diff --git a/thys/Buchi_Complementation/Ranking.thy b/thys/Buchi_Complementation/Ranking.thy
--- a/thys/Buchi_Complementation/Ranking.thy
+++ b/thys/Buchi_Complementation/Ranking.thy
@@ -1,480 +1,479 @@
 section \<open>Rankings\<close>
 
 theory Ranking
 imports
   "Alternate"
   "Graph"
 begin
 
   subsection \<open>Rankings\<close>
 
   type_synonym 'state ranking = "'state node \<Rightarrow> nat"
 
   definition ranking :: "('label, 'state) nba \<Rightarrow> 'label stream \<Rightarrow> 'state ranking \<Rightarrow> bool" where
     "ranking A w f \<equiv>
       (\<forall> v \<in> gunodes A w. f v \<le> 2 * card (nodes A)) \<and>
       (\<forall> v \<in> gunodes A w. \<forall> u \<in> gusuccessors A w v. f u \<le> f v) \<and>
       (\<forall> v \<in> gunodes A w. gaccepting A v \<longrightarrow> even (f v)) \<and>
       (\<forall> v \<in> gunodes A w. \<forall> r k. gurun A w r v \<longrightarrow> smap f (gtrace r v) = sconst k \<longrightarrow> odd k)"
 
   subsection \<open>Ranking Implies Word not in Language\<close>
 
   lemma ranking_stuck:
     assumes "ranking A w f"
     assumes "v \<in> gunodes A w" "gurun A w r v"
     obtains n k
     where "smap f (gtrace (sdrop n r) (gtarget (stake n r) v)) = sconst k"
   proof -
     have 0: "f u \<le> f v" if "v \<in> gunodes A w" "u \<in> gusuccessors A w v" for v u
       using assms(1) that unfolding ranking_def by auto
     have 1: "shd (v ## gtrace r v) \<in> gunodes A w" using assms(2) by auto
     have 2: "sdescending (smap f (v ## gtrace r v))"
     using 1 assms(3)
     proof (coinduction arbitrary: r v rule: sdescending.coinduct)
       case sdescending
       obtain u s where 1: "r = u ## s" using stream.exhaust by blast
       have 2: "v \<in> gunodes A w" using sdescending(1) by simp
       have 3: "gurun A w (u ## s) v" using sdescending(2) 1 by auto
       have 4: "u \<in> gusuccessors A w v" using 3 by auto
       have 5: "u \<in> gureachable A w v" using graph.reachable_successors 4 by blast
       show ?case
       unfolding 1
       proof (intro exI conjI disjI1)
         show "f u \<le> f v" using 0 2 4 by this
         show "shd (u ## gtrace s u) \<in> gunodes A w" using 2 5 by auto
         show "gurun A w s u" using 3 by auto
       qed auto
     qed
     obtain n k where 3: "sdrop n (smap f (v ## gtrace r v)) = sconst k"
       using sdescending_stuck[OF 2] by metis
     have "gtrace (sdrop (Suc n) r) (gtarget (stake (Suc n) r) v) = sdrop (Suc n) (gtrace r v)"
       using sscan_sdrop by rule
     also have "smap f \<dots> = sdrop n (smap f (v ## gtrace r v))"
       by (auto, metis 3 id_apply sdrop_smap sdrop_stl siterate.simps(2)
         sscan_const stream.map stream.map_sel(2) stream.sel(2))
     also have "\<dots> = sconst k" using 3 by this
     finally show ?thesis using that by blast
   qed
 
   lemma ranking_stuck_odd:
     assumes "ranking A w f"
     assumes "v \<in> gunodes A w" "gurun A w r v"
     obtains n
-    where "sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))) \<subseteq> Collect odd"
+    where "Ball (sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v)))) odd"
   proof -
     obtain n k where 1: "smap f (gtrace (sdrop n r) (gtarget (stake n r) v)) = sconst k"
       using ranking_stuck assms by this
     have 2: "gtarget (stake n r) v \<in> gunodes A w"
       using assms(2, 3) by (simp add: graph.nodes_target graph.run_stake)
     have 3: "gurun A w (sdrop n r) (gtarget (stake n r) v)"
       using assms(2, 3) by (simp add: graph.run_sdrop)
     have 4: "odd k" using 1 2 3 assms(1) unfolding ranking_def by meson
-    have "sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))) = sset (sconst k)"
-      unfolding 1 by rule
-    also have "\<dots> \<subseteq> Collect odd" using 4 by simp
-    finally show ?thesis using that by simp
+    have 5: "Ball (sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v)))) odd"
+      unfolding 1 using 4 by simp
+    show ?thesis using that 5 by this
   qed
 
