fix k
     have "sdescending (sdrop k w)" using sdescending_sdrop assms by this
     then obtain a b v where 2: "sdrop k w = a ## b ## v" "a \<ge> b" by rule
     then show "w !! k \<ge> w !! Suc k" by (metis sdrop_simps stream.sel)
   qed
 
   lemma sdescending_stuck:
     fixes w :: "'a :: wellorder stream"
     assumes "sdescending w"
     obtains k
     where "sdrop k w = sconst (w !! k)"
   using assms
   proof (induct "w !! 0" arbitrary: w thesis rule: less_induct)
     case less
     show ?case
     proof (cases "w = sconst (w !! 0)")
       case True
       show ?thesis using less(2)[of 0] True by simp
     next
       case False
       obtain k where 1: "w !! k \<noteq> w !! 0"
         using False by (metis empty_iff eqI_snth insert_iff snth_sset sset_sconst)
       have 2: "antimono (snth w)" using sdescending_snth_antimono less(3) by this
       have 3: "w !! k \<le> w !! 0" using 1 2 by (blast dest: antimonoD)
       have 4: "sdrop k w !! 0 < w !! 0" using 1 3 by auto
       have 5: "sdescending (sdrop k w)" using sdescending_sdrop less(3) by this
       obtain l where 5: "sdrop l (sdrop k w) = sconst (sdrop k w !! l)"
         using less(1)[OF 4 _ 5] by this
       show ?thesis using less(2) 5 by simp
     qed
   qed
 
 end