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
@@ -1,556 +1,556 @@
 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
 
     lemma path_alt_def: "path A w p \<longleftrightarrow> set w \<subseteq> alphabet A"
     unfolding lists_iff_set[symmetric]
     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]
     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_target) *)
+  (* TODO: create analogous locale for DFAs (automaton_path) *)
   locale automaton_trace =
     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> 'state stream \<Rightarrow> bool"
   begin
 
     definition language :: "'automaton \<Rightarrow> 'label stream set" where
       "language A \<equiv> {w. run A w (initial A) \<and> test (condition A) (trace A w (initial A))}"
   
     lemma language[intro]:
       assumes "run A w (initial A)" "test (condition A) (trace A w (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) (trace A w (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
 
   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> 'state 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> 'state pred gen"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'state 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> 'state degen pred"
   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) p k))
         (degen (condition\<^sub>1 A))"
 
     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) p k)"
       "condition\<^sub>2 (degeneralize A) = degen (condition\<^sub>1 A)"
       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)) (butlast (p # 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)) (p # 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)) (p ## a.trace A w p) k"
       by (coinduction arbitrary: w p k) (auto)
 
     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_trace =
     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 +
     a: automaton_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_trace 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> 'state 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> 'state pred gen"
     and test\<^sub>1 :: "'state pred gen \<Rightarrow> 'state stream \<Rightarrow> bool"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'state 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> 'state degen pred"
     and test\<^sub>2 :: "'state degen pred \<Rightarrow> 'state degen stream \<Rightarrow> bool"
     +
     assumes test[iff]: "test\<^sub>2 (degen cs) (r ||| sscan (count cs) (p ## r) k) \<longleftrightarrow> test\<^sub>1 cs r"
   begin
 
     lemma degeneralize_language[simp]: "b.language (degeneralize A) = a.language A"
       by (force simp del: sscan_scons)
 
   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_trace =
     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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_trace 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> 'state\<^sub>1 stream \<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> 'state\<^sub>2 stream \<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> ('state\<^sub>1 \<times> 'state\<^sub>2) stream \<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) (u ||| v) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 u \<and> test\<^sub>2 c\<^sub>2 v"
   begin
 
     lemma combine_language[simp]: "c.language (combine A B) = a.language A \<inter> b.language B" by force
 
   end
 
   locale automaton_union_trace =
     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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
     c: automaton_trace 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> 'state\<^sub>1 stream \<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> 'state\<^sub>2 stream \<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> ('state\<^sub>1 \<times> 'state\<^sub>2) stream \<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) (u ||| v) \<longleftrightarrow> test\<^sub>1 c\<^sub>1 u \<or> test\<^sub>2 c\<^sub>2 v"
   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)
         (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"
       "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)
     lemma combine_nodes_length:
       assumes "pp \<in> b.nodes (combine AA)"
       shows "length pp = 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)"
       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_trace =
     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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_trace 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> 'state stream \<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> 'state list stream \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
     +
     assumes test[iff]: "test\<^sub>2 (condition cs) w \<longleftrightarrow>
       (\<forall> k < length cs. test\<^sub>1 (cs ! k) (smap (\<lambda> pp. pp ! k) w))"
   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
       by (auto simp: combine_trace_smap iff: in_set_conv_nth)
 
   end
 
   locale automaton_union_list_trace =
     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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
     b: automaton_trace 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> 'state stream \<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> 'state list stream \<Rightarrow> bool"
     and condition :: "'condition\<^sub>1 list \<Rightarrow> 'condition\<^sub>2"
     +
     assumes test[iff]: "test\<^sub>2 (condition cs) w \<longleftrightarrow>
       (\<exists> k < length cs. test\<^sub>1 (cs ! k) (smap (\<lambda> pp. pp ! k) w))"
   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
       by (auto simp: combine_trace_smap iff: in_set_conv_nth) (metis INT_subset_iff in_set_conv_nth)
 