   lemma ranking_language:
     assumes "ranking A w f"
     shows "w \<notin> language A"
   proof
     assume 1: "w \<in> language A"
     obtain r p where 2: "run A (w ||| r) p" "p \<in> initial A" "infs (accepting A) (p ## r)" using 1 by rule
     let ?r = "fromN 1 ||| r"
     let ?v = "(0, p)"
     have 3: "?v \<in> gunodes A w" "gurun A w ?r ?v" using 2(1, 2) by (auto intro: run_grun)
 
-    obtain n where 4: "sset (smap f (gtrace (sdrop n ?r) (gtarget (stake n ?r) ?v))) \<subseteq> {a. odd a}"
+    obtain n where 4: "Ball (sset (smap f (gtrace (sdrop n ?r) (gtarget (stake n ?r) ?v)))) odd"
       using ranking_stuck_odd assms 3 by this
     let ?s = "stake n ?r"
     let ?t = "sdrop n ?r"
     let ?u = "gtarget ?s ?v"
 
     have "sset (gtrace ?t ?u) \<subseteq> gureachable A w ?v"
     proof (intro graph.reachable_trace graph.reachable_target graph.reachable.reflexive)
       show "gupath A w ?s ?v" using graph.run_stake 3(2) by this
       show "gurun A w ?t ?u" using graph.run_sdrop 3(2) by this
     qed
     also have "\<dots> \<subseteq> gunodes A w" using 3(1) by blast
     finally have 7: "sset (gtrace ?t ?u) \<subseteq> gunodes A w" by this
     have 8: "\<And> p. p \<in> gunodes A w \<Longrightarrow> gaccepting A p \<Longrightarrow> even (f p)"
       using assms unfolding ranking_def by auto
     have 9: "\<And> p. p \<in> sset (gtrace ?t ?u) \<Longrightarrow> gaccepting A p \<Longrightarrow> even (f p)" using 7 8 by auto
 
     have 19: "infs (accepting A) (smap snd ?r)" using 2(3) by simp
     have 18: "infs (gaccepting A) ?r" using 19 by simp
     have 17: "infs (gaccepting A) (gtrace ?r ?v)" using 18 unfolding gtrace_alt_def by this
     have 16: "infs (gaccepting A) (gtrace (?s @- ?t) ?v)" using 17 unfolding stake_sdrop by this
     have 15: "infs (gaccepting A) (gtrace ?t ?u)" using 16 by simp
     have 13: "infs (even \<circ> f) (gtrace ?t ?u)" using infs_mono[OF _ 15] 9 by simp
     have 12: "infs even (smap f (gtrace ?t ?u))" using 13 by (simp add: comp_def)
     have 11: "Bex (sset (smap f (gtrace ?t ?u))) even" using 12 infs_any by metis
 
     show False using 4 11 by auto
   qed
 
-  subsection \<open>Word not in Language implies Ranking\<close>
+  subsection \<open>Word not in Language Implies Ranking\<close>
 
   subsubsection \<open>Removal of Endangered Nodes\<close>
 
   definition clean :: "('label, 'state) nba \<Rightarrow> 'label stream \<Rightarrow> 'state node set \<Rightarrow> 'state node set" where
     "clean A w V \<equiv> {v \<in> V. infinite (greachable A w V v)}"
 
   lemma clean_decreasing: "clean A w V \<subseteq> V" unfolding clean_def by auto
   lemma clean_successors:
     assumes "v \<in> V" "u \<in> gusuccessors A w v"
     shows "u \<in> clean A w V \<Longrightarrow> v \<in> clean A w V"
   proof -
     assume 1: "u \<in> clean A w V"
     have 2: "u \<in> V" "infinite (greachable A w V u)" using 1 unfolding clean_def by auto
     have 3: "u \<in> greachable A w V v" using graph.reachable.execute assms(2) 2(1) by blast
     have 4: "greachable A w V u \<subseteq> greachable A w V v" using 3 by blast
     have 5: "infinite (greachable A w V v)" using 2(2) 4 by (simp add: infinite_super)
     show "v \<in> clean A w V" unfolding clean_def using assms(1) 5 by simp
   qed
 
