diff --git a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Graphs.thy b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Graphs.thy
--- a/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Graphs.thy
+++ b/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Graphs.thy
@@ -1,147 +1,147 @@
 section \<open>Connecting Nondeterministic Büchi Automata to CAVA Automata Structures\<close>
 
 theory NBA_Graphs
 imports
   NBA
   CAVA_Automata.Automata_Impl
 begin
 
   no_notation build (infixr "##" 65)
 
   subsection \<open>Regular Graphs\<close>
 
   definition nba_g :: "('label, 'state) nba \<Rightarrow> 'state graph_rec" where
     "nba_g A \<equiv> \<lparr> g_V = UNIV, g_E = E_of_succ (successors A), g_V0 = initial A \<rparr>"
 
   lemma nba_g_graph[simp]: "graph (nba_g A)" unfolding nba_g_def graph_def by simp
 
   lemma nba_g_V0: "g_V0 (nba_g A) = initial A" unfolding nba_g_def by simp
   lemma nba_g_E_rtrancl: "(g_E (nba_g A))\<^sup>* = {(p, q). q \<in> reachable A p}"
   unfolding nba_g_def graph_rec.simps E_of_succ_def
   proof safe
     show "(p, q) \<in> {(p, q). q \<in> successors A p}\<^sup>*" if "q \<in> reachable A p" for p q
       using that by (induct) (auto intro: rtrancl_into_rtrancl)
     show "q \<in> reachable A p" if "(p, q) \<in> {(p, q). q \<in> successors A p}\<^sup>*" for p q
       using that by induct auto
   qed
 
   lemma nba_g_rtrancl_path: "(g_E (nba_g A))\<^sup>* = {(p, target r p) |r p. NBA.path A r p}"
     unfolding nba_g_E_rtrancl by blast
   lemma nba_g_trancl_path: "(g_E (nba_g A))\<^sup>+ = {(p, target r p) |r p. NBA.path A r p \<and> r \<noteq> []}"
   unfolding nba_g_def graph_rec.simps E_of_succ_def
   proof safe
     show "\<exists> r p. (x, y) = (p, target r p) \<and> NBA.path A r p \<and> r \<noteq> []"
       if "(x, y) \<in> {(p, q). q \<in> successors A p}\<^sup>+" for x y
     using that
     proof induct
       case (base y)
       obtain a where 1: "a \<in> alphabet A" "y \<in> transition A a x" using base by auto
       show ?case
       proof (intro exI conjI)
         show "(x, y) = (x, target [(a, y)] x)" by simp
         show "NBA.path A [(a, y)] x" using 1 by auto
         show "[(a, y)] \<noteq> []" by simp
       qed
     next
       case (step y z)
       obtain r where 1: "y = target r x" "NBA.path A r x" "r \<noteq> []" using step(3) by auto
       obtain a where 2: "a \<in> alphabet A" "z \<in> transition A a y" using step(2) by auto
       show ?case
       proof (intro exI conjI)
         show "(x, z) = (x, target (r @ [(a, z)]) x)" by simp
         show "NBA.path A (r @ [(a, z)]) x" using 1 2 by auto
         show "r @ [(a, z)] \<noteq> []" by simp
       qed
     qed
     show "(p, target r p) \<in> {(u, v). v \<in> successors A u}\<^sup>+" if "NBA.path A r p" "r \<noteq> []" for r p
       using that by (induct) (fastforce intro: trancl_into_trancl2)+
   qed
 