   end
 
 end
\ No newline at end of file
diff --git a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA.thy b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA.thy
--- a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA.thy
@@ -1,100 +1,100 @@
 section \<open>Nondeterministic Büchi Automata\<close>
 
 theory NBA
 imports "../Nondeterministic"
 begin
 
   datatype ('label, 'state) nba = nba
     (alphabet: "'label set")
     (initial: "'state set")
     (transition: "'label \<Rightarrow> 'state \<Rightarrow> 'state set")
     (accepting: "'state pred")
 
   global_interpretation nba: automaton nba alphabet initial transition accepting
     defines path = nba.path and run = nba.run and reachable = nba.reachable and nodes = nba.nodes
     by unfold_locales auto
-  global_interpretation nba: automaton_trace nba alphabet initial transition accepting "\<lambda> P w r p. infs P r"
+  global_interpretation nba: automaton_run nba alphabet initial transition accepting "\<lambda> P w r p. infs P r"
     defines language = nba.language
     by standard
 
   abbreviation target where "target \<equiv> nba.target"
   abbreviation states where "states \<equiv> nba.states"
   abbreviation trace where "trace \<equiv> nba.trace"
   abbreviation successors where "successors \<equiv> nba.successors TYPE('label)"
 
   instantiation nba :: (type, type) order
   begin
 
     definition less_eq_nba :: "('a, 'b) nba \<Rightarrow> ('a, 'b) nba \<Rightarrow> bool" where
       "A \<le> B \<equiv> alphabet A \<le> alphabet B \<and> initial A \<le> initial B \<and>
         transition A \<le> transition B \<and> accepting A \<le> accepting B"
     definition less_nba :: "('a, 'b) nba \<Rightarrow> ('a, 'b) nba \<Rightarrow> bool" where
       "less_nba A B \<equiv> A \<le> B \<and> A \<noteq> B"
 
     instance by (intro_classes) (auto simp: less_eq_nba_def less_nba_def nba.expand)
 
   end
 
   lemma nodes_mono: "mono nodes"
   proof
     fix A B :: "('label, 'state) nba"
     assume 1: "A \<le> B"
     have 2: "alphabet A \<subseteq> alphabet B" using 1 unfolding less_eq_nba_def by auto
     have 3: "initial A \<subseteq> initial B" using 1 unfolding less_eq_nba_def by auto
     have 4: "transition A a p \<subseteq> transition B a p" for a p using 1 unfolding less_eq_nba_def le_fun_def by auto
     have 5: "p \<in> nodes B" if "p \<in> nodes A" for p using that 2 3 4 by induct fastforce+
     show "nodes A \<subseteq> nodes B" using 5 by auto
   qed
 
   lemma language_mono: "mono language"
   proof
     fix A B :: "('label, 'state) nba"
     assume 1: "A \<le> B"
     have 2: "alphabet A \<subseteq> alphabet B" using 1 unfolding less_eq_nba_def by auto
     have 3: "initial A \<subseteq> initial B" using 1 unfolding less_eq_nba_def by auto
     have 4: "transition A a p \<subseteq> transition B a p" for a p using 1 unfolding less_eq_nba_def le_fun_def by auto
     have 5: "accepting A p \<Longrightarrow> accepting B p" for p using 1 unfolding less_eq_nba_def by auto
     have 6: "run B wr p" if "run A wr p" for wr p using that 2 4 by coinduct auto
     have 7: "infs (accepting B) w" if "infs (accepting A) w" for w using infs_mono that 5 by metis
     show "language A \<subseteq> language B" using 3 6 7 by blast
   qed
 