   subsubsection \<open>Removal of Safe Nodes\<close>
 
   definition prune :: "('label, 'state) nba \<Rightarrow> 'label stream \<Rightarrow> 'state node set \<Rightarrow> 'state node set" where
     "prune A w V \<equiv> {v \<in> V. \<exists> u \<in> greachable A w V v. gaccepting A u}"
 
   lemma prune_decreasing: "prune A w V \<subseteq> V" unfolding prune_def by auto
   lemma prune_successors:
     assumes "v \<in> V" "u \<in> gusuccessors A w v"
     shows "u \<in> prune A w V \<Longrightarrow> v \<in> prune A w V"
   proof -
     assume 1: "u \<in> prune A w V"
     have 2: "u \<in> V" "\<exists> x \<in> greachable A w V u. gaccepting A x" using 1 unfolding prune_def by auto
     have 3: "u \<in> greachable A w V v" using graph.reachable.execute assms(2) 2(1) by blast
     have 4: "greachable A w V u \<subseteq> greachable A w V v" using 3 by blast
     show "v \<in> prune A w V" unfolding prune_def using assms(1) 2(2) 4 by auto
   qed
 
   subsubsection \<open>Run Graph Interation\<close>
 
   definition graph :: "('label, 'state) nba \<Rightarrow> 'label stream \<Rightarrow> nat \<Rightarrow> 'state node set" where
     "graph A w k \<equiv> alternate (clean A w) (prune A w) k (gunodes A w)"
 
   abbreviation "level A w k l \<equiv> {v \<in> graph A w k. fst v = l}"
 
   lemma graph_0[simp]: "graph A w 0 = gunodes A w" unfolding graph_def by simp
   lemma graph_Suc[simp]: "graph A w (Suc k) = (if even k then clean A w else prune A w) (graph A w k)"
     unfolding graph_def by simp
 
   lemma graph_antimono: "antimono (graph A w)"
     using alternate_antimono clean_decreasing prune_decreasing
     unfolding antimono_def le_fun_def graph_def
     by metis
   lemma graph_nodes: "graph A w k \<subseteq> gunodes A w" using graph_0 graph_antimono le0 antimonoD by metis
   lemma graph_successors:
     assumes "v \<in> gunodes A w" "u \<in> gusuccessors A w v"
     shows "u \<in> graph A w k \<Longrightarrow> v \<in> graph A w k"
   using assms
   proof (induct k arbitrary: u v)
     case 0
     show ?case using 0(2) by simp
   next
     case (Suc k)
     have 1: "v \<in> graph A w k" using Suc using antimono_iff_le_Suc graph_antimono rev_subsetD by blast
     show ?case using Suc(2) clean_successors[OF 1 Suc(4)] prune_successors[OF 1 Suc(4)] by auto
   qed
 
   lemma graph_level_finite:
     assumes "finite (nodes A)"
     shows "finite (level A w k l)"
   proof -
     have "level A w k l \<subseteq> {v \<in> gunodes A w. fst v = l}" by (simp add: graph_nodes subset_CollectI)
     also have "{v \<in> gunodes A w. fst v = l} \<subseteq> {l} \<times> nodes A" using gnodes_nodes by force
     also have "finite ({l} \<times> nodes A)" using assms(1) by simp
     finally show ?thesis by this
   qed
 