   lemma nba_g_ipath_run:
     assumes "ipath (g_E (nba_g A)) r"
     obtains w
     where "run A (w ||| smap (r \<circ> Suc) nats) (r 0)"
   proof -
     have 1: "\<exists> a \<in> alphabet A. r (Suc i) \<in> transition A a (r i)" for i
       using assms unfolding ipath_def nba_g_def E_of_succ_def by auto
     obtain wr where 2: "run A wr (r 0)" "\<And> i. target (stake i wr) (r 0) = r i"
     proof (rule nba.invariant_run_index)
       show "\<exists> aq. (fst aq \<in> alphabet A \<and> snd aq \<in> transition A (fst aq) p) \<and> snd aq = r (Suc i) \<and> True"
         if "p = r i" for i p using that 1 by auto
       show "r 0 = r 0" by rule
     qed auto
     have 3: "smap (r \<circ> Suc) nats = smap snd wr"
     proof (rule eqI_snth)
       fix i
       have "smap (r \<circ> Suc) nats !! i = r (Suc i)" by simp
       also have "\<dots> = target (stake (Suc i) wr) (r 0)" unfolding 2(2) by rule
       also have "\<dots> = (r 0 ## trace wr (r 0)) !! Suc i" by simp
       also have "\<dots> = smap snd wr !! i" unfolding nba.trace_alt_def by simp
       finally show "smap (r \<circ> Suc) nats !! i = smap snd wr !! i" by this
     qed
     show ?thesis
     proof
       show "run A (smap fst wr ||| smap (r \<circ> Suc) nats) (r 0)" using 2(1) unfolding 3 by auto
     qed
   qed
   lemma nba_g_run_ipath:
     assumes "run A (w ||| r) p"
     shows "ipath (g_E (nba_g A)) (snth (p ## r))"
   proof
     fix i
     have 1: "w !! i \<in> alphabet A" "r !! i \<in> transition A (w !! i) (target (stake i (w ||| r)) p)"
       using assms by (auto dest: nba.run_snth)
     have 2: "r !! i \<in> successors A ((p ## r) !! i)"
       using 1 unfolding sscan_scons_snth[symmetric] nba.trace_alt_def by auto
     show "((p ## r) !! i, (p ## r) !! Suc i) \<in> g_E (nba_g A)"
       using 2 unfolding nba_g_def graph_rec.simps E_of_succ_def by simp
   qed
 
   subsection \<open>Indexed Generalized Büchi Graphs\<close>
 
   definition nba_igbg :: "('label, 'state) nba \<Rightarrow> 'state igb_graph_rec" where
     "nba_igbg A \<equiv> graph_rec.extend (nba_g A)
       \<lparr> igbg_num_acc = 1, igbg_acc = \<lambda> p. if accepting A p then {0} else {} \<rparr>"
 
   lemma acc_run_language:
     assumes "igb_graph (nba_igbg A)"
     shows "Ex (igb_graph.is_acc_run (nba_igbg A)) \<longleftrightarrow> language A \<noteq> {}"
   proof
     interpret igb_graph "nba_igbg A" using assms by this
     have [simp]: "V0 = g_V0 (nba_g A)" "E = g_E (nba_g A)"
       "num_acc = 1" "0 \<in> acc p \<longleftrightarrow> accepting A p" for p
       unfolding nba_igbg_def graph_rec.defs by simp+
     show "language A \<noteq> {}" if run: "Ex is_acc_run"
     proof -
       obtain r where 1: "is_acc_run r" using run by rule
       have 2: "r 0 \<in> V0" "ipath E r" "is_acc r"
         using 1 unfolding is_acc_run_def graph_defs.is_run_def by auto
       obtain w where 3: "run A (w ||| smap (r \<circ> Suc) nats) (r 0)" using nba_g_ipath_run 2(2) by auto
       have 4: "r 0 ## smap (r \<circ> Suc) nats = smap r nats" by (simp) (metis stream.map_comp smap_siterate)
       have 5: "infs (accepting A) (r 0 ## smap (r \<circ> Suc) nats)"
         using 2(3) unfolding infs_infm is_acc_def 4 by simp
       have "w \<in> language A"
       proof
         show "r 0 \<in> initial A" using nba_g_V0 2(1) by force
         show "run A (w ||| smap (r \<circ> Suc) nats) (r 0)" using 3 by this
         show "infs (accepting A) (smap (r \<circ> Suc) nats)" using 5 by simp
       qed
       then show ?thesis by auto
     qed
     show "Ex is_acc_run" if language: "language A \<noteq> {}"
     proof -
       obtain w where 1: "w \<in> language A" using language by auto
       obtain r p where 2: "p \<in> initial A" "run A (w ||| r) p" "infs (accepting A) r" using 1 by rule
       have 3: "infs (accepting A) (p ## r)" using 2(3) by simp
       have "is_acc_run (snth (p ## r))"
       unfolding is_acc_run_def graph_defs.is_run_def
       proof safe
         show "(p ## r) !! 0 \<in> V0" using nba_g_V0 2(1) by force
         show "ipath E (snth (p ## r))" using nba_g_run_ipath 2(2) by force
-        show "is_acc (snth (p ## r))" using 3 unfolding infs_infm is_acc_def nba.trace_alt_def by simp
+        show "is_acc (snth (p ## r))" using 3 unfolding infs_infm is_acc_def by simp
       qed
       then show ?thesis by auto
     qed
   qed
 
 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
@@ -1,742 +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 *)
-  (* TODO: remove all trace_alt_def *)
+  (* TODO: this should be _stream or _run *)
   locale automaton_trace =
     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
 