   lemma simulation_language:
     assumes "alphabet A \<subseteq> alphabet B"
     assumes "\<And> p. p \<in> initial A \<Longrightarrow> \<exists> q \<in> initial B. (p, q) \<in> R"
     assumes "\<And> a p p' q. p' \<in> transition A a p \<Longrightarrow> (p, q) \<in> R \<Longrightarrow> \<exists> q' \<in> transition B a q. (p', q') \<in> R"
     assumes "\<And> p q. (p, q) \<in> R \<Longrightarrow> accepting A p \<Longrightarrow> accepting B q"
     shows "language A \<subseteq> language B"
   proof
     fix w
     assume 1: "w \<in> language A"
     obtain r p where 2: "p \<in> initial A" "run A (w ||| r) p" "infs (accepting A) r" using 1 by rule
     define P where "P n q \<equiv> (target (stake n (w ||| r)) p, q) \<in> R" for n q
     obtain q where 3: "q \<in> initial B" "(p, q) \<in> R" using assms(2) 2(1) by auto
     obtain ws where 4:
       "run B ws q" "\<And> i. P (0 + i) (target (stake i ws) q)" "\<And> i. fst (ws !! i) = w !! (0 + i)"
     proof (rule nba.invariant_run_index)
       have "stake k (w ||| r) @- (w !! k, target (stake (Suc k) (w ||| r)) p) ##
         sdrop (Suc k) (w ||| r) = w ||| r" for k
         by (metis id_stake_snth_sdrop snth_szip sscan_snth szip_smap_snd nba.trace_alt_def)
       also have "run A \<dots> p" using 2(2) by this
       finally show "\<exists> a. (fst a \<in> alphabet B \<and> snd a \<in> transition B (fst a) q) \<and>
         P (Suc n) (snd a) \<and> fst a = w !! n" if "P n q" for n q
         using assms(1, 3) that unfolding P_def by fastforce
       show "P 0 q" unfolding P_def using 3(2) by auto
     qed rule
     obtain s where 5: "ws = w ||| s" using 4(3) by (metis add.left_neutral eqI_snth snth_smap szip_smap)
     show "w \<in> language B"
     proof
       show "q \<in> initial B" using 3(1) by this
       show "run B (w ||| s) q" using 4(1) unfolding 5 by this
       have 6: "(\<lambda> a b. (a, b) \<in> R) \<le> (\<lambda> a b. accepting A a \<longrightarrow> accepting B b)" using assms(4) by auto
       have 7: "stream_all2 (\<lambda> p q. (p, q) \<in> R) (trace (w ||| r) p) (trace (w ||| s) q)"
         using 4(2) unfolding P_def 5 by (simp add: stream_rel_snth del: stake.simps(2))
       have 8: "stream_all2 (\<lambda> a b. accepting A a \<longrightarrow> accepting B b) r s"
         using stream.rel_mono 6 7 unfolding nba.trace_alt_def by auto
       show "infs (accepting B) s" using infs_mono_strong 8 2(3) by this
     qed
   qed
 
 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
@@ -1,89 +1,89 @@
 section \<open>Nondeterministic Büchi Automata Combinations\<close>
 
 theory NBA_Combine
 imports NBA NGBA
 begin
 
-  global_interpretation degeneralization: automaton_degeneralization_trace
+  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"
     defines degeneralize = degeneralization.degeneralize
     by unfold_locales auto
 
   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_trace
+  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 intersection_list: automaton_intersection_list_trace
+  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_trace
+  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_trace
+  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/NBA/NGBA.thy b/thys/Transition_Systems_and_Automata/Automata/NBA/NGBA.thy
--- a/thys/Transition_Systems_and_Automata/Automata/NBA/NGBA.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/NBA/NGBA.thy
@@ -1,25 +1,25 @@
 section \<open>Nondeterministic Generalized Büchi Automata\<close>
 
 theory NGBA
 imports "../Nondeterministic"
 begin
 
   datatype ('label, 'state) ngba = ngba
     (alphabet: "'label set")
     (initial: "'state set")
     (transition: "'label \<Rightarrow> 'state \<Rightarrow> 'state set")
     (accepting: "'state pred gen")
 