   lemma find_safe:
     assumes "w \<notin> language A"
     assumes "V \<noteq> {}" "V \<subseteq> gunodes A w"
     assumes "\<And> v. v \<in> V \<Longrightarrow> gsuccessors A w V v \<noteq> {}"
     obtains v
     where "v \<in> V" "\<forall> u \<in> greachable A w V v. \<not> gaccepting A u"
   proof (rule ccontr)
     assume 1: "\<not> thesis"
     have 2: "\<And> v. v \<in> V \<Longrightarrow> \<exists> u \<in> greachable A w V v. gaccepting A u" using that 1 by auto
     have 3: "\<And> r v. v \<in> initial A \<Longrightarrow> run A (w ||| r) v \<Longrightarrow> fins (accepting A) r" using assms(1) by auto
     obtain v where 4: "v \<in> V" using assms(2) by force
     obtain x where 5: "x \<in> greachable A w V v" "gaccepting A x" using 2 4 by blast
     obtain y where 50: "gpath A w V y v" "x = gtarget y v" using 5(1) by rule
     obtain r where 6: "grun A w V r x" "infs (\<lambda> x. x \<in> V \<and> gaccepting A x) r"
     proof (rule graph.recurring_condition)
       show "x \<in> V \<and> gaccepting A x" using greachable_subset 4 5 by blast
     next
       fix v
       assume 1: "v \<in> V \<and> gaccepting A v"
       obtain v' where 20: "v' \<in> gsuccessors A w V v" using assms(4) 1 by (meson IntE equals0I)
       have 21: "v' \<in> V" using 20 by auto
       have 22: "\<exists> u \<in> greachable A w V v'. u \<in> V \<and> gaccepting A u"
         using greachable_subset 2 21 by blast
       obtain r where 30: "gpath A w V r v'" "gtarget r v' \<in> V \<and> gaccepting A (gtarget r v')"
         using 22 by blast
       show "\<exists> r. r \<noteq> [] \<and> gpath A w V r v \<and> gtarget r v \<in> V \<and> gaccepting A (gtarget r v)"
       proof (intro exI conjI)
         show "v' # r \<noteq> []" by simp
         show "gpath A w V (v' # r) v" using 20 30 by auto
         show "gtarget (v' # r) v \<in> V" using 30 by simp
         show "gaccepting A (gtarget (v' # r) v)" using 30 by simp
       qed
     qed auto
     obtain u where 100: "u \<in> ginitial A" "v \<in> gureachable A w u" using 4 assms(3) by blast
     have 101: "gupath A w y v" using gpath_subset 50(1) subset_UNIV by this
     have 102: "gurun A w r x" using grun_subset 6(1) subset_UNIV by this
     obtain t where 103: "gupath A w t u" "v = gtarget t u" using 100(2) by rule
     have 104: "gurun A w (t @- y @- r) u" using 101 102 103 50(2) by auto
     obtain s q where 7: "t @- y @- r = fromN (Suc 0) ||| s" "u = (0, q)"
       using grun_elim[OF 104] 100(1) by blast
     have 8: "run A (w ||| s) q" using grun_run[OF 104[unfolded 7]] by simp
     have 9: "q \<in> initial A" using 100(1) 7(2) by auto
     have 91: "sset (trace (w ||| s) q) \<subseteq> reachable A q"
       using nba.reachable_trace nba.reachable.reflexive 8 by this
     have 10: "fins (accepting A) s" using 3 9 8 by this
     have 12: "infs (gaccepting A) r" using infs_mono[OF _ 6(2)] by simp
     have "s = smap snd (t @- y @- r)" unfolding 7(1) by simp
     also have "infs (accepting A) \<dots>" using 12 by (simp add: comp_def)
     finally have 13: "infs (accepting A) s" by this
     show False using 10 13 by simp
   qed
 