-    (* TODO: move this up in the hierarchy *)
-    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)
-    lemma language_alphabet: "language A \<subseteq> streams (alphabet A)"
-      unfolding language_def by (auto dest: run_alphabet)
+    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 =
     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 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)
+      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 =
     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
     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 pp. listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA pp))
+        (\<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 pp = listset (map2 (\<lambda> A p. transition\<^sub>1 A a p) AA pp)"
+      "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 pp = length AA"
-      assumes "b.run (intersect AA) (w ||| r) pp"
-      assumes "qq \<in> sset r"
-      shows "length qq = length AA"
+      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> qq. length qq = length AA) r"
-        using assms(1, 2) by (coinduction arbitrary: w r pp)
+      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 pp = length AA"
-      assumes "b.run (intersect AA) (w ||| r) pp"
-      obtains rr where "r = stranspose rr" "length rr = length AA"
+      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 rr where "rr \<equiv> map (\<lambda> k. smap (\<lambda> pp. pp ! k) r) [0 ..< length AA]"
-      have "length qq = length AA" if "qq \<in> sset r" for qq using intersect_run_length assms that by this
-      then show "r = stranspose rr"
-        unfolding rr_def by (coinduction arbitrary: r) (auto intro: nth_equalityI simp: comp_def)
-      show "length rr = length AA" unfolding rr_def by auto
+      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 intersect_run:
-      assumes "length rr = length AA" "length pp = length AA"
-      assumes "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rr ! k) (pp ! k)"
-      shows "b.run (intersect AA) (w ||| stranspose rr) pp"
+    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 rr pp)
+    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) pp"
+        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 rr ! k) (map shd rr ! k)"
+        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_elim:
-      assumes "length rr = length AA" "length pp = length AA"
-      assumes "b.run (intersect AA) (w ||| stranspose rr) pp"
-      shows "k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rr ! k) (pp ! k)"
+    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 rr pp)
+    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) (pp ! k)"
+        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 rr)) (shd (stranspose rr))"
+        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 "pp \<in> listset (map a.nodes AA)" if "pp \<in> b.nodes (intersect AA)" for pp
+      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 =
     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
     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)"
+      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 pp where 2:
-        "pp \<in> initial\<^sub>2 (intersect AA)"
-        "b.run (intersect AA) (w ||| r) pp"
-        "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w r pp"
-        using 1(1) unfolding b.language_def by auto
-      have 3: "length pp = length AA" using 2(1) by (simp add: listset_member list_all2_conv_all_nth)
-      obtain rr where 4: "r = stranspose rr" "length rr = length AA"
-        using intersect_run_stranspose 3 2(2) by blast
+      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 "pp ! 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 ||| rr ! k) (pp ! k)" using intersect_run_elim 2(2) 3 4 5 by auto
-        show "test\<^sub>1 (condition\<^sub>1 A) w (rr ! k) (pp ! k)"
-          using 2(3) 3 4 5 by (simp add: list_all_length)
+        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 unfolding a.language_def by auto
-      obtain rr pp where 3:
-        "length rr = length AA" "length pp = length AA"
-        "\<And> k. k < length AA \<Longrightarrow> pp ! k \<in> initial\<^sub>1 (AA ! k)"
-        "\<And> k. k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rr ! k) (pp ! k)"
-        "\<And> k. k < length AA \<Longrightarrow> test\<^sub>1 (condition\<^sub>1 (AA ! k)) w (rr ! k) (pp ! k)"
+        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 "pp \<in> initial\<^sub>2 (intersect AA)"
-          using 3 by (force simp: listset_member list_all2_iff set_conv_nth)
-        show "b.run (intersect AA) (w ||| stranspose rr) pp" using intersect_run 3 by auto
-        show "test\<^sub>2 (condition\<^sub>2 (intersect AA)) w (stranspose rr) pp"
-          using 3 by (force simp: list_all_length)
+        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 =
     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
     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 union_run:
+    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 run_union:
+    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 =
     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
     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"
     +
-    (* TODO: why do we not need wellformedness here but we do in the intersection_list case? *)
-    assumes test[iff]: "test\<^sub>2 (condition cs) w (sconst k ||| r) (k, p) \<longleftrightarrow> test\<^sub>1 (cs ! k) w r p"
+    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: run_union)
+        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!: union_run)
+        using assms unfolding a.language_def b.language_def by (force elim!: run_union)
     qed
 
   end
 
 end
\ No newline at end of file