   global_interpretation ngba: automaton ngba alphabet initial transition accepting
     defines path = ngba.path and run = ngba.run and reachable = ngba.reachable and nodes = ngba.nodes
     by unfold_locales auto
-  global_interpretation ngba: automaton_trace ngba alphabet initial transition accepting "\<lambda> P w r p. gen infs P r"
+  global_interpretation ngba: automaton_run ngba alphabet initial transition accepting "\<lambda> P w r p. gen infs P r"
     defines language = ngba.language
     by standard
 
   abbreviation target where "target \<equiv> ngba.target"
   abbreviation states where "states \<equiv> ngba.states"
   abbreviation trace where "trace \<equiv> ngba.trace"
   abbreviation successors where "successors \<equiv> ngba.successors TYPE('label)"
 
 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
@@ -1,734 +1,734 @@
 theory Nondeterministic
 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 set"
 
   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 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
 
-  (* TODO: create analogous thing for NFAs (automaton_target) *)
-  (* TODO: this should be _stream or _run *)
-  locale automaton_trace =
+  (* TODO: create analogous thing for NFAs (automaton_path) *)
+  (* TODO: adjust deterministic theory to the same test type *)
+  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> 'state 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> 'state pred gen"
     and automaton\<^sub>2 :: "'label set \<Rightarrow> 'state degen set \<Rightarrow> ('label, 'state degen) trans \<Rightarrow> 'state 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> 'state degen pred"
   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). transition\<^sub>1 A a p \<times> {count (condition\<^sub>1 A) p k})
         (degen (condition\<^sub>1 A))"
 
     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) = transition\<^sub>1 A a p \<times> {count (condition\<^sub>1 A) p k}"
       "condition\<^sub>2 (degeneralize A) = degen (condition\<^sub>1 A)"
       unfolding degeneralize_def by auto
 
     lemma run_degeneralize:
       assumes "a.run A (w ||| r) p"
       shows "b.run (degeneralize A) (w ||| r ||| sscan (count (condition\<^sub>1 A)) (p ## 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 = sscan (count (condition\<^sub>1 A)) (p ## r) k"
     proof
       show "rs = smap fst rs ||| smap snd rs" "pk = (fst pk, snd pk)" by auto
       show "a.run A (w ||| smap fst rs) (fst pk)"
         using assms by (coinduction arbitrary: w rs pk) (force elim: b.run.cases)
       show "smap snd rs = sscan (count (condition\<^sub>1 A)) (fst pk ## smap fst rs) (snd pk)"
         using assms by (coinduction arbitrary: w rs pk) (force elim: b.run.cases)
     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) p 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_trace =
+  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 +
-    a: automaton_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
+    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> 'state 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> 'state pred gen"
     and test\<^sub>1 :: "'state 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> 'state 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> 'state degen pred"
     and test\<^sub>2 :: "'state degen pred \<Rightarrow> 'label stream \<Rightarrow> 'state degen stream \<Rightarrow> 'state degen \<Rightarrow> bool"
     +
     assumes test[iff]: "test\<^sub>2 (degen cs) w (r ||| sscan (count cs) (p ## 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_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
 
-  locale automaton_intersection_trace =
+  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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
-    c: automaton_trace automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
+    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_trace =
+  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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
+    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 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 run_union:
       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_trace =
+  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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2 +
-    c: automaton_trace automaton\<^sub>3 alphabet\<^sub>3 initial\<^sub>3 transition\<^sub>3 condition\<^sub>3 test\<^sub>3
+    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 dest: run_union_a run_union_b elim!: run_union)
 
   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_trace =
+  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_trace automaton\<^sub>1 alphabet\<^sub>1 initial\<^sub>1 transition\<^sub>1 condition\<^sub>1 test\<^sub>1 +
-    b: automaton_trace automaton\<^sub>2 alphabet\<^sub>2 initial\<^sub>2 transition\<^sub>2 condition\<^sub>2 test\<^sub>2
+    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