   lemma remove_run:
     assumes "finite (nodes A)" "w \<notin> language A"
     assumes "V \<subseteq> gunodes A w" "clean A w V \<noteq> {}"
     obtains v r
     where
       "grun A w V r v"
       "sset (gtrace r v) \<subseteq> clean A w V"
       "sset (gtrace r v) \<subseteq> - prune A w (clean A w V)"
   proof -
     obtain u where 1: "u \<in> clean A w V" "\<forall> x \<in> greachable A w (clean A w V) u. \<not> gaccepting A x"
     proof (rule find_safe)
       show "w \<notin> language A" using assms(2) by this
       show "clean A w V \<noteq> {}" using assms(4) by this
       show "clean A w V \<subseteq> gunodes A w" using assms(3) by (meson clean_decreasing subset_iff)
     next
       fix v
       assume 1: "v \<in> clean A w V"
       have 2: "v \<in> V" using 1 clean_decreasing by blast
       have 3: "infinite (greachable A w V v)" using 1 clean_def by auto
       have "gsuccessors A w V v \<subseteq> gusuccessors A w v" by auto
       also have "finite \<dots>" using 2 assms(1, 3) finite_nodes_gsuccessors by blast
       finally have 4: "finite (gsuccessors A w V v)" by this
       have 5: "infinite (insert v (\<Union>((greachable A w V) ` (gsuccessors A w V v))))"
         using graph.reachable_step 3 by metis
       obtain u where 6: "u \<in> gsuccessors A w V v" "infinite (greachable A w V u)" using 4 5 by auto
       have 7: "u \<in> clean A w V" using 6 unfolding clean_def by auto
       show "gsuccessors A w (clean A w V) v \<noteq> {}" using 6(1) 7 by auto
     qed auto
     have 2: "u \<in> V" using 1(1) unfolding clean_def by auto
     have 3: "infinite (greachable A w V u)" using 1(1) unfolding clean_def by simp
     have 4: "finite (gsuccessors A w V v)" if "v \<in> greachable A w V u" for v
     proof -
       have 1: "v \<in> V" using that greachable_subset 2 by blast
       have "gsuccessors A w V v \<subseteq> gusuccessors A w v" by auto
       also have "finite \<dots>" using 1 assms(1, 3) finite_nodes_gsuccessors by blast
       finally show ?thesis by this
     qed
     obtain r where 5: "grun A w V r u" using graph.koenig[OF 3 4] by this
     have 6: "greachable A w V u \<subseteq> V" using 2 greachable_subset by blast
     have 7: "sset (gtrace r u) \<subseteq> V"
       using graph.reachable_trace[OF graph.reachable.reflexive 5(1)] 6 by blast
     have 8: "sset (gtrace r u) \<subseteq> clean A w V"
       unfolding clean_def using 7 infinite_greachable_gtrace[OF 5(1)] by auto
     have 9: "sset (gtrace r u) \<subseteq> greachable A w (clean A w V) u"
       using 5 8 by (metis graph.reachable.reflexive graph.reachable_trace grun_subset)
     show ?thesis
     proof
       show "grun A w V r u" using 5(1) by this
       show "sset (gtrace r u) \<subseteq> clean A w V" using 8 by this
       show "sset (gtrace r u) \<subseteq> - prune A w (clean A w V)"
       proof (intro subsetI ComplI)
         fix p
         assume 10: "p \<in> sset (gtrace r u)" "p \<in> prune A w (clean A w V)"
         have 20: "\<exists> x \<in> greachable A w (clean A w V) p. gaccepting A x"
           using 10(2) unfolding prune_def by auto
         have 30: "greachable A w (clean A w V) p \<subseteq> greachable A w (clean A w V) u"
           using 10(1) 9 by blast
         show "False" using 1(2) 20 30 by force
       qed
     qed
   qed
 
   lemma level_bounded:
     assumes "finite (nodes A)" "w \<notin> language A"
     obtains n
     where "\<And> l. l \<ge> n \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k"
   proof (induct k arbitrary: thesis)
     case (0)
     show ?case
     proof (rule 0)
       fix l :: nat
       have "finite ({l} \<times> nodes A)" using assms(1) by simp
       also have "level A w 0 l \<subseteq> {l} \<times> nodes A" using gnodes_nodes by force
       also (card_mono) have "card \<dots> = card (nodes A)" using assms(1) by simp
       finally show "card (level A w (2 * 0) l) \<le> card (nodes A) - 0" by simp
     qed
   next
     case (Suc k)
     show ?case
     proof (cases "graph A w (Suc (2 * k)) = {}")
       case True
       have 3: "graph A w (2 * Suc k) = {}" using True prune_decreasing by simp blast
       show ?thesis using Suc(2) 3 by simp
     next
       case False
       obtain v r where 1:
         "grun A w (graph A w (2 * k)) r v"
         "sset (gtrace r v) \<subseteq> graph A w (Suc (2 * k))"
         "sset (gtrace r v) \<subseteq> - graph A w (Suc (Suc (2 * k)))"
       proof (rule remove_run)
         show "finite (nodes A)" "w \<notin> language A" using assms by this
         show "clean A w (graph A w (2 * k)) \<noteq> {}" using False by simp
         show "graph A w (2 * k) \<subseteq> gunodes A w" using graph_nodes by this
       qed auto
       obtain l q where 2: "v = (l, q)" by force
       obtain n where 90: "\<And> l. n \<le> l \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k"
         using Suc(1) by blast
       show ?thesis
       proof (rule Suc(2))
         fix j
         assume 100: "n + Suc l \<le> j"
         have 6: "graph A w (Suc (Suc (2 * k))) \<subseteq> graph A w (Suc (2 * k))"
           using graph_antimono antimono_iff_le_Suc by blast
         have 101: "gtrace r v !! (j - Suc l) \<in> graph A w (Suc (2 * k))" using 1(2) snth_sset by auto
         have 102: "gtrace r v !! (j - Suc l) \<notin> graph A w (Suc (Suc (2 * k)))" using 1(3) snth_sset by blast
         have 103: "gtrace r v !! (j - Suc l) \<in> level A w (Suc (2 * k)) j"
           using 1(1) 100 101 2 by (auto elim: grun_elim)
         have 104: "gtrace r v !! (j - Suc l) \<notin> level A w (Suc (Suc (2 * k))) j" using 100 102 by simp
         have "level A w (2 * Suc k) j = level A w (Suc (Suc (2 * k))) j" by simp
         also have "\<dots> \<subset> level A w (Suc (2 * k)) j" using 103 104 6 by blast
         also have "\<dots> \<subseteq> level A w (2 * k) j" by (simp add: Collect_mono clean_def)
         finally have 105: "level A w (2 * Suc k) j \<subset> level A w (2 * k) j" by this
         have "card (level A w (2 * Suc k) j) < card (level A w (2 * k) j)"
           using assms(1) 105 by (simp add: graph_level_finite psubset_card_mono)
         also have "\<dots> \<le> card (nodes A) - k" using 90 100 by simp
         finally show "card (level A w (2 * Suc k) j) \<le> card (nodes A) - Suc k" by simp
       qed
     qed
   qed
   lemma graph_empty:
     assumes "finite (nodes A)" "w \<notin> language A"
     shows "graph A w (Suc (2 * card (nodes A))) = {}"
   proof -
     obtain n where 1: "\<And> l. l \<ge> n \<Longrightarrow> card (level A w (2 * card (nodes A)) l) = 0"
       using level_bounded[OF assms(1, 2), of "card (nodes A)"] by auto
     have "graph A w (2 * card (nodes A)) =
       (\<Union> l \<in> {..< n}. level A w (2 * card (nodes A)) l) \<union>
       (\<Union> l \<in> {n ..}. level A w (2 * card (nodes A)) l)"
       by auto
     also have "(\<Union> l \<in> {n ..}. level A w (2 * card (nodes A)) l) = {}"
       using graph_level_finite assms(1) 1 by fastforce
     also have "finite ((\<Union> l \<in> {..< n}. level A w (2 * card (nodes A)) l) \<union> {})"
       using graph_level_finite assms(1) by auto
     finally have 100: "finite (graph A w (2 * card (nodes A)))" by this
     have 101: "finite (greachable A w (graph A w (2 * card (nodes A))) v)" for v
       using 100 greachable_subset[of A w "graph A w (2 * card (nodes A))" v]
       using finite_insert infinite_super by auto
     show ?thesis using 101 by (simp add: clean_def)
   qed
   lemma graph_le:
     assumes "finite (nodes A)" "w \<notin> language A"
     assumes "v \<in> graph A w k"
     shows "k \<le> 2 * card (nodes A)"
     using graph_empty graph_antimono assms
     by (metis (no_types, lifting) Suc_leI antimono_def basic_trans_rules(30) empty_iff not_le_imp_less)
 
   subsection \<open>Node Ranks\<close>
 
   definition rank :: "('label, 'state) nba \<Rightarrow> 'label stream \<Rightarrow> 'state node \<Rightarrow> nat" where
     "rank A w v \<equiv> GREATEST k. v \<in> graph A w k"
 
   lemma rank_member:
     assumes "finite (nodes A)" "w \<notin> language A" "v \<in> gunodes A w"
     shows "v \<in> graph A w (rank A w v)"
   unfolding rank_def
   proof (rule GreatestI_nat)
     show "v \<in> graph A w 0" using assms(3) by simp
     show "k \<le> 2 * card (nodes A)" if "v \<in> graph A w k" for k
       using graph_le assms(1, 2) that by blast
   qed
   lemma rank_removed:
     assumes "finite (nodes A)" "w \<notin> language A"
     shows "v \<notin> graph A w (Suc (rank A w v))"
   proof
     assume "v \<in> graph A w (Suc (rank A w v))"
     then have 2: "Suc (rank A w v) \<le> rank A w v"
       unfolding rank_def using Greatest_le_nat graph_le assms by metis
     then show "False" by auto
   qed
   lemma rank_le:
     assumes "finite (nodes A)" "w \<notin> language A"
     assumes "v \<in> gunodes A w" "u \<in> gusuccessors A w v"
     shows "rank A w u \<le> rank A w v"
   unfolding rank_def
   proof (rule Greatest_le_nat)
     have 1: "u \<in> gureachable A w v" using graph.reachable_successors assms(4) by blast
     have 2: "u \<in> gunodes A w" using assms(3) 1 by auto
     show "v \<in> graph A w (GREATEST k. u \<in> graph A w k)"
     unfolding rank_def[symmetric]
     proof (rule graph_successors)
       show "v \<in> gunodes A w" using assms(3) by this
       show "u \<in> gusuccessors A w v" using assms(4) by this
       show "u \<in> graph A w (rank A w u)" using rank_member assms(1, 2) 2 by this
     qed
     show "k \<le> 2 * card (nodes A)" if "v \<in> graph A w k" for k
       using graph_le assms(1, 2) that by blast
   qed
 
   lemma language_ranking:
     assumes "finite (nodes A)" "w \<notin> language A"
     shows "ranking A w (rank A w)"
   unfolding ranking_def
   proof (intro conjI ballI allI impI)
     fix v
     assume 1: "v \<in> gunodes A w"
     have 2: "v \<in> graph A w (rank A w v)" using rank_member assms 1 by this
     show "rank A w v \<le> 2 * card (nodes A)" using graph_le assms 2 by this
   next
     fix v u
     assume 1: "v \<in> gunodes A w" "u \<in> gusuccessors A w v"
     show "rank A w u \<le> rank A w v" using rank_le assms 1 by this
   next
     fix v
     assume 1: "v \<in> gunodes A w" "gaccepting A v"
     have 2: "v \<in> graph A w (rank A w v)" using rank_member assms 1(1) by this
     have 3: "v \<notin> graph A w (Suc (rank A w v))" using rank_removed assms by this
     have 4: "v \<in> prune A w (graph A w (rank A w v))" using 2 1(2) unfolding prune_def by auto
     have 5: "graph A w (Suc (rank A w v)) \<noteq> prune A w (graph A w (rank A w v))" using 3 4 by blast
     show "even (rank A w v)" using 5 by auto
   next
     fix v r k
     assume 1: "v \<in> gunodes A w" "gurun A w r v" "smap (rank A w) (gtrace r v) = sconst k"
     have "sset (gtrace r v) \<subseteq> gureachable A w v"
       using 1(2) by (metis graph.reachable.reflexive graph.reachable_trace)
     then have 6: "sset (gtrace r v) \<subseteq> gunodes A w" using 1(1) by blast
     have 60: "rank A w ` sset (gtrace r v) \<subseteq> {k}"
       using 1(3) by (metis equalityD1 sset_sconst stream.set_map)
     have 50: "sset (gtrace r v) \<subseteq> graph A w k"
       using rank_member[OF assms] subsetD[OF 6] 60 unfolding image_subset_iff by auto
     have 70: "grun A w (graph A w k) r v" using grun_subset 1(2) 50 by this
     have 7: "sset (gtrace r v) \<subseteq> clean A w (graph A w k)"
       unfolding clean_def using 50 infinite_greachable_gtrace[OF 70] by auto
     have 8: "sset (gtrace r v) \<inter> graph A w (Suc k) = {}" using rank_removed[OF assms] 60 by blast
     have 9: "sset (gtrace r v) \<noteq> {}" using stream.set_sel(1) by auto
     have 10: "graph A w (Suc k) \<noteq> clean A w (graph A w k)" using 7 8 9 by blast
     show "odd k" using 10 unfolding graph_Suc by auto
   qed
 
   subsection \<open>Correctness Theorem\<close>
 
   theorem language_ranking_iff:
     assumes "finite (nodes A)"
     shows "w \<notin> language A \<longleftrightarrow> (\<exists> f. ranking A w f)"
     using ranking_language language_ranking assms by blast
 
 end